A Framework for Dynamic Oligopoly in Concentrated Industries

Transcription

1 A Framework for Dynamic Oligopoly in Concentrated Industries Bar Ifrach Gabriel Y. Weintraub August 2016 Abstract In this paper we introduce a new computationally tractable framework for Ericson and Pakes (1995)- style dynamic oligopoly models that overcomes the computational complexity involved in computing Markov perfect equilibrium (MPE). First, we define a new equilibrium concept that we call momentbased Markov equilibrium (MME), in which firms keep track of their own state, the detailed state of dominant firms, and few moments of the distribution of fringe firms states. Second, we provide guidelines to use MME in applied work and illustrate with an application how it can endogenize the market structure in a dynamic industry model even with hundreds of firms. Third, we develop a series of results that provide support for using MME as an approximation. We present numerical experiments showing that MME approximates MPE for important classes of models. Then, we introduce novel unilateral deviation error bounds that can be used to test the accuracy of MME as an approximation in large-scale settings. Overall, our new framework opens the door to study new issues in industry dynamics. 1 Introduction Ericson and Pakes (1995)-style dynamic oligopoly models (hereafter, EP) offer a framework for modeling dynamic industries with heterogeneous firms. The main goal of the research agenda put forward by EP was to conduct empirical research and evaluate the effects of policy and environmental changes on market outcomes in different industries. The importance of evaluating policy outcomes in a dynamic setting and the broad flexibility and adaptability of the EP framework has generated many applications in industrial organization and related fields (see Doraszelski and Pakes (2007) for an excellent survey). Despite the broad interest in dynamic oligopoly models, there remain significant hurdles in applying them to problems of interest. Dynamic oligopoly models are typically analytically intractable, hence numerical methods are necessary to solve for their Markov perfect equilibrium (MPE). With estimation methods, We would like to thank Vivek Farias for numerous fruitful conversations, valuable insights, important technical input, and careful reading of the paper. We have had helpful conversations with Lanier Benkard, Jeff Campbell, Allan Collard-Wexler, Dean Corbae, Pablo D Erasmo, Uli Doraszelski, Przemek Jeziorski, Ramesh Johari, Boyan Jovanovic, Sean Meyn, Ariel Pakes, John Rust, Bernard Salanié, Minjae Song, Ben Van Roy, Daniel Xu, as well as with seminar participants at various conferences and institutions. We thank Yair Shenfeld for exemplary research assistance. We are grateful to the editor Stéphane Bonhomme and three anonymous referees for constructive feedback that greatly improved the paper. Airbnb, Barifrach@gmail.com Graduate School of Business, Stanford University, gweintra@stanford.edu 1

2 such as Bajari et al. (2007), it is no longer necessary to solve for the equilibrium in order to structurally estimate a model. However, in the EP framework solving for MPE is still essential to perform counterfactuals and evaluate environmental and policy changes. The complexity of this computation has severely limited the practical application of EP-style models to industries with just a handful of firms, far less than the real-world industries the analysis is directed at. Notably, such limitations have made it difficult to construct realistic empirical models. Thus motivated, in this paper we propose a new computationally tractable model to study industry dynamics in settings with a few dominant firms and many fringe firms; in applications, dominant firms would typically be the market share leaders. A market structure in which a few firms holding significant market share co-exist with many small firms is prevalent in many industries, such as supermarkets, banking, beer, cereals, and heavy duty trucks, to name a few (see, e.g., concentration ratios from the U.S. Economic Census). These industries are intractable in the standard EP framework due to the large number of fringe firms. Although individual fringe firms have negligible market power, they may have significant cumulative market share and may collectively discipline dominant firms behavior. Our model and methods capture these types of interactions and therefore significantly expand the set of industries that can be analyzed computationally. In an EP-style model, each firm is distinguished by an individual state at every point in time. The industry state is a vector (or distribution ) encoding the number of firms in each possible value of the individual state variable. Assuming its competitors follow a prescribed strategy, a given firm selects, at each point in time, an action (e.g., an investment level) to maximize its expected discounted profits. The selected action will depend in general on the firm s individual state and the industry state. Even if firms were restricted to anonymous and symmetric strategies, the computation entailed in selecting such an action quickly becomes infeasible as the number of firms and individual states grows. This renders commonly used dynamic programming algorithms to compute MPE infeasible for many problems of practical interest. The first main contribution of this work is the introduction of a new framework and equilibrium concept that overcomes the computational complexity involved in computing MPE for EP-style models. 1 In an industry with many competitors it may be reasonable to assume that firms are more sensitive to changes in the dominant firms states. Further, it may be unrealistic to expect that firms have resources to keep track of the evolution of all rivals. Thus motivated, we postulate a plausible model of firms behavior: firms closely monitor dominant firms, but do not monitor fringe firms as closely. Specifically, we assume that firms strategies depend on: (i) their own individual state; (ii) the detailed state of dominant firms; and (iii) a small number of aggregate statistics (such as a few moments) of the distribution of fringe firms. We refer to the fringe firms distribution or state interchangeably. We call these strategies moment-based strategies, where we use the term moments generically, as firms could keep track of other statistics of the distribution of fringe firms, such as un-normalized moments or quantiles. Based on these strategies, we introduce an equilibrium concept that we call moment-based Markov equilibrium (MME). In MME, firms beliefs about the evolution of the industry satisfy a consistency condition, and firms employ an optimal moment-based strategy given these beliefs. One challenge in defining this concept, however, is that fringe moments may not be sufficient statistics to predict the future evolution of 1 We focus on the important class of capital accumulation games with out investment spillovers that EP originally introduced. 2

3 the industry. For example, suppose firms keep track of the first moment of the fringe state. For a given value of the first moment, there could be many different fringe distributions consistent with that value, from which the future evolution of the industry could vary greatly. Technically, the issue that arises is that the stochastic process of moments may not be a Markov process even if the underlying dynamics are. Therefore, to define MME we introduce a Markov perceived transition kernel to approximate the dynamics of the (non- Markov) moment-based state; we provide natural and concrete examples for such a kernel. Notably, this construction allows for fringe firms to become dominant as they grow large and vice-versa, endogenizing the market structure in equilibrium. We introduce MME as an approximation in EP-style games of symmetric information. However, because MME limits firms information sets (by allowing dependence on the fringe state only through few moments), it could also be cast as an equilibrium concept in a game of asymmetric information even if the underlying economic model is not. Hence, while their model and motivation is fundamentally different from ours, MME is related to the notion of experienced-based equilibria of Fershtman and Pakes (2012) that weakens the restrictions imposed by perfect Bayesian equilibria for dynamic games of asymmetric information. We discuss this relation in more detail in Sections 5.3 and 8.3. After defining MME, our second main contribution is a series of guidelines to use MME in applied work. We introduce an algorithm that efficiently computes MME even in industries with hundreds of firms, provided that firms keep track of a few moments of the fringe state and there are a few dominant firms. In addition, we discuss important implementation issues for the use of MME in practice, such as how to select moments and the number of dominant firms allowed. To further illustrate the applicability of our approach, we show how it can be used to endogenize the market structure in a fully dynamic model with hundreds of firms for which MPE would be impossible to compute. In particular, we perform numerical experiments motivated by the long trend towards concentration in the beer industry in the US during the years Over the course of those years, the number of active firms dropped dramatically, and three industry leaders emerged. One common explanation of this trend was the emergence of national TV advertising as an endogenous sunk cost (Sutton, 1991). We build and calibrate a dynamic advertising model of the beer industry and use MME to determine how a single parameter related to the returns to advertising expenditures critically affects the resulting market structure and the level of concentration in the industry. We also show how MME generates rich strategic interactions between dominant and fringe firms, in which, for example, dominant firms make investment decisions to deter the entry and growth of fringe firms. 2 While MME may be appealing intuitively, moment-based strategies may not be close to a best response, and therefore, MME may not be close to a subgame perfect equilibrium in general. The reason is that because moments may not be sufficient statistics to predict the future evolution of the industry, they may not induce a sufficient partition of histories and may not summarize all payoff-relevant history in the sense of Maskin and Tirole (2001). Indeed, observing the full distribution of fringe firms may provide valuable information for decision making. Our third main contribution is a series of results that addresses this difficulty, 2 As further evidence of the usefulness of our approach, Corbae and D Erasmo (2014) has already used our framework to study the impact of capital requirements in market structure in a calibrated model of banking industry dynamics with dominant and fringe banks. 3

4 providing support for using MME as an approximation. First, we explore the performance of MME as a low-dimensional approximation to MPE. We carry out a large set of numerical experiments comparing MME and MPE outcomes in industries with a few firms and a few individual states per firm, for which MPE can be computed. We consider two important dynamic oligopoly models: a quality ladder model and a capacity competition model. We show that MME is able to provide accurate approximations to MPE across a wide range of parameter values even when firms keep track of a single and natural fringe moment. For example, in the capacity competition model this moment is the total installed capacity among all fringe firms. Alternatively, MME can be thought of as an appealing heuristic and behavioral model on its own. To formalize this interpretation, we define the value of the full information deviation that measures the extent of sub-optimality of MME strategies in terms of a unilateral deviation to a strategy that keeps track of the full industry state, similar to the notion of ɛ-equilibrium. If this value is small, MME may be an appealing behavioral model, as unilaterally deviating to more complex strategies does not significantly increase profits, and doing so may involve costs associated with gathering and processing more information. Measuring the value of the full information deviation is intractable in large-scale applications with many firms; in these instances, computing a best response is almost as computationally challenging as solving for a MPE. In these cases we could not compare MME with MPE either. To address this limitation, using ideas from robust dynamic programming, we propose a novel computationally tractable upper bound to the value of the full information deviation. This bound is useful because it allows one to evaluate whether the state aggregation is appropriate or whether a finer state aggregation is necessary, for example, by adding more moments. We provide numerical experiments showing how the error bound works in practice. Our moment-based strategies are similar to and follow the same spirit of the seminal paper by Krusell and Smith (1998), which replaces the distribution of wealth across agents in the economy with its moments, when computing stationary stochastic equilibrium in a stochastic growth model. While our main focus has been on dynamic oligopoly models, our methods can also be used to find better approximations in stochastic growth models, as well as in macroeconomic dynamic industry models with an infinite number of heterogeneous firms and an aggregate shock (see, e.g., Khan and Thomas (2008) and Clementi and Palazzo (2016)). In particular, in Section 8.3 we discuss how our error bounds can be applied to this class of models as well as to Fershtman and Pakes (2012)-style dynamic games of asymmetric information. In summary, our approach offers a computationally tractable model for industries with a dominant/fringe market structure, opening the door to studying novel issues in industry dynamics. As such, our framework greatly increases the applicability of dynamic oligopoly models. The rest of the paper is organized as follows. We discuss connections to other related literature in Section 2. Section 3 describes our dynamic oligopoly model. Section 4 introduces important elements to define MME. Then, we define MME and an algorithm to compute it in Section 5. Section 6 provides a numerical comparison between MME and MPE and discusses important implementation issues. Section 7 describes the application to the beer industry and Section 8 introduces our approximation error bounds. Finally, Section 9 provides conclusions. 4

5 2 Related Literature Previous work has already recognized the challenges involved in applying dynamic oligopoly models in practice. For example, researchers have used different approaches to overcome the burden involved in computing equilibria in empirical applications. Some papers empirically study industries that contain only a few firms in which exact MPE computation is feasible if the number of individual states assumed is moderate (e.g., Benkard (2004), Ryan (2012), Collard-Wexler (2013), and Collard-Wexler (2011)). Other researchers structurally estimate models in industries with many firms using approaches that do not require MPE computation, and do not perform counterfactuals that require computing equilibria (e.g., Benkard et al. (2010)). Our work provides a method to perform counterfactuals in concentrated industries with many firms. Another set of papers perform counterfactuals computing MPE but in reduced-size models compared to the actual industry. These models include a few dominant firms and ignore the rest of the (fringe) firms (e.g., see Ryan (2012) and Gallant et al. (2016)). Other papers coarsely discretize the individual states (e.g., see Collard-Wexler (2014) and Corbae and D Erasmo (2013) that assume homogenous firms). Finally, some papers allow for richer heterogeneity but assume a simplified model of dynamics in which a certain process of moments that summarize the industry state information is Markov (e.g., see Kalouptsidi (2014), Jia and Pathak (2015), Santos (2010), and Tomlin (2010); Lee (2013) use a similar approach in a dynamic model of demand with forward-looking consumers). Our methods can help researchers determine the validity of these simplifications. Our work is related to recent methodological work that tries to overcome the complexity involved in computing MPE. For example, Pakes and McGuire (2001) proposes a stochastic approximation algorithm that solves the model only on a recurrent class of states, and reduces the computation time at each state through simulation. Doraszelski and Judd (2011) suggests casting the game in continuous time, which greatly reduces the computational cost at each state by reducing the number of future states reachable from each current state (see also Jeziorski (2014) for an application of this technique). Other papers propose different computational approaches to approximate MPE. Farias et al. (2012) develops a method based on approximate dynamic programming with value function approximation (see also Sweeting (2013) for an empirical application using value function approximation). Santos (2012) introduces a state aggregation technique based on quantiles of the industry state distribution. A stream of empirical literature related to our work uses simplified notions of equilibrium for estimation and counterfactuals. In particular, Xu (2008), Iacovone et al. (2015), and Qi (2013), among others use the notion of oblivious equilibrium (OE) introduced by Weintraub et al. (2008), in which firms assume the average industry state holds at any time. OE can be shown to approximate MPE in industries with many firms by a law of large numbers, provided that the industry is not too concentrated. Our approach extends this type of approach to industries that are more concentrated. Our work is particularly related to Benkard et al. (2015), which extends the notion of oblivious equilibrium to include dominant firms. In that paper, there is a fixed and pre-determined set of dominant firms, which all firms in the industry monitor. Firms assume that at every point in time the fringe state is equal to the expected state conditional on the state of dominant firms. Our paper builds on and generalizes Benkard 5

6 et al. (2015) s idea of considering a dominant/fringe market structure. 3 In particular, our approach is more general in at least two important dimensions: (i) in that firms not only keep track of the dominant firms states but also of statistics that summarize the fringe firms states; and (ii) that in our model the set of dominant firms arises endogenously in equilibrium. That is, we allow for firms to transition from the fringe to the dominant tier and vice-versa, and therefore, we endogenize the market structure and explicitly model how firms grow to become dominant. We provide more details about this connection in Section Dynamic Oligopoly Model and Solution Concept In this section we formulate a model of industry dynamics in the spirit of Ericson and Pakes (1995). Similar models have been applied to numerous settings in industrial organization such as advertising, auctions, R&D, collusion, consumer learning, learning-by-doing, and network effects (see Doraszelski and Pakes (2007) for a survey). As in the original EP paper, our model is a capital accumulation game with out investment spillovers, for which (i) decisions that determine static profits (e.g., prices) do not affect a firm s state; and (ii) a firm s individual state is only affected by its own dynamic action. 3.1 Model We introduce the main elements of our dynamic oligopoly model. Time Horizon. The industry evolves over discrete time periods and along an infinite horizon. We index time periods with nonnegative integers t N (N = {0, 1, 2,...}). All random variables are defined on a probability space (Ω, F, P) equipped with a filtration {F t : t 0}. We adopt a convention of indexing by t variables that are F t -measurable. Firms. Each firm that enters the industry is assigned a unique positive integer-valued index. The set of indices of incumbent firms at time t is denoted by S t. State Space. Firm heterogeneity is reflected through firm states that represent the quality level, productivity, capacity, the size of its consumer network, or any other aspect of the firm that affects its profits. At time t the individual state of firm i is denoted by x it X, where X is a finite subset of N q, q 1. We define the industry state s t to be a vector over individual states that specifies, for each firm state x X, the number of incumbent firms at x in period t. Note that because we will focus on symmetric and anonymous equilibrium strategies in the sense of Doraszelski and Pakes (2007), we can restrict attention to industry states for which the identities of firms do not matter. The integer number N represents the maximum number of incumbent firms that the industry can accommodate at every point in time and we let n t be the number of incumbent firms at time period t. Exit process. In each period, each incumbent firm i observes a nonnegative real-valued sell-off value φ it that is private information to the firm. If the sell-off value exceeds the value of continuing in the industry, then the firm may choose to exit, in which case it earns the sell-off value and then ceases operations permanently. 3 In their analysis of a stylized model of dynamic mergers, Gowrisankaran and Holmes (2004) also provide an earlier example of the dominant/fringe dichotomy. 6

7 We assume the random variables {φ it t 0, i 1} are i.i.d. and have a well-defined density function with finite moments. Entry process. We consider an entry process similar to the one in Doraszelski and Pakes (2007). At time period t, there are N n t potential entrants, ensuring that the maximum number of incumbent firms that the industry can accommodate is N (we assume n 0 < N). Each potential entrant is assigned a unique index. In each time period each potential entrant i observes a positive real-valued entry cost κ it that is private information to the firm. We assume the random variables {κ it t 0 i 1} are i.i.d., independent of all previously defined random quantities, and have a well-defined density function with finite moments. If the entry cost is below the expected value of entering the industry then the firm will choose to enter. Potential entrants make entry decisions simultaneously. Entrants appear in the following period at state x e X and earn profits thereafter. 4 As is common in this literature and to simplify the analysis, we assume potential entrants are short-lived and do not consider the option value of delaying entry. Potential entrants that do not enter the industry disappear and a new generation of potential entrants is created in the next period. Transition dynamics. If an incumbent firm decides to remain in the industry, it can take an action to improve its individual state. Let I R k + (k 1) be a convex and compact action space; for concreteness, we refer to this action as an investment. Given a firm s investment ι I and state at time t, the firm s transition to a state at time t + 1 is described by the following Markov kernel Q: Q[x x, ι] = P[x i,t+1 = x xit = x, ι it = ι]. Uncertainty in state transitions may arise, for example, due to the risk associated with a research and development endeavor or a marketing campaign. The cost of investment is given by a nonnegative function c(ι it, x it ) that depends on the firm s individual state x it and investment level ι it. Both the kernel and the investment cost are assumed to be continuous functions of investment. We assume that transitions are independent across firms conditional on the industry state and investment levels. We also assume these transitions are independent of all previously defined random quantities. Aggregate shock. There is an aggregate profitability shock z t that is common to all firms and that may represent a common demand shock, a common shock to input prices, or a common technology shock. We assume that {z t Z : t 0} is a finite, ergodic Markov chain, and independent of all previously defined random quantities. Single-Period Profit Function. Each incumbent firm earns profits on a spot market. We denote by π(x it, s t, z t ) as the single-period expected profits garnered by firm i at time period t, which depends on its individual state x it, the industry state s t, and the value of the aggregate shock z t. Timing of Events. In each period, events occur in the following order: (1) Each incumbent firm observes its sell-off value and then makes exit and investment decisions; (2) Each potential entrant observes its entry cost and makes entry decisions; (3) Incumbent firms compete in the spot market and receive profits; (4) Exiting firms exit and receive their sell-off values; (5) Investment shock outcomes are determined, new entrants enter, and the industry takes on a new state s t+1. Firms objective. Firms aim to maximize expected discounted profits. The interest rate is assumed to 4 It is straightforward to generalize the model by assuming that entrants can also invest to improve their initial state. 7

8 be positive and constant over time, resulting in a constant discount factor of β (0, 1) per time period. Finally, we assume that for all competitors decisions and all continuation values, a firm s one time-step ahead optimization problem to determine its optimal investment has a unique solution Markov Perfect Equilibrium The most commonly used equilibrium concept in such dynamic oligopoly models is that of symmetric pure strategy Markov perfect equilibrium (MPE) in the sense of Maskin and Tirole (1988). To simplify notation we re-define the industry state to incorporate the aggregate shock, that is s t = ( s t, z t ). Hence, there is a function ι such that at each time t, each incumbent firm i invests an amount ι it = ι(x it, s t ), as a function of its own state x it and the industry state s t. Similarly, each firm follows an exit strategy that takes the form of a cutoff rule: there is a real-valued function ρ such that an incumbent firm i exits at time t if and only if φ it > ρ(x it, s t ). Weintraub et al. (2008) show that there always exists an optimal exit strategy { of this form even among very general classes of exit strategies. We define the state space S = ( s, z) N X Z x X }, s(x) N that is, S is the set of industry states including the aggregate shock. Let M denote the set of exit/investment strategies such that an element µ M is a pair of functions µ = (ι, ρ), where ι : X S I is an investment strategy and ρ : X S R is an exit strategy. 6 Similarly, each potential entrant follows an entry strategy that takes the form of a cutoff rule: there is a real-valued function λ such that a potential entrant i enters at time t if and only if κ it < λ(s t ). It is simple to show that there always exists an optimal entry strategy of this form even among very general classes of entry strategies (see Doraszelski and Satterthwaite (2010)). We denote the set of entry strategies by Λ, where an element of Λ is a function λ : S R. We define the value function V (x, s µ, µ, λ) to be the expected discounted value of profits for a firm at state x when the industry state is s, given that its competitors each follow a common strategy µ M, the entry strategy is λ Λ, and the firm itself follows strategy µ M. Formally, V (x, s µ, µ, λ) = E µ,µ,λ ] β k t (π(x ik, s k ) c(ι ik, x ik )) + β τi t xit φ i,τi = x, s t = s, [ τi k=t where i is taken to be the index of a firm at individual state x at time t, τ i is a random variable representing the time at which firm i exits the industry, and the subscripts of the expectation indicate the strategy followed by firm i, the strategy followed by its competitors, and the entry strategy. In an abuse of notation, we will use the shorthand, V (x, s µ, λ) V (x, s µ, µ, λ), to refer to the expected discounted value of profits when firm i follows the same strategy µ as its competitors. An equilibrium in our model comprises an investment/exit strategy µ = (ι, ρ) M and an entry strategy λ Λ that satisfy the following conditions: 5 This assumption is similar to the unique investment choice admissibility assumption in Doraszelski and Satterthwaite (2010) that is used to guarantee the existence of an equilibrium to the model in pure strategies. It is satisfied by many of the commonly used specifications in the literature. 6 We restrict attention to states (x, s) X S, such that s(x) > 0 (recall that s(x) is the the x-th component of s). 8

9 1. Incumbent firms strategies represent a MPE: sup V (x, s µ, µ, λ) = V (x, s µ, λ), (x, s) X S. µ M 2. For all states, the cut-off entry value is equal to the expected discounted value of profits of entering the industry: 7 λ(s) = β E µ,λ [V (x e, s t+1 µ, λ) s t = s], s S. Doraszelski and Satterthwaite (2010) establish the existence of an equilibrium in pure strategies for a similar model. With respect to uniqueness, in general we presume that our model may have multiple equilibria. Doraszelski and Satterthwaite (2010) and Iskhakov et al. (2015) also provide examples of multiple equilibria in related models. A limitation of MPE is that the set of relevant industry states grows quickly with the number of firms in the industry and individual states, making its computation intractable when there are more than a few firms. This motivates our alternative approach of defining a new equilibrium concept that overcomes the computational complexity involved in solving for MPE. 4 Main Elements of Moment-Based Markov Equilibrium In this section we introduce important objects that will serve as foundations to define moment-based Markov equilibrium (MME) in Section Dominant and Fringe Firms We focus on industries that exhibit the following market structure: there are a few dominant firms and many fringe firms. Let D t S t and F t S t be the set of incumbent dominant and fringe firms at time period t, respectively. The sets D t and F t are common knowledge among firms at every period of time. These sets can change over time and our model incorporates a mechanism that endogenizes the process through which fringe firms become dominant and vice-versa. We let X f X and X d X be the set of feasible individual states for fringe and dominant firms, respectively. We define the state or distribution of fringe firms f t to be a vector over individual states that specifies, { for each fringe firm state x X f, the number of incumbent fringe firms at x in period t. We define S f = f N X f } x X f f(x) N to be the set of all possible states of fringe firms. Let d t be the state of dominant firms, specifying the individual state of each dominant firm at time period t. The set of all possible dominant firms states is defined by S d = Xd D, where D is the maximum number of dominant firms the industry can accommodate. Hence, the dominant firms state is represented by a list of individual states. 8 Note that by defining an inactive state, the number of active dominant firms could be 7 Hence, potential entrants enter if the expected discounted profits of doing so are positive. Throughout the paper it is implicit that the industry state at time period t + 1, s t+1, includes the entering firm in state x e whenever we write (x e, s t+1). 8 Because we focus on equilibrium strategies that are anonymous, we can restrict attention to a set S d for which the identity of dominant firms does not matter. For example, if the state is one dimensional, we can define S d = {d Xd D d(1) d(2),..., d(d)}. We choose this state representation for dominant firms as opposed to a distribution over individual states, because we will 9

10 anywhere between 0 and D in a given time period. With some abuse of notation, we define the state space S = S f S d Z, where a state s = (f, d, z) S is given by a distribution of fringe firms, a state for dominant firms, and a value for the aggregate shock. The division between dominant and fringe firms will depend on the specific application at hand. Typically, however, dominant firms will be market share leaders or, more generally, firms that most affect competitors profits. For concreteness, suppose that the state of the firm represents its ability to compete in the market and firms in higher individual states have larger market shares, as in the two examples we introduce below in Section 6.1. It may then be natural to define dominant firms as those that have a sufficiently large individual state. In our numerical experiments, we separate dominant firms from fringe firms by an exogenous threshold state x, such that i D t if and only if x it x. We note that in this case, X f X d =, which we assume throughout the rest of the paper to simplify notation. 9 More generally, one could consider a relative threshold that depends on some statistics of the current fringe state; for example, a fringe firm becomes dominant if it is certain number of standard deviations above the current average size of fringe firms. The threshold could also depend on the current state of dominant firms. Note that under all these specifications the transitions among the fringe and dominant tiers are naturally embedded in the firms transitions. Our approach could also accommodate alternative specifications on how firms become dominant depending on the context. In the applications we have in mind, dominant firms are a few and have significant market power. In contrast, fringe firms are many and individually hold little market power, although their aggregate market share may be significant. This market structure suggests that firms decisions should be more sensitive to the state of dominant firms than to the state of fringe firms. Moreover, given that the fringe firms state is a highly dimensional object, gathering information on the state of each individual small firm is likely to be more expensive than doing so for larger firms that not only are few, but also usually more visible and often publicly traded. Consequently, as the number of fringe firms increases it seems implausible that firms keep track of the individual state of each one. Instead, we postulate that firms only keep track of the state of dominant firms and of a few summary statistics of the fringe state. 4.2 Moment-Based Strategies Our approach requires that firms compute best responses in strategies that depend only on a few summary statistics of the fringe state. A set of such summary statistics is a multi-variate function θ : S f Θ, where Θ is a finite subset of R n. For example, when the fringe firm state is one-dimensional, θ(f) = x X f x α f(x) is the α th un-normalized moment with respect to the distribution f. If α = 1, the statistic θ(f t ) corresponds to the first un-normalized moment, which is equivalent to the sum of the individual states of all incumbent fringe firms, i F t x it. Note that with a finite number of fringe firms (N) and a finite number of individual states per firm (X f ), this and all other fringe statistics that we use in the paper take a finite number of values. typically consider applications with a small number of dominant firms. 9 Note that more generally we can always make the assumption that X f X d = by appending one dimension to the individual state in order to indicate whether the firm is fringe or dominant. This construction is useful as it allows, for example, for fringe and dominant firms to have different model primitives and strategies. 10

11 For brevity and concreteness, we call such summary statistics fringe moments, with the understanding that they could include normalized or un-normalized moments, but also other functions of the fringe distribution such as quantiles or the number of fringe firms that are about to become dominant. Accordingly, we define a set of moments of the distribution of fringe firms: θ t = θ(f t ). (1) We introduce firm strategies that depend on their own individual state x it, the state of dominant firms d t, the aggregate shock z t, and fringe moments θ t as defined in (1). We call such strategies moment-based strategies. We also define S θ as the set of admissible moments defined by (1); that is, S θ = {θ f S f s.t. θ = θ(f)}. In light of this, we define the moment-based industry state by ŝ = (θ, d, z), and the moment-based state space by Ŝ = S θ S d Z. A moment-based investment strategy is a function ι such that at each time t, each incumbent firm i S t invests an amount ι it = ι(x it, ŝ t ), where ŝ t is the moment-based industry state at time t. Similarly, each firm follows an exit strategy that takes the form of a cutoff rule: there is a real-valued function ρ such that an incumbent firm i S t exits at time t if and only if φ it ρ(x it, ŝ t ). Let ˆM denote the set of moment-based exit/investment strategies such that an element µ ˆM is a pair of functions µ = (ι, ρ), where ι : X Ŝ I is an investment strategy and ρ : X Ŝ R is an exit strategy.10 Each potential entrant follows an entry strategy that takes the form of a cutoff rule: there is a real-valued function λ such that a potential entrant i enters at time t if and only if κ it λ(ŝ t ). We denote the set of entry functions by ˆΛ, where an element of ˆΛ is a function λ : Ŝ R. We assume that all new entrants become part of the fringe. Note that moment-based strategies and the moment-based state space are defined with respect to a specific function of moments (1). Relative to MPE strategies, moment-based strategies do not keep track of the full fringe state; they only keep track of a few fringe moments. 4.3 Transition Kernel An important construct to define MME is a perceived transition kernel as introduced in this subsection. With Markov strategies (µ, λ) the underlying state, {s t = (f t, d t, z t ) : t 0}, is a Markov process with a transition kernel denoted by P µ,λ. In addition, we denote by P µ,µ,λ the transition kernel of (x it, s t ) when firm i uses strategy µ, and its competitors use strategy (µ, λ). Note that given strategies, both these kernels can be derived from the primitives of the model, namely, the distributions of φ and κ, the kernel of the aggregate shock, the kernel Q, and the sets X f, X d. We emphasize that the underlying industry state s t should be distinguished from the moment-based industry state ŝ t = (θ t, d t, z t ). Defining our notion of equilibrium in moment-based strategies will require the construction of what can be viewed as a Markov approximation to the dynamics of the moment-based state process {(x it, ŝ t ) : t 0}, where i is a generic firm. Note that this process is, in general, not Markov even if the dynamics of the underlying state {(x it, s t ) : t 0} are. To see this, consider for example that the individual state represents a firm s size and that firms keep track of the first un-normalized moment of the fringe state, 10 We restrict attention to states (x, ŝ), such that if the firm in state x is a dominant firm, then its individual state needs to coincide with one of the components of d. 11

12 θ t = θ(f t ) = x X f xf t (x). Suppose the current value of that moment is θ t = 10; this value is consistent with one fringe firm in individual state 10, but also with 10 fringe firms in individual state 1. These two different states do not necessarily yield the same probabilistic distribution for the first moment in the next period. Therefore, θ t may not be a sufficient statistic to predict the future evolution of the industry, because there are many fringe distributions that are consistent with the same value of θ t. In the process of aggregating information via moments the resulting process is no longer Markovian. Assuming that firm i follows the moment-based strategy µ, and that all other firms use moment-based strategies (µ, λ), we will describe a kernel ˆP µ,µ,λ[ ] with the hope that the Markov process described by this kernel is a good approximation to the (non-markov) process {(x it, ŝ t ) : t 0}. The Markov process described by ˆP µ,µ,λ are firm i s perception of the evolution of its own state in tandem with the momentbased industry state. We will describe this joint kernel by decomposing it into few sub-kernels. In particular, in Section we introduce the kernel ˆP µ,λ [θ ŝ] that describes the evolution of a hypothetical Markov process over moments, that is, the kernel that specifies perceived transition probabilities from moment-based states to next period moments. Then, in Section we introduce transition probabilities from fringe into dominance. Finally, in Section we collect them together to define the joint perceived transition kernel Examples of Perceived Transition Kernels Over Moments There are many possible specifications for the perceived transition kernel over moments ˆP µ,λ [θ ŝ]; however, some yield better approximations. We provide some useful examples next. To construct this kernel we let the industry evolve for a long time horizon T under strategies (µ, λ) and we record the moment-based states visited, {ŝ t = (θ t, d t, z t ) : t = 0,..., T }. Example 4.1 (Observed transitions). One possible definition for ˆP µ,λ [θ ŝ] is the kernel that coincides with the long-run average observed transitions from the moment-based state in the current time period to the moment in the next time period under strategies (µ, λ). More specifically, for each moment-based state ŝ that is visited, we observe a transition to a new moment θ. We count the observed frequency of these transitions and set the kernel ˆP µ,λ [θ ŝ] to be the observed distribution corresponding to these frequencies. A similar construction is used by Fershtman and Pakes (2012) in a setting with asymmetric information. We now formalize the definition of this kernel. The evolution of the underlying industry state is described by the kernel P µ,λ. We let R µ,λ be the recurrent class of moment-based states the industry will eventually reach. For all θ S θ and ŝ R µ,λ, we define: ˆP µ,λ [θ ŝ] = lim sup T T t=1 1{ŝ t = ŝ, θ t+1 = θ } T t=1 1{ŝ, (2) t = ŝ} and ˆP µ,λ [θ ŝ] is arbitrarily defined for states ŝ / R µ,λ. In other words, perceived transitions for fringe moments coincide with their observed transitions in the recurrent class of states, and are arbitrary outside this class. To define our equilibrium concept firms need to have perceived transitions in the entire state space. In Section 6.3 we provide numerical experiments showing how different specifications of the transitions outside the recurrent class affect equilibrium outcomes. 12

13 To make the example more concrete, suppose that θ t = θ(f t ) = x X f xf t (x), the first un-normalized moment of the fringe state. To simplify the exposition, we ignore dominant firms and the aggregate shock. We let the industry evolve for a long time under given strategies (µ, λ). Suppose that in the course of this simulation, we observe that one half of the time periods in which θ t = 10, the moment the next time period is given by θ t+1 = 10, one fourth is given by θ t+1 = 12, and the other fourth is given by θ t+1 = 8. Then, we set ˆP µ,λ [θ = 10 θ = 10] = 1/2, ˆPµ,λ [θ = 12 θ = 10] = 1/4, ˆPµ,λ [θ = 8 θ = 10] = 1/4. We would set ˆP µ,λ [θ θ] similarly for all moment states in the recurrent class. We note that for all model instances that we consider in our numerical experiments, the industry state process admits a single recurrent class for given strategies. In this case, one can show that the right-hand side of equation (2) converges almost surely (see equation (14) in the appendix). For the capital accumulation games without investment spillovers that we study there are simple sufficient conditions that ensure a single recurrent class. 11 Alternatively, in other settings that yield more than one recurrent class, to construct the perceived transition kernel we would need to assume that all firms agree on which recurrent class the industry will eventually transition into. 12 Another possible construction of the perceived transition kernel has been successfully used in stochastic growth models in macroeconomics (Krusell and Smith, 1998) and subsequent literature. Example 4.2 (Parametric linear transitions). This specification for ˆP µ,λ [θ ŝ] assumes a parameterized and deterministic evolution for moments given strategies (µ, λ). That is, starting from industry state ŝ t = (θ t, d t, z t ), the next moment value is assumed to be θ t+1 = G(θ t ; ξ(d t, z t )), where ξ(d t, z t ) are parameters. For example, this could represent a linear relationship with one moment, θ t+1 = ξ 0 (d t, z t ) + ξ 1 (d t, z t )θ t. In this case the goal would be to choose functions ξ 0 and ξ 1 that approximate the actual transitions best, for instance by employing linear regressions with the data of the simulated trajectory {ŝ t : t = 0,..., T }. Then, for all θ S θ and ŝ R µ,λ, we define: ˆP µ,λ [θ 1, if θ = G(θ; ξ(d, z)), ŝ] = 0, if θ G(θ; ξ(d, z)). In addition, ˆP µ,λ [θ ŝ] is arbitrarily defined for states ŝ / R µ,λ. Relative to the kernel in Example 4.1, this perceived transition kernel has the disadvantage of imposing strong parametric restrictions and assuming deterministic transitions. On the other hand, these same restrictions significantly reduce the computational burden of solving for the equilibrium. Both of these kernels are meant to capture the long-run evolution of the industry. We could also introduce a perceived kernel meant to capture the short-term dynamics of the industry starting from a given initial state, 11 For example, a sufficient condition is that the sell-off value has support on R Fershtman and Pakes (2012) face this issue in the models they consider and assume that the recurrent class firms agree on is the one selected by a reinforcement learning real-time dynamic programming algorithm that computes their equilibrium concept. (3) 13

14 for example, after a policy or an environmental change. For this, we could consider the average observed transitions from the current moment to the next, over many finite (and short) trajectories that start from the same state; this state describes the initial condition of the industry (see Appendix A for more details) Fringe to Dominance Transitions In general, moments in θ( ) may not contain sufficiently detailed information to precisely calculate the transition probabilities of a competitor into dominance. For example, consider the setting used in our numerical experiments in which firms are dominant if and only if they are above a pre-determined state x. Suppose θ(f) is the first un-normalized moment of the fringe state. Then, the probability that a fringe competitor becomes dominant depends on the fringe state beyond that moment. Therefore, in this case we need to specify firms perceived probabilities that a fringe competitor will transition into dominance given the moment-based industry state. We provide one such specification next. 13 To simplify the dynamics we impose the constraint that at most one firm becomes dominant in a time period; this constraint was typically never binding in our numerical experiments. Let the event (f d) correspond to a fringe competitor becoming a dominant firm in the next period. With a similar motivation to Example 4.1, we assume that for a given moment-based state, firms perceived probability of this transitioning event occurring is equal to its long-run average observed frequency. Therefore, with some abuse of notation, we define for all ŝ R µ,λ : ˆP µ,λ [(f d) ŝ] = lim sup T T t=1 1{ŝ t = ŝ, (f d) t } T t=1 1{ŝ t = ŝ}, (4) where (f d) t corresponds to the event of a fringe competitor transitioning into dominance in time period t. In addition, ˆP µ,λ [(f d) ŝ] is arbitrarily defined for states ŝ / R µ,λ. Again, if the industry state process admits a single recurrent class, the right-hand side of the equation above converges almost surely Joint Perceived Transition Kernel Having defined the kernel for fringe moments, ˆP µ,λ [θ ŝ], and for transitions into dominance, ˆP µ,λ [(f d) ŝ], we now define the perceived kernel ˆP µ,µ,λ[ ] over states (x it, ŝ t ), given strategies (µ, µ, λ), according to: P µ,µ,λ[x, d, z x, ŝ] ˆP µ,λ [θ ŝ] ˆP µ,λ [(f d) ŝ], if ŝ includes a new dominant competitor ˆP µ,µ,λ[x, ŝ x, ŝ] = associated to the event (f d) P µ,µ,λ[x, d, z x, ŝ] ˆP ( µ,λ [θ ŝ] 1 ˆP ) µ,λ [(f d) ŝ], if not. (5) We note that if the number of dominant firms in ŝ is equal to the maximum allowed, D, then the transition probabilities associated to the event (f d) are omitted. With some abuse of notation P µ,µ,λ[x, d, z x, ŝ] above denotes the marginal distribution of the next state of firm i at state x (including its own chances of 13 In Section 6.3 we also study how MME performs when including an additional moment that encodes the probability a fringe firm becomes dominant the next time period. 14

15 becoming dominant in the next time period), the next state of current dominant firms, and the next value for the aggregate shock, conditional on the current moment-based state, according to the kernel of the underlying state P µ,µ,λ. The definition above makes the following facts about this perceived process transparent: 1. Were firm i a fringe firm, the above definition asserts that this fringe firm ignores its own impact on the evolution of industry moments. This is evident in that x is distributed independently of θ conditional on x and ŝ. This seems reasonable in applications with many fringe firms, for which changes in a fringe firm s individual state have little impact on the value of the moments. However, in the smallscale experiments we run below to compare MME with MPE, we also tried a specification in which a fringe firm that becomes dominant removed itself from the moment. With a small number of fringe firms this had an important effect in improving the approximation to MPE, specifically in the capacity competition model described below Given information about the current moment-based state, the firm correctly assesses the distribution of its next state, the next state of current dominant firms, and the next aggregate shock. Note that because firms use moment-based strategies, the moment-based state (x, ŝ) is sufficient to determine the transition probabilities of (x, d, z) according to the transition kernel of the underlying state P µ,µ,λ. 3. However, it should be clear that the Markov process given by the above definition remains an approximation since it posits that the evolution of moments θ and dominance transitions (f d) are Markov with respect to ŝ as described by ˆP µ,λ [θ ŝ] and ˆP µ,λ [(f d) ŝ], respectively. In fact, the distribution of moments and the transition into dominance at the next time period potentially depend on the full distribution of the fringe firms and not only its moments. 4. To construct the perceived transition kernel we assume that the events that a fringe competitor becomes dominant and vice versa are both statistically independent of the moment transitions. This may be particularly reasonable if generally the value of the individual state from which fringe firms transition in and out of dominance is small relative to the current value of the moment. Our numerical results in Section 6.3 suggest that MME provides accurate approximations even under this independence assumption. Moreover, this simplification is computationally useful, because without it we would need to store the joint transition kernel of dominant firms and moment transitions, requiring significantly more memory; this requirement could be relevant in large-scale applications like the one presented in Section Since we assumed that at most one firm becomes dominant in a time period, in principle, one should explicitly consider and rule out the event that firm i and a competitor become dominant in the same time period. Even though we explicitly consider this event in our numerical experiments, it turned out not to be important in practice, because the constraint of having at most one fringe firm becoming dominant in a time period was typically never binding. Hence, for simplification, we ignored this effect in the kernel described above. 15 Under our simplification, to build the transition kernel we need to store the following elements in memory: the perceived transition kernel for moments, the tier transitions probabilities, and the strategy of dominant firms. This roughly requires memory space proportional to Ŝ S θ + 2 Ŝ (assuming no aggregate shock). Instead, the joint transition kernel of dominant firms and moment transitions would require memory space roughly proportional to Ŝ2. Recall that Ŝ = S θ S d. For example, if S θ = S d = 100, then the first expression is equal to 1,020,000 and the second to 100,000,