Markov Perfect Industry Dynamics with Many Firms

Abstract

We propose an approximation method for analyzing Ericson and Pakes (1995)-style dynamic models of imperfect competition. We develop a simple algorithm for computing an ``oblivious equilibrium,'' in which each firm is assumed to make decisions based only on its own state and knowledge of the long run average industry state, but where firms ignore current information about competitors' states. We prove that, as the market becomes large, if the equilibrium distribution of firm states obeys a certain ``light-tail'' condition, then oblivious equilibria closely approximate Markov perfect equilibria. We develop bounds that can be computed to assess the accuracy of the approximation for any given applied problem. Through computational experiments, we find that the method often generates useful approximations for industries with hundreds of firms and in some cases even tens of firms.

Markov Perfect Industry Dynamics With Many Firms

“…t |μ (m) λ (m) )] is small, MPE strategies and entry rates at relevant states should be well approximated by oblivious ones. Weintraub, Benkard, and Van Roy (2007) presented computational results that support this point. 13 Since OE will only have the ability to approximate MPE strategies in states that have a significant probability of occurrence, OE will not be able to approximate equilibria in which behavior depends on off-the-equilibrium-path values, as is typically the case in tacit collusive schemes.…”

“…As stated in the Introduction (see also Weintraub, Benkard, and Van Roy (2007)), OEs are simple to compute because they only involve one dimensional dynamic programs. However, how well do OEs approximate MPEs?…”

“…Since the main result of this paper is a limit theorem, it is natural to wonder whether there are any real world industries in which OE can be shown to approximate MPE well. We address this issue more directly in a companion paper (Weintraub, Benkard, and Van Roy (2007)), where we provide a computational algorithm that solves for OE and approximation bounds that can be computed to provide researchers with a numerical measure of how close OE is to MPE in their particular application. We find in that paper that OE approximate MPE better in some industries than others, depending on industry concentration and the nature of competitive interactions.…”

“…Finally, Section 6 presents conclusions and a discussion of future research directions. All proofs and mathematical arguments are provided in the Appendix and the Supplemental material (Weintraub, Benkard, and Van Roy (2008)).…”

Continuous time finite state space mean field games — A variational approach

“…There is also a growing interest in numerical methods for these problems [ALT], [AD10]. A concept called oblivious equilibrium, in which players are assumed to make decisions based only on its own state and knowledge of the long-run average industry state and stationary equilibrium models were introduced and studied in detail in, respectively, [WBR08] and [AJW11]. These equilibria are easier to compute than are Markov perfect equilibria and provide good approximations for many important models.…”