Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2016 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions. Answer one question from each section. Before turning in your exam, number each page of your answers in sequential order, and identify the question (for example, III.2 for section III and question 2). Indicate, by circling below, the questions you completed. If you answer more than one question in a section and do not circle a question corresponding to that section, the first of the two that appears in your solution paper will be graded. *********************************************** STUDENT ID LETTER: (Fill in your code letter) Answer one question from each section. Indicate the one you answered by circling: Section I: Question 1 Question 2 Section II: Question 1 Question 2 Section III: Question 1 Question 2 Section IV: Question 1 Question 2 *** TURN IN THIS SHEET WITH YOUR ANSWER PAGES ***
Part I Answer at most one question from Part I. 2
Question I.1 Consider the following utility function, which is a function of 2 goods, x 1 and x 2 : u = θx1 1/ε 1 1 1/ε + (1 θ)x1 1/ε 2, 1 1/ε where 0 < θ < 1. The consumer has wealth w to purchase these two goods, where the price of x 1 is p 1 and the price of x 2 is p 2. (a) What restrictions, if any, are needed on ε to ensure that utility is increasing in both x 1 and x 2? (b) For this utility function, work out the demand for x 1 and for x 2. These demands should be functions of θ, ε, w, p 1, and p 2. You can assume interior solutions for both. (c) Are these demands homogenous of degree 0 in wealth and prices? Do they satisfy Walras law? Show your answer rigorously. (d) What is the price elasticity of x 1 with respect to p 1? What is the price elasticity of x 2 with respect to p 2? Use your answers to part (b) to answer this question. (e) Show the indirect utility function for this consumer. Does it satisfy the homogeneity requirement of an indirect utility function? Is it strictly increasing in w? Show your answer rigorously. 3
Question I.2 (Expected Utility.) Expected utility theory begins with the assumption that an individual has preferences over lotteries. Consider a simple lottery in which there are three possible outcomes, and each outcome has an associated probability, denoted by p 1, p 2 and p 3, where p 1 + p 2 + p 3 = 1, and p 1 0, p 2 0, and p 3 0. The set of all possible lotteries is all possible values of p 1, p 2, and p 3 such that p 1 + p 2 + p 3 = 1. Recall from class that this set of lotteries can be depicted as a triangle (a 2-dimensional simplex). Each of the three possible outcomes has an associated utility, and these are denoted as u 1, u 2 and u 3. Note that there is no assumption here that these outcomes are amounts of money, they are just outcomes. Assume throughout this problem that the individual has a von Neumann- Morgenstern expected utility function. (a) To get warmed up, suppose that u 1 = 4, u 2 = 5, and u 3 = 6. Consider two lotteries. The first, denoted by L 1, has p 1 = 1/2, p 2 = 0, and p 3 = 1/2. The second lottery, denoted by L 2, has p 1 = 1/6, p 2 = 2/3, and p 3 = 1/6. What is this person s expected utility from each of these two lotteries? Which lottery does he or she prefer? (b) The triangle (2-dimensional simplex) that constitutes the set of all possible lotteries has utilities associated with each of those lotteries. For any given point on that set the expected utility, which can be denoted by EU, is u 1 p 1 + u 2 p 2 + u 3 p 3. We saw in class that a set of lotteries that have the same expected utility will be a line on this triangle. That is, for any fixed value of utility, which we can denote by EU 0, this line of constant EU has the property that EU 0 = u 1 p 1 + u 2 p 2 + u 3 p 3 for all points on that line. Using the requirement that p 1 + p 2 + p 3 = 1, express p 1 as a function of EU 0, u 1, u 2, u 3, and p 2. [Note that from here on it is no longer assumed that u 1 = 4, u 2 = 5, and u 3 = 6.] (c) Suppose that there are two points in the triangle that have an expected utility of EU 0. The first point has p 1 = 1/2 and p 2 = 1/4 (and thus p 3 = 1/4), and the second point has p 1 = 3/4 and p 2 = 0 (and thus p 3 = 1/4). Your solution for (b) should have a slope term and an intercept term. Use these two points, and your answer to (b), to solve for numerical values of each of those two terms (d) Consider a third point on the triangle, which has p 1 = 1/4 and p 2 = 3/8 (and thus has p 3 = 3/8). Does this point have the same expected utility, namely EU 0, as the two points in part (c)? Provide a rigorous justification for your answer. (e) Consider your answer to (c). Does it allow you to make any comparison to the relative sizes of u 1, u 2 and u 3? That is, for any two u s, say u i and u j, can you determine whether u i < u j, u i = u j or u i > u j? Explain your answer. 4
Part II Answer at most one question from Part II. 5
Question II.1 Consider the production possibility set (PPS): { PPS = (q, z) R + R 2 : z1z θ 2 θ ( ) } q1 2 + q2 2 1 2, where z 1 and z 2 are inputs; q 1 and q 2 are outputs; and θ > 0 is a constant parameter. This PPS is nonempty, strictly convex, closed, and satisfies weak free disposal in inputs and outputs. (a) Under what conditions on θ (if any) will this technology exhibit non-increasing returns to scale? Justify your answer. (b) Derive the input distance function for this PPS assuming q 1 > 0 or q 2 > 0 and the marginal rate of technical substitution between z 1 and z 2 using this input distance function. (c) The cost function for this PPS is C(r 1, r 2, q 1, q 2 ) = 2r 1 2 1 r 1 2 2 (q 2 1 + q 2 2) 1 4θ, where r 1 > 0 and r 2 > 0 are input prices. Use this cost function to find the producer s conditional input demands. (d) The cost function in part (c) could be used to derive the profit maximizing unconditional supplies and profit function, but you do not have to do this. This profit function will be a convex function of input and output prices. Show that this result holds generally for any profit function π(p, r) derived from a PPS with M outputs and N inputs such that p R M ++ and r R N ++, and that is nonempty, convex, closed, and satisfies weak free disposal in outputs and inputs. 6
Question II.2 Consider a producer s revenue cost function for a world with two-inputs denoted by n = 1, 2 and two outputs denoted by m = 1, 2 in two states denoted by s = a, b: ( ) C(r, p, R) = ( r 2 1 + r 2 2 ) 1 2 R a p a 3 4 1 p a 1 4 2 + R b p b 1 4 1 p b 3 4 2 where r R 2 ++ is a vector of input prices, p R 4 ++ is a vector of state-contingent output prices, and R R 2 + is a vector of state-contingent revenues. The PPS used to derive this revenue cost function is nonempty, strictly convex, closed, and satisfies strong free disposal in outputs and inputs. The producer s preferences for profit in each state are described by the utility function W (π) = ( (απ a ) 2 + (βπ b ) 2) 1 2, where π R 2 +. This utility function is derived from a preference relation that is rational, continuous, and strictly monotonic. (a) Derive the producers conditional input demand for the first input and conditional output for the first output in state a (Hint: These should be conditional on some or all of the input prices, and state contingent output prices and revenues). (b) Derive the producer s certainty equivalent revenue and subjective probabilities. (c) What is the producer s production risk premium? (d) For the general revenue cost function C(r, p, R), where r R N ++ is a vector of input prices, p R++ M S is a vector of state-contingent output prices, and R R S + is a vector of revenues, show that C(r, p, R) is homogeneous of degree zero in p and R., 7
Part III Answer at most one question from Part III. 8
Question III.1 Consider the two-player extensive form game in the figure below. (a) What is each player s pure strategy set? (b) Construct the normal form game for this figure. (c) Find all of the pure-strategy Nash equilibria in this normal form game. Which of these equilibria are subgame perfect? Justify your answer. (d) Find the mixed-strategy equilibrium for the subgame starting after Player A chooses R 1 and Player B chooses E 1. How might Player A s and B s strategies in the first part of the game (i.e., the choices over R 1 versus L 1 and W 1 versus E 1 ) change if this mixed-strategy equilibrium is played in the last subgame? Player A L 1 R 1 Player B W 1 E 1 W 1 E 1 (20, 20) (8, 30) (30, 8) Player A L 2 R 2 Player B W 2 E 2 W 2 E 2 (15, 10) (0, 0) (0, 0) (10, 15) Figure 1: (Player A s Payoff, Player B s Payoff) 9
Question III.2 Suppose there are two neighboring farmers denoted by i = 1, 2. Profitability for the ith farmer, assuming no pest infestation, is π i > 0. These profits are reduced by L i when there is a pest infestation. The probability of an infestation is 1 > p(x 1, x 2 ) > 0, where x i is the amount of effort farmer i devotes to preventing an infestation such that p(x 1, x 2 )/ x i < 0 and p(x 1, x 2 )/ x 2 i > 0. It is important to note that either there is an infestation that affects both farmers or no infestation. As such, the ith farmer s expected profit can be written as Eπ i (x i, x j ) = p(x 1, x 2 )(π i L i rx i ) + (1 p(x 1, x 2 )) (π i rx i ) = π i rx i p(x 1, x 2 )L i, where r > 0 is the unit cost of a farmer s pest infestation prevention effort. (a) Derive the condition for the ith farmer s optimal effort assuming it is interior. Use this condition to characterize the ith farmer s best response in terms of π i, L i, r, π j, L j, and x j. (b) Again assuming an interior solution, what conditions must the farmers Nash equilibrium efforts satisfy? (c) Suppose the government wants to reduce the likelihood of a pest infestation by reducing the cost to farmers of preventing an infestation (i.e., reducing r with some sort of subsidy). What has to be true for the government s program to unequivocally work? Explain the intuition of your result. 10
Part IV Answer at most one question from Part IV. 11
Question IV.1 Answer all three of the following questions related to exchange economies. (a) State and prove the first welfare theorem. For the next two questions, consider the following 2 2 competitive exchange economy, with consumers j = 1, 2 and goods x and y. The consumers preferences are given by U 1 (x 1, y 1 ) = ln x 1 + 2 ln y 1 and U 2 (x 2, y 2 ) = 2 ln x 2 + ln y 2. Endowments are ω 1 = (4, 2) and ω 2 = (2, 4). (b) Derive the contract curve for this economy. Construct a carefully labeled Edgeworth-box diagram depicting the economy, including the endowment, an indifference curve for each consumer, and the contract curve. (c) Derive the two consumers offer curves. Find a Walrasian equilibrium for the economy. Show that the equilibrium allocation is also Pareto optimal. Add the equilibrium allocation and equilibrium price line to your diagram. 12
Question IV.2 Consider the usual social-choice setting with a finite set of m alternatives X and a finite set of individuals J = {1,... n}. Preferences P j are strict on X and we say that a profile is {P j } n j=1. Answer parts (a) and (b) and either (c) or (d). (a) State Arrow s theorem. Include definitions of each of the four axioms. (b) Let r j (x) denote the rank that individual j gives to alternative x X. Assign a score to each alternative equal to the worst rank that is received by any of the individuals. More formally the score for x is W (x) = max[r 1 (x),..., r n (x)]. Consider a SWF that orders alternatives according to scores. Which axiom (or axioms) of Arrow s theorem is (or are) violated by this SWF? Answer either (c) or (d) (c) Show that there is some pair a, b X such that some person j J is semidecisive (SD) over a and b. (d) Prove that there is an extremely pivotal voter who can, by unilaterally changing her vote, move some alternative from the bottom of the social ranking to the top. 13