Equilibrium Models with Interjurisdictional Sorting

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Equilibrium Models with Interjurisdictional
Sorting

• Presentation by Kaj Thomsson
• October 5, 2004

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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

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Estimating Equilibrium Models of Local
Jurisdictions

• Dennis Epple
• Holger Sieg
• Journal of Political Economy, 1999

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Background

• Previously Models characterizing equilibrium in
  system of jurisdictions (Tiebout models)
• Assumption on preferences gt strong predictions
  about sorting
• Predictions not empirically tested

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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

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Basic framework (2) Equilbrium

• Budgets balanced
• Markets clear
• Housing markets
• Private goods markets
• No household wants to change community (SORTING!)

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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

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Formal Framework

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

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Households problem
Note they also optimize w.r.t. community
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Slope of indifference curve in the (g,p)-plane

• Assume M( ) monotonic in y,a gt
• Single-crossing in y (for given a)
• Single-crossing in a (for given y)
• which is used to characterize equilibrium (A.1)

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What does single-crossing mean?

• For given a, individuals with higher income y are
  willing to accept a greater house price increase
  to get a unit increase in level of public good

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Also assume

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

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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,a) between two communities that are
  indifferent as to where to choose to live
• Stratification
• For each a, individuals in community j are those
  with y s.t.
• yj-1 (a) lt y lt yj (a) , i.e. y is between
  boundaries from (1)
• Increasing Bundles Property
• if pigtpj, then yi (a )gtyj(a ) lt gt gigtgj

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Parametrization/Assumptions

• Assume (ln(a ), ln(y)) bivariate normal
• Assume indirect utility function
• a gt 0 differs between individuals
• ?lt0, ?lt0, ?gt0, ?gt0 same for all individuals

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gt Indifference Curve

• is monotonic, so the single-crossing property
  is satisfied
• note ?lt0 required, which gives us a test of the
  model

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Boundaries in y,a-space

• Set up boundary indifference
• V(gj,pj,a,y)V(gj1,pj1,a,y)
• gt ln(a) constant ?h(y) (10)
• with ?lt0, h(y)gt0
• i.e. a as function of y defines boundary between
  communities j, j1

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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)

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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,?)
• note ln?i(q,?) is implicitly defined by

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Summary (so far)

• Part III Theoretical analysis gt
• Equilibrium characteristics (Proposition 1)
• Part IV Parametrization gt
• computationally tractable characterization
  s (Lemma/results 1-3)
• i.e. we now have a number of model predictions
  and we can test these predictions

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Estimation Strategy

• Step 1 Match the quantiles predicted by the
  model with their empirical counterparts
• gt identification of some parameters
• Step 2 Use the boundary indifference conditions
  gt identification of the rest of the parameters

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Step 1 Matching Quantiles

• Let q be the quantile (data for 25, 50, 75)
• Let ?i(q,?) be the income for that quantile,
• A minimum distance estimator is then

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where e1N(?) is defined by
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Step 1

• The procedure above allows us to identify

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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)

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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

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Step 2

• The procedure above allows us to identify

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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

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Descriptive Results 1 Quantiles

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

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Descriptive Results 2 Prices

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

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Descriptive Results 3 Public Goods

• Prop. 1 if pigtpj, then
• yi (a )gtyj(a ) lt gt gigtgj
• Holds well

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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)

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Conclusions

• What have we done?
• Built structural model gt 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

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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

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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

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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