United States
Environmental Protection
Agency	
Policy, Planning
and Evaluation
(2127)	
EPA-230-10-89-069
October 1989
Measuring the  Benefits of
Water Quality Improvements
 Using  Recreation Demand
 Models: Part I

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           MEASURING   THE  BENEFITS

       OF  WATER  QUALITY   IMPROVEMENTS

      USING   RECREATION  DEMAND   MODELS



                  VoIume  I I
                      of

           BENEFIT  ANALYSIS  USING

     NDIRECT  OR  IMPUTED  MARKET  METHODS
           Prepared  and   Edited  by

             Nancy  E.  Bockstael
           University  of  Maryland

             W.  Michael   Hanemann
          University  of  California

            Ivar  E.  Strand,  Jr.
           University  of  Maryland
           Principal   Investigators
Kenneth  E.  McConnell  and  Nancy  E.   Bockstael
    Agricultural   and   Resource  Economics
           University  bf   Maryland
       EPA  Contract  No.   CR-811043-01-0
                Project  Officer

                Dr.  Alan  Carl in
          Office  of  Policy  Analysis
 Office  of  Pol icy  and  Resource  Management
   U.  S.   Environmental  Protection   Agency
           Washington,   D.C.    20460

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The information in this document has been funded wholly or in
part by the United States Environmental  Protection Agency
under Cooperative Agreement No. 811043-01-0.   It has been
subjected to the Agency's peer and administrative review, and
has been approved for publication as an  EPA document.  Mention
of trade names or commercial  products does not constitute
endorsement or recommendation for use.

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                             ACKNOWLEDGEMENTS
    Kenneth E.  McConneI I,  Terrence P. Smith,  and  Catherine L.  Kling  were
major contributors to  this  volume, providing both original  contributions and
beneficial  comments.   The authors have also benefited from  comments  of EPA
staff  members,   including  Peter Caul kins  and George  Parsons,  and  from
reviewers Edward  Money and Clifford Russell.  Technical  assistance  was
provided by Jeffrey Cunningham and  Chester  Hall. Additionally,  both  credit
and appreciation are due Alan  Carl in,  our  project  officer  in  the Benefits
Staff of EPA.  Throughout the research, Diane  Walbesser  supplied invaluable
secretarial  and technical  assistance.   Finally,  Linda Griffin of  ADEA
Wordprocessing and Patricia Sinclair  of the University  of Maryland  deserve
special thanks for undertaking  the arduous task of typing  this manuscript.

    All opinions  and  remaining errors are  the sole responsibility  of the
editors. This effort was funded  by  US EPA Cooperative Agreement number CR-
811043-01-0.

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                                  FOREWARD
    This  is  the second of  two volumes constituting the final  report for
budget period  I  of Cooperative Agreement  #811043-01-0,  which was  initiated
and supported by the Benefits Staff  in the Office of Policy Analysis at the
U.S. Environmental  Protection  Agency (EPA).  The two volumes,  while encom-
passed under the same cooperative  agreement,  are distinct  in nature.  The
topic of Volume 11  is  the use of recreational demand models in estimating
the benefits of water quality  improvements.

    The research reported  here  is  the result  of  interaction among  the
principal   investigators  of  the project,  the  editors  of  the  volume,
individual   contributors  at  the  University  of Maryland,  and  outside
reviewers.     In  addition to  the team of  editors,  Kenneth  E. McConneI I,
Terrence  P.  Smith,  and  Catherine  L.  Kling  were  major  contributors,  providing
both original research  and invaluable  review.

    The editors benefited considerably from  comments by outside reviewers,
Edward  Morey  of University  of Colorado  and Clifford Russell,  now of
Vanderbilt University.    Important contributions were also made by  EPA staff
including Alan  Car I in, Peter  Caul kins,  George  Parsons  and  Walter Mi I on. It
would be impossible to cite all the  individuals who had an influence on the
ideas presented here,  but two of these must be mentioned, V.  Kerry Smith of
Vanderbilt  University and  Richard  Bishop of  the  University of Wisconsin.

    Progress made  in this volume  toward the  resolution  of the  problems and
dilemmas which  plague  the assessment of environmental  quality   improvements
must be attributed to a  wide range of sources.    In  large part  the  work
reflects the  cumulative efforts  of  a decade or two  of  researchers  in  this
area.   And,  it is itself merely a transitionary stage in the  development and
synthesis of the answers  to those problems.   More progress has already been
made on many of these  issues  -  both  by  the authors and  by other economists
working in  the field.   This  new work will  be reflected  in  future  cooperative
agreement  reports.

    Also,   included  in the next budget period's  report will be discussion and
analysis  of survey data  collected  during budget period  I.   The survey,
designed  by Strand, McConneI I  and  Bockstael in conjunction with Research
Triangle  Institute  (RTI), was administered  by  RTI.   It  includes a telephone

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survey of  households  in the Baltimore-Washington SMSA's  and  a field survey
conducted  during the  summer of 1984 at public  beaches  on the Western shore
of the Chesapeake Bay.  The survey  provides data on swimming behavior which
 is being analyzed  using some of the developments discussed in this volume.
The survey  instrument,  the data,  and the analysis  will  be presented in the
next cooperative agreement  report.

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                              EXECUTIVE SUMMARY

      In  an  era of  growing Federal  accountability,  those programs which
cannot substantiate returns commensurate  with  budgets  are  severely disadvan-
taged.    Expressions such as Executive Order 12291  require  an.  account  of  the
benefits of  public  interventions.    Inability to provide, or  inaccuracy  in
the provision of,  those estimates undermines the credibility of programs  and
may cause their untimely demise.

     The public provision  of improvements  in water quality  is an activity
endangered by the complexities  involved  in the accounting of benefits.   The
lack of  markets and observed prices  in  water-related  recreational activity
has necessitated the  use of surrogate prices  in benefit assessment.    More-
over,  a  formal regime (i.e.  The Principles and Standards  for Water Quality)
articulates the assessment  procedure.   Unfortunately,   the regime still con-
tains ambiguities,   inconsistencies  and slippage sufficient to  raise  poten-
tial   controversy    over  any   estimate  of benefits   from  water   quality
improvements.

     The purpose of Volume 11  is to address  some  of  those ambiguities  and
inconsistencies and,  in so  doing,  provide a more comprehensive,  credible
approach to the valuation of  benefits  from water  quality  improvements.
Substantial  progress  is made in  improving valuation  techniques by  linking
the  fundamental concepts of the  "travel  cost"  model with  cutting-edge
advances  in  the labor  supply,  welfare,  and  econometrics  literature.

     At the  heart  of the  research  is the study of  individual  recreation
behavior.      As water   quality   improves,    individual   behavior   changes,
reflecting  improvements  in welfare.   Misconceptions and  inaccuracies may'
arise  if benefit evaluations  are  based on  inappropriate aggregation of
individual's  behavior.    An analysis of the "zonal" (an aggregate) approach
represents  one contribution of Volume  II.     Alternatives  to the zonal
approach are  offered.    The  new approaches are  based on advances  in  the
statistical  analysis of  limited  dependent variables.

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     The  realities of  recreational  choice  encompass more  dimensions than
traditional  demand analysis.   Time is critical  - over 50% of respondents  in
a  recent  national  survey  replied  that "not enough time"  was the reason they
did not participate  more  often  in their favorite recreation, while only 20%
replied  "not  enough  money."  Drawing  on  labor  supply literature,  an exten-
sion of traditional  demand  analysis to  include time constraints  is developed
 in Volume XI.   The  extension,  which  is made operational,  captures the true
nature of recreational  decisions  which  are affected  as much by  individuals'
time constraints as their money constraints.

     Statistical analysis  is emphasized throughout the  volume.  One example
 is an  examination  of the properties of welfare  estimates.   Because typical
welfare estimates are derived from numbers with  random components, they have
random components themselves.   Thus  it  is  important to study the statistical
properties of typically used estimators for welfare  measures.  These proper-
ties, such as biasedness,  are shown  to be undesirable in several instances.
More credible estimators are  provided.  Another  statistical  issue,  causes  of
randomness  in  estimates,    is shown  to influence  the magnitude of welfare
estimates.   Ways  in  which  information  about the  source of randomness can be
used to improve accuracy are discussed.

     Part  I I  of Volume  addresses problems specificaIly associated with
introducing aspects of  water quality  into the fundamental model   developed  in
Part I.   The desire to  incorporate environmental characteristics  (such as
water quality) has prompted  the treatment  of an  additional  dimension to the
recreational  model.   Data  collected for  one recreational  site do  not,  by
their nature,  exhibit variation in the  quality characteristics of that site,
preventing the  researcher from deducing  anything about how demand  changes
with changes   in quality characteristics.    The only reliable means of
incorporating  quality  is to model  the  demand  for an array  of sites of
differing  qualities.    However,  the need to  develop  models of multiple site
decisions has  been a blessing  in disguise,  for it  has forced  modelers to
recognize that  recreational  decisions  are  frequently  made  among an array of
competing,  quality-differentiated  resources.

     A major share of Part  II of this volume  is devoted to the discussion of
models which can incorporate quality characteristics   in multiple site recre-
ational demand  decisions.    While a theoretically consistent  model  can be
developed,   it  is not empirically  feasible,  and several second best models
are presented.    Criteria for evaluating these  alternative  models  includes
their ability to capture the nature of  recreational decisions and to respond
to the research goal  of valuing environmental  quality  changes.
                                     VI

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                                 TABLE  OF CONTENTS


                                       Part  I

   ADVANCES IN THE  USE  OF  RECREATIONAL DEMAND  MODELS  FOR BENEFIT  VALUATM
Chapter  1
 INTRODUCTION
Chapter  2
Appendix  2.1
Appendix  2.2
Chapter 3
Nonmarket Benefit Evaluation and the Development of ...
   Methods
Making Benefit Measures More Defensible  	
The Empirical  Foundation of Recreation Demand Models:  .
   The  Traditional  Travel Cost Model
The Theoretical  Foundation of Recreation Demand  Models:
   The  Household Production Approach
The  Plan of Research for  Part  I  .

SPECIFICATION OF THE RECREATIONAL DEMAND MODEL:  	
   FUNCTIONAL FORM  AND  WELFARE  EVALUATION
               The Integrabi I ity  Problem and Estimated Demand  ,  ,  ,
                   Function  Estimation
               Exact Surplus  Measures  for Common Functional  Forms
               Evaluating the El imination of a Resource 	
               Functional Form Comparison.. .     , ,, .
               Estimating a Flexible Form and Calculating Exact  .
                   Welfare  Measures
               An I  I lustration	
               Footnotes  to  Chapter 2    	
                                                                                  10
DERIVATION OF SOME UTILITY THEORETIC MEASURES FROM
   TWO  GOOD  DEMAND SYSTEMS
                                          .12

                                          .13
                                          ,16
                                           20
                                           21

                                           24
                                           27

                                          .29
COMPUTER  ALGORITHM FOR OBTAINING COMPENSATING
    EQUIVALENT  VARIATION  MEASURES  FROM ESTIMATED
    MARSHALL I AN  DEMAND  FUNCTIONS
                                          .33
AGGREGATION  ISSUES:
   APPROACHES
THE  CHOICE
EST
                A Review  of Past  Literature
                   1.   The  Zonal  Approach .
                   2,   The Individual Observation Approach
,35


 36
 36
 40
                                         VI I

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                           TABLE  OF  CONTENTS  (continued)
                Models of  Individual  Behavior  and  Their Implications	42
                    for Estimation
                    1.   A Simple Model  of Individual Behavior  	43
                    2.  A  Model  of Behavior  When Different Variables	46
                        Affect  Participation  and the Demand  for Trips
                    3,   Estimation When the Sample Includes Only	47
                        Participants-The  Truncated  Sample
                 Implications for  the  Estimation of the Zonal  Approach	49
                Cone I us i ons	51

Chapter 4      SPECIFICATION OF  THE  INDIVIDUAL'S DEMAND FUNCTION	53
                    THE TREATMENT OF  TIME

                Time  in Recreational  Decisions	54
                Time as a  Component of Recreational  Demand: A  Review	55
                Time in the Labor Supply Literature:  A Review	57
                A Proposed  Recreational Demand Model	60
                Considerations  for Estimating Recreational Benefits	64
                Estimating the  Model:   The Likelihood  Function	68
                An  III ustrat i on .  .	 69
                Observat i o.n.s	75
                Footnotes  to Chapter 4	77

Appendix 4,1 A COMPARATIVE STATICS ANALYSIS OF THE TWO	78
                    CONSTRAINT  CASE

                Utility Maximization with Two Linear Constraints	79
                The  Two Duals and the Two Slutsky Equations	81
                A Summary of  Resu Its	       	86
                Footnotes  to  Appendix  4.1 	  88

Chapter 5      THE  CALCULATION OF  CONSUMER  BENEFITS	89

                Sources of  Error  in the Recreation Demand   Model	90
                True  Consumer  Surp I us	92
                    1,   Omitted  Variables Case.  . .   . .       	93
                    2,   Random  Preference and Errors  in Measurement	93
                Graphical  Comparison of Surplus Computation and an	94
                    Empirical  Demonstration
                Calculating Expected Consumer Surplus	97
                Consumer Surplus  from Estimated Parameters	99
                Properties of the  Consumer  Surplus Estimator	104
                Minimum Expected  Loss  (MELO) Estimates  	 107
                Conclusion...  .	    ...    	 .    	
                Footnotes  to  Chapter 5...    .    ...   	110

Appendix 5.1   DERIVATION  OF DIFFERENCES IN ESTIMATED CONSUMER	112
                    SURPLUS USING THE  SEMI-LOG   DEMAND  FUNCTION
                                          VI I I

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                                 TABLE OF  CONTENTS
                                      Part  I I

       MULTIPLE  SITE  DEMAND MODELS  AND THE  MEASUREMENT  OF  BENEFITS  FROM
                            WATER  QUALITY  IMPROVEMENTS

Chapter  6     RECREATION DEMAND MODELS AND THE BENEFITS FROM	115
                   IMPROVEMENTS  IN WATER  QUALITY

               Valuing Quality Changes in Demand Models	16
               Extending the  Single Site  Model  to  Value  Quality Changes. .. I 17
               Plan of Research for Part        ,  ,  ,                                     ,119

Chapter  7     EVALUATING ENVIRONMENTAL  QUALITY IN  THE CONTEXT	I20
                   OF RECREATIONAL  DEMAND MODELS: AN  INTRODUCTION
                   TO MULTIPLE  SITE  MODELS

               The Nature  of Recreation  Demand	121
                Introducing Qua ity into the Demand Function	122
               The Specification of Demand Models for Systems  of	126
                   Alternatives
                Introducing Quality into Multiple  Site  Demand Models	132
               Footnotes to  Chapter 7.....  .....   .  ...   ...  ..  ~. .  —-...-136

Appendix 7.1   SOME TRANSFORMATION MODELS FOR  INCLUDING  QUALITY	137
                   IN DEMAND FUNCTIONS

Chapter  8     THE PROPERTIES OF  THE MULTIPLE SITE RECREATION DECISION  .  ...140

               Theoretical  Models  of  Corner Solution  Decisions  	141
                   The Extreme  Corner  Solution
               Theoretical  Models  of  Corner Solution  Decisions  	144
                   The General  Corner  Solution  Problem
               Estimating  General Corner Solution Models	148
               Concluding  Comments	   .-   ..   ...   ..e ...„,..,.-  152

Appendix 8.1   PROPERTIES OF THE  UNCONSTRAINED AND PARTIALLY	154
                   CONSTRAINED  PROBLEM

Appendix   8.2 ESTIMATION OF GENERAL CORNER  SOLUTIONS USING	158
                   KUHN-TUCKER  CONDITIONS

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                          TABLE OF  CONTENTS  (continued)
Chapter  9      A REVIEW AND DEVELOPMENT OF MULTIPLE SITE	161
                  MODELLING  TECHNIQUES

               Demand Systems in a Multiple Site Framework	I62
                  1.   Gravity Models. .	   ...     .    .    	162
                  2,   Systems of Demand Equat i ons	165
                  3,   Varying Parameters Models	167
                  4,   Hedonic Travel  Cost.       	 168
               Share  Models		171
                  1.   The Theory  of Share  Models	171
                  2,   The Morey Model	174
                  3,   Discrete Choice Models.  .      .  .      	178
               Welfare Measurement  Given the  Nature of Recreational	187
                  Decisions
               Concepts of Welfare  Evaluation  in  a Stochastic Setting	189
               Footnotes  to  Chapter 9	195

Chapter  10     ESTIMATION OF A MULTIPLE SITE  MODEL	196

               Measurement of Water Qual ity  Change	196
                  1,   Objective  Measures and Perceptions	196
                  2*   The  Correlation Between Perceptions and	I98
                       Obj ect i ve  Measu res
                  3, Quality in the Proposed Model	200
               Specification of the  Discrete/Continuous Choice	20!
                  Model  of Recreational   Demand
                  1,   The Micro AI location Decision  	20!
                       Choice Among Sites
                  2,   The Macro  AI location Decision  	205
                       Participation  and Number  of  Trips
               The  Data and  Model  Estimation	207
                  1,   Micro-Allocation  Model	207
                  2,   Macro-Allocation  Model  .     	210
               Results of the  Estimation	    ...    	212
                  1,   The Micro  Allocation Model 	 212
                  2,   The Macro  AI I ocat i on Mode I ....      	216
               Benefit Measurement  in the  Context of the	2I7
                  Multiple Site Model

Chapter  11     CONCLUSIONS	222

               The Traditional Single  Site/Activity  Model	223
               Water Qua ity and the Multiple Site Model	224
               The  Future	,	

Bibl iography      	228

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                                   LIST OF TABLES
Table  2.1       Utility Theoretic  Measures Related  to  Common Demand	15
                   Specifications

Table  2.2      Welfare Estimates Calculated from Different	26
                   Functional  Forms

Table  4.1       Mean  Estimates, Biases,  Standard  Deviations and Mean	70
                   Square  Errors  of Estimated  Parameters

Table  4.2      Mean  Estimates,  Biases,  and  Standard  Deviations of	74
                   Compensating  Variation   Estimates

Table  10.1      First Stage  GEV Model Estimates of Choice Among	2I3
                   Freshwater and Saltwater Beaches,  Boston-Cape Cod

Table  10.2      Second Stage GEV Model  Estimates of Choice Between	2I4
                   Saltwater  and  Freshwater Sites,  Boston-Cape  Cod

Table  10.3      Estimates of the  Tobit Model  of Boston Swimming 	2I6
                   Participation and  Intensity

Table  10.4      Average Compensating Variation  Estimates  of	22
                   Reductions in  Specific  Pollutants  at  Boston
                   Area  Beaches

Table  10.5      Average Compensating Variation Estimates  of Water	22!
                   Quality  Improvements for Bostonn  City  Beaches
                   and AI I  Boston Area  Beaches
                                          XI

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                                   LIST OF  FIGURES
Figure  1.1     The  Recreation  Demand Curve	

Figure  4.1     First  Generation  Budget  Constraint,

Figure  4.2     Second Generation Budget Constraint	

Figure  5.1     Two Different Procedures for Calculating  ,  ,
                   Consumer  Surplus

Figure  6.1     Benefits from Water  Qua I ity Change	
. 7

,58
,116
                                          XI I

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          PART
  ADVANCES IN THE  USE  OF
RECREATIONAL DEMAND  MODELS
   FOR  BENEFIT  VALUATION

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                                  CHAPTER 1

                                INTRODUCTION
     Volumes  I and  II  of this report are the result of one year's  research
conducted  under  EPA Cooperative Agreement  CR-811043-01-0.    The  particular
methods  designated  by EPA  to be of primary interest  in  this cooperative
agreement are  "imputed or indirect  market methods," i.e.  methods which de-
 pend on observed behavior in  related  markets rather than direct hypothetical
questioning.   Despite their similar themes,  the two volumes are distinct  in
many respects.   Volume I  addresses a specific technical  issue  (the  identifi-
cation problem)  associated  with  the hedonic method of  valuing  goods.   The
second volume discusses  a wider range of  technical  issues associated with
the  use of  recreational   demand  models to value environmental  quality
changes.    The primary purpose  of  the agreement has been to develop and
demonstrate  improved methods  for  estimating  the regional  benefits from
environmental  improvements.

     Within  this  volume  dedicated  to  recreation demand models,  Part I   is
restricted to  a  set of  issues which arise  in  benefit  valuation  using the
conventional  single site  recreational  model.    The  topic  of Part  11  is the
application of recreation demand  models  for the specific  task of measuring
the  benefits associated  with  changes in  the  quality of the recreational
experience.    Attention is given, in particular,  to water quality  improve-
ments.    In this  spirit,   Part  II  explores  a broad range of models  based on
individual  behavior which can be used to reveal  valuations of environmental
improvements.    These models attempt to  establish the  relationship between
use  activities (specifically  recreation)  and  water  quality  and can be used
to devise welfare measures to  assess  benefits.

     The emphasis this  volume gives  to recreation  behavior  is  not mis-
placed.   A 1979 report by  Freeman  (1979b)  to the Council  of Environmental
Quality estimated that over fifty percent of the  returns from air and water
quality  improvements would accrue  through recreational uses of  the envi-
ronment.   When considering water  quality  improvements alone,  the  percentage
was  even higher.    One of the earliest studies attempting to  quantify such

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effects  (Federal Water  Pollution  Control,  1966)  estimated that  recrea-
tion ists  would  receive  more than  95% of the benefits derived  from water
quality  improvements in  the  Delaware estuary.  These  sentiments were  further
supported by the U.S.   National  Commission on  Water Quality  (1975) which
maintained that  water  based  recreators would be  the  major beneficiaries of
the 1972  Federal  Water Pollution Control  Act.

     Thus, the emphasis  in these two volumes is on recreation, but the tasks
are wide-ranging.    The initial charge  in  the  Cooperative Agreement was a
broad one,  including the development of improved methods,  the demonstration
of new techniques,  the col lection of primary data and the assessment of the
usefulness of the  resulting  benefit estimates.    The emphasis in this first
year of work has been where  it must be, on the first  items in this I ist,  al-
though progress  has  been make on each  task.

Nonmarket Benefit Evaluation  and the Development  of Methods

     Despite the near consensus which currently exists  in  market-oriented
welfare theory  (i.e. welfare changes  in private markets), economists are far
from  embracing  a complete methodology for  valuing  public  (often environ-
mental),   non-market goods.      It hardly seems necessary to document this
contention.    One need only consider  some of the many recent  conferences
which have attempted to  resolve difficulties and  increase consensus on these
issues,   (e.g. Southern Natural  Resource Economics Committee,  Stoll, Shulstad
and Smathers,  1983;  Cummings,  Brookshire and Schulze, 1984;  EPA Morkshop on
the State of the Art  in  Contingent Valuation, and AERE  Workshop on Valuation
of Environmental  Amenities,  1985.)     In essence  "Nonmarket  valuation has a
long way yet to go  before a I  I the problems wi I I  be solved and its acceptance
by economists will  be unequivocal  (SNREC,  p.4)."

     The valuation exercise has been viewed by many economists as an attempt
to bring  nonmarket  goods into  policy  considerations  on  a comparable footing
with private marketed  goods.    However,  to  be  accurate,  some economists and
many non-economists have questioned the  relevancy of the market analogy for
public good  valuation.    Arguments by philosophers  include  reference to a
social ethic and contend that  societies may have collective  values indepen-
dent  of  individual  preferences.     Not  so well  articulated  are our  own
concerns about  how people  think about  public goods  and how they  relate
public goods to  private expenditures.    To what extent can a change in a
public good be translated into an effect on an individual such that an indi-
vidual's willingness to pay  is a meaningful  concept?

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     The existence of rival  theories and the lack of consensus  we  see  in  the
non-market  benefits literature  is  not unlike the early  stages of the  de-
velopment of other  fields of economics  and  of other  sciences.   In the early
stages of a science or  a  subfield of a science,  Thomas  Kuhn  has argued that
competition exists  among  a  number  of distinct views all  somewhat arbitrary
in  their  formulation.    Eventually a  set of theories,  Kuhn's  now familiar
"paradigm,"  emerges which provides  focus to future  work.    The paradigm is
the set of fundamental  concepts and  theories which all  additional  work takes
as given.    The eventual  acceptance of a paradigm  allows,  and in fact  en-
courages,    research to  become more  focused,   more   refined,   and more
detailed.    This body of  accepted thought  provides the necessary structure
and standards of judgement without which research becomes  confusion.   Kuhn's
essential  point was that  the  science could  only be  advanced  in the  context
of the paradigm.

     Whether we wish to view  it as  a pre-paradigm stage or a  crisis  in  the
neoclassical  paradigm,   the development of what has become "traditional"
welfare economics  (i.e. welfare measurement in  private markets)  provides a
case  in point.   Welfare economics has  a long history of controversy,  begin-
ning  with  loosely  defined  and imprecisely measured concepts of rent  and
consumer surplus  extending  as far  back as  Ricardo and  Dupuit.    The  estab-
I  ishment of these concepts as  foundations  of a  theory of economic welfare
was a  long and uphill  battle  involving  attacks by new welfare  economists on
the old welfare economics and the development of the  compensation princi-
ple.    For a very  long period the  state of welfare economics was one of
crisis,  with applied economists pursuing empirical  studies which theoreti-
cians condemned.   Over time, and with theoretical developments  by  economists
such  as Wi I I ig, Hausman,  Just et a I.,  Hanemann,  and others,  a theoretical
foundation for feasible empirical practices has  emerged in the form  of  the
"willingness  to  pay" paradigm.

     With  the recognition that public policies   frequently produce benefits
and  losses  outside of  markets comes  a  new controversy and  an attempt to
stretch the existing "willingness to  pay"  paradigm  to  cover new  ground.  To
many  established  economists,   the  problem seems  straightforward:      the
valuation  of nonmarket  benefits through benefit-cost analysis, under ideal
procedures for  extracting value measures,    is assumed  to provide the same
answer that the market  mechanism would  provide.    The major difficulties  lie
in  defining those  ideal  procedures.    Some  question whether these measures
exist, or  are meaningful,  in the context in  which we wish  to  use them  -  i.e.

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can  the wi I I ingness-to-pay paradigm  really be stretched  and modified  to
resolve the anomalies which public good valuation  present?

     This  subfield  of economics,  the valuation of  public goods,  is in a
period of  crisis  in  its  development,  but it is not unlike periods of crisis
which have arisen in other areas of economics or in the natural  and physical
sciences.    Kuhn describes these periods as marked by debates over  legitimate
methods, over relevant experiments,  and over standards by which results can
be judged  -  a description which fits closely the current activities in non-
market valuation.    In  these  periods  of crisis,  Kuhn argues,  many  speculative
and  unarticulated  theories develop  which eventually  point  the way to dis-
covery.

     The  implication  of Kuhn's  thesis is  that  more  refined and  precise
analysis either establishes a closer match between theory and observation or
provides more evidence that such a  match does not exist.    The only way to
determine whether standard welfare economics can be stretched to resolve the
public good valuation  problem  is  to  explore  nonmarket valuation problems in
a rigorous  welfare  theoretic framework.    If  the  anomalies can  not be re-
solved,  even with increasingly careful  modelling and  precise  measurement,
then the balance wi  I I  tip  In  favor  of seeking a new paradigm.   But  it is
only  in the context  of some carefully conceived  theoretical  structure that
progress can be made.     "Truth emerges more  readily from error than from
confusion  (Kuhn,  1969)."

Making Benefit Measures More Defensible

     An attempt  to  apply scientific methods to nonmarket benefit analysis
immediately  raises problems.    Our approaches  provide  estimates  of welfare
for which  we have no direct observations for comparison.   The absence of
direct  observation  on welfare changes directly only suggests  that welfare
measures should  be defined on  models  of behavior which  can be observed.

     Starting, as they do,  from models of economic behavior, one would think
that welfare  measures  derived  from models of observable behavior  in markets
related to  environmental  goods  (e.g.  recreational  demand models) would be a
popular approach.   Certainly,  the travel cost approach,  a  specific variant
of more general models of economic behavior,  has  produced many  benefit  esti-
mates in its  long life.   Yet  this approach's credibility has been  challenged
on two counts.

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     First, policy makers argue  that  many amenities of  interest can  not  be
associated closely enough with a market or with observable behavior to allow
for the use of  related market  methods.   This criticism has some very impor-
tant  implications.    On the  pragmatic  side,   it  is  useful  to  note recent re-
sults  in  contingent  valuation  assessment.    Contingent  valuation,  the prin-
ciple  alternative method,  has  been  pronounced  quite reliable  as long as the
good to be valued  is closely  related  to  a  market experience.  What  is  more
germane to the  argument here  is  that  when valuation is  unrelated to  observ-
able behavior,   it  is impossible  to  test the predictions of theories  against
observations  -   and as a consequence we can  have  no confidence in those  pre-
dictions.   In fact,  it is unclear that economic valuation has any meaning  in
a context where there exists  no related observable economic  behavior.  We
are reminded of Kuhn's  warning  "measurements  undertaken  without a paradigm
seldom lead to  any  conclusions  at all."

     The second criticism  of market  related  valuation approaches is that the
same  valuation problem can generate  a vast array of radically  different
benefit estimates.    How  can one trust a method which  appears capable  of
generating a  number  of  very  different  answers to  the same  question?

     If we examine the  literature or  conduct experiments ourselves,  we  in-
evitably encounter this embarrassing  problem:    benefit  estimates  seem  very
sensitive to  specification,  estimation  method, aggregation, etc.   It is the
contention of the current work, however, that valuation  methods  based on  be-
havioral  models allow the potential  for resolving inconsistencies,  since  the
apparent arbitrary choices we  make about specification,  etc.  are really  im-
plicit but testable  hypotheses about  individual  behavior.    By being  more
precise about  the behavioral  assumptions  of our  models,  more defensible
benefit estimates can be  defined.

     The  philosophy inherent in our research  agenda  is that if  benefit
measures  are to be taken  seriously by  pol icy  makers they must  be based  on
defensible, realistic models of  human  behavior.  Perfect  measures can  not  be
defined and will  always  be  inaccessible.     But  arbitrariness in estimating
human behavior can be reduced by careful model  specification  and estimation,
so  that  we know ultimately what assumptions are  implicit   in  the  benefit
est i mates  as  we I I  as  the  d i rect i on of  poss i bIe  b i ases  i n  these est i mates.

     This  phi losophy requires that we first assess the state of  benefit
estimation using  indirect market  methods  and then  attempt  to  make  im-
provements  in  those  areas which seem  either the most confused  or the  most

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vulnerable.    A goal  of the current  research is to bring together the many
recent advances  in recreational demand estimation, specifically, and applied
welfare economics,  more generally, to  further  the development of defensible
models of measuring water  quality improvements.

     One  comment  needs to  be  made with  regard  to alternative  benefit
measurement  techniques.    The  arguments in this Chapter  are  not intended to
champion the  cause of recreational  demand models  over  contingent  valuation
techniques.    The purpose  of this as we I I as other studies should  be  to im
prove the credibility of  techniques  for  valuing environmental amenities. It
is our opinion that the science wi I I  be advanced  if contingent valuation and
indirect market  methods are  considered as complements.    To  the extent  that
the two approaches can be made comparable,  their  conjunctive  use can only
strengthen benefit estimation.   While many  studies  have compared  estimates
derived from  the two approaches  (e.g.   Knetsch and Davis 1966; Bishop and
Heberlein  1979;   Thayer 1981),  few have  tried  to  relate the  approaches con-
ceptually and none have attempted to ensure  that  the underlying assumptions
of the models are  consistent.   The two approaches applied to the  same cir-
cumstances can potentially be  made comparable since they are  both the reali-
zation of  individual's  preferences  subject  to  constraints.    Just as  there
are assumptions  about behavior  implicit  in  the way  in which  we specify and
estimate recreational demand  models,  there  are similar  if  less conspicuous
assumptions implicit  in the way contingent valuation experiments are framed
and the way  benefit estimates  are  derived  from  the hypothetical  answers.
While a means  for  making  the  two approaches comparable  is beyond  the  scope
of this year's project,  future efforts  in  this  direction  will   be  rewarding.

The Empirical Foundation  of Recreation  Demand Models:    The Traditional
Travel  Cost Model

     The recent  research   in  environmental  valuation has had a foundation
upon which to build.   The earliest work focused on the valuation of a single
recreation  site,  using  aggregate "zonal" data.

      "Let concentric zones be  defined around each  park  so that the
      cost of travel  to the  park from a I I   points in one of these
      zones  is  approximately  constant.     .  ..If  we  assume that the
      benefits are the  same no matter what  the distance, we have,
      for those  living near the park,  consumer's  surplus consisting
      of the differences  in transportation costs.  The comparison of
      the cost of  coming  from a zone with the  number of people who
      do come  from it,  together with  a  count of the population of
      the zone,  enables  us  to plot one  point for  each zone  on  a
      demand  curve  for the service of the  park (Hotel I ing 1948)."

                                     6

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 In  fact  the development of methods of estimating the  demand  for recreation
so  closely  paralleled the use of zonal models that the so-called travel  cost
method is often considered  synonymous  with the use of zones.

     The concept  of this original travel cost  model  took advantage  of the
fact that unlike other goods,  recreational  sites are immobile and users must
 incur specific costs to  access  a  site.   Thus,  travel  costs were proposed as
a proxy  for market price,  with consumption  of  the  recreational  opportunity
expected   to  decline as distance  from the site and travel  costs rose.
Clawson,   in 1959,   and  Clawson and Knetsch,  in 1966,  developed  the  travel
cost  idea  into an operational model  by  estimating  demand for a  recreation
site and measuring the  total  value  or  benefits of the site.

     This basic model  has been widely replicated and extended to account for
various complexities of  the  recreation experience.   The procedure is recom-
mended for  project benefit  estimation  in  the  1979 revision of the  Water
Resources Council's  "Principles and  Standards."   Thus a  long  evolutionary
process  has establ ished a precedent  for the use of travel  cost models  in
valuing  aspects of recreation activities.

     The  essence of the traditional  travel  cost  approach  to valuing  benefits
is shown in Figure 1.1.   The sum  of travel  costs  and entrance fees  act as a
surrogate for the price of the recreational trip.    The demand  curve  of a
"representative"   individual  is  estimated by regressing trips per capita  in
each zone against average travel  cost per  trip and other average charac-
teristics of  each  zone.   An  aggregate demand curve is then  formed by  com-
bining the  representative demand  curve with zonal  characteristics of the
population.    The  shaded area between  the  aggregate  demand  curve and  the
actual entrance  fee is viewed as a measure of  the  consumers' surplus  from
the  site.
    Price
 (traveI  cost &
entrance fee)
                        Recreation  trips/time  period

                  Figure 1.1:   The Recreation Demand  Curve

                                      7

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     The fundamental  problem  with  using the simple  travel  cost approach as
shown  above  is  that it  is defensible only  in  certain  rather restrictive
circumstances.    Much  of the research since 1970 has  expanded the travel  cost
model to a more  general recreational demand model, making  it more defensible
in a wider variety of circumstances.   In  addition,  because  its  role has  been
benefit estimation,  a closer correspondence to  axioms of welfare economics
has been established.   Development of  increasingly  sophisticated estimation
techniques  is present throughout this period.

The  Theoretical  Foundation  of  Recreation Demand  Models:    The Household
Production  Approach

     While the travel cost method has been applied to  empirical problems for
decades,   its connections with the  theory of welfare economics  have only
recently been  articulated.     With the   increased acceptance of  benefit
measurement by the economics  discipline  in the  1970's  came the need to  link
travel  cost valuations to welfare theory.   The travel  cost  method  had  rested
mainly on the presumed analogy between travel  costs  and  market  prices. In
the 1970's more  general models of individual  behavior,  such as  the household
production  function,  established the link between travel  cost and individual
utility maximizing behavior giving greater credibility to existing empirical
practices.

     The household production framework is not an approach to estimation but
a general  model  of  individual  decision  making.    Its antecedent can be  found
in the economics literature on the allocation of  household time among market
and  nonmarket  employment  (Becker,  1965;  Becker  and Lewis, 1973).   The ap-
placability of the  household production  framework  for recreation decisions
was  first  noted by Deyak and  Smith  (1978)  and  later  explored  by Brown,
Charbonneau  and  Hay,  (1978).

     The household  production  function takes a  broader  view  of household
consumption   than traditional  market approaches.    Commodities,   for which
individuals possess  preferences  and  from  which  they derive utility,  may not
be directly purchasable in the  marketplace.   In  fact some goods which can be
purchased may not yield  uti I ity directly but may need  to be  combined with
other purchased  goods and time to generate uti I ity.   Rarely are goods  com-
bined by the  household  rather than  by  firms  unless they require substantial
time inputs.   Thus,  time  is a critical  feature of the model.

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     One can then view the  household as  a  producer,  purchasing  inputs,  sup-
plying  labor,   and producing commodities  which  it then  consumes.   This makes
for  a  perfectly defensible utility theoretic decision  model  which can  be
expressed as

     (la)     max u(zj_,...,zn)

     (Ib)     s.t.   2 = f{xlf...,  x,,,,  tx)

     (Ic)           Y(tw,  w) +  R - ^  Pi  xj  = 0

     (1d)           T - ^ - tx =  0

where z1s are  commodities,  x's are market  goods,  and p their prices,  tx is
time spent  producing commodities,  tw  is time spent working,  w  is the  wage
rate, Y is  wage income,  R is nonwage  income,  and T  is total  time endow-
ment.     Included  in the above  series  of expressions  is the usual utility
function (la),    a budget  constraint (Ic),  a production function  for the z's
(Ib), and a time constraint (1d).    If  one of the z's  represents recreational
trips with  inputs of time, transportation,   lodging,  equipment,  etc., then  we
have the makings of  a recreational  demand model.

     A  major contribution of this framework is that  it  provides  a justifi-
cation   for  using the travel cost  model  in  certain instances,  as well  as a
way in  which to generalize  the traditional  model  to  incorporate  other  ele-
ments.    While  the  household  production framework  provides  a general  and
flexible way of presenting  the  individual's (household's)  decision problem,
restrictions are required  to  make the  model  empirically tractable.    One
difficulty  inherent in the general form  is that the marginal  cost of  pro-
ducing  a Zj   is I ike Iy to be  nonI inear.     The imp I icat ions of this for
estimation  and  welfare evaluation are explored  in  Bockstael  and McConnell
(1981,  1983) and an application can be found  in Strong  (1983).  If the  pro-
duction technology  is Leontief and there  is no joint production, however,
the marginal cost of  producing  a  z,  (e.g. a recreation trip)  is constant and
thus functionally analogous to  a market price.   Interpreting "travel"  as the
principal  input and ignoring  the  time dimension equates this model to the
traditional travel  cost model.    Travel  costs no  longer depend for their
credibility on being a "proxy"  for market price.    They are a  legitimate
component of the marginal  cost  of  producing  a trip.

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      It  is  important to  note that this model,  as  we I I  as a I I of welfare
theory,  is grounded  in  individual  behavior.  For  this reason, and other more
practical ones,   researchers have tended to  move  toward using  individual
observations  rather than zonal averages in  more recent applications.    The
zonaI -individuaI   observation  controversy will  receive  greater attention  in
Chapter 3.

     The  general   model  also offers a framework  from which other aspects  of
recreational  demand, such as the opportunity cost of time, can be  introduced
(Desvousges,  Smith  and  McGivney,  1983).    As far back as Clawson,   research-
er's knew time costs were an  important  determinant  of recreational demand.
However,   these  costs  have often  been  ignored or treated  in an  ad  hoc
fashion.   A treatment of  time, which  is  theoretically  consistent and empiri-
cal tractable,  is the subject  of  Chapter  4.

The Plan of Research for Part  I
     The conceptual problems which are  addressed  in  Part I  have been chosen
because benefit estimates have turned out to be extremely sensitive to their
arbitrary treatment.    In each  case attempts have been made to show the sen-
sitivity by  citation  to  existing  literature,  by use  of existing data sets,
or by  simulating behavioral experiments.    Also  we demonstrate,  by using
existing data  or simulation results,   the application  of each improvement
wh i ch we deve I op.

     Two criteria are used  in  the  development  of improved techniques: theo-
retical acceptability  and empirical  tractabi I ity.    Improvements  are  proposed
only  if  they can be  implemented with  accessible  econometric techniques and
with data which can reasonably  be collected with manageable  surveys.

     Part  I  makes  substantive  contributions to the  single  site or activity
recreation demand  model.    Several  issues  - such as  the treatment of time,
specification and functional form,  aggregation and  benefit estimation - are
explored.    This work forms the  foundation for the multiple site modelling
techniques  discussed  in  Part II.
                                      10

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                                  CHAPTER  2

               SPECIFICATION  OF  THE  RECREATIONAL DEMAND MODEL:
                   FUNCTIONAL FORM AND WELFARE EVALUATION
      In  the period  of only  a  few years,   a  number of theoretical  papers
concerning precision  in welfare  measurement  and  the  relationship  among wel-
fare  measures  has emerged.    Perhaps  the most  often  cited of these  is by
Wi I I ig  (1976),  who has shown that  the differences among ordinary consumer
surplus, compensating variation,  and  equivalent  variation  are within bounds
which are  determined by the  income elasticity  of demand  and the ratio of
ordinary surplus to total  income.   The  issue of  the accuracy of the approxi-
mation  has become  less consequential since  the work by  Hanemann  (1979,
1980b, 1982d),  by Hausman  (1981),  and  by Vartia  (1983). The  first two have
shown how  to recover  exact we I fare measures  from some common functional
forms of demand functions.    The  latter has developed algorithms yielding
numerical solutions  which  provide arbitrarily  close approximations  to true
welfare measures  for  functional  forms  which  have no closed  form  solutions.
The first part of this chapter provides a review of this I iterature on inte-
grability and exact welfare  measures.

     The second part  of  the chapter  addresses the choice  of functional
form.   While a  particular functional form may be consistent with some under-
lying preference  function,  it may  not  be  a  preference structure consistent
with  actual  behavior.    That is,  arbitrary choice  of functional  form  may
imply too specific a preference  structure and one which is  inappropriate for
the sample  of individuals.

     The sensitivity of benefit  estimates  to functional  form has frequently
been cited  in the  literature and may be far greater than differences between
Hicksian variation  and  ordinary surplus measures of benefits.  This chapter
suggests one means of addressing the choice of functional  form.   We show how
close approximations  to  compensated  welfare  measures can be derived from
flexible forms  of the  demand  function.    Emphasis  is given  to the choice of
functional  forms which are both  consistent with utility theory and supported
by the data.
                                     11

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The Intergrabi I ity  Problem  and  Demand  Function  Estimation

     There are  two general ways  to  develop utility  theoretic measures of
consumer benefits.   The first employs an assumed utility function  from which
demand  functions are derived  through the appropriate  constrained utility
maximization process.   The  other begins  with a demand  specification and
 integrates back to  a utility  function.

     The preferable approach  depends on whether the problem in question
 involves  a  single good or a  vector  of related  goods.    In general,  it  is
desirable to  begin with a demand function and integrate to derive welfare
measures.    As Hausman  points  out,   the only observable  information  is the
quantity-price  data,  data which can be used to fit demand  curves not  utility
functions.    Good  econometric  practice would suggest we  choose the best
fitting form  of the demand function among theoretically acceptable candi-
dates.    The demand  function  approach  is preferable  because  it allows the
researcher  to include as  choice  criteria how  closely  the  functional form
corresponds to observed behavior.    For these  reasons this  approach will be
used  for  single site models.  Unfortunately,  multiple good models pose
severe  integrabi I ity problems.   As such we are forced  in  the latter  half of
this volume to  employ  the alternative approach of first  choosing a  prefer-
ence structure and  then deriving demand  functions from that  structure.

     The conditions  for  integrating  back  to an  indirect utility function
from demand functions are  now well known.    Integrabi I ity  depends  on  solving
the system of partial differential  equations:

 (1)                           ani/Bp,. =  x,. (p,m)

where m  is  income,   p  is  the price vector,  and  xi  and  pi are the quantity
demanded  and  price  of the  itn  good.   The solution  is  called the  income
compensation function m(p,c),  where c  is the constant of  integration.   This
function  is identical  to our concept of the expenditure function,  if c  is
taken as an  index  of utility.   The indirect  utility function can  be  derived
by  inverting m(p,u)  to  obtain  u=v(p,m).    Hurwicz (1971)  has shown that  par-
tial  differential  equations of the  type  in (1)  have  solutions  if a) the
Xi(.)  are   single  valued,   d ifferent iable functions and  b)  the  Slutsky
symmetry conditions  hold:

                  3X../3P- + X-3x.j/3m = 3Xj/3p.j + X^S
                                     12

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      If the problem of  interest  involves just one good, the convention is to
assume that the prices  of  all  other  goods  (those not of immediate interest)
either are constant or move together so that these goods can be treated as a
Hicksian composite  commodity  with  a single price. This  price  can be repre-
sented by a price  index,  or set to  one  when  price  is unlikely to vary over
the sample.   The  problem  is now reduced to the  two  good case:  x  and a com-
posite commodity.    Since  a system of N partial differential  equations can
always be  replaced by a system  of N -  1  such equations by normalizing on
the price of one  good,  the two  good case  requires the solution of only one
differential  equation.   There  is only one element to the Slutsky matrix now,
so there is no question of  symmetry, and any function which meets regularity
conditions is mathematically  integrable  (although a closed form solution for
the expenditure  function  may not  always  exist).

     Mathematical  integrabi I  ity does not  necessarily  imply economic inte-
grabi I ity,  i.e.    that the  implied utility function be  quasi-concave.
Economic integrabi I ity conditions require that a) the adding-up restrictions
hold,   i.e.   p'x=m,   and  the functions are homogeneous of degree  zero in
prices and  income  and  b) the Slutsky matrix is negative semi-definite,  i.e.

                           3x./9p--  +  x.  ax./3m <  0.

Hanemann (1982d)  has shown that for the two good case the adding-up property
implies the homogeneity property,  so that  for  this case one need  only check
that  the  negative  semi-definite  condition holds.    However,   this  latter
condition is nontrivial;  its  violation may cause anomol ies to arise in the
calculation of welfare  measures.    Violation  of negative  semi-definiteness
conditions implies  upward  sloping  compensated  demand functions and meaning-
less welfare  measures.

Exact Surplus  Measures  for Common Functional  Forms

     Closed form  solutions to (1) are  possible for several  commonly used
functional  forms.    The  procedure  discussed  above and  outlined  in  the
Appendix 2.1 to this  chapter  has  been  used to  derive  parametric bivariate
utility models consistent with tractable ordinary demand functions.    In  what
follows,  the results of this  procedure when applied to the  linear, semi-log,
and  log-linear demand  functions  are presented  (for  reference  see Hanemann,
1979, 1980b,  1982b; Hausman 1981).
                                      13

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     Consider the three specifications

(2)                      X1 = a + Bpj/Pg + Ty/P2

(3)                      Xj = exp(a + Bpj^/pg + ry/p2)

                                                  2
(4)                      X  - exp(a)  (P/p)B  (y/p)Y
where a, B, and Y are  parameters,  p^ is  the  price of  the  good in question,
p2 is  the  price of the Hicksian bundle,  and y  is  income.   Henceforth p will
designate normalized price, Pi/p2* and m  normalized income, y/p2.
     The expenditure  functions (denoted m(p,u)) which  result from integrat-
ing back from each  of  the  above forms are presented in Table 2.1.  Inverting
the  expenditure  functions  yields indirect  utility functions,  v(p,m),  also
presented in Table  2.1.   It is also  possible  in the  simple two  good case to
retrieve the  bivariate direct  utility function,   utility as a  function of
goods  rather  than  prices  and  income.   For the  simple  two good  case,  the
Marshal li an demand  function for  xj  together with  the  budget constraint can
be solved for y/p2  and pj/p2 as  functions  of x^  and  x2.   Substitution  into
the  indirect  utility  function  yields the  direct one.    Knowledge of  the
direct utility  function  implied  by  an estimated  demand  function is partic-
ularly useful  as  it provides  insight into  the  properties of the preference
structure implicitly assumed.
 o
     The  compensating  and  equivalent  variations  for  price  changes  from
pV to p] can be  derived by calculating  the  change  in  the  relevant expendi-
ture function when price changes.   Thus

                    CV = m(p°, U°) - m(p , U°)

and

                    EV = m(p°, U  ) - m(p , U  ),

where U1  takes the value  of the  indirect utility function evaluated at  p1
and m°.   The expressions for  CV  and  EV as  well  as that for ordinary surplus,
i.e. the Marshal I ian  consumer surplus,  are also recorded  in Table 2.1.

     Not  all  estimated demand  functions corresponding to  the functional
forms  in  (2),   (3) and (4)  can  be integrated  back to  well  behaved  (i.e.
quasi-concave)  utility  functions.    The  negative  semi-definiteness  condition

                                      14

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for these functions  translates  into restrictions on the  functions'  co-
efficients.    These restrictions are  given  in Table 2.1.    While frequently
ignored,   the  conditions are critical.    If,  in a given empirical  problem,
estimated coefficients  violate  these  conditions,  then one can  presume that
the model is misspecified  in some way.   That is,  the estimated coefficients
imply  an upward  sloping  compensated  demand  function and  are therefore
inconsistent with  utility maximizing  behavior.

Evaluating the  Elimination  of  a  Resource
     The formulas  in  Table 2.1  presume  interior solutions,   i.e.   x, and x2
strictly  greater than zero.    Frequently,  however,   we are interested in
evaluating  situations when  x,=  0.    For example,  we  may  wish  to calculate
the  lost benefits  associated with elimination  of access  to  a resource.
Alternatively  the  conditions at  the axis may  be  important  in assessing  a
change in a quality  aspect  of a  good (more  on this  in Part  II.)

     Typically,   economists have  evaluated the  losses associated  with  the
el imination of a resource  in the same way that they have evaluated the gains
or  losses of a  price change.    The price  is simply assumed  to  increase
sufficiently to drive demand to zero.  This  practice can generate anomolies,
since resource elimination  really involves a restriction on  quantity rather
than a de facto^ change in price.    For  many functional  forms,  the price which
drives the Marsha I  I ian  demand  to  zero  is different  from the price which
drives the  corresponding compensated demand  to zero.   When  the two cut-off
prices do coincide,    it  is generally because the cut-off price  is infinite.
An  infinite cut-off price  frequently (although not always)  implies  that an
infinite  sum is necessary to compensate for elimination  of  the good.

     Consider  first the  linear  demand function,  an  example of  a  form for
which a  finite cut-off  price  exists.    If  the Marshallian  function  is ex-
pressed  as  xj= a +  BP + Ym tnen   its  cut-off price  is Pm  =  -(a + yml/p. The
corresponding Hicksian demand  is   x^ = ^exp(Yp)u - B/T W1th  a cut-off price
Ph = Infl  - 2 lnY -  In u ^   For  purposes  Of  comparison with  the Marshallian
demand curve,  it is useful to substitute  V(p°,m°)  for u in the expression
for pn so that we identify the particular  compensated  curve  which  intersects
the Marshal Han demand at the initial point (x  {p°,m°),pu).  This  gives  us

                    -h _ Ins - Inv  - 1n(x° + 6/v) + Do
                                     16

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The difference between  Ph and Pm is (Ing - lny - ln(X° + B/Y))/Y - X°/B .

     There is some ambiguity  as  to  which P  should be used in calculating the
compensating  variation  associated  with  the  elimination of  the  resource.
Thinking  about  the  problem as one of a  quantity  not a price change suggests
the  question  "How much compensation would leave you as well  off  if  your
access  to  the  good  were denied?"  This  Implies  a movement along the compen-
sated demand function  to  its intersection with the  axis.  It  is this latter
interpretation  which  is  advocated  by Just,  Hueth  and Schmitz  (1982),  and
which seems the most convincing.

     The  implication of the  choice of p in  calculating CV  is  a potentially
important one.   All usual  comparisons of CV,  EV and  OS are made on identical
effective  price  changes.    When  considering  the  elimination of  a resource,
the usual  relationship  between  CV,  EV,   and OS  is now distorted.   OS  is  the
area  behind  the  ordinary  demand  function  between    p°  and pm   (the  price
which drives ordinary  demand to  zero).   CV  is  the  area behind  the compen-
sated demand curve which  passes  through  p°,   but not between the same bounds
as  the  ordinary samples.   Instead  we must  integrate between pu and p" (the
price which  drives  the Hicksian demand  to  zero).  EV must  logically  be  de-
fined as  the area  between p° and pm, behind  that compensated  demand  curve
which passes through pm .

     Because the bounds of  integration for CV are  not  the same  as for  OS  and
EV, the usual  relationship between the   latter and  former   is destroyed  and
Willig's  bounds no  longer hold.     Whether  or not  the difference  is  of
practical  significance  depends  on the relative  sizes of the parameters  and
can only  be  determined empirically.   Unfortunately  the greater  the differ-
ence between ordinary surplus and  compensating  variation,  the greater the
difference in the two  CV measures.

     For some functional  forms,  there exists  no  finite price at which demand
is  zero.   This  does not,  however,  mean  that the area  behind  the respective
demand  curve  is necessarily  unbounded.    In  some  cases, the  limit of  the
demand  for  xj  is zero  as  p1  + » and  thus,  the area  behind  the  demand curve
converges  to  some  finite  value.   In other cases the  limit of  xj  does  not
equal zero as p1 •»•  »,  and the area behind the demand  curve is infinite.
                                      17

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     To understand  this phenomena,   one needs to consider  the concept  of
essentiality.   Marty equivalent definitions of essentiality exist but perhaps
the most  intuitive and descriptive is the following:

           A good, Xj,  is essential  if,  given an  initial consumption

           vector  (x°, Xg, ..., x°)  there exists no subvector

           (x,, xJ,	x') such that
             L.   o        n
           II ( X°  X°       X° \ - U (0  X'  X'      X' \
           U \ A« 9 AA 9 • • • 9 A )   U V U , An y AO 9 • • • 9 A / *
              A.   £.        n          ^   *3       n

An equivalent  definition  is that there  exists no  finite sum which  can  com-
pensate for the elimination of  Xj.   These  definitions  are  both  equivalent  to
the condition  that for  x, to  be  essential

                                lim  xj(p.u)  ?  0
and for x to be  nonessential
                               lim xj(p,u) = 0.
 It should  be noted that these  definitions are  in  terms of the compensated
not the ordinary  demand  function.    In fact,  there  is not a  perfect  corres-
pondence between  the  limiting conditions for the  compensated demands  and
those  for  the ordinary demands.    There  exist  preference structures  which
 imply  ordinary  demand  functions which do  converge  but compensated  functions
which do not.

     An interesting example for  illustration  is  the general  CES form  for the
direct  utility  function,   u  =  (xj +  x£) 'p,  which   generates   the  following
functions:
                                      18

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                                       1
                    v =  (p1~a + p2~°) °" y

                                       1
                         / l-o  .   l-ax l-o
                    m =  (p1   + p2   )    u
                        P2
                              "I
where  o = l/{l-p).
Note that Tim xm = 0    for all values of a.   Correspondingly  lim  x   =  0
          PI+"                                                 P!*-

and lim m(.) is finite if o> 1 but lim x  7* 0  and  lim  m(«)  = », otherwise.
    PI-H»                           PJ-H-            pj-xo

     All of  this  is of importance not  only in calculating losses associated
with elimination of  a  resource,  but  also in assessing the  relative merits of
different functional forms.   Essentiality  is a property of preferences which
may  not be  very  applicable  when dealing  with recreational  goods.   It is
difficult  to  conceive of  a  recreational   experience which  is  indeed  es-
sential,  i.e.  its  elimination   would  reduce  utility   to  zero.    Thus,
functional forms which imply essentiality are  probably poor choices.

     In  this  light, let  us  examine  the last  two  "popular" functional forms
to  see  what they  imply  about the essentiality property.  For  the  semi-log
demand  function,   x1 = exp(a  + Bp +  ym),   there is  no  finite  price at which
the demand  for  xj  is  zero.   However,  the  limit of compensating variation is
finite  as  PI-**,  and  thus xj^  is  non-essential.

     For  the  log-linear  demand function, x^ - eap^, the  price  that  drives
the demand  for  xj  to  zero is  also infinite.   For  relatively elastic demands
compensating  variation converges  to a  finite quantity  as P}-*»  .   However,
when 0>B>-1 the  compensating  variation  associated  with  elimination  of  the
resource is infinite.  This implies  that x^ is  an  essential good,
                                      19

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Functional  Form Comparison

     While  there  are  no previous  studies  where compensating variation
measures are  compared  across  functional  form,   there are  some  which  document
the  potential differences  in ordinary  surplus estimates  which arise when
different functional  forms  are estimated on the  same  data and  others  which
simply address the  issue  of choice  among  functional form  in  recreational de-
mand models.     In a  study of warm water  fishing  in Georgia, Ziemer, Musser
and  Hill  (1980) assessed  the    importance of the  functional  form on  the size
of ordinary  consumer surplus estimates.   They chose to  consider  linear,
semi-log and  quadratic forms  and  found average  surplus per  trip  estimates  of
$80, $26  and  $20  respectively.   The researchers estimated a Box-Cox trans-
formation to  discriminate among  the three functional  forms and determined
that the semi-log was  preferable.3

     Two other  papers  of note  identified the  semi-log  function  as most ap-
propriate.     Both  papers addressed functional form  in the context of the
heteroskedasticity  issue  (a more  detailed discussion  of these papers can  be
found  in Chapter 3).   Vaughan, Russell  and Nazi I la (1982) tested for appro-
priate  functional  form and  heteroskedasticity, simultaneously.    They used
the  Lahiri-Egy  estimator  which is  based  on  the Box-Cox transformation, but
also incorporates a test for nonconstant variance.     They concluded that
both the  linear heteroskedastic and  linear homoskedastic  models were  inap-
propriate.   The  semi-log form which  did  not  exhibit heteroskedasticity was
found to be preferable.    In  a second paper  Strong (1983a)  compared the  semi-
log  model with the  linear model  based on the  mean squared error  in  predict-
ing  trips.   She also found that the semi-log function  performed  better.

     Another  consideration  of  the  functional  form  issue  can  be   found  in
Smith's  (1975b)  analysis of  visits  to  the   Desolation  Wilderness area  in
northern  California,    He  examined the  linear,  semi-log  and  double-log
functional  forms  for  wilderness   demand  using  the zonal  approach  with  64
origin zones  from California,  Nevada  and  Oregon.   While  the R2  is not an ap-
propriate test to compare specifications  with  different  dependent variables,
the  linear model exhibited such a low R   that it was not considered further.

     To try to  establish more  conclusively which  functional  form was more
appropriate,   Smith chose  to use a  method  suggested  by Pearsan which  discrim-
inates between non-nested competing regression  models.   Smith found  that  in
his  sample of wilderness  recreators he  was able to reject  both  the  semi-log
and  the double-log  functional  forms based on  this criteria.   His conclusion
                                      20

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that the travel  cost model  may be  inappropriate for  wilderness recreation
modelling may be correct but is too extreme a conclusion to be supported by
this analysis.    Even  if the  Desolation Wilderness area  is  representative of
other wilderness recreation  problems,  the alternatives tested in this study
are by no means exhaustive.   The functional  forms chosen  are but three among
a vast  array of choices.     Additionally,   Smith's  poor statistical  results
could well  be  a  reflection  of  other specification  problems inherent in his
conventionally designed  zonal  travel cost model.      (See  discussions in
Chapters 3  and  4.)

Estimating  a  Flexible  Form and  Calculating Exact Welfare Measures

     Each  of the above studies was  concerned with  calculating  ordinary
surplus   measures   from commonly estimated  functional  forms using zonal
data.    These studies either implicitly  assumed or  explicitly demonstrated
that consumer surplus estimates would differ  depending  on  the  choice of
functional  form.    Not surprisingly, compensating  (or equivalent)  variation
measures derived  from different functional forms may also exhibit  vast
d ifferences.

     In the  previous  literature,   the focus seems  to have been one of
 identifying  a  means  of choosing which of the popular functional  forms was
preferable.     If  it  were  possible  to select one,  then the  exact  welfare
results of the previous section  could  be directly applied.    Many of the
articles  appear to  point to the semi-log  as  a desirable form,  yet the
evidence is  far  from  conclusive and there  is no  reason to believe that the
same form would necessarily  be  appropriate for all  situations.

     It would  be far preferable  to consider a wider array of functional
forms than the three  discussed above  and  to  allow  the data to choose among
them.    One way to access a slightly broader  range of  functional  forms is to
estimate a flexible  form  such  as the  Box-Cox transformation.   However,   Box-
Cox forms  do not in  general  integrate back  to closed form expressions for
the expenditure or  indirect  utility functions.   A solution to  this problem
can be  found in  the  recent  work  by Vartia (1983),  among others, who demon-
strates a  means  of obtaining extremely close  approximations to  compensating
variation when exact measures are not possible.   The  procedure  uses a third
order numerical  integration  technique to  obtain  an  approximate  solution to
the differential  equation.
                                     21

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     The Vartia  algorithm,   and  others  like  it,  is based on an  intuitively
appealing  proposition.   The  ordinary and compensated  demand  curves are very
close  in the  neighborhood  of their  intersection.    The difference  in the
curves which  occurs  with a  movement  away from  that intersection  reflects an
adjustment in consumption   in  response to  additional   (compensations in)
income.    Therefore,  it  should  be  possible  to trace  out  the  compensated
curve,  approximately, by  a)  starting at the intersection point of the two
demand curves, b)  considering a  very small  (incremental) change In price, c)
calculating  the  approximate money compensation  associated  with that change.
d)  awarding  that  amount  of income to  the  individual  and  then shifting his
ordinary demand  function, e)  designating  this new  consumption level  at the
first  price  increment to  be  a point on the  compensated demand function and
f)  starting  the  process  once  again  with  a   new  price  increment.    This
procedure  is described graphically In Figure  2.1.

     The only  step of any  difficulty in this  procedure is {c),  calculating
the  approximate  money  compensation  leaving  utility  unchanged,  which  is
associated with  the  small price change.  Of  course this is the very problem
we  set out to solve, since this is the  definition  of  variation.   We can not
calculate  this number directly but we have  information on the  bounds of this
compensation.   The  compensation  for any  price change will  presumably be
greater than  (or equal to)  zero  and less than (or equal to) the total market
value of the  lost  (or gained) consumption, (p*  - p°)x°  .   The latter number
is  an  upper bound which  would equal  the  compensation  if, for example, the
given  quantity of  x  consumed  were  essential  and x  had  no  substitutes.   Ap-
proximation  algorithms  employ  iterative  techniques   to  calculate  income
adjustments  using  an  interpolation  of  this  upper   bound   as  a  starting
point.  While  Vartia's procedure  will  handle  systems of demands and multiple
price changes, we  describe heuristically the one  equation,  one price change
case here.

     The Vartia  procedure requires  the  following initial  information:   the
specific form  of the ordinary demand function(s),   the  income  level  and the
initial  and  final  values  of the price(s).    To Implement  the  procedure one
must also  choose  the number of steps,  M, one wishes  to make  in  moving from
the initial to the final  price.   The approximation will, in general, improve
with more  steps  (and thus smaller increments)  but  rising computer  costs and
rounding error will eventually take their toll.

     As pointed out  above,  the difficult task in the  procedure is the calcu-
lation of  the  appropriate income compensation  to  accompany each  price  step.
                                      22

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This  is  accomplished through an  iterative interpolation scheme of  the fol-
lowing sort.  For  any  given  price step  (p. to Pi+1)  define the first guess
at the income compensation for that step by

                                  ) + x{p  y )
                                J                     -
where PJ+I and PJ are the upper  and  lower  bounds of the price step and yj is
the income associated with  the  starting point for this step (i.e. the income
associated with  the ordinary  and compensated demand intersection as at point
A  in  Figure  2..1).   If  this income adjustment were  awarded then compensated
demand  would  be x(pj+1,y,- + Ay^), out  we  know  that  this  would  be  an over-
adjustment,  i.e.   utility   will  have  increased   rather   than  been  held
constant.  The second guess  at the income  adjustment will be  based  on  the
average  of  the  two  new  consumption  levels  *(pj+i»yj) which  implies  no
compensation  (and  thus   is  a  lower  bound)  and x(pj+js yj + Ayi) which  is
based on too much compensation.  Thus
             Ay2  =
The  iterative  procedure  progresses  with  each  new  guess   at  the  income
adjustment for this price increment equalling

               xfp.  .,y.  + Ay.  .)  +  x(p.  ..y.,  + Ay  ?)
        Ax/   _     J+1   J      K"1        J+i   3	ll£jf fn    -  n  )
        A	                5                     ip      p  ),
until  the  Ay^  converge,  i.e. Ay^ - Ay^,^ < convergence  criteria.    Once  the
convergence criteria  has  been  met at Ay^, we  shift  to  a new ordinary demand
curve  and  a new point  has  been  identified  on  the old  compensated demand
function, x{pj+1, yj+i) where yj+1 = yj + Ayk.    The  compensating  variation
of  the total"  price" change is approximated  by summing Ayk over all  N  price
steps.   This  will  be  equal  to yN  -  y0 in the  above notation.  A computer
algorithm  developed by Terrence  P. Smith to  implement Vartia's procedure is
presented  in Appendix  2.2.

     The Vartia  approximation  was  tested for  a functional form for which ex-
act  compensating variation expressions  exist.   The  Vartia  measure improved
with  the number  of steps  chosen in the  algorithm,  but quickly came within a
                                      23

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half  of one percent  of the true measure.  Thus  the  approximation  would seem
to  meet  an  acceptable tolerance  criteria  at low computing costs.

      In  what follows,  we  will  demonstrate how  this  approximation procedure
can be used with  the Box-Cox transformation.     The  approach  is equally ap-
plicable to  other  forms (flexible or not)  for a  single  equation  or system of
equations.    It should  be  noted,  however,   that the Vartia approximation does
not circumvent either mathematical  or  economic integrabi I ity conditions.
These conditions must  hold for the  results of the procedure to  have  mean-
ing.   The Vartia  technique provides a  close approximation  to  compensating
and equivalent variation  measures  when  no closed form solution to  the dif-
ferential equation  in  (1)  exists  or can easily be found.

An  I I lustration
     To  illustrate the application of this method for choosing functional  form and
calculating welfare  measures, the Box-Cox transformation was  estimated for a set of
sportfishing data.   The Box-Cox approach  was  chosen because of its wide  familiarity
and ease of estimation.   However,  as  noted above,  the  procedure  for  deriving welfare
measures is equally applicable to other  less restrictive functional  forms.

     All  individuals in the group took at  least  one trip of greater  than 24
hours on  a  party/charter boat.   This is  a subset of a  sample  of 1383 sport-
fishermen who  responded to a mail  questionnaire asking  details of their 1983
sportfishing activities in Southern  California.    A complete  description of
the  data  can be  found in National  Coalition for Marine  Conservation (1985).

     For  purposes  here, an  individual's demand  for party/charter  trips  (x)
 is  considered to  be a  function of  costs of  the trip  (c) ,   income  (y)  and
catch of target  species  (b) .

     Three  models were  estimated  using the same data  set.    The  first con-
strained  the functional form to be  linear,   the second employed  a semi -log
function  and  the third  used  the more flexible Box-Cox transformation  on  the
dependent variable  so that the  regression took the form:
                                  A
                                        - Cz.
where  x  is trips and z  is  the vector of  explanatory variables.    The param-
eters  to be  estimated  included the usual  coefficients (the 3 vector) and the
Box-Cox parameter,  y .
                                      24

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     The  linear  model  produced  the  following estimated equation  (t-statis-
tics in parentheses):

             x = 6,57 - ,0045 c + ,0000189 y + ,179 b    R2=S17S
                (6.38) (-4.08)     (1.67)     (1.54)

In contrast, the estimated semi-log demand function looked like

           In  x - 1.66  -  ,00102  c + .0000034 y +  ,0325 b    r2=,29,
               (10.66)  (-6.09)      (1.96)     (1.85)

Finally, the Box-Cox estimation produced the following results


            x'14 - 1    1.91  -  .0012 C +  .0000042 y  +  .04 b R2=27.
               rn(9.79)  (-5.83)     (1.93)      (1.82)
     In  this  particular  example,  the  Box-Cox produced  a  A  close  to  zero.
This  result  is  somewhat  consistent with  the  fact  that the  result of  the
semi-log  function  appear  superior to  that of  the  linear  equation.   This
should  not  be  construed as  a general  endorsement  of  the semi-log  demand
function,  since  other  applications  of  the  Box-Cox  transformation  have
provided a wide range of values for \.

      In  Table  2.2,  the results of this  experiment are presented.  The esti-
mated  coefficients from the  linear and semi-log  models have  been  used   in
conjunction with the expressions   in  Table 2.1 to  calculate  estimates of
ordinary surplus,  and  compensating and equivalent variation.    The compu-
tation  process  is  explained  in Appendix 2.1.   The Vartia algorithm has been
used to obtain  "approximate"  measures  of compensating  and equivalent varia-
tion and ordinary  surplus for the Box-Cox model . The algorithm is presented
 in Appendix  2.2.

     Some important points are  worth noting.   First,  these welfare measures
seem large.    It  should  be remembered that the  sample included only those who
took  longer  than one day trips  and are therefore likely to  be rather wealthy
 individuals.   In  fact,  the mean income  of  this  group  is $58,000.  Addition-
ally,   there are   reasons why  welfare measures  calculated  from  estimated
coefficients may  produce overestimates  of  the true  values.   These consid-
erations will  be discussed in  Chapter 5.
                                     25

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     The  important  point for consideration  here is that  if  one were arbi-
trarily   to  choose  between   the  linear and  semi-log  specification   in
estimating the  demand function, widely divergent   benefit estimates would
emerge.    In the case  above there  is only a 3 to  5% difference across welfare
measures  (CV,  EV, OS)  for any one functional form,  but  a  16  to 19% differ-
ence  between the two most commonly  used functional  forms.    The  Box-Cox
transformation offers a  means of choosing  among a continuous  range  of
functional forms.     In the example  above,  it  seems  to support the semi-log
function.    In other  cases we have  tried,  where neither the  I inear nor the
semi-log  results  appear  superior,   the Box-Cox ana lysis  often selects an i
significantly different from  either zero or one.   Then the Varita  routine  is
necessary  to calculate compensating  and equivalent variation approximations.

     While definitional  differences  in welfare  measures will  be of greater
concern in  problems  with  larger  income  elasticities (Wi I I ig,  1976),  bounds
on these  differences are well developed,  at least for simple models.    The
potential   differences from  functional form,  however,  may not be so  well
appreciated.
                                  Table  2.2


                              Welfare  Estimates

                 Calculated from  Different  Functional Forms

    (annual average estimates for a  sample of Southern  California sportfishermen)


                                       Functional  Form
                             Linear          Box-Cox         Semi-log
Compensating
   Variation                   8339            6950            6999
Ord inary
   Surplus                    8042            6812            6877
Equ ivalent
   Variation                  7899            6779            6763
                                      26

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                           FOOTNOTES  TO CHAPTER  2


1     LaFrance and Hanemann  (1985)  describe the process of  obtaining  direct
     uti I ity functions from estimated demand functions for systems of demand
     equations.

o
     There is some disagreement in the  literature as  to the  precise  form  of
     the compensating and  equivalent  variation  expression.   All  agree  that
     compensating and equivalent variation must  be  of the same sign.  How-
     ever,   differences of opinion  exist as to whether the  variational
     measures  have  the same  or  the opposite sign  as the  uti I ity change.
     Here we adhere  to the convention used by Just,  Hueth  and Schmitz  (1982)
     which seems most  closely aligned with the original  description  of
     Hicks.    Compensating  and equivalent variation  are positive  (negative)
     for price changes which  generate  increases  (decreases) in utility.

     The  Box-Cox functional  form  Is  a   limited  flexible   functional  form
     developed by Box and Cox (1962) using a  transformation of the dependent
     variable.   The  transformation  is  defined  as
     so  that the regression  equation  can  be written as

                               y(x)  - xS  + e,

     The interesting  feature  of the  Box-Cox  transformation is  that  when x
     takes  the value of 1, the above expression is just a linear function of
     y in x. When X = 0, y^x'  is not strictly  defined but y(x'  is continuous
     at
                       x = 0 since lin y-^ = log y.
                                   x-»0    x
     Box Cox models  are estimated by  maximizing  the maximum  liklihood
     function with  respect  to  the B'S and the x.   Thus  the  functional  form
     is   not strictly  imposed  and  one  can  establish  confidence  intervals
     on  x  which allows testing hypothesis  about functional  form.
                                     27

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The Lahiri-Egy  estimation   is  an extension of the  Box-Cox transfor-
mation.    It  introduces an additional parameter which allows one to test
for the  presence of heteroskedast icity joint ly with  functional  form.
The estimator assumes  that the  error  in  the  model
is distributed such that the expected value of  E.J  is Z     u^  where Ui
is normal with mean,  0 and variance, o2, and Z^ is some variable which
varies over  observations  (and  is likely  related  to one  of the x's).
The variance of e,-  is  then c2 Z* .  Consequently, if 6 = 0  then the  var-
iance of E is  homoskedastic;  if 6 i  0  then there  is heteroskedasticity
in the model .

Thus  the  Lahiri-Egy estimator  uses  a maximum  likelihood procedure to
estimate  the  Box-Cox  transformation   under   conditions   of  potential
heteroskedasticity.  The  likelihood  function  is maximized with  respect
to  e, A,  6,  and  a .
                                28

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                                 APPENDIX  2.1

          DERIVATION OF  SOME  UTILITY THEORETIC MEASURES FROM TWO GOOD
                                DEMAND  SYSTEMS
As Hausman has so bluntly,  and some what unkindly,  suggested

          From an estimate of the demand curve,  we  can  derive
          a measure of the  exact consumer's surplus, whether
          it   is   the   compensating    variation,   equivalent
          variation,   or  some measure  of uti I ity change.  No
          approximation   is  involved.    While this  result  has
          been known for a  long time  by  economic  theorists,
          applied economists have only  a  limited awareness of
          its  appl icat ion.

     a)    Following  Hausman's example,   we can begin with  a  demand  function
where quantity is a function of  price  and  income both deflated  by the price
of the  other  good.    Letting p and m stand for the "deflated" price and
income,  and using Roy's   identity  then

(A1)                        xi  =
Now, this partial  differential equation must  be  solved.    Hausman uses the
method  of  "characteristic curves".     Using the notion  of compensating
variation,   one  can consider  paths (designated by t)  of price  changes and
accompanying income changes,  such that utility is  left unchanged  as in the
fo I Iow i ng:
                 sv(p(t),m(t))    dp_  _   -av(p(t),m(t))    dm
                     apTET        Ht          &mTtnUt
                       v  -          -          _   dm
                       xl  ' • W7m  '  "dpTHf  "  lp
                                     29

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This  gives an ordinary  differential equation  which  in  many cases can  be
solved with  fairly  standard  techniques.   As  Hausman  shows,  the solution  to
the differential  equation

                    dm/dp = a +  pp +  y"1        (linear case)
1s
                        m(p)  = ceYP  - 1 (pp +1+  a).
                                       Y       Y
The only confusion is  in dealing  with, c,   the  constant of  integration.
Clearly  c wi I I  not  be a function  of any of the  parameters in the  demand
function but it will certainly be  a function of the  uti I ity level.    In a
sense it doesn't matter what function as  long  as  it  is   increasing  and
monotonic, since we have no way of measuring or interpreting  absolute levels
of uti I ity.  As a consequence  Hausman  simply substitutes U°for c which  is
fine  as  long as  everyone uses  u° only  for  ordinal  comparisons  and  does  not
try  to  interpret  the  absolute   level  of  u°.     In  some circumstances
interpreting c as  equal  to u° will  lead to  confusion because  utility will
appear to be negative.   There  is no  fundamental  problem,  however,  as  long  as
ac/au >  0, the  scaling of u"  is arbitrary.

      b)   Once the  expenditure function  is obtained from solving the  differ-
ential equations the  indirect  utility function  is  usually easy  to  obtain  by
solving  m(p,u°) for utility  giving   u =  v(p,m).   For some demand  functions,
it  is easier to integrate back to  the  indirect utility  function first,  in
which case the expenditure function  is  obtained by solving v(p,m) for income
as a  function of  utility and price.   The  three examples below demonstrate
how straightforward this can be when there are closed form  expressions  for
both  indirect utility and expenditure functions:
(A4) m = exp(Yp)u°- -( pp+ - +a)  =>  u=exp{-yp)(m+ -(pp+a+p/y))    (linear)
(A5) m . . I 1n(-YU°- Jt exp(0p+a))  =>  u=  "exP^) -  exP(6P+s)   (semi-log)
(A6) m = [(l-Y)(u0 +— )]  T^7  =>   u  =  "p   +             (log-linear)

     c)    Once  the expenditure function  is derived, the Hicksian  demand
function together with compensating and equivalent variation measures are of
course quite accessible:
                                     30

-------
                                            ov
                                VH    am(p,u ).
                                AI    "•""""^S^^^™^^""
                                 1       Sp
Compensating and equivalent variations are, by  definition



(A8)             C = m(p°,u°)  -
(A9)             E = mtpO.u1) - mfpi.u1)   =   mtp0^1)  -  m°.
Thus  they  can be  solved  for directly from the  expenditure function.   (Note
that  Hausman  defines C and  E with  reversed  sign.   The  above definition is
more  in  keeping  with the original  Hicksian  definitions and has the  property
that  the  sign of  C  =  sign  of  the welfare change  associated  with the  price
change.)   To  simplify expressions  and  to obtain actual  values  for C and E,
u° = v(p°,m°)  and  u's v^-(p^,m ) must  be evaluated.

     An example  is presented for the  linear demand, where

                                      i
                     m = exp(Yp)u° -  - (a +  8p + P/Y)  ;


      c =  exp(Yp°)u° - -(a + pp° + P/Y) - exp(Yp1)u0 + -(a +  ep1  + -£)


  =  exp(YP°)  exp(-YP°)(a *  Pp° * ym + -fy - -  (a + ep° + -£)
                                                   - + —yi  + (— +  —K]  - m
                                                       Y<-      Y    Y*-


                       (7-+-^)  -  expEYfp1  -  P°)3(-f +4) .
     d)    The  one  remaining  function  of  interest  is  the  direct  utility
function,   u(x^,X2),   which  is of  interest  because  it  best  portrays  the
properties  of the preference function  being assumed.   The task is to convert
                                     31

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a utility function  in  (normalized  price)  and income  into a  utility  function
in terms  of x^  and  X£-    Since we have two functions which relate the  x's
with p  and  m,   i.e.  the  Marsha I I ian demand  function  for  x, and the  budget
constraint,   it  is conceptually possible to make the transformation.    One
must first  solve XT  = f(p,m)  and  m =  pxj  +  X2  for  p  =  g(x1,X2)  and  m =
h(x^,X2), and  then the substitution into the  indirect utility function  is
straightforward.

     As an example,  consider the linear case where
                               x,  = a + 0p +
then
                  P =
                       x,  -  a  -  yni      x,-  a -
                                     X.  —  ct -
                            =>  P =   -TT
                               Xl  " aXl " YX2X1 +
               m =
                                xl   ~  aXl
By substitution


                                               +   B  +   2>
    u =  exp(-Yp)(m +  -(ep  +  a + -) =  exp(	     	^4.
                      Y          Y              r     -

                                      y( a + Y^O " xl  '
                                  exp[.
                             2      r      RT
                            Y
                                     32

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                                      APPENDIX 2.2


                     COMPUTER ALGORITHM FOR OBTAINING COMPENSATING AND
                       EQUIVALENT VARIATION MEASURES  FROM ESTIMATED
                              MARSHALL I AN DEMAND  FUNCTIONS*


******************************************************************************.
*A  COMPUTER  ALGORITHM  FOR APPROXIMATING  CV  AND  EV FROM  ESTIMATED  DEMAND
* FUNCTIONS.   CALCULATES   NUMERICAL   SOLUTION  FOR  SYSTEM   OF  DIFFERENTIAL   EQUATION
*
* BASED  ON  ALGORITHM  BY  VARTIA  (ECONOMETRICA,  VOL  51,   NO  1,  1983)
*  WRITTEN  IN  VS/FORTRAN  (FORTRAN   77  - ANSI(1978))
*       T.   P.  SMITH,  UNIVERSITY OF  MARYLAND,   COLLEGE  PARK,  MD
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ _|_
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Hs^^_|_
* PROGRAM  REQUIRES   STATEMENT  FUNCTIONS  (IN  LINES   10-200)  WHICH  CORRESPOND
*  TO  MARSHALL I AN  DEMAND  SYSTEM.   FOR  EXAMPLE,  IF   X1=BO+BI*P+B2*Y  AND  B0=2,
* Bl=-5,  B2=6,  THEN  THE  FOLLOWING  SHOULD BE  ENTERED
*     10          X1 (P1, INCOME)=2-5*P1+6*INCOME
* A  SYSTEM  OF  UP TO  20  EQUATIONS  CAN BE  ENTERED   IN  THIS  WAY.   THE FUNCTION
* CALLS  THROUGHOUT  THE  PROGRAM  MUST  BE  MODIFIED  TO  REFLECT THE  APPROPRIATE
* ARGUMENT  LIST  FOR  THE  FUNCTIONS  BEING  USED.  THE  #  OF  EQUATIONS AND  THE
*#  OF  STEPS FOR  THE  PRICE  PATH  MUST BE  SUPPLIED.   AVOID A LARGE #  OF  STEPS
+  (>500)  AS  ROUNDING  ERRORS  CAN  BECOME  SERIOUS.
+  SAMPLE  PROGRAM  BELOW  DEMONSTRATES  TWO  GOOD, ONE  PRICE  CHANGE  CASE.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ _|_
       DOUBLE   PRECISION   P(20,500),Y,XC(20),INCOME,P1,P2,P3,P4,P5,P6,
      *P7,P8,P9,P10,P11 ,P12,P13,P14,P15,P16,P17,P18,P19,P20,X1,X2,X3
      *X4 , X5 , X6 , X7 , X8 , X9 , X I 0 , X11 , X1 2 , X1 3 , X1 4 , X1 5 , X1 6 , X1 7 , X1 8 , X1 9 , X20 ,
      *PSTEP(20),XT(20),TERM(20),DIFF,EPS,SUM+NEWY,YO
####################             STATEMENT  FUNCTIONS           ##################"
  10   X1(P1, INCOME)=EXP(3.56-.019*P1-. 027*INCOME+.00026*PI * INCOME)
*20   x2(PI,P2,INCOME)=(P1/P2)*(INCOME/(PI/P2))
*30   ETC.
* * * * * * * * * * * * * * * * * * * *             CONVERGENCE   CRITERION           #################
       EPS=O.OOOI
* * * * * * * * * * : : # # # # # # # # #                  PROBLEM   SIZE              sit******************
       WRITE  (6,1)


             This algorithm was   developed  by Terrence P. Smith, Department  of
             Agricultural  and Resource Economics,  University  of Maryland,  College
             Park, Maryland.

                                           33

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 1     FORMAT  ('  ENTER  THE #  OF  EQUATIONS  IN  THE  SYSTEM',/,
     * •  AND THE  #  OF STEPS  FOR  THE PRICE PATH')
      READ  (5,*)  NEQ,N
      WRITE  (6,2)
 2     FORMAT ('  SPECIFY  THE  INITIAL  AND  FINAL  VALUES FOR  EACH',/,
     * '   PRICE,  IN  ORDER.    IF  A  PRICE  DOESNT  CHANGE,  SPECIFY',/,
     * '   SAME  'INITIAL AND  FINAL  PRICE.')
      READ  (5,*)  ((P(I,1),P(I,N)), I=1NEQ)
      WRITE   (6,3)   ((I,P(I,1),PIIUN))I=1NEQ),      (               ..
 3     FORMAT  ('   INITIAL  PRICE   FINAL PRICE     , /,20 (1 H , I 2,2F1 0 . 4 ,/))
      WRITE (6,4)
 4     FORMAT  ('  NOW ENTER  THE  INCONE LEVEL')
      READ  (5  *)  YO                                            ***********
*********** *v********    CALCULATE  THE PRICE  STEPS AND  PATHS
      DO  1000   I=1,NEQ
      PSTEP(I)=(P(I,N)-P(I,1)/N
      DO  1000  J=2,N-1
      P( I ,J)=P(I ,J-1)+PSTEP( I )
 "I 000  CONT NUE                                                 :^:^:^:^:^:^:^:^:^:^:^:^
********************        CALCULATE THE   INITIAL  VALUES
      DO  2000  1=1 ,NEQ
      IF  (I.EQ.D  XC(I)=X1(PU,1) ,YO)
*     IF  (I.EQ.2)  XC(I)«X2(P(1,1>,P(2,1),YO)
*     ETC.
 2000  CONTINUE                        Aimm-run                 ***********
********************                   ALbUKI  I MM
      ITIMES=0
      Y=YO
      DO 3000 J=2,K
 500  ITIMES=ITIMES+1
      OLDY=Y
      DO 4000  1=1,NEQ
       IF (I.EQ.l)  XT(I)=X1(P(l,J>,Y)
 *      IF (I.EQ.2)  XT(I)=X2CP(1,J),P(2,J),Y)

 *      ETC.
       TERM(I)=((XT(I)+XC(I))/2)*PSTEP(I)
  4000  SUM=SUM+TERM(I)
       NEWY=SUM+YO
       SUM=0

       l=FNE"llTIMES.EQ.500)  STOP   'ENDLESS  LOOP -  NOT CONVERGING'
       DO  5000  I=I,NEQ
  5000  XC(I)=XT(I)
        IF  (DABS(NEWY-OLDY).GT.EPS)   GO  TO  500
        ITIMES=0
       YO=NEWY
  3000  CONTINUE
       WRITE  (6,5)
       WRITE  (6 6)   (XC(I),1=1,NEQ),Y
  5    FORMAT   (I HO  'COMPENSATED   DEMANDS', 13X-  'COMPENSATED  INCOME')
  6    FORMAT   (1H  ', 5X, FI 0.4,17X,F10.4)
       STOP
        END

                                           34

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                                 CHAPTER 3

                            AGGREGATION ISSUES:
                   THE  CHOICE  AMONG  ESTIMATION APPROACHES*
     Our  ultimate  use of the  recreational  demand  model  is to  derive
aggregate welfare  measures of the effects of environmental changes.    How-
ever, the means by which these aggregate measures should be devised depends
upon the level of aggregation of observations  and  the  treatment  of users and
nonusers  in  the estimation  stage.    Thus ,  the appropriate aggregation of
welfare measures depends very much on the initial  decisions as to the types
of observations  used  and  the general  sampling strategy employed.

     Problems of aggregation plague applications of  macroeconomics.    The
theory  is derived from postulates of individual  behavior,  yet data is often
more readily accessible in  an  aggregate form.    In many  types of micro-
economic problems,  market  data  is  so much  easier  to  obtain that rarely are
cross sectional, panel  data used.    However,  in  recreational demand studies,
where markets do not usually exist,  survey  techniques are necessary to gen-
erate data.    Even  in  such surveys, however,   data are often collected in
aggregated  form  (by zone  of residence).   To many,  the  travel cost method is,
in fact, synonymous with  the "zonal  approach",  which  employs visit rates per
zone of origin  as  the  dependent variable and values  for  explanatory vari-
ables which  represent averages for each zone.

     In its  current  state,  the  travel  cost approach to  valuing nonmarket
benefits  is  the  product  of two  legacies.    One  dates  back to  Harold
Hotel I ing's extraordinary suggestion for estimating recreational demand. It
has  become  intimately  linked  to  the zonal approach  and dependent  on the
concept of average  behavior.    The  other  legacy  is  the axioms of applied
welfare economics which provide defensible  means of developing benefit
* This  Chapter   is  the work  of Kenneth  E.  McConneI I,  Agricultural  and
Resource  Economics  Development,  U.  of Maryland,  and  Catherine  Kling,
Economics,  U.  of Maryland.

                                     35

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measures based on  individual  behavior.   The two  come  in conflict over this
 issue which we broadly define as aggregation.

     This chapter  explores the relationship  between  the traditional  zonal
approach and  a model  based on  individual  behavior.  A central  theme in this
discussion  is the treatment of both recreational  parti  cipants and nonpartic-
 ipants.     The implications   for  estimation  and benefit calculations  are
d iscussed.

A Review of Past Literature
     Before addressing the  issues  anew,   it  is  useful  to put in perspective
the various discussions of aggregation problems found in the existing liter-
ature.    The  term "aggregation" has been  applied  in what we shall  call  the
"national benefits"  literature.    These  types  of studies attempt  to  value
widespread improvements in water quality due to  changes  in national  environ-
mental  regulations.    In  this  literature,  the "aggregation problem"  involves
estimating benefits over a  vast  number  of widely divergent water bodies,
geographical   regions,  and  recreational   users.     Vaughan and  Russell  have
developed methods to  evaluate comprehensive policy changes  in  this context
(see,  for example,   Vaughan  and Russell,  1981  and 1982;  Russell  and  Vaughan,
1982 ) .    Perfecting  these  methods  for obtaining  approximate  "value  per user
day"   figures  is of considerable  importance and is being  pursued  under
another  EPA  Cooperative  Agreement.

     The research reported  here,  however,   is not designed  to  address  these
issues.    The aggregation  issues  in question in this study are those  which
arise in all  studies which  attempt to use travel cost (or  its  more general
form -  household production)  models to  evaluate  benefits to  all individuals
affected by  an  environmental  change.    The following brief  review  offers  a
menu of the  problems  which have been raised  concerning aggregation within
the context of the zonal  and  individual  observation approaches  to the travel
cost method.

1.   The  Zonal  Approach
     Travel  cost models that employ  the zonal  approach generally regress
visits per  capita  in  each zone of residence on the  travel  cost  from  the
associated zone to the resource  site and on other explanatory variables.
The literature  on these zonal models has  addressed two types  of problems.
The appropriate size  and  definition of the zones and heteroskedasticity
problems  in  estimation.
                                     36

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     Sutherland  (1982b) questioned the degree to which the size of the zones
affected demand and benefit estimates and whether it was more appropriate  to
use concentric  zones  or population  centre ids.   He  estimated  demand  curves
for boating using ten and twenty mile  wide concentric zones as we I I  as
twenty  population  centre ids.    The  study revealed  larger consumer surplus
estimates  when  concentric  zones  were  used  as compared to population
centre ids,   suggesting  that benefit  estimates  obtained from a travel  cost
model  will  be sensitive  to  the  zone  definition.     However,  Sutherland
lamented  the absence of  clear criteria for  choosing  either population
centre ids  or concentric zones.

      In a  recent  paper, Wetzstein and McNeely (1980) discussed a related
issue of aggregating observations.    They  argued that if it is  indeed  neces-
sary to use aggregate  data  (i.e.  zonal  rather than  individual observations),
it  is  more efficient  to aggregate the observations  by  similar  travel  costs
rather  than  by  the more traditional  method  of similar travel   distances  to
determine  zones.   Aggregating the zones by travel cost would provide  "a more
efficient  estimate of  the  coefficient associated with  cost and  thus  improve
the confidence  in  the  value of the coefficient"  (p.  798).

     Wetzstein and McNeely estimated demand  equations for ski areas  under
the two alternative aggregation  schemes.   When the  data were  aggregated  by
costs,   both  the distance and  cost coefficients  were  significantly different
from zero.    However  when  the data  were aggregated  by distance, only the
distance coefficient was significant.    The  paper suggests that estimated
coefficients,  and thus benefit estimates,  may be highly sensitive to  varia-
tion  in explanatory variables  within  zones.

     The final   issue that  has arisen concerning the  determination of zones
has to  do with the spatial   limits  of the  travel  cost model ,   Smith  and Kopp
(1980)  pointed out that  including  zones  far  from the site being valued  will
likely violate some basic assumptions implicit  in the travel cost model.  As
the distance between origin zone and  site  increases,  it  is less likely that
the primary  purpose of the trip  is  to  visit the site  in question.    It  is
also  less  I  ikely  that the amount of time spent  on site and the form  of
transportation  will remain  constant.   Smith and Kopp proposed  the use  of a
statistical  test  to determine which  zones should  be included  in the model
and which should not.    This  test  was developed by Brown, Durbin and  Evans
(1975)  and  is based  on the  fact  that observations  inconsistent with the
assumptions  of  the  travel cost model  will  produce nonrandom errors.
                                     37

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     Smith and Kopp used 1972 United States  Forest Service data  on  visitors
to the Ventana area to  illustrate the  impact that the  spatial  limits  of the
travel cost  model  can have on benefit  estimates.   They had  information  on
visitors from 100 zones encompassing 38 states.   Applications  of the  Brown,
Durbin and Evans procedure suggested that a spatial  limit to  the  model  could
be established  at a distance of about 675 miles from the  site. The esti-
mated per trip consumer surplus  lost if the area were  destroyed  was  $14.80
when  all  observations  were included,   but only $5.28 when  the apparent
spatial  limits of the model were respected.   Thus the  definition  of zones
and the  I imitation of the  number of zones  are  important issues and  can  have
a significant impact  on  the size  of  benefit measures.

     Another  issue that has arisen  in  applying the  zonal travel   cost model
concerns possible heteroskedasticity in the error term.   This issue  has  been
 integrally  related  to the assumed  functional  form of the demand equation.
Bowes  and Loom is  (1980)   were  among the  first to warn of the potential
heteroskedasticity problem  which  zonal  data may create.   When  the defined
zones encompass  different  size  populations,  the variance of the dependent
variable, average number  of trips  in  each zone, will  vary  with zones.  If
the  variance of each  individual's visitation rate  is the  same,   i.e.
Var(v-ji)  = cr2 for all  Individuals i  in  all  zones j,  then the variance of the
mean visits per capita  from zone  j  will  be   Var(?v-jj/Nj)  =  a'V/Nj   where
Nj is zone j's population.   This is  a classic heterosskedasticity  problem for
which the correction  procedures are well  understood.    One simply  needs  to
weight all  variables  by  the square root  of  the  zone's population.

     To  illustrate the  potential importance of this  correction, Bowes  and
Loomis  estimated a  linear demand  equation for per  capita trips down  a
section of the Colorado River in Utah.   Using the unweighed OLS  estimates,
total benefits were calculated as $77,728.    When weighted  observations  were
used  to correct for  the apparent heteroskedasticlty,  only  $24,073  in
benefits  could be attributed to the  users of  the  Westwater Canyon.

     Another  possible  source  of  nonconstant  variance  is   suggested  by
Christiansen and Price (1982).    They argue that the variance  in  individual
visitation  rates is not  likely  to  be  constant  across zones.     Individuals
 located at different distances from  the site will exhibit different  partici-
pation rates and can  be  expected  to  have different  individual  variances.
The source of heteroskedasticity is  the unequal  visit  rates across zones.  If
both types of heteroskedasticity exist,  the authors  suggest  that the  proper
weighting scheme  would be  (N-j/E(V^))i/tL  where  Nj is   again  the population

                                     38

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in zone  j  and V^ is mean visitation  rate per capita  in zone j .   This pro-
cedure causes the dependent variable to appear on the right hand side of the
equation  and thus would seem to generate further statistical  problems.

      In  her  response  to  Bowes and Loom is,  Strong  (1983b)  made the case for
the use of a nonlinear function (specifically the semi log form) as an alter-
native to the  Bowes and Loom is correction  for heteroskedasticity.    Linear
and semi  log demand equations for steel head fishing were estimated using data
from  zones  around twenty-one  rivers  in Oregon, and  a  Goldfeld-Quandt test
was employed to  test  for the  existence  and  size of heteroskedasticity.   The
semi log  model   did not require a heteroskedasticity correction,  but the
linear model  did.  After correcting the linear model  for heteroskedasticity
(applying the  appropriate weights),  this model  was  compared to the semi log
model  by the mean squared error  in  predicting trips.    The semi log form per-
formed better than the corrected  linear  model  in  this  test.

     Vaughan,  Russell  and  Hazilla  (1982),  in another  comment  on  the Bowes
and Loom is article,  argued  that an alternative  to  assuming  a  linear demand
equation  and heteroskedasticity is to test  for  both  in the  data rather than
impose them as  assumptions.    To  do this,  they tested  the Bowes and Loom is
data  for  appropriate  functional  form and  heteroskedasticity simultaneously
by applying  the Lahiri-Egy  estimator which  utilizes  a maximum likelihood
procedure to estimate  the  appropriate functional form  with  a  Box-Cox trans-
formation under  conditions of  potential  heteroskedasticity.   As a result of
this  procedure,  they were able to  reject the linear  homoskedastic  and the
linear heteroskedastic models.   The appropriate functional  form for the data
appeared to  be nonlinear and  with a  nonlinear form  heteroskedasticity ap-
peared not to  be a  concern.    The benefit  estimate obtained with  a semi log
functional form  (and  no heteroskedasticity correction, since  none  was war-
ranted)  was  only $14,000 as  compared to the Bowes and Loom is estimate of
almost twice the size.   Vaughan  et al.  concluded  from their analysis that
the heteroskedasticity issue can not  be  separated  from the  choice  of appro-
priate functional form and that it is likely  that a non-linear  specification
is superior  to a linear one.

      In their  study of partyboat  fishing in California, Huppert and Thomson
(1984) suggested another  cause of heteroskedasticity  that can  not  be miti-
gated with the  semi log functional form.   They argued that,   in  practice, the
sampling scheme  used  to  collect  data  for a travel cost model  may  give rise
to heteroskedasticity.    The  semi log transformation suggested by  Vaughan
                                     39

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et a I ,   and Strong  will  not eliminate  the  problem,  unless  the number  of
visitors surveyed  from  each  zone  is  the  same.

     In their view,  heteroskedasticity  arises from the construction of  the
dependent  variable from sample  data.    The trips per capita variable  is
calculated as t-j = nit/p-jn where DJ = number of respondents sampled at  the
site from  zone j,  n  =  total number of  respondents sampled at  the site,  t =
total  number of  trips made to the site  in 1979,  and Pj=  population  in
zone j .   They argued that  it  is only n,,   the number  of sampled  respondents
from zone  j,  that is random  and  that n  can be thought of  as  a  binomial
variate since  it  is  equivalent to  the  number of "successes"  in n drawings.
The variance formula is then S2 = n(n^) (1 - nj) where Hj is the probability
that  an angler sampled  wi I I   be  from zone j .     The variance  for tj is
then (t/np^^and  thus varies with zone.    On the basis of this variance
formula, Huppert and Thomson concluded  that  "variance due to  sampling  error
depends  inversely  upon both sample size and zonal  population" (p.  8).  The
authors also  showed that the  use  of the  semi log  transformation would  not
eliminate  this heteroskedasticity.

     The discussions of the zonal  approach  in  the literature have focused
attention  on  practical or,  perhaps more  correctly,  statistical  problems
which zonal aggregation may generate.    By  using zonal  data,  researchers  are
more likely to encounter multicolI  inearity and  heteroskedasticity problems.
Additionally,  they are likely  to lose precision  in estimates  whenever  zones
lack homogeneity and explanatory variables exhibit  large variability within
zones.

2,   The Individual  Observations Approach
     The initial argument  to  use individual  observations  instead  of zonal
averages in the travel  cost model  can  be  traced  to  Brown  and Nawas (1973)
who sought  to combat multicolI  inearity  difficulties arising  from more aggre-
gated data.   They  wished  to include the opportunity cost of time in travel
cost demand  models but found  that since  zonal  money  and  time costs were
likely to  be  highly correlated, multicolI inearity became a  serious problem.
Brown  and  Nawas suggested  using observations on  individuals rather than
grouped or averaged data as a  solution.   The  authors offered an illustration
on a data  set consisting  of 248 big game  hunters in the northeast area  of
Oregon.     In  a model including money cost  and distance (as a surrogate  for
time),  the coefficient on money costs was  significantly different from zero
only when  the model was estimated on individual  observations.
                                     40

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     Some years  later,  Brown, Sorhus, Chou-Yang and Richards (1983) reversed
this position  on the zonal versus  individual  observation  question with the
following argument.   "The problem with  fitting  a  travel  cost-based outdoor
recreational demand  function  to unadjusted  individual  observations is that
such a  procedure does  not properly account for cases  in which  a  lower per-
centage of the more distant  population zones participates in the recrea-
tional   activity.   In such cases,  a biased estimate of the  travel  cost coef-
ficient  results"  (p.  154).   The fact that  more individuals choose  not  to
participate from more distant zones  holds  important  information  for the re-
searcher,   and   if  such  information  is  ignored,  bias  is  likely to  result.
Zonal  data  implicitly  incorporates this  information,  in  a way,  by  using
trips  per capita.    Brown et a I.  suggested that one might use  individual
observations without losing important participation data by transforming the
left hand side  variable  to individual  visits per capita (i.e.  the dependent
variable would  be  defined  as visits  by  individual  i  in zone j/population  in
zone j).

     While  detailed discussion awaits  the  subsequent   section of  this
chapter,  the underlying problem  here  is  one of  truncated  or  censored
samples.   A few authors  have attempted  to  deal  with  the problem  of partici-
pation  rates  (numbers  of participants versus  nonparticipants) using  econo-
metric techniques designed to handle this type of phenomenon.  Wetzstein and
Ziemer  (1982)  illustrated  Olsen's  method of correcting for  the  bias  intro-
duced by the use of a  truncated sample  with permit data for Dome Land and
Yosemite wilderness  areas  in  1972-1975.    The Olsen method  is  a  diagnostic
tool which can determine the relative importance of the bias associated with
omitting non-participants from a  sample.     It  also offers an approximate
correction for  this  bias  using  OLS parameter estimates.   The  impact of the
truncation on the parameter values is determined by comparing the unadulter-
ated OLS parameter  estimates with the "Olsen"  estimates.

     The OLS and Olsen regression models were  estimated  for Yosemite and
Dome Land.    The Olsen correction  was  found  to have a smaller  influence  on
the  Yosemite  data than  on the Dome Land  data based  on  similarity of the
Olsen estimates to the standard OLS estimates.    This result  is consistent
with the underlying theory, since  more  zero  visitor  days were  observed from
Dome Land than from Yosemite.    The authors  also  compared the OLS to the
Olsen estimates  based on  forecast  performance  through  the  use  of root-mean-
square-error,  mean error,  and mean absolute  error  determined from predicted
and observed visits  in 1975.   Again, the Dome  Land  OLSestimates  fared less
we I  I than the Yosemite OLS estimates as compared to the Olsen estimates, and
                                     41

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the  authors concluded  that the severity of the  bias  is dependent  on the
nature of the data set.

     Desvousges,  Smith  and  McGivney  (19831, recognizing the problem inherent
 in  a sample which only included observations  on the behavior  of partici-
pants, also  employed  01 sen's method to evaluate  the  importance  of the bias
 introduced by the omission  of nonparticipants.   They found that for several
of their sites this truncation  greatly biased their results.   To  compensate
for  the  bias  in  their final  model,  they chose to  use  two samples,  one which
 i ncIuded all  of  the  s i tes  and  one  wh i ch  om i tted  those  s i tes that exh i b i ted
 large biases from the effects of nonparticipants.

Models of  Individual  Behavior and Their Implications for  Estimation

     The controversy  in travel  cost  literature  surrounding the  use of zonal
vs.   individual   data  focuses principally on  data  oriented  problems.    The
zonal  approach   may  be  particularly susceptible  to  muIticolI inearity and
heteroskedasticity.    However,   individual  observations are expensive to col-
 lect and may be  more vulnerable to  severe errors  in measurement.   Discus-
sions of substantive conceptual  differences  in  the two  approaches have been
 less frequent and less well  developed.   Recent work  leaves one with the
vague  impression that welfare  measures may  be more difficult to  define  in
the  zonal  approach but  that,  in some  way, this approach better handles the
problem of nonparticipants.

     It  is useful  at this  point to  sort out  some  of  these issues.   One of
the  difficult problems  in calculating total  welfare changes as William Brown
has  pointed out,   is accounting  for  the  participation rate  in the population.
 It turns out  that this  consideration plays an important  role  in the estima-
tion stage as we I I as  in  the welfare calculations.   Nonetheless, the  proper
perspective  is  still  to think  of  the  problem  in terms  of the  individual.
Throughout this  report  we  have argued that the assumptions  implicit  in the
estimation of any recreational  demand  model must  be consistent  with logical
models of  individual  behavior.    To  model  individual   recreational  demand
adequately, one must allow an individual to choose not to participate.   That
 is,   a  model  of behavior must accommodate both positive and zero  levels  of
demand.     In  what follows,   a standard model  of  individual behavior  which
allows  for zero levels of demand  is  presented and  its  implications for
estimation  using  individual  observations are explored.
                                     42

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     The problem can be described  as follows.   For any  recreational  site,
groups of  sites,  or activities,  there will  be many nonusers  in  the popula-
tion.   While corner solutions of this  sort  (x = 0 for  some  individuals for
some goods)  can  be  handled deftly in  abstract  models,  they present complicat-
ions  for econometric estimation.     These  complications,  and  the  biases
resulting  from  ignoring  the  problem,  are proportional  to the rate of non-
participation  in  the problem.   Unfortunately recreational demand  studies  -
no matter how broadly defined -  frequently  encounter  low rates of participa-
tion  in the  population at large.

1.   A Simple Model  of  Individual Behavior
     The following  might be  conceived as a general model  of  an  individual's
demand for  recreation trips

                              xj = h(zi, 3, ej)

where Xj =  quantity demanded by individual i, z^  is  a  vector of explanatory
variables,  6  is  a  vector of parameters and  e^ is a random disturbance term.
Unless this  model  is modified, though,  it  implicitly  suggests the possibil-
ity of negative  trips.   For  many functional  forms (e.g. linear), a z^ vector
could be faced and  a disturbance term drawn  from the distribution, such that
x.j is less  than  zero.  For  other functional  forms (e.g. semi-log), it may be
impossible   to  generate  negative values for  x^  but  equally  impossible  to
generate zero  which is a  very legitimate and frequently  observed value for
x^.   The  model must incorporate assumptions  about  individual  behavior such
that  both   positive  and  zero,  but  not  negative,  values  of   xi  will  be
generated.

     The most popular assumption  (and the one attributable to Tobin)  is the
fo I Iow i ng:

(1)               X1 = htz^S) + ei       if h^-) + E.> 0

                  x.j =0                  if h^f-) + e. < 0,

where E  ~ N(0,c2).    Presuming  that  the demand function  is generated  by
utility maximizing  behavior, this  model seems to  imply  that preferences are
defined over both  positive  and negative values  of x,   but  reality prevents
the consumption of  negative  quantities.   Thus when the  demand function would
imply a negative quantity, a  zero quantity is  consumed.
                                      43

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     Assuming that  model  (1)  generated the  behavior  which is observed,  let
us consider what happens when conventional methods of estimation are employ-
ed.    When individual  observations are  available,  the customary practice  is
simply to estimate a demand function on data gathered from users.   There  are
two  problems with this  approach.    The  first  is that nothing  is learned
either about nonusers or about the factors which affect the decision to par-
ticipate.   There  is,  as  a  consequence,  no way to predict changes  in numbers
of participants  when parameters  in  the  system change.

     The second problem  is that  if nonparticipation is due to the  underlying
decision  structure of the sort described  in  (I),  then  estimating demand
functions from only users  will  generate biased coefficients.  If behavior  is
described  by model  (1)   and  the  e's    in  the  population are  distributed
as N(0,a2), then the e's   associated with the  sample  of users will not meet
Gauss-Markov assumptions.     They  will,   by definition,   be   those e's such
that £.,-> -h(z.j,e).

     When only users are  observed,  the  sample  is said to be truncated.  When
the entire population is sampled but the value of the dependent variable  (in
this case,  trips)   is bounded (as  in model  (I)),  the sample  is  said  to be
censored.    Methods are well  developed for consistent estimation of models
from either type of sample (see G.  S.  Maddala,  1983,  for a recent and exten-
sive treatment),  and some of these  will  be discussed  below.

     Both Wetzstein and Zeimer and  Desvousges,  Smith  and  McGivney  recognized
the  presence of this problem in their recreational  demand models.   These
studies employed 01 sen's  technique  to make an approximate correction for the
bias when only user data were available.    It  is  useful,  however,  to explore
other econometric  techniques for  eradicating the problem,  some  of  which
handle more general models of nonparticipation.   We  shall see that  consist-
ent  parameter  estimates can   be obtained whether the  sample  is composed
solely of users or  drawn from the  population as  a whole.  The latter type of
sample will  generate more efficient estimates,  however.

      If behavior  is described by model (I), then  the standard  Tobit can be
used to estimate the parameters of  the model.    From  (1),  an individual  i
will participate if e^>  -h-j(§).    Providing e^ is distributed normally with
mean 0,  the  transformed  variable,  e^/a, has a standard normal  distribution,
and

                   Prii  recreates) = Pr{e../0 > -h..{•)/
-------
This  probability  equals  1   - F(-hj{ • )/o),  or F(h^(-)/o)  where  F(')  is  the
cumulative  distribution function of the standard  normal.    The probability
that i does not recreate is F(-hi/o).

     To form  the  likelihood  function  for the  sample,  we  need  an expression
for the probability  that  i  chooses  x  days  given that x,- > 0.  This is given
by

,2)
where f(*)   is the  density function of the  standard normal.     Thus the
 likelihood function  for the sample  is

            L,  =   n  Pr{x,> 0} Pr{x. |  x,> 0}  n Pr{x.  = 0}
 (3)                US
                   n  f(e./o)/a   E  F(-h./a)
                                 ies     1
where  s is the  set of  individuals  who participate.    The  parameters 6
and o} can be estimated from (3)  using maximum likelihood methods.

     There is a second procedure  (attributable to  Heckman)  which  uses a two
step method  in addressing the non-participation  problem.    Considering the
same model,  one  can  express the  expected  value  of  individual  i 's  trips,
given that i  is  a user  as

                 E(x. |  Xi> 0)  =  Mzj.B)  + E(e.| £i> -hjM).

From the previous derivations,  it can be seen that the second  term  is

                                                      of(-h,Ar)
E(£I
                                -                             } .

The demand for recreational  trips  can  then  be  rewritten  as

 (4)                     x1  = h(z^,B) + a\i  + v,. ,

where x-j equals f(-hi/o)/F(hi/o) and v^ is  a normal  error with  zero  mean.

     From this expression   it  is  easy to see why OLS estimates of a  model
such  as (1) are  unsatisfactory.     The  denominator  of A, F(hi/a), is  the
probability that an  individual  participates at the site.   If there  is  a very
high rate of  participation among  the  population,  the x's will be  small  and

                                     45

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OLS estimates not too bad.   The sample selection problem  is most severe when
there is a  very  low participation  rate and the x's are large.   The presence
of x allows  for  the possibility of  considerable  misspecification,  and its
omission will  cause  the estimates  of 3 to be  biased  where  x is correlated
with any dimension  of z.

     Equation  (4)  can  be  estimated  with ordinary  least  squares  if X-j is
known.    One way of  obtaining an estimate of  Xj  is  to estimate  a  discrete
choice  model  of the participation decision.    Such  a model  would simply
explain the yes/no decision.    The  logical  choice for the qualitative re-
sponse model is probit  with a  likelihood  function  expressed as

(5)                  L   =  n  F(h./a)   n   F{-h./o).
From the  earlier  discussion, we  know  that  F(h^/cr)  is  the  probability  of
participating  and   F(-h-j/a)    is   the  probability  of  nonparticipation.
Maximum  likelihood  estimates  of the B's  and o will  allow construction  of
estimates of the x^'s  to  be  used  in  the  estimation of (4).

     One characteristic of  this  approach  is that  two sets of s's and a are
produced; one  from  each  stage of the estimation.  This may at first  appear
to be an unfortunate  feature of the approach.    However,  two sets  of esti-
mates may be appropriate  if the  demand function is  discontinuous or  kinked
at zero  (see Kill ingsworth,  1983) .

2,   A Model  of Behavior  When Different Variables  Affect  Participation and
    the Demand  for  Trips
     A  logical  extension  of the discontinuity of the function  at  zero  is the
 idea that different  variables  may affect  the dichotomous  participation
decision and the continuous demand  for trips decision.    This may  occur  if
factors such as good health  or  the ownership  of an automobile or recrea-
tional  equipment are  necessary for an  individual to become a participant.
Along these  lines,   a  final  model  is offered  which  employs Heckman's estima-
tion technique but begins  with  a model of  behavior which is more general
than model (1).   Consider  a latent variable w*  which  is an  indicator  of
participation
 (6)

                                     46

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where the  individual  participates (*.j  = 1)  if **>  0 and the individual does
not participate (^j = 0) if *-,• $ 0.  The number of  trips taken by  individual
i given that i participates is


                                                       *
Because ** 1s   an   index   denoting   participation, n^ is   observed  only
when IT.,- > 0.  The  vectors  zjj and Z2-,- may or  may  not have elements  in com-
mon, and the covariance matrix of the e's may or may not be diagonal.

     The Heckman estimation  technique  is  particularly suitable for this
model ,    If information on  nonusers as well  as users  is available,  one can
first estimate a probit model  of the form

(8)               U =  n   F(gi/on)   n  F(-g,/a  .)
                   J   ies     L  1X   *—     *  X1
where s  is the  set  of participants and an  is  the variance of  the e^-'s.
Note that  this  likelihood  function  is  based  only on the participation deci-
sion and requires a sample of the entire population.

     Using Heckman's  results,

         E(x.j| Z21,**> 0)  =

so that

(9)                   x. =

where Xi = f(-gii/ail)/(-F(-gi-j/erii))  and 012 is   the   covariance   between
e^  and eg-j.  Again an estimate  of  XT  can  be obtained from  the  probit model
in (8).

3,   Estimation  When  the Sample  Includes Only Participants  -  the Truncated
    Sample
     The  above models are all  we I I  and good,  but what  happens when  the
sample  of observations includes  only  participants?  This  is a common occur-
rence  in specific recreational demand studies where the  incidence of partic-
 ipation in the  population at  large  is exceedingly low.     In  such cases,
extremely  large, and thus  expensive,  household sampling procedures would be
necessary  to produce   sufficient  observations on  users.    As a  result,
researchers sample on  site and collect  data  only on participants.
                                     47

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     While samples which  include only  participants  preclude  the  use of some
of the methods  described above,  it is sti I I  possible  to obtain consistent
although not particularly efficient  estimates  of the  parameters of  the
demand for trips equation.   To do this,  we must  refer  back  to the  model  of
behavior  presented  in  equations (1).     It  should be obvious that  any more
general   model,  such  as those estimated with  the  Heckman  technique, require
 information about non-participants  and  thus can  not  be used on  a truncated
sample.    Model  (1) however assumes  that the  same  function  determines whether
 individuals participate  and  if so,  how  much  they participate,    If this is
true  it  is  straightforward  to estimate  the demand  for  trips conditional  on
participation .

     Referring  back  to equation  (2),  the  probability that  individual  i 's
demand equals some xn-  conditioned on the fact that  he participates  is  given

^                         ,   ,   nt   f(-h4/o)/0
                          Pr x, x,>0  =
                             11 '
The appropriate  likelihood function  for the  sample is  then  simply
                                  ff-h.cOa
(10)                       L   «.n
                                    F(h.j/a)
Because the  added  information about nonpartici pants is missing, the  esti-
mates  produced by  this  conditional  maximum   likelihood  will  be  less
efficient.     Nonetheless  the method corrects  for  truncated sample  bias
without requiring  very  expensive  data  collection.

     Perhaps the greatest cost of a truncated  sample  is the paucity of
information  about the participation choice which  it offers.   Although  it is
technically  feasible to use the coefficients generated by  (10) to predict
whether an  individual drawn randomly  from the population  would  participate
in the activity or not,  such predictions are dangerous.    They rely on con-
siderable confidence both  in the estimated coefficients and  in the  model  of
behavior  postulated  in  (1).    Thus if other variables which are all-or-
nothing threshold  sorts of factors (e.g. health,  equipment, etc.)  affect
participation,   we  will  never  learn  much about the participation  decision
from a truncated sample.

     Ultimately, the participation  decision may be more  or  less important to
capture.    If the  sorts  of policy changes  being considered  (access,  environ-
mental  quality,  entrance fees)  are  likely to alter participation  rates,  then
it  is  crucial for  welfare  evaluation that good predictions of participation
                                     48

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be possible.    Fortunately, the situations  in which other discrete conditions
affect  participation may be just the cases where  policy changes  (such  as
environmental quality changes) are  less  likely  to affect the participation/
nonparticipation  choice.

     One final caveat is in order here.   Throughout this discussion,  there
has been an  implicit restriction on the form of  the demand  function.  While
we have not required the demand function to  be  linear, we have assumed
errors to be additive.   Forms such as the semi-log  do not have this proper-
ty, and as we  noted they have the additional  problem  of not admitting zero
values  for the dependent variable.   As  such the  semi-log form is logically
 inconsistent with the notion  of  nonparticipation  and the models of behavior
presented  above.   More  general  functional  forms,  such as  the Box-Cox trans-
formation,  do allow  for  nonparticipation.   However, the error structure may
not always  be  additive.    In these cases  the above  results will hold  in
spirit but  not  in detai I .

Implications  for  the Estimation of the  Zonal Approach

     While researchers  have recognized  the advantages of using  individual
observations to  estimate recreational  demand models,  there has  been some
suspicion that the zonal approach avoids the  types of participation rate
problems encountered above.     In truth,  the zonal approach  is  plagued with
similar and sometimes additional  problems which  become apparent when  a  model
of behavior  such  as  (1)  is postulated.

     Assume  that the simple  model in expression (1) reflects the actual
behavior of  individuals,  but that only  zonal data   is available.  The  zones
in our  discussion will  be assumed to  be distinct and  well-defined,  whether
determined by political  boundaries such as counties  or by distance from site
as originally conceived by Hotel  I ing.    Suppose  that there are M such zones,
and in  each  zone j(j  =  1, M)  there are P^  people (the level  of population),
, •  of whom visit the site at  least once.   The  individuals, i = 1, I, within
trie  zone  may  differ  with   respect   to  explanatory  variables,  z^,  error
term, e..,  and chosen  number of  trips,  x^   The model  in (1)  is rewritten
using this  notation

                   *1J  = h(zij'e)  + eij     if hij  +  Eij >0

(11)                x   = 0                 if h    +  E   < 0.
                                     49

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     From  the  above  definitions, nj/Pj  is  the proportion  of the population
who  visit  the  site  at  least once.   Both nj  and  nj/Pj  are  random because nj
is  the  realization  through  expression  (11)  of random drawings  of  the dis-
turbance  term.    Denote the  first nj  people  as  participants and  the last
Pj - nj as  nonparticipants.   Defining  x.  as  the  zonal  average for zone j,
the  x.  are as follows:
      J

                         Pj           "j            PJ
                         V            V             T
                        i-            i-             -n^+
                         I  (Mz   6) + e  )/P
                        i=l      |J        iJ   J
(12)                 =  I  h(z..,6)/P. +  I  e  /P
                       •=      '<*     J   -=    «  «»
     Let  us  employ  two  assumptions  for the  moment, one  of which  in fact
favors the zonal  approach.   We assume that h(-) is linear in the explanatory
variables  and  that  each  zone  is  sufficiently  homogeneous  such  that the
assumption that z^j = z^j for all  i ,k is reasonable.

     Then (12) becomes
                                        n .
(13)                    x
This expression  reflects  the nature of the zonal  data  observed when expres-
sion (11) describes the  individual's decision process.

     Two problems  are  encountered when one attempts  to  apply OLS techniques
to  zonal  data.   The  first problem is  that the error term  1n  (13)  does not
possess the prescribed  properties.   If it is assumed that Hj  is distributed
normally with mean  zero  and variance  o2.  then e-jj/Pj  is distributed normally
with mean  zero,  variance o2/P? which  implies  a  heteroskedastic  problem of
the sort  discussed in  the  literature, but  easily  corrected.  The error term
in  (13)  is  actually  a  sum of  such  terms.    While  the sum  of independent
normals is  itself  a normal, the error term here is not the sum of  independ-
ent normal s=  The term
                                      50

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                                  n.

                                  S3  eij/pj
                                 1=1   1J  J

reflects  the  same  sort  of  selection   bias  encountered  in   the  previous
section,  because  it is  the  sum of  errors  conditional  on h(*) + E-JJ> o, and
its expected value will not be zero.

     The  second problem  relates to  the fact  that,  in general,  nj/Pj  will
vary over observations (i.e.  over zones).   The participation rate per zone,
ise= nj/Pj  , will  not  tend to be constant,  since as  distance between origin
and  site  increases,  participants   take  fewer  trips  and  there are  fewer
participants.

     Consequently  a  regression  of   x.   on  the Zj will not yield estimates
of B (even  up  to  a proportionality  constant).   To assume however implicitly
that fij/Pj  is  constant violates  the assumptions of the model.   The partici-
pation  rate cannot be  constant  and non-random;  because  it  is  determined in
part by  random  errors  and in  part by systematic variation in factors such as
travel  cost.

     If  the n-j/Pj  were  known,   however,  it  would be  possible  to  estimate
the B's in  (13) by  weighting  the explanatory  variables by the participation
rate.   This would  not, however, resolve the  problem  with  the error term and
a technique such as Heckman's would  be needed  to estimate  the zonal model.

Conclusions
      In  principle,  models estimated  on  individual  observations are  prefer-
able to those based on zonal aggregates.    Inferences about parameters of the
preference  function are  more  directly  revealed and thus welfare measures
easier to define.    Individual  observations also provide more information and
may help  avoid  multicolI inearity and heteroskedasticity  problems  aggravated
by the  zonal approach.    Perhaps the chief drawback to using  individual
observations  is that  they are  more  likely  to embody severe errors  in  meas-
urement.   Also  it may be more difficult to extrapolate welfare  measures for
the entire population from models based on  individual data.

     All of this  abstracts from  the  overriding  aggregation issue implicit in
estimating recreation  models -  the  treatment of nonparticipants.    There is
some  indication in the  literature  that the  zonal  approach may  be superior in
dealing with this problem.   As  we show  in  this chapter,  this  supposition is
incorrect.    In  fact  the  participation issue arises  in the estimation  of both

                                      51

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 individual and zonal based models.   Both models will yield biased parameter
estimates  if the problem  -  one of truncated or censured samples - is ignor-
ed.    The  key  point is that  individual data-based  models which  take  this
problem  into  account are well  developed.    Methods exist  for  estimating a
wide selection of models  of  individual  behavior  which  allow for nonpartici-
pation or  which use  truncated  samples  and  are  conditioned on participation.
While more flexible models and more efficient estimates are possible  when
both users and nonusers are sampled,  methods for obtaining consistent esti-
mates exist for samples of users  only.   In contrast,  zonal models actually
confound  the  problem of participation.   It is never quite clear what  such
models are estimating and  how they can be adjusted to recover the parameters
of interest to  us.

     In the next chapter,  we provide  an example  of  the application  of some
of the methods for  taking account of  the  participation  decision  when indi-
vidual  data is available.   This is pursued  in conjunction with a development
of the treatment of  the value of  time,  so that a more complete model can be
presented.
                                     52

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                                  CHAPTER 4

              SPECIFICATION OF THE  INDIVIDUAL'S DEMAND FUNCTION:
                            THE TREATMENT OF TIME
     Economists,   especially those working in the area  of recreational de-
mand, have  long  recognized  that  time spent in consuming a commodity may,  in
some cases,  be an  important determinant of the demand  for  that commodity.
 It remains true,  however,  that even though the potential  importance of time
has  been  discussed at some  length  in the  literature  it is  only relatively
recently,  and  in a fairly smalI set of papers, that the  problem of expl icit-
 ly  incorporating time  into the   behavioral framework  of the consumer has
been addressed.

     This  chapter provides  a  discussion of  the  ways  in  which researchers
have traditionally  incorporated  time costs  into  recreational  demand  models
and attempts to  develop  a  more complete  and general  model.   Improvements  in
both specification  and  estimation of the  model  are  achieved by  integrating
recent  labor  supply and recreational  demand  literature.   The new model  of
 individual  decision making  is characterized by two constraints.    Insights
 into the dual  constraint model  are offered.

     The treatment of time  is  one of the thorniest issues in the estimation
of recreational benefits.   A number of approaches (e.g.  Smith et a I.,  1983;
McConnelI  and Strand,  1981; Cesario and  Knetsch,  1970)  to valuing time are
currently  in vogue,  but no method  is  dominant  and researchers often  impro-
vise as they see  fit.    Unfortunately,  the  benefit estimates associated with
changes in public recreation policy are extremely  sensitive to these  improv-
 isations.   Cesario  (1976),   for example,  found  that annual  benefits from park
visits  nearly doubled depending  on  whether time was  valued at some function
of the  wage  rate or treated  independently  in  a  manner suggested in Cesario
and  Knetsch  (1970).   More recently,   Bishop  and  Heberlein  (1980) presented
travel   cost estimates of hunting  permit values which differed four-fold when
time was valued  at  one-half the  median income and when time was omitted al-
together from the model.
                                      53

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     Recreational  economists have understood the applicability of  the  clas-
sical  Iabor-Ieisure  trade-off to  this problem.     In his  1975  article,
McConnelI  was the first to discuss the one vs.  two  constraint  model.   Recog-
nizing that time remaining for recreation may be traded  off for work time or
 it may be fixed, he shows how the nature of the decision problem is affected
by the nature of the time constraint.    This chapter begins within  this
context and develops  a  general  framework for incorporating  time.    After
discussing  the  wide  range  of complex labor constraints which  the  model  can
handle, we  turn to making  the  model operational.    The  approach  developed
below  not only  incorporates a defensible  method for treating the  value of
time but also permits  sample selection bias  (Chapter 3) to be addressed and
exact measures of welfare (Chapter 2)  to  be derived.

Time in Recreational  Decisions
     Despite the  general  acceptance  that time plays  an  important role  in
recreational decisions  (e.g.  Smith,  et a I . ,   1983),  no universally accepted
method for  incorporating time  into  recreational demand  analysis  has emerged
and methods for  "valuing"  time  in  recreational demand  models are  numerous.
While many  methods have been  developed from  assumptions based on utility
maximizing behavior,  there  is no consensus  as to which  is  the "correct"
method.    In  actual  applications,  researchers  have  often been  forced to  take
a relatively ad hoc view of the problem by incorporating travel  time  in  an
arbitrary fashion as  an  adjustment  in a demand function  or,  alternatively,
by asking people  what  they would  be  willing to  pay to reduce travel  time.

     Ad  hoc  econometric    specifications or  general   wi I I ingness-to-pay
questions are  particularly problematic  with  respect to time valuation be-
cause time is such a complex concept.    Time,   like money,   is  a scarce
resource, for which there  is  a constraint.   Anything which uses time as  an
 input consumes a  resource for which  there are utility-generating  alterna-
tives.    While  time  is an  input into virtually  every  consumption  experience,
some  commodities  take especially  large amounts of  time.     These  have
frequently been modeled  in  a household production framework to  reflect the
 individual's  need  to combine   input purchases   with  household  time  to
"produce" a commodity for consumption.   Because time  is  an essential  input
 into the production of any commodity which we  might call  an "activity",  time
 is frequently used as a measure  of  that  activity as well.   Thus,  while  time
 is formally an input  into the production of the commodity,  it  may also serve
as the unit of measure of the output.
                                     54

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     The complexity  of  time's role  in  household  decisions  has implications
for  both travel  and on-site  recreational  time.    Both represent  uses of a
scarce  resource  and  thus  have positive  opportunity costs.    However, on-site
time, and sometimes  travel time, are used as units of measure of the uti I ity
generating  activities themselves.   Economists often measure the recreational
good  in terms of time,  i.e.  in hours or days spent at the site.  Travel time
may  also be  a measure  of a  utility generating  activity,   if the  travel  is
through scenic areas or  if  it involves other  activities  such as visiting
with  traveling  companions.    Hence,  direct questioning or  poorly conceived
econometric estimation  may yield  confusing results because the  distinction
between time as  a scarce  resource  and time  as a  measure of the uti I ity gen-
erating activity is  not  carefully  made.

     Both travel time and on-site time  are uses  of the scarce resource and
must  both appear in a time constraint  to  be properly accounted  for  in the
model.   The exclusion of  either will bias results.  But, does time belong in
the utility function?   Viewed as  a scarce  resource,  time  by itself does not
belong  in the  utility function.   What  does enter the  utility  function is a
properly conceived  measure (perhaps in units  of time) of  the quantity and
quality of the  recreational activity.   This does not present major problems
when the commodity   is defined  in terms of fixed units of  on-site time and
when travel  does not in itself  influence utility  levels.   When time per trip
is a  decision  variable,  an appropriate  and tractable measure  is  not  easi ly
conceived.   This Chapter focuses solely  on time as a scarce  resource.

Time as a Component  of  Recreational  Demand: A Review

     The fact that time costs  could  influence the demand  for recreation was
recognized  in  the  earliest travel cost literature (Clawson,  1959; Clawson
and Knetsch,   1966), although no  attempt was made to explicitly  model  the
role  of time  in consumer behavior.   The problems which arise when time is
left  out of the demand  for recreation  were first discussed by Clawson and
Knetsch (1966).   Cesario  and  Knetsch (1970) later argued that the estimation
of a  demand  curve which  ignored  time  costs would overstate the  effect of
price  changes  and  thus understate the consumer  surplus  associated with  a
price  increase.

      In practical  application,  both travel  cost and travel time  variables
have  usually been calculated  as functions of distance. As  a result,  includ-
ing  time as  a separate variable  in the demand  function  tended to  lead  to
muIticol I inearity.     Brown and Nawas  (1973)  and Gum  and Martin (1975)
                                     55

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attempted to deal with  the  mul ticol I ineari ty  issue by suggesting the use of
 individual  trip observations rather  than zonal  averages.     In contrast,
Cesar io  and Knetsch  (1976)  proposed  combining all  time  costs and  travel
costs   into   one  cost   variable   to eliminate   the  problem  of  multi-
col I  inearity .    These  papers had a primarily  empirical  focus,  with emphasis
given  to obtaining estimates.    Demand functions were  specified  in an
arbitrary way,  with no  particular utility theoretic underpinnings.

     Johnson  (1966) and McConnell (1975)  were among the first  to consider
the  role of time in the context  of  the  recreational ists's uti I ity maximi-
zation problem  (although others had considered  it  in other  consumer decision
problems).    McConnell  specified  the problem  in the  framework  of the clas-
sical  labor- leisure decision.   The individual  maximizes utility subject to a
constraint on  income  and time.   The  income constraint  is  defined  such  that

(la)                    E + F(TW)  = pxN +  C'XR

where E  is  non  wage income,  TW is work time, F(TW) is wage  income, p is the
price of  a  Hicksian  good XM, x^  is  a  column  vector of recreational  activi-
ties  and  c  is  the  corresponding  vector of money  costs  for  each unit of x^.
His time constraint is

(Ib)                          T =  iajAj T  ,w

where 3j  is  the time cost  of a  unit  of  Xj.    When  work  time is not fixed,
(Ib)   can  be  solved   for  Tw  and  substituted  into (la)  yielding  the maxi-
mization problem

                   max U(x)  - x(pxN+  zCjXj  -  F (1-ia.jXj)),
so that the  time  cost  is  transformed  into  a money cost at the implicit wage
rate.

     McConnell  (1975)  also  noted  that  if  individuals  were  unable to choose
the number of hours worked, the direct  substitution of (la)  into  (1b)  is not
possible.    tie suggested that  in this case one should still  value time in
terms  of money   before  incorporating  it  in the  demand  function.  This is
conceptually   possible,   since  at any given  solution there  would be an amount
of money which the individual  would be just wi I I ing to exchange for an extra
unit of time  so as to  keep  his  utility level  constant.    Unfortunately,  this
                                     56

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rate of  trade-off between money and time, unlike  the  wage rate,  is neither
observable nor  fixed.    It is itself a product of the  individual's utility
maximizing  decision.

     Much of the recent  recreation demand  literature follows the line of
reasoning which related the opportunity cost of time in some way to the wage
rate.    Of the  many  models of this sort,  the one offered by McConnel I  and
Strand (1981)  is  one of  the  most recent.  (See also Cesario,  1976;  Smith and
Kavanaugh, Nichols et a I  . ,1978).   Their work  demonstrates  a methodology from
which a factor  of proportionality between the wage  rate  and the unit cost of
time can be estimated within  the traditional travel  cost  model.

     More recently,  Smith, Desvousges and McGivney  (1983)  attempted to modi-
fy the  traditional  recreational demand model so  that more general  con-
straints on  individual use of time  were  imposed.    They considered two time
constraints,  one for work/non-recreational goods and another for recreation-
al goods.    The  available recreation time  could not  be traded  for work
time.    The   implications  of  their model  suggest  that when time and   income
constraints  cannot be reduced  to  one constraint,  the  marginal  effect of
travel  and on-site time  on recreational  demand is related  to  the  wage rate
only through  the  income  effect and  in the  most indirect  manner.   Unfortu-
nately,  their model   "does not suggest an empirically  feasible approach for
treating these time  costs" (p.   264).   For  estimation,  they confined them-
selves  to a modification  of a traditional  demand  specification.

     Researchers are thus  left with  considerable confusion about the role of
the wage rate  in  specifying an  individual's value  of time.   But there is an
important body  of economics  literature, somewhat better  developed,  which has
attempted to deal with similar  issues.   Just as the early literature on the
labor-leisure  decision provided  initial   insights   into the modeling of time
in recreational demand,   more  recent  literature  on labor supply  behavior
provides further refinement.

Time in the  Labor Supply  Literature: A Review

     The first  generation  of labor  supply models  resembled  the  traditional
recreational  demand  literature  in  a number  of ways.   These models treated
work time as a  continuous choice variable.  A budget constraint such as that
depicted in  Figure  4.1    was assumed for  each  individual, suggesting the
potential for  a continuous trade-off between money  and  leisure time  at the
wage rate,  w.   In this graph, E is  non-wage  income and T  is total  available
                                     57

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hours   Participants  in the labor force were assumed to be  at  points in the
open  interval  (BC) on  the budget line,  equating  their  marginal  rates of
substitution between  leisure and  goods to the wage  rate.   Those who did not
participate were  found at the corner solution B.
           Income
          E + wT
                            T-40
                                                Leisure time
             Figure 4.1:   The First Generation Budget Constraint

     Other  researchers  argued  that work time  may not be a  choice  variable.
 Individuals might  be  "rationed" with respect  to  labor supply  in  a  "take-it-
or-leave-it" fashion;  that  is they may  be forced to choose between  a given
number  of work hours  (say  40  hours/week) or none at  a I I  (Per I man,  1966;
Moss in and  Bronfenbrenner,  1967).    In this context,  there  is  no  opportunity
for marginally adjusting work hours,  and all   individuals are found  at one of
two corner solutions (A or B in Figure 4.1).

     While  useful  in  characterizing the general  nature  of  a time allocation
problem,first  generation  labor  supply models  were criticized  on both theo-
retical   and econometric grounds.  These  concerns fostered a  second generation
of  labor supply  research  which  made  improvements  in  modeling  of constraints
and  in  estimating parameters as  well  as making  models  more consistent with
utility  maximizing assumptions  (see Ki I I ingsworth,  1983,  p.  130-1).    Each of
these areas of development  have  implications for  the  recreation problem.
                                      58

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     The  second  generation  of  labor supply  I iterature  (see for  example
Ashenfelter,   1980;  Ham, 1982; Burtless  and Hausman, 1978)  generalized the
budget  line  to reflect more realistic assumptions about employment oppor-
tunities.    As Ki I I  ingsworth states  in his survey, "...the  budget I ine may
not be a  straight line:   Its  slope may change  (for  example,  the wage a moon-
 lighter  gets when  he  moonlights may  differ from  the  wage he gets  at his
 'first' job),  and  it may  also  have  'holes'  (for  example,  it may not be pos-
sible to  work between zero and  four  hours)".

     To appreciate this point,  consider an example:  an  individual  whose pri-
mary  job  requires  Tn  hours per  week within a  total  time constraint of T
hours per week.   The^relevant  wage rate  at  this  primary job  is wn  and  is de-
picted in Figure 4.2 as the slope of the implied line segment between A and
B,   This  individual  can earn more wage income  only by moonlighting at a job
with a lower  wage  rate (depicted  by the  slope of the segment between A and
c).    His  relevant  budget  line  is  segment AC and  point  B.   Depending on his
preference for goods and  leisure,  he may choose  not to  work and be at B; he
may work  a fixed work  week at A;  or he  may take a second job  and be along
the segment AC.   Consideration  of more realistic  employment constraints such
as these  have  implications for model  specification.   Only  those  individuals
who choose to work jobs with flexible work hours (e.g. self employed profes-
sionals,   and  individuals  working  second  jobs or  part-time  jobs)  can adjust
their marginal rates of substitution of  goods for  leisure to  the  wage rate.
All  others can be found at corner solutions where no such equi-marginal con-
d it ions hold.

          Income
E+wpTp


C
. 	 	 A
t *"" ^
t
1
T-TP

";
T
                                             Leisure time

              Figure  4.2:   Second  Generation  Budget  Constraints

     Two  other aspects  of the second  generation  labor supply models are
noteworthy.    The  first generation studies  estimated functions which were
specified in a relatively ad  hoc manner.    By  contrast,  second  generation
models have  tended to be  utility-theoretic.    This  has been accomplished by
deriving  specific labor supply  functions  from direct  or  indirect utility
                                     59

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functions (Heckman,  Ki I I ingsworth, and MacCurdy, 1981; Burtless and Hausman,
1978; Wales  and  Woodland,  1976,  1977).   Such utility-theoretic models  have
particular  appeal  for  recreational  benefit  estimation  because  they allow
estimation of  exact welfare measures.    Additionally,  first generation  re-
search was concerned either with the discrete work/non-work decision  or  with
the continuous hours-of-work decision.   Second generation empirical  studies
recognized the potential  bias  and inefficiency of estimating the two prob-
lems  independently  and employed estimation  techniques  to  correct for this.

A Proposed Recreational  Demand  Model

      It  is clear that the nature  of an  individual's labor  supply decision
determines whether  his wage  rate will yield  information  about the marginal
value of his time.    In the recreational  literature,  researchers have  conven-
tionally viewed  only two  polar  cases:   either  individuals  are  assumed  to
face  perfect substitutabi I ity  between  work  and  leisure time or work  time  is
assumed  fixed.    The choice between these two cases  is less than appealing.
Few people can be considered to have absolutely fixed work time, since part-
time  secondary jobs are  always possible.   On the  other hand, only some  pro-
fessions allow free choice of  work hours at a constant wage  rate. Addition-
ally  no  sample of  individuals is  likely to be homogeneous  with  respect  to
these labor  market  alternatives.    A workable  recreation demand model  must
refIect  the  i mpI i cat i ons wh i ch  Iabor dec i s i ons have on  t i me vaIuat i on  and
allow these  decisions  to vary  over individuals.

     In   developing  a  behavioral  model that includes  time as an input it is
useful  to  broaden   the  description  of  the nature of  the decision  problem
beyond  the   simple   travel   cost   framework.    The   more  general   household
production model  depicts  the  individual  maximizing utility  by  choosing  a
flow of recreational services, XR, and a vector of other commodities, x^.  A
vector of  goods,  SR,  is  combined with  recreation  time,  TR,  to produce  xp.
Both time, TN, and  purchased inputs,  SN,  may be  required to produce XN.

     The  individual's constrained    utility   maximizing   problem   can  be
represented  as

(2)                      Max U(xR,xN)
                         S,T

         subject  to       XR =
                                      60

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                         E + F(TW) = vK'SN + VR'SR,
and
                         T = Tw - TR - TN»

where  U(...)  is  a  quasi-concave,  twice-differentiable  utility  function,
f(...)   and   g(...)  are   vectors  of   quasi-convex,   twice-differentiable
production functions, E  + F(TW) is the sum of  the individual's  non-wage  and
wage  income,  VR  and VN are the price vectors associated with  the  vectors of
recreational  and  non=r8cr§ational  inputs  respectively,  Tw  is  labor  time
supplied, and T is the total time available.

     We  reduce  the problem by  assuming  (as  do Burt  and  Brewer,  1971,  and
others  before  us)  a  Leontief,  fixed-proportions  technology.    This   is
equivalent to  assuming  that the  commodities,  I.e. the x's, have  fixed  time
and money costs per unit  given  by t and p,  respectively.   For  the  recreation
good,  XD,  it  implies  that a  unit of XD  (e.g.   a  visit) has  a  constant
marginal cost (pR) and fixed travel and on-site time requirements  (tR).   All
other  commodities  are subject  to  unit  money or  time  costs and the  general
problem becomes

(3)                     Max    U(xR,xN)
                       XR,XN

subject to              E + F(TW) - PR'XR - PU'XN = 0,

and                    T - Tw - t^Xp -  tN'xN = 0,

were p and t are the unit money and time prices of the x's.

      In  order to  characterize  an  individual's solution to the problem posed
in  (3),  it  is  necessary  to  know  the  nature of the  labor market  con-
straints.    For  any individual,   it is possible  that an  interior solution  is
achieved, such as  along  line  segment AC  in Figure 4.2.    The  individual  can
adjust work time such that his marginal  rate of substitution between  leisure
and  goods  equals  his effective  (marginal)  wage rate.    As Ki I I ingsworth
points out,  this  is most  likely to be true for  individuals who work overtime
or secondary jobs,  but may also  be  true for those  with  part-time jobs  and
those  (e.g.  the  self-employed) with discretion  over their work  time.  An
individual  may,  alternatively,   be at  a  corner solution such as point A or  B
in Figure 4.2.    Point B  is  associated with  unemployment, while an  individual
at point A works  some fixed work week  at wage  wp  and  has  the  opportunity to
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work  more hours  only at  a  difficult  wage.     In  neither case  is there a
relationship between the wage rate the  individual faces and his valuation of
time.

     Strictly speaking, the  problem  in  (3) requires the simultaneous choice
of the x's and  the  individual's  position  in the labor market (i.e.  interior
or corner solution).    It  is,  however,  beyond  the  scope  of most  recreation
demand studies  to model  the entire  labor decision.   Labor market  decisions
may well  be  affected  by individuals'  recreational  preferences  and the type
of recreational opportunities  available  to  them.   However,  the sort of day
to day  and  seasonal  recreational choices about  which  data is collected and
models developed  can  reasonably be treated as short  run decisions  con-
ditioned on  longer  run labor choices.    Since  there are  high  costs  to
changing jobs,  adjustments  in labor market situations are not  made contin-
ually.    Thus,  recreational  choices are considered  to  be  conditioned on the
type of  employment which the individual  has  chosen. Of course  if the  indi-
vidual chooses  an employment situation with  flexible work hours,  then  time
spent working is treated as endogenous  to  the  model.

     The problem as posed  in (3)  is  restated  and the first-order conditions
provided, given  alternative solutions to  the  labor supply  problem.    For
individuals at corner  solutions  (such  as  B  or A in Figure 4.2), the problem
becomes

(4)               Max  U(x) + MY - £p1xi)  + u(f - ZtfXj)


where Y is effective  income  (including  the  individual's  wage  income  if he
works  and  nonwage income  which  may  include  the  individual's  share of the
earnings  of  other  household  members).    The  variable T  is  time  available
{after job market activities) for household  production  of commodities, in-
cluding recreation.

     First order conditions are
 (4a)                       Y - ZpiXl -o,

                          T - HX  = 0.
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Note that since work time cannot be adjusted marginally, the  two  constraints
are not  collapsible.    Solving (4a)  for the demand for xi  yields a demand
function of the general form

(4b)                      x1 = hc(p1-,ti)p°,t°,Y)T) + e

where p° and  to are the vectors of money and  time costs of all  other  goods
and e is the  random element  in the  model.    (The properties of  this demand
function are detailed  in the Appendix to this Chapter.)

     For an  interior  solution in the  labor market,  however, at  least some
component of work  time is discretionary and time can be traded for  money at
the margin.   Thus,  the time  constraint in problem  (3) can be  substituted
 into the  income constraint,  yielding the  one constraint
                         Y  + wDT  - Z(p1 + wDti)x1  =  0

where WD is the wage rate applicable to discretionary employment.

     The maximization   problem  conditioned  on  an  interior  solution to the
 labor supply decision  is

 (5)              Max U(x) + 6(Y + wDT - i(pi + Wg^lxj).
                  A
First order conditions are

 (5a)                    3U/9X.J - 6(pi + wDt-}) = 0

                        Y + wDT - i(pi + Wjjt^Xj = 0.

Solving  for  the  general  form  of  a  recreational  demand  function  for  the
 interior solution yields

 (5b)           Xf = hl(p1 + wDt1f p° + wDt°,Y + wDT) + e.

 Note  that,  for empirical  purposes,  Y  + wDT can  be  re-expressed  in  terms  of
 variables  easily  elicited  on   a  questionnaire.    The  term Y + WDT equals
 Y + wDtQ +  wD(T-tD)  where tD is  discretionary work  time,  Y  is total income,
 and T-tD is  the  time  available  for household production (or total time minus
 all hours worked).
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     Consideration of demand functions  (4b)  and  (5b)  suggests that the data
requirements of  estimation  are not  overly  burdensome.    In addition to the
usual  questions  about  income,   and the time  and  money costs of  the recre-
ational activity, one need  only ask a)  the individual's total work time and
b) whether  or  not he has discretion over any part of  his  work time.  If he
does,  his discretionary  wage must be elicited.

     In problem  (5)  the recreational demand  function  is conditioned on the
individual having chosen an interior  solution  1n  the  labor  market.  The wage
rate (WQ) reflects  the  individual's  value  of time because work and leisure
can  be traded-off marginally.   However,  when this  1s not  the  case  as 1n
problem (4),  the marginal  value of the individual's time  in other uses is
not equal  to the wage rate  he  faces.  This does not Imply  that the opportu-
nity cost of  time  Is  zero  for such  an  individual.   It  is only that his
opportunity  cost  is  not equal   to  an observable  parameter.    The opportunity
cost of an individual's time will  be  affected by  the  alternative uses of his
time.

Considerations  for Estimating  Recreational  Benefits

     In order  to estimate  recreational  demand functions and thus derive
benefit estimates,   it  is  necessary to define a specific form for the demand
equation  and to postulate  an error  structure.

     This task is complicated  by  the fact that  the individual's decision
problem,  as formulated  in  this Chapter,   is  not  the classical  one.    The
problem is  now the  maximization of utility subject to  both an  income  and a
time constraint.    The  comparative statics  and  general duality  results of
utility maximization in the context of  two  constraints  are developed in the
Appendix  to this  Chapter.   There,   it is demonstrated rigorously that maxi-
mization   under  two  linear  constraints  yields  a  demand  function  with
properties  analogous to the one constraint case.   The demand function is
sti I  I  homogeneous of degree zero,  but in a larger I 1st of  arguments - money
prices, time prices,  income and time endowments.    It also satisfies  usual
aggregation  conditions.    In addition,  two duals  are shown  to exist  -  one
which  minimizes  money costs subject to  utility and time constraints and the
other   which   minimizes   time  costs   subject   to  uti I  ity   and   income
constraints.    Associated with each dual  is  an  expenditure  function  and a
compensated  demand.   Both  income and time compensated demands are own price
downward  sloping  and  possess symmetric,  negative semi definite substitution
matrices.
                                     64

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     Despite the  analogies which exist  between the one and  two constraint
models,   integrating  a  demand function back to  an  indirect utility function
 is not  straightforward  in  the two constraint case.    In addition, it  is not
altogether  obvious how the  Vartia numerical  approximation  techniques de-
scribed  in  Chapter 2 can  be  applied  when the demand function  derives from
utility maximization subject  to two constraints.   Consequently  it is useful
to begin  with  a direct utility function and solve for recreational  demand
functions by maximizing  utility  subject to the  appropriate  constraint set.
The  form  of the  demand  functions and  the  indirect  utility function  will
depend  on which  constraint  set  is  relevant.     Rather than deal with the
general  model,  a specific  case is  shown  here.

     The utility function  chosen  for   illustration  is


             (Yi + Vxi  + e      r
-------
 (7)               X} = a

for  individuals at corner solutions in the labor market,  and

 (8)               x: - a + Yi(Y + WDf) + 6>Y1(P1+ WQtj) + e

for  individuals at interior solutions  in  the  labor market.

     A  word  about the  particular  utility   function  chosen  is  in order.
Because  there are  potentially two  constraints,  the  utility  function must
accommodate three  goods.   Only x, is  of interest-,  however s and in  order  to
avoid  the  more complex  problem of integrating  back  from systems of  demand
equations,  a  bivariate  direct  utility  function  with useful  properties was
modified to include three  goods.   The modification  involves the  inclusion  of
x2 and x3  in  such  a way  as to Imply  that they are perfect substitutes.  This
procedure  has two  unfortunate  ramifications.   For an  interior  solution, when
the  two  constraints  collapse into one,  this  form implies that either xo  _or_
x3 is  chosen  (but not both).   Which  one  is  chosen  depends on  the  relative
sizes of the  prices and  parameters.   It  turns out that if x3  is  chosen, then
the  coefficient ^i must  be replaced  by ^2 in   (8).    Another   unsatisfactory
feature is  that for  corner solutions, when the two constraints  are  not col-
lapsible,  the functional  form implies a  constant trade off between  time and
money, equal  to  -fg/fj-   This  is a  direct  result of the perfect  substitut-
ability between x2 and x3 which  produces linear iso-utility curves in time
and income space.

     Despite  the somewhat  restrictive properties  of the utility function  in
 (6),  its  maximization subject to the  two  constraints  allows us  to make
operational a demonstration of the suggested approach.    It is interesting  to
note  that  equations  (7)  and  (8),  being   linear in the  respective variables,
could  easily  have been specified  as ad  hoc demand functions,  without ref-
erence to  utility theory.    This would   not  have altered  the  implicit re-
strictions on preferences  implied  - no one would  have understood  their
 implications.   Additionally,  one  would have  no way of  properly  interpreting
the  parameters or of calculating estimates of compensating  and equivalent
variation.

     Since  the  two constraint problem possesses  two  duals and thus two ex-
penditure  functions,compensating variation can  be measured  in  terms  of
either of   two standards - time  or money or   a combination of both.   The
                                      66

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anomalies which  this can  cause are discussed  elsewhere (see Bockstael  and
Strand, 1985).    Here  compensating  variation measures  of  the price change
which drives  the  demand for  x  to zero in terms of both time and money  are
presented.    For  the interior solution,  the money  compensating variation is
given by
                   V               n    *1   A 8
(9a)              CV* = exp[Tl(prPi  >] ( 1  y*   )
                                            y

for the  interior  solution, where  (Pl°,  Xj0) 1s the  initial  observed point.
The time compensating variation for individuals  at interior solutions is
                                         o +
 (9b)           CVJ - exp [Tl(~Pl - pj)] (
Compensating variation for the two constraint  case  can  be  specified by first
substituting  demand  functions   into  (7) to  obtain the  indirect utility
function
                                 v  ,
  V(p,t,Y,T) =   exp(-nPl - Y2t!)  (
                                      a
and  inverting to obtain the  money  expenditure function
    my=   ^    U°exp


where  U°  is  the initial level of  utility.   The money compensating variation
for  a  loss of  the  recreation good  conditioned on a  corner  solution  in the
labor market is then
 (I0a)            CVJ = exp[Yl(Pl - pj)]
The time  expenditure function for this group  equals

       Y      Y2


 and  the  associated  time  compensating  variation  equals


 <10b)             CVJ  = expfyCp*!-  pj)]  (  X   ^^}  - ~


                                      67

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Estimating the Model:   The Likelihood Function

     As discussed  in  Chapter  3,  a random sample of the  population will  pro-
duce a significant portion of nonparticipants.   To  correct for  the truncated
sample problem  which  nonpartlcination would  generate,  the Tobit model  dis-
cussed in  the  previous  chapter  is employed.  The jth  individual  is observed
to  take  some  positive number of  recreational  trips,  x,  if and only  if  the
cost of  the trip,  p,  is less than his reservation  price  p, where the  reser-
vation price  is  a function  of   other  factors  influencing the  individual.
Thus

               Xj = hj(«) + EJ     if and  only if hj(«)  + EJ> 0
               Xj = 0             otherwise

where h,(*) is  the  systematic portion  of  the  appropriate demand function
evaluated for  individual  j  (eq. 4b or 5b).

     Referring back to the deviation of the  likelihood function presented  in
equation  3 of Chapter 3,  if the sample of persons is  divided so  that the
first m  individuals recreate and  the last n - m do not, then the likelihood
function  for  this sample  is
(11)               Li  - !  f(e,/c)/o    n   F(-M-)/a).
                      j=l    J       j=m+l     J
This general  form of the  I ike I ihood  function wi I I  be true  for  each labor-
market group.    However,   account must  be  given to  the difference in the
demand functions  for each group.   Thus, for our entire sample of persons
with  interior and  corner solutions  in  the labor market,  the  likelihood
function  is
          m              n_                nij             nj
(12) L* = n  f(eS/a)/o   n    F(-h?(-)/<0  n  f(eJ/o)/a   n  ,  F(-hJ(-)/o)
         j=l    J      j=mc+l     J       j=l    J      j=mi+l      J
where the subscripts c and  I refer to numbers of  individuals with corner and
 interior solutions respectively.

-------
     Should  only observations  on  participants  exist,  one can  still  avoid
sample  selection  bias  by employing  a form  of the conditional  likelihood
function as  presented  in  equation  (10)  of Chapter 3.  The conditional proba-
bi I ity  of an  individual j  taking xdvis its given that  Xjis positive  is
g i ven by
                       mc  f(ej/01/0     ml    f(e$/0)/o
t13'            L  =  4?i  Tntr;
                           F(h(O/0)
An  I I lustration
     The purpose  of this section  is  to demonstrate the appl i cat ion of the
proposed  approach  for estimating   recreational demand  functions  and for
calculating   recreational     losses   associated   with   elimination  of  the
recreational  site.    In a Monte Carlo  exercise, comparison of this model with
those  generated  by traditional  approaches  is  made.    The  exercise gives an
example of how the traditional   approaches  can  produce  biased parameter
estimates  and  inaccurate benefit measures.    For an application to actual
survey data see Bockstael,  Strand,  and Hanemann (1985).

     To have  a  standard by  which  results  can be compared,  we begin with a
direct  utility  function of  the  form  in (6), choose  parameter values {see
Table  4.1,  true  model),  and  generate  ten  samples of individual  observa-
tions.  Each sample or replication  is  composed of 240  drawings, one  third of
which are  consistent  with  each of the following situations:   a)  an  interior
solution in  the  labor market, b) a fixed work  week  solution, and  c)  unem-
ployment.    Two   hundred  forty  values  for  wage  income,  non-wage   income,
secondary  wage  rate,  travel  cost and  travel  time  are  randomly  drawn from
five  rectangular   distributions  R($0S$25.000)S  R(SOS$1000)S  R($2.5.  $5,0)s
R($0,  $60)  and  R(0,4),  respectively,  and  these  values  for  the exogenous
variables are repeated  in  each  replication.   The replications  are different
in  that  independent  error  terms  are  drawn  from  a  normal   distribution,
N(0,25), for each of the 2400 individual observations.

     Total  recreational  time is taken  to  be the sum of travel  and  on-site
time.    While  it  is  assumed on-site  time  is exogenous, fixed at six hours per
trip  for all  individuals,   it is still  necessary  to  include  this fixed  amount
since  in  the collapsible time model  it  wi I I  be  valued differently by  indi-
viduals with different time values.
                                     69

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                                Table 4.1

               Mean  Estimates,  Biases,  Standard  Deviations
             and Mean  Square  Errors of  Estimated  Parameters
                 (10  replications  of 240 random drawings)

True
Mean
Estimates
Y -4.00
e1 -120.48
Y/ .50
Y2+ .33
a 5.00
Bias
Y
6'
YI
Y2
a ...
Standard Deviations
Y
0 1
p ...
T , ...
1
Y2
o ...
Mean Square Errors
Y
3 ...
Y 1 ...
YI ...
1 2
O • • •

OLS-I


3.66
-104.68
.38
. . .
3.88

7.66
15.80
-.12

-1 .12

1.26
44- 66
,06
, , ,
,21

60.26
2244.00
.02


1.30

OLS-C


5.04
-196.03
,22
2.05
3.78

9.04
-75.55
-.28
1.72
-1.22

3.57
110.76
,06
2.05
1.77

94- 47
17975.00
.08
7.16

4.62
MODEL
ML-I


-6.45
-166.30
.60
...
5.38

-2.45
-45.82
.10
...
.38

5.38
60.34
.15
* • •
.74

34.95
5741.00
.03


.69

ML-C


-.56
-204.28
.53
.06
5.15

3.44
-83.80
.03
-.27
.15

3.09
96.86
.16
1.31
.73

21.38
16404.00
.03
1.79

.56

CML*


-6.11
-118.22
.57
.77
4.95

-2.11
2.26
.07
.44
-.05

5.67
54.72
.15
.91
.70

36.60
2999.00
.03
1.02

.49

ML*


-4.72
-113.65
,52
.43
4.65

-.72
6.83
,02
.10
-.35

2.01
30.87
.05
,74
,33

4.56
999 . 00
,00
,56

,23
Because  of scaling  differences,  estimated values  for Y! and Y2 are  one  one-
thousandth of the  values  shown  in the table.

                                    70

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     The true  demand models have  three forms,  conditioned on the labor
supply choice:

(14a)  x = -3.22 - .06 p + .0005 Y - .0$ t + e  (fixed work week)
          (a+Y2T)   0).

     For comparison purposes, estimates  for  the  parameters   a,  & ', r^,  ana ^
are  obtained using  five  different  procedures.   The  first two  procedures
(OLS-I  and   OLS-C)  approach  the   problem  in  the traditional  manner:    all
individuals  are treated  identically  with respect to  time  valuation  and only
participants are  included in the  sample.   Ordinary  least  squares  estimates
of  parameters  are obtained  for both  models.  The  two models  differ in  the
way  time is  incorporated in  the  model.   In the  OLS-I  model, everyone  is
assumed  to  value  time at his wage rate.   In  OLS-C, time  and money  costs  are
introduced  as  separate  variables  for all  individuals.   To  distinguish  the
biases  which may  arise  due to  model  misspecification  from those attributable
to  samole selection  bias,  a  second set of estimates are obtained from a max-
imum  likelihood  formulation  (ML) which corrects  for the  truncated sample
probTem  but  not  the misspecification.    All  individuals  are  incorrectly
presumed  to  be  at interior  labor  market solutions  in ML-I, and all  individ-
uals  are  incorrectly  presumed  to  be at  corner  solutions  in ML-C.   The  final
estimation  represents  the  "correct"  approach  in  that  both  the  truncated
sample  problem  and the specification problem are  addressed.


                                      71

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     CML* uses  exactly the same  data  set as OLS-1,  OLS-C,  ML-I and  ML-C;
that  is,  only  participants  are included  in the sample.  Similar  to  ML-1  and
ML-C, the CML*  approach corrects  for the  truncation  problem  by maximizing a
conditional   likelihood function,  conditioned  on participation  (see  eq.  13).
Unlike ML-I   and ML-C,  this  approach  also  conditions the recreational  demand
function on the labor market decision.   Finally ML*  is  estimated by maximiz-
ing the  likelihood function in (12).   The  difference-between  CML* and  ML*  is
that  the  ML*  approach  includes  nonparticipants.  This  is the  preferred
approach  when  possible,   but  information on nonparticipants  is often  not
available.    It  should be  noted  that  ML*, by  definition,  is  based on  a
slightly different sample since  it  includes nonparticipants.   To facilitate
some manner  of  comparison,  the sample  sizes  upon  which the  parameter esti-
mates are based are kept the  same even  though  some  of  the  observations
differ across approaches.

     In Table 4.1,  statistics on the parameter estimates from the experiment
are presented.   The  "true"  parameters  (denote these 9"), those used to gen-
erate the data,  are  recorded  in  the first row.    These are  followed  by the
average  parameter  estimates for  each  technique.    (EQ.j/10,  where  6-,-  is  the
estimated value of a  parameter  on the  ith repetition).   The parameter esti-
mates are averaged over the ten  rep I icat ions;  consequently,  these numbers
represent the sample  means of the estimators for each parameter and  each ap-
proach.   The second part of the table presents the estimated biases for each
parameter and each approach.   These are  the  differences between the  "true"
parameters  and  the   sample  means  of   the  estimates  (i.e.  /He   - 6-) ).
Finally,  mean-square errors are  provided for purposes of comparison  Iwhere
mean-square  error  is defined as bias2   + variance).   A comparison of mean
square errors shows  the ML*  approach  to be superior to  all  others  with  re-
spect to  all parameters  including the  standard  deviation of the  disturbance
term.   On the basis of mean square errors,  the  CML*  approach would appear to
be second best.   OLS-I provides estimates  of  6' and T1 with  slightly smaller
mean square  errors  (although  the biases are  larger),  but the  mean  square
error of  the OLS-I  estimate of cr is considerably  larger than  that of CML*.
Both  OLS approaches  produce  large  MSE's  for  a  and both  approaches  which
presume  everyone   is  at  a corner solution (OLS-C  and ML-C)  produce  large
MSE's for the  preference  parameters  -  particularly for e'.  OLS-C  is the
poorest  performing approach uniformly.    This  is  the approach  which ignores
the truncated sample problem and  includes time and money costs  separately in
the  regression.     It  is  important to  note here that no  correlation between
these  costs was  introduced  when generating the  data.    The  correlation
between  money and time prices which  is  usually  found  in travel cost data
would likely increase the  variance in these estimates.

                                     72

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     In addition to  estimating  parameters  for each procedure, estimated wel-
fare measures  for  hypothetical  price  increases sufficient  to  eliminate the
recreational activity are  provided.   First the "true" compensating variation
for each  participant in each of the ten replications  is  calculated.   These
are calculated  using the  formulas  in equations (9) and (10) from data on the
individual's  number  of  visits, costs,  etc.  together  with  the set of true
parameters.   These  "true"  compensating variations  for  the i"" individual  in
the jth replication  are  denoted CV*jt   The  CV^j  is the  standard by which one
can compare the results of the  six estimation approaches.

      In Table 4.2  are the results of compensating variation calculations.
For each  individual, six  estimated compensating  variations  were  calculated
using  the  estimated  parameters from each  of the six estimation approaches.
For comparison  purposes  the  ML* parameter  estimates  are applied  to  exactly
the same  sample of  individuals as the other  parameter estimates.    This  is
actually to the disadvantage  of the ML* approach  because  the parameters for
this approach were estimated  from a  slightly  different  sample.

     The numbers in  the  table  represent the  averages of the CV calculations
over all  individuals in all  replications.   For each approach,  the  bias re-
ported in this  table is  the average (over  all individuals in all samples)  of
the difference  between  the "true"  compensating variation  for  an  individual
and his estimated  compensating  variation.   For  the entire sample, including
all participants from  the  ten  replications,  the  average  "true" compensating
variation per  participant is  $428.85.   This figure reflects  the following
calculation: IjIic''ij/!:j™j» where  Nj equals  the  number  of  participants  in
the jth  replication.  The average CV's can  be transformed  to  "per  capita"
values by multiplying by .46.

     In comparing the average CV's  calculated from the  estimated parameters,
it  is  clear that the OLS  estimates are by far  the worst.   These estimates
are between two and three times as  great as the "true" average CV.    The
results are  consistent  with  the  a   priori   reasoning   that ignoring the
truncated sample problem will  bias welfare  measures upward.

     Interestingly,   the ML estimates  which take  account of the truncation
problem but which do not  incorporate the  individual's labor market decisions
both appear to  be  biased downward.   Also  of interest is the  fact that,  at
least  in this example,  if one  misspecifies the  demand  by ignoring the  labor
market decision,  it  does not  seem to matter  very  much  which  of the  two con-
structs (corner or interior solution) is applied to the sample.
                                      73

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                                  Table  4.2


               Mean Estimates, Biases,  and  Standard  Deviations
                     of  Compensating  Variation  Estimates
               Average
            Compensating   Average Deviation   Standard Deviation Standard  Deviation
   Model       Variation       From True  CV          around  Col.  1        around CV*
  True      $428.85                ..                624.57

  OLS-I      1169.00             740.13              2225.64              1137.37

  OLS-C      972.31              543.46              1487.73               892.57

  ML-I        311.03           -117.82               453.12               275.60

  ML-C       306.26           -122.59               441.39               277.86

  CML*       557.13             124.28               938.60               280.18

  ML*        495.75              66.91               716.80               206.36
     The  ML*  approach produces a CV estimate which, while  larger than the
true average CV,  is by far the best.   The  CML*  estimate  is  larger,  but  sti I I
is within 25% of the  "true"  value.    It  is  of importance  that  both preferable
approaches yield estimates  larger than  the  "true"  average  compensating  vari-
ation.    In  the next  chapter the reasons why  an  upward  bias may  be  expected
are explored.

      It  would  be  helpful at this point to  present measures of the  variance
of these compensating variation  estimates.   However  useful measures  of  vari-
abi I ity  are difficult to define in this  case.    When  examining  parameter
estimates from each approach,   sample  variances of  the  estimates were
calculated.    However in the case of the estimated compensating  variation,
sample variances might  be misleading.    In  the parameter  case  the  true  param-
eters were  fixed;  increasing variation  in  estimates of these parameters was
obviously undesirable.    However the true  values  of compensating variation;
                                      74

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the  CV*j's  vary  over all  observations;  thus  the  CV*'s themselves  have  a
nonzero  sample  variance.    Expressed  in  another way,  one no longer  has  the
desirable  circumstance of  observing  several  estimates  drawn  from the  same
distribution as was true with the estimated parameters.

     Also  in  Table   4.2   two   statistics  which   reflect  variability   are
presented.  The  first is  the simple  standard  deviation,  calculated for  each
replication and  averaged  over replications.   The second  statistic captures
elements of both bias  and  variability.   For each replication the square  root
of  the  sum  of  squared deviations  of CV^  from CV^ is calculated.    The
number reported in the table  is  the  average'of these statistics over  the ten
replications.     The   number  will   increase   with   increasing   bias   and/or
Increasing variability around the true CV.

     From Table 4.2 one can see  that the OLS estimates are  once again quite
dismal.   The standard  deviations around  their own means are between  two  and
four times  as  great as the variation in the  "true"  compensating  variations
in the sample.    In contrast, the  variation in ML*   is only  slightly  greater
than the variation  in  the  CV*'s.  Both ML-1 and ML-C  produce estimates with
smaller variances than the  actual  variance  in  the sample.   This is no doubt
related to the  fact  that  these  estimators  under-predict  CV.    Thus the  same
percentage variation around  the  mean  will  translate into a  smaller standard
deviation.

     The second half  of the table I ists the standard  deviations around  the
true values of CV.     Note  that  the ML* approach is sti I I  superior  to  a I  I
others.   The poor performance  of  the  OLS  models is once again apparent.

Observations
     At this point  it  is useful to summarize the key aspects of  this  chapter
and elaborate on  some  points not fully developed  in the  text.    Perhaps  the
major contribution of the chapter  is the  integration of the  labor  supply  and
recreational  demand  literature.    In so doing an attempt was made  to  provide
a coherent and  general  approach to the treatment  of time in the  context of
recreational  demand models used to value natural resources and environmental
improvements.

     The essential  property of the  generalized demand  model  incorporating
time  is that it  is derived from  a utility  maximization problem with  two
constraints.   The  details  of the two constraint problem are explored in  the
                                      75

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Appendix to this chapter.   The presence of two constraints causes theoreti-
cal difficulties  in  moving from a demand function to  a  utility function to
obtain exact welfare measures,  and as such the results of Chapter 2 can not
be applied directly.   While models from Chapter 2 could be modified to serve
our limited purposes here, an  examination of the Vartia approximation method
 in the  presence of two  constraints would  likely allow greater generality in
the demand function,  yet preserve the ability  to obtain Hicksian  measures.

     The two  constraint case  also has  interesting  implications for welfare
measurement.   The  utility  maximization problem now admits of two duals,   i.e.
two expenditure  functions and two compensated  demand  functions.     This
 implies that  the  welfare effects of a policy  change can  now  be measured in
either  (or a combination) of two standards  - money  or  time.    The impli-
cations of this dual  standard  are investigated elsewhere (see Bockstael  and
Strand,  1985).

     The illustration in this  chapter  focuses  on  the  traditional  money  com-
pensating  variation measures and  explores  the  biases  which  can  arise in the
estimates  of preference  parameters and compensating  variation  by  using a
misspecified demand  function.    While Monte Carlo  type examples  are  never
completely  conclusive,   the experiments  suggest wide  disparities in CV esti-
mates when different estimation  approaches  are  used.     Compared to  the
correctly  specified  approaches which also account for the  truncated sample
problem (the ML* and CML*  approaches),  the conventional  OLS approaches pro-
duce upwardly biased  estimates of CV with  large variances  around  their  own
mean and around the true CV values.    Maximum  likelihood  estimates  which
account for truncation  but not misspecificat ion of the  time-price variable
appear to  be  downwardly biased.   The ML*  estimate is much  preferred  with
relatively  small variance  and  deviations from the true  value of  CV.

     Both  ML* and  CML*, although calculated from presumably  consistent  pa-
rameter estimates,  produce CV estimates which on  average exceed  the  true
CV's.    In  the next chapter it  is  demonstrated why  compensating variations,
even when  calculated from unbiased  parameters,  may themselves  be upwardly
b i ased
                                     76

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                      FOOTNOTES TO CHAPTER 4


 In fact, the wage rate may not even serve as an upper or lower bound on
the  individual's marginal  valuation of time when labor time is institu-
tionally restricted.    That  is, an  individual  who chooses to be unem-
ployed  may  simply value his marginal  leisure hour more than the wage
rate,  or he may value  it  less but not  be  better off accepting a job
requiring  40 hours  of work per  week.     If restricted to an all-or-
nothing  decision,  40 hours may be  less desirable  than 0.  An   individual
at a point  such as A,  however, may value  the  marginal  leisure hour at
more than wp  but  choose 40 rather than  0  hours.   Alternatively he may
value leisure time at less than Wp but more  than  the  wage he  could earn
for additional  hours  by  working a  secondary job.
                                77

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                                APPENDIX  4.1

         A  COMPARATIVE STATICS ANALYSIS OF  THE  TWO  CONSTRAINT CASE*
     The subject of this  Appendix is the consumer  choice problem with two
constraints.   As we saw  in  Chapter 4,   labor  market  restrictions  and  labor-
leisure  preferences  cause  individuals  to  be either at  interior  or corner
solutions  in  the labor market.    Classic comparative  statics and welfare
evaluation  is directly applicable to  interior  solutions as  the time and
income constraints collapse  into one.   However the  comparative statics and
duality results associated with the corner  solution  case  (i.e.  utility maxi-
mization  subject to  time  and  income   constraints) have  received  little
attention.

     The first treatment of  the problem  was by  A.  C.  DeSerpa  (1971).
Suzanne Holt's  (1984)  paper is the only other which explicitly deals with
comparative statics  of the time and income  constraint.   Both Holt's  approach
and that of DeSerpa's involves  inversion of the Hessian,  a tedious and dif-
ficult task for problems  with  large dimensionality.   The Slutsky equation
derived from this approach  includes  cofactors of  the Hessian and, as such,
is a  complex  function of the  decision  variables  in the system.     In what
follows,  a more modern approach  is employed based on the saddle point theo-
rem,   as  proposed by  Akira  Takayama  (1977).    Making use  of the envelope
theorem,  this approach is simple to  apply  and  far more  revealing.    From  it
can be derived Slutsky equations  containing elements with  clear economic
interpretations.

     This Appendix goes beyond the previous work by examining duality re-
sults and  demand function properties   in  the  context of  the  two  con-
straints.   Several  new time analogs to the  well known results  in traditional
demand theory  are presented.   Specifically,  we derive a time  analog to Roy's
Identity and  two  generalized Slutsky equations.    These Slutsky equations
* This appendix is the work of Terrence P.  Smith,  Agricultural  and  Resource
Economics  Department,  University of Maryland.

                                     78

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which describe the effect  of a change  in a money price are si mi lar  to the
traditional  Slutsky  equation  but  contain additional  income  (time)  effect
terms which describe  how  demand responds indirectly to income (time)  changes
through the trade-off between time and money  in  producing utility.

Utility Maximization with Two  Linear  Constraints

     Consider  the  household who maximizes a utility  function,  U(x),  where x
is a  vector  of activities  that produce utility.   These  activities need not
be actual  market  commodities.   The link  to  the market  is  through a  set of
household  production  functions.  Suppose  that  the household produces these
activities,  x, according  to  the  non-joint production  functions,  f^s^.v^)
where s^  and v^ represent  a  vector of purchased  goods and  time  inputs into
the production of x^.  The  problem, then is to

(A1)       max U(x) subject  to  x,- = f^s^v,-) for all  i, and
                           Y  =  R  +  wTw = Er^,  and

                           T=T+T=T  + Iv.,
                               w       w     i'
where Y  is  total  income, the sum of  nonearned  income R and wage income wTw,
and  ri  is  a  vector  of money  prices corresponding  to  the  vector  s-j.   To
proceed to  specific  results,  a  fixed coefficients Leontief technology is as-
sumed, that is,  a technology with  no  substitution possibilities between the
purchased inputs  and  time.   This assumption implies  that the activities, x^,
have fixed  money and time costs, representable as  the scalars, p^ and tn-.

     As has been  explained in the  body  of this chapter,  the problem in (1)
can take two forms.   If work time  is an endogenous variable,  i.e.  a  decision
variable of  the  individual   who   can   choose  Tw   freely,   then the  two
constraints in the problem collapse  to one:

                           R + wT = z(p. + wt.j)x.j.

 In  this  case  the  problem is  structurally similar to any other one constraint
problem.     If,  as wi I I   be assumed  in this appendix, work time  is institu-
tionally  constrained,  then Tw  can  be treated as  fixed  and two  relevant and
separate constraints remain.   The problem then can  be rewritten
                                      79

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                                        I
(A2)           max U{x) subject to Y = p x and T = t x.

where T = T -T .

     U(x)  Is  a  twice  continuously differentiate  concave utility  function
with x  an  n-dimensional  vector of  commodities.   The consumer behaves so as
to  maximize  this  utility  function.   There  is  a  commodity,  say xk, which
represents savings  such  that  the  income constraint is always  satisfied,  and
there  is  another commodity, say  x^,  which is uncommitted leisure time  such
that the time constraint is effective.

     Since the objective function  is differentiable and  concave in x,  the
constraints  differentiable and  linear  in  x  and  b,  where b=(p,t,Y,T),   the
constraint qualification and  curvature conditions are  met.    This  implies
that,  if a solution exists,  then the quasi-saddle  point (QSP) conditions of
Takayama  (1973)  will  be both  necessary and  sufficient.    Also,  note that,
given  the  assumption  of  the existence of  slack variables, savings  and
uncommitted  leisure time,  the constraints  are  effective,  and  if a solution
exists  it will be an interior one.   Collectively, these  conditions allow  the
application  of the envelope theorem  to our problem.

If a solution to  (2) exists,  it will  be of the form x{b),  e(b), <(.(b).  Hence
we may  substitute these solutions  into the original  Lagrangian  to  obtain

(A3)       L(b) = U(x(b)) + 4>{b)  [Y - px(b)] + 6(b)  [T -  tx(b)].

Now U(x(b)) may be written as V(p,t,Y,T) and  interpreted in the usual way as
the  indirect  utility  function.    Note  that,  in addition to the traditional
parameters affecting  indirect utility  (prices, p,  and   income, Y),  the  time
prices,  t, and time endowment,  T,  are  also  relevant parameters.    Applying
the envelope theorem to the above  we obtain

(A4a)                   3V{p,t,Y,T)/SY = +(p,t,Y,T)


(A4b)                   aV(p,t,Y,T,)/3T = e(p,t,Y,T)

(A4c)                   8V(p,t,Y,T)/3Pi = -
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Combining (4a) and  (4c)  gives  ROY'S Identity,  viz.,
/  N                       3V(p,t,Y,T)/3Pi               .
(A5)                       	;—:	 = x-j  for all  i.
1  ;                       av(p,t.Yj)/8Y      1
Likewise,  combining  (4b)  and (4d) gives analogous  identity, viz.
 (A6)                         'r>  '  '  "  * = x,  for all  i.
 1  ^                       3Y{p,t,Y,T)/3T     n

Note  that (6) gives an  alternative way  to  recover  the  Marshallian demand
from  the  indirect  utility function.  However,  both differential equations may
be required  to be  solved to  recover the  indirect utility  function from the
demand function, since it will be shown  that there are two  expenditure functions.

     These envelope  results  can be  manipulated in other ways to demonstrate
time  extensions to  traditional  demand analysis.  For example, combining  (4a)
and (4b) with (4c) and  (4d) we obtain

               3V(p,t,Y,T)/n _ e(p>t,Y,T) _ avtp.t.Y.D/Hj
(A7)            av(p,t,Y,T)/3Y " *(p,t,Y,T) " aVfp.t.Y.T)/^


which is  McConnelTs my or "the opportunity  cost of scarce time measured in
dollars of income."   Multiplying (4c) by pi, (4d) by tj and summing over all
i yields

 (A8)               £p.j3Y/ap.j + Jt-jaV/at^ =  -^^p-jX^-e^t^x^

which by  (4a) and  (4b)  implies

 (A9)              £p-av/ap-+ itjBY/^tj + Yav/aY  + Tav/ai  = 0,
so that the  indirect utility function V(p,t,Y,T)  is  homogeneous of degree 0
in money and time prices,  income, and time.

The Two Duals and the Two  Slutsky Equations

      In  this  section  the  dual  of  the  utility  maximization problem  is
explored.     Since there  are two constraints,  there are  two duals  to the
problem.    The first  is  (money)  cost minimization  subject  to  constraints on
                                      81

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time  and  utility;  the second is time  cost minimization subject  to con-
straints  on  income and utility.    This exploration yields two expenditure
functions,  an  income compensated  function and  a  time  compensated function.
The existence  of two expenditure functions  allows one to  compute  welfare
changes either  in the traditional  way as  income  compensation  measures or,
alternatively,  as time compensation  measures.

      In addition,  these expenditure functions are combined with the envelope
theorem to  reveal two generalized  Slutsky  equations.    The  first of these
describes  how  Marsha I I ian  demand responds to  money price changes  and the
second how the ordinary demand changes  with  a  change   in time  prices.   The
manner of proof  is  in the style  of the  "instant  Slutsky equation" as first
introduced by Cook (1972).

     The  duals  to the utility maximization  problem  (2)  are

(A10)              min px   subject  to T =  tx and  U° = U(x)
                   x
       and
(All)              min tx   subject  to Y =  px and  U° = U(x)
                   x
where  U°  is some reference level of utility.

     Notice that  (10) and  (11)  can  be cast in  the  notation  of  our original
maximization  problem, where  the objective functions,  px and tx,  are  linear
and hence concave  in  x and  p or t,   and  the  constraint functions are quasi -
concave since the first constraint  is  linear  (either T  - tx  = 0 or Y - px =
0) and the second, concave.   It follows then,  as in our ear Her analysis of
the primal problem,  that  if a solution  exists,  the QSP conditions  will  be
both  necessary  and  sufficient.    Furthermore,   maintaining  the  existence of
the slack variables,   savings  and  freely  disposable time, and requiring that
the reference  level  of uti I ity be  maintained ensures  that  the constraints
are effective,  that  we  have an  interior solution,   and  hence, that the
envelope  theorem  may be  applied.

     Consider,  then,  the  two Lagrangians,
(A12a)           min LY{p,t,T.U°) « px + MT - tx)  + y(U°  - U(x))
       and
(A12b)           min LT(p,t,Y,U°) = tx + u(Y - px)  +
                                     82

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     Solutions to these minimization problems,   if they exist, are given  by,

(A13a)                        xY(p,t,T,U°)
       and
(A13b)                        xT(p,t,Y,U°).

The  first  of these  is the  "usual"  Hicksian income compensated demand, while
 (13b)  is an analogous time compensated  Hicksian demand.  Of course,  both
depend (in general)  on all  money  (p) and time  (t) prices.

     Solutions (13a) and  (13b), when  substituted back into the  objective
functions,    imply the existence  of  two  expenditure functions.   The first of
these,

(A14a )                EY(p,t,T,U°) = pxY(p,t,T,U°)

is the well  known classical  expenditure function with the exception that the
time prices,  t, and the time endowment, T,  appear as  arguments.

     The second,

(A14b)                ET(p,t,Y,U°) = txT(p,t,Y,U°),

is a time  compensated measure of the minimum  expenditure level  necessary to
maintain U°.   Either (14a)  or  (14b)  may be used  to  measure welfare effects
of a change  in money  or  time prices or both.  The novelty of using  (14b) for
welfare analysis is  that  it measures  the amount of time  compensation, rather
than income  compensation,  necessary to  maintain a reference  utility level in
the  face of, say, a money  price  change  for  one  of the commodities.

     Since  the two  expenditure   functions,  Ey  and  Ej,   are  concave,  then  if
these expenditure functions are  twice  differentiate  the matrix of  second de-
rivatives is negative semidefinite.  Therefore  the slopes of  the  (compensated)
own  money  price  and own   time  price   demands  are  necessarily  non positive.
Also, define S^j =3x.j/3pj  = 32EY/3pj3p.j, as the money substitution  effect for
good i, given  a  price  change for good j, and   T^j =  3x.j/3tj  = 3  Ej/3t^9tj   as
the  time  substitution  effect.    Then   it  follows that  S and T  are negative
semidefinite and symmetric,  and  since XY and XT are homogeneous of  degree 0  in
p and t, respectively, then
                                      83

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 (A15a)                 Z  p ax /3p  =  0  =  I  S   p
                      i   i   i   j       i  ij i
       and

 (A16b,                 i.o.T
That  is,   the aggregation conditions hold.    Finally,  note that  by the
envelope theorem

(A16a)                   3EY/9p.j  =  xY1(p,t,T,U°) and

(A16b)                   aEj/dtf  =  xT1  (p.t.Y.UO),

(Shepard's Lemma) .

     The above serves to formal \ze the equivalence of  several  of the we I I
known  properties of Hicksian demands in the classical  and two constraint
systems.   The Slutsky relations  that follow from the  present problem  are  now
derived.    Although  our  results  show structural  similarity to the classical
equations,  our derivation results  in two Slutsky  equations,  each of which
has a time effect as well  as  an  income  effect/

     Consider the solution  to  the primal  problem posed  in  the preceding
section.    This  solution  is the set of Marshal I ian demands which may  be
written,

(A17)                    xm = m(p,t,Y,T).

Now recall that  the solution to our money  minimization  problem, Y,  is just
p'xY(p,t,T,U)=EY,  and   likewise,   the  solution  to  the  time  minimization
probem, T, is defined as t'xT(p,t,Y,U)  =  ET,  Hence

(A18)          x =m[p,t,Ey(p,t,T,U), ET(p,t.Y,U}].

Note  that  (18)   now  defines  the set  of  Hicksian demands.   Differentiating
(18) with respect to the jth  price,  PJ, gives
(A19a]     ax^Sp  = 3m/3p  +  (3m/3EY)  (aEY/3j) +  (3m/3Ej)
                                     84

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using the  chain  rule.   Consider  also  how the  demand  for x-,- changes with  a
change in one of the time  prices,  say  tj.   Differentiating (18)  with  respect
to t  yields,
(A19b)     axj/atj* am/atj + (am/3EY) OEY/atj) + (am/aET) (3ET/atj).

These  are  the two  generalized  Slutsky equations  that  result from the  dual
constraint problem.   To  cast  them in more  familiar  terms use the envelope
theorem applied to equations (12) to obtain,
(A20a)

(A20b)                      3ET/nj = xj

(A20c)                      3Ey/3T = A

(A20d)                      3ET/3Y = p

(A20e)                      9EY/3tj = 'X

(A20f)
Substituting  (20a)  and  (20f) into  (19a)  and rearranging, obtains  the  money
price Slutsky equation,
(A21a)        ax/apj = 9x  /3Pj - xj C3x

where   x^"1   denotes   Marshallian  functions   and   x,-    denotes   Hicksian
functions.  This Slutsky equation  is  identical  to the classical  version  with
the  exception  of  the  additional  term   yXjax^/BT, which  is  the  indirect
effect of income through time.   If xn- is an income normal, time normal  good,
then   3x^/3 Y and 3x^m/3T are  both  positive,  and  since p.the  Lagrangian
multiplier  on  the  income  constraint  in the  time minimization  problem,  is
necessarily  non-jpositive,  1t  follows that  for a  "normal-normal" good  the
"income"  effect^ is enhanced relative to the classic income effect.

     Proceeding in exactly the  same way, the time price  Slutsky equation can
be  derived.   Substituting (20b)  and  (20c)   into the  second  Slutsky equation
(19b) and rearranging, yields

(A2lb)      ax^/at  = ax^/at  - X  [sx^/aT - x 3xm /aY].
                                      85

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Notice  that  in  addition  to  the  "pure"  time  effect, XjSx^/sT, there  is  an
additional  indirect  effect,  AXj8x.j/3Y,  which,  using  the  same  argument  as
above,  is  an indirect  time effect  through income, converted to time  by  the
marginal  (time)  cost  of  income (X), Again  the  two  terms  will augment  one
another for  a  "normal -normal"  good,  and, of course, offset one  another for a
"normal-inferior" good, where  "normal-inferior" is  taken  to  represent  a com-
modity which is income normal  and time inferior or vice versa.

     Utilizing the results that u = 3ET/3Y = 3T/3Y and x = 3EY/3T = 3Y/3T,
an equivalent way of writing (21) is
(A22a)      3x^/3 PJ = ax^/ap-j - xj [3x^/3 Y - <3x!f/3T) (3T/3Y)]

(A22b)       3x^/3 tj = SX^/Stj - Xj[3x!f/3T - (3xm/3Y) (3Y/3T)].
This version makes clear the substitution between  income and time  in the two
constra i nt mode I .

A Summary of Results

     The  "usual"  properties of classical demand  functions still  hold when
one solves the two  constraint  problem.   The demand functions that solve our
maximization problem are  homogeneous  of degree 0  in  money and  time prices,
 income and time,  and satisfy  the  aggregation  and integrabi I i ty conditions.
The compensated demands,  be they  income  or  time  compensated,  are own price
 (money or time)  downward sloping.    The "substitution"  matrix is negative
semi definite,  where  the substitution  matrix must  be  interpreted  as the ma-
trix  which describes a  response to a  money (time)  price change holding
utility and the time (income)  endowment constant.    Finally,  we  can partition
the ordinary demand  response to a change  in money  (time) price as  made up  of
two effects, a utility  held constant effect,  i.e.  a  movement along  an  indif-
ference surface,   and an income (time)  effect,  remembering the complication,
however,  that this  income (time) effect  is made up of a "pure"  income (time)
effect and an indirect effect of time  (income)  converted  to money  (time)
terms.

     These new demand functions contain additional arguments relative to the
 "classic"  demand  function.   That  is,  the ordinary demands  are functions  of
not only  money prices and  income,  but also of time  prices and of the time
endowment.    Likewise,   the  money and time expenditure functions depend not
only on  money prices and utility,  but  also upon  time  prices,  and the time

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endowment  (for the  money  expenditure function)  or income endowment  (for the
time expenditure  function).    Therefore,  welfare analysis  may be done  in a
straightforward  way using these expenditure functions  provided we account
not only  for  money  and income changes but also  for  time price and time en-
dowment changes.

     One  final  result  is of particular  interest.     The Slutsky equations
(22a)  and  (22b)  indicate  a two  term income  effect for  the money  price
version and a  two term time effect  for the  time price equation.    Restating
the Slutsky equation for our own money price  change,

                  ax1!1        ax..
                  ap.        3p.

The left  hand  side  variable  is the Marshallian  price  slope.   The  first  term
on the  right  is  the Hicksian price  slope.   The total income  effect is  made
up of   the  usual  income  effect  term -x1-3xm/9Y   and the  effect of  income
through time effect x1-(ax1-m/3T)(aE1-/9Y).   Both  terms are negative  if x.,-  is
normal   with  respect to Y and T,  because aEj/aY is  negative  and  represents
the change  in  time  costs necessary  to achieve  a  given  level  of  utility  if
the individual is given more income.

     From  this expression   it  is  clear  that  the  total   (combined)  income
effect  is  greater in absolute  value than the  conventional  (direct) effect.
This has  the  interesting  result  of  pushing  compensated  and ordinary  demand
functions farther away from each other.

     This  divergence  between  the  Marshallian  and  Hicksian demands implies
that the  consumer  surplus   measure  will   be decreased  and the  compenstir.g
variation  measure  increased.    Hence  the  use  of   the  consumer's surplus
welfare  measure  to   approximate   the theoretically   correct  compensating
variation  is  made  less defensible.    It  would  seem  useful  to  reexamine  the
Willig  bounds  on  using the  consumer's'surplus as a  welfare measure  in light
of these implications.

     Whether  a good   is  time  normal or time  inferior is not  altogether
obvious.    One could  develop examples which would  suggest either case.  It
seems,   that this is  likely to be  an important question  for recreational
goods along with  the  question  of whether  or  not  an  individual's work time  is
fixed.
                                      87

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                          FOOTNOTES TO APPENDIX 4.1

    The solution to,  and sensitivity analysis of,  a more general  problem,
    i.e.  maximization of an objective function subject to multiple,  possibly
    nonlinear,  constraints has appeared  in the mathematical  economics liter-
    ature.
0
    The similarity  can also be  seen in the approach  of DeSerpa and  Holt.
    Unfortunately,   that approach,  which  relies on the  inverted  Hessian,
    tends to obscure the detail  of the time and income  effects.

3    The  interpretation  of y is  the  marginal  (money)  cost  of  time,  hence
    y  converts  the  time  effect into  income  units,  and therefore the second
    term in brackets may be interpreted  as an additional  income effect.

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                                 CHAPTER 5

                    THE CALCULATION  OF  CONSUMER  BENEFITS


     Until  this point,  emphasis has been placed on  obtaining unbiased and
consistent parameter estimates of the structural  model  of behavior.   Devel-
opments have  been  made in the  creation of models consistent with  utility
theory, in  introducing realistic time constraints on recreational  behavior,
and  in establishing appropriate  estimation  techniques.   These efforts have
all been directed to obtaining the relevant parameters of recreational  pre-
ference functions.     It  has  implicitly  been  presumed  that consistent
preference parameter estimates  together  with  correct formulas for ordinary
surplus  and   Hicksian   variation   measures   will   automatically  produce
unambiguous,   consistent estimates of  these  welfare measures.     In this
chapter two aspects of the calculation of  welfare measures  from  estimated
preference  parameters are examined.

     Despite the scores of  articles  containing surplus estimates, only a few
(e.g.   Gum  and Martin,  1975) have  devoted even modest attention to the
procedure  for calculating benefits from  estimated equations.   Most studies
presumably follow the  process outlined  by Gum and Martin, although Menz and
Hilton  (1983) indicate other ways  of  calculating  benefits from a zonal
approach.    This  "procedure"  for calculating welfare efforts from estimated
coefficients is the first  aspect of consideration.    The second  is the
explicit recognition  of the fact that  benefit estimates are computed from
coefficients with a  random component  and  therefore  possess statistical
properties in their own right. To our knowledge,  no  one  in the recreational
demand literature has  been concerned with this.

     The beginning of this  chapter considers the  common sources of re-
gression error  and the statistical  properties of benefit  estimates which
arise because  of that  error.   Three common sources are considered: omission
of some explanatory variables,  errors  in measuring  the dependent variable,
and randomness  of  consumer behavior.   For each,  the  procedures  one would
employ to  obtain estimates  of ordinary  consumer surplus and  examine the
statistical properties of  estimates  derived following  these procedures are

                                     89

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outlined.    Similar  results would be  true of CV and  EV measures,  but the
derivations are  considerably more difficult.   The two familiar  functional
forms  referred  to frequently  in the  last few chapters, the  I inear  and the
semi-log specification,  are used for  illustration.

     The general  results  are at first  alarming.   The expected value of
consumer surplus seems to depend on the source of the error.    Error from the
common  assumption of omitted  variables  leads to higher expected  benefits
than that from other error  sources.   Secondly, benefit estimates calculated
in the conventional way are generally upwardly biased when they are based on
small  samples.    The expected  value  of consumer surplus based  on  maximum
likelihood  estimates exceeds  the  true surplus values.    All  is not  lost,
however.    The  benefit estimates  are,  at  least,  consistent.    Perhaps of
greater  importance,  minimum expected  loss (MELO)  consumer surplus estimators
with superior  small  sample  properties  are available.

     The mathematical  derivations are  specific to the unbiased,  maximum
likelihood estimators  and  ordinary surplus calculations.   Nonetheless,  the
specific results  of this  chapter 'are  supported by more  general  theorems, and
the message remains  relevant whenever the welfare measures  of interest are
nonlinear functions  of  estimated parameters.

Sources of Error in the Recreation  Demand  Model
     Discussions of the  sources  of error  in  recreation  demand  analysis are
common  in the  existing  literature.    The most traditional  line  of thought
(e.g. Gum and Martin,  1975)  considers the error component in predicting the
individual's  recreation  behavior to arise  from unmeasured  socio-economic
factors.  Others (e.g.  Hanemann,  1983a) attribute at  least some of this error
to fundamental  randomness in human behavior.    Applied  statisticians (e.g.
Hiett and Worrall,  1977) on the  other hand,   suggest  that  recall  of annual
number  of  recreational  trips  (i.e.   the quantity  demanded) is  subject to
substantial  error.    Still  others  (e.g.  Brown et a I . , 1983)  have argued that
recall  of explanatory variables,  such  as  travel  expenses,  contains  error.

     The several explanations for  the stochastic term  in econometric models
which have  been proffered by  econometricians  are  made explicit below:

(1)  Omitted  variables:     factors which  influence recreational  demand have
     not been  introduced and,  thus,   error-free explanation of recreation
     demand is not possible.
                                     90

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(2)  Human indeterminacy:     behavior,   even with all  explanatory variables
      included  and  measured perfectly,  cannot  be predicted  because of in-
     herent randomness in preferences;
(3)  Measurement error I:   exact measurement of the  dependent variable  is
     not possible;  and
(4)  Measurement  error II:   exact measurement of  the  independent variable  is
     not poss i bIe.

     Each explanation  has  a particular relevance for  welfare analysis. Yet
only the  first three sources of  error conform to the Gauss-Markov assump-
tions, and then  only  if the omitted variables are assumed to  be uncorrelated
with  included variables.   Thus, the same estimation procedure  (e.g.  ordinary
least-squares analysis) will  be appropriate if the  error is associated with
(1)  through  (3)  but not  with  (4).   The fourth explanation  violates the as-
 sumed independence between the error  and  explanatory  variables.    When such
violations are  expected,  estimation techniques such  as  instrumental  vari-
ables  are frequently employed.     However,   these methods will  generate
different coefficient  estimates from  the  other three.   As such,  meaningful
comparisons between cases (1)  through  (3)  on the one hand and  (4)  are nearly
impossible to make.   Discussion  is thus  restricted  to consideration of (1)
through (3) and throughout most of the chapter the error is assumed  indepen-
dent of included variables.

     Two functional forms of  individual demand  are  postulated here, each  of
which  is consistent with  utility  maximizing behavior  (see  Hanemann,  1982d):

(i)                         xi = a + BP.J + Yy^ ui

and

(2)                       In xi  =  o  + Bp.j  + Yy-j + u.,-.

In  each  specification, Xf  is  the  ith individual's  demand  for  the good  in
question,  p^ is  the price he  faces  for the good,  and y,-  is  his income.  Both
p and y are  normalized   on  the  price of the numeraire  good.   The   parameters
a, 6, and Y are  preference  function parameters.  As  is usual, all  individ-
uals  are  assumed to face different explanatory variables but to  possess  the
same  general  form  of preferences,  except for random differences, so that
preference function parameters are constant over the population.
                                     91

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     The u-j  in  (1)  and  (2)  are  the disturbance  terms  which arise from  the
sources of error described above.   Consistent  with  Gauss-Markov  assumptions,
u^  is  assumed to be  distributed  normally with a mean  of zero  and constant
variance which is denoted by a ,  irrespective  of  the  source  of error.   Addi-
tionally ECu^U) = 0 for i f j, EtpiU-j) = 0, and Efy^-) = 0.
     In most econometric applications,  the  source  of the disturbance term or
"error" is  immaterial  as  long as Gauss-Markov assumptions  hold.   These  con-
ditions  are  sufficient  to  produce  unbiased   and  efficient  estimates  of
a, B, and Y.  However,  if the ultimate  purpose of  the  estimation  exercise is
to compute consumer surplus estimates, then the story does not end here.

"True" Consumer Surplus

     In  this  section,  expressions   for  the  value of  consumer surplus  are
derived under the  competing assumptions that the  randomness  is due  to omit-
ted  variables,  randomness  in preferences,  or errors in  measurement.  These
expected values  are determined on  the  premise that the  coefficients of  the
demand equations are  known  with  certainty,  so that these coefficients do  not
embody any  random  element.    In  a  later section,  the  discussion  is  extended
to the case when consumer surplus is calculated from estimated coefficients.

     Suppose  that  one  knows  with  certainty  the  coefficients, a,  0, and i ,
which  are  common  to  all  individuals  and  wishes  to calculate the  consumer
surplus associated  with a change  from p^ to pi  (the  price which drives indi-
vidual i's  demand  to  zero).1  For  each  individual,  consumer  surplus will be
determined  by  his relevant demand curve  and  his  initial  circumstances.
Clearly an  individual's observed  price-quantity  combination (p^.x^)  will  not
in general  lie  on  the systematic portion of  demand  function   x*=  a  + Bp  + yy
because of  the  random component.   The  question  is:   does one calculate  con-
sumer  surplus  based  on  a  demand curve drawn through  the observed  x and p
combination (p°, x°)  with slope 0 or  do  we  base  it on  the systematic portion
of the  demand  curve evaluated at (p°, x^)?   It  would  seem these  two methods
have been  used somewhat   interchangeably  in  practice,  without too much
thought.    Does  the method make a  difference in  consumer surplus  calcula-
tions?   If so,  what  explanations  of the error source  are consistent with
each usage?
                                      92

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1.   Omitted Variables Case
     Consider first  the  case in which  the  randomness  across individuals de-
rives from a relevant  variable  being  omitted from the equation.  (This vari-
able is not  correlated with the other  explanatory  variables).   In  this case
it would  make  sense  to  use the  demand curve  drawn  through (p°, x°.), since
the random  term  will  represent  an  unknown component of  the price  intercept
and thus  will  shift the systematic portion  of  the  demand curve sufficiently
to  pass   it  through  the  observed  price-quantity  point.   Gum  and  Martin's
procedure seems  consistent  with  this  as  it  "utilizes the  actual  number of
trips taken  by a  household and  the  actual  average variable costs per trip to
define the  household's individual  demand curve."   An  implicit assumption is
that the  omitted variables  remain  the  same  as price  drives  the individual
from the  market.    Thus,  the individual's  true  error,  u^,  remains  constant.
The  individual's  "true" consumer surplus,  if values  of  necessary  variables
and parameters are  known with certainty, is
(a

+ 6P° + yy° +
-28
ui _ i
-26
(3a)           CS^ /  x.(p(


for a linear demand curve and

                     p,                {« + ep° + Yy° + u.)   x°
                 *
(3D)            CS2i= /  X1(p1)dpl
                     po                        -6             -B

for a semi -log demand curve.

2,    Random Preferences and Errors  in  Measurement
     Two  other  explanations   for  error  in  regression  analysis are  con-
sidered:   a) the  individual's preferences vary randomly and b)  the dependent
variable  (trips)  is measured  inaccurately.   The first explanation  has been
used extensively  in  the literature (see, for example,  Hausman  1981)  and the
 latter  has  been studied  by  professional sample-gathering  firms  (e.g.  Hiett
and Morral I ,  1977) .

     In both  of these  cases,  it is  the value of  the  systematic portion of
the  demand  function  (x*  = a  +  Bp° + yy°}    instead  of  the observed  value
(x°j)  which  is relevant to the measurement of  surplus.   If the consumer has
random  preferences,  then  one  cannot  be certain  that the  observed  value of
x° will  be  chosen  by  the ith  individual  each  time the  same price-income
 i
                                      93

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situation arises.   The  "best guess" at the level of x, consumed by the indi-
vidual  facing  the price-income  situation  (p^, y^)  is  the systematic portion
of demand x^   When the error occurs  because  the individual cannot remember
the  exact number  of  trips  or  intentionally  misrepresents  his consumption
level,  then once  again  the  "best guess" of the actual number of trips is the
systematic demand x^.

     For  the  case  in  which these  types  of errors  completely  dominate, the
individual's  quantity  demanded  can  be expected  to  be  x.. = a + gp^ + Yy^ .
The consumer surplus for the linear demand  is mathematically  represented  by
            *
(4a)      CS3.= /  (c.
                po         ...         _2B


For the semi-log  demand  function,  the prediction of x-j is not an unambigious
issue, but  for the time  being,  let us use  the  systematic portion  of demand
given by exp(a + Bp° + yy°), such that the  consumer  surplus  is measured  by


(4b)         CS, .= J1 expfa + Bp? + Yy°)dp,  = 	n         = — •
              4l   J_           1      1    1             „             a
                  po                                 -,           -6

Graphical  Comparison of Surplus Computation  and  an  Empirical  Demonstration

     Figure 5.1  is presented  to  recapitulate the argument  and  also to dis-
play  visually the  process of  computing  surplus  with different  sources of
error.    Although  the   disturbance   term may   include  all   three   types  of
"error",  the  procedure  chosen,  to calculate the consumer  surplus  implies a
specific  interpretation  of the  error term.     When all  error  is  implicitly
assumed  to  be due  to  omitted variables  (that  is,  when  consumer  surplus is
calculated  from  a  demand curve which  is drawn through  the observed price-
quantity  point (x°,p°)j, the  residual   is  treated  as  part of  the  constant
term.   In  contrast, consumer  surplus  calculated from the demand curve  which
passes through (x*sp°) implies an  error  in measurement or random preferences
interpretation.    In  this  case,  the  error  term  represents  the   correction
factor in the observed xi  value.

     Two  individuals  facing  the same price  but with opposite and equal dis-
turbance  terms  (u-j and uk) are  depicted in the graph.   The points  (x.. ,p )
and (x?,p°) represent  their  observed quantity-price points  as  well  as  their


                                      94

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actual quantity-price  combinations  if  omission  of variables created the dis-
turbance.   To  obtain  the  surplus,  the price slope  coefficient (6)   is used
to determine Pi and  PL and  the  surpluses  Ap°Bp.  and &p°Cp,.   On  the other
hand,  the  point {x  ,p )  represents the  appropriate quantity-price  for both
individuals if  the  disturbance  term is  generated entirely by mismeasurement
                                                                 f\ ***
of x  or  random preferences.  The  appropriate  surplus  is then Ap Ap for both
individuals.   It may seem  that  these  two alternative procedures will produce
the  same  consumer  surplus, on  average.   However,  the  graph illustrates,  at
least  in  the  linear case,   that they will  not.    The average of surpluses  at
x< and XL is larger  than the surplus at x*.

     To demonstrate  the  different computation methods and to illustrate the
degree  to  which the error assumptions  actually cause  differences  in esti-
mates  of  consumer  surplus, consumer  surplus for a  sample of sportfishermen
is estimated.   The  data  set is  the same one used in Chapter 2 to demonstrate
differences due to functional form.

     Because appropriate wage information  for  a more complex model  incorpor-
ating  treatment of  time  and nonparticipation such as the one in Chapter 4 is
not  contained  in  this data set,  the  same  model and parameter estimates  as
shown  in  McConnell   and  Strand  are presented.   The individual  is  viewed  in
this  model  as being unaffected  by institutional  constraints  in the labor
market and therefore at  the margin in labor-leisure decisions.   Thus, fish-
ermen  are assumed to choose the hours they  work and to  make marginal  trade-
offs  between leisure and labor time.

     The  McConnell-Strand  model  yields   the   following  estimated  demand
function (p. 154):

            x.  = 9.77 - .0206 pj. - .0126 w^ +  1.90 si +  .157 mi
                         (3.89)     (-2.00)     (2.50)      (5.06)

where  the  numbers  in  parentheses  are t-ratios,  x.,-  is  the  number  of annual
sportfishirig trips for the  i*"  angler, p.j  is the i -  angler's trip expenses,
t.j is  his round trip travel time (computed as round-trip  distance/45 mph), w^
is   his  hourly  income  (computed as annual  personal income/2080 hours), s^  is
a site dummy for the Ocean  City resort,  and mi  is the length of the angler's
boat.   The  standard error  of the estimate  (a)  is  6.00  trips/person, the  F-
statistic (4,411) is 12.8,  and the R 2 is .10.
                                      95

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     The  process  depicted  in Figure  5.1 is  used to  compute  the competing
surplus estimates.   The  first estimate, CS3 calculates the predicted surplus
as  the  area behind  the  estimated demand function  and above observed price.
The  line  passes  through predicted  trips (x^  = x^ - Uj).  The  second  esti-
mate, C$i, represents  the  area  behind  the  regression Jine_  after  it  is
shifted to pass through  the observed price and  quantity  (x^, pu.).

     For  the  entire  sample,  the  omitted variable  estimate (C§i) is  calcu-
lated to  be $801,274 or  an  average of $1,931 per  fisherman.   The error in
measurement estimate  (083) is  calculated  to  be $450,086 for the sample or an
average of  $1084  per fisherman.   Thus  the assumption  of  omitted variable
error increased  the estimated average  surplus by  $847  (or  78%) relative to
the measurement error assumption.
P.
                                   X. = a •*• Bp + YY  +  U,
                                                         Trips
                                  Figure 5.1
          Two Different Procedures for  Calculating  Consumer  Surplus

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Calculating  Expected  Consumer  Surplus

     The graphical analysis and  the  empirical  example demonstrate that con-
sumer surplus calculations for  an  individual  will differ  depending on the
error assumption.    The analysis  also suggests  that these  differences  in
consumer  surplus calculations  may not cancel  out (as do  the  errors them-
selves)  when  aggregated over  the sample.    In  order to  determine the  general
conditions under which these differences  in surplus arise it  is necessary to
consider expressions for expected  consumer surplus (conditioned on explana-
tory variables),   since  the  expected  value is  conceptually  equivalent to  the
average  over the  sample.

     Once again  assume  that the parameters (a, B, and Y) are known and  that
the expected  surplus  conditioned on  values of  p  and  y is  to be  calculated.
It  is  obvious but  nonetheless  worth  noting  that the expected  consumption
level  E[x] must  be equal under  the competing  error source assumptions if  the
regression  equation  is  linear.   Consider  first  the  expected  consumption
level  if the error is assumed to arise from omitted variables:

(5a)                E[x] = E[a + Bp + Yy + u] = a + Bp + Yy;

and if the error  arises  from measurement or random preference:

(5b)                E[x] = E[a + Bp + Yy] = a + Bp + Yy.

Clearly  the two are equal.

     While  expected  consumption  levels  are  equal,  the expected  value  of
consumer surplus  will not be.   Denote  f(x) as  the consumer  surplus  operator;
then equality in  expected consumer surplus requires that

(6)                 E[f(a + Bp + Yy + u)] = E[f(a + Bp + Yy)].

Note that a + Bp + Yy does  not  include  a stochastic term,  so that  the  right
hand   side    of   (6)   equals  f(a +  Bp  +  Yy).       Also   since a + Sp + Yy =
E(a + Bp + Yy + u),   the  right  hand  side  of  (6)   could  be  written   as
f(E[o + Bp + Yy + u]) so that the condition in (6) can be rewritten  as

{7}                 E[f(a + Bp + Yy + u)] = f(E[a + Bp + Yy  + u]).
                                     97

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     Jensen's  inequality  (Mood, Graybi I I and  Bees,  1963)  states  that if q  is
a random  variable  and  f(q)  is  a  convex function, then E[f(q)] >  f(E[q]).  It
 is  expected therefore that if the  consumer surplus operator  is a convex
function  then the omitted variable assumption  will  lead to   an  estimated
surplus at  least as great as the measurement error assumption.

     This  is borne out by the derivation  of expected surplus in the  linear
case for  the omitted variables explanation

(8a)                EECSj] = E[(a + Bp + yy + u)V(-2B)]
                           = (a + ep + yy}2/(-26) + o2/(-2B)

and for the errors  in  measurement explanation

(8b)       E[CS3] = E[(a + 6p + yy)2/(-2e)] = (a + Up + yy)2/(-2s).
                                         Q
The difference in the two expressions,  a /(-2e),  increases  with  the variance
of the true error  and  decreases with price responsiveness.

     For any consumer  surplus  function  which  is convex i n x,  the above  dis-
cussion demonstrates that there will  be a  difference  in calculated  consumer
surplus  depending  on   the  implicit assumption   about  the  source  of the
error.    One commonly used functional  form for  demand,  the semi -log, gener-
ates a consumer  surplus function  which  is I i near  i n  x.    However,  the  semi-
log has  problems of its own,  because the  conditional expectation on x  (the
dependent variable)  is now a convex  function of the error.    Unlike the
linear case, the conditional  mean  of x for the semi -log function is not the
systematic portion of the demand  function.   That  is

          E[x]  =  E[exp(a + Bp + yy +  u)] = exp(a +  6p  + ry)exp(a2/2)
                       * exp(a  + 6p + yy) = E(x)


because the mean of exp(u)  is  (a2/2),   if u is distributed  N{0,a2).   It  is
solely because of  this  result  that a  difference arises in  the semi-log's ex-
pected values of consumer surplus for the two error source interpretations.
                                                              2
(9a)     E[CS ] • EEexP(° * BP +  ?y + ")] = exp(a +Sp + yy + o /2)
             2             -8                       -6

                                         Bp + yy)] = exp(c    »p + ryl
(9b)                 * E[CS4] -


                                     98

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     Econometricians have suggested adjusting  the  constant term so that the
expected value of predicted x's  wi I I  be  equal  to the observed x's; that is,
the  adjustment  would force the  distribution  of x^  to have mean  xi.    This
adjustment would involve defining a new constant

                                a1  = a +  o2/2

and using  a' to calculate x.

     There  is a  subtle inconsistency in  the   logic of the above adjustment
however.    If  the researcher  believes  that the error (u) is due  to errors  in
measurement, then there is no  reason  to  desire  E(x) = E(x).  The errors  in
measurement explanation suggests no particular credence  should be given the
observed values  of  x.     In fact,  the semi-log specification  implicitly as-
sumes the errors in measurement of x  are skewed.   It may be this property  of
the  semi-log which explains  its frequent success at fitting  recreational
data.   Surely errors in recall of  x:will  be  larger with  larger x's.

     Interestingly,  calculations  for  consumer surplus under  the two  error
sources would be identical if  the constant were adjusted  in  calculating the
consumer surplus from   x.  However the calculation of consumer  surplus  from
x implies the error  source  is errors in measurement  and it is  in just  this
case  that  adjustment  of  the  constant term   is  ill  advised.   Without the
                                                     
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     Suppose the  parameters  of a linear  demand function have  been  estimated
on  the  basis of  a  sample of  observations  on x, p,  and y.  These  parameter
estimates are  denoted a,  6,  and y.   Analogous  to  (3a)  and (3b) the  consumer
surplus estimates for the individual,  if  the error is presumed to  be  due  to
omitted variables, are given by
                                            X2

(lOa)                                 CS. =  -ir
                                        ii   -26

for the  linear case and

                                             x
(lOb)                                 CS


for the semi -log.   If one believes  the errors in measurement or random pre-
ference explanation,  the  individual   estimates analogous to  (4a) and  (4b)  are

                                       A n     A   A     A    t\
                                       x^    (a +  6p,+ yy.)
(lla)                           CS,=-i =  - I - —
                                 Ji  -26         -26

and
                                       A         A    A    A
                                       x.    exp(a  + 6p.+ yy.)
(lib)                           CS, = -J- =  - ^-! - -
                                 Hi  -6          -3

respectively.

     Comparing  the  estimates  associated  with  the linear  demand  function
under  the two  error  source assumptions (i.e.  (lOa)  and (lla)). the  following
difference arises for the  individual


                                *2.    (a +  Bp.+  yy.)2
(12)               CS. -  CS,=
                           J
                               -26          -26

                                AA     A     A9     AA    Art
                               (a +  Bp.+ yy^  u)     (a + ep^ yy1 )

                                    -26                  -26

                               u^ +  2  (a +  BP.J+ yy^  u^
For  any  specific  individual,  this expression cannot  be  signed,  but the aver-
age  for the  sample can be.    Summing  the  difference  in consumer  surplus
estimates over the  sample  and  dividing by N yields
                                      100

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                     N
 (13)
               z(csn - "3i   z«5
                         ^_^_ s  .
                                                         *  •      2
because by definition of the least  squares  estimators,  Ex-u^ = 0.   Thus  for
any sample of data  and  linear model,  the  method  for calculating consumer
surplus  which implicitly  assumes omitted  variables will produce  a   larger
estimate of  average  consumer surplus than  will  the method which  implicitly
assumes all  error  is due to errors  in measurement.   The difference will be
equal  to

                                  (N-k)s2
                                   -2N6

       0
where  s  = variance  of the  residual  and k  is the number of parameters  in  the
equation .

     Taking these results a bit further,   it is useful  to  examine the  proper-
ties of  (13).    Equation  (13)  is the expression  for the difference  between
the two calculations  of  consumer  surplus  for a given sample.    Its  size will
vary,  of course,  for different samples,   since  it  is  itself  a random vari-
able.    The  expression for  the  expected value  of the difference suggests
something about the problems in which this difference will  likely be large.

     Equation  (13),  which  is the expected value of a  ratio  of random vari-
ables,   does not have an  exact representation.    However,  an  approximation
formula for  such problems  exists. 'Applying the  approximation to  this case
g i ves the f o I I ow i ng :

             l(CSn - cLn       ZUi/N     EdUi/N)      var  I
(14)      E [ - li- - il] = E [— T— ] -- T— (1 +
                                   -20       -2E(g)        (Eg)
 If the model  is correctly  specified so that the coefficients are unbiased
estimates of the true parameters,  then (14)  can  be expressed  as
                                     101

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                   *     *       kl
                r(CSir CS3.)
 (15)         E  [ - ^ - — ]~
     The  first  term of (15)  is  simply the ratio^of the expected values of
the  numerator  and  denominator  in  (13).    Since  6  is  an unbiased estimator
of 6 ,  the  denominator Js -2e.   The  numerator  of this first term is  simply
the  expected  value of u  .  The  second term  in  (15)  reflects the fact  that
the  expected  value of a  ratio of two random variables  is  not the ratio of
the  expected values,   but must be weighted by the  population analog to the
samp I e  stat i st i c
                                   (t-ratior

This weight  will  be  greater  than  one since  l/(t-ratio)2  is positive.   The
important point  is  that when one  takes  into  account the fact  that  consumer
surplus  estimates  are  der-ive-d   from   estimates   of  the   demand  function
parameters,   a  difference  still  remains  between the  omitted  variables  and
errors  in  measurement  consumer  surplus  estimates.   The above  demonstrates
for the case of unbiased coefficients  that the difference  can be expected to
be larger than  if the coefficient were known with certainty.

     Returning briefly to the semi -log function, a  comparison of expressions
 (I Ob)  and (11b)  depend  on  whether  an adjustment in the constant  term  of  the
expression  is employed.   The  econometric procedure  of adjusting the  constant
term would now involve defining an estimate of o2/2 , i.e.  s2/2 where
If  this  adjusted  constant  were  used  in  calculating  x,  then  the  expected
value of  the difference  in consumer surplus estimates would  disappear, since
the  adjustment  is made  such  that E(XJ)  =  E(X-J).   However,  consistent  with
the  earlier arguments,  the  adjustment  is  considered  to  be  inappropriate
here.   The  non symmetrical  pattern of errors around x values  implicit  in  the
semi -log  specification may  represent reality better and  may be one  reason
why  the semi -log  often appears to provide  a better  fit.     Thus for the
 individual
                                     102

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                       x. - x.   exp(a + Bp. + iry.)
(16)     CS?. - CS,. =  ^ A  1 =	^	1- (exp u, - 1).
                         -e             -e                n

     If  the  constant term  1s not adjusted,  the difference in  the  individ-
ual's consumer  surplus  is  given in (16).  To evaluate  the  expected  value of
the difference,  it  is  easier to evaluate the expected  value of each expres-
sion first.   The  expected value of CSg-j (omitted  variables  interpretation)
is
(17.)            E[CS21] - E[i] =
                             -B                -B
Calculating  the expected consumer  surplus under the  errors  in measurement
assumption  yields
(17b)                        "                "9
                          exp (a + 6Pi + Yy^ exp (ko^/ZN)       var 8
where the derivations can be found in the Appendix to this Chapter.

     A comparison  of equation  (14)  and (15) demonstrates  the  expected dif-
ference  between  the estimates  obtained from  the  same  data  set with  two
different  error  source  explanations  when  a  semi-log  function  is  fitted,
Once  again  omitted  variables  will  lead to  a  larger expected  surplus  esti-
mate,  because exp(a2/2)  = exp(Na2/2N) > exp(ka2/2N) for  any  data  set  which
will support estimation of the k parameters.
                                     103

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Properties of  the  Consumer  Surplus Estimator

     In  the  last  sections  consumer  surplus estimates  were  shown  to differ
depending  on  the procedure used  to calculate them  which  In turn implied as-
sumptions  about the  source of the disturbance  term.   In  this  section it 1s
demonstrated  that,  irrespective  of  the  source of error,  the  conventional
consumer  surplus estimators  (those presented above) will be  biased.

     The  process  by which surplus  estimates  are conventionally derived {i.e.
the  procedure  employed  in  the previous  section)  1s to replace  the true pa-
rameters  in  expressions  such  as  (3)  and (4) by their  regression estimates.
Taking  as an  example the linear,  omitted variables  case,  from  (3a)  we see
that consumer  surplus is  given by
                                    =  2
                                    = x£/(-2B)
                                  A

and the conventional  estimator  (given in (10a)) is

                                CSj = x2/(-2&).

     If 6  is  a  maximum likelihood estimator of g,  then  CSj will  be the maxi-
mum  likelihood  estimator of CS^  (Zellner and Park, 1979).    However,  this
maximum  likelihood  estimator  has  some  undesirable properties.   As  can  be
seen  from the  derivations  in  the  previous section  (or  derived from similar
expressions  in  Zellner and  Park),  the expected  value of CSj  is not equal  to
the expected  value of CSj.

That is                          .
                       E(c§  ) ~     +  BP H
(18)                           ~         -2^
                                          -26

The conventional  consume/ surplus  estimator is biased.   It is biased upward
by a factor of  (1 +  var  6/B2)  .

     Likewise  in  each  case -  linear or semi-log,  omitted variables or errors
in measurement - one  finds that the conventional estimator is  biased upward.  In each
case  the  bias  is related to the  term  var e/&2.  *   This  is  because in each case  the
estimator for consumer surplus  is a function of the reciprocal  of B-
                                      104

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     Note that  the bias decreases  with the price slope  and increases with
the variance  of the estimated price  coefficient.   The  latter  suggests  that
the bias will  increase  with  a)  increasing variance of u, b) decreasing dis-
persion  in  price across the  sample,   and  c)  increasing correlation  between
price  and  other explanatory variables  in  the  equation. All  of these bode
 ill for  the travel  cost method which  depends  on cross  section data, fre-
quently explaining only a smaI I  portion of the variation  in  trips, and  is
often plagued by multicolI inearity  problems particularly with respect  to the
treatment of the vaIue of t i me.

     While  the  conventional  consumer  surplus estimators  can be  shown  to  be
biased, they  appear  to  be consistent estimators.  One  can  see  this  from  the
formula for var g which, in the general case is

(19)                            var $ = a2m6B

       RR          -                      _   _      _'_  -1
where m"" is  the  element  on  the  diagonal  of  the  (Z Z)  "  matrix associated
with   the  e coefficient   (Z   is   defined   as   the   vector   of  exogenous
variables).   For  our particular  case,  this term can be  written  more  intui-
tively as

(20)                   var I = a2(z{prp)2 (1-r^))'1,

where  r_y  is  the  correlation between  price  and income.  As sample  size  in-
creases,  the  only term  which changes  is  the  dispersion   in price.    In  the
limit as  N -> », r(p.,--p)2 •*• - and var 6 + 0.

     There  are  more  general  principles upon which  both the  biasedness  and
consistency  properties  rest.   Referring once  again  to  Jensen's inequality
helps  establish  the  biasedness  property  for a  broader  range  of cases.   If
the  estimated  consumer  surplus,  designated g(e),  can  be shown   to  be  a
strictly   convex function  of 6 then,  by  Jensen's  inequality,  the  expected
value of the  estimate (E[g(e)])  should  be  greater  than  g(E[g]).  This latter
term  equals  the   true   consumer  surplus,  gig),  if  B  1s   unbiased.     Thus
while B is  an  unbiased  estimator  of  the  true B, strict  convexity in  the
estimated surplus implies upwardly  biased consumer surplus estimates.

     Zellner  (1978)  has shown that,  indeed,  when one  calculates a  function
of  the reciprocal of a  maximum likelihood estimator,  then the expected value
of  the function will be  an  upwardly biased estimate of  the function  of the
                                     105

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expected  value  of the  parameter.    Additionally,    the estimator of the
function  will  not possess finite  moments and,  when using a quadratic  loss
function,  has infinite risk.

     However, the consumer surplus estimators are  consistent.  Mood,  Graybill
and Boes  show  that if e is  a ML estimator  for  e,   then  f(e)  is  an  ML  esti-
mator for f(e),  if there  is  a one to one mapping  between e  and f(0).   Zehna
has extended  these results such  that  the property  holds for  any f{.)  which
is  a  function of  e.   Maximum likelihood estimators  may  be  biased  but  they
generally  can  be  shown  to  be  consistent,   except in unusual  circumstances
(Chandra).   As  a  consequence,  the  consumer surplus estimators will  be  con-
sistent estimators,  if they  are  functions  of maximum likelihood estimators
of the parameters: a, B, and Y.

Minimum Expected Loss (MELO) Estimators

     Consistency  is  certainly  a  desirable property  for an estimator, but  it
is a  large sample  property.   That is,  it is not of  great practical  value  if
the estimates of interest  are usually generated  in  the context of relatively
small  samples.   Given  the  scarcity of  large  samples  in recreational  studies,
it is the small  sample  properties of  consumer  surplus estimates which are  of
particular interest.

     Zellner  (1978)  and Zellner  and  Park  (1979)  have proposed  a  procedure
for  correcting   for   the   bias  which   arises when  we are  interested  in   a
function  which   is the  reciprocal of  a  maximum  likelihood  parameter.   The
core of their  argument rests on providing an estimator that will  minimize a
loss function.

     As an  example  of  the  technique,  consider the  function for  consumer
surplus in the  linear-omitted variables  case C$i=  x2/(-2s).  Its ML estimator
is CS±= x2/(-2g).  Zellner's  loss function  for  the estimated  surplus  would
be [(CSj - CS^/CsJ]2.  Minimizing this  function  implies  a surplus estimator
defined as:
 (21)                 UV-2B) (-
                                       A AO
                               1 + var B/6


which  is  the ML  estimator   of  CS^  times   a  "shrinking  factor"  (Zellner,
p.  185).    Interestingly,  the  shrinking factor  is  the  ML  estimator of  the
inverse of the multiplicative bias factor arising in (18).
                                     106

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     Unfortunately, even  (21)  is  of limited value  to  us  because  it presumes
knowledge of  var e (and hence  
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     To give greater  insight into how  large these  differences might be  in
practice,  estimates of consumer surplus from a sample  of  sportfishermen are
derived.    The  sample  yielded relatively  high t-statisties on  independent
variables  although it did not predict very accurately (R*  = .10),  implying a
rather  large variance of the error.   These characteristics are  fairly typi-
cal of cross-sectional  data.   The results  show a  substantially  higher value
(78%)  for the omitted variable  error assumption than for the measurement
error/random  preference  explanation.

     This  is only half  the problem,  however.     Surpluses  computed  as
functions of regression parameters will  likely be  upwardly biased, even when
these  parameter estimates are themselves  unbiased.  When  surplus estimates
are non-linear  in  the  parameters,  their  expected value  is  larger than the
surplus when the true parameters are used.   The  degree of biasedness  is
positively related to  the variance  in the  price  parameter and the  inelas-
ticity of  demand.

     Large samples do,  however,   provide consistent measures for surplus.
Thus,  there are pay-offs from having  large samples and  confidence in param-
eter estimates.     ML  estimators  of  consumer  surplus will  have poor small
sample properties  (Zeliner,  1978; and  Zeliner  and  Park,   1979).    However,
Zeliner offers  us  MELO (minimum  expected  loss) estimators with far better
properties.    Since recreational  surveys are costly, these MELO estimators
are a valuable alternative  to increased sample sizes.

     What  implications  do  the  results  of this   chapter  have for  the
researcher active  in measuring benefits?   There are  a  lot of forces  at work
to confound  benefit estimates,  and it is  difficult  to  treat all  of then?  at
once.   This chapter shows that the source  of error wi I I  make a difference  in
consumer surplus values.

      If the  researcher attributes all  of  the error to  omitted variables
(i.e.   draws  his demand curve through the  observed  (x°^,p°^)) when at  least
some of the error  is due to measurement  error,  he may  be  substantially  over-
estimating consumer  surplus.      If the  researcher  employs the alternative
practice  of  calculating  surplus  behind  the estimated  regression  line, then
he will surely  be  underestimating surplus  since omitted variables  are always
a source of some error.
                                     108

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      In the past,  the source of error has been considered  of  little  conse-
quence.    Yet,  it  is shown that  improved  estimates of consumer surplus  can
result  if one can  a)  reduce  the  variance  of  the error  in  the regression  and
b) provide  information  as  to the source of the error.  Survey  designs which
reduce measurement error,   for example,  by limiting recall  information, will
be helpful on both counts.    Another approach is to collect  more  in the  way
of potential  explanatory variables.    The  marginal  cost  of additional  infor-
mation may be low,  but its pay-off may be great  if  it  reduces the variance
in the  error of  the  regression.   Thus,  even  though precision in travel  cost
coefficients is not gained, there is a decrease  in the  potential  error aris-
ing from wrong assumptions  concerning  the  error term.

     A warning is  offered  against  the usual  practice of  assuming  all  error
is associated  with omitted  variables.    The  practice can lead to  upward
biases  in benefits when either  random preferences or measurement  error  are
present.    At a minimum,   the researcher  should explicitly  acknowledge  the
likelihood of  upwardly biased estimates.    A bolder approach would be  to
offer estimates of benefits  under competing  assumptions about  the  source of
error.

     The second  imp I icat ion  of the results  is that the care and  attention
spent by   researchers  in  obtaining  statistically   valid  estimates  of
behavioral parameters must carry over to  the derivation  of benefits.  Esti-
mates  of consumers  surplus  have,   by  construction,    random  components.
Knowledge of how the  randomness  affects estimated  benefits  may  be  as  impor-
tant  to policy makers  as  knowledge  of the  statistical  properties  of  the
estimated behavioral  parameters.   At a minimum,   researchers  should  assess
whether  their  consumer surplus  estimates are  likely to  be badly biased.
Since  Ze liner's  MELO estimators for the  linear  and semi-log (as well  as
other) functional forms are straightforward to calculate,  MELO  estimators of
consumer surplus  would be simple  to  provide.
                                     109

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                      FOOTNOTES TO CHAPTER 5


Since  everything  in this  chapter   is  demonstrate in  terms of  the
ordinary  demand  curve  and  ordinary  consumer  surplus, p 1s  the  price
which drives Marshal I ian demand to zero.    Of  course p  in the semi-log
case depends on the  limiting properties of the  function.

The following  approximation   is  necessary to derive expected values
throughout the  chapter:

    E(x/y) ~ E(x)/E(y) - cov(x,y)/(E(y) )2 + E(x)  var(y)/(E(y))3.

The expected value  of the  ratio of two  random variables  does not have
an exact equivalence.

Should the  coefficients  not be unbiased   (that is,  should the equation
be at  least slightly  misspecifled),  then expression (14)  will still  be
true but  it wi  I I  not simplify to  (15).   Given  that  the  misspecif icat ion
is due  in some way to  the correlation  between  included and omitted
variables,   it  is  not  possible  to  determine a  priori  ,  whether  the
existence of such correlation  will  increase or decrease the difference
in surplus estimates.

Suppose that I*  and e were  correlated where Zj  is  the j   explanatory
variable.    The  expected values of each  of the  terms  in  (14) would no
longer  be  as  simple, reflecting  the fact that E(Zjt)   is  no   longer
equal  to zero.

Using  matrix notation  for efficiency and  labelling the explanatory
variable matrix,  Z,  the  first term in  (14)  now  becomes
        r(u'G)     K-k  2    F  .u'KZ'Zj^Z'u,  .  (N-k)g2   E(u'Z)  (Z'Z)"1E(Z'u)
      = E~~	0  "  L	
                                110

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where the second term above no  longer disappears but reflects whatever
correlation  exists  between  included  and  omitted variables.

     The expected values of the  estimated coefficient 6, now become

                E(e> = e + EKz'z^Z'u) =  e + (z'z
where E(ej) will  exceed 0j  if the correlation  between  Zj  and u  is  positive
and vice versa.   (Of course  if there is also correlation  with other explan-
atory variables everything  becomes more  complicated.)

     F ina I ly,

       var e = E(e - E(e))2 = a2(Z'Z)'1  - (Z'Zi^EtZ1 ujEtu'ZMZ'Z)'1.
The second term  is positive,  so correlation  between  Z  and  u  wi I I  reduce  the
variance of  e.

     As  a  consequence  of the  above three  derivations,  the  presence of
correlation  can  not  be determined a  priori  either to  increase or  decrease
the difference  in the consumer surplus measures.
                                     Ill

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                                 APPENDIX 5.1
            DERIVATION  OF DIFFERENCE IN ESTIMATED CONSUMER SURPLUS
                      USING  THE SEMI-LOG DEMAND FUNCTION
     The  following is the  derivation  for the expected  value  of the differ-
ence  in  consumer surplus  estimates for the semi-log  demand  function.    When
omitted  variables causes the error,  the  expected  value of the  individual's
consumer  surplus  estimate  is

                           x.        exp(Z.e + u.)
(AD                    E(-l) =  E  (	^—	4
                           -P              -B

where Z-j  is  the  ith row of the  matrix  of  explanatory  variables  =  [1  pi  yi]
and 9 is  the vector of coefficients [a  6 y]'.

Then, using  the  approximation formula  for the ratio of  two  random variables
yields

        exp(Z.e +  u.)     exp(Z.e)E[exp(u,)]       var p    cov(exp{Z.e + uH),-6
(A2)   E[	1	—] = 	i—s	—  [1 + —9—] +	
            -B                  -e                r               e

                                                                       2
Given  that u^ is  distributed as a normal  with mean 0 and  variance  a ,  then
                                                                  2
exp(u^)  is distributed as a  lognormal  with expected value  exp(a /2).    Not-
ing  that the covariance  term equals zero,  expression  (A2) can  be  rewritten
as

                                        2
                     x.     exp{Z.e)exp(a /2)       var e
(A3)               E[4]	—.	 (1+—5—}•
                     -B           -B                 6"

     The  expected value of  the individual's consumer  surplus estimate  when
errors  in  measurement  is the  principal  cause  of the disturbance  term  is
                                      112

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             x.       exp(Z.e)       exp{Z.e + Z.U'zru)
(A4)       l[-L] - E [ - -i_] = E [ - ] - 1 - ].
             -$          -$                    -B
Applying the  approximation formula  and noting  the  covariance term  is  zero
gives
              x.    expU.eJEfexpfZ.tZ'zrVu)]       var e
(A5)       £[-!-] -- ^ - - - 2 - -(1+ - 2-).
              -3                 -3                      B

 oting  that exp[Z.(Z'Z)   Z'u]   is  simply exp[Au]  where  A  is  a vector  of
 on-random  terms,  we draw  on  the  result that the  expected  vaue of  exp(w)
 hen w  is  normally  distributed  is  equal  to exp( {variance w)/2).  The vari-
 nce of Z-jt'Z'Zj^Z'u can be expressed as

 A6)        varU^Z'Z^Z'u) = Etu'ZU'Zl^ZjZjU'zr^'u].

 he  vector  Zn-   is   simply  the   ith  individual's   vector  of  explanatory
 ariables.   So  that the formula reflects  the  average  values  of the  explana-
 tory variables,  the matrix  ZjZ^ can  be  rewritten as(l/N)Z'Z,   Making  this
 ubstitution gives  us  an idempotent matrix  and  allows the following simpli-
 'ications:
 A7)        EC

 loting that expression  (A7) is a scalar and equal to  its own trace,
(A8)           TT- E[u'Z(ZlZ)""Ziu] = jj-tr(Z(Z!Z)~~Z!) E(UU!)

                                  = - a2


because  the  trace of  an idempotent  matrix equals  its  rank, which  in  this
case is k (or 3 in our example).

     NOW since,
                                          2
                   var (Z-itZ'Z)" Z'u) = -§•
                                     113

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then
                       -6
                                         114

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