AMERICAN UNIVERSITY
Department of Economics
Comprehensive Examination
Winter 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 TWO (2) SHORT ANSWER QUESTIONS. All short-answer questions are
equally weighted. (Time allotted: 30 minutes each.)
1. 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.
2. Players 1 and 2 take turns making offers about how to divide a pie of size one. Time
runs from t = 0, 1, 2, ... At time 0, player 1 can propose a split (x0 , 1 − x0 ) (with
x0 ∈ [0, 1]), which player 2 can accept or reject. If player 2 accepts, the game ends
and the pie is consumed (with player 1 getting x0 and player 2 getting 1 − x0 ). If
player 2 rejects, the game continues to time t = 1, when player 2 gets to propose a
split (y1 , 1 − y1 ). Once player 2 makes a proposal, player 1 can accept or reject, and
so on. If player 1 accepts, then the game ends, and the pie is consumed (with player
2 getting y1 and player 1 getting 1 − y1 ). If player 1 rejects, the game continues to
time t = 2, when player 1 again proposes a split (x2 , 1 − x2 ). If player 2 accepts,
then the game ends, and the pie is consumed according to (x2 , 1 − x2 ). If player 2
rejects, then the game ends, and both players get nothing. We assume that both
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players want a larger slice, and also that they both dislike delay. Thus, if agreement
to split the pie (x, 1 − x) is reached at time t, the payoff for player 1 is δ t x and the
payoff for player two is δ t (1 − x), for some δ ∈ (0, 1). Find the Subgame Perfect
Nash Equilibrium of this game.
3. Consider the following game with imperfect information. There are two players,
each with two actions {L, R}, and there are three possible states, α, β and γ. The
signals, τ , and prior beliefs p are as follows:
τ1 (α) = a τ1 (β) = τ1 (γ) = b
τ2 (γ) = g τ2 (α) = τ2 (β) = c
p(α) = 9/13 p(β) = 3/13 p(γ) = 1/13
The state-contingent utilities are as follows
L
R
L 2, 2 0, 0
R 3, 0 1, 1
state α
L
R
L 2, 2 0, 0
R 0, 0 1, 1
states β and γ
Interpret this game in terms of the players’ information. Find the Bayesian Nash
equilibrium of this game. Explain your result.
Part B. DO ONE (1) LONG ANSWER QUESTION. (Time allotted: 1 hour.)
1. Describe the Quantal Response Equilibrium and Level-K models in detail. Compare
and contrast them from a theoretical point of view, e.g. their assumptions and the
aspects of strategic interaction that they capture. Which would you anticipate to
provide the best predictions for anonymous strangers in a completely novel strategic
game? Would your answer differ if the players were a married couple engaged in
a resource allocation problem? Explain and cite specific experimental studies to
support your discussion where relevant.
2. Consider an economy with two dates t = 0, 1, two equally probable states of nature
s = 1, 2, two goods l = 1, 2 and two consumers i = 1, 2. The consumers have
endowments of the goods at the second date only (there is no consumption at the
first date). Consumer i’s endowment is denoted by ωsi = (ω1si , ω2si ) in state s and
defined by
ωs1
(
(1, 2) if s = 1
=
(3, 4) if s = 2
and ωs2
2
(
(4, 3) if s = 1
=
(2, 1) if s = 2
Any non-negative bundle of goods is feasible for the consumers. Consumer is preferences are described by a von Neumann Morgenstern utility function usi (x1si , x2si ),
where xlsi is i’s consumption of good l in state s. Assume that usi (·) = ui (·) for all
s, i.e., i’s utility function does not depend on the state of the world. Also assume:
u1 (x1s1 , x2s1 ) = ln x1s1 + 2 ln x2s1
and
u2 (x1s2 , x2s2 ) = ln x1s2 + ln x2s2
a. Suppose the economy described above has a complete set of markets for contingent commodities at date 0, that is, it is an Arrow-Debreu economy. Give
a formal definition of the competitive equilibrium for this Arrow-Debreu economy, including a complete set of first order conditions. (time permitting: solve
for the Arrow-Debreu equilibrium.)
b. Now suppose that there are no markets for contingent commodities at date 0.
Instead, consumers can trade two real assets k = 1, 2 at date 0 and can trade
good 1 on spot markets in each state s at date 1. Let rsk be the returns of
asset k (units of good 1) in state s. The asset returns are defined as follows:
1 2
r11 r12
=
R=
r21 r22
4 1
The assets are assumed to be in zero net supply, so all portfolios must be selffinancing. Show that the asset structure is complete. Give a formal definition
of a Radner equilibrium for this economy, including a complete set of first
order conditions.
c. Briefly discuss the connection between the economies in parts (a) and (b),
focusing on the ways each economy handles uncertainty about the state in
time 1.
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MICRO POLITICAL ECONOMY 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 - Problems. Choose one (1). [1 hour]
1) A worker in a capitalist firm has a utility function U=U(w,e), so that expected utility
V = U(w,e) + f(e)V + (1-f(e)) Z
1+i
where
w = wage rate
e= effort per hour by the worker
f= probability of retaining one’s job, i.e. the level of supervision in the firm
I = interest rate
Z= worker’s fall back position, i.e. what s/he would earn if fired
1.1 If we set p = 1, the firm has the production function Q = Q(he), where h = hours
hired.
Assume the firm maximizes profit and that f is costless. Use the firm’s first order
conditions to calculate h*, w* and e*. Maximize profit subject to the workers’ effort
function, choosing w*. Explain the FOCs.
1.2 Explain how w* differs from the standard neoclassical wage. What assumptions
of political economists produce the differences in outcome? What are the
economic implications of the differences in outcomes?
1.3 Graph the equilibrium wage and effort levels chosen by the employer and worker.
1.4 Using the worker’s BRF and also the FOCs from the full equation in from part b.,
outline two ways that the firm could reduce unit labor costs (labor costs per unit of
output) without reducing the wage (given that Qe > 0), linking these directly to
variables in the BRF. Graph the new labor extraction functions.
1.5 Carefully explain three ways that these dynamics change if the firm were workerowned. Compare the expected equilibrium to that graphed in (c) above.
2) Invidious Consumption: Consider two people, Upper and Lower, who are part of the
same society, and whose consumption levels affect their own and each other’s utility
in the following way: each derives positive (marginal) utility from their own
consumption, and negative (marginal) utility from the consumption of the other
(invidious means having envy). They also have negative marginal utility from hours
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of work and, the more the one consumes, the greater for the other is the marginal
utility of his/her own consumption.
Let c and C be the consumption levels, h and H be the hours of work expressed as a
fraction of each day, and u and U be the levels of utility enjoyed by each. You may
assume that each may survive, if necessary, with 0 consumption and zero utility, and
consumption is possible only by working, with an hour of work producing an hour of
consumption, so c=h and C=H..
Suppose the utility function of Lower is u = a(c-bC) +gcC+dh2 , and analogously for
Upper, where a, b, g and d are constants, and a=A, b=B, g=G, and d=D.
Assume the agents make their choices non-cooperatively, and a=A=1, g=G=1, d=D=2, and b=B=0.5.
2.1 Write down the optimizing problem must each solve and derive their best
response function, defined for h, H.
2.2 Give the Nash equilibrium values of h and H, labeled h* and H*. Graph it.
2.3 Can you say anything about whether Upper and Lower arrive at the Pareto
Optimal level? (Explain carefully.)
2.4 Now assume that the state (having full knowledge of all relevant information)
wants to implement at Pareto Optimal solution using a tax, which will in turn be
distributed as a lump sum (equally) to the two people. Does there exist a tax which
will implement the Pareto Optimal equilibrium? If so, what is it?
2.5 Describe one alternative (non-governmental) approach to improving outcomes
and explain why it might, or might not, be preferable.
3) 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.
3.1 What is the landlord’s optimal effort level (farming his own land)?
3.2 Assuming that the landlord has the power to determine α, what value would she
select? (Give both FOCs and a numerical value.)
3.3 Indicate the levels of utility achieved by all three individuals.
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3.4 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 land lord’s 3 acres, including the former landlord. As a result,
how much does each farmer work and what is their level of utility?
3.5 Explain this result.
Part B - Essay. Choose one (1) of the following. [1 hour]
1) Explain the important 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.
2) Drawing on Bowles, Knight, and Greif, carefully explain a “Bowlesian” theory of
underdevelopment and offer relevant examples to support the theory.
3) How can power be exercised in voluntary exchange? Discuss three alternative
explanations based on recent work in heterodox microeconomic theory. In each case,
be clear about how power is defined, and about any policy implications resulting from
the theory.
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