Cooperative Ph.D. Program in Agricultural and Resource Economics, Economics, and Finance QUALIFYING EXAMINATION IN MICROECONOMICS

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1 Cooperative Ph.D. Program in Agricultural and Resource Economics, Economics, and Finance QUALIFYING EXAMINATION IN MICROECONOMICS June 13, :45 a.m. to 1:00 p.m. THERE ARE FOUR QUESTIONS ANSWER ALL FOUR QUESTIONS. You must complete the examination within four hours. You will have 15 minutes to read over the questions before starting (8:45-9:00). This exam is closed book. Calculators and paper will be provided. Read the question carefully. Allocate your time carefully. Parts within questions will often vary in difficulty and weight. Be sure to do all parts of each question chosen. If necessary, it is permissible to make clarifying assumptions, but be sure to label them explicitly. (Grades will not take unstated assumptions for granted.) Also, label graphs and define notation. Number your answer sheets consecutively. Begin your answer to each question on a new page and identify the questions number. Write your exam ID number on EACH page of the exam.

2 Microeconomic Theory Washington State University Ph.D. Comprehensive Exam June 2011 Instructions: This exam has four (4) questions. Please read the questions carefully, and answer all of them in a formal and concise manner. Include all your steps, since this will allow you to obtain partial credit. Please start every question in a di erent page, and ensure all pages are in the correct order before submitting your exam. You have 4 hours to complete the exam. Good luck!!

3 1. [Taxes versus Subsidies] The trucking industry has recently complained about the high price of gas. Consider a representative trucker with utility function u(q 1 ; q 2 ) = ln q 1 + q 2, where q 1 denotes gallons of gas and q 2 is a numeraire representing all other goods. The price of q 2 is therefore normalized to one, p 2 = 1, while the price of gas is p 1 (1 + t), where t 2 [0; 1] represents a speci c tax per gallon of gas. The income of this trucker is given by m > 0. a) Find the Walrasian demand for q 1 and q 2, denoting them as q1 W and q2 W, and distinguish the case in which m > 1 and that when m 1. b) Find the associated indirect utility function, v q W 1 ; q W 2. After months of lobbying and violent demonstrations from the union of truckers, the government is considering implementing either of the following policies: (1) reduce the tax on gas, from t to t 0 = t ; or (2) maintain the tax at t but give a subsidy of S dollars to the trucker equal to the tax revenue collected by the tax on gas. c) Let us rst consider that the trucker s income satis es m > 1, i.e., the trucker is relatively rich. Find the trucker s indirect utility function if the government implements the rst policy, v I q1 W ; q2 W, and if the government implements the second policy, v II q1 W ; q2 W. Under which conditions does the trucker prefer the rst policy? d) Let us now consider that the trucker s income satis es m 1, i.e., the trucker is relatively poor. Find the trucker s indirect utility function if the government implements the rst policy, v I q1 W ; q2 W, and if the government implements the second policy, v II q1 W ; q2 W. Under which conditions does the trucker prefer the rst policy? 1

4 2. [Uncertainty] A farmer uses his own labor, x, to produce wheat, q, with the production function q = f(x). At the end of the harvesting season, the farmer sells its production at the given world price of wheat. The price he receives is a random variable, in particular p p = H with probability, and p L with probability 1 where p H denotes a high price, whereas p L represents a low price of wheat, and p H > p L. The farmer s preferences are given by utility function u(w k ) f(x), where w k represents the farmer s income when the realization of the random variable is k, where k = fh; Lg. Consider that u() is increasing and concave in w k > 0 < 0. Assume that the cost of e ort is increasing and > 0 2 > 0, and denote by w 0 the farmer s initial income The government has just created a price-guarantee program in which the farmer is guaranteed a price p G, where p H > p G > p L, for the proportion of his production that he includes in this program at the beginning of the season, that is, before knowing the particular world price for that year. Let 2 [0; 1] denote the proportion of the farmer s production that he chooses to include in the price-guarantee program. Answer the following questions: a) Describe the farmer s income level when world prices are high and low. Set up the farmer s expected utility maximization problem. b) Take rst-order conditions with respect to x and, and combine them to characterize the (implicit) solution for x and. For simplicity, assume interior solutions. c) Consider now that the price-guarantee program produces zero expected pro ts for the government. Explain how your implicit solution for x and is a ected. 2

5 3. [General equilibrium] Consider an exchange economy in which consumers utility function are and their initial endowments are u a (x a1 ; x a2 ) = 2x a1 x a2 for consumer a, and u b (x b1 ; x b2 ) = 4x b1 + 2x b2 for consumer b! a = (1; 0) and! b = (0; 1). (a) Calculate the set of Pareto optimal allocations. What is the core? (b) Draw the Edgeworth box for the economy (c) Solve for a competitive equilibrium. 3

6 4. [Signaling game] Consider the signaling game Get a II. The sender wants to buy a car. He may be honest or dishonest. Getting a co-signer on the loan is a costly signal. If the buyer is honest, it is easier for him to get a co-signer than if he is dishonest. The dishonest type will not pay back the loan. However, if there is a co-signer, the bank (receiver) can recover part of the money. The honest type pays back the loan. In this problem, use the following notation: Sender s strategies are (a; b), where he sends the message a if honest and b if dishonest. Receiver s strategies are (x; y), where he takes action x if he receives message no co-signer (NC) and y if he receives message get co-signer (C). 2,2 No co signer.25 Get co signer 1,3 honest 0,0 3, 2 R No co signer S dishonest Get co signer R 1,0 1,1 0,0 U s, U r.75 Get a II 3,0 U s, U r (a) Find all the pure-strategy Bayesian Nash equilibria in this game. (b) Find all the pure-strategy Perfect Bayesian Equilibria (PBE) in this game. (c) Find all the pure-strategy PBE in this game that satisfy Signaling Requirement 5. 4