The homework is due on Wednesday, December 7 at 4pm. Each question is worth 0.8 points.

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1 Homework 9: Econ500 Fall, 2016 The homework is due on Wednesday, December 7 at 4pm. Each question is worth 0.8 points. Question 1 Suppose that all firms in a competitive industry have cost function c(q)= Q The demand function is Q D (P)=1, P. (a) Suppose that there are currently 10 firms in that industry. Determine equilibrium prices and profits. (b) Now suppose there is free entry, and firms will enter until equilibrium profits are zero. How many firms will operate in equilibrium? (c) Suppose we are again in (a) with 10 firms, but these firms manage to lobby policy makers to introduce a law that prevents new entry. Suppose that there 1,000 consumers with the same utility functions u(q, m) = 100q 50q 2 + m, where q is the firm s product and m money spent on other goods. Each consumer has an income I = 200. You can check (but you don t have to) that aggregate demand for the firm s product is then Q D (P)=1, P. Since m enters linearly, we can interpret utilities as monetary amounts, i.e., we can add utilities of consumers together and compare them to firm profits. Determine the utility loss of each consumer from the new law, i.e., compare the utility from (a) to that from (b). Don t forget that purchasing q units of the firm s product reduces consumption of other goods by qp, where P is the price charged by the firms. To get the total utility loss just add up the losses of all consumers. Now determine the profit gains of the 10 firms if the law is introduced and new entry is prevented, i.e., compare profits in (a) to those in (b) for each firm, and then multiply this by the number of firms, i.e., 10. Is this law a good idea? What are the total gains or losses to society if the law is introduced? Question 2 A monopolist sells a product in two different markets. Demand in the first market is Q=100 P. Demand in the second market is Q=100 2P. The firm s cost function is given by C(Q) = 10Q. (a) Suppose the firm can charge different prices in the two markets. Determine the prices and the firm s profit. (b) Now suppose that the firm must charge the same price in both markets. What is the firm s profit now. (c) Determine the the total change in surplus (consumer + firm) between (a) and (b). 1

2 Question 3 Suppose there are 100 households whose demand for electricity is given by Q=20 P. The local power company has a constant marginal cost of 2, and a fixed cost of 10,000. (a) Suppose the power company charges a price per unit of electricity that maximizes profits. Determine the price, the demand of each household, and the company s profit (Recall that you must use aggregate demand of all 100 households. Q=40 Pis the demand of a single household). (b) Now suppose that the power company uses two part pricing. It selects a price P per unit and a fixed fee F that maximizes profit. Determine P, F, and the firm s profit (Now you must use individual demand Q = 40 P to determine P and F. Of course, in order to determine profits you must use the fact that there are a total of 100 such households.) (c) Determine the efficiency gain from using two-part pricing instead of a price per unit. This efficiency gain is measured by the sum of the following: (a) the change of firm profits; (b) the change of consumer utility. (d) Now suppose that the government wants to regulate the power company. In particular, the government wants (i) production to be efficient and (ii) firm profits to be zero. This objective can be achieved by charging a fixed fee F in addition to a price per unit P. Determine F and P. Question 4 There are 200 firms in an industry. Half of the firms use newer technology resulting in a cost function c(q)=200+q 2, while the the remaining firms cost functions are c(q)=200+2q 2. Market demand is Q D (P)=3, P. Thus, P(Q) = Q. The industry is competitive (i..e., P = MC.) (a) Determine the equilibrium price and quantity and the profits of both types of firms. (b) Now suppose that the firms with the inferior technology exit the market (their profit in (a) should be negative). Determine the new equilibrium price, quantity and firm profits. (c) Determine the loss to the consumers when the high-cost firms exit the market (recall that the area underneath P(Q) between two values Q 1 and Q 2 measures the benefit to all consumers from increasing consumption from Q 1 to Q 2 ). (d) Taking the effect of firm profits into account determine by how much welfare increases or decreases when the high-cost firms exit the market. (e) Should the government subsidize the industry such that all producers will remain in the market? Your answer should be based on the argument in (d). Does your answer change if the high-cost firms are domestic firms and the low-cost firms are foreign, and all the demand for the product is from domestic consumers only? Question 5 Consider a Cournot Oligopoly. One firm has costs C 1 (Q 1 )=10Q 1 while the other firm s cost function is C 2 (Q 2 )=12Q 2. The demand for both firms products Q=Q 1 + Q 2 is Q D (P)=200 2P. 2

3 (a) Determine the equilibrium price P, the market shares s 1, s 2, and the quantities Q 1, Q 2 produced by both firms. (b) Suppose two more firms with the newer technology enter the market. What are the profits of both types of firms? Should the firm with the old technology exit? Question 6 Suppose a market demand function is given by Q D (P)=1, P. The product can be produced with a cost function c(q) = 10, Q. (a) Determine Q and P and the firm s profit if there is a single firm. (b) Determine total output Q, the equilibrium price, and the profit of each firm if there are two firms (i.e., a Cournot oligopoly). (c) Determine Q, the equilibrium price, and the profit of each firm if there are 10 firms (again assume a Cournot oligopoly). (d) Determine the welfare gain or loss between situations (a) and (b). Question 7 Suppose there are two types of consumers, type 1 and 2, with demand functions P(Q 1 )=50 Q 1 and P(Q 2 )=50 4Q 2. Suppose that the firm s cost function is C(Q)=5Q. The firm wants to offer two pricing schedules: (F 1, Q 1 ), (F 2, Q 2 ), where F i is the price the consumer must pay to get Q i units of consumption. The pricing schedules must be selected such that type 1 consumers choose (F 1, Q 1 ) and type 2 consumers (F 2, Q 2 ). One can show that the optimal Q 1 and Q 2 solve P 1 (Q 1 ) = MC and P 2 (Q 2 ) = MC+ n 1 n 2 (P 1 (Q 2 ) P 2 (Q 2 )), where n 1 and n 2 are the number of type 1 and type 2 consumers, respectively. (a) Suppose that n 1 = 200 and n 2 = 600. Determine Q 1, Q 2, F 1 and F 2 and the firm s profit. (b) Now suppose that the firm offers the efficient level of consumption also to type 2 costumers (n 1 and n 2 remains the same as in (a)). Determine Q 1, Q 2, F 1 and F 2 and the firm s profit (Note that type 1 consumers should still be better off choosing contract 1). (c) Now suppose that n 1 = 200 and n 2 = 100. Determine Q 1, Q 2, F 1 and F 2. What happens to the distortion in Q 2 does it increase of decrease compared to (a)? Determine which types are better off and which are worse off compared to (a). Question 8 Suppose that demand is given Q D (P)=40 2P. A firm s cost function is given by C(Q)=Q 2. Determine the equilibrium price and quantity, and the price elasticities for supply and demand at the equilibrium. Using the attached grid, do the following graphically. 1. Graph the marginal revenue curve if the firm is a monopolist. 3

4 2. Determine graphically the monopolist s profit maximizing choice of P and Q. 3. Now suppose that this firm sets price equal to marginal costs. Determine P and Q. 4. Suppose the firms uses two-part pricing. Because such a firm should set P = MC, the price is the same as in the previous question. Determine the area that corresponds to the fixed fee, F. 4

5 P Q