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 work—Vaughan and Russell 1982—in 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
control—mimicked as the making available of previously unavailable
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sites—was 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
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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 theory•can
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
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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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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.
-------
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
-------
'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 * p»xa
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
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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
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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 space—the region southwest of bliss—which 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 estimation—an
interesting but impractical alternative. This option is particularly
impractical since when choice models are estimated from real world survey
-*rr'TK±g~«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 model—the 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 model—a 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 - t»i»*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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Ben-Akiva, Moahe and Thawat Watanatada. 1931. "Application of a
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Desvouges, William H., V. Kerry Smith and Matthew P. McGivney. 1982. A
____ Comparison of Alternative Approaches for Estimating Recreation and
Related Benefits of Water Quality Improvements. Draft. Research
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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 i«1 ®
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
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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 area—the
county—the 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
-------
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 P»l 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
-------
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
81
-------
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
82
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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
83
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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
au
-------
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
-------
Muellbauer, 1980, p. 74), but it is simple to implement empirically. While
we do not implement the AIDS approach in the present study—its estimation
can be computationally burdensome—we 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
36
-------
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
method—AIDS—was 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.
87
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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
-------
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
89
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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) -
90
-------
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
91
-------
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.
92
-------
(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
-------
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
-------
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
95
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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
-------
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
-------
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
-------
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
>4»gj * 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
-------
"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.
-------
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.
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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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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
-------
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
-------
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
'•—So—facilitate 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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
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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
-------
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
-------
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
-------
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.
-------
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
-------
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 Ia«d? - 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
-------
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
21«667
-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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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.
P£»P£ • 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 x»1 . 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
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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.
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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
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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.
-------
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
of—participation 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).
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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
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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
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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).
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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.
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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?
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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 n»1 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.
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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
-------
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
-------
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 < p°ql, (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,r—p*), 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
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(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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McKenzie,. George W, 1983. Measuring Economic Welfare; Mew Methods (Mew
York: Cambridge University Press).
Mendelsohn, Robert and Gardner M. Brown, Jr. 1983. "Revealed Preference
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vol. 23 (July), pp. 607-618.
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193
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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(p»y) *
+• (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
-------
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 3«J different PAGG—F, PAGG—C, 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
-------
• 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(p»y) * 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
-------
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
-------
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
-------
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
-------
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
-------
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 * (
-------
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
-------
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 Yi°3t of
post policy outcomes generated by a second simulation based on the
same initial conditions as (1) except for random number seeds, so
that Xi°3t is orthogonal to Xi°3t.
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
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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
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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
-------
Figure A.I
Consequence of Failure to Use Reservation Price in Welfare Calculations
224
-------
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York: John Wiley and Sons) Chapter 6.
Davidson, .Rusaell and James G. Mackinnon. 1962. "Some Non-Nested
Hypothesis and the Relations Among Them," Review of Economic Studies,
vol. 49, pp. 551-565.
Dhrymes, Phoebus J., E. Philip Howrey, Saul H. Hymans, Jan Kmenta, Edward
E. Leanerr Richard E. Quandt* James 3. Ramsey, Harold T. Shapiro, and
Victor Zarnowitr. 1972. "Criteria for Evaluation of Econometric
Models," Annals of Economic and Social Measurement, vol. 1, no. 3, pp.
291-324.
Ouan, Naihua, Willard G. Manning, Jr., Carl N. Morris, and Joseph P.
Newhouse. 1982. A Comparison of Alternative Models of the Demand for
Medical Care, Publication R-2754-HHS. (Santa Monica: Rand
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, , and 1983. "A Comparison of Alternative Models for
the Demand for Medical Care,1* Journal of Business and Economic
Statistics, vol. T,. no. 2 (April) pp. 115-126.
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
Regression Equation: Some Empirical Results" xerox. University of
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10 (May), 31 PP.
Granger, C. W. J. and Newbold, P. 1973* "Some Comments on the Evaluation
of Economic Forecasts", Applied Economies, vol. 5 (March), pp. 35-47.
Griffin, J. M. 1977. "Long-Run Production Modeling with Pseudo-Data:
Electric Power Generation", Bell Journal of Economics, vol. 8
..(Spring), pp. 389-397.
Hale, Douglas R. n.d. "The Evaluation of Economic Forecasting Models"
Unpublished xerox, 36 pp.
Harvey, A. C. 1981. The Econometric Analysis of Time Series (New York:
John Wiley and Sons) Ch. 5.
Hoel, Paul G. 1947. "On the Choice of Forecasting Formulas," Journal of
the American Statistical Association, vol. 42, pp. 605-611.
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Hollander, Myles and Douglas A. Wolfe. 1973- Nonparametric Statistical
Methods (New York: John Wiley, and Sons) Chapter 10.
Klein, Lawrence R. 1971. An Essay on the Theory of Economic Prediction
(Chicago: Markham Pub. Co.) 140 pp.
Kmenta, Jan. 1971. Elements of Econometrics (New York: Macmillan).
Kotz, Samuel and Normal L. Johnson eds. 1982. Encyclopedia of Statistical
Sciences, vol. 2 (New York: John Wiley and Sons) 613 pp.
Learner, Edward E. 1983. "Model Choice and Specification Analyses", in
Handbook of Econometrics, vol. 1, Zvi Griliches and Michael D.
Intriligator, eds. (Amsterdam: North Holland) pp. 286-325.
Maddala,. G. S» and R. B. Roberts. 1979. "An Evaluation of the Pseudo-Data
Approach*, Final Report EPRI EA-1108 (Palo-Alto: Electric Power
Research Institute) 63 pp.
_ 1983. Limited-Dependent and Qualitative Variables in Econometrics
(New York: Cambridge University Press), 401 pp.
Mincer, J. and V. Zarnowitz. 1969. "The Evaluation of Economic
Forecasts1*, in Economic Forecasts and Expectations, J. Mincer, ed.
(New York: National Bureau of Economic Research).
Pindyck, R« So and D. L. Rubinfeld. 1976. Econometric Models and Economic
Forecasts (New York: McGraw-Hill) 573 pp.
Platt,. Robert B. 1971. "Some Measures of Forecast Accuracy1*, Business
Economics, vol. 6, no-. 3 (May), pp.30-39.
Quandt, Richard E. 1965. "On Certain Small Sample Properties of !c-Class
Estimators", International Economic Review, vol. 6, no. 1 (January),
^ pp. 92-10*.
Schmidt, Peter. 1974. "Choosing Among Alternative Linear Regression
Models," Atlantic Economic Journal, vol. 2 (April) pp. 7-13.
Smith, V. Kerry and William J. Vaughan. 1979. "Some Limitations of
Long-Run Production Modeling with Pseudo-Data", Journal of Industrial
Economics, vol. 28, no» 2 (December), pp. 201-207.
______ and . 1981. "Strategic Details and Process Analysis Models for
Environmental Management", Resources and Energy, vol. 3. pp. 39-54.
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.
"Sample Selection Bias in Analysis of Consumer Choice: An Application
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
-------
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
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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
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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
-------
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~~ub3erv«U
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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RECSII'I
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S9.2236
(continued)
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9.44S96
4.10002
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6,946402
0.782085
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111. "84 7
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17.4067
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111.61
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244.411
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424.209
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61.0146
44.1412
620.
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-10.4021
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ACCP6033
1.60«2C«46
6!ail2f«32
ACCC50I2
ACCP403I
ACCP6014
4CCP6014
l.0900t*47
3.9194EM4
4. 492 IE -22
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0
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711.696
111.551
420.149
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-242.386
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7.801
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17.4061
-92.6322
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8,1256E*13
126654
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1611.65
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0.806154
1.9*01-4
0.460005
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2.9991E*I1
-632.433
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-4. '11451
10.1.116
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211.489
1.1140ft
90.5209
0.272211
-1.43176
ACO'5013
ACCP5032
ACCP602I
ACCP6015
5.290flE»40
6.42I9E*46
*.20I2E*36
?.155BE*40
I
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244.126
-565.416
2465.84
1*5.334
-62.7786
442.4
-(.96.732
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-44.112
•92.6862
20.4459
•5.00591
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11.1399
6.8949
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-II.IB
11.5519
•5.41119
1.03125
6i'.5512
•11.610578
-1.11461
•0.1.47703
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-7.17365
AGGF50I4 A6CP50I5
ACCP5033 ACCP5034
ACCP6022 ACCP6023
3.54B9C«38 i.5403E*ia
I.£60BE»JO 6.825IE-20
2.20061*33 3.B114E«39
0
6
1
5038.06
606.349
7J6.461
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041.49
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•141.329 -5.85?3E*12
-68.898
-73.322
34.4278
54.9995
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2.11454
i. 37935
-19.4321
19.0661
32.9469
36.2007
-0.574M1
-1.64934
-9,. 611714
2718.44
1.96351*12
-90.1223
-912.048
0.10351*11
-2.63122
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6.2569C*!!
3.04362
294.298
2.5B22t«ii
8.71815
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1II.6H1
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Tabla 0.9.
Ricsu*! WAnnw
«r.GP*021
-320.549
-14.9871
-20. «»793
fT» -0. '10101062
h.*66l
|o.00|3
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-1.74IH5
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117.4)0
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117.47*
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ACCP5022
AKCP60I1
•154.143
1.0272465
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15.1525
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-2.050*8
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-0.0541455
-0.0718176
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0.345312
0.200134
-5.91331
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-9.92B9
521. IM.
-1.17445
1,00974
15.1282
86.7937
-j. 12074
•1.47617
(1.334986
-0.71882
4.418*5
•0.' 16 7 P2
0.0125068
0.432)6)
0.35627|
-O.OI2A5I9
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-7.23904
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-3.4245E»|2
4,f916l
21.9092
-559.740
-0.630660
-1.39
2.2H46
-«». S49
2.43332
-0.102219
-6.19469
6.868)1 j
0.42405
0.340409
0.303421
0.90096.1
0,0210403
-O.Q31690I
-O.I37749
-Q. 1 25644
-9.01*84
•0.701354
If .6644
-3.12454
0.00901304
t.oolllAit
0.00735349
0.00674197
117.470
117.478
II f!4 71
119.40
106.0)2
A6GP50I3
AGGP6021
AGGP603S
u. eu?
-96.9764
149.236
91.9662
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20.9319
19,109
0.491229
P. 009990 95
-0,731034
O.I 11905
-0.040064
0. (-180 70
0.209004
0.409209
0.174135
9,251169
•0.0652501
0.0284002
0.130913
-0.0196
-0.1«690i
-1.52312
•3.12976
0.210951
0.00032840
P. 0074 7405
0.00762005
0,00471049
117.478
117.478
117.478
117.470
117. 47B
IB7.47H
117.478
117.478
AGGP&014
AGGP5033
A6CP6022
485,039
-20.004
26,2769
11:2!!!
19.3169
0.101714
{•S6726
-0.91716
0.426729
0.0855557
0.19488
.0.514747
0. 0925|22
0.0204412
0.0044594
-0.161207
6.960806
•9.62609
•0*102699
0.00310202
6.66924111
O.C063S453
117,470
117.470
117.470
117.478
117.478
117.470
AGGPSOI5
AGGP5034
AGGPf.023
* . --
-4248.58
59.0152
mm
15.4619
2.21924
-I.Uboa
-10.501
0.109121
-0,586.591
0.903421
0.9P0961
6.310143
•0.131140
-0.125644
0.010016
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-2.50629
0.460103
0.00135349
5.866I45S)
0.00661034
111.410
n 1.410
7.4tO
IIV.OOI
121.001
111.470
(oonllniiad)
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• 6024
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121 .771
hi. 773
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117.994
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4.2>i52?
4.2«>f.??
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77.5611
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77. (341
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121.773
12) .773
121.773
121.773
121.694
III. 609
116.579
4.29527
4l2952i
4.2952?
4.206*5
-29.7352
-*. 778*4
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77.5611
77.5611
77.5611
77.561 |
77.63*1
77.63*1
77.43*1
77.63*1
77.4911'
.901909
-999999
-999909
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-*9!'f394
-?. 17617
-?.21lrt9
ll.45?5
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17. tt>n
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ACCP5023
A OOP 60 12
Ar.r.P603l
121.77)
121.773
121.773
121.773
U6.097
4.041
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116.215:
4.29527
JllJIII
4 12952?
-|.1flOB9
-7.63626
161.515
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77.5611
77.5611
77.5611
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7?.63*|
77.6341
77.6341
77.6341
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506«6|
6.IA104
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•*.mi
-.'.7 70? 3
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SHPAVPJIV
Ar,r>Rf.U24
ACCP60I3
Ar,OP6032
121.173
J3».??3
121.773
121.773
121.968
A7, 7841
311.462
120,95?
4.20527
4,29527
4.2«52?
4. 4«»08I
-29.6935
193.985
3.47*^
77.5611
77.5611
7?.f.6ll
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77.63*1
77.63*1
77.6341
77.6341
-9999*9
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-019999
14.0*9
666 ,*04
7.41401
11.76?'.
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S57.6J7
6,21301
I?.9|OH
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612.071
ft.HM^i
Af.CP50ll
ACCP5025
ACCP60I4
ACCP6.033
121.773
121.773
121.773
121,773
150.036
119.572
123.0^?
158.19
4.2992?
4,295??
4.2952?
4.295??
32.5563
-1.90533
5,5798?
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77.5611
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77.5611
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77.6341
77.63*1
77.63*1
77.6341
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15.402
113.334
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14.1441
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4CC05Q3I
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121.773
121*773
121.773
121.773
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104.31?
16.65*4
I.4320E»13
4.295*7
4! 2952?
4.2952?
0
-13.1606
77.5611
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77.5611
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77,61*1
77.61*1
77.6341
77.6141
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4.3707EM3
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-18.6792
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3.456Ht?«|3
0
-20.502*
-151. r.|?
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ACCP5013
ACCP5032
ACCP6021
ACGP6035
121.773
121,773
121.773
121.773
l?9*??i
8U.II5
175.249
136.106
4.2952?
4*.2952}
4.2952?
11.7441
-37.3625
57.771
le.ftJai
77.5611
77.5611
77.5611
77.56||
77.6341
77.6341
77.6341
77.6341
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37.545|
-70.5089
164.141
51.252
31.4125
-•5B.992
137. 3i
42.8805
3*.* 7118 •
-6*.75U*
150.735
47.061.2
ACCP5014
AC6P5033
ACCP6022
121.773
121.773
121.773
222.,!?!
97.514
130,252
4.2952?
4.2952?
4.2952?
104.694
-19.9636
12.77,41
77.5611
77.6611
77.5611
7?,634|
77.6341
77,6341
-999999
-999999
-999999
301.295
fc$3.645S
40.9029
•
252.0C2
-44.. 4229.
34.221?
276.689
-48.7*92
37.5622
ACCP50I5
ACCi'5034
ACCf'6023
12 i.m " "" '
I2J.773
121,773
.
-113.^3?
3.40l6f*ll
159.877
4.29527
4^.2952?
4.HS2f
-7.0929?
9.40l6E«li
41.3993
77.9611
77.5611
77.56J1
77.6341
77.6341
77.6341
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-21.9402
1.0675^*12
120.485
-18.3665
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100.805
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9.803|F«II
l|0.6*5
(oonttnued)
-------
ACS2PKFII
SMAPOMP
.4C.C.P5035
45GPA024
HO SCR VHP
Ar.CI'5022
ACGP60H
•M62.939
-31.4284
31.31
-366.634
18.0201
-«»9.3639
72.5002
-»23.?77
7.1M37
-329,4S8
-0.0790748 -<).00882*11
-17.3215 -34.0304
-3.67064* -fulfill?
-P.19361
18.889
•>.5«619
21.4564
0.01*2*1
1.l3
1.M*1I7
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0.10*74179
7.071*2
0.0*231!i«
0.759935
0.71195*9
0.00720319
«. 8*441
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-74.7471
-140. *2
-167.301
•HO.4373
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ACGP6QI3
AGGP6032
-429.071
3J33..69
77.2224
-4io.&98
3314,64
49.4216
O.|9*i539
-O.A16277
6.2220?
34.6792
A.9AI29
0.931U36
O.I«34U97
506.44
0.987009
-169.047
AHA.943
-70.1471
02.677
174.503
594.09
86.32*9
n.7*3!»/7
O.H71H33
AGCP40I1
Af.r.P5025
Ar.GP60|«
AGGP6033
4GCP5012
4GCP5011
ACGP601S
4GGP6034
129,676
497.106
459.06)
-32.76H
95.9491
700.006
20.2631
-354.64t
2.4268C«14
0
•226.304
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2.4614C»14
36,4179
49.6792
10,2042
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4.29427
23.7729
60.9A29
0.029214 1
0.677416 0.010440
0.544544 0.937309
O.A03020 I
29.4039
27.0109
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35.1729
132.934
8.05755.
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2.6624M24
-71.5611
-99.0071
•244.71
4.3707E»1»
77.5611
4J6.?H
245.A6
4.3707EU3
0.694245 • 1
0.112(1906 4.737042
H.111166} ').S33517
0,552431 1
1
1.46106
0.91! 7 15
0.fl(JJ*3I
1.0'>3(2
AGGP5013
AGCP5032
AGGP6021
AGGP6035
214.321
•662.344
915.161
446.350
-6*2o*i2
991.400
320.125
7.4460
-41.6378
14.131
14.0606
71.75
65.6624
22.5766
0.653465
0.925115
0.054775
0,679919
1.00033
66.4007
57.3456
7.52667
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06.5795
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102.5MB
204.74
173.691
112.422
0.761674
0.790434
0.479756
0.591151
0.996205
3.96)74
1.70522
O.n?<»4!tt5
AGCP50I4 ACGP5015
ACCP5033 ACGP5034
«G6P6022 ACCP6021
•211,912 6.3000E«12
221.101 961.56)
1000.27 -121.960
•141.266 5.049|Hi2~
219.659 729.002
100.190 -11.1002
3.40I6(«11
is; i 64"
100.424 14.146
20.1140 3.40iff«li
46.970
0,905751 O.030747
0.902515 I
0.56351 0.000001
194.732 • 12^*«AI
22.7247 l.6IO»«^tr
2.23677 . 10.1669
423.734 -99.5013
•110.657 1.06751*12
•16.6504 42.9235
243.508 106.449
i42.i>27 i.oiHi*li
09.4712 151.257
0.59b902 0.761170
0.046007
i.6925l5 0.52490
3.06*48 I.41975
1.99012 4.963IEH9
0.01607 I 1.10308
jcontinued)
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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?
5.5628f»11
" ~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
»»I1P»WPXI
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
|.2902fc«2)
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.
MM»rniCF.
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 |