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CAS EC 201. Intermediate Microeconomic Analysis. Problem set 9 (due on Thursday April 30, 2020) 1. In this exercise we examine a duopoly market. Firm one produces quantity y1 and firm two produces quantity y2. The market inverse demand function is p(y) = 300 − 20y. Firm one’s cost function is C1(y1) = 40y1 and firm two’s cost function is C2(y2) = 245 + 20y2. 1.1 Assuming firms compete à la Cournot, find the equilibrium quantity of output for each firm. Find the equilibrium market price and the profit earned by each firm. 1.2 Firm two has a lower marginal cost than firm one, but there is the constant 245 that appears in firm two’s cost function. Suppose this reflects the fact that firm two has a new technology that requires paying a licensing fee for a patent (the fixed cost) which then allows firm two to produce at a lower marginal cost. Hence, even in the long run, firm two cannot avoid paying the cost 245 if it is going to produce. Suppose firm one gets to choose its quantity first, and after seeing this quantity firm two gets to choose its quantity. What quantity maximizes one’s profits, and what profit does one earn? What profit does firm two earn in this case? 2. Assume now that the market demand function is p(y) = A − y, for some A ≥ 0 and both firms’ cost function is C(y) = y2 . Plot the best-reply function in a graph with y1 on the horizontal axis and y2 on the vertical axis. Find the Cournot-Nash equilibrium quantities. 3. Suppose a monopolist produces Soma which is used as an input by two industries producing two different goods (say Amos and Mosa). In turn, these two firms are monopolists in their respective industries. Assume that both firms convert one unit of Soma into one unit of output. The demand for Amos has a constant elasticity of demand of 3/2, and for Mosa the elasticity of demand is 2. That is, the inverse demand function for Amos is pAMOS(y) = y−2/3 and the inverse demand function for Mosa is pMOSA(y) = y−1/2. Finally, assume that the monopolist’s constant marginal cost of production is $1. Suppose the monopolist is able to charge different prices for the same good (Soma) to the two firms, an example of third-degree price discrimination. We want to determine the price it will charge to each of them. 3.1 Write the profit of the producer of Amos as a function of the quantity of Amos produced and the price charged by the producer of Soma, pSOMA. (Hint: the cost function of this firm is C(yAMOS) = pSOMAyAMOS. Why?) Find the profit- maximizing quantity for this firm. You have just derived the demand function of the producer of Amos for Soma. Why? 1 3.2 Using the demand you derived in the previous step, compute the optimal quantity the producer of Soma should sell to the producer of Amos and the optimal price it should charge. 3.3 Repeated the previous steps for the producer of Mosa. Which of the two firms is charged a lower price? Why? 4. From 1988 through 2005, over 46,000 intentions to merge were filed with the US Fed- eral Trade Commission and Department of Justice in accordance with the HartScott- Rodino Act. In reviewing a merger between two firms in the same industry anti-trust authorities must trade off the costs of monopoly power with the benefits of efficiency gains, usually lower costs, that may be passed through to consumers. To examine this trade-off formally, suppose two firms face a demand function p(y) = 1 − y. 4.1 Pre-merger, each of our firms has constant marginal cost c of production, and the two firms compete à la Cournot. Compute the Cournot-Nash equilibrium of the industry and the equilibrium price. 4.2 By merging, the firms could reduce their marginal costs of production. Post- merger, the merged firm will have a constant marginal cost of αc, where α < 1. the merged firm will act like a monopolist. what is the profit-maximizing quantity for the merged firm? what is the profit-maximizing price? 4.3 when does the merger increase consumer surplus? (for what values of α?) 2 1.="" the="" merged="" firm="" will="" act="" like="" a="" monopolist.="" what="" is="" the="" profit-maximizing="" quantity="" for="" the="" merged="" firm?="" what="" is="" the="" profit-maximizing="" price?="" 4.3="" when="" does="" the="" merger="" increase="" consumer="" surplus?="" (for="" what="" values="" of="" α?)="">