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CHAPTER 27 OLIGOPOLY 27.1 Choosing a Strategy Oligopoly: There are a number of competitors in the market, but not so many as to regard each of them as having a negligible effect on price. Duopoly: Two firms only. 27.1 Choosing a Strategy Sequential game: Players take actions in a given order. Quantity leader: The firm who chooses its quantity first. Quantity follower: The firm who chooses its quantity following the leader. 27.2 Quantity Leadership The follower chooses its output, given the output of the leader. max p( y1 y2 ) y2 c2 ( y2 ) y2 F.O.C. MR2 p( y1 y2 ) p( y1 y2 ) y2 MC2 Reaction curve: the optimal y2 as a function of y1. y2 f 2 ( y1 ) 27.2 Quantity Leadership Linear demand: p( y ) a by Zero marginal cost for simplicity. The profit function of firm 2: 2 ( y1 , y2 ) [a b( y1 y2 )] y2 Isoprofit: the locus of (y1, y2) that yield a constant level of profit. ay2 by1 y2 by22 2 Reaction curve: a by1 y2 2b 27.2 Quantity Leadership Higher profits for inner isoprofits. The reaction curve attains the maximal profit for a given y1. The reaction curve passes though the top of the isoprofits. 27.2 Quantity Leadership The leader chooses y1 in recognition of the follower’s reactions. max p( y1 f 2 ( y1 )) y1 c1 ( y1 ) y1 Profit function: 1 ( y1 , y2 ) p( y1 y2 ) y1 ay1 by12 by1 y2 F.O.C.: a b 2 y1 y1 2 2 a * y1 2b 27.2 Quantity Leadership The leader knows that the system will settle on firm 2’s reaction curve. The leader choose the lowest isoprofit on firm 2’s reaction curve. 27.3 Price Leadership The leader has set a price p; The follower take this price as given and set its output S(p): max py2 c2 ( y2 ) y2 The demand faced by the leader is the residual demand R(p)=D(p)-S(p); The leader’s problem: max( p c) R( p) p 27.3 Price Leadership 27.3 Price Leadership Follower’s cost function: c2(y)=y2/2 Follower’s supply curve: y2=p Linear market demand: D(p)=a-bp Residual demand: R(p)=a-(b+1)p The leader’s problem: 1 max (a y1 ) y1 cy1 y1 b 1 The leader’s output: a c (b 1) y 2 * 1 27.5 Simultaneous Quantity Setting Each firm forms a belief about the other firm’s output; Each firm chooses its profit-maximizing output according to this belief; Nash equilibrium: the belief is consistent with the outcome and no firm has incentives for further adjustment. 27.5 Simultaneous Quantity Setting Firm 1 expects that firm 2 produces y2e units; Firm 2 expects that firm 1 produces y1e units; Firm 1’s problem: max p( y1 y ) y1 c( y1 ) y1 e 2 Firm 1’s output: y1=f1(y2e); Firm 2’s problem: max p( y1e y2 ) y2 c( y2 ) y2 Firm 2’s output: y2=f2(y1e). 27.5 Simultaneous Quantity Setting The equilibrium satisfies y1*=f1(y2*) and y2*=f2(y1*). The equilibrium is given by the intersection of the two reaction curves. 27.6 An Example of Cournot Equilibrium Linear market demand: p(y)=a-by; Zero marginal cost; a by2e a by1e y1 y2 The reaction curves: 2b 2b Equilibrium conditions: y1=y1e, y2=y2e a by2 y1 2b a by1 y2 2b Equilibrium outputs: a y y 3b * 1 * 2 27.7 Adjustment to Equilibrium 27.8 Many Firms in Cournot equilibrium Suppose that there are n firms; Total industry output: Y=y1+…+yn; Firm i’s F.O.C.: p p(Y ) yi MC ( yi ) Y p Y yi p(Y ) 1 MC ( yi ) Y p(Y ) Y 27.8 Many Firms in Cournot equilibrium Firm i’s market share: si=yi/Y si p(Y ) 1 MC ( yi ) (Y ) With a large number of firms, each firm’s market share is negligible, and the Cournot equilibrium is effectively the same as pure competition. 27.9 Simultaneous Price Setting Firms set their prices and let the market determine the quantity sold; Assuming constant marginal cost c; Prices can never be lower than c; If ph>c, the low bidder can always increase its profits by charging a slightly lower price; Both firms charging p=c is the unique equilibrium. 27.10 Collusion Cartel: Firms set prices and outputs so as to maximize total industry profits. Cartel’s problem: max p( y1 y2 )[ y1 y2 ] c1 ( y1 ) c2 ( y2 ) y1 , y2 F.O.C.: * * * * * p( y y ) p ( y1 y2 )( y1 y2 ) MC1 ( y1 ) * 1 * 2 p( y1* y2* ) p( y1* y2* )( y1* y2* ) MC2 ( y2* ) 27.10 Collusion Marginal profits of firm 1: * * * * MR1 ( y ) p( y y ) p ( y1 y2 ) y1 MC1 ( y1 ) * 1 * 1 * 2 p( y1* y2* ) y2* 0 Each firm has incentives to deviate from the cartel solution. Cartel is unstable in the short run. Punishment mechanism is needed to maintain the cartel in the long run. 27.10 Collusion 27.11 Punishment Strategies Punishment strategy Coorperation stage: produce the Cartel quantity until a deviation is observed; Punishment stage: produce the Cournot quantity for ever. The firm being punished will choose the Cournot quantity in response. 27.11 Punishment Strategies Present value of cartel behavior: m m m m m 2 1 r r Present value of cheating: c c c d d 2 1 r (1 r ) (1 r ) r Cheating won’t happen if m c m d r m c r d m r