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## PowerPoint Slideshow about 'Equilibrium Models with Interjurisdictional Sorting' - brigham

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### “Estimating Equilibrium Models of Local Jurisdictions”

### Background

### Basic framework (1): Setup

### Basic framework (2): Equilbrium

### Epple & Sieg (ES) test:

### Formal Framework:

### Household’s problem:

### Slope of indifference curve in the (g,p)-plane:

### What does single-crossing mean?

### Also assume:

### Proposition 1:

### Parametrization/Assumptions

### => Indifference Curve:

### Boundaries in y,α-space :

### 2 key results (& 3 Lemmas)

### 3rd (out of 2) key results

### Summary (so far)

### Estimation Strategy

### Step 1: Matching Quantiles

### Step 1

### Step 2: Public-Good Provision

### Step 2: Public-Good Provision

### Step 2

### Data

### Descriptive Results 1: Quantiles

### Descriptive Results 2: Prices

### Some empirical results

### Conclusions

### Comments (1)

### Comments (2)

### The 2 Extensions

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”

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

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

Budgets balanced

Markets clear

Housing markets

Private goods markets

No household wants to change community (SORTING!)

Predictions about distribution of households by income across communities

Whether the levels of public good provisions implied by estimated parameters can explain data

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

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

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)

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

Agents are price-takers

Mobility is costless

Equilibrium existence

Shown in similar models

Found in computation examples

… but not formally shown here

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

Assume (ln(α ), ln(y)) bivariate normal

Assume indirect utility function:

α > 0 differs between individuals

<0, <0, >0, >0 same for all individuals

… is monotonic, so the single-crossing property is satisfied

note: <0 required, which gives us a test of the model

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

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)

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:

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

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

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:

The procedure above allows us to identify:

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):

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

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

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

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

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

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

… 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

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

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