United States
Environmental Protection
Agency
Research and Development
Air and Energy Engineering
Research Laboratory
Research Triangle Park NC 27711
EPA/600/S7-85/034 Dec. 1985
Project Summary
Inferring Willingness to Pay for
Housing Amenities from
Residential Property Values
Joel L Horowitz
The standard hedonic model of con-
sumers' choices in housing markets,
which forms the basis of much envi-
ronmental benefits analysis, fails to
take into account three important
characteristics of the bidding process
through which houses are sold. First,
the prices of houses are established
through sequential bidding, so sellers
do not necessarily accept the highest
bids that might be made for their
houses. Second, ignorance of the mar-
ket may lead potential buyers to make
bids that are either higher or lower than
the ones they would make if they had
complete information. Finally, the dis-
tribution of bids for a house is truncated
at the seller's asking price. These char-
acteristics of housing markets imply
that buyers cannot choose houses so as
to maximize a deterministic utility func-
tion subject to a deterministic budget
constraint as required by the standard
model. This report describes a new
model of consumers' choices in housing
markets that incorporates the foregoing
market characteristics. The new model
has been tested econometrically against
the standard model, and the standard
model has been strongly rejected. The
new model gives estimates of buyers'
willingness to pay for housing amenities
that are significantly different from the
estimates produced by the standard
model.
This Project Summary was developed
by EPA's Air and Energy Engineering
Research Laboratory. Research Triangle
Park, NC. to announce key findings of
the research project that is fully docu-
mented in a separate report of the same
title (see Project Report ordering infor-
mation at back).
Introduction
Econometric estimation of households'
willingness to pay for environmental and
other residential amenities often is based
on observations of the effects of envi-
ronmental and other factors on residential
property values. The standard estimation
procedure is based on the hypothesis that
each household occupies a house that
maximized its utility subject to a budget
constraint. It can be shown that, subject
to regularity conditions, utility maximiza-
tion occurs when the household's mar-
ginal willingness to pay for each housing
attribute (e.g., house size, air quality at
the residential location) equals the mar-
ginal price of that attribute.
In the standard procedure, willingness-
to-pay estimation takes place in two steps.
First, a hedonic price function describing
the relation between the observed market
prices of houses and attributes such as
size and air quality is estimated. The first
partial derivative of the hedonic price
function with respect to an attribute level
yields the marginal price function of this
attribute. The numerical marginal prices
pertaining to a particular household can
be obtained by substituting the values of
the attributes of that household's house
into the estimated price function. The
resulting marginal price of the attribute is
then treated as an observation of that
household's marginal willingness-to-pay
for the attribute. In the second estimation
stage, the relation between marginal
willingness-to-pay and relevant explan-
atory variables is specified up to a finite
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set of constant parameters. The param-
eter values are inferred from the "ob-
served" marginal willingness-to-pay val-
ues (i.e., the estimated marginal prices)
using suitable econometric methods.
This hedonic pricing approach has been
used in a variety of studies of willingness-
to-pay for environmental improvements.
However, the method has some serious
difficulties. Three that are particularly
relevant to the research described in this
report are:
1. The assumptions underlying the
standard hedonic model imply that
houses are sold to the households
that are willing to pay the highest
prices for them (i.e., houses are sold
to the highest bidders). However, it
is unlikely that houses are sold to
the highest bidders in real housing
markets. In real markets, houses
normally are sold through sequen-
tial bidding processes. Bids must be
accepted or rejected as they are
received, and a seller is not able to
assemble all bids that might be
made before deciding which one to
accept. Therefore, the seller does
not necessarily accept the highest
bid that might be made. Substantial
transactions costs tend to prevent
the rectification of erroneous de-
cisions regarding the rejection and
acceptance of bids.
2. Potential buyers bid for houses with
little or no knowledge of the bids
made by others. Therefore, buyers
do not know the price at which a
house will be sold in advance of the
sale, and buyers have incomplete
information about the house price
components of their budget con-
straints. This makes it impossible
for buyers to choose houses by
maximizing utility subject to deter-
ministic budget constraints. Trans-
actions costs tend to prevent buyers
from rectifying any resulting errors
in bidding or in choosing houses.
3. The sale price of a house rarely
exceeds the seller's asking price.
Thus, the asking price effectively
truncates the distribution of bids. If
ignorance of the market or other
factors cause sellers to establish
asking prices that are below the
prices buyers are willing to pay,
then truncation of the distribution
of bids at the asking prices will
prevent identification of the buyers
willing to make the highest bids.
Therefore, houses will not neces-
sarily be sold to these buyers.
Moreover, a process in which
buyers choose houses by maximiz-
ing utility subject to budget con-
straints will not assign unique
buyers to houses.
This report describes a model of con-
sumers' choices in housing markets that
incorporates the market characteristics
just discussed. The new model and the
standard model have been estimated and
tested econometrically to determine
which provides a better description of the
operation of these markets. The formula-
tion and testing of the new model and the
implications of this model for willingness-
to-pay estimation are summarized in
report Chapter 1. The remaining chapters
are more detailed discussions: Chapter 2
gives detailed mathematical formulations
of the new and standard models; Chapter
3 presents econometric tests of the
models; and Chapter 4 presents the
willingness-to-pay analysis.
Formulation of the Models
In this research, the new and standard
models have been formulated as bidding
models. Bidding models are housing
market models in which the dependent
variable is the sale price of a house,
modeled as a function of attributes of the
house and the buyer. It is assumed in
these models that sales occur as results
of a bidding process.
The standard model used in this re-
search is the well-known bid rent model
of housing markets. Roughly speaking,
this model assumes that houses are sold
to the buyers who bid the most for'them,
and a house's sale price is the amount the
highest bidder is willing to pay for that
house. Under the customary assumptions
of hedonic analysis, the bid rent model is
equivalent to the model in which buyers
choose houses to maximize utility subject
to a budget constraint. Moreover, under
the same assumptions, willingness-to-
pay analysis based on the bid rent model
is equivalent to analysis based on a utility
maximization approach. Thus, the stand-
ard market model can be tested using
either the bid rent or the utility maximi-
zation form. The bid rent form is more
convenient in the research discussed
here.
In the version of the bid rent model
used here, the probability density that a
house with observed attributes X has sale
price p8 is:
Density (p.) = f[pjW(X)],
(1)
where W(X) denotes the buyer's willing-
ness to pay for a house with observed
attributes X, and f is the function giving
the probability density that ps is the
highest bid made for the house.
In the new model, the probability
density that a house with observed attri-
butes X and asking price Pa has sale price
Ps is:
Density (ps) =
Q(pa|Pa)g[pa|W(X)]|P(sale) (2)
if ps < Pa, where g is the probability
density that the buyer makes a bid of ps,
Q(ps|Pa) is the probability that the seller
accepts a bid of size ps, and P(sale) is the
probability that the house is sold at any
price. The probability that ps =Pa is:
ProbfPa) = {1 - G[Pa| W(X)]}/P(sale), (3)
where G is the cumulative probability
distribution function corresponding to g.
The probability that ps > Pa is zero. P(sale)
is given mathematically by:
P(sale) =
/£• Density(p)dp + {1 - G[Pa|W(X)]}. (4)
Equations (2-4) constitute a bidding
model of consumers' choices in housing
markets that is analogous to the standard
model but that is based on different
assumptions concerning the bidding pro-
cess that operates in housing markets. In
particular, Equations(2-4) incorporate the
hypotheses that sellers process bids
sequentially, buyers do not have certain
knowledge of the bids made by other
buyers, and the distribution of bids for a
house is truncated at the sellers' asking
price. These hypotheses are inconsistent
with the standard model's assumption
that houses always are sold to the buyers
willing to pay the most for them. As a
consequence, it is possible to distinguish
empirically between the new model [i.e.,
the model of Equations (2-4)] and the
standard model and to test each against
the other empirically to determine which
better explains the available data.
Data
The new and standard models were
tested and compared empirically using
data describing a sample of 1196 houses
sold in Baltimore City and County, Mary-
land, during 1978. The data were as-
sembled from records of the Central
Maryland Multiple Listing Service, Balti-
more City and County tax records, the
U.S. Census Bureau, the Maryland
Bureau of Air Quality, and the results of
previous investigations of school quality
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and crime levels in the Baltimore area.
The data include the sale price, asking
price, and annual property tax liability of
each sampled house as well as 49
attributes of the house and its neighbor-
hood. Environmental quality is character-
ized in these data by three air pollution
variables: annual geometric mean partic-
ulate concentration, annual average N02
concentration, and annual maximum
hourly Os concentration.
Results
The new and standard models were
both estimated (i.e., fit to the data) using
standard econometric methods. These
methods make it possible to test the
models against one another by comparing
how well each model fits the data. If the fit
of one model is significantly worse than
the fit of the other, then it is possible to
conclude with a high degree of confidence
that the worse fitting model is incorrect.
Figures 1 and 2 showthe results of this
comparison process. In Figure 1, the
observed sale prices of 50 randomly
selected houses are plotted against the
predicted sale prices obtained from the
new model. In Figure 2, the observed sale
prices of the same 50 houses are plotted
against the predicted sale prices obtained
from the standard model. The predicted
sale prices were computed as expected
values conditional on attribute levels and,
in the case of the new model, asking
prices. Figure 1 shows that the new
model fits the observations well: there is
no indication of serious errors in the
model. Figure 2 shows that the standard
model fits the observations much more
poorly than does the new model.
The differences in fit shown in Figures
1 and 2 are reflected in the root-mean-
square (RMS) errors of the two models'
predictions of'houses' sale prices. For the
1196 houses in the full data set, the new
model has a RMS prediction error of
$2,830; whereas, the standard model has
a RMS prediction error of $11,360.
These results suggest strongly that the
standard model is erroneous. It is possible
to perform a formal statistical comparison
of the new and standard models. This
comparison leads to the conclusion that,
if the standard model is correct, the
observed goodness-of-fit results are
events whose probabilities of occurrence
are less than 0.0001. Thus, it is virtually
certain that the standard model is er-
roneous and that the new model provides
a better explanation of the operation of
the Baltimore housing market than does
the standard model. Since the bid rent
model is equivalent to the utility maxi-
70
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Joel L Horowitz is with the University of Iowa, Iowa City, I A 52242.
J. David Mobley is the EPA Project Officer (see below).
The complete report, entitled "Inferring Willingness to Pay for Housing Amenities
from Residential Property Values," (Order No. PB 86-103 082/AS; Cost:
$ 11.95, subject to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Air and Energy Engineering Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27.711
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Official Business
Penalty for Private Use $300
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