# Economics 384 B1. Intermediate Microeconomics II. Assignment 2. S Landon Winter 2007

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1 Economics 384 B1 Intermediate Microeconomics II Assignment 2 S Landon Winter 2007 Due: By 4:00pm, 2 April 2007 This assignment will be marked out of 100. The maximum number of marks that can be earned on each question is given in brackets next to the question. Your answers should be concise and to the point. All diagrams should be clear and easy to read. For numerical questions, you must show your calculations. Late assignments will not be accepted. 1. Consider a society with only two people, A and B. The marginal willingness to pay of person i (P i ) for a pure public good is: Person A: P A = 10 1G, Person B: P B = 20 2G, where G represents the number of units of the public good. Suppose the marginal cost of producing the pure public good is 15. (5) a) What is the optimal quantity of the pure public good? Show your work. (5) b) What is the total net benefit (that is total benefit total cost) at the quantity found in (a)? Show your work and also show that total net benefit falls if one more unit of G is provided. (5) c) How many units of the pure public good would each individual choose to purchase? Explain and show your work. (5) d) Suppose the good is non-rival, but excludable. If the government provided the optimal quantity of the pure public good, but charged a user fee equal to the average cost per user, would both individuals pay the user fee and use the good? Explain and show your work. S Landon

2 2. Suppose the marginal benefit (MB) function of an individual is: MB = 50 2Q where Q is the quantity of units consumed. (2) a) Suppose the marginal cost (and market price) of good Q is 10 dollars. How many units of the good will the individual consume? Show your work. b) Suppose that, for each additional unit of the good that this individual consumes, the members of society, other than the individual consuming the good, receive a benefit of 10 dollars. (5) i) Derive a function that gives the marginal social benefit (the marginal benefit to the individual plus the marginal benefit to the rest of society). Show your derivation and illustrate the marginal benefit and marginal social benefit curves. (5) ii) What is the socially optimal quantity of the good? Show your derivation. 3. Consider a monopolistically competitive firm. The total cost (TC) function of the firm is: TC = Q, where Q is the quantity produced by the firm. The initial demand function faced by the firm is: Q = 10 P, where P is the price charged by the firm. (4) a) Derive the firm s average cost and marginal cost functions. (4) b) Derive the firm s total revenue (TR), where TR=P Q, as a function of Q only. (4) c) Derive the firm s marginal revenue function (MR= TR/ Q). (4) d) Derive the short run quantity produced by the firm, the price it charges, its average cost, and its profits. (4) e) The positive profits found in (d) induce other firms to enter the industry. This causes a fall in the intercept of the demand function faced by the firm (as entry of other firms causes its demand curve to shift down). In the long run, a monopolistically competitive firm earns zero profits, but also maximizes profits. If, following the entry of other firms, the demand function faced by the firm is: S Landon

3 P = X Q, what is the value of X? That is, what is the value of the intercept associated with the demand function faced by the firm in the long run? Show your work. Hint: Two conditions must hold in the long run equilibrium and these allow you to solve for two unknowns, X and Q. The solution for X is not an integer. 4. Consider an industry with two identical firms, 1 and 2. These firms produce a homogeneous product. The inverse demand function faced by both firms is: P = 10 (Q 1 + Q 2 ), where P is the per unit price, Q 1 is the number of units of output produced by Firm 1 and Q 2 is the units of output produced by. The total cost function of Firm 1 is: TC 1 = 5Q 1. has an identical total cost function. Each firm chooses its quantity to maximize its profits under the usual Cournot assumption. Both firms choose output simultaneously and there is only one period to the model. The profits of Firm 1 (Π 1 ) are Π 1 = PQ 1 TC 1. The reaction function of Firm 1 gives the Firm 1 profit maximizing value of Q 1 as a function of Q 2. This reaction function can be derived directly from the equation: Π 1 / Q 1 = 0. The reaction function of can be derived from the equation: Π 2 / Q 2 = 0. (6) a) Find the reaction function for each of the two firms. Show your work. When answering this question, it is important to remember that P is a function of Q 1 and Q 2. (6) b) What are the Nash equilibrium values of Q 1, Q 2 and P? Show your work. (4) c) Suppose the two firms decided to collude. How much would the two firms produce jointly and what would be the market price? Show your work. (2) d) What would be the competitive total market quantity and price? Show your work. S Landon

4 5. Consider the following payoff matrix for two firms that have the choice of pricing high (H) or pricing low (L): H L Firm 1 H 2,2-2,4 L 4,-2 3,3 (2) d) Does the equilibrium change, relative to your answer to (b), if Firm 1 chooses first 6. Consider the following payoff matrix for two firms that have the choice of expanding capacity (E) or not expanding capacity (DE): DE E Firm 1 DE 4,4-2,5 E 5,-2 3,3 (2) d) If the agreement in (c) is not binding, what will happen? Explain. (2) e) Does the equilibrium change, relative to your answer to (b), if Firm 1 chooses first S Landon

5 7. Consider the following payoff matrix for two firms that have the choice of expanding capacity (E) or not expanding capacity (DE): DE E Firm 1 DE 4,4-2,2 E 2,-2 3,3 (2) d) Does the equilibrium change, relative to your answer to (b), if Firm 1 chooses first (2) e) Suppose the two firms choose simultaneously and do not collude. The government subsidizes the production of, but only if expands its capacity. The subsidy raises the payoff of by 3. What impact does the subsidy have on the behaviour of the two firms? Explain. (2) f) How would your answer to (e) change, if at all, if Firm 1 moves first? Explain. S Landon