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# notes_11 - Game Theory and Oligopoly 1 Oligopoly Models...

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Game Theory and Oligopoly April 26, 2009 1 Oligopoly Models Exercise 1 The demand is given by P = 100 - Q where Q = q 1 + q 2 (there are only two ﬁrms in the market). The two ﬁrms have similar marginal costs MC i = 20 + q i for i = 1 , 2. a) Find the equilibrium quantities and price in the Cournot equilib- rium. The ﬁrms choose quantities individually, taking the action of the rival as given. For ﬁrm 1 the residual demand given q 2 is ˜ P = (100 - q 2 ) - q 1 and the marginal revenue is MR 1 = (100 - q 2 ) - 2 q 1 Optimality requires MR 1 = MC 1 , so (100 - q 2 ) - 2 q 1 = 20 + q 1 = q 1 ( q 2 ) = 80 - q 2 3 For the second ﬁrm (by symmetry) q 2 ( q 1 ) = 80 - q 1 3 Since the problem is symmetric (ﬁrms have the same marginal cost) we know that they will choose the same quantity in equilibrium, therefore q 1 = q 2 = q c . Now from the best response, we can substitute q c = 80 - q c 3 = q c = 20 = q * 1 = q * 2 The price will be P = 100 - (20 + 20) = 60 1

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b) Find the optimal quantities and price in the case the two ﬁrms formed a proﬁt maximizing cartel. A cartel would coordinate individual ﬁrms’ decisions to maximize industry proﬁts, therefore max q 1 ,q 2 [100 - q 1 - q 2 ]( q 1 + q 2 ) - TC 1 - TC 2 Optimization implies the two conditions 100 - 2( q 1 + q 2 ) = MC 1 = 20 + q 1 100 - 2( q 1 + q 2 ) = MC 2 = 20 + q 2 Assuming simmetry, q 1 = q 2 = q , 100 - 2(2 q ) = 20 + q = q = 16 = q 1 = q 2 for a price of P cartel = 100 - (16 + 16) = 68 > P cournot c) What would the price be in perfect competition? The equilibrium price would be at the intersection of demand and supply: we need aggregate supply. To do so, we need the individual supplies (we assume no ﬁxed costs): S i ( P ) = MC i = P = 20 + q i = P = q i = P - 20 and the aggregate supply is just S ( P ) = S 1 ( P ) + S 2 ( P ) = 2 P - 40 = P = Q 2 - 20 The equilibrium price is given by D ( P ) = S ( P ) = 100 - Q = Q 2 - 20 so that Q * = 80, and by symmetry each ﬁrm will produce half of that quan- tity q * 1 = q * 2 = 40. The competitive price is P = 100 - 80 = 20 Exercise: Stackelberg The demand is P = 100 - ( q 1 + q 2 ); each ﬁrm has marginal cost equal to MC = 10. Find the Stackelberg equilibrium in which ﬁrm 1 is the leader and ﬁrm 2 is the follower. 2
Since ﬁrm 1 is the leader, ﬁrm 2 is the last to decide; sequential models have to be solved backward, so the problem of the follower is the ﬁrst one that we need to solve. Firm 2 takes q 1 as given, therefore its residual demand will be ˜ P 1 = (100 - q 1 ) - q 2 and the marginal revenue MR 1 = (100 - q 1 ) - 2 q 2 Optimality requires MR 1 = MC 1 , so MR 1 = (100 - q 1 ) - 2 q 2 = 10 = q 2 ( q 1 ) = 90 - q 1 2 is the best response.

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notes_11 - Game Theory and Oligopoly 1 Oligopoly Models...

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