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Empirical Welfare Analysis with Hedonic Budget Constraints
\title{Empirical Welfare Analysis with Hedonic Budget Constraints}
\author{Debopam Bhattacharya\thanks{
Financial support from Economic and Social Research Council (ESRC) is
acknowledged. We are grateful to Weida Liao for helpful conversations
regarding partial differential equations, and to Whitney Newey for feedback.
Email for correspondence: [email removed]} \\
University of Cambridge \and Ekaterina Oparina \\
LSE \and Qianya Xu \\
University of Cambridge}
\date{31st October, 2024 \\
\vspace{-1cm}}
\maketitle
\begin{abstract}
We analyze demand settings where heterogeneous consumers maximize utility
for product attributes subject to a nonlinear budget constraint. We develop
nonparametric methods for welfare-analysis of interventions that change the
constraint. Two new findings are Roy's identity for smooth, nonlinear
budgets, which yields a Partial Differential Equation system, and a
Slutsky-like symmetry condition for demand. Under scalar unobserved
heterogeneity and single-crossing preferences, the coefficient functions in
the PDEs are nonparametrically identified, and under symmetry, lead to
path-independent, money-metric welfare. We illustrate our methods with
welfare evaluation of a hypothetical change in relationship between property
rent and neighborhood school-quality using British microdata.
Keywords: Hedonic model, nonlinear budget, nonparametric identification,
welfare, compensating/equivalent variation, partial differential equation,
Slutsky symmetry, Roy's Identity, Path Independence.
JEL code: C14, I30, H23
\end{abstract}
\section{Introduction}
Nonlinear budgets arise in a variety of economic applications. A leading
example is hedonic modelling of markets for differentiated goods with a
large number of available varieties, but where each variety can be viewed as
a distinct bundle of a limited number of attributes (Rosen, 1974). Examples
include cars, houses, hotels, etc. An important characteristic of such
markets is that in equilibrium, the marginal price of an attribute typically
varies with quantity, making budget frontiers nonlinear (Diewert, 2003;
Ekeland et al 2004). For example, Goodman (1983) records that for new cars,
the willingness to pay for additional mileage per gallon (MPG) typically
decreases as MPG increases. In empirical models of labour supply, the income
tax rate is often progressive, causing potential workers to face piecewise
linear budget constraints. Since presence of bunching at kink points is rare
in the data, MaCurdy et al. (1990, Section II.D) propose replacing the
piecewise linear budget-constraints with a smooth budget frontier to reflect
optimization and/or measurement error.\footnote{
We are grateful to Whitney Newey for making us aware of this work.} The
present paper develops econometric methods for welfare analysis of policy
interventions in such settings. In contrast to previous research analyzing
this problem, we allow for nonparametric, unobserved preference
heterogeneity across consumers, and perform exact analysis, as opposed to an
approximate one based on linear interpolation of nonlinear budget frontiers
(Palmquist, 1988, 2005).
As a motivating example, consider the well-known relationship between
housing costs and neighborhood school quality (Sheppard, 1999). To enable
children to attend nearby schools, local governments often mandate
`catchment area' rules, which restrict school access solely to neighborhood
children. This, however, means that the presence of a good school makes its
adjoining residential neighborhood attractive, and raises housing costs.
This leads to wealthier families moving in from worse school districts,
aggravating existing socioeconomic segregation. One potential way to stop
this vicious cycle is to relax catchment area restrictions. This would lead
housing costs to become less entangled with school quality, and change
choices in equilibrium. For example, Machin and Salvanes (2016) find that
relaxing catchment area boundaries in Oslo caused significant weakening of
the price-school-quality relation. However, the overall welfare effects of
such policy interventions are likely to be heterogeneous, depending on both
household preferences over consumption and neighborhood school quality, and
on their income. The question is: how can we calculate the distribution of
these heterogeneous welfare effects, using microdata on housing and schools.
The present paper provides a framework and corresponding econometric
methodology to achieve this objective.
In what follows, we present an economic model of choice for a heterogeneous
population of consumers, each facing a convex budget set characterized by a
nonlinear, smooth frontier. We then derive the analog of Roy's Identity for
this setting, which yields a system of linear partial differential equations
(PDEs). When unobserved heterogeneity is a scalar, and a single-crossing
condition is satisfied by preferences, the coefficient functions of the PDEs
can be identified from quantiles of demand. We show that these quantile
demand functions must satisfy a Slutsky like symmetry condition which is
different from the standard case of linear budget frontiers (Hausman and
Newey, 2016). Furthermore, welfare at the quantile can be expressed as a
line integral whose value is path-independent under the above symmetry
condition. These steps can be repeated separately for each quantile of
unobserved heterogeneity to obtain the entire distribution of welfare
effects. We emphasize here that the key purpose of this paper is to derive
welfare measures, assuming that the budget frontiers are already identified.
This means that the relation between price and the attribute of interest
(e.g. property rent and neighborhood school quality) is assumed to be known,
or consistently estimable from the data. We do not discuss -- and indeed, do
not contribute to solving -- well-known issues of omitted variable problems
that jeopardize the identification of this relationship (Black, 1999).
Our work is substantively related to Heckman et al. (2010), who show how to
nonparametrically identify consumers' marginal utility for a single
continuous attribute in a hedonic setting, where unobserved heterogeneity is
a scalar, preferences are quasilinear in consumption and satisfy a
single-crossing condition. For the present paper, we borrow the set-up in
Heckman et al. (2010), except that utilities are not assumed to be
quasilinear in consumption, and our focus is on welfare effects, which we
show to be obtainable \textit{without} identifying the underlying marginal
utilities, the focus of Heckman et al. (2010). In fact, our paper continues
a line of research started by the seminal article of Hausman (1981),
subsequently refined in Hausman and Newey (2016), who show that in a demand
setting with one continuous inside good, \textit{linear} budget frontiers
and general heterogeneity, welfare distributions resulting from a price
change are not point-identified. In contrast, our setting allows hedonic
budget frontiers to be nonlinear, but (A) restricts heterogeneity to be
one-dimensional, and (B) imposes a single-crossing condition, analogous to
Heckman et al. (2010). We later show how to include additional attributes
into the analysis. Blomquist and Newey (2002) and Blomquist et al. (2021)
have investigated identification of demand with general heterogeneity when
budget constraints are continuous and piecewise linear, with the slope
changing at finitely many kink points.\footnote{
Once demand distribution is identified for hypothetical linear budget
constraints, welfare analysis would resort to methods developed in Hausman
and Newey (2016).}\medskip
The rest of the paper is organized as follows. Section 2 describes the
set-up and states the key assumptions. Section 3 presents the nonparametric
analysis of the problem where functional forms of utilities and how
unobserved heterogeneity enter them are not specified. In particular, we
show how to obtain the analogs of Roy's identity and Slutsky symmetry in
this setting, and how to use the resulting system of PDEs to obtain welfare
measures, using data from a large number of markets, each characterized by
its own budget frontier. Section 3.3 extends the nonparametric analysis to
include additional attributes. Section 4 presents the empirical
illustration, and finally, section 5 concludes with directions for future
research. All figures and tables are collected at the end of the manuscript,
and additional descriptive statistics are reported in an Appendix.
\section{Set-up}
Denote the key product characteristic by $S$, a generic value assumed by $S$
to be $s$, and the hedonic price schedule describing the relation between
price and $S$ is given by $P\left( S\right) \equiv P\left( S,\theta \right) $
where $\theta $ is a finite-dimensional parameter. In our empirical
illustration, $P\left( S,\theta \right) $ is the annual rent for a property
whose neighborhood school quality is $S$. We have data from multiple
markets, each with its own $\theta $. For individual consumers, consumption
(of the numeraire) is given by $C=Y-P\left( S,\theta \right) $ where $Y$ is
individual income. Individual preferences are described by the utility
function $U\left( S,C,\eta \right) $ where $\eta $ represents unobserved
preference heterogeneity, and $c$ is consumption. A household maximizes its
utility by choosing $S$ optimally, subject to the budget constraint $
Y=P\left( S,\theta \right) +C$. For the purpose of this paper, viz.
identification of welfare effects, we assume that the function $P\left(
S,\theta \right) $ in each market is known to the analyst,\footnote{
Indeed, $\theta $ will typically be estimated from the data, but at a
parametric rate, and these estimated $\theta $s will be used subsequently as
regressors, leading to standard measurement error issues. However, variance
of the measurement error in $\theta $ is of order $O\left( n^{-1}\right) $,
where $n$ is the number of observations in each market used to estimate $
\theta $ in that market. Hence replacing $\theta $ by its estimate will lead
to a very small attenuation bias when $n$ is large. For related discussions,
see Heckman et al, (2010), Section 5.} and the marginal (or conditional on
observables) distribution of $\eta $ is identical across markets.
We impose the following assumptions on the utility functions. Let $
U_{sc}\left( s,c,\eta \right) $, $U_{cc}\left( s,c,\eta \right) $, etc.
denote the second order derivatives of $U$.\smallskip
\begin{assumption}
(i) $U\left( \cdot ,\cdot ,\eta \right) $ has continuous second-order
derivatives in its first two arguments; (ii) $\eta $ is a scalar,
distributed independently of $Y$, and is identically distributed in each
market, (iii) $U\left( \cdot ,\cdot ,\eta \right) $ is strictly increasing
in each argument for any fixed $\eta $, (iv) the cross-partial derivatives
satisfy $U_{s\eta }\left( s,c,\eta \right) >0$ and $U_{c\eta }\left(
s,c,\eta \right) \leq 0$ for all values of $s,c,\eta $ on the support of $
\left( S,Y-P\left( S,\theta \right) ,\eta \right) $; (v) $P\left( s,\theta
\right) $ is smooth in both $s$ and $\theta $ and is increasing in $s$ for
fixed $\theta $; (vi) for all $s,y,\theta ,\eta $, we have that
\begin{equation}
\left\{
\begin{array}{l}
U_{ss}\left( s,y-P\left( s;\theta \right) ,\eta \right) -2\frac{\partial }{
\partial s}P\left( s;\theta \right) \times U_{cs}\left( s,y-P\left( s;\theta
\right) ,\eta \right) \\
+\left( \frac{\partial P\left( s;\theta \right) }{\partial s}\right)
^{2}\times U_{cc}\left( s,y-P\left( s;\theta \right) ,\eta \right)
\end{array}
\right\} <0\text{.} \label{12}
\end{equation}
\smallskip
\end{assumption}
The smoothness assumption (i) enables us to obtain the key analytical steps
for calculating welfare effects; (ii) is the key substantive restriction on
unobserved heterogeneity,\footnote{
It allows for $\eta $ to be a single index of multi-dimensional underlying
heterogeneity} and implies rank invariance, i.e. the ordering of any two
different consumers' demand remains identical across budget frontiers; (iii)
is non-satiation in $S$ and consumption, which is intuitive and is a key
sufficient condition for our welfare measure, viz. the compensating
variation, to be well-defined. Assumption (iv) says that the marginal
utility w.r.t. $s$ is strictly increasing, and marginal utility w.r.t. $c$
is decreasing in $\eta $. Intuitively, this means that higher $\eta $ types
have higher marginal utility w.r.t. $s$ and lower marginal utility w.r.t. $c$
. This implies the so-called `single crossing' condition, i.e. that the
marginal rate of substitution between $S$ and $C$ is increasing in $\eta $:
\begin{equation*}
\frac{d}{d\eta }\left( \frac{\frac{\partial U\left( s,c,\eta \right) }{
\partial s}}{\frac{\partial U\left( s,c,\eta \right) }{\partial c}}\right) >0
\text{.}
\end{equation*}
It will be shown below that assumption (iv) implies that for fixed budget
line, demand for $S$ is strictly monotone in $\eta $. This creates a 1-to-1
map between quantiles of \textit{observed} demand and quantiles of \textit{
unobserved} preference, which is helpful for identifying welfare. Heckman et
al. (2010) assume utilities are quasilinear in consumption, so that $\frac{
\partial ^{2}U\left( s,c,\eta \right) }{\partial \eta \partial c}\equiv 0$;
assumption (iv) is therefore a generalization required to cover the more
general non-quasilinear case. Assumption (v) says $S$ is a `desirable'
attribute, i.e. consuming more $S$ costs more. Finally, assumption (vi) says
that the hedonic budget frontier should be `less convex' to the origin than
the indifference curves,\footnote{
The slope of the indifference curves in the $S-C$ axes are given by $-\frac{
U_{S}}{U_{C}}$, whereas the budget curve has slope $-p^{\prime }$. Then (\ref
{12}) is equivalent to the difference between $-\frac{U_{S}}{U_{C}}-\left(
-p^{\prime }\right) =p^{\prime }-\frac{U_{S}}{U_{C}}$ being strictly
negative, i.e. the indifference curves are more convex than the budget
frontier.} which guarantees that utility is maximized uniquely at an
interior point on the budget frontier. In particular, (\ref{12}) holds if
the budget frontier is strictly concave and indifference curves are strictly
convex to the origin.
\begin{remark}
Note that, other than smoothness, we make no functional form assumption on
utilities, on how they depend on $\eta $, or the marginal distribution of $
\eta $. In particular, we do not require utilities to be increasing in $\eta
$.
\end{remark}
The policy intervention we wish to evaluate is one that changes the hedonic
price frontier. In our empirical illustration, an important case of interest
is where school choice becomes less or more restrictive, which would weaken
(respectively, strengthen) the relationship between rent and school-quality
(Machin and Salvanes, 2016). The pre- and post-intervention situations are
depicted via Figure \ref{fg:compensation_variation} where, for ease of
exposition, $\eta $ is held fixed.
In Figure \ref{fg:compensation_variation}, consumption is measured on the
vertical axis, and school quality along the horizontal axis. The original
budget frontier $C=Y-P\left( S,\theta \right) $ is depicted by the steeper
blue curve OD. Utility is maximized at C where the indifference curve,
convex to the origin is tangent to OD. Now, due to a policy intervention
(e.g. relaxed school choice in our example), the hedonic price schedule
changes, and the budget frontier shifts to the flatter orange curve AE,
whence optimal choice is B, representing a fall in utility relative to C.
We wish to compute the welfare effect of this intervention via the
compensating variation, which calculates how much would a household need to
be compensated, so that its maximized utility with the additional income in
the post-intervention situation equals its maximized utility in the
pre-intervention period with the original income. To see this graphically,
consider the curve depicted by the dashed curve GF, which is the AE
translated vertically up and is tangent to the original indifference curve
at F. Then the compensating variation, GA$>$0, is the income
supplement needed for the individual facing the blue budget curve so that
she can reach utility equal to what she was enjoying initially.
Given the position of the indifference curves, the CV is positive,
indicating that the consumer is losing as a result of the change, and hence
needs to be compensated by a positive income transfer to restore her utility
to its pre-intervention state. However, if the original indifference curve
were tangent to OD at a point below its intersection with AE, then the shift
of the budget line to AE would lead to a \textit{gain} in utility. Such a
consumer would benefit from the change, and the CV will be negative.
Intuitively speaking, the former type of households value school quality
less relative to consumption, and so were initially consuming relative lower
quality schooling. After the intervention, housing costs rise for lower
quality school areas, and therefore these households can afford less
consumption than before. The latter type of household values school-quality
relatively more, and choose higher school quality. The intervention makes
housing costs lower for areas with good schools, and hence expands the
budget set of these types of consumers. This reasoning illustrates that
welfare effects of a shift in the budget frontier can be heterogeneous in
both magnitude and sign; hence it is of interest to find the distribution of
welfare as the heterogeneity varies across consumers. We now turn to
developing the methods for these nonparametric calculations.
\section{Demand and Welfare Analysis\label{Nonparametric}}
In this section, we first derive the analogs of Roy's Identity and Slutsky
symmetry, and then move on to show how to identify welfare effects of a
change in the budget frontier. We start with the case where there is a
single attribute $S$, and then extend the analysis to include additional
attributes.
\subsection{Roy's Identity and Slutsky-Symmetry}
For ease of exposition, consider the case where $S$ is the only attribute of
interest, the known hedonic price function is given by $P\left( S,\theta
\right) $, where $\theta $ is an unknown vector of parameters. We will
introduce additional attributes later in Sec 3.3. The utility of an $\eta $
-type consumer is given by $U\left( s,c,\eta \right) $ where $y$ represents
disposable income, $s$ is the amount of $S$ chosen, $c$ is consumption of
the non-$S$ numeraire, and $\eta $ is unobserved heterogeneity. Utility
maximization and nonsatiation (assumption (iii)) imply that at the optimal
choice $S^{\ast }\left( y,\theta ,\eta \right) $, we must have that
\begin{equation}
\left. U_{s}\left( s,y-P\left( s;\theta \right) ,\eta \right) -U_{c}\left(
s,y-P\left( s;\theta \right) ,\eta \right) \frac{\partial }{\partial s}
P\left( s;\theta \right) \right\vert _{s=S^{\ast }\left( y,\theta ,\eta
\right) }=0 \label{3}
\end{equation}
Finally, the indirect utility function is given by
\begin{equation}
V\left( y,\theta ,\eta \right) =U\left( S^{\ast }\left( y,\theta ,\eta
\right) ,y-P\left( S^{\ast }\left( y,\theta ,\eta \right) ;\theta \right)
,\eta \right) \text{.} \label{10}
\end{equation}
Then for fixed $\theta ,\eta $, and given assumption 1(i), we have that $
V\left( \cdot ,\theta ,\eta \right) $ is differentiable, and the envelope
theorem\ condition holds, i.e.
\begin{eqnarray}
&&\frac{\partial V\left( y,\theta ,\eta \right) }{\partial y} \notag \\
&=&U_{c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \notag \\
&&+U_{s}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \frac{\partial
S^{\ast }\left( y,\theta ,\eta \right) }{\partial y} \notag \\
&&-U_{c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \times \frac{\partial
}{\partial s}P\left( S^{\ast }\left( y,\theta ,\eta \right) ;\theta \right)
\frac{\partial S^{\ast }\left( y,\theta ,\eta \right) }{\partial y} \notag
\\
&=&\underset{=0\text{, by (\ref{3})}}{\underbrace{\left[
\begin{array}{l}
U_{s}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \\
-U_{c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \times \frac{\partial
}{\partial s}P\left( S^{\ast }\left( y,\theta ,\eta \right) ;\theta \right)
\end{array}
\right] }}\frac{\partial S^{\ast }\left( y,\theta ,\eta \right) }{\partial y}
\notag \\
&&+U_{c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \notag \\
&=&U_{c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( S^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \label{4}
\end{eqnarray}
Therefore, $V\left( \cdot ,\theta ,\eta \right) $ is strictly increasing, by
assumption 1(iii).
Further, letting $P_{j}\left( s^{\ast }\left( y,\theta ,\eta \right) ;\theta
\right) =\left. \frac{\partial P\left( s;\theta \right) }{\partial \theta
_{j}}\right\vert _{s=s^{\ast }\left( y,\theta ,\eta \right) }$, we have by
the envelope theorem that
\begin{equation}
\frac{\partial V\left( y,\theta ,\eta \right) }{\partial \theta _{j}}
=-U_{c}\left( s^{\ast }\left( y,\theta ,\eta \right) ,y-P\left( s^{\ast
}\left( y,\theta ,\eta \right) ;\theta \right) \right) \times P_{j}\left(
s^{\ast }\left( y,\theta ,\eta \right) ;\theta \right) \text{.} \label{5a}
\end{equation}
From (\ref{4}) and (\ref{5a}), it follows that for each $j=1,2,...,\dim
\left( \theta \right) $, it must hold that
\begin{equation}
-\frac{\frac{\partial V\left( y,\theta ,\eta \right) }{\partial \theta _{j}}
}{\frac{\partial V\left( y,\theta ,\eta \right) }{\partial y}}=P_{j}\left(
s^{\ast }\left( y,\theta ,\eta \right) ;\theta \right) \label{Roys}
\end{equation}
which can be interpreted as Roy's identity for a nonlinear budget frontier.
Now, suppose we want to measure welfare-effects resulting from a change in $
\theta $ from $a$ to $b$. A common money-metric measure is the compensating
variation $C\equiv C\left( y,\eta \right) $, which solves
\begin{equation}
V\left( y+C,b,\eta \right) =V\left( y,a,\eta \right) \text{.} \label{8}
\end{equation}
There is a unique solution in $C$, since $\frac{\partial V\left( y,\theta
,\eta \right) }{\partial y}>0$ with probability 1, by (\ref{4}).
To find the distribution of $C$, suppose, initially, that we know the value
of $\eta $, then we can learn $P_{j}\left( s^{\ast }\left( y,\theta ,\eta
\right) ;\theta \right) $ from the hedonic price schedule in the data. Now,
equation (\ref{Roys}) can be rewritten as a system of linear, first-order
partial differential equations of order 1
\begin{equation}
\frac{\partial V\left( y,\theta ,\eta \right) }{\partial \theta _{j}}+\frac{
\partial V\left( y,\theta ,\eta \right) }{\partial y}\times P_{j}\left(
s^{\ast }\left( y,\theta ,\eta \right) ;\theta \right) =0\text{.} \label{9}
\end{equation}
Therefore, the goal is to solve for (\ref{8}), where $V\left( \cdot \right) $
satisfies (\ref{9}). The key difficulty in calculating welfare effects
nonparametrically is that $\eta $ is unobserved. To address this problem, we
use the single-crossing condition and scalar heterogeneity to implement a
quantile-based analysis, as follows.\medskip
\textbf{Quantile-based Analysis}: If $\eta $ is a scalar and $S^{\ast
}\left( y,\theta ,\eta \right) $ is strictly monotone and invertible in $
\eta $, then we can interpret the observed $\tau $th quantiles of $S^{\ast
}\left( y,\theta ,\eta \right) $, conditional on $y$ and $\theta $ (cf.
assumption (ii) above) as the demand of the individual who is located at the
$\tau $th quantile of the distribution of $\eta $. This is identified by the
$\tau $th quantile of demand for those at income $y$ on the budget frontier $
P\left( s;\theta \right) $, i.e.
\begin{equation*}
S^{\ast }\left( y,\theta ,F_{\eta }^{-1}\left( \tau \right) \right)
=F_{S^{\ast }\left( y,\theta ,\eta \right) }^{-1}\left( \tau \right) \equiv
q^{\tau }\left( y,\theta \right) \text{,}
\end{equation*}
where $\tau \in \lbrack 0,1]$, and $q^{\tau }\left( y,\theta \right) \equiv
F_{S^{\ast }\left( y,\theta ,\eta \right) }^{-1}\left( \tau \right) $ equals
the $\tau $th quantile of demand for those with income $y$ and facing a
budget frontier characterized by $\theta $. Further, since the indirect
utility function
\begin{equation*}
V\left( y,\theta ,\eta \right) =\max_{s,c}U\left( s,c,\eta \right) \text{
s.t. }c=y-P\left( s,\theta \right) \text{,}
\end{equation*}
by the envelope theorem, we have that
\begin{equation}
\frac{\partial }{\partial y}V\left( y,\theta ,\eta \right) =\frac{\partial U
}{\partial c}\left( S^{\ast }\left( y,\theta ,\eta \right) ,y-P\left(
S^{\ast }\left( y,\theta ,\eta \right) ,\theta \right) \right) >0 \label{15}
\end{equation}
when utility is strictly increasing in consumption, i.e. assumption (iii).
Now, differentiating the LHS of (\ref{3}), we get that
\begin{equation*}
\left.
\begin{array}{l}
\left\{ U_{ss}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta
\right) -U_{sc}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta
\right) \frac{\partial P}{\partial s}\right\} \frac{dS^{\ast }}{d\eta } \\
+U_{s\eta }\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
-U_{c}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\frac{\partial ^{2}P\left( S^{\ast };\theta \right) }{\partial s}\frac{
dS^{\ast }}{d\eta } \\
-\frac{\partial }{\partial s}P\left( S^{\ast };\theta \right) \times
U_{cs}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\frac{dS^{\ast }}{d\eta } \\
+\left\{ \frac{\partial }{\partial s}P\left( S^{\ast };\theta \right)
\right\} ^{2}\times U_{cc}\left( S^{\ast },y-P\left( S^{\ast };\theta
\right) ,\eta \right) \frac{dS^{\ast }}{d\eta } \\
-\frac{\partial }{\partial s}P\left( S^{\ast };\theta \right) \times
U_{c\eta }\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\end{array}
\right\vert =0\text{,}
\end{equation*}
implying
\begin{equation*}
\frac{dS^{\ast }}{d\eta }=-\frac{
\begin{array}{l}
U_{s\eta }\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\\
-\frac{\partial }{\partial s}P\left( S^{\ast };\theta \right) \times
U_{c\eta }\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\end{array}
}{
\begin{array}{l}
U_{ss}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right) \\
-U_{c}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\frac{\partial ^{2}P\left( S^{\ast };\theta \right) }{\partial s^{2}} \\
\left\{ \frac{\partial }{\partial s}P\left( S^{\ast };\theta \right)
\right\} ^{2}\times U_{cc}\left( S^{\ast },y-P\left( S^{\ast };\theta
\right) ,\eta \right) \frac{\partial }{\partial s}P\left( S^{\ast };\theta
\right) \\
-2U_{sc}\left( S^{\ast },y-P\left( S^{\ast };\theta \right) ,\eta \right)
\frac{\partial }{\partial s}P\left( S^{\ast };\theta \right)
\end{array}
}
\end{equation*}
The denominator of this expression is negative by (\ref{12}). The numerator
is positive by assumptions (iv) and (v). Hence $\frac{dS^{\ast }}{d\eta }>0$
with probability 1. Note that Heckman et al. \ (2010) derived an analogous
result for the case where utility is quasilinear in consumption, so that $
\frac{\partial ^{2}U\left( s,c,\eta \right) }{\partial \eta \partial c}=0$,
which is a special case of our set-up. In any case, the monotonicity of $
S^{\ast }$ w.r.t. $\eta $ will be used below for identifying the
distribution of the compensating variation.
In order to implement our method of welfare analysis, it is also useful to
introduce a Slutsky-symmetry type result. This result is of independent
interest, as it characterizes demand when budget frontiers are nonlinear.
Toward that end, define
\begin{equation}
Q_{\tau }\left( y,\theta \right) \equiv V\left( y,\theta ,F_{\eta
}^{-1}\left( \tau \right) \right) \text{,} \label{1}
\end{equation}
i.e. the indirect utility obtained by an individual with income $y$ and
located at the $\tau $th quantile of unobserved heterogeneity, i.e. whose
value of $\eta $ equals $F_{\eta }^{-1}\left( \tau \right) $, when the price
function is characterized by the parameter $\theta $.
\begin{lemma}[Slutsky-symmetry for nonlinear budget-sets]
Suppose the price-attribute relationship is given by $P\left( s,\theta
\right) $ where $\theta $ is of dimension $d\geq 2$. Let $q^{\tau }\left(
y,\theta \right) $ denote the demand at the $\tau $th quantile of $\eta $ \
when income is fixed at $y$. Then for each $j,k\in \left\{ 1,2,...d\right\} $
, it holds that
\begin{equation}
\begin{array}{c}
\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta \right)
}{\partial \theta _{j}\partial q}\left\{ \frac{\partial q^{\tau }\left(
y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{k}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{k}}\right\} \\
=\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta \right)
}{\partial \theta _{k}\partial q}\left\{ \frac{\partial q^{\tau }\left(
y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{j}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{j}}\right\}
\end{array}
\label{S6}
\end{equation}
\end{lemma}
\begin{proof}
Let $e\left( \mathbf{\theta },u\right) $ denote the expenditure function,
i.e. the solution to
\begin{equation*}
Q_{\tau }\left( \mathbf{\theta },e\right) =u\text{.}
\end{equation*}
This function is well defined since $Q_{\tau }\left( \mathbf{\theta }
,e\right) $ is continuous and strictly increasing in $e$. Let $j=1$ and $k=2$
WLOG. Then, by definition
\begin{equation*}
\begin{array}{l}
\frac{\partial }{\partial \theta _{1}}\left\{ Q_{\tau }\left( \mathbf{\theta
},e\left( \mathbf{\theta },u\right) \right) \right\} =0 \\
\Longrightarrow \frac{\partial }{\partial \theta _{1}}Q_{\tau }\left(
\mathbf{\theta },e\left( \mathbf{\theta },u\right) \right) +\frac{\partial }{
\partial e}Q_{\tau }\left( \mathbf{\theta },e\left( \mathbf{\theta }
,u\right) \right) \frac{\partial e\left( \mathbf{\theta },u\right) }{
\partial \theta _{1}}=0 \\
\Longrightarrow \frac{\partial e\left( \mathbf{\theta },u\right) }{\partial
\theta _{1}}=-\frac{\frac{\partial }{\partial \theta _{1}}Q_{\tau }\left(
\mathbf{\theta },e\left( \mathbf{\theta },u\right) \right) }{\frac{\partial
}{\partial e}Q_{\tau }\left( \mathbf{\theta },e\left( \mathbf{\theta }
,u\right) \right) }\overset{\text{by (\ref{7})}}{=}\left. \frac{\partial
P\left( s,\theta \right) }{\partial \theta _{1}}\right\vert _{s=q^{\tau
}\left( e\left( \mathbf{\theta },u\right) ,\theta \right) }
\end{array}
\end{equation*}
Similarly,
\begin{equation*}
\frac{\partial e\left( \mathbf{\theta },u\right) }{\partial \theta _{2}}
=\left. \frac{\partial P\left( s,\theta \right) }{\partial \theta _{2}}
\right\vert _{s=q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) }
\end{equation*}
Thus we have that
\begin{eqnarray}
\frac{\partial e\left( \mathbf{\theta },u\right) }{\partial \theta _{1}}
&=&\left. \frac{\partial P\left( s,\theta \right) }{\partial \theta _{1}}
\right\vert _{s=q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) } \label{S1} \\
\frac{\partial e\left( \mathbf{\theta },u\right) }{\partial \theta _{2}}
&=&\left. \frac{\partial P\left( s,\theta \right) }{\partial \theta _{2}}
\right\vert _{s=q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) } \label{S2}
\end{eqnarray}
Now, differentiating (\ref{S1}) w.r.t. $\theta _{2}$, we get
\begin{eqnarray}
\frac{\partial ^{2}e\left( \mathbf{\theta },u\right) }{\partial \theta
_{2}\partial \theta _{1}} &=&\frac{\partial }{\partial \theta _{2}}\left\{
\frac{\partial P\left( q^{\tau }\left( e\left( \mathbf{\theta },u\right)
,\theta \right) ,\theta \right) }{\partial \theta _{1}}\right\} \notag \\
&=&\frac{\partial ^{2}P\left( q^{\tau }\left( e\left( \mathbf{\theta }
,u\right) ,\theta \right) ,\theta \right) }{\partial \theta _{2}\partial q}
\left\{
\begin{array}{l}
\frac{\partial q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) }{\partial y}\frac{\partial P\left( q^{\tau }\left( e\left( \mathbf{
\theta },u\right) ,\theta \right) ,\theta \right) }{\partial \theta _{1}} \\
+\frac{\partial q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) }{\partial \theta _{1}}
\end{array}
\right\} \notag \\
&&+\left\{ \frac{\partial ^{2}P\left( q^{\tau }\left( e\left( \mathbf{\theta
},u\right) ,\theta \right) ,\theta \right) }{\partial \theta _{1}\partial
\theta _{2}}\right\} \label{S3}
\end{eqnarray}
Similarly, differentiating (\ref{S2}) w.r.t. $\theta _{1}$, we get
\begin{eqnarray}
\frac{\partial ^{2}e\left( \mathbf{\theta },u\right) }{\partial \theta
_{1}\partial \theta _{2}} &=&\frac{\partial ^{2}P\left( q^{\tau }\left(
e\left( \mathbf{\theta },u\right) ,\theta \right) ,\theta \right) }{\partial
\theta _{1}\partial q}\left\{
\begin{array}{l}
\frac{\partial q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) }{\partial y}\frac{\partial P\left( q^{\tau }\left( e\left( \mathbf{
\theta },u\right) ,\theta \right) ,\theta \right) }{\partial \theta _{2}} \\
+\frac{\partial q^{\tau }\left( e\left( \mathbf{\theta },u\right) ,\theta
\right) }{\partial \theta _{2}}
\end{array}
\right\} \notag \\
&&+\left\{ \frac{\partial ^{2}P\left( q^{\tau }\left( e\left( \mathbf{\theta
},u\right) ,\theta \right) ,\theta \right) }{\partial \theta _{1}\partial
\theta _{2}}\right\} \label{S4}
\end{eqnarray}
Equating (\ref{S3}) and (\ref{S4}), and evaluating at $y=e\left( \mathbf{
\theta },u\right) $, we get the symmetry condition
\begin{equation}
\begin{array}{c}
\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta \right)
}{\partial \theta _{1}\partial q}\left\{ \frac{\partial q^{\tau }\left(
y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{2}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{2}}\right\} \\
=\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta \right)
}{\partial \theta _{2}\partial q}\left\{ \frac{\partial q^{\tau }\left(
y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{1}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{1}}\right\}
\end{array}
\label{S5}
\end{equation}
\end{proof}
Equation (\ref{S5}) is the analog of Slutsky symmetry when budget
constraints are nonlinear. Indeed, in the standard case with linear budgets,
where $P\left( q^{\tau }\left( y,\theta \right) ,\theta \right) =\theta
q^{\tau }\left( y,\theta \right) $, (\ref{S5}) would reduce to
\begin{equation*}
\frac{\partial q^{\tau }\left( y,\theta \right) }{\partial y}q^{\tau }\left(
y,\theta \right) +\frac{\partial q^{\tau }\left( y,\theta \right) }{\partial
\theta _{2}}=\frac{\partial q^{\tau }\left( y,\theta \right) }{\partial y}
q^{\tau }\left( y,\theta \right) +\frac{\partial q^{\tau }\left( y,\theta
\right) }{\partial \theta _{1}}
\end{equation*}
which is the textbook case of Slutsky symmetry with linear budget
frontiers.\smallskip
\subsection{Calculation of Money-metric Welfare}
We now outline the steps for obtaining welfare effects. Toward that end,
first note from (\ref{15}) \ and (\ref{1}) that
\begin{equation*}
\frac{\partial }{\partial y}Q_{\tau }\left( y,\theta \right) >0\text{,}
\end{equation*}
and from (\ref{3}) that for all $\tau \in \left[ 0,1\right] $, and $
j=1,...,J $,
\begin{equation}
\frac{\partial Q_{\tau }\left( y,\theta \right) }{\partial \theta _{j}}
+\left. \frac{\partial P\left( s,\theta \right) }{\partial \theta _{j}}
\right\vert _{s=q^{\tau }\left( y,\theta \right) }\times \frac{\partial
Q_{\tau }\left( y,\theta \right) }{\partial y}=0\text{.} \label{6}
\end{equation}
Suppose the parameter $\theta $ characterizing the budget frontier changes
from $a$ to $b$, and we wish to calculate the compensating variation
corresponding to this change for this individual with $\eta =F_{\eta
}^{-1}\left( \tau \right) $. The compensating variation is defined as the
income supplement required to maintain the utility of the consumer, i.e.
solve for $C=C\left( y,\tau \right) $ which satisfies
\begin{equation}
Q_{\tau }\left( y+C,a\right) =Q_{\tau }\left( y,b\right) \text{, i.e. }
C\left( y,\tau \right) =Q_{\tau }^{-1}\left( Q_{\tau }\left( y,b\right)
,a\right) -y\text{.} \label{23}
\end{equation}
Now, note that the function $P\left( s,\theta \right) $ and the quantile
demand $q^{\tau }\left( y,\theta \right) $ are identified from the observed
data. Suppose, for concreteness, that the hedonic price function is given by
\begin{equation}
P\left( S,\theta \right) =\theta _{1}+\theta _{2}\ln \left( S\right) \text{.}
\label{p}
\end{equation}
The values of $\theta _{1},\theta _{2}$ vary across markets. Our goal is to
find $C$ which solves
\begin{equation*}
Q_{\tau }(y+C,b)=Q_{\tau }\left( y,a\right) \text{,}
\end{equation*}
where $Q_{\tau }\left( \cdot ,\theta \right) $ is strictly increasing, and
satisfies the system
\begin{equation}
\left.
\begin{array}{l}
\frac{\partial Q_{\tau }\left( y,\theta _{1},\theta _{2}\right) }{\partial
\theta _{1}}+\frac{\partial Q_{\tau }\left( y,\theta _{1},\theta _{2}\right)
}{\partial y}=0\text{,} \\
\frac{\partial Q_{\tau }\left( y,\theta _{1},\theta _{2}\right) }{\partial
\theta _{2}}+\ln \left\{ q^{\tau }\left( y,\theta \right) \right\} \times
\frac{\partial Q_{\tau }\left( y,\theta _{1},\theta _{2}\right) }{\partial y}
=0\text{.}
\end{array}
\right. \label{7}
\end{equation}
The term $\ln q^{\tau }\left( y,\theta \right) $ can be identified from the
data by running a quantile regression of (natural log of) the demanded
attribute on individual income and the market level $\theta $, when there
are multiple markets, each with its own $\theta $. It is natural to start
with the simple linear specification for the quantile regression function
(which may be generalized to a higher order polynomial, spline, etc.)
\begin{equation}
\ln q^{\tau }\left( y,\theta \right) =r_{0}+r_{1}y+r_{2}\theta
_{1}+r_{3}\theta _{2}\text{,} \label{d}
\end{equation}
where the $r$ coefficients are obtained via a $\tau $-quantile regression of
$\ln S$ on individual $y$ and market-level $\theta _{1}$ and $\theta _{2}$.
Before proceeding further, it is important to verify that (\ref{d}) is
indeed a valid specification for demand.
\begin{proposition}
In order for (\ref{d}) to be a valid specification for demand, it is
necessary that $r_{1}+r_{2}=0$.
\end{proposition}
\begin{proof}[Proof of proposition 2]
From (\ref{S1}), (\ref{p}), (\ref{d}), we have that
\begin{equation}
\frac{\partial e\left( \mathbf{\theta },u\right) }{\partial \theta _{1}}
\overset{\text{by (\ref{p})}}{=}1\Longrightarrow \frac{\partial ^{2}e\left(
\theta _{1},\theta _{2},u\right) }{\partial \theta _{2}\partial \theta _{1}}
=0\text{.} \label{zero}
\end{equation}
On the other hand, by (\ref{S2})
\begin{eqnarray}
&&
\begin{array}{l}
\frac{\partial e\left( \mathbf{\theta },u\right) }{\partial \theta _{2}}
\overset{\text{by (\ref{p})}}{=}\ln q^{\tau }\left( \theta _{1},\theta
_{2},e\left( \theta _{1},\theta _{2},u\right) \right) \\
\overset{\text{by (\ref{d})}}{=}r_{0}+r_{1}\times e\left( \theta _{1},\theta
_{2},u\right) +r_{2}\theta _{1}+r_{3}\theta _{2}
\end{array}
\notag \\
&\Longrightarrow &\frac{\partial ^{2}e\left( \theta _{1},\theta
_{2},u\right) }{\partial \theta _{1}\partial \theta _{2}}=r_{2}+r_{1}\frac{
\partial e\left( \theta _{1},\theta _{2},u\right) }{\partial \theta _{1}}
\overset{\text{by (\ref{p})}}{=}r_{2}+r_{1}\text{.} \label{z}
\end{eqnarray}
Now (\ref{zero}) and (\ref{z}) imply that $r_{1}+r_{2}=0$.
\end{proof}
Now suppose the value of the parameter $\theta $ characterizing the price
frontier changes from $a$ to $b$. We wish to find the compensating
variation, which is the hypothetical income transfer that an individual
needs when $\theta =b$ to be able to reach the utility level she had
attained when $\theta $ equalled $a$. Given heterogeneous preferences, the
compensating variation for the same change in $\theta $ is heterogeneous,
and we wish to obtain its distribution. To do this, we fix a value of $\tau
\in \left[ 0,1\right] $, and develop a method to compute the CV for
individuals whose $\eta =F_{\eta }^{-1}\left( \tau \right) $. We then vary $
\tau $ to generate the CV for different quantiles of $\eta $.
Toward that end, consider a price path $\theta \left( t\right) $, with $t\in
\left[ 0,1\right] $, such that $\theta _{1}\left( 0\right) =a_{1}$, $\theta
_{2}\left( 0\right) =a_{2}$, $\theta _{1}\left( 2\right) =b_{1}$, $\theta
_{2}\left( 1\right) =b_{2}$. By definition, the compensating variation at a
generic value of $t\in \left[ 0,1\right] $ is given by $C\left( t,y\right)
=e\left( \theta \left( t\right) ,\bar{u}\right) -y$, will satisfy
\begin{equation}
Q_{\tau }\left( \theta \left( t\right) ,y+C\left( t,y\right) \right) =\bar{u}
\text{ for all }t\text{,} \label{2}
\end{equation}
where the initial utility level $\bar{u}=Q_{\tau }\left(
a_{1},a_{2},y\right) $. Differentiating (\ref{2}) w.r.t. $t$ we have that
for all $t$,
\begin{equation*}
\frac{d}{dt}Q_{\tau }\left( \theta \left( t\right) ,y+C\left( t,y\right)
\right) =0\text{, i.e.}
\end{equation*}
for all $t$,
\begin{equation*}
\sum\limits_{j}\frac{\partial Q_{\tau }\left( \theta \left( t\right)
,y+C\left( t,y\right) \right) }{\partial \theta _{j}}\frac{d\theta
_{j}\left( t\right) }{dt}+\frac{\partial Q_{\tau }\left( \theta \left(
t\right) ,y+C\left( t,y\right) \right) }{\partial y}\frac{dC\left(
t,y\right) }{dt}=0\text{.}
\end{equation*}
This implies
\begin{eqnarray}
\frac{dC\left( t,y\right) }{dt} &=&-\sum\limits_{j}\frac{\frac{\partial
Q_{\tau }\left( \theta \left( t\right) ,y+C\left( t,y\right) \right) }{
\partial \theta _{j}}}{\frac{\partial Q_{\tau }\left( \theta \left( t\right)
,y+C\left( t,y\right) \right) }{\partial y}}\frac{d\theta _{j}\left(
t\right) }{dt} \notag \\
&&\overset{\text{by (\ref{6})}}{=}\sum\limits_{j}\frac{d\theta _{j}\left(
t\right) }{dt}\times \left. \frac{\partial P\left( \theta ,s\right) }{
\partial \theta _{j}}\right\vert _{\theta =\theta \left( t\right) ,s=q_{\tau
}\left( \theta \left( t\right) ,e\left( \theta \left( t\right) ,\bar{u}
\right) \right) } \label{ode}
\end{eqnarray}
The second term on the RHS of (\ref{ode}) is identifiable from the data
across many markets, since $e\left( \theta \left( t\right) ,\bar{u}\right)
=y+C\left( t,y\right) $. So finding the compensating variation for a change
in $\theta $ from $a$ to $b$ for an individual at income $y$ and whose $\eta
$ equals its $\tau $th quantile is equivalent to finding $C\left( 1,y\right)
$, where $C\left( t,y\right) $ solves (\ref{ode}) with the initial condition
$C\left( 0,y\right) =0$. Observe that
\begin{eqnarray}
C\left( 1.y\right) -C\left( 0,y\right) &=&\int_{0}^{1}\frac{dC\left(
t,y\right) }{dt}dt \notag \\
&=&\int_{0}^{1}\left\{ \sum\limits_{j}\frac{d\theta _{j}\left( t\right) }{dt
}\times \frac{\partial P\left( \theta \left( t\right) ,q_{\tau }\left(
\theta \left( t\right) ,e\left( \theta \left( t\right) ,\bar{u}\right)
\right) \right) }{\partial \theta _{j}}\right\} dt \notag \\
&=&\int_{\mathcal{G}}\sum\limits_{j}\frac{\partial P\left( \theta ,q_{\tau
}\left( \theta ,e\left( \theta ,\bar{u}\right) \right) \right) }{\partial
\theta _{j}}d\theta _{j}\text{.} \label{l}
\end{eqnarray}
The final expression on the RHS is a line integral of the vector field $
\left( \frac{\partial P\left( \theta ,q_{\tau }\left( \theta ,e\left( \theta
,\bar{u}\right) \right) \right) }{\partial \theta _{1}}\text{, }\frac{
\partial P\left( \theta ,q_{\tau }\left( \theta ,e\left( \theta ,\bar{u}
\right) \right) \right) }{\partial \theta _{2}}\right) $ along the path $
\mathcal{G=}\left\{ \theta _{1}\left( t\right) ,\theta _{2}\left( t\right)
\right\} $, $t\in \left[ 0,1\right] $ connecting the points $\left(
a_{1},a_{2}\right) $ and $\left( b_{1},b_{2}\right) $. The symmetry
condition (\ref{S5}) and the gradient theorem for line integrals (cf.
Spiegel 2010, Sec 10.6; Courant and John 1989, Sec 1.10) then imply that the
value of (\ref{l}) is path-independent, i.e. its value does not depend on
the path $\mathcal{G}$. Thus the compensating variation $C\left( 1.y\right) $
is well-defined.
In particular, given the specification (\ref{p}) and (\ref{d}), equation (
\ref{ode}) reduces to the ordinary differential equation
\begin{eqnarray*}
&&\frac{dC\left( t,y\right) }{dt} \\
&&\overset{\text{(by \ref{d})}}{=}\frac{d\theta _{1}\left( t\right) }{dt}+
\frac{d\theta _{2}\left( t\right) }{dt}\times \left( r_{0}+r_{1}\left(
y+C\left( t,y\right) \right) +r_{2}\theta _{1}\left( t\right) +r_{3}\theta
_{2}\left( t\right) \right) \\
&&\overset{\text{by }r_{2}+r_{1}=0}{=}\frac{d\theta _{1}\left( t\right) }{dt}
+\frac{d\theta _{2}\left( t\right) }{dt}\times \left( r_{0}+r_{1}\left(
y+C\left( t,y\right) -\theta _{1}\left( t\right) \right) +r_{3}\theta
_{2}\left( t\right) \right) \\
&\Leftrightarrow &\frac{dC\left( t,y\right) }{dt}-r_{1}\frac{d\theta
_{2}\left( t\right) }{dt}\times C\left( t,y\right) \\
&=&\frac{d\theta _{1}\left( t\right) }{dt}+\frac{d\theta _{2}\left( t\right)
}{dt}\times \left( r_{0}+r_{1}\left( y-\theta _{1}\left( t\right) \right)
\right) +r_{3}\frac{d\theta _{2}\left( t\right) }{dt}\times \theta
_{2}\left( t\right)
\end{eqnarray*}
This implies
\begin{eqnarray*}
&&\frac{dC\left( t,y\right) }{dt}-r_{1}\frac{d\theta _{2}\left( t\right) }{dt
}\times C\left( t,y\right) \\
&=&\frac{d\theta _{1}\left( t\right) }{dt}+\frac{d\theta _{2}\left( t\right)
}{dt}\times \left( r_{0}+r_{1}\left( y-\theta _{1}\left( t\right) \right)
\right) +r_{3}\frac{d\theta _{2}\left( t\right) }{dt}\times \theta
_{2}\left( t\right)
\end{eqnarray*}
This linear ODE can be solved using the method of integrating factors as
\begin{eqnarray*}
&&\frac{dC\left( t,y\right) }{dt}e^{-r_{1}\theta _{2}\left( t\right) }-r_{1}
\frac{d\theta _{2}\left( t\right) }{dt}e^{-r_{1}\theta _{2}\left( t\right)
}\times C\left( t,y\right) \\
&=&\frac{d\theta _{1}\left( t\right) }{dt}e^{-r_{1}\theta _{2}\left(
t\right) }+e^{-r_{1}\theta _{2}\left( t\right) }\frac{d\theta _{2}\left(
t\right) }{dt}\times \left( r_{0}+r_{1}\left( y-\theta _{1}\left( t\right)
\right) \right) \\
&&+r_{3}\frac{d\theta _{2}\left( t\right) }{dt}\times \theta _{2}\left(
t\right) e^{-r_{1}\theta _{2}\left( t\right) } \\
&=&\frac{d\theta _{1}\left( t\right) }{dt}e^{-r_{1}\theta _{2}\left(
t\right) }-e^{-r_{1}\theta _{2}\left( t\right) }\frac{d\theta _{2}\left(
t\right) }{dt}\times \theta _{1}\left( t\right) \\
&&+\left( r_{0}+r_{1}y\right) e^{-r_{1}\theta _{2}\left( t\right) }\frac{
d\theta _{2}\left( t\right) }{dt}+r_{3}\frac{d\theta _{2}\left( t\right) }{dt
}\times \theta _{2}\left( t\right) e^{-r_{1}\theta _{2}\left( t\right) }
\end{eqnarray*}
This implies
\begin{eqnarray*}
\frac{d}{dt}\left\{ C\left( t,y\right) e^{-r_{1}\theta _{2}\left( t\right)
}\right\} &=&\frac{d}{dt}\left\{ \theta _{1}\left( t\right) e^{-r_{1}\theta
_{2}\left( t\right) }\right\} \\
&&+e^{-r_{1}\theta _{2}\left( t\right) }\frac{d\theta _{2}\left( t\right) }{
dt}\left( r_{0}+r_{1}y\right) \\
&&+r_{3}\frac{d\theta _{2}\left( t\right) }{dt}\times \theta _{2}\left(
t\right) e^{-r_{1}\theta _{2}\left( t\right) }
\end{eqnarray*}
Integrating both sides, we get that
\begin{eqnarray}
C\left( t,y\right) e^{-r_{1}\theta _{2}\left( t\right) } &=&Cons+\theta
_{1}\left( t\right) e^{-r_{1}\theta _{2}\left( t\right) }-\left(
r_{0}+r_{1}y\right) \frac{e^{-r_{1}\theta _{2}\left( t\right) }}{r_{1}}
\notag \\
&&-r_{3}\theta _{2}\left( t\right) \frac{e^{-r_{1}\theta _{2}\left( t\right)
}}{r_{1}}-\frac{e^{-r_{1}\theta _{2}\left( t\right) }}{r_{1}^{2}}r_{3}\text{,
} \label{X}
\end{eqnarray}
where \textquotedblleft $Cons$\textquotedblright\ denotes a constant. This
implies
\begin{equation*}
C\left( t,y\right) =Cons\times e^{r_{1}\theta _{2}\left( t\right) }+\theta
_{1}\left( t\right) -\left( r_{0}+r_{1}y\right) \frac{1}{r_{1}}-r_{3}\theta
_{2}\left( t\right) \frac{1}{r_{1}}-\frac{r_{3}}{r_{1}^{2}}
\end{equation*}
Applying the boundary condition $C\left( 0,y\right) =0$, and $\theta
_{1}\left( 0\right) =a_{1}$, $\theta _{2}\left( 0\right) =a_{2}$, we get
\begin{equation*}
Cons=e^{-r_{1}a_{2}}\left\{ -a_{1}+\left( r_{0}+r_{1}y\right) \frac{1}{r_{1}}
+a_{2}\frac{r_{3}}{r_{1}}+\frac{r_{3}}{r_{1}^{2}}\right\} \text{.}
\end{equation*}
Replacing in (\ref{X}), and evaluating at $t=1$, using $\theta _{1}\left(
1\right) =b_{1}$, $\theta _{2}\left( 1\right) =b_{2}$, we get
\begin{eqnarray}
C\left( 1,y\right) &=&e^{r_{1}\left( b_{2}-a_{2}\right) }\left\{ -a_{1}+
\frac{r_{0}}{r_{1}}+y+\frac{a_{2}r_{3}}{r_{1}}+\frac{r_{3}}{r_{1}^{2}}
\right\} \notag \\
&&+b_{1}-\frac{r_{0}}{r_{1}}-y-\frac{r_{3}b_{2}}{r_{1}}-\frac{r_{3}}{
r_{1}^{2}} \label{CV}
\end{eqnarray}
It is clear that $C\left( 1,y\right) $ does not depend on the path from $
\left( a_{1},a_{2}\right) $ to $\left( b_{1},b_{2}\right) $ because the
exact form of $\theta _{1}\left( t\right) ,\theta _{2}\left( t\right) $ as
functions of $t$ were never used to derive (\ref{CV}). We state the above
derivation as a proposition.
\begin{proposition}
Suppose the price function $P\left( \theta ,S\right) $ and the quantile
demand function $q_{\tau }\left( \theta ,y\right) $ are defined on connected
open sets, and are continuously differentiable on their domain. Suppose all
assumptions on preferences stated in assumption 1 are satisfied.
Additionally, suppose (\ref{p}) and (\ref{d}) hold with $r_{1}+r_{2}=0$.
Then the compensating variation due to a movement of $\left( \theta
_{1},\theta _{2}\right) $ from $\left( a_{1},a_{2}\right) $ to $\left(
b_{1},b_{2}\right) $ for an individual at $\eta =F_{\eta }^{-1}\left( \tau
\right) $ is independent of the path along which $\left( \theta _{1},\theta
_{2}\right) $ changes, and is given by
\begin{eqnarray}
C\left( 1,y\right) &=&e^{r_{1}\left( b_{2}-a_{2}\right) }\left\{ \frac{r_{0}
}{r_{1}}+y+\frac{a_{2}r_{3}}{r_{1}}+\frac{r_{3}}{r_{1}^{2}}-a_{1}\right\}
\notag \\
&&+b_{1}-\frac{r_{0}}{r_{1}}-y-\frac{r_{3}b_{2}}{r_{1}}-\frac{r_{3}}{
r_{1}^{2}} \label{cv}
\end{eqnarray}
\end{proposition}
An analogous exercise can be done for every other quantiles, which produce
different values of the $r$'s in (\ref{d}) and, correspondingly different
values of the compensating variation (\ref{cv}).
\begin{remark}
Note that $Q_{\tau }\left( y,\theta \right) $ defined in (\ref{1}) need
\textit{not} equal the $\tau $th quantile of the indirect utility $V\left(
y,\theta ,\eta \right) $ because $V\left( y,\theta ,\eta \right) $ need not
be monotonic in $\eta $. Nonetheless, as $\tau $ varies over $\left[ 0,1
\right] $, we can trace out the distribution of $V\left( y,\theta ,\eta
\right) $. In particular, for each specific quantile, say, $\tau =0.1$ or $
\tau =0.5$ etc., we get the value of the compensating variation for
individuals with income $y$ and who are at the lowest decile or the median
of unobserved heterogeneity, respectively. These need not equal the the
lowest decile or median respectively of the marginal distribution of the
compensating variation for people with income $y$.
\end{remark}
The previous proposition can be generalized in the obvious way for general
price and quantile functions, as follows.
\begin{proposition}
Suppose $\theta \in R^{J}$; the price function $P\left( \theta ,S\right) $
and the quantile demand function $q_{\tau }\left( \theta ,y\right) $ are
defined on connected open sets, and are twice continuously differentiable on
their domain. Suppose the Slutsky symmetry condition is satisfied, i.e.
\begin{eqnarray*}
&&\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta
\right) }{\partial \theta _{j}\partial q}\left\{ \frac{\partial q^{\tau
}\left( y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{k}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{k}}\right\} \\
&=&\frac{\partial ^{2}P\left( q^{\tau }\left( y,\theta \right) ,\theta
\right) }{\partial \theta _{k}\partial q}\left\{ \frac{\partial q^{\tau
}\left( y,\theta \right) }{\partial y}\frac{\partial P\left( q^{\tau }\left(
y,\theta \right) ,\theta \right) }{\partial \theta _{j}}+\frac{\partial
q^{\tau }\left( y,\theta \right) }{\partial \theta _{j}}\right\}
\end{eqnarray*}
for all $j\neq k$. Then for any path $\theta \left( t\right) $, $t\in \left[
0,1\right] $, with $\theta \left( 0\right) =a$ and $\theta \left( 1\right)
=b $, the compensating variation is given by $C\left( 1,y\right) $, where $
C\left( t,y\right) $ is the solution to the ordinary differential equation
\begin{equation}
\frac{dC\left( t,y\right) }{dt}=\sum\limits_{j=1}^{J}\frac{d\theta
_{j}\left( t\right) }{dt}\times \frac{\partial P\left( \theta \left(
t\right) ,q_{\tau }\left( \theta \left( t\right) ,y+C\left( t,y\right)
\right) \right) }{\partial \theta _{j}}\text{,} \notag
\end{equation}
and this solution is independent of the path $\theta \left( t\right) $.
\end{proposition}
The proof of this result is completely analogous to that of the previous
proposition, and uses the fundamental theorem for line integration that line
integrals of gradients are path-independent (cf. Courant and John 1989, Sec
1.10).
\subsection{Multiple Attributes\label{Multiple}}
To incorporate additional hedonic attributes into the above analysis,
consider the additively separable utility function
\begin{equation*}
U\left( s,x,c,\eta \right) =U_{1}\left( s,\eta \right) +U_{2}\left( x\right)
+U_{3}\left( c\right) \text{, with }c=y-P\left( s,x\right) \text{,}
\end{equation*}
where $s$ is the key attribute of interest, $x$ represents the other
attributes, $y$ is income, and the hedonic price function is given by
\begin{equation*}
P\left( s,x\right) =P_{1}\left( s,\theta \right) +P_{2}\left( x,\delta
\right)
\end{equation*}
We want to measure the distribution of the compensating variation $C$ that
solves
\begin{equation*}
V\left( y,\theta ,\delta ,\eta \right) =V\left( y+C,\beta ,\delta ,\eta
\right) \text{,}
\end{equation*}
where
\begin{equation*}
V\left( y,\theta ,\delta ,\eta \right) =\max_{s,x}U\left( s,x,y-P_{1}\left(
s,\theta \right) -P_{2}\left( x,\delta \right) ,\eta \right)
\end{equation*}
If the attributes are continuous and the utility function is differentiable
in each, then the first order conditions for maximization are given by
\begin{eqnarray}
\frac{\partial U_{1}\left( s,\eta \right) }{\partial s}-U_{3}^{\prime
}\left( y-P_{1}\left( s,\theta \right) -P_{2}\left( x,\delta \right) \right)
\times \frac{\partial P_{1}\left( s,\theta \right) }{\partial s} &=&0
\label{14} \\
\nabla _{x}U_{2}\left( x\right) -U_{3}^{\prime }\left( y-P_{1}\left(
s,\theta \right) -P_{2}\left( x,\delta \right) \right) \times \nabla
_{x}P_{2}\left( x,\delta \right) &=&0 \label{13}
\end{eqnarray}
while a sufficient second order condition for an interior maximum is that
the matrix
\begin{equation}
H=\left[
\begin{array}{cc}
\begin{array}{c}
\frac{\partial ^{2}U_{1}\left( s,\eta \right) }{\partial s^{2}}
-U_{3}^{^{\prime \prime }}\times \left( \frac{\partial P_{1}\left( s,\theta
\right) }{\partial s}\right) ^{2} \\
+U_{3}^{\prime }\times \left( \frac{\partial P_{1}\left( s,\theta \right) }{
\partial s}\right) ^{2}
\end{array}
& \nabla _{x}P_{2}\left( x,\delta \right) \times U_{3}^{\prime \prime
}\times \frac{\partial P_{1}\left( s,\theta \right) }{\partial s} \\
\nabla _{x}P_{2}\left( x,\delta \right) \times U_{3}^{\prime \prime }\times
\frac{\partial P_{1}\left( s,\theta \right) }{\partial s} &
\begin{array}{c}
\nabla _{xx}U_{2}-U_{3}^{\prime }\times \nabla _{xx}P_{2}\left( x,\delta
\right) \\
-U_{3}^{^{\prime \prime }}\times \nabla _{x}P_{2}\left( x,\delta \right)
\times \nabla _{x}P_{2}\left( x,\delta \right) ^{\prime }
\end{array}
\end{array}
\right] \label{11}
\end{equation}
is negative definite for all $s,x$.
Now, evaluating the first-order conditions (\ref{14})-(\ref{13}) at the
optimal choice and differentiating w.r.t. $\eta $, we have that
\begin{eqnarray}
\frac{\partial U_{1}\left( S^{\ast },\eta \right) }{\partial s}
-U_{3}^{\prime }\left( y-P_{1}\left( s^{\ast },\theta \right) -P_{2}\left(
x,\delta \right) \right) \times \frac{\partial P_{1}\left( S^{\ast },\theta
\right) }{\partial s} &=&0 \notag \\
\left[
\begin{array}{l}
\frac{\partial ^{2}U_{1}\left( S^{\ast },\eta \right) }{\partial s\partial
\eta }+\frac{\partial ^{2}U_{1}\left( S^{\ast },\eta \right) }{\partial s^{2}
}\frac{\partial s^{\ast }}{\partial \eta } \\
+U_{3}^{^{\prime \prime }}\left( y-P_{1}\left( S^{\ast },\theta \right)
-P_{2}\left( x,\delta \right) \right) \times \left\{ \frac{\partial
P_{1}\left( S^{\ast },\theta \right) }{\partial s}\right\} ^{2}\times \frac{
\partial S^{\ast }}{\partial \eta } \\
+U_{3}^{^{\prime \prime }}\left( y-P_{1}\left( S^{\ast },\theta \right)
-P_{2}\left( x,\delta \right) \right) \times \frac{\partial P_{1}\left(
x^{\ast },\theta \right) }{\partial s}\times \nabla _{x}P_{2}\left( x,\delta
\right) \frac{\partial x^{\ast }}{\partial \eta } \\
-U_{3}^{\prime }\left( y-P_{1}\left( S^{\ast },\theta \right) -P_{2}\left(
x,\delta \right) \right) \times \frac{\partial ^{2}P_{1}\left( S^{\ast
},\theta \right) }{\partial s^{2}}\frac{\partial S^{\ast }}{\partial \eta }
\end{array}
\right] &=&0 \label{17}
\end{eqnarray}
Similarly
\begin{equation}
\nabla _{xx}U_{2}\left( x\right) \times \frac{\partial x^{\ast }}{\partial
\eta }-U_{3}^{\prime \prime }\left\{ \nabla _{x}P_{2}\left( x,\delta \right)
\nabla _{x}P_{2}\left( x,\delta \right) ^{\prime }\right\} \frac{\partial
x^{\ast }}{\partial \eta }-U_{3}^{\prime \prime }\left\{ \nabla
_{x}P_{2}\left( x,\delta \right) \right\} \frac{\partial P_{1}\left( s^{\ast
},\theta \right) }{\partial s}\frac{\partial s^{\ast }}{\partial \eta }=0
\label{18}
\end{equation}
Equations (\ref{17})-(\ref{18}) can be written in matrix notation as
\begin{equation*}
H\times \left[
\begin{array}{c}
\frac{\partial s^{\ast }}{\partial \eta } \\
\frac{\partial x^{\ast }}{\partial \eta }
\end{array}
\right] =\left[
\begin{array}{c}
-\frac{\partial ^{2}U_{1}\left( s^{\ast },\eta \right) }{\partial s\partial
\eta } \\
0
\end{array}
\right]
\end{equation*}
where $H$ is defined in (\ref{11}). Therefore,
\begin{equation*}
\left[
\begin{array}{c}
\frac{\partial s^{\ast }}{\partial \eta } \\
\frac{\partial x^{\ast }}{\partial \eta }
\end{array}
\right] =H^{-1}\left[
\begin{array}{c}
-\frac{\partial ^{2}U_{1}\left( s^{\ast },\eta \right) }{\partial s\partial
\eta } \\
0
\end{array}
\right]
\end{equation*}
implying
\begin{equation*}
\frac{\partial S^{\ast }}{\partial \eta }=-H^{11}\times \frac{\partial
^{2}U_{1}\left( s^{\ast },\eta \right) }{\partial s\partial \eta }
\end{equation*}
where $H^{11}$ is the $\left( 1,1\right) $th entry of the matrix $H^{-1}$.
Now since $H$ is negative definite, so is its inverse. Therefore $H^{11}$
must be strictly negative. Therefore, if $\frac{\partial ^{2}U_{1}\left(
S^{\ast },\eta \right) }{\partial s\partial \eta }<0$, then it follows that $
\frac{\partial V^{\ast }}{\partial \eta }>0$. That is, for given $y,\theta
,\delta ,$ we have that $S^{\ast }\left( y,\theta ,\delta ,\eta \right) $ is
strictly increasing in $\eta $. Therefore, we have that for each $\tau \in
\left[ 0,1\right] $,
\begin{equation*}
S^{\ast }\left( y,\theta ,\delta ,F_{\eta }^{-1}\left( \tau \right) \right)
=F_{s^{\ast }\left( y,\theta ,\delta ,\eta \right) }^{-1}\left( \tau \right)
\text{,}
\end{equation*}
i.e. the value of $S^{\ast }\left( y,\theta ,\delta ,\eta \right) $ at the $
\tau $th quantile of $\eta $ equals the $\tau $th quantile of $s^{\ast }$
for fixed values of $y,\theta ,\delta $.
For measuring the welfare effect of a change in $\theta $, holding $\delta $
fixed, we follow the essentially the same steps as outlined above. In
particular, we have that
\begin{equation*}
V\left( y,\theta ,\delta ,\eta \right) =U_{1}\left( s^{\ast },\eta \right)
+U_{2}\left( x^{\ast }\right) +U_{3}\left( y-P_{1}\left( s^{\ast },\theta
\right) -P_{2}\left( x^{\ast },\delta \right) \right)
\end{equation*}
so that, by the envelope theorem, one gets
\begin{equation*}
-\frac{\frac{\partial V\left( y,\theta ,\delta ,\eta \right) }{\partial
\theta }}{\frac{\partial V\left( y,\theta ,\delta ,\eta \right) }{\partial y}
}=\left. \frac{U_{3}^{\prime }\times \frac{\partial P_{1}\left( s,\theta
\right) }{\partial \theta }}{U_{3}^{\prime }}\right\vert _{s=S^{\ast }\left(
y,\theta ,\delta \right) ,x=x^{\ast }\left( y,\theta ,\delta \right)
}=\left. \frac{\partial P_{1}\left( s,\theta \right) }{\partial \theta }
\right\vert _{s=V^{\ast }\left( y,\theta ,\delta \right) }\text{.}
\end{equation*}
This last simplification, i.e. that the RHS depends only on $S^{\ast }\left(
y,\theta ,\delta \right) $ and not on $x^{\ast }\left( y,\theta ,\delta
\right) $, results from the additive separability of the hedonic price
function.
Evaluating this at $\eta =F_{\eta }^{-1}\left( \tau \right) $, we get (\ref
{6}) replaced by
\begin{equation}
\frac{\partial Q_{\tau }\left( y,\theta ,\delta \right) }{\partial \theta }
+\left. \frac{\partial P_{1}\left( s,\theta \right) }{\partial \theta }
\right\vert _{s=F_{s^{\ast }\left( y,\theta ,\delta \right) }^{-1}\left(
\tau \right) }\times \frac{\partial Q_{\tau }\left( y,\theta ,\delta \right)
}{\partial y}=0\text{,} \label{16}
\end{equation}
where $F_{S^{\ast }\left( y,\theta ,\delta \right) }^{-1}\left( \tau \right)
$ is the $\tau $th quantile of the optimal (i.e. chosen) $s$ across
individuals with income $y$ in markets characterized by $\left( \theta
,\delta \right) $. Therefore, we can apply the method outlined in the
previous subsection, holding $\delta $ fixed, and obtain the value of the
compensating variation for each type defined by a quantile of $\eta $.
\section{Empirical Illustration}
\subsection{Data}
The dataset used for our illustration comes from the restricted-access
version of Wave 2015 of English Housing Survey (DCLG{\normalsize , 2018}),
which is a nationally representative survey on housing stock, conditions,
and household characteristics. We use the data on rented properties. For
each property, we observe the annual rent as well as a range of property
characteristics, including floor area, number of floors, dwelling type
(terrace, detached, flat, etc.), age, number of bathrooms, bedrooms, and
living rooms, and an index of local economic deprivation, measured at the
level of the so-called `Lower Layer Super Output Area'. We also observe a
set of property and household characteristics, including structural features
of the property (e.g. number of bedrooms, floor area etc.), net annual
income of the household, whether the household receives housing benefits,
and tenure type, i.e. whether renting privately, via local authority
provision, or through housing associations.
To proxy for school quality, we use the average point score per pupil for
secondary schools. The point score comes from the Key Stage 4 data in the
School Performance Tables, commonly known as \textit{league tables}.
\footnote{
Key Stage 4 represents the two years of education for students aged 14 to
16, corresponding to Years 10 and 11 in the English education system.} These
data are publicly available via the UK government's official website,
GOV.UK. We exclude independent, i.e. private, schools and schools for
children with special educational needs, i.e. special schools, because they
follow different admission procedures and cater to a distinct population.
Each property is matched to the nearest school based on postcode proximity.
The matching process was carried out using the open-source geographic
information system software, QGIS. We start with a total of 6,611
property-school matched observations. We then exclude 110 cases where
household incomes are negative after accounting for rent. To remove the
outliers, we further drop the households with rent-to-income ratios above
the 95th percentile and those within the top or bottom 5 percent of the
income distribution, leading to excluding 866 observations. The final sample
includes 5,635 properties, each matched to the nearest school.
Table \ref{tab:desc} of the Appendix presents the descriptive statistics for
the dataset. On average, households in our sample have a post-tax weekly
income of \pounds 396, with around \pounds 111 allocated to rent. Around 28
percent of households rent privately, while the remainder are social
renters, either from local authorities or housing associations.
Approximately 56 percent of the respondents receive housing benefits, and 55
percent reside in areas within the three most deprived deciles. For the
purpose of empirical application, we reduce the dimensionality of the
property and household characteristics by using its first principal
component.\footnote{
Appendix Table \ref{tab:pc_cor} reports the correlations between the
original variables, while Appendix Table \ref{tab:pc_loadings} provides the
loadings on the first principal component.}
Table \ref{tab:main_res} presents the results of the hedonic regression for
property rental prices for the whole of England. The rents are positively
and significantly associated with the quality of the nearest school: an
increase in one standard deviation in school quality is associated with
extra \pounds 7.5 or 7 percent of weekly rent.\footnote{
One standard deviation of logarithm of total average point score per pupil
equals 0.20.} The literature on the relationships between school quality and
rental prices is scarce, making it hard to compare our result to earlier
findings.\footnote{
To the best of our knowledge, Bayer, Ferreira, and McMillan (2007) is the
only paper that includes an analysis of rental prices. They use a dataset
that combines rental and purchase prices to study the association with
elementary school quality. They found that households are willing to pay
less than 1 percent more in house prices when the average school performance
increases by 5 percent. We estimate a moderately higher relationships of 1.7
percent, focusing solely on secondary schools and renters.} Our estimate is
at the upper bound of what is typically found in the larger literature that
focuses on purchase property prices (Machin, 2011).
\subsubsection{Computation Steps\label{Calculation_nonp}}
The computation of welfare is done through the following steps, where for
simplicity, we use $\tau =0.5$ for illustration.
\begin{enumerate}
\item Construct the scalar index $X$ which equals the first principal
component of all non-$S$ attributes (STATA command pca). This is done to
reduce the dimension of the covariates.
\item Divide locations into $M$ markets. For each market, estimate the price
function by regressing price of unit on $\ln S$ and $X$; call the
coefficients $\alpha _{1m},\alpha _{2m},\delta _{m}$, $m=1,...M$
\begin{equation*}
P_{mi}\left( S,X\right) =\alpha _{1m\left( i\right) }+\alpha _{2m\left(
i\right) }\ln S_{i}+\delta _{m\left( i\right) }X_{m\left( i\right) }
\end{equation*}
\item Run a linear median regression (qreg in STATA), using all
observations, of $\ln S_{i}$ on an intercept and $y_{i},\alpha _{1m\left(
i\right) },\alpha _{2m\left( i\right) }$ and $\delta _{m\left( i\right) }$
\begin{equation*}
med\left( \ln S_{i}|y_{i},\alpha _{1m\left( i\right) },\alpha _{2m\left(
i\right) },\delta _{m\left( i\right) }\right) =r_{0}+r_{1}\left(
y_{i}-\alpha _{1m\left( i\right) }\right) +r_{3}\alpha _{2m\left( i\right)
}+r_{4}\delta _{m\left( i\right) }
\end{equation*}
where $m\left( i\right) $ is the market in which $i$ lives
\item Fix a value of $y=y_{0}$, $\delta =\delta _{0}$ (say, median values of
$y$ and $\delta $ in the data)
\item Then consider the change in $\alpha _{1},\alpha _{2}$ from $\left(
a_{1},a_{2}\right) $ to $\left( b_{1},b_{2}\right) $ (say, from the bottom
quartile to top quartile)
\item Calculate compensating variation as
\begin{eqnarray*}
CV &=&e^{r_{1}\left( b_{2}-a_{2}\right) }\left\{ \frac{r_{0}}{r_{1}}+y_{0}+
\frac{a_{2}r_{3}}{r_{1}}+\frac{r_{3}}{r_{1}^{2}}-a_{1}\right\} \\
&&+b_{1}-\frac{r_{0}}{r_{1}}-y_{0}-\frac{r_{3}b_{2}}{r_{1}}-\frac{r_{3}}{
r_{1}^{2}}
\end{eqnarray*}
\item Replace median in Step 3 to other quantiles, e.g. $\tau =0.25,0.75$
etc. and repeat steps 4-6.
\end{enumerate}
\subsection{Results}
In this section, we report the results obtained by applying the methods
outlined in Section \ref{Calculation_nonp} with a single attribute, viz.
school-quality. For the estimation we split the dataset into 9 markets,
represented by English regions: North, East Yorkshire and the Humber, North
West, East Midlands, West Midlands, South West, East England, and South East
London. Table \ref{tab:reg_by_region} presents the results of hedonic
regressions for rental prices estimated for different regions. The model is
described in Step 2 of Section \ref{Calculation_nonp}. The estimates for the
relationship between rental prices and school quality vary from 2.92 in
North East to 40.48 in London.
Table \ref{tab:q_iv} reports the results of a linear quantile regression for
the logarithm of the school quality presented in Step 3 of Section \ref
{Calculation_nonp}. To control for endogeneity, the term $y-\alpha _{1m(i)}$
in the model is instrumented with $s-\alpha _{1m(i)}$, where $s$ is the
level of past year savings reported by the household, and using the `ivqreg'
command in STATA, implementing the method of Chernozhukov and Hansen 2004.
The model produces the following results. The estimate of $r_{1}$ is
positive and decreasing across quartiles: income affects the demand for
school quality less for the households who value schools more. The estimate
of $r_{3}$ is negative and increasing from 25th to 75th percentile, \textit{
i.e.} its magnitude decreases: when the relationships between school quality
and prices grow stronger, demand decreases less for those who value schools
more. Both coefficient are significant for the 25th and 50th percentile, but
not for the 75th percentile of the distribution of school quality.
We now introduce a hypothetical policy change that increases the sensitivity
of rental prices to school quality. An example of such policy would be
introducing more strict distance criteria for school admission. In
particular, we change the relationships between the school quality and
rental prices, $\alpha _{2m(i)}$, from the 25th percentile of the
distribution across the markets (18.30 in the North West) to the 75th
percentile (28.56 in the East), and analogously exchange the constant term.
By design, rental prices near better schools should increase, while the
prices near worse schools should go down. Figure \ref{fig:np_policy_change}
presents the change in the relationship between school quality and rental
payments as a result of the policy change. In our sample, rents increase for
the majority of school quality levels, with only houses near the worst
schools becoming cheaper to rent.
The resulting estimates of compensating variation for different quantiles of
$\eta $ evaluated at specific quantiles of income are presented in Table \ref
{tab:cv_nonp1} and Figure \ref{fig:np_cv}. The policy change results in a
universal welfare loss. Compensating variation at the median preference for
school quality ($\eta =F_{\eta }^{-1}\left( 0.5\right) $), evaluated at $
y_{0}$ equal to the median income,\ equals \pounds 14,\ which constitutes
13\ percent of average weekly rent in England. Households at the same income
that live near schools of higher quality (i.e. have higher $\eta $)
experience a greater welfare loss that those living near worse schools. This
is to be expected, as their rents increase more, see Figure \ref
{fig:np_policy_change}. For households at identical percentiles of $\eta $,
those with higher incomes experience higher loss. This is again to be
expected, as comparatively richer households are likely to live near better
schools, where housing costs rise after the policy change, leaving lesser
funds for consumption.
As a robustness check, we also estimated the compensating variation for
deciles of preference for school quality between the 20th and 80th
percentiles, and find that the same patterns hold for the compensating
variation. These additional results for deciles are presented in Table \ref
{tab:cv_nonp2} and Figure \ref{fig:np_cv2}.
\section{Conclusion}
In this paper, we analyze individual demand of attributes in markets
characterized by smooth, nonlinear budget constraints. We provide an
econometric method of computing welfare effects of policy interventions that
change the budget frontier. The method works by deriving the analog of Roy's
identity when preferences are nonsatiated and budget-constraints are
nonlinear. This analog takes the form of a system of PDEs that involve
partial derivatives of the indirect utility function with respect to income
and with respect to parameters characterizing the price function. We show
how dimension restrictions on unobserved heterogeneity and a single-crossing
property of preferences enable one to identify the coefficient functions in
these PDEs, and then derive and use a set of Slutsky-like symmetry
conditions to calculate welfare effects resulting from a change in the
budget frontier. We provide a practical illustration of our methods to
evaluate welfare effects of a hypothetical change in relationship between
property rent and neighboring school quality in England.
Two issues are left to future work. The first is to develop methods for
flexible calculation of the budget frontier, correcting for potential
omitted variable bias, and possibly using penalized regression with many
covariates. The second is to develop methods of inference for the estimated
welfare thereof.
\newpage
\onehalfspacing
\section*{Tables}
\begin{center}
\begin{table}[h]
\caption{Hedonic regression results for England.}
\label{tab:main_res}\centering
\begin{tabular}{lc}
\hline
& Weekly rent \\ \hline
Logarithm of total average point score per pupil & 37.384*** \\
& (2.752) \\
First principal component of property characteristics & 3.977*** \\
& (0.279) \\
Constant & -125.187*** \\
& (16.243) \\ \hline
R-squared & 0.06 \\
Number of observations & 5,635 \\ \hline
\end{tabular}
\newline
\end{table}
{\footnotesize \textit{Note}: Standard errors in parentheses. * p$<$0.10, **
p$<$0.05, *** p$<$0.01}
\newpage
\bigskip {\footnotesize
\begin{landscape}
\begin{table}[h]
\centering
\footnotesize
\caption{Hedonic regression results by region.}
\label{tab:reg_by_region}
\begin{tabular}{lccccccccc}
\hline
& North East & North West & Yorkshire and & E. Midlands & W. Midlands & East & London & South East & South West \\
& & & the Humber & & & & & & \\ \hline
Logarithm of school quality & 2.927 & 18.297*** & 16.274*** & 25.836*** & 18.563*** & 28.556*** & 40.484** & 26.747*** & 30.954*** \\
& (3.173) & (4.857) & (4.626) & (6.574) & (6.021) & (7.064) & (16.111) & (7.858) & (9.162) \\
First principal component & 4.451*** & 4.035*** & 5.239*** & 5.249*** & 4.880*** & 5.900*** & 8.911*** & 5.238*** & 5.216*** \\
& (0.527) & (0.436) & (0.512) & (0.616) & (0.522) & (0.684) & (1.262) & (0.801) & (0.761) \\
Constant & 53.490*** & -28.164 & -25.139 & -76.485** & -30.314 & -73.695* & -119.299 & -52.004 & -92.781* \\
& (18.361) & (28.671) & (27.252) & (38.342) & (35.359) & (41.310) & (96.190) & (46.566) & (54.412) \\ \hline
R2 & 0.16 & 0.10 & 0.15 & 0.15 & 0.14 & 0.13 & 0.07 & 0.07 & 0.10 \\
Observations & 385 & 829 & 666 & 497 & 596 & 681 & 672 & 782 & 527 \\
\hline
\end{tabular}
\end{table}
{\footnotesize \textit{Note}: Standard errors in parentheses. * p$<$0.10, ** p$<$0.05, *** p$<$0.01}
\end{landscape}}
\end{center}
{\footnotesize \pagebreak }
\begin{center}
\begin{table}[h]
\caption{{}Quantile instrumental variable regression estimates for school
quality demand at different quartiles.}
\label{tab:q_iv}\centering
\begin{tabular}{lccc}
\hline
& 25th percentile & 50th percentile & 75th percentile \\ \hline
$y-\alpha _{1m(i)}$ & 0.00066** & 0.00047*** & 0.00022 \\
& (0.00027) & (0.00016) & (0.00017) \\
$\alpha _{2m(i)}$ & -0.00401* & -0.00252** & -0.00105 \\
& (0.00211) & (0.00122) & (0.00135) \\
$\delta _{m(i)}$ & 0.01553*** & 0.01136*** & 0.01119*** \\
& (0.00464) & (0.00337) & (0.00293) \\
Constant & 5.46753*** & 5.66965*** & 5.85488*** \\
& (0.07841) & (0.04600) & (0.04970) \\ \hline
Observations & 5635 & 5635 & 5635 \\ \hline
\end{tabular}
\end{table}
{\footnotesize \textit{Note}: To control for endogeneity, the term }$
{\footnotesize y-}\alpha _{{\footnotesize 1m(i)}}${\footnotesize \ in the
model is instrumented with }${\footnotesize s-}\alpha _{{\footnotesize 1m(i)}
}${\footnotesize , where }${\footnotesize s}${\footnotesize \ is the level
of savings reported by the household. F-statistics for the first stage from
a linear regression model for the mean equals 47.97. Standard errors in
parentheses. * p$<$0.10, ** p$<$0.05, *** p$<$0.01}
\newpage
\bigskip
\begin{table}[h]
\caption{Compensating variation estimates for quartiles of preference for
school quality, GBP.}
\label{tab:cv_nonp1}\centering
\begin{tabular}{lccc}
\hline
& \multicolumn{3}{c}{Income percentile} \\
School quality percentile & 25th percentile & 50th percentile & 75th
percentile \\ \hline
25th percentile & 11.8520 & 12.4455 & 13.2846 \\
50th percentile & 13.6383 & 14.0606 & 14.6575 \\
75th percentile & 15.0347 & 15.2322 & 15.5114 \\ \hline
\end{tabular}
\end{table}
\end{center}
\newpage
\bigskip
\begin{table}[h]
\caption{Compensating variation estimates for deciles of preference for
school quality, GBP.}
\label{tab:cv_nonp2}\centering
\begin{tabular}{lccc}
\hline
& \multicolumn{3}{c}{Income percentile} \\
School quality percentile & 25th percentile & 50th percentile & 75th
percentile \\ \hline
20th percentile & 11.6858 & 12.2037 & 12.9359 \\
30th percentile & 12.2099 & 12.7434 & 13.4976 \\
40th percentile & 13.0697 & 13.4612 & 14.0146 \\
50th percentile & 13.6383 & 14.0606 & 14.6575 \\
60th percentile & 14.1730 & 14.5000 & 14.9621 \\
70th percentile & 14.7370 & 14.9303 & 15.2035 \\
80th percentile & 15.4711 & 15.6454 & 15.8919 \\ \hline
\end{tabular}
\end{table}
\newpage
\bigskip
\section*{Figures}
\bigskip\ \
\begin{figure}[hptb]\centering
\includegraphics[height=3.9064in, width=6.9444in, height=3.9548in, width=7.0093in]{graphics/SMA5SA00__1.pdf}
\caption{Compensating variation when the budget frontier changes.}
\label{fg:compensation_variation}
\end{figure}
\newpage
\begin{center}
\begin{figure}[hptb]\centering
\includegraphics[height=3.3806in, width=4.6535in, height=3.4264in, width=4.7063in]
{graphics/Figure_NP1_price_vs_z_nonparam__2.png}
\caption{Change in the relationships between rents and school quality as a
result of policy change.}\label{fig:np_policy_change}
\end{figure}
\end{center}
\newpage
\begin{center}
\bigskip
\begin{figure}[ptb]\centering
\includegraphics[height=8.3472in, width=11.4726in, height=3.6625in, width=5.0237in]
{graphics/fig2_np__3.png}
\caption{Compensating variation estimates for quartiles of preference for
school quality, GBP.}\label{fig:np_cv}
\end{figure}
\end{center}
\newpage
\begin{center}
\bigskip
\begin{figure}[ptb]\centering
\includegraphics[height=3.3806in, width=3.806in, height=3.7455in, width=4.2142in]
{graphics/Figure_NP2_CV_d__4.png}
\caption{Compensating variation estimates for deciles of preference for
school quality, GBP.}\label{fig:np_cv2}
\end{figure}
\end{center}
\newpage
\begin{center}
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\end{center}
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\newpage