Dynamic and Stochastic Model of Industry. Class: Work through Pakes, Ostrovsky, and Berry

Transcription

1 Dynamic and Stochastic Model of Industry Class: Work through Pakes, Ostrovsky, and Berry Reading: Recommend working through Doraszelski and Pakes handbook chapter Recall model from last class Deterministic All firms have constant returns (with adjustment costs) technology Get a result: convergence to equal shares in the long run

2 More stuff in here Framework delivers very little in the way of analytic results... This work is mainly motivated to be used in empirical Some papers take the model and try to simulate it (Dorazelski has papers, probably talk about this on Monday)

3 My Special Case of the Pakes, Ostrovsky, and Berry special case of the Ericson-Pakes Model 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. This is the way this industry goes. Take reduced form profit function. Perhaps one that is estimated. Incumbent firm draw an exit value φ each period from the

4 standard exponential distribution, so the density and c.d.f. is f(φ) = σe φ σ 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.

5 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.

6 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.

7 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

8 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]

9 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

10 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 σ, i =1, 2 (1) and solve the four equations in four unknowns {ˆφ 1, ˆφ 2,F 1,F 2 }.

11 Empirical Work With this Kind of Model Overview (Details in Later Courses) Data: history of industry. Standard nested fixed point approach (e.g. Rust). Suppose know π 0, π 1 and π 2 (from later courses) and β and want to estimate σ. 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.)

12 Stage 1. Use data to estimate reduced-form policy functions of market participants. So say have π 0, π 1,andπ 2. Let s say know β as well. Then we can use realizations to ˆF 1 and ˆF 2. Note given knowledge of π 0 see everything that the firm sees. Stage 2. Now find parameters consistent with these policies. Estimate VC 1 and VC 2 and σ, finding all three to solve four 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] F i = 1 e VC i σ,i=1, 2 (from : F i =1 e ˆφ i σ, ˆφ i = VC i

13 Lot of the motivation for all of this is an extreme pessimism of the usefulness of cost data In reduced form profit functions, often back out marginal costs from demand estimates. (If know a firm s elasticity of demand Infer entry cost distribution from revealed preference in entry decisions.

14 Quick Overview of Ericson-Pakes Full Model Each firm is in discrete state ω i Have some stage game with reduced form profits π i (ω i,ω i ). e.g. if have differentiated products oligopoly with demand q i (p i,p i,ω i,ω i )costc i (ω i,ω i ), and some reduced from equilibrium price function p e i (ω i,ω i 1 ), then π i (ω i,ω i )=[p e i (ω i,ω i 1 ) c i (ω i,ω i )] q i (p i (ω i,ω i ),p i (ω i,ω i ),ω i, Make decision to exit and take φ or invest at x and move state for next period.

15 Ericson Pakes particular investment model: and Pr(ν x i )= αx i 1+αx i if ν =1 = 1 1+αx i if ν =0 ω 0 i = ω i + v η (where η market) Incumbent s problem V (ω i,ω i,φ)=π (ω i,ω i )+max ½ φ, max x x i + βe[v (ω 0 i,ω0 i,φ0 ) ω i,ω i, x i Entrant (suppose for similicity): V e (ω, φ e )=max ( 0, max x e i φ e x e i +... )

16 A Symmetric Markov-Perfect Equilibrum: Investment Policy functions x i (ω i,ω i 1 ), Entry rules, Entry(ω, φ e ), Exit rules Exit(ω i,ω i,φ), and value functions V (ω i,ω i,φ), satisfying the Bellman equations and optimality conditions above. Symmetry has bite, all differences in behavior coming through differences in states.

17 Example of Courtnot Model with Fixed Cost Look at Cournot example, P =12 Q and suppose have a fixed cost equal to F (paid if q>0, otherwise if q =0is avoided). Given output q 2, firm 1 solves the problem: Either q 1 > 0andq 1 =argmax q 1 of q 1 = 0 and π =0. So if q 1 > 0, then reaction is So compare max (given positive) = 6 q 2 2 pq 1 =[12 q 1 q 2 ] 6 q 2 2 [12 q 1 q 2 ] q 1 F = 6 q 2 2 2

18 with fixed cost Let 6 q = F 6 q 2 2 = F 1 2 ˆq 2 = 12 2F 1 2 So R 1 (q 2 ) = 6 q 2 2, q 2 < 12 2F 1 2 = 0, q 2 < 12 2F 1 2, If F = 0, then unique equilibrium is q 1 = q 2 =4. earn 16. Bothfirms

19 Suppose F = 15: Then have three equilibria, one symmetric, two asymmetric Suppose F =17?

20 Stategies for Dealing with Curse of Dimensionality Dorazeslski and Judd: Try continuous time: 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. 2N different things can happen. Key point, measure zero event that two change states the same time. Obvlious Equilibrium (Weintraub, Benkard, and Van Roy)