AGGREGATION PROBLEMS IN BENEFITS  ESTIMATION:
            A SIMULATION APPROACH
             William J.  Vaughan
                John Mullahy
               Julie A.  Hewitt
               Michael Hazilla
                     and
             Clifford S. Russell
     Cooperative Agreement CR  810M66-01
               Project Officer

             Dr.  George Parsons
          Office of Policy Analysis
    U.S. Environmental Protection Agency
           Washington, D.C.   20460
          Resources for the Future
           Washington,  D.C.  20036

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               AGGREGATION PROBLEMS IN BENEFITS  ESTIMATION:
                           A SIMULATION APPROACH
                             Executive Summary
Background
     This project was undertaken in response to  several  concerns about the
potential perils of aggregating and disaggregating  in  the context of
pollution control benefit estimation.   The oldest of these concerns
involved the fairly common practice of using results from national-level
studies as the basis for regional benefit estimates.   (For example, earlier
RFF workVaughan and Russell 1982in which national  participation
equations for recreational fishing were estimated,  was used by another EPA
contractor to assess damages to northeastern states from acid deposition.)
The mirror image of this practice that is the "blowing up" of regional
studies, which are often seen as cheaper or easier  pieces of research, to
obtain national benefit estimates, was also to be investigated.
     As the research proceeded, however,  it became  clear that a prior
aggregation practice cried out for examination;  that is  the use of average
aggregate resource-availability measures as explanatory  variables in
benefit estimation in either national  or regional studies.  These measures
have in the past been used routinely,  if without much  formal justification,.
because a link was necessary between measured participation behavior and
the results of pollution control.   The resource  availability variables
served conveniently as such a link,  because a reasonable case could be made
that availability must matter to recreators and  that at  least rough account
could be taken of pollution control by showing availability increasing as a
result.  As it turns out,  however, it  is possible to prove that
availability, measured by density of recreation  acres  per acre of total
area, is a conceptually correct proxy  for the expected price of obtaining a
recreation experience for the recreator around whose home the density is
measured.  In practice, state or county-level density  measures are the only
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ones available,  however,  and  must  be used for every resident.  The
aggregation question,  therefore, is how are  the  results affected by using
the average density over  some politically or geographically defined area as
a price proxy for every resident of that area.
     Because of  the ubiquity  of these  several practices, it was seen as
important by EPA to have  some idea of  how much violence they each do to
results.  Unfortunately,  for  no actual cases could this question be
answered.  While it is clear  that  dome of the practices involve the
introduction of  errors in variables problems into statistical analysis,
there is no benchmark  against which to compare the results of an
aggregation exercise in any specific setting.  This lack is not an
inevitable one or the  result  of a  problem of principle.  It is simply the
reflection of the practical problems,  generally  data problems, that drive
researchers toward the aggregation approaches in the first place.  But in
the absence of real-world benchmarks,  this project was designed around a
simulation model from  a hypothetical world.

Approach
     To create a simulation model  appropriate to the examination of the
aggregation problems just described required at  a minimum that the
following features be  captured:
         consumer/recreators, located  in space,  with known utility
         functions defined over the included recreation and consumption
         activities
         recreation sites, located in  space, and initially either available
         or unavailable because of pollution
         an overlay of jurisdictional  boundaries that would define the
         areas over which recreation site density would be averaged.
     On the basis of the  "data", generated by the reaction of the
(utility-maximizing) consumers to  the  relative prices of general
consumption and recreation at particular sites,  the activities going on in
the regions could be calculated.   The  situation  in the pre-pollution
control situation could be compared to that  obtaining after pollution
controlmimicked as the  making available of previously unavailable
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siteswas implemented.   The "true"  results  then would form a benchmark
against which the results of various aggregation practices could be
compared.  It was also possible to establish intermediate situations to
provide additional comparisons, as for example by supplying each consumer
with a correct recreation-site price proxy as well as  the true travel-cost
prices of the nearest recreation site.
     The practices the simulation model was  designed to address were:
         the use of jurisdictionally-averaged travel price .proxies instead
         of individually correct travel-cost prices of recreation
         the use of models estimated for one region to project benefits for
         a "nation" of differing regions
         the use of a model estimated for the "nation" to project benefits
         for specific subnational regions.

Model and Calculations
     A sample of five hundred consumer/recreators,  each with a known
quadratic utility function, with two kinds of recreation and a composite
other consumption activity as arguments, were assigned randomly to
locations on a rectangular plane representing a "nation."  For one set of
runs the distribution of the consumers was uniform within subnational
regions that would later be taken to be the  smallest "political"
jurisdictions.  For another set the distributions used were truncated
normal, designed to mimic the peaking of population density in urban areas.
One of the recreation activities was taken to depend on "water" which was
in turn subject to pollution.
     Recreation sites for each type  of recreation were also distributed on
the plane on the basis of uniform distributions associated with subnational
regions.  (These regions were intended to represent different geologic
provinces.)  For the water-based recreation  activity the sites may be
thought of as equal-sized lakes.  Each lake  was assigned to either the
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class "available for recreation"  or  the  class "unavailable by virtue of
pollution" for the base case.   (Two  different extents of pollution were
assumed for alternative cases.  Three  and  30 percent unavailable.)
     The price of the composite consumption good was taken to be one, in
the units of consumer income.   Travel  cost was  taken to be 0.10 per mile
and no costs were assigned to  site access.
     The set of consumer maximization  problems  were solved for true
equilibrium choices of consumption and recreation, in days, in the
with-pollution situation.   The correct travel-cost price for each
individual calculated from his or her  location  relative to unpolluted sites
was used for this benchmark.   In  addition, for  the with-pollution
situation, several other pieces of data  were recorded:
         the correct recreation site density for each individual based on
         actual availability around  the  individuals home local
         the average density of available  sites within each jurisdiction.
     Aggregation of the average travel cost proxies was further explored by
combining the smallest jurisdictional  elements  into larger states of a
variety of average sizes,  on the  basis of  a randomized choice routine.  The
average sizes used were 5, 10  and 18,  where the entire nation consisted of
36 elemental jurisdictions.
     To produce the benchmark  benefits of  pollution control for the
simulated situation required the  following steps:
         applying the pollution control  policy  by making available for
         recreation all the sites placed in the polluted/unavailable class
         in the base case.
         re-solving the recreator-consumer optimization problems for the
         correct new travel cost  prices
         calculating the correct  measures  of the welfare change resulting
         from the policy (as equivalent  and compensating variation and
         several approximations thereto).

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     For the jurisdictionally averaged price  proxies  (recreation site
densities), alternative estimates of  the  welfare  changes were established
using a version of the participation  method.  That  is, participation
equations were estimated from the participation data  generated in the
with-pollution case,  with the key independent variable being the recreation
site density for the  jurisdiction of  residence  of the individual.  Such
equations were estimated for each level of jurisdictional aggregation.  The
change in recreation  days due to the  increase in  availability caused9 by
pollution control was projected in the usual  way.   This change could itself
be compared to the change in days of  recreation from  the equilibrium
solution to the consumer maximization problems.   In addition, the welfare
measures were approximated using the  average  value  of willingness to pay
for recreation days based on the results  of the individual optimizations.
(This willingness to  pay figure may be thought  of as  representing the
results of a separate willingness-to-pay  (contingent  valuation) survey.)
     To explore the effect of using national  studies  on the subnational
level, the national participation equations for alternative average
jurisdiction sizes were used to project changes in  participation in each
region taken separately.  These results were  compared, for both days and
willingness to pay values, to the true changes  for  that region.
     Exploring the effect of going the other  way, blowing up regional study
    
results to the national level was slightly more complicated than simply
reversing the national-to-regional chain  of calculation.  This was the
result of the small number of individuals within  the  average elemental
jurisdiction.  To obtain enough observations  to do  a  subnational equation
estimation it was necessary effectively to make up  a  new, larger nation
from four of the original nations. Each  of these original nations became a
region in the new conglomerate.   For  each such  region, the participation
equation was used to  estimate national participation  change due to
pollution control.

Results
     The results of using RECSIM to explore the aggregation problems of
central concern are interesting but more  than a little disquieting.  First,
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in using aggregated price proxies  it  appears  that we run very great risks
indeed.  While time and budget  constraints  did not permit a fine-grained
search for the level of aggregation at  which  the situation deteriorates
markedly, it is clear that by the  time  aggregation involves five elemental
units or more, we are in trouble.   At this  point, sizes and even signs of
participation change projections have become  unreliable.  While for some
individual trials, the aggregated  proxies produce equations with excellent
predictive ability, these are clearly random  events.  More likely are
events from the same distributions that produce predictions very wide of
the mark.
     This, it should be emphasized, is  not  a  problem with the proxies per
se, for when the correct proxy  is  assigned  to each observation, the results
are very close to those based on actual travel-cost prices.  It is a
problem arising from assigning  an  average (or aggregated) proxy value to
each observation in a jurisdiction.   As such, it is not surprising.  But
since data sets on recreation participation have never been rich enough to
allow calculation of individually  tailored  prices and since it is common
practice to use average price proxies (availability measures) this
particular manifestation of the aggregation problem must be viewed with
great concern.
     The exploration of the practice  of applying models at different levels
of aggregation than those from  which  they were estimated leads to similar
concerns.  This applies both to "scaling up"  from a regional case study to
a national participation (or benefit) estimate, and to "scaling down" from
a nationally estimated model to attack  a regional problem.  Again, both
procedures are commonly suggested.
     Overall, then, this study  suggests that  benefit estimation is even
harder than is commonly assumed.   While defensible methods for doing
participation-based benefit studies are available, the data necessary to
support those methods usually are  not.   This  implies that if benefit
estimation is to become a long  term and believable part of the policy
choice process, some substantial investment in data generation will be
required.  Such investments do  not come cheaply, but this study suggests
just how large their payoffs could be.
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                                CONTENTS

Executive Summary	    i

Chapter 1  - Aggregation Problems in Benefit Estimation:
 Simulation for Better Understanding  	    1
     RECSIM Model Structure  	 . 	    2
     Problems to be Explored  	    4
         Alternative Methods  of Approximating Welfare Changes	    4
         Aggregated Proxy Price Variables	    4
         Using Regional Case  Study Results to Estimate
          National Benefits	    5
         Using Nationally Estimated Relations at a
          Regional Level 	    6
     Plan of the Report	    6
     Anticipating the Results  	    7
     References  	    9

Chapter 2 - Some Theoretical  Background  	   10
     General Considerations in Designing Simple Simulation
      Models of Recreation Choice	   10
     Recreation Decisions and the (New) Theory of
      Consumer Behavior  	   12
     The Utility and Production Functions: Wants
      Versus Characteristics  	   15
         Wants or Characteristics	   15
         Indirect Utility,  Demand Functions for Wants and Goods,
          and the Structure of the B  Matrix of the Lancaster Model .  .   23
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     The Utility Function:  Selection of a Specific
      Formulation for Simulation  	    29
         The Indifference Map and Marshallian Demand
          Curves from the Additive Quadratic	    30
         Demand Functions from  the Additive
          Quadratic  Function  	    38
         The Indirect Utility Function from the Direct
          Additive Quadratic Utility Function	    40
     The Treatment of Time and  Visits in the
      Recreation Simulator 	    41
     Concluding Remarks	    46
     Appendix 2.A -  The Lancaster Model: An Overview	    48
     References	    55

Chapter 3 ~ Econometric Considerations 	    59
     Situations Where a Subset  of the Regressors are
      Observed Only  as Group Averages 	   59
         Distinction Between Classical Errors-in-Variables
         Problem and the Disturbances with Nonzero Means Problem. ...   61
         Parameter Bias in Mixed Models Using Individual-Specific
          and Group  Average Regressors: The McFadden and Reid
          Approach	*.....	62
         Another View of the Parameter Bias Problem in Mixed
          Models: The Theil Approach	66
         Implications and Obstacles	69
         Unknown Geographic Regions 	   72
         The Value of Additional Information  	 	   77
     Methods for Analyzing Demand and Hence Welfare Changes 	   79
         Single Equation Methods  	   81
         The Dual System-Wide Approach	84
     Concluding Remarks  	   36
     Appendix 3.A -  The Role of Recreation Resource Availability
      Variables in Participation Analysis 	   88
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         Relating Density  and Distance   	 	   89
         Some Implications for Aggregation: Measuring the
          Proxy for X	96
     Conclusion	97
     Appendix 3-B - The AIDS Model	98
         Estimation of AIDS - Some Specific Examples	103
         Calculating Welfare Changes: Exact, Almost Exact
          and Approximate  Measures   	  110
     References	119

Chapter 4 - RECSIM Model Design: The Data-Generating Modules 	   123
     Creating Information  on Individuals		125
         Elemental People  and Geographical Grids	126
     Poisson Module: Geographical Grid Placement of the Universe
      of Recreation Sites	128
     Steps to Place Fishing Sites	  .   128
     Policy Module: Selecting a Subset of Pre-Policy Fishable
      Sites from the Universe of Post-Policy Sites 	   134
     Steps for Policy	134
     People Module: Locating Individuals in Space  	   135
     Steps for People	135
     Passive Module: Locating Passive Recreation Sites in Space. ...   139
     Euclid Module: Connecting Recreation Sites and Individuals. ...   140
     Socio Module: Assigning Socioeconomic Attributes
      to Individuals	142
     Aggregate Module: Combining Elementary Political Units  to
      Form Aggregated  Political Units	145

Chapter 5 - Model Design:  Optimization and Welfare Calculation
 in RECSIM	152
     Preliminary Structure of Optimize 	   152
         Zero Consumption  Levels 	   157
         Scaling the RECSIM Utility  Maximization Problem 	   159
         Application to RECSIM 	   160
         Basic Data.	161
     Welfare Changes 	   162

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     Golden Section Search	165

     Approximations to Average Values Per Recreation Occasion 	  168

     Concluding Comments	174
     Appendix 5.A - Pitfalls  in Applied Welfare Analysis with

      Recreation Participation Models 	 	  175

         Origins of the Two-Step Method 	  178

         Valuation Issue	;	180
            Valuation with Marginal Unit Values	 .  . .   182
            Valuation with Average Unit Values 	   185
   Concluding Remarks	189
   References	191

Chapter 6 - Model Design: The Estimate, Evaluate and Compare
 Modules	194
   The Estimate Module	   194
        Functional Form	194
        Variables	195
   The Evaluate Module: Predicting with the Estimated
    Demand Models	201
        Single Demand Equation Model in Prices 	   201
        Single Demand Equation Model in Proxies for Price	204
   The Compare Module	205
        Criteria for Evaluation of Econometric Model
         Performance	206
        The Argument for  Non-Parametric Model Evaluation
         Procedures	209
        Non-Parametric Model  Evaluation Criteria 	   211
        Some Formal Non-Parametric Tests of Homogeneity	217
   Concluding Remarks	221
   Appendix 6.A - The Correct Calculation of Welfare Changes
    from the Estimated Single Demand Equation"Models 	   223
   References	225

Chapter 7 - Results and Discussion	228
   Aggregation of Price Proxies	230
   Aggregated and Disaggregated Use of Estimation Results	237
   Summary and Conclusions	239
   Appendix 7.A	242

Chapter 8 - Summary and Implications for Future Work	404
   Summary of Results	'	405
   Further Possible Research  Using RECSIM   	   406
        Aggregation of Different Response Relations
         for Same Activity	406
        Aggregation of Different Activities Under
         a Single Name	406
   A Possible Focus of Longer Run Work	407
                                      XI

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                                  CHAPTER  1

                 AGGREGATION PROBLEMS  IN BENEFIT ESTIMATION:
                     SIMULATION  FOR  BETTER UNDERSTANDING
     If we wish to estimate the benefits  of a public policy that accrue-to
society via routes involving reactions  by individuals to prices, we are on
conceptually familiar economic ground.    If we have data on the individuals
involved  their characteristics,  the  prices they face before and after
policy implementation,  and the consumption choices they made  we can
estimate the relevant set of demand equations and extract the individual
                     4
changes in consumer surplus attributable  to the policy.  The sum of the
changes over all affected individuals will be a proper measure of the benefit
of the policy.   Even in cases of policies affecting market goods, however, we
are never in such an excellent position.   Often we lack data on the
characteristics of individuals,  or  on the price changes faced.  Or our data
on all- other prices and choices may not be comprehensive enough to support a
complete demand system.
     Since these problems interfere with  our*ability to correctly estimate
policy benefits in market situations, it  should hardly be surprising-to find
that the difficulties are enormously greater when policies affect unpriced
activities or resources,  as they do in  the case of pollution-control
policies..  Then we cannot find market prices to attach to particular choices.
Rather, for example, travel costs of available recreation alternatives must
be calculated.   Neither are data on choices actually made thrown up
automatically by market operation but must be collected by special surveys.
Further, it may be impossible to be sure  that the categories of choices we
presume are appropriate,  and therefore  gather data on, are in fact the
categories used by consumers.
     Additional complications often arise when we try to match the benefit
estimates we can produce with the needs of our policy analysis.  For example,
we may, because of budget constraints,  do a regional study of a policy's

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impact though what we want is a national benefit estimate.  Or a national
study may have been accomplished when a subsequent  question calls for answers
specific to a regional problem.  Many of these problems  that arise as we move
away from the idealized benefit estimation situation can be thought of as
involving aggregation, whether of data or of  the estimates themselves.  This
broad compass of the word can easily lead to  misunderstanding, as one
person's aggregation problem need not be another's.   (Several of these
aggregation issues in the travel demand context are discussed in Koppelman,
et. al., 1976.)  Beyond* that, it remains an open question how important any
one of the aggregation problems may be relative to  the others .and relative to
other departures from the ideal.
     These questions cannot in general be answered  using actual data, because
the benchmark for comparison does not exist.   We have no actual data
sufficiently detailed and comprehensive to support  idealized benefit
estimation.  This, in a nutshell, is why we have undertaken to design and
build a simulation model reproducing the essential  characteristics of a
pollution control policy context, in which benefits accrue via individuals'
reaction to prices sensitive to the policy choice.   This model, RECSIM was
designed to produce true, if hypothetical, benchmarks against which to
compare results reflecting the operation of one or  another methodological
compromise, the use of surrogates for prices, or the existence in front of
the available data of one or another aggregation "veil." In this brief
introduction we acquaint the reader with the  structure of RECSIM.  Then we
describe the problems, of both methodology and aggregation, that RECSIM can
address and  rank these problems .according to our a priori judgement of their
importance and the extent to which simulation as opposed to theory is
necessary in examining them.  The judgement on importance reflects a logical 
ordering of the problems.  If the first ones, model  structure and variable
specification, cannot be shown to be handled  satisfactorily by the available
data and approaches, the latter ones concerning aggregation of results are
merely curiosities.
RECSIM MODEL STRUCTURE
     The recreation simulation model is designed to generate data on the
choices made by hypothetical individuals faced with alternative consumption
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P033ibilltie3 that include active recreation with the characteristics of
"fishing" and "camping," an urban recreation alternative one might think of
as movies, and a generalized alternative for all other ways of spending
money.  The fishing and camping activities take  place only at sites to which
travel is necessary (in general) from place of residence.  Fishing is
sensitive to an hypothesized "pollution" which makes  some sites unavailable
for the activity in the base case.  Camping is not sensitive to this factor,
and site availability does not change when pollution-control is hypothesized
to occur and increase the availability of the sites for  fishing.  The urban
activity also takes place at specific places, and these  are located within
areas of higher than average population density.  The composite activity is
site-less.
     The distribution of water "sites" and of consumers  is the initial
problem tackled by the model.  These are placed  randomly on a plain in
accordance with density function parameters that may  vary over sections of
that plain.  The sites and people are then contained  within jurisdictions,
which are artificial subdivisions of the plain at a finer level than" the
divisions used in the assignment problem.  One of the model's key
capabilities is the production of larger jurisdictions out of the initial
units in random ways so that shape and size vary across  model runs.
     Each consumer is endowed with income and time constraints and each faces
a set of travel costs depending on residential location  relative to sites.
The consumer's problem is to maximize utility, under  a quadratic utility
function and subject to the constraints.
     Pollution control policy, as suggested above, has the effect of
increasing the number of available sites and thus decreasing Cor at least not
increasing for any person) the travel cost to the closest fishing site.
Exact before and after welfare calculations are  possible in the simulation
format, and they may be compared with a variety  of approximate measures based
on methods in general practical use.
\.- The welfare measures of interest are dollar equivalents of utility  changes
 the compensating and equivalent variations; the approximation represented
by simple consumer surplus; and a variety of two-step method for aggregate
change, based on participation and unit values.
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     The model is structured to 'allow comparison of  welfare measures in a
variety of contexts, including:  model specification or general variable
speciftcation, average level of jurisdictional aggregation, and level of
study contrasted with level at which an answer is being sought.  The
comparisons are made using formal tests to reduce the impressionistic
component.
PROBLEMS TO BE EXPLORED
     We' intend to use RECSIM to investigate the seriousness for benefit
estimation of four common second-best practices.   It appears that theorycan
tell us what is correct in each case but cannot tell us  in advance the
direction or size of the error introduced by approximate methods.  RECSIM
will provide some of that missing information.
Alternative Methods of Approximating Welfare Changes
     The most important because most fundamental  question we shall put to
RECSIM is, what penalty do we pay for the use of  such ad hoc methods of
benefit estimation as the participation equation approach?  Because we shall
have both true benefits (compensating and equivalent variation measures) and
a "correct" approximation via a Marshallian demand function, we can be quite
precise about the impact of going the participation  route (often the only
practical route).  Further, we can see how this impact varies with the
simulation model's initial conditions, for example:   the severity of the
pre-policy pollution problem, the structure of the household production
"technology" matrix, and the extent of variation in  the surrogate distance
prices (availability densities).  This last opportunity is closely linked t'o
our second major question for RECSIM.
Aggregated Proxy Price Variables
     In recreation benefit studies actual individual-specific micro data on
recreation choices along with incomes, ages,  sex,  and other relevant
variables is usually available.  But other data,  especially data on the
prices of recreation alternatives Is usually not available.  This is true,
for example, of the hunting and fishing data set that we used in an earlier
study (Vaughan and Russell 1982), the 1975 MSHFWR survey.  In this survey,

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Individuals and their choices are finely characterized but virtually no
information is available on the relative price  sets  each individual faced in
making the reported choices.   Surrogates for prices, such as density
variables, must be found if econometric estimation is  to be performed.  (See
below, appendix A to chapter 3).
     Only by (1) locating each person on a very fine grid; and (2),
characterizing each person's recreation choice  set (at vast expense in time
and effort) can we improve on the use of a more aggregated proxy for the
price set  something like the density of relevant  recreation opportunities
in the general neighborhood (state or county).   But  a  priori, we would expect
the utility of the method to depend on whether  the aggregation level of the
available surrogate price data matched the actual  choice "horizon" of the
recreationista.  If, for example, recreationists tend  to make choices among
sites within 30 or 50 miles of home while the available  density data is for
averages by'state, we should expect the match to be  poor.  RECSIM will have
the capability of providing data at any level of spatial aggregation from the
smallest grid square within which average density  is constant up to two or
three major- subnational regions.  The method for aggregating regions will be
random, so that the benefit estimate comparisons can be  carried out for a
number of different arrangements all having the same average aggregation
1evel.
     RECSIM- was designed primarily with this type  of data aggregation problem
in mind.  This reflects its intellectual roots  in  our  recreation benefits
work and the fact that, for recreation,  data on individual choices but not on
the price vector behind those choices, are commonly  available.
Using Regional Case Study Results to Estimate National Benefits
     A form of aggregation of the benefit estimates  themselves involves going
from-regional caae studies to national totals.   Interest in this possibility
might be- said to arise partly because of historic  accident and partly because
of. research budget constraints.  In particular,  regional case studies have
long been popular with funders and researchers  because they were seen as
cheaper and more manageable.   But once a regional  study  is done, how can it
be used to get a national benefits?

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     For acme, this aggregation problem raises the moat interesting
conceptual economic problems although it is not the most challenging
econometrically.  This is because we are dealing here with "edge effects'*
across regional boundaries that are complicated by simultaneous but
presumably differential change on both sides of each boundary.  In other
words, we are flirting with questions of the importance of general
equilibrium models.
     RECSIM could be used to begin exploration of this problem by performing
simulated regional studies for differently defined regions and estimating
national total benefits based on regional per capita benefits (or regional
per capita participation effects) and national populations.  The effect of
using more or less care to attach per capita effects to income/age/sex strata
could be examined at the same time (eg: the use of simple per capita averages
could be contrasted with use of a projection equation having the key
socio-economic variables as independent variables and per capita benefits as
dependent variables.)
Using Nationally Estimated Relations at a Regional Level
     At the other extreme from the previous problem, existing national
studies, like our freshwater fishing participation model, seem to promise
money saving routes to benefit estimates for less than national situations.
For example, in the fisheries case, PIEC used our national model to estimate
benefits from acid rain control affecting the northern and eastern tier of
states.  Investigation of the penalties to be expected from this procedure
could also be undertaken using RECSIM.
     The "best" national participation equation (for any particular level'of
choice-set aggregation) could be obtained and then applied to "regional"
benefit questions.
PLAN OF THE REPORT
     After this brief introductory chapter, we lay the foundations for the
simulation model in chapters on theory (chapter 2) and econometrics (chapter
3).  In the former, we discuss consumer theory for situations in which zero
consumption is a possibility to be allowed for; describe the utility function
and household production approach to be used; and problems in the treatment

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of time in such optimization problems.  The latter  chapter deals with the
major problem for practical analysis raised by having seme variables
available only in aggregated form.   That  is,  the second of the aggregation
problems noted above is dealt with  on a theoretical level in chapter 3.  We
then go on to discuss alternative approaches  to- the estimation of demand
functions and their corresponding econometric implications.  Two technical
appendices to chapter 3 deal in more depth with (A) the role of recreational
availability variables in participation analysis; and (8) the multi-equation
demand system referred to as "Almost Ideal Demand System" or AIDS.
     There follow three chapters describing the design and construction of
RECSIM in more detail.  Chapter 4 provides an overview and then goes into
data generation.  In chapter 5 we discuss the optimization routine and
related correct welfare calculations.  Chapter 6. is concerned with the
estimation of demand functions, the evaluation of approximate welfare
measures, and the techniques for comparison of answers arrived at by
different routes.  A discussion of  results from RECSIM runs is provided in
chapter 7.  Finally, in chapter 8 we summarize, our  findings and discuss
additional research possibilities for the model.
ANTICIPATING THE RESULTS
     The results of using RECSIM to explore the aggregation problems of
central concern are interesting but more  than a little disquieting.  First,
in using aggregated price proxies it appears  that we run very great risks
indeed.  While time and budget constraints did not  permit a fine-grained
search for the level of aggregation at  which  the situation deteriorates
markedly, it is clear that by the time  aggregation  involves five elemental
units or more, we are in trouble.  At this point, sizes and even signs of
participation change projections have become  unreliable.  While for some
individual trials, the aggregated proxies produce equations with excellent
predictive ability, these are clearly random  events.  More likely are events
from the same distributions that produce  predictions very wide of the mark.
     This, it should be emphasized, is  not a  problem with the proxies per se,
for when the correct proxy is assigned  to each observation, the results are
very close to those based on actual travel-cost prices.  It is a problem
arising from assigning an average (or aggregated) proxy value to each
                                     7

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observation in a jurisdiction.  As such,  it is not surprising.  But since
data sets on recreation participation have never been rich enough to allow
calculation of individually tailored prices and since it is common practice
to use average price proxies (availability measures) this particular
manifestation of the aggregation problem  must be viewed with great concern.
     The exploration of the practice of applying models at different levels
of aggregation than those from which they were estimated leads to similar
concerns.  This applies both to "scaling  up" from.a regional case study to a
national participation (or benefit) estimate, and to "scaling down" from a
nationally estimated model to an attack in a regional problem.  Again, both
procedures are commonly suggested.  (To name only one example, we were asked
for help, in an attempt by PIEC to apply the national models in Vaughan and
Russell, 1982 to the analysis of the recreational fishing benefits in the
northeast to be expected from acid rain control.)
     Overall, then, this study suggests that benefit estimation is even
harder than is commonly assumed.  While defensible methods for doing
participation-based benefit studies are available,  the data necessary to
support those methods usually are not. This implies that if benefit
estimation is to become a long term and believable part of the policy choice
process, some substantial investment in data generation will be required.
Such investments do not come cheaply, but this study suggests Just how large
the-ir payoffs could be.

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                                REFERENCES


1.   Haitovsky,  Yoel;   1973.  Regression Estimation from Grouped
     Observations,  New York: Hafner Pre33.

2.   Koppelman,  Frank  S., Moshe Ben-Akiva and Thawat Watanatada.  1976.
 j	Development of an Aggregate Model of Urbanized Area Travel Behavior,
     Phase 1  Final. Report to U.S. Department of Transportation, Cambridge:
     MIT Center  for Transportation Studies.

3.   Stewart, Mark B.   1983.  "On Least Squares Estimation When the Dependent
     Variable ia Grouped," Review of Economic Studies, vol. 50, pp. 737-753.

4.   Vaughan, William  J. and Clifford S. Russell.  1982.  Freshwater
     Recreational Fishing, Washington, D.C., Resources for the Future.

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                                   CHAPTER  2
                          SOME THEORETICAL  BACKGROUND

     Before a useful  simulation model  of recreation participation  can  be
constructed it is necessary to clarify some underlying  theoretical issues.
In particular, we shall wish to explore the related questions of how to
represent the utility function presumed to lie  behind observed decisions and
how to allow for the  possibility of  zero consumption of some subset of
available goods and services.   Along the way the  issue  of how to include time
in the problem may usefully be addressed.   These  background matters are taken
up here.
GENEBAL CONSIDERATIONS IN DESIGNING  SIMPLE SIMULATION MODELS OF RECREATION
CHOICE
     In order to exploit  the calculus,  conventional utility theory makes the
implicit assumption that  the consumer's optimal consumption bundle will
represent an interior solution in the  space of  available alternatives.  That
is, the maTrtimmt of the consumer's utility  function  occurs at an interior
point, of the budget plane where all  goods  are consumed  in positive amounts,
not at a corner where one or more commodities are not consumed at  all
(Russell and Wilkinson, 1979,  p.  36).
     Quandt (1970) observed that this  implicit  assumption is unrealistic in
travel-oriented applications-,  since  consumers do not "undertake a  little bit
of travel by every mode on every link  in a network" (p. 5).  The same
observation could be  made about leisure activities  since it is a rare
individual indeed who dabbles  in each  and  every possible leisure pursuit
across the spectrum of possibilities from  sky diving to bird watching.
Rather, individuals pick  and choose, and engage in  some recreation activities
at the expense of others;  an observation which  is repeatedly confirmed by
surveys of recreation participation.
                                       10

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     Thus, the implicit interior solution assumption of conventional utility
theory must be relaxed, or .the theory itself reformulated, in order to
incorporate the phenomena of non-participation (i.e., zero consumption) in a
simulation context.
     The first alternative is to remain within the confines of traditional
utility theory, relaxing the interior solution assumption.  The corner
solution rationale for zero consumption in leisure pursuits is made by Ziemer
et. al., (1982) based on the Kuhn-Tucker conditions (see Silberberg, 1978,
Ch. 12).  Essentially this means ruling out the class of utility functions
where the marginal rates of substitution between pairs of goods are
everywhere defined and equatable to the respective goods price ratios.  For
example, members of the Bergson. family of utility functions which are all
transformations of the additive (in logarithms) homothetic utility function
U - Hx  are ruled out, since their indifference curves never cut the goods
axes,'and corner solutions cannot occur.
     Another route to explain the same phenomena, rather than restricting
attention to utility function formulations which allow for corner solutions
and excluding those which don't, is to reformulate neoclassical utility
theory along household production lines.   Lancaster's consumption theory,
which was initially brought to bear on travel demand problems in the 1970
Quandt volume, is such a route.
     The general form of the Lancaster model sketched in appendix 2. A
guarantees zero consumption of some goods, independent of the class of
utility function specified.  But conventional utility theory can be regarded
as a special case of the general Lancaster model.  In this instance, corner
solutions can be- produced by an appropriate formulation of the utility
function.  Therefore, the flexibility of the Lancaster model to represent
either, the neoclassical case with corner solutions or the "pure" Lancaster
case makes it an obvious choice for simulation, since one need not "believe"
in either model.
     But, as explained below, the general form of the Lancaster model
introduces complications of its own for the econometric analysis of the
outcomes it generates.  The econometric analysis becomes more tractable when
the Lancaster model is structured to represent the neoclassical utility

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maximization problem as a special case, with corner solutions allowed by
selection of the appropriate type of utility function.
RECREATION DECISIONS AND THE (NEW) THEORY OF CONSUMER BEHAVIOR
     In addition to the conventional theory of consumer behavior presented in
most micro texts, where utility is defined in goods space, there are at  least
two general lines of theoretical attack on the consumer's choice problem in
the context of the optimal selection of market goods and leisure activities,
including recreation.  The first, and perhaps most popular theoretical
construct in the recreation literature, is the Becker (1965)  household
production model (Deyak and Smith, 1978, Desvouges et. al., 1982, for
example).  The second, perhaps more appealing but less utilized approach is
the Lancaster household production model (Rugg, 1973, Mak and Honour, 1980,
and Greig, 1983* for example).
     Both of these theoretical models (reviewed in a general  context in
Deaton and Muellbauer, 1980 and in the recreation context by  Cicchetti and
Smith, 1976) are particular variants of the general approach  to consumer
behavior called household production theory.  In this "new" approach to
consumer theory, the household does not obtain utility directly from goods
purchased in the market.  Rather it employs these goods, along with its  own
time, to produce output of utility yielding, non-market goods over which the
utility function is defined.
     The new theory of the consumer may be decomposed into three basic
components:  A utility function, a production function, and  resource and
time constraints.  The utility function has as its argument a vector of
entities which may be variously defined to be those processes, events or
objects from which the individual or household directly derives utility.  The
production function is the technical relation which depicts the manner in
which market good inputs and time are combined to produce the vector of
utility generating entities.  The resource constraint may be  a simple
function of market goods, prices and household income or may  incorporate
additional constraints on household time.
     Becker's version of the new theory of the consumer assumes neoclassical
production functions and smooth convex utility surfaces.  Such assumptions
concerning the shape and differentiability of the functions permit the use of

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classical Lagrangian optimizing techniques as opposed to the programming
approach of Lancaster.  It is interesting to note that the connection between
either of these theoretical models and applied econometric work in recreation
analysis is often somewhat loose.  But, it is also true that the different
theoretical models appear to yield roughly equivalent equations to be
estimated from survey data explaining recreational trips.  Particularly, the
inclusion of income, site characteristics, and trip expenses is commonplace
(See McConnell and Strand, 1981, Rugg, 1973, and Ziemer et. al., 1982 for
superficially comparable "trips" equations derived from different theoretical
models).  Thus each theoretical model leads, generally speaking, to a roughly
similar estimating equation.
     The principal difference between the two new approaches appears in
practice to be that estimating equations are often derived from the Becker
model under the highly restrictive assumptions of non-Jointness in household
production (Pollak and Wachter, 1975) and a Cobb-Douglas type utility
function.  Taken together, these two assumptions imply that the only "site
prices," or proxies thereto, appearing in the reduced form participation
equation to be estimated are the prices of sites supplying the particular
service flow of interest (fishing for example) and not the site prices for
sites supplying complementary or substitute services.  Particularly, the
utility function defined on service flows must be of the sort that produces
demand functions which are independent of the (shadow) prices of other types
of service flows.  Similarly, the marginal cost function for a particular
service flow derived from the total cost function must be independent of the
level of output of any other service flow (Deyak and Smith, 1978).  Another
feature distinguishing the Becker approach from the Lancaster approach is
that the most general form of the Lancaster model posits that each input to
the consumption technology produces a set of Joint service flow outputs over
which the utility function is defined.
     For the purpose of constructing a simulation model of recreation choice
the Lancaster version is preferable as a manageable way to represent
hypothetical consumers and the universe of spatially distinct choices.  It
can be set up as a programming problem to generate realistic outcomes in the
sense that some consumers will choose not to recreate at all and others will

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choose some subset of available recreation activities- such as fishing and
swimming or just fishing alone.
    . For this simple model the optimal pattern and level of consumer choice
of goods (sites indexed by activity category) is a function of income, the
nature and particular parameters of the utility function (where in general
the latter may be functions of socioeconomic variables like educational
attainment, sex, age* race and the like), all goods prices, and goods
characteristics.  Thus a "loose" econometric specification of equations to be
estimated explaining choice of visits or participation intensity in broad
activity classes, which are viewed as goods, can be obtained directly, as in
Rugg (1973).
       There is a more elegant route to the same end.  However complicated
the structure of the model to be estimated, choice theory in general states
that the probability of selecting a particular alternative is proportional to
the representative utility of that alternative.  For a particularly complex
model involving a sequential recursive structure for trip destination choice,
trip duration choice and trip frequency choice, see Hensher and Johnson
(1981, pp. 312-316).
     The direct utility function in the Lancaster model can be written in
terms of goods to produce the individual choice problem:
     Max  U - u (Bx)
     S.t. y >. px
          x > 0
     The solution to this problem provides the equilibrium values of the x.
elements in.the x vector (i - 1,...,n), which are functions of all prices,
income and the parameters of the B matrix.  If we substitute this system of
demand equations g1(p, B, y) for the x.'s in the direct utility function an
alternative representation of the preference ordering in terms of the
indirect utility function with prices, income, and the B matrix as arguments
is obtained.  This formulation suggests the arguments in an estimated
probability-of-participation choice function:
     U* - u* (p, B, y).
     We turn next to some complications and particulars:  the meaning of
characteristics and the related coefficients in the consumption technology

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and utility function;  the form and properties  of  the  utility function assumed
for simulation;  and finally,  the role of  time  in  our  version of the
recreational choice model.
THE UTILITY AND PRODUCTION FUNCTIONS:  WANTS VERSUS CHARACTERISTICS
     In the Lancaster model,  market goods and  time are  the inputs in the
production of joint outputs of characteristics.   The  characteristics possessed
by a good are assumed to be the same for  all consumers  (Lancaster, 1966a).
Market goods themselves yield no direct utility.  Instead utility is a function
of the characteristics, which are assumed to be measurable on a cardinal scale
(Lipsey and Rosenbluth, 1971), and are "in principle  intrinsic and objective
properties of consumption activities"  (Lancaster, 1966b, p. 15).
Wants or Characteristics
     Lancaster's "characteristics"  are identified in  the literature with
Becker's "commodities" (Pollak and Wachter, 1975).  Both are outputs of
household production.   Generally,  in reference to the Lancaster-type model
the term characteristics is uniformly employed for the  outputs (Gorman, 1930,
Lancaster, 1966a and b) while when reference is made  to Becker's household
production model the terms "basic commodities" (Becker, 1965, 1971) "basic
goods" (Muellbauer, 1974), "underlying" or "non-market" goods .(Deaton and
Muellbauer, 1980) or "service flows'* (Deyak and Smith,  1978) are used
interchangeably for the outputs.  (An exception is Muth, 1966, who labels
inputs into household production "commodities" and outputs "goods").  The
essential feature of such an entity is seemingly  that,-while it is produced
from market goods  whose qualities  can change, its quality is constant
(Muellbauer, 197U).
     The several authors writing in this  area  seem to have some difficulty
reaching a consensus regarding the  practical definition of "commodities" or
"characteristics".  Indeed, such a  practical definition is in part a
philosophical question (Edwards, 1955, Ch. 3), and in a way it is irrelevant,
since such entities are often immeasurable in  the absence of clever
definitional legerdemain.   A  sample of attempts to capture this illusive
concept suggest the difficulty:
                                      15

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The main difficulty in the traditional theory is the assumption
that goods purchased in the market place,  food,  clothing,  theater
tickets, heating fuel, medical care,  and so forth are the  objects
of choice that directly enter the preference system.   Obviously,
this assumption is not literally true; for example,  food does  not
directly give utility, but only contributes to the "production"
of meals that do give utility.  Preparation time, shopping time,
stoves, refrigerators, knowledge of cooking, and many other
inputs are also used in producing a meal,  and food no more
directly produces utility than do these'other inputs.   (Becker,
1971, p. 44).

A meal (treated as a good) possesses nutritional characteristics,
but it also possesses aesthetic characteristics, and different
meals will possess these characteristics in different relative
proportions.  Furthermore a dinner party,  a combination of two
goods, a meal and a social setting, may posess nutritional
aesthetic and perhaps intellectual characteristics different from
the combination obtainable from a meal and a social  gathering
consumed separately.  (Lancaster 1966a, p. 133).

If we eat an. apple, we are enjoying a bundle of
characteristics-flavor, texture, juiciness.  Another apple may
have the same flavor but associated with a different texture, or
be more or less Juicy.  (Lancaster, 1966b, p. 15).

Suppose that the eggs have certain measurable characteristics A,
B, ... and that each egg of type i yields  a. units of A, b, units
of B, and so on.  Suppose moreover that these characteristics are
additive and do not interact, so that his total  consumption of A
is a - la.x. and of 3 is b - b.x. and so on. We can envisage
these as quantities of Vitamin A, Vitamin B. . ; (Gorman,  1980,
p. 843).

The term "characteristics1' was chosen for its normative
neutrality; in my earliest draft of this idea I  called them
"satisfactions," but that" has too many connotations.   (Lancaster,
1966b, p. 14).

The consumer desires to have satisfying experiences  and can
achieve satisfaction by using commodities [in this context,
purchased goods].  The great number of wants that he wishes to
satisfy arise from physiological and psychological needs...
Springing from the idea of differences between commodities and
the idea of differences between wants, comes the idea that
commodities have different want-satisfying powers.  Some
commodities satisfy certain wants but will not do at all for the
satisfaction of other which, in turn, may easily be  satisfied by
the consumption of some alternative commodities.  Essentially
this idea of want-satisfying powers is the notion that
commodities have different qualities.  These qualities are partly
due to the technical nature of the commodities themselves  and
partly due. to the nature of the wants they serve and the

                           16

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     .consumer'3 valuation of their effectiveness in serving these
     wants.   It is impossible to separate these two determinants of
     qualities since, although the technical characteristics of
     commodities are objective, each consumer's evaluation of them in
     relation to his own wants is subjective.  Nevertheless these
     subjective qualities are the links between commodities and wants.
     To  begin with, these qualities will be regarded as fixed and they
     completely specify the consumer's technology.  (Ironmonger, 1972,
     p.  15).

     Calories and alcholic content, for food and beverages, and
     distance travelled and travel time for transportation services,
     are examples of easily measurable characteristics, while
	 qualities like tastiness in food would have to be translated into
     a number of measurable, characteristics such as salt content,
     liquid content, etc.  Italics added (Lipsey and Rosenbluth, 1971,
     p.  133).

     A reading of the above quotations makes it obvious that no precise

 definition of what it is that enters the utility function of the consumer is

 available.  Nor can some of these entities be easily observed * some are

 objectively measurable characteristics such as vitamin and protein content,

 but others are vaguely defined, immeasurable (or difficult to measure)
 entities such as good health, flavor, pleasant climate, and beautiful

 scenery. Lipsey and Rosenbluth (1971) refer specifically to this measurement
 problem,, while Hensher and Johnson (1981) present a particularly confusing

 set of distinctions among objective characteristics of commodities - which

 they call features- * and the attributes of these commodities which are

 arguments- in  the utility function - which they call consumption services.
 For Hensher and Johnson, consumption services are functions of objective
 characteristics and there need not be a one-to-one mapping between them.

     But, there is another way to view all of this, which is presented in a

 particularly  lucid way in Ironmonger (1972) and ties into the Henscher and

 Johnson  view.

 Want Power-
     In  Ironmonger's lexicon, purchased goods are called commodities.  In the

 conventional  theory of the consumer, happiness (utility) is a direct function

 of the. quantities of commodities consumed.  But for Ironmonger, purchased

 commodities do not directly produce happiness (utility).  Rather, happiness

 (utility) is  determined by the degree of satisfaction of separable "wants"
                                 17

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which stand between utility, and commodities^  Separate wants (warmth,
shelter, entertainment, companionship, variety, distinction, knowledge,  etc.)
exist independent of goods  and services (commodities), but these commodities
possess differential want-satisfying powers.  Thus the objective
characteristics of commodities do not directly enter the utility function,
which instead is defined in terms of wants, which any reasonable person  would
admit are immeasurable.  Often all we can measure are the quantities of
market goods purchased which, in the recreation.case, are visits to sites,
each with its own "generalized cost".
     This view recasts the  Lancaster consumption technology and utility
functions in terms of wants which are no less vague than characteristics.
The game then involves how  the redefined B matrix is structured;  that  is,
which goods supply which want or combination of wants.  Specifically,  the b
                                                                          * J
activity coefficients in the consumption technology matrix are no longer
regarded as units of measurable objective characteristic i obtained from one
unit (a site visit) of purchased good j.  Rather, the b..  are redefined  as
coefficients representing satisfaction of want i obtained from consuming one
unit of good J.
     Ironmonger views the want satisfaction coefficient values as
immeasurable or subjective, being partly determined by the objective
characteristics of the commodity and partly by the consumers subjective
evaluation of those characteristics.  This suggests a complex model with
random coefficients in the wants consumption technology matrix, an
interesting but unnecessary complication for our purposes.
     If we assume that all  consumers translate objective characteristics into
wants in the same way (i.e., are members of what Edwards (1955) calls  the
same taste community) then  the want satisfaction coefficients are related to
measurable objective characteristics, albeit in an unknown way.  Then, if
sites are classified into groups which share roughly the same objective
characteristics, we can reasonably assume they share the same want
satisfaction coefficients.  For example, uncrowded coldwater game fishing
sites can all be considered to satisfy wants in a similar way, which is
different from, in general, the way crowded warmwater rough fishing sites
satisfy the same wants.  Additionally, one category of activity may satisfy

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an overlapping intersection of wants with another activity,  while both may at
the same time supply wants particular to each.   Or,  finally, subsets of  the
want space defined over activity categories may be mutually  exclusive.
Alternative B matrix structures are discussed below, based on these
possibilities.
The Nature of the Wants Matrix Representing the Consumption  Technology
     Lancaster originally assumed the B matrix for any individual contained
no linear dependencies, so its rank is the number of columns or  the number of
rows, whichever is less, implying that no characteristic (or want)  be
redundant.  Juxtaposed to this theoretical consideration is  the  practical
econometric dicta that spatial alternatives (destinations satisfying leisure
related wants) and their associated choices (fishing, boating, etc.) must  be
defined to encompass geographic zones within which the elemental alternatives
(sites) are homogenous or at least have equal across-zone variances in their
utility measure (Ben-Akiva and Watanatada, 1981). As Hensher and Johnson
(1981, p. 73) remark:
     In practice, it becomes necessary to limit the  number of
     alternatives in a choice set, regardless of the knowledge of the
     full set of relevant alternatives.  In other words, it  is
     necessary to group alternatives by design or randomly select a
     subset of relevant alternatives.  Furthermore,  data availability
     is such that often it is not possible to empirically specify a
     choice set containing elemental alternatives but only a lesser
     number of alternatives with some implicit  relationship  between a
     "group" alternative and a subset of elemental alternatives.
     In practice this means that the elemental  alternatives  (sites) must be
grouped in some way into sets of assumed identical alternatives. In our
simulation model this is done in two ways by construction.   Both methods
assume we know a priori the groupings of sites  which keep within-group
variation: in. the want coefficients to be nearly zero but permit  between-group
variation.  Essentially we assume that all fishing sites are alike, all
camping sites are alike, all tennis courts are  alike but that the groups
differ in their want-satisfying abilities.  Note that sometimes  the same site
can supply wants in boating, swimming and fishing at effectively the same
travel-cost based price for all three activities to  a certain individual.

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This means that some multipurpose sites appear as repeat entries in the B
matrix.
     Second, several partitions of the wants matrix are possible, apart from
the most general case of linearly independent columns, each column pertaining
to a site (e.g., each site a unique entity).  This possibility is ruled out
by the practical necessity of creating sets of assumed identical
alternatives.
     The first partition of the B matrix is the neoclassical case, where all
sites in a broadly distinct activity category similarly satisfy a unique set
of wants which does not overlap with the wants provided by another group of
sites in any other activity category.  In this special case utility can just
as easily be defined over goods as it can over wants.  Particularly, in the
recreation case, the consumer will select the closest site from each activity
category (all other sites in the activity category being inefficient) and
optimize over sites, since goods (sites) map uniquely into wants, and the
utility function can be defined over either quite simply.
     Given the general wants consumption technology z  Bx the neoclassical
variant of the general modal can be written in terms of the 3 matrix
partitioned as block diagonal;
            I" z,!
            [ Z2]
where
               xt ... x

                        sites unique to satisfaction of want  column
                        vector Zi of row dimension p.
      X- - x  ,... x    sites unique to satisfaction of want  column
            m+i    . m+n
                        vector Z2 of row dimension q.
and, for Btl and B22:
                                          b1m'
                                           2m

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              '22
                             b(p-H)(m+1)  "  b(p+l)(m+2)
                              (p+2)(m+1)
'(p-MKm+nj
'(p+2)(m+n)
                                         *  b
                                            (p+q)(m+2)
 (p+q)(m+n)
     Within any B. .  sub-matrix the elements in any  row are equal,  so  the
sites (x's) associated with that submatrix are perfect substitutes in
provision of the associated wants.
     For example, suppose p-q-2, m-2'and n-3.  Then  we have,  dropping  the
column subscript to  denote want coefficient equality:
              bi bt  0  0   0
              b2 b2  0  0   0
              0  0  b, bj  b,
              0  0  b,, b,,  b,.
     Performing the matrix multiplication gives:
     zz - bjX; * baxa
     z, - b,x, * b,x% * b,x,
     Take want zl and let a be any number between zero and one.   Then,
obviously a given level of want zlt say zlt  can be achieved either using
at fixed level x\ with x2 equal to zero;  or  with x2 set at the same fixed
level xz with xt equal to zero, or any weighted combination:
                                      21

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                          X2)
     Thus xt and x2 are perfect substitutes in production of wants and are
the same good since they, and they alone, supply wants zt and z2 in fixed
proportion (b^bj).  The same perfect substitution relationship in production
exists among goods x,, x* and x, which alone supply wants z, and z* in fixed
proportions (b,/b%).  All wants but one in each Zl and Z2 sub vector partition
of the z column vector are therefore redundant, since each row in the
corresponding submatrices of the partitioned B matrix can- be obtained as  a
linear combination of the elements in, say, the first row of each submatrix
BH and B22.  The consumer's problem simplifies to choosing one least cost
good (site) from each sub-vector Xt and X2 of the X vector partition, since
all goods within a partition satisfy wants in the same way but at different
costs.  Then, since all wants but one are redundant within the corresponding
partition of the 3 matrix the reduced B matrix is square with zeros on the
off-diagonal and row (and column), dimension equal to the number of X matrix
partitions, because the number of sub-vectors in the partitioned X vector
define the number of distinct goods (unique activity categories).  From the
example, then, any x in Xj is the same as any other, since they are all
representative of the same homogenous good, as is true of any x in.X2.  The
problem is then easily written either in terms of wants or goods, since the
former are directly proportional to the latter.  Normalizing wants on,  say zt
and z, to eliminate redundancy we can write two equivalent problems based on
the example:
     I.       Max U (zlt z,)
              S.t.
     II.
z,
-
o b,

 *
y - Ri*i * pxa
Max U (blxl., b,x2)
S.t. y - ptx, * p2x2
                                      22

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     The second problem is clearly the neoclassical case, but for the trivial
(and unnecessary) conversion of goods quantities into want quantities via ^
and b,.                                            .
     When the production technology matrix cannot be partitioned as block
diagonal, we have the more general Lancaster model.  In our simulation model
of the more general case we still maintain the assumption that all sites
within an activity category have the same want coefficients as- a matter of
computational convenience.  This assumption simplifies the process of
selecting sites on the efficiency frontier and reduces the size of the
                                             
programming model to be solved, since.for each individual only the sites
closest to his location (i.e., lowest "priced") in each activity category
need be considered in setting up the problem.  This within-category
homogeneity of sites assumption is adopted only to isolate the aggregation
issue front other complications which, along with increased realism, would
introduce additional computational costs.
     In passing, however, it should be noted that the conceptual link between
wants and site quality attributes is consistent with studies of angler
attitudes (Moeller and Engelken, 1972,  Sports Fishing Institute, 1974,
Spaulding:,-1971, Hendee, 1974, Hampton.and Lackey, 1975) which have confirmed
the notion that there are important dimensions of the sport fishing
experience other than catching and eating fish.
     Similar observations could be made about other general categories of
recreation activity such as boating, swimming,  and camping.  Our simulation
model is rather skeletal in regard to these complications, one of which,
congestion, has received some attention in Oeyak and Smith (1978).
     The next section discusses a problem of more immediate importance - the
nature of the goods (site) demand functions and the indirect utility function
produced by either the general or specifically neoclassical alternative
formulations of the Lancaster model, which has  important implications for
econometric estimation.
Indirect Utility, Demand Functions for Wants and Goods,, and the Structure
of the B Matrix of the Lancaster Model
    .The form of the econometric model  to be estimated from the simulation
data .depends on the structure of the Lancaster  model itself, particularly the
                                      23

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B matrix. Econometric details are discussed at length in chapters 3 and 6.
For now it is enough to know that it is necessary to arrive at expressions
for the indirect utility function, wants demand functions, and goods (trip)
demand functions from the Lancaster model in order to properly specify the
econometric model(s) to be estimated.
     A manageably small general Lancaster model with two wants and three
goods is adequate for illustrative purposes.  The problem is the-familiar
maximization of utility defined over wants subject to the consumption
technology and a budget constraint..  The x's represent trips to sites in
various activity categories and the p's the costs of travel:

     Max   U - F (zlvzs.)            .                    .
     S.t. zl - bii*i + blzxa + bj,x,
          za - baix! * baaxa * ba,x,
          y  - ptxt  * pzxa  * p,x,
          bij 1  pj >  XJ 1 

     Suppose the optimal solution to this problem involves a choice of a (non
vertex) combination of xt and x2, and a zero quantity of x,.  The shadow
prices of the wants zl and. za can be related to the column vector of goods
prices (trips to sites in various activity categories) actually consumed, p,
through the matrix B made up of the same goods (i.e., with the columns
pertaining to goods not consumed deleted by (Klevmarken, 1977):

     p - B'ir
           where ir is a r x 1 column vector of shadow prices.
         From the example:
        fPO -
1. The rtn shadow price is equal to 3U/3z  converted into a monetary
equivalent by multiplication by the marginal cost utility, 1/X,  (Deaton and
Muellbauer, 1980, p. 250).
                                      24

-------
or
         Pi " bl1  ffl * ^21*2
         Pi " &12  *2 * &22*2


     Also, if r S n, the vector IT is equal to:
         ir -
    .The inverse of Bf can be found by first calculating its determinant
|B'|: ^
     |Bf|   btl baa - bla bat
     Then, construct the cofactor matrix of all the elements of B and take
the transpose to get the adjoint of B:
and
      adj B
(Bf)
                -b
                  1 2
                   adj B' -
b    -b
 22    21
so
or, the shadow prices of the z's are, at the optimum:
                1
     it* -
                1
     If the individual's direct utility function is Cobb Douglas, we can
write it as:
     Ui  zraizaaa ,      a + a, - 1

The associated Marshallian demand functions for wants are:

-------
     z^ifi , ira, y) -  (o
     z2(irt , ira, y) -  (
-------
where  K - (bl2b2J - bllb2a)(olar)(aaa*)

     This illustrates the problem that the indirect utility function itself
gets redefined in terms of B matrix coefficients if goods prices change
because it is defined in terms of goods shadow prices, which are endogenous
to the model.  If we observe only the p's and the x's but not the IT'S or the
b's, the estimation problem is severe since the parameters of the indirect
utility function cannot be identified, the function itself is inherently
nonlinear, and any 'approximation'1 will be misspecifled.  To drive the point
home, suppose we insert the consumption technology matrix into the direct
utility function and maximize utility over the x. in the three good case.  If
we could derive the demand functions for goods (days spent by activity
category, an observable), these functions could possibly be captured in
estimation by a tobit-type model.                  
     To simplify matters, suppose the utility function is multiplicative
(tabia " bl2b2J)
Pi(baabia - blabaj)
                                                       Ps(bllb22 - bl2b2l)
                                   nba,, - blsb2l)
      i, - bl2b2J)
                                          - bx,bai)  * P3(bltbaa - bl2b2l)
                               y(bltbaa - biabal)
          Pi(baab1, - b12ba,)  * Pa(btlb2, - blsbal)  + Ps(bub2a  - bl2b2l)

     There are several unsavory features of this result.   The trips demand
functions are Inherently nonlinear.   Although each function has  a negative
own price response, the prices of all goods appear in each function in  the
demand system, which is due to the B matrix structure.  Finally, with more
                                      27

-------
complex utility functions, the goods demand functions will be even more
complex functions of B matrix parameters, utility function parameters,  and
goods prices.  From this exposition, it is clear that the. correct
specification of an estimable t obit- type model does not immediately fall out
of the theoretical Lancaster model, as some articles in the literature  might
suggest (Rugg, 1973, for example).
     If, however, there is a one-to-one relationship between goods and
characteristics so every good is "unique, " the B matrix becomes diagonal.
This just brings us back to a conventional demand theory case with no joint
  A
production of wants .and redefines the utility function in terms of days of
recreation of each generic type (and the composite commodity) .  Non-jointness
in the consumption technology greatly simplifies the problem, resulting in
Mar shall ian demand functions for the wants, z, of the form:
     z2 - a2y/(p2/l>22) - a2y/ir2

     In this instance, a naive tobit-type specification for the intensity of
participation in any of the recreation activities is straightforward.   So is
a multinomial logit approach in terms of the logarithmic indirect utility
function, -which in the simple Cobb-Douglas case of an additive- in-logs direct
utility function, with a diagonal B matrix (r wants equal to n goods) is:
     In goods:
                  N
     InVCp  , y) - I.a.lna  *. Iny - I a p
                 n-1 "            n-1
2. For an exposition of a sophisticated tobit-type approach to systems of
demand equations in terms of shares of expenditures on goods, see Wales and
Woodland (1983).  We do not pursue this elaboration here, but assume
estimation in the simple single equation tobit context.
                                      28

-------
     N                      .
     I . - 1
    n-1
     In wants:
                 N                N
     InVdr ,y)  - I a Ina  * Iny - I a p
          p     n-1  n   n        n-1 n n

Such a formulation is consistent with the multinomial/conditional logit  model
(Hensher and Johnson, 1981, p. 123).
THE UTILITY FUNCTION:  SELECTION OP A SPECIFIC FORMULATION FOR SIMULATION
     In previous sections, we have used simple multiplicative Bergson utility
functions for expository purposes.  But a specification of the utility
function which is computationally convenient for implementing a
recreation-choice simulation model of the Lancaster type is the additive
quadratic.  The individual consumer's choice problem then becomes a  quadratic
programming problem which can be solved by readily available computer
algorithms.  Further, the additive quadratic can be formulated to give zero
consumption of some goods even when the Lancaster model is given a
neoclassical structure, not allowing corner solutions.
     The additive quadratic direct utility function reflects the strong
assumption that the marginal utility provided*by one good is independent of
the consumption of any other good, so the second order  cross partial
derivatives of the utility function are all zero (Phlips, 197"*f p. 58).   This
does not imply, however, that the change in the price of the good leaves the
demand for any other good unaffected (Phlips, 197^, p.  63).
     Both-Phlipa (1971*) and Pollak (1971) remark that the additivity
assumption is often used in econometric work and is defensible if the
arguments of the utility function are taken to be broad aggregates of goods.
Our model is compatible with this notion if each "want" is regarded  as an
aggregate.  Further, Oeaton and Muellbauer (1980), as well as Ironmonger
(1972) noted that the paramount role of wants in the utility function and the
idea of independent wants owe their origins to the founders of consumer
theory.
                                      29

-------
     The additive quadratic utility function is not globally quasiconcave and
nondecreasing, so a satiation point (bliss) can be reached, marginal utility
can be negative, and the own Slutsky substitution effects can become positive
(compensated, demand curves can be come upward sloping beyond bliss).  Yet the
additive quadratic utility function is quasiconcave and nondecreasing. over a
subset of the commodity spacethe region southwest of blisswhich is the
region of the "economic" problem of choice.  Finally, the ideas of bliss and
disutility, although mathematically inconvenient, need not be an unrealistic
depiction of a consumer's preference structure.  In fact, indifference curves
of the sort generated by the quadratic utility function are discussed in at
least one elementary principles text (Watson and Holman, 1977), and the idea
of (partial) satiation is discussed at length in Ironmonger (1972).
     Specifically, the additive quadratic utility function is (Pollak, 1971):
                 R
     U(z) - y* - I ap(dr - zr)2
                rM
where we arbitrarily scale such that
          R
     y* - I d  - bliss
         r-1   ,
and
     a . d  - positive parameters
     zp  wants, r - 1.....R

     The cardinal properties of the additive quadratic are linear  marginal
utilities (3U/3z, - 2aid ~ 2aizi^ wnich can become negative for sufficiently
large z.; diminishing marginal utility (32U/32z.  - ~2a.);  and independence
O2U/3z13z  - 0).
The Indifference Map and Marshallian Demand Curves from the Additive
Quadratic

     To generate an indifference map for two wants from the additive
quadratic, the utility function can be rearranged in the general quadratic
form axa. * bx * c - 0:

     alzl* - 2aIdlzl * a^f - (y* - aa(d, - z2)*  - U)  0
                                     30

-------
     Given values for U, at,  a2,  di,  d2,  alternative values  can be  assumed
for z2 and the corresponding  value for zt  found by finding the positive root
of the quadratic:
     zt - (2atdt + CUafdf + Ua^y* -  a2(d2 - z2)2  - U *  ald?)]1/2)/2al
     Utility surfaces, indifference curves and Marshallian demand curves for
zl are plotted in figures 1 through 6 for two parameterizations of  the
additive quadratic.  The indifference curves are restricted  to the  positive
         3                                  v
quadrant.   From these plots  it can be seen that the dfs determine  the  '
position of the indifference  map and  the  a's determine its shape.   To allow
for corner solutions (zero consumption),  the positive quadrant restriction
can be removed.
     Further, the bliss homotheticity of  the indifference map is confirmed.
Specifically, for any indifference curve,  homotheticity  with respect to the
origin implies that the slope of the  indifference  curve  evaluated along a
radial expansion of an initial point  is identical  to the slope at the initial
pointr i.e., the marginal rate of substitution is  constant along a  given ray
from the origin.  That the indifference maps depicted are homothetic with
respect to- the bliss point, and not the origin, is easily shown.  Therefore
any ray from the origin that  does not pass through bliss violates the
necessary condition that MRS,  , be constant, and the only ray from  the origin
                             4
which satisfies the homotheticity condition is the line  that passes through
bliss from the origin with slope da/dt.
3- For the indifference curves to be  restricted to the  positive  quadrant
U 2 0 at zt - dt,  z2  0
U  0 at z2  d2,  zl - 0
To achieve this we select bliss (dltd2)  and adjust at and a2  to  restrict
UiO.  For zl - 0,  z - dz, for example,  and y* - dt  + d2:
0 - y* - at (dt -  zt)2 - a2 (d2 - z2)a or 0 -  dt * d2 - a^f
so
at & (dl  dz)/df.  Similarly for z2  - 0,  zt - dt:
a2 & (
-------
                                         Figure 1



                   Utility Surface for Quadratic Direct Utility Function

                 with Parameters a^O.16, a2 - 1.20, dt - 10;  dt  dz -  10.
  <  rtn
  tf\
" i c. J 
                         13-5
                                                                           A 11-33
                                             32

-------
                              Figure 2

      Indifference Map for Quadratic Direct Utility Function
      with  Parameters ai - 0.16, aa - 1.20, dt.- 10, d2 -  10.
19.2*.-
 S.SG-
I3.7S-
il.OC-
      '.CS2.75     5-53    S.:S    il-OC    ',3.75    iS-'^O    '.
                                  33

-------
                        Figure 3

      Marahallian Demand Curve for Zl with P2 - 1.
2   3
5   S
9   3
1
0
t
3
4
1
5
t
5
1
7
1   1
3   3
                            31*

-------
                       Figure U

 Utility Surface for Quadratic Direct Utility Function
with Parameters at  - 0.60, a2 - 0.60, dt - 10, d2  10.

-------
                              Figure 5  -


      Indifference Map for Quadratic Direct Utility Function
      with  Parameters at - 0.60, aa - 0.60i  dt - 10, d2- 10.
19.25-
16.50-
13-75-
II .00
 3-25 =
 5.SO-
 2.TS
 0.00
       .00     2.75     5.50
            il-OS    -.3.75    iS-'jS    13.2V
       LECtNC; U
 2
10
 4
12
 S
U
                                    36

-------
                Figure 6


Marshallian Demand Curve for Zt with Pz - 1
PI
20-
19-
iS-
13-
12-
11-
10-
9-
,.
;

L

:
.
'
"
1
r

L
,
,.
1
,
I
I
;
i
i
i
;'
t
t
t
t
t
\
I
0   1
  557390
1
2
1
3
i    1   i
4    5   S
1
7
1    1   2
930
                      37

-------
Demand Functions from the Additive Quadratic  Utility Function-
     To derive a ays tan of Marshall ian demand functions for wants from the
additive quadratic utility function,  utility  defined over wants is maximized
subject to the budget constraint.  In this section, we treat want prices as
if they were exogenous to arrive at a demand  function specification.  This
procedure is only legitimate if the consumption technology matrix is diagonal
(the neoclassical case) so that want shadow prices can be directly obtained
                                                           *
fron goods prices.  Then it does not matter for the analysis whether utility
is defined over goods or wants.
     Take the simple case of two wants zl  and z2  whose prices (or shadow
prices) pt and p2 are exogenously given.   Then, ignoring the nonnegativity
constraints and confining the analysis to the southwest region of the
commodity space below bliss, the ordinary first order conditions for a
constrained maximum can be invoked to produce the demand functions (for an  
extensive treatment based on the Kuhn-Tucker  conditions, see Wegge, 1963).
The problem iss

     Max   U - y* - ax(dt - zt)2 -
     S.t.  ptzt * paza - y
     y* - bliss utility level
      y * actual income

     To find the. first order conditions for a local maximum, form the
Lagrangian:

     L - y* - at(d! - zt)a - aa(dz - za)a  - \ (plzl + p2z2 - y)
Differentiating the above with respect to  zlt z2, and \ yields:
     3L/3zt - 2al(dl - z,) - ;pl - 0
or   zl - dt - X(pl/2al)
     3L/3za - 2az(d2 - za) - Apa  0
                                      38

-------
OP   z2 - da - A(pa/2a2)
     3L/3X - Pjzt + p2z2 - y - 0
     To  find  the equilibrium  value  of  \,   labeled  \0,  substitute  the
expressions for zl and z2 into the budget constraint:
                                2 - Xpa/2a2)
so
          2a2pf + 2a.rf}
                                      2 - y)
     Inserting the value for \0 expressed in terms of the parameters,  prices,
and income into the first order conditions expressed in terms of zt  and zz
produces the demand equations:
     za  - da -
     Cancelling terms:
                                      i  * P2d2 - Y)
                                         P2d2 - Y)
Pi
aT

 Pz
2a2
        /     aapv    \          /   a2pt    \          /  aap!      \
T-*II -  - pa + . Oa I  d^!  -  I    a  . .  Oa|   
-------
                             n
      h (P, y.) - d, - Y.(P) .1 d  pk +  Y.(P)y
                       i    k-1 k K    i

where  P  is the price vector (plf ....  pn) and Yi(P) is:

               pi/ai
     Y,(P)
              n
                PkVAK
The  Indirect  Utility Function from the  Direct Additive Quadratic Utility
Function
      The  indirect  utility function defined over prices and income associated
with the  additive  quadratic utility function defined over goods has been
derived by  Pollak  (1971).
      The  general result  is:
                R                  R      R
      V(p.y) - y(I  a ~1p  *f 1/2 *  (I
                       r
where    drp  is "bliss"  income,  y*.
     This  indirect  utility function  is  intrinsically nonlinear, i.e., not
linear with respect to the parameters.   Thus,  if the choice model to be
estimated  from participation data requires a specification of the indirect
utility  function, and extant maximum likelihood algorithms are to be.
employed,  a second-order approximation  to any  indirect utility function must
be used  which is linear  in the parameters (e.g., Christensen, Jorgenson and
Lau, 1975).  The alternative of  specifying a choice model in terms of an
intrinsically nonlinear  indirect utility function  (an intrinsically nonlinear
demand function) requires a tailor-made solution algorithm for estimationan
interesting but impractical alternative. This option is particularly
impractical since when choice models are estimated from real world survey
-*rr'TKg~qiiivalence is trivial because In  the  two good cases, for example,
 finding the common denominator in Pollak's expression for Yt(P):
               P/aa
                            aia        40

-------
 data--not  data generated by a simulation modelthe form of the utility
 function is  unknown  to  the analyst.  The most that can be said in such cases
 is that the  specification of a strictly linear utility index function in
 terms  of the untranaformed explanatory variables involves a gross
 approximation.   Such a  practice  is common in the literature.
     Having  discussed the consumption technology and the utility function,
 one final  component  needs to be  added to complete the modela leisure time
 constraint.   The introduction of a leisure time constraint assumes that
 working time is  institutionally  fixed and not a matter of personal choice
.(Sherman and Willett, 1972).
 THE TREATMENT1 OF TIME AND VISITS IN THE RECREATION SIMULATOR

     The distinctions among days of recreation, visits to recreation sites,
 and hours  of on-site recreation  can be important in certain recreation
 planning contexts.   For our purposes, however, there is little to be gained
 and much to  be lost  by  maintaining these distinctions.  Most importantly,
 differentiating  visits  from time on-site and allowing for the choice of both
 over some  consumer planning horizon creates a nonconvex optimization problem
 for each of  our  individual consumers (illustrated in Rugg, 1973).  This would
 vastly complicate the task of finding each person's optimum in a simulation
 model  context because multiple local optima would in general exist.  Either
 multiple-starting points or some search procedure over the feasible corner
.solutions  would  be- necessary, and computational costs would rise while
 reliability  (of  the  model's response to exogenous change) would decline.  In
 this section, we describe the simple method we propose to use in our
 simulation model and describe how the more complicated method gives rise to
 problems.
     The simplest method of handling time in the recreation simulator is to
 assume that  a "visit" is not of  variable length.  Then, when a person decides
 on a number  of visits to site j, say v., he is choosing to pay a total cost
 c.v,  (where  c, is the cost per visit to site j and to obtain amounts of the
  J J          J
 "wants" z.,  given by z,  bnv<  ^or ^^ *  Tne &,, are wants supplied per
 visit.. We can think of each visit to site J as being of fixed and prechosen
 length 31,. so that wants supplied per hour are b. ./s,.  But these refinements

                                      41

-------
 add nothing because the s. cannot be chosen in the optimization but are fixed
 in  advance.
      The  structure of the resulting problem is simple and mathematically
 convenient.   To illustrate this structure,  we  can use the case of two wants
 and two sites.   The consumer's problems is  to
      maximize     U(zlt zz)
                                .  biiVt  * &iaV2 - z.
    -  subject to   z - x'B  or
      and          CtVj  * czva $ f
                   tiVr  + tav, S T
 where c.  is cost per visit v to i;  Y  is  income;  t,  is time required per visit
 (including travel and on-site time) to i;  and T  is  total time available for
 recreation over the consumer's planning  period.  The income and time
 constraints are stated  in v space and can  be graphed as in figure 7  The
 feasible set of visits  is indicated by the hatched  border.  The constraints
 in  v space translate into corresponding  linear constraints in z with an
 analogous convex frontier, also indicated  by a hatched border in figure 3.
 It  is straightforward to show convexity in  z  space algebraically.  The common
 sense of the illustration can be  seen from  observing that all the b. . must be
                  C     T
  0 so that if Vt  > vt , for example, then:

        P1         C        T    T1
      Zi    - ti*i  > biiV - zi

             C1         C        T      T1
      and  z    - biV   > bv   -  z
 The individual's optimization problem in the  z's  is therefore conveniently
 convex.
________ The sensible notion that it is time on site that  produces want
 satisfaction,  suggests attempting to separate the  decisions about number of
 visits to and  time on site at site J .  Let  us call the former v  and the
 latter s , .  Then total time on site over the  planning  horizon is v.s. .  If

-------
                     Figure 7
           Problem Constraints in v Space
<  ?r
*r
                     Figure 8
           Problem Constraints in z apace

-------
                                    T--V,C,-C2

                                _J   	f~ "* 'w ,
                                  Figure 9
              When Travel and Site Time are Decided Separately

costs are still assumed to be incurred per  visit,  the total costs are c,v. 
                                                                      J J
     But we must also take account of the time used to get to and fron the

site.  The new consumer problem looks like  this (again with 2 wants and 2

sites):

     Maximize     UCzjZ,)

                         tVi3t * blav2sa
     Subject to   zt -
                   ll
where t'  represents travel time onl/o
5. Further complications are introduced if we assume  that there are costs per
unit time on site-  But the problems we want to avoid creep in even without
this.

-------
     The easiest  way  to  see  how this  formulation creates  a problem of
nonconvexity is to look at the second constraint  in v.3  -space,  the  analog
of v-space in the simpler  formulation.   From figure 9 we  can see  that the
distinction between travel time,  on-site time and  number  of  visits  implies
that the total hours at site 1  when only site 1  is visited is T - vlsl, but
as soon as site 2  is visited at  all, the time available  for on-site  use is
immediately reduced  by ta.   Thus, giving  up an hour at site 1 does not
imply that  an hour  is available to use  at site 2, only that an hour is
available for the  trip to and from site 2, which may or may  not leave any
time  to spend  at  site  2.   The  disconnected  points  must  be  examined
separately from the linear constraint for combinations  of sites.
     Actually, the problem is even more difficult  in the day-trip  context.
The complete problem specification would require  numbers of days and time
per  day available  for travel and recreation  to  enter the  constraints,
giving  numerous  discontinuities   in  the  set of  feasible  combinations of
trips and visit times.   For example,  if  the planning horizon were only 3
days, the  amount of each day available for  recreation  10  hours,  and the
choice between two  sites,  the  first at  round trip distance  1  hour and the
second  at  round  trip distance  2 hours,  the  possible  choices can  be
summarized as below and illustrated in figure 10.
Possibility                                 Max  (vlsl,VjS2)
3 Visits to site 1                                 27,0
2 Visits to site 1  > 1  to  site 2                   18,8
2 Visits to site 2, 1 to site 1                     9.16
3 Visits to site 1, 1 to site 2             (25, 0 to 18, 7)
2 Visits to site 1, 2 to site 2             (16, 8 to 9,  15)
3 Visits to site 2, 1 to site 1              (7,  16 to 0,  23)
3 Visits to site 2                                0,21
     Any of the disconnected points or some point  on on one  of the
combination lines could be chosen under  certain  shapes  of the indifference
curyest.   But all must be explored individually since no myopic optimization
algorithm can "find its way"  through the maze.

-------
                                  Figure 10
     When Travel and Site Time for Several Sites are Decided Separately
 CONCLUDING REMARKS
      The theoretical  foundations  of one particular theory of recreation
 activity choice have  been laid in this chapter.  There are many other ways  of
 describing the problem.   Several  are catalogued in a travel demand context  by
 Bruzelius (1979).   But,  the Lancaster model as sketched above is adequate as
 a practical vehicle for  exploring the aggregation question associated with
"availability1* variables, despite its naivete in the time dimension and its
 assumption of  site homogeneity within any activity category.  To isolate one
 problem, others must  be  given minimal attention, and those choices have been
 made explicit  here.  However, it  is our belief that this same type of model,
 made richer in the time  and site  attribute dimensions, could also be
 fruitfully used to explore other  questions in recreation participation
 analysis.  The general issues of  the nature and specification of econometric
 models fit to  participation data,  and the link between systems of travel  cost

-------
site demand models and models of individual participation choice are
particular examples.

-------
                               Appendix 2.A
                     THE LANCASTER MODEL: AN OVERVIEW
                                                             *
     The Lancaster type model (Gorman, 1980, Lancaster 1966a, 1966t>,  1968,
Ironmonger, 1972), has at its heart the consumption technology matrix B whose
                                         A.W  *
elements b.  define the quantity of the i   characteristic possessed  by a
                      th
unit quantity of the J   market good, and each good j has its own vector of
characteristics associated with it.  Thus each consumption activity (each
column of the 3 matrix) has a single input - a purchased market good  - and
several Joint outputs - the characteristics.  If the number of rows in the B
matrix is less than the number of columns, and the rank of the B matrix is
the number of rows then no characteristic can be redundant in the sense that
it can be obtained as a linear combination of other characteristic rows.
     If there are r characteristics and n goods (and r < n) the column vector
of characteristics, z, can be written in terms of the r x n consumption
technology matrix B and the n x 1 column vector of purchased goods
quantities, x:
     z  Bx.
     The consumer's choice problem is to maximize utility defined on
characteristics space subject to the consumption technology relationship and
a budget constraint.  (A time constraint can also be added, but this
introduces special problems which are discussed in a separate section below).
The consumer's optimal choice set is given by the solution of the nonlinear
programming problem:
                        Max  U - u(z)
                        S.t. z - Bx
                        y  px
                        x i 0
                        where y - income.
                    p - 1 x n row vector of market goods prices.
                    x -'n x 1 column vector of market goods quantities.
                                      U8

-------
     This is the simplest Lancaster model.  Note first that if the
consumption technology matrix is square with zeros on the off-diagonal,  each
good has a single characteristic associated with it which is unique to it.
This special case is the traditional representation of the consumer's choice
problem.  Second, the simple model can be generalized to cover
complementarity in consumption by a two-matrix consumption technology,
(Lancaster, 1966a) an elaboration we ignore.
     The optimal solution to this problem is either- given by a point of
tangency between a facet of the production characteristics surface and the
indifference surface (assuming preferences are convex) or a vertex optimum.
At a facet optimum (Klevmarken, 1977) the individual consumer will never
consume more than r goods*   In this case a column vector ir of goods shadow
prices of dimension r x 1 exists and is related to the column vector of
                                               A
observed market prices of the r goods consumed p'  by:
     pf  B'TT
where B is the consumption technology matrix made up of the columns
pertaining: to the optimal subset of r goods chosen from the set of n possible
goods.. So, the budget constraint is satisfied at the optimum either as  the
product of the quantity of goods optimally chosen and their associated market
prices or as the product of the characteristics levels selected and their
shadow prices (Klevmarken, 1977).
     The optimal solution can be displayed graphically for two
characteristics- zt and z2 and several market goods  The graphical
representation depicts the two components of the choice problem:
     1..  The efficiency choice determining the frontier of the attainable
         characteristics set given a price vector and income.
     2.  The personal choice determining the preferred characteristics
         combination on the characteristics frontier.
6. With such a theory of consumer behavior,  if all consumers were to have
identical preferences the number of goods available in the market would
always equal the number of characteristics,  since each Individual consumer
optimum implies the consumption of no more than r goods.   But Klevmarken
(1977) notes that if indifference maps differ across consumers,  there may be
more goods in the market than there are characteristics.

-------
     An example consumption technology for five goods, common to all
consumers, can be written as:
  z2 - b2l xt * b22 x2 * b2, x, + b2, x,, + bas x,

     Suppose recreation site visits are regarded as goods, and sites xt  and
x2 are fishing sites, x, and x,, are.camping sites, while xs is a Hicksian
composite non-recreation good above subsistence requirements.   Further  for
simplicity assume all camping sites yield the same characteristics per visit,
as do all fishing sites so blt - bia, b2I - baa and b13 - bl(,, b2, - b2%.
Every consumer considering a day trip to any site faces a different site
price constellation determined by his location relative to the locations of
each recreation site.  Thus each consumer's set of site prices can be
different over the j - n-1 sites.  For the i   consumer the "price" of a
visit to the J   site, assuming on-site costs are zero for all consumers and
ignoring the value of travel time is:
  PJ " dU I
            where
                 d.. - Round trip travel distance from individual i's
                   **                    ^H
                       location to the j   site.
                  c. - Variable travel cost per mile for individual i.
     The budget constraint for the i   consumer with the price of the
composite commodity normalized to one is therefore (for y  committed
expenditures for subsistence, and yQ discretionary expenditures equal to
y-yc):

     yQ 2 Xj dt c * x2 d2 c * x, d, c + x% d,. c + x,
7. The artifice of subsistence expenditures is introduced in this example to
avoid the unrealistic possibility of a consumer engaging in recreation
activities alone with zero purchases of other commodities (the composite).
When a total leisure time constraint is added to the model, the possibility
is no longer feasible.
                             '         50

-------
From the budget constraint, the efficiency frontier for individual i can be
obtained by calculating the maximum quantity of each good obtainable if all
of the budget were allocated to it, and then translating these quantities
into z space.  Suppose dx < d2 and d, < 
-------
                                 Figure A, 1
                       Zero Consumption Possibilities
     Moreover, indifference curves which are not homothetic to the origin but
which: ar identical across a subset of  consumers (see the more detailed
section on the utility function below)  add additional richness to the model.
As income expands, given fixed market-goods  prices and consumption technology
parameters, the characteristics frontier will expand linearly and
proportionally with the increase in income (Lancaster, 1966a, p. 140), which
reflects constant returns to expenditure. But, if the indifference curves
  -                                 *   
are homothetic to the bliss point zlt za of,  say, a quadratic utility
function, the optimal characteristics combinations selected and hence the
optimal goods combinations required will depend on income levels, even when
individuals face the same goods prices  and share the same utility function.
Thus, as shown in figure A.2 for individual  2, at lower incomes such
individuals may not recreate (corner solution at Gs) while at higher income
levels, they may.  In figure A.2 all C points are associated with Y which
is less than Y*.  The higher income Y*  produces G* points.  The
bliss-homothetic indifference map implies equal marginal rates of
                                      52

-------
                                Figure A.2
                      The Effect of Bliss Homotheticity
substitution in consumption along any ray from bliss.  At 7 the point Gj is
optimal, and the consumer  purchases only the composite commodity but does not
recreate.  But, with higher income G, which is along the same ray from the
                                         *
origin as Gj, is not optimal.   Instead, Gl  is chosen, implying a combination
of composite commodity purchases and camping site visits, as given by the
tangency of indifference curve  Ij (utility at 112 > utility at l) and the
line segment Gj - Gj.
     Finally, income and the parameters of the utility function can be fixed
for all individuals to demonstrate the effect of variation in goods prices
across space.  The parameterization isolates the location effect from income
and taste effects.  Suppose that of two individuals, one is located
                                     53

-------
sufficiently close to fishing sites so that his characteristics frontier  is
as given in figure A. 1,  with preferences It.  But, another individual  may be
located such that the site price of x,  is sufficiently high to move the
maximum attainable point along the 0 -  G, ray inside the line segment  Joining
G! and Gj, so combinations of fishing and camping or camping and the
composite good became inefficient, and that consumer will not engage in any
camping at all, whatever the nature of  his utility function.  This result is
shown in figure A.3.
                                 Figure A.,3
                             Ruling Out One Good

-------
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McFadden,  Daniel  and  Fred  Reid.     1975.    "Aggregate   Travel  Demand
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McConnell, Kenneth E.  and  Ivar Strand.   1931.   "Measuring the Cost of  Time
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                                      56

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 Moeller,  George H. and John H.  Engelken.   1972.  "What Fishermen Look for
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 Muellbauer,  John.   1974.   "Household Production  Theory,  Quality,  and the
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 Muth,  Richard   F.    1966.    "Household  Production   and  Consumer  Demand
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 	.	J9J1.   "Additive Utility  Functions  and Linear  Engel Curves," Review
      of Economic Studies,  vol.  38,  no. 4,  (October), pp. 401-414.

 	__-  and Michael L. Vfachter.   1975.   "The  Relevance of  the Household
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 Quandt. Richard E.,  ed.   1970.   Demand  for Travel; Theory and Measurement,
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 Silberberg,  Eugene.    1978.     The   Structure  of  Economics  (Mew  York:
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 Small,  Kenneth A. and Harvey S.  Rosen.   1981.  "Applied Welfare Economics
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      pp.  105-130.
                                      57

-------
Smith,  V.  Kerry  and William  J.  Vaughan.   1981.   "Strategic Detail  and
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Spaulding,   Irving   A.     1971.     "Occupation,   Recreation  and   Physic
     Communication:  Selected Rhode  Island  Sport  Fishermen,"  Bulletin  No.
     405, University of Rhode Island Agricultural  Experiment Station.

Sports Fishing Institute.  1977.  SFI Bulletin, no.  286 (July)

Vaughan,  William   J.   and  Clifford  S.  Russell.     1982.     Freshwater
     Recreational   Fishing,   the  National   Benefits   of  Water   Pollution
     Control.  (Resources for the Future:  Washington,  D.C).

Wales,  T.  J. and A.  0. Woodland.   1933.   "Estimation of  Consumer  Demand
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     4th ed. (Boston: 'Houghton-Mifflen) pp. 92-94.

Wegge,  L.   1968. "The Demand Curves from a  Quadratic Utility Indicator,"
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Ziemer, Rod, F., Wesley M. Musser, Fred C. White and R. Carter  Hill.   1982.
     "Sample Selection Bias  in  Analysis of Consumer Choice: An Application
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     (April) pp. 215-219.
                                      58

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                                  Chapter 3
                          ECONOMETRIC CONSIDERATIONS
      In this  chapter,  we discuss one major econometric problem arising in
.participation modeling of the sort  to be explored with the simulation model.
 Then we go  on to a more general consideration of alternative routes to
 capturing welfare changes and their econometric implications.
 SITUATIONS  WHERE A SUBSET OF THE REGRESSORS ARE OBSERVED ONLY. AS GROUP
 AVERAGES
      The essence of the problem of  estimating recreation participation
 equations from survey  data on individuals' participation choices is that such
 data usually  contain no information on the vector of  (travel-cost based)
 prices for  the choices facing each  individual in the  sample.  If
 individual-specific activity price  data were available, participation
 equations based on a. cross-section  of such data could be conceptually
 regarded as. equivalent to demand equations for the activities, since once an
 individual  has- fixed the location of his residence, travel-cost based
 recreation  activity prices are exogenous, and no identification problems
 arise (assuming a neoclassical structure of the 3 matrix discussed in Chapter
 2).
      In the absence of such data, practitioners have  traditionally
 substituted proxy "resource availability1* variables measured at some level of
 spatial aggregation beyond the individual.  The practice has not been
 formally justified,  but rather has  been explained as  a reasonable way to
 account for the effect on participation patterns of (the regionally
 differentiated) supply of recreation-related resources such as lakes,
 campgrounds,  natural forests and the like.  But, the  presence of
 "availability1'' variables as regressors in participation equations bears only
 a loose link  to economic theory in  most of the studies surveyed, despite
 apparent reasonableness.   One might think that the only problem for
 econometric estimation raised by the appearance of such proxies is the
                                     59

-------
well-known bias in parameter estimates introduced by their inclusion,  as
covered in standard econometrics texts (Maddala, 1977,  for example).   But,
when the more fundamental question of just what " unobservable1*  the proxies
are intended to measure, and what role this unobservable plays  in  the
theoretical model is asked, some confusion arises.
     In appendix 3.A we make the argument that, if  correctly measured,
availability variables are really not proxies in the usual sense,  but  instead
bear a close relationship to the price variables that belong in the true
theoretical model.  Indeed, they represent the expected value of such  prices
within particular geographic boundaries, assuming individuals are  uniformly
distributed in space.  This latter assumption is critical, as is explained
below.
     In the version of the errors-in-variables problem commonly appearing in
econometrics texts, the proxy variable x* reflects  the true value  of the
variable x with measurement error u so x* - x + u.   In multiple regression
models the question usually addressed in this context is whether or not the
proxy variable should be an included regressor, or,  would one be better off
without it?  The evaluation criterion (in a simple  2 independent variable
model) often is whether or not the mean square error of the parameter
estimate attached to the regressor measured without error is reduced by
inclusion of the proxy measured with error (Aigner,  1974, for example).
Under certain circumstances* inclusion of the proxy is not recommended.
     But, our situation departs from the usual proxy explanatory variables
problem in two respects.  First, for recreation participation analysis
concerned with benefit estimation of water quality  improvements, the effect
of which works through the proxy, the option of excluding the proxy in
estimation to reduce bias in other-parameter estimates is not open.  Second,
the usual econometric analysis treats the unobservable variable as fixed and
the proxy as stochastic.  In our case, the proxy (A., proportional to
expected site price in region i) is fixed within any region i and  its
divergence from the unobservable variable, person-specific site price  for
each individual in that region, is nonstochastic, being completely determined
by the latter.
     Thus, the question we want to answer is, broadly,  under what  conditions

                                     60

-------
la the use of proxies for site price justified in the econometric estimation
of recreation participation equations or, indirectly,  to produce benefit
measures.  More narrowly, it is which, if any,  proxy price variable,  measured
at alternative levels of spatial aggregation,  yields models whose performance
(in terms of unbiased parameter estimates and prediction accuracy)  is as good
as models estimated from the correct price measures.  Econometric theory can
reveal a good deal about the bias question, as shown in the sections  that
follow.  The predictive performance question will be assessed in a simulation
context using our RECSIM model, and is not addressed here.
Distinction Between Classical Errors- in- Variables Problem and the
Disturbances with Nonzero Means Problem
     The classical errors-in-variables problem (EIV),  with error measurement
on a (set of) r.h.s. variable(s) can be described as follows.  Assume the
true model is
          Yt - o + Xtft + et'

where X   is (1 x 1).  We observe only X*, however,  and know that X*  - X.  +
ik, where u, is generally assumed to be N(u,-
  L;.  However, in the present case (zonal averages),  the N(0,  a2) assumption
does not characterize appropriately the state of  affairs.  In our case, there
is no stochastic aspect of the errors-in-variables,  i.e. for each individual
L, knowledge of the X  and the proxy X^ (which proxy is analogous to  the X*
above) is sufficient to identify the u  characterizing that individual.
Repeated draws of the same individual would yield identical values of u ,  but
u  will in general, be nonzero.  Thus we are left with a degenerate case of
EIV, with ^ - N(u, aj), but u-0 and a*-0.
     In the linear model, however, such a case can be  cast in terms of the
problem of "disturbances with nonzero mean" (DNZM),  (see Schmidt,  1976,
pp. 36-39).  We can write our EIV model as:
          *L - a * 3 (*L + ut) * e.t
             - a + SXL * (&UL + et)                                  (1)
             - a * SX  + v
                                     61

-------
 Mote here that E(\i)  - BU^  Var(Vl)  - Var(ei)  -  a*  because  a* - 0 by
 assumption.   Ordinary least squares  (OLS)  applied to  (1) will yield biased
 and inconsistent estimates of (a,  8), but  using  the Theil-McFadden-Schnidt
 methodology outlined below the "correct" estimates  can be backed-out.
      The degree of bias in the parameter estimates  of models specified to
 include a zonal average regressor  (our A.  variable) along with other
 regressors measured at an individual-specific  level can be  derived
 theoretically,, following McFadden  and Reid (1975).  (The problem is common in
 transportation demand analysis,  since the  procedure of collecting some
 individual-specific survey information and supplementing it with information
 on other variables measured as group or zonal  averages often reduces the cost
 of data collection.)  The demonstration of the bias resulting from such
 procedures in McFadden and Reid (1975)  is  both ingeniously simple and
 intuitively  obvious.  We display it in full below, followed by a simple
 numerical example to clarify the argument.
 Parameter Bias in Mixed Models Using Individual-Specific and Group Average
 Regressorst   The McFadden and Reid Approach
      Assume  the classical multivariate linear  regression model satisfying the
 standard assumptions.   In matrix form:
           (Y|X) - X8 + u                                            (2)
 where Y is a vector of n observations on the dependent variable conditional
 on the observed values of the independent  variables, X, X is a n x k matrix
 of "observations on the independent variables,  u is a vector of unobservable
 stochastic disturbance terms and C(u)  - 0,  E(uu') *  
-------
 G                                           n
 I n  - n.  Assign the group mean x  - 1/n   5x.   to all observations in each
(7 1                                85 i1   

group rather than the individual observations x.   originally in the vector X2.
                          ^                     
Call the resulting vector X2.   The partitioned model using a mixture of

individual observations (the matrix Xx) and group means ()C2) is (assuming no

interaction terms between the two partitions):

          Y - alXl + a2X2 * u                                        (4)


     The least squares parameter vector for the mixed model is:

                                      -1

                         XJX,
                                           XJY.
                                                                     (5)
     Substituting, the true expression for I from (3)  above in  (5):
                                           xixt  I  xj

                                           ??Y  I  7t
                                           Al  .1  1  *Z
                         x*xx
                 X'X2
                                      -1
Xju


Xju
                                                                     (6)
If the stochastic error term is independent of  the explanatory  variables  in

the mixed models so E(uXx)  - E(uXa)  - 0  the last term on the r.h.s. of  (6) is

zero in the limit.  Then in large samples the biaa of the a  vector is:

                                     -1
            S
                                                                     (7)
     Because X2X2 - X2X2 the above expression simplifies  to
                                                           1
                                                        g
1. Within any group the matrix product X2X2  equals  x    * x.  .  Within

the same group,  where every observation i  is assigned the value x  , the
                           n          _                    '     *
product  X2X"2 is equal  to J8?*  or n^x*.  which can also be written as


v1   KX-
   ag i-i
               jr  "  "tf a'

   "g      1H                                 "g     1    "g
   I x1(-).  Cancelling, this is equivalent to (I x,  )(-   I  x. J
'. J 4  5"                                      *  1   5  ^**^ 4  1   3
g i-1             n                           i-1       g i-1
or the expression for 5C2X2  of  x
                                     63

-------
                                      -1
0
0
X[(X2-X2)
0

-!L
                                                               	   (8)
      Note  that  If the  sub-matrix  XJ(X2-X2)  (analogous  to a  covariance
 matrix)  is zero,  then the  parameter vector Cot:a2]'  is unbiased.   This
 means  that if the matrix  of variables measured as zonal  averages  (or the
 column  vector  in our case)  is  orthogonal  to  the matrix of  variables
 measured  at  the  individual level,  no  bias in  parameter  estimates  is
 introduced by the averaging  process.
      Even  if  X[(X2-X'2) is not  a null matrix, McFadden and  Reid  note  that
 consistent parameter  estimates  can be obtained by  expressing all variables
 as zonal averages and estimating on the group means.    Alternatively, prior
-outside-of-sample information on the matrix X{(X2-X2)  can be used for a bias
 correction using individual  information, since X2X2. in the first matrix on the
 r.h.s. in  (3) is known (equal to X2X*2) and the matrix XJX2 can be obtained as
 the sum of the outside of sample matrix XJ(Xa-X~2) and the within-sample matrix
 y y
  * *
      But,  If orthogonality does not hold, then the parameter vector [Stsa2]'
 estimated from the mixed model will be biased and inconsistent,  as will be s2,
 the estimate of 
-------
                           Table  1.  Example Data


Group Obs.
1
(2x1)
2
I
3
*.
5
6
II
7
3
Intercept
(Xt) X2
1 -30

'.I',. -20

1 -10
1 0
1 10
1 20

1 30
1 0

x,
10

-20
*
20
20
__-
-10
-30

10
0

x,
7.5

7.5

7.5
7-5
-7.5
-7.5

-7.5
-7.5

y
90

40

130
140
90
60

150
100

    -The partitioned matrix to  be  inverted  in (3) above is easily calculated

from the data.  The appropriate sub-matrices are:
X'X
(2x2)
XJX2 -
(2x1)
~3 0

0 2300
"o
-500

          X2Xt   -  [   0  -900   ]
          (1x2)
X2X2  -
(1x1)
                      450
          The second sub-matrix  in  (3) above is
          X'(X2-X2)   -
          (2x1)
                 0

                 400
                                    65

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     In terms of (8) we have
100
1
2



~100
1.8
3.6

-

                                  80      0
                                  0   2800  -500
                                  0  -900    450
-1


000
0 0 400
o "o o
 m



r -\
100
1.8
.3-6
 
     Or, after inversion and multiplication
100
1
2

-

100
1 .8
- 3*6-

-

                                  0.125  0         0
                                  0      0.000556  0.000617
                                  0      0.001111  0.003457
     So, as expected from the McFadden and Reid derivation:
   0
1440
   0



B 
100
1
2_

-

100
1.8
3.6

-.

0
0.8
1.6



Another View of the Parameter Bias Problem in Mixed Models: The Theil
Approach
     Theil (1971) takes another route but reaches the same conclusion as
McFadden and Reid (1975).  The McFadden and Reid proof is preferable in the
practical sense that it expresses the appropriate bias correction factor  in a
lucid way which makes it possible to use out side-of-sample information to
remove the bias inherent in averaging.  But, Theil's demonstration is
instructive, and some may find it easier to follow, so we summarize it here
for completeness.
     Theil treats the problem in a model specification framework,  with the
true model being the same as (2) above:
          Y - XS
    (9)
     If a specification error is committed by replacing the last  column's
elements with group averages the new matrix X0 is identical to the original
                                     66

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matrix X but for that column,  K.   The parameter vector of  the incorrect
regression, a is then:

          a - (XJXor1XJY                                            (10)
Again, substituting the true relationship for Y from (9) in  (10) above under
the assumption that X and X0 consist of nonstochastic elements and  taking
expected values
          E(a) - E((xjx;rx;(xs + u))  - p,s                          (11)
where Pa - (XJX0)~TX;X.   Equation (11)  says that a linear relationship exists
between the expectation of the parameter vector produced by estimating the
misspecified mixed model,  a,  and the true but  unknown parameter vector 3.
     The transformation matrix P0 is the key to forming. Theil's auxiliary
regressions, because the (K x K) matrix Pa can be regarded as  the coefficient
matrix of the regression of the correct (but not completely known)
explanatory variables X on those used in the mixed model,  X.  The auxiliary
regressions are a didactic device demonstrating the unknown relationship
between the correct and misspecified variable  matrices:

          X  X0P + matrix of residuals                             (12)
     If X and X0 are both of order n x  K but differ only in the Ktn column
due. to- replacement of tfte x   with their group means x^  ,  the  hth element of
E(ct) - P08 can be written in terms of the p..  elements of  P0.  When only one
             A.W                            l"&
column,  the K  , differs between X and  X0, the matrix P0 can be partitioned
along the row h - K-1  to form an upper  matrix  Pj of row dimension K-1 by
column dimension K.  The lower matrix P is a  row vector (the  fC   row) with
column dimension K.  The upper matrix PJ has unity elements for p   along the
diagonal where h - k and zeros elsewhere in the partition of Pa dimensioned
as h.  1, ..., K-1,  k - 1, ..., K.   The remaining k - 1,  ..., K elements of
P, in its K   row (the partition PJ) can be estimated from the auxiliary
regression on the i * 1, ..., N observations:
                K-1
                                       re3i<1Ual                      (13)
           iK *   ,  "hk'ih * pKK*igK *

                                     67

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where x... is the true value of observation i and x, K is its group mean.
     Given the results of the auxiliary, regression in (13). the expectation
of the K parameters of the mixed model can be written in terms'of the
(unknown) parameters of the true model and the p   as:

          E(ah) " 8h * phk8K          ' h - 1	K-1               (1U)

          E(ah) - >KK8K                h - K

     So, in general, if the vector of the K   column of X is orthogonal to
the K-1 columns of all other variables the p_  will be zero, and the
coefficients of these latter variables will not be subject to any bias due to
estimating the mixed model.  In the special case of using group averages  in
the K   column, this means that the coefficient of the K   variable will  also
be unbiased, since orthogonality implies pvv  1.
                                          tvK
     Particularly, it is easy to show that if p^ - o for all h - 1,  ..,  K-1
the vector of original individual observations is always proportional to  the
equivalently dimensioned vector of n observations where the group means have
been substituted, in place of the original observations*  The factor of
                           "3             t*ta
proportionality equals 1.O.3  So if the K   column of the matrix X of
individual observations is perfectly orthogonal to its remaining K-1  columns
3. The relationship x,  PKK* ,, where x .  represents the i   individual's
group mean, leads to a parameter estimate for p of
The numerator and denominator of p,-. are equal because:
                                  tux
                                G      n              G   n_      n
                                1      ** "  l   l  x    l
and
      G            G   n
      ^  xiA "  &   
          1S is   g-T i-1
                                     68

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the same column of X0 will be also,  and no damage at all  is  done  by
                          i
                           4
substituting group averages for individual  observations  in the Kth column and
estimating the mixed model.
     Returning to the example data where K - 3  and X,  (indexing the vector of
ones for the intercept as Xt) was replaced with its group mean, we can estimate
Theil's auxiliary regression:
          x, - 0 * 0.4xz + 1*8*g3                                    (15)
                                                                   *
So PU - 0; pa,  0.4; and p,, - 1.8.   From this information  the relationship
between the a and 8 parameters is, from (14):
          E(ot) - fft + PuBj - 100 f 0(2)  - 100
          E(aa) - 8a + pa,8, - 100 * 0.4(2)  -1.8                    (16)
          E(a) - PsaS, - 1.8(2)  - 3.6

Implications and Obstacles
     The results above implies that use of a density proxy for price  can
indeed bias the parameter estimates of  OLS-type participation models, but
that the bias could be removed, if somehow the  appropriate "correction"
matrix can be obtained from, say, a separate survey sample, or,
alternatively, if a two-step probit/OLS model could be estimated from
averaged data.
     But., use of a density based proxy  for price introduces additional
complications.  First, even if the correct geographical delineation of the i
regions with distinct \,  population parameters  could be found, four problems
remain.
No Variation in A Values Across Regions
     This problem is trivial.  That is, if there is no spatial variation in A
so X  - X, for all i,J, then there will be no spatial  variation in X.  In
this-case, a model in prices could be estimated but a  model in A could not.
4. Another way to see this is that in a simple  linear regression model with an
intercept and one independent variable, y.    80 *  B1x,  + e., substituting group
means in place of x^ will never have an effect  on  the estimate of 8, provided,
of course, we have more than one group.
                                     69

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Sites of Varying Size
     As discussed in appendix A to this chapter,  if sites  are not  equi-sized,
an acres per acre proxy is not directly derivable from a sites per acre-
proxy, and the neat connection between site density and expected price will
be broken.
Edge Effects-
     Each region-specific \  can itself be a biased measure of the expected
value of region i's travel-cost based price vector because of edge effects.
Since individuals in geographic region i can travel across regional
boundaries to recreate, and will in fact do so if they can find a  site in  .
another geographic region with a lower travel-cost based price than the
closest site in their own region, we can expect in general that

          2c(1/2A~1/2)  2cE(dji)                                    (17)
where:      2c - round trip cost per unit distance travelled
                 expected distance from any arbitrary point 1
                 the closest site in region i,  based  on the  density measure
     1 /2
1/2A.     - expected distance from any arbitrary  point in region i to
        E(d. .) - expected value of the true distance to the closest site for
                 recreation for all individuals j  living (but  not necessarily
                 recreating) in region i.
     This systematic error in measuring the expected value of  travel distance
will be transmitted into a systematic bias in the parameter estimate attached
to the travel-cost based price variable.  The effect is analogous to a simple
change in the units of measurement of the variables in linear  regression
(Griepentrog et. al., 1982).  If the ratio

          2c(1/2xT1/2)
                 1      -  k > 1                                     (18)
            2cE(di )
is constant across all regions, then the parameter attached to the  density
proxy (a*) will be smaller than the parameter attached to the expected value
(group mean) of travel-cost price because:
                                     70

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          a* - ~-a                                            '     (19)

But, while systematic error leads  to biased parameters  It will have no effect
on goodness of fit or "t" statistics.   Edge effects  will have little  or no
effect in prediction of participation changes,  in the sense that the  model
predicting with the biased a vector (before application of the McFadden and
Reid or equivalent Theil correction to get the  correct  parameter vect.or, 8)
will predict equivalent!/ to the model with the biased  a* vector reflecting
edge effects, provided these effects are constant across regions.
     Of course, neither model will be  correctr  since neither provides an
unbiased and. consistent estimate of the true parameter  vector 8.  Moreover,
if  the severity  of  the edge effect   is not constant  across  regions so k
4 k  for some i,j  the problem cannot be described so simply.  Rather, when
the edge effect factor k. varies across regions the  situation becomes
analogous to that discussed below  under the problem  category of unknown
geographic regions.  But, before turning to that problem, a fourth difficulty
remains, even when the correct geographic boundaries are known by the
analyst.
The Distribution of Individuals in Space
  -"The fourth problem is perhaps the most severe.  If a large number of
individuals are uniformly distributed across a  geographic region
characterized, by \, the expected  value of the  vector of individual
site-visit distances will be T/2X~1/2.  As long as all  individuals face the
same travel cost per mile, c, (or  the distribution of travel costs per mile
is Independent of the distribution of  distances) the expected value of the
vector of individual one-way site-visit prices  will  be  c1/2X.~    (or
         1 /2
E(c)1/2X,    ).  However, if individuals are not uniformly distributed in
space, so that the probability of  an individual being located at any  randomly
chosen point of latitude and longitude is not equal  to  the probability of His
being located at any other randomly chosen set  of grid  coordinates
(individuals are clustered) then the expected value  of  the vector of
individual site-visit distances (or prices) conditional on the distribution
of individuals in space will not correspond to  1/2X  ~1/2.  The probability of
observing any particular distance  is not Independent of the probability of
                                     71

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observing any particular individual location.  So, when people are not
uniformly distributed in apace the conditional distribution of the vector of
individual site-visit prices is not uniform, with every possible individual
location (and hence site-visit price) receiving equal weight in the expected
distance calculation.  Instead, some individual locations (and therefore
prices) get more weight than others in the expected value calculation.  Under
                         -1 /2
these circumstances 1/2X,     is a biased measure of the expected value of
the vector of individual distances, but the direction of the bias is, in
general, unknown.  Even using the technique of regression on the group means,
this bias in the proxy will be transmitted to the parameter estimates.
     Up to now, we have assumed the geographic regions delimiting, the various
population X. values reflecting the geological process of water body
formation are known a priori.  Yet this assumption is almost never satisfied
in practice, when political jurisdictions at some arbitrary level of spatial
aggregation (state, county, etc.) dictated solely by data availability are
used to define the regions over which separate X values are computed.  This
question is addressed next.
Unknown Geographic Regions
     It is rarely possible to demarcate geographic regions distinguished by
separate population X parameters, especially when density data are reported
by political units  states, counties and the like.   But, if we assume Wiat
X values pertain to elemental spatial units which are no smaller than, say,
counties, something constructive can be done, at least in the OLS framework,
even in the absence of exact geographic demarcation of unique areas,  each
distinguished by its population density of water bodies per unit l-and area.
     Specifically, suppose X values are computed at the county level.  At
this fine a level of spatial disaggregation, it is doubtful that much harm
will be done by not knowing the exact geographic agglomerations of adjacent
counties which constitute a unique geographic region in terms of X.   The
county-specific X values can.be regarded as sample estimates of the
population X value attached to the correct agglomerated geographic region the
counties belong to.  If such an argument Is plausible, the principal
confounding problems are those already discussed.  The county average density
values are only estimates of the unknown X of the unknown super-county
                                     72

-------
geographic region (population)  to which they belong,  but  in  expected value
terms the X's in each county equal the super-county  \.  Although the extent
of distorting edge effects is likely to be  exacerbated  by using sub-area
(county) X density measures in  lieu of the  unknown super-county A, It is an
unavoidable consequence of the  recommended  procedure.
     A procedure that is definitely not advisable  (but  was used by the
authors in a former study due to data deficiencies)  is  to use very large
political units (states, or even census regions) as  the.unit over which
density measures are computed.   If the chosen political unit happens not to
coincide exactly with the super-county geographic  unit  characterized by a
unique population \, great harm in terms of parameter bias can be done by
measuring density over the large political  unit.
     The rationale for this claim is simple.  The  correct state density
measure could be built up from  the density  measures  of  the counties belonging
to it as a population-weighted  average.  The county  density  data must be
population weighted to reflect  the relative composition of the sample of
state residents facing different expected densities  within the state.  As
such, the state measure will be the "true"  expected  density  at the state
level, conditional on the distribution of individuals across counties within
the state.  By choosing a reasonably small  unit of spatial areathe
countythe problem of severe nonuniformity of population distribution within
the- spatial unit referred to previously is  mitigated somewhat (while edge
effects are exacerbated somewhat).  By using a population-weighted density
measure to aggregate up from the county to  the state level,  the correct
conditional expectation of density at the state level is  obtained.
     But, if density is measured directly-at the state  level as the number
(or acres) of water bodies per  unit land area, it  is equivalent to an
area-weighted average of the county expected densities, and, as sucH, is not
consistent with the expected density conditional on  the distribution of
individuals in space, unless county area and county  population are perfectly
proportional.  Likewise, a simple average of county  densities within a state
(i.e., a mean of county means)  also is not  the expectation of state density
conditional upon the distribution of population within  the state.
     An example will perhaps illustrate the effect of using  large spatial

                                     73

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aggregates (e.g.., states) and computing the mean of an explanatory  variable
as the average of the mean values of the variable in each sub-unit  within the
aggregate (e.g., counties).  Suppose we have the errorless data in  table 2
for counties A, B, C and D generated from the function y.  - $l  + 82x2. +
SjXSi where i indexes the 1, ..., 12 data points in the sample  and  8t - 100,
8Z - 1. 83-2.  County A has 2 observations, county B,  3,  county C, 5, and
county D, 2.                            .
     With our errorless data, a regression on the 12 individual  data points
(or any three of them, for that matter) would reproduce the true 8  vector,
with an R2 of 1.00.  But, what if counties are combined into "states" for
purposes of measuring X, as a state average, and that average substituted
for each true X, observation in the 12 element column vector for X, in a
regression?  Does it matter how the averages are computed?
     In table 3 we show mixed model regression results for  a few possible
four-state, three-state, and two-state county combinations.  (In the
four-state case, each county is itself a state).   In situations  where two or
more counties form a state, we show the regression results  under two methods
of computing the expected value of X, for estimating a (biased)  mixed model.
The correct method computes the expected value of X, conditional on the
county population distribution within a "state" and is the average  of the
individual observations for the counties whose union forms  the state.  The
incorrect method is analogous to what happens when a density measure is
constructed inappropriately for a large political jurisdiction in space, and
involves computing the state average for X, as the mean of  the county means,
a biased measure of the true mean when there is an unequal  number of
observations per county.
     Inspection of\the initial regression results and Theil's auxiliary
regressions in table 3 reveals:
       All averaging methods produce biased parameter estimates and biased
        measures of s2 (and hence R2).  The R2 measure always falls due to
        averaging, as expected from Schmidt's results on the upward bias in
        s2.
5. Note that if each county had equal populations  of  individuals (i.e., equal
sample sizes) the problem discussed in this section would vanish.
                                     7U

-------
Table 2.   County Data from Relation y  - 100 * 1x?t * 2x-t

County 
114
A
104
1 48
B 152
126
90
' 510
C 1100
815
1325
460
0
780
X2
10

-20
20
20
-10
-30
10
0
15
25
20

30
Xs X,
2

12
14
16
18
10
200
500
350
600
170

325
'
I 7.00

V
 16.00



332.00



, 247.50

                           75

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     Table 3.  Regression Results for Different "State" Combinations

Initial Regression
Groups/Averages*
1.
2.
3.
4.
5.
6.
7.
8.
True Model
I,B,Cf,D"
AB.CT
AB,CD
C.ABD
C,ABD
D'.AB.C
ff,AB,c
QI ctj oij R
100
54.
-318.
-1588.
-45.
-78.
52.
54.
18
55
73
75
99
32
64
1
9.26
15.88
15.88
13.25
13.25
9.33
9.33

1
3
12
2
2
1
1
2
.91
.66
.66
.29
.39
.92
.91
1 .00
0.68
0.48
0.48
0.67
0.67
0.68
0.68
Auxiliary Regression
PlS Pl P$J
n.
-22.
-209.
-844.
-72.
-89.
-23.
-22.
a.
9T
28
36
87
49
59
63
n.a.
.4.13
7.44
7.44
6.12
6.12
4.17
4.16
n.a.
0.96
1.83
6.33
1 .15
1 .20
0.96
0.96

The union or elements in two or more groups such as AUB is denoted as  AB.
Correct means calculated from individual observations for X,  in the union  of
two or more groups are denoted with- a bar, e.g. AB.  Incorrect  (i.e.,
unweighted) means calculated as the average of the group means  for X,  are
denoted as a tilde, e.g. A.
      The extent of parameter bias is never reduced by forming arbitrary
      "super- groups" (states) as agglomerations of counties,  whatever
      averaging method is used to form the average variable X, for
      observations in each super-group.
      The degree of parameter estimate bias of the models using the correct
      mean vector of X, is always less than that using the incorrect
      "mean- of -means" vector, although both models produce a  parameter vector
      a whose expectation is not the true parameter vector, 8.  An R2 measure
      cannot be used to distinguish between models estimated  using a correct
      averaging method versus those using an incorrect method, however,  when
      the grouping schemes are identical for both models (e.g., AB.CI) versus
      AB.CD).
                                   76

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     From these results we conclude that density measures  (Xrs) representing
the expected distance to the nearest recreation site of a  certain kind are
best measured at a reasonably fine level of  spatial disaggregation, such as
the county, rather than at the state level.  This intuitively obvious point
can be given formal justification by arguing that  the finer the level of
spatial disaggregation used for averaging (in our  case, calculating expected
distance), the closer the auxiliary regression parameters  p  , h - 1, ...,
                                                          ilK
K-1 will be to zero, and the closer the parameter  pKK will be to 1.0.  Hence,
the extent of parameter bias in participation equations appears to be an
increasing function of the degree of spatial aggregation involved in .
constructing the density variable which stands in  for expected travel
distance.
The Value of Additional Information
     How can we apply lessons learned from our simulation model to estimation
using recreation participation survey data?   First, our simulation will allow
the assumption of uniform distribution of  people in space assumption to be
relaxed, allowing individuals to be clustered in,  for example, a bivariate
normal distribution.  In populations that  are uniformly distributed within
their geographic area, the density-based distance  proxy for that region will
always- be an upper bound on expected distance because of the edge effects
discussed earlier.  In populations with bivariate  normal distributions of
people, how-the density-baaed distance proxy is related to the true expected
distance depends oh additional information regarding the placement of water
bodies and the center of population.  If the population centroid is
relatively close to one or more water bodies in the same region, the true
expected distance can be much less than that indicated by-the density
measure, especially if the water bodies constitute a relatively large
percentage of the region's water area or the population is tightly centered
about the centroid.  Alternatively,  suppose  that the population centroid is
close to the border of the region.  In this  case,  edge effects could produce
a true expected distance much lower than the density-based distance proxy.
     In theory, if the population spatial  distributions of people and water
bodies are known, it would be possible to  compute  the exact population
weighted distribution of distances to the  nearest  water body in a region.
                                     77

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Having this information of course also allows one to calculate the expected
value of distance to the nearest water body,  from which the density measure
pertinent to this population can be calculated.   However,  the expected
distance multiplied by a unit travel cost yields  the expected value of the
travel cost measure we are in fact trying to approximate.  If such
information were available for all geographic regions (however defined) then
expected distances to the nearest water body (regardless of which region it
falls in) for people within each region, could be  calculated.  The obvious
advantage to having this information for all geographic regions is the
ability to correct for edge effects in calculating expected distance.
However, while this information is in fact available to us in our simulation
(where the population is equivalent to the sample), there  is no systematic
way to derive correction factors to be applied to real world models which
obviously violate the simplistic assumptions of our simulation.  Thus to
correct for edge effects in models estimated on data- from  typical recreation
participation surveys, we need a priori information on the distributions of
people and water bodies.
     Note that three other recognized problems associated  with using a
density-based distance proxy would be solved if this distribution information
were- available.  While there may well be no variation in densities in an
area, the true expected distances will vary as long as either the
distribution of people or water bodies changes.   In addition, since no
assumptions need be made about the sizes of water bodies,  or how people are
distributed, the expected distance will not be biased as the density-based
distance proxy would be.  However, there would still be an irremediable
problem with applying this distributional information on people and water
bodies to survey data.  This is much like the previously discussed problem of
unknown geographic regions.
     With information on the distribution of people and water bodies across
the country, say, it would be possible to overlay a set of area boundaries
such that the variance of the expected distance for people in each area is
minimized.  This would of course reduce the parameter bias which is due to
the expected value nature of the distance measure.  However, the likelihood
of being able to overlay such a set of area boundaries to  which surveyed

                                     78

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individuals could actually be assigned is very remote.   Thus  we must  still
live with using the given boundaries of the smallest political units  to which
we can assign surveyed individuals,  noting that even when we  can assign
people to particular counties,  the expected distance measure  may be a poor
approximation to an individual's true distance to  the nearest water body.
Though we are constrained by the spatial level to  which individuals can be
located, it might still be possible to use information  on the
outside-of-sample (or population) distributions of people and water bodies  in
calculation of a parameter bias correction vector,  if data on the other
relevant independent variables was also part of the outside-of-sample
information.  In such a case, the submatrices of Equation 8 which are unknown
could be calculated as XJXZ - XJXa and XJX2 - X{(Xa - X,)  * X[X2 where X[(Xa
- X2) is from the outside-of-sample information.
     Another potential advantage of using outside-of-sample information is
that the effect of a pollution control policy might be  more carefully
determined since both pre- and post-policy expected distances will be better
approximated if it is known which water bodies are unsuitable for recreation
due to pollution."  In application of outside-of-sample  information to
participation survey data, it should be noted that the  calculation of
expected travel distance at the county level for the 48 contiguous states
would be no trivial task even if the distributions of people  and water bodies
were somewhat simplified.  Also, the more complicated the model in terms of
other relevant regressora (ie., the larger the number of columns in Xlf or
k-1), the richer the outaide-of-sample data must be in  order  to be useful.
     Next we turn to more general questions of demand function representation
and estimation.
METHODS FOR ANALYZING DEMAND AND HENCE WELFARE CHANGES
     The aim of this study is ultimately to compare the accuracy of various
econometric approaches to estimating monetary measures  of  welfare change
occasioned by the impact of water pollution control on  water-based
recreation.  The extent of approximation error inherent in various lines of
empirical attack must be evaluated,  assuming negligible random error  in the
pseudo data.  Some of these empirical approaches require a good deal  more
price and quantity information than others,  and thus are less feasible given

                                     79

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the quality of most extant real world data seta.  A menu of  some  fairly
simple approaches we propose to consider in this study and apply  to RECSIM's
pseudo-data appear in table 4.  They are discussed below from  bottom  to top,
in order of increasing information requirements.
     While the Lancaster model is a reasonable way to structure the
consumer's optimization problem, it is empirically intractable to attempt to
disentangle the household's production and taste parameters  in econometric
estimation.  Wants, as we have defined them are unobservable,  so .as a
practical matter all that can be used in estimation is information  on inputs
                                                        
- site visits and site visit prices.  Further, as Bockstael  and McConnell
(1983) have pointed out, welfare measures in the context of  the
household model are properly made in terms of the household's  derived input
demand functions, not its output demand functions.  For these  reasons, we
adopt, the viewpoint of the purely neoclassical econometrician  throughout and
do not attempt to invoke Lancaster's household production model in
estimation.  Instead, we estimate conventional demand models on observables:
site visits and site visit prices.  Such an approach is not  inconsistent with
the bulk of the empirical work undertaken even by proponents of the
theoretical household production model in their analysis of  family  labor
supply, health, and leisure.  (For an exception, see Moray,  1981).  In fact,
although Barnett (1981) has suggested how the structural parameters of
household production models could conceivably be estimated,  in general it is
qui-ta difficult (personal communication from William A.  Barnett), and to our
knowledge only one attempt by Rosenzweig and Schultz (1983)  using health
data, has appeared in the literature.  By ignoring the complications
introduced in attempting to disentangle utility and-production parameters,
evea in the moat complex of the models discussed below,  we feel we  are
reflecting what is practical in the context of recreation participation
analysis.  If the production technology matrix is indeed diagonal,  moreover,
doing so is wholly appropriate.  Whether or not our benefit  estimates diverge
substantially from the true benefits when the production technology matrix is
not diagonal remains to be explored.
     Second, we made no distinction, either in generating our  data  or in
analyzing it, between the household and the individual.   All of our

                                     30

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               Table- U.   Methods to Estimate Welfare Changes
General
Method

I.  Dual System-
    wide Approaches
II.Single Demand
   Equation
   Approaches

   a.  Full price
       variant* no
       price proxies
       Own price/
       proxy price
       variant
Information
Requirement

All goods prices,
income, shares
of expenditure on
all goods categories
All goods prices,
income, quantity of
j   good consumed.
Homogeneity imposed..

Own-price j,  proxies
for all prices 4 j,
income, quantity of j
good consumed.
th
       All-proxy price  Proxies  for all  prices..
       .variant
income, quantity of j
good consumed.  Proxies
measured at alternative
levels of spatial
aggregation.
         Welfare
         Measure

         Direct calculation
         of CV and EV from
         approximation to
         expenditure function.
         Marshallian surplus
         or CV and EV via pseudo-
         expenditure function.
         Marshallian
         surplus.
         Change in quanity  of j
         good consumed valued
         at an average willingness
         to pay.
observations are on one-person households,  and are treated as such in

econometric estimation following  conventional  practice.

Single Equation Methods

:    iSingle equation methods  are  just  that  - the estimation of the demand

equation for a single good or category of leisure activity.  The methods

reviewed under this general heading  only differ in terms of independent price

variable specification and hence  the extent to which the theoretical

restriction of homogeneity of degree zero in income and prices can be

imposed.  Specifically, if proxies are used for price which bear only an

approximate relationship to price, the accuracy of which can itself differ
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 across goods  categories,  it does not seem  reasonable to normalize all proxy
 prtces-and income by one  of the proxies  to impose homogeneity.  Of course,
 the estimation of single  equation demand functions, however specified, makes
 tt impossible to impose adding up or symmetry, which require system-wide
 estimation.
      The crude method of  valuing quantity  changes predicted from a single
 equation by an average consumers'  surplus  appears as the bottom entry II.c in
 the table. It is discussed in detail in appendix A to chapter 6.  This
 welfare measure bears no  general relation  to either Marshallian consumer's
 surplus (OS)  or the theoretically correct  compensating and equivalent
 variation (CV.EV) measures of  welfare change.  It can either be a reasonably
 good or quite poor approximation to Marshallian consumer's surplus depending
 upon the (unobserved) form of  the true Marshallian demand function for the
 good involved (assuming only a single price change), and the accuracy of the
 statistically predicted quantity change.   The method requires no price
 information.   Indeed, it  is a  roundabout way of overcoming the absence of
"slichrpFice~ihformation by employing price  proxies in a single participation
 (pseudo-demand)  equation.
      The single  demand equation method II.b requiring only own-activity price
 information (and proxies for other goods  prices) is used in a recreation
 context by Ziemer et.  al.,  (1982).  Those authors ignore all other prices
 except own-activity price (travel  cost) in the demand equation specification
 and. estimate a truncated regression model using sample observations only on
 individuals who  engaged in the activity,  as travel costs are often
 unavailable for  no n-participant a.   There  are  two problems here; one
 theoretical and  one empirical.   First,  unless consumer's utility functions
 are, say* Cobb-Douglas, one would  not expect  own-price to be the only price
 argument in the  single equation Marshallian demand specification, so bias due
 to:specification error is a distinct possibility.  Second, the omission of
 non-participant  data in estimating the truncated regression model involves a
 loss of information (Maddala, 1983). and  Ordinary Least Squares (OLS)
 estimates will be biased.   If own-price information were available for all
 individuals, including participants and non-participants, a single equation
 tobit model would be preferred, perhaps including proxy variables for "other

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activity" prices,  OP even better,  all  prices  as  in the full price variant
II.a in table 1.   The advantage of the single equation full price variant is
that homogeneity  of degree zero in prices  and income can easily be imposed.
     In either II.a or II.b the integral under the estimated Marshallian
demand curve between the pre^policy and post-policy own price levels would be
a Mar shall ian consumer's surplus (CS)  measure of  the welfare change in a
particular recreation category attributable to water pollution control, if
that category were the only one whose  travel-cost-based price was affected.
     If the change in the marginal utility of income is assumed to be
negligible and a  single- price changes, an  approximation to the consumer
surplus integral  based on Simpson's rule is suggested - but not recommended -
by McKenzie (1983,  p. 122)  and is easily calculated without analytical
integrals.  In fact, Willig (1976,  p.  592) demonstrates how, when income
elasticity is constant (even if it is  unequal to one) the exact CV and EV
measures can be obtained directly as a function  of the Marshallian consumers
surplus measure.   Willig further (1976) showed that even when the income
elasticity of demand is not constant over  the region of price change, bounds
on the percentage error of the change  in consumer surplus as a proxy for CV
or E3 can be derived.  These bounds can be used  to adjust the CS measure to
produce: approximations to CV or EV given information on the income elasticity
of demand and the base Income level.   In instances where there is only a
single price change, that change is small, and the good involved absorbs a
small proportion  of base expenditure*  the  bounds are often so tight that the
use of an unadjusted Mar shall ian consumer's surplus as a benefit measure is
Justified.  All we need is the estimated Marshallian demand curve.  But the
argument is not easily extended to the case of multiple price changes, when
some prices increase and some prices fall, in which case the Mar shall ian
measure is path dependent (see the extensive  discussion in Just, Hueth and
Schmitz 1982, Appendix B, and Willig,  1979).
     To overcome  the problem of large  discrete single price changes one can
still employ information contained ia  the  Marshall ian demand curve to get
exact CV and EV measures by employing  Hausman's  (1931) method.  Hausman
suggests using the observed market demand  curve  to directly estimate CV or EV
by invoking Roy's identity to integrate and recover the indirect utility

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function  (or "pseudo"  function  in the multiple price case).from which exact
CV  and EV measures  can easily be calculated.  This ingenious method allows CV
and EV to be reported  free of numerical approximation error, and confidence
intervals around CV and EV can  be established reflecting only statistical
estimation  error.   Analogously, for multiple price change case McKenzie
(1983) presents  a numerical  route to an exact EV measure once an estimated
system of Marshallian  demand curves is in hand.
     While  either the  Willig, Hausman or McKenzie methods are appealing,  they
all assume  the correct specification of the demand equation (or system of
equations)  is known, or at least that a close approximation to it can be
estimated.  Thus the CV and  EV  measures derived from the Marshallian demand
equations are exact conditional on the demand specification being correct,  a
maintained  hypothesis  of the theoretical argument.  This implies that
non-nested  statistical tests must be employed to choose among a plethora  of
alternative demand  specification before CV and EV can be calculated from  the
preferred specification.  We therefore propose only to report Marshallian
consumer's  surplus  measures  for single equation specification II.a and b.
Although  either  the Hausman  or McKenzie methods could be used, our principal
mission is  to evaluate Method II.c of table 1 vis-a-vis the "true" welfare
measures.  So, methods II.a  and II.b are merely sidelights which, under
certain circumstances  of data availability, could feasibly be estimated.
     But, with complete price and quantity data there is another, perhaps
preferable  method - direct estimation of a complete system or demand
equations.
'The Dual  System-Wide Approach
     Instead of  estimating a single demand equation, a complete system of
demand equations can,  in principle, be parameterized, and the" additivity,
homogeneity and  symmetry restrictions of demand theory can either be imposed
or  tested for statistically.  There are two general approaches to system-wide
estimation. The dual  approach  begins with a specified functional form for
the direct  or indirect utility  function (or expenditure function) and derives
the appropriate  demand or budget share equations assuming utility
maximization subject to a budget constraint.  The demand equations
automatically satisfy  adding up, and homogeneity and symmetry can easily  be
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imposed.  The second method is to specify the functional form of  the system
of demand equations directly without  reference to the functional  form of the
direct or indirect utility functions.   An excellent  review of the formulation
and estimation of complete systems of  demand equations appears in Barten
(1977), though considerable empirical  work involving the dual approach has
appeared since that time.
     The dual approach involves the estimation of the parameters  of either a
specified utility function (Stone-Geary,  for example) or, more often, a
second-order local approximation to an arbitrary  direct or indirect utility
function using market data.  The dual  approach exploits the relationships
between direct utility functions, indirect utility functions, and expenditure
functions (Deaton and Muellbauer, 1930).   Particularly, if  one can specify a
mathematically tractable direct or indirect utility  function or a local
approximation thereto, systems of either  Marshallian demand functions with
prices and nominal expenditure as arguments or systems of Hicksian demand
functions with prices and real expenditure as arguments can be derived; the
former by application of Roy's identity to the indirect utility function and
the latter by differentiation of the  expenditure  function with respect to
goods prices.  The parameters of the  system of demand equations can then be
estimated using familiar statistical  techniques.   When based on a
second-order approximation the approach is elegant,  theoretically appealing,
fairly "general" and internally consistent, but it requires the most
information, of all methods listed in  table 4.  However, once the  appropriate
parameters have been estimated, exact  CV  and EV calculations using the
indirect utility function are straightforward. Although the Hicksian demand
curves can be derived, they are not needed to calculate CV and EV.
     There are many contenders under  the  general  category of dual system-wide
approaches, since there are many alternative flexible functional  forms which
can provide a second order local approximation to an arbitrary twice
differentiable direct or indirect utility function.  (Particularly, see
Berndt, Darrough and Olewert, 1977).   We  note only one contender, the Almost
Ideal Demand System (AIDS), used, for  example, by Lareau and Darmstadter
(1983).  Under certain circumstances AIDS may not provide the "beat"
approximation, nor can it  stand in for any utility function (Deaton and

                                    85

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Muellbauer, 1980, p. 74), but it is simple to implement empirically.  While
we do not implement the AIDS approach in the present studyits estimation
can be computationally burdensomewe present a thorough discussion of  the
method in appendix B to this chapter with the view towards  implementation in
future extensions of this research.
CONCLUDING REMARKS
     In the first part of this chapter, we discussed the problems  raised by
the use of proxy price variables that are themselves average  values taken
over arbitrary geographic units and applied to all individuals  residing in
those units.  Although the results derived pertain only to  the  classical
ordinary least squares regression model, they are revealing in  themselves and
suggestive of the dangers inherent in bringing more sophisticated  maximum
likelihood estimators (tobit, etc.) to bear on data where at  least one  of the
columns of the X vector is available only as a group mean.
     In the OLS context, the following assumptions have to  be satisfied if
the estimated parameter vector, a, from a mixed model is to be  unbiased, so
that its expectation is the true parameter vector gs
       Individuals are Toniformly distributed in space or,  if they are  not,
        population-weighted density variables can be constructed;
       Recreation sites visited for a particular type of recreation activity
-_-_.    are of equal size and have homogenous characteristics;
       Edge effects are minimalr so the expected value of  the  true
        visit-price vector over all individuals in a geographic region  is
        equal to the cost-equivalent of the density-based expected distance
                      1 /2
       .measure (d/2!V,   ) in that same region;
       Density measures vary across regions;
     e  The true price vector for all individuals is orthogonal to the  matrix
        of observations on all other independent variables  included in  the
        model.
     This is a daunting list of assumptions, all of which are unlikely  to be
met in any real survey data set on recreation participation and participant
characteristics, supplemented by "supply" variables measuring density.  But,
if a sufficiently fine density measure can be constructed and models
estimated in OLS on, say, the county means of all variables,  then  the most
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severe bias problems remaining unconnected will  be  edge effects, and the
irremediable blurring introduced in expected distance  by  sites of widely
varying sizes.   With rich out side-of-sample data, the  problem of unknown
geographic regions persists,  while  the other problems  are mitigated.
     The second major part of the chapter amounted  to  a summary of options,
available in principle,  for estimating demand equations and hence,
ultimately, welfare changes.   Several  single-equation  methods were discussed
and their advantages and disadvantages compared.  A multiple equation
methodAIDSwas briefly mentioned in the text  and discussed at some length
in an. appendix.  The application of the ideas developed within the RECSIM
model context will be taken up in chapter 6,  when the  "Estimate" module is
described.
     We now turn to the description of RECSIM beginning with data generation
and working our way through to comparisons of results.
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                               APPENDIX 3.A

          THE ROLE OF RECREATION RESOURCE AVAILABILITY VARIABLES
                        IN PARTICIPATION- ANALYSIS*
    Suppose a decision on providing or not providing sane general addition to
recreation resourced hinges on what impact the addition is projected to have
on participation in the activities to which they are relevant.   For example,
suppose a decision about expanding camping areas across the U.S. is to be
made on the basis of the projected addition to camping activity  attributable
to the addition of resources.  This problem setting allows us  to postpone
until later consideration of the problems of valuation within  the
participation model context.
    To address the problem a cross-sectional data set reflecting individual
leisure-time pursuits and the socio-economic characteristics of  the same
individuals is required, so that population leisure participation can be
estimated econometrically as a-function of these characteristics, as in
Settle (1980).  It also seems necessary to have variables measuring the
supply of recreation resources appear as arguments in the equations to be
estimated, so that the effect of alterations in supply can be  appraised
directly.  But, a question arises at this point:  Do such supply variables
belong in recreation participation equations, in the sense that  the equation
specification is consistent with economic theory?
    A hint of the answer is given by the travel-destination/modal-choice
literature, where relevant independent variables in the empirical model of
choice are the variables that would appear in the consumer's indirect utility
function--for example travel cost (analogous to goods prices), site
attributes, consumer income and consumer characteristics (Hensher and
Johnson, 1981, Rugg, 1973 Small and Rosen, 1981).  Unfortunately, few, if
any, recreation participation surveys from a broad sample of the population

*A version of this appendix has been published in The Journal  of
Environmental Management, vol. 19, 1931, pp. 185-191T

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 contain detailed individual-specific information on travel and other costs
 incurred in  going from place of residence to the recreation site or sites
 chosen, let  alone other potential sites not chosen.  Nor do the surveys
 normally identify the location of individuals or sites at all precisely.
 Thus,  if a correctly specified recreation participation equation is to be
 estimated econometrically from such survey data, a proxy variable must be
 developed which  can stand in, however crudely, for the expected site prices-
 associated with  an individual's participation in one or more recreational
 activities.   Fortunately, this variable is indeed a resource supply variable.
     Previous empirical analyses of population recreation participation in
 broad activity categories (rather than site-specific travel cost studies)
 have either  employed a measure of average variable travel cost consistent
 with theory  (Ziemer and Musser, 1979; Ziemer et. al., 1982) or, when such
 measures were unavailable from survey data, substituted aggregate "supply"
 variables as proxies (Davidson, Adams and Seneca, 1966; Chlcchetti, 1973;
 Deyak and Smith,  1978; Smith and Munley, 1978; Hay and McConnell, 1979;
 Vaughan and  Russell, 1982) or even ignored the problem entirely (Settle,
 1980.)--  The  rationale for such proxy recreation resource supply variables has
 generally been vaguely asserted rather than clearly established.  Yet it
 makes intutitive sense to link participation to the "availability" of
 recreation alternatives measured in terms of quantity (number of facilities
-in a geographic  region) or quality (number of facilities per capita to
 account for  congestion) (Cicchetti, Fisher and Smith, 1973).  In fact, it is
 possible to  go beyond intuition and provide a firm rationale for the
 inclusion of explanatory physical supply quantity variables in recreation
 participation equations.  We do so below, using the case of a water-based
 recreation activity (eg., fishing).
     A version of the theory of distance estimators of density (or in our case
 density estimators of distance) developed in the statistical ecology
 Literature can be applied to show that expected travel cost should be
 functionally related to the number of water bodies per unit land area in a
 region.
 RELATING DENSITY AND DISTANCE
     The idea behind this link is intuitively appealing, the more objects
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there are randomly arranged in a given space, the closer will  be  the nearest
such object on average, to any randomly chosen point.   If we knew the
parameters of the process that put the objects in their places, we could
obtain an exact expression for the expected distance.   However, we will
usually not know-either the exact process behind the location  or  the
parameter appropriate to an approximate process.  In those circumstances,
which characterize the analyst looking at actual water bodies  in  regions and
wondering about a proxy for travel cost, observed density of the  bodies may
be used either directly or after transformation as a proxy for expected
distance.
    To tie dowa the Intuitive idea with a bit more rigor, assume  that a
region can be divided into H equal-size squares.  These squares will be taken
to be units.  Some number, n of "tiles" representing water bodies and also of
unit size, will be placed on the grid by a random process such that the
probability of a "tile" falling on a square is 1/N P.   More than  one tile can
land-on a square, so that after all tiles have been placed, the observed
number of "lakes" will be wSn*  If N is large (p small) the resulting
probabilities of a particular number, m, of tiles falling on any  chosen grid
square can be approximated by the Poisson density function:
                            -np
                P(m;np) - -&
                              mi
The expected number of water bodies, allowing for multiple tiles  per square,
is Nd-e"0*1) - w.  Because e"1* can be approximated by the first  few terms of
the series
and because np - n/N <1 by assumption, it is also true that w/N,  the observed
density, of lakes, Is an approximation for np, the Poisson parameter (often
written as \) .
    Thus, w/N - l-e'"5  1-(1-np) - np - n/N
    This approximation result is Important when the objects on a grid may be
assumed to have been distributed according to a Poisson density function with
parameter X.  Then it is possible to show that the expected distance E(r)
front a randomly chosen point to the nearest such object is given by E(r) -
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The derivation of .this expected distance formula is reasonably
straightforward.  By the Poisaon distribution the probability of no objects
in a circle of radius r is:
     PUirr2 - 0) - e
If the nearest object appears at distance r from the center of this circle,
we can define an annular ring of width dr within which it is the only such
object.  The area of the annular ring is
     ir(r + dr)2 - irr2 - ir(r2 > 2rdr + dr* - r2)
                      -.ir(2rdr * dr2)
Ignoring terms in (dr)2 we can approximate the probability that the band
contains the one object by
using the reasoning developed above.  Note, however, that
     xe X s  x(1-

                      -    -
                     27   31
Since x. * A2irrdrr and ignoring: terms of order 2 and higher in dr, we have

                    = 2irrdrx
     If the two events (no objects within the area irr2; one object within the
annular ring with area ir(2rdr + dr2)) are assumed to be independent their
joint probability is the product of their individual probabilities.  Thus the
Joint probability density function of distance r is the product of the
Poisson probability expressions for finding zero objects out to r and 1
object in the narrow- band at  r

     f(r) - 2Trr\e"Xirr2dr
     Thus, the expected value of r, or the average distance to the nearest
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object from random pointa In the apace, aa a function of the density
parameter ia:
     E(r) - J r 2irrAe~7rp  dr - /
            00
Thia definite integral can be shown to produce:
     E(r) -   X

which ia to aay that the expected distance from a randomly chosen point to
the nearest object depends on the Poiaaon parameter.   Thus,  if we can
approximate \ by w/N, we can approximate E(r) by 1/2(w/N)     ao that
expected distance fall a with increaaing density.  Thia relation ia shown in
figure A.T.
     The variance in expected distance (VAR r) can be obtained by recognizing
(Laraen and Marx 1981, p., 114) that VAR(r) equals E(r2)  -  (E(r))2.  The
expected value of r i a already known to be 0.5\     ao the second term in
VAR(r) ia this quantity squared, equal to 0.25/T .   To obtain the expected
value of r2, we take the definite integral:

     E(r2)
                0
                  T(2)     2irA
                          2(irX)a   \ir
     So,

               ~^r - ir
     Putting thia expression in terma of a common denominator and aimplifying

      YAR(r)  (}   =0.068x"1.
and the variance of the expected distance also falls with density.
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(0
01
O (0
4J 41
  -*
at -H
(A
o
0)
u
a>
(X
        so 4-
       40 f
30  t
       20  +
       10  +
                             .001                .002               .003

                                X ~w/N = (Objects per square mile)
                                                                                  .004
                             Figure A.I:   Density Distance Relationship

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     While these relations have both intuitive appeal and formal
justification, there are several possible pitfalls associated with  using a
measure of the density of water bodies (acres per acre)  in the region of
interest as an inverse proxy for distance and hence travel cost.
     First, the relation between measured density as a point  estimate of
expected density and \ is better the smaller X.  This may be  seen by
inspecting the series approximation for e""1* given above.  The smaller n
relative to N the more rapidly the terms with exponents greater than one
approach zero.  Thus, the more richly endowed the region the  less reliable
                                                                 a
the approximation.
     Second, in the real world water bodies do not come as discrete unit area
pieces, or indeed as pieces of any common size across a single region let
alone across several regions.  Thus, the assumptions underlying the
derivation will be violated in actual regions.  Particularly,  data  on surface
acreage (rather than the number of lakes) is the most common  measure of the
availability of water for recreation, and surface acreage is  composed of
lakes of varying sizes as well as rivers and streams.  So the Poisson forest
analogy does not translate perfectly in application.
                            A
     To see the problem let A, measured as the square miles, covered by the
objects (lakes) per square mile of regional surface area, be  the available
data.  Suppose that all objects have the same size, m, so that \ (number of
         A
units) - A/m.  Then,

     E(r) - 0.5A~-5 - 0.5 (i/m)"0'5 - (0.5A~5) On0'5)
                                     A
So, if m is constant across regions, A can be used as a proxy for A as an
explanatory variable in estimating activity participation relationships,
since the constant term (m   ) will merely scale the estimated availability
parameter.  A plausible assumption is that large lakes are composed of
                                                           A
clusters of equal radii objects, so proportionality between A and A is
maintained.  It is however, implausible to think that m will  be constant
across regions; and finding a set of region specific average  m is  neither
practically non-theoretically appealing.
     Third, even if the objects of interest are of uniform size across the

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regions, but their locations were generated by a  heterogenous, nonrandonx
process rather than a homogenous  Poisson process  (i.e., the objects' centers
were not uniformly and Independently distributed) the  expected distance '
formula will not hold (Ripley,  1981, Ch 7,  8).
     Finally, if the intensity  parameter varies from place to place but the
manner in which it varies is unknown a priori, spatial groupings cannot be
established which uniquely reflect the variation  in the several population
Afs associated with the different regions.   All one can do is to produce
different area-weighted mean density proxy  measures for A for different
levels of aggregation across space.
     For example, in a 100 by 100 grid, we  generated two samples with 400
objects (A - .04) and two samples with 200  objects (A  - .02).  The distance
to the nearest object was computed from 31  points systematically located at
the intersection of lines of latitude and longitude ten units apart.  (Border
intersections were excluded).  The expected value of distance to the nearest
object is 2.5 miles for A - .04 and 3.54 miles for A -.02.  The sample
outcomes for expected distance and the associated standard errors of the
means fran this simple experiment show that in these cases the sample means
are all within one- standard error of the population expectation given by
0.5A-0-5:

                        A  - .02                       A  - .04
                   Sample 1        Sample 2         Sample 1      Sample 2
Sample Mean
 Distance           3.38           3.61             2.41          2.59
 Std Error of Mean  0.19-           0.20             0.16          0.13
Theoretically
 Expected Distance        3*54                           2.50

     Note, however, that if we were to sample over  both  grids believing that
both-belonged to the same population (i.e.,  shared  the same  A) our estimate
of A would be (200 + 400)/2(10,000) or 0.03  and our expected distance would
be 2.89.  Although this expected distance would perhaps  be realistic for
individuals located on or around the border  delineating  the  regions

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(particularly the geographic centroid of the two regions together) it would
not be for individuals located acme distance from that border, who more
properly should be assigned their respective region -  specific expected
distances.
SOME IMPLICATIONS FOR AGGREGATION: MEASURING THE PROW FOR  X
     With aggregate real world data we do not pick a set of randan points in
space and. mark off the distance from each of those points to the closest
"object" (i.e., water body), to estimate a value for \ from the inverse of
expected distance formula.  Rather we use acres of objects  per acre of total
area as a proxy for A. and hence for expected distance.  The question is how
to demarcate the relevant boundaries of total regional areas?  Counties,
combinations of counties, or fixed areas around each individual could be
used, but the cutoff distance over which our proxy for X should be measured
is unknown.
     However, a University of Kentucky Vater Resources Institute survey
(Bianchi, 1969) of over 3,000 fisherman reported that  only  slightly more than
3 percent .travelled over 30 miles to fish.  Similar calculations of
the percent of days fishing by travel -distance can be  made  from U.S.
Department of the Interior, 1982:

                   One-Way
                   Distance                 Frequency
                   (miles)                     (g)
                   0-5                         19-
                   6-24                        26
                  25-49                        17
               .   50-99           '             14
                 100-249                       10
      	    250-499   '                     3
                 500-999                        1
                   >1000                       Nil

     The median travel distance from this data is 32 miles. The Davies test
of skewness (Langley, 1970) suggests this data is approximately logarithmic
in-distribution, so the geometric mean is appropriate, yielding a value of
31 .6 miles.  It also appears that 250 miles would be a generous upper limit
for the radius of the region whose characteristics determine recreation!at
                                     96

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behavior.  Two alternatives,  then,  suggest themselves.  One  is  to  use density
data only from an individual's county of  residence.   At the  other  extreme,
circular regions around the centroid of the individual's  county of residence
could be constructed and weighted density data from  all the  counties
represented in this region used to construct a measure of  \.
CONCLUSION
     It is appropriate to include two "availability1* variables  in  the
econometric analysts of recreation participation choice;  one to capture the
distance or travel cost influence via the number (or acres)  of  recreational
resource facilities per unit land area and one to capture the- (expected)
congestion influence via the number (or acres)  of such facilities  per capita.
     Further, it is reasonable to maintain that individuals  base their
recreation participation decisions on expected (travel-cost  based) prices
across the gamut of alternative types of  recreation  activities  rather than
actual prices, since the latter cannot always be known with  certainty for a
broad array of activities.  In this case  availability variables are not just
proxies introducing errors-in-variables problems into the econometric
analysis (Maddala 1977, Ch. 13).  Ratherr these observed  variables are the
true price variables which we desire to measure based on  the theoretical
model.  In this context errors-in-variables problems would occur only if the
degree of spatial aggregation involved in constructing a  measure of \ was too
coarse, encompassing several areas  which  belonged to separate populations,
each with its own particular X.   In such  a situation it is likely  that the
estimated parameter reflecting the relationship between participation and
average availability will be a biased measure of the true effect.
                                     97

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                               APPENDIX 3.B
                              THE AIDS MODEL

     The AIDS .model (Deaton, 1978, Deaton and Muellbauer,  1980)  begins  with a
parameterization of the cost or expenditure function as a logrithmic second
order Taylor's series approximation:
     In c(c
,p) -<*+ a.In p.   1/2 H Y, ,ln p.ln p.
       o   j  j    j        1J   ij     i    j
                    * UBOJHPW
                   where i, j - 1	n goods.
                           p. - price of the j*  good
                            u - ordinal utility index
                           a   logarithm of subsistence expenditure.
                                Equal to the sum of the product of
                                committed subsistence quantities
                                and their associated prices.
                           a. - budget share equation intercepts
                           8, - Engel curve parameters reflecting change
                                        th
                                in the i   budget share with  a change
                                in real income.
                           a  - arbitrary constant
                                           teU
                          Y.. - change in i   budget share with a change
                                        th
                                in the j   price, holding real Income
                                constant.
     The .expenditure function represents the minimum cost of  attaining a
given utility level u given prices p .  This is the dual to the consumer's
original problem of maximizing utility for a given expenditure.  (Deaton and
Muellbauer, 1980, Chapter 2).  While the indirect utility function  derived
from the original problem is defined over price and income variables,  the

                                     98

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expenditure function representing the solution to  the  dual problem is defined
over prices and utility.   The properties  of  the cost function are (Deaton,
1978, Deaton and Muellbauer,  1980):
       Total expenditure (income) at any time equals  the current value
        of the cost function,
       Increasing in u and the vector p,
     *  Homogenous of degree 1  in the vector p
      \Concave in the vector p
       Continuous in p with first and second derivatives.
     Not all of these properties (especially continuity in p) are consistent
with RECSIM's quadratic utility maximization problem.  But, if these
properties are assumed, they translate into  the following share equation
parameter restrictions in the AIDS model:
       Adding-Up-.  Budget shares must sum  to one since the sum of
        expenditures on each good exhausts income. So  the following column
        sum restrictions hold:
                                 - 0
                           I B.  -
        Homogeniety of Degree Zero  in  Price and Income.  If prices and income
        double, for example,  the quantity demanded q  and hence its share in
        total expenditure will remain  unchanged.  So for each share equation,
        i, the following row  sum restrictions  hold:
        Symmetry.   The Hickslan demand functions are the price derivatives of
        the expenditure function, and the cross price derivatives of the
        Hicksian demands must  be symmetric for all  i * J price pairs so:
                                    99

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     By Shepard's Lemma, the partial derivatives of the cost function with
respect to the prices produce the system of Hlcksian compensated demand
functions, so 3c(u,p)/ p^ - q^.  Multiplying both sides by p,/c(u,p)  produces
the budget share equations, w :

                        3 In c(u,p)      p.q.
                          3 In p         c(u,p)
                                                 wi
     Logrithmic differentiation of the AIDS cost function thus  yields  the
share equations as a function of prices and utility:

                        w, - a, * I Y, ,ln p. * a.uSoHp,
                         i    i   j  ij    J    i   j  J

     Because the utility index, u, appears in the above share equation, a
system of such share equations (which are first order approximations)  cannot
be estimated in this form.  But, we know from the first property of  the cost
function that a utility maximizing consumer will be minimizing  costs to reach
a particular utility level so c(u,p) will equal observed income, y.  Thus we
can invert the AIDS cost function and express u indirectly as V(p,y) in terms
of income y, and prices ps

                        In y-a0 - I a.ln p. - 1/2  Yijln Piln Pj
     S0u - 80[v(p,y)] - 	^	3	^	
                                      J
     It is obvious in this form that utility is proportional to the logarithm
of observed real discretionary income in the AIDS model given a fixed set of
prices, a proposition known as Bernoulli's hypothesis.   Qlven the definition
of the price index P on the following page utility can be written as  V(p,y) 
                                31
4>ln(y/P) where $ equals 1/(B0Hpi  ).  This hypothesis is not inconsistent
with the survey results of Van Herwaarden and Kapteyn (1979).
     Substituting the indirect utility function expression for V(p,y)  in
place of u in the budget share equations produces an estimable set of share
equations - one for each of the i goods consumed:
         wi * ai * I Yu In p, * s{ In (y/p)
                             J
                                     100

-------
     where P is a price Index defined by
         InP - 
-------
         * 2~ H YiJ ln Plln PJ " B

     Uncompensated price n^, HJJ and expenditure nly elasticities from AIDS
associated with the Marshallian demands can be expressed (Christensen and
Manser, 1975) in terms of the shares as:
Expenditure
     niy
Own Price
                          Y

Cross Price
            31n w
                 L.-ii-   -i
     Compensated elasticities can be recovered as  well  employing the Slut sky
equations, as can Allen elasticities of substitution (see Christensen and
Manser, 1975 and Henderson and Quandt, 1971, pp. 31-39).
     Specifically, the Slutsky equations are:
where the uncompensated cross-price elasticity n,,  equals  the compensated
response to a price change with all other prices held fixed but allowing
total  expenditure to  adjust to  maintain the initial  utility level (the
w 
-------
     Compensated Own Price Elasticity
     nU ' nii * Viy
     Compensated Cross-Price Elasticity
                 Viy
ESTIMATION OF AIDS - SOME SPECIFIC  EXAMPLES
     The performance of the AIDS Model  can be  illustrated by fitting it to
consumption data generated from  the constrained maximization of two simple
and well known utility functions -  the  Stone-Geary utility function which
produces a system of demand functions known as the Linear Expenditure System
(LES) and the Constant Elasticity of Substitution (CES) utility function.
The utility functions, along with the demand and share equations they imply,
are:
I.   LES
         Utility Function

           u -  b  In (q.  - g )
               i
              where
                 q.  - quantity consumed
              >4gj  * parameters
              and
              by 0  <  b   <  1,  b  - 1 , q  >
                                    103

-------
         Demand for ith Good
           qi " gi
         Budget Share for 1   Good
           wi " 
-------
         Budget Share for 1   Good

              t - Pi/I PJ
                      j
     The constant marginal rate of substitution in the CES  function equals
(1-r). The properties of these functions are discussed at length in Varian
(1978), Phlips (1974), and Powell (1974), so we do. not cover them in depth
here.  The elasticity formulas appear in table B.1.
     To generate quantities demanded and budget shares for  estimation a
                                  
sample of 125 consumers and four goods was employed.   Goods prices were
independently drawn from a uniform distribution over  the 0,1  interval along
with 125 observations on consumers with incomes between 5 and 75,  again from
a uniform distribution.  Quantities demanded were calculated exactly from the
above formulas assuming LES parameters bt - .1, ba -  .2,  ba - .3,  b,,   .4, gt
- .5, g2 - 1.0, g, - 1.5,  g* - 2, where the sum of the g, terms  implies a
subsistence income of 5 when all prices equal 1.0. For the CES  case, a value
of 0.28 for r was arbitrarily chosen, so the results  lie between the
CobtrDouglas unitary elasticity of substitution and Leontief zero elasticity
of substitution bounds.  Budget shares for all 125 consumers were then
calculated and a normally distributed error appended  which  was sufficiently

                Table B.1.  Uncompensated Elasticity  Formulas
                   Expenditure         Own Price          Cross Price
Utility            Elasticity          Elasticity         Elasticity
Function             (r\  )                (n^)                (nlj)

1.  CES               1 .0              r(1-wt)-l           -rw
2.  LES              bi/wi
3.  AIDS             1  * a,/w,
                                     105

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"small" (standard deviation of 0.005) to avoid violation of  adding  up.
     There are two routes to estimating the AIDS system of share  equations.
The first is to approximate the price index P by an expenditure-share
weighted average of the logarithms of the actual prices,  whereupon  the share
equations become linear and can either be estimated one at a time using
ordinary least squares (OLS) or as a system with additivity,  homogeneity and
symmetry restrictions imposed using the Zellner (1962)  seeming unrelated
regressions (SUR), or iterated SUR estimator.  The index In  P* suggested by
Deaton and Muellbauer (1980) to linearize the share equations bears a close
(but not exact) resemblance to the Fisher-.Tornquist index (Diewert, 1975).
The Stone index suggested by Oeaton and Muellbauer is In P*  -  w.  In p..
                                                               i  j     J
     If we normalize all prices to unity so their logarithms are  zero it
becomes apparent that the index P is related to the index P* by a factor of
proportionality represented by subsistence income:

  exp(lnP) - exp(a0 * In P*)"

Thus the parameter a0 cannot be identified when the linear share  equation
version is estimated, and the a. parameters are only identified up  to a
scalar multiple of 3..  The- linearized share equations  are:
     wi " (ai ~" 8iao) * ^YiJ ln pj * Si ln (y/p*)
So, only a^  (a^-g-a.) can be estimated.
6. In other words, errors are not introduced directly into the optimization,
but appended almost as an afterthought to keep things simple.   Note  that
problems of censored dependent variables and zero observed shares  are avoided
by specifying a tight error distribution around the true values.   These
problems can appear in actual data, either due to the nature of the  utility
function, the error generating process, or both.  Of course, holding o0
constant in estimation is incorrect, since in reality it is a function of
prices and therefore varies across individuals.  Properly,  a0  is a parameter
to the consumer's optimization problem which cannot be estimated.
                                     106

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     The preferred approach when collinearity  is  not  a  problem is to estimate
the nonlinear system of seemingly unrelated nonlinear (SUNK) equations using
either an iterative or. non-iterative variant,  assuming  additive disturbances
with a joint normal distribution (Gallant,  1975).  One  equation is deleted
from the system to produce a non-singular covariance  matrix since adding up
means that only n-T of the equations are independent.
     Our sample data are consistent with demand theory  so the homogeneity and
symmetry restrictions are maintained rather than  testable hypotheses,  this
being the case, there are three free o parameters, three free S parameters,
and six free Y   parameters in the four equation  system.  The remaining ten
Y. . parameters can be identified from adding up (4) homogeneity (3) and
symmetry (3).
     Yet it is still the case that a, cannot be identified if it is constant
across all observations.  But,  advantage can be taken of our prior knowledge
of exp(a0), which we set to 1  in the CES case  and 5 in  the LES case, so in
the former instance each nonlinear share equation has a fj. (in y) term and in
the latter each has a '8. (In y-aa)  term.
     The results of estimating, the system of share equations in their
                                                              7
linearized and nonlinear forms is reported  in  tables  B.2 and 3.   Only the
asymptotic- "t" statistics on the free parameters  are  reported.  The implied
standard errors on the restricted parameters could, however, be easily
obtained using a second order'Taylor's series  error propagation formula,
given knowledge of the variances and covarlances  of the parameter estimates.
     Some general remarks about the results can be made, without undertaking
an exhaustive discussion of the own and cross  price elasticity estimates
across different price vectors,-as  would be required  in a full-scale
econometric investigation.
     First some general observations.   The  AIDS model estimated with the CES
data- provides, a much better fit in  terms of R2 than the AIDS model estimated
7. Iterative estimation provided results were unconsequentially different
from the results using the non-iterative linear and nonlinear system
estimators, so only the latter are  reported.  This is not too surprising
because the cross-equation covariances are negligable due to the way to data
were generated.
                                     107

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                                                                     Table B.2.  AIDS Model Fit to LES Data
O
00

.
,
Parameter
Alpha 1
Alpha 2
A 1 pita 3
Alpha 4
Beta 1
Beta 2
Beta 3
Beta 4
Camua 11
Gamma 22
Gamma 33
Gamma 44
Gamma 12-21
Gamma 13-31
Gamma 14-41
Cumma 23-32
Cdwutt 24*~42
Gamma 34-43
EQI K~
EQ2 8<
EQI 8'
CPU Sec
SU8 Estimator*
Parameter
Estimate
0.253569
0.249580
0.250119
0.246732
-0.000834
-0.000037
0.000122
0.000759
0.046985
0.049669
0.048010
0,049500
-0.015374
-0.015288
-0.016323
-0.016920
-0.017375
-0.015802
0.97
0.97
0.98
2.2
'
t
Statistic
55.25
36.85
63.74
1
0.80
0.04
0.14
1
62.77
61.18
72.39
1
26.29
30.80
i
31.63

1




SUNB Estimator**
Parameter
Estimate
0.252612
0.250604
0.249306
0.247478
-0.000609
-0.000267
0.000297
0.000579.
0.047170
0.049676
0.047907
0.049391
-0.015371
. -0.015376
-0.016423
-0.016934
-0.017371
-0.015597
0.97
0.97
0.98
8.8

t
Statistic
55.20
57.79
62.23
1
0.60
0.28
0.33
1
65.91
63.33
71.86

27.46
31.54

32.47
1






                      8    SU8 Model estimated using proxy price indea P so Alpha(i) parameters identified only up to a scalar multiple of  Beta(i).
                           Since subsistence income is theoretically zero in the CES model,  the difference between the SDR and SUNK Alpha estimates is
                           negligible, unlike the LES case.                                                               .  


                      *   Alpha zero parameter set to zero in estimation.


                          Restricted Parameter.

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                                               Table B.3,  AIDS Model Fit  lo US Data



Parameter
Alpha 1
Alpha 2
Alpha 3
Alpha 4
Ucta 1
Beta 2
Ik-la 3
Bel a 4
Gamma 11
Gamma 22
Carnua 33
Gamma 44
Camua 12-21
Caiaoa 13-31
lUimau 14-41
Uurna 23-32
Cawiiia 24-42
CUIDIUI 34-43
EQl 8*
EQ2 8,
EQ3 IT
CPU Sec.
SUB Estimator"
Parameter
Estimate
0.101134
0.185014
' 0.337825
0.376027
-0.000282
0.002836
-0.007828
0.005274
0.003550
0.005887
0.012293
0.012848
-0.000913
-0.000793
-0.001844
-0.002735
-0.002239
-0.008765
0.26
0.51
0.56
' 2.3

t
Statistic
27.10
47.56
56.81
i.
0.35
3.42
6.16
t
6.64
9.82
11.53

2.21
1.45
1
4.60
1
1




SUNK Estimator"
Parameter
Estimate
0.100675
0.189477
0.325132
0.384716
-0.000278
0.002855
-0.007761
0.005184
0.003604
0.005791
0.012086
0.012768
-0.000885
-0.000835
-0.001884
-0.002637
-0.002269
-0.008614
0.25
0.48
0.52
8.9

t
Statistic
39.71
69.81
78.06

0.34
0.29
5.82
i.
6.75
9.43
10.51
i.
2.14
1.47
i
4.19







    SDK Model estioated using proxy price index P* so Alpha(i) parameters identified only up to a scalar Multiple of Beta(l).

*   Foreknowledge of Alpha 0 parameter used in estimation (Alpha 0  In 5).                          *
                                                                                                                  i
    Restricted parameter.

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with the LES data, and thus is true for both the SUR and SUNR  estimates.  The
difference between the SUR and SUNR parameter estimates  is  not great,
suggesting that linearization with P*,  at least in this  particular case, is
an adequate shortcut which saves about  75 percent on CPU time.
     Second, the AIDS  parameters on the income variable are significantly
different from zero with the LES data and insignificantly different from, zero
with the CES data.  This is to be- expected since the CES utility function is
homothetic (unitary expenditure elasticity) and the LES  utility, function is
only marginally homothetic (linear Engel curves with non-zero  intercepts).
     Third, evaluating the two AIDS functions at unitary prices and the
sample mid-point of 35 for income produces reasonable share and
own-elasticity estimates, displayed in  table B.4.  When  appraised at these
same price and income values all CES cross-price elasticities  are -0.07, and
                  a
the AIDS estimates   are close at -0.062 which implies a true  AES of 0.72
from the Slutsky formula and an AIDS approximation of about 0.75.  Without
going into details, it turns out the AIDS cross price elasticities from
estimation on the LES data are further  away from the true values than in the
CES case at this single evaluation point.
     Although elasticity comparisons and the like have some value in their
own right all of this is mere prelude  since the important  matter is not the
rate good qt substitutes for good q2 in consumption or the  like.  Rather, it
is how knowledge of the system of demand functions can be used to estimate
the welfare effects of price changes.  The next section  addresses this
question.
CALCULATING WELFARE CHANGES; EXACT, ALMOST EXACT AND APPROXIMATE MEASURES
     For concreteness, suppose the same two data sets used  for estimation of
AIDS in the previous section are at hand. If it were possible  to know the
form of the utility function generating the data with certainty, it would of
8. The equality of all cross price elasticities is an artifact of all prices
being equal to 1.0, setting all shares to be equal.   This  is not generally
the case.  Equality of the estimated Y   parameters (i>j)  with the CES data
produces the near constant cross elasticities at this evaluation point, since
n. . * ^i/w setting all S^ to zero.
                                     no

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                  Table B.4.   Exact and AIDS Predicted Shares
                      and Own Price Elasticities for  All
                          Prices - 1.0  and Income - 35
CES Data
 Exact. Share

 AIDS Predicted
 Share
 Exact Own Price
 Elasticity*
 Aids Predicted
 Elasticity**

LES Data
 Exact Share
 AIDS Predicted
 Share

 Exact Own Price
 Elasticity*
 AIDS Predicted
 Elasticity**
                        1
 0.2500


 0.2504


 -.79

 -.81


 0.1000

 0.100T


-0.87

-0.96
*  Equal tar(1-wi)-T.
** From formula in text.

*  Equal to C(1-bi)(glpl/y)]
                                               Good
 0.2500


 0.2497


 -.79

 -.80


 0.2000

 0.1950


-0.88

-0.97
 0.2500


 0.2504


 -.79

 -.81


 0.3000

 0.3100


-0.90

-0.97
 0.2500


 0.2495


 -.79

 -.80


 '0.4000

 0.3949


-0.91
                               - 1
                                     111

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course be appropriate to fit a system of CES demand equations to the CES data
and a system of LES demand equations to the LES data.  Then, with knowledge
of the indirect utility function parameters, exact  CV  and EV calculations
could be made for single or multiple price changes  in  straightforward fashion
using the following:
CES Data
     Indirect Utility Function
                   v(p.y) - y"(I p)~1/P
     Expenditure Function
                   C(u.p) - (I p[)1/r
LES Data
     Indirect Utility Function

                   V(p,y) - (y - I gjpj) n (bj/Pj)bj
                                         J
     Expenditure Function
                   e('u>          "
                              SCb./p )b        * jKj
                              t  J  J  J

     But, if the functional form of the utility function which generated the
data is unknown, as always is the case, the appropriate- course of action is
not quite so clear.
     A set of direct utility functions could be selected which provide simple
analytical solutions for the indirect utility function and  its inverse, the
expenditure or cost-of-utility function.   Then, we would proceed to estimate
each of them using a given data set and let the data indicate which is
"best"-.  Unfortunately, the competing models will most likely be non-nested,
and statistical discrimination among them, either by an information criterion
or a non-nested hypothesis testing procedure is an uncertain undertaking.
Choice of the "wrong1* model could lead to erroneous  but seemingly exact
welfare calculations.
                                     112

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      Another, similar course of action would be to forget about the
 parameterization of the preference system and instead estimate either a
 system  of demand equations or a single demand equation and use the McKenzie
 (1983)  or Hausman  (1981) procedures to derive "exact" CV and EV welfare
 measures or, even  more crudely, compute'Marshallian surpluses, comforted by
 the security of the Willig (1976) bounds (at least for a single price
 change).  Again, however, there exist a plethora of equally reasonable
 competing,  non-nested demand specifications.  Again, artful model selection
 criteria must be implemented, for careless inspection of a limited subset of
 reasonable  systems which to do contain the "true" system or selection of the
 "wrong1* system even after exhaustive and sophisticated application of model
 selection criteria to a large array of reasonable systems can again lead to
 erroneous,  though  apparently "exact" welfare calculations.  Put most simply,
 the McKenzie, Hausman and Willig arguments all begin with the maintained
 hypothesis  that the "true" demand function or system of functions or a very
 close approximation thereto has been estimated from the data.
' :    The third alternative is to rely on duality theory and approximate
 either  the  expenditure or indirect ability functions with a flexible
 functional  form.   Again, alternative flexible functions are available which
 are-non-nested, and one muat either be able somehow to discriminate among
 thea-given  a data  set or to place complete but wavering faith in one of them.
-But-,-it is  hard to see why the proponents of "exact" welfare calculations
 based'on demand ays tern a not derived from any underlying preference structure
 can be  sure that the accuracy of their "exact" calculations is any greater
 than "exact" calculations based on an arbitrary approximation to the
 preference  structure.  The model selection problem remains, whichever course
 is  pursued.  If convenience in implementation is a primary consideration, the
 dual approach exemplified by AIDS is a good deal more straightforward.
-   -r How well does the AIDS system estimated from our CES and LES sample data
 perform in  terms of "exact" monetary measures of welfare change?  Table B.5
 compares- the true  CV and EV measures when the utility function is assumed
 known exactly to the AIDS approximation, computed for a change in the price
 pt  from 1.0 to 0.5 evaluated at an initial expenditure of 35.  Exact
 Marshalllan consumer's surplus appears in the table as well. For this

                                     113

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                 Table B.5.  CV, EV and Marshallian Consumers
                    Surplus:  AIDS Versus the True Measures
                                       CV*                 EV*
I.  CES Data
     True CES Welfare Change           5.201               6.106
     AIDS Approximation                5.242               6.163
        (% error)**                    (0.79)              (0.93)
     Marshallian CS+                   5.637               5.637
        (% error)**""      .             (8.38)              (7.68)

II. LES Data
     True LES Welfare Change           "2.259               2.421
     AIDS Approximation                2.319               2.482
        (% error)**                    (2.66)              (2.52)
     Marshallian CS++                  2.339               2.339
        (% error)**                    (3.54)              (3.39)

*   Reported as absolute value.  For welfare improvements,  CV and EV are
    negative.  All other prices iAj equal to 1.0.
**  Reported as absolute value, using true measure as the base.
_*   Assumes parameters known.  Integral is equal to

r                        ->*
                                P!
                                p?
   Assumes parameters known.  Integral is equal to

          * Pi In pt
example, the AIDS approximation does adequately well in both areas, but is
better for the CES data.  In both instances, it out performs the
Marshallian measure, which, as expected, lies between the true  EV and CV
measures.
     But, what if the true form of the utility function is unknown, and
casual empirical work is performed or specification errors are  committed?
In that case, we shall see that the AIDS approximation becomes  far superior
in capturing the exact welfare change, at least for our contrived example.
     As an extreme case of casual empirical work,  suppose two reasonable
nested single equation demand specifications are fit to the CES data, the
                                     114

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beat one chosen on the basis of  R2,  and the Marshallian consumers surplus
measure for a fall in p:  of 0.5,  calculated given  income of 35.  Will the
results be close to the true values? Not  necessarily.
     Two demand models which could easily  be found in "quick and dirty"
empirical work are a second-order Taylor's series  approximation in prices
and, to parsimoniously capture nonlinearity, a model in the logarithms of
prices and income.  The results  of fitting these models on the quantity of
good 1 consumed generated by of  the  CES data appear in table B.6.  In both
cases, homogeneity of degree zero in prices and incomes has been imposed by
expressing prices and incomes relative  to  p*.  Since the dependent
variable, quantity consumed, is  measured in the same units in both
regressions,, adjusted R2  can be  used to select between them.  The model in
logarithms is preferred with this criterion, and looks fairly good in terms
of the signs and significance of the parameter estimates. But, the
Marshallian consumer's surplus estimated for a change in pt from 1 to 0.5
is definitely not reasonable. While the true Marshallian surplus is 5.637,
the surplus from the misspecified Model II in table B.7 is nearly double
this value: 10.345!  Clearly, confidence in Marshallian surplus measures
depends on confidence in  having  estimated  the correct single equation
demand function specification.
     This example nay appear overdrawn  but similar things can happen with
even more sophisticated approaches.   Suppose, for  instance, that the
analyst- displays a distinct preference  for the Linear Expenditure System,
and imposes it on any data set he acquires. It could be his  misfortune to
acquire a data set like our CES  data, for  if he commits a specification
error by estimating the LES system on the  CES data, table B.7 could result.
But for the fact that subsistence income when prices are unity bulks
unreasonably large in total income and  that the b  parameters are
suspiciously equal to each other the results in table B.7 pass casual
inspection.  Equation fits are not disastrous, and individual parameters
are all properly signed and significant.   But, again, the welfare measures
for a decrease in P! of 0.5 are  wildly  erroneous:
                                    115

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                Table B.6.   Two Naive Single  Equation Demand
                       Functions for Good  1,  CES Data

Model I
Parameter
Variable Estimate*
Intercept 40.47
(3o79)
P/P,, -58.55
- (6.63)
Pj/P,, 14.15
(1.53)
Pj/P,, 18.67
(2.45)
(Pi/Pj* 5.17
(3o56)
(P2/PJ2 -3.08
(1.72)
(Pj/Pj* 0.10
(0.11)
(Pt/PjCPa/PJ 2.51
(1.11)
(1.60)
(Pj/P,,)(Pj/PJ -1.73
(0.82)
y/P^. 0.28
(3.52)
ADJ R* 0.3120
Model II
Parameter
Variable Estimate*
Intercept -114.25
(4.80)
Infpypj - 46.13
(12.24)
ln(Pa/Pj 2.58
(0.52)
ln(P,/Pj 8.18
(2.06)
In (y/PJ 33.97
(6.47)

ADJ Rz 0.5921











*  Absolute value of "t" statistics  in parentheses.
                                     116

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                    Table B.7.  LES Model Fit to CES Data
                            Using SUNR Estimator
     Parameter
        Si
        S*
        s,
     EQ.1 -R*
     EQ.2  R2
     EQ.3  R2

*  Restricted Parameter
Parameter
Estimate
0.2U3
0.253
0.249
0.255
2.543
2.877
2.370
2.647
Statistic

  53.73
  63.23
  56.39
    *
  15.48
  13.75
  16.04-
   8.71
0.4523
0.5215
0.4656
                                    117

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                                        CV       .      '    EV
     True CES Model                   2.259        "  '     2.M82
     False LES Model fit to CES Data  5.075               5.98U
                (% error)             (125)               (1M1)
     Note: Absolute values of welfare ore as urea  and errors.
     In conclusion, if one were to pick a single system model to fit to the
data, AIDS would have been the safer choice.
                                     118

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                                     121

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                                     122

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                                 Chapter 4
             RECSIM MODEL DESIGN: THE DATA-GENERATING MODULES'
     With the background from chapters 2 and 3  and their  appendices  in mind,
we now take up a piece by piece (or module by module)  description of  the
simulation model put together for this project.  An overall  schematic of  this
model is provided in figure 1 .
     Initially we construct two elemental grids in space,  one  to  distribute
people and one to distribute sites.  We shall refer to the former as  the
ELEMENTAL PEOPLE GRID (EPG) and the latter as the ELEMENTAL  GEOGRAPHICAL  GRID
(EGG).  After the operation of POISSON, PASSIVE,  and PEOPLE  and the
application of POLICY and EUCLID these grids allow us  to  produce  travel
cost-based recreation activity site prices.   These site-prices, along with
the aocioeconomic information (SOCIO) go into the OPTIMIZE module, which
solves the consumer's choice problem pre- and post-policy.   The WELFARE
module computes the- exact CV and EV monetary measures  of  welfare  change
accruing to each individual as a result of the  pollution  control  policy.  The
outcomes represent samples of individuals with observations on  their
aocio economic characteristics (sex,  income,  and total  available leisure time)
along with their optimal pre- and post-policy recreation participation
patterns by activity category,  site  visit prices,  and  welfare measures.
Because of the complexity of OPTIMIZE  and WELFARE,  they are discussed in a
separate chapter (5).
 -  ""Since one of our purposes is to explore the effect of  data aggregation
(averaging) in the price proxies, a  major part of  the  model is  the set of
modules^labelled AGGREGATE,  SURROGATE  and ESTIMATE.  The AGGREGATE module,
discussed in this chapter, blurs the degree of resolution in site visit
prices- in the simulation data by creating site visit price  proxies measured
at alternative levels of spatial aggregation.   The proxies  are  to be the
number of sites per land area of any arbitrarily drawn  set of
"jurisdictions" composed of  subsets  of EPG squares.  These  subsets are

                                    123

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fouev
                     Figure 1



              Flow Diagram for RECISM
                         era si*
                 Crito for
    Sites aa4 Faopl*
saezo
                Aetivicy
                          L_
                        12U

-------
 identified by a subroutine  in  the AGGREGATE module.  This subroutine accepts
 a specification of  mean  subset size and  then randomly groups adjacent EPG
 squares to form subsets  of  size drawn from a distribution with that mean.  It
 will also  be  necessary to compute the number of recreation sites pre- and
 post-policy per AGGREGATE jurisdiction in the camping, fishing and movies
 categories and assign that  value to all  individuals living in the particular
 jurisdiction.   These AGGREGATE computations have no link to OPTIMIZE and
 WELFARE but merely  serve the purpose of  distorting the proxy price variables
 to be- used later In estimation.  The ESTIMATE module is described in chapter
 6.                                              '
      The results of exact welfare calculations based on individual
 optimization  must be compared  with results from econometric manipulation of -
 various forma of pseudo  survey data.  These comparisons are performed in the
 COMPARE module described in chapter 6.   These comparisons are.the meat of the
 exercise,  telling us what is lost, by necessity, when aggregated data is used
 or when models estimated for one level of aggregation are used at another
 level.
 CREATING INFORMATION OK  INDIVIDUALS
      The RECSIM model has six  modules which create information on
^individuals'  characteristics and their locations relative to recreation site
=locations*  before and after  water pollution control policy implementation in a
 hypothetical  country.  Briefly, these six modules are:
      1.  POISSON:  Randomly locates sites in space for two types of active
.	... recreation:  A  water-based activity influenced by pollution control
      :   policy (FISHING);  and nonwater  based activity (CAMPING) which is
         unaffected by the  policy.
      2.  PASSIVE:  Randomly locates in space sites for a passive, nonwater
         based urban recreation activity (MOVIES).  Sites are not affected by
         water pollution control policy.  The locations of these sites depend
         on population density, an outcome of the PEOPLE module below.
      3,  POLICY:  Randomly  selects, from the universe of potential fishing
         sites generated in POISSON, a subset of sites which are suitable for
         recreation prior to pollution control.  (That is, are unpolluted)

                                    125

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          and thus  the nature and extent of  possible  improvement in
          availability due to pollution control.
      4.   PEOPLE:  .Randomly locates individuals  in space according to one of
          two alternative distributions as specified  by the user.  Individuals
          can either be spread uniformly over any particular elemental grid
          square or spread by a truncated bivariate normal distribution to
          produce clustering around a randomly chosen population center of
          each elemental grid.
      5.   EUCLID:   Computes the travel distance  and two-way travel cost to the
          closest FISHING,  CAMPING and MOVIE site for each individual in a
          sample draw.
      6.   SOCIO:   Randomly assigns individuals a sex designation, an income,
          and an annual leisure time constraint.
      Below we discuss placement of points  in the elemental PEOPLE and
GEOGRAPHICAL GRIDS,  which involves the POISSON, PASSIVE, POLICY, PEOPLE and
EUCL&-me4ules.   Then we describe the  assignment of income, sex, and leisure
time characteristics through SOCIO.. Finally, we .sketch the random grouping
.of adjacent EPGs to create aggregate jurisdictions, the AGGREGATE module.
Elemental People and Geographical Grids
	'Sofacilitate aggregation of  elemental  PEOPLE grids under various
 arbitrary political boundary demarcations  and  to produce sufficient variation
 in the mean number of recreation sites  per unit of elemental land area, we
 have adopted the grid layout in figure  2.  This layout has 9 ELEMENTAL
-GEOGRAPHICAL grid (EGG)  squares and 36  ELEMENTAL PEOPLE grid (EPG) squares;
 with 4 EPG squares contained in each EGG  square.  The sizes are summarized in
 table?-?If we want to combine countries to  form a larger super-country we
 can stack countries of the above size and solve a larger problem.
      Before getting down to the mechanics of  generating points, a few remarks
 on the rationale for this sizing of  the EGG  are  in order.  Our desire is to
 allow for a "reasonable* expected one-way travel distance from residences to
 sites.   We take "reasonable" to be between 15  and 30 miles, as shown by
 actual  survey data.  From the POISSON  expected distance formula, expected
 distance E(r) equals:

                                     126

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                                  Figure 2

                          Elemental Grids for RECSIM
36

30


34

18


.'-.
t 
-
."..
6



6 1
1
i
1
P l P
1 [ 2
1
r
F3 P4


P13 P14
I
i
t .
9 t O
P15 P16
t
1
i
P25 I ?26
1
-   . G-     -
i
1
P27 [ P28
1
  _ i
2 18 2
1 
P5 P6
r
""""""2 "" ~
P7 P8


P17 P18
1
P19 P20



P29 P30
i
.   G -   -
1
1
P31 P32
i
4 30
1
1
p 1 p
i
T
Pll | P12
1
1
i
P21 j P22
I
1
j
P23 P24



P33 P34

	 G9 	
P35 P36



in


74

18
Ix
1
1
17


ft



                                                                              L*titud<
                                                                               Units
                                                                              [10 mil.
                                                                               per un
0,0
    12           18           24

Longitude Units  (10 miles per unit)
30          36
                                      127

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                                  Table 1
                   SIZES OF ELEMENTS OF THE RECSIM SPACE
t
Unit Length x Width
ELEMENTARY POL. UN IT . 60 x 60
ELEMENTARY GEO.UNIT 1 20 x 1 20
. 	 COUNTRY 360 x 360
Area
3,600
14,400
129,600

          E(r) - ,,>.,.

          where X.  -  number of objects per unit land area in a particular
                      geographic grid with index i -  1,  ..., 9.
With G. dimensions of 120 by 120 miles, or a total area of  14,400 square
miles, thus, the ranges for expected distance shown in table  2 pertain.
Reasonable expected travel distances will be produced by site densities in
the neighborhood of 5 or 10 per geographical unit.
POISSON MODULE:  GEOGRAPHICAL GRID PLACEMENT OF THE UNIVERSE  OF RECREATION
SITES.
     To make things simple to execute, let us assume the universe of sites
initially placed is itself the post policy situation.  The steps necessary to
place FISHING sites in the EGGs are as follows:
STEPS TO PLACE FISHING SITES
     1.  Specify the average number of FISHING sites desired per GRID SQUARE
         as exogenous input from the user.
     2.  Using the value specified in step 1  as  the mean (and variance) of a
         POISSON distribution, select 9 random integers from a POISSON random
         number generator.  (As \ grows beyond 5 the POISSON tends  to
         approximate the normal.  Other discrete distributions are  possible
         but this one is convenient).  This will produce the number of
                                    128

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                          Table 2
        Site Density and Expected Travel  Distances
                      (from POISSON)
  Number of
Sites Per EGG
         1
         2
         3
         4
         5
         6
0.00006944
0.00001389
0.00020833
0.00027778
0.00034722
0.00041667
    Expected
 One-Way Travel
Distance (miles)
       60
       42-
       35
       30
       27
       24
        10
0.00069444
       19
        20
0.00138889
       13
       100
0.00694444
       200
0.01388889
      1000
0.06944444
      2000
0.1388889
                        129

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fishing sites, NF^ to be placed in each GL ELEMENTAL GEOGRAPHIC GRID
(i - 1, ..., 9), as follows:
EGG Square
Number
Gi
G1
G2
Number of Sites
From POISSON Draw
NFt*
NF1
NF2
                           G9                           NF9
         *Sub3cript3 on  same line as alphabetical indicator for typing
         convenience.
     3.  Given NF. from step 2 randomly select the latitude and longitude of
         each site in an EGG-square from an independent bivariate uniform
         distribution.  Since we have assumed independence this is quite
         simple.  The latitude and longitude values can each be drawn in
     =   order, independently of each other, and the x,y coordinate outcomes
     =   paired to produce each site's latitude/longitude coordinates.
     	Specifically, we need 3 distinct uniform distributions from which to
         draw latitudes and 3 distinct uniform distributions from which to
         draw longitudes.  Sinee the uniform probability density is defined
         by its upper (b) and lower (a) limits (f(x) - 1/(b-a) for a < x < b)
         specifying these limits for each grid square defines the density
         function for that square.  The limits of the separate uniform
         distributions required to initiate the draw are shown in table 3-
         Let the longitude/latitude pairs for site i be denoted x, ,y,.  For
     	later use in the AGGREGATION module we want to assign each site to
         an elemental people grid square, EPG.  Because the POLICY module
         eliminates specific sites from recreation use in the pre-policy
         setting, we also need a convenient identifier for sites within grid
         squares.  This latter consideration suggests a simple numbering

                                    130

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       Table 3
Limits of EGG Squares

Limits for
EGG Square
Number.
Number of Latitude
and Longitude Pairs Latitude
per Square ' Lower
Uniform Distribution
Limits Longitude Limits
Upper
Lower
Upper

G1
G2
G3
G4
G5
G6
G7
G8
... G> . .
NFT
NF2 LAT. I.
NF3
NF4
NFS LAT. II.
NF6 .
24
24
24
12
12
12
NFT ( 0
NFS LAT. III. 7 0
NF9 f 0
36
36
36
24
24
24
12
12
12
LON. I.
LON. II.
LON. III.
LON. I.
LON. II.
LON. III.
LON. I.
LON. II.
LON. III.
0
12
24
0
12
24
0
12
24
12
24
36
12
24
36
12
24
36

scheme in which the NF.  sites within  an  EGG, G, square are first
sorted by EPG square and then numbered within each EPG square.
Since we know the boundaries  of  these squares the sort routine is
easy.  For any x.ry. we can first use longitude to choose the two
possible EPG squares within the  G.  which could contain x,, y_
The latitude sort chooses one of dthese  two, producing a unique
P..  Schematically this is shown in figure 3.
The logic is as follows:
     If LON  <
< LONk * 6 then
                    is in  P  or
         131

-------
       If LONk f 6 < xt < LONk * 12 then NFt  is In ?k or
       If LATm * 6 < Y, < LAT  + 12 then NF,  is In P. or P.
             in        i      m          -1         K     L
                           Figure 3
            Locating a Given Point in a Grid Square

LAT  + 6
L4Ta + 12
- 
*





pi
- ~ ' 	 1
P
a
r




i
P
 G
,.






Pl
t 	
Pn



UDK, * 6
~



.


LO!^
    LAT	 -
                      LOK,
  Together these statements assign NF.  uniquely to an EPG square.  The
  numbering problem can be solved in either of  three ways.  A number
  can be assigned when the latitude and longitude are drawn,
  independent of the results of the draw.   Or we can assign numbers
  primarily by longitude or primarily by latitude with ties "decided"
  by the other dimension.  The latter two alternatives are illustrated

                             132

-------
4.
in figures 4.A and 4.B for the same set  of  points in an CPG. square.
That is, the sites NFj. within EGG^ and  EPG,  are arrayed in order
either of x. or y. with ties decided  by  the other dimension.
We must also place CAMPING sites,  but the logic for doing so is
analogous to the fishing procedure, so we do  not repeat it.
After all sites have been placed and  numbered they should receive an
alphanumeric ID number containing  the following information:
    TYPE OF ACTIVITY

    PEOPLE UNIT WITHIN GL


    SEQUENTIAL SITE NUMBER


    LATITUDE LOCATOR

    LONGITUDE LOCATOR
                              F (fishing)  or C  (camping)
                              P.  people grid square number
                              (2  digits 01	36)
                              NF^ or NCk within P,
                              (4  digits 0001,...,9999)
                              /  (yyy.y)
                              x  (xxx.x)
    The outcome* of POISSON for any  particular run represent the
    universe of possible FISHING and CAMPING sites in our hypothetical
    region.  But these are not the USABLE  sites, since we must account
    for the pollution effect rendering some sites in the region
        A.  Longitude B*ed
                 |      iii 7
               -,--*----*'
           I   I
          *r
           1
                                         Latitude B*cd
                                                   43
                                                     12
                                        :i
                                                       7
                               133

-------
          unsuitable for fishing prior to implementation of water pollution
          control.  We assume implementation of water pollution control in any
          .pun makes all sites generated by POISSON suitable for recreation
  	post-policy.  To get the usable sites pre-policy, we need to
          construct a POLICY module.
 POLICY MODULE:   SELECTING A SUBSET OF PRE-POLICY  FISHABLE SITES FROM THE
 UNIVERSE OF POST-POLICY SITES
      Water pollution control affects only fishing sites, bringing sites which
 were unfishable pre-policy up to fishable quality post-policy.  In this
 module we.determine which sites are  unfishable pre-policy.  For now, the
-camping" sites are assumed to be unaffected by operations in the POLICY
 module.
      A simple and efficient way to delete sites from consideration in a
 pre-policy run is as follows:
 STEPS FOR POLICY
      1.  We know that en toto we have ZNF.  si tes  in  the whole region
                                       i  l
          (country).
      2.  From a uniform distribution over the 0,a interval (where 0 < a < 1
          is specified by the user) draw one random number, p .  Call p the
          fraction of all sites which are not fishable pre-policy.
      3.  Front a uniform distribution, assign a random number drawn from a 0,1
          interval to each site in the country (which means ZNF, random
                                                             ^
          numbers to be assigned). Select out as  unfishable all sites with
          random numbers less than or equal to p drawn in step 2.
      Thus,  from POLICY we get the subset of the universe of fishing sites .
 which cannot be used for fishing prior to implementation of the policy, as
 well as the subset that can.  Schematically:
                                                               Site random
                                                                  numbers
Unfishable
pre-policy
               0       p a                     1
Post-policy:  All sites assumed ftahable.
                                     134

-------
PEOPLE MODULE:  LOCATING INDIVIDUALS IN SPACE
     There are many conceivable alternatives for generating the locations  of
individuals in space.   A primary consideration for us  was  to be able  to
control our total sample size of individuals exactly over successive
pseudo-data runs.  First, because it will be convenient to  keep sample sizes
fixed in econometric,estimation under varying initial conditions and
aggregation schemes.  Second, because we want to be able to control  the cost
of generating any particular pseudo-data sample with RECSIM since the  number
of optimizations,, as well as the number of WELFARE calculations (CV, EV)
depends on the number of individuals located in space,  not  the number  of
sites.  A method that accomplishes this goal is the following.
STEPS FOR PEOPLE
     1.  Choose a total sample size for people, NP, for the region as  a
         whole.  We want to find NP , when NP is exogenous  input chosen by
         the user and J is the index of EPG squares.
     2.  Choose a set of -36 random- numbers from a uniform distribution over
         the 0,1  interval.
     3.-  Normalize the numbers drawn in step 2 so they  sun  to one and  each
         normalized number now represents the proportion of the total
         regional sample of size NP allocated to each PEOPLE GRID square.
         Refer to these random proportions as k , j * t, ..., 36.  The i
         value of k. is computed as:
              k   RANDOM NUMBER./I RAND CM NUMBERS.
               J                 J 4              J
         Then Ek   1.            J
     4.  Multiply the total desired sample size NP by each  of the k  to get
         the 36 EPG square populations NP .  Round to whole numbers  to get:
V. One possibility, for example,  would be to generate PEOPLE in  the same
manner that we generated sites in POISSON.  That is,  if  we have  36 EPG
squares and want an approximate sample size of,  say,  500, the average
number of people per grid used to initiate a draw of  36  square-specific
numbers of. people from a POISSON distribution (see step  1 of POISSON) would
be 14 (i.e., 14 x 36 - 504).  This route will not give exactly the same
sample size each time it is executed,  however.
                                    135

-------
               IMT(MP  )  = (k.)(NP)
          while maintaining ZNP.  -  default  size of 500 exactly.  Calculate
          NP/129.600 and NP./3600 (population densities) and output to PASSIVE
                           J
          module.
      4a.  In lieu  of step 4,  which requires rounding to maintain ENP. - 500
          exactly,  the multinomial  distribution can be used instead.  In the
          multinomial  distribution  we have* an experiment performed M times,
          and each time  the outcome must belong to one of k mutually exclusive
          and exhaustive categories,  each with probability p (0 < p. < 1).
          The sum  over i - 1 , ...,  k  of the p equals 1 .  In our context, k
          equals 36 EPG  squares,  the  p come from step 3 above, and N equals
          total sample size (HP  in  our notation with a default value of 500).
-------  We want to draw x.  (M   in  our  notation) which is the number of
          outcomes (people)  that  belong in  each category (EPG square) j.  Then
          the random, vector  x * (x ,  .... x. )' has a multinomial distribution
          and any particular  outcome  vector can be produced given the
          parameters p.  and  the total sample size, N.  The resultant x 's (MP.
          in our notation) will automatically sum to NP.  We prefer 4a to 4.
      5.   Proceed: to draw- the latitude and  longitude for each individual in
          each EPG square.   This  step is  analogous to step 3 in the POISSOM
          model, but the subroutine contains two options.  The first is to
          draw from- an independent bivariate uniform distribution, just like
          POISSOM.  This will not produce spatial clustering of individuals.
          To achieve a clustering effect, there is an alternative in which
          coordinates will be drawn from  the independent bivariate normal
          distribution.
      So,  under PEOPLE we have two submodules as shown in figure 5.
      PEOPLE;   SUBMOD 1 .  The steps under SUBMOD 1 are exactly analogous to
 step 3  in POISSOM,  so they  are not repeated here.
      PEOPLE;   SUBMOD 2.  The SUBCD  2 option of PEOPLE places people in apace
"USTng the independent bivariate  normal distribution.  This allows for dense
 concentrations of individuals  around the  chosen center of population mass of
 each EPG square.
                                     136

-------
     The bivariate normal Joint p.d.f.  of two random variables x,y which are
independently distributed is:
                         1
                               expC-1/2((x - u,)*/a* * (y -  u )a/
-------
                           (y--u)
                                    2
                        .
     fv(y) -   expC-1/2
                   expC-1/2
The steps for placement of points are therefore:
1 .  To begin the process, we must have values In  each EPG square for u ,
                                                                     &
    u , a r 
-------
       where  u   - 30 + RAN1 .   So we just solve for
               / 1
Given a. trial y.  we must check to see if  it  is contained within the
latitude limits of EPG square P1  of  30 to 36.  If it falls in this
interval, keep it and move on to another  individual.  If it does
not, keep redrawing standard .normal  variates until an acceptable
       
value is found.  This is Just a crude way of truncating, the
distribution at the grid boundaries  and means that while the mode of
the population distribution will be  at u  ,  u   the mean in general
                                       7i   xi
will not be.  Do this for all NP individuals making sure to obtain y
values conditioned on the appropriate grid square specific values
of u  and o  from steps 1  and 2 above.
Obtain NP acceptable longitude values to  be  paired with the NP
latitude values following the procedures  of  step 3 above.  The
pairing, can be accomplished by simple sequencing.  The first NPt
latitudes are paired with the first  NPt longitudes and both have
been drawn using; the rules specific  to EPGt.  After individuals have
been located using either submodule  1  or  2 of PEOPLE, each person
should be assigned an 10 designator  containing information on
location and Jurisdictional membership.   The following format is
suggested:
     PERSON DESIGNATOR            I (Individual)
     EPG SQUARE NUMBER            P.  (2  digits 01, .... 36)
     INDIVIDUAL SEQUENTIAL        NP.  (3 digits 001 ..... 999)
       NUMBER WITHIN P               J
                      J
     LATITUDE LOCATOR             ILAT (yyy.y)
     LONGITUDE LOCATOR            ILONG  (xxx.x)
                           139

-------
 PASSIVE MODULE:   LOCATING PASSIVE RECREATION SITES IN SPACE
      The PASSIVE module places  a MOVIES recreation site In a EPG square,  P.,
                                                                          J
 if  the population density of  that grid is greater than the average density of
 the entire region.  Otherwise,  the square does not get a MOVIES site.   The
 steps in PASSIVE depend on whether SUBMOD 1 or SUBMOD 2 was selected in
 PEOPLE:
      I.  Steps in PASSIVE:  SUBMOD 1 from PEOPLE selected
           1.   Decide whether  or not a P  grid gets a MOVIES site:
                  YES,   if NP./3600 2 NP/129600 (- 500/129600 in the example
                                               used, here)
                  NO,   if NP./3600 < NP/129600 ( 500/129600 in the example
                                                used here)
           2.   If yes in step  1, place the MOVIES site in the center of  P..

      II.  Steps in PASSIVE:   SUBMOD 2 from PEOPLE selected
           1.   Decide whether  or not a P. square gets a MOVIES site.  Same
                                       J
               rule as step 1  in I above.
           2.   If yes In step  1r place the MOVIES site at the grid coordinates
               given by u  r Uy  chosen in PEOPLE, SUBMOD 2.            .
 In  either case,  give the MOVIES sites ID designations analogous to those  for
 FISHING and CAMPING sites:
           TYPE OF ACTIVITY                 M (movies)
           EPG SQUARE     "                 P. (2 digits 01, ...,  36)
:  .      ;  LATITUDE LOCATOR                 Lat (yyy.y)
           LONGITUDE LOCATOR                Long (xxx.x)
 EUCLID MUDULE:   CONNECTING RECREATION SITES AND INDIVIDUALS
      The aim of EUCLID is to compute the two-way travel distance from each
 individual's location to the closest fishing, camping and urban leisure site.
 The fishing calculations must be  done twice  once pre-policy and once
 post-policy.  The camping and urban leisure distances are unaffected by the
 policy and only have to be done once.
      There is obviously no need to search over the whole "country" and
 compute the distance from every residence to every site before being able to
                                     140

-------
choose the closest site in each category.  "Sorting rec tangles" centered on
each -individual for initial search and distance computations would be more
efficient.  If no sites are found within the initial sorting rectangle, its
size can be enlarged and the search repeated until closest sites in each
category are found.
     For instance, suppose we let the size of the initial sorting rectangle
have a (maximum) base of 6 longitude units (60 miles) and a (maximum) height
of 6 latitude units (60 miles).   (The word maximum is used to account for *
edge effects which could truncate the sides).  Then each individual's sorting
rectangle will be located around his LAT and LONG coordinates, and distances
to all sites in each category (FISHING, CAMPING, MOVIES) calculated and the
minimum, distance found.  If no  sites in a category are found the size of the
sorting rectangle can be enlarged to, say, 12 latitude and longitude units
and the computations repeated.
     The steps for EUCLID in general will then be:
     1 .  Calculate the size of  each individual's irs sorting rectangle from
         his individual (I) ILAT,  ILONG values as
               ESORTLNGI - LOHG  * 3 if LONG  * 3 * 36
                         - 36  if LONGL + 3 > 36
               WSORTLNGI  -  LONGj^ - 3  if LONGj^ - 3 2 0
                         -  0  if LONG  - 3 < 0
               NSORTLTI   -  LA^ * 3 if LATt + 3 S 36
                         -  36  if LATt + 3 > 36
               SSORTLTI   - LAT^- 3 if LA^ - 3 2 0
                         - 0  if LAT1 - 3 < 0

     2.  Find all sites  within the individual's sorting rectangle, which will
         encompass all sites  with longitudes greater than or equal to
         WSORTLNGI but less than or equal to ESORTLNGI and latitudes greater
         than or equal to SSORTLTI but less than or equal to NSORTLTI.  If
         there are no sites in a category in the sorting rectangle, enlarge
                                   141

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         it by 2 units of latitude and longitude in each direction,
         maintaining the aide constraints.
     3.   Calculate the distance from each individual to every site in a
         recreation category within his sorting rectangle by the Euclidian
         formula based on knowledge of the J   sites (S.) latitude and
         longitude (S - F, C or M)
               North-South     MS  - JILAT. - SLAT.|(10)
                                                J

               East-West       EW   |ILONG  - SLONG.|(10)
                                         *        
-------
the OPTIMIZE problem's constraints and SEX implies a sex-specific B matrix in
OPTIMIZE.
     The steps in SOCIO are:
     I. INCOME
          1.  Generate INCOME  for each individual in the sample by assuming
              that income,  c,  is lognormally distributed (its logarithm is
              normally distributed).  This implies that c* - Inc is N(u, a*)
                                   gn
                              (1/2) a2
         c*
and c  e   has the lognormal  distribution with mean equal to:
                    E(c)  -  e
              and variance:

                    V(c)  -
              Suppose we want  the  expected value of per capita income to be a
              fairly representative value, and set it at $15,000.  At the
              same time we  want  to keep the odds of being a millionaire
              fairly low, say  one  in one million.  With these two factors in
              mind we can solve  (numerically) for the appropriate u and a
              values, which are  (approximately) u - 9.1334 and a - 0.985.
              Using these values,  we then draw c* from a normal distribution
              N(u - 9.1334, 9*  0.970) and create c - ec  for our income
              values which  are passed to OPTIMIZE.  The distribution of c
              based on these values will in general be as in table 4, which
              is in line with  the  income distribution of employed persons in
              the U.S.  The income distribution in this table makes it fairly
              easy to pick  an  upper limit income value to be used in scaling
              the utility function .parameters to avoid exceeding bliss.  Any
              value over $50,000 can be used to keep more than 95 percent of
              any sample below bliss, and a value of $100,000 will keep
              nearly everyone  from complete joy most of the time.
     II.  LEISURE-TIME CONSTRAINT
          1.   The leisure time constraint value selected in this module is
              passed as a r.h.s. constraint value on total leisure time to
              the OPTIMIZE  module,. There are three options:
              OPTION 1.   Set everyone's leisure time constraint to 365.  This
              is the DEFAULT.
                               U3

-------
                        Table 4

     Income Distribution:  Statistical Information

Income
Level
c
1,000
5,000
10-, 000+_
15,000
20,000
25,000
30,000
35,000
40,000
45,000
50,000
100,000
500,000
1,000,000
Log of
Income
c*=logc
6.9078
8.5.172
9.2103
9.6158
9.9035
10,1266
10.3090
10.4631
10.5966
10.7144 -
10.3198
11.5129
13.1224
13.8155
Standard
Normal
Deviate*
Z
-2.2595
-0.6256
0.0781
0.4898
0.7818
1.0084
1.1935
1.3500
1.4855
1.6051
1.7121
2.4158
4.0497
4.7534
% of Sample
Below Income**
Level c
1.19
26.58
53.11
68.78
78.28
84.34
88.36
91.15
93.13
94.58
95.66
99.21
99.99+
99.99+
" . . 
 Z = (c* - 9.1334)70.985
From cumulative normal distribution
 (Median income equals $9259 is the DEFAULT.
                       144

-------
              OPTION 2.   Select  each individual's leisure time
              constraint randomly from  a  uniform distribution over the
              interval 104 to 365.
              OPTION 3.   Select  each individual's leisure time
              constraint randomly but contingent on income.
                    If income is less  than or equal to the median value,
                     9259, select leisure time constraint from uniform
                     distribution over  interval 104 to 132.     
                    If income is greater than 9259, select leisure time
                     constraint  from uniform distribution over interval 104
                     to 365.
     III.  SEX
         "1.   The sex indicator   0 if male, 1 if female  is used to
              select the appropriate B  matrix in OPTIMIZE.  If SEX - 0 use
              the base B matrix.   If SEX  - 1 decrease the b . values in the
              columns J  - 1  (FISH) and  j  - 2 (CAMP) respectively by 30 and 25
              percent and increase the  b. . values in the columns j - 3
              (MOVIES) and j   4 (HICKS)  respectively by 10 and 20 percent.
              We need two options for SEX:
              1. OPTION 1.. Set  SEX  equal to 0 for all individuals, implying
                 no difference in the household production technology between
 *                males and females.
              2. OPTION 2. Assign random members drawn from a uniform
                 distribution over the  0,1 interval to all individuals.
                 Then, if an  individual has a random number less than or
                 equal to 0.5, let SEX  equal 0.  Otherwise, SEX equals 1.
AGGREGATE MODULE:  COMBINING  ELEMENTARY POLITICAL UNITS TO FORM AGGREGATED
POLITICAL UNITS                 '
     The AGGREGATE module draws  an artificial veil of ignorance over the
elementary political and geographic  boundary demarcations to reflect the
aggregation problem encountered  when working with real survey data.  That
is, the appropriate geographical  boundaries over which density-based
measures_of expected site - visit prices  should be calculated are generally
unknown (in our context the EGG  boundaries are unknown, as is the mapping
                                   145

-------
of EPG grid squares into EGG grids).  So,  all that can be done is to
compute density measures from politically drawn boundaries  in ignorance of
the relationship between these political units and the geographic units
defined by distinct population   measures.  The AGGREGATE subroutine of
RECSIM builds up the elemental population units into larger jurisdictions.
The basic operations of AGGREG are as follows.
     First, AGGREG selects a target jurisdictional size (where size is the
number of contiguous EPG cells) via a single trial from a multinomial
distribution with parameter vector P1.   The dimensionality  of PI is equal
                      ^             
to the maximum acceptable size minus one (MAXSIZ).  The elements of P1  are
declared by the user such that PI (!),...,  P1(k)  correspond  to the
probabilities that the target jurisdiction size will be 2,....,(k+1).  If
the user wishes the target jurisdiction to be nonstochas.tic and therefore
equal with certainty to, say, j, then the user specifies P1 (j-1)-1.000 and
all other elements of P1(.)-O.ODO.  We refer to a  "target"  jurisdiction for
the following reason: AGGREG operates so as to "attach" up  to (MAXSIZ-H)
contiguous cells to each other to form the aggregated political
jurisdiction.  However, the algorithm can "run out of room," as it were, by
encountering a boundary or a cell from which all directions (N,E,W,S) are
either boundaries or are already occupied.  AGGREG does not retrace its
steps in order to attain the jurisdiction size (MAXSIZ+1),  but will, when
confronted by the situation described above,  search for a new starting
point for the (n*1)at jurisdiction.
     AGGREG searches for jurisdiction starting points as follows.  The user
specifies a probability vector P2 of dimension MGSQ,  where  MGSQ is the
number of cells in the grid.  MGSQ equals MAXGRD**2,  where  MAXGRD is the
row/column dimension of the grid.  Given the elements of P2, which for
convenience might each be set equal to 1 .ODO/XXX.ODO (where XXX is the
double precision representation of MGSQ) AGGREG draws one trial from a
multinomial distribution, the realization of which becomes  the count
variable (say mK  Then, from the (1,1)  grid position,  AGGREG proceeds
across columns by row until m empty cells have been encountered.  Upon
encountering the m-th empty cell, AGGREG begins to form its political
jurisdictions.  Should AGGREG encounter the (MAXGRD,  MAXGRD)-th cell before
                                    146

-------
encountering m empty cells,  the subroutine returns  to the  (1,1) position
and continues counting until the m-th empty cell  is found.
     Once having found the starting point,  AGGREG draws  one  trial from a
multinomial distribution with four-dimension parameter vector PP, where
PP(1)-PP(2)-PP(3)-PP(4)-0.25PO in order to determine in  which direction to
move- to trace out the jurisdiction (North,  East,  South,  West).  AGGREG also
draws a Bernoulli trial with p-(1-p)-0.5DO  in order to determine whether to
spin clockwise or counterclockwise should the destination  be either outside
the grid boundary because the starting point is already  on the edge of the
grid or the destination is already occupied.
     When either the target jurisdiction size has been met or all of the
moves North, East, South, and West are "blocked," then AGGREG increments by
one the jurisdiction identification number, redraws the  starting point
counter, and continues as above until all cells are identified as members
of jurisdictions.
     AGGREG is accessed as follows:
     Call AGGREG (P1, P2, MAXGRD, MGSQ, MAXSIZ, DSEED, JMAT, JOUT)
     Where the arguments are:
    . P.T.........Probability vector as described above,  dimensioned P1 (MAXSIZ),
       and declared as REAL*8.
    ;.P2	Probability vector as described above,  dimensioned P2(MGSQ),
       and declared as REAL'S.
   _. MAXGRD.....Integer variable equal to the number of  rows (or columns)
       of the square grid.
     MGSQ	Integer variable equal to the number  of cells  in the grid
       (-MAXGRD**2).
     MAXSIZ	Integer variable equal to the cell-size  of the maximum
       acceptable target jurisdiction minus one.
     DSEED	REAL*8 seed variable in the inclusive range 1.000 to
       2147483647.000.
     JMAT	Integer-dimensioned matrix of dimensions (MAXGRD, MAXGRD).
    . JOUT.....Integer-dimensioned matrix of dimensions (MAXGRD, MAXGRD).
       This is the matrix returned from AGGREG that contains the
       jurisdiction identifiers.  For example,  if MAXGRD-5 and it happens
                                    147

-------
       that the target Jurisdiction size in a particular run equals  4,  then
       JOUT might look like;
1 1 1
1
1
5
: 5
7
6
5
222
3 3 3
7|
6
5
2
6
6
4
; 4
I 4- 4
AGGREG  requires  that use of the FORTRAN subroutine MLTNOM that returns  the
realization of one trial from a multinomial distribution.  Via MLTNOM,  the
returned value of OSEED from AGGREG will be different than the value
supplied as argument.  Note that each separate run of AGGREG should produce
a  unique grouping scheme, given a different initial seed.
     Once AGGREG has chosen a grouping scheme, density measures based on
that scheme can  be calculated.,  The entire menu of alternative ways of
measuring density* including two "correct" measures, is:
     1.   Exact  Unweighted Population Density:  True \ values based on  the
.   _ '    .EGG'S.
 L    2.   .Sample Unweighted Density:  Estimate of true \  values based on
          observed number of sites per EPG area.
     3.   AGGREG Unweighted Sample Density:  Estimate of  number of sites
          contained per unit area of an AGGREG jurisdiction.
     4.   AGGREG Population Weighted Sample Density:  Estimate of number of
          sites  contained per unit area of each EPG contained in an AGGREG
          jurisdiction, weighted by that EPG'3 share of total AGGREG
                     2
          population.
                                    th
 2.  Let f.  be  the  population in the i"" EPG contained in an AGGREG
 jurisdiction, and \   be the sample estimate of its density measure.  Then
 the weighted  measur   "~
e X  summing over all  EPG's  in an AGGREG jurisdiction
        (Footnote continued)
                                    148

-------
We have elected to specify three base case vectors for the  target size
probability vector P1.   These three instances allow for a fairly rich variety
of target jurisdiction sizes while at the same time keeping computational and
data handling burdens to a minimum.  The cases can be  described as follows:
     Case 1;  Equally likely targets, maximum acceptable target size - 18.
          Specifications:   MAXSIZ  17

                      P1(1) - 1.0DO/17.0DO
                      P1(17)  - 1.ODO/17.0DO
     Case 2;  Bulk of target  probability distribution at size four, with
              maximum acceptable target size  -6.   The probabilities
              corresponding to the target size  outcome vector (2,3,4,5,6) are
              (1/10,  2/10,  4/10,  2/10,  1/10).
          Specifications:   MAXSIZ - 5
                           PUD  - 0.1DO
                           P1(2)  - 0.2DO
                           PU3)   0.4DO
                           PI (4)  - 0.2DO
                           PUS)  - 0.1DO
     Case 3;  Bulk of target  probability distribution at eight, maximum
              acceptable target size -  .10, minimum acceptable target size *
              6. The probabilities corresponding to the target size outcome
              vector (2,3.4,5,6,7,8,9,10)  are (0,0,0,0, .1, .2, .4, .2, .1).

          Specifications:   P1(1)  - O.ODO
                           PU4)  - O.ODO
                           PUS)  - 0.1DO
                           P1(6)  - 0.2DO
2. (continued)
is:
                 *  *   ff
                               i
                                    149

-------
                           P1(7) - O.UDO
                           P1(8) - 0.2DO
                           P1(9) - 0.1DO
 - -The number of passes through AGGREG for each of  the  three baselines
above is specified by the user, and determines the total  number of potential
econometric models to be estimated using the aggregated proxy variables
(i.e., 3 probability measures x n passes per probability  vector oc 2
aggregated proxy X measures per individual per pass - 6 n models to be
estimated).  This completes the data-generation components of RECSIM.
                                               o
Particulars about solving each consumer.'s optimization problem pre and post
policy (the OPTIMIZE and WELFARE modules) are discussed in the next chapters.

                     Summary User Supplied Starting Values

     1.  Number of Regions Desired.  (The text assumes this a default value
         equal to 1).
     2.  Average number of fishing sites desired per  EGG  square (POISSON
         MODULE).
     3.  Average number of camping sites desired per  EGG  square (POISSON
         MODULE).
     4.  Lower limit, p of fraction sites not fishable for uniform
         distribution drawn in POLICY MODULE.  Assume default of 0.5.
     5.  Total sample size of individuals NP, in PEOPLE MODULE.  Default is
         500.
     6.  Grid square coordinate selection procedure desired to locate
         individuals in space.  Options are SUBMOD 1  (bivariate uniform) or
         SUBMOD 2 (bivariate normal) in the PEOPLE MODULE.  Default is
 "      SUBMOD 1 .
     7.  Degree-of-clustering parameter for use in bivariate normal
         distribution of people (PEOPLE MODULE).
     8.  Travel cost per mile traveled (EUCLID MODULE).  Default is $0.10.
     9.  Leisure time constraint value in SO (HO MODULE.  Default is 365 for
         all individuals.  Options assign values randomly.
                                    150

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10. Sex indicator for  individuals in SOCIO used to select the
    appropriate B matrix  in OPTIMIZE.  Default is 0 (no sex distinction)
    for all individuals.
                               151

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                                 Chapter 5
       MODEL DESIGN:  .OPTIMIZATION AND WELFARE CALCULATION IN  RECSIM

     Two closely linked modules in RECSIM are the utility optimization
(OPTIMIZE) and benefits calculation (WELFARE) routines.   The first solves
the consumer's utility maximization problem and the second calculates
welfare changes attributable to a chosen policy.  The following discussion
covers implementation of the theoretical groundwork laid  in chapter 2 and
appendix 2.A in the simulation model context.  Thus, our  concerns here are
largely practical matters of structure, scaling, and computational method.
PRELIMINARY STRUCTURE OF OPTIMIZE
     The optimization problem in the simulation model,  in which an
individual maximizes utility subject to time and budget constraints and the
relation between purchased goods and "wants", takes the following form:
     Maximize         U - u(z)
     Subject to       z. - Bx
                      P'x S y
                      j'x S t
                      x p, J  0 and B s 0
where z   is the (mxl) vector of wants (or characteristics) from which
          utility is actually derived;
      B   is the household production technology matrix,  with  the elements
          b   representing the amount of want i produced  by a  unit
          consumption of good j;
      x   is the (nx1) vector of decision variables, where the first (n-1)
          variables refer to the days of recreation consumed,  and the nth
          is the Hicksian composite good;
      p   is an (nx1) vector of observed prices of the goods;
      y   is the individual's income;
      j   is the (nx1) vector of time Input per unit consumption of the
                                    152

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          good with no leisure time  input  implied by consumption of the
          Hicksian composite good;
      t   is the individual's total  leisure.
     Further the utility function is specified to be of the additive
quadratic-form, such that,

          U - u* - Eai 0, z^ & Q~  This problem is easily transformed into a form which can be
solved by a quadratic programming (QP) algorithm.  We use the Lemke
complementary pivot algorithm,  as put forth in Ravindran (1972).  The
general form of the problem  solved by this algorithm is:
          Minimize    U - c'z + z'Qz
          Subject to  Az  b
                      z & 0
where a solution can be found only if U  is a convex function.   This is
equivalent to restricting Q  to be a  positive semidef inite matrix, that is,
that z'Qz  0, for all z.
 ^    The additive quadratic  utility  function from RECSIM can be expanded as
follows:
                                
          U - u- Ta.(d, -  z,)*
1. The objective function is sometimes described in the form c'z +
(1/2)z'Qz (Houthakker,  1960)  where the 1/2 term is factored out of Q for
ease in differentiation.
                                   153.

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The RECSIM maximization problem may then be converted to a minimization
problem which has for its objective function:
          Minimize    -U - c'z * z'Qz
where,
                                            at 0   .   .  .0
         c 
               Q -
                                            0  .
                                                     .   0
                                            0  .   .   0   a
                                                         m
The Q matrix is positive 3emidefinite if all a.  5 0,  which is  true by
definition in our characterization of the individual's utility function.
     To take advantage of the Lemke algorithm in RECSIM,  a driver routine
was constructed which converts the maximization problem into the equivalent
minimization problem and constructs the A and b matrices  of constraint
coefficients.  A trial problem (in terms of goods only) due to Houthakker
and described in Boot (1964, p. 108) was solved initially to insure that
our implementation of the QP algorithm, worked.  The inputs to  the
conversion routine, for the trial problem, were very simple:   the nun be r of
goods r the number of constraints (together defining the problem's size),
and the coefficients of the utility function and constraints,  as expressed
in standard quadratic form (c, Q, A, and b).  The problem was  solved for
incremental levels of income for the budget constraint (Boot,  1964, pp.
148-9).
     The Houthakker problem may be written in standard quadratic form as,
-        Maximize:   U - c'z * z'Qz
          Subject tos Az S b   
                       z  0
where
      c 
"is"1
16
22
20

Q -


3.0  0.5  4.0  0.0
0.5  5.0  0.5  2.0
4.0  0.5  8.5  1.5
0.0  2.0  1.5  5.5
A -
 1111

 5   0  10   0

LO   4   0   5,
 2

L3.
                                    154

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 The  z.  represent diet inputs (food) in Houthakker's problem and < is the
 money  income level.  Food is bought using both money and ration coupons,
 where  the  latter come in two varieties, subject to two allocation
 constraints.  The money prices of the goods are unitary, but the ration
 coupon prices are not, the final two constraints representing the rationing
 scheme.  The output of the program,' at this stage, included the optimal
 values of  the decision variables (goods), the optimal utility level from
 the  objective function, and the marginal utility and slack variable
 associated with each constraint.  Our implementation of the Lemke algorithm
 successfully replicated the results presented for the Houthakker problem
 for  seven  discrete values of income.  With confidence in the central
 algorithm, we expanded it to solve a problem more typical of RECSIM, where
 we introduced the transformation of goods into wants, and an incidental
 protection against a solution with consumption greater than that producing
 bliss.
     We address the bliss consumption possibility first.  An objective
 function of the form c'z * z'Qz will yield the same utility index  as the
 function Iad? - Ia,(d, -> z. )* as shown above, but the optimization
 problems-using these objective functions are not the same.  If all q. . are
 nonnegative, the maximum- value  of  o'z + z'Qz  is positive  infinity,  with
 the  z.  approaching infinity, while the unconstrained  optimal  point  of
 consumption under  the la.df " Iat^di " zt^2 objective function is the z
 vector which describes the bliss point.  To restrict the optimization
 problem of the form c'z + z'Qz to the economic region below (southwest of)
 bliss, we  had to add (m) constraints of the form, z, S d,, since z, > d.,
 tor  any i, violates the notion of bliss. There are now (2+m) constraints:
 the  budget constraint, the leisure time constraint, and the (m) constraints
 restricting the individual to be below bliss.
:.    One inconsistency still remains between the RECSIM optimization
 problem and the problem solved by the Lemke QP algorithm.  This
 inconsistency is between the decision variables in the objective function
-and  the decision variables in the constraints.  While z - Bx describes how
 to obtain  the optimal want levels from the optimal goods consumption, the
 objective  function does not yet capture this transformation of goods into
                                    155

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wants.  Thus we had to change the objective function to put  it  in terms of
goods, as follows:

          U - u(z) - u(Bx) - c'(Bx) * (x'B')Q(Bx)
                           - (c'B)x * x'(B'QB)x
where c and Q are as originally defined.  The (m)  constraints which
restrict the individual to the  economic  region below  bliss  had to be
transformed to Bx  d, to be consistent.  The final  problem is then:
          Maximize
          Subject to
                       (c'B)x * x'(BfQB)x
                       p'x S y                         rp'~|      |~y
                       J ' x S t      or Ax S b where A   J' , b -  t
                       Bx S d                          LB J      Ld.
                       x i 0
     We used the Houthakker diet problem again as  a benchmark to insure
that the conversion routine, including the transformation from goods to
wants, was properly implemented.  If we treat the  coefficient matrices
(linear and quadratic) of the Houthakker problem,  c and Q, as really being
the matrix products c'B and B'QB, then if we have  a household production
technology matrix B, we can find the values of c and Q themselves, such
that the conversion routine should reproduce the original Houthakker
results when given c, Q and B. Suppose that
      B -
-.3
1
2
. 3
5
2
4
7
2
1
1
5
6 '
1
5
3 -
                                                             -22.333
                                                              12.667
                                                              21667
                                                              -2.333
     and  Q
- 15.694
-9.139
-17.639
. 0.944
-9.139
30.778
6.528
=5.639
-17.639
6.528
20.528
-0.389
0.944 '
-5.639
-0.389
1 .194 -
Then the coefficient matrices of the utility function are, as expected.
                                    156

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c - (B'c)'-
-18.000
-16.000
-22.000
-20.000 J
Q - B'QB -
3.000
0.500
4.000
-0.000
0.500
5.000
0.500
2.000
4.000
0.500
8.500
 1 .500
o.ooo-
2.000
1.500
5.500-
and the solution to this expanded problem is identical  to the solution from
the original Houthakker problem.   The information produced by the
optimization problem now includes the optimal levels  of goods (x*), wants
(z* - Bx*),. and the utility index (u), given the global parameters that
all individuals- are assumed to face (a , d , B).    We also know the
marginal utility of income (A*), leisure time, and the  marginal  utility of
increasing the bliss level of each want, as well  as the slack in each of
the constraints.
Zero Consumption Levels
     If an individual consumes zero days of any recreation good  at his/her
price vector, then we.would like to know the price at which that individual
would just start to consume that recreation good.  This will be  important
information when estimating demand curves.  To find these reservation
prices-.for the (n-1) recreation goods, we add (n-1) additional constraints.
We will restrict the consumption of each recreation good to be at least
soae-smair positive amount.  To an individual who would indeed consume some
of that, recreation good, this is a redundant constraint.  For an individual
who would not consume any of the recreation good  at his/her observed price,
the small amount of forced consumption has a very small effect on the
consumption of other goods and thus the utility level.   We chose 1x10
2. We may ignore the economic interpretation of  and justification for
negative elements of c and Q,  since the products c'B and B'QB contain
nonnegative elements.
3. Lancaster (1966)  indeed states that  all  consumers will face the same 3
matrix.  We restrict all individuals to have the same  utility function in
the base case so that demand equations  may  later be estimated.  The only
distinction allowed for in alternative  cases is  that utility function
parameters are distinguished by gender.
                                    157

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days  as  the  small  positive  amount  that an individual must consume of the
recreation goods.  We  simply augment the A, b constraint matrices with
(n^l)  constraints  of the  form -x.   -0.000001 and solve this new problem.
The additional  information  we gain from adding these (n-1) constraints is
for each recreation good  (5^.
      To  find the reservation price, we need, to know how much the observed
price must be decreased to  induce  the individual to consume the good
without  the  positive consumption requirement.  This price change can be
extracted fron  the solution with the positive consumption constraints
included. The  marginal utility  of relaxing the positive consumption
requirement  times  the  marginal cost of utility (the inverse of the marginal
utility  of income, Deaton and Muellbauer, 1980, p. 250 and Varian, 1979, p.
209),  is the marginal  cost  of relaxing the positive consumption
requirement. This value  will be positive and is equivalent to the price
decrease which  would allow  the individual to optimally consume the
recreation good in the same small  anount.  Thus we apply the price decrease
(Ap.)  to the observed  price (p.) to generate the reservation price (rp.)
'by,
                       rpi " pi ~ Api " pi ~ (5i/x)

Note  that although we  need  only  calculate the reservation price for those
who do not otherwise consume a recreation good, the above method would
yield the observed price  for an  Individual who indeed recreates, since
his/her  marginal utility  of relaxing the positive consumption requirement
is zero.  This  method  (which involves adding one constraint for each
recreation good) is more  efficient than the brute force method of
incrementally decreasing  the prices of the recreation goods not otherwise
consumed to  find the actual price  at which consumption becomes positive,
each  iteration  of  which involves resolving the QP problem (though without
the additional  constraints).  Note that the brute force method requires
additional QP solutions to  provide each additional decimal place of
accuracy.
      Calculation of the reservation price, of course, depends on having a
strictly positive  marginal  utility of income.  If an individual's marginal
                                     153

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utility of income is zero,  and he/she does  not  consume  a particular
recreation good, then no additional  income  or equivalent price decrease
would serve to increase his/her consumption of  the  good. .Alternatively,
one could say that this individual's reservation price  is negative
infinity, and for these individuals  we set  the  reservation  price  (for the
appropriate good) to a large negative nunber (-999,999.0) as a proxy for
negative infinity and flag this individual  for  the  estimation stage.
     Another problem arises for individuals whose marginal  utility of
income is zero.  The budget share spent on  a particular good for  such
individuals is misleading due to the existence  of "excess1*  income (the
slack variable of the budget constraint).   Since we added constraints to
insure that an individual does not consume  more than the bliss mix of
goods, no one can exceed the bliss utility  level.  The  presence of seme
positive amount of excess income instead signals this person's capacity to
be-beyond bliss consumption, were it possible.   For such individuals,
income minus excess income is the correct income value  to use in  the
estimation of share equations.
Scaling the RECSIM Utility Maximization Problem
    .Since all individuals face the  same utility function in our  model, we
will scale the a. and d,  parameters  and 3 matrix to produce results that
seem- representative of the real world.  A description of the economic
meaning of these parameters follows.
     The household production technology is characterized by the  B matrix,
representing a linear production transformation from goods  to wants.  Each
b~, is the contribution to satisfaction of  want i produced  by a unit's
 * w
consumption of good j.  The b. . values will be  nonnegative, which means
that goods can only contribute positively to want satisfaction.   The value
of- these elements matters only relative to  other values in  the B  matrix.
The values within a col mm of B signify the contribution good J makes to
each of the wants..  This is independent of  the  relative rankings  of wants
in producing utility.
                                    159

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     In wants space, the bliss saturation point is described by  (Q  c)/2
              h
(Wegge, 1968).   Using Q and c as defined for our additive quadratic
utility function, the bliss point is described by (Q^c/a)'  - (-dT  ... -d,
... ~4 ). which is found in the negative orthant since c and Q were
constructed to represent a minimization problem. .The d.  parameters are the
basic determinants of the bliss level of the wants.
     The a^ parameters describe the relative weight  attached to  each want
in terms of satisfying, utility.  If the a.  are all equal to seme constant
k, then the utility index is k times the square of the straight,  line
distance to the bliss point from the tangency point on the highest  utility
contour attainable by the individual, subtracted from the utility index
value at bliss, Ia.d*.  Thus as this distance decreases,  utility increases.
Application to RECSIM
     We may solve the RECSIM utility maximization problem for an individual
with the Lemke QP algorithm by providing the following information  to the
conversion routine:
     n      the number of goods, (n-1) being recreation goods
     m      the number of wants over which utility is defined
     k      the number of constraints in the problem
     a. ,d,   the utility function parameters
     B      the household production technology function matrix
     y      the individual's income
     t      the individual's leisure time
     p      the vector of observed prices the individual faces
4. Wegge actually states that the bliss point is defined by Q  c, but he
uses the functional form in which a 1/2 term is factored out of Q.
5. If k - 1, the individual is subject to the budget constraint only; k - 2
adds the requirement that the individual use no more than his/her available
leisure time.  If k  2+m, the individual is also constrained to be below
bliss.  If k - 2+m+(n-1), then the individual is also subject to the
positive consumption of recreation goods requirements.   This k represents
the highest value of k possible in RECSIM as it now exists.
                                    160

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 The first  three  values above define the problem size.  The next three
 matrices are  constant over all individuals (except for the possibility of a
 gender difference).  The y, t, and p values are specific to individuals.
      There are three possible relationships between n (goods) and m
 (wants).   If  m >'n there are more wants then goods, which Lancaster (1966)
 suggests is a simple society.  If m - n and B is a diagonal matrix then
 there is a one-to-one relationship between each want and respective good,
 which is very similar to traditional utility theory*  Alternatively, n > m,
 which Lancaster  suggests describes a complex society, where many different
               
 goods vectors can translate into the same wants vector.  The optimal
 consumption vector translating to a particular wants vector is chosen by
 minimizing costs, given a vector of prices.  Thus a realistic
 representation of the world should have m S n, allowing for either the
 traditional case in which goods and wants are identical, or the inclusion
 of  the household production technology in some fashion.
     .The current version of RECSIM operates with four goods, four wants,
.and nine constraints (from k-2+m+(n-1)).  The four goods are fishing days,
 days of urban leisure (movies, etc.), camping days, and the Hlcksian
 composite1  good.   There- are four wants, not specifically defined, from which
 utility is derived, so 3 is a square matrix.  If we structure B to have
 nonzero elements only on the diagonal, then each good uniquely satisfies a
 want, as in traditional utility theory.
 Basic. Data
  ;   To produce  data that are plausible, we scale the a., d.  parameters and
 the 3 matrix  as  follows:

a -


T
1
1
i

d -


140~
150
145
588

B -


*0.77 000
0 0.32 0 0
0 0 0.71 0
0 0 0 0.012.
 6.  The other  possibilities include: a B which is diagonal but not equal to
.the-identity  matrix, a square B with nonzero elements off the diagonal, or
 a rectangular B matrix.
                                    161

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In particular, note that the d and B parameters on the Hicksian composite
'commodity vary significantly from those for the recreation goods.   Since B
here is a nonsingular matrix we may solve Bx S d for the maximum x values
which keep the individual from going beyond bliss.  This vector is x' *
(132, 183. 204, 49000), which will restrict most individuals to be below
bliss with no excess income, when considering that the median income
($9259) and days of leisure (125 days) constraints are well below these
       7
values.   With a constant price on the Hicksian commodity of $1, the large
value for dv, given that all a. fs are equal, means that an individual must.
initially spend his/her income on the Hicksian composite good so as to
minimize the 'distance' from the bliss point.  As income increases,  ceteris
paribua, the individual will begin to recreate.  The similarity of the d
and B values for the recreation goods makes the individual's recreation
choices sensitive to recreation prices, which may vary significantly by
individual.
WELFARE CHANGES
     Since a pollution control policy would increase the availability of
fishing sites, the travel coat for certain individuals would decrease,
while not rising for the remaining individuals.  No one would suffer a
welfare loss, though seme enjoy no gain either.  (If an individual is at
bliss pro-policy, then he/she will also be at bliss post-policy, regardless
of the price decrease.)  An individual may have a welfare gain and be at
bliss post-policy, if he/she is below bliss pre-policy.
     For individuals who do experience a welfare gain, we want to find the
theoretically correct measures of welfare, or compensating variation (CV)
7.  Based on a lognormal distribution for income, Z * (c* - 9.1334)70.985
(see  chapter 4), where c* - In (c), the probability of having an income
greater than $49.000 is 0.0455.  For an individual endowed with $49,000,
all income  would have to be spent on the Hicksian composite good to reach
bliss.  Thus, 4.55} represents an upper bound on the percent of individuals
capable of  reaching bliss in RECSIM.  Also, with a maximum of 365 days of
leisure time, no individual can consume the 51 las level of all these
recreation  goods, a feat requiring 435 days.
                                    162

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and equivalent variation (EV).
     Suppose that each individual's optimization problem  is  solved  twice -
once facing the base (pre-policy)  price vector p9 and once facing the
post-policy price vector p1,  where in general  the only  price potentially
unaffected by the policy is the price of the Hicksian composite commodity.
The optimal solutions produce the  optimal utility indexes u and u1 given
the consumer's available income, y.   In oir model,  for simplicity  the more
specific case of a single price change -t for water based  recreation -< is
the only possibility considered.
     Policy benefits can be calculated both in compensating  variation (CV)
and equivalent variation (EV) terms,  (Oeaton and Muellbauer, 1980).  Define
     CV - aCp'.u") - e(p9,u)
     EV - e(pl.u) - e(j>9,ul)
where7 e() represents the minimum  expenditure  required  to reach the stated
utility level, given the price vector.  Obviously, e(p*,u)  - eCp1,^1) - y
if the consumer's income constraint is binding (he is not beyond bliss).
     .In words, CV is the minimum monetary amount that a consumer would have
to be taxed or compensated after a price change in order  to  be as well off
as he was- before- the- change.   It measures the  offsetting  change in  income
necessary to make* the individual indifferent between the  original situation
(p) and the new price vector (p1).  Equivalent variation measures  the
change.in- income, to be spent at the original  prices p9,  which would allow '
hint to attain the utility level ul occasioned  by the new  price vector p1.
     To calculate CV in RECSIM  e(pl,u) must be known.  It represents the
income level that, under the post--policy price vector p1, would produce the
pre-policy level of utility u. Analogously,  to calculate EV, e(p,ul)
must be determined, where it represents the income that would produce the
post-1 policy utility level given the base price vector.
     The situation as shown diagrammatically in Figure  1, where the
relationship between utility, u (which expressed in  indirect terms  is a
function-of income and prices as V(p,y)), and  expenditure, e(p,u),  is shown
for two price vectors, p9 and p1 (See Varlan,  1978 for  details).  Note that
in: the figure we assume a price decrease so plSp, and  CV and EV are
negative, which logically meana that if welfare after- the change is higher
                                    163

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                       Figure 1 .   CV and EV  Measures
than it was originally, income must be taken away  (Deaton and Muellbauer,
1980, p. 188).  For a price fall |EV|  will exceed  |CV| , which is a
theoretically demonstrable result for normal goods  (Varian, 1978, p. 211).
To obtain the measures in principle, we would have  to:
        Solve the consumer's optimization problem  subject to his income
         constraint, deriving the Marshall ian demand  functions for goods
         quantities.
        Substitute the demand functions into the direct utility function,
         producing the indirect utility function, with utility as a
         function of prices and income.
        Invert the indirect utility function to obtain the expenditure (or
         cost) function, with expenditure as a function of prices and
         utility.
     Once we had the expenditure function we could  proceed directly with
the welfare calculation, and would not need to work with the area under the
Hicksian demand functions to produce welfare measures.
     However, analytical expressions for the expenditure function cannot be
easily derived for the household production model with a quadratic utility
function if the B matrix is not diagonal, and would be quite messy even in
                                 164.

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 the neoclassical  cas.e.   Therefore, we  cannot calculate the true CV and EV
 measures analytically in RECSIM  but oust  resort  to approximate numerical
 methods .
      Two approximations  to each  CV and EV welfare measure (see Deaton and
 Muellbauer,  1980,  Varian,  1973 for details) can  also be calculated:
      CV  Approximations

           CVi  - I q'(p'-p?)
                          L
      EV Approximations
           Wi  -  I  q{(p'-p)
         where
           q?,qj  -  Base  (pre^policy) quantity of the i   good consumed (0)
                                                      f h
                   or  post  policy (t) quantity of the i   good consumed.
           PP    Base  (0) or  post-policy  (1) price of the i   good.
           u*,ul  -  Base  (0) or  post policy  (1) utility level.
           \",\l    Marginal utility of income Ou/3y) evaluated at base (0)
                   or  post-policy (1) utility level.
--:  -  In the context of  the household production model with a quadratic
 utility function defined over  wants, Cv'1 and EVt cannot be calculated
 because of the distinction between unobserved wants (with their attendant
 shadow: prices) and goods (site visits).  However, (5v2 and EV2 can always be
 calculated in RECSIM, given the  shadow price on the income constraint from
 the quadratic programming  model.  These calculations will be useful, as we
 show in the next section.
 GOLDEN  SECTION SEARCH
 '     The problem of finding the  correct CV and EV values in RECSIM can be
 treated as one of  single variable- optimization, where the search takes
 place over alternative  values  of  income, y, minimizing the difference
 between the known  utility  level aimed at (either u or ul) and the one
 actually achieved  for a given  y value in any iteration (based on the exact
                                    165

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CV and EV definitions).  One efficient approach to this problem  is via
golden section search, since the relative size of the interval of
uncertainty regaining after a f ixed number of iterations can be  pre-set and
will always be achieved.  The golden section search requires unimodality of
the optimum in the search area.  (See Biles and Swain,  1980).
     Since CVa is usually a maximum lower bound for CV we can start our
search over the interval y - C*V2, y"
     Define for the 1th iteration:                                          *

         L.  -  Golden section number equal to (1/1.613)  (b^aj)
         a,  -  Lower limit of search interval
         b,  -  Upper limit of search interval
         x.  -  Independent variable, iteration 1, obtained from upper
          i f i
                limit as b. - L,
         x,  -  Independent variable, iteration i, obtained frcm lower
          if 2
                limit as a, + L,
         f(x. .,)- Objective function value given by x.  .
            1 1 1                                      i , i
         f (x. _) Objective function value given by x.  ,
            *&                                      it*
     Then if we deliberately arrange our search along the real line so that
x.   < x. , for all i the Golden Section algorithm can be written as:
 *  '    * ^  ,
     Maximization Problem:
         If f(xu
            Then
         If      >
            Then
     And, we only have to. find one new objective function value  at each
iteration, i, because:
         If f(xlfl) >fUlf2)
            and
         If
                                    166

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            Tnenxi>1,1 - Xi,2
            and
      For  our minimization problem, simply reverse the signs of the  '
 inequalities in all steps above and proceed* defining the objective
 function  to be minimized as:
 CV Problem
      OBJ  - min  |u-uj|
           where  u - Utility from pre-policy solution with price vector p
                      and income y
                  u?  Utility from i   golden section iteration with price
                      vector p1 and income level y> < y
      For  the CV problem, we set at - y - CV2 and bt - y.  For the EV
 problem,  we can set the upper bound at y plus some reasonable factor, k,
 times the actual  CV obtained in CV optimization.
 EV Problem
      OBJ  - min  |ul-u*|
           where  ul * Utility level frcm post policy solution with price
                      vector p  and income level y
--'--.              u*  Utility level frca lfc  golden section iteration with
                      price vector p and income level y. > y.
   --  Heuristically, the golden section search procedure works by successive
 interval  elimination.  For example, take a maximization problem with an
 initial search  interval along the real line from ax to blf where any x. 1
                                                                      *  i
 is by construction less than any x. ,.  Figure 2 shows successive
                                  i &
 iterations of the search, for three iterations involving discovery of four
 unique objective  function values, f(xlt), f(x12), f(x22), f(xsl).  This
 number of unique  search points produces an area of uncertainty around the
 final x values of (1.618)~^ (100) - 1U*.
-      In table 1 six iterations of golden section search involving seven
 unique trial values for y are used to find the maximum of the function y -
 xe x which has  an analytical optimum y-0.3679 at x1 .  In table 5.2, six
-iterations of golden section search are used to find the minimum of the
 function  y - x*-Ux+5 which has an analytical minimum y-1 at x-2.  In both
                                    167

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cases the x value found after the seventh unique  point is less than 5
percent away fron the correct, value,  as expected  using the rule that the
interval of uncertainty after n trials of golden  section search is less
than or equal to p n, where p is the  golden ratio 1.613034 (Biles and Swain
1980, p. 188).  In our example, (1.618)-   100  equals 3.4 percent.
APPROXIMATIONS TO AVERAGE VALUES PER  RECREATION OCCASION
     The exact CV and EV measures discussed above take into account all
reactions to perturbation of the initial price vector due to policy-induced
water quality improvements, but do not isolate benefits by water-based
recreation activity category - fishing, boating,  swimming etc.
     In general, for the two-step method of benefit estimation (discussed
in appendix A to this chapter) to be  at all consistent with the exact
surplus measures, all possible benefit categories must be explored.  That
is, changes in participation in each  broad category must all be valued in
the second step at some category-specific average surplus, and each result
summed to approximate CV or EV.  Because we assume only one water-based
recreation activity in RECSIM, this complication  need not be considered,
although in general it should be dealt with.
   -_ There are at leaat two ways to produce crude values per occasion fron
RECSIW.  The first is analogous to a  willingness  to pay survey, and the
second is analogous to a travel-cost  approach.
   :  The willingness- to pay approach  mimics the response to the sort of
question asked in the 1975 NSHFVIR, "Having thought about how much this
activity cost you in 1975, how much more money would you spend annually on
your favorite activity before deciding to stop doing it because it is too
expensive?"  To replicate this sort of response in RECSIM it would be
necessary to uniformly increase all elements in the (travel-cost based)
price vector associated with a particular broad activity category (eg.
fishing) holding other prices constant until participation in that category
fell to zero, and then compute the usual exact CV or EV measures by the
search process discussed above.  Summing across individuals and dividing
the result by the sum of base level participation in the particular
activity category would produce an approximate category-specific average
                                    168

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             Figure 2

Golden Section Search for 3 Interactions
                      ^r
                      *-

                      * * *l T   ** 
                      *!	*
                X.    X. =
                               r-
               169

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                  Table 1.  Maximizing y - xe   by Golden
                              Section Search
Iteration i    x Values       f(x) Value
               xlx-0.7639     ylt-0.3559       1.2361         0
               xxj-l.2361     yx2-0.3591
                              ya 1-0.3591        0.7639     0.7639       2
               xa 2-1.5279     y 2 2-0.3315.

               xsx-1.0557     y,x-0.3673        0.4721      0.7639      1.5279
               x, 2-1.2361     y, 2-0.3591
                              y.n-0.367287     0.2918     0.7639       1.2360
               x^2-1.0557     /a-0.36329
               x9l-T,0557     ysl-0.3673       0.1803     0.9443        1.2360
                              y 52-0.3652
    6 (STOP)

    OPTIMUM.-   x,2-1o0132     y82".3678       0.1114      .9443        1.1246
* L.  (1/1.618)1 (bj-aj) where b^aj-2.
                                    170

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                    Table 2.   Minimizing y - x*-4x+5  by
                           Golden Section Search
Iteration      ix Values     f (x)  Value        L^*           a.
                              yti-1,562        1.8541         0
              xla-1.854       y,a-1.021
                   .    
              xal-1.854       yai-1.021        1.1459      .1.146
              x-,,-2.292
              xsl-1.584       ysl-1.173
              x,a-1.854       y,a-1.085        0.7082       1.146       2.292
                              y,, i-1.085
              x,,a-2.022        y%2-1.0005       0.4377       1.584       2.292

    5       .  xsl-2.022        ysx-1.0005       0.2705       1.854       2.292
              x,a-2.124        ysa-1.0155

    6 (STOP)

    OPTIMUM*  XH-T.957        xtl-1.00l8       0.1672       1.854       2.124
* L  - (1/1.6l8)l(bl-al)  where bt-at-3
                                   171

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 value.   However,  there, is a rough approximation to such a result which is
 computationally much less expensive  involving the CV2 and EVa
 approximations  discussed above.   In  particular, the utility level
 attainable without  any  participation possible in the 1   activity category
 of  interest  can be  obtained by masking all columns pertaining to that
 activity category in the B matrix.   Then the difference between that
 utility level  (u?33 ) -and the base utility level (u) can be converted into
 monetary values using,  say,  a weighted average of the marginal utilities of
 income  in the two situations:
 --- activity category  i surplus/ per son - (u^u^^/Cl/ax'U'+X0331*)]
      Again,  summing these surpluses  over all persons in the sample and
 dividing by  the sum of  their base participation levels under the initial
 conditions produces an  approximate average surplus per recreation occasion
 of  the  i  type (fishing,  camping, etc.).
      The alternative approach involves econometric estimation of input
 (site visit  or  days of  participation) demand curves by activity category*
 Indeed, this is the correct econometric specification of participation
                    category i all relevant site prices (travel costs)
 could be identified.   It is also Just a travel- coat model writ large.
 Since all sites' in any particular recreation category are assumed
 homogenous,  instead of  estimating site-specific demand curves for site
 services via separate travel  cost models, all site visitation data from
 RECSDt can Just as well be pooled and a single visitation function
 estimated across the entire sample.  The estimated function would contain
 the same arguments, along with own visit price, that either the
 participation or travel cost  models include:  income, prices for
 substitutes, and socioeconomic variables.  The only difference is that the
 visitation data could either  be  indexed by site and distance zone or
 estimated on a persoR->speclf ic basis using in dividual-specific prices.
      Fixing  all variables other  than own price at the sample mean, the
jLntegral under such an  aggregate site visit function for the i   activity
 type can be  found between the existing average travel cost-based price and
 the price driving aggregate visitation to zero.  Again, dividing by the
 estimated base total aggregate participation level provides an average
                                     172

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 surplus estimate similar to that reported in Vaughan and Russell (1982),
 chapter 5.
      Although it would also be possible to use the difference between the
 integrals under pre-policy and post policy aggregate input demand curves
 for seme assumed path of integration as a Marshallian surplus measure, we
 do not consider that  possibility for a fairly compelling practical reason:
 visit price da.ta for  all alternatives facing all individuals in large
 national survey samples are generally not available, so such visit demand
 functions, could rarely be estimated in practice.  We do propose to estimate
 this function for two different reasons:  first to get approximate average
 values of recreation  days and second to show how ouch better participation
 changes could be predicted using these functions as opposed to models
 estimated using proxy "availability" data.
      There is one other possibility we do not touch upon, although it
 could, under very particular circumstances, produce useful approximations.
 It.involves using RECSIM to mimic the response to contingent valuation
 survey questions which ask individuals in general or participants in a
 specific activity how ouch they would be willing to pay for a particular
 environmental change-  (Daubert and Youngv 1981), and expressing that value
 in: per day terms. This would involve calculating values like CV discussed
 above from RECSIM and dividing by the incremental amount of participation
: occasioned by the policy.  One general problem with this approach is that
 the value* cannot be attached to a category specific change in days, unless
 the question is directed at category-specific participants.  Otherwise it
 reflects the welfare  benefits under simultaneous participation changes in
-all water-based recreation categories.  If the entire value were pro-rated
 over one particular type of participation change, it vould have to be
 interpreted as a proxy for benefits in all categories.   This is not a
-problem, in RECSIM, which only has one water-based activity.  Second, such
 incremental values are specific to the degree of change stipulated and may
 not apply_to greater  or lesser policy-Induced changes.   This is in contrast
 to. the way we apply average surplus calculations, which are only done once
  in: the case of our model, they are obtained from a benchmark run and
 applied to value all  participation changes produced by all possible RECSIM
                                    173 .

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model configurations.  Again, this is not an "ideal"  procedure, but one
that reflects usual practice.
CONCLUDING COMMENTS                                      0
     In this chapter we have described the optimization method used in
RECSIM for solving the consumers' constrained maximization problem,
including transformation of this original problem to one  of minimization.
           
Further, we have discussed the derivation, and more important, practical
calculation of correct welfare change measures.  Finally,  we described our
method of obtaining measures of average welfare per unit  of consumption;
such measures being- required for the common two->step method or benefit
estimation.
     One of the goals of the project is to obtain estimates of how well or
poorly the two step method approximates the correct measures, and how this
varies with the characteristics of the problem.  In order to carry out this
task, we require participation equations, estimated to mimic the usual
procedure.  To this and related matters for demand function estimation we'
turn in the next chapter.

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                               APPENDIX  5. A
                 PITFALLS IN APPLIED  WELFARE  ANALYSIS WITH
                      RECREATION PARTICIPATION MODELS
     Applied economic models of  consumer  recreation decisions have
historically proceeded along two parallel tracks,  the  parallelism seemingly
dictated: by the nature of the data available for model estimation
(Cicchetti. Fisher and Smith 1973).
     The "macro" track has been  characterized by the recreation
participation equation approach,  which involves  estimation of a
cross-sectional relationship explaining the pattern and intensity of
individual participation in specific  recreational  activities.  This
approach is at a national or regional level of spatial analysis, ignoring
the sites where the activities took place,  hence modeling the demand for
the activity (not sites).  The "micro1* track,  in contrast, is characterized
by travel cost models attempting to econometrlcally capture the demand
       '  .                                                      Q
relationship for the services of a known  site or group of sites.
     The micro travel cost model,  being a structural representation of a
demand function (or system) can  be employed directly to produce site
value's.  It can also be used to  assess the welfare change occasioned by
adding or deleting a site from a pre^specif led system, or answer other
welfare- related questions, such  as the benefits  of upgrading site quality.
     In comparison with the micro approach, the  macro  approach suffers from
a distinct lack of specificity.   Particularly, since prices .do -not often
appear as independent variables  in the macro model specification - due
primarily to data deficiencies -> direct welfare  analysis with the macro
8. For a discussion of aggregation bias  in  the micro travel cost model
context, see McConnell and Bockstael  (1984); Dwyer, Kelly and Bowes (1977)
                                    175

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                                                                                  Q
          participation model is much more tenuous than with  the travel cost model.

          Yet the macro models are used indirectly for welfare analysis.  Indeed,

	their, piain practical purpose has been the prediction of changes in levels

	ofparticipation over time or across space under  alternative hypothetical

          policies affecting the supply of recreational resources.    These changes,

          and the policies engendering them, have often been  valued using a unit

          value which Freeman (1982) graciously refers to as  an "activity shadow

          price".  The monetary welfare measures assigned to  the possible policy

          alternatives are usually obtained as the product  of the predicted change in

          days of participation, summed over the population,  and the unit day

          value.11

               To those familiar with the site-specific travel cost approach, the

          unit day value method may seem no more than an irrelevant curiosum, but in

          fact its use is commonplace.  It is a practice that was recommended, until

          recently, by the Water Resources Council and was  cited recently as an

          alternative when other methods were not available (WRC, 1979).  It has been
          9. Some studies (USDI 1973) have used "trip costs" constructed from
          population survey information in participation  equation estimation.  If an
          unconditional demand function specification is  intended, trip costs must be
          collected on all sites and all possible recreation activities that every
          consumer can choose among.  It is doubtful  that trip costs variables
          constructed by averaging over several trips to  many sites in one particular
          activity category are adequate, and the problem of missing substitute
          activity costs because participation in such substitute pursuits is zero is
          usually impossible to overcome.  A more sophisticated but similar example
          of the above approach which allows consumer surplus to be computed
         -directly,-thus avoiding the two-step method iS'Ziemer, Musser and Hill
         JL19J3PJL.	A more theoretically consistent approach is exemplified in the
          work of Morey (1981, 1983) who estimates conditional demand functions.

          10. There may be two effects which occur simultaneously as a result of
          recreation resource supply changes:  movement along a demand curve and
          shifts of the demand curve.  Demand curve shifts are possible if the
          quality of supply is a factor in the utility function (Maler, 1974).  See
          Ziemer, Musser and White (1982) and Bouwes  and  Schneider (1979) for
          examples of this method of analysis.

          11. Another legitimate use of participation models is in the prediction of
          future participation "demand."  Examples of this purpose are the work of
          Cicchetti (1977) and Rausser and Oliviera (1976).
                                              176

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used to value an entire recreational  fishery in British Columbia (Pearce
Bowde'n, 1971), and to estimate the recreational benefits of the Illinois
river in Oklahoma under the Wild and  Scenic Rivers Act (U.S. Department of
the Interior, 1979).   Other agencies  like  the Corps of Engineers continue
to use this method.  Moreover,  the Forest  Service, in responding to
requirements of the multiple  use and  sustained yield legislation has
incorporated the equivalent of  unit day values in their.programming models
(Sorg et. al., 1984).  Finally,  the jnethod has found frequent application
in analyses of the national recreation benefits of water pollution control
programs/as catalogued by Freeman (1982).
     The- unit day value method is particularly convenient when no
information other than prediction of  a policy's impact on days of
participation is available from a macro participation model.  But, there
are three problems with the macro modeling approach:  differential site
quality characteristics are often not accounted for; prices are often
omitted in estimation;  and unit values are employed to monetize predicted
quantity changes.  The first  two problems  lead to biased predictions of
quantity change while the first and third  distort the estimate of welfare
change,, even if the quantity  predictions are accurate.  Only rarely are
these limitations acknowledged (an exception is Sorg et. al., 1984).
     This paper focuses on the error  implications of bifurcating the
estimation of the benefits of recreational resource enhancement into two
unrelated steps - quantity change and valuation.  Throughout we assume the
absence of systematic error in predicting  quantity change, since such error
can either offset or  compound the error attributable to valuing that
change.  This discussion is confined  to macro participation models of the
aggregate level of recreation activity service flows enjoyed at an unknown
site or set of sites, rather  than travel cost models of the demand for the
services of a site or system  of  sites,  because in the latter case the
demand functions, being estimable, obviate the need for unit values.
     After a brief review of  the genesis of the two-step valuation method
the valuation problem is addressed.   It is shown that the two-step
valuation method is questionable on theoretical grounds and is not likely
to; provide a reasonably accurate monetary  measure of the welfare change
                                   177

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associated with participation quantity change stimulated by a policy of
recreation resource enhancement.
ORIGINS OF THE TWO-STEP METHOD
     Traditionally* data on the regionally differentiated availability of
recreational resources (acreage or number of lakes,  Campgrounds, natural
forests etc., contained in seme region, state etc.)  has  been obtained to
supplement the data in population recreation surveys and included among the
set of relevant regressors in macro participation model  specifications.
Inclusion of these variables has seme basis in common sense; it is
intuitively appealing- to anticipate that recreation  resource availability
must have seme role in Influencing recreation participation and intensity.
One would expect individuals living in areas amply endowed with water to be
more likely to engage in water sports, and at higher intensities, ceteris
par!bus, than individuals living, in poorly endowed areas.  But more
importantly, inclusion of availability regressors in the model is
absolutely necessary if it is to be a useful tool for evaluation of
potential broad policies of supply alteration.  If there are no supply
variables in. the participation equation,, then prediction of participation
changes due to supply changes is obviously not possible  with the model.
     Initially, the inclusion of quantity-type availability variables in
macro models was justified by somewhat vague allusions to "supply" factors
(Cicchetti, 1973) > although it was never clear what  sort of supply function
was implied.  Later, to help dispel the confusion, Oeyak and Smith (1978)
invoked household production theory to explain supply in terms of the
household's marginal cost of "producing" recreational service flows.  For
these authors, marginal cost itself was a function of "characteristics"
variables, which happened to be availability variables in disguise as
facilities per capita, a measure of expected congestion. The consequence
of this paradigm was the essential endogeneity of (self-supplied) price,
which meant that only reduced forms could conveniently be estimated, as the
household's internally determined shadow price is never  observed.  Hence
the requirement of a second valuation step for welfare analysis.
     The elaborate household production model is really  not necessary to
justify the inclusion of a measure of the quantity of recreational
                                    173

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 resources per unit  land area (not, note, per capita) in econometric models
                             12
 of recreation participation.    Drawing on the statistical ecology

 literature (Pollard,  1971),  it  is possible to show (Vaughan and Russell,

 1984)  that,  if correctly measured, physical availability measures such as

 the number of lakes per unit land area in a region are inversely related to

 the expected travel cost from any arbitrary point in the region to the

 closest recreation  site.  This  fundamental relationship between physical

 availability and expected  travel cost may be what Cicohetti (1973) had in

 mind,  but it has not been  acknowledged in past participation analysis.

 This notion conforms to intuition and offers an explicit Justification for

 using: availability  regressors as proxies for "average" travel--cost based

 activity prices in  the direct estimation of an aggregate structural demand
 12. A cursory reading- of Deyak  and Smith  (1978), in both the theoretical
 and applied sections,  leaves  the  impression  that direct travel expenses
 play no role in the reduced form  participation models desired from
 household production theory.  However,  such  an interpretation is apparently
.incorrect, since Deyak and Smith  specify  the marginal cost (shadow price)
 of service flows- as a function  of the  prices of "recreational market goods"
 which: presumably should include travel  cost  as a measure of "site price",
 although they do not so state.

 Notably,- the empirical analysis in Deyak  and Smith includes no such measure
 or proxy for it, focusing instead on congestion-type variables measured as
-the acres of recreational facilities per  capita.  Thus, their econometric
:model-specification appears to  be distinct from their theoretical model.
 Our previous work,  which followed Deyak and  Smith's empirical (not
-theoretical) specification appears deficient in this regard (Vaughan and
 Russell, 1982), as  do several other  empirical analyses of recreation
 participation in the literature.

 The omission has rarely been  explicitly addressed until, recently, when
 Mendelsohn and Brown (1983) observed that "In order to assess the
 usefulness of the household production function it is important to remember
'that the fundamental purpose  of recreation analysis is to determine the
-value of the quality and quantity of the  public good, the recreation site.
 The recreation site is a good which  enters like other goods as an input
 into-the household  production function.  The critical issue is to value the
 site or its objective qualities in terms  of  the price of the site or the
 price of each quality.  Although  the household production function may be
 able to provide insights about  why people exhibit certain tastes for goods
-(sites) the tool is an unnecessarily cumbersome approach to measure the
 value of sites or their qualities" (pp. 610-611).
                                    179

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equation, rather than a reduced form.   Due to the expected value nature of
the proxy, paraneter bias is1 the penalty imposed by not  using the correct
indi vidual-specif ic. activity prices (McFadden and Reid,  1975).
     Yet the problem of placing a unit dollar value on the participation
change occasioned by a particular policy of recreation resource enhancement
to produce a monetary benefit measure is equally difficult, whether we
believe we have estimated a reduced form activity participation equation as
a function of individual characteristics and site characteristics measured
by some availability measure or a structural activity  quasi-demand equation
with availability as a proxy for activity price.  In neither case do we
observe individual prices directly, and the best that  can be done is to
predict a quantity change conditional  on a hypothesized  change in
availability, and value it arbitrarily in a second step.
VALUATION ISSUE
     The conceptually correct Marshall!an measure of benefits arising from
increased resource- availability may be written in terms  of structural
activity service flow supply and demand equations.  Consider any
individual, whose marginal cost of obtaining the recreation experience is a
function of recreation resource availabilitys
     me9 - marginal cost at pre-policy availability
                a* particular to individual j  and,
     me1 - marginal cost at post-policy availability
                a1 particular to individual j.
     Suppose a policy of supply augmentation so that a1  > a.  The
individual's marginal willingness to pay for the experience is the demand
price, p, an (inverse demand) function of the service  flow quantity .  For
the j   individual the net benefit of a policy of supply augmentation,
NB.U'.a1) can be written as:13

     NB.(a,a1) - /mc<> q(p) dp .                                    (1)
       J           me1
13. Hereafter, all welfare gains from a price decrease are defined for
convenience to be positive.
                                    180

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The aggregate net benefit of the policy is  the sum over all j-1 ,...,J

individuals of the net benefits in (1):


           J
     NB -  I MB (a9,a1).                                             (2)
          J-1      -

     As we have noted, however, most  often  the detailed individual data

necessary to perform the calculation  in (2)  is unavailable.  A common
situation is to have data allowing a  prediction of the total increase in

participation via a macro model,  E(qj-qj)r  and, from an independent source,
a unit value to assign to the quantity  change.
     For instance Cicchetti,. Fisher and Smith (1973) suggest:
         ...,   the  reduced-form  participation  equation can
         also  be  used, as  we have suggested, to  derive a
         measure of  the  benefits from  a new facility.   The
         amount of  participation  in  an activity  ia first
         forecast under changing conditions of supply, i.e.,
         without and then with  the  new facility.    Then a
         measure of  value  or willingness  to  pay must  be
         imputed to each unit (recreation day) of additional
         participation.  Such measures  have in the past been
         set for federal  projects- by water-resource agencies
         and  approved  by  the  U.S.   Senate.    Aggregate
         benefits- are given by multiplying,  the imputed value
         per  unit of  participation  by  the change  in the
         level  of  participation  occasioned  by  the  new
         facility,  (p. 1011).

But-^n&rdistinction ia made by Cicchetti, Fisher and Smith (1973) between
unit values which are conceptually equivalent to marginal willingness to

pay (I.e.,  individual-specific activity  prices or, in the household model,

unobserved shadow prices)  and unit values representing average willingness
to pay over all units consumed (i.e., average consumer's surplus for the

activity),  although they  seem to  have had the former in mind.

     A survey of the unit value literature  reveals that most reported

values are approximations to average, not marginal,  willingness to pay

(Dwyer, Kelly and Bowes,  1977).  If so,  the direction of the valuation bias
can be derived, and we do so below.  But first, if we assume marginal unit

values are available, can the procedure  be  justified?
                                   181

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Valuation with Marginal Unit Values
                                 
     As McKenzie (1983) observes, there are two routes  to welfare
measurement; the familiar one where demand functions  are known, allowing
direct computation of the Marshallian surplus measure;  and alternative
index-nun be r approximations based on "only the prices and quantities that
hold in alternative situations but not information about the shape of
preferences or the consumer demand functions'* (p.  101).  The two-step
valuation method is a particularly simplistic version of this second route.
     While the adjective marginal may evoke a subconsciously sympathetic
response, valuation of a quantity change with marginal  unit values (prices)
does not guarantee a close approximation to the Marshallian consumer's
surplus measure, let alone the desired measures the latter approximates,
compensating and equivalent variation (CV, EV). To demonstrate, begin with
the most general case where a single price changes.  Although the
consumer'3 demand function for the good whose price has changed is unknown,
assume that the quantity changes for all goods in  the consumer's choice set
are known, as are the initial and final price vectors.
     When a single price changes the product of the 1 n vector of all goods
prices (measured at pre-policy levels, p*, post-policy  levels, pl, or an
average of the two) and the n1 vector of quantity changes can be used to
produce welfare approximations if the demand function for the good whose
price has changed is unknown (McKenzie, 1983* Ch.  6.; D eat on and
Muellbauer, 1980, Ch. 7).  These measures are known respectively as- the
Laspeyres and Paasche quantity variation indices (LQV,  PQV), and
Harberger's consumer surplus (HCS).  Representing  the marginal utility of
expenditure as X and the utility index as U:

     LQV - Ip'Aq1 - AU/X                                           (3)

     PQV - Ip^Aq1 - AU/X1                                           (4)

     HCS - 1/2UQV+PQV) - 1/2(AU/X9 * AUYX1)                        (5)

where i  T,...,n goods.
                                    182

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     It can be shown that the HCS measure is an approximation to
Marshallian consumer surplus, since it simply takes  the short-cut of
assuming the Marshallian demand curve is linear in the region of the price
change (Deaton and Muellbauer, 1980,  p.  188;  McKenzie, 1983, pp. 109-111).
     However, the two-step valuation method,  lacking information on the
own-good demand function and the quantity changes taking place outside the
market of direct interest, is more restrictive than  the general case of
(3), CO and (5).  It deals only with the product of quantity change and
price for the good whose price changes,  ignoring, quantity changes for all
other-goods.  So, the macro participation model's partial welfare measures
analogous to (3), CO and (5), where the i   good changes price are:

     LQV - pjAQ1 * AU/X"                              .             (6)

     PQV - p^ * AUA1                                            (7)
     HC"S - 1/2UQV + PQV) * 1/2 (AU/\  * AU/X1).                     (8)

     The partial index-number measures assume,  perhaps incorrectly, zero
cross-price effects.  Except, for unusually restrictive demand systems (eg:
Cobfr-Douglas),. when the price of a single good,  i, changes, the quantities
of some other goods J * i will change as well.   But  if other goods'
quantities change, the partial LQV, PQV and HCS measures which ignore the
sums of p-.Aq, for all J  i are unlikely to bring us>  reasonably close to the
        J  J
change in the ideal welfare measures, CV and EV, or  even to the
approximation they bound, Marshall ian consumers surplus (CS).
     The only case where quantity changes in  other goods induced by a
change in the price of the i   good can be ignored in calculating LQV, PQV
and HCS is when the elasticity of demand for  the ith good is unitary in
absolute value over the region of interest.   To prove this, arrange the arc
price elasticity of demand formula (where e represents the absolute value
of the arc elasticity) as:                              ""   .
     1/2(pi+pJ)(qJ-qJ))   [l/2(qj*qj)(pj-pj)]e  .                     (9)
                                    183

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     The l.h.a. of (9) is the definition of  the partial Harberger consumer
surplus measure, HCs. Expansion of the r.h.s.  reveals  that it represents
the arc elasticity measure, e, times an approximation  to the Marshallian
consumer surplus integral CS obtained by linearizing the (unknown) demand
curve between q and q|:

     CS - 1/2(qJ+q)(p-pp - q(pj-pp + 1 /2[(q{-q) (pj-pj)] .      (10)
The two expressions following the second equality  in  (10) represent the
                                    
familiar welfare rectangle and triangle measures of Marshallian surplus.
     So, the l.h.s. or (9) representing the partial measure HCS either
understates, overstates, or equals the approximate Marshallian consumer
surplus measure on the r.h.s. depending upon whether  the absolute value of
the arc price elasticity of demand for the good whose price has changed is
respectively less than* equal to, or greater than  one.
     But from (9) and (10), there is obviously no  reason to compute the HCS
measure if q  q*  p and p are all known (or q, p*, q and e are known,
permitting calculation of pM.  In these circumstances the approximation CS
can be obtained directly by linearizing the unknown demand function between
p|, q| and p, q and: applying (10).  Of course the more nonlinear the
demand function and the larger the price change the poorer the quality of
the approximation CS to the correct measure CS.  More often in macro
models,, only p? and the quantity change are known  and no assumption is made
about e; the welfare change being approximated instead by LQV.
     Under what circumstances will LQV be a good approximation of CS?  From
(6) and (10) construct the ratio:

     CS/LQV - (1/2)[(qJ+qJ)(pJ-pJ)]/[pJ(q'-q)]  .                    (11)

If the functional form of the demand equation is known to be q - q(p), then
substitution for q* and. q* in the above equation yields an expression in
terms of p and pJ and other function parameters.  Since p? is exogenous! y
given, the function (1 - CS/LQV) * can minimized with  respect to p|.  So
                              *i                      *  
doing readily shows that the p,  which minimizes (1 - CS/LQV)*, and its
                                    184

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associated q.  is not necessarily a p,q combination in  the economic region,
or one that might be an outcome of the policy under evaluation.
     However, fron (6)"and (7),

     PQV - (pJ/pJ)LQV                      .       .                 (12)

which substituted into (8) yields

     HCS - (1/2)[LQV * (pj/pJ)LQV].                                  (13)

Also, (90 shows, that HCS - CS e, therefore

     LQV - {2e/{l * (pj/pj)]j CS.                                   (14)

In general (from Eq. 14)  then, the accuracy of LQV  as a measure of CS
.depends on the magnitude of the relative price change and the elasticity in
the-region of interest.   For instance,  when the relative price change is
small* and the elasticity is near 1, LQV will be very close to CS.  This
confluence of favorable circumstances is surely a very  special case.
   -  In conclusion, it normally will not be possible to compute the full
LQV,  PQV or HCS measures using the participation equation method of
recreation benefits analysis because changes in the consumer's entire
consumption bundle renal n unquantif ied.  Without -knowledge of the demand
function, partial measures are unlikely to be representative of even a
crude? Mar shall ian consumers surplus  measure of individual welfare changes,
unless the function is such that P/P  - 2e - 1.  While this condition
salvages the HCS measure, if it is not  met it is uncertain whether the
total welfare change measured as the sum of the unadjusted HCS measures
across individuals will  or will not  be  a useful aggregate.  Can anything be
salvaged by using an average willingness to pay unit value rather than a
marginal one?' The answer, unfortunately,  is not encouraging.
Valuation with Average Unit Values
 ._..  In the usual case,  only a measure  CS  of  individual J's average surplus
for the quantity of recreation activity undertaken  before a price change in
                                   185

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activity i is available.  In terms of the (inverse) deaand expression this
may be written as:
           i
                  dq
                                                                    (15)
            0
Under what circumstances, then, is the following approximation for net
benefits of the aggregate quantity change q*-q  a good one?
     NB - CS(ql-q)    .                           .                  (16)
     We previously examined this question theoretically, using a
representative consumer's situation, for a linear Inverse demand function,
p * a+bqr and for a constant elasticity demand function with elasticity n,
P - A1/n q~1/n.  If k is the ratio q(al)/q(a),  the following expressions
were obtained (see Vaughan and Russell 1982 for  full derivationOctober 29,
1985):14
      inear demand:
                    NB   1-Hc
     linear demands ~ -  < -                                    (17)
                          ~*   I       ( - 1 )V
     constant elasticity    -|1  - nq n    I	1-&L )           (18)
                                       ,1
                              -fl  - nq''
        demand.            NB

where NB is the change in Marshallian.consumers  surplus.
     But if the demand function is of the s en i- logarithmic form, q -
exp(a*bp), CS evaluated  at q is -(q"/b)r where b < 0.  This yields an
average surplus, CS of -1/b.  It can easily  be shown that NB, the product
of this average surplus and a quantity change  given as exp(a)[exp(bp1) -
exp(bp)] is exactly equivalent to the Mar shall ian consumers surplus change
NB from the definite integral of /p? exp(a * bp)dp.15
14. When the constant elasticity demand curve exhibits unitary elasticity
formula (18) is indeterminate.  But, the limit of NB/NB as n approaches one
can be calculated by L'Hopital's rule as (k-1) (1-lnq)/(-lnk).
15. All of these results can be verified through simple numerical examples.
An example based on an arbitrarily par an atari zed quadratic utility function
is available from the authors.
                                    136

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     Once confined to the two-step method by survey data limitations, our
goal becomes to value the predicted  quantity change with a unit value that
produces the correct level of  net benefits.  Call this value the average
consumer surplus and define it as:
     ACS - NB/(ql-q)  -
? f(p)dp
/(ql-q9)                        (19)
where f(p) is the aggregate demand curve and p, p1 are the expected travel
costs under pre- and post-policy supply implementation (plSp).  Since the
demand curve is unknown,, we cannot make use. of  (19) to calculate ACS. So,
suppose we turn our attention to deriving bounds on its value.
     Simple graphical analysis of areas under a downward sloping curve with
two known points shows that:

     (p-pl)q + plql < MB  * prql < pql,                            (20)
and
     (p"-pl)q9 < MB < (p-pl)ql.                                     (21)
                                              *
How, by the Mean Value Theorem, there is some p , between p9 and p1, where
                                  
the slope of the demand curve at  p  is equal to the slope of the line
between the two known points, (q,p)  and  (ql,pl), or

     (p-pl)/(q-q!)  - f'(p*) .                                      (22)

Thus, using (22), the bounds  on MB in (21) can be rewritten as:  ~~

     f'(p*)(q9-qMq9  < MB < f' (p*) (q^q1 )ql.                         (23)

For a price decrease,

     -f'(p*)q8 < NB/(ql-q) < -f'(p*)ql,                             (24)
                                    187

-------
providing the bounds on ACS.  Unfortunately,  the  value of p  is unknown and.
furthermore only useful in our situation in calculating these theoretical
bounds.  It is not calculable or observed in the  context of the macro
model.  However, using the equation of a line through two known points, it
can be shown that the bounds on ACS in (24)  are equivalent to the interval
(b-p,b-pl), where b is the intercept of the line through the known
       16                                           *
points.    While the Mean Value Theorem proves that p  is in the interval
(pl,p), it is in fact a point in the interval (b-p,b-pl) which we become
interested in as the correct average surplus value.  Note that the bounds
on ACS do not vary with the assumed functional form of the demand curve.
     Unfortunately, the correct (but unknown)  average surplus value,. ACS,
is conceptually distinct fron the measure CS defined in (15) which is the
value typically reported in the literature.   If r denotes the true
reservation price according to the actual demand  curve, then CS as defined
will be somewhere in the interval (0,rp*),  depending on the assumed
functional form.  There is no guarantee that these bounds contain ACS.
     In summary, if individuals' demand functions are all linear (or nearly
linear in the relevant range) valuation with a CS type of average surplus
always understates the total Marshallian CS measure of the welfare change
by at least one half.  If the demand function is  of the constant elasticity
sort, the approximation can either be correct, understate, or overstate the
individual's surplus change.  Of the three forms  of demand relationships
examined, only the semi-logarithmic form produces- the correct result using
the two step method.  So, applying an average unit value to an aggregate
quantity change is dangerous, with unknown risks  a positive or negative
valuation bias, depending on the nature of the demand function.
16. Using the equation of a line with slope m through  (0,b), two
relationships based on our known points can be formulated; p - mq+b  and
p1 - mql*b  which can be equivalently expressed as -mq - b-p8 and -mq1 -
b-pl.  By the Mean Value Theorem, m - f'(p ), 39 the l.h.s. quantities of
these two relationships are equivalent to -f'(p )q  and -f'(p )ql, and can
be calculated from known points.
                                    188

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CONCLUDING REMARKS
     The few cautions about  the unreliability of the general class of two
step evaluation procedures noted above  that can be found in the literature
(Freeman, 1979, p. 227;  1983,  pp.  142-143) are well taken.  This paper has
attempted to formally demonstrate the mechanisms Justifying Freeman's
(1983) warning that "The application of average values does not give
sufficient weight to the concept of  consumer surplus and total willingness
to pay" (p. 143)   We might  even rephrase Freeman to say that the procedure
tries to capture these concepts, and is indeed linked to them, but is
likely to fail unless the demand function is semi-logarithmic*  Similarly,
the use.of marginal values (prices)  is  also unreliable, unless the demand
function is such that Pj/p?    2e - 1 in the region of interest.
     Summing up,  the two-step  valuation route is dictated by the lack of
accurate data on individual  marginal willingness to pay for the spectrum of
recreation activities.  Applied welfare analysis for these non-marketed
entities is therefore caught in a vicious circle.  If surveys of population
recreation participation were  to contain individual-specific marginal
willingness to pay information for potential (as opposed to actual) visits
to all available sites for all leisure  purposes, the two step approach
would be, unnecessary.  Instead, the  net benefits could be obtained directly
as the. change in the area behind the estimated compensated (or Marshallian)
                                 
unconditional demand function  for visits of a particular sort (Bockstael
and-ttcConnell* 1983; Morey,  1983;  Ziemer, Musser and Hill, 1980) Just as we
would da with a marketed good.  But  when such price data are not available
at the level of the individual, prices  cannot be used in estimation.
Instead group average unit values, which are perhaps prices but most likely
are'not, have to be found to arbitrarily value a quantity change, however
estimated.
     Therefore, there are two  potential distortions arising out of the
conventional two-step macro  participation method for approximating a
welfare change due to recreational resource enhancement:
     (1) Mis-prediction of the change in quantity demanded post-policy, due
         to use of availability proxies for price or site attributes.
                                    189

-------
     (2)  Error in valuation due to use of  either a marginal unit value or
         an average surplus.
     To hope that all of these errors will cancel in the aggregate, or to
try to Justify the procedure for "small" changes, is, in our view,
extremely sanguine.
                                    190

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 Bouwes,  Nicolaas W.  Sr.  and Robert  Schneider.   1979.   "Procedures  in
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 Cicchetti,  Charles  J.,  Anthony C.  Fisher  arid  V.   Kerry Smith.    1973
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 Daubert, John T. and Robert A. Young.   1981.  "Recreational Demands  for
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    _.   1982.   Air and Water Pollution Control (New York: John  Wiley  and
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Houthakker, H. S.   1960.   "The Capacity Method of Quadratic Programming."
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Lancaster, Kelvin J.   1966.   "A Mew Approach to Consumer Theory," Journal
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    Future).

McFadden,  Daniel  and  Fred  Reid.     1975.    "Aggregate  Travel  Danand
    Forecasting  from  Disaggregated   Behavioral  Models,"  Transportation
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McKenzie,. George W,   1983.  Measuring Economic Welfare;  Mew Methods (Mew
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Mendelsohn, Robert  and Gardner  M.  Brown,  Jr.   1983.   "Revealed Preference
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Morey,  Edward R.    1981  .    "The  Demand  for  Site-Specific  Recreational
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	.   1983-   "Characteristics, Consumer Surplus,  and  Mew Activities?
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Pearce  Bowden Economic Consultants Limited.   1971.   "The  Value of  Fresh
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 -  .Department of Recreation and Conservation).

Pollard,  J.  ff.   1971.    "On Distance Estimators  of  Density  in Randomly
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Rausser, Gordon C. and Ronald A. Olivier a.   1976.   "An Econometric Analysis
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Ravindran A.   1972.   "Algorithm 431: A Computer Routine for Quadratic and
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United  States Department  of  the Interior, Bureau of Outdoor Recreation.
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Varlan, Hal.  R.  1978.   Microeconomic Analysis  (Mew York:  W. W. Norton).

Vaughan,__William  J.   and  Clifford S.   Russell.     1982.     Freshwater
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	  and 	.  1982.   "Valuing a Fishing  Day:    an Application  of a
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                                 Chapter 6
          MODEL DESIGN:   THE ESTIMATE,  EVALUATE AND COMPARE MODULES

      In the estimate module of RECSIM, alternative demand models are
 estimated using various specifications of explanatory variables as
 generated in the modules described in  chapter 4.  One of these models is a
 demand function in the usual meaning of the phrase; that is recreational
 fishing consumption related to the travel cost prices facing each consumer.
 The other model is more familiarly known as a participation equation.  In
 it, recreational fishing consumption is related to the price proxies based
 on site density as developed in  appendix A to chapter 3  For both sorts of
 model,  welfare measures are calculated in the EVALUATE module of RECSIM.
 THE ESTIMATE MODULE
 Functional Form

          functional specification  of the single equation Marshallian demand
 function models we elect  to implement here is a Taylor series expansion to
 approximate any nonlinear function (Kmenta, 1971).  We use such an
 approximation Lit all single equation demand models.  Since the price of the
 HicksIan composite commodity is normalized to unity, the specification with
 homogeneity imposed is:
      F(p,y) - So * 8iPt * 8aP2  * 8,P, +  S^y
             * s^pf + e.pf * 87pJ * say*
    -    * 8,(piPa)  *  Slo(piP,) * 8n(p,P,)
	         * 8l2(piy)  *  8ia(py)  *
             +  (Remainder)
      where
                  Pt - price of the iC   good  (i-1-Fishing, i-2-Canping,
                       i-3-Urban Leisure) whether measured at the individual
                       level as an observed or shadow price or as a
                       density-based expected value proxy.
                                     194

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     The single equation approximation requires that fifteen parameters be
 estimated.  This is one more parameter than the number to be estimated if
 the Marshallian demand curves based on a second-order expansion in logs
 rather than levels were to be used.  The latter offers no distinct a priori
 advantage in the single equation framework.

 Variables

     Each RECSIM data set has several alternative versions of the price
 vectors which can be used in estimating the demand models.  These are
 docunented in tha list in table 1.  While all of the general models in
 table 3.4 could conceivably be estimated using each alternative price
 measure, for realism we confine our attention to the single equation demand
 model estimated using either actual (observed or shadow) prices or -
 density-based proxies for expected prices.
     Estimating a single equation model using a combination of actual
 own-price and density-based proxies for other prices might be interesting,
 since it would represent an alternative to the specification error of
 omitting: other prices entirely in estimation, as in Ziemer at. aJL. (1982).
 But, in the interests of computational economy, we did not explore this
 variant.  The menu of models we do estimate is given in table 2.
     Zero observations on consumption of a particular commodity introduce
 problems for estimation.  Normally,, when the response variable is
 stochastic, censored regression methods such as the Tobit maximum
 likelihood estimator (see Maddala, 1983) can be employed.  But, our data
 contain no stochastic component.  So, for a fixed regressor vector X.,  the
 response variable Y.. is characterized by a fixed point rather than a
 probability density function.  In this case, each individual with a given
 set of characteristics (income, sex, etc.) facing a given relative price
-set for-all goods has a reservation price for a particular good above which
 consumption will be zero.  The relationship is exact, and the demand
 function is discontinuous at the break point defined by the reservation
 price, as shown in figure 1, panel A, below.  Defining p  as the
 reservation price commodity i, the demand relation is akin to a switching
 regression model:
                                    195

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  Table 1.  Variables for Share Equation and for Demand Equation Estimation
INC


PTF


PTC


PTU


PSF
PSC
PSU
OGGF
DGGC
DGGU
Demand Model

Both I
and II (all)

Both I
and II (all)

I 3
I 3
I B
I A
                              Dependent Variable

                                     Description
I A
I A
Intermediate
variables
                                   Values
DPGF
DPGC
Intermediate-
variables
Days spent fishing.                 0
If at 0.000001, set to 0

Income actually spent on utility
producing goods.

Observed travel-cost based site price for
fishing

Observed travel-cost based site price for
camping

Observed travel-cost based site price for
urban leisure

Shadow price for fishing.   Equals  observed
travel-cost based site price for fishing
(PTF) if FISHD  > 0.000001.  If FISHD -
0.000001 PSF equals the reservation price
for fishing, RPF.

Shadow price for camping.   Equals  observed
travel-cost based site price for camping
(PTC) if CAMPD  > 0,000001.  If CAMPD -
0.000001 PSC equals the reservation price
for camping, RPC.

Shadow price for urban leisure. Equals
observed travel-cost based site price for
urban leisure (PTU) if URBD > 0.000001.
If URBD - 0.000001  PSU equals the
reservation price for urban leisure, RPU.

Density-based (DEN) expected price measure
based on known population X for a
Geographic Grid (GG) measuring number of
fishing (F) sites per unit GG area, number
of camping sites (C) per unit GG area, or
number of urban leisure (U) sites  per unit
GG area.  Note for urban leisure,  this
measure is computed over the GG area.

Density-based (DEN) expected price measure
computed from observed nunber of sites per
                                    196

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Table 1  (Continued)
Name
Demand Model
Dependent Variable

       Description
Values
DPGU
                      unit area of an elemental  People Grid  (PG)
                      in the fishing (F),  camping (C) and urban
Ai.jC
AiJU
Intermediate
variables
WAiJF
WAi.jC
WAi.JU
Intermediate
variables
PDGGF
PDGGC
PDGGU
II A
PDPGF
PDPGC
PDPGU
II B
       leisure (U) categories.   These are sample
       estimators of the true population values
       DGGF,  DGGC. DGGU.

       Aggregated (A)  unweighted density measure
       from AGGREG subroutine.   Based on PI
       vector i (i - 1,.,fi3)  and pass j  through
       AGGREG with the J   seed,  where
       j  1,...JMAX number of  seeds  and hence
       passes.  Again F indicates fishing, C,
       camping, and U, urban leisure.

       Population weighted (W)  aggregated (A)
       density measure from AGGREG subroutine.
       Based on P1 vector i (i  - 1,...31 and
       pass j through AGGREG with the j-   seed,
       where J  1....JMAX nunber of  seeds and
       hence passes.  Weights are based  on the
       population of each elemental people grid
       contained in an AGGREG Jurisdiction.
       Again F indicates fishing,  C,  camping,
       and U, urban leisure.

       Expected two-way travel  cost based price
       (P) obtained from expected distances
       DGGF,  DGGC and DGGU with a travel cost of
       $0.10 per mile.  Specifically:
                                PDGGF -  (DGGF)
                                PDGGC -  (DGGC)"
                                PDGGU -
                                                                  (0.10)
                                                                  (0.10)
                                                                  (0.10)
       Expected two-way travel  cost  based  price
       (P)  obtained from expected distances
       DPGF,  DPGC,  and DPGU  with a travel  cost of
       0.10 per mile.   Specifically:
                                              PDPGF
                                              PDPGC
                                              PDPGU
                                       (DPGF)'I'?  (0.10)
                                       (DPGCT '?  (0.10)
                                       (DPGU) 17
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  Table 1 (Continued)

                                Dependent Variable

  Name           Demand Model          Description                        Values
      jF         II C                  Expected two-way travel coat based price
  PAi.jC                               (P) obtained form expected distances Ai,
  Pfti.JU                               JF; Ai,JC; and Ai.JU; with a travel cost
                                       of 0.10 per mile.  Specifically:
                                                PAi.jF - (AiJF)
                                                PAi.jC - (Ai.JC) J; (0.10)
                                                PAiJU - (A^JU)"1 '* (0.10)

"  PWAi.JF        II D                  Expected two-way travel cost based price
 PWA-iv;J -                           (P) obtained from expected distances
  PWAi.jU                              WAiJF; WAi.jC; and WAi.jU;  with a travel
                                       cost of 0.10 per mile.  Specifically:
                                              PWAi.jF - (WAiJF)     (0.10)
                                              PWAiJC - (WAi.JC) \'% (0.10)
                                              PWAiJU - (WAiJU)   ^ (0.10)
                                      198

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   Table 2.  Alternative Price Measures Used for Estimating demand Models
	 . 	 .- -
DemamTModel Prices Price
I. Taylor Series form, A. Shadow
full price variant
B. Observed
II. Taylor Series form, A. Population.
density-based price- averages
proxies
B. Sample
averages
C. Unweighted e.g.
aggregates
: D Weighted e.g.
aggregates
Variables *,**
PSF
PSC
PSU 
PTF
PTC
PTU
PDGGF
PDGGC
PDGGU
PDPGF
PDPGC
PDPGU
PAi.jF
PAi,jC
PAiJU
PWAiJF
PWAiJC
PWAijU
Model Code
DEM1
DEM2
DEM1
DEM2
' DEMI, j 3
DEM1.J4

     * - For  the  aggregate measures   i  indexes  the  probability  vector
specified in AGGREa and j  1,...,J the number of times AGGREG is run with
a? di-fferent seed  given a probability  vector.   The value of J is specified
by the user and i  - !,...! where  1-3-

     This  means that  there are  3J  different  PAGGF,  PAGGC,  PAGG>U
matrices of prices, so each model is estimated 3J times, once with each ij
combination.

   : -. *If a zero is calculated for any density measure expected distance la
incalculable.   In such cases,  the density measure computed over the entire
country is substituted.
                                    199

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                              0 if pt >
                             - f (x) if p.
The demand relation will be incorrectly specified if  the observed price p.
is used as a regressor for both positive and zero consumption observations,
as shown in panel B.
                                       \
            Fan*l A
Panel 5
                                 Figure 1
     Alternatives when Zero Consumption Observations  Exist in the Data

     To avoid the potential bias in slope  and intercept parameters involved
in using observed travel-cost based prices aa, regressors as a demand
relation when consumption is zero several  routes  are  possible.  Two simple
alternatives are:
        Estimate only positive consumption observations.
        Obtain the reservation price vector from the quadratic program for
         individuals with zero consumption of one or  more commodities.,
         Substitute these shadow prices for the observed travel-cost-based
         prices and estimates.
The second route seems preferable and is used here.
                                    200

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THE EVALUATE MODULE:  PREDICTING WITH THE ESTIMATED DEMAND  MODELS


     Predicted changes in days of fishing and consumer welfare have  to  be

obtained in the EVALUATE module in order to make the model comparisons  in

the COMPARE module .discussed below.  This section outlines the prediction

procedures used in EVALUATE.

Single Demand Equation Model  in Prices


     The single equation demand model in (observed or shadow)  prices is

estimated for good 1  (fishing) as a Taylor's series approximation

normalized by the price of the Hicksian composite (pt) which always  equals
1 so:
     
-------
       A         A      A       A       A      A    ty

      3ql/3pl  -  8t *   8,,P2 +  BiPi.+ 2B7pt  + 8loy <
Quantity Change 

     In estimation, if any individual consumes a zero amount of good 1, we

maintain the option of using either observed or shadow prices as

regressors.  In evaluation of the function for changes in  qt due to p}
-------
negative.  So,  the estimated demand function is really discontinuous at the
reservation price for good 1  and must  be  evaluated recognizing this
discontinuity,  even if it is not explicitly  accounted for in estimation
when observed rather than reservation  prices are used as regressors.
Observed Price Regressors 
     When observed prices are employed as regressors and an IF check shows
negative predicted pre-policy quantity, the  following quadratic equation
   *                          V                                
must be solved for the (estimated)  reservation price value of p? which
drives (estimated) consumption of good 1  to  zero; given base observed
prices for all other goods:

     0 - c + bp? * a(pj)*; p? - (-b(b-4ao)1/2)/2a
where
              A    A      A       A            A     .
         c -  So * S2PZ * SjP3 + 8(p2p3) * 89(P2)2
              J%         A          A         A      J%
           +  8,(p,)2 * Slt(p2y) *  Bla(py)  * s^y + sl->
     In the case of the single demand  equation estimated using shadow
prices as regressors and IF  check must again be performed to identify
negative predicted pre-policy demand.  If demand is negative the estimated
demand function must also be evaluated using the quadratic formula to find
the estimated reservation price for pt that  drives consumption to zero,
since this value need not coincide  with the  true reservation price frcm the
                                   203

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individual's QP solution.  The rule for the price of  good 1  to be used in
the quantity change formula is the same as before:
     IF: p> p > pj use p
     Otherwise: Use pj
All other pre-policy prices substituted in'the solution for  p; and the
change in quantity formula should be in shadow prices,  not observed prices.
Welfare Measure 
     The simple welfare measure obtainable from the single demand'equation
in prices is a Marshallian consumer's surplus (CS).  To obtain CS, we
evaluate the definite integral of the estimated Taylor's
series demand function:

           P.               A                             A
     CS - I    f(pt,y)dp1 - S,Pl * 1/2  MPi)* + 3a(PiPa) * BsCPiPj)
        01  D
                              A                   A
                      +  1/2  Bfe(p,)*(p,) *  1/2  3(pi)a(p,)
                         8, (?,)*(?!) * 1/2
     As before, observed prices of goods 1,  2,  and 3  or  the predicted (if
relevant) reservation price of p? are used to get the CS from the model
estimated on observed prices.  For the model estimated on shadow prices
(which equal observed prices for positive consumption),  the predicted
reservation price of good 1 (if relevant) is used along  with the actual
shadow prices of goods 2 and 3  (See appendix 6. A).
Single Demand Equation Model in Proxies for Price                     :
     The density-based models are similar to the price model in terms of
quantity change predictions, but differ in the welfare measure used.
                                    204

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Quantities --'
     Here again, an IF check is required to ensure that  predicted demand is
greater than or equal to zero at the reigning pre-policy value of the
density-baaed proxy for the price of a fishing day.   If  it is not, the
quadratic formula must be solved for the density*based reservation price.
The prediction of quantity change is performed exactly as for the single
equation demand model in prices.
Welfare 
             .                                     
     The welfare measure for these models is just  the product of each
individual"'3 quantity change and an average value  per day of fishing.  (See
appendix 5.A).  The average value can be defined in three ascending levels
of approximation.
     (1) An individual->specific average value per  day from the WELFARE
         calculations.
     (2) A sample^specific average value per day obtained as the sum of the
         i individual values in sample j  divided by sample size, 500.
     (3) A constant across samples and individuals average value per day
         obtained arbitrarily from the first sample created - i.e. the
         first value from (2)  above.
THE COMPARE MODULE

     The RECSIM model produces a sample of exact individual outcomes, given
its initial conditions.  This sample of outcomes,  along  with the values of
the likely regressors which influence them,  become a simulated sample data
set amenable to statistical estimation.  Alternative types of models
(single equation, systems)'with alternative functional specifications and
regressor measures (especially for the site visit  price  regressor) can be
fit to the data as described above.   The question  remains, however:  How
can we discriminate among the alternative models?
     In this section we review, very briefly,  some general classes of
econometric model evaluation procedures,  and discuss why non-parametric,
out side-of-sample procedures are preferable for our  purposes.  The set of
criteria chosen for the COMPARE module are subsequently  described in
                                   205

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detail.  Fixing the functional specification for all quantity (days)
functions to be a second order Taylor's series approximation, however  the
price variable (true or proxy) is measured, prevent us confounding  the
effects of functional form and variable specification.
Criteria for Evaluation of Econometric Model Performance

     Errors in variables problems plague almost all econometric models, be
they simultaneous equation macro models of the economy or ordinary  least
squares (OLS) models of visitation to a wilderness site in Montana.
Whether explicitly recognized or not, measurements on the variables of even
the most sophisticated econometric models are often imprecise, in the  sense
that they can differ from the true values by a constant bias  with random
fluctuations.  So, the problem is not unique to recreation participation
analysis, although, it is least easily concealed or ignored there, since the
concept of a quantity proxy for price is a foreign one.
     For example, Douglas R. Hale, Director of the Quality Assurance
Division of the U.S. Energy Information Agency notes possible sources  of
error in the data used in estimating complex energy models (Hale, undated):
         Despite  Morgenstern's cautions  common practice  is to  take
         descriptions of published numbers at face value and treat the
         numbers  as  exact  measurements.   The  April  producer  price
         index, for examplep has nothing to do with prices  realized in
         April.   Instead,  the prices refer to March.  Motor gasoline
         as reported  by the Federal  Highway Administration is a  hodge
         podge  of fifty^one different chemical  substances  as defined
         by  the  individual  states   and  the  District  of  Columbia.
         Similarly, industrial consumption of electricity (and natural
         gas)  refers not  to  consumption by plants  in selected  SIC
         codes, but rather  to "large"  consumers  as  defined by utility
         rate commissions.   These definitions  are inconsistent across
         states,  within states,  and over  time.   A different sort  of
         problem  is  illustrated  by  industrial  use  of   fuel   oil.
         Industrial   use  of   residual  fuel   oil   is  not  measured.
         Instead,  industrial  use is  a   composite  of  refiners'   best
         guesses  as to where residual fuel is ultimately burned.
         (P. 7.)
     Such problems, coupled with the increasing influence of econometric
models in the policy making process,  have led to a reawakening of interest
                                    206

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 in the validation of econometric models which are used for policy impact
 analysis, or even for the formulation of policies.
      It is well known that measurement error problems bias the parameter
 estimates of econometric models.  But, if the intended use of the models is
 for forecasting (predicting outcomes outside of the sample used for
 estimation), the influence of such measurement errors on the desired result
 -> an  accurate forecast - may matter more than parameter bias problems per
 se.   the two issues are distinct.                                      y
      In general, the emphasis in most econometric work, narrowly defined,
 is on testing hypotheses concerning parameter estimates, while in the
 statistical literature, emphasis is placed more heavily on selecting models
 which predict well (Schmidt, 197*0   For instance, if the parameters of a
 true  multiple linear regression model are known, the true model minimizes
 the mean square, error of prediction (MSEP) of a future value of the
 dependent variable.  But, when the model parameters must be estimated, a
 false model with, biased parameters can produce smaller MSEP than the true
 model (Schmidt, 1974).
      Generally, model evaluation can either be undertaken using only the
 sample data from which the model(s) was estimated, or using post-sample
 data. The model evaluation criteria can either be parametric - relying on
 formal statistical tests based on the stochastic specification assumed to
 apply to the econometric model - or non-parametric '- relying on performance
 statistics with unknown probability characteristics.  (Dhrymes et. al.,
 1972).
      Within-sample parametric and non-parametric evaluation procedures are
 familiar to almost all econometric!ans who practice the art of model
 selection with secondary emphasis on forecast accuracy.  Often, economic
"theory suggests several econometric models which are plausible and
 consistent with it, but not nested within each other.  When all the
 plausible (possibly nonlinear) models use the same (possibly imprecise)
 sample "data in estimation, choosing among them on the basis of
 within-sample information either involves the use of an information
 criterion if one of them is regarded as "true" (Klein, 197U Schmidt, 197U)
 or non-nested tests, which admit the possibility that all of the plausible
                                    207

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models can be deemed inconsistent with the data (Dhrymes  et. al., 1972 and
particularly, Davidson and Mackinnon, 1982).  Frequently, even these tests
are ignored, with functional specification treated as  a maintained
hypothesis in order to allow parameter restrictions to be tested..
     Less familiar, but increasingly popular, is the use  of only a part of
the available sample data for estimation, reserving the rest to  assess
forecast accuracy by predictive tests or non->parametric methods.  The
bootstrap and cross-validation, for example, are non-parametric  methods
which attempt to empirically establish forecast, standard  errors  from the
sample at hand without reliance on the hypothesized theoretical
distribution of the error term used in predictive tests (Learner, 1983;
Freeman and Peters, 1982, Ouan et. al.,  1983).
     Waiting for new outside of sample data to be generated also permits
outside^of sample model evaluation.  With only a single model to evaluate,
parametric tests of structural change such as the Chow test, or  parametric
tests based on recursive residuals are popular (Harvey, 1981).   If several
non"nested models are to be evaluated with the hope of selecting one as
"beat" an out aide-'of sample analogue to the parametric Davids on-Mackinnon
(D-M) test can be invoked.  This obscure test, the Ho el test (Hoel, 1947),
actually predates the D- test, but is similar to it in spirit and
construction, a fact which has unfortunately gone unremarked in  the
literature.
     The battery of non-parametric, out side-* of-sample  evaluative criteria
are perhaps better known, and are relevant in a policy context because
there "the main criterion is how well the model performs  with respect to
conditional forecasts based on particular configurations  of policy options'*
(Dhrymes et. al., 1972).  Because this is just the.sort of question we are
seeking to answer, in this chapter a battery of norr*parametric,  outside^of
sample criteria are suggested.  These criteria are intended to indicate
which econometric model, using either true prices or a density-based proxy
for the correct site price variable, (measured at different levels of
spatial aggregation) performs best.  Best is defined in terms of predictive
accuracy of changes in levels, both for days of participation and monetary
                                    208

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measures of welfare occasioned by the price-reducing e.ffeet  of  a
hypothetical water pollution control policy.
The Argument for Non-parametric Model Evaluation Procedures

     The descriptive model evaluation criteria discussed in  this chapter
are model validation tools,. which should help us discern whether
participation models estimated using a proxy  price variable  as  a regressqr
fulfill their stated purpose of producing reliable predictions  of
recreation participation changes and welfare  changes occasioned by water
pollution control policy implementation, "regardless of  the  strict
faithfulness of their specification1*.  (Dhrymes et.  al.,  1972,  p. 310).
     In our simulation context, there is compelling reason to perform our
econometric model evaluation on the basis of  non-parametric  rather than
parametric criteria.  The argument against parametric procedures has its
origins in pseudo-data analysis.   This methodological tool, originated by
Griffin (1977), fits neo-classical econometric cost of profit functions to
exact sample data produced by the repeated solution of industrial process
analysis models.  Statistical tests on parameter estimates produced by
econometric models estimated frcnt exact pseudo-data have often  been
undertaken in the past (for example, Smith and Vaughan,  1981),  usually with
the caveat that such tests must be interpreted in heuristic  terms given the
non^stochastic nature of the data.  (Smith and Vaughan,  1979).
     But the validity of statistical testing  of parameter estimates of
functions fit to errorless data has been strongly questioned by Maddala and
Roberts (1979), and their argument makes a great deal of  sense.   If the
data on the outcomes (responses) of interest  are non-stochastic,  then it is
impossible to derive any maximum 'likelihood estimator, because  stochastic
errors do not exist, hence it is logically impossible to specify the
functional form of the probability distribution of the disturbances.
Consequently parametric statistical tests are logically  impossible.
2. Particularly,  it is the assumption of  normally distributed error terms
which distinguishes the linear regression model  from  the normal linear
regression model, and results  in the  equivalence of the OLS and maximum
                           (Footnote  continued)
                                   209

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     In using errorless data, estimation mere!/.involves curve (or response
surface) fitting - that is, finding a continuous  function that fits the
data adequately - and nothing more.  Adequacy can be  defined in terns of
the least squares norm of paramaterizing a specified  function such that the
sum of squares of the residuals is minimized, but other norms are possible
(for example, minimum absolute deviations) although computationally more
burdensome.  The point, however, is that the only source of "error" in
fitting errorless data is the approximation error caused by the analyst's
inability to specify the correct functional form  for  the response surface.
The estimated function is.just a shorthand way of summarizing the complex
mechanism which is the underlying model generating the data.  It is not an
econometric model amenable to parametric testing  because the sole source of
error is approximation error, and such errors cannot, a-priori, be
reasonably hypothesized to follow a normal (or any other) distribution.
     For our purposes it does not make much sense to  add randan errors to
the outcomes of the deterministic process just to be  able to conduct
classical parametric within-sample or outside-of  sample hypothesis tests on
our estimated recreation participation functions. That would only add
another layer of obfuscation to an already difficult  problem.
     Instead the- least squares norm will be Invoked to fit a second order
Taylor's series approximating function which is linear in parameters to the
simulation data for all situations where fishing  days of participation is
the dependent variable.  The approximation is used because we know that the
demand function we are approximating is inherently non->linear (Kmenta,
1971, p. 399, 453).  The predictive ability of  the fitted functions under
different variable specification regimes' (true prices versus proxies) will
then be assessed in terms of the non-parametric methods discussed in the
following section.
2. (continued)
likelihood estimators given the assumptions of the normal linear regression
model.
                                    210

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Non-Parametric Model Evaluation Criteria

     In this section a subset of the battery  of  nomparanetric criteria
outlined in Dhrymes et.  al.,  1972 and elsewhere  are discussed, along with
one criterion of our own design - the Sign  test.  The criteria are all to
be applied to assess the accuracy of forecasts of policy outcomes in terms
of changes in levels, pre to  post policy of:
        The number of fishing days  chosen  by each individual in the sample
        The resulting welfare (monetary benefit) in the fishing category.
     Because our principal concern is only  with  one activity affected by
the policy, fishing, we do not propose to evaluate the models in terms of
their demand forecasts for other goods (camping, urban leisure, and the
Hicksian composite commodity).  Similarly,  since the level of total welfare
is not of direct interest, we evaluate forecasts of changes in welfare
accruing via consumption of the fishing good, as influenced by the policy,
not total pre or post policy  welfare across all  commodity categories.
   '  The criteria we employ to assess the predictive accuracy of our
alternative econometric models are listed below.  The first two "simple
criteria" are appropriate when a model's ability to predict total changes
over all individuals is assessed, while the rest of the criteria assess the
accuracy of a model 10 prediction of individual  changes.
     I.  Simple Criteria
         T.  Mean Prediction  Error (MPE)
         2.  Mean Absolute Prediction Error (MAPE)
         3.  Concentration Coefficient (C)
     II. Forecast Error Decomposition Criterion
         1.  Theil's Ut  and U, statistics and the decomposition of mean
             square prediction error into coefficients of inequality
             between actual and predicted outcomes due to unequal. central
             tendency, unequal variation, and imperfect covariation.
     These measures are all covered  below,  where we use the notation P. for
the prediction of the change  in the  i   response and A. for the actual
(true) outcome change.  All measures are discussed under the assumption
that our predictions are unconditional, which ore ana that the values of the
                                    211

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explanatory variables in the prediction cross-section are known with
certainty.
Simple Criteria --
     In what follows, accuracy is taken to refer to a compound error
measure influenced by both bias and precision.   In purely statistical
terms, the trade-off between low bias and low variance (high precision) can
be formalized as ah accuracy measure which is a weighted average of the
square of the bias and the variance of the statistic used as an estimator
of a population parameter.  The'accuracy measure then encompasses the
notions of bias and precision (variance), not Just bias  alone, where bias
                                                A
is the difference between the expected value (E(9))  of a statistic and the
value of a population parameter 9 it is intended to measure precision, on
the other hand, refers.to the size of the deviations from the expected
                                              A
value of the distribution of sample means, E (9)  obtained by repeated
application of the sampling procedure.  (Note here that the context is one
of summarizing repeated measurements by a statistic, not single
measur eaients).
     Define mean square error (MSB) to be a measure of dispersion of the
          A                                        A
estimator 9 around the true value of the parameter 9.   Then, following
Kmenta (1971, p. 156), it can be shown that MSE is equal to the variance in
A                                                              A
9 measuring the dispersion of the distribution of the estimator 9 around
                                        4
its mean., E(9), plus the square of bias.
3. While this is indeed the case in the context of  RECSIM, it is not the
case in actual practice, where the effects of water pollution control
policy are almost never known with certainty.
4. By definitions
MSE (9) - E (9 - 9)a
Adding and subtracting E(9l to MSE(9):
MSE (9) - E [8 - BO) * E(9) - 9]a
    - E [(8 - E(9)) * (E (9) -> 9)]2
    - E (J - E (9))* * E (E(9) - 9)2
                           (Footnote continued)
                                    212

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     MSE(9)  - E(9 * E (9))2  +  (E(8) -  9)2

     Therefore,, if the estimator  is unbiased  (no statistical bias due to
the algebraic form of the estimator and no  bias due to systematic
                                                 <*
measurement error) the measure of the  accuracy of 9 will be equal to the
measure of precision (variance) .   Note that it is possible for a biased but
relatively precise method to be more accurate in the above overall sense
than an unbiased but relatively imprecise one.
     Concentrate first solely  on  the bias component.  Two simple prediction
error measures which are commonly used as criteria for ranking models in
terms of freedom from bias (see,  for example, Duan et. al. , 1982, 1983;
Pindyck and Rubinfeld, 1976; Platt, 1971) are mean prediction error (MPE)
and mean absolute prediction error (MAPE) .  Mean prediction error for a
sample of size n is defined  as:
MPE
           1   n
           1  I  (P.  -  A  )  '                                        (1)
           n       l
     Mean prediction error can be- close to zero if large positive errors
cancel large negative errors.   To remove  this possibility the same measure
can be calculated in terms of  absolute  values of the differences in actual
and predicted changes:

     MAPE - I  I  |P  -,  Aj                                          (2)
            n 1-1    l
4. (continued)*
        * 2E((e - E(9))  (E(9) -  9))
Since the last tens in the final  expression above is zero, MSE (9) equals
variance plus the square of bias.
                                   213

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Both of these latter measures can be expressed as  percentages to be more
readily interpretable.
     One of these simple measures, mean prediction error  (MPE) is of
critical importance in the context of our problem,  which  is the prediction
of the total (over the set of individuals) change  in  days of participation
and welfare occasioned by a water pollution control policy.  One way to
obtain a predicted total is to evaluate the econometric model as many times
as there are- individuals in the population, substituting
individual- specific values each time, and then sum the results.  Another
way is to compute; the change predicted by the model for the average or
representative individual and multiply the result  by  the  number of
individuals in the population.  Using this second  method, a prediction of
the total change over the population will be better,  the  smaller the
difference between the means of the predicted and  actual  series, P - A.
The difference of averages (P - A) equals the average of  differences
defined by the MPE criterion.  Therefore this criterion reflects the degree
of bias in the prediction series, and is appropriate  for  assessing a model
where the information desired is the total predicted  change.
Theil's U Statistics and Error Decomposition Criteria -*
     Mean square prediction error (MSPE) was mentioned in the previous
discussion of simple model evaluation criteria. It is defined as:
MSPE
            ,  a      '
          - ;-  I  (P. - A.)*                                        (3)
            n      ' *    r
This measure is an estimate of the expected squared forecast error.
     Theil's original inequality coefficient,  Ult  employs  the square root
of MSPE in its numerator and a denominator such that the ratio, U  , is
                                                                i
bounded between zero and one.  The Ut coefficient  is a non-monotoni c
function of the MSPE defined above:
5. Klein (1971) remarks that "The mean absolute percentage  deviation is so
intuitively obvious that I would prefer it on grounds  of  simplicity and
ease of understanding" (p. 40).
                                    214

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U1
( 1

n
i-1 l i
n 2 1/2
/n 1 P ) * <
i-1 l
1/2

n 2 1/2
[1/n 7 A. )
L. I
i-1 l
                                                                     (4)
     The coefficient ux  la bounded between 0,  the  perfect forecast, and 1 ,
the "maximum of Inequality."   For instance,  if P_  - -  B  A. for B > 0 then
there will be non- positive proportionality between P and A and U will equal
1.0, which can be proved by substituting P.(1  + 1/b)  for P, - A, in the
expression in the numerator (Theil,  1958,  pp.  32-33).  A naive forecast of
no change (P.-O)  also drives  Ut  to 1.
     Another variant of  the inequality coefficient, U2,  was proposed by
Theil (1966):

               n
          1/n  I  (P  -  A.)2
     U2 --  - -                                         (5)
          1/n    A.
                  .
              1-1.
The numerator in U2  is again MSPE while  the  denominator is the mean square
error that would apply if a naive forecast of  no  change (P  - 0) had. been
made.  Uz is bounded between 0  and <* , not 0 and  1 , and
        U2  0 represents a perfect prediction
        U2 - 1  means the model predicts no  better than a naive no- change
              prediction
        U2 > 1  means the model is worse than  a n exchange prediction.
     The population  equivalent  to the sample expression for MSPE in the
numerator of Ut  and  U2 can be expressed  as a function of the mean of the
predictor series,  its standard  deviation,  and  the correlation between the  .
actual and predicted outcomes (Theil, 1958;  Granger and Newbold, 1973).
Several mathematically equivalent decompositions  of MSPE (and U1 or U2) are
possible:

     E(MSPE) - E(A-P)2 - (UA -  Up)2 + aj * a2  - 2p
-------
or
     E(MSPE) - (UA - up)2 * (ffp -; aA)2 * 2 (1-p)ffAap.                 (7)
or,
     E(MSPE) - (UA - Up)2 * ( - A)1)1'2 and  p with
r, equal to J (Pt - P) (At ^ A)/nspsA).6
     In the second decomposition above (Eq. 7)  the three terms on the
r.h.s. are respectively the bias component, the variance component, and the
covariance component.  (Theil, 1958; Platt, 1971).
     The bias component measures the systematic error  of unequal central
tendency of the two series, and would be zero only if  actual  and predicted
changes were equal, on average.
     The variance component is another source of systematic error
reflecting the possibility that the variations in the  predictions of
changes around their average might fluctuate more (or  less) than the
variations in. changes in actual outcomes around their  average-.  Or in
Platt'a (197D words "The existence of this type of error  suggests that the
forecasts are either too sensitive (if s  > sj or not sensitive enough (if
3_ < 3.) to changes in the underlying casual variables" (p. 58).
     The covariance component is what Theil (1958) calls imperfect
covariation.  If A is stochastic, perfect correlation  between A and P is
impossible, and the less perfect the correlation, the  larger  the variance
component.  So, even if the prediction model is not flawed (is perfectly
specified), this final component is unavoidable, which presumably is  why
Theil refers to this component as "unsystematic1* (p. 37).   Yet to the
extent that specification error (incorrect functional  form, for example)
leads to further reductions in r below 1 over and above that  due to the
6.-Note there are no degrees-of -freedom corrections  in  Theil's sample
formulas.
                                    216

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influence of randomness in A,  the third component is not entirely
unavoidable and unsystematic.
     The third decomposition,  in (3)  above,  also leads  to a natural
interpretation.  That is, to the extent that the deterministic  part  of  the
relationship between A and P is perfectly predicted, the regression  A on  P
will have a zero intercept and unit slope.  Then u.  ~ u0 will be  zero since
                  *%       A              A        A  A  *
with a unit slope a - T - b P" and a zero a and unit  b implies A" - "?  is  also
                                                                    A
zero.  Likewise, in the regression of A on P the slope  coefficient,  b,  is
by definition equal to r(3./3p).  So, Lf b equals 1, r  equals 3p/3..  Then
the second term in the decomposition of (8)  above will  also equal zero
because of a perfect prediction of the deterministic part of A..  All that
remains after perfect prediction of the deterministic part of the
relationship is the third term, which represents the variance of  the
disturbances in the regression of A on P.   Therefore, there is  not much new
information contained in the error decomposition that is not already
reflected in the regression criterion.  For this reason,  we only  use Ul and
(J2 (variants of. mean square error) as criteria.
Some- Formal Non-Parametric Tests of Homogeneity

     Homogeneity refers to sameness.   Tests of homogeneity attempt to infer
whether the populations involved share a single  common  attribute  - the
equality of central tendency,  for example -  or are indeed completely
equivalent* having the same central tendency and dispersion.  In  our case
the populations we wish to make inferences about in  pairwise comparisons
are the population of actual outcomes (or changes in outcomes)  and the
population of predicted outcomes (or changes in  outcomes)  obtained from one
or another statistically estimated prediction formula.
     Formal non-parametric tests of equivalence  in terms of location,
spread on both are preferable  because of our ignorance  about the  form of
                                                                 7
the probability distribution of the actual and predicted changes.
7. Any Individual actual outcome  does not  have  a  probability distribution
in our deterministic model.   But  at  any set  of  fixed values that the X
vector can assume, the outcome itself is a random variable with an
(unknown) distribution function,  being a linear combination of the X
variables, which are random  variables over repeated trials.
                                    217

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 Non-parametric methods  require only minimal and general assumptions in
 order to be  valid,  and  can be  employed when the distribution function of
 the random variable producing  the data is unknown.  (See Conover, 1980).
      While for purposes of parsimony we  elect in the present study not to
 implement the norriparametric of  tests of homogeneity, we present a brief
 discussion of some  such, tests  which might prove useful in future extensions
 of this line of research.
      The problem with most tests of homogeneity is that two independent
 samples are  required to perform  them.  This means that if we have two
                            
 samples, A and B, the first representing predicted changes and the second
 representing actual changes, for independence to hold the probability of
 any outcome  in sample A given  that any outcome in B occurs must be the same
 as the probability  of any  outcome in sample A without information of the
 occurrence or norfoccurrence of  an event in sample B.  Representing
                      *
 predicted outcomes  as Y and actual outcomes as Y this means the conditional
                A
 probability  of Y. given Y, is  the same as the unconditional probability of
 -                i       J
 V
      Prob (YjY.) - Prob (Y^)
 But,   since  the X vectors   are random  variables in  repeated trials  and
i^ -  SX1>t Y. - SX.  this means:

      Prob (8 XjsXj)   Prob (SX^

 So, for any  individual, for independence to obtain the X vector attached to
 him in the prediction sample cannot be. the same as the X vector attached to
 him in the true sample. An obvious necessary condition for independence to
 hold, then,  is that the correlation between actual and predicted outcomes,
 r, be zero.
      It is not impossible  to ensure independence in the simulation context.
 Doing so implies extra  computer  costs, or split sample analysis with
 smaller sample sizes.  The steps, if two separate samples are generated,
 would be:
                                     218

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     1)   Create a sample for estimation and prediction by generation of one
                  Pi* 9                              Pr*s
         matrix,  Xt    of regressors  and one vector Yt   of outcomes
         pre'-policy.
     2)   Estimate 3  fron X^re,
     3)   Predict ?,oat from 8,
     4)   Compare result in (3)  to another  equi-> dimensioned vector Yi3t of
         post policy outcomes generated by a second simulation based on the
         same initial conditions  as  (1)  except for random number seeds, so
         that Xi3t  is orthogonal to Xi3t.
Of course,, the same  effect could  be  achieved by reserving half of the
original sample frcnr step (1)  above  for estimation and prediction and the
other half for the validation comparison of  step 4.
     As noted above, creation of  two independent samples, one for
estimation/prediction and one for verification, guarantees the absence of
correlation between  actual and predicted changes.  Then, from the mean
square prediction error decomposition,  that leaves just the bias and
variance components  of MSPE to be analyzed.  Under these circumstances it
is appropriate to test whether the two distribution functions - one for the
actual changes and one for the predicted changes - are associated with two
identical populations.  The Smirnov  test (Conover, 1980, Ch. 6) is an
omnibus norr*parametric test designed to test all possible deviations
(differences in location and/or spread)  from the null hypothesis of
                                               8
homogeneity and would be- ideal in this context.
     However, it is  not clear that the  extra Information provided by a test
like the Smirnov test would be  worth the computational effort.  Indeed, the
null hypothesis requires strict equality,  while even minor departures from
homogeneity reside in the alternative hypothesis.  But tests like the
Smirnov test have too much power  with large sample sizes in the sense that
they tend to reject  the null hypothesis for even slight deviations from it
with increasing probability as  the sample  size increases.  The problem, of
rejecting- the null with near certainty  for non^zero but negligible  	
8. Another possible test is the squared ranks statistic, which is useful
for testing scale shift under asymmetric  alternative distributions with
mass confined to the positive axis  (Duran,  1976).                     	
                                   219

-------
departures frcm it can only be dealt with by careful  determination of
sample sizes and definition of alternatives "close1* to the null for which
rejection is not desired.
     Without such sophisticated manipulations,  it is  possible to invoke the
sign test.  The sign test is a less powerful norr- parametric test which does
not require Independence between observation pairs in the prediction
series, P, and the true series, A, although independence across pairs is
required (that is, Pt, Ax independent of P2, Aa,  and  so forth).  Moreover,
the sign test does not require that the distribution  of the differences
between observation pairs be symmetric, a requirement of the more powerful
signed rank test (Hollander and Wolfe, 1973).
     For the sign test, we consider a draw of n mutually independent pairs
of predicted and actual values, (Plt At), (Pz,  A2), ..., (P , A ), from two
cumulative distribution functions F(P) and G(A).   On  the basis of these n
elements we test the null hypothesis.
     H-os F(P) - G(A)
against the alternative
     H1? F(P) *  G(A)
which can be regarded as a test of the equality of medians between the
predicted: and actual change series.
     Implementation of the test is simple and involves computing the test
statistic T as the number of pairs for which D   p.  - A, is greater than
zero out of all pairs for which ties are disregarded  (n & total numbers of
pairs, n).  In large samples, the statistic, T, is compared to the
(approximate) test value t (Conover, 1980) based  on the normal
approximation to the binomial distribution of r

     t - 1/2 (n * Z    n172)
where Z  ._ is the standard normal variate at significance level cs.  For a
       0/2
two sided test of the null hypothesis that the median of 0   is 0 against
the alternative that the median is not equal to zero, H  is  rejected at the
a level of significance if (for proofs see Conover,  1980 or  Hollander and
Wolfe, 1973):
                                    220

-------
     T S t or T   n - t

Essentially this test says that T differs from its expected value  of  1/2 n
(pluses are as equally likely as minuses if H  is true)  if T does  not fall
in the interval t, n-t, where t is the largest interger  for which

     P(tSTSn-t)>1-a

CONCLUDING. REMARKS

     The first two major sections of this chapter reviewed the mechanical
details of the ESTIMATE and EVALUATE modules of the RECSIM model.  The
conceptual background for these was provided in chapter  3, with additional
background on approximate welfare measures in appendix A to chapter 5.
     The COMPARE module was given more space because no  such background had
been provided earlier.  A major lesson of the resulting  discussion seems to
us to be that a fuzzy, but nevertheless real, distinction can be made
.between criteria which evaluate model performance in terms of accuracy in
predicting: individual outcomes, and criteria which focus almost exclusively
on the ability of the model to accurately predict an average or
representative outcome.
     The former criteria are relevant in a policy evaluation context  if an
assessment of the policy's impact on a subset of the population -
individuals either in a particular locality or belonging to a particular
3ocio-ieconomic stratum - is desired.  Specifically,  these criteria yield
indirect evidence of the ability of a model estimated frcnr a national
cross-section to predict localized policy impacts in one or another regions
of the nation, given individual-specific information on  the inhabitants of
those regions.
     The second set of criteria focus on the prediction  of an average, or
representative,, outcome, be it changes in days of recreation or changes in
welfare due to the policy.  As such, these criteria provide evidence  on the
ability of the model to yield acceptable national estimates of policy
                                    221

-------
impact, but say nothing about .the applicability of  the model to localized
situations.
     From this perspective, the criteria applied  in RECSIM can be
reshuffled into two groups:
                         PREDICTION MODEL ACCURACY


     REPRESENTATIVE                      INDIVIDUAL
       PREDICTION                        PREDICTIONS

     Mean Prediction Error               Theil'a Ul and Ua

     Mean Absolute Prediction Error
                                    222

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                               Appendix 6. A

            THE CORRECT CALCULATION OF  WELFARE CHANGES FROM THE
                  ESTIMATED SINGLE DEMAND  EQUATION MODELS
     The predicted welfare change calculation for  those Individuals
responding to a price decrease by consuming positive  predicted quantities
of a good; when they had formerly consumed none Is  tricky.  This appendix
discusses the evaluation of the relevant  predicted welfare change measures
for such Individuals In the context  of  the single  equation demand models.
     Predicted Marshalllan surplus Is the area under  the-estimated demand
curve between the Initial and post-policy price levels.  It can only have
meaning in the north-west quadrant of price-quantity  space because by
assumption the individuals in our model cannot consume negative quantities
                                                                         #
of any good.  For all Individuals with  an initial  pre-policy price below pa
(the reservation price) in figure A.6.1 the estimated demand function will
predict positive quantities pre-policy, and no difficulty arises.
     But, any individual initially at p,  > p  in the  figure would be
predicted to consume a negative quantity  pre-policy if the discontinuity in
                                                  
the demand curve at a quantity of 0  and a price of p, were not recognized.
Whereas the correct Mar shall ian surplus is the definite integral of the
                                  
demand function over the interval p,  to p,  shown as Area II in the figure,
the incorrect (overstated) surplus over the interval  pa to plt is
equivalent to areas I plus II,  Hence, it  is necessary to solve the
estimated demand function for the price which drives  consumption to zero
(the reservation price) before quantity changes or surpluses are evaluated.
                                   223

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                              Figure A.I



Consequence of Failure to Use Reservation Price in Welfare Calculations
                                 224

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Davidson, .Rusaell  and James  G.  Mackinnon.    1962.     "Some   Non-Nested
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Dhrymes, Phoebus J.,  E. Philip Howrey,  Saul H. Hymans, Jan Kmenta,  Edward
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Ouan,  Naihua,  Willard  G. Manning,  Jr., Carl  N.  Morris,  and  Joseph  P.
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     ,     ,	and        1983.  "A  Comparison of Alternative Models for
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Duran>  Benjamin  S.   1976.   "A  Survey of  Nonparametric  Tests  for  Scale.
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Freeman, David  A. and  Stephen C.  Peters.    1982.     "Bootstrapping  A
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Granger, C. W. J.  and Newbold,  P.  1973*  "Some  Comments  on the  Evaluation
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Hale,  Douglas  R.  n.d. "The Evaluation  of Economic  Forecasting Models"
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Hoel, Paul G.   1947.   "On the Choice of Forecasting Formulas," Journal of
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Hollander, Myles  and Douglas A.  Wolfe.   1973-   Nonparametric  Statistical
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Kmenta, Jan.  1971.   Elements of  Econometrics (New York:  Macmillan).

Kotz, Samuel and Normal L. Johnson eds.  1982.  Encyclopedia of  Statistical
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Learner,  Edward E.   1983.    "Model  Choice and Specification Analyses",  in
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Maddala,. G. S and R. B. Roberts.  1979.  "An Evaluation of the Pseudo-Data
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 _        1983.   Limited-Dependent and Qualitative  Variables in  Econometrics
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Mincer,  J.  and  V.  Zarnowitz.    1969.    "The  Evaluation of  Economic
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Pindyck, R So and D. L. Rubinfeld.  1976.  Econometric Models  and Economic
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Platt,.  Robert  B.   1971.    "Some Measures of Forecast  Accuracy1*,  Business
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Quandt,  Richard  E.   1965.  "On Certain  Small Sample Properties of  !c-Class
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Schmidt,  Peter.   1974.    "Choosing Among  Alternative Linear  Regression
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Theil,  Henri.    1958.   Economic Forecasts  and  Policy   (Amsterdam:   North
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Ziemer,  Rod,  F., Wesley M. Musser, Fred C. White and R. Carter Hill.   1982.
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     to  the Warmwater Fishing  Deaand,"  Water Resources Research vol.  18,
     no. 2 (April) pp. 215^219.
                                   227

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                                  Chapter 7
                            RESULTS AND DISCUSSION

      This chapter presents a summary of the RECSIM estimation,  evaluation,
 and comparison results.
	In_all instances, the dependent variable for estimation is the pre-policy
 level or fishiny days  (PREFISHD), generated for each observation as  the
 utility-maximizing outcome at the pre-policy price-income vector by  the
 RECSIM optimization algorithm.  This pre-policy level of  participation is
 used to mimic the approach taken in a typical econometric analysis of
 recreation participation, in which prevailing levels of participation and
 their presumed determinants are obtained from survey or other data.  These
 data, in turn, are used to estimate the participation models that serve as
 the basis of the predictions of the effects on participation of
 hypothetical policy measures expressed as hypothetical changes  in one or
 more of the participation determinants.  The estimated models vary
 according to the price or price-*proxy measures used as regressors, and it
 is the question of the sensitivity of the participation and welfare
 predictions to the choice among alternative price-*proxies that  is central
 to this phase of the analysis.
      Because of the nonstochastlc nature of the data (recall that the
 equation "errors" arise solely because of misspecification of the true
 functional form), no standard statistical results (e.g. ^statistics or
 likelihood ratios) will be presented.  As suggested in the previous
 chapters, the econometric analysis here is best likened to a
 response-surface fitting exercise, so that the only estimation  results of
 interest are the estimates of the parameters of the demand or participation
 1. Recall from above that the "econometric1* models estimated as  the  bases
 of this analysis are single-equation, linear, second-order Taylor series
 approximations to the "true" models.  All estimation is performed by
 ordinary least squares (OLS).
                                     228

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functions.   Of major interest is the quality of  the predictions generated
by the various specifications.
     Pre- and post-policy participation are predicted in the manner
described in detail in chapter 6.   Briefly, the  models estimated using the
pre-policy price-income vector are used to predict the pre-policy levels of
participation by evaluating the estimated function at the pre-policy
price-income vectors, and the post-policy levels of participation using the
post-policy price-income vectors.   The  post-policy vectors differ from the
pre-policy vectors only in the elements involving- the price of the fishing
activity (or its proxy).
     In the cases where observed or shadow prices are used, the relevant
fishing activity prices are the actual  travel-cost-based prices on which
the consumers'  utility-maximizing decisions are  based.  Where the density
proxies., are used, the post-policy "prices" are calculated using the
density-to-price transformation described in chapter 3-  The measure of
density used is the number of fishable  sites in  an area, with the
percentage of fishable sites available  post-policy taken to be one hundred
percent..  The pre-policy "prices"  on which estimation and prediction of
pre-policy participation are based are  calculated using, the post-policy
density (i.e. the density of all sites, fishable and not), corrected by the
percentage of water not fishable pre-policy,. i.e.:
price pre-policy  (cost/mi)*
                   (post-policy density*percent fishable pre-policy)
   ~ The detailed results of  the estimation, evaluation, and comparison
exercises are presented in the  tables  in Appendix A to this chapter.  These
results are for the thirty-four sets of regressors representing the
thirty-four sets of the price or price-proxy variables in each of the
simulated datasets.  There are  twenty  such datasets, of which ten assume a
bivariate normal distribution of individuals in space, and ten assume the
distribution is bivariate uniform.  Of these twenty, five of the normal and
five of the uniform datasets  are generated assuming that the share of total
                                   229

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 water not fishable pre^policy is .30,  while the other five datasets in each
 case assume a .03 pre-policy share of  unf ishable water.
- AGGREGATION OF PRICE PROXIES

      The large mass of output precludes detailed discussion of all results.
 However, because seme consistencies emerge across  the various datasets and
 proxy measures, it is possible to summarize-the results of Interest without
 substantial information loss.  First it should be  noted that the negativity
 checks (displayed in the "NEC CHEK* rows in the appendix tables) indicate
 that in not all instances is the partial derivative of the estimated
 participation function with respect to the owrfprice (i.e. the price or
 price-^proxy for fishing) negative.  This indicates that it is possible for
 decreases in the owrr>price to elicit decreases in  the participation measure
 of interest.  Insofar as the microeconomic laws of demand are concerned,
 this is an unappealing result.
      It must, however, be recognized that  these derivative results are only
 local results, and that the derivative itself depends on the value of,
 inter alia, the fishing price or price-proxy (i&e. the fishing price or
 price-proxy tens enters linearly into the expression for the derivative of
 predicted fishing days with respect to own-price).  Thus, it is not
 necessarily the case that the sign of the mean predicted difference in pre-<
 and pos^pollcy fishing days will be in accord with the MEG CHEK result, as
 the former relies on discrete changes in the fishing price or price-*proxy
 while the latter depends solely on the derivative  properties of the
 estimated demand functions as evaluated at the base^level price (or
 price-proxy) vectors.  So, while in most instances the  direction suggested
 by the MEG CHEK indicator will agree with the realized  sign of the change
 in fishing days summary measure, such agreement is not  necessary and in
 fact does not occur in numerous instances.
      Because of the differences in the sample levels and variances of the
 explanatory variables, comparison of the magnitudes of  the estimated
 coefficients is uninformative.  It might be noted  from  the detailed tables
 that the magnitudes of the fl. . estimates vary widely across the
 specifications of price and price proxy.  This reflects to a considerable

                                     230

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extent the fact the the sample variances  of  the explanatory variables
themselves vary widely across the price/price-proxy specifications.  The
means of the explanatory variables vary relatively much less widely.
     As expected, regardless of the price or price-proxy specification
used, the predictions of base level,  or pre-policy, participation (FISHDOP)
are generally identical to the sample mean of  the actual participation
data.  This is nothing more than a manifestation of the ordinary least
squares estimation property in which the  mean  prediction of the dependent
variable equals its sample mean because the  sum of the prediction errors
must be zero when an intercept is included as  a regressor.
     In some instances, however, insufficient  sample variation in one or
more of the price or price-proxy variables precluded estimation of the
parameters.  In such instances, a small value of DET(XTX) will be noted in
the detailed results tables as well as  a  discrepancy between the means of
the actual and predicted participation  series.  Although the estimation
algorithm used here attempts to estimate  the parameters in these instances,
and does in fact produce numbers, the results  in such instances should be
disregarded.  In seme but not all of these instances, the detailed tables
will show OFISHP and some the welfare change predictions based thereon
equaling exactly zero and some the of comparison criteria therewith
associated equaling* exactly one.  More  discussion of these estimation
problems is found below.
     The results become interesting when  we  begin to examine the
post-policy predictions of participation  and other measures of interest.
So far as prediction of post->policy participation (FISHD1P) is concerned,
both inspection of the average predicted  magnitudes (when juxtaposed with
the true outcomes) and consultation of  the various comparison criteria
(MPE, MAPE, and Theil'a U1  and U2 statistics)  presented.in the detailed
tables suggest the following.  First, the models estimated using.the true
and shadow prices, when evaluated at the  post-policy price sets, outperform
all the other specifications.  Second,  the models based on
population-average and sample-average proxies  (columns 3 and 4) in turn
generally perform- better than the other specifications of price-proxy
considered.  These results obtain for the predictions of the mean
                                   231

-------
  differences In pre- and post-policy participation (DFISHP) as well.
       In terms of assessing the weighted and  unweighted aggregated proxies
  for prediction of participation,  it can fairly be said that their
  performance is generally quite poor.   Predictions of average post-policy
  participation often have the wrong sign, and the comparison criteria tend
  to support the hypothesis that the quality of the predictions based on such
  measures will be dominated by the true/shadow price models as well as the
  models based on sample and population averages of the proxy variables.  The
  comparison criteria are- not uniformly supportive of this contention,
  however.  In seme instances models based on  the aggregated availability
  proxies appear to outperform the models based on true/shadow prices and
  aaapte/population average proxies. (As conjectured in chapter 6, the mean
  prediction error statistics appear to be especially peculiarly behaved in
  this respect, this likely due to to large negative prediction errors
  cancelling large positive ones.)  However, these instances appear to be far
  more spurious than suggestive of any systematic tendencies, and the
  variance of the quality of the predictions based on the aggregated
  availability proxies appears to be quite large.
       For the model estimates that were attempted, those for which
  estimation was precluded by the lack of sufficient sample variation in one
  or more of the regressors almost exclusively consisted of the price proxy
  regpessors for which the target aggregate size was either ten or seventeen
  (see table 1).  In only seven of the thirty  cases using the target size of
  five did insufficient variation in one or more of the aggregated price
  proxies preclude estimation.  In no instance were such problems encountered
wt-ttr-the-true prices, or with the sample or  population average proxies.
       Such results, while in one sense unfortunate, do provide valuable
  insights into the essence of some problems inherent in using aggregated.
  density measures aa price proxies in applied recreation analysis.  Indeed,
  on reflection the fact that the measures formed using the more
  highly-aggregated target sizes (i.e. ten and seventeen) often exhibited
  insufficient sample variation is not surprising given the design of the
  aggregation analysis as described in preceeding chapters.
                                      233

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     While the inability to estimate sane specifications for some dataseta
introduces ambiguity into the comparisons of model performance across the
candidate price/price-proxy measures, certain unambiguous findings do
emerge.  The specifications estimated using .actual prices outperform, by a
considerable degree, and  on the  basis  of all comparison criteria used,
those estimated using sample average proxy measures.  The estimates based
on the sample average proxies,  in turn, outperform the estimates based on
the population average proxies, although the superiority of the former over
the latter in this instance might be characterized as marginal.
     Excluding from the comparisons those seven cases with target sizes of
five where estimation was impossible, the comparison criteria also indicate
that the models based on the true prices, and sample and population average
proxies uniformly outperformed the specifications estimated using the
aggregated price proxies for the  target size of five.  Table 7.2 below
shows the average of the U2 statistics  for the predicted change in fishing
days (DFISHP) for the true prices, sample and population average proxies,
and aggregated proxies with the target  size of five (with the seven cases
for which estimation was not possible excluded from the averages).
     Table 2 also shows that for  target sizes of ten and seventeen,
averages of the comparison criteria over those few cases where estimation
was not precluded by insufficient sample variation irr the regressors are
neither uniformly superior nor inferior to those based on the target size
of five.  However, the true prices and  sample and population average
proxies are uniformly superior to the aggregated densities with targets of
ten and seventeen.  Yet, because  the averages of the comparison criteria
over the feasible ten and seventeen target models are typically calculated
over very few points, considerable caution should be exercised in
interpreting these results.
     The estimates of Marshallian consumer surplus are obtained only for
the two models estimated using- true- and shadow prices.  Recall frcnt chapter
5 that this estimate is obtained  by integrating the estimated demand
function with respect to own-price, and evaluating the definite integral at
the pre- and post-policy price sets.  The detailed tables show that the
estimates of Marshallian surplus  obtained in this manner are generally very
                                   233

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                                  Table 1
              SUMMARY OF MODELS FOR WHICH (NEAR-)SINGULAR  X'X
                        MATRIX PRECLUDES ESTIMATION
                (Cell Entries are the Number of Instances)
                             Distribution/Rho
              Normal/.30    Normal/.03    Uniform/.30     Uniform/.03
Aggregated
  Proxy
  Measure
  (target aggregate
   size)

Unweighted (5)

Weighted (5)

Unweighted (10)

Weighted (10)

Unweighted (17)

Weighted (17)
0033

0010

5554

6554

871010

771 010
                                    234

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close to the "true" measures of  welfare  change,  i.e. compensating and
equivalent variation,  obtained from the  RECSIM optimization algorithm and
given knowledge of the true utility functions.   Because the compensating
and equivalent variations in these instances  are fairly close, estimates of
Marshallian surplus based on a demand function that mimics well the .true
demand function would be expected to be  reasonably accurate.  It appears
from our results that such seems to be case.  While it is of course true
that the magnitudes of the compensating  and equivalent variation measures
are larger in those instances where the  amount of water not fishable
pre-policy is -30 than in the cases where this amount is .03, the estimates
of Marshallian consumer surplus  in both  instances are remarkably close to
the "true" CV and EV measures.
     Because the Laspeyres,. Paasche,  Harberger,  and average consumer
surplus measures of welfare change are calculated using the predicted
differences between pre- and post-policy fishing participation levels, it
will necessarily follow that when this predicted difference (DFISHP) is
negative (i.e. decreases in price elicit decreases in participation), all
these welfare measures will have the "wrong*  sign.  Recall that the true
welfare measures, compensating and equivalent variation, are defined for
this- exercise such that positive values  represent the increased (or at
least not->decreased) participation that  actually results frco the decline
in the true relative price of participation in the RECSIM optimization
algorithm.
     Insofar as comparison of welfare results across the specif ications of
prices/proxies is concerned, a fair summary statement based on all the
comparison criteria utilized (MPE,  MAPE,  U1,  and U2) is that these welfare
measures are typically quite poor approximations to the true compensating
and equivalent variation measures.   Because the  predicted differences based
on the models estimated using true/shadow prices always have the correct
sign, the predicted welfare measures based thereon will accordingly have
the same signs as do the true measures,  although the predicted magnitudes
are typically quite wide of the  mark.  A similar finding obtains in those
cases where the sample and population average proxies are used, although in
these instances there is an occasional Incorrect prediction of the sign of
                                   235

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                                Table 2

          AVERAGE ACROSS FEASIBLE SPECIFIC AXIOMS  OF  THEIL'S U2
         STATISTIC FOR PREDICTED DAIS OF FISHING  PARTICIPATION*
                     Population Distribution/Rho**

              Normal/.30     Normal/.03     Uniform/.30
  (target number of
   jurisdictions per
Uniform/.03
Price/
Price- Proxy
True Price 0.01 O.OT
Sample Avg. 1.01 0.90
Popul. Avg. 1.14 3.18
Aggregated
Proxies

0.02 0.01
1.01. 0.88
K05 1.05

Unweighted (5)
Weighted (5)
o
Unweighted
Weighted (1
Unweighted
Weighted (1

(10)
0)
(17)
7)
2
3

12
.00
.32

.53
1T.69
13
8
.76
38
39.76
81.57

23
29
32
55

.71
.74
.21
.35
2
2

1
12
6
3
.94
.84

.50
.49
.46
.24
7
10

23
9
16
4
.27
.54

.09
.00
.48
.89

*  Theil's U2 statistic is defined in Chapter 6.

** Rho is the- parameter indicating the fraction of total recreation
   sites assumed unflshable pre-policy.
                                    236

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DFISHP,  these yielding in turn incorrectly^signed welfare change
predictions.   Interestingly,  in many  instances where the predictions based
on these measures are correctly signed,  the average magnitudes of the
predicted welfare changes obtained using these sample and population
average proxies and the ACS1  and ACS2 welfare measures are nearer to the
averages of the true CV and EV measures  than are those based on the true
and/or shadow prices, although the comparison criteria other than MPE 
which are more sensitive to observation-by-observation deviations >
suggest generally that the- true/shadow price models perform better.
     In a nutshell, the welfare change predictions baaed on the two-step
method and aggregated availability proxy measures are quite poor, as judged
by the four comparison criteria and by simple inspection of the details of
the predicted magnitudes in the appendix tables.  Occasionally, it happens
that one or more such measures in a given dataset will produce a model
which subsequently generates a welfare change prediction superior to those
obtained from either the true/shadow  price models or frcm the
population/sample average proxy measures;  again MPE and, in some less
frequent Instances, Thell's U1  statistic appear to behave somewhat
peculiarly in this regard.  Such proximity, however, seems spurious, and.
inspection of the detailed tables in  fact reveals that the variance of the
prediction quality for the aggregated availability proxy models is ia all
instances rather large.
     Finally, it does not appear that the models and predictions based on
the datasets in which individuals are distributed in the bivariate normal
fashion either dominate or are dominated by those in which the distribution
of individuals in space is bivariate  uniform.
AGGREGATED AND DISAGGREGATED USE OF ESTIMATION RESULTS

     The RECSIM data and the estimates obtained in the manner discussed
above allow other interesting evaluations of the performance of estimated
models in predicting recreational participation.  The "real-world" analogs
to such evaluations would be two:  first, a situation where models of
participation estimated for a local/regional area on the basis of locaL
data are used to make global/national predictions of recreation
                                   237

-------
participation or other outcomes of interest; and,  second,  the converse
situation in which participation models estimated on the basis of
global/national data are used to make local/regional participation
predictions. .                 *
     The approach used here is to view each of the twenty generated samples
as. a "country1* and to view any four of these "countries'* as  a RECSIM
"continent," where country and continent refer respectively  to the local
and global Jurisdictions described above.  The first exercise uses the
parameters of the participation models estimated in the- manner described
earlier for each "country" to predict participation for the  "continent"
using both the pre- and post-policy levels of the explanatory variables as
they prevail in the continent.  The second exercise concatenates the data
from, the four "countries," estimates models using the pre-policy data for
all the observations in the continent, and then predicts pre- and
post-policy participation in each of the four constituent countries.
     Because of the problems discussed above in estimating the models based
on the more highly aggregated price proxies, this phase of the analysis is
less- ambitious- than we had originally hoped.  Yet,  in order  to provide seme
insights into these problems, we undertake the fallowing limited exercise.
     First, we restrict the analysis to a comparison of the  true prices,
sample and population average proxies, and aggregated price  proxies having
a target .density of five.  Second, we exclude from- the candidate
"countries'* those for which models based on these aggregated price proxies
could not be estimated, this resulting in a set of sixteen candidate
countries.  Then, ten continents are constructed by drawing  ten random
samples (with replacement) of four countries from this universe of sixteen
and designating each realization of four countries as a single continent.
     Then, based on the models estimated above, the first exercise obtains
four participation predictions for each continent by evaluating the
estimated models of each of the four constituent countries at the pre- and
post-'policy price sets for all observations in the continent.   These
predictions are then compared for each of the 10 samples with the actual
participation changes using the U2 comparison technique discussed earlier.
The second exercise first uses for each of the ten trials  the continent
                                    238

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data to estimate a continent model,  uses these 10  continent models to
predict participation in each of the four- constituent  countries, and then
the U2 technique is used to assess the quality of  the  predictions in each
of the four countries by comparing the continent-based predictions with the
actual participation levels in each count.  We again restrict attention in
these exercises exclusively to measures of  participation, with the not less
interesting issues of welfare comparisons set  aside for future research.
     The results of these exercises  are summarized in  table 3 below.  The
tabla entries are- the- averages of the U2 statistics across the ten trials.
For both exercises (i.e. continent models predicting country participation
(RC) and country models predicting continent participation (CR)) the table
reveals a pattern similar to that seen in the  earlier  analysis.  That is,
the models based on the true prices  outperform by  a substantial margin
those based on population and sample average proxies.   Again, the sample
average proxy outperforms the population average proxy.  It might also.be
noted that for the true prices and tha sample  and  population average
proxies, the- CR exercises, i.e. where country  models predict continent
participation, appear to perform quite a bit better than those where the
continent models are used for predicting country participation.
     As expected, and much like the  earlier analysis,  the performance of
the aggregate price proxies is quite poor vis-a-vis the other three
measures.  While the magnitudes of seme or  the U2  statistics for the
individual predictions are suggestive of some  possible outliers, even when
these are discarded the relative performance of the aggregate price proxies
is clearly inferior to the other measures.  We conjecture that the price
proxies based on more highly aggregated measures (i.e.  target sizes of 10
and 17) would tend to perform even more poorly.
SUMMARY AND CONCLUSIONS
     The results discussed in this chapter  corroborate fairly well -- in
those instances where unambiguous comparison across model and data
specifications is possible ->- the a  priori  expectations about- the problems
attending aggregation in empirical recreation  analysis.  That is, models
estimated using the true prices,  on  which the  utility^maximizing decisions
of the recreationers were based,  uniformly  outperformed the models based on

                                    239

-------
 sample and population average proxies.  The latter,  in turn,  tended to
 outperform the models that used the density-baaed or aggregated price
 proxies in those instances where comparison of the latter with  the former
 was feasible.  These results were consistent across  the experiments
 performed: (1) using basic predictions within datasets or "countries"; (2)
using- "country" models to predict "continent" participation measures; and
 (3) using "continent" models to predict "country" participation measures.
      The inability to estimate all the density-baaed or aggregated price
'proxy models was admittedly disappointing.  While it was fairly unambiguous
 that the models based on the aggregated price proxies exhibited inferior
 performance to those based on any of the other measures (true or proxy),
 the relative performance across the different levels of'aggregation could
 not be ascertained without ambiguity.  While we conjecture  that the
 performance of the models based on the more aggregated density  measures
 .would be worse than that of those using less-aggregated measures, empirical
 corroboration of this conjecture must be left for some future analysis.
      Nonetheless* the lessons of this research must  be seen as  cautionary
 and as relevant to the assessment of a wide range of applied  studies.
 These include studies of recreation per se9 of pollution control benefits,
 and of yet other subjects for which a participation  decision  can be
 hypothesized to depend oa a price for which only Jurisdlctionally
 -aggregated* proxies are available.  Further, the tempting notion of doing a
 small-area "case study" and then blowing up the results by using national
 values for the Independent variables has been shown to be dangerous.
 Similarly for the obverse idea of taking national equation results and
 applying them to a regional problem.  And it must be stressed that the
 danger is not merely that we will miss the true answer by seme small amount
 with a known bias direction.  The results of these common exercises can be
 right on or wildly off in either direction.  We cannot predict in advance
 the size or direction of these misses and thus can never be sure how
 seriously to take our results.
                                     240

-------
                               Table 3

            SUMMARY OF COUNTRY CONTINENT MODEL PERFORMANCE
                      USING THEIL'S U2  STATISTIC
          (Table entries are averages  across  the ten trials;
    "CR" denotes exercises where country models  predict continent
     participation;  "RC" denotes exercises  where cdntinent models
                    predict country participation)

True prices
Sample Average
Proxy
Population Average
Proxy
CR
0.01
0.99
3.08
RC
0.18
16.35
18.33
Unweighted Aggregate
  Proxy (aggregate
  target size - 5)      312.05                     24.77

Weighted Aggregated
  Proxy (aggregate
  target size - 5)      522.81                      56.70
                                    241

-------
                                  Appendix 7.A
The column and row designations in each of these appendix tables are as
follows (table columns are models, rows are summary measures):
  COLUMNS
  1.  SHADOW* - Model estimated using shadow prices (PSF,  PSC,  PSU).
  2o  OBSEHIDP - Model estimated using observed prices (PTF,  PTC,  PTU).
  3  POPAVPTCf - Model  estimated  using population  average density proxy
           measures for price (POGGP, POGGC, PDGGU).
  4.  SMPAVPXY.  - Model  estimated  using  sample average  density proxy
           measures for price (PDPGF, PDPGC, PDPGU).
  5"19. AGGP501J  -  Models estimated using unweighted proxies for price
           from  AGGREG  subroutine, with  ie{1B2,3}  and Je{U2,3.4.5}
           denoting*  respectively,  the  inde* for  the i-th   target
           aggregation  scheme  and  the  J-th  loop  for  that  target.
           1-7,2,3 corresponds- to target aggregation densities  of  17, 5,
           and 10, respectively.
  20-34.  AGGP601J   Models  estimated  using weighted  proxies for price
           from  AGGREG  subroutine, with  ie{1,2,3J  and je{1 ,.2,3,4,5}
           denoting,  respectively,  the  index for  the i-th   target
           aggregation scheme and the J-th loop for that targe to
  -ROWS
  1.  DIT(XTX)  The determinant of the X'X matrix,  used here as  a check
           for  singularity.     In some   instances*  small  -values  of
           Det(X X), which  largely reflect cases of small variation in
           one or more of  the regressors, indicate that the  tabulated
                                                                    T
           parameter estimates should be  ignored.   Values of Det(X X)
           smaller than approximately l.OE+25 have- proven troubles one in
           practice.
                                    242

-------
 2.   MEG CHEK - A  check  for  own-price, or own-proxy-price, negativity
          of  the  estimated demand  functions,  -1  if negative,  -0  if
          positive.
                           A   A    <*_        **
 3H7.  Bij -  The estimated  8-(S01,8Q2	815) parameters.-       -  -
 18.  FISHDO - Sample mean  of  actual  PREFISHD (i.e. pre-policy fishing
          participation),  generated by the optimization  algorithm  as
          the pre-policy  participation  rate.
 19.  FISHDOP  -  Sample  mean  of  predicted  PREFISHD,  from  estimated
          participation equations.
 20.  PISHDt * Sample mean of actual PSTFISHD (i.e. post-policy fishing
          participation),  generated by the optimization  algorithm  as
          the post-policy participation rate.
 21.  FISHDtP  -  Sample  mean  of  predicted  PSTFISHD,  from  estimated
          participation equations.
 22.  DFISH -  Sample mean  difference of  actual (PSTFISHD-PREFISHD).
 23.  DFISBP - Sample mean difference of predicted  (PSTFISHD-PREFISHD).
 24.  CV - Sample mean of  actual compensating variation (defined here to
          be  positive for post-*policy welfare improvements).
 25*  EV  Sample mean of actual  equivalent  variation (defined here to
          be  positive for post-policy welfare improvements).
 26.  MCSPRBD    Sample mean  predicted Mar shall ian  consumer  surplus,
	     defined  only  for models estimated  in  alidduv*or~~ub3ervU
          prices, and --999999 otherwise.
 27.  PLQVPRED -   Sample   mean  predicted  partial  Laspeyres  welfare
          estimate.
 28.  PPQVPRED - Sample mean predicted, partial Paasche welfare estimate.
 29.  PHCSPRED -   Sample   mean  predicted  partial  Harberger  welfare
          estimate.
 30.  ACS1PRED -  Sample  mean  predicted  in dividual-specific  average
          surplus welfare estimate.
 31.  ACS2PRED -  Sample mean predicted  sample-specific  average surplus
          welfare estimate.
 32^End. The prefixes of these rownames  indicate  the comparison tests
          used:
                                  243

-------
MPE  Mean prediction error.
MAPE - Mean absolute prediction error.
(J1 - Theil'a U1 statistic.
02 - Theil's U2 statistic.

    Affixed  to these are  the descriptors for the comparisons  of
     interest:

DPS&-  Actual   versus  predicted  difference  between  pre-  and
     post-policy fisnin; participation,
PLQC- Predicted Laspeyres versus actual CV welfare measures.
PLQB- Predicted Laspeyres versus actual EV welfare measures,
PPQC- Predicted. Paasche versus actual CV welfare measures.
PPQB- Predicted Paasche versus actual EV welfare measures.
PHCC- Predicted Harberger versus actual CV welfare measures.
PHCE- Predicted Harberger versus actual EV welfare measures.
AC1O Predicted  individual-specific  average surplus  versus actual
     CV welfare measures.,
ACTS* Predicted  individual-specific  average surplus  versus actual
     EV welfare measures.
                                                 
AC2C- Predicted  sample-specific average surplus  versus actual  CV
     welfare measures.
AC2E- Predicted  sample-specific average surplus  versus actual  EV
     welfare measures.
                              244

-------
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                                                              AGGP6033
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                         ACGP6035
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                                        -2.60310
                                         |5. 3059
                                                              -WJ.9586
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            -6.75869
            18.2316
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             2.38037
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                                                                                                                         (continued)

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 6.40137
 6.00639
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 6.24707
 8.34855
 7.21944
0.943628
0.697243
0.929729
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0.957749
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 1.01225
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0.982369
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 6.2017)
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 6.0M77

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             76.26166
37.6551
12.9835


0.76133

1808599
                                                                                                    354.61
             -4.14151
              14.3999
              -6.2791
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                                      37.6592
                        0.761233
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                         6.56217
                         354.025
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                          17.012
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           -6.8232*1
           -41.7993
           18.4055
 l|.53?2
 45.6087
 29.1623
0.120055
 0,9518.1
0.929593
            3.34634
            489.531
            160.506
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                          488.641
                          160.215
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                          15.0684
                          35.4869
                          24.7696
                                                                                                                       (continued)

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0.86*92?
0.99)002
0.722756
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 26.1603
 1.14524
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 16.2169
 19.0600
 36.1
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0.691937
0.722765
 26.1176
 8,14469
 6,65661*
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 6.720940
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 -5.29203
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  16.7257
 0.656643
 0.692802
 6.729037
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6.695013
0.653690
0.406235
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0.695901
0.653654
0.606359
 1.10745
 B.77617
 1.03057
A6CP5015
ACCP5034
ACPI'6023


0.727356
0.754924
0.730504
             1.21002
               3.194
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            6.12623
            26.3304
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0.754900
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 3.19110
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     1.08052
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      33.871
    
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    '171,268
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     171.268
     116.803
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    -37.9298
     72.9911
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      11.08?
     96.2714
     60.6668
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    0,912934
    1,98617?
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     1.14909
     1.9174?
    31.9*1)
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    -134.192
    
     30.1149
     96.2*02
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    U.910262
    0,030269
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     1.24966
     92.2960
    0.991478
     5144.29
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     4264.11
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     99.1350
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     9144.22
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                -49.826
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    2196.9
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    6.862949
    5344.83
    296.233
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    72.1687
    191.928
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                                                                                                                          (continued)
    

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                                                                                         -0.0066106
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                                                                -6.66?i?oi2
                                                                -0.00635232
                                                                -0.00645789
    f I VUO
                     
                  ilB.165
                  UU.155
                  lift. 165
                   f .
                    .iA
                 I.U-.IW
                    III .1
    
                    l-'l-.Ti
                                          III.U.%
                                            .!/>>
                                          lit. 749
                                          MO.
                                             11.165
                                            110.165
                                            IIH.I6S
    
                                            I M. 165
                                            !|fl>l6*
                                            11C.165
                                                      I.70.5I6
                  J.'".SM.
                                            14.165
    
                                           118.'165
    
                                           119.759
                                                      lie
                                           120.f16
                                           120.&I6
                                           (20.516
                                           120.5.
                               in-lfl
                               111.165
                               111.165
                               111.165
    
                               IIM.I65
                                 1.165
                               116.169
    
                               124.516
                               120.516
                                                                              I2'I.M6
                               118.165
                               IIH.165
                               110.165
    
                               IIU.I65
                               (I&.I65
                               1(6.165
                               IIP.165
    
                               120.516
                               120.516
                               |?0.5I6
                               120.516
                                                                                                165
                                                                                            118.165
                                                                                          II IS 165
                                                                                          j.?fc5
                                                                                          118.165
                                                                                          120.516
                                                                                          120. ^16
                                                                                          120,516
                                              nf,|65
                                              8,1(5
                                            MO. 165
    
    
                                            IIH.165
    
                                              fli65
                                                                                                        120.516
                                                                                                        120.516
                                                                                                        120.516
                                                                                                                          (continued)
    

    -------
                                                               Tbl  4.9.
         RCfSUM
         r i sinn i'
         OF I Till
           I 'HI*
                           POPAVPIV
         cv
                     *r,r,reo24
                      l?3.443
                      114.086
                      119.225
    2.3*131
    2.15m
    
     ?.. 21
    5.27861
    I.OM178
    
    
    V!5H4
                122.412
                 216,69
                119.021
                2.35131
    
                2.30816
               O.Mf652?
    
                39.7544
                            117.3
                           -22.7235
                            110.074
      2.15111
    
    
    1.401801
      31.1584
    
      39I75A4
                                       SltPAVI'IV
                                       A6GP1024
                                       AGGP60I*
                  Il9.46
                  1*1.091
                  121.137
    
                  2.35)37
                   7 .  f , 
                                                          2,3f|31
                                                          2.3f|37
                                                          2.92646
                                                          d;4l505
                                                          4.97263
                                                           31.7504
                                                           39.75*4
               AGGP60I4
               AGGP6033
    
                161.446
                119.2*4
                123.606
                 IIA.71
    
                MM?
                2.35|31
                2.35137
                43.6011
                  09934
                39.15P4
                39.1564
                39.1564
                                         AGGP50I2
                                         ACCP5031
                                         4GGC60I5
                S3 .181*
                120.676
                120.674
                  76.18
                2.35|31
                2.15137
    
               -64.1832
    J
                39.7504
                39.7504
                39.7584
               AGGP50I3
               AGLP5032
               AGCP6021
               AGGP6Q35
    
                111.951
                119.2Q|
                122.63)
                118.219
    
                2.35137
                2.35|3i
                2.35131
    
               -P.21361
                             0.605003    -39.948B   0.0540293
                39.1584
                39.1584
                39.1584
                                       AGGP50|4
                                       AGGP5033
                                       AGCP6022
                 119.464
                 114.898
                 (22,229
                                                                                         2..35U1
    
                                                                                         2^35131
                 1.29975
                -3.26646
                 4.0648}
                 39.7584
                 39.7584
                 39.7584
                            AGGP50I5
                            AGGP5034
                            AGGP6023
     114.726
     92.6213
     118.713
                             ?.?m
                             2.35137
                             2.35137
    -3.43866
    -21
                                                                                                     39.15P4
                                                                                                     39.7564
                                                                                                     39.7504
         MC "il'K HI
         P9VI'RFO
                       9.128
                       J9.128
     19.028
    
    V.H3I2
    -r, 90999
    l.Ollhl
    1.04')2t
    
     7,44'.6
                      2.51916
                      >.30135
    
                      l'.'-271
                 V.-.83*
    
                 JV.B28
                D.664A
                |5.06'I6
                279.626
                 /.47H2
                             19.028
                             19.874
                             11.028
                            -991019
      -110999
    
     -tA.AMV
     -7.4A2I5
     -62.1946
      II .4619.
     -11.
     -6.HM2
                           -|5.?77'J
                           -*.. 71 nc
                                               21.
                           -.".041
                           -.II.
                   19.828
                   19.H28
    -999999
    .i)i)')9
    -------
                                                            T.bl
    lUMfllFSM
    tllf "ill
    iirri'l.oc
    HIPl-K.
      ?i'i ir.
    ICf
     .7466
                 V'.'MII
                 '.4.1004
    
                 4?'.3114
                 -". 3111
    
                 - ( .  IM
                            Ar.CF.5a22
                            AOGI-6011
                            ACUI'6039
                                         *r.CP60l2
                                         AiT.GP603l
                             '6.61)94
                             -I.474H6
                              4.
                            0.632303
    
                            o'67||r.4
                              l.l
                              -2I..SI2
                              0.1B33
                                 SOU
                               lu .
                                         -46) .062
                            -11.0939
    
                            -.><>. 76.16
                             0.99m
                             tt.UU*
                             11.7149
    
                            0.771349
                              O.T175
                             3.94114
    
                             16.070^
                             6.3949
                             -47.2403
                             -101.95J
                             -fl.3-1049
                             60.7466
    
                             ri (1.8*3*2
    
                             1.471201
                                         0.72.'619
                                          0.61162
    
                                          8.14115
    SNPAVPXV
    
    Ar.GI'6013
                                                                 AGCP^Otl
                                          31.1)64
                                          74.3427
                                          10.1466
    -I..9MU
     2.63126
    
     3.24117
     4.^6915
     6.42646
    0.644691
    Q.22364
    0.713611
     1.11067
     1.37461
    -10.8841
    -21.0155
    -36.0020
       41.14
     M.11M
    
    0.6ril62
                                         .1*'0*2
    
                                          '1.0114
                                         - t0.r-'i)o
    
                                         -l^.f1/4
    ir-GP6084
    4GGP6033
    
     MI6.i2
     21.9946
     118.2H
                                                                  41.3363
     9.06956
    -1.14631
    
     41.3646
     ^.6J6H
      4.92
    
    0.639626
                                                     0.452258
                                                     6.14dli9
                                                                  1,61195
                                                                   1.2361
     102.013
     -31.842
    -24.8469
    14.6339
    
     113.134
     f.4.1101
                                                                  4H.0373
                                                                 O.P66676
                  1,00(69
    
                  102.POT
                 -11.2I6'
                 -/4.2K.5
                AGGP5012
                4GGP503I
                                                                             AGGI'5013
    4GGP6034
    
       -1636
     I89
                             -)7.6490
                            ACGP5014
                            AG6P5033
                            AGGP6022
                                                      91.6212
                                                     -63.0022
                                                       103.29
                                                                                                        1.1139
     4.60431
     1.9)618
    0.692326
    0.145653
    p.flfc?M
                                          1,26654
                                          2.31262
                                         0.991215
     35.144
    -46.0111
    25.9212
     46.6)56
     96.6311
     41.1564
                C>.6.61994
                6 .6 32 38 8
                0.632051
                                                       1.0041
                                                      1.09333
                                                     (1.921511
                             -35.2136
                             -46. CHOI
                             -25.9966
                AGGP501S
                AGGP5034
                AGGP6023
                 -87.376
                -626.591
                 13.9335
                5.19003
                -2i.06'69
                1.60303
     6.21252
     30.9411
     i.?$)2i
                                                                                                                  0.149041
                                                                                                                  0.903551
                                           2.1055
                                          30*6492
                                         0.955851
    48.9227
    118.966
     -36.413
     91.1389
     131.283
     42.4639
                0,64.9605
                0.626037
                0.939976
                              I.IOM9
    
                             0.9ir>634
                             48.992)
                             -119.056
                             -33.4P26
                                                                                                                         (oontlnued)
    

    -------
                                                                   I '   I
                                                                   M.
    RFC VIM
    tui>ri>i.oi
    upcni'oc
    uipi"ir.
    MPfilM'Of
    IIMTi'Pjr.
     siunnNp
    r,cpM>2i
    4FI.P9036
                            PPI'AVPXIf
     2I-.M162
     *4.I45|
     41.7466
    :09|
    0.734146
                ft.
                n.ooiooi
                 U.^577
                 4>,03M
                0.764248
                0.12M>75
                            Af.crno??
     fl.0417
    
     46,5205
    
    fV?62|
    0.7I'9|26
    0.057442
    
    0.426751
                -92.2802
                -26.4817
    32..? 002
    49.07'..
                             102.02)
                            fl,"?:?!!*
    
                            (l!72261.f
      1.14164
    -lauB
     0.616739
     -53.6715
     46.0164
     -11.794}
     -11.42*8
      55.0674
      ii'.74i
                            O.DRAI76
                            0.1067ll
                0.
    -V.W4.
    
    -171*091
                 0.97706
    -12.1491V
    
     |94.l?4
                 1.11174
    
                 o!"2192
    
                ."*?'*'
    
                11,6631
                 12 .
                 4V.I37)
                 .'>. l I
                                        Ar.P60l3
     48.2017
     51.22A9
    
    0,695314.
                            0.750206
    
                            O.740211
                                          ft 7. 12,31.
                             ^.''4111
                             '.T677.
                                    S
                                    J
    0.676966
    
    -32.3136
                                         40.9401
                                         46.41*2
                                         46.7011
                            0.73,1061
                            O.A70U44
                            O.P77991
                            0.7M0337
                                        0. <>49417
                                         -12.40)3
                             -26.7314
    
                              41.0064
                              46.47(44
                                         U.CP6P14
                                         ACPP60.1)
                                                                  ^4.^>9
                                                                  4,6469
                                                                  40.9931
    0,791925
    Q. 795006
    0.666762
    
     2.04376
    0,962565
    0.69277!
     1.60657
    
     76.9062
    -32.5494
    -?*"*?
                                                     Hi. 119
                                                     2,1669
                                                      41.914
                                         1!. 46429
                                        0.616143
                                        0.021733
                                        0.664410
                0.077136
    
                  ' 1.0022
    
                  70.6366
                 -26.7665
                 -V.*i405
    
                 116.326
                 *<2.2422
                 47.9736
                 46.6541
                                                    O.II2I9I7
                                                    0.84Mf-
                                                   AGGP5012
                                                   4GGP503I
                                                   4GGP601S
                                                   AGCP6034
    
                                                      ?42.5
                                                    50.7061
                                                    43.7146
                                                    194.199
    
    
                                                   "Will
                                                   0.671096
                                                    P.84059
    
                                                    7,11845
                                                    9.45199
    
                                                    -209.126
                                                    -21.2606
                                                                              12.7645
                                                                              lU.
                                                                      ^ei
                                                      20<>.126
                                                      48.7614
                                                      42.9963
                                                   9.657699
                                                   0.760165
                                                   0.691156
                                                    0.81299
    
                                                    5.51696
                                                   0,660159
                                                   1.942436
                                                    4,29975
    
                                                   -209.196
                 -12.6141
                 -MI.052
    
                 209.196
                 48.6161
                 41,36.17
                 161.797
                                                     OI.H0196
                                                     O.f>')1476
    AGGP50I1
    AGGP6012
    AGGP602I
    AGGP6035
    
     47.1613
     47.6977
     47.2596
     48.5419
    
    0.867961
     6.66467
    0.615611
    0.689164
    
    0.961416
    0.972341
     6.69403
     1.02612
                                                                            -39.3411
                                                                            -17.2044
                                                                            -27,343
                                                                  45,6738
                                                                  46.3209
                                                                  45.9091
                                                     0.685901
                                                     0.683275
                                                     0.640603
                                                     0.904327
    
                                                     0.961126
                                                     0.974124
                                                     0.907791
                                                       1.0193
    
                                                     -39.4129
                                                      -1}.2U
                                                     -27.4126
                                                     -16,0056
    
                                                      45.9365
                                                      46.1978
                                                    . 4*>.962
                                                      47,0759
    
                                                     II.(if! 6021
                                                     0.1103421
                                                     O.M077I
                                                      V.V044I
                                                                             AC6P50I4
                                                                             ACCP5031
                                                                              46. (999
                                                                              56.6954
                                                                               47.222
                                                                                                     6.612503
                                                                                                     0.032233
                                                                                          1.00411
                                                                                           .09306
                                                                              35.6977
                                                                              46,6631
                                                                              26.1664
                                                      45.5055
                                                      55.6324
                                                     .45.6996
                                                     0.896163
                                                     0,651476
                                                     0.654982
                                           1.6669
                                          1.06912
                                        -35.9673
                                        -46.73*7
                                                                                          45.5699
                                                                                          55.69P4
                                                                                          45.9633
                                                                             C, 696293
                                                                             0.6^1595
                                                                              0,t!>M4
                                                                                         AG6P50I5
                                                                                         AG6P5034
                                                                                         AGGP6023
                                                     57.7041
                                                     111.152
                                                     42.5276
                                                                                         0.8497|8
                                                                                         6.826072
                                                                                         Q.940039
                                                                                                      1.10568
                                                                                                      2.67159
                                                                                         -47.4266
                                                                                         -106.047
                                                                                         -16.6126
                                                      64.6661
                                                      116.126
                                                      42.0141
                                                     0.667141
                                                     0.626662
                                                     0.949194
                                                                                          i.oflloe
                                                                                          2.41747
                                                                                         0.901914
                            -1r.?*
                            -i06.116
                            -16.7024
                                                                                          54.7535
                                                                                          ||6.397
                                                                                          42.0H01
                                                                 0.667251
                                                                  0.^2674
                                                                 0.949247
                                                                                                                         (oou^lnuad)
    

    -------
    RFCSUM
    Mi'fii'iir.c
    u_p"r.f.
    !" 4L If
                 MUtHIHP
                0, 76052
                -.'!.; 306
                -'1.0411
                -16. 9925
                -If., 1 32*
                 12.0211
    
                O.MM34
                0.14*664
                -20.11102
                -.'1.1109
                 17. .!'
                -v.,'022
                 I."?. HSftS
                 0.927??
    
                O.f
                  "i . ?><
                  !< .*')
                  '. '?, J
                            0,672}!%
                            O.)770|9
                              ?!?. Oil
                             P0.026J
                             0.t'l>OA72
                             0.6} 1322
                            n.!.3752^
                            0.9)9771
                              O.OR'>7
                              230.139
                                   !
                              -?>.!*.
                              IS* rt.*<
                              ' . 1 !'.
                                         PRPAVPXV
                                         4r,GPi023
                                         AHCP/.OI2
                                          ii.iim
    0.79-19 3.1
    0.71990%
                                          l.lft.Sl
                                          I.IJ'O*
                                          J.43232
                                         -"ft. 9* 51
                                         -10.9202
                                         -II1*. 267
                                                     AKCffrfll)
                                                     0.106 |i9
                                                     0.9^J*IH
                                                       1.00*4?
                -10.6499
                -36.3U9S
                                                         47. on
                                                      4I.4IM
                                                      >0.|909
    
                                                     O.I 1.1089
                                                      n.A6B6|2
                                                      b^i.UH
    
                                                      0.792173
                                                      0.946042
                                                        1.00*3
                 -31.6786
                 -30.M55
                 -16,3791
                 -2S.S079
    
                 41,1146
                                                      MI.24A3
    
                                                     O.?|1244
                                                     0.1i>623
                 -HI.
                   llo..v.
                 -):i,4U44
                 -!, IS
                               I.-.79I
                             0.976980
                              0.90545
                              (.00211
                                                                   U4.U9
                                                                    47, Jit
    
                                                                  0.?I2?9
                                                                    0.8096
                                                                  0.807991
                              1.80098
                             0.979549
                             V.89U789
                                                                   90.4199
                                                                  -31.9152
                                                                  -21.49IS
                                                                   124,184
                              48.109?
                              47.3724
    
                             0.i7IP92
                             0.80f>72&
                             0. WOP If 4
                                                                   l.7l)7fm
                                                                  ."< 4
                                                                   1.004lb
    
                                                                   604.746
                                         ACCP5012
                                         AGGP5031
                                                                  ICCP6033    AGGP6034
                                                                               9.12726
                                                                              1.860427
                                                                              0.942486
                                                                               4.29286
    
                                                                              -221.878
                                                                              -28.6*04'
                                                                              92 .0818
                                                                              "157.449
    
                                                                               221.878
                                                                               49,7093
                                                                                           ACGP50I3
                                                                                           ACCP5032
                                                                                           ACCP602I
                                                                                           ACCP6035
    
                                                                                            0.98172
                                                                                           0.914177
                                                                                           0.907872
                                                                                            1,01921
    
                                                                                           39.3028
                                                                                           -96.95*1
                                                                                           26.131)
                                                        46.486
                                                       46.9159
     181.418
    
    0.861829
    0.143354
    9.812054
    6.8U6B9
    
     6.30506
    a.851922
     0.9)767
      4 .8595
      47.74)
    
    0.876712
    0.873836
    0.827799
    0.896614
    
    0.981447
    0.973075
    0.900698
     1.02264
                                                                              221.948
                                                                                -31.72
                                                                              "32.1514
                                                                              157.919
    
                                                                               J21.944
                                                                               49.7576
                                                                               43.3066
                                                                               181.487
                                                                              4.N121BI
                                                                              0.836668
    
                                                                               6.?9397
                                                                              0.937721
                                                                               4.W132
    
                                                                              -1913.79
                                                                                1 10.62
                                                                               9.12136
                                                                              -tli 1.602
                 -39.3724
                 -37.0247
                 -26.2009
                 -37.8271
    
                 46.5508
                 47,0428
                  46.608
                 47,8084
    
                 0.87684
                 6.011993
                 0. H2 79 76
                 0.896766
    
                 0.981434
                 O.V73I29
                 0. 960? 82
                 1.022*>4
    
                 2.*9974
                 -86.941
                                                                                           -t-7.9409
    AGGPS014
    AGGP5033
    AGGPt022
    1.00092
    g. 06892
    0.931172
    -47l))71
    27.0569
    46.0706
    *J??4?
    46.3*9
    0.889944
    0.84161
    0.843324
    1.00237
    1.08078
    0,926154
    -35.5905
    -47.4067
    -27.1264
    46.1344
    57.2929
    46.5926
    0.890083.
    6.84il3i
    0.843492
    1.00239
    1.06055
    0.92622
    -132.78
    30.4465
    -2U.9795
    AGGP50I5
    AGGP5034
    AGGP6023
    i coeoej :
    2.41482
    0.981914
    -iii.sia
    -38.5229
    56.1636
    a). 805
    .'i)il
    0.858099
    0.825964
    0.944558
    1.09323
    2.63896
    0.9K0356
    48.2438
    -112.587
    . -38.5925
    96.I28B 	 "" - 	 -"
    123.874
    42.30J9
    0.858211
    6.8*66!
    0.944619
    1.093
    2.63583
    0.9803S&
    -24.4337
    -1301.58
    -2H.60I3
                                                                                                                           (oouttnued)
    

    -------
                                                            TbU
    IIITAf. If
    
    
    
      UJ
    
      VJ1
    UIM'le
                 117.636
                 106.611
                1.4)1049
                0.42491?
                 140.5
    
                -.71.4259
                 117.
    4'i7.671
    
     |i'o!l9
                  44.371
                -U..MI?
                  176.
                i.i.'.i.isn
                            no SIR wop
                            AfCr60||
                            ArCI'60?*
                             1)1,029
                            0.937079
                            0.3I28Q4
                             t'O'-'rV"?
                             11*11.4
                             .'03.511
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    XGGP60I9
    4GCP6Q34
    110,9)6
    n 1,736
    1.734
    119.736
    
    I12*,9ii
    122.797
    hrt.tQ?
    3.AOOB?
    J'IPfl8?
    i, ooftA ?
    3.000B?
    12.8030
    3.4|?31
    6.07006
    -|i2dH
    59.6720
    55.6720
    95.6720
    99.6720
    99.6949
    55.6"49
    95.6949
    55.6'49
    
    -414^94
    -999999
    -999999
    -999999
    
    39.1730
    -5.694S)
    13.370?
    -16.2-ill
    32.7752
    4.16774
    II.IQ34
    -I3.60M
    
    3*.'74*
    rS . ? 33 1 4
    12 ..?R6I
    -I4.9?96
    
    
    ACGP5013
    A(iGi*5032
    ACGP602I
    AGGP6035
    119.7.16
    119.736
    119.736
    119.736
    
    139.266
    114.565
    124.072
    3.00007
    3,0088?
    J. 00661
    3.00887
    -17.8608
    22.5385
    -2.16182
    I6.4)fi
    55.6728
    59.6720
    99.6720
    55.6728
    55.6949
    55.6949
    59.6949
    55.6949
    
    -499999
    -999999
    -999999
    999999
    
    -62.131
    9.|}48
    0.10449
    3>.529?
    -!>I.9Q25
    t>?.li?25
    -6.711071
    29.7263
    
    -!>?.U*(iB
    <3.*>2|6
    7.4426
    32.628
    
    
    AGGP50I4
    AGGPSQ33
    ACGP6022
    
    119, 7l6
    119.736
    119.736
    
    
    },6794i*|3
    113.76
    118.41
    
    3.0000?
    3.0000?
    J.OOflftt
    
    |.6?94(*I3
    -2.96746
    1.^0255
    
    55,6728
    59.6720
    55.6728
    
    59.6944
    95.6949
    95.6949
    
    
    -444444
    -999999
    -999999
    
    
    9,e620El3
    -IB. 4411
    1.94109
    
    4.t542f*l3
    -15.434
    1.62404
    
    
    ^ . )DB5C *I 3
    -16.4406
    I.?t256
    
    
    
    AG6P5015 , ;.
    AGGP*)034 ;
    AGGP6023
    1
    ii9.?36: ' " " 	 ~
    119.736
    119. 736
    
    
    109.924
    114.426 	
    119.43 ,
    .
    '
    3.00007
    3.0088?
    ' J.ooee) 	 	
    
    -6.00291
    -2.36136
    2.70253 '
    t
    95.6720
    59.6720 ;
    55.6720
    
    .
    55.6949
    59.6949
    59.6949
    
    \
    -449494 	 	 " 	
    -999999
    -999999
    '
    
    1
    -11.3856 '
    4.05486 	 	 ?
    12.5897
    i
    -9.62584 '
    -7.97586 .1
    (O.sJii \
    \
    
    -10.4557
    8.31537 |
    11.9615
    
    
    (oontlnuad)
    

    -------
                                                          TabU   4.10.
                                                                                                                                                  I
                                                                                                                                                  I v
                                                                                                                                                  I '
                                                                                                                                                  ;
     IV)
     o
    hPEPLOC
    MAPFPI.OC
    UIPIOC
    uzpiqc
     sitAiuiMp
    r,r.p.o2
    KfCSlIM
    ACSIPHH*     22.9214
                -I0.95A9
                 24.9504
                            AT.CP601I
                                               I     Slip A VPIV
    
                                        Ar.f.Pf.012     Af,l.p(.013    ACGPf.014
                                                                            f.l. '(>?
                            -I32.IM
                             II,
      7.1575
    -III.
                             9.92129
    HPEUfSH    0.0251319   O.0273031
                             2.01011
    0.158416
     7.36426
     IA.M51
                             If.914I
    
    
                            0.019609
    0,7M36?
    0.600063
     1.416*4
     2,4*941
     9.|*i55|
    
    39;0049
    -61.4326
                -..
    3'J. 004 9
    to;?9C5
    77.0488
    91.4729
                          0,00i|23(.!>
                 5.97862
                0.9&I961
    
                -30.9916
                            -52.3856
    
                             3U.9916
                                         44.1H4I
                                          93.634
                                         114.6?!
                                         77.|HI4
                                        -I4.1tf.4
                                          429.91
                                                                -450.708
                                                     34.64.1?    -461.521
                           30I.?*>|
                            60.4096
    
                            2.74619
    
                           iVUizi
                                         44S.S99
                                         4?.01|9
    
                                        0,770939
                                        tiHW
                                         40.0716
                                                                40.3492
                                         6. 19005
                                         14.9636
                                        it. 702002
    
                                        0.639078
     tt.60407
     41.6386
     li.oiii
    
    0.631749
    0.745571
    01024590
    0.64f06l
                                                                 48.S492
                                                                  &.13M
                                                     12.87
    
                                                    0,906190
                                                    0,004709
                                         4.63091
                                         i?. ,1764^24
                 1,2015$
    
                -205 ."999
                -59.0276
                0.494625
                0.892BI1
                   r.032i
                 103.460
                 Ml.8314
    
                0.4943SS
                0.71967?
                             67,1301     64.24H8
                                                    -W.M14
    
                                                     205.999
                             74.69M
    
                             0.84869
                              0.7191
                                                     152.419
                                                      90.471
    
                                                    0.869699
                                                    0,8it9
                   05,09
    
                0.01349g
                0.92961J
                       9
    0,l>592f>|
    
    0.461341
                 i.2179?
                            0.4.L30I7
                             6.04914
                              I.33M3
    >,??4?|9
    
    0.93U742
     0.1939I
     1.1199 76
    0.443206
                                        0.940868
                                         1.04026
                                           ,44654
                                                                  8. 0090 B
                            4CKP6015
                            4PCP6014
    
                             124.273
                            -li.9045
                             52,1696
                                                                            10.9742
                                                                             62 .Q954
                                                                            -16.9691
    
                                                                              9.6749
                                                                             5.068
                                                                            -4., 6296)
    
                                                                             11.6621
                                                                             20.0261
                                                                             Si*?l8a
                                                                            4.812747
                                                                            0.792377
                                                                            0.63363?
                                                                 9.17914
                                                                  2.5909
                                                                 6.10951
    
                                                                 -16.499
                                                                -61.1711
                                                                -42.2941
                                                                71,9101
                                                                 61.2005
                                                                 IOI.4SI
    
                                                                9.696995
                                                                0.731314
                                                                0.165631
                                                                 .49357
                                                                 . 17926
                                                               O.*964?6|
                                                                            ACCP5U13
                                                                            ACf.piOl?
                                                                            ACCP6021
                                                                            ACQP6035
    
                                                                             -202.37
                                                                             220.71?
                                                                            -20.3909
                                                                             126,996
    
                                                                            -102.595
                                                                             210.416
                                                                            -22.100?
                                                                                         AI'CPtOI4
                                                                                         ACCP&OJ1
                                                                                      |.?090M14
                                                                                       -10.639?
                                                                                        11.111?
                                                                                                   lt?69M|4.
                                                                                                     -30.1369
                                                                                                       17.20)
                                                                             20.069?   |,6794^ll
    
                                                                             4^7069
                                                                             ?.9|846
                                                                            20.669?   |,t?94iU
                                                                            21.6230      15.9071
                                                                             i|M
                                                                            106060
                                                    0.051916
                                                    0.726900
                                                    0.04V449
                                                    0.593702
                                                                                                1
                                                                                         0.771752
                                                                                         Q.60609S
                                                                                          9,05244
                                                                                          1. 24101
                                                                                           4.6943
                                                    -117.604
                                                      |1.49b
                                                    -63.7771
                                                      61.649
                                                     79.6705
    
                                                    0.016000
                                                    0.619014
                                                     6.'96216
                                                                              1.69577
                                                                            0,943486
                                                                                                     -74,1199
                                                                                                     -51.731?
                                                  5.5628f11
                                                  " ~94.424i
                                                      76.161?
                                                                                                     0.034114
                                                                                                     0.634tOJ
                                                                                                   1.2908(21,
                                                                                                      1.19472
                                                                                        0.003085
                                                                                                    ACCP50I&
                                                                                                    4CCP5034
                                                                                                    ACCP6023
                                                                                                     -54.4051
                                                                                                     -20.7419
                                                                                                      41.1605,
                                                                                                                 -21.52
                                                                                                                  27,6284
                                                                 -9.8U76
                                                                 -StMPM
                                                                -0.306340
                                                                                                      2|t1669
                                                                                                      5.05514
                                                                                                      10.^429
                                                                                                    0.023422
                                                                                                    0.010950
                                                                                                    0.664911
                                                                                                     7.05630
                                                                                                     1.19174
                                                                                                     M266?
                                                                                                                  67.0501
                                                                                                                  64,727?
                                                                                                                  43.0631
                                                                                                                    0.942
                                                                                                                    19109
                                                                                                                  75.8192
                                                                                                    0.604981
                                                                                                    0.941666
                                                                                                     6:2h64
                                                                  1.299?!
                                                                  1.05454
                                                                 0.970909
                                                                                                                           (oontlnuad)
    

    -------
                                                       TabU  A. 10.
    ft<-csim
    MPEPI Of
     SIIAOOMP
                                        I1PWPXI
    UIPIHF
    MPCPPOC
     U)
     ro
    MPCPPGt
                            Af.CP60tl
    Ar,r.ft024
    
    -VI. 02 71
                -so. n?)
                'ft ft. hOt 6
                 V>,02
                 70 .4|
    -19.3U-H
    -44.9015
    -99.nl1>
    -S3.4.07?
    
    
     oo.'biu?
      10).49
                            -.U.3S26
                             ^27.009
                             n9.af.66
      91.495
    
    0.494578
    0.09*1)6
    O.III4J1
    0,059))?
    
    A,46))95
      1.1279
    0.904709
                JsfKrfVr.
                0.654)90
    -50.I735
    -62,08|6
    -51.00(2
                 50,1735
                 6A.1205
                 7J.527?
                O.A444V)
                0.906B09
                (1.837089
                0.07,196)
    
                . II52S 7 7
     1.04932
    
    0.9BJ764
    
      -50.1?
    
    -zlooai
      74.209?
    
     0.04A915
     **!..
     1,114)6
    0.961941
     1.1445?
    
    -22,9198
    -60.462?
    -44 15015
    -6.9,29*8
    
     121.04?
     98.5169
     76.9201
     9)!64?7
    ACCP50I3
    ACCP5012
    AtCI'6021
    AfCI-6035
    
    r||7.626
     13.471.9
    -i1.?9V4
    r20.l652
    
     )l?,826
     V7.6025
     6).86?)
     75.8711
    
    0,616779
    0,6)9626
    6.952166
                                                                                        AK.P6034
                                                                                        AUPt022
                                                                                        -74.1421
                                                                                        -53.75)9
                                                                                         94.4344
                                                                                                     0.6)47)5
                                                                                      |.2902fc2)
                                                                                         1*19471
                                                                            0.94))?)
                                                                             1,033)4
                                                                            0.66)06?
                -107.655  <
                 2.1996)    -71.1068
                -62.45)5    -54.0467
                                                                              10?,655
                                                                              90.0946
                                                                                         67.9336
                                                                             72.418)
    
                                                                            0.829176
                                                                            0.667655
                                                                            0.959282
                                                                            0.?6))65
    0.916029
     1.02726
    0.69)500
     I  .
    -107.67?
    
    -ls!&l
                                                                                        0.85121?
                                                                                        0.655064
                                                                                      9.0)5?|*22
                                                                                      "l~; 149*5
                                                                                      4.6542it|)
                                                                                                     -7|,|269
                                                                                                     -54, .
                                                                                            0)09
                                                                  )0?.6?7  4.t542f|)
                                                                  90.0924     67.94)7
                                                                  62,5323     72.924?
                                                                                        ACCI'5034
                                                                                        ACCP602)
                                                                                                     -64.7496
                                                                                                     -4). 1052,
                                                                                                                  75.6.159
                                                                                                    0.605015
                                                                                                    0,9416?)
                                                                                                     0.01166
                                          |.299t6
                                          1.0545)
                            -65.1966
                            -63.2466
                            -45.1)95
                                101.6
                             64.7406
                             72,4025
                            0.621609
                            0.950061
                            0.6)5055
     l:Ii!R
    0.9 (.9401
    -65.2200
    -64.2706
    45.1616
                                                                              101.615
                                                                              64.7626
                                                                              72.4992
                                                                                                                         (ooatlnuad)
    

    -------
                                                                  A. tO.
    RECSUM
    UIPP'IF
    MPfotirc
    HAPEPHCC
     K
    uii'iicr.
    u2Piir.r.
    MMrniCF.
    Ill PMC F
     MIAIUIHP
    4QGP502 i
    IIP UK VHP
                            AFCP601I
                0.1144493
                0.90*814
                            o.oi.ao'i)
                n. '174021
    1,
     l.l*?2a
    
    -44.V192
    -6?. 70