AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Spring 2012
Exam Page Total: 6
ADVANCED MICRO THEORY
This examination has two Sections, (Microeconomic Analysis II and Micro Political Economy). You must answer both sections; be sure to follow the directions in each part carefully. Each part receives equal weight in the overall grading. Therefore, you should plan
to spend an equal amount of time (i.e., about 2 hours) on each section, Microeconomic
Analysis II and Micro Political Economy, regardless of the number of questions in each.
Please make sure that all math is intuitively explained, all diagrams are clearly labeled,
and all answers are responsive to the specific questions asked. The time limits should
suggest the expected length and depth of your answers. The standard for passing this
exam is demonstration of “mastery” of the material.
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 THREE (3) SHORT ANSWER QUESTIONS. All short-answer questions
are equally weighted. (Time allotted: 30 minutes each.)
1. Consider an economy with two firms and two consumers. Firm 1 is entirely owned
by consumer 1. It produces guns from oil via the production function g = 2x. Firm
2 is entirely owned by consumer 2. It produces butter from oil via the production
function b = 3x. Each consumer owns 10 units of oil. Consumer 1’s utility function
is u1 (g, b) = g .4 b.6 and consumer 2’s utility function is u2 (g, b) = 10 + .5 ln g + .5 ln b.
a. Find the market clearing prices for guns, butter and oil.
b. How many guns and how much butter does each consumer consume?
2. Consider the normal form game below.
H
T
H
T
2,1 0,0
0,0 1,2
Find the MSNE (p, q) where p is the probability that player 1 plays H and q is the
probability that player 2 plays T . Assume errors are distributed Log Weibull so that
the QRE is of the logistic form. Write the system of equations that characterizes
the QRE π. What if urow (H, H) changes from 2 to 3? Describe the effect on the
equilibrium and compare it to NE. Interpret.
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3. Find conditions on the discount factor, δ, such that tit-for-tat is a NE of the infinitely repeated general prisoner’s dilemma below, where a > b > c > d. Interpret
this result in terms of cooperation, punishment and patience.
C
D
C
D
b,b d,a
a,d c,c
Is tit-for-tat an SPE of this game? Explain.
4. Consider the Level-K model. Give an example of a 2x2 normal form game such that
when level 0 is the uniform random strategy, the Level-K model converges to a NE
as k goes to infinity. Also give an example that doesn’t converge to NE. Explain
both.
Part B. DO TWO (2) LONG ANSWER QUESTIONS. (Time allotted: 1 hour.)
1. Consider a second-price sealed-bid auction with two bidders in which each bidder’s
valuation is drawn independently from the uniform distribution from 0 to 1. Suppose that the seller imposes the reserve price r. That is, if both bids are less than
r, the object is not sold (and neither bidder makes any payment), if one bid is less
than r and the other is at least r, the object is sold at the price r, and if both bids
are at least r, the object is sold at a price equal to the second highest bid. Show
that for each bidder, a bid equal to her valuation weakly dominates all her bids.
For the equilibrium in which each bidder submits her valuation, find the reserve
price r that maximizes the expected price at which the object is sold.
2. Consider the two-player Bayesian game below, where 0 < < .5. The states ω1 and
ω2 , which occur with probability p and 1 − p respectively, correspond to different
payoff functions for player 2.
L
M
R
T 1,2 1,0 1,3
B 2,2 0,0 0,3
L
M
R
T 1,2 1,3 1,0
B 2,2 0,3 0,0
Table 1: ω1 occurs with probability p
Table 2: ω2 occurs with probability 1 − p
a. Suppose neither player knows the state. Characterize the Nash equilibrium of
this game in terms of p?
b. Suppose it is common knowledge that player 2 is informed of the state and
that player 1 is not informed of the state, i.e., player 2 discovers his payoff
function before the start of the game but player 1 only knows the probability
distribution p over states ω. Characterize the Bayesian NE of this game in
terms of p.
c. Discuss how information affects player 2’s payoff.
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3. Describe both the Median Voter Theorem and Arrow’s Impossibility Theorem. Explain the relationship between them.
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HETERODOX MICRO SECTION
(2 hours and 30 minutes total)
Part A – Problems. Choose one (1). [1 hour and 15 minutes]
1) Consider a production function Q = qE, a utility function U = C – E2 (there are only variable
inputs of labor effort, E, and the producer consumes the good she produces in amount C) and
an economy composed of 3 individuals with the above functions. Two are tenants and the
other their landlord, whose income is his own production (governed by the above production
function) plus a share, α, of the two tenants’ crops. For each of the tenants, C = (1–α)Q. Let
q = 1.
1.1 What is the landlord’s optimal effort level (farming his own land)?
1.2 Assuming that the landlord has the power to determine α, what value would she select?
(Give both FOCs and a numerical value.)
1.3 Indicate the levels of utility achieved by all three individuals.
1.4 Since none of the tenants can obtain capital to buy land from the landlord, protests ensue,
as a result of which there is a land redistribution (costless!), in which everyone gets one
acre of the landlord’s 3 acres, including the former landlord. As a result, how much does
each farmer work and what is their level of utility?
1.5 Explain this result and its relevance to the Second Fundamental Theorem of Welfare
Economics.
2) Two fishers, Eye (i) and Jay (j), fish in the same lake, using their labor and their nets. They
consume their catch and do not engage in exchange. They do not make agreements about
their actions, yet each one’s action affects the other. Specifically:
yi = αi (1–Bej)ei , where yi = amount of fish caught by Eye over some period
αi>0 = constant which varies with the size of Eye’s nets
ei and ej = respectively the share of the 24-hour day each spends fishing,
B > 0 is the impact of one agent’s actions on the other.
Fishers have analogous production functions, yi, and utility functions such that:
Ui = yi – ei2, Ue < 0, Uy > 0
2.1 Solve for the best response functions of the two actors and graph them.
2.2 Indicate any Nash equilibria. Are they/is it Pareto Optimal? What accounts for
this?
2.3 Suppose the state wishes to determine the Pareto Optimal outcome. Write down
the state’s optimization problem and find the new FOCs for determining the
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optimal levels of ei and ej. Compare these to the FOCs from the individual
decisions. Explain the difference.
2.4 If the state wants to use a tax to get individuals to choose the optimal outcome,
what should the tax rate be?
2.5 Show how the existence of altruism might affect this outcome, and link this to at
least one relevant empirical paper.
3) Let a standard prisoner’s dilemma illustrate the exchange problems created where incomplete
contracts create the possibility of cheating (Defect). Note: a > b > c > d and a + d < 2b.
Cooperate
Defect
Cooperate
Defect
b, b
a, d
d, a
c, c
Some argue that, in this context, institutions (like the use of repetition and retaliation) can
permit decentralized interactions to generate more efficient results. Define the probability, ρ,
that the game (exchange interaction) will be terminated after this round, and the share of the
population = τ that will be playing tit-for-tat. All others play defect.
3.1 Write down payoff functions for the two types of agents.
3.2 Find the equilibrium share of tit-for-tatters mixed population, τ*. Are there any pure
population equilibria?
3.3 Which equilibria are stable? Show both graphically and mathematically. Explain the
relation between the two methods.
3.4 Suppose ρ increases. Show how this affects τ*.
3.5 Discuss some empirical evidence about the use of such institutions in facilitating
exchange.
3.6 Could τ be a quasi-parameter, in the Greif-ian sense? If an increasing share of tit-fortatters in the population causes the probability of termination to fall (say exchange
becomes more stable in a population where more people play tit-for-tat), what would
Greif say about the long run stability of τ*. Explain.
Part B – Essay. Choose one (1) of the following. [1 hour and 15 minutes]
1)
Explain the importance of the “Constitutional Conundrum” to economic analysis. Define
institutions and explain how these relate to this problem. Outline 3 explanations of why
institutional change might be difficult, even when institutions fail to implement a Pareto
Optimal outcome.
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2)
Drawing on Bowles, Knight, and Greif, carefully explain a “Bowlesian” theory of
underdevelopment and offer relevant examples to support the theory.
3)
Bowles offers a theory of class. Explain the fundamental economic problem underlying
(causing) the emergence of classes. What might have determined who ended up in which
class position as exchange expanded? What is the role of credit markets and rationality in
this process? What implications does this have for the link between income distribution
and efficiency in this theory?
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