AMERICAN UNIVERSITY
Department of Economics
Comprehensive Exam
Preliminary Theory
June 2012
Total Exam Pages: 4
COMPREHENSIVE EXAM IN PRELIMINARY THEORY
Directions: Complete all questions in both Part I and Part II. Read the questions carefully. Please
show all of your work, and make sure to „explain‟ if the question indicates. Your grade depends on
the quality of your explanations. Answers should be expressed in your own words.
Part I. Microeconomics
A. Short Answers
1. An oligopolistic market has n firms producing a homogeneous good. Let q i denote the
quantity produced by firm i and c i its constant marginal cost. The market demand curve is
n
given by Q  P  , where Q   q i is the total quantity sold. Find the price P in the
i 1
Cournot equilibrium (assume that the c i ‟s are such that all firms produce a positive amount
and that the second order conditions are satisfied). If all firms have the same marginal cost
ci  c , what is the price when n  1 and when n   ? Discuss.
2. Farmer Greenjeans owns 10 acres of land, on which he can grow wheat or corn. If the
weather is good, each acre of land yields a profit of $20 if planted with wheat and $10 if
planted with corn. If the weather is bad, an acre of wheat yields $10 in profit and an
acre of corn yields $17.50. Good and bad weather are equally likely. Mr. Giant is an
expected utility maximizer with utility function: U = ln(Y), where Y is total profit.
a. How many acres should Mr. Giant devote to wheat production and how many
to corn production?
b. Suppose that Mr. Giant can buy insurance against bad weather. For every $1 in
premium he pays, he will receive $2 if the weather is bad and nothing if the
weather is good. How much insurance should Mr. Giant buy, how many acres
should he plant with wheat, and how many with corn?
B. Long Answers
1. A consumer has utility function
u( x1 , x2 )  ln( x1 )  x2
Consumer income is w.
a. Find the Marshallian demand for good 1 and good 2 as a function of p1 and p2.
Calculate the consumer‟s indirect utility function.
b. Calculate the consumer‟s expenditure function and Hicksian demand for good 1.
c. Assume that the price of good 1 increases to q1. Illustrate on a graph and
calculate the consumer‟s equivalent variation associated with the price increase.
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2. Glee! The Concert is scheduled to perform at a theater with seating capacity 80. All
tickets for this concert are to be sold in advance at a single uniform price per seat.
Because too many empty seats can make the concert less enjoyable, the demand curve for
the performance has the following form:
Q = 100 - P
for Q < 56
Q = 100 - P + ½ (Q - 56)
for 56 < Q <81
The owner of the theater sets the price for concert tickets, and pays all costs. These costs,
including Glee’s fee, can be expressed as a constant marginal cost of $c per ticket.
a. Graph the demand curve for this concert, and the associated marginal revenue curve.
Show and explain your derivation.
b. If c = 0, what price and quantity should the owner choose?
c. Find the critical value of c, (call it c*), that determines whether the owner will set the
price so as to get an audience that fills more than 70% (56 seats), or less than 70% of
the arena. (A precise expression for this may be complicated – in that case, it is
sufficient to show the condition(s), which would determine the answer). Explain
intuitively how the owner‟s profit-maximizing decision depends on whether c is
above or below c*.
d. If the owner currently sets the level of Q at 55, what must the value of c be? Explain.
(If there is no such value of c, say so, and explain why not).
e. Suppose that consumer preferences change in such a way that demand is now equal to:
Q = 100 - P
for Q < 56
Q = 100 - P + (Q - 56)
for 56 < Q <81
How does the owner‟s profit-maximizing decision change as the level of c changes?
(An intuitive explanation is sufficient).
Part II. Macroeconomics
A. Short Answers
1. Lifecycle saving
a. Lay out the key equations of the basic lifecycle model of consumption and saving, in
which wage earnings and lifespan are known with certainty.
b. Assuming logarithmic utility, set up the Lagrangian and solve the model for optimal
consumption at time t. What determines the level of consumption in any given period?
c. In this model, how do increases in interest rates affect consumption? Explain,
drawing implications for monetary policy.
2. What is the „Lucas critique‟? Briefly explain the simple model developed by Lucas, in which
producers can‟t readily distinguish „news‟ from „noise‟ in incoming information on changing
economic conditions. How does this model suggest that the Phillips curve should be reinterpreted?
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B. Long answers
1. Exchange rates and the small open economy.
Under the “small country assumption,” the country is so small relative to the world economy
that it can take all foreign variables [*] to be given. The following set of equations describes
the economy:
Aggregate supply
yt = -b1 pt + b2(pt - Et-1[pt]) + et
Aggregate demand
yt = a1pt – a2 rt + ut
UIP:
pt = rt* - rt + Et [pt+1]
Real exchange rate:
pt = st + pt* - pt
Consumer prices:
qt = h pt + (1-h) (st + pt*)
Fisher relationship:
rt = it – { Et(pt+1) – pt }
Expected depreciation:
Et [st+1] – st = it - it* where it* = r* + Et [pt+1*] – pt*
Real money demand:
mt – qt = yt – cit + vt
yt = real output
pt = real exchange rate
st = nominal exchange rate
pt = prices of domestically-produced goods
qt = consumer prices
it = nominal interest rate
rt = real interest rate
mt = nominal money
[et,ut,vt] are normally distributed shocks to supply, demand, and money demand respectively.
They are assumed to be: mean zero, serially uncorrelated, and uncorrelated with each other.
a. Assume initially that wages and prices are flexible so that the „price surprise‟ in the
aggregate supply equation equals zero. Find the equilibrium exchange rate pt by using the
interest rate parity condition to eliminate rt from the aggregate-demand equation, then
equate aggregate demand and supply and solve for pt.
b. Taking Et [pt+1] to be exogenous, how will pt be affected by: an increase in the world
interest rate? a shock to aggregate demand? a shock to aggregate supply? Explain.
c. Use the expression derived in (a) and solve forward to derive an expression for pt as a
function of future values of {rt*,ut, et}.
d. Using your answer in c., explain how an unanticipated increase in the world interest rate
would be expected to affect the domestic currency: will it appreciate, depreciate or remain
unchanged? Indicate whether this is differs from the expected result.
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2. Capital and growth
Consider an economy in which output is a function of two types of capital: physical kt and
human ht. Output per worker is:
yt = A ktα ht1-α
The equations of motion for the two types of capital stock are:
Δkt+1 = itk – δkt
Δht+1 = ith – δht
where itk and ith are levels of investment in physical and human capital respectively.
Population growth is assumed to be zero. The national resource constraint is as follows:
yt = ct + itk + ith
The representative consumer is infinitely lived and maximizes discounted utility as follows:
∞
U = Σ βs ln (ct)
s=0
a. Set up the Lagrangian using the representative consumer‟s utility maximization problem
and the national resource constraint, and derive the first-order conditions with respect to
ct+s, kt+s, and ht+s.
b. Show the Euler equation and the optimal ratio of the two capital stocks.
c. Derive the optimal steady-state rate(s) of growth of consumption, output, and the two
types of capital.
d. In this model, can growth in output per capita be positive in the steady-state, even though
there is no technological change and returns to capital diminish? Explain.
e. In this model, would a one-time positive shock to A affect the steady-state growth rate
temporarily, permanently, or not at all? Explain.