Download Public Finance Seminar

Public Finance Seminar
Spring 2013, Professor John Yinger
Department of Public Administration
The Maxwell School, Syracuse University
Hedonics
These notes are provided for teaching
purposes only and are not to be quoted
without the permission of the author.
Comments are welcome.
Class Outline

What Are Hedonic Regressions?

The Rosen Framework

Methods for Separating Bidding and Sorting

Hedonic Vices
Hedonics
Hedonic Regressions

A regression of house value or rent on
housing and neighborhood traits is called a
hedonic regression.

Hedonic regressions appear in markets for
other products with multiple attributes, such
as automobiles or computers.
◦ But these notes focuses on the application of
hedonic analysis to housing markets.
Hedonics
Hedonic Regressions, 2

Many of the outcomes of interest in public finance, such as
public services (e.g. education) or neighborhood amenities
(e.g. air quality) are not traded in private markets.
◦ As a result, we cannot directly observe demand for these
outcomes or determine the value households place on changes
in these outcomes (as in benefit-cost analysis, for example).

However, households have to compete for access to
locations with desirable attributes, and their bids reveal
something about the “missing” demand.
◦ Thus, hedonic regressions provide the main way for scholars to
study household demand for public services and neighborhood
amenities.
Hedonics
Hedonic Regressions, 3

Hedonic regressions have been used, for example, to study
household demand for:
◦ School quality
◦ Clean air
◦ Protection from crime
◦ Neighborhood ethnicity
◦ Distance from toxic waste sites
◦ Distance from public housing
Hedonics
Hedonic Regressions, 4

As we know from the theory of local public finance,
different household types also compete against each
other for entry into the most desirable neighborhoods
(see Ross and Yinger’s Handbook chapter, 1999)

This competition takes place through bids on housing.

So hedonic regressions also can, in principle, tell us
something about the way different household types
sort across locations—and therefore can give us
insight into the nature of our highly decentralized
federal system.
Hedonics
The Rosen Framework

A famous paper by Sherwin Rosen (JPE 1974)
gives a framework everyone uses based on a bid
functions and their mathematical envelope.
◦ A bid function is the amount a household type would
pay for housing at different levels of an amenity, holding
their utility constant.
◦ The hedonic envelope is the set of winning bids; each
point on the envelope is tangent to one of the
underlying bid functions.
Hedonics
The Rosen Picture

Hedonics
Here is the the Rosen picture, where z1 is a product
attribute, θ indicates a bid function, and p is the
hedonic price function, that is, the envelope.
Household Heterogeneity

Note that Rosen’s framework is designed to
consider heterogeneous households—that is,
households with different demands for the
amenity as indicated by utility, u*, in the bids.

If all households are alike, the hedonic price
function is the same as the bid function.

But with heterogeneous households, we only
observe one point—the tangency point—on each
household type’s bid function.
Hedonics
Rosen and Housing

Rosen’s paper is about hedonics in general, not
just about housing.

But it is consistent with the bidding and sorting
framework developed at about the same time in
local public finance (as Rosen mentions).

However, many housing applications make two
changes in the Rosen framework:
◦ They ignore the supply side.
◦ They look at bids per unit of housing services.
Hedonics
Rosen’s Supply Side

In Rosen’s framework, each firm has an offer function,
which is the amount it would pay to supply a given
level of an attribute, holding profits constant.

With heterogeneous firms and households, the
hedonic price function is the joint envelope of firm
offer and household bid functions.

Because amenities are not supplied by firms (and
because most house sales are of existing housing), the
supply side is not central in most housing applications
and is simply ignored.
◦ That is, the distribution of amenities is taken as given.
Hedonics
Housing Services

Rosen treats all attributes the same way.

But many housing scholars distinguish between
housing services, H, which are a function of
structural housing traits, X, and the price per unit of
housing services, P, which is a function of public
services and neighborhood amenities, A.
◦ The value of a house is V = P{A}H{X}/r
◦ The index of housing services, H{X}, may be assumed not
to vary across households.

Hedonics
You need to look for this distinction when reading the
literature.
The Rosen Challenge

The Rosen framework is valuable because it brings
together bidding and sorting.

Just as in the local public finance literature, households with
steeper bid functions end up at places with higher
amenities.

But this framework also reveals an enormously difficult
challenge:
◦ The relationship we can observe—the hedonic price function—
reflects both (a) the underlying bid functions and (b) the factors
that determine sorting across household types.
Hedonics
◦ Because we only observe one point on each bid function, there is
no simple way to separate these two phenomena, that is, no
simple way to study either the determinants of household
demand for amenities (bidding) or the determinants of
household sorting across locations.
Interpreting the Envelope

Many studies simply estimate the hedonic, that
is, the bid-function envelope.
◦ For example, there is a large literature on school
quality “capitalization” in which most studies simply
regress house values on a measure of school
quality (and controls).

Hedonics
Two types of information can be gained from
these studies.
Interpreting the Envelope, 2

First, a positive, significant coefficient for an
amenity indicates that people value that amenity.
◦ If people do not care about an amenity, their bid
functions will be flat—and so will the envelope,
regardless of sorting.

Results must be interpreted with care.
◦ One can say that a 1 unit increase in school quality
leads to an x% increase in house values, all else equal.
Hedonics
◦ One cannot say that people are willing to pay x% more
for housing when school quality increases one unit,
because the result does not refer to any particular
household type and it mixes bidding and sorting.
Interpreting the Envelope, 3

Second, Rosen shows that a utility-maximizing household sets its
marginal willingness to pay (MWTP) for an attribute equal to the
slope of the hedonic price function.
◦ This slope is called the implicit price of the attribute.
◦ In our terms, it is (∂V/∂S) = (∂PE/∂S)H/r.

Hence, we can observe every household’s MWTP at the level of the
attribute they consume—and we can calculate the average MWTP.

But this average MWTP has a limited interpretation.
◦ It indicates only what people would pay on average for a small, equal
increase in the amenity at all locations, starting from the current
equilibrium.
◦ This average cannot be compared across places or across time
because the equilibrium is not the same.
Hedonics
Separating Bidding and Sorting

To go beyond this limited information from the
hedonic itself, scholars must separate bidding and
sorting.

As just noted, this is inherently a very difficult
issue because we only observe one point on each
bid function.

Thus, there is no general solution to this
problem.
◦ It is impossible to separate bidding and sorting
without making some strong assumptions!
Hedonics
Separating Bidding and Sorting

Scholars have come with 5 different approaches to
separating bidding and sorting, based on different
assumptions with different strengths and weaknesses.
◦ 1. The Rosen two-step method.
◦ 2. The Epple et al. general-equilibrium method.
◦ 3. Fancy econometric methods (often linked to Heckman
and co-authors).
◦ 4. Discrete-choice methods.
◦ 5. The Yinger “derive the envelope” method.
Hedonics
Method 1: The Rosen Two-Step

Rosen proposes a two-step approach to
estimating hedonic models.
◦ Step 1: Estimate a hedonic regression using a
general functional form (the envelope) and
differentiate the results to find the implicit or
hedonic price, VS, for each amenity, S.
◦ Step 2: Estimate the demand for amenity S as a
function of VS (and of income and other things).
Hedonics
Rosen’s First Step

The idea of the first step is to use as general an
estimating method as possible to approximate
the hedonic envelope.
◦ Early studies used a Box-Cox form, which has linear,
double-log, and semi-log forms as special cases. (We
will return to this form later.)
◦ Some more recent studies use a nonparametric
technique, such as “local linear regression.”
◦ Some studies still just use linear, log, or semi-log, which
are undoubtedly not non-linear enough.
Hedonics
Endogeneity in the Rosen Second Step

As later scholars pointed out (e.g. Epple, JPE
1987), the main problem facing the 2nd step
regression in the Rosen framework is that the
implicit price is endogenous.
◦ The hedonic function is almost certainly nonlinear, so
households “select” an implicit price when they select
a level of S.
◦ Households have different preferences, so the level of
S, and hence of VS, they select depends on their
observed and unobserved traits.
Hedonics
Addressing Endogeneity in the 2nd Step

Scholars have identified two ways to deal with this
endogeneity in the second-step of the Rosen procedure.

The first is to find an instrument, which is, of course, easier
said than done.
◦ Some early studies pooled data for metropolitan areas and used
construction cost as an instrument; however, hedonics operate at
the level of a metropolitan area and this pooling is not
appropriate.
◦ Other studies (see the Handbook chapter by Sheppard in 1999)
use nearby prices as an instrument; however, nearby prices may,
because of sorting, reflect the same unobservables that cause the
problem in the first place.
◦ No general solution through instruments has been identified.
Hedonics
Addressing Endogeneity, 2

The other approach is to assume that the price elasticity of
demand for the amenity equals -1.0. See, for example,
Bajari and Kahn (J. Bus. and Econ. Stat. 2004).

In this case the demand for S can be written as follows:
S  KY  (VS ) 1 or S (VS )  KY 
where Y is income and K is other demand factors.

Because the implicit price now appears on the left-side,
nothing endogenous remains on the right side and
endogeneity bias disappears.

The problem with this approach, of course is that the price
elasticity is the main thing we are trying to estimate!
Hedonics
Method 2: General Equilibrium

Epple and Sieg (JPE 1999); Epple, Romer, and Sieg
(Econometrica 2000).
◦ These scholars derive a general equilibrium model of bidding,
sorting, and public service determination with a specific
functional form for the utility function.
◦ Their model includes an income distribution and a taste
parameter with an assumed distribution.
◦ They solve for percentiles of the income distribution (and other
things) in a community as a function of the parameters and then
estimate the values of the parameters that best approximate the
income distribution in the communities in the Boston area.
◦ The second paper includes the demand for public services
through voting.
Hedonics
Method 2: General Equilibrium, 2

This approach obviously requires considerable technical
skill on modeling and econometrics.

It also makes some strong assumptions.
◦ For our purposes, the most important weaknesses are (1) that
the model contains a single index of public services and
amenities and (2) that it requires a certain distribution of
preferences and income.
◦ Their index includes school quality and the crime rate, with an
estimated weight—a very restrictive approach.
◦ So far, the only applications of this method are to communitylevel data.
Hedonics
Method 3: Fancy Econometrics

One approach is to use models of random coefficients (i.e. different
bid-function slopes) with endogenous variation (due to sorting)—
but no theory; the required instruments are hard to find!
◦ The models were developed by Heckman and Vytlacil (JHR 1998), among
others.
◦ A hedonic application is Chay and Greenstone (JPE 2005) on air pollution.

Another approach is to use non-parametric techniques that
recognize the difference in curvature between bid functions
and their envelope.
◦ Ekeland, Heckman, and Nesheim (JPE 2004)
◦ This approach has not yet been applied to housing, so far as I know, and is,
like the Epple et al. approach, limited to a single amenity index.
Hedonics
Method 4: Discrete Choice

Several scholars have pointed out that a hedonic price
function can be estimated with a discrete-choice model
of the allocation of household types to housing types as
a function of housing traits.

A recent article by Bayer, Ferreira, and McMillan (JPE
2007) takes this logic a step farther.
◦ They estimate a multinomial logit model of the allocation of
heterogeneous households to individual houses.
◦ This model directly addresses sorting—and makes it possible to
simulate new sorting equilibria with other assumptions, such as
equal income distributions for black and white households.
Hedonics
Method 4: Discrete Choice, 2

This method is clever, but requires strong
assumptions.
◦ They assume that utility functions are linear!
◦ They assume that housing prices 3 miles away are a good
instrument for actual housing prices.
◦ They assume that the hedonic price function is linear.
Hedonics
Method 5: Derive the Envelope

Yinger (2012) derives a new form for the hedonic
equation using the standard bidding model with
constant-elasticity demand functions for the
amenity and housing.

This paper draws on the standard theory of
sorting to derive a bid-function envelope
across households with different preferences.

This approach can be generalized to any number
of amenities and can be applied to the housingcommuting trade-off in a standard urban model.
Hedonics
Method 5: Derive the Envelope, 2

This approach estimates the price elasticity of demand
for S, μ, in the hedonic equation itself.
◦ Thus the main parameter of interest, μ, can be estimated
without encountering the standard endogeneity problem.

This approach allows for a general treatment of
household heterogeneity, but then integrates out the
determinants of this heterogeneity in deriving the bidfunction envelope.
◦ So this approach accounts for household heterogeneity
without requiring data on household characteristics to
estimate μ.
Hedonics
Method 5: Derive the Envelope, 3

Yinger’s first key assumption is that households
have constant-elasticity demand functions for the
amenity and housing:
 

S  KS N Y W

   
H  K H M Y  P 1   
  r 



ˆ
 KH M Y P


where the “^” indicates a before-tax housing
price.
Hedonics
Method 5: Derive the Envelope, 4

The bid functions that result take a Box-Cox
form.

The Box-Cox form is
X
( )


( X  1)

if   0
and
X (  )  ln{ X } if   0
Hedonics
Method 5: Derive the Envelope, 5

To be more specific, the bid functions are:


Pˆ{S }
(1 )
 C  S
 (1  )/  
where C is a constant of integration and

   KS N
Hedonics

 1/ 

KH M Y
( /  ) 

1
Method 5: Derive the Envelope, 6

The next step is to bring in sorting.
◦ A fundamental theorem is that sorting depends on
the slope of the bid function: a household class
with a steeper slope sorts into a jurisdiction with a
higher value of S.
ˆ
ˆ ; ψ contains all non-shared terms in P
◦ The slope is P
S
S
and is thus an index of this slope.
◦ So a steeper slope (= higher ψ) is associated with a
higher S as illustrated in the following graphs.
Hedonics
Bid Per Unit of Housing Services
Figure 1: Bid Functions for School Quality
0
10
20
30
40
50
Percent Passing
Hedonics
60
70
80
90
100
Figure 2. Supply and Demand for School Quality
Envelope
Bid Per Unit of Housing Services
Offer function m
Offer function k
Bid function i
Bid function j
0
10
20
30
40
50
60
70
80
90
100
Percent Passing
Slope of Bid or Offer Function
Slope of bid function i
0
Hedonics
Slope of bid function j
ψ{S}
Slope of
envelope
Slope of offer
function k
10
20
Slope of offer function m
30
40
50
60
Percent Passing (S)
70
80
90
100
Method 5: Derive the Envelope, 7

Yinger’s approach is to assume that the
equilibrium relationship between ψ and S can be
approximated with the following equation:
S   1   2 
3

The σs are parameters to be estimated.
◦ The sorting theorem predicts that so long as σ3 > 0, as
expected, σ2 > 0, which can be tested.
Hedonics
Method 5: Derive the Envelope, 8

This assumption leads to the following form
for the pre-tax hedonic envelope:
 
Pˆ
E
( 1 )
1 ( ) 1 ( )
 C0 
S 
S
2
2
2
3
where
1  1  ; 2 
Hedonics
(1   )

; 3  2 
1
3
Extension to Multiple Amenities

So long as Si is not directly a function of Sj,
this approach can be extended to multiple
amenities, with two terms in the bid function
for each amenity.

This approach assumes that amenity space is
dense enough so that we can pick up bidding
for Si holding other amenities constant.

Highly correlated amenities may need to be
combined into an index.
Hedonics
The Final Hedonic Equation

The final estimating equation is
P E {S , t}H { X } Pˆ E {S}H { X }
V

r
r  

To estimate this (nonlinear!) equation:
◦ Use the form just derived for Pˆ .
◦ Assume a multiplicative form for H{X}.
◦ Introduce the degree of property tax capitalization.
E
Hedonics
Special Cases

This general Box-Cox specification includes most of
the parametric estimating equations in the literature as
special cases,

On the left side, the assumption that the price
elasticity of demand for housing, ν, equals -1 leads to a
log form, which is used by most studies.
◦ Studies that use this form do not recognize that they are
making this assumption about ν .

Hedonics
On the right side, a wide range of functional forms are
possible depending on the values of μ and σ3.
Special Cases, 2
Value of μ
Value of σ3 Formula
= -0.5
=∞
= -1
=∞
= -∞
=1
= -∞
=∞
< 0; ≠ -1
=∞
 1  1   1 

    inverse

 2  S 
 1  1 

 ln{S}  log

 2 

1 2
 1 S
S  quadratic
2
2 2
 1  1 

 S  linear

 2 
 1  1  S (1  )/   1 


  Box-Cox

(1


)
/


 2 
Note: μ = -∞ implies a horizontal demand curve;
σ3 = ∞ implies no sorting
Hedonics
Hedonic Vices

Despite the fame of the Rosen article, many
scholars have forgotten some of its key
messages.

What follows is my guide to “hedonic vices,”
that is, to approaches that are not consistent
with the Rosen framework and related
literature.
Hedonics
Functional Form Vices

1. The use of a linear (or semi-log or loglinear) form for the hedonic equation.

It is difficult to come up with assumptions that
yield these forms.

With constant-elasticity demands, a linear
form arises, for example, with no sorting
based on bid-function slopes and the
assumption that μ = -2; see the following
figure.
Hedonics
Functional Form Vices, 2
PE{S}
Figure 1. Example of a Linear Hedonic Envelope
0
10
20
30
40
50
60
70
School Passing Rate (Percent)
Hedonics
80
90
100
Functional Form Vices, 3

2. Contradictions between Rosen’s 2 steps.

The envelope is mathematically connected to the bid
functions.

It makes no sense to estimate a hedonic based on one
assumption about the price elasticity and then to
estimate the price elasticity in the next step.
◦ For example, a quadratic form assumes that μ = -∞, so it
makes no sense to get implicit prices from this form and
then to estimate μ.
Hedonics
Control Variable Vices

1. Using demand variables as controls.

The Rosen framework implies that bid
functions depend on demand traits, such as
income, but the hedonic envelope does not.

Adding demand variables therefore changes
the estimation into a bid-function estimation,
not an envelope estimation.
Hedonics
Control Variable Vices, 2

There is an exact analogy here to cost functions, as
pictured on the next slide.

The family of short-run average cost functions has
plant size as a variable.

The long-run average cost function, which is the
envelope of the short-run functions, does not have
plant size as an argument.

Short-run average cost functions are analogous to bid
functions; plant size is analogous to demand factors;
and long-run average cost functions are analogous to
the hedonic envelope.
Hedonics
Control Variable Vices, 3
Short- and Long-Run Average Cost Curves,
CES Production Function with Increasing Returns
0.35
0.3
Average Cost
0.25
0.2
0.15
0.1
0.05
0
0
20
40
60
80
100
120
Quantity
SRAC1
Hedonics
SRAC2
SRAC3
SRAC4
SRAC5
SRAC6
LRAC
Control Variable Vices, 4

This change in interpretation has two critical
implications.
◦ First, included demand variables are endogenous; they
bring the 2nd stage endogeneity into the 1st stage.
◦ Second, included demand variables must be interacted
with the amenity variables; otherwise the coefficients
of the amenities do not vary with demand traits and
there can be no sorting!
Hedonics
Control Variable Vices, 5

One cannot avoid this problem by using
neighborhood-level demand traits instead of
household-level demand traits.
◦ These two types of variables are highly correlated.
◦ Neighborhood level traits remove the transitory
component in household income and may
therefore be a better approximation to household
permanent income than is actual household
income.
Hedonics
Control Variable Vices, 6

One cannot avoid this problem by arguing that
neighborhood-level demand traits, such as income, are
neighborhood amenities and therefore need to be included.
◦ Neighborhood income, education, and other demand traits might,
indeed, be viewed as amenities by house buyers (although these
traits cannot be observed directly).
◦ But this possibility does not alter the fact that including them
changes the meaning of the regression.

This leaves researchers with two choices:
◦ Leave out these traits and estimate a (possibly biased) hedonic
regression
Hedonics
◦ Include these traits, treat them as endogenous, and interact them
with amenities, and interpret the regression as a bid-function
regression.
Control Variable Vices, 7

2. The use of neighborhood-level fixed effects.

One strategy for addressing omitted variable bias in
Rosen’s first step is to include geographic fixed effects,
such as census tract fixed effects.

The problem is that these fixed effects pick up the
impact of demand factors, at least to some degree.

As a result, the use of small-area fixed effects runs into
the same interpretation issues as the use of
neighborhood-level demand variables.
Hedonics
Interpretation Vices

1. Misinterpretation involving the mean
MWTP.

Some studies estimate a linear regression and
assume that the coefficient is the mean
MWTP.

But a mis-specified regression does not yield a
unbiased estimate of this mean.
Hedonics
Interpretation Vices, 2

Example from Nguyen-Hoang/Yinger (2013); linear
specification understates true mean MWTP by 46 percent.
ln{PE}
Figure 5. Error Using Linear Prediction for Mean MWTP
0
10
20
30
40
50
60
70
School Passing Rate (Percent)
Actual
Hedonics
Linear Prediction
80
90
100
Interpretation Vices, 3

Other studies compare the mean MWTP from a
study in one location (or at one time) with the
mean MWTP in another location (or at another
time).

These comparisons are not warranted, because
one cannot assume that the underlying equilibria
are the same at the two locations (or at the two
times).

The hedonic mean MWTP is a very limited
concept!
Hedonics
Interpretation Vices, 4

2. Misinterpreting Difference Regressions

One common strategy for estimating the impact
of an amenity that changes is to obtain panel data,
identify double sale data, and then look at the
change in house value, ΔV, as a function of the
change in the amenity, ΔS.

This strategy is equivalent to the use of a fixed
effect for each house, and therefore eliminates
bias in the coefficient of ΔS from all timeinvariant house and neighborhood traits.
Hedonics
Interpretation Vices, 5

The problem is that the coefficient of the ΔS variable could
reflect any of three things:
◦ 1. The willingness to pay of existing residents for the change in S
(which is what most people assume).
◦ 2. The willingness to pay of new residents (due to re-sorting) for
the new S minus the willingness to pay of previous residents for
the old S, which is neither group’s willingness to pay.
◦ 3. Shifts in the distribution of households that have nothing to do
with the change in S, such as those due to immigration.

Hedonics
No method now available makes it possible to separate
these possibilities.
Interpretation Vices, 6

Hedonics
Example from Bogin (2012).
Interpretation Vices, 7

One way to summarize this issue is to remember
that the estimated coefficients in a hedonic
regression reflect the current equilibrium in the
area.

The coefficient of a change in an amenity variable
reflects the impact of changes in the
equilibrium—not just of changes in that variable.

The change in the equilibrium cannot be
accounted for simply by including control
variables.
Hedonics
References
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Hedonics
Bajari, Patrick, and Matthew E. Kahn. 2005. “Estimating Housing Demand with an Application to Explaining Racial
Segregation in Cities.” Journal of Business and Economic Statistics 23 (1): 20–33.
Bayer, Patrick, Fernando Ferreira, and Robert McMillan. 2007. “A Unified Framework for Measuring Preferences for
Schools and Neighborhoods.” Journal of Political Economy 115 (4): 588–638.
Bogin, Alex. 2011. “The Impact of a No Child Left Behind ‘Failing’ School Designation on Local Property Values”.
Working Paper. Syracuse University.
Chay, Kenneth Y., and Michael Greenstone. 2005. “Does Air Quality Matter? Evidence from the Housing Market.”
Journal of Political Economy 113 (2): 376–424.
Ekeland, Ivar, James J. Heckman, and Lars Nesheim. 2004. “Identification and Estimation of Hedonic Models.” Journal
of Political Economy 112 (S1): 60–109.
Epple, Dennis. 1987. “Hedonic Prices and Implicit Markets: Estimating Demand and Supply Functions for
Differentiated Products.” Journal of Political Economy 95 (1): 59–80.
Epple, Dennis, and Holger Sieg. 1999. “Estimating Equilibrium Models of Local Jurisdictions.” Journal of Political
Economy 107 (4): 645-681.
Epple, Dennis, Thomas Romer, and Holger Sieg. 2001. “Interjurisdictional Sorting and Majority Rule: An Empirical
Analysis,” Econometrica 69 (6): 1437-1465.
Heckman, James J., and Edward J. Vytlacil. 1998. “Instrumental Variables Methods for the Correlated Random
Coefficient Model: Estimating the Average Rate of Return to Schooling When the Return Is Correlated with
Schooling.” Journal of Human Resources 33 (Fall): 974–87.
Rosen, Sherwin. 1974. “Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition.” Journal
of Political Economy 82 (1): 34–55.
Ross, Stephen, and John Yinger. 1999. “Sorting and Voting: A Review of the Literature on Urban Public Finance.” In
Handbook of Urban and Regional Economics, edited by Paul Cheshire and Edwin S. Mills, 3:2001–2060. North
Holland.
Sheppard, Stephen, “Hedonic Analysis of Housing Markets,” In Handbook of Urban and Regional Economics, Volume 3,
Applied Urban Economics, edited by Paul Cheshire and Edwin S. Mills (Amsterdam: North-Holland, 1999), 15951635.
Yinger, John, and Phuong Nguyen-Hoang. 2013. “Hedonic Vices.” Paper presented a the annual meetings of the
Association for Education Finance and Policy, March 16, 2013.
Yinger, John. 2012. “Hedonic Markets and Explicit Demand: Bid-Function Envelopes for Public Services and
Neighborhood Amenities.” Working Paper, Center for Policy Research, Syracuse University, December.