Summer 2009 examination EC202 Microeconomic Principles II 2008/2009 syllabus Instructions to candidates Time allowed: 3 hours. This paper contains nine questions in three sections. Answer question one (section A) and THREE other questions, at least one from section B and at least one from section C. Question one carries 40% of the total marks; the other questions each carry 20% of the total marks. Calculators are NOT permitted clse 2009/EC202 1 of 9
Section A 1. Answer any ve questions from (a)-(h). Each question carries eight marks. (a) Consider a rm producing a single output from m > 1 inputs. Suppose that for two subsets of input space Z and Z 0 the following are true: z 2 Z ) (tz) > t (z) z 2 Z 0 ) (tz) < t (z) where is the production function, z is an input vector and t is a number greater than 1. Show that average cost must be strictly decreasing for z 2 Z and strictly increasing for z 2 Z 0. (b) In the context of standard consumer theory which of the following statements (if any) are true? In each case give brief reasons for your answers: i. The own-price substitution e ect will be negative if and only if the consumer s indi erence curves are convex to the origin. ii. The e ect on the compensated demand for good i of a fall in the price of good j must equal the e ect on the compensated demand for good j of a fall in the price of good i. iii. The e ect on the ordinary demand for good i of a fall in the price of good j must equal the e ect on the ordinary demand for good j of a fall in the price of good i. iv. The demand for a good must be a decreasing function of its price. (c) Explain how (i) Walras law and (ii) the fact that individual demand and supply functions are homogeneous of degree zero simplify the problem of nding general-equilibrium prices in a model with n commodities. clse 2009/EC202 2 of 9
(d) In an uncertain world where there are exactly two possible states i = 0; 1, an individual has preferences represented by the function exp ( y 0 ) exp ( y 1 ) where y i is the payo in state i and ; ; ; are parameters. i. What restrictions on the parameters are required to ensure that these preferences can be represented by a von-neumann- Morgenstern utility function? ii. What restrictions on the parameters are required for the individual to display risk aversion? iii. Suppose y 0 = 0, y 1 = ln (9) and = = = = 1. What is certainty-equivalent income in this case? (e) Two people enter a bus. Two adjacent cramped seats are free. Each person must decide whether to sit or stand. Sitting alone is more comfortable (worth 5) than sitting next to the other person ( 3) which is more comfortable than standing ( 0). i. Suppose that each person cares only about her own comfort. Model this as a strategic-form game. ii. Is this game the Prisoner s Dilemma? Why or why not? Find its Nash equilibrium. iii. Are payo s ordinal or cardinal? Explain why. (f) In the following game, there are 2 players: an entrant and an incumbent. The type of the incumbent rm is either high-cost with probability 3/4 or low-cost with probability 1/4. Both players choose their strategies simultaneously. The payo s are given below, where the rst coordinate is the incumbent s payo : High-Cost Incumbent Low-Cost Incumbent Enter Don t Enter Don t Fight 0, 1 2,0 Fight 3, 1 5,0 Acquiesce 2,1 3,0 Acquiesce 2,1 3,0 i. Write down the extensive-form game with a move of Nature. ii. What are the pure strategies of the incumbent and the pure strategies of the entrant? iii. Show that there is no pure strategy Nash equilibrium for this game where the entrant chooses not to enter. Explain why. clse 2009/EC202 3 of 9
(g) Suppose the demand curve facing rms in a duopoly is given by: q = A p: Marginal costs are constant at c and assume that rms can change prices only in discrete increments of 1p. Consumers always go to the lowest priced rm and in case prices are the same, then the two rms share demand equally. Show in a graph using the best response functions, how Bertrand competition between rms (i.e. rms choose prices simultaneously) leads to zero pro ts in the Nash equilibrium. (h) Explain how adverse selection might a ect the market for health insurance. clse 2009/EC202 4 of 9
Section B. [Answer at least one question] 2. In the market for a single homogenous good there are N rms. Each rm has the cost function 16 + qi 3 where q i is hthe output of rm i. The market demand for the good is given by N A p i p=3 where p is the market price and A is a positive parameter. (a) Find average cost and marginal cost for rm i. What is the smallest positive amount that the rm would supply to the market? (b) Suppose that N = 1 but that the rm acts as a price-taker. i. If A = 6, show that the equilibrium price is 27 and the rm supplies 3 units of output. ii. Explain what happens in the market in the cases A = 2, A = 3, A = 4. (c) Now suppose that N is a very large number but that other aspects of the problem remain the same. Explain how the answers to part (b) would change, if at all, in the cases corresponding to the di erent values of A. 3. A consumer has the utility function x 1 + p x 2, where x 1 ; x 2 are quantities of two goods. The consumer has income y and faces market prices p 1 ; p 2 where p 2 1=p 2 < y. (a) Find the demand for the two goods in terms of prices and income and show that both goods will be consumed in positive amounts. (b) Find the indirect utility function and the consumer s cost function. (c) Assume that the prices p 1 ; p 2 include a sales-tax component. Suppose the government proposes to cut the tax on good 1 and is interested to know how much better o it will make the consumer. What is the compensating variation of this tax cut? What is the equivalent variation? Is the CV smaller or larger than the EV? (d) Suppose instead that the government cuts the tax on good 2. Again compute the e ect on the consumer s welfare. In what way is the answer di erent from the case in part (c)? Brie y explain why the answers are di erent. clse 2009/EC202 5 of 9
4. In a two-good exchange economy there is an equal number of two 1 types of traders. Type a s preferences are given by 2 [xa 2 1 1] 2 [xa 2 2] and type b s preferences are given by ln(x b 1) + ln(x b 2) where x h i denotes consumption of good i by a trader of type h. Each a-type owns an equal share of the total endowment of good 1 (but no good 2) and each b-type owns an equal share of the total endowment of good 2 (but no good 1). (a) Find the excess demand function for good 1 and good 2. (b) How many competitive equilibria does this economy have? (c) What is the core allocation of this economy if the a-types are each endowed with 10 units of good 1, the b-types are each endowed with 32 units of good 2, and there is a large number of each type? 5. In a two-good economy the endowment of the two goods is (0; 1). Good 1 can be produced from good 2 as follows z k if z > k q = 0 otherwise where q is the amount of good 1, z is the amount of good 2 used as input and k is a parameter such that 0 < k < 1. Persons in the economy are identical and can be represented by a single consumer with utility function e x 1 + x 2 where x i is consumption of good i and is a parameter satisfying 0 < e 1 k : (a) Draw the attainable set of this economy and the indi erence curves for a given value of. (b) If also satis es the condition ln () > k + 1: (*) show that Pareto e ciency requires that the consumption of the two goods be at the point x := (ln () ; 1 k ln ()) : (c) If does not satisfy condition (*) what is the Pareto-e cient allocation? (d) If satis es condition (*) could x be supported by a competitive equilibrium? (e) Now suppose that good 1 were to be provided by a private monopoly. Brie y explain why this might lead to a Pareto-ine cient outcome. clse 2009/EC202 6 of 9
Section C [Answer at least one question] 6. There is a cake of size 1 to be divided between Ann and Helen. In period t = 1 Ann o ers Helen a share of the cake 1 s A and keeps a share s A for herself. Helen can accept or reject. If she rejects, the game ends and they both get payo s of 0. However, if she accepts then they get shares (s A ; 1 s A ) of the cake. (a) Show that all the payo s (; 1 as a Nash equilibrium. ); 0 1, can be supported (b) What are the subgame perfect Nash equilibria of the game? Explain your reasoning. (c) Consider what happens if instead of the game ending in period 1, it is extended to two periods: if Helen rejects then the game moves to period t = 2, and Helen makes an o er 1 s H to Ann, keeping s H for herself. This time, Ann can accept or reject. If she rejects they both get 0 and the game ends. Otherwise they get the shares (1 s H ; s H ) of the cake. The discount factor for both Ann and Helen is given by 1 > > 0. Find the subgame perfect Nash equilibria for this extension of the game. Explain your reasoning. 7. Two rms compete in selling a good: both know that the cost function of rm 1 is 2q 1 but only rm 2 knows its own cost. Firm 1 believes that rm 2 s cost function is 2q 2 with probability 1 2 and 4q 2 with probability 1 2. The inverse demand function is P = 100 Q where Q = q 1 + q 2 when 100 Q and otherwise P = 0. (a) De ne a Bayes-Nash equilibrium and nd one Bayes-Nash equilibrium of the game. (b) Compare the output of rm 1 with its output in the full information case (i.e. when rm 2 s cost function is common knowledge). Explain the intuition behind this result. clse 2009/EC202 7 of 9
8. Consider a competitive market for insurance (where insurance rms make zero expected pro ts) where there are two types of consumers: high risk, and low risk. Let starting wealth be given by y 0 and a potential loss be denoted by `. The prospect they face is y 0 = 200 in the Good state, G and y 0 ` = 150 in the Bad state, B. The high-risk consumers have a probability H = 3 of the bad state while the lowrisk types have a probability L = 1 of the bad state. Let p 3 i, i = H; L 4 denote the price of insurance for the high-risk types and low-risk types respectively, and let Z i, i = H; L denote the corresponding cover that they buy. Hence the consumer purchases insurance cover for p i Z i, and the insurance company pays out Z i in the bad state. The nal wealth of the consumer in the states G and B are X G and X B respectively. U i (X j ) = p X j for i = H; L; j = G; B. The proportion of low-risk types in the population is 1. All insurance companies are risk neutral. 3 (a) Write down the expression for the expected utility of each type and sketch the indi erence curves in the (X G ; X B ) space. What is the expression for the slopes of the indi erence curves for both types? Check that the single crossing condition is satis ed. (b) In the full information case, what are the prices p H ; p L? Show that in this case, both types fully insure. (c) Now, suppose that the types are unknown to the insurance companies. Show that a pooling equilibrium where both types buy positive insurance cover does not exist in this market. clse 2009/EC202 8 of 9
9. The manager of a rm chooses to exert e ort e 2 f1; 2g. Gross pro ts 2 f200; 80g depend on e ort and a random shock. The probability of getting high pro ts increases with e ort. So = 200 with probability 3 if e = 2 and = 200 with probability 1 if e = 1. The manager s 4 4 utility is given by u m = p w e and his reservation utility is v = 0. The owner of the rm is risk neutral and only cares about expected net pro ts. He o ers a contract to the manager which speci es a wage contingent on realized pro ts: so a contract is a pair (w; w). (a) Characterize the set of full information contracts and depict in the Edgeworth box (with dimension 200, 80) using the manager s and owner s indi erence curves. Does the owner prefer high or low e ort from the manager? (b) Suppose now that the owner cannot observe e ort levels but wants the manager to choose e = 2. What additional constraint must be satis ed by the optimal contract in this case? Show in the Edgeworth box that both constraints must bind at the optimum. Characterize the optimal contract using the result that both constraints are binding. clse 2009/EC202 9 of 9