Equilibrium Models with Interjurisdictional Sorting

Presentation by Kaj Thomsson

October 5, 2004

Set of 3 papers:

1. Epple & Sieg (1999): “Estimating Equilibrium Models of Local Jurisdictions” (MAIN PAPER)

2. Epple, Romer & Sieg (2001): “Interjurisdictional Sorting and Majority Rule”

3. Calabrese, Epple, Romer & Sieg (2004): “Local Public Good Provision, Myopic Voting and Mobility”

“Estimating Equilibrium Models of Local Jurisdictions”

Dennis EppleHolger 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

1. Budgets balanced

2. Markets clear• Housing markets

• Private goods markets

3. No household wants to change community (SORTING!)

Epple & Sieg (ES) test:

1. Predictions about distribution of households by income across communities

2. 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:

1. Agents are price-takers

2. Mobility is costless

3. 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:

1. 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

2. StratificationFor 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)

3. Increasing Bundles Propertyif 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)

1. The population living in community j can be obtained by integrating between the boundary lines for community j-1 and j (L. 1)

2. 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

3. 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:

1. Myopic Voting behavior2. ”Utility-taking” framework

In general, mixed support for the models ability to predict and replicate data