Recovering Preferences from a Dual-Market Locational Equilibrium

Transcription

1 Recovering Preferences from a Dual-Market Locational Equilibrium Nicolai V. Kuminoff Ph.D. Candidate Department of Economics North Carolina State University nvkumino@ncsu.edu Abstract: This paper develops a new structural estimator that uses the properties of a market equilibrium, together with information on households and their observed location choices, to recover horizontally differentiated preferences for a vector of local public goods that differentiate communities. The estimation process is consistent with equilibrium capitalization of local public goods and recognizes that ob and house location choices are interrelated. By using the logic of set identification to distinguish the identifying power of structural restrictions on the indirect utility function from the identifying power of maintained assumptions on the distribution of preferences, the estimator provides a new perspective on characteristics-based models of the demand for a differentiated product. The estimator is used to recover distributions of the marginal willingness-to-pay for improved air quality in Northern California s two largest population centers: the San Francisco and Sacramento metropolitan areas. Preliminary estimates for the marginal willingness-to-pay for ozone reduction in change substantially when ob opportunities are included as a dimension of location choice. November, 2005 Draft: Please do not cite I thank Paul Fackler, Ray Palmquist, Dan Phaneuf, and especially V. Kerry Smith for their help and support. I also thank Jaren Pope and participants at Camp Resources XIII for comments and suggestions. Support from the National Science Foundation is gratefully acknowledged.

2 There is no way in which the consumer can avoid revealing his preferences in a spatial economy. Spatial mobility provides the local public-goods counterpart to the private market s shopping trip. --Charles Tiebout (956) I. Introduction Markets thrive because people are different. Access to differentiated products allows heterogeneous consumers to satisfy their idiosyncratic tastes. Over the past decade, microeconometric models have sought to use the logic of revealed preferences to exploit the ways in which this diversity influences market outcomes. These models start by specifying the relationship between consumers and their choices in the marketplace. Then, using the observable attributes of consumers and their choices, they invert this relationship to infer tastes for product characteristics (e.g. Berry, Levinsohn and Pakes [995], Baari and Benkard [2005]). A similar logic forms the basis for sorting models of households location choices. These models demonstrate how the properties of equilibrium in the housing market can help to identify and estimate households preferences for the local public goods that differentiate locations. Epple and Sieg (999) offered the first illustration of this logic. In their analysis, people have different tastes for an index of local public goods but they all evaluate its constituent elements in the same way. This feature, labeled vertical differentiation, implies all households agree on a single ranking of communities by an index of the public goods they provide. Relaxing vertical differentiation is important because it is reasonable to expect households to evaluate components of a vector of local public goods differently. For example, households with school age children may be more concerned about school quality while retirees may place more emphasis on climate and other environmental amenities. When households are allowed to differ in their relative preferences for these and other local public goods (i.e. horizontal differentiation) the properties of a housing market equilibrium provide less information to identify households preferences from their observed location choices. While several strategies have been proposed for this situation, none have used the properties of a market equilibrium to recover preferences so that it is consistent with equilibrium capitalization of local public goods (Starrett [98] and Scotchmer [985]). This paper describes a new structural estimator that meets this obective. It is based on the information provided by location choices in a market equilibrium derived from households that have horizontally differentiated preferences for public goods and differ in their ob skills. 2

3 The estimator recognizes that observed location choices provide set identification of the heterogeneous preference parameters (Baari and Benkard, [2005]). That is, it recovers a set of values for the parameters that describe how local public goods contribute to sorting behavior. To attach values from this set to the population of households requires additional assumptions about the distribution of each preference parameter. A key feature of the new estimator is that it uses the set identification logic to distinguish the identifying power of structural restrictions on the indirect utility function from the identifying power of maintained assumptions about the distribution of preferences. The estimator is applied to Northern California s two largest population centers: the San Francisco and Sacramento Consolidated Metropolitan Statistical Areas. This region is divided into 23 housing communities and 8 work destinations, and each (community, worksite) pair is assigned a price of housing, a set of public goods, a set of wage rates, and a commute time. The model explains the location choices made by households in each of 22 occupational categories, where wage options differ for each category. The application demonstrates that it is feasible to identify and estimate a horizontally differentiated sorting model in the housing and labor markets. Moreover, the preliminary estimates for the marginal willingness-to-pay for ozone reduction in change substantially when ob opportunities are included as a dimension of location choice. The next section describes Tiebout sorting and discusses how structural restrictions allow preferences for public goods to be inferred from observed house locations. Then the labor market is added as a second dimension of the choice set and a single-crossing restriction is used to characterize sorting across the urban landscape. Section III briefly summarizes alternative strategies to infer preferences from observed location choices. Then the empirical model and estimator are discussed in section IV. Section V describes the application to northern California and discusses results, and section VI concludes. Two appendices provide greater detail on the computational model and the data. II. Theoretical Model Tiebout s locational sorting model assumes, ceteris paribus, heterogeneous households select a community based on its local public goods. Suppose the urban landscape can be divided into a finite set of J housing communities, each of which differs in its price of housing ( p ) and in its 3

4 exogenous provision of local public goods such as school quality, crime, and environmental amenities. Households differ in the relative importance they assign to each public good. Let γ represent relative preferences for public goods, and ( γ ) g represent composite provision of public goods in community as perceived by a γ type household. Each household chooses the community that maximizes its utility, given its exogenous income ( y ) and its preferences (α ) for the composite public good relative to private goods. For heuristic purposes, utility maximization can be depicted as a two-stage problem, where each household first determines the optimal quantities of housing and numeraire in every community and then chooses the community that maximizes its utility. The first stage is shown as equation (). () maxu ( h, b) [ g( γ ), h, b,α ] subect to ph = y b. Conditional on a community, households choose quantities of housing (h) and a composite private good (b) to maximize their utility subect to the budget constraint. Assume that zoning does not constrain housing construction. Then households can purchase any quantity of housing at the market price in each community, in which case preferences can be restated using the indirect utility function in (2). [ α ] (2) V [ g( ), p, α, y] U g( γ ), h( g( γ ), p, α, y), y ph( g( γ ), p, α, y) γ =,. Assuming households are price-takers and can move freely between communities, each household will choose the community that maximizes its well-being, given income and prices. (A) Identifying Heterogeneous Preferences from Structural Restrictions Two types of structural restrictions are required to point-identify households preferences based on their observed location choices. First, a parametric indirect utility function must be selected. Second, a distribution must be specified for each preference parameter in that function used to characterize household heterogeneity. Each restriction makes a different type of contribution to the identification. Distributional assumptions are necessary due to the discreteness in the choice set. When household i chooses from a finite set of communities, utility maximization is characterized by 4

5 the set of inequalities in equation (3). (3) V g ( ) [ p, α, y ] V [ g ( γ ), p, α, y ], k J γ. i, i, i i i, k i k i i =,..., Given a parametric form for the indirect utility function, the inequalities provide set identification of the heterogeneous preference parameters. It must be the case that ( i, γ i ) A i, { } where A ( α, γ ): ( α, γ ) ( 3) α,, = i i i i satisfies. In words, the choice of community reveals only i that household i s preferences lie somewhere in the allows the analyst to identify the density of preferences within A,. A, set. Imposing a distribution on ( α, γ ) To illustrate the role of each type of restriction in identifying preferences, consider a specific example using the following CES indirect utility function : i i V.0 { [ ] }. 0 i i, i [ g( ), p, α, y] α ( g ) +.79 exp( 4 y p ) γ, = where g i, γ i, air AIR + γ i, school = SCHOOL. The first term represents utility from public goods, and the second term represents utility from the private good component of housing. Households differ in their income and in their preferences for a linear index of two public goods that differentiate communities, air quality and school quality. There are two components of preference heterogeneity. Households differ in the relative weights they place on each public good in the index ( γ i, air,γ i, school ) and in the overall strength of their preferences for public goods relative to private goods ( α i ). Suppose households maximize their utility by sorting among the following four communities: Community Public Goods* Air quality School quality * Higher values indicate higher quality. Price This CES function provides the basis for the subsequent structural model. Specifically, it is the indirect utility function from equation (2) with β = 2, η =. 963, ν =. 75, and ρ =. 0. If the weights in the public goods index are constant, it reduces to the indirect utility function in Epple-Sieg (999). 5

6 To see how the form of the indirect utility function provides set identification of preferences, first consider the simplest form of preference heterogeneity vertical differentiation. In a vertically differentiated model such as Epple and Sieg (999), all the variation in tastes can be condensed into a single heterogeneous preference parameter that ranks locations by quality. The CES utility function simplifies to this case when households are constrained to have the same relative preferences for the two public goods. For example, let the weights be: ( γ ) ( 0.48, 0.52), i i, air, γ i, school. With constant weights, all households agree on a common ranking of communities by the public goods index, and sort according to their income andα i. By conditioning on income, the system in (3) can be solved for the bounds of the α i sets that rationalize each location choice. At y=$50,000, the partition of α corresponds to: α 0 A i, A i,2 A i,3 A i, The figure illustrates two critical limitations of set identification. First, preferences are not point identified within the bounds of a set. The choice of community 2 reveals only that the household s preferences lie somewhere in A i, 2 :.09 α. 28. Second, the preference set that corresponds to the highest (lowest) provision of public goods is not bounded from above (below) by the revealed preference logic in (3). These two limitations require that a distribution be specified for α i. This added information transforms the observed location choices by a population of households into a distribution of preferences. When vertical differentiation is relaxed, observed location choices are required to setidentify more heterogeneous preference parameters. Returning to the CES example, horizontally differentiated preferences imply households differ in their relative preferences for the two public goods. This generalization complicates the partitioning process. For example, at y=$50,000 the inequalities in (3) can be solved for a large number of ( α, γ air, γ ) points that would make a household exactly indifferent between each pair of communities. Figure A shows the indifference loci formed from these points in R 2 +. A household with preferences anywhere i school 6

7 along the locus from (0., 0) to (4, 0.49) would be exactly indifferent between communities 2 and 3. Households above (below) this line have stronger (weaker) relative preferences for air quality, and therefore prefer community 2 (3). The indifference loci delineate 2 areas, each of which corresponds to a different ranking of communities by utility. The rankings (from highest to lowest) are shown in the figure. Notice that adacent areas often have the same utility maximizing community. Community maximizes utility throughout the region defined by [(0,0), (0,), (0.87,), (,0.47), (0.87,0)], for example. By taking the union of areas that share the same utility-maximizing community, figure B partitions preference space into regions that rationalize each of the four community choices. This partitioning process illustrates how the identifying power of the indirect utility function differs under vertical and horizontal differentiation. In the vertical case the choice of community 2 indicates that the household s preferences belong to the set: ( =. 48,.09 α. 28 ), which appears in figure B as the dashed line in the lower left γ air corner of the A i, 2 region. A i, 2 is the preference set identified by the choice of community 2 in the horizontal case. This comparison illustrates a general principle: preference sets revealed by vertically differentiated sorting are subsets of their horizontally differentiated counterparts 2. In an empirical analysis, distributional assumptions will influence estimated welfare measures for policy changes. The marginal-willingness-to-pay (MWTP) for public goods is a function of ( α, γ air, γ ) which means every point in figure B corresponds to a specific school MWTP. Suppose we want to infer the distribution of MWTP for air quality for households living in community 3. The choice of community 3 reveals only that households living there have preferences somewhere in A i, 3. Thus, to calculate their distribution of MWTP we must first specify a function for the preference parameters over the A i, 3 region. Two extreme cases provide bounds for the MWTP. The first case is where every household has preferences at the point (*), which corresponds to the lowest MWTP of any point in A i, 3. The opposite extreme is where every household has preferences at (**), which corresponds to the highest MWTP. Thus, [MWTP(*), MWTP(**)] spans the range of possible measures for individual MWTP. The wider 2 Assuming preferences are truly horizontally differentiated, the vertical/horizontal modeling choice is not unlike a bias/variance tradeoff. By restricting relative preferences, vertical differentiation biases welfare measures. Horizontal differentiation eliminates the restriction that causes bias, but the added dimensionality of preferences increases the scope for distributional assumptions to influence results. 7

8 this range the greater the sensitivity of welfare effects to the distributional assumptions made in order to move from set to point identification. The figure also illustrates there are limits to what can be learned from revealed preference analysis, ust as there are limits to the extent of capitalization of public goods. Consider community 4. Because it provides the most public goods, it will attract households with the strongest preferences and the highest MWTP. These households may make the largest contribution to summary measures of the average MWTP. In this case, revealed preference analysis is limited because there is no upper bound on α i in region A i, 4 of the partition. To recover a MWTP distribution for community 4, either an absolute upper bound must be imposed on α i or a distribution that limits weight in the tail. This identification problem parallels Starrett s (98) demonstration that housing prices may not capitalize increased provision of local public goods in communities that already have extreme levels of public goods. (B) Preference Heterogeneity and Substitution Patterns Structural restrictions on the utility function control the scope of substitution patterns. With vertical differentiation each community has at most two substitutes, the adacent communities in the ranking by public goods 3. With horizontal differentiation the total number of substitutes for each community falls between 2 and J, depending on the number of choices relative to the number of public goods (Anderson, DePalma and Thisse [992]). The communities that are substitutes will share borders in the partition of preference space. Community 2, for example, shares borders with each of the other three communities in figure 2. Consider a marginal increase in the price of housing in community 2. Households that currently reside in 2 but have preferences on the border between 2&4 will respond to the price increase by moving to community 4. Likewise, households on the borders between 2& and 2&3 will move to communities and 3. The substitution patterns that arise from horizontal differentiation have intuitive properties. Locations that are similar in terms of prices and public goods are more likely to be substitutes than those that are not. Notice that in figure 2 the two communities with intermediate 3 The definition of substitution used here is defined as strong gross substitution in Anderson, DePalma and Thisse (992), where k is a substitute for iff h P > 0. k 8

9 levels of public goods, 2 and 3, share borders with each of the other three locations while the most and least expensive locations, and 4, do not share a border. Because locations and 4 are furthest removed in terms of prices and public goods, it seems natural to expect that there are few, if any, households that consider them as substitutes 4. (C) Introducing the Labor Market For working households there are two dimensions of location choice the choice of a house and the choice of a ob. Intuition and recent empirical research suggest these two choices are interrelated (Rhode and Strumpf [2003]). This section expands the theoretical model to allow households with heterogeneous ob skills to simultaneously sort among communities and labor markets. Under these conditions, the levels of public goods will affect behavior in both markets (Rosen [979], Roback [982]). Thus, one might expect ob locations to convey additional identifying information about preferences. A single crossing restriction on preferences leads to three properties that must characterize sorting behavior for every household type in any locational equilibrium. These properties guarantee that housing and labor market choices convey sufficient information to recover preferences. The primary difference between these properties and the ones derived in Epple and Sieg (999) arises because a multiplicity of types implies sorting behavior that is less restrictive and less informative. Let the urban landscape be divided into K labor markets that differ in the wage paid to workers of each ob skill. With J housing communities and K labor markets, each (,k) pair represents a unique ob-house combination, which will be referred to as a location and denoted by L, k. Each location requires a specific commute. For a household that commutes between and k, let ( θ ) w, represent wage earnings less the value of time spent commuting. In a slight k abuse of notation θ indexes both ob skill and the shadow value of time. Then, a household s income equals ˆ + ( θ ) y w, k, its exogenous non-wage income ( ŷ ) plus its effective wage income. Utility maximization is similar to ()-(2), except that households now optimize over two dimensions of location choice and a budget constraint that varies across locations. Equation (4) 4 If it is important for the model to characterize substitution patterns for every pair of communities, idiosyncratic location-specific preference parameters can be added as in Bayer, McMillan and Reuben (2005). 9

10 shows the utility maximization problem for household i. [ ] * (4) = max V g ( γ ), p, α y, where y y w ( θ ) L, k i i, i,, k, k, = ˆ +. Holding the community fixed at, a utility-maximizing household will always choose to work in the labor market that provides it with the highest effective wage income, given its ob skills. Let ( θ ) ŵ represent the maximum effective wage income that can be obtained by a household living in community. Then (4) can be rewritten as (5), with k optimized out of the expression. [ ] i, k * (5) = max V g ( γ ), p, α, y, where y yˆ wˆ ( θ ) L i i i, i, i + i, k =. For each ( γ, θ ) type household, the relevant choice set can be further reduced to a subset of the J communities. This is because, conditional on values for γ and θ, some communities may be dominated. A community is dominated if there is another with more public goods and either a sufficiently lower price, a sufficiently higher effective wage, or both. For example, given ( ) ( γ ) γ g2 g >, community dominates community 2 if prices and effective wages are defined such that: P2 w ˆ θ > wˆ 2 θ. No utility-maximizing ( γ, θ ) type would ever locate in community 2. Let R denote the total number of communities that are not P < and ( ) ( ) dominated. Then equation (6) shows how the relevant choice set for each ( γ, θ ) type relates to the set of all communities in (5), and to the set of all locations in (4). (6) { L L γ, θ} { L,..., L γ, θ} { L,..., L γ, θ},...,,,. R J J K Imposing a single crossing restriction on preferences makes it possible to characterize how, in equilibrium, households of each ( γ, θ ) type must be sorted across the R communities that are not dominated for that type. Equation (7) shows the slope of an indirect indifference curve in ( g, p) space. i i (7) M [ g( γ ) p, α, yˆ, w( θ )] [ g( γ ), p, α, yˆ, w( θ )] g [ g( γ ), p, α, yˆ, w( θ )] p dp V, = V = V =. dg V Assuming M is monotonically increasing in ( ŷ α, γ, θ ) and ( α ˆ, γ, θ ) y, indifference curves in 0

11 the ( g, p) plane satisfy single crossing in ŷ and α conditional on relative preferences, ob skills, and the shadow value of time. This restriction has an intuitive interpretation. Roy s Identity must equal the marginal utility of income, λ = V () y, times the implies that V () p Marshallian demand for housing, h [ g( γ ) p, α, yˆ, w( θ )],. (8) M () = V V () () g p = V λ () g = h() h() () V g (). V y The term in brackets in equation (8) is the Marshallian virtual price of public goods. Therefore, the single crossing restriction implies that the Marshallian virtual price, per unit of housing, is strictly increasing in income and in preferences for public goods relative to private goods 5. The single crossing property implies that, in equilibrium, three properties characterize sorting by each household type: boundary indifference, stratification, and non-decreasing bundles 6. Without loss of generality, let the R locations be ordered according to their perceived provision of public goods, g ( γ ) <... < g ( γ ) the border between two locations in (, ŷ) R. Boundary indifference requires a household on α space to be exactly indifferent between those locations. Equation (9) defines the set of border individuals. It must hold for all r =,..., R. { } (9) ( ˆ γ, θ ): V [ g ( γ ), p, α, yˆ, wˆ ( θ )] V [ g ( γ ), p, α, yˆ, w ( θ )] α, r r r = r+ r+ ˆ r+ y. The non-decreasing bundles property requires that for any two locations in the ordering, ( r, r + ) equation (0) must hold. + r pr+ > pr or > or both. wˆ (0) g ( γ ) > g ( γ ) r r+ ( θ ) wˆ ( θ ) r 5 This property is related to the Willig condition that is often applied together with weak complementarity to identify the Hicksian willingness to pay for changes in public goods. The Willig condition requires the willingness-to-pay per unit of the weak complement to be constant at all levels of income. See Smith and Banzhaf (2004) or Palmquist (2005) for details. 6 Boundary indifference and stratification follow from the proof of proposition in Epple and Sieg (999) because income is separable in non-wage income and effective wage income. To see why non-decreasing bundles must hold, suppose equation (0) fails for some (r,r+) pair. Then r must have fewer perceived public goods, more expensive housing, and lower effective wage income. If so, r+ dominates r, which implies r R, a contradiction. In the special case where wage income is exogenous to location choice and households are vertically differentiated, the three sorting properties reduce to the ones derived in Epple and Sieg (999).

12 The equation implies that households must pay for the additional public goods provided by higher ranked locations through housing prices, effective wage income, or both. The third property, stratification, requires that households of each type are stratified across the R ordered locations by ( α ŷ) and by ( α ) ŷ, as defined in (). () ( yˆ α, γ, θ ) < ( yˆ α, γ, θ ) < ( yˆ α, γ, θ ) r ( α yˆ, γ, θ ) < ( α yˆ, γ, θ ) < ( α yˆ, γ, θ ) r r r r+ and. r+ Boundary indifference, stratification, and non-decreasing bundles describe the partition of preference space that rationalizes equilibrium location choices. Non-decreasing bundles identifies locations that have adacent regions in the partition. Boundary indifference defines the borders that divide those regions, and stratification guarantees that each region is connected in ( α ˆ, γ, θ ) y. While the three conditions are necessary for a locational equilibrium to exist, they are not sufficient. Any locational equilibrium must also be characterized by a set of housing prices and wage rates such that no household could increase its utility by changing locations, and all locations are occupied. The estimator in this paper assumes an equilibrium exists and focuses on recovering values for the preference parameters that ustify observed (equilibrium) location choices 7. III. Background Tiebout s early analysis recognized that a household s location reveals something about its preferences for the differentiated bundle of public goods that communities provide. Negligible moving costs and exogenous income are important in his reasoning. Subsequently, Rosen (979) suggested that local public goods influence spatial variation in rents and wages which, in turn, affects households location choices. Recent evidence reinforces the importance of this insight. Proximity to employment is the reason most frequently cited by households for choosing to live in their current neighborhood (American Housing Survey, ). Moreover, as noted earlier, Rhode and Strumpf s (2003) comprehensive study of sorting across U.S. communities 7 While a locational equilibrium has not been proven to exist for this model, Epple and Platt (998) and Sieg et al. (2004) demonstrate existence numerically when income is exogenous and preferences are vertically differentiated. 2

13 over the last 50 years finds that ob opportunities may be a critical determinant of sorting behavior. At least four recently developed structural models have the capability to translate observed location choices into household-specific preferences for public goods: Baari and Kahn (2005), Bayer, McMillan and Reuben (2005), Epple and Sieg (999), and Ekeland, Heckman, and Nesheim (2004) 8. The models differ in their depiction of preference heterogeneity and in the way they use equilibrium choices to recover preferences. Epple and Sieg and Ekeland et al. both depict an equilibrium in the entire housing market, and use the features of that equilibrium to recover the distribution of preferences for a single attribute (i.e. vertical differentiation). The two approaches differ in how they recover this information. Epple and Sieg recover preferences from location choices, given equilibrium housing prices. Ekeland et al. recover preferences from the hedonic price function that reflects equilibrium location choices. In contrast, Bayer, McMillan, and Reuben (2005) depict sorting by horizontally differentiated households. Horizontal differentiation allows wider substitution patterns at the cost of transferring identifying power to the distributional assumptions. Bayer et al. provides a somewhat extreme example of this tradeoff, allowing households to have idiosyncratic tastes for every possible location 9. Their model predicts a positive probability that every household will choose every location and equilibrium prices are defined by the assumption that each location is occupied in probabilistic terms 0. This formulation precludes interpretation of welfare effects in a way that is consistent with the previous literature on capitalization theory (Starrett [98] and Scotchmer [985]). Moreover, the model can predict multiple equilibria in response to a non-marginal change in public goods (Bayer and Timmins [2005]). Baari and Kahn (2005) also recover horizontally differentiated preferences but avoid distributional assumptions entirely by adding more structure to the problem in the form of a 8 Reduced form models that consider both housing and labor market choices usually follow Roback (982) who demonstrated that spatial variation in rents and wages can be used to calculate the marginal willingness-to-pay for a spatial amenity in a multi-market hedonic framework. Subsequent studies found that rents and wages both appear to reflect a substantial share of implicit prices (e.g. Blomquist et al. [987]; Graves and Waldman [99]). 9 A household might have a strong idiosyncratic taste for their chosen location because their family and friends live there, or because it provides a familiar environment. One could also argue that tastes proxy for moving costs. 0 The assumption that each taste parameter follows a Type I extreme value distribution allows them to express the probability of choosing a location as a function of the structural parameters. Then, the estimator from Berry, Levinsohn and Pakes (995) is used to find values for the parameters that minimize the difference between predicted and observed location choices. 3

14 hedonic price function. This allows them to recover preferences by setting marginal implicit prices equal to marginal rates of substitution. IV. Empirical Model To simplify notation in what follows, let locations k (, ),...,( J, K ), = be indexed by z =,..., Z. Working households are assumed to possess one of S different observable occupations and every household may differ in its preferences ( α, γ, θ ), so households are indexed by both i and s. i Then the indirect utility obtained by household i,s in location z can be expressed as (2). ν y ν ρ i, s, z (2) ( ) exp z = + exp, V i, s, z α i g i, z i i η+ βp + η ρ ρ where g i, z = γ i,g, z γ i, N g N, z + γ i, Nξ z, and y i, s, z yˆ i + θi,ws, z ( θi,2ts, z ) =. The first term in the CES function represents utility from public goods, and the second represents utility from the private good component of housing. All households are assumed to share the same (constant) elasticity of substitution between public and private goods, ρ, as well as the same housing demand parameters: price elasticity (η ), income elasticity (ν ), and demand intercept (β ). The signs of these parameters provide a test on the consistency of the theoretical model. With η < 0, ν > 0, and β > 0, the single crossing restriction implies ρ < 0. goods, Households have horizontally differentiated preferences over a linear index of public g i, z ( g N z = ξz. Of the N public goods in the index, N- are observable. The N th public good, ) is not observed by the econometrician. Households differ in the weights they place on each public good in the index ( ) γ i,,...,γ i, N and in their overall preferences for public goods relative to private goods( α i ). The weights are assumed to sum to, allowing α i to be identified separately as a scaling parameter on the strength of preferences. As in the theoretical model, a household s income equals the sum of its exogenous nonwage income and its effective wage income. The primary earner of each household is assumed The function in (2) nests the vertically differentiated parameterization in Epple and Sieg (999). To recover their parameterization, simply eliminate the i subscript on ( γ, γ 2,..., γ N ) and the s,z subscripts on y. 4

15 to possess skills that qualify them for a certain occupation (e.g. biomedical engineer, locksmith, etc). This is the observable component of ob skill indexed by s. In the labor market represented by z, the average wage for that occupation is w s, z. However, a worker s ability to collect that wage if they were to move from their current ob depends on unobservable components of their ob skill (e.g. quality of education, experience, people skills, etc.) and on unobservable attributes of the ob. All the unobservables are reflected in a single parameter, θ i,, that represents the worker s labor market mobility. θi, equals one at the worker s current ob and may be greater or less than one in alternative labor markets. The wage in each ob location is adusted for required commute time. t s, z is the ratio of commute time to work time, and θ i, 2 represents the shadow value of time as a share of the wage rate. Ifθ 0, effective wage income equals actual wage income. At the other extreme, if θ, the worker s shadow value i, 2 = i, 2 = of time equals their wage rate. Unobserved heterogeneity in ob skill is an important feature of the model. The idiosyncratic ob skill parameter, θ i,, plays a role that is similar, but not identical to that of the idiosyncratic tastes for locations ( ε i, ) in Bayer, McMillan, and Reuben (2005). The similarity is that both exploit unobserved heterogeneity to help explain observed location choices. There are three important differences in its use here. First, the number of household-specific idiosyncratic taste parameters in Bayer et al. equals the number of locations, while in the current model each household has a single idiosyncratic skill parameter. Second, the distribution of the ε ' s is specified by the researcher while the distribution of θ i, can be informed by data on variation in wages within each occupation. Finally, the ε ' s are unbounded due to the type I extreme value assumption, while θ i, has natural bounds. Assuming wages are nonnegative, θ i, must be nonnegative. Likewise, it is bounded from above by the logic of revealed preferences. Because it is bounded, θi, has less ability to rationalize observed location choices than the ε ' s, and is therefore less capable of undermining predicted welfare effects in the manner discussed by Petrin (2002). 2 2 The ε ' s ensure that the model explains observed location choices, no matter how poor the explanatory power of the other variables. In a case where the variables of interest have poor explanatory power, Petrin demonstrates that the ε ' s dominate welfare predictions. 5

16 A second feature that differentiates this empirical model from previous work is the absence of maintained assumptions on the distribution of preferences. The econometric strategies used in Epple and Seig (999) and Bayer, McMillan, and Reuben (2005) rely on a priori parametric assumptions on the distributions of heterogeneous preference parameters. The observation that those parameters are set identified by the partitioning logic described in the theoretical section suggests an alternative strategy for estimation: first recover the region of preference space that ustifies each location choice, then sample from that region according to various assumptions about the distribution of preferences. The logic of the estimation process is described in figure 2. It can be decomposed into two stages. The first stage estimates the price of housing in each community ( p,..., pj ), and the homogeneous housing demand parameters ( β, η, ν ). These results are treated as known constants during the second stage of the estimation, which recovers a composite unobserved public good for each community ( ξ,...,ξ J ), the homogeneous CES parameter ( ρ ), and a partition of preference space for the heterogeneous parameters A ( α, γ, θ ). All of the second stage parameters are estimated simultaneously, following the iterative process depicted in figure 2. The remainder of this section discusses each component of that estimation. (A) First Stage Estimation: The Price of Housing ( p,..., ) As in most locational equilibrium models, housing is treated as a homogeneous commodity that can be consumed in continuous quantities. Under this assumption, the price of housing reflects the cost of consuming the public goods provided by each community. Of course, in practice housing is not homogeneous. Its structural characteristics (e.g. bedrooms, bathrooms, sqft.) vary within and between communities, and these differences will be reflected in observable sale prices. This can be addressed if we are prepared to assume that housing construction is not constrained by local zoning regulations so that a house with any combination of structural characteristics can be built in any community for the same cost 3. This implies the equilibrium locus of housing expenditures defined by a hedonic price function will be separable in the structural characteristics of houses and the effect of public goods, as shown in (3). p J 3 Technically, this also assumes that construction costs are constant within the study region. See Sieg et al (2002) for more details. 6

17 (3) e, n f( z, n, π ) + f 2[ p ( g,,..., g N,, ξ )] + ε, n =. The left side of the expression represents expenditures on house n in community. The first subfunction on the right side depends on a vector of structural characteristics ( z, n ) and their marginal implicit prices (π ). The second sub-function represents the price of a homogeneous unit of housing in community, which depends on the public goods it provides, observed and unobserved. Given a parametric form for (3) and data on housing transaction prices and their structural characteristics, the price of housing in each community can be recovered as a community-specific fixed effect. Seig et al. (2002) demonstrate that the fixed effects represent a consistent index of prices for a homogeneous unit of housing in each community. (B) First Stage Estimation: Housing Demand Parameters ( β, η, ν ) Having recovered housing prices, the next step is to use them along with data on housing expenditures and household income to estimate the homogenous housing demand parameters ( β, η, ν ). Equation (4) displays an individual household s demand for housing, which follows from the indirect utility function in (2). (4) η ν h = βp y. Aggregating across individuals in each community and taking logs, total demand for housing in community can be written as equation (5), where aggregate demand is assumed to be measured with error. ln( y ) (5) ln H = ln( ) + η ln( ) + νµ β + ε p In the equation, ln ( y ) µ is the mean of log income in community. Similar equations can be derived for other quantiles of the housing demand distributions. After multiplying both sides of (5) by housing price, the demand intercept and the price and income elasticities in (6) can be estimated by regressing quantiles of the distribution of housing expenditures, r housing and quantiles of the income distribution, N y. N, on the price of 7

18 N N (6) ( r ) = ln( β ) + ( η + ) ln( p ) + ν ln( y ) + ε ln. The housing prices are estimated as fixed effects in a hedonic regression. As such, they will be measured with error. The observable public goods will be used as instruments for price in a 2SLS estimation of (6). Throughout the second stage of the estimation the first stage estimates are treated as known constants 4. To reduce notation in the following discussion, let δ represent the first stage results plus all the data on attributes of locations: [ p, β, η, ν ; g, w, t] δ =. (C) Second Stage Estimation: Composite Unobserved Public Good ( ξ,...,ξ J ) Communities are usually differentiated by some public goods that matter to households but are unobservable to the econometrician. Their influence is revealed in the data by the presence of seemingly inferior communities which, compared with other plausible locations, have higher housing prices and lower provision of every observed public good. Without accounting for unobserved public goods, the estimator cannot explain why a household would choose an inferior location. Fortunately, the way that households react to the supply of unobserved public goods helps to identify composite provision of unobserved public goods in each community. If unobserved public goods influence households location choices, they should also influence the price of housing. Under the maintained assumption that households have nonnegative preferences for public goods, the price of housing will be strictly increasing in unobserved public goods as in (7.a). 5 (7.a) p ( g,...,, ξ ) g N > ξ 0. (7.b) ξ g,..., g N. If (7.a) holds, the price of housing in each community that was recovered as a fixed effect in 4 Alternatively, β, η, ν, ρ, ξ could be estimated simultaneously in a GMM framework. However, any additional identification of β, η, ν would come from the independence restriction used to identify ρ in the second stage. 5 Baari and Benkard (2005) prove a hedonic price function for housing exists and is strictly increasing in ξ if utility satisfies differentiability, continuity, and nonsatiation in ξ and the numeraire. These conditions are satisfied for the indirect utility function in (2). 8

19 (3) should contain information about the provision of public goods in that community. More specifically, after controlling for the variation in the price index due to observed public goods, the remaining variation can be attributed to unobserved public goods. However, theory does not suggest a functional form for the relationship between p and g,...,, ξ. Importantly, the g N function need not be separable 6. Given this indeterminacy, the strategy used here is to impose the additional independence restriction in (7.b) which allows ξ to be recovered nonparametrically whether the price index is separable or nonseparable in the public goods. When (7.a,b) hold, Matzkin (2003) demonstrates that the quantiles of the distribution of the unobserved public good will equal the quantiles of the price distribution, conditional on observed public goods. This result is shown as (8). (8) F ( ξ ) F ( p ) F [ f ( g ξ )] =. ξ p g= g = p g= g, A variety of methods can be used to map the price of housing in each community into its corresponding quantile in the distribution of prices, conditional on observed public goods. Examples include the kernel estimator in Matzkin (2003) and the local linear method used by Baari and Benkard (2005). In either case, the estimated quantiles are a monotonic transformation of the unobserved characteristic itself since, assuming ξ has a continuous distribution, it can be normalized such that its marginal distribution is U[0,]. The estimated values of ξ must permit the indirect utility function to explain every observed location choice, including the choice of seemingly inferior communities. This requires a certain degree of smoothness in the relationship between the price of housing and the unobserved public good 7. In practice, the minimum bandwidth that delivers this smoothness may exceed the bandwidth that would otherwise be chosen to address the bias/efficiency tradeoff from estimating (8). In the estimation, this is treated as a constraint on the bandwidth. The estimator starts with the optimal bandwidth and then increases it until it can find values for the 6 The parameterization of the indirect utility function implies that MWTP for a change in an observed public good is a nonseparable function of the unobserved public good. Access to tennis courts, hiking trails, and other outdoor amenities (which are unobserved) may have higher value in locations with better air quality (which is observed). This relationship may or may not lead to nonseparability in the price index. 7 This is a common feature of characteristics models (Feenstra and Levinsohn [995], Epple, Peress, and Sieg [2005]). 9

20 heterogeneous parameters that ustify every observed location choice. This process is shown in figure 2 by the arrows that link estimation ofξ to the process of partitioning preference space 8. Appendix A proves there is some threshold bandwidth above which every location can be ustified by nonnegative values of the heterogeneous parameters. (D) Second Stage Estimation: CES Parameter ( ) ρ Given the data, the first stage estimates, and values for the unobserved public goods, households location choices can be expressed as a function of the elasticity of substitution parameter, preferences for public goods, the opportunity cost of time, and unobserved ob skill ( ρ, α, γ, θ ). Unfortunately, observed location choices are not sufficient to simultaneously identify ρ and ( α, γ, θ ) without some knowledge of the relationship between preferences and income. Previous locational equilibrium applications have supplied this information by specifying a parametric form for their oint distribution (Epple and Sieg [999]) or assuming they are independent for a subset of households (Epple, Peress, and Sieg [2005]). The later approach is used here. All else constant, the interaction between ρ and ŷ in the CES indirect utility function dictates how income shocks affect the desired bundle of housing and public goods. This relationship can be inverted to identify ρ from the location choices made by households that are identical except for their non-wage income. Put differently, the observed stratification by income of (otherwise) identical households reveals the extent to which they substitute public goods with the private good component of housing. Let ( α, γ, θ ) F denote the distribution of the heterogeneous parameters for a subset of s households, s, for which F s yˆ. Suppose this subset can be further divided into two groups with non-wage income ŷ and ŷ 2. The Gibbs algorithm outlined in the next section can be used to partition preference space for each group. Sampling over the resulting partitions will produce ~ two approximations to F s, F s, and F ~ s, 2. These conditional distributions will equal the unconditional distribution only when the partitioning process is performed at the true value of the CES parameter, ρ = ρ0, as depicted in equation (9). 8 In contrast, in Epple and Sieg (999) the assumed from of the oint distribution of income and preferences allows them to identify ξ from observed population shares. 20

21 ~ ~ = s, s,2 y, for yˆ ˆ y2. (9) F ( α γ, θ ) F ( α, γ, θ yˆ, ρ, δ, ξ ) F ( α, γ, θ ˆ, ρ, δ, ξ ) s, = 2 For other values of the CES parameter, ρ ρ0, the first equality cannot hold and the second equality is unlikely to hold. This follows from the observation that the boundary indifference loci defining the partition of preference space are nonseparable in (, ŷ) ρ. 9 A movement in ρ away from its true value will distort the boundaries of the partition to a different extent for ŷ and ŷ 2, leading to predictions for F s, and F s, 2 that differ from the true distribution and from each other: ~ F ~ s, Fs, 2 F. The estimator applies this logic to recover the value of ρ that s minimizes the predicted difference between F s, and F s, 2, as shown in equation (20). ~ ~ min s, s,2 y2. (20) F ( α, γ, θ yˆ, ρ, δ, ξ ) F ( α, γ, θ ˆ, ρ, δ, ξ ) ρ If location choices can be observed for s-type households at more than two income levels, the efficiency of the estimation may be improved by minimizing the difference between the predicted distributions for all pairwise combinations of income. In general, evaluating the obective function requires partitioning preference space at each of the d =,..., D income levels ~ ~ and then sampling from those partitions to obtain,...,. This process must be repeated, F s, F s, D updating ρ on each step, until the relevant convergence criteria are satisfied. Figure 2 depicts this iteration with arrows linking the estimation of ρ to the partitioning process. (E) Second Stage Estimation: Partitioning Preference Space A ( α, γ, θ ) Given the demand for housing, a value for the elasticity of substitution between public and private goods, and all the attributes that differentiate communities and obs, location choices can be expressed as a function of preferences for public goods, the opportunity cost of time, and unobserved ob skill. The partitioning process inverts this relationship, using the logic of revealed preferences to recover values for the heterogeneous parameters that rationalize observed location choices. This step of the estimation manifests Tiebout s logic that location choices 9 An expression for the boundary indifference loci is provided in appendix equation (A5). 2