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 equation—the  relationship  between
housing prices and housing  attributes—give 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

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       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

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    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

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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                                  (3.1€b)

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

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 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

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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

-------
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

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    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

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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

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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

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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

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             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

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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

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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
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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.
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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)

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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
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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

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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

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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

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    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

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"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

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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

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    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
                                   90

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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


                                    91

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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


                                   92

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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)
                                   94

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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
                                   96

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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.
                                   98

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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
                                    99

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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.
                                   100

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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.
                               102

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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.
                                    103

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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


                                     104

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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
actors—the 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.
                                    105

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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
                                    106

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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.
                                    107

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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
^^t  ,..„...• K...VJ1.-  ,,„!.,«:,,,  •„.   n..^   ;^.,.,f !?;:..,f; ..   .,;._.«. v,,,   j»   ;.  fv i;,,n.,i;v  jc
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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