Lecture Pakes, Ostrovsky, and Berry. Dynamic and Stochastic Model of Industry

Transcription

1 Lecture Pakes, Ostrovsky, and Berry Dynamic and Stochastic Model of Industry Let π n be flow profit of incumbant firms when n firms are in the industry. π 1 > 0, 0 π 2 <π 1 π n =, forn 3. Incumbent firm draw an exit value φ each period from the standard exponential distribution, so the density and c.d.f. is f(φ) = σe φ σ

2 F (φ) = 1 e φ σ. where the expected value of φ is σ. If leave get this tomorrow. One possible entrant in each period. It draws an entry cost of κ = 0 with probability γ and cost κ = with probability 1 γ. β discount factor.

3 Markov-perfect Equilibrium Let VC n be the continuation value (i.e. return starting next period in next period dollars) when remain in the industry and there are n firms. Let V n (φ) be the discounted value given n firms today and given exit value of φ, V n (φ) =max{π n + βφ, π n + βv C n }. Let ˆφ n be a cutoff rule such that an incumbent exits if φ>ˆφ n when there are n firms. Since π 1 > 0, 0 π 2 <π 1 and π n =, forn 3, it is immediate that there will never be entry if n =2.

4 Since the entry cost draws an entry cost of κ =0withprobability γ and cost κ = with probability 1 γ, itisclearthat the entrant comes if κ =0andn 1 and otherwise doesn t enter. Taking this entry behavior as given, a MPE is a list {ˆφ 1, ˆφ 2,VC 1,VC 2 } such that ˆφ n is the optimal policy rule at state n taking as given that other firms obey (ˆφ 1, ˆφ 2 )and(vc 1,VC 2 )arethe continuation values given behavior according to these rules.

5 Derivation of MPE Let F 2 = F (ˆφ 2 ), the probability an incumbent stays in when there are two firms. Then VC 2 = (1 F 2 ) EV 1 + F 2 EV 2 VC 1 = (1 γ) EV 1 + γev 2 Next V n (φ) = π n + βv C n,ifφ<vc n = π n + βφ, if φ VC n

6 Observe that given the exponential assumption on φ, (has mean 1). E{φ φ >VC n } = σ + VC n. Hence EV n = π n + βf n VC n + β (1 F n )(σ + VC n ) = π n + βv C n + β (1 F n ) σ Results in two equations: VC 1 = (1 γ)[π 1 + βv C 1 + β (1 F 1 ) σ] +γ [π 2 + βv C 2 + β (1 F 2 ) σ] VC 2 = (1 F 2 )[π 1 + βv C 1 + β (1 F 1 ) σ] +F 2 [π 2 + βv C 2 + β (1 F 2 ) σ]

7 Rewrite as VC 1 = (1 γ)(π 1 + β (1 F 1 ) σ) +γ (π 2 + β (1 F 2 ) σ)+(1 γ) βv C 1 + γβv C 2 VC 2 = (1 F 2 )(π 1 + β (1 F 1 ) σ) +F 2 (π 2 + β (1 F 2 ) σ)+(1 F 2 ) βv C 1 + F 2 βv C 2 or or = Ã!Ã! 1 (1 γ) β γβ VC1 (1 F 2 ) β 1 F 2 β VC 2 Ã (1 γ)(π1 + β (1 F 1 ) σ)+γ (π 2 + β (1 F 2 ) σ) (1 F 2 )(π 1 + β (1 F 1 ) σ)+f 2 (π 2 + β (1 F 2 ) σ) Ã a11 a 12!Ã VC1!! = a 21 a 22 Ã! b1 b 2 VC 2

8 Use the fact that ˆφ i = VC i and substitute out for VC i above and add the two equations F i =1 e ˆφ i, i =1, 2 (1) and solve the four equations in four unknowns {ˆφ 1, ˆφ 2,F 1,F 2 }.

9 Estimation Overview Data: history of industry. Suppose know π 0, π 1 and π 2,andβ and want to estimate σ. Standard nested fixed point approach (e.g. Rust). Take a set of parameters, θ =(σ, π 0,π 1,π 2,β). Solve for equilibrium. Then write down the likelihood function. Here easy, but usually hard. Pick σ to maximize likelihood. Note need to recalculate equilibirum at every iteration. Two-Stage Approach (POB, Hotz-Miller, Bajari-Benkard-Levin.) Stage 1. Use data to estimate reduced-form policy functions.

10 Use realizations to estimate VC 1 and VC 2. (since can see π 0 ) and ˆF 1 and ˆF 2. Note given knowledge of π 0 see everything that the firm sees. State 2. Now find parameters consistent with these policies. No nest. Estimate VC 1 and VC 2 once and for all.

11 Estimation Implementation for the Monopoly Case Assume π 2 = (so n 1) V 1 (φ) =max{π 1 + βφ, π 1 + βv C 1 }. Then VC 1 = EV 1 but (remember trick from above) EV 1 = π 1 + βf 1 VC 1 + β (1 F 1 )(σ + VC n ) = π 1 + βv C 1 + β (1 F 1 ) σ

12 so or VC 1 = π 1 + βv C 1 + β (1 F 1 ) σ VC 1 = π 1 1 β + β 1 β (1 F 1) σ Now solution is obtained by solving F 1 = F (ˆφ 1,σ) ˆφ 1 = VC 1 Estimation. Recall parameters π 1, γ (entry), β, andσ where distribution of φ is F (φ, σ) =1 e φ σ. Say that π 1, β, andγ are known. Want to estimate σ.

13 Nested Fixed point approach Take a given value of σ and solve the dynamic programming problem. Pins down ˆφ 1 (σ) andf 1 (ˆφ 1 (σ),σ). Now take data. Suppose have n periods of data and let n x be the number of periods where firm exits and n s be number where firm stays, The likelihood is n = n x + n s L = k(n x,n s ) F 1 (ˆφ 1 (σ),σ) ns h 1 F 1 (ˆφ 1 (σ),σ i n x ln(l) = n s ln F 1 (ˆφ 1 (σ),σ)+n x ln h 1 F 1 (ˆφ 1 (σ),σ i

14 Simple Alternative Let F 1 = n s n For a given value of σ, let V C 1 (σ) = π 1 1 β + β ³ 1 F 1 σ 1 β Note there is no nested fixed point here. Directly estimating from observed payoffs (inthenestweknowσ) Can use a moment condition V C 1 (σ) =ˆφ 1 F 1 = F (V C 1 (σ),σ)

15 Straightforward to see how this generalizes to the duopoly case from last class. Now as add more to the model, the simple alternative gets no more complicated. But the nested fixed point case? Have to solve the fixed point. Have to worry about perhaps multiple equilibria.

16 But full power of this approach is really with the duopoly case...