1102 the question is complete

There are N > 2 profit-maximizing firms that compete as Cournot oligopolists in a market with inverse demand given by P(Q) = a Q, where P is price, Q = PN i=1 qi and qi is the output of firm i for i = 1, . . . N. Assume that all firms have the same cost function: C(qi) = cqi . (a) Solve for Cournot equilibrium. Show that the profit for each firm in Cournot equilibrium, C i , is equal to the square of output: C i = q 2 i . (b) Suppose that prior to choosing quantities, two firms have the option to merge into a single firm. Compare the profit of the single merged firm in this new Cournot equilibrium, with a total of N 1 firms, to the sum of the two firms' individual profits in the original Cournot equilibrium, where there were N firms. Show whether the firms should merge or not. (c) Now suppose N firms compete in an infinitely repeated game in which in each period they simultaneously choose qi . Let be the discount factor between periods (0 < < 1). Let t = 0, 1, 2, . . . represent the time period and let QM denote the monopoly level of output. Suppose that each firm plays the following simple trigger strategy: Set qi = QM/N in period t = 0 and in all t > 0 for which aggregate output was QM in all preceding periods, and Set qi equal to the Cournot equilibrium output for all t > 0 for which aggregate output was not QM in all preceding periods. For what range of the discount factor can the trigger strategy support collusion at the monopoly output level as a subgame-perfect equilibrium outcome?

A painting is to be auctioned in a sealed-bid auction in which all bidders simultaneously submit a bid and the highest bid wins the painting. Suppose there are only two bidders, i = 1, 2. Let bi be the bid of player i. The value of owning the painting for bidder i is vi . Assume that each vi is drawn from a uniform distribution on [0, 1]. Each bidder knows her own value but not the other bidder's value. (a) In a first-price sealed-bid auction the bidder with the highest bid wins the painting and pays her bid to the auctioneer. Show that bidding bi = vi/2 is a Bayesian Nash equilibrium. (b) In a second-price sealed-bid auction the bidder with the highest bid wins the painting and pays the bid of the second-highest bid to the auctioneer. Show that it is a dominant strategy for each bidder to bid bi = vi . (c) The auctioneer's goal is to maximize the expected revenue from the auction. Show whether the auctioneer should choose the first-price sealed-bid auction or the second-price sealed-bid auction, or whether the auctions generate the same expected revenue. (Hint: A potentially useful mathematical fact is that the expected value of the maximum of two independent random variables uniformly distributed on [0, x] is 2x/3.)

Consider an exchange economy with two consumers, j = 1, 2, and two goods x 1 and x 2 . Consumer 1 has an endowment of 1 = (6, 0) and consumer 2 has an endowment of 2 = (0, 6). Subscripts denote consumers and superscripts denote goods. Utility functions are given by U1(x 1 1 , x2 1 ) = x 1 1x 2 1 if x 2 1 2x 1 1 x 1 1 + x 2 1 if x 1 1 /2 < x2 1 < 2x 1 1 x 1 1x 2 1 if x 2 1 x 1 1 /2 and U2(x 1 2 , x2 2 ) = x 1 2x 2 2 . (Consumer 1's indifference curves have a flat portion between the rays along which x 2 1 = 2x 1 1 and x 2 1 = x 1 1 /2.) (a) Derive the offer curves for the two consumers. Solve for a Walrasian equilibrium allocation and prices (x , p ). You may include a carefully labeled diagram as part of your answer if you wish, but this is not required. (b) Derive the contract curve for this economy. (This is the set that Mas-Colell calls the Pareto set.) Prove that the Walrasian equilibrium allocation you found in part (a) is Pareto optimal. (c) Consider an endowment vector R 4 ++ with the property that the aggregate endowment is the same as for : for each good i, i 1 + i 2 = 6. Derive expressions for (i) the associated equilibrium price vector p() and (ii) the associated equilibrium allocation x().

. Consumer 1 has an endowment of 1 = (6, 0) and consumer 2 has an endowment of 2 = (0, 6). Subscripts denote consumers and superscripts denote goods. Utility functions are given by U1(x 1 1 , x2 1 ) = x 1 1x 2 1 if x 2 1 2x 1 1 x 1 1 + x 2 1 if x 1 1 /2 < x2 1 < 2x 1 1 x 1 1x 2 1 if x 2 1 x 1 1 /2 and U2(x 1 2 , x2 2 ) = x 1 2x 2 2 . (Consumer 1's indifference curves have a flat portion between the rays along which x 2 1 = 2x 1 1 and x 2 1 = x 1 1 /2.) (a) Derive the offer curves for the two consumers. Solve for a Walrasian equilibrium allocation and prices (x , p ). You may include a carefully labeled diagram as part of your answer if you wish, but this is not required. (b) Derive the contract curve for this economy. (This is the set that Mas-Colell calls the Pareto set.) Prove that the Walrasian equilibrium allocation you found in part (a) is Pareto optimal. (c) Consider an endowment vector R 4 ++ with the property that the aggregate endowment is the same as for : for each good i, i 1 + i 2 = 6. Derive expressions for (i) the associated equilibrium price vector p() and (ii) the associated equilibrium allocation x().