This paper supplies fast, effective, and simple computational methods for important special cases of Ericson and Pakes’ (1995) model of dynamic oligopoly. These cases feature aggregate uncertainty, sunk entry costs, and stochastic firm-specific technological progress; but they exclude investment decisions other than entry and exit. This simplification facilitates a range of equilibrium characterization, existence, and uniqueness results that are not available for the more general framework. Moreover, it enables the development of algorithms that calculate equilibria by finding the fixed points of a finite sequence of low-dimensional contraction mappings. These results can be used to explore some key aspects of Ericson and Pakes’ model with very low computational cost. This is often useful in itself, and can serve as a first stage of a richer analysis with a more complex specification.
Substantial methodological progress in the computation of Markov-perfect equilibria followed Ericson and Pakes’ original presentation of their framework. Nevertheless, Doraszelski and Pakes (2007) note that these methodological developments are only in their infancy and applications remain rare. This paper contributes to this literature by developing relatively rich analytical results and effective computational methods for a comparatively simple model. It shares this approach with Abbring and Campbell’s (2010) analysis of last-in first-out oligopoly dynamics.
They consider a dynamic extension of Bresnahan and Reiss’ (1990) static entry model that can naturally be applied to the empirical analysis of market level entry and exit data (Abbring, Campbell, and Yang, 2010). Timing and expectational assumptions simplify its equilibrium analysis: Otherwise homogeneous firms move sequentially, oldest first; and older firms never exit expecting to leave a younger firm behind. The present paper contributes more directly to the analysis of Ericson and Pakes’ framework and its potential applications, because it allows for idiosyncratic technological progress in a model with simultaneously moving incumbent firms.
Our results leverage one key insight into the structure of payoffs in a symmetric Markov-perfect equilibrium: If any firm chooses to exit with positive probability, then all identically situated firms must have an expected continuation value of zero. This allows us to calculate firms’ expected continuation values at some nodes of the game tree without knowing everything about how the game will proceed thereafter. Our results demonstrate how to use these initial calculations to recover all equilibrium payoffs and actions. For this task, it is very helpful to know beforehand that adding an active firm to an industry weakly reduces all other firms’ continuation values.
We prove that this intuitive property must hold if at most two firms can serve the industry at one time. For the more general oligopoly case, we show that if a Markov-perfect equilibrium with such monotonicity exists, then it is essentially unique. In this case, the algorithm we propose always computes it. If no such equilibrium exists, then our algorithm can be easily adapted to find all equilibria satisfying a desirable property we call “one-shot renegotiation proofness”.
