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date: 23 August 2019

# Use of Property Values and Location Decisions for Environmental Valuation

## Abstract and Keywords

The economics literature has developed various methods to recover the values for environmental commodities. Two such methods related to revealed preference are property value hedonic models and equilibrium sorting models. These strategies employ the actual decisions that households make in the real estate market to indirectly measure household demand for environmental quality. The hedonic method decomposes the equilibrium price of a house based on the house’s structural and neighborhood/environmental characteristics to recover marginal willingness to pay (MWTP). The more recent equilibrium sorting literature estimates environmental values by combining equilibrium housing outcomes with a formal model of the residential choice process. The two predominant frameworks of empirical sorting models that have been adopted in the literature are the vertical pure characteristics model (PCM) and the random utility model (RUM). Along with assumptions on the structure of preferences, a formal model of the choice process on the demand side, and a characterization of the supply side to close the model, these sorting models can predict outcomes that allow for re-equilibration of prices and endogenous attributes following a counterfactual policy change.

Innovations to the hedonic model have enabled researchers to more aptly value environmental goods in the face of complications such as non-marginal changes (i.e., identification and endogeneity concerns with respect to recovering the entire demand curve), non-stable hedonic equilibria, and household dynamic behavior. Recent advancements in the sorting literature have also allowed these models to accommodate consumer dynamic behavior, labor markets considerations, and imperfect information. These established methods to estimate demand for environmental quality are a crucial input into environmental policymaking. A better understanding of these models, their assumptions, and the potential implications on benefit estimates due to their assumptions would allow regulators to have more confidence in applying these models’ estimates in welfare calculations.

# Introduction

For the fiscal years 2005 through 2015, the cost of major federal regulations totaled between $74 and$110 billion annually (in 2014 dollars), of which environmental and public health issues constitute a large share (OMB, 2016). Recommendation and assessment of these policies rely on cost-benefit analysis (CBA), an economic approach that compares the benefits of proposed policies, including those relating to the environment, to their costs. As such, the accurate measurement of these costs and benefits is a crucial input into environmental policymaking. This practice was first signed into law by President Ronald Reagan in 1981 through Executive Order 12291, which stipulates that “regulatory action shall not be undertaken unless the potential benefits to society from the regulation outweigh the potential costs to society.” For the environment, direct compliance with these policies (e.g., reporting, pollution abatement) make up most of the costs. While these can be difficult to measure, cost estimation is more straightforward in comparison to the task of quantifying the benefits of a cleaner environment. This is because most environmental attributes do not have a market price as they are not typically traded in a market. The economics literature has developed various methods to estimate the values for environmental commodities; this article reviews two such methods related to revealed preference that use property values and household location decisions. In particular, these strategies employ the actual decisions that households make in a related market, real estate, to indirectly measure their demand for such non-marketed goods.

The two specific models reviewed in this article are property value hedonic models and equilibrium sorting models. The hedonic model, formalized by Rosen (1974), recovers people’s values for attributes of a differentiated product that is not traded in a market. Applied to housing, for example, proximity to green space may be considered a desirable attribute of a residential property. Relying on the idea that individuals require compensation in the form of reduced housing costs to live in locations closer to green space, the hedonic model infers household demand for green space from housing price differentials between properties located at different proximities from this environmental attribute. Under the assumptions of zero mobility costs, full information, and no price discrimination, the hedonic method allows one to recover theoretically consistent welfare measures of environmental quality. There are, however, several drawbacks of the hedonic model due to its use of the observed, equilibrium outcomes of household decision processes. The more recent equilibrium sorting literature, by imposing additional structure on the model, formally characterizes the way in which households decide where to live and, in doing so, is able to deal with considerations in the household’s optimization problem that the hedonic model cannot. Importantly, these considerations, if ignored, have been shown to induce large biases in the estimation of environmental values.

Valuation of the environment from an anthropocentric perspective is important for policymaking. Humans can have significant impacts on the functioning of environmental systems. Since there is no market from which we can directly recover individuals’ values for environmental attributes, we must rely on alternative methods to recover these values. Property value hedonic and sorting models are among those established to recover valuation. As such, an understanding of these models is an important component in the broader field of environmental science.

# Property Value Hedonic Models

Ridker and Henning (1967) conducted the first study to apply hedonic pricing to the environment by estimating the impact of air pollution on property values. Along with other early work (Court, 1939; Griliches, 1971; Lancaster, 1966), this study decomposes the value of a differentiated good into its characteristics so as to recover the implicit, or “hedonic,” prices for these various characteristics. The seminal paper by Rosen (1974) provided a theoretical basis on which consumer preferences could be recovered from these prices. This was accomplished by formalizing the relationship between hedonic prices for the characteristics of a house (i.e., a quality-differentiated good) and the preferences of the consumer who purchases that house as result of utility maximization.

Suppose a house, our commodity of interest, can be completely described by a vector of characteristics, where characteristics include structural house attributes $Z$ (e.g., square footage) and environmental quality $q$. A consumer, with income $Y$, has preferences, described by $α$, for houses that are defined over these attributes so that choosing the combination of attributes is equivalent to choosing a house. The amount that a consumer pays for a house with a specific set of characteristics is set according to an equilibrium price schedule, $P(Z,q)$. Because it is assumed that characteristics of a house cannot be unbundled, the price schedule is generally thought to be non-linear in $Z$ and $q$. Given prices and income level, $Y$, the consumer’s problem is to maximize her utility, which is assumed strictly concave and monotone in attributes, by choosing the attribute vector, $Z$ and $q$, and all other consumption, x, subject to her budget constraint,

$Display mathematics$

Here, income and prices have been normalized to the price of the numeraire good, x, which takes a value of 1. The first-order conditions of this optimization problem are such that

$Display mathematics$
(1)

This says that, at the optimum, the rate at which a consumer is willing to trade off between environmental quality and the numeraire good to hold utility constant (i.e., the marginal willingness to pay for q) is equal to its implicit price. This was an important development for hedonic pricing models, as it related the estimated coefficients from a price regression (e.g., the marginal contribution of a bedroom to the house price) to structural parameters that characterized individual preferences.

To gain further insight into how hedonic prices relate to consumer preferences and how the hedonic price function is formed, Rosen (1974) represents the consumer’s problem using bid functions, $θ(Z,q,u0,α)$, which he implicitly defines according to

$Display mathematics$
(2)

The bid function gives the maximum amount a particular individual with preferences α‎ is willing to pay for $q$ to maintain utility at u0. Differentiating (2) with respect to the environmental good of interest, $q$, yields the following that equates the marginal rate of substitution between $q$ and x to the derivative of the bid function,

$Display mathematics$

Combined with (1), one sees that the marginal implicit price of $q$ is equal to the derivative of the bid function with respect to $q$, or the inverse demand for $q$, at the optimally chosen bundle of attributes, or

$Display mathematics$
(3)

To understand the formation of the hedonic price function, one has to include the production of housing on the supply side of the market and the characteristics of suppliers. Firms with characteristics β‎ produce housing with characteristics $(Z,q)$ by maximizing the following net-profit function, taking prices as given:

$Display mathematics$

where $C(Z,q;β)$ reflects the cost of building a house. Profit maximization implies that firms choose to supply q according to

$Display mathematics$

As with consumer bid functions, Rosen defines offer curves, $o(Z,q,Π,β)$, for an individual firm that describe the lowest amount that a firm is willing to accept for producing houses with characteristics $(Z,q)$ and earn profit $Π0$,

$Display mathematics$
(4)

Differentiating (4) with respect to environmental quality q yields

$Display mathematics$
(5)

which says that the marginal implicit price of q is equal to the derivative of the firm’s offer function with respect to q, or its supply function of q, at the optimum. As utility-maximizing buyers and profit-maximizing sellers interact in the marketplace, they are matched according to (3) and (5). In equilibrium, the quantity of attributes demanded is equal to that supplied for every value of the attribute, and the equilibrium hedonic price function is formed as the upper and lower envelope of buyers’ and sellers’ optimizing bid and offer functions.

Under the assumption that (1) there are no mobility costs associated with re-locating (e.g., financial or psychological moving costs), (2) households have full information about housing characteristics, and (3) everyone faces the same price schedule, the hedonic price gradient with respect to an environmental attribute returns the willingness to pay for a marginal change in environmental quality.1 Rosen’s paper sparked a large body of empirical work that estimated hedonic price functions to recover MWTP for environmental (and other non-marketed) goods. Estimation of such hedonic price functions, $P(Z,q)$, referred to as Rosen’s “first stage,” is performed by regressing data on real estate property transaction prices2 on their associated properties’ structural characteristics and nearby spatially and/or time-varying environmental attributes. The price of a house, j, for example, could be modeled as,

$Display mathematics$
(6)

where $uj$ captures all other unobserved attributes. There are numerous examples of first-stage hedonic analyses.3 The literature has identified important issues to account for when estimating these price functions. Issues that are specific to hedonics include the choice of the appropriate spatial and temporal extent of the market and functional form specification (Cropper, Deck, & McConnell, 1988; Kuminoff, Parmeter, & Pope, 2010; Parmeter, Henderson, & Kumbhakar, 2007). Discussion of these issues can be found in Palmquist (2006) and Taylor (2003).4 Other concerns, which apply to empirical work in economics more generally, include correlation between houses based on geographic space (Kim, Phipps, & Anselin, 2003; Leggett & Bockstael, 2000) and consistent estimation of implicit prices in the presence of omitted variables. The issue of consistent estimation, in particular, has received a lot of attention. Specifically, unobserved housing attributes may be correlated with the attribute of interest. That is, decomposing the error term in (6) as $uj=ξj+ej$,

$Display mathematics$
(7)

$ξj$ is potentially correlated with $qj$. To deal with this, researchers have creatively exploited “natural” experiments that induce random variation in the attribute of interest (e.g., air quality) to allow for identification of marginal prices. These “quasi-experiments” have been widely applied in other fields within economics. Greenstone and Gayer (2009) discuss quasi-experimental methods applied to environmental economics, and Parmeter and Pope (2012) review these designs applied within a property value hedonic framework. Both discuss the benefits as well as the limitations and assumptions required to identify MWTP from quasi-experimental variation.

That hedonic prices reveal MWTP at the point of consumption is extremely useful for evaluating the benefits of marginal changes in environmental quality; yet, if changes in environmental quality are not marginal, as is often the case, and MWTP for environmental quality is not constant, then approximation of benefits for an increase in environmental quality using MWTP will be biased. As such, Rosen proposes a procedure to recover a consumer’s entire inverse demand curve using a second step that follows the first stage estimation of the hedonic price function. From the first stage, one can recover implicit prices for the amenity of interest at the chosen bundles, $∂P(Z,q)^∂q=α^q,1+α^q,2q$, where the ^ denotes implicit prices based on the first-stage estimates. In Rosen’s second stage, one regresses these implicit prices of q on the levels of q that are demanded and supplied, along with consumer and firm characteristics, X, to estimate the following demand and supply functions,

$Display mathematics$
(8)

$Display mathematics$
(9)

There are two problems associated with this procedure: the first relates to identification of the demand parameters; second, assuming that the demand function is identified, the demanded attributes are endogenous to their implicit prices.

Brown and Rosen (1982) and Mendelsohn (1985) first showed that the parameters of the demand curve are not identified using transactions from a single market under a linear hedonic gradient. The intuition behind non-identification can be seen from the equilibrium condition in (3), which shows that a point on the hedonic price function reveals only one point on an individual’s bid curve. When moving to a different point on the hedonic price function, the observed choice is being made by an individual who is characterized by a different α‎. Thus, the hedonic gradient returns one point from many different individuals’ demand curves, when we need at least two observed choices made by one individual to recover her (linear) demand. One of the originally proposed solutions was to assume a non-linear hedonic gradient to achieve identification since, in this case, the non-linear gradient could only be explained by a single set of linear MWTP functions with different intercepts but the same slope. A second solution involved “seeing” at least two choices made by individuals with the same preferences by using transactions from multiple markets. Under the first case, one must be willing to make functional form assumptions that may not be obvious, a priori. Ekeland, Heckman, and Nesheim (2004) (discussed later) justify estimating the hedonic price function flexibly using non-parametric methods to deal with identification.5 With the latter strategy, one must assume that individuals who bear the same observable characteristics, but appear in different markets, have the same preferences.

Even if one were willing to impose non-linearity assumptions or use a multi-market strategy to achieve identification, Epple (1987) and Bartik (1987) both show that estimation of demand parameters will suffer from endogeneity in the sense that $ud$ will be correlated with $qd$ from (8). This is because consumers (and suppliers), in choosing the amount of a characteristic q are also choosing the implicit price they pay for the attribute, $∂P(Z,q)∂q$, based on where on the hedonic price function they are located. In equilibrium, we can see the nature of the endogeneity by setting the implicit price equal to the MWTP function,

$Display mathematics$

and reorganizing,

$Display mathematics$
(10)

Since observable demand shifters $Xd$ such as race are generally thought to be predetermined, this means that $ud$ must be correlated with q. The common approach of finding instruments from supply-side variables to deal with simultaneity is, however, invalidated as the matching of buyers with certain demands to suppliers who can cost-effectively fulfill those demands necessarily means that producer characteristics ($Xs$) will be systematically related to unobserved buyer characteristics in the hedonic equilibrium.6

While there have been efforts to carry out second-stage hedonic analyses,7 the strong assumptions required to identify demand (i.e., choice of a non-linear functional form or cross-market preference restrictions) combined with a lack of strong instruments associated with these older approaches have led most hedonic studies until more recently to focus on estimation of the first stage.

## Innovations to the Hedonic Model

There have been various innovations to the hedonic model. In this section we first discuss solutions to the identification and endogeneity concern with respect to estimating the entire demand function. Next we review complications that arise in recovering average MWTP using the traditional hedonic framework that ignores the discrete nature of various attributes and household dynamic behavior. Then follows a discussion about how using prices from multiple hedonic equilibria, often employed to yield causal estimates, could complicate the interpretation of the resulting estimate as a welfare measure. In conclusion, we overview some remaining weaknesses that hedonic models have been unable to deal with as a segue into equilibrium sorting models.

### Identification and Endogeneity of MWTP Function

Several papers have offered solutions regarding identification and endogeneity of demand for environmental quality in second-stage hedonic analyses. Recall that a non-linear hedonic gradient would resolve identification concerns in the second stage. Ekeland et al. (2004) show that the preferences and technology that produce a linear hedonic gradient function are very special cases and that, in general, theory would predict a non-linear hedonic gradient. To reduce reliance on a specific functional form, one can non-parametrically estimate the hedonic price function, recover implicit prices for each value of the attribute of interest, and then estimate the second stage by regressing the implicit prices on the chosen levels of the attribute and individual characteristics. Ekeland et al. (2004) then deal with the endogeneity concern by proposing an instrument for q using a function of demand shifters, $Xd$, where the instrument is excluded from the demand equation based on the assumed functional form of the MWTP function.8 Identification is based on an additive separability assumption in the MWTP function between individual heterogeneity and the amount of amenities chosen. In an extension to this work, Heckman, Matzkin, and Nesheim (2010) analyze conditions for identification allowing for non-separable unobserved heterogeneity.

Bishop and Timmins (2011) provide an alternative approach to deal with the endogeneity problem by reformulating the implicit price function in a way that avoids the use of instrumental variables altogether. They note that the endogeneity problem associated with second-stage hedonic estimation arises from using the equilibrium condition (where implicit price is set equal to MWTP) in estimation. Rather than form implicit prices to use as a dependent variable in the second stage, they base estimation on the reorganized condition in (10) where the endogenous variable q is isolated on the left-hand side. The amenity q is now a function of parameters that characterize the implicit price for attributes (which consumers take as given and can be recovered via the first stage), MWTP parameters, and predetermined individual observable and unobservable characteristics, all of which are exogenous. That such a relationship holds between the quantity of an amenity chosen by households and household characteristics is consistent with the notion of stratification first discussed in Ellickson (1971). Parameters associated with the MWTP function are then estimated via maximum likelihood while imposing a distributional assumption on unobserved individual heterogeneity ($ud$). Their estimator avoids strong parametric assumptions on the utility function and is able to relax the assumption of preference homogeneity across markets used to identify instruments as in Bartik (1987) and Epple (1987).

Various researchers have also begun to apply more structure on preferences to estimate demand parameters within a hedonic setting (Bajari & Benkard, 2005; Kumino & Jarrah, 2010; Yinger, 2015). Bajari and Benkard (2005) achieve identification of preference parameters by assuming that individual heterogeneity enters through the parameters of a specific parametric utility function. The assumed form of the utility allows one to directly solve for each household’s preference for a particular attribute by using the first-order conditions from utility maximization. This procedure relies on the assumed functional form of the utility to bypass estimating a second stage. Bajari and Kahn (2005) apply this method to analyze racial segregation across several metropolitan areas.

### Discrete Choice Sets

Aside from the issues of identification and endogeneity, researchers have recognized other weaknesses of the hedonic model. These can be seen through further examination of the household’s problem,

$Display mathematics$
(11)

where $U(.)$ represents the household’s indirect utility after having substituted the budget constraint into utility $u(.)$. Recall that the relationship between MWTP and implicit prices was derived upon taking a first-order condition with respect to the amenity of interest, q. This assumes that households can choose any combination of characteristics (i.e., the choice set is dense), even though this may not be realistic. Often housing attributes are discrete (e.g., q could represent whether a house is a single family dwelling or not) or they cannot be freely combined with all other attributes. As such, one can only infer a lower or upper bound on the MWTP. An example from Bajari and Kahn (2005) of such an attribute is whether a house is a single-family dwelling. If a person chooses to live in a single family home (i.e., single = 1), then

$Display mathematics$

That is, one could only infer that the individual’s MWTP to live in a single-family home is at least as large as the implicit price that she faces. On the other hand, if the individual chose to live in a non–single-family residence, then

$Display mathematics$

and we can only infer that her MWTP is less than or equal to the implicit price. Griffith and Nesheim (2013) extend the hedonic model to allow for discrete characteristics. They derive bounds on individual willingness to pay for discrete attributes using the mean value theorem. In an application that uses baskets of goods purchased on shopping trips from Nielsen scanner data, they use the price premium from a hedonic equilibrium to bound the MWTP for a product that is “organic.” They note that to point-identify MWTP estimates, one needs to place additional structure on consumer preferences. An example of this comes from Bajari and Kahn (2005), which recovers point estimates of the MWTP for several discrete characteristics by specifying a parametric utility as well as a distribution that characterizes consumer unobserved heterogeneity.

### Correlated Unobserved Characteristics

The variation (or lack thereof) in an attribute of interest also causes a second problem in empirical estimation. In search of variation in environmental quality to causally estimate MWTP, researchers have turned to changes in an environmental attribute over time, which often mixes prices from different equilibria (Kuminoff and Pope, 2014; Banzhaf, 2015). For example, researchers often estimate the MWTP using the price differential between houses with varying levels of the amenity q (e.g., $q=q0$ versus $q=q1$),

$Display mathematics$

However, because changes in environmental quality over time are often tied to policies and are likely non-marginal, this would induce people to re-optimize (e.g., move), where the matching process of (a different set of) buyers and sellers after the policy change would lead to a new hedonic equilibrium. If this were the case, then the housing price changes over time associated with the amenity change are no longer the MWTP of either those living in the area before the change or after. In order for hedonic analyses that combine prices before (P1) and after (P0) a change in an amenity to recover MWTP, one needs to assume that the hedonic equilibrium does not change over time, i.e., $P1=P0$, otherwise

$Display mathematics$

where the superscript 0 denotes the baseline and 1, the new equilibrium. The expression on the right side of the equation has been referred to as the “capitalization” of the amenity q into housing prices. Panel variation in the amenity of interest has often been exploited in quasi-experimental designs to control for neighborhood and house-specific unobserved contributors to price. The clear benefit of such designs is in their ability to arrive at a causal impact of changes in q on housing prices; however, the underlying hedonic theory that yields the prices given the variation in the amenity complicates the interpretation of capitalization as MWTP. Kuminoff and Pope (2014) make this point by decomposing the estimated capitalization effect (assuming time-varying hedonic price functions) into the structural parameters that characterize preferences of households and show that MWTP is generally not equal to capitalization unless the preferences of those living in the area before and after the change are the same and the implicit prices of all other housing attributes have remained the same after the change.9 In a similar vein, Kuminoff and Jarrah (2010) investigate the magnitude of the difference between MWTP and housing market capitalization with a calibrated model that predicts new hedonic equilibria via an iterative bidding algorithm, where households compete for housing units via second-price auctions. Using school quality data from San Joaquin county in California, they find that the wedge between MWTP and capitalization ranges between −20% and 24% depending on the initial value of the school quality, the policy change used to measure capitalization, and the sorting that occurs in response. As a result of this work, more recent hedonic applications have taken care to estimate MWTP without assuming a time-varying price function (e.g., Haninger, Ma, & Timmins, 2017; Kuminoff & Pope, 2014; Muehlenbachs, Spiller, & Timmins, 2016).

Even if one allowed for time-varying hedonic gradients, one still needs to be concerned with consistent estimation. Furthermore, natural experiments that induce exogenous variation in the amenity of interest are not always easy to come by. Using the previous characterization of the price function, this refers to dealing with the correlated, time-varying unobserved attributes, $ξ$,

$Display mathematics$

Bajari, Cooley, Kim, and Timmins (2012) propose an estimator that deals with unobserved heterogeneity in the absence of an available natural experiment. An added benefit of their estimator is that for time-varying attributes such as $qt$, they do not need to assume time-invariant hedonic price functions to estimate MWTP (i.e., parameters on qt are allowed to vary by time), which avoids the concern raised by Kuminoff and Pope (2014). They make three assumptions for the estimator. First, the housing price can be fully characterized as a function of its attributes (a standard assumption in all hedonic models). This assumption allows recovery of the residual $ξt$ from a hedonic regression, which includes information on unobserved attributes that contribute to price. Second, they assume that the evolution of the unobserved component can be parameterized by a Markov process, $ξt’=γξt+ηt,t’$, where $ηt,t’$ is the stochastic innovation in the unobservable. Finally, they assume that homebuyers are rational in the sense that, conditional on information known at a time t, they are unable to predict $ηt,t’$, i.e., $E[ηt,t’|It]=0$. This identifying assumption relies on the efficiency of the housing market, which they assess using tests from Case and Shiller (1989). In a hedonic regression of $Pt’$ on its characteristics, they then directly control for correlation with the unobserved characteristic, $ξt$ by substituting it out as a function of lagged prices and observed characteristics. The remaining error term becomes $ηt,t’$, which is uncorrelated with the attributes based on the rationality assumption. In an application to value marginal improvements in air quality, their estimator finds MWTP estimates that are in the higher range of what has been found in previous literature and that their estimator compares favorably to cross-sectional and fixed effects estimators in terms of the expected sign of MWTP.

### Dynamics

Lastly, there have been recent innovations within the hedonic framework to relax the assumption that households are static utility optimizers. Let the variables that a household observes when making its location decision at time t be summarized into the vector $sit=(Zt,qt,Pt,α,Yt)$. If households are forward-looking, then a household’s utility from consuming attributes $dit=(Z,q)$ at time t would be given by

$Display mathematics$
(12)

where $β$ represents the discount factor for consumption in the future. Note that the first component of utility is the static portion of utility from (11), except with time subscripts. The traditional hedonic model effectively sets β‎ = 0 and, in doing so, assumes that households only care about the amenities attached to the house in the current period. However, because switching in the housing context is likely to be costly, households would generally consider how the amenities attached to a house are likely to change in the future.10Bishop and Murphy (2011) incorporate this type of forward-looking behavior into a hedonic framework. They first derive the household’s first-order conditions using the dynamic utility in (12) and show that ignoring forward-looking behavior is like having an omitted variable in a first-stage hedonic regression. Next, they borrow from the dynamic discrete choice literature and approximate the future value of choice alternatives, allowing them to directly control for these confounding factors in the estimation of the hedonic price gradient.11

While researchers have come up with solutions to deal with the weaknesses in the hedonic model while maintaining a hedonic framework, others have turned to modeling the sorting process that underlies the hedonic equilibrium to recover the structural parameters that characterize individual preferences. The starting point of many of these models is that consumers choose among a discrete number of locations in which to reside. Immediately, these models assume that the choice set has “holes” in that only specific, discrete combinations are available to consumers, which relaxes the assumption of continuous choice sets from the hedonic model. These models can also accommodate dynamics by defining consumer preferences to incorporate the future stream of utilities, rather than current utility. Along with assumptions on the structure of preferences, a formal model of the choice process on the demand side, and a characterization of the supply side to close the model, sorting models can predict outcomes that allow for re-equilibration of prices and endogenous attributes following a counterfactual policy change. The literature on neighborhood tipping (Banzhaf & Walsh, 2013; Card, Mas, & Rothstein, 2008; Schelling, 1969) would suggest this could be particularly important for welfare calculations. We turn to this set of models next.

# Equilibrium Sorting Models

“Sorting” describes the way in which households choose where to live across a varied landscape. Sorting models use the observed choices of residential locations, along with the prices of those locations, their characteristics, and demographic information about households to reveal household demand for non-marketed attributes. Tiebout (1956) provides the foundation for sorting models, suggesting that, in a spatially varied landscape where households can choose between different communities that offer different qualities of locally provided public services, households will reveal their preferences for such public services by choosing a community of residence based on their preferences. This notion of “voting with your feet” applies in the environmental context: various papers have found empirical support that households migrate in response to air pollution and Superfund sites (Banzhaf & Walsh, 2008; Cameron & McConnaha, 2006).

Beginning with Ellickson (1971), a set of theoretical papers followed that formalized Tiebout’s intuition. Ellickson (1971) first provided restrictions on household preferences that would explain an equilibrium in which communities are stratified by income and no households could become better off by moving to a different location. Subsequent theoretical papers continued to examine the existence of sorting equilibria and their properties, extending the basic model from Ellickson (1971) in various dimensions (Epple, Filimon, & Romer, 1984; Epple & Platt, 1998; Epple & Romer, 1991; Westhoff, 1977).

The sorting process described by these models yields an equilibrium that is defined by a set of prices, attributes that characterize the landscape, and location choices such that every household occupies its utility-maximizing location and no one wants to move. Two predominant frameworks of empirical sorting models have been adopted in the literature to estimate valuation for environmental quality. They are the vertical pure characteristics model (PCM)12 and the random utility model (RUM). The fundamental difference between these models is in how they characterize consumer tastes for public goods of interest. The following two sections lay out the basic model for each. A comprehensive review can be found in Kuminoff, Smith, and Timmins (2013). For a practical guide on details of implementation, see Klaiber and Kuminoff (2014).

## Pure Characteristics Sorting Model

Epple and Sieg (1999) provide the first structural estimator of an equilibrium sorting model developed in Epple et al. (1984) and Epple and Platt (1998). Their framework begins with defining a metropolitan area consisting of J fixed jurisdictions, where each jurisdiction j is characterized by a linear index for the quality of public goods, gj, and the price of a homogeneous unit of housing services h, given by pj. Households are differentiated by their income y, and taste for the public good α‎, where the joint distribution of y and α‎ is characterized by $f(y,α)$. Household preferences are defined over housing, h, public goods, g, and a composite private good, b, where the household’s problem is to choose a continuous amount of housing and the composite good to maximize the following utility subject to a budget constraint,

$Display mathematics$

Assuming that households can choose any quantity of housing, one can solve the preceding problem for the optimal choice of housing, $h(p,y)$ and represent household preferences for a community with the following indirect utility based on public goods, prices, and income:

$Display mathematics$

With the optimal levels of housing and numeraire that would be chosen in each community determined, the household then decides the community of residence that maximizes its indirect utility. The paper next imposes a single crossing property in preferences where the slope of indifference curves in the $(p,g)$ plane, i.e.,

$Display mathematics$
(13)

is monotonically increasing in $α$ given y and y given $α$. This implies that, given $α$, indifference curves in the $(p,g)$ plane become steeper as y increases (and vice versa). Graphically, this is depicted in Figure 1, which maps indifference curves for households with different income levels (holding α‎ constant).

Click to view larger

Figure 1. Single crossing property.

In order for an equilibrium to exist, the single crossing property implies three necessary conditions that characterize the equilibrium: boundary indifference, stratification, and increasing bundles. If communities are ordered based on the provision of public goods, then boundary indifference says that for each set of adjacent communities, $(j,j+1)$, there are a set of individuals, characterized by $(α,y)$, who are indifferent between these communities, i.e.,

$Display mathematics$
(14)

Equation (14) implicitly defines a function $y(α)$. Stratification then says that for a given α‎, communities will be stratified by income, i.e., $yj−1(α). If households were homogeneous except in their income, i.e., α‎ is the same for everyone, then the single crossing assumption would imply that communities are perfectly stratified by income, as was the case in Ellickson (1971). Allowing for unobserved heterogeneity across households in a sorting equilibrium (introduced in Epple & Platt, 1998) implies the more relevant case in which there is imperfect stratification by income across jurisdictions. This can be seen from Figure 2, where households are partitioned into various communities (denoted by solid lines) in the (ln α‎, ln y) plane. At a given level of income, a household may live in any of the j communities, depending on the strength of its preferences for public goods (α‎). Lastly, the sorting equilibrium is characterized by increasing bundles, which says that for two communities i and $j,pi and $gi. This condition requires that the ranking of communities by amenity level must match that by price, which is needed for all of the communities to be populated so that they exist in equilibrium. Taken together, these conditions imply that households, given their income and preferences, are partitioned across communities based on the distribution of public goods and prices. Referencing Figure 2, this says that the set of households lying on the border between two communities (the solid lines) are indifferent between the two communities on either side of the border. Combined with a parameterization of the model and assumptions on the distribution of consumer heterogeneity, these necessary conditions can be used to make predictions for the way in which people locate across communities.13 Specifically, the conditions can be used to derive expressions for the share of people living in each jurisdiction as well as the income distributions within each, based on the deep structural parameters that characterize consumer preference heterogeneity. Epple and Sieg (1999) propose an estimator that recovers those structural parameters by finding the values that best match the predictions from the model to data describing population shares and income distribution from the census. These parameters can then be used to calculate the demand for various non-marketed amenities that contribute to the public good index.

Click to view larger

Figure 2. Stratification.

The parameterizations in Epple and Sieg (1999) are as follows: consumer heterogeneity, f(ln α‎, ln y), is characterized by a bivariate log-normal distribution and the indirect utility from the choice of a community is characterized by the following CES utility function:

$Display mathematics$
(15)

This gives rise to a particular housing demand, $h(p,y)=Bpηyν$, obtained using Roy’s identity, where the expected signs for the housing demand parameters are $η<0,ν>0$, and B > 0. The single crossing condition implies that the slope of the indifference curves based on the above indirect utility is increasing in Y and α‎.14 Combined with the expected signs for the housing demand parameters, the single crossing property then requires that ρ‎ < 0. Thus, the estimate for ρ‎ provides a test of whether preferences satisfy single crossing and, as such, whether the data can support this type of sorting equilibrium. Given the parameterization, the boundary indifference condition, $U(gj,pj;α,y)=U(gj+1,pj+1;α,y)$, can be rewritten as

$Display mathematics$
(16)

where

$Display mathematics$
(17)

Equation (16) corresponds to the boundaries that separate communities in Figure 2 under the current parameterization, where the Kj’s represent the intercepts on the ln α‎ axis. Integrating over all of the households within the boundaries of a community J as defined by (16) returns the share of the total population living in a community J, which can then be used to predict various percentiles of the income distribution for each community. These predictions from the model (e.g., the 25th, 50th, and 75th percentiles of the income distribution) are compared with their empirical counterparts from the data, creating a set of J × 3 moment conditions used to construct a GMM objective function. The estimation yields parameters that best match the data given the model. In an application, Sieg, Smith, Banzhaf, and Walsh (2004) include an additional moment condition based on the model’s prediction of each community’s level of public good provision. The public good provision in each community can be solved for recursively (up to a normalization) through (17) using observed population shares, housing prices, and a guess of the parameters,

$Display mathematics$

With predicted levels of public good provision for each location, a moment condition can be included based on the following assumed decomposition of $gj$ into observable ($qj$) and unobservable ($ξj$) attributes,

$Display mathematics$
(18)

Estimation of γ‎ provides the marginal utilities of various community attributes, which, along with an estimate of the marginal utility of income, can be used to recover values for non-marketed housing attributes. Equation (18) reveals an important restriction of this model: individuals are required to have the same weight, γ‎, for each component of the public good index. This implies that consumers rank the communities with different gj in the same way, even if they differ in how much they are willing to pay for these services.15 Agreement over community rank stipulates that everyone also agrees on the closest substitutes to a particular location.

The modeling of a formal choice process does not preclude dealing with potential correlation between unobserved and observed amenities, $ξj$ and $qj$. The authors propose an instrumental variables strategy. The instruments are based on the community ranking in income: the ranking is clearly correlated with public goods levels based on the single crossing assumption, but as long as the unobserved characteristic is unimportant enough (relative to $qj$) so that it does not affect the ranking of the communities, then it is uncorrelated with qj. Using this empirical model, Epple and Sieg (1999) are able to match predictions from their model with empirical income distributions reasonably well and find a statistically significant $ρ^<0$, an important test of whether single crossing conditions are satisfied.

## Random Utility Sorting Model

The random utility model provides an alternative framework to estimate equilibrium sorting models. This class of models was introduced by Bayer, McMillan, and Rueben (2004) within a residential sorting framework and derives much of its set up from horizontally differentiated models of demand in industrial organization. An important difference between the RUM and the pure characteristics model is the way in which consumer preference heterogeneity is introduced, where individual preferences for specific neighborhood attributes can vary based on the individual’s characteristics.

A household i, with K observable characteristics $zi$, chooses a location of residence from J house types that are available within a housing market. Household preferences for a specific choice location, J, are defined over its observable attributes, $Xj$, which include both structural characteristics (e.g., age) and neighborhood characteristics (e.g., environmental quality). Households also care about attributes that are unobserved (to the researcher), which are captured in $ξj$. Lastly, the utility from living in a location depends on an unobserved component that is idiosyncratic to the individual and the choice, $εij$. The household’s problem is to choose a location J to maximize the following,

$Display mathematics$
(19)

where

$Display mathematics$
(20)

The introduction of a choice- and individual-specific idiosyncratic component of preference, $εij$, implies that each location in the choice set has a non-zero probability of being chosen as the utility-maximizing location by each individual and thus allows households to rank communities differently. In addition, (20) shows that household preferences for location attributes depend on the individual household’s characteristics, $z ik$. That individuals can have distinct preferences over each choice and choice attribute is a key characteristic of the horizontal sorting models that contrasts from the vertical preferences specified in the PCM framework. This allows for more flexible substitution patterns between choice alternatives than that allowed in the PCM framework. A limitation from assuming a linear utility in (19) is that it requires attributes of communities and the numeraire to be perfect substitutes. Furthermore, the specification rules out income effects, since the marginal utility of income, $βp$, is constant. Various papers in the RUM framework have relaxed this assumption (e.g., Bayer, Keohane, & Timmins, 2009; Takeuchi, Cropper, & Bento, 2008; Tra, 2010). Income effects complicate welfare calculations as there is no longer a closed-form solution for compensating variation (assuming a Type I Extreme Value idiosyncratic error) and must be estimated via simulation.

Given the preferences set up in (19) and (20), a household chooses to reside in a location J if the utility it receives from living there exceeds that from all other available locations. Letting di denote household i’s choice, this is represented by

$Display mathematics$

Suppose the distribution of idiosyncratic taste is given by $f(ε)$, then the probability that an individual chooses a location J is given by

$Display mathematics$
(21)

where I[.] denotes the indicator function. If we aggregate over the set of individuals who choose a location J, then the aggregate demand for each residential location, $Dj$, is given by,

$Display mathematics$
(22)

The housing market clears when the demand for locations equals that supplied, $Dj=Sj∀j$. In the random utility framework, Anas (1982) shows that there exists a unique price vector that clears the market as long as the location choice probabilities are strictly decreasing in housing price. Bayer and Timmins (2005) additionally prove existence of a sorting equilibrium in the presence of local spillovers such as congestion or agglomeration.16 While they find that an equilibrium exists, situations with large agglomeration effects can lead to multiple equilibria. The magnitude of an agglomeration effect that can be allowed to sustain a unique equilibrium is increasing in the number of choice alternatives, the variation in choice attributes, and the heterogeneity in individual preferences.

Estimation of the model proceeds by making a distributional assumption on the idiosyncratic shock to utility; a common assumption is that the ε‎’s are distributed i.i.d. Type I Extreme Value. This allows the choice probability in (21) to take on the following closed-form expression (McFadden, 1974):

$Display mathematics$
(23)

It follows that the log likelihood function of the parameters given data on actual choices, location, and individual characteristics is given by

$Display mathematics$
(24)

In the case where there are no endogeneity concerns, one can estimate all of the parameters in one step by maximum likelihood. However, as before, attributes of a residence are often correlated with unobserved neighborhood attributes. This is especially the case with price, i.e., $E[pjξj]≠0$. The strategy that has been employed follows that from Berry (1994) and Berry, Levinsohn, and Pakes (1995), where one first rewrites the indirect utility to separate out the choice-specific components from those that vary by individual characteristics,

$Display mathematics$
(25)

where

$Display mathematics$
(26)

Here, the choice-specific component is denoted by $δj$, which is also interpreted as the baseline utility of choosing to live in location J. Berry (1994) shows that there exists a unique vector, $δ$, that maps the predicted shares of choices, $s(δ,X,βkX)$, into the observed shares in the market, σ‎. As such, the system of equations that map the baseline utilities into the observed market shares can be inverted and δ‎ can be identified using observed market shares, data on attributes, and the remaining parameters not subsumed into δ‎.17 Estimation then proceeds by recovering the parameters for individual-specific components ($βkX$) using a gradient-based search, where, for each guess of the individual-specific parameters, the δ‎j’s are solved using the inversion. As it is not always possible to compute the inversion analytically, Berry et al. (1995) show that the δ‎j’s can be solved recursively using a contraction mapping routine for each guess of household-specific taste parameters, $βkX$,

$Display mathematics$
(27)

where m indexes the iteration, σ‎j represents the observed shares for choice J, and sj (.) represents the predicted choice shares based on a guess of the parameters, i.e., $sj(m)=sj(δj(m),Xj,βkX)$.

Next, the δ‎j estimates from the first stage are decomposed into choice attributes, where the endogeneity of price can be dealt with using instrumental variables (IV) in a linear setting. The IV approach taken follows a strategy from the differentiated products demand literature, where an instrument for the price in a particular location J is constructed based on the exogenous attributes of distant communities. The identifying assumption is that while the cost of living in a community J depends on the substitutes that are available nearby, the utility of living in community J does not. Therefore, one can use the attributes of distant communities as instruments for price. Others have dealt with price endogeneity by either inferring marginal utility of income from other types of tradeoffs that are unrelated to neighborhood characteristics or by introducing exogenous variation in price through the use of a randomized experiment. Bayer, McMillan, Murphy, and Timmins (2016) use the marginal value of wealth to proxy for the marginal utility of income in order to avoid dealing with the endogeneity of price in the second stage baseline utility decomposition. Specifically, they reformulate the sales transactions data of residential properties to infer the moving decisions of the buyers of those properties over time. Next, using the inferred individual move–stay decision, which involves a one-time financial moving cost (e.g., 5% of the value of the house that is being sold), they estimate the marginal value of wealth in the first stage from (24). Assuming that the marginal utility of wealth is a proxy for that of income, (26) can be rewritten as

$Display mathematics$

They then price adjust mean utilities with β‎FMC and use the price-adjusted mean utilities in the second-stage decomposition:

$Display mathematics$

Alternatively, Galiani, Murphy, and Pantano (2015) estimate a neighborhood choice model that brings in experimental data from the Moving-to-Opportunity (MTO) program. The MTO experiment randomly generates variation across individuals in the amount of rent that is paid. This helps disentangle the part of rent that is correlated with neighborhood-level unobserved factors and allows for clean identification of the marginal utility of consumption important for recovering a MWTP. Applying the Berry, Linton, and Pakes (2004) result to the residential sorting framework, the estimator is consistent and asymptotically normal in the presence of the various sources of error from the data (e.g., sampling error from a draw of individuals from the population and that from the set of choice alternatives18) as long as the number of households increases faster than the number of choice alternatives (Bayer et al., 2004).

## Innovations in Equilibrium Sorting Models

Equilibrium sorting models have been applied to value various environmental amenities, including air quality (Bayer, Keohane, & Timmins, 2009; Sieg et al., 2004; Tra, 2010), open-space (Klaiber & Phaneuf, 2010; Walsh, 2007), landscape attributes such as proximity to parks and rivers (Wu & Cho, 2003), and climate (Fan, Klaiber, & Fisher-Vanden, 2016; Timmins, 2007). Non-environmental attributes include those such as school quality (Bayer, Ferreira, & McMillan, 2007; Klaiber & Smith, 2012) and crime (Bayer et al., 2016). Many of these applications have innovated upon the basic PCM and RUM frameworks to recover preferences that characterize behavior in the presence of market frictions.

### Non-Marginal and Endogenous Amenity Changes

Changes from environmental policies are often non-marginal and, as previously discussed, are likely to lead to new equilibria. The power of these types of models, given their assumed structure, lies in their ability to simulate a new equilibrium in the face of a counterfactual policy shock. This enables these models to examine welfare changes and/or redistribution as a result of counterfactual policy scenarios. Sieg et al. (2004) apply the PCM framework from Epple and Sieg (1999) to estimate preferences for air quality. Using a calibrated housing supply to close the model, they value large air quality improvements in southern California accounting for general equilibrium impacts on housing prices due to changes in housing demand initiated by air quality changes. Tra (2010) similarly values large air quality improvements, but within an RUM framework. Both papers find large differences between partial and general equilibrium welfare calculations for a reduction in air pollution. Over their entire study area, Sieg et al. (2004) find that average WTP under general equilibrium is 13% higher than that under partial equilibrium. Upon examining individual jurisdictions, the general equilibrium WTP is found to be anywhere between 55% smaller to 350% larger than the partial equilibrium counterpart! Tra (2010) finds a similar bias of 17% overall and large heterogeneity in the bias across various jurisdictions. These biases in partial equilibrium welfare measures result from ignoring the subsequent price increases in low-air-quality areas in the face of large air quality improvements.

Second, neighborhood characteristics determined in the sorting process constitute an important set of attributes that households care about. Inclusion of these endogenous choice attributes creates difficulties in estimation but, if ignored, may create biases in valuation estimates. Walsh (2007) extends the Epple and Sieg (1999) model to include an endogenously determined attribute of housing, share of open space within a community, as well as an estimated supply of housing that is used to predict the impacts of policies to protect land and prohibit development. The study instruments for open space using soil characteristics that predict suitability for residential development and non-developed land value, assuming that these measures only affect the supply of land. A key feature of the model is that changes in the amount of protected land (given exogenously by a policy) can impact household demand for private development (i.e., protected land and private lot size are substitutes). As a result, restrictions in land development can potentially cause a reduction in the overall amount of open space (protected and private) due to individuals responding by increasing private lot size. Furthermore, feedback effects19 due to the endogenous nature of open space are important and cause partial equilibrium welfare impacts to differ significantly from general equilibrium calculations. Allowing for endogenous open space adjustment causes the average compensating variation (CV) to be 42% larger under one policy experiment and the CV to flip signs from −$4.32 per household to$5.69 for another.

Two studies using the RUM framework that deal with endogenous attributes are worth discussing even though one does not have an environment focus and the other does not use housing markets. Bayer et al. (2007) similarly allow for endogenous school and neighborhood characteristics to estimate preferences for school quality. They allow the sociodemographic characteristics of the neighborhood (e.g., racial breakdown, educational attainment, average income) to enter into households’ preference functions to allow for peer effects. As households sort across school boundaries based on heterogeneous tastes for school quality, this induces changes in those endogenous neighborhood attributes for which households have a preference. To separately identify preferences for a neighborhood’s sociodemographic character from those for (unobserved) attributes correlated with sociodemographics due to sorting, they embed a boundary discontinuity design using school districts, pioneered by Black (1999), within their residential sorting model. They find that ignoring the endogenous neighborhood attributes overstates willingness to pay for school quality, reducing the WTP for a standard deviation increase in test scores from $33 per month to$17 per month. Additionally, their estimates show that inclusion of boundary fixed effects reduces the negative WTP for share of black neighbors from a statistically significant value of −$100 to$1.5 (not statistically significant), implying that the commonly found negative relationship between minority shares and housing prices is primarily driven by the former’s correlation with unobserved neighborhood attributes.

While Bayer et al. (2007) use quasi-experimental variation in school boundaries to identify WTP, IV strategies based on Bayer and Timmins (2007) have also been employed within the RUM framework. In an application to estimate recreational demand for fishing locations, Timmins and Murdock (2007) allow congestion, captured by the share of all anglers choosing a particular site, to impact demand for fishing alternatives. The instrument uses the predicted share of anglers based on the exogenous attributes of the choice set only (i.e., ignoring congestion and other attributes that may be endogenous). Monte Carlo simulations show that as long as the exogenous attributes used to construct this instrument are important determinants of choice, the instrument will have good predictive power (Bayer & Timmins, 2007).

### Migration Costs and Dynamics

A third area of concern in traditional hedonic analyses where equilibrium sorting models have been able to contribute is the inclusion of migration costs in valuation. In the residential sorting context, moving can entail both financial costs (e.g., realtor fees and closing costs) and psychological costs from being uprooted from one’s community. If individuals consider these moving costs in the decision of whether (and where) to move, then the observed price differentials across locations must not only compensate individuals for the difference in amenities but also the cost to move.

Bayer et al. (2009) theoretically relate the bias from ignoring migration costs to the framework in Roback (1982). They then empirically demonstrate the magnitude of the bias in air quality valuation using a residential sorting model that incorporates migration costs, which are incurred when an individual chooses to live outside of his or her state of birth.20 They find that the MWTP with respect to air quality that accounts for migration cost is almost triple that which ignores it. Geyer and Sieg (2013) also incorporate moving costs in their model of demand for public housing units among low-income households. Using data from the Pittsburgh housing authority, from which they observe entry and exit of households between public housing and the private rental market (as well as between various communities within public housing), they identify moving costs upon observing housing choices over time.

Related to migration costs, is the inclusion of dynamics. Moving costs can be large; for example, closing costs can range from 2 to 5% of the purchase price of the home. Thus, households are likely to be forward-looking with respect to neighborhood attributes in their residential choice so as to minimize costly re-optimization when attributes become less desirable compared to other housing options or when household characteristics are projected to change due to changes in age and family structure. Furthermore, housing is often considered an asset for which current choices may be justified by future expected price appreciation. Many papers acknowledge that ignoring forward-looking behavior can bias valuation estimates. Without modeling forward-looking behavior explicitly, Bishop and Murphy (2018) characterize the magnitude and direction of the bias from ignoring such dynamics using the time-series property of an amenity. They find that static models will understate (overstate) valuation of a mean reverting (diverging) amenity.

In an application to education, Epple, Romano, and Sieg (2012) build dynamics into the PCM framework within a two-period overlapping generations model. They allow household location decisions to reflect anticipated changes in demand for educational quality as a household ages from its child-rearing years, during which the household would vote for high educational spending, to old age, when it would prefer lower taxes and spending. In their model, households decide where to live over two periods of the life cycle: young and old. The location choice made in the first period affects the optimal consumption and housing choice in the second period. Furthermore, households take into consideration the costs of moving when deciding where to live. They calibrate their model and find that sorting by age in addition to income potentially reduces unequal distribution in education across jurisdictions. This occurs as older households, who are wealthier, move to low-educational-quality localities and subsequently increase the tax base in these areas, which then helps fund education expenditures, reducing gaps in educational quality.

Aside from Epple et al. (2012), few applications in the context of residential choice have introduced dynamics due to computational issues arising with the size of the state-space that needs to be considered. Recently, advances in the dynamic discrete choice literature (Arcidiacono & Miller, 2011; Hotz & Miller, 1993) have helped make dynamic problems more tractable.21 Bayer, McMillan, Murphy and Timmins (2016) use these methods to estimate a dynamic model of residential housing choice within an RUM framework, where they recover valuation of crime, air pollution (ozone), and neighborhood racial composition. Their model allows households to consider housing as a financial asset in addition to the amenity values that are attached, and lets wealth evolve endogenously. Their results find that the static model understates WTP for ozone and crime by 41% and 16%, respectively, and overstates WTP for neighborhood own racial composition by approximately 70%. The directions of these biases are consistent with the time-series properties of these attributes as noted in Bishop and Murphy (2018). These findings are important for recovering consistent willingness-to-pay estimates, as the magnitude of these biases can certainly alter the conclusions of cost-benefit test.

### Ongoing Work and Limitations

Additional progress is being made in sorting models to further account for factors that affect individual household optimization decisions and more accurately measure demand for environmental (and other types of non-marketed) amenities. This includes incorporating the labor market decision (Bayer et al., 2009; Bishop, 2012; Gemici, 2011; Kennan & Walker, 2011; Kuminoff, 2012), which can potentially impact valuation as job choice is intimately tied to choice of residential location. Labor market considerations would be especially important when the aim is to value amenities that only vary across large geographic space (e.g., climate), for which the choice set would need to be also defined over a large spatial area. There is also work to relax the perfect information assumption about the environmental amenities attached to choice locations (Ma, 2017) as well as research to better characterize the housing supply side, as that affects the way in which households sort across the landscape (Epple et al., 2010; Murphy, 2015). Continued innovation in these models will aid in policy evaluation of exogenous environmental changes that have not, but could potentially, occur.

While equilibrium sorting models could help generate policy counterfactuals without having to carry out costly (or, at times, infeasible) experiments, the credibility of these counterfactuals is potentially of concern given that identification of preference parameters is achieved via various modeling assumptions. In order for these types of models to be useful for cost-benefit analysis of federal regulations, additional clarity is needed to understand how the estimates from these models relate back to those estimated by reduced form and the extent to which the estimates depend on modeling assumptions. Bayer et al. (2007) relate RUM and hedonic estimates in their discussion, showing that the MWTP estimate from a hedonic regression will equal that from an RUM framework if preferences are homogeneous (up to the idiosyncratic shock, ε‎ij). The hedonic estimate reflects the preferences of the marginal buyer, which could deviate a lot from that of the average buyer depending on whether there is preference heterogeneity and whether the attribute is widely supplied across the landscape.

Both Banzhaf and Smith (2007) and Kuminoff (2009) investigate the implications of structural modeling assumptions. In particular, Kuminoff (2009) identifies a partition for the possible values that preferences can take based only on the observed choice that a household makes, an assumed CES preference function, a particular choice set from which a household chooses, and a form of preference heterogeneity. Imposing an additional assumption of the specific distribution of unobserved heterogeneity then point-identifies the preference parameters within each partition. Changing the assumed preference distribution and/or the choice set shifts the boundaries of the partition, which affects the estimated MWTP. Without imposing such a distribution, one can instead report a range of MWTP based on the implied partition that would still be useful for assessing whether a policy would pass a cost-benefit test and be less sensitive to a model’s assumptions. Alternatively, validating the estimates with a (non-random) sample that is not used in estimation potentially boosts credibility. With experimental data from the MTO program that randomized low-income households into two different housing assistance treatments (and a control group), Galiani et al. (2015) estimate household preferences using residents in the control group and only one of the treatment groups; with the residents from the other treatment group not used in estimation, they are able to perform out-of-sample validation by comparing their actual behavior to that predicted based on the model and its estimates. The ability to match the choices of individuals not used in estimation helps establish the extent to which the model’s estimates are externally valid and not reliant on the model’s assumed structure. Out-of-sample validation and work to clarify sources of uncertainty could bolster the use of sorting models for policy evaluation.

# Conclusion

Valuation of environmental amenities is a crucial input into policymaking and has progressed substantially since the early descriptions of how choice of residential location could reveal preferences for public goods (Ellickson, 1971; Tiebout, 1956) and how housing prices could be used to recover these preferences (Rosen, 1974). This article reviewed two models that have been employed in the economics literature to indirectly measure household demand for environmental quality using actual household decisions within the housing market: property value hedonic models and sorting models. Both methods estimate valuation from the same underlying choice process for the residential location decision but differ in how they recover these estimates. Hedonic models only use equilibrium outcomes in estimation, whereas sorting models combine these outcomes with a formal description of the choice process. These differences present trade-offs. Hedonic models are general without having to place specific functional form assumptions on preferences and are computationally light compared to sorting models. This potentially accounts for their widespread use: as of the writing of this article, Rosen (1974) has 10,368 citations on Google Scholar. Notably, however, these models are limited in their ability to yield policy predictions in the face of endogenous attributes and dynamics. Sorting models, on the other hand, are powerful in that they can incorporate these considerations to provide counterfactual policy analysis. However, they rely on additional modeling assumptions and can quickly become computationally intractable. It is important for researchers and practitioners alike to understand the strengths and limitations of each model when using either valuation method to estimate the benefits of a cleaner environment. A better understanding of these differences and their potential implications for benefit estimates allows regulators to have more confidence in applying these models’ estimates in welfare calculations.

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## Notes:

(1.) This framework was later expanded in Rosen (1979) and Roback (1982) to additionally include a labor market that yielded a wage gradient, incorporating the idea that individuals may also require increased wages to compensate for living near an undesirable neighborhood characteristic.

(2.) These data have recently been made more widely available to researchers either directly through county assessor offices or third-party data-collection companies. It is worthwhile to note that in addition to modeling advancements, recent improvements in access to data that better describe houses, their characteristics, and even the individuals that purchase those houses have also considerably aided consistent estimation in revealed preference methods.

(3.) As opposed to the nonlinear specification in (6), most of the literature makes a simplification by assuming a linear hedonic price function and a constant MWTP in order to avoid second stage concerns. Examples in the environmental context include Bajari, Cooley, Kim, and Timmins (2012), Bui and Mayer (2003), Chay and Greenstone (2005), Currie, Davis, Greenstone, and Walker (2015), Gamper-Rabindran, Mastromonaco, and Timmins (2011), Gamper-Rabindran and Timmins (2013), Gayer, Hamilton, and Viscusi (2000), Gayer and Viscusi (2002), Greenstone and Gallagher (2008), Hallstrom and Smith (2005), Haninger et al. (2017), Leggett and Bockstael (2000), Michaels and Smith (1990), Muehlenbachs et al. (2016), and Palmquist, Roka, and Vukina (1997). This list is by no means comprehensive.

(4.) For a full textbook treatment of valuation, see Freeman, Herriges, and Kling (2014) and Phaneuf and Requate (2016).

(5.) Several others have also suggested nonparametric or semiparametric estimation of hedonic price functions in order to avoid imposing the functional form, e.g., ., Anglin and Gencay (1996), Bajari and Benkard (2005), Pace (1993), Parmeter et al. (2007), and Stock (1991).

(6.) One can see this by setting $qd=qs$ in equilibrium, which leads to unobserved demand shifters, $ud$, being necessarily correlated with supplier characteristics, $Xs$.

(7.) Chattopadhyay (1999) provides an example of second-stage valuation in a single market, whereas Palmquist and Israngkura (1999) and Zabel and Kiel (2000) use the multi-market strategy.

(8.) For example, assuming the MWTP is linear in both q and $Xd,(Xd)2$ will be correlated with q via sorting but does not belong in MWTP.

(9.) In their paper they refer to these, respectively, as a price effect and a substitution effect.

(10.) An example using an extreme case can be seen with a locally undesirable land use (LULU) area that will be converted into a park within the next month. A household who is evaluating a house near this LULU area will evaluate it based on its future characteristics as a park rather than the current LULU. If one had ignored the forward-looking nature of households, then one would infer that all households purchasing houses near the LULU (before it was converted) do not care as much to avoid the LULU than they actually do (since they know that this LULU will be gone within the next month).

(11.) In particular, they use a combination of conditional choice probabilities (Hotz & Miller, 1993) and finite dependence (Arcidiacono & Miller, 2011) to approximate the future stream of values associated with living in a particular location.

(12.) The vertical PCM referenced in this article refers to a vertically differentiated model in which all individuals rank choice alternatives in the same manner. A “pure characteristic model” in general, as defined by Berry and Pakes (2007), can refer to a horizontally differentiated model that allows for individual-specific preferences for choice attributes but does not allow for an individual- and choice-specific taste parameter (e.g., a logit error).

(13.) In the application, Epple and Sieg (1999) define a community as a school district.

(14.) The slope is given by

$Display mathematics$

(15.) This follows vertically differentiated models of demand from the industrial organization literature (Bresnahan, 1987).

(16.) Local spillovers can be captured by modifying the household utility to be

$Display mathematics$

where σ‎j denotes the share of individuals choosing location J, and its inclusion allows for congestion (α‎ < 0) or agglomeration effects (α‎ > 0).

(17.) In the case where household heterogeneity appears only through the idiosyncratic shocks, i.e., utility is given by

$Display mathematics$

one can analytically invert the shares equation as follows

$Display mathematics$

to solve for the vector of baseline utilities, given a normalization that sets the mean utility of one choice to 0, e.g., J = 0,

$Display mathematics$

(18.) Where preference parameters are random, i.e., $βik~N( β¯,Σ)$, one needs also to be concerned with simulation error.

(19.) Initial changes (e.g., in response to an exogenous policy) lead to changes in the demand for lot size and community choice, which then not only causes prices to adjust but also determines a new level of the open space attribute (aggregated from new, individual decisions).

(20.) The paper uses micro data samples from the U.S. Census at most able to observe location choices—current state of residence and birth state. In general, micro data on migration decisions in the United States are difficult to come by.

(21.) For an overview of these methods, see Arcidiacono and Ellickson (2011).