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.
-------
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.
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
61
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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.
62
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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).
63
-------
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.
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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.
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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.
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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;
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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
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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
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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.
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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
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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 = -
-------
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
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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) +
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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
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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)
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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].
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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
-------
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.
-------
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
-------
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
-------
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.
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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-
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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).
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Unfortunately, even (21) is of limited value to us because it presumes
knowledge of var e (and hence
-------
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.
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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.
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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
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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
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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) = -§•
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then
-6
114
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