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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