Note on Oblivious equilibrium

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1 Note on Oblivious equilibrium November 29, Introduction Ericson and Pakes(1995) introduced a framework for dynamic oligopoly equilibrium with entry and exit, which can be used for policy analysis. However, solving for equilibria of this type of model is burdensome, especially when the number of firms is not small. Suppose that the state of each firm is summarized by an integer, ϕ. If there are 20 firms ( and ) there are 40 possible state ϕ {1, 2,..., 40}, the number of possible N + K states of world is = K 15. If we want to store the binary policy function (0 or 1) on this space, then we need at least 8 million gigabyte memory. According to Weintraub et al.(2015), it took days to solve a MPE with 7 firms, months to solve a MPE with 10 firms using massive parallelization, and it is infeasible to solve a MPE with 12 firms. Weintraub et al.(2008) introduced a new notion of equilibrium concept, called Oblivious equilibrium. In this equilibrium, firms respond only to their own state and the long run average industry state, and thus computation of the equilibrium do not face the curse of dimensionality. This approximation works well if the number of firms is large. 2 Environment A firm i in period t is endowed with it own state, ϕ it N. If it exits, it is assigned ϕ = and stay exit forever. The future profit is discounted by β (0, 1). The industry space is summarized by µ t = (µ t (ϕ)) ϕ=0, where µ t (ϕ) is the number of firms in state ϕ. Here we implicitly assume anonymity: firms only care about the number of firms in each state, not the state of each firms. This allows us to reduce the state space. If ϕ {0, 1} (binary state), two firms are in 0 and three firms are in 1, then the industry state is µ t = (µ t (0), µ t (1), µ t (2),...) = (2, 3, 0,...). The total number of firms at the beginning of the period can be computed from the industry state by N t = µ t (ϕ). ϕ=0 1

2 The state space is given by { } S = µ N µ(ϕ) < How large is this space when the number of firm is given by N and the upper bound of state variable is K? We have to count the number of the solutions (µ(ϕ)) K ϕ=0for the following problem: ϕ=0 K µ(ϕ) = N ϕ=0 The number of integers which satisfies this equation is given by N+K C N, which is the number of K permutations of N + K. If N = 20 and K = 40, then 60 C 20 = Let µ i,t S denote the state of the competitors of firm i, defined as { µ t (ϕ) 1 if ϕ = ϕ it, µ i,t (ϕ) = µ t (ϕ) otherwise. We assume that the price of products is determined by the static Nash equilibrium. The reduced form static profit of an incumbent is given by static profit = π(ϕ it, µ i,t ). Assumption 1 For all ϕ S, π(ϕ, µ) is increasing in ϕ. Furthermore, sup ϕ,µ π(ϕ, µ) <. Example of static profit function: Production function: F (ϕ, q) = ϕq α, α (0, 1). The demand function is given by p(q) = d Q. p (Q) = 1, where Q is the aggregate output, define as N Q = j=1 ϕq α j Marginal cost is constant at 1: cost function is given by q. There are 2 state, ϕ {1, 2}. Given competitor s production strategy {q j } N j 1, the profit maximization problem of a firm i is given by max q i pϕ i q α i q i, subject to p = d ϕ i q α i N ϕ j qj α j i The first order condition is 0 = d ϕ i qi α 1 We could define the industry state by N ϕ j qj α αϕ i q α 1 i j i 1 αϕ 2 i q 2α 1 i. µ t = (µ it ) N t i=1 where N t is the number of incumbents. If N = 20 and there are K possible realizations of s, then the size of state space is = , which is much larger than the anonymous case. 2

3 We will focus on the symmetric Nash equilibrium, q i = q(ϕ i, µ i ). Then q(ϕ, µ) satisfies 0 = d ] µ(ϕ)ϕq(ϕ) α αq(1, µ) α 1 1 αq(1, µ) 2α 1, 0 = ϕ=1,2 d ϕ=1,2 µ(ϕ)ϕq(ϕ) α ] α2q(2, µ) α 1 1 α4q(2, µ) 2α 1. Given µ, this is two equation for two unknown, q(1, µ) and q(2, µ). We can solve this either numerically or analytically if possible to obtain the Nash equilibrium strategy. Suppose α = 0.5. Then the first order condition is 0 = d ] µ(ϕ)ϕq(ϕ) q(1, µ) , 0 = ϕ=1,2 d ϕ=1,2 µ(ϕ)ϕq(ϕ) 0.5 ] q(2, µ) After some algebra, the Nash equilibrium, which is the solution to the above system of equations, is ] 2 d q(1, µ) = µ(1) + 2µ(2) q(2, µ) = 1 ] 2 d. 2 µ(1) + 2µ(2) Then we can plug these Nash equilibrium strategy into the profit to obtain the reduced form static profit function as ] ] 2 d π(ϕ, µ) = ϕ d (µ(1) + µ(2)) q(ϕ, µ) α q(ϕ, µ). µ(1) + 2µ(2) The conditional distribution of the state of individual firm is given by ϕ i,t+1 F (ϕ i,t+1 ϕ i,t ) Every period firms draw a sell off value φ it from a distribution G. They will obtain this if they exit. Assumption 2 1. {{φ it } Nt i=1 } t=0 are i.i.d and have finite expectation and well-defined density function with support R {{ϕ it } Nt i=1 } t=0 are i.i.d and independent of {{φ it } Nt i=1 } t=0. There are many potential entrant. In order to enter into the market, they have to pay a fixed cost c e > 0. When an entrant decided to enter, it can start operation from the next period. Once they enter, their state is given by ϕ e. One way to think of this in terms of Hopenhyan is that the distribution of state for entrants is degenerate, { 1 if ϕ = ϕ e, η(ϕ) = 0 otherwise. 3

4 Under the assumptions that entrants use a mixed strategy for entry, the number of entrant is given by an i.i.d Poisson random variable with mean λ(s t ). For later use, let M t denote the number of entrant in period t. The entry profit of an entrant is given by βepresent discounted value of profit at ϕ = ϕ e ] c e, where the present discount value of profit will be defined later. Assumption 3 1. The number of firms entering during period t is a Poisson random variable that is conditionally independent of {{φ it, ϕ it } Nt i=1 } t=0, conditional on µ t. 2. c e > βφ, where φ is the expected net present value of entering the market, earning zero profits each period, and then exiting at an optimal stopping rule. Timing: 1. Incumbents observe its sell-off value φ it, decide whether to exit. 2. Entrants enter by paying a fixed cost of entry, c e. 3. Incumbents receive profits. 4. Exiting firm exit and receive φ it. 5. ϕ i,t+1 is realized. New entrants enter, and the industry state for next period µ t+1 is determined. 3 Markov Perfect Equilibrium 3.1 Definition of Markov perfect equilibrium From the timing, firms decide where or not it exists after the sell off value φ is observed. Suppose that the competitors use the function σ c = (x c (ϕ j, µ j )) to determine their strategy. Given the competitor s strategy σ c and entrant s strategy λ, the firm i chooses its exit strategy χ to solve 2 Ŵ (ϕ i, µ i, φ σ c, λ) = π(ϕ i, µ i )+ max χ {0,1} { χφ + 1 χ]βeŵ (ϕ i, µ i, φ σ c, λ) ϕ i, µ i ] } (1) From this, optimal exit policy is a cutoff rule: exit if the sell off value is higher than the continuation value, and stay otherwise. { 1 if φ βew (ϕ i χ(ϕ i, µ i, φ) =, µ i, φ σ c, λ) ϕ i, µ i ] 0 otherwise. This formulation includes φ as a state variable. Once we know that the optimal exit policy is characterized by the cutoff rule, we can reduce the state variable by rewriting the Bellman equation in terms of W (ϕ i, µ i σ c, λ) = Ŵ (ϕ i, µ i, φ σ c, λ)df (φ): W (ϕ i, µ i σ c, λ) = π(ϕ i, µ i )+ max Eφ φ > r(ϕ i, µ i )]1 x(ϕ i, µ i )] + x(ϕ i, µ i )βew (ϕ i, µ i σ c, λ) ϕ i, µ i ] (2) r where x(ϕ i, µ i ) = P φ < r(ϕ i, µ i )] and r(ϕ i, µ i ) is the threshold value for exiting. 2 Note that this is the value function before firms observe their sell-off value φ. We could write down the value function after they observe φ, but since we know that the exit decision is characterized by the cutoff rule, adding φ into the state just increases the computational cost. 4

5 For notational convenience, I will call the exit threshold r and implied probability of staying in the market x as the strategy (σ = (r, x)) while what firms actually choose is exit decision χ. For later use, define the choice specific value function as W (ϕ i, µ i σ, σ c, λ) = π(ϕ i, µ i ) + Eφ φ > r(ϕ i, µ i )](1 x(ϕ i, µ i )) + x(ϕ i, µ i )βew (ϕ i, µ i σ c, λ) ϕ, µ i ] (3) In order to compute the expectation in this Bellman equation, we have to compute the conditional distribution of the industry state. To do so, suppose that all competitors use a common strategy σ c. Then the conditional distribution of individual competitor s state is given by { P σc (ϕ 1 x c (ϕ i, µ i ) if ϕ i i ϕ i, µ i ) = = x c (ϕ i, µ i )F (ϕ i ϕ i) if ϕ. (4) i. Since there is a finite number of agent ( finite draws of ξ j, φ j), the industry state is a random variable. Given φ (φ 1,..., φ N ) and the number of entrant M, µ is given by µ i(ϕ) = N F (ϕ j = ϕ ϕ j )1 x c (ϕ i, µ)] + Mη(ϕ) (5) j i where M is a Poison random variable with mean λ. Note that if ˆµ = lim N µ/n and ˆM = lim N M/N exists, then (??) can be written as ˆµ i(ϕ) = F (ϕ = ϕ ϕ)1 x c (ϕ, µ)]ˆµ i (ϕ) + ˆMη(ϕ). (6) ϕ=0 This looks like the law of motion in Hopenhyan (equation (12) in the handout Hopenhayn Model of Firm Heterogeneity ). The perceived conditional distribution of competitor s state µ i (probabilistic analogue of (??)) is given by N ] M P σc,λ(µ i ϕ i, µ i ) = P σc (ϕ j ϕ j, µ j ) P λ (M µ) η(ϕ e) j i M=0 e=1 (7) The first term represents the distribution of incumbent, while the second term represents the distribution of entrants. Given (??) and (??), the expectation in (??) can be written as EW (ϕ i, µ i σ c, λ) ϕ i, µ i ] = µ i S ϕ i =0 W (ϕ i, µ i σ c, λ)f (ϕ i ϕ i )P σc,λ(µ i ϕ i, µ i ) Given existing incumbent s strategy σ c and other entrant s strategy λ, the entry profit is given by where expectation is taken over µ i. W e = βew (x e, µ i σ c, λ) µ i ] c e (8) Definition 4 The Markov Perfect equilibrium of this model is a list of functions, value function W, policy function σ = (r, x ) and entry decision λ, such that 5

6 1. For all (ϕ i, µ i ) N S, σ (ϕ i, µ i ) is the mutual best response: σ (ϕ i, µ i ) = arg max σ Eφ φ > r]1 x] + xβew (ϕ i, µ i σ, λ ) ϕ i, µ i ] (9) where W is the fixed point of the Bellman equation W (ϕ i, µ i σ, λ ) = π(ϕ i, µ i ) + max σ Eφ φ > r]1 x] + xβew (ϕ i, µ i σ, λ ) ϕ i, µ i ]. (10) 2. For all µ S, Either zero profit condition is satisfied, or entry rate is zero: where inequality holds if λ (µ) = Computation of Markov perfect equilibrium W e = βew (x e, µ i σ, λ ) µ] c e 0 (11) Pakes and Mcguire(1994) proposed the following backward solution algorithm. 1. Make an initial guess, W 0, σ 0, and λ Set σ c = σ 0, λ = λ 0 and compute the conditional distribution of competitor s state using (??). 3. Solve the Bellman equation (??) with W = W 0 on the right hand side. From this, we obtain an updated strategy σ 1 and value function W Given W 1, compute the entry profit using (??) and obtain updated entry strategy λ Keep iterating on 2-4 until the value function and strategies converge. There are several problems which makes the application of this algorithm difficult. 1. The state space. As we discussed earlier, even if we assume annonimity, there are N+K C N possible states of world. 2. Time to compute the expectation. Since we have large state space, it is costly to evaluate the expectation over µ i in the Bellman equqation: we have to sum up over N+KC N possible states. These computational costs increase rapidly as the number of possible state/number of firms increases. Later Pakes and Mcguire(2001) proposed the stochastic approximation algorithm. Let V (ϕ i, µ i ) = µ i S W (ϕ i, µ i σ c, λ)p σc,λ(µ i ϕ i, µ i ) (12) In this algorithm we only use this expected value function. Then the Bellman equation can be written as W (ϕ i, µ i V ) = π(ϕ i, µ i ) + max Eφ φ > r](1 x) + xβ V (ϕ i, µ i )F (ϕ i ϕ i ) (13) σ ϕ i =0 Start the algorithm from some location (µ 0 i ) and initial guess V Substitute V l into (??) and compute the optimal strategy at µ l, σ(ϕ i, µ l i ). 2. Using σ(ϕ i, µ l i ), generate new location µl+1 i. 6

7 3. Let α(µ i, l) denote the number of times the state µ l i has been visited prior to iteration l.ãeeupdate expected value function at µ l i by V l+1 (ϕ i, µ l i) = α(µ i, l + 1) W (ϕ i, µ l+1 i V ) α(µ i, l) α(µ i, l + 1) V l (ϕ i, µ l i) (14) We can think of (??) as the Monte Calro integration of (??) over µ i : instead of using P σ c,λ(µ i ϕ i, µ i ) to compute the expectation, we draw µ i ranodomly from the strategy and take the average of them. This algorithm update the value function only on the recurrent state. In some cases the number of recurrent state increases linearly. In addition, we don t need to evaluate the expectation. Doraszelski and Judd(2012) uses the continuous time model where the state of only one firm changes in a given time to reduce the cost to evaluate the expectation. In this case, the number of future possible state is K(N 1). 4 Oblivious Equilibrium 4.1 Definition of Oblivious Equilibrium Since it is hard to compute the MPE directly, approximate the equilibrium by assuming µ = µ and dropping µ from the state space, as in Hopenhyan. In the model, however, there is no stationary distribution. What should we use for µ = µ = µ? use long run average industry state. Let σ(ϕ) ( r(ϕ), x(ϕ)) and λ denote the strategies in the OE, respectively. We call it Oblivious strategy because this decision is made without full knowledge of the state of the world. Note that under the Oblivious strategy, the individual state ϕ follows a Markov chain whose transition matrix is given by { P σ (ϕ 1 x(ϕ) if ϕ =, ϕ) = x(ϕ)f (ϕ ϕ) if ϕ.. Let P k σ (ϕe, ϕ) denote the k-step transition probability from ϕ e to ϕ under the Oblivious strategy σ. If competitors follow the oblivious strategy ( σ c, λ), the long run average industry state can be obtained by µ σc, λ (ϕ) = Eµ t(ϕ)] = λ P k σ c (ϕ e, ϕ), (15) Here, the transition probability P σ (ϕ ϕ) is a matrix, and the industry state is a vector. So we can write the law of motion for the industry state µ as k=0 µ = P σc µ + Mη (16) where M is the number of entrant, which follows a Poisson distribution with mean λ. If we take the unconditional expectation, then Eµ ] = P σc Eµ] + λη (17) 7

8 Note that Eµ ] = Eµ] because the expectation is unconditional. We can obtain Eµ] by iterating on (??) µ σc, λ (ϕ) = Eµ t(ϕ)] = λ Eµ] = P σc P σc Eµ] + λη] + λη = P 2 σ c P σc Eµ] + λη] + P σc λη + λη = = λ P k σ c η k=0 P k σ c (ϕ e, ϕ) k=0 where the last line follows from η(ϕ) = 1 ϕ = ϕ e. If I P σc ] is non-singular, then we can simply take the inverse to get µ σc, λ = I P σ c ] 1 λη To get an intuition, suppose we are in period t. How many firms are in state ϕ on average? Every period we have λ new entrant with state ϕ e. The probability that the entrant which entered in the market in t k 1 (and became incumbent in period (t k)) will be in state ϕ is P k σ c (ϕ e, ϕ). So, the expected number of firms which is born in period t k and now in state ϕ is λp k σ c (ϕ e, ϕ). Sum up this for all k to get the expected number of firms in state ϕ. Example: Suppose ϕ {, 0, 1, 2}, ϕ e = 1. x(0) = 1 and x(1) = x(2) = 0, λ = 5. Also, suppose that σ c is chosen so that P σc is given by P σc = P σc (ϕ = j ϕ = i) = (ϕ, ϕ ) /3 1/3 1/ /2 1/2 Suppose a firm entered in period t 3 as a new entrant and became incumbent in t 2. Then the distribution of its state in period t, ϕ t, is drawn from P 2 σ c (ϕ e, ϕ t ) = (third raw of P σc P σc ) = ( ) No firms stay in the market forever: For k high enough (here k = 60), ϕ t = with probability 1. P 60 σ c (ϕ e, ϕ t ) = (third raw of (P σc ) 60 ) = ( ) So the long run average industry state exists. Under this transition matrix, the long run average industry state is µ σc, λ (0) = 5 µ σc, λ (1) = 15 µ σc, λ (2) = 10 8

9 However, since there is only a finite number of agents, this is not the stationary distribution of this model. That is, even though µ t = µ σc, λ, it s possible that µ t+1 µ t = µ σc, λ. Consider the following situation: all firms in state 1 gets a high shock and go to state 2, all firms in state 2 remains in state 2. 5 entrants in state 1. In that case, the industry state in period t + 1 is given by µ t+1 (0) = 0, µ t+1 (1) = 5, µ t+1 (2) = 25 In the next state, since there is no firm in state 0, no firm exits while 5 firms enter (note that under the oblivious strategy entry does not depend on current industry state), so the number of firms will be 35. This happens with very small probability (1/3) 10 (1/2) 10 = , but it s possible with a finite number of agents. Given the competitor s oblivious strategy σ c and entrant s strategy λ, the Oblivious Bellman equation is given by where the expectation is computed as W (ϕ i σ c, λ) = π(ϕ i, µ σc, λ ) + max σ Eφ φ > r]1 x] + xβe W (ϕ i σ c, λ) ϕ i ] (18) E W (ϕ i σ c, λ) ϕ i ] = ϕ i =0 W (ϕ i σ c, λ)f (ϕ i ϕ i ). Here, since firms think that the industry state is constant at µ σc, λ, we don t take expectation over µ i. This way we can reduce the state space and computational cost dramatically. Definition 5 The Oblivious Equilibrium in this model is a list of functions, value function W (ϕ), policy function σ = ( r, x ), entry rate λ, and the long run distribution µ σ, λ such that 1. For all ϕ N, σ is the mutual best response: σ (ϕ) = arg max σ Eφ φ > r]1 x] + xβe W (ϕ σ, λ ) ϕ] (19) where W is the fixed point of the Bellman equation W (ϕ σ, λ ) = π(ϕ, µ σ, λ ) + max σ Eφ φ > r]1 x] + xβe W (ϕ σ, λ ) ϕ]. (20) 2. Either zero profit condition is satisfied, or entry rate is zero: where inequality holds if λ = µ σ, λ (ϕ) is consistent with the policy function σ, λ. W e = β W (ϕ e σ, λ ) c e 0 (21) What is the difference between this equilibrium and the stationary equilibrium in Hopenhyan? in this model, the price is determined in the static Nash equilibrium. 9

10 in Hopenhyan the price is determined by the market clearing condition (computationally it is determined by the zero profit condition, though). In addition, in this model the profit function depends on the long run average industry state for each λ, we need to recalculate the policy function, while in Hopenhyan we don t need to. What is the different between this equilibrium and the Markov perfect equilibrium? In MPE, we have to solve incumbent s problem for each industry state µ. Here, firms believe that the industry state is constant at µ σ, λ. As a result, industry state is not a state variable, which reduce the computational cost significantly. 4.2 Computation of Oblivious Equilibrium Computational algorithm 3 1. Make an initial guess of λ, λ 0. Given this, compute σ. (a) Make an initial guess of σ, σ 0. (b) compute µ σ 0, λ (ϕ) using (??). 0 (c) Given µ σ, λ(ϕ), compute the Oblivious value function using (??). Then compute the best response to σ 0, which is the policy function in (??). Call it σ 1. If sup ϕ σ 1 σ 0 < ɛ 1, set σ = σ 1 and go to step 2. Otherwise go to step (b) with σ 0 σ Check the zero profit condition. If β W (ϕ e σ, λ 0 ) c e < ɛ 2, done. Otherwise, change λ and go to step 1. 5 Asymptotic results and Error bounds 5.1 Asymptotic results As the number of firms increases, the uncertainty in the industry state decreases, so OE will be close to MPE. Since the number of firms is endogenous, in order to do this asymptotic analysis, we have to pick up a parameter which increases the number of firms. Here we use the market size m, which is a parameter in the demand system. Suppose that as m, N (m). If the profit function is such that there is a very large firm in the equilibrium, 5.2 Error bounds Since OE is easy to compute, we want to use it instead of MPE if the approximation error is small. How close is the OE strategy to the MPE strategy? If σ = σ, then σ should solve the MPE Bellman equation (??). So, we will use the following comparison to assess the approximation error Esup W (ϕ, µ σ, σ, λ) W (ϕ, µ σ, σ, λ)] σ where expectation over the invariant distribution of µ. 3 For more detailed and efficient algorithm see chapter 5 of Weintraub et al(2010,or). 10

11 Since the state space is large, it is difficult to evaluate W (ϕ, µ σ, µ, λ). Instead, Weintraub et al(2008) constructed the upper bound on the approximation error. Let (µ) sup y π(y, µ) π(y, µ σ, λ). Then Esup W (ϕ, µ σ, σ, λ) W (ϕ, µ σ, σ, λ)] 2 E (µ)] (22) σ 1 β In order to understand how we can derive this, let σ = arg sup σ W (ϕ, µ σ, σ, λ). The approximation error can be written as EW (ϕ, µ σ, σ, λ) W (ϕ, µ σ, σ, λ)] = EW (ϕ, µ σ, σ, λ) W (ϕ σ, σ, λ)] + E W (ϕ σ, σ, λ) W (ϕ, µ σ, σ, λ)] We will focus on the first term in the bracket. The same argument applies to the second term. Let M and M denote the set of Markov strategy and the Oblivious strategy, respectively. In the Oblivious value function firms believe that µ t = µ σ, λ so, there is no gain even if we extend the set of strategy from M to M: Hence, W (ϕ σ, σ, λ) sup σ M = sup σ M W (ϕ σ, σ, λ) W (ϕ σ, σ, λ) W (ϕ σ, σ, λ) W (ϕ, µ σ, σ, λ) W (ϕ σ, σ, λ) W (ϕ, µ σ, σ, λ) W (ϕ σ, σ, λ) From the definition, the value function can be written in the sequential form as τ ] W (ϕ i, µ i σ, σ, λ) = E β k t {π(ϕ it, µ i,t ) di } + β τ t φ ϕ it = ϕ i, µ i,t = µ i k=t τ ] W (ϕ i σ, σ, λ) = E β k t {π(ϕ it, µ σ, λ) di } + β τ t φ ϕ it = ϕ i k=t where τ is the stochastic exit time implied by x : P (τ = T ϕ it = ϕ i ) = ϕ i,t+1,...,ϕ it 1 x (ϕ it )] T k=t+1 x (ϕ ik )F (ϕ ik ϕ i,k 1, I (ϕ i,k 1 ))] If we take the difference between these two value functions and take expectation over µ, then τ ] EW (ϕ i, µ i σ, σ, λ) W (ϕ σ, σ, λ)] = E β k t {π(ϕ, µ) π(ϕ, µ σ, λ)} where we used the triangle inequality in the last line: k=t ] = E β k t 2 (µ) k=t = 2 1 β E (µ)] π(ϕ, µ) π(ϕ, µ σ, λ) π(ϕ, µ) π(ϕ, µ σ, λ) + π(ϕ, µ) π(ϕ, µ σ, λ) Here the term di and β τ t φ disappear from the equation because the same strategy is used and we take the expectation over µ. The RHS of (??) is easy to evaluate. 11

12 6 Computational exercise 6.1 The computational model Finite number of agents Suppose that each consumer has a following demand function: q(p) = e p d If there are m consumers, then the aggregate demand function is given by ( ) e Q(p) = m p d which implies the inverse demand function of the form p(q) = e Q/m + d Firms use labor input to produce the homogeneous goods. The production function is There are two possible states: ϕ {ϕ H, ϕ L }. The wage rate is normalized to 1. f(ϕ i, n i ) = ϕ i n α i, α (0, 1). Firm i chooses its labor input to maximize its static profit, subject to the aggregate demand function and taking the labor input of competitors (n j ) N j i as given: The first order condition is 0 = p(q) f(ϕ i, n i ) n i max p(q)ϕ i n α i n i, subject to n i e p(q) = Q/m + d N Q = ϕ j n α j j=1 1 + p (Q) Q n i f(ϕ i, n i ) eαϕ i n α 1 i = ( N ) 1 1 m 1 j=1 ϕ jn α j + d m eαϕ 2 i n2α 1 i m 1 ( N j=1 ϕ jn α j ) ] 2 + d This condition tells us that if lim m m 1 Q = Q <, then as m increases the effect of individual behaviour on price decreases. In the limit, the first order condition is 0 = p(q)αϕ i n α 1 i 1 Let n(ϕ i, µ i ) denote the strategy in the symmetric Nash equilibrium. Then it satisfies eαϕ i n(ϕ i, µ i ) α 1 0 = ( ) 1 1 eαϕ 2 i n(ϕ i, µ i ) 2α 1 ( m 1 ϕ j=ϕ H,ϕ L ϕ j n(ϕ j, µ j ) α µ(ϕ j ) + d m ) m 1 ϕ j=ϕ H,ϕ L ϕ j n(ϕ j, µ j ) α µ(ϕ j ) for all ϕ i. which is the first order condition in Hopenhyan. ] 2 + d 12

13 In the Oblivious equilibrium, firms do not observe the true industry state µ but observe the long run state µ σc, λ. Therefore, we compute the labor input in the Oblivious equilibrium ñ(ϕ i), by solving the following equation numerically: eαϕ i ñ(ϕ i ) α 1 0 = ( ) 1 1 m 1 ϕ j=ϕ H,ϕ L ϕ j ñ(ϕ i ) α µ σc, λ (ϕ j) + d m eαϕ 2 i n(ϕ i, µ i ) 2α 1 ( m 1 ϕ j=ϕ H,ϕ L ϕ j ñ(ϕ i ) α µ σc, λ ) ] 2 + d Once we compute ñ(ϕ i ), we can compute the profit function as π(ϕ i, µ σc, λ ) = p( Q)ϕ i ñ(ϕ i ) α ñ(ϕ i ), where Q = ϕ j ñ(ϕ i ) α µ σc, λ (ϕ i). ϕ j=ϕ H,ϕ L The sell-off value φ are i.i.d exponential random variables with mean K. That is, φ has the cumulative distribution function of the form Firms stay in the current state with prob θ, where θ > 0.5. F (ϕ = y ϕ) = G(φ) = 1 exp( Kφ). { θ if y = ϕ, 1 θ if y ϕ Given a guess W, we know the continuation value. Firms exit if and only if its continuation value is below the sell off value, φ. So the probability of exit is given by x(ϕ i ; W ( ) = G βe W (ϕ i σ c, λ) ϕ ) i ] = 1 exp 1 ( βe K W (ϕ i σ c, λ) ϕ i ]) ]. (23) With (??), it is easy to update the value function. Entrants receive ϕ H with prob η H and ϕ L otherwise. Then the entry profit is W e = βη H W (ϕh σ c, λ) + (1 η H ) W (ϕ L σ c, λ)] c e Continuum of firms: Hopenhyan model If there is a continuum of agents, then firms take price as given. So the profit maximization problem of a firm in state ϕ is The first order condition is Then the profit function is max n pϕn α n 0 = pϕαn α 1 1 n(ϕ; p) = pαϕ] 1/(1 α) π(ϕ; p) = pϕn(ϕ; p) α n(ϕ; p) 13

14 Parameter Value Description α 0.33 Production function parameter d 1 Demand function e 10 Demand function ϕ H 1 High productivity ϕ L 0.6 Low productivity η H 0.5 Prob ϕ e = ϕ H. β 0.96 Discount factor c e 35 Entry cost θ 0.9 transition probability K 10 mean of sell-off value Table 1: Parameter values The Bellman equation of a firm which receives the scrap value φ is Again, this leads to the cutoff rule: ˆV (ϕ, φ; p) = π(ϕ; p) + max χ {0,1} χφ + (1 χ)βe ˆV (ϕ, φ ; p) ϕ] χ(ϕ, φ) = { 1 if φ βe ˆV (ϕ, φ ; p) ϕ] 0 otherwise. Let V (ϕ; p) = ˆV (ϕ, φ; p)dg(φ) Then r(ϕ) = βev (ϕ ; p) ϕ], and Entry profit is given by Free entry requires V e (p ) = 0. x(ϕ; p) = G (βev (ϕ i; p) ϕ i ]) = 1 exp 1 ] K βev (ϕ i; p) ϕ i ]. (24) V e (p) = βη H V (ϕ H ; p) + (1 η H )V (ϕ L ; p)] c e When the measure of entrant is M, the law of motion is µ (ϕ H ) = µ(ϕ H )θx(ϕ H ) + µ(ϕ L )(1 θ)x(ϕ L ) + Mη H µ (ϕ L ) = µ(ϕ L )θx(ϕ L ) + µ(ϕ H )(1 θ)x(ϕ H ) + M(1 η H ) Once we compute the stationary distribution µ, we can compute the aggregate supply as The aggregate demand is given by Q s (M) = µ (ϕ H )ϕ H n(ϕ H ; p ) α + µ (ϕ L )ϕ L n(ϕ L ; p ) α ( ) e Q d = m p d. Even though we have market size here as well, this doesn t change the equilibrium price. It is because the price is determined by the free entry condition, and the market size doesn t change the entry profit. It only changes the measure of entrants. The equilibrium entry is determined by the market clearing condition: Q s (M ) = Q d. 14

15 6.2 Results Table 1 shows the equilibrium quantities. Variable Hopenhyan Oblivious (m = 1) Oblivious (m = 10) Value in high state Value in low state x(ϕ H ) x(ϕ L ) n(ϕ H ) n(ϕ L ) Normalized Equilibrium entry Normalized Number of firms Aggregate output Fraction of firm in high state Table 2: Comparison of Hopenhyan and Oblivious equilibrium. 6.3 Check consistency In Krusell-Smith, we checked if the perceived law of motion is consistent with the actual policy function with simulation. Here we can also do the similar exercise: Compute entry/exit strategy assuming firms believe that the industry state is constant over time. We can use this entry/exit strategy to simulate the industry state and see if it is actually constant over time. Constant industry state means the price is also constant. Only Q/m matters for price in this case. So firms only care about the industry state normalized by market size, µ σc, λ /m. The industry state and price may be less volative in MPE because in MPE firms respond to the industry state. In OE, even if no firm is in market, entrants do not change their entry strategy. If there is no firm in the market, p(0) = e/(0/m + d) = e/d = 10. Figure?? shows the simulated number of firms with m = 1. In this case, the number of firms in each state can be from 0 to 8. As a result, the simulated price (figure 2) is very volatile. So the oblivious equilibrium does a poor job approximating the true equilibrium. On the other hand, as we increase the market size, the normalized number of firms becomes close to the long run average. As a result, the simulated price becomes close to constant. (Figure 3-4 for m = 1000 and figure 5-6 for m = 10000). 15

16 8 # firms in high state, simulated # firms in low state, simulated # firms in high state, Long run average # firms in low state, Long run average number of firms/market share Time Figure 1: Simulated number of firms in each state, m = Simulated price Belief on price 8 7 price Time Figure 2: Simulated price, m = 1. p = 10 when there is no firm in that period 16

17 # firms in high state, simulated # firms in low state, simulated # firms in high state, Long run average # firms in low state, Long run average # firms in high state, simulated # firms in low state, simulated # firms in high state, Long run average # firms in low state, Long run average number of firms number of firms/market share Time (a) Simulated number of firms in each state Time (b) Normalized number of firms in each state Figure 3: m = Simulated price Belief on price price Time Figure 4: Simulated price, m =

18 # firms in high state, simulated # firms in low state, simulated # firms in high state, Long run average # firms in low state, Long run average # firms in high state, simulated # firms in low state, simulated # firms in high state, Long run average # firms in low state, Long run average number of firms number of firms/market share Time (a) Simulated number of firms in each state Time (b) Normalized number of firms in each state Figure 5: m = Simulated price Belief on price price Time Figure 6: Simulated price, m =

19 6.4 Error bound Since OE is easy to compute, we want to use it instead of MPE if the approximation error is small. How close is the OE strategy to the MPE strategy? If σ OE σ MP E, then σ should solve the MPE Bellman equation (??). So, we will use the following statistics Esup W (ϕ, µ σ, σ, λ) W (ϕ, µ σ, σ, λ)] σ where expectation over the invariant distribution of µ. Weintraub et al(2008) constructed the upper bound on the approximation error. Let (µ) sup y π(y, µ) π(y, µ σ, λ). Esup W (ϕ, µ σ, σ, λ) W (ϕ, µ σ, σ, λ)] 2 σ 1 β E (µ)] In OE firms believe that the industry state is constant. If the number of firms is small, the industry state is very volatile. As long as the profit function is not linear in the industry state, then there are some approximation errors. This is upper bound of the error, not the error itself. True error could be lower than this bound Error bound Market size Figure 7: Error bound (%) 19