Cooperative Ph.D. Program in Agricultural and Resource Economics,
Economics, and Finance
QUALIFYING EXAMINATION IN MACROECONOMICS
June 10th, 2010
8:45 a.m. to 1:00 p.m.
THERE ARE FIVE QUESTIONS –ANSWER QUESTION 1
THEN ANSWER ANY THREE OF FOUR QUESTIONS.
If you answer all problems 2-5, then specify which problems you want scored
RESPOND TO A TOTAL OF FOUR QUESTIONS.
 You must complete the examination within four hours. You will have 15
minutes to read over the questions before starting (8:45-9:00).
 This exam is closed book. Calculators and paper will be provided.
 Read the question carefully. Allocate your time carefully. Parts within
questions will often vary in difficulty and weight. Be sure to do all parts of
each question chosen.
 If necessary, it is permissible to make clarifying assumptions, but be sure to
label them explicitly. (Grades will not take unstated assumptions for
granted.) Also, label graphs and define notation.
 Number your answer sheets consecutively. Begin your answer to each
question on a new page and identify the questions number.
Write your exam ID number on EACH page of the exam.
Instructions
Do problem 1. Do three of problems 2-5. (If you attempt all of problems 2-5, specify which
three you want scored.) Each of the four problems is worth 25% of your total score.
Problem 1
Consider a Diamond overlapping generations economy in a steady state. There are two identical
countries, labeled 1 and 2. The government in each country imposes a corporate tax to finance a
public good that does not affect utility or production, for simplicity. Firms produce the one
private good using capital and labor. Capital is perfectly mobile across countries, while labor is
immobile. Each period there is one consumer born in each country who lives for two periods.
The consumer is endowed with one unit of labor in the first period and none in the second. The
consumer earns a wage (w) in the first period of life and can consume it (c1) or save (s) it. In the
second period of life she simply consumes (c2) her savings income. Savings is allocated in the
world economy and earns the world interest rate r. She has preferences over consumption in the
two periods U = c1 + u(c2). (We have omitted a country superscript for brevity.)
The question are equally weighted.
a.
Pose and solve the representative consumer’s decision problem who is living in the
representative country and find the savings function Sj( ), where the j superscript denotes the
country j = 1, 2. What is savings a function of?
b.
How does savings respond to the interest rate? How does it respond to the wage? Prove
your answer.
Output per worker is produced by competitive firms using capital per worker. Firms maximize
profit, f(k) – (r + t)k – w, where k is capital per worker and t is the corporate tax rate.
c.
Find the profit maximizing condition and solve it for the capital demand function Kj( ) for
country j. What is capital per worker a function of?
d.
How does capital per worker respond to the interest rate? How does it respond to the
corporate tax rate? Prove your answer.
e.
What is the equilibrium condition in the world capital market using your answer of parts
A and C? (You’ll have to use the country superscripts to clarify.)
f.
Suppose country 1 lowers its corporate income tax rate t1. How does the world interest
rate respond? Suppose you are an advisor to country 2. What will you tell the government in
country 2 about the impact on its capital stock when country 1 lowers its tax on capital?
Problem 2
In this problem, you will consider how a possible occurrence of "rare disasters" in
aggregate endowment can a¤ect asset pricing, as explored by Rietz (1988) and Barro
(2006). Consider a closed, endowment economy with a representative consumer, with
a single perishable consumption good and a complete …nancial market. The consumer
maximizes at period 0
#
"1
X Ct1
t
;
E0
1
t=0
where 0 <
growth is
< 1,
> 0 and
Yt+1
=
Yt
6= 1; and Ct is the consumption at period t. The endowment
exp(g) with probability 1 p ("no disaster")
exp(g)B with probability p ("disaster")
Here, Yt is the endowment at period t. Also, g, B and p are constant. Notice that B
re‡ects an endowment decrease in case of a disaster. For example, if B = 0:7, then
the endowment falls by 30%. A disaster occurs with probability p at each period. The
occurrence of the disaster is independent and identically distributed over time.
All parts are equally weighted.
a. What is your economic interpretation of g? Write in plain English.
b. The fundamental equation of asset pricing is P0j = E0 [m1 X1j ], where P0j is the price
of one-period asset j at period 0, m1 is the stochastic discount factor between periods 0
and 1 for any one-period asset, and X1j is the payo¤ of asset j at period 1. Obtain m1 for
the given preferences.
c. Obtain a formula for a one-period risk-free rate between periods 0 and 1, denoted
by r0f . Obtain the value of r0f by calibration, using B = 0:7, p = 0:008, exp(g) = 1:0202,
= 10, and = 0:95.
d. The wealth portfolio provides the endowment as dividend at each period. That is,
if one pays P0W units of consumption goods to purchase a unit of this asset, it delivers
Y1 units of consumption goods at period 1, Y2 units at period 2, etc. Calibrate the pricedividend ratio of this asset at period 0, i.e., P0W =Y0 , using the numbers given in part
c.
e. Calibrate the expected return on the wealth portfolio, E0 [r1W ], where 1 + r1W
(Y1 + P1W )=P0W , using the numbers given in part c. Discuss whether an introduction
of rare disasters can improve the model’s predictions on the risk-free rate and equity
premium.
2
Problem 3
Time is continuous. There is no uncertainty. Consider a closed, endowment economy
with a representative consumer. The consumer solves at time 0
Z 1
max 1
e t [log[H(t)] + C(t)] dt;
fI(t);C(t)gt=0
0
where I(t) is the residential investment at period t, H(t) is the stock of housing, C(t) is
the non-residential consumption, and 0 < < 1 is constant. (Notice that the logarithm
is for H(t) only.) The constraints are
Y (t) = I(t) + vH(t)
_
H(t)
= I(t)
_
for all t > 0, where H(t)
I(t)
H(t)
2
+ C(t);
H(t);
dh(t)=dt. Here, Y (t) is exogenous income (or endowment),
2
I(t)
and v > 0 and > 0 are constant. Notice that the term vH(t) H(t)
is the adjustment
cost, re‡ecting the imperfect short run supply of resources to construction activities.
All parts are equally weighted.
Hint:
maxfu(t)g1
t=0
R1
0
e
t
h(x(t); u(t))dt
s.t. x(t)
_
= g(x(t); u(t)); x(0) given.
H(x(t); u(t); (t)) = h(x(t); u(t)) + (t)g(x(t); u(t)):
First-order conditions: Hu = 0 and Hx =
(t)
_ (t):
a. Eliminate one of the two constraints and set up the Hamiltonian function. Introduce
(t) as a co-state variable (as in the "Hint" above).
b. Obtain the …rst-order conditions. Write these conditions as
(t) =
_ (t) = (t)
?
?
+
?
:
c. What is your economic interpretation of (t)? Write in plain English.
d. Consider a "steady state" in which all variables (including, of course, Y (t) and
(t)) are constant. This is like a balanced growth path in which all growth rates are zero.
Recall that Y (t) is exogenously given, and hence, the consumer knows its steady-state
level in advance. Solve for the steady-state levels of I(t) and H(t).
3
Problem 4
Consider an endogenous growth model. The representative consumer has the utility function
U = ∑ t =0 β t log ct ,
∞
where 0 < β < 1 . The consumer is endowed with k0 units of initial capital. The resource
constraint for this economy is
ct + kt +1 − (1 − δ )kt =
θ kt ,
where 0 < δ ≤ 1 . The tax rate on capital income is τ . Tax revenue is rebated to the consumer as
a lump sum.
All parts are equally weighted.
a. Define an Arrow-Debreu equilibrium.
b. Define a sequential markets equilibrium.
c. Reduce the equilibrium conditions to a second-order difference equation in kt , kt +1 , and kt + 2 .
d. Calculate the growth rate of the economy on a balanced-growth path. How is the growth rate
affected by an increase in the capital income tax?
e. Set up the social planner’s problem as a recursive dynamic programming problem and
characterize the solution.
Problem 5
Consider an economy with two islands and measure one of workers. Each worker has the utility
function
U = ∑ t =0 β t ct ,
∞
where 0 < β < 1 . Island i , i = 1, 2 , starts with measure mi of workers. Each island produces the
same consumption good, the price of which is normalized to one. Output on island i is given by
yi = θi αi ,
where θi is a technology shock with Markov transition function q (θi , θi′) ,  i is the measure of
workers that are employed on island i , and 0 < α < 1 . Each employed worker is paid a
competitive wage, which is spent on consumption. Profits go to agents outside of the model.
After learning the current technology shocks, a worker can either stay on his/her current island
and work or switch islands. Travel between islands takes one period.
The Markov process is such that the technology shock is either low, θ L , or high, θ H , and the
probability of switching shocks is 1 − π . Shocks are independent across islands.
All parts are weighted equally.
a. Define a recursive competitive equilibrium.
b. Set up a social planner’s problem in recursive form such that the planner’s allocation of
workers is the same as the recursive competitive equilibrium allocation.
c. Calculate the steady-state allocation of workers for the following two cases: (i) π = 1/ 2 and
(ii) π = 1 , θ1 = θ H , and θ 2 = θ L .
d. Suppose that 1/ 2 < π < 1 . Initially, m
=
m=
1/ 2 , θ1 = θ H , and θ 2 = θ L . Show that the size
1
2
of the first migration is increasing in π .