Lecture Doraszelski and Judd. Dynamic and Stochastic Model of Industry

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1 Lecture Doraszelski and Judd Dynamic and Stochastic Model of Industry Discrete Time Model Time discrete, horizon infinite, N players States ω t Ω, finite stet Action of player i at tx i t a set that may depend upon ω t. x i t =(x 1 t,x 2 t,..x i 1 t,x i+1 t,...,x N i ) Pr(ω 0 ω t,x t )=Π N i=1((ω 0 ) i ω i t,x i t), case where evolution of indidual i s position independent of what others do. Payoffs π i (x t,ω t ) current profit Φ i (x t,ω t,ω t+1 ) return in next period dollars when move to ω t from ω t+1. Markov Perfect Equilibrium value function V i (ω), X i (ω) a policy function such that X i (ω) = arg max π i (x i,x i (ω),ω)+βe ω 0 Φ i (x i,x i (ω),ω,ω 0 )+V i (ω 0 ) ω, x i,x i (ω) ª x i 2

2 Continuous Time Model Path peicewise-constant, right continuous function of time. At time t, the hazard of a jump is φ(x t,ω t ) Probability moves to ω 0 f(ω 0 ω t,x t ) where ω t and x t is the state and action right before the jump. π i (x i,ω t ) flow of dollars per unit of time Φ i (x t,ω t,ω t ) shance in stock of wealth in dollars. ρ>0. discount rate Bellman equation ρv i (ω) = maxπ i (x i,x i (ω),ω)+ i φ(x i,x i (ω),ω) V i (ω)+e ω 0 Φ i (x i,x i (ω),ω,ω 0 )+V i ω 0 ω, x i,x i (ω) ª 3

3 Computational Strategies Discrete Time Start with X i (ω), V i (ω). From these define ˆX i (ω) = arg max π i (x i,x i (ω),ω)+βe ω 0 Φ i (x i,x i (ω),ω,ω 0 )+V i (ω 0 ) ω, x i,x i (ω) ª x i and ˆV i (ω) is the value of this. Then X i (ω) ˆX i (ω) V i (ω) ˆV i (ω) How update, PM1 pre-gauss-jacobi method. Go through each ω, then update. Block Gauss-Seidel. Go through each ω. But then update. 4

4 Computational Strategies Continuous Time ˆX i (ω) = argmaxπ (x i,x i (ω),ω) φ(x i,x i (ω),ω),v i (x i,x i (ω),ω) x i +φ(x i,x i (ω),ω)e ω 0 Φ i (x i,x i (ω),ω,ω 0 )+V i (ω 0 ) ω, x i,x i (ω) ª ˆV i (ω) = 1 ³ ˆX(ω),X ρ + φ( ˆX(ω),X i (ω),ω) πi i (ω),ω 1 + ρ + φ( ˆX(ω),X i (ω),ω) n o Φ i ( ˆX(ω),X i (ω),ω,ω 0 )+V i (ω 0 ) ω, ˆX(ω),X i (ω) E ω 0 5

5 Point about contractions. Individual problems are contractions. But entire systems not contractions. Curse of dimensionality Look at special case where can stay the same, go up one, or go down one. For a given ω, (suppose no-one at the bound). Then suppose N guys. Look at expectation. There are 3 N different possibilities. So have to sum over a mess of things. Look at continuous case. 2N. Key point, measure zero event that two change states thesametime. Storage issue. Remembers still a curse of dimensionality. Suppose there are M states ω i {1, 2, 3..., M}. Then whether continuous or discreate, still have Ω with N M states. Can pare these down. Impose symmetry V i (ω) =V 1 (ω i,ω 2,...ω i 1,ω 1,ω i+1,...) Then can look at representative firm. Anonymity, exchangeability. Only care about distribution of the other ω, not the identifies of which have it. Soforplayer1,(1, 1, 3) same as (1, 3, 1). Business about the address matching. (Key point do as much work as possible before hand and store it). 6

6 Example, Pakes and McGuire 1 Demand Caplin and Nalebuff U ik = g(ω i ) p i + ε ik = δ i + ε ik outside g(ω i ) = 3ω i 4, ω i 5 = 12+ln(2 exp(16 3ω i ) and for the outside good U 0k = ε 0k Suppose ε ik is i.i.d. extreme value then Consumers pick max i {U 0k,...U Nk } q i (p 1,..., p N exp(δ i ),ω)=m 1+ P N j=1 exp(δj ) Show a picture easy case of monopoly (with outside good. Price competition Bertrand competition max q i (p 1,...p N ; ω) p i c p i 0 The FONC is 0= p i qi (p 1,...p N,ω) p i c + q i Law of motion Random depreciation of δ invest x i then advance with probability αxi 1+αx i 7

7 Pr i ((ω 0 ) i ω i,x i )= = δ 1+αx i (1 δ), (ω 0 ) i = ω i 1. completement if ω i = ω i. αxi (1 δ), (ω 0 ) i = ω i αx i Butbangupatboundof1orM. Now for continous time φ i (x i,ω i )= αxi 1+αx i + δ f i = αx i 1+αx i αx i 1+αx i + δ, (ω0 ) i = ω i +1 8