IDENTIFICATION OF PREFERENCES
IN HEDONIC MODELS
Volume I
of
" BENEFIT ANALYSIS USING
INDIRECT OR IMPUTED MARKET METHODS
Prepared and Edited by
Kenneth E. McConnell
University of Maryland
Maureen Cropper
University of Maryland
Robert Mendelsohn
Yale University
Tim T. Phipps
Resources for the Future
Nancy E. Bockstael and Kenneth E. McConnelS
Agricultural and Resource Economics
University of Maryland
EPA Contract No. CR-811043-01-0
Project Officer
Or. Alan Carl in
Office of Policy Analysis
Office of Policy and Resource Management
U. S. Environmental Protection Agency
Washington, D. C. 20460
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PREFACE
This report on identification in hedonic models represents the first year's
work on the hedonic portion of the Cooperative Agreement between EPA and
the University of Maryland. It will be followed by additional work on
hedonics which investigates more fully the empirical issues associated with
using the hedonic model to value environmental amenities.
In addition to the authors, a number of other people contributed to the
ideas of this report. Both Kerry Smith and Michael Hanemann were influential
in the development of Chapters 4 and 6.
Thorough review of reports is a characteristic of EPA Cooperative
Agreements. This report benefited from the detailed comments and criticisms
of the following individuals:
Raymond Palmquist
North Carolina State University
George Parsons
Environmental Protection Agency
Walter Milon '
University of Florida
(on leave at EPA at the time of the review)
A number of graduate students helped draw figures, proofread, and
otherwise assist in the preparation of the report. They include Douglas Orr,
Terry Smith, Bruce Madariaga, Utpal Vasavada, Chester Hall .-and Laurence
Crane.
Our contract officers on the research, Alan Carlin and Peter Caulkins,
have been supportive and patient.
Finally, it is worth noting that this report represents the initial year's
work on hedonics in a Cooperative Agreement that is designed to last four
years. Additional work now under way will confront the conceptual questions
with numbers.
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EXECUTIVE SUMMARY
IDENTIFICATION OF PREFERENCES
IN HEDONIC MODELS
EPA Cooperative Agreement CR-811043-01-0
University of Maryland, College Park, Maryland
Voluuie I
N. E. Bockstael and K. E. McConnell
Principal Investigators
This volume reports on the research of our project under the EPA
Cooperative Agreement with the University of Maryland. The purpose of this
project is "to solve the identification problem in hedonic models." .The
purpose of the research is thus quite specific and rather theoretical in
nature. This volume describes those circumstances under which the problem
is solved and analyzes other issues consistent with the use of the hedonic
model in benefit-cost analysis.
The results of the project, while relating to technical issues, can be
expressed intuitively. The hedonic model is a method of assessing the
economic costs of pollution. Its use in environmental economics stems from the
fact that when people buy homes, .their willingness to pay for the attributes
of the house is reflected in the sale price. The attributes of the house
include not only its size and number of rooms, but also neighborhood
characteristics and various dimensions of environmental quality, including air
quality. Hedonic analysis connotes various approaches to the empirical study
of the price of goods, when those prices reflect the characteristics of goods.
For example, consider two houses which are located next to one another and
differ only in that one house has an extra bathroom. Then when the housing
market is in equilibrium, the difference in the housing prices reflects the
additional bathroom. This basic principle allows us to impute housing price
differences to differences in several attributes of houses, including
environmental quality. Further, we can say the difference in the home price
reflects a household's willingness to pay for the. attribute. Consider two
difference in the home prices reflects a household's willingness to pay for.
reductions in ozone.
The identification problem concerns the difficulties researchers encounter
in trying to find the household's schedule of willingness to pay for various
levels of attributes, not just a small change in the attribute. The
identification problem stems from the fact that observed hedonic prices reflect
not only on the value of the attribute to the household but also on the
distribution of households of various types, the scarcity of houses, and the
distribution of housing characteristics in the stock of housing.
11
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In the context of benefit-cost analysis, the identification problem makes it
more difficult to infer the benefits of non-marginal changes in attributes.
Hcdonic prices show what households would pay for small changes in housing
traits, not their schedules of willingness to pay for various levels of the
attributes. In measuring the value of various kinds of goods and service in
the economy, we typically find that the more of a good a person has, the less
he would be willing to pay for additional units of the good. Consequently, it
would be wrong to compute how much a person would pay for 10 gallons of
milk per week by finding what he pays for one gallon and multiplying by 10.
The same holds for attributes of houses, including environmental attributes.
The solution to the identification problem would therefore permit more accurate
measurement of the benefits of the non-marginal changes in environmental
amenities reflected in housing prices.
The basic finding concerning the solution to the identification problem
when housing prices come from only one housing market is negative. Chapter
3 and 4 address the issue in detail. These chapters differ in how they
address the problem, but both demonstrate that identification of the
household's functional relationship between attribute levels and willingness to
pay can be achieved only when the hedonic prices obey curvature patterns
significantly different from the curvature of the individual willingness to pay
function. Further, it is shown that the curvature properties which permit
identification are not testable, but must simply be assumed. We are therefore
in a position of solving the identification problem, but of not being able to
test whether households behave in a way compatible with the assumptions that
allow identification.
When we combine housing prices from different markets, for example, from
different cities, the situation is not quite so pessimistic. If we are willing to
believe without testing that households from different cities value attributes
of houses approximately the same, then we may be able to identify the hedonic
model (Chapter 4, Section 4).
Is it worthwhile to proceed with attempts to identify hedonic models?
The answer depends on several factors. First, can we be satisfied that
housing markets work approximately as hedonic analysis specifies? Second,
does the estimation of the hedonic price equationthe relationship between
housing prices and housing attributesgive an accurate reflection of what is
going on in the housing market? Third, are there serious damages using
,...,,...,» . .;...., * ^, ' ..' '. rr . ?...* ". ' ' ' '' ' f..-T
Chapters 5 though 7 explore these issues. Chapter 5 asks whether the
identification problem which plagues the recovery of information about
willingness to pay for environmental attributes also confuses us about the
term hedonic price equation. The answer is basically no.
Chapter 6 explores how much difference it makes to use marginal prices
to calculate the benefits of non-marginal changes. The conclusion is that
errors from using marginal prices are less serious than errors from other
sources, such as specification of the hedonic relationship.
in
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Chapter 7 investigates the structure of choice in hedonic models. It
recognizes that residential locational choice can be viewed as a choice of two
dimensions on a plane. If air pollution is tied systematically to either or both
of these dimensions, then differences in housing prices will not reflect
differences in willingness to pay for tied attributes. This chapter suggests
that we may achieve more reliable results for the economic costs of pollution
by developing a more realistic model of individual bids.
The conclusion of this volume is that while it is conceptually possible to
identify the hedonic model, it is not a good use of research resourcesr-
Further research into how the housing market works, the accuracy of marginal
prices, and other issues which logically precede the identification problem
should be pursued.
IV
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TABLE OF CONTENTS
Section Page
Preface i
Executive Summary ,. ii
Figures . vii
Tables . viii
Chapter 1 Introduction
1.1 Benefit Cost Analysis and the Hedonic Model 1
1.2 Overview of the Volume 5
1.3 Some Conclusi-Shs 5
Chapter 2 Hedonic Models: Current Research Issues
2.1 Introduction 8
2.2 Choice of Quality and the Hedonic Model 8
2.3 The Hedonic Model in Environmental Economics . 10
2.4 The Basic Rosen Model 11
2.5 Some Research Issues 13
2.6 The Charge of the Research . 18
Chapter 3 Identification of Hedonic Models
Robert Mendelsohn
3.1 Introduction 20
3.2 Simultaneous Supply and Demand 22
3.3 Predicted Prices 23
3.4 Nonlinearity and the Simultaneity of Shift and
Price Effects 25
3.5 Conclusion 31
Chapter 4 Identification of the Parameters of the Preference
Function
K. E. McConnell and T. T. Phipps
4.1 Introduction 34
4.2 The Structure of the Problem 35
4.3 Single Market Approaches to Identification 39
4.4 Multiple Markets 51
4.5 Conclusion 53
Appendix to Chapter 4 57
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Chapter 5 The Structure of Preferences and Eslimaiion of the
Hedonic Price Equation
K. E. McConnell and T. T. Phipps
5.1 Introduction . 61
5.2 The Structure of Preference and the Equilibrium Conditions . 62
5.3 Estimation of the Hedonic Price Equation . 63
5.4 Some Monte Carlo Results on the Identiflability of the
Hedonic Price Equation 65
5.5 Conclusion 69
Chapter 6 The Formation and Use of the Hedonic Price Equation
K. E. McConnell and T. T. Phipps
6.1 Introduction 71
6.2 Preference, Income Distribution and the Functional Form
of the Hedonic Price Equation 71
6.3 The Welfare Effects of an Exogeneous Change
in Attributes 77
6.4 Conclusion 86
Appendix to Chapter 6 89
Chapter 7 Should the Rosen Model Be Used to Value Location-
Specific Amenities?
Maureen L. Cropper
7.1 Introduction 90
7.2 Consumer Choice in an Hedonic Model 91
7.3 Applying the Hedonic Model to Residential Location Choice. . 94
7.4 Discrete Choice Models of Residential Location Choice. ... 96
7.5 Conclusion 98
Appendix to Chapter 7 103
Chapter 8 Summary and Assessment
8.1 Introduction 104
8.2 The Identification Problem: Summary and Resolution 104
8.3 Suitability of the Rosen Model for Valuing Environmental
Amenities 107
8.4 Future Research 108
References 109
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FIGURES
Number page
1.1 The Links Between Regulatory Actions and the Net Economic
Benefits of Environmental Improvements . .. 2
2.1 The Basic Hedonic Model 12
2.2 Welfare Measures for Increasing an Attribute 16
6.1 Alternative Welfare Measures .82
6.2 Welfare Measures in a Concave Neighborhood for h(z) 83
7.1 Bid Functions and Hedonic Price Equations . 93
7.2 Locational Restrictions on Choice 95
7.3 Observed TSP and Distance from CBD, Baltimore 97
vii
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TABLES
Kumber Page
5.1 Monte Carlo Rnsults for Hedonic Parameters Diagonal
Covariance Matrix 5.12 63
5.2 Monte Carlo Results for Hedonic Parameters Diagonal
Covariance Matrix 5.13 68
6.1 Transformation Parameter for Quadratic Box-Cox Hedonic
Price Functions 76
6.2 Alternative Measures of Welfare for Exogenous Changes
in an Attribute 81
6.3 Further Comparisons of Welfare Changes 85
6.4 Calculating Welfare as Changes in the Rent of Affected
Sites Only 86
vin
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CHAPTER 1
INTRODUCTION1
1.1 Benefit Cost Analysis and the Hedonic Model
This report deals with one approach to inferring the value of
environmental improvements the hedonic method. It is part of the accepted
wisdom of economics that environmental quality is a public good. Hence
improvements in environmental quality will tend to be provided in less than
optimal quantities by decentralized decisions. A corollary to this tenet is that
government intervention may be required to provide optimal quantities of
environmental improvements. To determine optimal quantities, the costs and
benefits of environmental improvements are needed. In practice, optimal
quantities cf environmental improvements .are almost never directly sought,
Instead, government intervention for environmental improvements comes in the
form of new rules or changes in rules. Benefit cost analysis can be applied
to changes in rules to determine whether they are in the right direction. If
enough rule changes are evaluated, then optimal quantities of environmental
improvements can be achieved / indirectly.
The hedonic method is one of several widely used approaches to measure
the benefits of environmental improvements. It relies on individual choices in
markets when the quality of the environment is one dimension of the quality
of the good for sale. The basic approach of the hedonic method is to infer
willingness to pay for environmental quality from market prices reflecting
quality differences. This method is typically practiced by gathering data on
the sales of goods, for example housing, and then showing with statistical
methods the relationship between sales price and all the characteristics of this
good, including practical measures of the quality of the environment. This
relationship is called the hedonic price equation and the specific effects of
pollutants on the sales price, as shown by statistical methods, have provided
an important link in determining the benefits of environmental improvements.
The role of benefit cost analysis in general and the hedonic method in
'
net benefit changes in Figure (1.1) (adapted from Desvousges, Smith and
McGivney, 1983, page 1.2). A rule change or regulatory action is designed to
force households or firms to reduce emissions. In cases of any consequence,
the reduction of emissions requires changes in behavior which are costly to
households and firms. Hence the initial economic effect of rule changes is to
impose costs on economic units. If the rule changes are effective at reducing
emissions, then they will improve the ambient environmental quality.
Improvements in environmental quality will be valued by society.
Improvements in environmental quality which are perceived lead some
households and firms to change their behavior. Implicit market methods of
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FIGURE 1.1
THE LINKS BETWEEN REGULATORY ACTIONS AND
THE NET ECONOMIC BENEFITS OF ENVIRONMENTAL IMPROVEMENTS
Regulatory
Act ions /Rule
Chanff.es
*
Impacts on
Households and
Firms
1
Reduction
in
Emissions
1
Improvement
in Ambient
Quality
I
Costs
of
Actions
Perceived Impact on
Households and Firms
(e.g., greater visibility)
Imperceptible Impacts
on Households and Finns
(i.e.. health effects);
Benefit Analysis:
Hedonic Models
Ol
Mortality and Morbility
Others _
.-it. t \
I of Actions!
1
Net Benefits of
Regulatory Action =
Benefits - Costs
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benefit measurement, such as the hedonic method, attempt to measure the
changes in benefits by recognizing that rational, consistent behavior reveals
information about preferences. When we reveal information about choices
involving environmental quality which are explicitly or implicitly costly, then
under some circumstances we can infer what people will be willing to pay for
changes in environmental quality. Consider air quality improvements. As
households perceive different air quality in different locations, they will
change their behavior in a directly economic way by bidding up the price of
sites which have improved air. The role of hedonic analysis is to use such
information on behavior to infer the willingness to pay for improvements in air
quality. The purpose of this volume is to assess the potential of the hedonic
method for measuring the benefits of changes in environmental quality.
There are both administrative and economic reasons for wanting to
improve benefit estimation techniques in general and the hedonic method in
particular. The administrative impetus is provided by Executive Order 12291,
which requires agencies of the Federal government to estimate the benefits
and costs of major regulatory actions (with impacts greater than $100 million).
Good benefit estimation techniques can help make the E012291 a productive
order. Bad techniques will make it a charade. " .
While the administrative procedures under which the Federal government
operates are important and certainly should influence research in benefit-cost
methods, there are additional cogent reasons for improving benefit estimation
techniques. There is a compelling logic to benefit-cost analysis. Whatever its
fault, it is the only fully consistent method available for assessing resource
allocation. Hence it will tend to have influence, implicitly or explicitly, in the
public decision process. In the use of benefit-cost analysis for environmental
rule changes, benefits seem less plausible than costs because they come from
intangible or aesthetic services that are not traded on the market. Costs tend
tc be incurred directly for purchases of physical capital goods or as higher
operating costs and indirectly as higher prices for consumer goods. Further,
the direct costs of environmental improvements tend to be borne by well
represented groups. For example, air quality improvements may require
expensive alterations of fossil fuel power plants. For any region we can
describe the impact of rules about the sulphur content of coal or the
installation of scrubbers on the stacks of power plants. We can also rest
assured that the co^ls of such rule changes will find their way into the
public debate over rule changes for they are incurred by small groups. But
benefit estimates are far harder to introduce into the debate because they
V. . .-1J
defend. The benefit estimates are at a disadvantage because of the
metaphysical nature of benefits and the difficulties with techniques which
estimate such benefits. So from the perspective of making the best use of
our resources, we would do well to learn more about methods of estimating the
benefits of environmental improvements.
The logic of economics in benefit-cost analysis is clear. Computing money
measures of the benefits and costs of regulatory changes provides a common
unit of analysis, and under the right circumstances, enables researchers to
suggest when changes in rules are socially worthwhile. Yet, as Figure 1.1
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shows, there is more to benefit cost analysis than simply measuring benefits.
To determine economic benefits, the impact of rule changes must be traced
through a variety of environmental and technical relationships. Further, as
study of environmental decisions shows, there is more to the decision process
in evaluating rule changes than the simple logic of calculating benefits and
costs. These changes in economic welfare play a role in the decision process
but so does information about who gets the benefits, and who incurs the
costs, information about the effects of rule changes on emissions, emissions on
ambient quality, and ambient quality on humans. Descriptive information about
all the links in Figure 1.1 improves the cogency of analysis in part by
reducing apparent uncertainty. Further, not all benefits and costs of equal
magnitude carry equal weight in the decision process. It is the whole picture,
from rule change to net benefit-cost analysis, including all the intermediate
links, which determines whether proposed rule changes are enacted. Those
analyses which appear more certain and which tell a more plausible story will
be more convincing. Studies which communicate their results to a broader
audience will be more effective, as will studies which provide a richer picture
of the course of events.
What are the implications of such a pluralistic decision process for
research on methods of benefit estimation? Should we abandon the attempt to
develop logically consistent and plausible models of economic behavior for
benefit measurement? We believe not, for two reasons. First, models which
are logically consistent must help explain how people respond to changes in
external circumstances, including changes in the economic rules of the fame
and changes in the natural environment. Such "responses play a critical role
in the link between rule changes and net benefits in Figure 1.1. Thus the
effort to explain behavior in a consistent and plausible way, which is the
essence of economic models, will help establish the framework not only for
calculating benefits but also for describing the environmental links. Second,
while benefit analysis works within the limited truth of logically consistent
behavior, it is nevertheless our only tool for thinking systematically about
scarce resources, whether environmental or other.
When we take a broad view of assessing the worth of rule changes, the
hedonic method shows especial promise. At best, this approach would allow
researchers to infer the value of changes in environmental amenities which
result from the workings of a market. The potential advantage of thin method
over other methods, such as travel cost models or contingent valuation^, is
the presence of market prices which reflect differences in environmental
^VT^^^,'4,-^- * ,,,.( »1-- t. .- J ...-V «!..-.» ... ....!.* ..,. . . . VI -1 ...... IV .
environmental changes influence behavior, and adverse changes may make
people worse off. Such scientific evidence can help establish the intermediate
links in Figure 1.1. Evidence that environmental changes influence behavior
is perhaps the weakest link in Figure 1.1, as we can learn from the General
Accounting Office (1984) and Freeman (1982). Epidemiological studies do not
always provide unambiguous evidence that air pollution affects human health.
The adverse effect of water pollution on recreational activity is easy to
imagine but there is little hard scientific evidence to document it. Thus part
of the attraction of the hedonic method is its direct use of evidence. It
shows in a way that noneconomists can appreciate how pollution affects
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well-being. If researchers can find a way to make the method yield measures
of -willingness to pay for changes in air quality, they will have an
exceptionally valuable tool. If.-all we can salvage is evidence that air pollution
affects housing values, we at least have evidence that pollution matters, which
is often more than can be said now.
In the right circumstances, the hedonic method can be used to
determine benefits of changes in public rules. There are several unsolved
practical and conceptual problems involving the use of the hedonic models.
The purpose of this report is to investigate the conceptual and practical
problems of using hedonic models. The impetus for the research in this
volume comes from the so-called identification problem in hedonic models.
Solving the identification problem means developing the hedonic method so
that it will tell us something about the preferences of individuals for
environmental quality, and how individuals respond to changes in
environmental quality. Without such information, the hedonic method can tell
u.s only what emerges in the market, which reflects only one piece of
information about preferences, the value of quite small environmental changes.
Solving the identification problem means pushing the hedonic method to tell us
more about the preferences of individuals behind the market, so that we know
how to value large changes in environmental quality.
1.2 Overview of the Volume
. »
The chapters in this volume are prepared by different authors or
combinations of authors. While they all contribute toward the goal of the
research, they may nevertheless be read independently of one another. Chapter
2 gives an assessment of the hedonic method as it is currently practiced,
discussing the variety of its applications as well as its unsolved problems.
Chapter 3 reviews current solutions to the identification problems and offers
an interpretation of identification in a single market setting. Chapter 4
develops the structural system of which the hedonic equation is one part, and
states the conditions for identification in a traditional econometric Betting.
Chapter 5 provides some evidence on estimation of the hedonic price equation
in the form of Monte Carlo results. Chapter 6 creates a model which- simulates
the workings of a housing market and explores welfare measurement and
choice of functional form in the hedonic price equation. Chapter 7 deals with
the question of whether the hedonic model is appropriate for housing choices,
and proposes several alternatives to current practices.
rrM f < . V . 1 « - < . . i . ~f - . r
- . , - ' ' ' -<- ' "ii'"> ' '
Chapter 2 through 7 are rather diverse. Chapter 8, the conclusion, attempts to
distill what has been written in the previous chapters as well as what has
been learned on the project to provide an understanding of how to make the
best use of hedonic models for measuring the benefits of environmental
improvements.
1.3 Some Conclusions
The identification problem cannot be solved through empirical research.
The identification problem deals with how much prior information one needs to
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bring to empirical analysis in order to recover the parameters related to
preferences for environmental quality. In the hedonic case, we are concerned
with the amount of prior information needed to identify the parameters of the
preference function. Thus it is in the nature of our charge from EPA that
our results are conceptual, not empirical. Empirical support, where provided,
comes in the form of Monte Carlo or simulated markets, which allows the use
of prior information.
Our findings with regard to identification are positive although heavily
qualified. Chapters 3 and 4 demonstrate that identification of the preference
parameters from single market data is possible, but only through the choice of
functional form which is largely untestable. Chapter 5 is concerned with
consistency in the estimation of the parameters of the hedonic price equation.
Our findings concerning the applicability of the Rosen version of the
hedonic model are negative. Chapter 7 shows that the hedonic model is not
well suited for locational choice. Chapter 6 demonstrates that applying
different benefit measures from the Rosen model to changes in locational
attributes can lead to vastly different results, a consequence of the disparity
between choice in the hedonic model and locntional choice. These conclusions
relate to the use of hedonic models for valuing locational amenities, but not
necessarily other uses of the hedonic model. Even when the hedonic model is
not used for valuing locational amenities, one must still deal with the
identification problems.
/
These conclusions suggest that environmental research which attempts to
impute the benefit of improvements in air quality from the relationship
between property values and air pollution should pursue new methods. In
particular, methods which characterize the process of bidding for discrete
bundles of attributes under uncertainty may prove fruitful.
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CHAPTER 1
FOOTNOTES
Chapters with no authors listed (1, 2 and 8 and appendixes) were written
by K. E. McConnell.
The travel cost method is an approach for evaluating recreation resources.
It is useful also for valuing environmental amenities when they influence
the quality of recreation. The method works by observing how people
change their visits to a site as their costs increase. The contingent
valuation approach works by asking an individual how much he would pay
for hypothetical changes in environmental amenities. A thorough discussion
of each can be found in Freeman (1979a).
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CHAPTER 2
HEDONIC MODELS: CURRENT RESEARCH ISSUES
2.1 Introduction
The purpose of this chapter is to provide a brief introduction to hedonic
models and to outline the chief research issues currently facing practitioners.
The chapter will not attempt a survey of the literature, nor an exhaustive
catalogue of issues raised by the hedonic method. The emphasis here will be
on the use of hedonic models for measuring benefits of environmental
improvements, especially through the relationship between housing values and
air quality.
i
2.2 Choice of Quality and the Hedonic Model . '
This research investigates the hedonic method, yet this method
encompasses a fairly broad range of approaches. In practice, the term
hedonic has come to mean any method valuing the quality of a good through
measuring its demand. In /the context of environmental research, hedonic
tends to mean any method which values the public good environmental
quality through information on purchases of a private good. Our focus will
be narrower, specifically on the Rosen model, but it will be useful to survey
briefly the origin of various approaches which go by the name of hedonic.
Models of quality may be examined along several different lines. For
example Hanemann (1981) distinguishes between the "differentiated" and
"generalized" approaches to demand analysis, depending on whether goods
with different quality characteristics are treated as separate commodities or
the same generalized commodity. In the current discussion, we will consider
two types of quality models: those in which the consumer chooses quality.-in
a vector of n dimensions and those for which quality may be measured as a
sralar. While the distinction may occasionally appeared blurred on close
examination, it will serve our purpose for the analysis to follow.
- -, < » i« t, .«.i § * ,t
originated with the work of Houthakker (1952) and Theil (1952), though
Houthakker only analyzed the case where each commodity has only one
dimension of quality. (Houthakker cites the prior work of Court (1941)).
Work by Adelman and Griliches (1961) is a direct descendent of the
Houthakker work and provides the initial theoretical basis for the use of
hedonic price indexes. Adelman and Griliches posit a preference function of
the form
U = U(x1,...,xn,z1,...zin)
8
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where xj. (i=l,m) are commodities purchased on the market and z* (i-l,m) are
dimensional vectors measuring the attributes of commodity i. All elements of
the preference function are subject to choice, and the price of the
commodity is also a function of its vector of characteristics:
The hedonic method as an index number practice was originally applied to
automobiles by Grilichea (1961). Additional applications may be found in
Griliches (1971). Work by Becker (1965) and Lancaster (1966) is similar in the
sense that it involves quality choice in a large number of dimensions, but
does not directly tie into the hedonic practices.
The hedonic models differ from the Becker-Lancaster models of
household-produced commodities by. having a market interposed between
household choice and prices reflecting quality. This market was typically
assumed to exist, in the sense that prices reflect quality but there was no
formal demonstration of why market prices reflect quality. This gap was filled
by Rosen (1974) who showed how buyers and sellers of a good with
measurable attributes establish a price locua reflecting these attributes.. This
locus can be taken as a given by any single buyer, who then chooses the
kind of good to buy by choosing the optimal quantity of each attribute.
The choice along one dimension, or the exogenous scalar influencing the
quality of a private good, represents the alternative modeling approach. This
approach seems to have been developed independently by several different
people. Maler (1971, 1974) developed the theoretical conditions for measuring
the value of a public good by examining purchases of private goods.
Quantities of the public good influence the quality of the private good, as for
example, water pollution might measure the quality of recreation trips.
Stevens (1966) provided an application, without the theoretical qualification.
Bradford and Hildebrant (1977) provide theoretical results similar to Maler.
These results are extended by Willig (1978). Fisher and Shell (1968) developed
a model which' is also relevant because, while they were interested in price
indices, they limited their analysis to one dimension.1
The distinction between the number of dimensions is especially crucial
when we consider the location decision. By its nature it is limited to two
dimensions, and typically converted to one dimension, the distance from the
center of the city. Thus, for example, the location model of Alonso (1964) is
. .. !,... «.^ «i.- .- ' . ^-.,l-.V- --< -TO- ? t»O-t. «..,»*. ,.4 , .1. TT:I,I.~VV-~,- ^f r>»irt
Willig.
Models for estimating the effect of the quality of a commodity cover a
broad spectrum. These models, have all come under the rubric n hedonic",
broadly interpreted. We are interested in a narrow segment of hedonic
models, the Rosen model. In the following section we discuss its use in
environmental economics.
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2.3 The.Hcdonic Model in Environmental Economics
In concept, hedonic models provide information on the willingness to pay
for public goods because preferences revealed for private goods in part
reflect the demand for public goods. Private goods which provide .better
access to public goods, such as cleaner air or more quiet, will be valued more
highly by households, and private transactions will reflect the value of public
goods. The hedonic model is both a theory and an empirical method which
attempts to separate the effect of qualities such as access to public goods
from other influences on the price of private goods. Like several methods for
assessing the benefits of environmental improvements, the hedonic method of
valuing the environment began as an empirical approach. Ridker and Henning
(1967), Nourse (1967), and Anderson and Crocker (1971) analyzed the effect of
air pollution on housing values. Their empirical results and analytical efforts
to understand their empirical results spawned a lengthy debate over the
method. The development of the Rosen model played an Important role in
settling some of the issues debated.
The initial applications of the hedonic method to environmental quality
attempted to infer willingness to pay for changes in air quality from housing
prices. In current environmental work, applications of the hedonic method to
the air quality-housing price case predominate. However, the first application
of hedonic models was to automobiles, with subsequent applications of the
hedonic method to labor services (hedonic wages), and other goods and
services. /
The promise of the hedonic method can be gauged by the number and
variety of applications in the current literature. Under the rubric of air
pollution, a number of different pollutants have been valued. For example,
Palmquist (1983a) investigates the effect of total suspended particulates,
nitrogen dioxides, sulphur dioxide and ozone on property values in 14 cities.
Bender et al. (1980), Li and Brown (1982), Schulze et al. (1983), and Harrison
and Rubinfeld (1978) (among many others) have also estimated the
relationships between housing prices and air pollution.2 In addition, other
environmental effects have been measured using the hedonic model. Noise
(Nelson, 1978; Li and Brown, 1982), accessibility to shore line, (Brown .and
Pollakowski, 1977; Milon et al., 1983) and water pollution (Epp and El-Ani,
1979; Rich and Moffit, 1982) have all been shown to influence housing prices.
Work to determine the effect pf proximity to hazardous waste sites on housing
values is also proposed or under way. The hedonic model has been used or
* f .' r " ' * V ~ V V '«.- . -.-1 ,~1-.
* *
account for the attraction of the house, for example, schools and crime (Jud
and Watts, 1981; Bartik and Smith, 1984), threat of earthquake (Brookshire et
al., 1984), climate (Freeman, 1984) and many kinds of urban amenities (Bartik
and Smith, 1984). Of course, all aspects of the house itself have shown to be
influential in determining housing prices, for example, size and number of
rooms, presence of air conditioning, swimming pool, fireplace, detached garage,
number of bathrooms, age, type of construction, etc. (Palmquist, 1983b; Li and
Brown, 1982).
10
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The .consistency of findings, especially with regard to air pollution, has
been as impressive as the variety of applications. While several papers
skeptical of the relationship between housing prices and pollution appeared in
the 1970's (Wiuand, 1971; Smith and Deyak, 1975), recent work has supported
the relationship. Published research tends to show that higher levels of air
pollution are correlated with lower housing prices, cet. par., though it may be
that positive or inconclusive findings are less likely to get published.
Somewhat more surprising is the result from hedonic wage models that wage
premia are associated with higher air pollution (Bayless, 1983; V. K. Smith,
1983). Thus the hedonic models show their promise through the variety of
applications and the consistency of findings. Perhaps most important, the
basic model is intuitive and easy to explain to noneconomists.
The two types of models discussed in the previous section are useful for
examining some work which occassionally goes under the rubric hedonic.
Polinsky and Shavell (1976) and Polinsky and Rubinfeld (1977) have developed
empirical models where the bid for each location depends on the attributes of
that location. These models involve optimization in one dimension, and hence
are similar in spirit to the second category of models, the single public good
of Maler, Bradford and Hildebrandt, and Willig. Thus, the work of Polinsky
and Shavell and Polinsky and Rubinfeld may be considered hedonic, but
because it involves only one dimension of choice, it is different from the
Rosen model.
2.4 The Basic Rosen Model ,
Despite the promise of the hedonic method, there remains a number of
problems which arise in its application. Before spelling out the nature of
these problems, it will be useful to give some structure to the Rosen version
of the hedonic method. The following gives a skeletal version of the hedonic
model, which was given its conceptual framework by Rosen.
Suppose that a market exists for a good with several attributes of quality.
Wine may have sugar content, hue, and bouquet, or many more chemically
measurable attributes. A house has windows, lot size, rooms, square feet,
carports, etc. Cars have horsepower, length, acceleration. Sellers are aware
of the costs of producing the good with different attributes. Buyers know
thnt units of tho good with different, attributes bring different utility levels.
When the market is relatively dense, that is, almost any level of attribute is
technically feasible and may be supplied, and demanded, then we can equate
' > «'. > : * 1.1 »...« rpi .. '.. ...-i ;.- - ' ';-..1 ;
Figure 2.1
Assume that there is only one attribute of the good, and it is measurable.
Consumers come to the market willing to pay more for a unit with more of its
attribute. This information is revealed by their bid functions, B0i B», Ba,
which differ if they have different preference functions or different incomes.
Sellers know the extra cost of producing the good with more of the attribute,
and because there are sellers with different characteristics, they offer dif-
ferent quantities of the attribute at different prices, denoted by the schedules
S0r S,, S2. The market equilibrium yields the hedonic price equation denoted
11
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Price of
good
B: buyers' bid schedules
S: sellers' offer schedules
h: Locus of equilibria
the hedonic equation
Quantity of the attribute
/
The Basic Hedonic Model
Figure 2.1
h, which is a locus of equilibrium points of various quantities of the attribute.
Individual buyers or sellers take the hedonic price relationship as given and
make their marginal selling or buying decisions according to its implicit trade-
offs. Buyers choose goods which equate the marginal value of the attribute
with its marginal cost, given by the hedonic equation. Sellers produce goods
which equate the marginal cost of production with the marginal returns, also
given by the hedonic equation. This model will be the source of much greater
scrutiny late in this volume.
The structure of the model given above was developed persuasively by
Rosen. The estimation methods were also codified by Rosen in the following
«.,-« T-- v .... ' : .,-,-. : ''. .. 'i
. .' . . .1- .-/»''' '..- jft-. .. - L~
attributes. For example, housing price depends on the site-specific attributes)
neighborhood characteristics and environmental quality. The resultant
relationship is the hedonic price equation. Second, compute the partial
derivative of the hedonic price with respect to the i^n attribute, and use this
as an endogenous marginal price in a model of supply and/or demand.
It will aid our discussion of the hedonic method to be more specific about
the two step approach. Let us assume that we analyze buyers' choices, and
12
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hence are interested in parameters of preferences. Let
P = h(z;r) (2.1)
be the hedonic price equation, where z is a K-diTnensional vector of attributes
of the good and j is a vector of parameters describing the function. Using
best fit methods, we estimate (2.1). In equilibrium, the consumers' marginal
bid for the attribute will equal the marginal cost of the attribute, as given by
the hedonic price equation. Then we use the predicted derivative as a
dependent variable, marginal price, in the following equations:
ah/azi = n>i(z,y;0) i = 1.....K (2.2)
where mj is the marginal bid function (marginal to the functions B0, Ba and B*
in Figure 2.1), y is income and P is a vector of parameters describing tastes.
Expression (2.2) is the equilibrium condition for individual buyers in the
hedonic market.
The economic framework created by Rosen has been rather widely accepted
as providing a plausible explanation of the effect of amenities on the price of
private goods. While there have been many questions about procedures for
applications, there have been few about the theoretical structure. Especially
in the areas of urban and environmental economics, it has become part of the
accepted theoretical structure.
/
2.5 Some Research Issues
While the Rosen model of hedonic pricing has served well in its positive
role, questions arise when we try to use the model for normative purposes.
For example, the hedonic equation may do well in predicting the cet. par.
effect of another bathroom on the price of a house, but it is less clear what it
reveals about the welfare effects of a decrease in total suspended particulates.
Further, there are some ambiguities about the applicability of the Rosen model
to the choice of housing location. In this section we survey several questions
currently debated in the literature. These questions are important because
they relate to the use of the hedonic method for measuring the changes in
environmental amenities, but they in no way exhaust current research topics.
A discubbion of these issues will help in understanding the focuu of this
volume.
nw Ociil Aiauoti cico ouj t t,iib i«JbU*ilCll ull cCLlOUb Ujr L CjttU i'lilg to tllu
model and to expressions (2.1) and (2.2) and to figures similar to Figure 2.1.
We divide the research topics into five areas:
1. What practical problems arise in estimating the hedonic price
equations?
2. Can the parameters (p) of the m function in equation (2.2) (typically
called the inverse demand function or marginal bid function) be
identified, and if so are there serious estimation problems which then
arise?
13
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3. jHow can the welfare changes induced by exogenous changes in
attributes be measured?
4. Does the hedonic model capture all the welfare change associated with
changes in an environmental attribute?
5. Are the structure and assumptions underlying the hedonic model
appropriate for the issues relating to choice of location by
households? That is, when households choose the location of their
residence, is the hedonic model working?
Considerable effort has been directed to problems encountered in
estimating the hedonic price equation, the first topic. Pour of the problems
that arise in fitting the hedonic price equation are multicollinearity, selection
of functional form, measurement of the amenities or attributes, and the
aggregation issue. The collinearity problem is especially severe. Bigger
houses typically have more of all kinds of attributes - bath rooms, lot size,
garage space, and a higher likelihood of having amenities which come in
discrete units - pool, air conditioning, a scenic view. Amenities within a
community tend to be highly correlated. Localities with good schools tend to
have nice park systems as well as high tax rates. Different air pollutants are
particularly likely to be correlated. Weather patterns and location close to
common emission sources cause some areas to have more of all pollutants than
other areas. Collinearity is probably most severe for the characteristics
specific to the house. It would be wrong, however, to argue that
multicollinearity is always a problem. Palmquist (1983a) has shown that for
one set of 14 cities, collinearity is not a problem for pollutants.
The choice of functional form for the hedonic price equations is .-a critical
one, in that it determines how marginal prices behave. Yet by the nature of
the model, we can expect little or no theoretical guidance for choosing among
alternative functional forms. As can be seen from Figure 2.1, the hedonic
equation is a locus of equilibria, and has embodied in it the structural aspects
of buyers and sellers. Best fit methods, such as Box-Cox approaches used by
Bender, Gronberg and Hwang (1980), Halvorson and Pollakowski (1981) and
others seem appropriate, but these methods may not result in well-defined
maxima for households with quasi-concave preference 'functions. Closely
related to the choice of functional form is the problem of complete
specification. It is virtually impossible to specify a hedonic equation which
includes all the attributes which influence price. The exclusion of collinear
attributes can have two affects. First, it can bias the coefficients of the
hedonic equation. Second, when combined with nonlinearity, such
v««4 «> »»-»«> -."'**«»4,.*«T-* «» »* o 1-14 /s «- /-n-1-,-*---. ?»- 4 V *-N V-'-«^'"n~i/^ T>^"i/^f> <*-.**»i^-'**^'T> . T^^%f*rvif?r» **f 4 »*f*
nonlinearity these errors are transmitted to the estimated marginal price, and
are quite likely to be correlated with any instruments (such as income) used
in the estimation of demand relations. (See Epple, 1982, and Bartik, 1983).
The measurement of pollution variables is an important issue. The theory
requires that all market participants respond to the same attributes, but
perception of air quality .may vary substantially across households. And
perceptions may not be closely linked with actual measures of pollutants,'
available for example, from monitors. The problems of multicollinearity and
amenity measurement complicate one another, because it is doubtful that a
14
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single air pollutant can capture households' perceptions of air quality. Yet if
the pollutants are highly correlated, it will be quite difficult to separate their
effects. The work by Palmquist (1983a) on creating an air pollution index is
quite promising in this regard, because it is a first attempt to compute an
index which might replicate households' perceptions. Further, Palmquist has
shown for at least one set of 14 cities that collinearity is not especially severe
for the pollutants. Bartik and Smith (1984) have highlighted the problem of
perceptions.
Finally, there is the question of aggregation of observations. Early work
buch as that by Ridker and Henning used median sales price of
owner-occupied housing, where the census tract was the unit of observation.
But recent empirical research has relied predominantly on housing sales data
or homeowner opinion surveys. The question of when and whether parameters
of the hedonic equation can be recovered from aggregate data has received no
formal attention.
The second issue_~m the research list is the identification problem. The
nature of this problem can be understood by rewriting equations (2.1) and
(2.2) as
P = h(z;7) (2.3)
ah(zj,7)/*zi = nii(z,y;0) i = 1.....K (2.4)
where 7 is a vector of parameters describing the hedonic price equation and f
is a vector of parameters describing the marginal bid function. The Rosen
two step approach estimates (2.3) first, and then uses the predicted derivative
to estimate (2.4). The identification problem in an intuitive sense comes from
having estimates of ft actually be combinations of 7 and f. Brown and Rosen
(1982) give the best illustration of this particular problem. The issue is
currently receiving as much attention as any other issue in hedonic models.
As we show next, the identification problem is important to the extent that
information about preferences for environmental amenities is needed. It is
possible, however, that benefit measures can be computed without such
information.
The third issue deals with the way welfare, measure can be derived from
the hedonic method. There has been surprisingly little research on this topic,
especially since for environmental matters, welfare analysis plays such a
government actions can cause the attributes of housing to be improved by.
reducing air pollution. How should the hedonic model be used to measure the
economic benefits of better air? The problems surrounding this issue can be
addressed with Figure 2.2. This figure shows the bid function (B) for an
individual, and the market hedonic price function, h(z). Suppose a
government rule results in an increase in the amenity better air which
is experienced by the individual as an increase from z to z*. Assuming the
individual to be in equilibrium at z, we can discuss three measures of welfare
changes commonly used in the literature:
15
-------
Price of
house
I 2'
Clean air
Welfare Measures for Increasing.an Attribute
Figure 2.2
(i) the household's increase in willingness to pay for the site: ab
(ii) the predicted increase in the price of the site, based on the hedonic
price equation p(z): ad
(iii) a first order Taylor's series approximation of (i) and (ii). L is
tangent to the equilibrium at e, so that its slope is equal to the
common marginal price - marginal willingness to pay, and an estimate
of (i) or (ii) based on a linear extrapolation is given by ac.
i i . .- , -
- <- - . .- - ', - .'--
A hedonic price >
linear extrapolation of A hedonic price >
A willingness to pay
The consumer's willingness to pay is a superior measure, but requires
knowledge of the parameters f of m in (2.4) and requires successful
completion of the Rosen two-step approach. The linear expansion of B or h is
most often used and most criticized. Its accuracy can be seriously impaired
by two possibilities:
16
-------
a) Only a few combinations of the z's are available in practice so that
there is no equilibrium of marginal price and marginal willingness to
pay. In fact, unless, the available z's are quite dense, a negative
marginal bid is quite possible for z.
b) The hedonic price function need not be convex; for equilibrium
purposes it need only be less concave than the bid surface. In thai
case, linear extrapolation of p will exceed the prediction made by p(z*)
- P(Z) given in (ii).
Thus it scents that benefit measures will be improved by recovering the
parameters of the bid function, but this requires solution of the identification
problem. If the hedonic price equation is not "too" convex, then it may
provide a decent estimate of the value of changes in z. At least we know that
whether we use the prediction from the hedonic price or its linear
extrapolation, we will have overestimated the change in willingness to pay.
Another difficulty in welfare measurement becomes apparent when we look
more closely at Figure 2.2. The household equilibrium requires tangency
between the hedonic price equation h and the bid function B. The tangency
exists at z, but not at z*. Hence the measures described above are; in the
phrase of Bartik and Smith (1984), restricted partial equilibrium measures.
They are restricted because they do not allow the market to adjust to
changing conditions. When the z's are changed exogenously, the initial supply
conditions no longer hold, and a new hedonic price equation must be
established. The appropriate welfare measures require comparing an old
equilibrium with a new equilibrium, something which the "restricted partial
equilibrium" measures do not do.
The fourth topic given above also involves welfare measurement. The
essence of this problem concerns potential double counting of benefits from an
environmental improvement. To what extent does the hedonic method applied
to property values measure benefits that might also be captured by other
methods? Roback (1982) has investigated the case when wages are influenced
by environmental attributes. Other cases remain to be investigated. The
economic use of epidemiological studies attempts to measure the benefits of
improving air quality, which is also the role of housing value studies.
Location near a clean water site may be capitalized into land prices, and hence
measure in part, the demand for travel to the clean water. A separate but
related issue concerns the purchase of attributes which reduce the effect of
pollution, for example, air conditioning. Because people spend a majority of
4 "»T» «% *»-i^l^j->w. J,-J*"»C,-..-. - m .^. r. .-]*!..*. -, -»*- »n - ,. 1 * , *~ * ? -. rJ.v.". ,. -- *- .- «*<*>-
conditioning can avert some effects of air pollution. These issues must be
worked out in concept before we can investigate their practical importance.
Research on the fifth topic has addressed two questions, both especially
problematic for the real estate market. The hedonic model assumes that the
goods are sold at auction with buyers and sellers having full information.
The housing market, in fact, is one of sequential bids and substantial
uncertainty about the hedonic locus. Work by Ellickson (1981), Lerman and
Kern (1983) and Horowitz (1983) is designed to model the housing market to
reflect more accurately the way transactions are made. Another important
17
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assumption in the hedonic model is the continuity of the hedonic price
function in attributes of the good. Continuity assumptions are routinely made
and violated in economics, usually with little impairment of conceptual or
empirical analysis. Continuity assumptions may not be so innocuous in hedonic
models. Housing attributes such as rooms, air conditioning, and swimming
pools not only are not continuous but are typically available in only a few
combinations. Further, because of the limited number of bundles available,
choices may tend not to equate marginal bids with marginal costs. The lumpy
aspect of housing, implying discrete choices, is modelled initially by McFadden
(1978). This particular aspect of hedonic models is a fruitful area for
research.
For purposes of this volume, the issues raised above fall into two
categories. On the one hand there are the important practical problems
involving estimation of the hedonic equation, determining what benefits can be
calculated from the hedonic model, and the accuracy of various restricted
measures of welfare changes. These problems are not different from the
problems one confronts in any kind of empirical work in economics. They are
primarily the consequence of less than perfect data. On the other hand, there
are the issues of identification and whether the hedonic model is appropriate
for the choice of residential location. These issues have the common aspect
that their solution does not hinge on better data. The problem of identifying
parameters of preference functions when households have nonlinear budgets is
severe even with perfect data. Further, if the hedonic model is not the .right
model for choice of location of residence in concept, no amount of data will
make it so in practice. This volume is concerned with problems of the second
sort. That is, we will investigate those issues which in principle may prevent
the method from providing useful input to benefit-cost analysis.
2.6 The Charge of the Research
This research was undertaken as a part of research project on implicit
market methods of measuring the benefits of environmental changes. The
explicit charge for the hedonic research is to "develop solutions for the
underidentification of hedonic demand curves for environmental public goods
and demonstrate, using suitable pollution problems" (EPA Request for Pro-
posal, April 1983).
This charge has been the driving force of our research. But we have
expanded our research to those topics which in principle prevent the
.. , , .. . ,, . ... . , rr, , ^
interpreted the identification problem here as the problem of recovering the
parameters of .a function which yields willingness to pay by households for
changes in attributes of a good. That is, we wish to ascertain under what
circumstances we can recover the parameters of the m(z,y;l) function in
equation (2.4) because recovering these parameters may help improve welfare
measurement. The following two chapters explore directly the identification
problem. These chapters are quite different in approach but have in common
the idea that identification is solved in concept. Other chapters, too, are
concerned with whether the hedonic method works in concept.
18
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CHAPTER 2
FOOTNOTES
Muellbauer (1974) indicates how the distinction between types of models
can become blurred. He increases the quality dimension of the Fisher-
Shell model to make it a choice of several attributes and reduces the
dimension of the Houthakker model to make it a one dimensional choice.
As we shall argue later, the choice of model ultimately depends on the
technical and institutional characteristics of the problem.
For further works on pollution and property values in the hedonic model,
see the references at the end of this volume, Bartik and Smith's (1984)
references, and those provided by Rowe and Chestnut (1982).
19
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CHAPTER 3
IDENTIFICATION OF HEDONIC MODELS
Robert Mendelsohn^
3.1 Introduction
Although the theory and econometrics for understanding markets for
homogeneous goods have been understood for decades, the problems of
modelling markets for heterogeneous goods has received attention only
recently. One fruitful approach to dealing with heterogeneous goods has been
the hedonic model. The heterogeneous good is envisaged as a bundle of
homogeneous attributes. For example, a residence is composed of attributes
such as the number of rooms, lot size, school quality, air quality,, and other
characteristics. From the work of Court (1941) and Griliches (1971), it is now
commonplace to estimate the implicit prices of these attributes by regressing
expenditures on the bundle (the price of the heterogeneous good) upon the
observed attributes. As noted by Rosen (1974), the resulting marginal price
gradient is the locus of market prices which equilibrate demand and supply.
For marginal valuations, this locus is all that is needed. However, for
nonmarginal valuations where the observed price gradient is expected.' to
change in response to some policy of interest, it is necessary to uncover the
underlying structural equations of the model. The purpose of this chapter is
to discuss when and how the demand and supply curves for characteristics
can be identified with available data and econometric techniques.
The first discussion of the identification problem with hedonic markets was
raised by Rosen (1974) in his development of the basic hedonic market model.
Rosen perceived the hedonic structural equations to be no different from
traditional market models. He consequently asserted that the identification
issue was just the familiar problem of sorting out supply from demand.
More formally, suppose the hedonic price function for the good is:
where z is a vector of attributes. Then the price gradient (of marginal
prices) for each attribute zj is:
PI(Z) * -^T" (3>1)
i
The underlying inverse supply (g) and demand (f) functions for the attributes
are:
20
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= f(z,y) + «t
(3.2)
where y and w are exogenous demand and supply shift variables, respectively,
and c£ are random error terms. Rosen recommended that the hedonic price
function be estimated by OLS in a first step. Taking the derivative of the
hedonic price function, the appropriate marginal price for the observed
purchased bundle z would then be the dependent variable in the estimation of
the structural equations (3.2). The identification problem, according to Rosen,
is the separation of demand from supply effects.
Brown and Rosen (1982) offer an alternative identification problem. They
are concerned about the use of the' predicted marginal price from the price
regression (3.1) in the structural equation estimation (3.2). They note that
with linear functional forms, the variation in z captures all the variation in
the predicted price. "'That is pj is constructed:
Thus, to estimate demand by regressing p on z and other shift variables y
such as: .
(
p. = /?o t PS + fij
one should expect 10 = 7oi P\ - 7n and pa = 0 because there is no random
variation in p that cannot be perfectly explained by z. Furthermore, at least
with a linear marginal price model, the linear structural equation will always
be the best fitting functional form because it provides a perfect fit. The
structural estimation consequently just reproduces the original marginal price
equation. The structural equations remain unidentified.
A third perspective is voiced by Mendelsohn (1980), Bartik (1983) and
Diamond and Smith (1985). These authors note that maximization of profits or
utility subject to the nonlinear budget constraint of a single price gradient
results in only one observation for each actor in the market. Each of the
,H » i » . i r . . ii i . . ..........?',}..,.».,..., i .... -i, . pr~
substantively different. The identification problem in hedonic markets is not
between demand and supply per se but rather between the response of one
demander to one price versus a different demander to another price.
There are consequently three potential identification problems with single
market hedonic models. (1) The "garden variety" simultaneity of demand and
supply; (2) the use of estimated prices in structural equation estimation; and
(3) the separation of price effects from shift effects across consumers or
across suppliers. Corresponding to each of these problems, authors have
recommended specific solutions.
21
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In Section 3.2, we- discuss solutions to the "garden variety" identification
problem and demonstrate that traditional solutions are not adequate because of
the simultaneity of shift and price effects; In Section 3.3, we review the use
of predicted marginal prices in the structural equation and show that the
Brown and Rosen critique can be generalized to any structural equation where
the exogenous shift variables are additive. . We further show that the
estimation of prices is not the central problem. In Section 3.4, we address the
special identification problem of hedonic markets, the untangling of price and.
shift effects. In this section, we show how nonlinearity in the price gradient
and restrictions on the structural equations can lead to identification. The
identification problem in the Brown and Rosen linear model can disappear in
nonlinear models.
Finally, in Section 3.5. we discuss how observations from multiple markets
(either inter temporal or cross sectional) can overcome the identification
dilemma in certain circumstances.
3.2 Simultaneous Demand and Supply
If the inherent identification problem of hedonic models is the simultaneity
of supply and demand equations, there are several plausible solutions. As
recommended by Nelson (1978), Linneman (1981), and Rosen (1974), one could
use econometric techniques such as instrumental variables or two stage least
squares to separate demand from supply. For example, suppose the under-
lying model is: <
2 = f(P,y) + *i (3.3a)
2 = *(p,w) + « (3.3b)
where f(') is demand, g(*) is supply, y and w are shift variables, and tt and
e2 are error terms. Marginal price p is endogenous in this model, being the
result of both supply and demand factors. Consequently p is affected by
both t, and *2 and so is correlated with both. OLS regressions with p would
be biased. To correct this problem, one regresses p on the exogenous shift
variables y and w. The resulting predicted level of price, p, can then be
entered into either structural equation for second stage estimation.
An alternative way to control for the simultaneity of supply -and demand is
to assume one of these structural equations is fixed. For example, Harrison
un<_i iiuui«uuiu VAJIO/ a&auuie mat. cue t»uppi> oi oiemt «ii i<> uu* ed^oiifeive to
the price of clean air. As Nelson (1978) and Freeman (1979a) note, the level of
air quality in each area may indeed be insensitive to the prices charged in
each housing market. However, the supply of clean air is the amount of
housing available with clean air, not the amount of acreage available.
Consequently, builders could provide more housing per acre in clean air
locations if the price of clean air were sufficiently higher. Thus, it may often
be inappropriate to assume that supply functions are perfectly inelastic in
hedonic markets.
22
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Parsons (1983) and Epple (1982) demonstrate that the identification
problem in hedonic single markets deals with more than the traditional
separation of demand and supply. Both these authors show that the
traditional methods used to untangle demand from supply will not -work in the
single market context. Along a nonlinear price gradient, suppliers and
demnndcrs arrange themselves according to their underlying shift parameters
y and w. This sorting procedure means that certain types of suppliers will
tend to match up with particular demanders to transact special bundles along
the gradient. For example, with housing, builders of homes in the outlying
suburbs will tend to supply the attribute clean air. Demanders of clean air,
possibly asthmatics, will tend to purchase these outlying homes. The
introduction of the variable, asthmatics, will represent the builders of outlying
homes just as much as the domaitders for these clean air homes. The single
market results in a one-to-one correspondence between particular demanders
and suppliers, making it difficult to identify either structural equation. Thus,
the identification problem is clearly more than "the garden variety" found in
traditional goods markets. The untangling of supply and demand is just at
the surface of the problem.
3.3 Predicted Prices
Brown and Rosen (1982) show that when both the price gradient and the
structural equations are linear, the predicted marginal prices cannot be used
to identify the structural equations. The linear estimation of the structural
equation simply reproduces the coefficients of the hedonic price gradient.
Brown and Rosen's proof can be generalized. Regardless of the shape of
the price gradient, any structural equation which is additive in the exogenous
shift effects will merely reproduce the price gradient. For example, suppose
the price gradient is
z = ^fJ- (3.4)
^
Any structural equation which additively includes pj(z) will reproduce (3.4).
For example:
Pi ~ Piz * ° Z + ° y*
That is, the estimated coefficients on z° and y would be zero.
To surmount this problem, analysts have restricted the family of
structural equations so that none of the members can have the above
properties. Brown (1983) suggests omitting particular expressions for z in the
structural equation which are in the hedonic equation. For example, one could
leave out the log z term found in (3.4). Alternatively, one could omit a
23
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particular attribute z^ in the structural equation. Finally, one could adopt a
different functional form (log linear, linear, or Bcmitog) in the hedonic price
versus structural equations. This latter approach is used by Harrison and
Rubinfeld (1978), Nelson (1978), Linneman (1981), Wiite et aL (1979) and
Bloomquist and Worley (1981) in their hedonic models.
Although the alteration of functional form between the hedonic and
structural equation leads to different parameters between the two equations, it
is not clear whether the assumption has identified the true underlying
structural equations. After all, making different assumptions about the shape
of any of the curves leads to different parameters. Although the Brown and
Rosen (1982) model has touched the surface of an identification problem, the
paper provides little guidance to the underlying cause of the problem or to
its appropriate solution.
In order to show that the problem with hedonic markets is not the use of
estimated prices, let us reproduce the Brown and Rosen model and show that
the structural equations are not identified even when the price gradient is
observed (hot estimated). To keep the notation simple, let us assume the
marginal price is a linear function of a single attribute:
. ah(z
The structural equations are also assumed to be linear:
pi = ^o + ^i2 * ?2y (demand)
P4 = GO + Gfz + Gaw (supply).
Because the price gradient is the locus of equilibrium points between supply
and demand, for each z, it must be true that:
7o + TiZ = ^o + PI* + P2y = Go + Gi2 * GaW '
Solving for y and w respectively:
70 - P0 (7. - f.)z
v - -5 - ^ + _! - !_ <
'V
x» ~. »
(3.5b)
24
-------
If we can observe z, y, and w for all pairs of demanders 'and suppliers, then
we could estimate:
y = AO + A^z (3.6a)
w = Do + DiZ (3.6b)
Suppose we also could observe the marginal prices so that we could know TO
and 7,. The issue is whether the P and G parameters of supply and demand
could be identified. If the problem is only with the use of estimated prices,
the equations should be identified.
For the data to be consistent with both (3.5a) and (3.6a), it must be true
that:
70 - P0
A = 2 (demand)
O 0
and A = ; .
i p
Similarly using (3.5b) and (3.6b), it follows that
7 ~ G
_ o o , , .
Do= G (supply)
a
7 ~ G
'"V-
For both demand and supply, there are three unknowns and two equations.
The parameters of the structural equations are not recoverable. The
identification problem posed by Brown and Rosen (1982) is not a result of the
need to estimate marginal prices. Identification, in this case, remains a
problem even when the price gradient is known. The identification problem is
deeper, lying in the amount of nonlincarity in the hcdonic and structural
equations.
Of the three potential identification problems facing hedonic models, we
have shown that the first two are merely surface reactions to the third. The
simultaneity of demand and supply and the use of estimated prices in the
structural equations are special problems in hedonics only to the extent that
they reflect the problem of simultaneity between price and shift variables.
The problem with single market data is that prices and exogenous structural
shift variables vary together throughout the sample. In this section, we
explore the assumptions about functional form which are necessary and
sufficient to identify structural equations with data from a single market. By
25
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restricting the permitted functional form of the structural equations, the
nonlinearity of marginal prices can be used to identify the price and shift
parameters of both demand and supply. The identification approach must be
used with great caution, however, because the true shape of supply and
demand functions are often unknown and so the necessary restrictions may be
unjustified.
Let us assume we observe a set of constant marginal prices pj(z) for a
single good or characteristic z. The characteristic could be a typical measure
. of quality such as the number of bedrooms in a house or the horsepower of a
car. As discussed by Rosen (1974), we assume the price gradient is the
equilibrium of a multiplicity of supply and demand curves. Each actor is
assumed to be a price taker (more precisely, a price gradient taker) in that
the price gradient is determined exogenously to the actor. Consumers are
assumed to maximize well-behaved utility functions subject to . the budget
constraint imposed by their income and all market prices (including the price
gradient). Similarly, suppliers are assumed to maximize their profits subject
to technology, input prices, and the price gradient (output price schedule).
In addition to observing the price gradient, let us assume we observe \he
demand (y) and supply (w) shift variables of, respectively, each purchaser
and producer interacting in this market. :
As Hall (1973) has shown, maximization of utility subject to a nonlinear
budget constraint is equivalent to maximization of utility with respect to a
linear budget constraint which is tangent to the nonlinear constraint at the
optimum bundle z*. Assuming second order conditions are satisfied, the
behavior of the consumer can be described in terms of a set of simultaneous
equations:
p = F(z*,y) (3.7)
P = Pi(z*)- (3.8)
The first equation is a traditional inverse demand function defined over a
linear budget constraint. The second equation adjusts marginal prices to keep
the individual upon the nonlinear budget constraint.2 Together, these
equations characterize a consumer's behavior subject to the price, gradient
Pj(z). A parallel construction is clearly possible upon the supply side
generating:
_ _ r*f..ic ,..\ ' (? 0)
p = Pi(z*) (3.10)
where G(z,w) is the inverse supply curve assuming constant output prices and
(3.10) is the same price gradient as (3.8).
For the demanders and. suppliers represented by (3.7) and (3.9) to have
produced the price gradient (3.8) or (3.10), it must be true that
Pi(z«) = F(z*,y) = G(z*,w). (3.11)
26
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For each observed level of z, the buyers should have the characteristics, yf
which would generate a marginal willingness to pay of Pi(z). Similarly, the
sellers should be observed to have characteristics, w, for a marginal
willingness to sell equal to pi(z). This consistency requirement (3.11) ia the
source of the identification problems inherent in a single market.
Given heterogeneous actors in the market and a single price gradient,
the only consistent reason agents choose different bundles is because of their
shift variable.3 Let us assume that each shift variable y or w has a
monotonic effect on demand or supply, respectively. Holding the price
gradient constant and simply varying y (or w) would result in a monotonic
relationship between z and the level of y (or w). For example, as income
increases, consumers purchase more of each normal good and less of each
inferior good throughout the range of observed incomes. Let us describe this
expansion path in terms of a function () and X(') for demand and supply
respectively:
_. z = «(y) (3.12a)
z = X(w). (3.12b)
Because «() and X(«) are monotonic functions, their inverse must exist. Let us
define this inverse as:
y = A(z) (3.13*)
w = D(z). (3.13b)
The solution to (3.11) is (3.13a) and (3.13b). The shape of A and D
depend upon both the shape of the price gradient and the functional form of
the underlying structural equations. Substituting (3.13a) and (3.13b) back
into (3.11) provides a framework to analyze the identification issue:
Pi(z) = P(z,A(z)) = G(z,D(z)), (3.14)
Intuitively, the problem with single market data is that exogenous shift
variables and prices are functionally related. It is as though one chose a
sample design BO that for every increasing level of price there would be an
increasing (or decreasing) level of the shift variables. Separating out the
effect of prices from that of shift variables becomes difficult. For example,
by a single monotonic curve in three dimensional good, price, and shift
variable space. An infinite number of structural equation surfaces could fit
this single nonlinear curve. Further, even in the neighborhood of the
observations, the set of consistent structural equations can have widely
differing properties,
27
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In order to analyze how nonlinearity can yield identification, it is
necessary to characterize nonlinearity in concrete terms. Let us assume,
therefore, that each function is a polynomial:
1 i-1
* * 1 l l
p(z-5" = A
»» i_-t
G(z,w) = I g z1 1 w
where ?{, lj,k» and gi,m are a11 constants and I, J, K, L, and M represent the
number of nonzero terms in each expression. Let us further assume that A(z)
and D(z) can be written:
N -1
y = A(z) = Z a z
J x ' n n
w = D(z) = z d zq .
q q
Substituting the above expressions into (3.14) yields:
1 i i J(K --1 rN n-n11'1
r r.z = .1. p. , zj U a z
11 JfKj.k vn n i
(3.15)
,Q ,,B-1
I . , L,M . , .Q ,.1
I T.z1'1 = .i g. z1-1 E d z^1
i i i,a *i,n lq q J
Since the above equations must hold for all levels of z, the coefficient for
each term z*~l on the left-hand side must equal the sum of the coefficients
AWA «>A«w ^W* A WW^^AAltlilg VW A *i* \^i ^ WAi CAAW 1 1^ il if " I i«A jl li t^A^IW OJ. ^O.XO^A/ «iA114 \ *S A. O O / .
For example, associated with z*~l:
*"
There is a separate demand and supply side equation for each power of z.
Compressing this information in matrix notation yields:
28
-------
7 = A0 and (3.16a)
7 = DG (3.1b)
where 7 is a Ixl vector, p is a JKxl vector, A is a matrix IxJK, G is a vector
LMxl and D is a matrix IxLM.
The clue to the identification role of nonlinearity lies in (3.16a) and
(3.16b). The parameters in 7 are observable; they simply reflect the price
gradient. The parameters in A and D are also known since these reflect the
observable expansion path between y or w and z. What is unknown are the
parameters in P and G. Solving (3.16a) and (3.16b) for f and G yields:
f - (A'A)-1 A'7 <3.17a)
G = (D'D)-1 D'7. (3.17b)
A necessary condition for solving (3.17a) and (3.17b) is that there be as many
equations as there are unknowns. Thus, for a unique solution, 1 > JK and 1 >
LM for the demand and supply side, respectively. The number of nonzero
terms in the price gradient must be equal to or greater than the number of
nonzero terms in the structural equation.
A sufficient condition tor' solving P and G in (3.17a) and (3.17b) is that
the number of linearly independent rows in A and D equal or exceed the
number of parameters to be estimated. That is, the number of linearly
independent parameters in the price gradient must exceed the number of
parameters which must be estimated in the structural equations.
These simple results can easily be extended to incorporate vectors of
characteristics or demand and supply shift variables. Correspondingly,
nonlinearity can be measured by the increased number of parameters in the
price and structural equations in these more complex models. For example,
instead of the demand parameters ft being JK, they could be expanded to
I JKj with N characteristics.
Adding interaction terms among the characteristics would complicate the
model further requiring even more parameters to be estimated. Interaction
structural equations. Each function could not only include single powers of
each characteristic but also multiplicative terms amongst the characteristics.
For example, the polynomial of each function could include all terms whose sum
of exponents does not exceed a parameter, r. As an illustration, a price
gradient with two characteristics and an exponent limit rp = 3 would include
the following terms:
V z\ ZI V zl z\ Z,V V, zl V
29
-------
For any -polynomial with N characteristics and r exponent limit, the number of
terms would be: rN + .|L (i-1) .§, (j-1).
The solution to this more difficult problem can be written in terms of
equations (3.16) and (3.17) by redefining the individual vectors and matrices.
A, it and f would have the following dimensions:
N r N r .
A: r7N + ^ (i-1) £ (j-1) Xr^N + ^ (i-1) jj* (j-1)
N r
r: 1 x rN + | (i-1) * (j-1)
N ra
P: 1 x rH + l (i-D (j-1)
where rr is the exponent power of terms in the price gradient and rp is the
exponent power of terms in the demand equation. A parallel transformation
would occur in the supply side. There would be a separate equation for each
of the N characteristics in z.
The solution for p and G can be characterized by (3.17). The necessary
condition is that the number of nonzero terms in the price gradient be equal
to or greater than the number of nonzero terms in the structural equation.
The sufficient condition is that the number of linearly independent nonzero
terms in the price gradient exceed the number of terms needed for estimation
in the structural equations.
To illustrate how nonlinearity can lead to identification, we reproduce the
Brown and Rosen model but allow the marginal price gradient to be quadratic:
Pi(z) = TO + 7»z
As shown in Section 3.3, suppose the demand and supply curves are:
Pi(z) = 0o + Piz + Pay (demand)
Utilizing (3.11) and the above equations, it is evident that A(z) and D(z) must
.be quadratic:
^-1 ^ 1 ft I "»
30
-------
Given observations about y, w and z, this quadratic expansion path could be
estimated:
y = q0 + q»z + Qaz2 (3.19)
w = d0 + dxz » daza.
For the data to be consistent with (3.18) and (3.19), it follows that:
T -» T ~G
o ,
p - =
-------
number .of linearly independent terms in the price gradient, the parameters of
the structural equation can be identified. Additional terms are possible from
higher powers of each characteristic and also from interaction terms among
the characteristics.
There is information about the behavior of consumers and suppliers in a
single market. The information, however, is not as complete as in the multiple
market case. Consequently, it is necessary to restrict the functional form of
the structural equations to permit use of single market data. If such
functional form restrictions can be justified (for example by being tested on
multiple market data), then single market data analysis could serve as a useful
supplement to multiple market analysis. All too frequently, however,
assumptions about functional form are made for convenience only. If the true
functional form has too many parameters to be identified with data from a
single market, arbitrary restrictions of functional form will produce arbitrary
results. No matter how well the unidentified functional form fits the data, the
results would not necessarily approximate the truth, even in the neighborhood
of the observations. Although the choice of functional form for estimation
purposes may or may not be a serious issue, the same choice of functional
form to justify identification is always critical. Given how little is known
about the true shape of structural equations and how important that
information is to single market analyses, practitioners should be highly
cautious about using single market data to reveal structural equations.
If possible, analysts should turn to multiple market examples, either
intertemporal or cross-sectional. By varying the .price gradients facing
individuals, one can break the functional relationships A(z) and B(z) between
prices and exogenous variables which plague single market data. In fact, it is
only the existence of exogenous variation of price gradients which prevents a
much larger set of papers in the labor, electricity, and urban literature from
falling prey to the identification problem discussed in this paper.
There are several papers which have utilized multiple markets to properly
estimate hedonic structural equations. Palmquist (1982) uses housing data
from several cities to estimate the demand for housing characteristics.
Mendelsohn (1980) uses workplace location to identify spatially separated
housing markets for estimating the demand for housing characteristics. Brown
and Mond^lsohn (1984) use residential users to estimate the demand for
recreation characteristics.
subdividing a single market into independent submarkets. For example, King
(1976) and Strazheim (1973) attempt to estimate the demand for housing
characteristics by assuming that different towns within a single metropolitan
area are different markets. Unfortunately, the choice of whether to live
downtown or in the suburbs is generally made precisely because of the
housing characteristics. The assumption that these are independent markets
will frequently be inappropriate. Single market identification cannot be
corrected by arbitrarily subdividing the market into smaller submarkets.
32
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CHAPTER 3
FOOTNOTES
School of Forestry and Environmental Studies and Department of
Economics, Yale University. Many thanks go to K. E. McConnell for his
administrative support and substantive comments. I would also like to
thank Michael Hanemann for his helpful criticisms.
In addition to the marginal price effect, there is also an income effect
associated with the change in inframarginal prices. For most examples,
this income effect is small and for expositions! simplicity it is omitted in
the following discussion.
If all consumers are alike, the price gradient would reflect a compensated
demand function. If ail suppliers are alike, the price gradient.- would
reflect an iso-profit supply function. If both consumers and suppliers are
alike, only one bundle would be transacted. Although perhaps extreme,
these assumptions provide an example of how demand and supply can be
estimated by restricting the model.
33
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CHAPTER 4
IDENTIFICATION OF THE PARAMETERS OF THE PREFERENCE FUNCTION:
CONSUMER DEMANDS WITH NONLINEAR BUDGETS
K. E. McConnell and T. T. Phipps*
4.1 Introduction
The hedonic approach has become widely accepted as a method of
modelling quality choice in a market .where prices reflect quality. A problem
which arises in practice with the hedonic technique is the recovery of
information about preferences for the quality of foods. Solutions to this
problemf the so-called identification problem, have evolved from the initial
suggestion by Rosen that exogenous market supply will solve the identification
problem to arguments by Diamond and Smith (1985) for the use of multiple
markets.
Despite the evolution of solutions to the identification problem, there is
still a good deal of uncertainty about the issue. This uncertainty exists in
part because there is little' agreement on criteria for identification, and
perhaps more fundamental, there is seldom explicit discussion of precisely
what is being identified. For example, Brown and Rosen (1982) tie the
identification problem to definitional links between marginal prices and quality
levels, but give little guidance as to the precise nature of the function being
estimated. Quigley (1982) derives the structural equations from explicit utility
maximization, but does not deal with the potential for underidentification in
this context. Thus, while there are many contributions on the identification
problem, they tend to be fairly diverse in their statement of the problem and
their approach to solutions.
The purpose of this chapter is to explore the problem of identifying the
parameters of hedonic models in a framework consistent with choice theory
and the structure of preferences. Our point of departure Is thnt empirical
hedonic analysis using observations on individual purchases (prices and
attributes of goods) is strictly a problem of consumer demand analysis with a
.... ..t: . .. . i... i ... ... . . v-. -.'." " ' ' .. ' . V _ ]0i :>_c
econometric model relating to consumers' choices. The advantage of deriving
the econometric structure from the household's utility maximization problem is
two-fold. First, by utilizing the household model, we see exactly what the
endogenous variables are and where they come from. Second, by requiring
the household's maximization system to fit into a traditional econometric model,
we avail ourselves of the use of traditional econometric criteria for
identification.
Deriving the econometric structure from the household's choice problem
provides considerable unifying insight into the identification problem. Among
34
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the insights this approach allows are:
- the Rosen two step approach requires restriction assumptions about
errors and preferences;
- parameters of the hedonic price equation as well as the preference
function are subject to underidentification;
- successful estimation by maximum likelihood is evidence of identification;
- the linear hedonic price equation can be used in some cases in a single
market setting.
An especial advantage of the approach of this chapter is that it allows us to
assess identification of parameters in single and multiple markets with the
same criteria.
The chapter proceeds along the following line. In the next section, we
develop the structure of choice for households with nonlinear budget
constraints. This section is crucial because there we show precisely what we
are seeking when we solve the identification problem. In section 4.3, we
explore identification in the single market, showing how various criteria for
identification can be used. In section 4.4, we address the use of multiple
market data. Section 4.5 concludes the chapter with pessimistic arguments
about the prospect of recovering parameters of preferences in hedonic
markets.
4.2 The Structure of the Problem
In this section, we attempt' to give a clear statement of the identification
problem in hedonic markets and show briefly how others have addressed and
solved the problem. The analysis assumes the existence of a hedonic market
where buyers and sellers compete for the purchase or sale of a good with
several attributes. Assumed measurable, these attributes are denoted z. The
existence of this market implies a hedonic price equation:
P = h(z;7) " (4.1)
where p is the price of a unit of the good, z is a K-dimensional vector of
attributes of the good, and 7 is a vector of parameters which describe the
function h. This equation gives the amount households expect to pay and
firms expect to receive for units of the good characterized by the attribute
vector z. Perfect competition is assumed, i.e., buyers and sellers treat the
hedonic price function as given. They cannot influence the parameters 7 but
they can influence the price by the selection of z.
Our interest is in preferences for attributes. It is assumed households
have a well defined preference function, designated U(x,z;/J) where x is a
Hicksian bundle with a unit price and f is a vector of parameters describing
preferences. Households choose levels of the vector z and the composite
commodity x to maximize U(x,z;0) subject to the budget constraint y = x +
h(z;r), where y is household income. Equilibrium conditions for the optimal
choice of the attribute vector by the household include
*h(z;y) _ .T < - i v (a.
" ' l
35
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This condition stales that the marginal price of the &b attribute equals the
marginal rate of substitution between the iih attribute and the numeraire
good. Much of the hedonic literature presumes that solutions to the
equilibrium condition (4.2) exist in the form of direct or inverse functions for
z and focuses on the estimation of the presumed demand functions.
We now have sufficient structure to give a clear statement of the
identification problem. Given observations on household purchasesf
Can we recover the vector of parameters, /?, which describes house-
hold preferences?
The identification problem is solved when we have enough of the parameters
of U(x,z;l) to calculate the change in a household's welfare from an exogenous
change in the attribute bundle, given income.
The identification problem in hedonic markets is different from the
problem typically encountered in simultaneously estimating supply and
demand functions. The general problem of estimating supply and demand
functions arises with three types of data sets. First, one can use aggregate
market data to estimate these functions. An identification problem arises
because market price is simultaneously determined with aggregate quantity.
Second, one can estimate parameters of supply and demand using individual
data on quantities and prices for firms or households when individual actors
are price takers. There is no identification problem in this setting because
price is exogenous to the individual quantities chosen. Third, one can
estimate behavioral functions from disaggregate data when individuals are not
price takers. In this case, where/ there are monopolistic or monopsonistic
elements, the same type of identification problem found in hedonic models is
encountered.
The consequence of the identification problem is that parameters of
preference (ft'e) are confused with parameters of the hedonic price equation
(r's). This is similar to the problem in separating tastes and technology in
the household production function.2 For econometric purposes, the structures
of hedonic models and household production models are quite similar. The
most important difference between the two structures is that the -budget
constraint from the household production function must be convex, because it
results from a household minimization problem. The hedonic price equation is
not constrained to be convex by any market forces, and as we discuss in
i . r- . . ;-. :.. «v ». .1, . T ... ... r,'*:..-. .-T. ,.r.,.,.r. .Ifff :r-,,V:..,.
when it comes to measuring welfare changes.
We have defined the identification in hedonic markets to be the recovery
of the parameters of the preference function. Since the literature typically
discusses identification of the parameters of demand functions, we explore
briefly the distinction.
Recall the equilibrium conditions for an optimum as:
36
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For ease of notation, denote
The equilibrium condition requires marginal price to equal the marginal rate of
substitution between the attribute and the numeraire good. We may denote
m|(x,z;l) as the marginal rate of substitution function. It is this function
which is the so-called demand function for attributes or the "hedonic demand
function".
The marginal rate of substitution function differs from the inverse demand
function. Further, when the budget constraint is nonlinear, neither the direct
nor inverse Marshallian demand functions exist as solutions to the consumer's
maximization problem. These points are crucial because they bear on
estimation, interpretation and welfare measurement. First, consider the
difference between the mj function for the hedonic problem and the i*-n
inverse demand function from a traditional linear budget constraint problem.
In the traditional problem, the consumer chooses levels of a K-dunensional
vector x at constant prices p in order to
' «
max{U(x)lpx - y = 0).
X
Then the inverse demand functions are (where Ui »U/*xj)
Pi
- = V./l U.x. i = l.K (4.4)
y i j J J
by Wold's theorem. This problem has prices as parameters and has been
completely solved. In contrast, the marginal rate of substitution conditions
for the same consumer are
Pi/Pj = Ui/Uj i = I.*-
These are equilibrium conditions which have not been solved to eliminate the
\ . .t . , 4 , \ i - . f ' . T! i ' ' - " :"~.,
demand functions have entirely different implications for estimation and
welfare calculation.
For the individual household, the nonlinear hedonic price function creates
a nonlinear budget constraint. There are two consequences of a nonlinear
budget constraint for the utility-maximizing or cost-minimizing household.
First, Marshallian and Hicksian demand curves as traditionally conceived,
where price taking consumers choose quantities (utility or income held
constant) do not exist. These demand concepts depend entirely on the linear
budget constraint or prices-as-parameters paradigm. Second, the solution of
37
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Ihe first .order or equilibrium conditions gives quantities demanded of the
attributes as a function of the parameters of the hedonic price equation as
well as income and other exogenous variables.
Traditional demand functions! both Marshallian and Hicksian, rely on the
happy coincidence that some of the parameters of the consumer's maximization
problem are the prices of the goods. It is always correct to solve for optimal
quantities of the goods (or attributes) as functions of parameters. But only
when these parameters are also per unit prices will the traditional demand
functions, with all their well known properties, emerge. The failure of these
traditional concepts to hold when the budget constraint is nonlinear can be.
shown intuitively in two ways. First, one can attempt the mental experiment
of asking: If the price were $p, how much would be consumed? It is clear
that asking this question when the budget constraint is nonlinear requires
one to know already how much is being chosen. The absence of a .traditional
Marshallian or Hicksian demand when the budget constraint is nonlinear is
analogous to the absence of a supply curve for a monopolist. Both concepts
require that prices be parameters. When prices are not parameters, neither
function exists. Second, one can construct examples, (as in the appendix to-
this chapter), given preference and cost functions, which show that quantities
depend on parameters and not on average or marginal prices. Similarly, one
can also show that well-behaved inverse demand functions are not defined for
nonlinear budget constraints.9 (These results on Hicksian and Marshallian
demand functions are developed in more detail in Bockstael and McDonnell,
1983.) ' ....
Consider the equilibrium conditions (4.3) again. In principle, when
combined with the budget constraint, these conditions can be solved for z and
x. If the derivative on the left hand side of (4.3) were constant, i.e., the
hedonic price equation is linear, then the solution of (4.3) would be a
traditional demand function. The existence of a traditional demand function,
with prices as parameters, is assumed in most hedonic work which pursues
Rosen's second step. However, when h(z;-y) is nonlinear, the parameters and
exogenous variables on which z depends are income and the parameters of the
hedonic price equation, so that the solutions for quantities chosen are:
z = V(y,r,f>) J(4.5)
x = Dx(y,Ttf) (4.6)
Marshallian only in that they depend on parameters. But they do not depend
on prices.
The solution for z is a demand function in that it describes how choices
depend upon parameters. Because prices are not parameters, they are not
arguments in (4.5) and (4.6). When the hedonic price equation changes, the
vector 7 changes, and households respond. Expressions (4.5) and (4.6) are
reduced form equations. Estimating these equations allows one to make
predictions of z and x. But successful estimation of (4.5) and (4.6) solves the
identification problem only when the parameters of the hedonic equation and
38
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the preference function can be deduced from the reduced form parameters.
The conclusion of this section concerns the question "what are we
seeking?" when we attempt to identify demand structure in hedonic models.
The answer is that we are seeking to identify the marginal rate of
substitution functions. These functions are demand relations only in the
sense that they equal marginal price at optimum. The true demand functions
can be solved for only rarely; hence, the equilibrium conditions must be
estimated. There are at least two practical consequences of this result. For
estimation purposes, the structural equation must integrate back to a
quasi-concave function, ruling out most polynomials. Further it will in general
include the hedonic price as an argument. Second, when computing welfare
changes, one must either start with a utility function and derive the implied
marginal rate of substitution functions or start with the marginal rate of
substitution functions and derive the appropriate welfare functions. We have
also explained why we have framed the problem as one of recovering the
parameters of the preference function. These parameters are embodied in the
marginal rate of substitution functions. Traditional Marshallian and Hicksian
direct and inverse demand functions do not exist as solutions when the
hedonic equation is nonlinear.
In the following sections we discuss the identification problem for two
kinds of hedonic models. The first deals with simultaneous estimation of the
hedonic price equation and the marginal rate of substitution functions for a
single market. This arises /when both hedonic parameters and preference
parameters are estimated from the same set of transactions data.
Identification criteria are derived for . models that are linear and nonlinear in
parameters. Within the single market, we consider two special cases: the case
of the linear hedonic equation and the case when the hedonic parameters are
available from an alternative source, and only the preference parameters are
estimated. The second kind of hedonic model concerns identification from
multiple markets.
The criteria we develop are based on the econometric theory of
identification of the parameters of linear and nonlinear systems. Hence, the
identifiability of a system will be determined by the restrictions we impose,
i.e., homogeneous and nonhomogeneous parameter restrictions, across-equation
parameter restrictions, and the specification of the functional form of the
hedonic and marginal rate of substitution equations. It is shown that, unlike
the traditional problem of identifying supply and demand equations by
,... i.. .4:..... ..,.:,. t i. f . i v. .. i. ' ...-.., . . . .. . > v.i . . ,
identify hedonic models. Identification of the parameters of hedonic models
generally involves the imposition of untestable restrictions on the functional
forms of the equations of the system.
4.3 Single Market Approaches to Identification
This section presents an approach to hedonic models which views the
marginal rate of substitution functions as part of a system. Here we are
interested in hedonic analysis of observations on prices and attributes of
goods which come from sales transactions. We can then be confident that
39
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when the JLraditional hedonic story is told, prices and attributes will be jointly
dependent. This section presents an approach to identification that brings us.
closer to the question of whether it is possible to identify the parameters of
concern.
*
Let us construct the econometric system. The maximization problem, when
z is a scalar,
max {U(x,z;£)ly-h(z;r) - x =0}
has three first order conditions
Ux(x,z;/») = X
y - h(z;-y) - x = 0
and three choice variables (x,z,X). The ratio of the first to the second yields
= n(x,z;l)
where m(") is the marginal rate of substitution function. Defining p y - x
and substituting x = y - p for x wherever it appears yields two unknowns
(ZIP) in two equations
p = h(z;r)
These are the two structural equations of the consumer's optimum. For the K
attributes case, there would be K+l equations and K+l unknowns, but the
basic arguments remain. In analyzing transactions data involving prices and
attributes of goods, we should treat these variables as jointly dependent.
This characterization of the structure is significantly different from the
standard hedonic literature. Typically, both a supply function and a demand
or marginal rate of substitution function are specified, with e and *h/*z as
v^iUOfiUJjOUto. CU 101 VAUUlplti, i>. IvOteUll, li><4, Ol £l OW11 U11U H. JiOUtUl, ii*b£.J
At the margin, the hedonic model is analogous to the typical market model of
supply and demand. While it is intuitively appealing to utilize the market
model of supply price and demand price, it is misleading in the household case
with nonlinear budget constraints. In the context of the individual consumer's
choice, consumers are price schedule takers and we may safely ignore the
modelling of sellers' decisions. In this context, knowledge of z and *h/*z does
not allow the computation of utility without further information. In fact,
knowledge of «h/0z is of no particular value to the consumer. It cannot be
used to predict consumer choices or utility.
40
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Accepting lhat z and either p or x are endogenous, one naturally asks:
why not solve for z and x in terms of the exogenous variables? The answer
is that it is easier said than .. done. Solving for z and x requires severe
restrictions on the preference functions and the hedonic price equation.
Moreover, it is often not possible to solve for z and x given even the simplest
preference function and nonlinear hedonic price equations. (See the example
in the appendix to this chapter.)
Because we generally cannot solve for the endogenous variables, we are
forced to estimate the equilibrium conditions. When we are analyzing
transactions data with observations on purchase price and attributes, we can
capture the econometric spirit of the choices facing the consumer by
specifying the following system:
p = h(z;r) + *i (4.7)
i = 1,K (4,8)
where 7 is the unknown vector of parameters of the hedonic price function
and f is the unknown vector of parameters of the preference function. The
endogenous variables are p and the vector z, and the functions mj are the
marginal rate of substitution functions. Because the hedonic model is not
customarily written as in (4.7) and (4.8), there is little discussion of the
probability densities of the *'s. Specifying the error structure in hedonic
models should be an integral part of model construction. The error term in
the hedonic equation may arise from errors in measurement, unobserved or
omitted variables, and approximation errors due to lack of knowledge of the
true functional form of the hedonic equation. The error term in the marginal
rate of substitution equations may arise from the same type of misspecification
encountered with the hedonic equation, though we have the additional problem
of unobserved variation in tastes across households. The errors are econo
metrician's errors rather than stochastic elements in household* behavior. No
prior restrictions are obvious, so it makes sense to specify them as having
mean zero and constant variance.
The general hedonic model to be estimated is the system (4.7) and (4.8).
For identification, it is necessary to determine whether:
- The parameters of the hedonic price equations are identifiable;
- The parameters of the marginal rate of substitution function are
identifiable;
The focus of the identification debate has been whether parameters relating to
individual behavior (here the marginal rate of substitution function) can be
identified. It perhaps makes more sense to ask whether the hedonic structure
or the system as a whole can be identified. Several dimensions of the hedonic
model warrant attention.
- The model almost certainly will be nonlinear in variables;
- For most preference functions, the model will also be nonlinear in
parameters;
- There may be shared parameters in different equations or
cross-equation parameter constraints.
41
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4.3A. Models Linear in Parameters But Not in Variables
A source of difficulty in identifying hedonic models is nonlinearity. For
the case of models which are linear in parameters, however, identification
criteria are well established. Suppose there are M equations and endogenous
variables. When there are K attributes, then the system (4.7) and (4.8) has M
= X + 1. Assuming there are no implied equations, we let the system be
written
Aq(w) = e
where A is the M by N parameter matrix and q(w) is the N element vector of
basic endogenous variables, exogenous variables, and functions of endogenous
variables, which are labelled additional endogenous variables. Let 4j be the
matrix of prior homogeneous restrictions for the it*1 equation. With no implied
equation in the system, the necessary condition for identifiability of the i*n
equation is
rank (*i) > M - 1 (4.9)
when a parameter has been normalized. The necessary and sufficient
condition is
rank (A»i) = M - 1. (4.10)
;
The caveat that the conditions hold for equations with a normalized parameter
is critical, for the marginal rate of substitution equation will be unnormalized
of necessity. Normalization of a parameter in the marginal rate of substitution
function in effect determines the relative value of coefficients in the utility
function, and in many cases places quite restrictive assumptions on tastes.
For example, for one attribute, when preferences are given by U(x,z) = /»,ln z
+ Pain x, the marginal rate of substitution function is (la/li) z/x. A
normalization of P2/Pi - 1 determines all of tastes. No estimation is then
necessary.
When there are no normalized parameters, the necessary condition for the
identification of the i*n equation is
rank («[) > M (4.11)
rank (At;) r M. (4.12)
Criteria (4.9) and (4.10) can be used for the hedonic price equation, while
criteria (4.11) and (4.12) are suitable for the marginal rate of substitution
equation.4 Observe that by characterizing the hedonic price function and the
marginal rate of substitution functions as the structure with p and z jointly
endogenous, we uncover the possibility that the hedonic price equation as well
as the marginal rate of substitution equation will be underidentified. This
topic is explored in the following chapter.
42
-------
It is revealing to utilize these criteria in the one attribute example
discussed by Brown and Rosen (1982). This example is inconsistent with the
spirit of Section 4.3 in that it does not integrate back to a quasi-concave
utility function, nor does it contain p as an argument. However, it is useful
because of its widespread consideration in the literature. Let
h(z;r) = To + 7»Z + 7aZ2/2 + e
and
m(y,z;/J) = P0
Then q(w) = [p z 1 y z2/2JT and
(4.13)
(4.14)
A =
1 ~7i
0 72-0
-To
-7a
0
with
and
*, = [ 0 0 0 1 0 ]
1 0 0 0 0
*2 [ 0 0 0 0 1 J
where the T indicates transposition. Both conditions (4.9) and (4.10) are
satisfied for the hedonic price equation:
rank (4,) = rank (A*,) = 1.
When we apply criterion (4.11) to the unnormalized rate of substitution
equation (4.14), we see that the rank (4a) = 2, so that the necessary condition
holds. However, applying the necessary and sufficient conditions yields
rank (Afa) = rank
1
0
= 1 < M = 2
so that in fact the marginal rate of substitution equation is not identified.
This application of the formal criteria for identification leads to the same
results as Brown and H. Rosen's analysis of S. Rosen's two-step approach.
The standard linear restriction criteria developed by Fisher and extended
to systems nonlinear in variables work as long as the constraints are simply
written. However, when more complicated information becomes available, these
criteria are not applicable. Such information becomes available when the
hedonic price function is known to be more complicated. For cases, which are
still linear in parameters, the work by Wegge (1965) provides the basis for
43
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identification. Wegge's criteria are similar in .spirit to those of Fisher, but
allow for cross-equation parameter constraints and nonlinear constraints.
Consider a one attribute example. Let the hedonic price function be
P = To + Tiz + raza/2 -f r3z3/3 + «,. . (4.15)
With the same marginal rate of substitution function as in (4.14), the
equilibrium condition is
= fo + Pi*i + Pay + ea- (4.16)
Utilizing the sufficiency criterion in (4.10) for (4.15) we find that rank (A*,) =
1 so it is identified. Applying criterion (4.12) to the unnorraalized (4.16) we
see that rank (A*2) = 1> so that it is not identified. We have added
information which should help us distinguish between the two structural
equations, but the standard criteria imply that the second equation is still not
identified.
The intuitive explanation of this result comes from observing that in the
system (4.15) and (4.16) there is an exact across-equation constraint. If we
write the A matrix
A _ -7i To -Ta -73 ( .
A " [ 0 72-Pi -ri-Po 2r, -t>* 0 j {*'L7)
we see that aa« + 2a16 = 0. The identifiability of .this system can be
determined by Wegge's criterion. Strictly speaking, Wegge's results apply to
systems linear in variables. In most cases we can harmlessly convert
nonlinear systems to linear systems by substituting polynomial functions of
exogenous variables for the additional endogenous variables. Our concern is
to determine whether the two equations are observationally equivalent. Let T
be any nonsingular M by M matrix and let vec (TA) be the vector created by
taking TA one row at a time. If the two equations are observationally
equivalent, constraints on A will also hold on TA. Let .
*i(vec(A)) =0 i = l.R
be the vector of constraints, including normalizations, across equation
^orv^mnf^r »»r>r»wt r*itr>rt«rortr>r»«'<5 T-n
of such constraints. Define the matrix J as
«* (vec(TA))
J =
Then a sufficient condition for the identification of the system is that
rank (J(I)) = Ma (4.19)
44
-------
where I is the M by M identity matrix (see Wegge, Theorem II, p. 71). For
Wegge's results, the constraints need not be linear or homogeneous.
The constraints that are implicit in the A matrix in (4.17) are
! - 1
*5:aj4
Computing J(I) gives
J(D =
= 0
= 0
= 0
= 0
= 0.
(4.20}
1
0
0
0
27,
0
0
0
0
-72
-73
0
0
0
27,
0
(4.21)
Denote by J* the matrix derived by deleting the first row of J. Then we find
that
del J* = -4r33
implying that the rank of J is 4 = M2. Hence the sufficient condition holds
for this system to be identified. Note that the requirement that 7s * 0 is
quite intuitive because when 7, ~ 0, we have the model given* by (4.13) and
(4.14), which we have already shown to be underidentified. An extension of
this system to'several attributes, while maintaining the basic functional forms,
will show that the hedonic price equation will no longer be identified, a result
discussed in Chapter 5.
The conditions can be usefully applied in practice and can be easily
generalized to the setting where there are several endogenous variables. The
restrictions needed for identifying the marginal rate of substitution equations
». . -I,,.
.Ml
This leads to
be specified empirically, typically using Box-Cox techniques.
nonlinearity.
4.3B Models Nonlinear in Parameters
While nonlinear analytic functions may be approximated as closely as
desired by polynomials linear in parameters, many models are inherently
nonlinear in the parameters. Further, specifying the functions as polynomials
obscures the basic concavity or convexity which economic functions typically
possess. Polynomials cannot in general be integrated back to quasi-concave
45
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preference functions. For example, the marginal rate of substitution function
for the preference function
U = /
is
m(y-p,z;/0 = /»,
which is nonlinear in P2.
Hence, it is important to examine the conditions for identifying this class
of model. The approach used is that of Rothenberg (1971) and Bowden (1973).
In addition to providing necessary and sufficient conditions for identification
of a wide class of parametric models, their approach links the existence of
maximum likelihood parameter estimates with identification, which may have
some practical applications.
The identification conditions have been stated most generally by Bowden.
Let be the vector of parameters to be estimated. (In the hedonic context
= (7il)>) A sufficient condition for local identification is that the information
matrix have full rank when evaluated at the true parameter point (*). (See
Bowden, section 3.) The necessary condition requires that, when * is a
locally identified regular point, s the information matrix possess full rank at *.
The nonsingularity of the information matrix is more useful in practice than in
testing for identification on an a priori basis.
The nonlinearity in parameters makes the criterion difficult to apply
analytically. When the model is nonlinear in parameters, it would be most
unlikely for the first order conditions to be linear in *. Hence solving for
* typically requires numerical methods. Without explicit solutions for *, it is
not generally possible to determine analytically the rank of the information
matrix.
The requirement that a locally identified system possess a nonsingular
information matrix has limited usefulness. From the perspective of maximum
likelihood methods, the ability to obtain unique parameter estimates is
sufficient to demonstrate local identifiability. When a well formulated model
has been estimated using maximum likelihood methods, one can argue that the
identification problem has been solved. However, the dimensionality and
4.3C The Linear Hedonic Price Equation: A Special Case
Research on hedonic models uniformly dismisses the case of a linear
hedonic price equation in a single market. There are good conceptual
arguments against linearity. It implies that repackaging is possible. There is
good reason to believe that two six-foot Cadillacs don't make a twelve-foot
Cadillac. Intuitively, it means that an individual can buy unlimited quantities
of a single attribute without raising its marginal price. Practically, a linear
hedonic price function implies no variation in marginal prices. When the
46
-------
marginal price is endogenous, there is no variation in one of the endogenous
variables. However, when we recognize that p and z are jointly endogenous,
the linear hedonic price equation is no longer a hopeless case.
In the following, we show that it is possible to recover preference
parameters from a single market's data, even when h(z) is linear. The
purpose of this example is not to provide new and practical approaches.
Rather it is presented as an illustration of potential gains from characterizing
z and p as endogenous.
Consider the system (which is again inconsistent with what utility
maximization tells us about the marginal rate of substitution function but is a
useful example)
P = 7o + 7»z + *,
+ Pa? +
(4.22)
(4.23)
The parameter matrix is
A =
1 r,
o -0!
To
In this model, there is an across equations parameter constraint (a12 - aaa =
0). Hence we can use Wegge's Theorem II (equation 4.19 above). There are
four constraints
*l'-*ll - 1
a: 8-t 4
= o
= o
= o
= o.
Computing J(I) as given in (4.19) yields
i
0
o
0
02
0
0
0
1
-7o
0
0
0
7i
which has rank = 4 = M* because det(J(I)) = Pa7i * 0. Hence, we can obtain
some information about preferences even when the hedonic price equation is
linear. This information exists because consumers with different incomes and
equal prices purchase different levels of attributes. Information about
preferences can come from observing income effects as well as price effects.
47
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4.3D Mods! Where the Parameters of the Hodonic- Price Are Known
For a variety of practical reasons, the researcher may wish to estimate
the hedonic price equation and the marginal rate of substitution functions
separately. As an example, one may have a hedonic price equation estimated
from the Annual Housing Survey. This source of information would be
different from the individual transaction data and would suggest a different
econometric structure. A reasonable structure would be
ah(z;7)/«zi = *i(y,z;0) + ^i i = 1,K (4.24)
where now the endogenous variables are zjt i = 1,K and p is taken as
exogenous. In expression (4.24) the 7*8 are known numbers.
The one attribute case is illustrative. Consider the quadratic hedonic
price function - linear marginal rate of substitution function which gives the
equilibrium condition
7i + raz = Po + Piz + P2y + t
Solving for z gives the reduced form equation:
Z = n0 + n,y + ft ,
where '
7T =
O
n =
i 7 - P '
a i
We have three coefficients (P0, Pi, Pa) to recover, but only two reduced form
parameters from which to find them. Hence we cannot identify the P'a as m is
specified. Prior information can obviously be useful in identifying the I's,
even in the single equation case. For example, suppose that the marginal rate
of substitution function is given by
m(y,z) = PiZ + 0ay * *a»
i.e. Po - 0. Then the reduced form remains the same but the Pi may be
recovered from the relationships
Pi =
Pa - «i(Ta - Pi)
Of course this method of identification requires belief in the maintained
hypothesis that P0 is zero, which is not testable nor does it have any obvious
48
-------
behavioral implications. It is thus a good example of the kinds of restrictions
needed in solving the identification problem. We can generally make
assumptions analogous to P0 - 0, but we will rarely have good economic
reasons for such assumptions. However, the approach is easy to use for one
attribute. As long as -we can solve for z, estimate the reduced form
parameters, and recover estimates of ft from the reduced form parameters TT
and 7, then we can identify ft.
The heart of the matter is of course the multi-attribute case, when the 7*8
are known constants. The system is
(4.25)
If both the hedonic price equation and the utility function are strongly
separable in the attributes and the errors uncorrelated (E/jyj = 0, i * j), each
equation in the system (4.25) can be treated separately. This would be
analogous to the one attribute case, but highly unlikely.
In general, we must treat (4.25) as a system of K equations in K
endogenous variables. As in /the previous analysis, it is useful to think of
two cases. First, when (4.25) is linear in the fa, some form of least squares
may be applied. Second, when (4.25) is nonlinear in P'B, ML methods are
required. In either case, what is the role of the 7*8?
Consider first the linear-in-parameters case. In that case, the hj are
nonlinear functions of endogenous variables, and may be considered additional
endogenous variables. As long as the hj are not linearly dependent, there are
K-l exclusion restrictions for each equation. Further assuming the coefficient
on hi is known (and equals unity) only K-l restrictions are required for
identification. Consider a case where K = 2 and and mj are linear in
parameters and endogenous variables:
hi(z;7) = "01 + *ii7i + ^iaza + P^y + "i (4.26a)
at' i aa a ' asj
Given z, h, can be computed because 7 is known. Hence its coefficient is
unity. Without changing the substance of the problem, we can divide each
equation in (4.26) by lij. This yields the coefficient matrix
49
-------
A =
ft*
01
ft*
02
*?,.
where
The restriction matrix for the first equation is
= £0
1 Oj
implying that rank(*a) = rank(Ati) = 1, so that both necessary and sufficient
conditions for identifying equation (4.26a) and (4.26b) are met.
The successful application of nonlinear 2SLS in practice depends upon
how linearly independent ht, ha and z are. Thus, for example, if the hedonic
price equations were quadratic, as in the Brown and Rosen case, the marginal
prices would be linear, hj would be perfectly correlated with the right hand
side of (4.26) and 2SLS not feasible. If they are quite collinear, then while
identification holds formally, actual parameter estimates will be imprecise.
Further, nonlinearity in h(z;r) is not sufficient to guarantee that hi and hj
are linearly independent. For example, suppose that we have a Box-Cox
transformation of a linear function of z's:
/ h(z) = [«rz-l] T
For this case rjp*"1 = hj so that
M =
That is, the hj are not linearly independent of each other, regardless of the
value of , which determines the nonlinearity of h. No restrictions would be
provided by this functional form.
. The hj also play a role in identifying the ft in non-linear systems, though
the role is less straightforward because ML methods are needed. To get some
insight into ML models, suppose that the PJ are distributed as independent
normals with mean zero and variance 0|2. Since our observations concern the
vector z, we must transform from A» to z. The log-likelihood for the
where
c -- In J(t) -
J(t) = det
h D
KK KK
-------
Note that hjj - m^j = - 7 (h.(z;r) - m.(z,y;0)") will depend upon ft as long as
J
the mj are not separable in the z. Hence the derivatives of the likelihood
function with respect to ft will depend on the h(j functions. The precise way
in which hj influences the log-likelihood can only be determined on a case-
by-case basis. But the essence of the matter is the choice of endogenous
variables. It can be shown that by designating Zj as endogenous and hj(z;r)
as nonbasic endogenous, the Jacobian of transformation has the effect of
moving the estimates of P'a away from those that minimize the squared error
(hi-mj)2. Thus, while the practical effects of the ML criterion, nonsingularity
of the information matrix, are not great, framing the problem as ML
demonstrates the role of the hj. The choice of endogenous variable influences
the parameter estimates. The endogenous variables which accord most with
consumer choice are the tj.
The situation where the 7' a are estimated with error is the case
considered by Epple (1982) and by Bartik (1983). In that case, we consider
the realistic situation where the hedonic price equation is misspecified by
omitting attributes of the good. By the solutions (4.5) and (4.6) we know .that
any attribute is a function of income, and hence . correlated with income.
Thus, for example, omitting the attribute view from a sufficiently nonlinear
hedonic price equation will cause error in the marginal price to be correlated
with the view, and hence with income. (In this case, income can stand for a
whole vector of socioeconomic characteristics without changing the argument.)
Thus misspecification of the hedonic price equation will make errors (pj)
correlated with income (y) and seriously undermine any attempt to recover the
P'a.
4.4 Multiple Markets
Several researchers (Diamond arid Smith (1983), Parsons (1985), Palmquist
(1984)) have concluded that the use of multiple market data holds the most
promise for recovering preference parameters for hedonic models. Multiple
markets might -exist in housing, for example, in different cities or perhaps in
different, areas of the same city. One might question this approach
immediately on the grounds that it requires preferences to be identical across
hedonic markets. Accepting equality of preferences for the sake of argument,
we investigate the conditions under which multiple market data will help solve
the identification problem.
hedoiuc models are estimated from other sources. Assume that we have G.
markets, and from each market we have a vector of hedonic parameters, 7%, g
- 1,...,G which we treat as known constants. We have two separate cases,
depending on the functional form of the hedonic price equation. First we
consider the case where all hedonic price equations are nonlinear.
4.4A Multiple Markets: Nonlinear Hedonic Prices
When h(z;-y£) is nonlinear, we cannot solve for the z's and are forced to
work with the equilibrium conditions. Suppose, as before, that there are K
51
-------
characteristics. Then the equilibrium conditions for the gt-h market for a
household with income yJ£, attribute vector z3S and hedonic price p3S are:
(4.27)
where, for convenience, we assume that ft ~ N(0,r). This model has K
endogenous variables (z) and K structural equations, given in (4.27). Unless
we make very restrictive assumptions about the utility function, the mj
functions will not be linear in parameters. Hence in general we can only
establish the id entif lability of the ft through maximum likelihood estimation.
To get more insight into the multiple market setting, ignore temporarily
the right hand side of (4.27). What is the role of the hi(z
-------
4.4B Multiple Markets: Linear Hodonic Prices ~
The utility of multiple markets is greatly enhanced by linearity in the
hedonic price equation. The estimation problem in (4.27) can be transformed
to a standard demand system when h(zj7) is additive and linear in z. In that
case the marginal prices are
h.(zj*,7*) = 7?
and the system becomes (ignoring the
r? =
7* =
K
We can then solve for zJ& as in (4.5) and (4.6):
^.^ (4 2g)
where now the r's play the role of prices in linear budget constraints. . If
there are enough markets, then the variation in ?&, being exogenous to the
individual household's behavior', will allow the estimation of a demand system.
The best example of this approach is. provided by Parsons (1985) who
estimates the almost ideal demand system for attributes using multiple city
data. As in other situations, we can make tradeoffs between price information
and the complexity of the model we estimate. For example, if we make the
preference function additive, we need variation in only one of the 7j*.
Further variation in relative prices can be gained from the requirement that
equation (4.28) be homogeneous of degree zero in yJ& and 7*.
Thus we see that multiple market data definitely aids in identifying
parameters of the preference functions. It can do BO only by maintaining a
specific hypothesis about the preference structure - that it not include the
hedonic price as an argument which in turn allows the testing .of the
iicccssary result that the hedonic price equation be linear.
4.5 Conclusion
This chapter has addressed the general problem of the identification of
the parameters of hedonic models. Three basic questions were addressed:
1. When we estimate a hedonic system, what are we seeking? It was
shown that the so-called hedonic demand function is really a marginal rate of
substitution function embodying the parameters of the preference function.
As long as the hedonic price equation is nonlinear, traditional direct or
inverse Marshallian and Hicksian demand functions do not exist as solutions to
the consumer's choice problem. Estimation of a hedonic system is therefore an
attempt to recover the consumer's preference parameters.
53
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2. Under what circumstances is it possible to identify the preference
parameters? Necessary and sufficient conditions were derived for the
identification of the parameters of recursive and nonrecursive single-market
hedonic models and multiple market models. Models linear in parameters and
models nonlinear were investigated. As with all econometric identification
problems, identification is dependent on prior restrictions imposed on the
parameters and functional form of the equations in the model. Unlike the
traditional problem of identifying supply and demand functions by exclusion of
variables, very few theory-based restrictions are available for hedonic models.
Identification instead requires the imposition of generally untestable
restrictions on .the functional form of the hedonic and marginal rate of
substitution equations. These restrictions often place unknown or unrealistic
limitations on the underlying preference or market structure. Our results on
identification my be summarized by the following:
i) Identification must be determined by prior considerations. In
particular, there are no circumstances where one can apply the Rosen
two-step approach without imposing prior constraints and be assured of
identification.'
ii) Successful estimation of a hedonic system by maximum likelihood
techniques is sufficient to demonstrate the existence of an identified model*
iii) When the parameters of the hedonic price equation are known (available
from another source) it may be possible to solve for the attributes' reduced
form equation. The system will then be identified if it is possible to derive
the preference parameters from the reduced form estimates.
iv) The use of data from multiple markets definitely aids in the
identification of the preference parameters, though it is still necessary to
impose severe restrictions on the underlying preference structure.
. v) The conditions for identification just discussed are technical, relating
to the application of traditional criteria to the rather special case presented
by hedonic markets. But the fundamental question of identification relates to
behavior: What kind of behavior must we assume to achieve identification and
are we likely to find such behavior in the real world? The answers to this
compound question are not very satisfying, mostly due to the nature of the
hedonic price equation. This equation, which is structural to the household,
reflects the combined influence of buyers, sellers and the distribution of
goods. Restrictions on the functional form of the hedonic price equation may
help satisfy the technical criteria, but restrictions cannot be translated into
information nhout t.h« b«b«vior of buyers and sellers. As we show in Chapter
b, ciiaracterutliutt ol Ouyert. and Kellers are iiKeiy to uc uidbKeu AU iiio iieuomc
equation. Of course, we also need restrictions on the marginal rate of'
substitution functions. The restrictions which are most likely to be useful are
separability restrictions on the utility function. For example, the elementary
rule of having the number of excluded exogenous variables exceed the number
of included endogenous variables is helped by separability, because it means
fewer endogenous variables in each marginal rate of substitution equation.
There are few tests of separability in the hedonic setting, but it seems a safe
bet that real world behavior does not support much separability. In sum, we
can describe behavior needed to support identification, but we cannot find
strong arguments to support the common practice of such behavior.
54
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-------
3. Is. the solution to the identification problem worth the restrictions we
must impose? The cost of identification come in the form of maintaining very
specific and restrictive hypotheses about preferences and the hedonic price
equation. The restrictions required for identification in the hedonic model are
especially disturbing because they involve functional form rather than the
exclusion of exogenous variables. Thus they lack the intuitive appeal of the
more traditional approaches to identification. For example, in supply and
demand models of agricultural commodities, we can identify demand by
excluding rainfall from the demand function. No such appealing restrictions
appear to be available in hedonic models. The benefits of recovering the
parameters depend on how they will be used and whether in fact the hedonic
model is suitable for valuing environmental amenities. In succeeding chapters,
we show that there is a number of serious problems in using the hedonic
model for measuring welfare effects, even when all parameters are known
perfectly. We will thus postpone until the concluding chapter a full response
to the questions of whether the solution is worth the cost.
55
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CHAPTER 4
FOOTNOTES
McConnell is with the Department of Agricultural and Resource Economics,
University of Maryland, and Phipps is with Resources for the Future.
And thus the exchange between Pollak and Wachter and Barnett is
especially relevant.
Properties of inverse demand functions are derived from the problem
where
min{V(p,y)lpx - y = 0} (i)
V(p,y) = inax{U(x)lpx - y = 0}
and where x and p are the vectors of goods and prices respectively and
V(») and U(-) are respectively the indirect and direct utility function*.
Suppose the nonlinear budget constraint is h(x,r) - y = 0, where 7 is a
vector of parameters. Then the indirect utility function becomes
V(r,y) = «ax{U(x)ih(x;7) - y = 0} .
'But there is no well-defined dual such as (i) which yields the inverse
demand functions in this case.
For the motivation of these criteria, see Fisher (1966), Chapter 6, and
Gold f eld and Quandt (1972) Chapter 8. They are analogous to the
conditions for linear-in-variables systems when the additional endogenous
variables play the role of exogenous variables.
The regularity assumption requires that the information matrix be of
constant rank in an open neighborhood of <>
56
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APPENDIX TO CHAPTER 4
1. A Cob b -Douglas Example
Consider the following example. Let the attribute vector be one-
dimensional. For simplicity, let preferences be given by the Stone-Geary
function:
U = |»tln z + 0aln(x - PJ (4.A.1)
and suppose that the hedonic price equation is given by
h(z;y) = r0z71
so that the budget constraint is given by
y = 70zri + x.
i
The goal is to solve for the choice variables z and x. The equilibrium
conditions are
h(z) = rozri
V,«(7l~° = )T A'2)
+ P*7^' (4.A.3)
These are demand functions in the sense that only exogenous variables are on
the right hand side. But they are not traditional because neither marginal
nor average price appears on the right hand side. The demand function
collapses to the traditional Marshallian demand function when 7\ - I, implying
that the hedonic price function is linear. This case is a linear expenditure
system demand, function because of the form of the preference function in
57
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(4.A.1): .
2 = Pt(y - 03)/>0 (4.A.4)
where it is assumed without loss of generality that pt + pa ~ I. This is of
course just the demand function in the linear expenditure system with a zero
level of the subsistence parameter for z. The expressions for z and x in
(4.A.2) and (4.A.3) can be written:
TT
z = n (y - TT ) (4. A.5)
x = noi + njay (4.A.6)
when
"It = P, (4.A.8)
» = l/7l (4.A.9)
= P P /(P + P 7 ) (4. A. 10)
02 13 1 2 1 ' ^
Although there are five reduced form parameters and five structural
parameters, we cannot solve for the J'B and P'a without more prior
information. Note that "os/"!! + n,a = 0 for all values of Ptt P2 and 7,, and
hence there is a redundancy. However, by imposing the prior constraint 0, +
/>, = 1 we can solve for the f's and 7*8 and hence solve the identification
problem.
In the case where the 7*8 are known with certainty (section 4.6), we have
reduced form equations for attributes only. Then we estimate (4.A.5), imposing
the prior constraint (4.A.9). This leaves two reduced form parameters, w0i
and "tt and two structural parameters (assuming Pi + P3 - 1): Pi and 0S.
Since (4.A.8) tells us where to get P*, we need only solve (4.A.7) for Pt.
Imposing Pi + Pa - 1 and solving (4.A.7) for Pi yields
P =-T/(l+7 -7"
ri i' i 001
58
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For this .purticular hedonic equation, knowing the y*s simply reduces the
estimating problem, with no fundamental change in the identif lability of the
f's.
This one attribute example shows the difficulty of solving for the
demands. Irish (1980) has developed cases which can be solved, but the
necessary simplifications show the difficulties involved.
2. A CES example
A separate example illustrates the contention that simply assuming a
utility function and applying the Rosen two-step approach is no guarantee of
identification.
Suppose the hedonic price equation is as before but that the preference
function is given by the GOBS:
U(z,x) = P^z* +
The equilibrium condition is
r.T1«(7t~1) = (P, VV«>«('*"1> (y ~ *>(1~*4)- (4.A.12)
;
We can use the Rosen two step on this expression (in logarithms, as in
Quigley, 1982) with errors added on. The model to be estimated from the
logarithm of (4.A.12) is
h. = S + « In z + 6 ln(y - p) + error (4.A.13)
101 2
when hi = ln(r«>7») + (r» - 1) In z
are parameters to be estimated. An application of OLS to (4. A. 13) yields
= 0.
59
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We simply reproduce the parameters of the hedonic price equation as in the
examples given by Brown and Rosen. Hence even though we recognize that we
should be estimating the marginal rate of substitution conditions, we still have
ample room to create a constructed marginal price problem.
The subtle nature of the constructed marginal price problem can be
illustrated if we impose the restriction that preferences are homothetic, BO
that the utility function becomes the CES. The logarithm of the equilibrium
condition becomes, on imposing ft, - /?4 = ft,
h = 6 + 6 ln(z/(y - P)) + error (4.A.14)
i o i
where <50 = ln(/?i//?3) and «t = (f - 1). Applying OLS to (4.A.14) does not
imply that the estimates of repeat the parameters of the hedonic price
equation even when the power function for h(z;r) is used. While this example
is perhaps too simple to consider for applications of the hedonic method, it
illustrates the difficulties of hypothesis testing in this approach. For the
GOES preference function, given the power function for the hedonic equation,
the constructed marginal price problem makes the structural estimation
meaningless. But when the CES preference function is imposed there is no
longer a constructed marginal price problem.
60
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CHAPTER 5
THE STBUCTURE OF PREFERENCES AND ESTIMATION
OF THE HEDONIC PRICE EQUATION
K. E. McCdnnell and T. T. Phipps1
5.1 Introduction
In the previous chapter, we tried to determine the circumstances under
which it is possible to identify the parameters of preference functions. In
Chapter 2, we surveyed the practical problems encountered in using ordinary
least squares on the., hedonic price equation. The joint problems of
multicollinearity, errors in specification and functional form plague the single
equation estimates of hedonic price equations in housing markets (see Bartik
and Smith, 1984 and Palmquist, 1983 for additional details). The issues which
have arisen in estimating the hedonic price equation are primarily of a
measurement nature, having little to do with simultaneity.
In the chapter 4, we developed the nature of the choice problem for the
household. We argued that in an econometric structure which models the
choice of the attributes and the, price of the commodity, it makes sense to
designate these same variables as endogenous. Then the hedonic equation is a
part of the structural equation: the household's nonlinear budget. If the
hedonic equation is in fact structural to the household, then it must be
subject to possible under-identification. In this chapter, we follow the logic
of Chapter 4 to investigate the circumstances under which the hedonic
equation will be identified. These circumstances relate to the structure of
preferences.
In this chapter, we will first show that the hedonic price equation may
reasonably be considered part of the simultaneous system, then derive the
circumstances when the hedonic equation can be consistently estimated with
OLS, and finally, develop some Monte Carlo results showing the effects of
simultaneity on OLS estimates of the parameters of the hedonic price equation.
~ . '. '»' . ' -'> I'i-.'I .<-- -i> ii|;i,'ir.,-ci '.. J ali-
en prices and attributes collected from market transactions. Hedonic price
equations fitted on housing prices which are household's own estimates will
obviously not be subject to any simultaneous equation issues because such
estimates will not have been jointly determined with the purchase of attribute
levels.
This chapter has two rather different purposes. First, it is designed to
explore simultaneity in hedonic markets by developing the logical consequences
of this simultaneity for the hedonic price equation. This chapter is not
designed to critique the practice of estimating hedonic price equations. It
61
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would be foolhardy to. assert that, in the midst of such pressing data
problems and with so many attributes, one should worry about identification.
Rather, we are trying to learn about choice in hedonic markets. The second
purpose is more practical. Multicollinearity is a serious problem in hedonic
models. But in nonlinear systems, the distinction between multicollinearity .and
under-identificatioh is blurred. We argue .that what is apparently
multicollinearity may be endemic to the system precisely because of
underidentification. In that case, the cure for multicollinearity of enlarging
the sample size may simply cause parameter estimates to converge on the
wrong values.
One conclusion of this chapter relates to the requirements for successful
estimation of the parameters of the hedonic price equation. For analyses
using market transactions, it will be shown that consistent estimates of these
parameters require the assumption of .restrictions on the form of the utility
function. These restrictions will, in general, be untestable. This conclusion is
quite similar in spirit to the received literature on identification of the
parameters of preferences. In concluding their paper on identifying
parameters relating to preferences, Diamond and Smith (1985) note
Consistent estimation of the structural parameters of
demand requires sufficient restrictions to identify
functions. The minimum requirements can be met through
the assumption of a utility function and hedonic function
which imply the presence in the marginal price function of
appropriate nonlinear transformations of the endogenous
variables in the demand function. However, this approach
relies heavily on the choice of utility function, while
providing no independent statistical means to test that
choice (p. 281).
We will argue that consistent estimation of the hedonic price equation by
ordinary least squares with market transactions also requires making
assumptions about the functional form of the hedonic price equation and the
preference function.
5.2 The Structure of Preferences and the Equilibrium Conditions
In Chapter 4, the following choice problem was described for the
household (section 4.3)
jax[U(x,z;0) j y-h(z;r) - x = 0].
When there are K attributes, this problem has K+2 first order conditions:
Ux = X (5.1)
Ui = X*h/*zi i = 1,K (5.2)
y - h(z;r) - x = 0 (5.3)
62
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where X is the multiplier on the income constraint and Uj « aU/izj. These
first order conditions yield solutions for the K+2 variables (x,X,z) of the
optimization problem. By the substitutions of 4.3, this system can be reduced
to K+l equations in the K+l variables natural to the consumer (p,z):
p = h(z;r)
where, as before mi(y - p,z;l) * iV(xiz\P)/3zi/il3(x,ztP)/3x evaluated at x -
l'-.p. We give this system an additive error structure which we consider to be
econometrician's error and which captures the spirit of empirical efforts in
hedonic modelling. Tien our system is
p = h(z;7) + «i (5.4)
) + eai i = 1,K. (5.5)
The purpose of deriving (5.4) and (5.5) is to make clear the origin of the
system. It is a structural representation of the household's optimization for a
nonlinear budget constraint. In general, hedonic models are concerned with
recovering the parameter vectors 7 and ft. The discussion of the identification
problem has focused on the difficulties of estimating the ft' a and how they can
be confused with the y'a, as for example, Brown and Rosen (1982) have shown.
However, we can also see that, it is possible in principal to confuse the 7*»
with the ft's. Our focus here will be on the problems of recovering the
parameters of the hedonic price equation. Specifically, how do the values of P
influence the identifiability of the 7*8?
5.3 Estimation of the Hedonic Price Equation
In this section we ask under what conditions we can estimate the
parameters 7 using single equation methods. While there have been numerous
efforts to use Box-Cox techniques (for example, Halvorsen and Pollakowski,
1981), we will assume linear-in-parameters models. Nonlinearity would
complicate the form but not alter the substance of the argument.
Let equation (5.4) be written as linear-in-parameters:
p = h(z)7 + «»
where h(z) is a vector of functions of the z's and h(z) and 7 are conforming
vectors of dimension J, where J is less than the number of observations. The
OLS estimates of 7 are
7 = T + (hThr'hT*!. (5.6)
Note that h, being a function of z's, depends on c2. Further, if p is in m,
tlien the z's depend on et. Hence, h is a random function of *2 ano< possibly
c,. The randomness of h and the nonlinearity of random terms in (hTh)~lhTe1
make it difficult to give general statements about the bias in 7. But we know
63
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that the consistency of 7 requires that
plin hT*, = 0. (5.7)
For expression (5.7) to hold, we must have the vector of z's uncorrelated in
the limit or distributed independently of *t. Since expression (5.5) can be
solved for zj, i = l(K in principle, we could have z as a function of p and ea,
or substituting for p, have z depending on *i and *3. Thus we see in
general that (5.7) will not hold, so we need to look closer at what assumptions
will make it hold.
Suppose first that m is independent of p. Then z, and hence h, are
functions of t2 only. Blips in «a will influence h, but h will move
systematically with «» only when *, and ea are correlated. Hence, correlation
between = «>i(x,z;0)
u. u - u.u
ix x i xx
:
u
x
64
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This condition must hold for i = 1,K for consistency. The restriction that the
numerator of (5.8) be zero is imposed on the preference function. It can be
satisfied by the restriction Ui/Ux = Ujx/Uxx. Or it holds when U is linear in
x. We can obtain it, for example, from the preference function U(x,z) = x +
U(z) where U is any quasi-concave function of z. The assumption of mj
independent of p imposes restrictions on the preference function, restrictions
which are as untestable as those needed to identify the marginal rate of
substitution function through nonlinearity in the hedonic price equation.
The practical significance of the use of y rather than y-p may be
tempered by the magnitude of y relative to p and by measurement errors in
y. The relationship between income as measured in most aurvey work and
income which constrains the household's budget must surely be prone to large
errors. One cause of the difference, for example, would be real wealth
holdings, which usually do not show up in current income figures. This would
be especially important in home purchases. When coupled with large y
relative to p, it seems intuitively plausible that such large errors would mask
the omission of p from the argument y-p.
There is less reason to be reluctant to assume that ct and «|a are
uncorrelated. At least we have no reason to argue for correlation in one
direction or another. But there is a strong tradition in demand systems
analysis for correlation of errors across equations. Depending on the data
source, one might argue for or against this correlation. Hence, it is the
structure of the preference function which is the strongest requirement in
obtaining consistent estimates of 7.
5.4 Some Monte Carlo Results on the Identifiability of the Hedonic Price
Equation
To some extent, the question of whether the hedonic price equation is
identified is an empirical one. That is, for some structures, the single
equation estimates may be good enough. To test, the degree to which OLS
estimates miss the true value of hedonic parameters, we have done some simple
Monte Carlo estimations for a model which we a priori know to be not
identified. The model contains two attributes. The preference, part of
structure of the model is consistent with a linear approximation of the bid
function. The hedonic equation is given by
aa"a'
and the equilibrium conditions are given by
First we demonstrate using traditional criteria that the first equation is
not identified. Let us write the system as in section 4.3 above:
65
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Aq(w) =
where q(w) = (p.
and
A =
92
-*,
-'.I
-',2
->2,
7 -P
'22*22
22
0
0
0
-P
91
Let *i be the matrix of restrictions on the parameters of the first equation.
In section 4.3 (equations 4.9 - 4.12), it is argued that the necessary condition
for identifying any parameter of the hedonic model is
rank (*i) * M - lq '
where M is the number of equations and basic endogenous variables in the
model, and kj = 0 (or 1) is the number of normalized parameters in the i*-*1
equation.
/
In the case of the model above, M = 3 and kj = 1, so the necessary
condition for identification of the hedonic price equation is
rank (+,) » 2.
The only restriction placed on the equation is that y is excluded. Hence,
*! = [ 0 0 0 0 0 1]T
rank (*a) = 1 < 2.
Thus the necessary conditions for identifying the hedonic price equation are
not met, and the equation is not identified according to the traditional
criterion. Applying OLS to (5.9) will result in biased estimates of r*s.
Further, as the sample size increases, the OLS estimates will not get closer to
" -.'.,..
To demonstrate further with this example, we show the results of OLS
applied to equation 5.9 using Monte Carlo methods. We have performed two
different sampling experiments with the basic structure as given in (5.9) -
(5.12). The experiments have the same distribution of the income variables
and one of two possible distributions of errors. The income variable is drawn
from a uniform distribution between 40 and 90. The errors are normally
distributed errors with mean 'zero and diagonal covariance matrix:
66
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I.
0
0
2
(5.12)
or the nonrtiagonal covariance matrix:
E
f 4 2 1
= 2 3-1
I 1 -1 2
(5.13)
The experiments use the coefficient matrix:
p
1
.075
.008
2t
.9
-.8
-.5-
Za
3.5
.006
-.5
1
15
4
6.5
wZ
.5
0
0
y
0
.075
.008
The experiments consist of estimating the model 20 times for the 50
observations and 20 times for the 500 observations. Various measures of the
performance of estimators are given.
From Tables 5.1 and 5.2 we can get some feel (though not proof) of the
properties of the estimates of 7. Consider first the diagonal error covariance
case (Table 5.1). The relative bias of the f0 is small and gets smaller as the
sample size increases. The relative bias of yl is also small but shows only a
barely perceptible change with the increase in sample size. The bias in f,2
grows with sample size. The bias in fa a is uncomfortably large, but decreases
marginally with the increase in sample size. When we consider the
nondiagonal error covariance case (Table 5.2), we find that the 7,, and fa2
have bigger relative biases with higher sample size. For f12 the bias
improves, though the relative error is eight percent. In both cases the
relative bias of 722 appears substantial.
Of course, these results simply confirm what theory tells us, but they do
also add some concreteness to theory. The basic result is that we cannot be
absolutely confident that when we regress the transactions prices on the
attributes of the good that we will recover the parameters of the hedonic
67
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TABLE 5.1
Monte Carlo Results for Hedonic Parameters
Diagonal Covariauce Matrix
5.12
A
To
;
7»
722
Parameters
Ta= 50
T = 500
T = 50
T = 500
T = 50
T = 500
T = 50
T = 500
Expected
Value
15.448
15.298
.8397
.8392
3.833
3.855
.0892
.1142
Variance
1.705
.2352
.0264
.0033
.6302
.0663
.1640
.0156
Bias
.448
.298
-.0603
-.0608
.333
.355
-.4108
-.3858
Relative
Bias
.0299
.0199
-.0670
-.0675
.0951
.1014
-.8216
-.7716
Mean
Squared
Error
1.906
.3240
.0300
.0070
.7411
.1923
.3327
.1644
a T = sample size.
TABLE 5.2
Monte Carlo Results for Hedonic Parameters
Diagonal Covariance Matrix
5.13
Parameters
7o
7i
72
722
ff
T
T
T
T
T
T
T
'= 50
= 500
= 6U
= 500
= 50
= 500
= 50
= 500
Expected
Value
15.194
15.15
.8377
.8252
3.905
3.814
.1784
.1202
Variance
2.511
.695
.8423
.0020
.0686
.0683
.1268
.0176
Bias
.194
.150
-.0623
-.0748
.405
.314
-.3216
-.3798
Relative
Bias
.0129
.01
-.0692
-.0831
.1157
.0897
-.6432
-.7596
Mean
Squared
Error
2.549
.717
.8462
.0076
.2326
.1669
.2302
.1618
a T = sample size.
68
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5.5 Conclusion .
In the estimation of hedonic models from data on market transactions, the
hedonic price equation and the' marginal rate of substitution function form a
simultaneous system. This chapter has undertaken to investigate the
relationship between the structure of the marginal rate of substitution
function and the consistency of OLS estimators of the hedonic price equation.
Specifically we have shown that for consistent OLS estimators of the hedonic
price parameters, the Hicksian bundle or income must not influence marginal
values of attributes. This is a strong but generally untestable assumption
which is not likely to hold in general.
Because most plausible preferences will violate the structure of
recursivity, it may be that the parameters of the hedonic price equation are
not identified. To test the nature of OLS estimates we performed some Monte
Carlo experiments on several linear-in-parameters hedonic models. Our results
showed that in some cases OLS estimators do not tend to get close to true
values as the sample size grows.
Our results may provide some insight into the multicollinearity problem in
the hedonic equations. Lack of identification shows up as perfect collinearity
in linear and nonlinear two-stage least squares estimation. Further, as Wegge
and Feldman (1983) have stated so succinctly, identification in nonlinear
systems may sometimes be a matter of data and not structure:
Instead of viewing the problem in a discontinuous fashion,
one should perceive that the interface between identifi-
ability, estimation, and prediction is a continuous
relationship. Long before we reach the point of a
discontinuous jump in the rank and its concomitant
requirement of more prior information, we would be in a
near singular moment matrix situation when the distinctions
between some parameters become very confused, indicating
that the parameter is close to not being identifiable (p. 253).
This description of the problem is quite apt for the hedonic price equation.
Attributes which provide utility will tend to increase together with income and
other socioeconomic measures. In this view, multicollinearity is simply a
symptom of underidenfrffication and may not be resolved as sample size
increases.
69
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CHAPTER 5
FOOTNOTES
McConnell is with the Department of Agricultural and Resource Economics,
University of Maryland, and Phipps is with Resources for the Future.
Note that this derivative, and not the more complicated version imposing
the first order conditions, is appropriate here.
70
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CHAPTER 6
THE FORMATION AND USE OF THE HEDONIC PRICE EQUATION:
A SIMULATION APPROACH
K. E. McConnell and T. T. Phipps1
6.1 Introduction
The purpose of this chapter is to look behind the veil of the hedonic
price equation and into the workings of the market. To do so, we create a
simulated market in which consumers choose housing locations, choosing
attributes only implicitly because they are tied to locations. The market
simulation allows us to explore two important issues in hedonic analysis: 1) the
empirical connection between the parameters of the preference function and
the hedonic price equation; and 2) the accuracy of four commonly used
"restricted partial equilibrium" welfare measures (Bartik and Smith's phrase)
in comparison to a true measure of welfare, given market adjustment. These
two issues are closely related. Their resolution requires knowledge of the
workings of the housing market: specifically, what is the nature of the
equilibrium process which allocates households to sites? Further, welfare
measurement directly or indirectly makes use of the hedonic price equation so
the way this equation is estimated strongly influences welfare calculations.
The simulated market provides a good with three attributes. The supply
of the good is fixed. For simplicity, the number of units of the good equals
the number of buyers. The fixed supply is allocated to households as in a
bid rent or utility maximization model. From this model, a price for each unit
of the good is established. The price varies with the exogenously given
attributes of the good, and hence is a hedonic price. In section 6.2a we
describe the equilibrium of location choices. In section 6.2b we try to
determine the effects of parameters of the preference function and different
income distributions on the estimates of parameters of the hedonic price
function. In section 6.3 we use the model to calculate partial and general
equilibrium welfare effects of exogenous changes in the attributes of the fixed
ei tT"ivOT r*f rrr*r^<4&
6.2 Preferences. Income Distribution and the Functional Form of the Hedonic
Price Equation
A component of current research in the implicit markets literature is that
the structure of preferences is embodied in the hedonic price equation. One
implication of this argument is that prior restrictions on the form of the
hadonic equation may be derived from preference theory.
71
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-------
Rosen .(1974) developed the theory in which the hedonie price function is
generated by the competitive behavior of suppliers and denianders of goods
containing a bundle of attributes z = (z,,...,zn). He argued that the hedonie
function, p(z),
can sometimes be obtained if sufficient structure is imposed on
the problem. However, it is not always possible to proceed in that
manner. In general, the differential equation defining p(z) is
nonlinear and it may not be possible to find closed solutions.
Moreover! a great deal of structure must be imposed. For
example, the distribution of income follows no simple law
throughout its range, making it difficult to specify the problem
completely. Finally, partial differential equations must be solved
when there is more than one characteristic (p. 48).
For these reasons, he recommended using the well known ' two step
estimation approach in which the hedonie function is estimated first and then
the calculated marginal prices are used to estimate what he calls the "marginal
demand and supply functions."
i
Quigley (1982) used a simple fixed supply housing market example to
demonstrate that the hedonie function may be derived by integrating the
marginal rate of substitution function for a single hedonie attribute and a
single Hicksian good. In his conceptual example, he assumed Cobb-Douglaa
preferences and -the existence of a monotonic mapping from consumer income to
the housing attribute.2
While Rosen and Quigley have demonstrated that the imposition of
sufficient structure on preferences and income distribution (and supplier
characteristics in the case of endogenous supply) in principle allows
calculation of the hedonie price function, the empirical relationship between
the structure of preferences and the form of the hedonie price function has
not been explored. In this chapter, a simulation of an open city housing
market, with given preference structure and a fixed supply of housing
attributes, is used to examine this relationship. Two different utility functions
(Stone-Geary and translog), and four different income distributions (uniform,
segmented uniform, Pareto and normal) are used in the simulations. Box-Cox
flexible forms are used in estimating the hedonie functions in each case. We
find no clear empirical relationship between consumer preference parameters
and the structure of the hedonie equation. Quite different mathematical
. »,,.. ~~~ ,..:_.,».J *~,. .1.. 1 1-. ,f. *,,. .1;.,, ,v« , ;«. ,. ,».,t,."vi';,;J. ?«f
income is varied, even with preferences and supply held constant. One
implication of the chapter is that when the researcher is merely interested in
estimating the hedonie function, use of a best fit approach, such as a Box-Cox
flexible form, without taking account of consumer preferences, will, in the rare
worst case, lead to a reduction in the efficiency of estimation. This case
occurs only when we know the exact form of the hedonie price equation.
72
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6.2a. The Allocation Model
In our model, consumers choose between locating within the city or on the
periphery. The periphery is assumed to be composed of an undifferentiated
agricultural plane. If all consumers have identical preferences and income,
the existence of the "agricultural bundle," denoted ZA, available in unlimited
quantities at a fixed price, pA, sets an exogenous utility level, UA, that may be
used to solve for the equilibrium housing price structure. When individual
incomes differ, U^ is still exogenous to households, but varies among
households according to income. The bidding process will ensure that house
prices within the city adjust such that all consumers achieve their exogenous
utility levels set by ZA, pA. This model is thus an open city model in the
sense that household well being is fixed by exogenous factors. We have used
this model because it makes the determination of equilibrium relatively simple.
Note that this model requires only open competition for sites among buyers
and sellers, with the potential for migration, to ensure equilibrium. No
migration need occur. The real alternative is not migration but commuting.
The equilibrium in " this model is determined by the adjustment of
households. Households move among sites with exogenously given attributes
until the households with highest incomes occupy the best sites, where best is
determined by a separable component of the preference function. Having the
preference function separable in z means that rankings among different
bundles of z are not affected by other arguments of the utility function, in
our case x or y - p. Thus if /U(x0,zt) > U(x0,z3) then U(x,Zi) > U(x,z2) for
any x. Then the allocation can precede the determination of the hedonic
price. Any ranking of sites based on the attributes will depend on the
preference function. Different preference functions may give different
rankings. Once household equilibrium is reached, the hedonic price is
determined as if a monopolist owned the site. The hedonic price is bid up
until each household, i, is just as well off as it would be with the agricultural
bundle:
U(y. - P., z ) = U(y - PA, zA) (6.1)
^ J «/ *
where pj is the hedonic price for bundle j. Expression (6.1) is the essence of
the bid rent model.
The utility maximization model yields the marginal conditions which derive
^»»/>TT> t Vt r> T"» t-r» V»lr*»r*
max{U(x,z)ly = h(z) + x}
x.z
or
max U(y-h(z),z)
z
73
-------
which of course yield the necessary conditions
bi = Ui/Ux (6*2)
where hi * *h/" .* . ' ' -"... ;. ^;
hedonic rent function" (van Lierop, 1982, p. 281). In practice, the equilibrium
hedonic price function would be solved numerically, and where there is no
separability, iterative methods will be needed.
One noteworthy conclusion emerges concerning the open city model.
Polinsky and Shavell (1976) have shown for a model with homogeneous
households, "in a small open city the rent at any location depends on the
level of amenities at that location" (p. 123). When incomes vary, this
conclusion no longer holds. A change in the amenities at one site which
changes the relative rankings of sites can cause a change in the whole
74
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hedonic gradient. This change occurs because even when the household's
utility level is pegged by exogenous factors, as in the open city case, the
assignment of households to sites must be done within the city. That is, we
need some mechanism to describe equilibrium within the city when households
are not identical. We have chosen the approach of allocating sites lo the
highest bidder. If the distribution of attributes among households changes in
the sense that the rankings change, the equilibrium must also change.
Imagine a change in the attribute vector that converts the worst site into the
best. Then the rankings of all sites will change, and the price at each site
must be recomputed. When households' preferences and incomes are the same,
the assignment does not matter.
6.2b. Simulations
In this section we simulate the market described above, where preferences
are identical, supply is fixed, and incomes vary. Our goal is to determine how
hedonic price functions vary. The steps used in the simulations are:
i) Rank each housing bundle using the subutility function; then assign
consumers to houses based on their income ranking. This is
equivalent to assigning housing bundles to the highest bidder.
ii) Compute the exogenously determined utility level (UA) each household
would receive if the bundle ZA were bought at pA.
iii) Calculate the price each household would have to pay for its
respective site to give it the same utility level (U*-) it would receive
if it chose the alternative bundle ZA at prices p\
iv) Estimate the hedonic equation, using a flexible functional form, by
regressing a transform of the calculated prices on the hedonic
characteristics.
Simulations were run using two different preference functions, the
Stone-Geary:
3
U(x,z) = ft ln(x - 6 ) -t- Vp ln(z - «.) (6.4)
O O j=J J 1 J
and the translog:
3 33
U(x,z) = p In x + I p .In z. + .5 I I 6. .In z.ln z.. (6.5)
o j=i j j i j ij i j
hundred and fifty house attribute vectors were generated using random
drawings from the uniform distribution. Each vector contained three
attributes. Four different distributions of income were generated: uniform,
segmented uniform (a combined sample composed of drawings from two
independent uniform distributions to simulate a segmented housing market),
Pare to and normal. (The parameters of these income distributions are also in
the appendix.) All incomes were scaled so that each distribution had a mean
of 20,000. Hence, under any distribution of income, aggregate incomes are
equal. Since both utility functions are separable, it was possible to rank each
bundle using the housing sub-utility function. Housing bundles were then
75
-------
matched with incomes, and hedonic prices were calculated based on the bundle
ZA = (5,15,20) available at pA = 2000.
One hedonic price function was estimated for each combination of
preference functions and income distributions using Box-Cox flexible forms,
similar to the approach of Halvorsen and Pollakowski (1981). The general form
of the hedonic equation was:
Table 6.1 gives values of for different models.3 In general, the fits
appeared excellent. T-statistics were very high and over 90% of the variation
of the transformed dependent variable was explained.
While the estimated values of do not tell the whole story about
functional form, they certainly play a big role. In these examples, the range
of the estimates of « is from -1.2 to .79. There are substantial differences in
the behavior of the hedonic prices as a function of attributes.
TABLE 6.1
Transformation Parameter for Quadratic Box-Cox
Hedonic Price Functions
Preference Function
Income
Distribution Stone-Geary Translog
Uniform .49 - .13
Segmented Uniform - .47 - .87
J-uuubu 1.4 1.06
Normal .79 .65
76
-------
As is apparent from the above results, the parameters of the hedonic
price function are sensitive to both the specific form of the preference
functions and the distribution of income. While it is difficult to generalize, it
seems that the hedonic function is more sensitive to variation in the
distribution of income. For example, the maximum variation in », given the
distribution of income, is .62 (uniform), whereas the maximum variation in *,
given the preference function, is 1.99 (Stone-Geary). This result is consistent
with the presentation in the last section which showed that the hedonic
function arises from the joint interaction of consumer preferences, income
distribution, market structure and the characteristics of the existing stock of
houses.
We conclude that our empirical ability to determine the influence of
preference parameters on the hedonic price equation is virtually nil. For
practical considerations, then, one may assume that the preference parameters
and the parameters of the hedonic price function are not intertwined in any
way that is not already obvious from examination of the consumer's equilibrium
conditions. From the perspective of an empirical description of the housing
market, when the desiderata are the parameters of the hedonic function, little
will be lost by direct estimation of the hedonic equation, without taking
preferences into account.
6.3 The Welfare Effects of an Exogenous Change in Attributes
We are ultimately interested in using the hedonic technique to determine
the welfare effects of changes in air pollution and other environmental pol-
lutants which influence the value of locations. Our simulation model provides
a laboratory for experimenting with changes in exogenous attributes. By
constructing the market, we can see precisely what happens as locations are
improved.
Calculating welfare measures in hedonic markets raises a number of issues.
These issues have been the focus of considerable and deserved attention.
Work by Freeman has been especially crucial here (especially 1971, 1974a and
1974b); in addition, papers by Polinsky and Shavell (1976); Polinsky and
Rubinfeld (1977); Scotchmer and Fisher (1980); Bartik and Smith (1984) and
Brookshire et a]. (1982) have dealt with the problem.
In this section, we appraise five welfare measures using the market that
,..^ V"---. ...., ^ « ,j TV - .»- -«-i -- ,-f n.J-- .,--.. -» ;.i'.i < ..11 . ]/
calculations before and after adjustment to an exogenous change in
environmental quality.
In the following section, we investigate the welfare effects on a change in
Zi. Using this attribute as an instrument requires some explanation because
Zi is, after all, an endogenous variable in all the models of attribute choice so
far investigated. However, z, is exogenous at the aggregate or market level,
since its physical distribution cannot be influenced by household behavior.
We can imagine the following events. A government agency institutes a policy
which improves air quality. With households remaining at their houses, this
77
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change in air quality is exogenous. Under, a variety of circumstances,
however, 'the change in this attribute will disturb households' locational
equilibrium. Households will then relocate according to the equilibrium
mechanism, and at the new equilibrium, according to hedonic theory, prices
\vill appear 'as if* households chose attribute levels.
Initially we calculate five kinds of welfare effects. The first four are
estimates of the benefits of an increase in Zj assuming that no relocation
occurs (partial analysis). The fifth is the change in the hedonic price at the
site after relocation and a new equilibrium is established. The five measures
are:
Ml: Suppose we have solved the identification problem, so that we have
the parameters of the marginal rate of substitution function. Then we can
compute the change in the area under the marginal rate of substitution
schedule, holding the marginal utility of the numeraire constant. The marginal
rate of substitution is given by
"VA«*" au(x,z)/«x x
where X is the marginal utility of income and the price of x is unity. Holding
X constant, we have
/
fz*
Ml = m (x,z) dz = (U(x,z*) - U(x,z°))/X. (6.7)
Jz° *
Note that Ml is in units of A$ = AU. For X approximately constant, Ml is
approximately equal to the compensating variation for a change in z,.
Compensating variation, denoted CV, is defined by the expression
U(y - p - CV, z*) = U(y - p, z«). (6.8)
With X constant, this expression can be written (via Taylor's series expansion
because X = ) as
Solving for CV gives
CV i (U(y - p, z*) - U(y - p, z°))/X
± Ml
when x is substituted for y - p. This measure is an exact measure of com-
pensating variation only if the marginal utility of income is constant. Ml is
typically the measure used when computing the area under a hedonic
78
-------
"demand", curve, as in Freeman's (1974a) equation (4). It requires that the
identification problem be solved because the parameters of the utility function
urc needed.
The exact measure of compensating variation is calculated by solving
equation (6.8) for CV, rather than solving the Taylor's series expansion. The
result, (where Uy1 denotes U inverted for y)
CV + p = y - Uy»[U
-------
The measures Ml through M4 assume that the households do not move in
response to the disequilibrium created by an exogenous change in attributes.
The final measure, M5, is calculated after households move:
M5 = p* - p, (6.13)
where p* is the price which emerges after relocation and p is the original
price. This calculation was made from the actual prices at the locations. It
accrues to landlords because, given the assumption of a small open city model,
the utility of all homeowners will remain constant. Thus, the increase in rent,
M5, is the maximum amount landlords are willing to pay rather than go without
the change in the attribute. This measure is the correct one for the benefits
of changing z, in this open city case, as stated by Polinsky and S ha veil
(1976): "In the open city, the change in the aggregate property values
corresponds to the total willingness to pay on behalf of all parties" (p. 125).
When aggregated across households, M5 correctly measures total benefits:
"Benefits ... equal the total of all changes in land rents, positive and negative
..." (Lind, p. 189).
The computation of M5, the change in rent, requires the following steps:
i. compute U0(z*), the separable part of the utility function, and rank
the bundles according to U0(z*);
ii. rank households according to their incomes;
iii. associate each household with the location of corresponding rank;
iv. calculate the hedonic price that would make the household indifferent
between its equilibrium site and the opportunity bundle. This gives
p* from which MS can be calculated.
For housing attribute improvements, M5 will exceed the exact measure of
the restricted partial equilibrium welfare change, the maximum sum of
households' bids for their current houses as given in (6.9). As long as only
improvements occur, adjusting the equilibrium will allow some households to
move to better houses, and none to worse houses. The open city assumption
insures that each household's utility is constant, so that households will
always pay their compensating variation.
The measures M1-M4 are calculated for each household experiencing a AZj
of 5 units, and summed across households for each distribution of
summed across sites for each distribution of income-utility function combin-
ation. These results are presented in Table 6.2.
80
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TABLE 6.2
Alternative Measures of Welfare for
Exogenous Changes in an Attribute8
Azt = 5
Income distribution
uniform
segmented uniform
Pareto
normal
Mlb
M2
M3
M4
Stone-Geary preferences
M5
85921
74727
64362
77076
219540
334641
521563
405051
121039
99735
85148
103817
291012
475685
630229
464901
85241
74968
63914
77621
Translog preferences
Income distribution
uniform 120162
segmented uniform 106154
Pareto 90109
normal 108316
193432
89266
8569
355053
165925
144197
121251
147425
319434
287106
11952
474126
118450
106523
88922
108985
a The initial range of supply is given in the appendix.
b The approximate measure calculated according to equation (6.7)
The calculations in Table 6.2 present some surprises which give insight
not only into welfare measures but also into the working of the hedonic
market. Order-of-magnitude errors are found in several different ways. M4
overstates Ml by almost an order of magnitude for the Pareto distribution and
e>« -.-../-........ .. . * . . .'' «"> - ' ! . r«« v , ' >-.. P
; ' - ° . * A
magnitude. M2 and M4 typically overstate the other more acceptable measures.
Let us look at the standard graphical analysis of Ml, M2, M3 and M4 at
equilibrium. Figure 6.1 shows the equilibrium as the tangency between h(z)
and the bid function at z,:
81
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hedonic
price
1 1
Alternative Welfare Measures
Figure 6.1
The best measure of the value of an increase in zlf assuming no
relocation, is Ml in Figure 6.1. At equilibrium, the marginal rate of
substitution between the numeraire and zt equals the slope of the hedonic
price equation. Hence M4 should equal M3, and with concavity of the bid
function,
M4 = M3 > Ml.
Further, when h(z) is convex, we have
(6.14)
M2 > M3 = M4.
(6.15)
Now let us look at Table (6.2). We find the following observations:
b. M4 > M2 implies A2,»h/*z, > h(z+Az) - h(z)
The result (a) violates the idea that each attribute is in equilibrium at the
margin. Result (b) contradicts the convexity of the hedonic price equation.
These results shed some light on the hedonic practices. They pertain
primarily to the use of the hedonic price equation. Consider (a). We know
that the equilibrium process ensures that at the margin, each household bids
82
-------
-------
its willingness to pay for the i*-n attribute. Yet when we complete the process
of estimating the equilibrium bids as functions of attributes, and calculating
the marginal hedonic prices, we find considerable differences between the
known marginal bid and the slope of the hedonic price function. There are
two explanations for these differences. First, the number of households is
finite, and we have only points on the hedonic price function, not the exact
function. Second, while all hedonic functions fit well, they still fit
imperfectly, and the nonlinearity of the hedonic slope will in general prevent
its expectation from equaling its true value. That is, the expectation of a
function of a random variable will typically not equal the function of the
expectation of the random variable, except when the function is a simple linear
one.
Result (b) suggests that we could draw the hedonic price equation as in
Figure 6.2. This shows the hedonic price equation to be concave in the area
of some z,'s. First, this does not violate optimality conditions because they
require only that h(z) be less concave than the bid function. Second, from a
practical econometric perspective, nothing about the choice of functional form
of the hedonic price equation restricts the chosen function to having the
right curvature. Thus, while the 3ox-Cox method may allow the researcher
statistical flexibility, it makes it harder to keep track of whether the apparent
household equilibria fulfill the appropriate convexity conditions.
Welfare Measures in a Concave Neighborhood
for h(z)
FIGURE 6.2
83
-------
These results have implications for the identification problem. If we can
calculate acceptable benefit measures from the slope of the bid function (M3),
and we are confident that the households' equilibrium at the margin holds
(M3=M4), then we could neglect the identification problem. From Table 6.2, we
can see that linear extrapolations of the marginal bid (M3) provide 'in the ball
park' approximations of Ml. M3 exceeds Ml by 35% - 40%. This result
depends on the parameters of the preference function, and cannot be
generalized. But what is more important is that the hedonic slope misses the
marginal bid considerably. Since in applications, our 'only knowledge about
marginal bids comes from the slope of hedonic equations, it would seem
somewhat premature to worry about the identification problem. Consequently!
one conclusion from this simulation is that we need to know more about the
distribution of the slopes of the hedonic price equation.
As a consequence of the discrepancies in welfare measures, we have
discarded M2, M3, and M4 for further experiments and will concentrate on the
restricted partial equilibrium measure of willingness to pay (Ml) and the
actual change in rents (M5).
Table 6.2 shows that the change in rents after the relocation is quite
close to the households' approximate willingness to pay in the restricted case.
In order to assess the potential magnitude of differences we have calculated
Ml and MS for three additional changes in z,:
i) A2l = 1
ii) AZ! = .2z!
..., . (8 for worst half of the sites (1-125)
111) Az, = |(
10 for other sites (126-250).
These results are presented in Table 6.3. In cases (i) and (ii) there is little
change in the equilibrium because all bundles are improved, and little reason
to expect differences in Ml and MS. Hence we have approximated Ml. as in
equation (6.7), keeping the marginal utility of income constant. In case (iii),
where there is considerable reshuffling, we calculate the exact Ml according to
equation (6.9). The two measures are quite close for the small changes in (i)
to ue 2% to IUA less than the change in rents, a result consistent with theory.
Finally, recall that mean household income and hence aggregate income are
the same in all models. Consequently, given the preference function, the only
reason for variation among the measures is the distribution of incomes. For
the case of substantial distributional change in the attractiveness of the sites
(iii), there is more than a two-fold difference in the extremes of the estimates
of changes in rents. This case occurs when we compare MS for the uniform
(62180) and Pa re to (30545) distributions of income. This result is one of
aggregation and while the qualitative aspect is not surprising, the size of
difference is. It suggests that the distribution of income is an important
84
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determinant of willingness to pay for changes in air quality, and that
substantial inaccuracies can occur by ignoring this distribution.
TABLE 6.3
Further Comparisons of Welfare Changes
Ml
4z, la .2z,«
f8b
10
M5
1 2z (o
Stone-Geary preferences
Income distribution
uniform
segmented uniform
Pareto
normal
Income distribution
uniform
segmented uniform
Pareto
normal
21917 54356 59661
18415 50347 34412
15720 46914 29614
19112 51150 42971
21879 53832 62180
18470 49940 39888
15688 46481 30545
19225 50778 47027
Translog preferences
30847
26924
22687
27514
72516 80176
67218 48659
60437 43281
68064 58637
30727 71243 84154
26949 66536 55889
22590 59711 44698
27612 67426 65247
a Ml is calculated according to (6.7), its approximate value.
b For this case, Ml is calculated according to (6.9), its exact value.
85
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TABLE 6.4
Calculating Welfare as Changes in the Rent
of Affected Sites Only
(8
lo
c
Sura of Sum of Rent Changes
Rent Changes at Affected Sites Only
(M5)
Stone-Geary preferences
Income distribution
uniform 62180 48047
segmented uniform 39888 28441
Pareto 30545 24204
normal 47027 33706
Translog preferences
Income distribution
uniform , 84154 68956
segmented uniform 55829 42367
Pareto 44698 37327
normal 65247 50577
As our last experiment, we calculated what the estimate of benefits would
be if, after relocation, we looked at the affected sites only. The only case
where not all sites are affected is the case where Azt = 8 for the Worst half
of the sites. We know from Freeman (1974b) and Lind (1973) that for this to
serve as an upper bound, the willingness to pay must be identical among
households. (This is directly related to the Polinsky-Shavell result that in a
small open city, housing prices at any area location are independent of other
i i;,. .. ;«? .11 i. ..i. .1 .1, ... -i .,,4 ,,,i \ »T ..« . > ^i * « -vrr ,
some rents go up and some rents go down when the equilibrium changes, but
all households' willingness to pay will go up, because everyone moves to a
better house. But in the open city case, we get the same result if we sum
households' bids or landlords' rents, and we know that the sum of households'
bids will increase if we allow adjustment. Therefore, looking at the rent
changes at the affected sites only will understate the welfare change in the
small open city when households differ by income. It is interesting to look at
the magnitude of these rent changes and their variation across preferences
and income distributions. The results are shown in Table 6.4, where the
complete measure (sum of rent changes) is compared with the sum of rent
86
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changes on affected sites only. This table again shows the considerable
variation in the measures across income distributions.
6.4 Conclusions
In this chapter we have simulated an open city housing market in order
to investigate the determination of hedonic prices. This simulation market has
allowed us to address two topics: (1) the influence of preference parameters
and the distribution of income on the estimated functional form of the hedonic
price equation and (2) the relationships among the various restricted measures
of welfare and the post-adjustment change in rent, all induced by an
improvement in the attributes of locations.
There are two principle findings with regard to the functional form of the
hedonic price equation. First the distribution of income plays as strong a
role in determining the functional form as preference parameters. Given any
preference function, we can induce substantial changes in the form of the
hedonic equation by changing the distribution of income. This result conforms
with results of Rosen and Quigley and supports the use of best fit techniques.
Further, one may take the hedonic equation as part of the household's
exogenous budget constraint. Second, care must be taken in applying best fit
techniques. While there is no necessity for the hedonic price equation to be
convex, gross departures from convexity seem unlikely. It is possible for
Box-Cox methods to yield many kinds of curvatures.
We have also learned some important lessons in the use of the hedonic
price equation for welfare measurement. Despite excellent fits, hedonic price
equations may not give good estimates of marginal bids. And Box-Cox
estimation techniques do not necessarily yield hedonic price equations which
have curvature appropriate for welfare measurement. This suggests a careful
look at the distribution of marginal prices. How does the distribution of the
marginal bid vary with parameter estimates from the hedonic "price equation?
This sort of question will be explored in detail in succeeding EPA work.
We have shown that for small changes in a single attribute, aggregated
households' restricted willingness to pay is only a modest underestimate of the
changes in rent. Further, we have shown that some attention must be paid to
the distribution of income (and other household characteristics) in computing
aggregate benefits.
TV . - ..i ; .i..i.. v . " <' 1 r ' . i " ., " , T- ' -.
of a simulation model in exploring the workings of hedonic markets. In
additional work for EPA, we will use this approach with much more realistic
data on housing markets to assess hedonic techniques.
87
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CHAPTER 6
FOOTNOTES
McConnell is with the Department of Agricultural and Resource Economics
of the University of Maryland. T. T. Phipps is with Resources for the
Future, Washington, D. C.
In his empirical work, Quigley used a GOES utility function.
The coefficients for the model (6.6) were estimated via maximum likelihood
using SHAZAM's 'BOX' routine.
88
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APPENDIX, CHAPTER 6
Parameters of Simulation Model
Stone-Geary
0
.8
1000
1
.06
5
Trans log
2
.04
15
3
.1
20
P0 = 2 ftl - .06 P2 = .04 /?3 = .1
i/j 12 3
1 -.3 .15 .2
2 .15 -.2 .5
3 .2 .5 -.1
The supplies of z were generated as follows:
zt: uniform (6,26)
za: uniform (16,26)
zs: uniform (21,31).
The distributions of income were generated as follows:
2. segmented uniform
a. 12*5 observations uniform [5,000, 15,000]
b. 125 observations uniform [20,000, 40,000]
3. Pareto generated as y = y0(l~u)* where = "Vi.at Yo = 4000, and
u is uniform [0,1]
4. normal (20,000, 225-10*).
Each distribution was transformed to have a mean of 20,000.
89
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CHAPTER 7
SHOULD THE ROSEN MODEL BE USED TO VALUE
ENVIRONMENTAL AMENITIES?
Maureen Cropper^
7.1 Introduction
There is a. large literature in both urban and environmental economics
which attempts to value site-specific amenities access to workplace or air
quality using data on residential property values.9 With few exceptions these
studies appeal for their theoretical justification to Rosen's model of hedonic
markets, and they follow his two-stage procedure in valuing amenities. In the
first stage property values are regressed on housing characteristics and
location-specific amenities to estimate an hedonic price function. The partial
derivative of this function with respect to an amenity is interpreted as the
marginal value which consumers place on the amenity. In the second stage
marginal amenity price, computed from the hedonic price function, is regressed
on the quantity of the amenities consumed and household characteristics to
estimate a marginal willingness to pay function.
The purpose of this chapter is to discuss why these procedures may be
inappropriate for valuing location-specific amenities, and why a discrete model
of location choice may be preferred to the Rosen model on theoretical grounds.
Reasons why the. Rosen model may be inappropriate fall into three categories.
First, some amenities are inherently discrete (whether a house has a river
view), implying that the individual cannot make marginal adjustments in the
amounts consumed. The assumption that marginal adjustments are possible,
which is crucial to the Rosen model, is therefore unwarranted and renders the
model inappropriate.
A second difficulty occurs when amenities which are in principle
continuous assume only a few values in an urban area due to economies of
scale in production. Examples of these include high school quality, which can
police force. The problem here is that local public goods, which require a
minimum population for efficient production, cause indivisibilities in the set of
amenities available (Ellickson, 1979). Thus, as with inherently discrete
amenities, the individual cannot make marginal adjustments in quantities
consumed.
These two problems, of course, are not unique to the attributes of
locations. In markets for differentiated products, such as automobiles, one
encounters inherently discrete attributes (the number of doors on a car) and
finds "holes" in the menu of choices caused by economies of scale. (Only a
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few engine sizes are available to consumers" due to the large amounts of
product-specific capital required for engine manufacture.) The third difficulty
with the Rosen model is, however, unique to the location choice problem.
A key assumption of "Rosen's model is that each characteristic of a product
can be varied independently of the others, subject only to a budget
constraint. In the location choice problem, however, the attributes of location
often cannot be varied independently of one another. This is because these
attributes are tied to geographic location, and the choice of geographic
location is a two-dimensional choice. Thus, if one wishes to model location
choice as choice in amenities space, one must add the constraint that the
choice of amenities l,...,g determines the amounts of amenities g+l,...,n
consumed. Constraints of this type destroy the main result of the Rosen
model, viz., that each amenity is consumed to the point where its marginal
value to the consumer equals its marginal price. The two-stage procedure
described above therefore cannot be applied. <
The foregoing problems are discussed at length below. Section 7.2
reviews the Rosen model and discusses whether the model should be applied
when some characteristics of goods are available only in discrete amounts. In
Section 7.3 the model is applied to the choice of residential location. This
means that geographic constraints must be added to the problem, and the
section explores the implications of these constraints for location choice in
amenities space. The difficulties discussed in Sections 7.2 and 7.3 can be
resolved in part by estimating' a discrete model of residential choice, in which
the objects of choice are geographical locations. The structure of such
models is outlined in Section 7.4. Section 7.5 concludes the chapter.
7.2 Consumer Choice in an Hedonic Market
" In the model developed by Rosen to explain product differentiation under
pure competition alternative brands of a product are indexed by an
n-dimensional vector, z, zRn, which describes the amount of each attribute
provided by the brand. In the special case in which the consumer purchases
only one unit of the brand his utility is a function of the vector z and the
quantity consumed of a numeraire good, x,
U = U(x,z). (7.1)
U is assumed to be strictly increasing in x, strictly quasi-concave in (x,z),
,! «.,..,. .fff. ...,! J..1.1,, -TV. I.T ' - .- ' ' ' .; i'° 1 v
' ': '- \ * t
subject to a budget constraint
p(z) + x * y, (7.2)
where y is income and p(z), the hedonic price function, gives the unit cost of
the differentiated commodity as a function of the attribute vector z. In
Rosen's presentation the set of z's available to the consumer is infinite and
p(z) is assumed to be differentiable.
For the present discussion two features of the model should be
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emphasized. One is that the consumer is free to choose each attribute of the
brand independently of the others, subject only to his budget constraint.
The other is that his choice set is infinite. Together these assumptions imply
that the consumer equates the marginal value of each attribute to its marginal
price,
7z7 * ..... ' '
Equation (7.3) implies that the derivative of the hedonic price function with
respect to amenity i equals the consumer's willingness to pay for that amenity
at the level he is currently consuming. It also justifies the second stage of
the Rosen procedure in which the coefficients of the marginal willingness to
pay functions (the left-hand-sides of (7.3)) are estimated.3 "' ~
In the notation of this section the problem of inherently discrete amenities
occurs when some of the zj's can assume only a countable number of values.
For example, in choosing an oven the characteristic "fuel type" can assume
only two values, gas or electric. Formally, suppose that z, can assume only
two values but that the other zj's are available in infinitely divisible quantities.
In this case the marginal rate of substitution of zt for x is, of course, not
defined, and (7.3) does not apply when i = 1. The choice of z is now a mixed
discrete-continuous choice problem. Conditional on zt, the remaining n - 1
equations in (7.3) can be solved together with (7.2) to yield conditional
demand functions for x and for amenities 2,...,n. Upon substituting these
functions in (7.1) one obtains an indirect utility function conditional on zt,
V(z,). The value of z, is selected which maximizes V(zj).
When Zi is inherently discrete one is still interested in measuring the
parameters of the utility function since willingness to pay for discrete
changes in z, is well defined. This can be done by simultaneously estimating
the last n - 1 equations of (7.3), the -hedonic price function, and an equation
for the probability of selecting z,. Applying the Rosen model to discrete
attributes, however, does not make much sense. The problem is not simply
that the marginal willingness to pay function for zt is literally not defined,
but that the notion of a marginal bid function assumes that the values of Ea
can be ordered. This is usually not the case with inherently discrete
amenities, e.g., "fuel type" or "river view"; thus the Rosen model cannot be
viewed as an approximation to reality in this case.
n ci(.b.i Moukcc*, buuii tttt ticitooj quttiily, happen to
be available only in discrete amounts is somewhat different.4 Although this
problem is formally equivalent to the problem of inherently discrete attributes,
and can be solved as a mixed continuous-discrete choice problem, it differs
from the foregoing problem in one important respect: with attributes such as
school quality the marginal willingness to pay function is a meaningful concept
which one can try to approximate using the Rosen model.
To illustrate, suppose that zlt the only amenity of interest, assumes three
values within an urban area. The smooth curve pictured in Figure 7.1 is
fitted to these three points, A, B and C, and the slope of the curve at each of
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$
Figure 7.1
Bid Functions and Hedonic Price Functions
93
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these points is interpreted as the marginal value of z, to the persons
consuming that amount of the amenity. In reality, however, the slope
evaluated at point C underestimates the marginal willingness to pay for zt by
person I, whose best choice of zl among the three alternatives is C. By
contrast, the slope of the estimated hedonic price frontier overestimates
marginal willingness to pay for person 2, whose optimal choice is point A. The
failure of (7.3) to hold for persons 1 and 2 biases estimates of the marginal
willingness to pay function; however, one suspects that this bias should
diminish as the number of values of zt available increases. In this sense, one
can justify the Rosen model as an approximation when there are "holes" in the
data. This is not true when the amenity in question is inherently discrete.
7.3 Applying the Hedonic Model to Residential Location Choice
While the problems discussed in Section 2 create difficulties in using
Rosen's model to measure preferences for attributes, they are not problems
unique to the choice of residential location. The problems discussed in this
section, however, have few counterparts in hedonic markets for manufactured
products.
The main point of this section is that when the model of equations (7.1)
and (7.2) is applied to residential location choice additional constraints must
be placed on the problem because of the two-dimensional nature of geographic
choice. These constraints prevent the household from independently varying
all n amenities and thus render (7.3) invalid. To emphasize that these
constraints do not arise because of the. discreteness of available choices, we
assume that all n location-specific amenities are available in infinitely divisible
quantities.9 Even when this is true, the choice of z is constrained by the set
of equations (7.4) which describes the vector of amenities available at each
point (u,v) in geographic space,
zj = fi(u,v) i = l,...,n. (7.4)
Since the amenity vector consumed can be altered only by changing locations,
the set of available z's is implicitly defined by (7.4).
To see intuitively why (7.4) may prevent the individual from .indepen-
dently varying nil n amenities suppose that two of the n amenities are access
amenities. Specifically, let zj = distance to the point (ui,vj), i = 1,2, where
(u,,vt) and (ua,va) are two points of interest, (e.g., the workplaces of a two-
' " \ f | ^ > * ; . / / . fi *
. - . . n' K ' -'-;
and since the circumferences of two distinct circles intersect in at most two
points, there are at most two points in the u-v plane corresponding to any
feasible (Zi,za) pair (see Figure 7.2).' This implies that once z» and za are
determined the individual has at most two choices for each of the remaining
n-3 amenities of interest.7
The necessary conditions for location choice in amenity space are
therefore not given by (7.3) if two or more amenities are access amenities.
For za and z2 defined as above the household would locate two points in
geographic (and amenity) space by choosing Zi and za to maximize (7.1)
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Figure 7.2
Locational Restrictions on Choice
u
95
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subject to (7.2), (7.4) and a feasibility constraint, (7.5),
A, n A2 * 0 (7.5)
where A = {(u,v)lz* = (u - uj)2 -«- (v - vj>a}, zi - 1,2. The household would
then locate at the point yielding the higher utility.
This example gives a specific and reasonable instance of the way in which
the two-dimensional nature of location choice limits choice in amenities space.
Suppose, however, that access amenities are not of interest to a household. Is
choice in amenities space still restricted by the two-dimensional nature of.
geographic choice? The answer to this question depends on the nature of the
functions fj(u,v), i.e., on the distribution of amenities over geographic space.
Consider the level curves of two amenities plotted in the u-v plane. If the
distribution of each amenity is tnonocentric and radially symmetric then its
level curves are concentric circles and the same result obtains as when the
amenities are access amenities: any feasible choice of the two amenities
restrict the household to two points in geographic space and hence to at most
two values for each of the remaining n-2 amenities.
If the distribution of an amenity is asymmetric or if it is multicentric,
then the number of possible intersections of any two level curves increases.
This is illustrated in Figure 7.3, which pictures level curves for total
suspended particulates and distance from the CBD in Baltimore, MD. It is
evident from Figure 7.3 that the choice of . 60 pg/m' of particulate matter and
five miles from the CBD no longer restricts the household to two locations;
however, only four points satisfy these two amenity values. For amenities
that occur in continuously variable amounts, it is clear that the choice of two
or three amenities restricts the choices available for remaining amenities to a
finite number of points.
At this point the reader may wonder how the foregoing argument is
altered if some site-specific amenities are discrete, e.g., if the relevant
pollution variable is an index which assumes only five values. In this case
the level curves are areas and no longer restrict the choice of other amenities
in the manner described above.*
It should, however, be borne in mind that for the two dimensions of
geographic space to restrict choice in amenities space it is necessary that
only two amenities be continuous, with spatial distributions that are
>»TM->»"^vi~-->»«sl»' nirr.- rn»» v'p -.-.J ,,-,,,.-.,.,4 rj , «TH,;.. '- - <-.,-.» - --JTI. :- ''.??'. !« ;-..
condition to satisfy in view of the importance of "distance to work" in house-
hold location decisions.* In a two-earner household it is certainly reasonable
that distance to each person's place of work is an important amenity in so far
as residential location is concerned.10
7.4 Discrete Models of Residential Location Choice
Although conceptually different, each of the three problems described
above has a similar effect on the household's choice of amenities: it causes
the choice set to become discrete (at least for some subset of amenities) thus
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violating the assumptions of the Rosen model.
This suggests thai one consider discrete choice models of residential
location as a method of valuing site-specific amenities. In a discrete model of
residential location the objects of choice are geographic locations, indexed it
where the set of all i is finite. To each location there corresponds .a vector
of amenities zj. As in the Rosen model, utility is defined over z and a
numeraire x. By making locations rather than amenities the objects of all
choice, geographic restrictions are incorporated into the problem ipso facto.
In this framework, household h chooses the location i for which
Uih = Uh(zih*vh ~ Pi)
is highest, where Pi is the price of location i. To make the model a statistical
one, it is usually assumed that utility is random from the viewpoint of the
researcher since he cannot observe all attributes of locations. Redefining zjh
to include only those attributes observable by the researcher, utility may be
written as the sum of a deterministic term, Vih(zih»yh ~ Pi)i and a random
term *£h« Vih> a*8° termed "strict utility," is usually written as a linear-
in-parameters function of yn - pj, zjh and interacts between these variable*
and household characteristics. The probability that household h selects
location i is given by
P(Vih + «ih * Vjh + *jh» all j * i). (7.6)
To value site-specific amenities given data on residential location choices
one maximizes a likelihood function with individual terms of the form (7.6). If
the {*ihl are assumed to be identically distributed for all i and h with a
Type I Extreme Value distribution, the resulting likelihood function
corresponds to the multinomial logit model. If choice of house is also
observed, a nested multinomial logit model is usually assumed (McFadden,
1978). Given estimates of the parameters of Vjj,, random counterparts of
compensating and equivalent variations can be constructed for changes in tho
z vector (see Hanemann (1984)).
7.5 Conclusion
The purpose of this chapter has been to explain why the Rosen model may
be inappropriate for valuing location-specific amenities, such as air quality
-,,» i .! ,-,.,»,. T".,- .. ,...*. *.-.., ... . ' - .. ?r ,.
». ' * ' ' **V
amenities are inherently discrete (e.g., z» = location has a view of the beach),
and if these discrete variables cannot be ordered, then the notion of a
continuous bid function for amenities is meaningless, even as an approximation.
In this case the Rosen model is clearly inappropriate. A second but less
damaging situation occurs when amenities which enter the utility function as
continuous variables are available only in discrete quantities for one reason or
another. In this case one can at least view the Rosen model as an
approximation to reality, which improves as the size of the discrete choice set
increases.
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Thje third situation emphasized in this chapter occurs when the
two-dimensional nature of location choice restricts choice in amenities space.
Here the amenities of interest enter the utility function as continuous
variables and are also available in infinitely divisible quantities; however, the
choice of two or more amenities restricts the number of choices available for
the remaining amenities to a few. Since bid functions for location-specific
amenities are defined in case three, it is tempting to use the Rosen model as
an approximation to reality, as one might do in case two. This, however, is
not possible. In case two, equation (7.3) at least may be viewed as holding
approximately (see Figure 7.1). In case three, however, the first-order
conditions of the Rosen model no longer apply since all n amenities cannot be
chosen independently of one another.
The three situations described above argue for the use of a discrete choice
model to value location-specific amenities. In the first and second situations
the case for a discrete choice model is obvious. In the third it has been
demonstrated that the choice of certain site-specific amenities restricts the
household to a few points in geographic space and, hence, to a finite number
of amenity vectors. The reader, however, may object that a discrete choice
model is awkward when the number of choices is large, and that a commonly
used discrete choice model, the multinomial logit, is flawed by the assumption
that the error terms are independently and identically distributed.11 There
are several responses to these criticisms.
The fact that the number of possible residential locations is large may be
considered a problem for two reasons, one computational and the other
behavioral. The computational problem has been treated by Me Fad den (1978)
who demonstrates that for purposes of estimating the multinomial logit model
each household's choice set can be obtained by sampling from the universal
choice set. Thus the existence of thousands of choices in the universal choice
set need not pose a barrier to estimation.
The more disturbing problem created by a large choice set is behavioral.
When the choice set is large it is unrealistic to assume that the individual
compares all possible alternatives according to each attribute of interest. This
limitation of discrete choice models can be overcome in two ways. If the
choice set has tree structure (e.g., the household selects an area of the city,
then a neighborhood, then a house), one can apply Tversky'a hierarchical
elimination-by-aspects model (Maddala). In this model the individual selects a
single branch at each level of the decision tree, thus eliminating all
^M^r.^^t; 0 ,^4 ,'^r,l,.^mj !._ «V-« '.v--,.- .' ). -I'. , -,« , .. - .' -'7«.'«-?
by Cha, is to assume that the individual ranks alternatives according to a
small subset of attributes and then compares only the k highest ranked
alternatives according to all attributes.
The assumption that the random component of utility is independently
and identically distributed across households and alternatives is most
objectionable when the objects of choice are individual houses rather than
large neighborhoods. For example, it is unlikely that the unobserved
attributes of a house are distributed independently of those of the house next
to it. Correlation between the unobserved attributes of alternatives on the
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lower levels of a decision tree is, however, allowed in Me Fad den's (1978)
nested logit model. Thus, the Independence of Irrelevant Alternatives
property need not destroy discrete choice models.
One final point. Although it would be foolish to pretend that discrete
choice models are not without econometric difficulties, these difficulties must
be judged in light of the econometric problem of the Rosen model, described
in earlier chapters. From this perspective discrete choice models are a
method of valuing environmental amenities worthy of consideration.
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CHAPTER 7
FOOTNOTES
Department of Economics, University of Maryland.
Portions of this literature have been summarized by Freeman (1979a),
Diamond and Tolley, and Bartik and Smith.
For the coefficients of the marginal willingness to pay functions to be
estimated efficiently, these functions must be estimated jointly with the
hedonic price function.
In the introduction the fact that some amenities are available only in
discrete amounts was motivated by economies of scale in the provision of
local public goods. An analogous problem occurs if attributes which are
available in infinitely divisible amounts are coded as discrete by data
collectors.
The consequences of relaxing this assumption are explored below.
There must be at least one point in the u-v plane corresponding to
or th° (zi»za) pair is not feasible.,
If there is a third point of interest in the city, (z3, defined analogously
to z, and z2) the above argument is even stronger. As long as the
three points of interest in the city do not lie on the same straight line it
can be shown (see Appendix) that any feasible choice of (zlfz2,zs)
uniquely determines the household's geographic location. Once location is
determined the levels of all other amenities are uniquely given by (7.4)
since there is only one value of zj at each point in geographic space.
In Figure 7.2, for example, the area between 45 and 60 pg/m* might
represent a single value of the pollution index.
Empirical studies of residential location choice (Anas, 1982; Lerman, 1979)
b^VP ^orisiqf r»«"l*1'V ^r^»ir>H rliit^""^ (or *t-f>T-r>' «!^^\ i.'- ...-,-,»- »n J. ., ..,
slaUbiically significant determinant of household location. One difficulty
in assessing the importance of distance to work within the Rosen
framework is that any amenity which varies with household as well as
location cannot be valued unless all households are similar. Thus, in an
urban area with many work centers, distance to work center i may not
have a statistically significant coefficient in an hedonic price function
area, even though distance to work is an important determinant of
residential location.
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This assumes, of course, that workplace location is fixed as far as the
residential location decision is concerned. If workplace location is
determined jointly with residential location then the argument of Figure
7.2 must be applied to each workplace location. As long as the number
of possible workplace locations is finite the choice of z,, z2, (unvt) and
(u2lva) still restricts the choice of amenities z3,...,zn to a finite number
of points.
This assumption together with the assumption that each error term has. a
type I Extreme Value distribution gives rise to the Independence of
Irrelevant Alternatives property of the multinomial logit model. This
means that the probability of selecting alternative i divided by the
probability of selecting alternative j is independent of the other
alternatives available.
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APPENDIX TO CHAPTER 7
The purpose of this appendix is to prove that any feasible choice of
amenities zt, z3 and z3 where zt = distance to the point (uj,vj), i = 1,2,3,
uniquely determines a household's location in the u-v plane, provided that all
of the points (ujjVi), i = 1,2,3, do not lie on the same straight line.. For any
zj the locus of points z{ away from (uj,vj) form the circumference of a circle
with radius z£. The result to be proved is that the circumferences of the
three circles which are z{ away from (u{,vj), i = 1,2,3, intersect in at most one
point, provided the points (u^vj), i = 1,2,3, do not lie on the same straight
line. If the three circumferences do not intersect in at least one point then
the choice of (z,,z2,za) is not feasible.
We begin by noting* that the circumferences of any two distinct circles
intersect in at most two points. Call these points A and B and let AB denote
the line joining A and B. (See Figure 7.2.) The line joining the centers of
the two circles must be perpendicular to AB. If A and B lie on the
circumferences of two circles then the center of each circle must be
equidistant from A and B. The locus of points equidistant from any two
points is a line perpendicular to the line joining the two points. Call this line
XY.
For a third circle to intersect the first two in more than one point it
must pass through points A and B. We show that this can happen if and only
if the center of this circle lies on the line XY. To see that this is possible
only if the center of the third circle lies on XY note that the circle whose
circumference passes through points A and B must, by definition, be
equidistant from A and B. However, the locus to points equidistant from any
two points is a line perpendicular to the line joining the two points. Thus,
the circumference of three circles can intersect in more than one point only if
their centers lie on the same straight line.
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CHAPTER 8
SUMMARY AND ASSESSMENT
8.1 Introduction
The purpose of the hedonic component of the Maryland-EPA Cooperative
Agreement, as originally defined, was to "solve the identification problem in
hedonic models." Our conclusions concerning the identification problem, based
on the reasoning of Chapters 3 and 4, in hedonic markets can be solved only
be assuming fairly specific functional forms for preferences and the hedonic
price equation, without the ability to test whether these forms hold. While
there may be occasions when household behavior conforms with the necessary
assumptions, the difficulties in statistically testing such assumptions make the
solution to the identification problem rather unsatisfactory. Because we have
concluded that identification of preference parameters is quite difficult, we
have also explored other issues in hedonic models and other methods of asses-
sing the benefits of environmental improvement from housing transactions.
8.2 The Identification Problem; Summary and Resolution
Two questions arise in addressing the issue of the identification problem.
The first pertains to whether a solution exists. The second relates to the
costs of the solution.
8.2A. Can We Do It?
The identification problem deals with the question: can we use the
hedonic model to recover information about preferences? In particular, can
the parameters of the preference function be identified and therefore used for
determining the benefits of non-marginal changes in attributes? The answer
to the basic question of identification is 'yea', we can identify the parameters
of preference functions under certain conditions. For participants in a single
market who face the same hedonic price equation, we can identify their
preference parameters in the following way:
\..f ., .f - ^..wi-o.i w^. «>~k.v4i\* Uii viio Aiiit*oix ill pclluiuUtCl'tJ iitOUGl «>O lilttL it
can be shown to be identified by traditional exclusion criteria
(Section 4.3A). The variables excluded will typically be nonlinear
transformations of endogenous variables.
(2) Successfully estimate the whole system of equilibrium conditions
using maximum likelihood methods (Section 4.3B). Successful
estimation implies that preferences and the hedonic price equation
have sufficiently different curvature to allow the maximum likelihood
estimates to converge.
(3) Estimate the reduced form with attributes as endogenous variables
and show that the preference parameters can be derived uniquely
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from the reduced form parameters (Section 4.3C, Appendix 4.A). This
can be achieved in a very limited number of cases.
(4) Estimate different linear hedonic price equations from segmented
markets or multiple markets and use the coefficients as prices in a
traditional demand system with prices as parameters (Section 4.4B).
Finally, for households in different markets, we have an additional approach:
(5) Use marginal prices from multiple-cities hedonic price equations, and
estimate the system as Rosen originally intended.
Of the five suggested approaches, only the last makes use of the traditional
Rosen two step model. Further while multiple markets may provide the basis
for identification, the numerical questions of how many markets one needs and
what additional structure must be imposed remain to be investigated. It is
worth emphasizing that regardless of the chosen functional form, there is no
way to determine identification from the simple application of the Rosen two
step approach in the single market setting. This holds even when we derive
the marginal value functions from an explicit utility function as, for example,
in Quigley (1982). (See Appendix 4.A, example 2.)
Identification of parameters in an equation is always derived from prior
information. In some cases the imposition of prior information is innocuous in
that it has no behavioral implications. For example, the normalization of the
parameter on the dependent variable in a single equation linear regression
model is necessary for the estimation of the model but has no behavioral
implications. On other occasions the imposition of prior information has
behavioral implications, but is quite plausible. For example the structural
parameters of a model of an agricultural commodity might be identified by the
plausible assumptions that demand is increased by increases in per capita
income and supply is increased by greater summer rainfall.
»
The resolution of the identification problem in hedonic models is less
satisfactory. In all of the five approaches to identification given above, there
are no simple and intuitive assumptions, such as rainfall influences supply but
not demand, to identify parameters. No such assumptions are available
because the basic equations which are simultaneous stem from the same
actorsthe individual households from which the data are taken. Instead, the
identification of preference parameters in hedonic models comes only as a
result of assumptions about functional form. We have shown, for example (in
rl^,,.-,4,'^-r,r- ft It; * O1* «%..-« «Vr ,,_,,,,f *,.-« ; . . f <>... \.,'fi .'. .. .1 ,.,... '.,.. -t. ...
cubic rather than a quadratic function will serve to identify a linear marginal
rate of substitution function. While in some cases such assumptions about
functional forms are subject to nested testing (for example when the hedonic
price equation is recursive), in most cases they are not. Most important, such
assumptions have none of the compelling plausibility that identifies the demand
for an agricultural commodity by omitting summer rainfall. In sum, we can
identify the parameters of preferences, but only by imposing assumptions
about functional form for preferences and for the hedonic price equation
which rarely have any intuitive appeal.
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In one. sense, this result does not make identification in hedonic models
quite as gloomy a prospect as it seems. Functional forms are not devoid of
economic content. The general requirement for household equilibrium in the
Rosen model is for the preference function to show more concavity than the
hedonic price equation. The second order conditions have a certain economic
force. However, such economic content typically requires functions nonlinear
in parameters, and thus ignores some fairly significant practical hurdles.
Identification of models nonlinear in parameters requires successful estimation
by maximum likelihood, an unrealistic requirement for the typical model with
many attributes. And converting to linear-in-parameter models by polynomial
approximation usually obscures the economic content of functional form. Thus,
practical reasons undermine the economic content of functions.
Thus we are in a position to identify the preference parameters of
hedonic models, by imposing structure on the marginal rate of substitution
functions and on the hedonic price equation. Typically the assumptions
needed to induce identification will be fairly severe and arbitrary, but if
identification gives us enough new information such assumptions may well be
worthwhile.
In sum, identification of parameters of preference functions in hedonic
models can be achieved through assumptions about functional form. Such
assumptions are commonly made in empirical work, but they are generally
testable. In the hedonic model, they are typically not testable. Further, the
gains in accuracy do not seem'to be worth it. If we use the hedonic model
for welfare changes, we may as well use the guidelines for approximations laid
by Freeman ten years ago (Freeman, 1974a).
8.2B. Is It Worth It?
Whether identification, when conceptually feasible, is worthwhile depends
in part on whether the implied behavior is plausible. Thus an important
question in the context of identification is not whether the appropriate
coefficients can be recovered, but whether the prior restrictions imply
plausible behavior. As noted above, standard commodity models are identified
typically by appealing to constraints on behavior: the level of rainfall does
not affect the demand for wheat. What sort of behavior is implied by the
methods of identification implied by this volume?
First consider the hedonic price equation. The results of both Chapter 3
-,.,,» 01 . .... « ».-..» « ««- ..».. '.*«<:... .'. !
price equation can help identify the marginal rate of substitution equations.
But other theory (Rosen, Quigley) as well as the empirical results of Chapter 6
demonstrate that no particular behavior can be deduced from curvature of the
hedonic price equation. As we showed in detail in Chapter 6, the preference
parameters, the distribution of household tastes, and the distribution .of
amenities determine jointly the functional form of the hedonic price equation.
Further, we typically have no strong prior beliefs about this functional form,
but are free to estimate best fitting functional forms. Thus, part of the
solution to the identification problem comes from the functional form of the
hedonic price equation, and only in rare instances can we ascertain the
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behavioral implications of such forms.
Results from Chapter 4 suggest that identification is likely to be enhanced
by separability assumptions. That is, identification of one marginal rate of
substitution function is easier when it excludes variables which appear in
other marginal rate of substitution functions. Such exclusion of variables
occurs when the utility function is separable. This result is in keeping with
the literature on the estimation of demand systems, where it has long been
recognized that various forms of separability would reduce the estimation
burden. There is a crucial distinction, however, between assuming separ-
ability to reduce the number of parameters to be estimated in demand systems
and assuming separability to identify parameters in hedonic markets. In
demand systems, we can test for separability. In hedonic markets, we cannot
typically test for the assumption of separability, for without it, we do not
even have preference parameters.
In the end, identification of preference parameters in hedonic markets
requires assumptions of unknown validity in the hedonic price equation and
separability in the preference function. Our knowledge of behavior is not
sufficient for us to argue that separability of the preference function is
plausible.
8.3 Suitability of the Rosen Model for Valuing Environmental Amenities
Chapter 7 has questioned , whether the Rosen model can be used for
environmental quality. In Rosen's model, which was developed to explain
product differentiation, brands are indexed by an n-dimensional vector of
attributes. In selecting a brand the consumer is faced with an infinite set of
attribute vectors and can choose each attribute of the brand independently of
the others, subject only to his budget constraint. Utility maximization thus
requires that the marginal utility of each attribute (*U/*Zi/«U/*x) be equated
to its marginal price. This justifies the interpretation of the partial
derivative of the hedonic price frontier as measuring the marginal value of an
attribute to some consumer.
Consumers in the land market, however, do not have as many degrees of
freedom as purchasers of manufactured products. Even when the set of
residential locations is infinite, so that marginal changes in location can be
made, the consumer cannot freely vary each of n attributes of the housing
site. This is because the consumer has only two degrees of freedom in
making marginal changes in latitude and longitude the consumer, must weigh
the effect of these changes on each of the n attributes of the housing site
and compare a weighted sum of marginal valuations to the marginal valuations
to the marginal cost of the move. The consumer is therefore unable to equate
the marginal value of each attribute to its price, and the slope of the hedonic
price frontier with respect to an attribute cannot be interpreted as the
marginal value of the amenity to the consumer. Since the consumer cannot
freely choose all elements of the attribute vector his demand (bid) functions
for various attributes will not correspond to those in Rosen's model.
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Since jnuch of the empirical work which attempts to value air quality
follows Rosen's approach, these studies must be re-evaluated. One way of
doing this is to be compare the results of these studies with the results of
alternative approaches suggested in Chapter 7.
8.4 Future Research
Our research on the hedonic model has focused on two issues: the
identification problem and the use of the Rosen model for environmental
amenities. In the second cycle of our Cooperative Agreement, we plan to
explore these issues in several different ways. First, we plan to pursue
approaches which emphasize discrete choices or bids for housing. The
bidding approach will follow the work of Ellickson (1981), Lerman and Kern
(1983), and Horowitz (1983). The discrete choice models will follow the work of
McFadden (1978) and Anas (1982). We.plan to develop and estimate a variety
of these models on several different data sets, with the emphasis on measuring
the benefits of improvements in air quality. In the process of developing new
approaches, it will be useful to compare these empirical results with empirical
results from the Rosen model. ,
While the departure from the Rosen model means a loss of some intuitively
appealing properties such as continuity and equilibrium at the margin, it also
gives us the opportunity to discard or at least test two maintained but
unrealistic hypothesis: perfect information and equilibrium. The discrete
choice or bid models of Horowitz, McFadden and others do not require an
equilibrium in the housing market. Further they do not require complete
information. Hence this research direction not only allows us to advance from
a model which does not seem to fit the residential housing market in concept.
It also allows us to model the actual purchase or rental of a housing unit in a
much more plausible way.
Second, we plan to use the simulation approach of Chapter to explore
more workings of the hedonic model; We will enrich the simulation approach
so that we are modelling the housing markets of discernible cities. In
particular, we will attempt to mimic the behavior of markets in Los Angles and
Baltimore. Further, we will develop markets in a fair number of cities to see
if we can determine in what circumstances the multiple markets approach to
identification will work.
Finally, we have concluded that because there is a feasible but perhaps
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proceed with benefit estimation cautiously using the slopes of the hedonic
price equation. We will explore the statistical characteristics of these elopes
for different forms of hedonic price equations.
108
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