Macro Candidacy Exam: August, 2013
Directions. There are …ve (5) questions worth a total of 180 possible points.
The points assigned to each question and each part of each question are given in
parentheses. (If you can proceed at the rate of a point per minute, then you can
complete the entire exam with time to spare.)
1. Stochastic dynamic programming (30 points)
Consider an oil producer with a given amount of reserves in the ground. The
producer can extract the oil at a cost which depends on a random variable, z. Suppose
that the extraction cost is increasing and convex in the amount extracted and the
marginal extraction cost is increasing in z. The producer faces a stochastic downward
sloping demand curve. The objective of the producer is to maximize discounted
expected pro…ts. The shocks to costs and to demand are serially correlated over
time but not correlated with each other.
(a) (10 points) Formulate the dynamic optimization problem of the oil producer
as a dynamic programming problem.
(b) (10 points) Set out the Euler equation of the producer and interpret it.
(c) (10 points) If you were going to compute a solution to the above dynamic
optimization problem, how would you do so? Would you be able to …nd a solution
using value-function iteration?
2. The Lucas model of the Phillips curve (45 points)
In 1986, a prominent person made the following claim:
The Lucas Journal of Economic Theory 1972 model of money non-neutrality
is incapable of matching the observation that shocks to the stock of money
have aggregate e¤ects on real variables beyond one period.
(a) (25 points) Explain in detail why the claim is true.
(b) (20 points) Describe one or more modi…cations of the Lucas model that would
make it consistent with the observation.
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3. An RBC model with distortionary labor taxation (40 points)
The representative household’s expected discounted utility is given by
Et
1
X
j
(log Ct + log Lt );
j=0
where Ct is consumption, Lt is leisure, 0 <
according to
< 1, and
Yt = At (Kt )1
> 0. Output, Yt , is produced
(Nt )
where Kt is capital, Nt is labor, and At is an aggregate random productivity shock
that follows
log At = log At
where
1
+ "t
2 (0; 1) and "t is iid with …nite mean and variance. Also, 0 <
< 1 and
Nt + Lt = 1:
The evolution of capital is given by
Kt+1 = (Yt
Ct
G) + (1
) Kt ;
where G is constant government spending (which does not enter household utility)
and 0 < < 1.
Assume that the representative household owns the capital stock and that the
government …nances G by a tax on labor income and runs a balanced budget every
period. That is,
t wt Nt
where
t
= G;
is the labor tax rate and wt is the pre-tax real wage earned by the household.
For a competitive equilibrium of the above model:
(a) (20 points) Find the non-stochastic (that is, where "t
function of the model parameters.
0) steady-state as a
(b) (20 points) Log-linearize the (non-linear) equilibrium conditions around the
(non-stochastic) steady-state described in (a). Present the log-linearized conditions
in state-space form.
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4. Optimal monetary and …scal policy under no commitment (30 points)
The government’s objective is to minimize the loss function
Et
1
X
j
2
t+j
2
y yt+j
+
+
2
T Tt+j
j=0
subject to
t
= Et
yt = Et yt+1
t+1
(it
+ yt ;
Et
t+1 )
+
t;
and
bt =
1
bt
1
1
t
+ it
'Tt :
Here, Et is the mathematical expectation operator conditional on period-t information
(inlcuding t ), t is in‡ation, yt is output, it is the one-period nominal interest rate,
Tt is taxes, and bt is the real value of one-period nominal government debt. In addition, t is a shock that follows an AR(1) process and y ; T ; ;
; and ' are
model parameters.
The government’s instruments are it and Tt , which are chosen after t is realized
at the beginning of the period. Suppose that the government cannot commit to
future actions and, therefore, chooses it and Tt every period.
Characterize, as completely as you can, the Markov-perfect (time-consistent)
equilibrium of the model.
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5. Noncontingent borrowing and lending in a model with idiosyncratic endowment shocks (35 points)
In a widely cited 1993 paper, Huggett studied the following pure-exchange model.
There is a nonatomic unit measure of people and one good per discrete date. Each
person maximizes expected discounted utility with discount factor 2 (0; 1) and
date-t period utility u(ct ). Here, u : R+ ! R and is bounded above, strictly increasing, strictly concave, and satis…es u0 (0) = 1. Let Y = fy1 ; y2 g where 0 < y1 < y2 .
Each person’s endowment of the good follows a Markov process over the set Y , with
realizations that are independent across people.
The only market is a price-taking market in noncontingent one-period loans.
Therefore, each person faces a sequence of budget constraints, where, after seeing
the date-t realization of the endowment, each person chooses (at+1 ; ct ) subject to
ct + at+1 qt
at + yt and at+1
a,
¯
where at a is the individual’s wealth at the beginning of date-t and a > 0 is the
¯
¯
borrowing limit.
For a class of u functions, Huggett did the following. He de…ned a steady state.
He proved that when faced with qt = q 2 (0; ) for all t, there exists a stationary
distribution for individual wealth, a distribution which he treats as the cross-section
distribution of wealth associated with that q. In other words, he showed that there
exists a one-to-one mapping from q to a stationary distribution of wealth. Then,
for given parameters, he numerically searched for a magnitude of q that clears the
market; namely, one such that the integral of at over the population is zero.
Here are some things that he did not do and that we want you to do.
(a) (5 points) At the start of a date, whether in steady state or not, specify the
state of this economy.
(b) (20 points) De…ne a recursive equilibrium for the above economy. (If we knew
such an equilibrium and if it were unique, then we would know how the economy
evolves from an arbitrary state as speci…ed in part (a).)
(c) (10 points) Given the mapping from q to a stationary distribution of wealth
that Huggett showed exists, what kind of argument could be used to prove that a
steady state exists?
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