Micro Theory Exam May 6, 2011 Part I. True‐False Questions Answer ALL of the following nine true‐false‐uncertain questions and, of course, explain your answers. 1. If a competitive firm operates at a profit‐maximizing output level y > 0, and has zero economic profit, then it must be facing constant returns to scale. 2. In a 2‐good world, the two goods must be net (Hicksian) substitutes. 3. A profit‐maximizing firm’s “derived factor demand” for factor j, that is, xj(w,p) where w is the vector of factor prices and p is the output price, can be upward sloping with respect to wj if there is a negative “output effect” which outweighs the “substitution effect.” 4. Consider a Cournot duopoly in which each firm has constant marginal costs, but one is higher than the other. If market demand is linear and if Firm 1's cost increases, then Firm 2's equilibrium production quantity will increase. 5. If a consumer’s utility function is of the quasi‐linear form U = a x1 + b ln x2, then for one of the goods, the Marshallian and Hicksian demand curves will have the same slope. 6. The “compensating variation” measure of the impact of a price increase will always be greater than the “equivalent variation” measure. 7. In the game below, (L,A,p=1) and (R,B, p≤1/4) are pure strategy perfect Bayesian equilibria. R
1 2,2 L M
[p] A 3,1 2
B [1‐p]
A
0,0 1,0
B
0,1
8. Consider an economy with 2 goods and 2 agents. Suppose u1(x, y) = 3/4log(x) + 1/4log(y) and u2(x, y) = min(2x, y). Let the endowments be e1 = (e1x, e1y) = (5, 5) and e2 = (e2x, e2y) = (10, 10). The Walrasian equilibrium involves agent one consuming 6.4 units of x and 12.8 units of y. 9. Suppose that three people have decided to rent an apartment, and have agreed to split the rent equally. They have found three possible apartments to rent: A, B, and C. The gross willingness to pay per month of each person for each apartment and its rent are as follows: A B C Person 1 $300 $350 $440 Person 2 250 200 400 Person 3 400 400 250 Rent 510 600 750 Under the demand‐revealing process and an assumption that rent is split equally three ways, Person 3 is pivotal and pays an incentive tax of 90. Part II Section A Answer ONE of the following two questions 1. In a competitive industry, the long‐run total operating cost (i.e., the cost of labor and capital) for each firm is given by TOC = 0.5q ‐ 10q2 . In addition, each firm needs an entrepreneur with the talent to oversee the whole operation. The supply of entrepreneurs is given by N = 0.25w where w is the annual salary paid to the entrepreneur. Therefore, the long‐
run total cost of a typical firm would be TC = 0.5q2 ‐ 10q + w. a. Suppose the market demand for the industry’s output is given by Q = 1,500 ‐ 50P. Find the long‐run equilibrium price, the quantity of output produced by a given firm and by the industry, and the number of firms in the industry. b. Suppose the demand for the industry’s output changes to Q’ = 2,428 ‐ 50P. Find the new long‐run equilibrium price, the quantity of output produced by the firm, and by the industry, and the number of firms in the industry. c. Illustrate your results in parts a. and b. in a diagram. d. Calculate Profit, Producer Surplus, Consumer Surplus, and Total Surplus for the equilibrium in part b. e. What would happen if this industry became monopolized? Be as specific about details as possible. 2. Suppose a consumer has preferences represented by the utility function U = ln X + ln H, where X is consumption goods and H is health services. Let prices be PX and PH and let income be I = 900. a If PX = 2 and PH = 2, what will be the consumer's optimal demand combination? Call these amounts X0 and H0. b. The consumer is offered a choice of two possible health care plans: Plan 1: The price of health care services is reduced from 2 to 1. Plan 2: A certain amount of health care service is offered free ‐‐ suppose that amount is equal to H0 / 2 ‐‐ and the price of additional service beyond that level remains at 2. Which plan will the consumer prefer? c. What will be the consumer's willingness to pay (compensating variation) for Plan 1 and for Plan 2? d. Discuss how your answers to parts b. and c. relate to each other. Part II Section B Answer ONE of the following two questions 1. Firm U is the world’s single supplier of widgets, which it produces at zero cost. Widgets are used as inputs to make gadgets. There are two firms, A and B, which convert widgets into gadgets. Each of these firms purchases a quantity of widgets from Firm U, coverts the widgets at no cost into gadgets and places the gadgets on the market. One widget makes one gadget. Market demand is such that if Q gadgets get placed on the market, the resulting price will be P(Q)=1‐Q. (a) Suppose that Firm U sets a fixed price r per widget. How many widgets will A and B purchase and what will be the equilibrium market price? (b) What is Firm U’s profit maximizing price for widgets? What are the resulting industry profits (for all firms combined) and how do these differ from a situation where U could transform widgets to gadgets itself and bypass firms A and B? Now suppose that firm U could offer a two‐part tariff for widgets. It could charge some amount F for the privilege of being a customer and then offer an unlimited amount of widgets at a per‐
unit price r. (c) What is Firm U’s profit‐maximizing two‐part tariff if it uses a tariff that will induce both firms A and B to produce positive quantities of gadgets? (d) Is there an alternative tariff that Firm U could offer that would result in the same or higher profits? 2. Mr. A has a car he would like to sell. He places zero value on it and he has a buyer who is interested in buying the car. The buyer’s willingness to pay depends on how long the car will last. If the car will last Θ months, the buyer is willing to pay a+b Θ, where a,b>0. Both parties know a and b, but only A knows precisely how long the car will last (i.e. knows Θ). The buyer believes Θ ~U[0,12]. To model the interaction as a game, suppose that A learns Θ, then makes a price demand p. The buyer then decides to accept or reject the offer. (a) Show that in any Perfect Bayesian Equilibrium (PBE) all seller types will sell at a single price. (b) What set of prices are consistent with this pooling equilibrium? What is the highest price consistent with this sort of equilibrium? Now, suppose that S comes onto the scene and suggests that A offer a warranty. With this possibility, A can now ask for a price p, but guarantee a full refund if the car does not last for some amount of time, say x, months. Thus an offer is a pair (p,x). (c) Construct a PBE in which A either offers a six month warranty at some price p6, or no warranty and a price p0. What are the highest prices p6, p0 that are consistent with such an equilibrium? What beliefs must the buyer have if he sees an unexpected offer of an x month warranty and price px?