THIS EXAM CONSISTS OF 9 PAGES INCLUDING THIS PAGE PLEASE ENSURE THAT YOU HAVE ALL OF THE PAGES

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1 THIS EXAM CONSISTS OF 9 PAGES INCLUDING THIS PAGE PLEASE ENSURE THAT YOU HAVE ALL OF THE PAGES VANDERBILT UNIVERSITY Department of Economics Microeconomic Theory Preliminary Examination Spring 2016 Time Limit: 4 Hours General Instructions: There are three sections to this examination. Please read the instructions and each question with great care. If you do not understand a question, ask the proctor to obtain a clari cation. Write legibly and budget your time carefully. (Suggested times for each section of the exam are given in italics.) 1

2 A. One hour. Answer all of the following three questions. Indicate whether each of the following is TRUE, FALSE, or UNCERTAIN. Justify your answer: if true, explain why (this might involve a proof, if appropriate); if false, give an example that shows why it is false; if uncertain, explain when and why the statement is true some of the time and when and why it is false other times. All points will be awarded based on the quality of your justi cation. 1. Consider an industry consisting of price-taking rms, operating in a perfectly competitive, homogeneous-goods market; each rm uses both capital and labor to produce output. The industry cannot be in long-run equilibrium if a rm s marginal product of capital exceeds its average product of capital. 2. Suppose that a pro t-maximizing monopoly, employing a constant marginal cost of production, uses third degree price discrimination. The rm charges p 1 in market 1 and p 2 in market 2 for its product. If p 1 > p 2, then demand in market 2 is more elastic than demand in market 1 at the chosen prices. 3. Consider an Edgeworth box economy for which one consumer has preferences given by u A (x; y) = x 1 2 y 1 2 and the other consumer has preferences given by u b = x + y: Then the set of Pareto optimal points is a straight line joining the origins of the box. Illustrate your answer with picture(s) 2

3 B. One hour. Answer two of the following three numbered questions; note that each question starts on a separate page of the exam. If you answer more than two, we will pick the lowest two to contribute to your score. 1. Consider an oligopolistic industry with N rms. The inverse demand curve of the market is given p = a q, where q is the aggregate supply. The consumers who buy the commodity are price-takers. Each rm i produces q i units of output; the cost to rm i of producing q i units is K + cq i if q i > 0 and 0 if q i = 0. All this information is common knowledge to everyone and the rms choose their output levels independently to maximize their own pro ts. (a) Suppose that there is a pure strategy Nash equilibrium wherein n rms produce positive amounts of output. Characterize this Nash equilibrium. Provide conditions on the exogenous parameters to guarantee the existence of such an equilibrium. (b) Ignore the NE that you characterized in a. above. Suppose that n ( 0) rms produce q (> 0) units of output in total. Compute the aggregate surplus, which is the summation of consumer s surplus and producers total pro t. What is the n that maximizes aggregate surplus of this economy given any speci ed level of q? Given this level of n, what is the q that maximizes the aggregate surplus? What are conditions on exogenous parameters for such a q to maximize aggregate surplus? (c) Check if there are any pure strategy Nash equilibria that maximize aggregate surplus. 3

4 2. Wanda works as a waitress and earns cash tips that are not reported by her employer to the Internal Revenue Service. Her tip income is variable. In a good year (G), she earns high income, so that her tax liability to the IRS is $50. In a bad year (B), she earns low income, and her tax liability to the IRS is $0. The IRS (and Wanda) know that the probability of her having a good year is 0.6, and the probability of her having a bad year is 0.4, but only Wanda knows for sure which outcome has resulted for her this tax year. In this game, rst Wanda decides how much income to report to the IRS. If she reports high income (H), she pays the IRS $50. If she reports low income (L), she pays the IRS $0. Then the IRS has to decide whether to audit Wanda. If she reports high income, they do not audit. If she reports low income, then the IRS can either audit (A) or not audit (N). When the IRS audits, it costs the IRS $10 in administrative costs, and also costs Wanda $10 in the opportunity cost of the time spent gathering bank records and meeting with the auditor. If Wanda reports low income and the IRS audits Wanda in a bad year (B), then she owes nothing to the IRS, although she and the IRS have each incurred the $10 auditing cost. If Wanda reports low income in a good year (G) and the IRS audits then she has to pay the $50 she owes to the IRS and, in addition, Wanda and the IRS each incur the auditing cost of $10. (a) To get started, suppose that Wanda has a good year (G), but she reports low income (L). Suppose the IRS then audits her (A). What is the total payo to Wanda, and what is the total payo to the IRS? (b) Write down the extensive-form game tree for the game. Be careful about information sets. (c) How many pure strategies does each player have? Brie y explain your reasoning. (d) Write down the strategic-form game matrix for the game. Find all of the Bayes- Nash equilibria of the game, specifying strategies and the equilibrium beliefs of the IRS (speci cally, specify the belief of the IRS that Wanda had a Bad year when she reported L). Identify whether the equilibria you nd are separating, pooling, or semi-separating. 4

5 3. Alice s preferences on R 3 + can be represented by the utility function u(x 1 ; x 2 ; x 3 ) = x 1 + ln(x 2 x 3 ): Alice has income m > 0 and faces given prices p 1 > 0; p 2 > 0; and p 3 > 0 for goods 1; 2; and 3, respectively. (a) Find a necessary and su cient condition on the parameters m, p 1 ; p 2 ; and p 3 such that Alice s optimal demands for all three goods are always positive. (b) In the following questions, assume the condition you found in (a) above holds. Provide mathematical answers to the following questions, showing your work. (c) Find the demand functions x i (p 1 ; p 2 ; p 3 ; m), for goods i = 1; 2; and 3. (d) Show that good 1 is both income elastic (i.e., a luxury good) and own-price elastic. (e) Find the indirect utility function v(p 1 ; p 2 ; p 3 ; m) and the expenditure function e(p 1 ; p 2 ; p 3 ; U). (f) Show that Roy s identity holds for all three goods. (g) Show that the Slutsky equation holds for good 2. 5

6 C. Two hours. Answer two of the following three numbered questions; note that each question starts on a separate page of the exam. If you answer more than two, we will pick the lowest two to contribute to your score. 1. De ne tic-tac-toe lite to be a version of tic-tac-toe that starts from the position shown below with some Xs and Os already lled in. It is now Player X s turn to move. For convenience, the boxes yet to be lled with Xs or Os are labelled 1,2, and 3. That is, Player X can put an X in any one of the boxes labelled 1,2, or 3. Then Player O can put an O in one of the boxes not already containing an X or an O. O X X X O 2 O 1 3 Apart from the starting position, the standard rules of tic-tac-toe are followed: (i) Two players (Player X and Player O) and a 3 3 board as shown. (ii) The players alternate in moving, with Player X going rst. When it her turn to move, a player writes X or O, respectively, in a blank cell on the board. (iii) The rst player to get three of her symbols in the same row, the same column, or the same diagonal, wins the game. If neither player wins, the game is considered a tie. We will assume that the player who wins receives a payo of +2 and the player who loses receives a payo of 1. If there is a tie, both players receive payo s of 0. Observation: After a choice by Player X and a subsequent choice by Player O, the outcome of the game is determined. Use this observation to keep the game tree in the following short that is, X moves once and then O moves, resulting in a terminal node. (a) Making use of the observation above draw the entire game tree of tic-tac-toe lite. Specify the strategy sets of each of the players. Find all Nash equilibria in pure strategies of the game. (b) Find all subgame-perfect equilibria in pure strategies of the game. (c) We now move on to consider tic-tac-toe lite repeated an in nite number of times with a discount rate of (0 < < 1). For the in nitely repeated game with discount rate, specify strategies that support a subgame perfect Nash equilibrium that, for close to 1, yield each player a payo close 1. 6

7 2. David is interested in buying a car, and he is thinking about buying either an Audi (A) or a Beamer (B). Let x A be the indicator variable for David s possible purchase of an Audi (i.e., x A is 0 if he does not buy an Audi and 1 if he does) and let x B be the indicator variable for David s possible purchase of an Beamer (i.e., x B is 0 if he does not buy a Beamer and 1 if he does); note that this means that David is not allowed to buy two or more of the same car. Of course, he also buys other goods (which we will model via a composite commodity denoted as C). Let the continuous, nonnegative consumption variable for the amount of this composite commodity consumed be denoted as x C. David s utility for an Audi is denoted as u A and for a Beamer it is denoted as u B, and his utility (function) for the composite good C is u(x C ). The utilities for the cars are strictly positive, though not necessarily equal, and the utility function for the composite good is twice continuously di erentiable, strictly increasing, and strictly concave on [0; 1), with u(0) = 0. David s income is denoted as m. Furthermore, assume that u A < u B < u(m): Assume that David s aggregate utility is additively separable and can be written as the utility of an Audi if he buys one plus the utility of a Beamer if he buys one plus the utility of the amount of the composite good consumed. Finally, the price of an Audi is p A, that of a Beamer is p B, and the price of the composite commodity, C, is normalized to be 1; the car prices and David s income are positive numbers. (a) Write a complete mathematical description of David s consumer choice problem, providing all relevant constraints. (b) Provide a complete interpretation of the assumption that u A < u B < u(m). In particular, what does u A < u B imply and what is the intuitive meaning of u B < u(m)? (c) In all the parts below, assume that the prices and income satisfy the following restrictions: maxfp A ; p B g < m < p A + p B : What do these restrictions imply about the feasible car choices that David can make? (d) Find an optimality condition such that, if Beamers did not exist, David would strictly prefer the consumption bundle involving purchasing an Audi to the bundle involving not owning a car at all. (e) Find an optimality condition such that, if Audis did not exist, David would strictly prefer the consumption bundle involving purchasing a Beamer to the bundle involving not owning a car at all. 7

8 (f) Assuming the availability of both types of car, nd an optimality condition such that David would strictly prefer purchasing an Audi to purchasing a Beamer. (g) Assume all the conditions in (d) through (f); the maximum amount David would be willing to pay for an Audi is called David s reservation price for an Audi, and is denoted as r A (p B ; m; u A ; u B ): What equation must David s reservation price for an Audi satisfy when both Audis and Beamers are possible to buy? Draw a completely-labeled diagram illustrating this demand function for an Audi. (h) Using the demand function in (g), derive the comparative statics results for changes in David s reservation price with respect to (one-at-a-time) changes in: 1) p B and 2) u A ; make sure you show how you get your results. 8

9 3. Suppose that a rm earns pro ts from two activities undertaken by a single worker. Activity 1: the pro t from activity 1 is deterministic. If the worker takes a high e ort H then the pro t is 5. If the worker takes a low e ort L, then the pro t is 0. Activity 2: the pro t from activity 2 is stochastic. If the worker takes a high e ort h, then the pro t is either 14 or 0 with equal probability of 1. If the worker takes a low 2 e ort l, then the pro t is 0. The possible combinations of e ort levels that the worker can take are (L; l), (H; l); and (L; h), but the worker cannot choose high e ort for both activities (i.e., (H; h) is not possible). The worker s utility at the wage w is p w if he chooses (L; l) and p w 1 if he chooses either (H; l) or (L; h). If the worker chooses not to work for the rm, his utility is 0. Assume that the rm is risk-neutral. Assume that the rm observes the worker s e ort levels for both activities. (a) Characterize the contract (wage for each combination of e ort levels) that the rm will o er. Which combination of e ort levels maximizes the rm s pro t? Note that the contract has to provide an incentive to the worker to work for the rm. Now, suppose that, for the questions below, the rm observes the pro t of each activity, but not the worker s e ort levels. (b) If the rm wants to induce the worker to choose (L; l), what is the pro t-maximizing wage scheme and what is the pro t? Similarly, if the rm wants to induce (H; l), then what is the pro t-maximizing wage scheme and what is the pro t? (c) If the rm wants to induce the worker to choose (L; h), then what is the (expected) pro t-maximizing wage scheme and what is the the expected pro t? (d) Using your answers from b. and c., what is the optimal contract that this rm can provide to maximize its (expected) pro t? If the optimal contract here is di erent from your answer in 1, explain why. 9