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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”

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equilibrium models with interjurisdictional sorting

Equilibrium Models with Interjurisdictional Sorting

Presentation by Kaj Thomsson

October 5, 2004

set of 3 papers
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

“Estimating Equilibrium Models of Local Jurisdictions”

Dennis Epple

Holger Sieg

Journal of Political Economy, 1999



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

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

Basic framework (2): Equilbrium

Budgets balanced

Markets clear

Housing markets

Private goods markets

No household wants to change community (SORTING!)

epple sieg es test

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

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

Household’s problem:

Note: they also optimize w.r.t. community

slope of indifference curve in the g p plane

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

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

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

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


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

=> 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 space

Boundaries in y,α-space :

Set up boundary indifference:


… => 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

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

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

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

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

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:

step 1

Step 1

The procedure above allows us to identify:

step 2 public good provision

Step 2: Public-Good Provision


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 provision26

Step 2: Public-Good Provision

Problem: (20)

g enters system (12), but is not perfectly observed


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

Step 2

The procedure above allows us to identify:



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

Descriptive Results 1: Quantiles

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

descriptive results 2 prices

Descriptive Results 2: Prices

Proposition 1: housing prices should be increasing in income rank. Holds ~well:

descriptive results 3 public goods

Descriptive Results 3: Public Goods

Prop. 1: if pi>pj, then

yi (α )>yj(α ) < => gi>gj

Holds ~well

some empirical results

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)



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

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

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

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