Ebook Markov Perfect Industry Dynamics with Many Firms
Ericson and Pakes (1995) (hereafter EP) introduced an approach to modeling industry dynamics in which entry, exit, and investment, together with idiosyncratic shocks, result in heterogeneity among firms. The analysis of such models — which we will refer to as EP-type models — relies on computation of Markov perfect equilibria (MPE) using dynamic programming algorithms (e.g., Pakes and McGuire (1994)). A great advantage of the EP framework is that it is easily extended to cover many important dynamic phenomena. A major shortcoming, however, is the computational complexity of solving for MPE. Methods that accelerate these computations have been proposed (Pakes and McGuire (2001) and Doraszelski and Judd (2003)). However, even with such improvements, in practice computational concerns have typically limited the analysis to industries with just two or three firms. Such limitations have made it difficult to construct realistic empirical models, and application of the EP framework to empirical problems has been rare (exceptions include Benkard (2004), Gowrisankaran and Town (1997), Jenkins, Liu, Matzkin, and McFadden (2004), and Ryan (2005)). More generally, model details are often dictated as much by computational concerns as economic ones.
In an EP-type model, at each time, each firm has a state variable that captures its competitive advantage. Though more general state spaces can be considered, we focus on the simple case where the firm state is an integer. The value of this integer can represent, for example, a measure of product quality, the firm’s current productivity level, or its capacity. Each firm’s state evolves over time based on R&D investments and random shocks. The industry state is a vector representing the number of firms with each possible value of the firm state variable. Even if firms are restricted to symmetric strategies, the number of relevant industry states (and thus, the compute time and memory required for computing a MPE) becomes enormous very quickly. For example, most industries contain more than 20 firms, but it would require approximately 22 million gigabytes of computer memory to store the policy function for an industry with just 20 firms and 40 firm states. As a result, it seems unlikely that exact computation of equilibria will ever be possible in many applied problems of interest.
With this motivation, in this paper we instead propose an approximation method, one that dramatically reduces the computational complexity of EP-type models in industries with many firms. The intuition behind our approach is as follows. Consider an EP-type model in which firm shocks are idiosyncratic. In each period, some firms receive positive shocks and some receive negative shocks. Now suppose there are a large number of firms. It is natural to think that changes in individual firms’ states average out at the industry level, such that the industry state does not change much over time. In that case, each firm could make near optimal decisions knowing only its own firm state and the long run average industry state. We call oblivious strategies, strategies for which a firm considers only its own state and the long run average industry state, and we will define a new solution concept, called oblivious equilibrium, in which firms use oblivious strategies. Computing an oblivious equilibrium is simple because dynamic programming algorithms that optimize over oblivious strategies require compute time and memory that scale only with the number of firm states, and not with the number of firms. Indeed, we will show that it is easy to compute oblivious equilibria for industries with thousands of firms and hundreds of firm states.
To formalize the intuition above, we prove an asymptotic result that provides sufficient conditions for oblivious equilibria to closely approximate MPE as the market size grows. It may seem that this would be true provided that the average number of firms in the industry grows to infinity as the market size grows. However, this is not sufficient. If the market is highly concentrated — for example, as is the case with Microsoft in the software industry — then the approximation is unlikely to be accurate. Instead, we show that, alongside some technical requirements, a sufficient condition for oblivious equilibria to well approximate MPE asymptotically is that they generate a firm size distribution that is “light-tailed,” in a sense that we will make precise. For example, if the demand system is given by a logit model and the spot market equilibrium is Nash in prices, then the condition holds if the average firm size is finite for all market sizes.
We provide an algorithm based on dynamic programming that computes oblivious equilibria. The algorithm is computationally light, often terminating within a couple minutes of run time on a common laptop computer even for industries with thousands of firms. It is also easy to implement, requiring, typically, fewer than two hundred lines of Matlab code. This represents a considerable savings over existing algorithms. Another distinguishing feature of the algorithm is that it places no a priori restrictions on the number of firms or the number of firm states. Instead, these are determined endogenously and computed alongside the oblivious equilibrium.
Contents
1 Introduction
2 A Dynamic Model of Imperfect Competition
- 2.1 Model and Notation
2.2 Model Primitives
2.3 Assumptions
2.4 Equilibrium
3 Oblivious Equilibrium
- 3.1 Oblivious Strategies and Entry Rate Functions
3.2 Oblivious Equilibrium
3.3 The Invariant Industry Distribution
4 Asymptotic Results
- 4.1 Assumptions about the Sequence of Profit Functions
4.2 Asymptotic Markov Equilibrium
4.3 Uniform Law of Large Numbers
4.4 A Light-Tail Condition Implies AME
4.5 Example: The Logit Demand System with Price Competition
5 Algorithm and Bounds
- 5.1 Computing Oblivious Equilibria
5.2 Error Bounds
6 Computational Experiments
- 6.1 The Computational Model
6.2 Numerical Results: Behavior of the Bound
6.3 Closeness to Markov Perfect Equilibrium
6.4 Example: The Evolution of Industry Structure as Market Size Increases
7 Conclusions and Future Research
A Proofs and Mathematical Arguments
A.1 Proofs and Mathematical Arguments for Section 2
- A.1.1 The Poisson Entry Model
A.1.2 Bellman’s Equation
A.2 Proofs and Mathematical Arguments for Section 3
A.3 Proofs and Mathematical Arguments for Section 4
- A.3.1 Preliminary Lemmas
A.4 Proof of Theorem4.2
- A.4.1 Proof of Theorem4.4
A.5 Proofs and Mathematical Arguments for Section 5
B Tables and Figures
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