1. Structural Static Models December 2008 Steven Stern

2. Introduction  Static Models of Individual Behavior  Static Models of Equilibrium Behavior  Modelling with Estimation in Mind  Estimation  Examples

3. Relevant Literature  Empirical IO Literature (Berry, BLP, Bresnahan & Riess,Tamer, Aguiregabiria & Mira)  Stern Long-Term Care Papers  Location Choice (Feyrerra, Bayer)

4. Static Models w/ Single Agents  Modelling  Estimation  Examples

5. Modelling  Utility function and budget constraint (possibly implied) with errors built into model  Compute Pr[observed choice] as statement that error is in range consistent with observed choice

6. Estimation  MLE or MOM with estimation objects implied by structure of the probability statements associated with model  May need simulation methods to integrate over relevant subset of error domain

7. Example 1: Kinked Budget Set Analysis

8. Model Specification  Hausman: h ik =βy ik +αw ik +Z i γ+u i  Wales & Woodland: specify utility w/ errors built into utility function → indifference curves  Simple example: U= βlogL+(1- β)logC, logβ~indN(Xα,σ 2 )

9. Example 2: Heckman Selection Model  Model:

10. Semiparametric Specification

11.  Estimate using Ichimura

12. Interpretation

13. Interpretation

14. Interpretation

15. Static Models w/ Multiple Agents  General Model Structure  Estimation  Examples

16. General Structure: What is an economy?  family in my work;  metro area in Feyrerra and Bayer;  Army unit in Arradillas-Lopez

17. Notation and Structure Define d ijk =1 iff ij chooses k, let d ij ={ d ij1, d ij2,.., d ijK }, and define d /ij to be the set of choices made by other members of the economy other than i. Objective function of each member i of economy j: U ij (d ij,d /ij ;β,x ij,z j,ε ij ), i=1,2,..,I j → Pr[d ij |d /ij,β,x ij,z j ]

18. Pr[d ij |d /ij,β,x ij,z j ] Define A ij (d ij |d /ij,β,x ij,z j ) = { ε: U ij (d ij,d /ij ;β,x ij,z j,ε)> U ij (d,d /ij ;β,x ij,z j,ε)  d≠ d ij } → Pr[d ij |d /ij,β,x ij,z j ] = Pr[ε  A ij (d ij | d /ij,β,x ij,z j )] Note importance of adding randomness to model

19. Role of Information  Full information: Ω ij ={ ε ij  i=1,2,..,I j } → issues in existence of an equilibrium or multiple equilibria  Partial information: Ω ij = ε ij → each member maximizes EU ij (d ij,d /ij ;β,x ij,z j,ε ij ) over the joint density of the other errors where d /ij becomes a random vector

20. One must be able to solve for an equilibrium and, when there are multiple equilibria, choose among them.

21. Estimation  Use Pr[error in appropriate area consistent w/ choice]  Much emphasis on Tamer (Heckman logical inconsistency property)  Use moments or likelihood

22. Tamer Problem

24. Moments Estimation  Define D jk =Σ i 1[ε ij  A ij (d ijk | d /ij,β,x ij,z j )] with conditional expected value Σ i Pr[ε ij  A ij (d ijk | d /ij,β,x ij,z j )]   Minimize quadratic form in deviations between D jk and its conditional moment

25. Moments Estimation  Issue: What does the deviation between the sample and theoretical moments represent? (What if added an error u j ?)

26. Example 1: My Long-Term Care Models  Economy is family with n children and n+2 choices  Value to family member i of choice k is V jik =Z j0 β k +X jk δ+Q jik λ+u jik  Equilibrium mechanisms → probabilities of observed choices

27.  In most recent paper, we model utility function of each family member as U ji = β 1 logQ j + (β 2 ε 2 )logX ji + (β 3 ε 3 )logL ji + (β 4 +ε 4 )t ji + u ji  Choices: X ji, L ji, H ji, t ji subject to a budget constraint.  Construct subsets of the domain of the errors consistent with each observed choice and the maximize the probability of errors being in those subsets.

28. Divorce Model w/ Private Information  U h =θ h +ε h -p; U w = θ w +ε w +p  θ j =Xβ j +e j, j=h,w  V j [U h, U w ]  Bargaining mechanism  Data: {X,H,D}

32. Feyrerra  Economy is a set of school districts in metro area  3 school types: public, private Catholic, private non-Catholic  Households differ in income, religious preferences, and idiosyncratic tastes for Catholic schools and neighborhoods  Public school choice depends on residence; private does not

33. Feyrerra  s=school quality  κ=neighborhood quality  c=consumption  ε=idiosyncratic preference for particular neighborhod/school choice  Utility:U(κ,s,c,ε) = s α c β κ 1-α-β e ε

34. Feyrerra  Budget constraint: c+(1+t d )p dh +T=(1-t y )y n +p n  Production of school quality: s = q ρ x 1-ρ q = y(S) where S is set of households who attend particular school, and y(S) is the average income of those attending. s kj =R kj s j

35. Feyrerra  Funding for schools: for private, x=T; for public, x=((t d (P d +Q d ))/(n d ))+AID d

36. Feyrerra  Household decision problem  Majority rule voting  Equilibrium  Estimation

37. Adding Dynamics  Issues w/ modeling dynamic equilibrium  Data needs much greater  Significant computation problems

38. Pitfalls of Ignoring Structure  Macurdy Criticism of Hausman  Feyrerra Errors Interpretation Problem  Linear probability model

39. Value of Thinking thru Structure  Policy Analysis  Discipline  Clarity  Fun