Sunk Costs, Entry, and Exit in Dynamic Oligopoly

Transcription

1 Sunk Costs, Entry, and Exit in Dynamic Oligopoly Jaap H. Abbring Jeffrey R. Campbell Nan Yang Preliminary and Incomplete Draft January, 2011 Abstract This paper develops a dynamic econometric framework for the analysis of entry, exit, and competitive conduct in oligopolistic markets. This framework only requires panel data on the demand and structures of geographically dispersed markets over time. It is a dynamic extension of Bresnahan and Reiss s (1990; 1991) framework for the analysis of static competition in a cross-section of markets. This extension facilitates the empirical analysis of the genuinely dynamic determinants of market structure and competition: sunk entry costs and uncertainty. Moreover, it is needed for the consistent measurement of static market primitives, such as the toughness of price competition, in such genuinely dynamic markets (Abbring and Campbell, 2010). Our framework is rooted in Abbring and Campbell s model of infinite-horizon Markov-perfect oligopoly dynamics with sunk costs and ongoing demand uncertainty. We extend their theoretical model to a non-degenerate econometric model by adding market-specific profitability shocks that satisfy a version of Rust s (1987) conditional-independence condition. Timing and expectational assumptions ensure that our model has a unique equilibrium that can be computed quickly by solving a finite sequence of low-dimensional Markov discrete decision problems. We develop and implement a version of Rust s nested fixed point estimator that exploits this close link to dynamic discrete decision processes. We apply our methods to the empirical analysis of sunk costs and the toughness of competition in the US market for dental services, using Bresnahan and Reiss s (1993) panel data on the number of dentists across geographical markets in the US. CentER, Department of Econometrics & OR, Tilburg University. J.H.Abbring@uvt.nl Federal Reserve Bank of Chicago. jcampbell@frbchi.org VU University Amsterdam and Tinbergen Institute. yang@tinbergen.nl JEL Code: L13 Keywords: Sunk/fixed costs, Entry and Exit, Markov-perfect equilibrium, Dynamic game estimation, Nested fixed point

2 1 Introduction This paper proposes an estimable model of infinite-horizon dynamic oligopoly markets with fixed cost, sunk cost, demand uncertainty, and market-level profitability shock. By focusing on the last-in first-out (LIFO) dynamics: firms are making decisions in the order of their ages, and hence a younger firm is never expected to produce longer than any older incumbents, our model predicts an essentially unique Markov-perfect equilibrium. Moreover, we can quickly compute this equilibrium by finding the fixed points of a sequence of contraction mappings. The LIFO equilibrium strategy implies a set of threshold conditions on the unobservable profitability shock: entry occurs when the shock is favorable enough to promise firms positive payoff after entry; and exit happens when unfavorable shock makes continuation unprofitable. With distributional assumption on the shock, we can construct the likelihood function from this set of conditions to recover firms fixed and sunk costs, using market-level panel data on demand indicators (size of the market) and number of firms. Fixed and sunk costs play important role in determining entry, exit and market structure on oligopoly markets, and directly estimating them using accounting data is usually infeasible due to data confidentiality. Quantifying these costs from market-level data has thus been a central issue on the agenda of recent empirical I.O. research. Using demand threshold rule to identify fixed and sunk cost has been first proposed by Bresnahan and Reiss (1990) and Berry (1992) in a class of static firm entry models. The logic behind their identification is that the number of entrants to a market reflects the revealed preference of rational firms that have been presented with the opportunities to operate on this market: Suppose that variable costs have been standardized to zero. Then, potential entrants start operation only when demand is high enough such that their expected revenue net of fixed costs can compensate the sunk cost of entry. Under this insight, multiple entry thresholds can be derived for various numbers of entrants. Subsequently, the difference in these entry thresholds provides information on the magnitude of sunk costs. Our threshold conditions are conceptually close to those in Bresnahan and Reiss (1993). Bresnahan and Reiss (1993) extend their earlier static entry model to a two-period one, and distinguish between entry and exit thresholds. With longitudinal observations on demand indicators and number of active firms, they estimate the magnitude of sunk costs from the difference between entry and exit thresholds. Our approach enables us to achieve the same goal, and avoid the initial conditions problem. By focusing on an infinite-horizon model s stationary equilibrium, any before-change market structure is our model is endogenously determined. Our econometric model is built on the theoretical model in Abbring and Campbell (2010). Their model features an infinite horizon dynamic oligopoly market with a random 1

3 number of consumers demands the industry s services, and this number evolves stochastically. Entry requires paying a sunk cost, and continued operation incurs fixed costs. In markets that are no longer profitable, incumbents choose to exit to avoid these per-period fixed costs. To avoid the common equilibrium multiplicity problem in this type of dynamic games, they exploits the sequentiality of firms actions and focus on LIFO dynamics. This assumption respects the sequentiality timing assumption made by both Bresnahan and Reiss (1990) and Berry (1992). It is a proxy to those empirical observations that young firms exit more frequently than their older rivals 1. Abbring and Campbell prove the uniqueness and existence of a LIFO Markov-perfect equilibrium and design an algorithm that can quickly compute this Markov-perfect equilibrium by solving for the fixed points of a finite sequence of contraction mappings. Abbring and Campbell s model is statistically degenerate, i.e., for any set of parameter values, the prediction of market structure is unique. To apply it to analyzing market data, we add profitability shocks that satisfy a market-level version of Rust s (1987) key conditional-independence assumption. This results in an estimable model describing the market structure dynamics for an industry s distinct markets in relation to these markets sizes. We extend Abbring and Campbell s results to show that an essentially unique LIFO Markov-perfect equilibrium exists in this model. The conditional-independence assumption ensures that we can generalize the algorithm developed in Abbring and Campbell to quickly computed this Markov-perfect equilibrium. The speed of our equilibrium computation gives us the choice to depart from the popular two-step approach (e.g., Bajari et al., 2007) and adapt the Nested Fixed Point (NFP) algorithm, a full-solution method to estimate the model. This is particularly useful if the data set is too small to deliver a good first-step identification in the twostep method. We shall also demonstrate that it is possible to incorporate unobservable time-invariant heterogeneity into the model, which substantially improves the model s explanatory power. To illustrate the feasibility of our estimation approach, we recollect Bresnahan and Reiss s (1993) panel data on dentists in U.S. local markets, and focus on reproducing their empirical results and contrasting these with our fully structural analysis. The remainder of this paper proceeds as follows. The next section presents the model s primitives. It introduces the equilibrium concept and also discusses the existence and uniqueness of the equilibrium. Section 3 discusses in detail how estimation can be carried out by using NFP algorithm. In the inner loop of the algorithm, the LIFO Markov-perfect equilibrium is computed quickly from a finite sequence of contraction mappings. In the outer loop of the algorithm, a likelihood function or a GMM objective function can be constructed from the implications of the equilibrium conditions. Section 4 presents the 1 For example, Dunne et al. (1988) 2

4 preliminary estimation results of one specification of our model. By assuming that the productivity shock follows normal distribution, we derive and estimate an dynamic order Probit model from the equilibrium conditions. Section 5 concludes. Proofs and tables are included in the Appendices. 2 The Model The model is an econometric extension of a simple and tractable model of oligopoly dynamics developed by Abbring and Campbell (2010). We introduce the primitives of the model in Section 2.1. In Section 2.2, we establish the existence and uniqueness of the model s LIFO Markov-perfect equilibrium. Proving these results requires straightforward extension to the their counter part in Abbring and Campbell (2010). Hence, in this section we only review the key steps towards the results. 2.1 Primitives In an oligopoly market, there is a countably infinite number of firms that are potentially active. We index these firms by assigning a name j N to each of them. At time 0, N 0 = 0 firms are active and J 0 is normalized to 1. Entry and exit decisions determine the number of active firms in each later period, N t for t {1, 2,...}. The level of demand in the market C t is bounded between ĉ 0 and č <. All active firms face a market-level profitability shock W t, which is a continuous random variable with support O R and is never observed by econometrician. The observable time-invariant market characteristics that affect firms value are collected in vector ζ. The joint transition density of {C t, W t } is denoted by f(c t+1, W t+1 C t, W t, ζ; θ 1, θ 2 ), in which θ 1, θ 2 denote the vectors of parameters. Figure 1 in Appendix B illustrates the sequence of events and actions within a period using a portion of the game tree. It begins with the inherited values of C t 1, W t 1 and N t and with the name of the first potential entrant, J t. Then C t and W t are realized and revealed to all firms. Afterwards, all active firms serve the market, earn a profits π(n t, C t, ζ; θ 3 ) κ exp(w t 1 ) and then decide whether they will remain in the market. θ 3 denote the vector of parameters of surplus function π, and κ > 0 is the fixed cost of production. We assume that the surplus function increases with the demand and decreases with the number of competitors. Assumption 1 (Monotone Producer Surplus). For any ζ, the surplus function satisfies (i). π(n, c, ζ; θ 3 ) π(m, c, ζ; θ 3 ) for all n, m N such that n > m; (ii). π(n, c, ζ; θ 3 ) π(n, d, ζ; θ 3 ) for all c, d [ĉ, č] such that c < d; and 3

5 (iii). lim n π(n, c, ζ; θ 3 ) = 0 for all c [ĉ, č]. Here and throughout, we use lower case letters to denote random variables realizations. Exit allows the firm to avoid future fixed production costs and unfavorable market-level shock W t, which has been revealed before the exit decision is being made. Exit is irreversible but otherwise costless. The rank of an active firm with name j is denoted as R j t. This equals one for the firm with the lowest name, and it equals N t for the firm with the highest name. The active firms continuation decisions proceed sequentially in increasing order of this rank. Firms can use mixed strategies. These strategies pure realizations also occur sequentially, so a firm later in the sequence conditions on its rivals realized (binary) continuation choices. After active firms continuation decisions, those firms that have not yet had an opportunity to enter make entry decisions in the order of their names, starting with J t. These continue until one potential entrant decides to remain out of the industry. The first potential entrant for the next period, J t+1, has this firm s name plus one. This way, firms do not have the option to delay entry. Because entry decisions proceed sequentially in increasing order of the firms names, any entrant j will have a rank R j t+1 greater than that of any surviving incumbent. The entrant starts to serve the market in next period, faces the realized productivity shock w t, and must pay a one-off rank-dependent setup cost ϕ R j on top of κ upon entry. Firms discount future payoff with a factor β < 1. t+1 The key feature of this model is that older firms are given priority in committing to continuation. Entry decisions proceed sequentially in increasing order of the firms names; so active firms values of R j t rank their ages as well as their names if they entered in distinct periods. In this case, the ordering of continuation decisions by R j t puts the oldest firm first and the youngest firm last. 2.2 Markov-Perfect Equilibrium A symmetric Markov-perfect equilibrium generates the market outcome. A Markovperfect equilibrium is a subgame-perfect equilibrium in strategies that are only contingent on payoff-relevant variables. When firm j decides whether to stay or exit, N t R j t (the number of active firms following it in the sequence), C t, R j t+1 (its rank in the next period s sequence of active firms), W t, and ζ are available and payoff-relevant. Collect these terms into H jt (N t R j t, C t, R j t+1, W t, ζ). Similarly, the payoff-relevant state to a potential entrant ] is H jt (C t, Rt+1, j W t, ζ). Note that H jt takes its values in H S Z + [Ĉ, Č N O Z for firms active in period t and in H E [Ĉ, Č] N O Z for potential entrants, where Z is the support for the market characteristics vector. Here and below, we use S and E to denote potential survivors and entrants. 4

6 A Markov strategy for firm j is a pair of functions a j S : H S [0, 1] and a j E : H E [0, 1]. The values of the functions represent the probabilities of being active in the next period given that the firm is currently active (a j S ) and given that the firm has an entry opportunity (a j E ). The symmetric Markov-perfect equilibrium requires all firms to follow the same Markov strategy. This anonymity allows us to omit the firms names from the superscript hereafter. When firms use Markov strategies, the payoff-relevant state variables determine an active firm s expected discounted profits, which we denote with v(h S ). In a symmetric Markov-perfect equilibrium with strategy profile (a S, a E ), these satisfy the Bellman equation v(h S ) = max a [0,1] aβe [π(n, C, ζ; θ 3 ) κ exp(w ) + v(h S) H S ]. (1) We adopt conventional notation and denote the variable corresponding to Y in the next period with Y. In Equation (1), the expectation is calculated using f(c t+1, W t+1 C t, W t, ζ; θ 1, θ 2 ) and all firms strategies conditional on the particular firm of interest choosing to be active. A dynamic model of this type usually admits multiple equilibria. To select a unique one, we restrict attention to equilibria in which firms entry and exit policies arise from a last-in first-out (LIFO) strategy. Definition 1. A LIFO strategy is a strategy (a S, a E ) such that a S is pure, with a S (n r, c, R, w, ζ) weakly decreasing in r. If all firms adopt a common LIFO strategy (a S, a E ), then an active firm with rank r 2 never stays if the predecessor in the sequence of active firms exits, because a S (n r, c, R, w, ζ) = 0 = a S (n r 1, c, R, w, ζ) = 0. As a consequence, if firms adopt a common LIFO strategy, they exit in the reverse order of their entry. Conversely, if firms use a common strategy and always exit in the reverse order of their entry, then the common strategy is a LIFO strategy. Under a LIFO strategy, if any r-ranked firm continues to next period, all of its competitors with lower ranks also survive, so R = r. Restricting to LIFO strategy, we still have one uninteresting source of equilibrium multiplicity. When multiple actions give that player the same equilibrium payoff, then we can construct multiple subgame perfect equilibria by varying that player s choice. To eliminate this type of equilibria, Abbring and Campbell requires that in equilibrium players choose inactivity whenever it gives the same payoff as entry or continuation. Under this restriction, we have the following proposition. 5

7 Proposition 1. There exists a unique symmetric Markov-perfect equilibrium in a LIFO strategy that defaults to inactivity. This equilibrium s payoff v(x, c, r, ζ) is weakly decreasing in x; the survival rule a S and entry rule a E are such that a S (n r, C, r, w, ζ) is pure, i.e., takes the value in {0, 1}, constant in n r, and weakly decreasing in r; a E (c, r, w, ζ) is pure and weakly decreasing in r; and a S (n r, c, r, w, ζ) a E (c, r, w, ζ). The equilibrium survival and entry probabilities in Proposition 1 weakly decrease with the firm s rank in the next period. Moreover, the survival probability is constant in the number of firms with unresolved continuation decisions, and weakly larger than the entry probability of a firm with the same prospective rank in the same demand state. The proof of this proposition requires a minimum adaption of the constructive proof in the original paper. Hence we only review the intuition behind the proof here. First, we note that the limiting conditions in Assumption 1 implies that the number of firms that remain active in a Markov-perfect equilibrium cannot exceed a upper-bound, which we denote by Ň. Next, we consider the exit decision of a firm with rank Ň. When making this decision, this firm rationally expects no further entry and none of its older competitors to cease production before it does. Therefore, the exit decision can be made independent of the knowledge of any other firm s strategy, which implies that the equilibrium value function, and the optimal exit and entry rule for this firm can be constructed by solving a simple dynamic programming problem. Then, we solve the exit decision problems in turn for firms with ranks r = Ň 1, Ň 2,..., 1. A firm with rank r forms its expectations about the behavior of firms with higher ranks using the obtained exit and entry rules of those firms. Therefore, this firms equilibrium value function, and the optimal exit and entry rule can be again constructed by solving a simple dynamic programming problem. With these solutions, we can subsequently verify that it satisfies the monotonicity conditions in Proposition 1. In LIFO equilibrium, the first-mover advantage implies that early entrants can creditably crowd out younger competitors and deter late entrants by continuation. As a result, their payoffs are never worse than their younger competitors under the same market conditions. Consequently, a firm can conveniently use the expectation that no younger firm would continue to next period when making entry/continuation decisions. The only case when this expectation is inconsistent with the actual equilibrium market dynamics is when this firm s younger competitor does continue to next period or enter the market. Under those circumstance, the payoff monotonicity guarantees that this firm will also decide to continue/entry, and this decision is still consistent with profit-maximization. The following proposition summarizes this property. Proposition 2. Suppose that in a market with characteristics ζ, current demand c, and future shock w, n firms continue to next period. Denote the r = n-ranked firm s expected 6

8 payoff excluding the shocked fixed cost and right before new (C, W ) is drawn as v S (c, r, w, ζ) E W,C [π(n, C, ζ) + v(0, C, r, W, ζ) C = c, W = w, ζ]. (2) If (a E, a S ) forms a symmetric Markov-perfect equilibrium, then for any r-ranked firm and any (x, c, w, ζ), it satisfies, a S (x, c, r, w, ζ) arg max a (v S(c, r, w, ζ) κ exp(w)) and (3) a [0,1].a E (c, r, w, ζ) arg max a ( ) v S (c, r, w, ζ) κ exp(w) β 1 ϕ r (4) a [0,1] Proof. See Appendix A. Other symmetric Markov-perfect equilibria that default to inactivity might exist, but in them the apparent advantage of early entrants to commit to continuation does not always translate into longevity. In our empirical analysis, we constrain our attention to the unique symmetric Markov-perfect equilibrium in a LIFO strategy that defaults to inactivity. 3 Estimation The structural parameters of this model are collected into the vector Θ (θ 1, θ 2, θ 3, κ, Ň, ϕ 1,..., ϕ Ň ). In this section, we discuss how to estimate these parameters. Because W is never observed, the estimation relies on matching the observations (N m,t, C m,t, ζ m ) with the market transition probability P r(n ; N, C, ζ) implied by the unique LIFO equilibrium. Proposition 2 ensures that payoff function v S dictates marginal firm s continuation/entry decision. Using v S, we can calculate the transition probability as P r(n = m N = n, C = c, ζ) = P r(v S (c, m, W, ζ) β 1 ϕ m > exp(w ), v S (c, m + 1, W, ζ) β 1 ϕ m+1 < exp(w )) P r(v S (c, m, W, ζ) > exp(w ), v S (c, m, W, ζ) β 1 ϕ m < exp(w )) P r(v S (c, m, W, ζ) > exp(w ), v S (c, m + 1, W, ζ) < exp(w )) if m > n if m = n if m < n (5) The intuition behind this formula is straightforward. (i) If the observed next period number of firms N exceeds the current observed n, then the unobserved shock has to be favorable enough for an potential entrant to become the N -th ranked active firm, and 7

9 sufficiently unfavorable to deter the next potential entrant; (ii) if N equals n, then the shock has to be favorable enough for the N -th ranked firm to continue, and sufficiently unfavorable to deter the next potential entrant; (iii) if N is below n, then the shock has to be favorable enough for the N -th ranked incumbent firm to continue, and sufficiently unfavorable for the N + 1-th ranked incumbent to continue. Based on Equation (5), we can construct a likelihood function l m (C m,t0 +1,..., C m,t0 +T, N m,t0 +1,..., N m,t0 +T, ζ m ; Θ) from the longitudinal observations (C m,t, N m,t, ζ m ) for all periods t {t 0,..., t 0 + T }, on any market m M. Consequently, the structure parameter Θ can be recovered as the one that maximizes the likelihood function Π m M l m. Computing v S is obviously the corner stone of this procedure. However, we face two difficulties. First, computing the LIFO equilibrium payoff v using Equation (1) requires us to resolve the strategy a S, a E used in the expectation. The strategy in turn depends on the payoff. This interdependence implies that we cannot reply on standard contractionmapping-based dynamic programming technique to solve v. Second, when computing the expectation in Equation (1), we have to integrate v over (W, C) simultaneously. This is a two-dimensional numerical integral. In addition, to obtain v S, we need to integrate v(x, C, W, R, Z) over (W, C) again. This dual task is computationally very cumbersome, especially if decently approximating W s distribution requires a large number of grid points. We explain how we solve these two issues first, before giving an parameterized example of the likelihood function. 3.1 Computing the LIFO Equilibrium To tackle the first difficulty, we explore the sequentiality natural of the LIFO equilibrium, following Abbring and Campbell. Because any active firm rationally anticipate those with higher ranks will remain active as long as it decides to continue, it only needs the equilibrium survival rules used by those with lower ranks to compute its payoff. This enables us to compute v sequentially by finding the fixed points of a sequence of contraction mapping, each representing a single-agent decision problem of a firm with a particular rank. We start with the firm of lowest possible rank Ň. In practice, we set Ň to the maximum of number of active firms over all markets, all periods 2. Let w Ň (x, c, w, ζ) denote this firm s value function when the number of younger competitors is x, the demand is c, and the shock is w, i.e., w Ň (x, c, w, ζ) = v(x, c, Ň, w, ζ). When wň is being computed, this firm 2 Alternatively, we can follow the model s implication that Ň = min{n N; π(n, c, w, ζ) < κ, (c, w, ζ)}. Normally, we get a higher value of Ň. 8

10 rationally expects any younger competitor to exit and no firm to enter before its exit. In other words, in equilibrium this firm either receive zero payoff from exiting, or positive payoff from continuing with all order competitors. Thus, its value function necessarily satisfies w Ň (x, c, w, ζ) = w Ň (0, c, w, ζ) = max β { 0, E W,C [π ( Ň, C, ζ ) C = c, W = w, ζ] κ exp(w) + Ew Ň (0, c, w, ζ) } (6) where Ew Ň (0, c, w, ζ) = E W,C [wň (0, C, W, ζ) C = c, W = w, ζ]. The right hand side of (6) does not involve any other firm s strategy, and hence defines a contraction mapping with w Ň as its unique fixed point. With w Ň determined, we can compute the sets of state variables that enable a ranked-ň firm to survive/enter. We refer to these as the entry and survival sets, E Ň,ζ {(c, w) w Ň (0, c, w, ζ) > ϕ Ň } and S Ň {(c, w) w Ň (0, c, w, ζ) > 0}. The rest of the computation proceeds sequentially for r = Ň 1,..., 1. For an r- ranked incumbent, r {1,..., Ň}, let w r(x, C, W, ζ) denote the LIFO equilibrium value function w r (x, c, w, ζ) = max{0,βe W,C [π (N r(x, c, w, ζ), C, ζ) + Ew r (N r(x, c, w, ζ) r, c, w, ζ)} C = c, W = w, ζ] κ exp(w) (7) where Ew r (N r(x, c, w, ζ) r, c, w, ζ) E W,C [w r(n r(x, c, w, ζ) r, C, W, ζ) C = c, W = w, ζ]. in which N r(x, c, w, ζ) is the number of firms in the following period, conditional on the r-ranked firm s survival. It depends on all younger-than-r firms equilibrium strategy. Note that we have uniquely computed entry sets E r+1,ζ,..., E Ň,ζ and survival sets S r+1,ζ,..., S Ň,ζ for the younger firms. The rank r firm rationally expects that these sets govern the younger firms entry and survival decisions, and that no firm will enter with rank larger than Ň. That is, Ň r N r(x, c, w, ζ) = r + [I {j x, (c, w) S r+j,ζ } + I {j > x, (c, w) E r+j,ζ }], (8) j=1 for x = 0, 1,..., Ň r. Because LIFO strategy is a pure strategy, given (x, c, w, ζ) and a S, a E, N r(x, c, w, ζ) is unique. Hence, the right-hand-side of (7) only involves firms 9

11 strategy that is known at this point, and it defines a contraction mapping with unique fixed point w r (X, C, W, ζ). Consequently, we can define the entry and survival sets for the r-ranked firm. By Proposition 2, they are E r,ζ {(c, w) βe W,C [π (r, C, ζ) C = c, W = w, ζ] κ exp(w) + (Ew r )(0, c, w, ζ) > ϕ r } S r,ζ {(c, w) βe W,C [π (r, C, ζ) C = c, W = w, ζ] κ exp(w) + (Ew r )(0, c, w, ζ) > 0}, for a firm with rank r. After we have computed w 1, eventually, the LIFO equilibrium value function is assembled by v(x, C, W, R, ζ) = w R (X, C, W, ζ). 3.2 Computing the Integrated Value Function The two dimensional integral over (W, C ) required to compute Ew r in each step of this procedure manifests the second difficulty that we have mentioned. We circumvent this problem by imposing the Conditional Independence Assumption, following Rust. Assumption 2 (Conditional Independent Assumption (CI)). The transition density of the {C, W } process factors as f(c, W C, W, Z; θ 1, θ 2 ) = f C (C C, ζ; θ 1 )f W (W C, ζ; θ 2 ). Under (CI), the dependence between W and W goes through only via C, and the density of C does not depend on W. The benefit of (CI) is twofold. First, note that under CI, (C, ζ) provides the sufficient statistics for W, hence Ew r is a function of w only via N r(x, c, w, ζ). When equilibrium payoff is being computed for each r = Ň,..., 1, N r(x, c, w, ζ) is predetermined by (8) for all (x, c, w, ζ). Therefore, Ew r (N r(x, c, w, ζ) r, c, w) is a known function independent of w, once conditional on N r(x, c, w, ζ). To stress this crucial independence, we henceforth drop w as the last argument and write Ew r (N r(x, c, w, ζ) r, c) instead. Second, (CI) implies that the integral of w r over (W, C ) with respect to their joint distribution can be computed iteratively as two one-dimensional integrals Ew r (N r(x, c, w, ζ) r, C, W ) = E C [E W [w r (N r(x, c, w, ζ) r, C, W, ζ) C = c ] C = c] Also, the expectation of π over C is independent of W. Therefore, the integrated value function E W [w r (x, c, W, ζ) C = c, ζ] can be computed as the solution of the functional equation below, by integrating both sides of (7) over W. E W [w r (x, c, W, ζ) C = c, ζ] = P r(w S r,ζ (c))β ( E C [E W [π (N r(x, c, w, ζ), C, ζ) κ exp(w ) +E W [w r (N r(x, c, W, ζ) r, C, W ) C = c, ζ] W S r,ζ (c), C = c, ζ] C = c, ζ] ) (9) 10

12 In this equation, because N r(x, c, W, ζ) r, depending on the realized w, takes its value in {0, 1,..., Ň r}, the integrated continuation value function is nothing but a weighted average over (Ew r )(x, c, ζ), x = 0,..., Ň r, with weight given by P r(n r(x, c, W, ζ) r = x; f W (W C, ζ; θ 2 )). Once conditional on N r, it is independent of W too. S r,ζ (c) is the c-specific survival set for this firm. It contains all possible values of w such that, for the given level of c, this r-ranked firm finds it profitable to survive. It is determined as follows { ( S r,ζ (c) w E C [π (r, C, ζ) C = c, ζ] κ exp(w) } +E C [E W [w r (0, C, W, ζ) C = c, ζ] C = c, ζ]) > 0, We can define the c-specific entry set E r,ζ (c) analogously. Despite equation (9) s seemingly involving expression, it forms a contraction mapping that can be solved rather easily. Proposition 3. Define W to be the space of all functions: ] [ g : Z + [Ĉ, Č Z 0, βπ(0, Č, w ] min, κ) 1 β and define the Bellman operator K r : W W with where K r (g)(x, c, ζ) = P r(w J g r,ζ (c))β( E C [E W [π (N r(x, c, W, ζ), C, ζ) κ exp(w ) J g r,ζ (c) {w +g(n r(x, c, W, ζ) r, C, ζ) W J g r,ζ (c), C = c, ζ] C = c, ζ]) ( E C [π (r, C, ζ) C = c, ζ] κ exp(w) ) } +E C [g(0, C, ζ) C = c, ζ] > 0, Then K r is a contraction mapping. Proof. See Appendix A. The unique fixed point of this contraction mapping is E W [w r (x, c, W, ζ) C = c, ζ]. This is an integrated value function, ] with its domain in a reduced state space Z + [Ĉ, Č] Z instead of Z + [Ĉ, Č O Z. This simplification significantly reduces the dimensionality of the problem. Note that under CI, Ew r (N r(x, c, w, ζ) r, c) is independent of W once conditional on N r. Thus, the expected continuation value E W,C [v(0, C, r, W, ζ) C, W, ζ] in equation (2) is also independent of W, and so is v S. Again, we omit W from v S s arguments and write v S (c, r, ζ) to visualize this independence. Because v S dictates firms entry/exit 11

13 strategy as stated in Proposition 2, its independence of W ensures that a S, a E are all weakly decreasing in w. This is an intuitive result: observing a less favorable market level shock, no firm is more likely to stay/enter into next period. In other words, given c, a threshold rule in w prescribes all firms entry/exit decisions, and those c-specific entry/survival sets are intervals. This result implies that we can rewrite E W in (9) as simple Riemann integrals. For a r-ranked firm, the anticipated number of active firms in next period N r is also weakly decreasing in w. Then, for any n such that r n Ň and give (x, c), we can determine the upper- and lower-bound of w, denoted by w(n, r, x, c, ζ) and w(n, r, x, c, ζ), such that for any value between them, N r(x, c, w, ζ) = n. { sup S n,ζ (c) if r n r + x w(n, r, x, c, ζ) = sup E n,ζ (c) if r + x < n Ň and w(n, r, x, c) = { inf S n,ζ (c) inf E n,ζ (c) if r n r + x if r + x < n Ň Note that w(n, r, x, c, ζ) = w(n + 1, r, x, c, ζ) if w is a continues variable. Consequently, we can rewrite (9) as E W [w r (x, c, W, ζ) C = c, ζ] [ Ň w(i,r,x,c,ζ) ( = βe C π(i, C, ζ) κ exp(w ) i=r w(i,r,x,c) +E W [w r (i r, C, W, ζ) C = c, ζ] ) ] f W (W C = c, ζ)dw C = c, ζ. (10) 3.3 A Likelihood Function: Dynamic Probit Model Under certain parameterizations, we can compute the integrals in (10) analytically. For instance, let W be independent of C, ζ and follow a normal distribution W N (0, σ 2 W ). Fixed cost κ is then perturbed by a multiplicative log-normally distributed error. Then, ( ( )) w(r, x, c, ζ) σ E W [w r (x, c, w, ζ)] = β W r (x, c, ζ) κ exp(σ 2 2 /2) Φ (11) σ where W r (x, c, ζ) = Ň ( ( ) ( )) w(i, r, x, c) w(i, r, x, c) Φ Φ σ σ i=r E C [π(i, C, ζ) + E W [w r (i r, C, W, ζ)] C = c, ζ]. 12

14 in which Φ is the c.d.f for standard normal distribution. Since (11) is a special case of (9), Proposition 3 ensures that we can obtain a unique E W [v R (,, W )] by solving (11) using value function iteration and then construct v S using equation (2) for r = Ň,..., 1. Under the normality assumption on W, we obtain an order probit model P r(n = m n, c, ζ) = Φ( ln max{v S(c,m,ζ) β 1 ϕ m,0} ln κ ) Φ( ln max{v S(c,m+1,ζ) β 1 ϕ m+1,0} ln κ ) if m > n σ σ ) Φ( ln max{v S(c,m,ζ) β 1 ϕ m,0} ln κ Φ( ln v S(c,m,ζ) ln κ σ Φ( ln v S(c,m,ζ) ln κ σ σ ) if m = n ) Φ( ln v S(c,m+1,ζ) ln κ ) σ if m < n (12) The likelihood function simply is: L = T 1 t=1 m=1 M P r(n m,t+1 N m,t, C m,t, ζ m ; Θ)f C (C m,t+1 C m,t, ζ m, θ 1 ) i.e., a market with N m,t firms in t period and N m,t+1 firms in t + 1 period contribute to the likelihood function the probability of transition defined by (12). With this likelihood function established, we employ the Nested-Fixed-Point Algorithm to find the values of structure parameters Θ that maximizes this likelihood. This algorithm iterates between an outer-loop and an inner loop. It starts with some initial values of Θ 0. In the inner loop, the unique LIFO Markov-perfect equilibrium is computed by finding the fixed points of a sequence of contraction mappings. Then, market transition probability P r(n m,t+1 N m,t, C m,t, ζ m ; Θ 0 ) is computed using Equation (5) for all t and m, and the likelihood is evaluated. Next, the outer loop searches for the new parameter values Θ 1 to increase the likelihood value. When the parameter values are updated, they are passed to the inner loop to repeat the contraction mappings and regenerate the likelihood value. The procedure stops when the likelihood value cannot be improved. 4 Results To demonstrate the estimation procedure of our model and compare our results to what Bresnahan and Reiss (henceforth BR) have obtained from their two-stage model, we replicate their American dentists data set and estimate the sunk/fixed costs for the dentists. 4.1 Data Description This data set records the number of dentists operating in some selected U.S. counties during the period from 1980 to 1988, and also includes variables for local demand of dental service. The counties are selected based on the following criterions. First, the 13

15 county population in 1980 was not in excess of 10,000. Second, in these counties, no town with a population in excess of 1,000 could lie within 25 miles of the 1980 population center of each county. Third, no town or city with a population greater than 1,000 within 125 miles of the 1980 population center could have a population to driving distance ratio in excess of 600. These criterions are set in order to minimize the possible interplay of dentists in adjacent markets, and ensures that observations from multiple markets are independent. With this selection procedure, we are left with 152 small and isolated counties as markets in our model. Arguably, in these markets, dentists primarily serve the local public. To find dentists serving these counties, we eliminate all small cities and towns with an unknown population. This leaves us with in total 445 towns. We then obtain dentist information for all dentists in these towns from the American Dental Association (ADA) dentist directories. These directories contained dentist information as of August 1980 through August For each listed dentist, the ADA directories contained a name, address, zip code, year of birth, ADA member status indicator, specialty indicator, dental school indicator, year of graduation, and a unique identifier, or ADA number. Beginning in 1986, the directories added a new classification system to extend the specialty indicator called occupation, which identified among other things dentists only practicing part-time, along with an indicator for whether the address listed was that of a home or office. We recorded each of these items for every dentist in all the 445 towns and aggregate the number of dentists to the county level. At this stage, we are merely able to collect the number of residents in each county as the only demand indicator. In BR s original estimation, a richer set of parameters was present, including the per capita income, in/out-flow of residents, land value, etc.. Another notable data limitation is that currently we do not have observations for the market-level characteristics ζ. Therefore we are restricted to estimate the model with homogenous markets. 4.2 Summary Statistics We analyzed the resulting sample to compare some key statistics against those found in BR. These comparisons can be found in Tables 1, 2, 3, and 4 in Appendix D. Table 1 presents summary statistics of the raw data as well as detailed break-downs of some of the data cleaning measures and the exit and entry patterns in the data. Tables 2 and 3 replicate tables 1a and 1b from BR in summarizing dentist exit and entry by market between 1980 and 1988 with and without adjustments made for retirements and possible deaths. These tables also present the marginal distributions of dentists by market for Table 4 offers further comparisons of our sample versus that of BR by comparing 14

16 the counts and age distributions of active and retired dentists along with market averages of dentists and town and county populations. In many ways, our sample is very similar to that of BR, though it does differ in some substantial ways. For instance, our raw and adjusted data counts are fairly similar for everything in Table 1 except the number of active dentists in This is a common theme in comparing our sample to that of BR. It appears again in Table 1 in the entry and exit details where we have significantly fewer dentists that leave the sample in 1990 than BR do. Accordingly, Tables 2 and 3 show joint and marginal counts of dentists by market and time which differ slightly from BR with this difference greatest for markets with 4 or more dentists. On a positive note, however, the marginal counts of dentists in Table 3 for markets with less than 4 dentists are very close to that found by BR after adjusting for retirements and possible deaths. One potential explanation for this fact is that there is a possibility that our list of counties differs slightly from that of BR. Table 4 shows that while we match the mean and standard deviation of towns they classify as population centers, we marginally fail to do so at the county level. In addition, the information in this table depicts that our sample of dentists are slightly younger than the BR sample. This could stem from the fact that our list of towns omits very small towns possibly included by BR which are more likely to have older dentists. The table also confirms the above observation in that the average number of dentists per market in 1990 is greater in our sample than BR s, while similar in 1980 with and without adjustments for retirements and possible deaths. 4.3 Estimation Results: Homogenous Model Because all markets are homogenous, we pool the 9-year panel from 1980 to 1988 together as a 2-period panel, resulting 1216 (152 8) observations on market transitions. C t is defined as the logarithm of number of county residents at time t, and it is discretized on a grid with 301 points. The grid covers all realized values of C t. We set f C ( C t 1 ) to equal a mixture over 51 reflected random walks in C with uniformly distributed innovations. The standard deviation of the innovation is the sample standard deviation of C t. In most of the markets, the number of operating dentists never exceeds 5. Therefore, we set Ň = 5 and convert any observation N t > 5 to 5. When counting the exits, we do not exclude retirement or death. The surplus function π is parameterized as π(n, c; θ 3 ) = α n exp(c) Because we can only identify the parameters up to scale, we normalize the standard deviation of W, σ W, to be 1. 15

17 We implement the estimation in Matlab. One execution typically takes minutes on an off-the-shelf laptop. In Table 5, we present the estimation results for two specifications of our model. In Specification 1, we assume that the sunk costs of entry are identical for all potential entrants and equal to ϕ. We allow α n to vary, but superimpose the constraint that α n is decreasing in n. In Specification 2, we allow both α n and ϕ n to be decreasing in n. More discussions on the estimation results, report on the standard errors, and the estimation with market-level characteristics are to come in future. 5 Conclusion In this paper, we develop an estimable dynamic oligopoly model to quantify the fixed cost and sunk cost using panel data on market size and number of firms. To tackle the common equilibrium multiplicity issue in this type of dynamic models, we impose the LIFO assumption. This helps us to essentially break down firms interdependent decision process into a sequence of individual dynamic discrete choice problems. From this point, we can readily use tested methods on dynamic discrete choice models to deliver a fast estimation procedure and a reliable result. We finish by briefly discussing the LIFO assumption. Admittedly, it is the crucial piece that renders our model tractable. However, it is by no means the only solution. First, some other sequentiality assumptions (e.g., first-in first-out ) are also sufficient to ensure the essential uniqueness of Markov-perfect equilibrium in our model. In general, a uniquenessadmissible sequentiality assumption needs not to be age-related at all, although we do require the order of actions to be determined exogenous or upon entry 3. For instance, we can extend our model to allow firms (observed) fixed effect dictating the order of action. Second, sequentiality is not necessary to guarantee dynamic model s tractability. Abbring et al. (2010) provide a model in which firms exit decisions are made simultaneously. In their model, although the uniqueness of symmetric Markov-perfect equilibrium is not guaranteed if number of firms exceeds 2, a set of refined Markov-perfect equilibria can still be quickly computed by finding the fixed points of a sequence of contraction mappings. 3 If, for instance, firms can make investment to overtake rivals in the order of action, equilibrium multiplicity emerges. Ericson and Pakes (1995) is a well-known example 16

18 References Abbring, J. H. and J. R. Campbell (2010). Last-in first-out oligopoly dynamics. Econometrica 78 (5), , 1, 2, 3, 5, 8 Abbring, J. H., J. R. Campbell, and N. Yang (2010). Simple markov-perfect industry dynamics. Mimeo, CentER, Department of Econometrics & OR, Tilburg University. 16 Bajari, P., C. Benkard, and J. Levin (2007). Estimating dynamic models of imperfect competition. Econometrica 75 (5), Berry, S. (1992). Estimation of a model of entry in the airline industry. Econometrica 60 (4), , 2 Bresnahan, T. F. and P. C. Reiss (1990). Entry in monopoly markets. Review of Economic Studies 57 (4), , 1, 2 Bresnahan, T. F. and P. C. Reiss (1991). Entry and competition in concentrated markets. Journal of Political Economy 99 (5), Bresnahan, T. F. and P. C. Reiss (1993). Measuring the importance of sunk costs. Annales d Economie et de Statistique 31, , 1, 2, 13 Dunne, T., M. J. Roberts, and L. Samuelson (1988). Patterns of firm entry and exit in U.S. manufacturing industries. RAND Journal of Economics 19, Ericson, R. and A. Pakes (1995). Markov-perfect industry dynamics: A framework for empirical work. Review of Economic Studies 62, Rust, J. (1987). Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica 55, , 2, 10 17

19 Appendices A Proofs Proof for Proposition 2. In any Markov-perfect equilibrium, a S necessarily satisfies in which a S (n r, c, r, w, ζ) arg max a [0,1] aβe W,C [π(n r(x, c, w, ζ), C, ζ) κ exp(w) + v(n r(x, c, w, ζ) r, C, r, W, ζ) C = c, W = w, ζ] Ň r N r(x, c, w, ζ) = r + [I {j x, (c, w) S r+j,ζ } + I {j > x, (c, w) E r+j,ζ }], j=1 When a S is determined by (3), the only case it may possibly disagree with (13) is when N r(x, c, w, ζ) > r. In such case, on one hand, it is resulted by (c, w) S r+1,ζ or (c, w) E r+1,ζ, which implies a S (n r 1, c, r + 1, w, ζ) = 1 or a E (c, r + 1, w, ζ) = 1. Either way, when the r + 1-ranked firm operates in the market, in LIFO equilibrium the r-ranked firm receives strictly positive payoff by continuation, βe W,C [π(n r(x, c, w, ζ), C, ζ) κ exp(w) + v(n r(x, c, w, ζ) r, C, r, W, ζ) C = c, W = w, ζ] > 0. Therefore, from (13) we get a S (n r 1, c, r + 1, w, ζ) = 1. On the other hand, because v(x, c, r, w, ζ) is weakly decreasing in x (Proposition 1), and π(n, c, ζ) is weakly decreasing in n, we have (v S (c, r, w, ζ)) βe W,C [π(n r(x, c, w, ζ), C, ζ) + v(n r(x, c, w, ζ) r, C, r, W, ζ) C = c, W = w, ζ] >βκ exp(w). This means that under (3), we also get a S (n r 1, c, r + 1, w, ζ) = 1. Similarly, we can prove that a E determined by (4) is the equilibrium entry strategy. Proof for Proposition 3. We prove that the Blackwell s sufficient conditions hold for K r. With the presence of β < 1, discounting is obvious. To show the monotonicity, consider w 1, w 2 W and w 1 (x, c, ζ) w 2 (x, c, ζ) for all (x, c, ζ). Use J w i r,ζ, i = 1, 2 to denote the J r,ζ set associated with function w i. By construction, J w 1 r,ζ J w 2 r,ζ. In addition, by the definition of J w 2 r,ζ (c), for any w J w 2 r,ζ (c), E C [π (r, C, ζ) + g(0, C, ζ) C = c, ζ] κ exp(w). 18 (13)

20 As we have shown in the proof for Proposition 2, if N r(x, c, w, ζ) > r, it is resulted by the entry or continuation decision from firms with rank larger than r, which implies that for any w J w 2 r,ζ (c), Therefore E C [π (N r(x, c, w, ζ), C, ζ) + g(n r(x, c, w, ζ) r, C, ζ) C = c, ζ] κ exp(w). (14) K r (w 2 )(x, c, ζ) = P r(w J w 2 r,ζ (c)\j w 1 r,ζ (c))β( E C [E W [π (N R(x, c, W, ζ), C, ζ) κ exp(w ) +w 2 (N r(x, c, W, ζ) r, C, ζ) W J w 2 r,ζ (c)\j w 1 r,ζ (c), C = c, ζ] C = c, ζ] + P r(w J w 1 r,ζ (c))β( E C [E W [π (N R(x, c, W, ζ), C, ζ) κ exp(w ) +w 2 (N r(x, c, W, ζ) r, C, ζ) W J w 1 r,ζ (c), C = c, ζ] C = c, ζ] 0 + P r(w J w 1 r,ζ (c))β( E C [E W [π (N R(x, c, W, ζ), C, ζ) κ exp(w ) +w 1 (N r(x, c, W, ζ) r, C, ζ) W J w 1 r,ζ (c), C = c, ζ] C = c, ζ] K R (w 1 )(x, c) For any w J w 2 R (c)\j w 1 R (c), (14) holds. So the first term after equality is non-negative. The first inequality uses this fact and w 1 w 2. With the monotonicity requirement satisfied, K r is proven to be a contraction mapping. B C D The Game Tree Within A Period Estimation Results Summary Statistics 19

21 R j Rank R Incumbent s Survival Choice Firm j s Entry Choice Start with (C t 1, ε t 1, N t, J t ) Nature chooses C t, ε t Incumbents earn π(n t, C t, ε t 1, κ) a = 0; Exit for sure Survival Choices for 2 R j t = 3,..., N t 1 1 a [0, 1] a = 1; Continue for sure Survival 2 Choices for R j t = 3,..., N t 1 N t N t a = 0; Pass and earn 0 J t a = 1; Pay ϕ(r ) and enter Continue to J t + t + 1 with 1 J t+1 = J t + 1 Continue to t + 1 with J t+1 = J t + 2 J t + 2 a [0, 1] Figure 1: The game tree within a period. 20

22 Table 1: ADA Directory Sample Sta Our Sample BR Sample Counts Counts Dentists 371 Dentists 370 Students 8 Students 8 Public Retired Health 6 Public Health4 Dentists Retired Dentists Active Dentists Active Dentists Firms Firms Group Practices 20 Group Practices 12 Total Entry76 Total Entry 75 graduations27 graduations 37 relocations36 relocations 19 w/in state from public 5 health unknown 7 Total Exit 95 Total Exit 117 retirement 31 retirement 40 retire and 4relocate retire and relocate 16 relocate 38 relocate 30 possible death 22 possible death 31 21

23 Table 2: Counts of Markets by Number o Our Sample or more Total or more BR Sample or more Total or more

24 Table 3: Counts of Markets by Adjusted Nu Our Sample or more Total or more BR Sample or more Total or more

25 Table 4: Dentist and Market Sample Age Our Sample Dentist Averages Number Age Incumbent Adjusted Number Active Active Retired Market Retired Statistics MeanStd. Dev. MeanStd. Dentists Incumbents MeanStd. Town Population Dev. Dev. County Minus Town 2644 Population 1727 BR Sample Dentist Averages Age Active Number Age Incumbent Adjusted Number Active Retired Market Retired Statistics MeanStd. Dev. MeanStd. Dentists Incumbents MeanStd. Town Population Dev. Dev. County Minus Town 2618 Population

26 Spec. 1 Spec. 2 κ θ θ θ θ θ ϕ ϕ ϕ ϕ ϕ Table 5: Estimation Results 25