Equilibrium Models with Interjurisdictional Sorting

Equilibrium Models with Interjurisdictional Sorting Presentation by Kaj Thomsson October 5, 2004

Set of 3 papers: • Epple & Sieg (1999): “Estimating Equilibrium Models of Local Jurisdictions” (MAIN PAPER) • Epple, Romer & Sieg (2001): “Interjurisdictional Sorting and Majority Rule” • Calabrese, Epple, Romer & Sieg (2004): “Local Public Good Provision, Myopic Voting and Mobility”

“Estimating Equilibrium Models of Local Jurisdictions” Dennis Epple Holger Sieg Journal of Political Economy, 1999

Background Previously: Models characterizing equilibrium in system of jurisdictions (Tiebout models) Assumption on preferences => strong predictions about sorting Predictions not empirically tested

Basic framework (1): Setup MSA = Set of Communities Competitive housing market price of housing determined by market in each community Each community: 1 public good … financed by local housing tax

Basic framework (2): Equilbrium Budgets balanced Markets clear Housing markets Private goods markets No household wants to change community (SORTING!)

Epple & Sieg (ES) test: Predictions about distribution of households by income across communities Whether the levels of public good provisions implied by estimated parameters can explain data

Formal Framework: MSA with: C = continuum of households J communities Homogeneous land Communities differ in: Tax on housing, t Price of housing, p ( p = (1+t)ph ) Households can buy as much housing as they want

Household’s problem: Note: they also optimize w.r.t. community

Slope of indifference curve in the (g,p)-plane: Assume: M( ) monotonic in y,α => Single-crossing in y (for given α) Single-crossing in α (for given y) …which is used to characterize equilibrium (A.1)

What does single-crossing mean? For given α, individuals with higher income y are willing to accept a greater house price increase to get a unit increase in level of public good

Also assume: Agents are price-takers Mobility is costless Equilibrium existence Shown in similar models Found in computation examples … but not formally shown here

Proposition 1: In equilibrium, there must be an ordering of community pairs {(g1,p1),…,(gJ,pJ)} such that 1-3 are satisfied: Boundary Indifference ~ There are individuals on the ”border” (in terms of y,α) between two communities that are indifferent as to where to choose to live Stratification For each α, individuals in community j are those with y s.t. yj-1 (α) < y < yj (α) , i.e. y is between boundaries from (1) Increasing Bundles Property if pi>pj, then yi (α )>yj(α ) < => gi>gj

Parametrization/Assumptions Assume (ln(α ), ln(y)) bivariate normal Assume indirect utility function: α > 0 differs between individuals <0, <0, >0, >0 same for all individuals

=> Indifference Curve: … is monotonic, so the single-crossing property is satisfied note: <0 required, which gives us a test of the model

Boundaries in y,α-space : Set up boundary indifference: V(gj,pj,α,y)=V(gj+1,pj+1,α,y) … => ln(α) = constant + *h(y) (10) with <0, h’(y)>0 …i.e. α as function of y defines boundary between communities j, j+1

2 key results (& 3 Lemmas) The population living in community j can be obtained by integrating between the boundary lines for community j-1 and j (L. 1) We have system of equations (12) that can be solved recursively to obtain the community-specific intercepts as functions of parameters (L. 2)

3rd (out of 2) key results For every community j, the log of the q-th quantile of the income distribution is given by a differentiable function ln[i(q,θ)] note: ln[i(q,θ)] is implicitly defined by:

Summary (so far) Part III: Theoretical analysis => Equilibrium characteristics (Proposition 1) Part IV: Parametrization => computationally tractable characterizations (Lemma/results 1-3) i.e. we now have a number of model predictions and we can test these predictions

Estimation Strategy Step 1: Match the quantiles predicted by the model with their empirical counterparts => identification of some parameters Step 2: Use the boundary indifference conditions => identification of the rest of the parameters

Step 1: Matching Quantiles Let q be the quantile (data for 25, 50, 75) Let i(q,θ) be the income for that quantile, A minimum distance estimator is then:

…where e1N(θ) is defined by

Step 1 The procedure above allows us to identify:

Step 2: Public-Good Provision Idea: Suppose housing prices available We solved system (12) recursively to obtain the community-specific intercepts as functions of parameters (L. 2) Use NLLS to estimate remaining parameters from (12):

Step 2: Public-Good Provision Problem: (20) g enters system (12), but is not perfectly observed Solution: Combine (12) and (20), and solve for j Can still use NLLS in similar way If endogeneity, find IV and use GMM instead of NLLS

Step 2 The procedure above allows us to identify:

Data Extract of 1980 Census Boston Metropolitan Area (BMA) 92 communities within BMA Smallest: 1,028 households (Carlisle) Largest: 219,000 (Boston) Poorest: median income $11,200 Richest: median income $47,646 … i.e. large variation

Descriptive Results 1: Quantiles Model predicts it should not matter which quantile we rank according to. Holds ~well:

Descriptive Results 2: Prices Proposition 1: housing prices should be increasing in income rank. Holds ~well:

Descriptive Results 3: Public Goods Prop. 1: if pi>pj, then yi (α )>yj(α ) < => gi>gj Holds ~well

Some empirical results In general, signs of parameter estimates compare well with empirical findings Income sorting across communities important, but explains only small part of income variance 89% of variance within community (heterogenous preferences) Rich communities do provide higher levels of Public Goods (prediction supported)

Conclusions What have we done? Built structural model => set of predictions Checked predictions against descriptives (data) Estimated structural parameters Analyzed the parameters E & S: The structural model presented is able to replicate many of the empirical regularities we see in data

Comments (1) Some assumptions questionable mobility costless? Can buy as much land as they want? Single-crossing: Do they assume the implications/predictions of the model? Evidence: Are the predictions really validaed? What is the relevance of the model? Does it add anything to just looking at descriptive data

Comments (2) … but still: a nice ’exercise’ shows that Tiebout models may have some predictive power (although says nothing about normative power, cf Bewley) maybe the framework can lead to answers to policy relevant questions

The 2 Extensions Use the same framework, but … introduce voting behavior in communities: Myopic Voting behavior ”Utility-taking” framework In general, mixed support for the models ability to predict and replicate data

Equilibrium Models with Interjurisdictional Sorting