Lecture Notes: Industrial Organization Joe Chen 64

Transcription

1 Joe Chen 64 6 Oligopoly Pricing Dynamic The simple Bertrand price game is one-shot in nature. As mentioned previously, the introduction of repeated interaction makes firms to take into account the possibility of price wars and long-run losses when they decide whether to undercut a given price. Thus, the time dimension itself can work to soften the intensive price competition in oligopolistic markets. In an oligopolistic setting, the threat of a vigorous price war can be sufficient to deter the temptation to cut prices. Hence, firms might be able to collude in a purely noncooperative manner. This opens a door to our discussion of tacit collusion. Wewill consider a variety of theories that explain tacit collusion. Before we dive into different ideas, we stop a little bit to talk about factors facilitating or hindering collusion (conventional wisdom), and static approaches to dynamic price competition. 6.1 Collusion hindering (facilitating) factors Conventional wisdom suggests several factors may hinder or facilitate collusion. The point is that there are (always) two opposite forces working behind tacit collusion: the temptation to undercut and making current (short-run) profits; the retaliations (price wars) and future losses associated with undercutting. Secrecy: detection lags and few buyers (lumpiness); Multimarket contact; The existence of large sales; Decreasing returns to scale; Asymmetry: difficult to coordinate; Market concentration (number of firms). We will see later on how to model some of these intuitions.

2 Joe Chen Static approaches to dynamic price competition Because dynamic pricing behavior is difficult to analyze and the tool to do so is developed only recently, there is a considerable literature that attempts to formulize the dynamic aspects in a static content. We briefly mention the kinked demand theory and the conjectural variation approach Kinkeddemandtheory This theory tries to explain the absence of frequent price cutting in oligopolistic markets. Let there be two firms,i=1, 2, with unit cost c. The demand is: q = D(p). Letp f denote the focal price, or, the steady-state (long-run) price. Each firms reasons as follows: If p p f,therivalwillnotfollowsuit;if,instead,p<p f,therivalwillmatch.thus,firms: max (p i c)d(p i )/2 s.t. p i p f. When p f [c, p m ], p f is an equilibrium. This equilibrium results in a demand curve with a kink Conjectural variation approach This approach proposes a conjectural variation function to answer the question whether firms are colluding. Suppose firm i believes that firm j reacts according to e R j (a i ). Hence, firm i: max a i Π i (a i, e R j (a i )). Let a i = q i.ifπ i and e R j are differentiable, we have: FOC yields: max q i P (q i + e R j (q i ))q i C i (q i ). P C 0 i + P 0 (1 + e R 0 j)q i = P C 0 i + q i P 0 + q i P 0 e R 0 j =0.

3 Joe Chen 66 Hence, zero conjectural variation: e R 0 j =0, corresponds to Cournot competition, and e R 0 j = 1 corresponds to competitive outcome. As to the collusive outcome, it requires positive conjectural variation, i.e., e R j 0 > 0. The traditional criticisms of industry models using conjectural variations are its lack of a theoretical framework and that each firm s conjecture about the output response of the other firms would not be confirmed if such a firm actually altered its output level from the equilibrium. As a result, there has been considerable interest in oligopoly models with consistent conjectural variations. A conjectural variation is consistent if it is equivalent to the optimal response of the other firms at the equilibrium defined by that conjecture. Perry (1982) showed that when the number of firms is fixed, competitive behavior is consistent when marginal costs are constant; when marginal costs are rising, the consistent conjectural variation will be between competitive and Cournot behavior. Finally, when free entry is allowed and redefine consistency to account for such, the only competitive behavior will be consistent.

4 Joe Chen Supergames Consider the standard setup in the Bertrand game: Two firms produce perfect substitutes with the same marginal cost, c. The difference is that we replicate the basic Bertrand game T +1 times. The game is then called a repeated game, or a supergame, with Bertrand game as the stage (constituent) game. Let Π i (p it,p jt ) be firm i s profit at time t, t =0, 1,...,T, when it charges p it and its rival charges p jt. Each firm maximizes the present discounted value of its profits, X T t=0 δt Π i (p it,p jt );where: δ = e rτ,withan instantaneous interest rate r and the time between periods τ. Denote the history of the the game at time t as: H t (p i0,p j0 ; p i1,p j1 ;...; p i,t 1,p j,t 1 ). The price strategy p it (H t ) depends on the history of the game. We look for strategies to form a perfect equilibrium: for any history H t, firm i s strategy from date t on maximizes the present discounted value of profits given firm j 0 s strategy from t on. If the horizon is finite, i.e., T<, the game is a finite replication of the simple Bertrand game. For finitely repeated games, to find a perfect equilibrium, it is easiest to proceed by backward induction. Given H T,whatarethefirms choices of prices at the last period? For all H T, p i,t = p j,t = c. H T does not affect the profits in period T ; Each firm maximizes: Π i (p it,p jt ). What will be the equilibrium prices in period T 1? Again, for all H T 1, p i,t 1 = p j,t 1 = c. Price choices at T does not depend on what happens at T 1; Everything is as if T 1 were the last period.

5 Joe Chen 68 Following the same argument, the outcome of the (T +1)-period price game is the Bertrand solution repeated T +1times. The dynamic element contributes nothing to the model. Things change dramatically when the horizon is infinite; i.e., T =. On the one hand, the above result: p it = p jt = c, forallt, is still an equilibrium. In particular, consider a strategy to price at the marginal cost regardless of the history of the game. On the other hand, an interesting feature arises: the repeated Bertrand result is no longer the only equilibrium. Let p m be the monopoly price, and Π m be the monopoly profit. Consider a trigger strategy as follows: p it (H it )= p m c p m if t =0 if t>0, andh t =(p m,p m ;...; p m,p m ) otherwise. The strategy is called a trigger strategy because a single deviation triggers a halt to the cooperation. Note that the temptation to deviate from p m is a current profit ofπ m /2, the retaliation is a loss of future profits of: (δ + δ )Π m /2; hence, as long as: Π m /2 (δ + δ )Π m /2; or, equivalently, δ 1/2, the trigger strategy is an equilibrium strategy. This equilibrium is collusive in that the equilibrium price p it = p jt = p m >c; it is a tacit collusion because the collusion is enforced through a purely noncooperative manner. Naturally, this raises the question that the results of the supergame framework are not robust to finite-length price interaction. Note that the infinite horizon assumption need not be taken too seriously. Suppose that at each period there is a (constant) probability x (1 x) that the market continues (ends). Everything is as if the horizon were infinite and

6 Joe Chen 69 the firms s discount factor were equal to e δ xδ. Hence, if both δ and x are sufficiently high, supergame collusion can be enforced. As a matter of fact, one can replace p m with any p [c, p m ],andπ i [0, Π m ],the above result still holds. This is a special case of a very general result, known as the Folk theorem. The Folk theorem asserts that any pair of profits (Π i, Π j ) such that: Π i, Π j > 0 and Π i + Π j Π m, is a per-period equilibrium payoff for δ sufficiently close to one. Thisis to say: there exist strategies {p it (H t ), p jt (H t )} that form a perfect equilibrium where firm i (j) makes a per-period payoff: Π i (Π j ). 6.4 Folk theorem Actually, several results are called by game theory people the Folk theorem. Here we are interested in the version (infinitely repeated games of complete information) given by Friedman (1971). Consider an infinitely repeated n-player game with the static game (the constituent game) definedbyactionspacesa i for all i =1, 2,...,n, and payoff functions for player i, Π i (a 1,...,a i,...,a n ),wherea j A j for all j =1,...,n. Denote: a i (a 1,..., a i 1,a i+1,...,a n ). Since the game is repeated infinitely many times and we allow the strategies to be functions of past history, the total payoff is: V i = X t=0 δt Π i (a 1 (t),...,a n (t)), with average payoffs, v i,defined as: for all i, v i (1 δ)v i. We can define player i s reservation payoff, Π i, as the worst outcome (on average) that she can be forced to take; i.e., Π i =minmax Π i (a i,a i ). a i a i Apayoff vector Π =(Π 1,...,Π i,...,π n ) is individually rational if: for all j, Π j Π j. It is feasible if: there exists strategy a =(a 1,...,a i,...,a n ), such that: for all j, a j A j ; and, Π j (a) = Π j. The individually rational profits in the Bertrand price game or the Cournot quantity game are zeros. Moreover, any set of profits whose sum does not exceed the monopoly profit is feasible.

7 Joe Chen 70 The Folk theorem states that: Any average payoff vector that is better, for all players, than a Nash equilibrium payoff vector of the stage game can be sustained as the outcome of a perfect equilibrium of the infinitely repeated game if the players are sufficiently patient. Mathematically, denote Π in = Π i (a N 1,...,aN n ), where: Π i (a N i,an i ) Πi (a i,a N i ),forall a i A i,andforalli =1,...,n. Let v =(v 1,...,v n ) be feasible, and for all i, v i Π in. Then there exists δ 0 < 1, such that: for all δ δ 0, v is an equilibrium payoff vector. The proof of the Folk theorem is not difficult. Construct a trigger strategy with Nash threats (threats to revert to Nash behavior forever). By deviating today, a players gains at most a bounded amount; on the other hand, she loses: (v i Π in )(δ + δ ). Clearly, the loss goes to infinity as δ goes to 1. When the Nash equilibrium payoffs arethesameasthereservationpayoffs for the stage game (e.g., the Bertrand price game), this version of the Folk theorem gives a full description of the set of equilibria for δ close to 1. Fudenberg and Maskin (1986) show that: Every individually rational and feasible payoff vector can be enforced in perfect equilibrium for δ sufficiently close to 1. The multiplicity of equilibria is too successful in explaining tacit collusion. The multiplicity of equilibria is an embarrassment of riches. In many applications, we need to pick one out of these equilibria. It is natural to focus on symmetric equilibrium if the game is a symmetric one. In addition, it is natural to assume that firms coordinate on the equilibrium that is Pareto-optimal. In our content, it is the monopoly price, p m.

8 Joe Chen A few direct applications Market concentration On the one hand, more firms in the market sharing the profits implies a higher profit from undercutting. On the other hand, a large number of firms reduces the profit perfirm and thus the cost of being punished for undercutting. Hence, concentration facilitates collusion. Mathematically, consider a homogenous good industry with n-firms facing the same constant marginal cost. The constituent game is the Bertrand price competition. Suppose the (perfect) equilibrium is the one where all firms charge the monopoly price and share the market. It is enforced by Nash threats. Then, the benefit of undercutting is: Π m Π m /n; and, the cost of undercutting is: (δ + δ )Π m /n. Hence, it requires: δ (n 1)/n =1 1/n. The higher the concentration is (a smaller n), the lower the value of δ that is required to sustain collusion Information lags and infrequent interaction The latter is straightforward because it decreases δ and thus the cost of undercutting. The former (an information lag) increases the temptation to undercut since it takes time for the rival to detect undercutting (if happened). Both factors hinder collusion. For the former, consider the setup of 2 firms producing perfect substitutes with the same marginal cost. Suppose prices are observed two periods after they are chosen. Note that collusion is sustainable if and only if: Π m /2+δΠ m /2 (δ 2 + δ )Π m /2,

9 Joe Chen 72 or, δ 1/ 2 > 1/ Multimarket contact Consider two identical and independent markets, and both firms participate in both markets. Assume further that market A meets more frequently than market B. To be concise, let market A meets every period and market B meets every even periods. The implicit discount factor of market B is: δ 2. Suppose δ 2 < 1/2 <δ. In the absence of multimarket contact, collusion is sustainable in market A but not in market B. With multimarket contact, full collusion on both markets is sustainable if: 2 Πm 2 (δ + δ2 +...) Πm 2 +(δ2 + δ ) Πm 2 ; or, when: δ Hence, for δ =0.6, full collusion in both markets can be sustained under multimarket contact, whereas no amount of collusion is sustainable in market B under single-market contact. The intuition is that the loss of collusion on market A can be so large so that it actually facilitates collusion in market B. The concern of a general price warfare in all markets discourages firms from undercutting in any of the markets that they participate. Technically (mathematically), the incentive constraints for these two markets are pooled into a single constraint. If (originally) they are both satisfied, then the pooled constraint is satisfied. However when one of them is highly satisfied while the other is just a little bit short, it can be the case that the pooled constraint is satisfied Demand fluctuation Rotemberg and Saloner (1986) propose a theory of price wars during booms. Consider the typical setup of two (symmetric) firms with constant marginal cost competing in prices in a market where demand is stochastic with i.i.d. shocks over time. Let D L (p) <D H (p) for all p. At time t, demand can be low, D L (p), with probability 1/2, or high, D H (p), with

10 Joe Chen 73 probability 1/2. In each period, firms learn the state of the demand (low or high) before they make their pricing decisions. We look for a pair of prices {p L,p H } such that: Both firms charge p s where the state s {L, H}; {p L,p H } is sustainable in equilibrium; The expected present discounted profit is not Pareto dominated. Let p m s arg max p (p c)d s (p), andπ m s =(p m s c)d s (p m s ). Thefirst question to ask would be: Can the fully collusive outcome, {p m L,pm H }, be sustainable in equilibrium? Note that the payoff V when both firms charge the price configuration {p L,p H } is: V = X ½ 1 δ t 2 t=0 (p L c) D L(p L ) = (p L c)d L (p L )+(p H c)d H (p H ) 4(1 δ) = Π L(p L )+Π H (p H ). 4(1 δ) (p H c) D H(p H ) 2 So, when the fully collusive outcome is sustainable, V m =(Π m L + Πm H )/[4(1 δ)]. Thecost of undercutting at anytime t is: δv m ; and, the benefit of undercutting is (note that firms already know the state s): Π m s /2. Hence, if the fully collusive outcome is sustainable, then we must have: Π m s /2 δv m,foralls = L, H. Since Π m L /2 < Πm H /2, weneedtoconsider only: Π m H /2 δv m. Substitute for V m,wederive: ¾ δ 2Π m H Π m. L +3Πm H Denote 2Π m H /(Πm L +3Πm H ) as δ 0,wehave: δ 0 (1/2, 2/3). Therefore, when: δ [1/2,δ 0 ), the fully collusive outcome is not sustainable. What would be the price configuration when: δ [1/2,δ 0 )?Thefirms problem is: Π L (p L )+Π H (p H ) max p L,p H 4(1 δ) s.t. Π L (p L ) 2 δ Π L(p L )+Π H (p H ) 4(1 δ) Π H (p H ) 2 δ Π L(p L )+Π H (p H ) 4(1 δ).

11 Joe Chen 74 Since δ [1/2,δ 0 ), it has to be the case that one of the two incentive constraints is binding. Intuitively, the temptation to undercut is higher when demand is high. So, the binding constraint should be the second one. This gives us: Π H (p H )= and, the whole problem reduces to: µ δ Π L (p L ) KΠ L (p L ); 2 3δ (Π L (p L )+KΠ L (p L )) 1+K max max p L 4(1 δ) p L 4(1 δ) Π L(p L ). Hence, p L = p m L and p H can be solved by: Π H (p H )=KΠ L (p m L )=KΠm L. The important question to ask is: p H R p m H?NotethatK is increasing in δ, andwhenδ = δ 0, K = Π m H /Πm L. So, when δ<δ 0, Π H (p H )=KΠ m L < Πm H. This implies p H <p m H. Observe also that the first incentive constraint, Π L (p L ) KΠ H (p H ),issatisfied: Π m L K2 Π m L (when δ 1/2, K 1). We can conclude: When δ [1/2,δ 0 ), although the fully collusive outcome {p m L,pm H } is not sustainable, some kind of collusion is still possible. In the low state of demand, p L = p m L ; in the high state of demand, p H <p m H. This is interpreted as price wars during booms i.e., in booms, firms are forced to lower the price to prevent undercutting. Countercyclical price movement is consistent with this theory (though not necessarily because: p H R p m L ).