AMERICAN UNIVERSITY
Department of Economics
Comprehensive Exam
Preliminary Theory
June 2014
Total Exam Pages: 4
COMPREHENSIVE EXAM IN PRELIMINARY THEORY
Directions: Complete all questions in Parts I - II. Please show all of your work, and make sure
to explain and discuss if the question indicates. No electronic aids are permitted.
Part I. Microeconomics
A. Short Answers
Q1. Suppose Jill A.’s indirect utility function is
V(p x , p y , I) =
I
,
px + py
where I denotes income, px the price of good x, and py the price of good y. Assume good x refers
to food and y to electricity. Jill lives in New York where px = py = 1 and her income is $150.
Suppose that food prices double while the price of electricity stays the same. However, the
Board of Directors of the New York Times, where Jill works, does not provide her with a raise.
Jill is outraged. She requests the Board to raise her income or else she will quit and work for the
New York Post.
a) Define compensating variation (CV) and equivalent variation (EV).
b) Calculate CV and EV from the information above.
Q2. Two combatants are about to face off: David and Goliath. Each can choose one weapon:
either a slingshot with a pebble or a sword with armor. David is nimble and fast and relatively
better at using the slingshot. Goliath is massive and very powerful at wielding the sword and can
be outfitted with heavy armor. Their preferences are asymmetric. Goliath would prefer that both
fighters use the same weapon. David would prefer that they use different weapons. Here are the
payoffs from this battle, conditional on weapon choices:
Goliath
Sling
Sword
Sling
(0, 0)
(3, -2)
David
Sword
(2, 0)
(1, 4)
where the first number in each pair is David's payoff, the second Goliath’s.
a) Define Nash Equilibrium. Does a N.E. exist?
b) Find the mixed strategy equilibrium, and briefly discuss the intuition behind each combatant’s
strategy.
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B. Long Answers
Q3. A competitive firm is initially operating in long run equilibrium. Its long run average cost
function is given by LRAC = y2 – 2y + 6. Assume the market demand curve is QD = 110 – 2P.
a) At what level of output, y, is the LRAC at its minimum? At that output level, compute the
value of the long run average cost, LRAC.
b) In the long run, what is the equilibrium price, P? Briefly explain why.
c) In the long run, what is the total (market-clearing) quantity of output produced, assuming that
all the firms in the industry are identical.
d) Suppose that market demand increases to QD = 210 – 2P. Assuming a constant cost industry,
what would be the new long run equilibrium output and price of the (typical) firm? What would
be the total market quantity of output produced?
e) Explain intuitively what is happening during the transition from one long run equilibrium
state to another. Specifically, what happens to the firm and industry once market demand
increases, and how is long run equilibrium restored?
f) Again, let’s suppose the new market demand is QD = 210 – 2P. But this time assume that we
are in an increasing cost industry; that is, as the industry expands, long run average costs rise to
LRAC = y2 – 2y + 7. What is the level of output at which this LRAC is at a minimum? What is
the value of LRAC at this level of output? What will be the new long run equilibrium price and
total market quantity of output produced?
Q4. Rhoda runs a small Bed & Breakfast business. She has only one employee, Carlton, whose
main job is to be a doorman. She hires him for 20 hours a week. Each week, Carleton spends
some of that time actually working and the rest of that time shirking (e.g., playing computer
games and talking on his smartphone annoyingly like passengers on United). Suppose that
Carlton’s utility function is u = ws, where w is the wage he earns and s the ‘hours’ of shirking on
the job. Rhoda’s revenue function is R = 120 – s and profit function is π = R – w.
a) From Rhoda’s perspective, what are the optimal hours of shirking by Carlton (subject to the
constraint that his utility must be no less than 100)? Specifically, what value of s would
maximize her profits subject to the constraint that u  100 . Find the equilibrium wage and
profits associated with that ‘optimal’ level of s.
b) Suppose that Rhoda pays Carleton a flat wage of w = 10 (i.e., he gets paid ten per week).
Given that he is employed for 20 hours a week, what amount of shirking, s, would maximize his
utility? What would be the firm’s profits, π?
c) Alternatively, since Rhoda is concerned about Carleton’s shirking and the adverse impact on
her revenues and profits, she decides to let him share in the profits. First, she will pay herself 50
(i.e., R = 50). Next, she will let Carleton keep the rest. Specifically, he is to choose the value of
s to maximize residual profits π’ = R – w – 50, subject to the constraint that his utility level is
100. Find s, w, and the residual profits, π’, that Carleton gets to keep.
d) Repeat the analysis in part c) where Rhoda decides to take more for herself, namely R = 100.
Find s, w, and the residual profits π’.
e) Show the results of your analysis from a) – d) in graph(s).
f) What is the ‘Principal Agent’ problem? What does this example suggest about solving that
problem?
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Part II. Macroeconomics
A. Short answers
Q1. What is the ‘Lucas critique’? How does the simple island model developed by Lucas – in
which producers can’t readily distinguish ‘news’ from ‘noise’ in incoming information on
changing economic conditions – suggest that the Phillips curve should be re-interpreted? Explain.
Q2. Why is population growth important for a PAYG public pension system to be welfareimproving? Explain, making reference to key equations of the overlapping generations model.
B. Long answers
Q3. The representative firm hires capital and labor to maximize profits as follows:
π t = At Ktα Lt1-α – rt Kt – wt Lt
where 0 < α < 1
(1)
Productivity in period t, At, is a function of the level of government expenditures, gt, as follows:
At = gt1-α
(2)
where the representative firm takes gt to be given. The representative consumer is infinitely lived
and has logarithmic utility, U(ct) = ln(ct), with a time discount factor of β. He/she allocates aftertax income between consumption and capital investment as follows:
ct + kt+1 = (1-τk) rt kt + (1-τw) wt
(3)
where ct and kt are expressed on a per-capita basis. Government expenditure on services is
financed by tax revenue:
gt = τk rt kt + τw wt
(4)
Finally, the national resource constraint is as follows:
yt = ct + kt+1 + gt
(5)
a) Use the firm’s profit-maximization to find expressions for rt and wt as a function of α, gt, and
kt.
b) Use these expressions for rt and wt and (4) to derive an expression for gt in terms of α, kt. τk,
and τw.
c) Write the Lagrangian for the consumer’s optimization problem, deriving an expression for
ct+1/ct=γ. How do τk and τw affect the consumer’s incentive to save? [Hint: a faster rate of
growth of consumption implies higher saving].
d) Define the steady-state rate(s) of growth of output, consumption and capital (on a per capita
basis). Is growth in this model exogenous or endogenous? Explain.
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e) Suppose the government sets τw=0; what value of τk maximizes γ? Next suppose the
government sets τk=0; what value of τw maximizes γ? Why is there a difference?
Q4. Consider a small open economy characterized by the following equations:
Money demand:
mt – pt = - η i t+1 + φyt
Purchasing power parity:
pt = et + pt*
Uncovered interest rate parity:
it+1 = it+1* + Et [et+1] – et
Variables are defined as follows (all variables are in logs except the interest rates):
mt
pt
pt*
yt
et
it
i t*
nominal money supply
price level
world price level
real output
exchange rate
nominal interest rate
world nominal interest rate
a) What does the small open-economy assumption mean for how this model is solved (say, in
contrast to the two-country Obstfeld-Rogoff model)?
b) Derive an expression for et as a function of mt, yt, it*, pt*, and Et[et+1] by substituting PPP and
UIP into the money demand function.
c) Use forward substitution to eliminate expected future values of et from (a). What would
cause the exchange rate to depreciate (i.e. et to rise)?
d) Suppose that yt, it*, and pt* are constant and ηi* - φy – p* = 0. Use your finding in (b) to
show how the expected depreciation of the exchange rate, Et(et+1) – et, depends on expected
future changes in money supply.
e) Suppose also that money supply follows the process:
mt - mt-1 = ρ(mt-1 – mt-2) + εt
where εt is a serially uncorrelated mean zero shock such that Et-1(εt) = 0. Using this
knowledge of money supply dynamics and your result in (c), show how a given change in
money supply between time t-1 and time t, mt - mt-1, affects expected depreciation between
time t and time t+1, i.e. Et(et+1) – et. What is the role of ρ?