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Additional Exercises

Additional Exercises - Economics 210C Exercises Spring 2011...

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Economics 210C Exercises Spring 2011 1. Consider a modification of the Cournot duopoly model, under which producers can revise their supply decision to match their rival’s supply if it exceeds their own. More specifically, the firms play a market game that has the following two stages: Stage 1: The firms choose simultaneously what quantity to supply. Stage 2: The firms are informed of each other’s quantities chosen in stage 1. A firm with a smaller quantity can either keep its quantity or revise it to match the quantity of the other firm. Let p ( Q ) = a - bQ be the inverse demand and C i ( q i ) = cq i the cost function of firm i , i = 1 , 2. Assume a, b, c > 0 and a > c . Can there be a subgame-perfect equilibrium for the above 2-stage game, in which each firm chooses 1/2 of the monopoly quantity? Support your answer. 2. Consider a Cournot oligopoly with three firms. Let p ( Q ) = 150 - Q, 0 be the inverse demand and C i ( q i ) = 18 q i + q 2 i the cost function of firm i = 1 , 2 , 3. (a) Show that any two firms would have profit incentives to merge into one. (b) Show that consumers become worse off when two firms merge into one. 3. Two firms, A and B, produce a homogeneous good which they sell in two markets, 1 and 2. Firm i = A, B has cost function C ( q i 1 + q i 2 ) = 1 2 ( q i 1 + q i 2 ) 2 , where q i 1 and q i 2 are the quantities firm i sells in market 1 and market 2, respectively. The inverse demand function is the same in each market and is given by P ( Q k ) = 20 - Q k , where Q k = q A k + q B k for k = 1 , 2. (a) Write down firm A’s profit (from sales in both markets) as a function of q A 1 , q A 2 , q B 1 , q B 2 . Answer: Denote firm A’s profit at quantities q A 1 , q A 2 , q B 1 , q B 2 by Π A ( q A 1 , q A 2 , q B 1 , q B 2 ). Note first that firm A’s total production cost at quantities q A 1 , q A 2 , q B 1 , q B 2 is 1 2 ( q A 1 + q A 2 ) 2 1
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and its total revenue is (20 - q A 1 - q B 1 ) q A 1 + (20 - q A 2 - q B 2 ) q A 2 . Thus, Π A ( q A 1 , q A 2 , q B 1 , q B 2 ) = (20 - q A 1 - q B 1 ) q A 1 +(20 - q A 2 - q B 2 ) q A 2 - 1 2 ( q A 1 + q A 2 ) 2 . Firm B’s profit function can be similarly determined. (b) Consider the case where the firms sell the good in market 1 and market 2 simultaneously. Write down the strategy set for each firm and find the Nash equilibrium. Answer: Note that firms’ production capacities are not limited. This means that each firm can supply any nonnegative quantity to either market. We conclude that when the firms sell the good in both markets simultaneously, a generic strategy for either firm consists of two nonnegative quantities, with one to be sold in market 1 and the other in market 2. Thus, firm A’s strategy set is S A = { ( q A 1 , q A 2 ) | q A 1 0 , q A 2 0 } and firm B’s strategy set is S B = { ( q B 1 , q B 2 ) | q B 1 0 , q B 2 0 } . The market game in this case is one in which the firms’ strategy sets are as above and their payoff functions are those in part (a). Let (( q * A 1 , q * A 2 ) , ( q * B 1 , q * B 2 )) be a Nash equilibrium. Then, ( q * A 1 , q * A 2 ) solves max ( q A 1 ,q A 2 ) S A Π A ( q A 1 , q A 2 , q * B 1 , q * B 2 ) and ( q * B 1 , q * B 2 ) solves max ( q B 1 ,q B 2 ) S B Π B ( q * A 1 , q * A 2 , q B 1 , q B 2 ) .
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