AMERICAN UNIVERSITY Department of Economics Comprehensive Examination

Winter 2011 Exam Page Total: 6

ADVANCED MICRO Carefully read the directions for each section.

MICROECONOMIC ANALYSIS II SECTION (2 hours total) This section has two parts, A and B. You must answer both parts, and there is some choice in each section. Part A. DO TWO (2) SHORT ANSWER QUESTIONS. All short-answer questions are equally weighted. (Time allotted: 30 minutes each.) 1. Each of two players receives a ticket t on which there is a number in [0, 1]. The number on a player’s ticket is the size of a prize that he may receive. The two prizes are identically and independently distributed according to a uniform distribution. Each player is asked independently and simultaneously whether he wants to exchange his prize for the other player’s prize. If both players agree, then the prizes are exchanged; otherwise each player receives his own prize. Fully characterize all Bayesian Nash equilibria of this game. 2. There is a single consumption good, two states and two consumers. Note that this allows the use of Edgeworth boxes. Utility functions are of the expected utility form. Bernoulli utility functions are identical across states. That is, U1 (x11 , x21 ) = π11 u1 (x11 ) + π21 u1 (x21 ) and U2 (x12 , x22 ) = π12 u2 (x12 ) + π22 u2 (x22 ) where xsi is the amount of s-contingent good consumed by consumer i and πsi is the subjective probability of consumer i for state s. We assume that every ui (·) is concave and differentiable. The total initial endowments of the two contingent commodities are ω ¯ = (¯ ω1 , ω ¯ 2 )  0. We assume that every consumer gets half of the random variable ω ¯ , that is (ω11 , ω21 ) = .5¯ ω and (ω12 , ω22 ) = .5¯ ω. a. Suppose that consumer 1 is risk neutral, consumer 2 is not, and the subjective probabilities of the two consumers are the same. Show that at an interior Arrow-Debreu equilibrium, consumer 2 insures completely. b. Suppose now that consumer 1 is risk neutral, consumer 2 is not, and the subjective probabilities of the two consumers are not the same. Show then that at an interior (ArrowDebreu) equilibrium consumer 2 will not insure completely. Which is the direction of the bias in terms of the differences in subjective probabilities? Argue also that consumer 1 (the risk-neutral agent) will not gain from trade. 1

3. Consider a country called Lakeland. Locations in Lakeland are points x(θ) = (cos(θ), sin(θ)) ∈ R2 around the perimeter of the circular lake. The citizens of Lakeland are distributed according to the continuous p.d.f. g(θ) where g(θ) > 0 for all θ ∈ [0, 2π). Lakeland will build a community center but must first decide on a location θc . Each citizen i’s utility of θc is given by the negative of the shortest walking distance from x(θi ) to x(θc ). Is there a Condorcet winner? What specific assumption(s) lead you to this answer? Does changing this assumption(s) produce the opposite result. Part B. DO ONE (1) LONG ANSWER QUESTION. (Time allotted: 1 hour.) 1. Find all pure PBE of the following game.

4,2

2,-1 A

2,0

F

0,1 A

F

Incumbent

Ready

Ready p

Challenger

Nature

Strong

Unready

1-p

Challenger

Weak

Unready

Incumbent

A 5,2

A

F 3,-1

5,0

F 3,1

The game has the following sequence. a. Nature chooses the challenger’s type, strong with probability p or weak with probability 1 − p. This is the central node. b. The challenger observes his type and chooses ready or unready. These are the left and right nodes connected by the central solid line. c. The incumbent does not observe the challenger’s type and chooses acquiesce(A) or f ight(F ). These are the remaining four nodes. How does the equilibrium set correspond to the equilibrium set in the education signaling game? 2. Consider an infinitely repeated Bertrand Duopoly. Both firms have factor δ < 1, and P∞ discount t−1 each firm attempts to maximize the discounted value of profits t=0 δ πjt , where πjt is firm j’s profit in period t. Suppose xt (pt ) is demand in period t at prices pt = (p1t , p2t ). Further suppose that both firms have identical unit costs c, and that these unit costs are less than the 2

monopoly price pm . Consider the following Nash reversion strategy. Firm j chooses pm in period t if both firms have chosen pm in every previous period or if t = 1. Otherwise, firm j chooses c. a. Assume that xt (·) = x(·), i.e. demand is constant across periods. Show that these strategies constitute a subgame perfect Nash equilibrium of the infinitely repeated Bertrand duopoly game if and only if δ ≥ 1/2. b. Give the conditions under which these strategies sustain the monopoly price under each of the following conditions: • Market demand in period t is xt (p) = γ t x(p) where γ < 0. • Assume xt (·) = x(·), and at the end of each period, the market ceases to exist with probability γ.

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HETERODOX MICRO SECTION (2 hours total) This section has two parts A and B; you must answer both parts and there is some choice in each part. Part A - Essay. Choose one (1) of the following. [1 hour] 1) Discuss under what conditions voluntary exchange by decentralized agents is likely to result in Pareto optimal outcome. Explain the potential role of power and altruism in competitive market exchanges between decentralized agents. 2) Drawing on recent empirical papers in the area of behavioral economics (experimental papers), discuss three outcomes which would difficult to explain based on Walrasian economic theory. Outline the alternative explanation offered by evolutionary political economy, identifying the specific assumptions of which allow an alternative explanation in each case. 3) According to Bowles, what fundamental economic problem results in the emergence of classes? (note: It is not the distribution of resources, which is the result.) Outline carefully how what happens in credit markets prevents movement by individuals between classes over time (economic mobility), carefully explain why the outcomes represent rational decisions for both parties to the credit transaction. Explain the link between this interaction and overall economic efficiency. 4) Drawing on Bowles, Knight, and Greif, carefully explain a “Bowlesian” theory of underdevelopment and offer relevant examples to support the theory.

Part B - Problems. Choose one (1). [1 hour] 1) Consider Chris and Pat, partners in household bargaining: Chris (C) and Pat (P) are partners in a household with one child. Chris earns an income of $400 per week and Pat earns an income of $200 per week. They are each trying to allocate 50 hours of free time weekly between play time with their child (K), and personal leisure (L) (time spent working in the market is exogenous). Thus, the time constraint is given by 50 = Ki + Li . Play time is a public good, i.e., Chris and Pat derive utility from the amount of time each parent spends playing with the child. In the event of divorce, social norms require that Chris and Pat individually spend 20 hours of play time with their child. Each no longer derives utility from their partner's play time with their child (i.e. play time is no longer a public good). Their utility functions take the following form:

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1 m 1 U C = ln 400 + ln K C + ln K P + ln LC , 4 4 2 1 m 1 U P = ln 200 + ln K P + ln K C + ln LP , 2 4 4 where m is an indicator variable which takes the value 1 if the couple stays married, and 0 if the couple decides to divorce.

1.1 Find the BRF for each and the non-cooperative equilibrium. (Hint: like Fishers) 1.2 Find the equilibrium utility levels for the two players. 1.3 How much would each get in case of divorce? 1.4 Set up, but do not attempt to solve, the Nash Cooperative Bargaining model for these two, using information derived in 1.3. Provide a graphical solution and explain it. How do their relative utilities differ from that in the non-cooperative solution? 1.5 Does power affect the outcome? Explain your answer in precise terms. What is the source of the power? (Discuss at least two sources.) Link this to the answer to part 1.4. 2) A population of individuals faces a prisoners’ dilemma-type situation, with payoffs as in the box below and a > b > c > d. One possible solution to this dilemma is to “inspect” the possible trading partner and use local information on player reputation to decide how to play (cooperate with cooperators and defect with defectors).

PLAYER 1

PLAYER 2 Cooperate

Defect

Cooperate

b, b

d, a

Defect

a, d

c, c

Players who inspect are called “Inspectors.” They pay a cost of δ to inspect and find a cooperator. The alternative behavior is to play defect. There are α Inspectors in the population. 2.1 Write down the expected payoffs for each type of individual (Inspectors and nonInspectors). 2.2 Find the equilibrium share of Inspectors in the population. 2.3 Is this equilibrium stable? How many total equilibria are there? Explain. 2.4 What can you say about the Pareto efficiency of this equilibrium? 2.5 What happens to the equilibrium if δ increases? What does this imply for which equilibrium is likely to be achieved and Pareto efficiency?

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2.6 Relate this problem to at least one real-world example described by Elinor Ostrom or Avner Greif.

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