Problem Set # 3. IO. Spring 2014 Note: This time the problem set has six questions. We will have to roll a dice to choose a question! 1. Cournot Consider the following problem. Two firms produce identical goods with a constant marginal cost of c > 0 and compete in quantities. Which of the following statements is correct? Reason your choice. (a) Firms do not increase their profits if they merge into a single firm and distribute the monopoly profit equally. (b) Both firms increase their profits if they merge into a single firm and distribute the monopoly profit equally. (c) If the marginal cost of one firm decreases, then the best response of the other firm is to increase the quantity it produces. (d) If the marginal cost of one of the two firms decreases, the aggregate quantity produced in equilibrium also decreases. 2. Cournot Model There are two firms producing cars: Ford and GM. The demand function for cars is given by D(p) = 10 − p, where p denotes price. Both firms have a marginal cost of 1 and a fixed cost of 0. (a) Write down the profit function of each firm. (b) Compute the first order condition to the problem of each firm. (c) Compute the best response functions. (d) Draw the best response functions (both in the same graph). (e) Compute the Nash equilibrium quantities and calculate the profits of each firm. (f) The EU’s environmental agency has decided to restrict per car emissions. In order to implement the change, the firms will have to use a more costly technology that has a marginal cost of 3. With the new technology, the emissions per car will be reduced in half. What will the aggregate effect be? Will total emissions be reduced by more or less than 50%? Justify your answer. Compute production and total emissions in both cases. 3. Bertrand Model Consider an industry in which there are two firms, A and B, producing a homogeneous good with marginal costs 0 ≤ cA , cB < 1, where cA and cB are the marginal costs of firms A and B respectively. Suppose the market demand is given by D(p) = 1 − p, for p ≥ 0. The firms only interact once and both firms choose the price simultaneously. Let pA and pB be the prices of the firms respectively. The produced goods are perfect substitutes, so the firm that charges the lowest price serves the entire market. If both firms sell at the same price, the demand is divided equally. There are no fixed costs of production. (a) Suppose cA = cB = 0 and there are no capacity constraints. What are the equilibrium prices? Calculate the profits. (b) Suppose now that 0 ≤< cA < pM < cB , where pM is the monopoly price, and there are no capacity constraints. What are the equilibrium prices? Compute the profits of each firm. (c) Suppose that 0 ≤ cA < cB < pM and there are no capacity constraints. Compute the equilibrium prices and profits. (d) Suppose that cA = cB = 51 and the capacity of each firm is 51 respectively. What are the equilibrium prices? Calculate the profits for each firm and compare your answer to the previous ones. 4. Cournot and Bertrand Suppose that there are N identical firms selling a homogenous good and with marginal costs of production equal to 2. The demand function for this good is given by D(p) = 10 − p. (a) Suppose the firms compete in quantities. Compute the price, the aggregated quantity and the profits of each firm. Show that the profit of each firm decreases in N . (b) Suppose the firms compete in prices. Compute the equilibrium prices, the aggregate quantity and the profits of each firm. (c) Suppose there are N = 2 firms and the 2 firms in prices. In order to reduce its deficit, the government imposes a tax t = 1 per unit sold by Firm 1. (Firm 2 does not pay taxes.) What will the government’s tax revenue be? (d) Assume, instead, the firms compete in quantities and compute the government’s tax collection. 5. Price Discrimination P&G is a firm producing laundry detergent. There are two distinct groups of consumers. We denote these groups 1 and 2, and assume they have equal proportions. Consumers have multi-unit demands given by: D1 (p1 ) = 1 − p and D2 (p) = 1 − 21 p. The cost function is equal to C(q) = 21 q. Answer the following questions: (a) The monopolist can use first-degree price discrimination. Calculate the optimal tariff.
(b) The monopolist can use third-degree price discrimination and separate the two demands. Compute the optimal tariff. (c) For all questions above, compute profits of the firm and the consumers’ surplus. What is the relationship between the dead-weight loss and the amount of information the firm has on the consumers’ willingness to pay? Explain. 6. Price Discrimination Consumers differ in their willingness-to-pay for flashiness when buying a car. There are two types of consumers in the population, types A and B, in proportion π and 1 − π, respectively. The following table summarizes the willingness-to-pay of the different types: Type A Type B
Flashy Car 100 80
Simple Car 50 40
Assume the marginal cost of production is equal to 0 and π = 1/2 and answer the following questions. (a) Suppose the firm can only offer a single product: a simple car or a flashy car. What product will it offer? At what price will it sell it? Compute its profits. (b) Now suppose the firm can price discriminate. Solve for the optimal prices and compute the firm’s profits. (c) Answer again the previous question by assuming that π can take any value in [0, 1]. How does your answer change with the value of π? Explain and derive.