Macroeconomics
Preliminary Examination
Part II
Spring 2012
Instructions: Answer two (2) questions in Part II. Each question is worth
fifty (50) points.
PLEASE MAKE YOUR ANSWERS NEAT AND CONCISE.
Make whatever assumptions you need to answer the questions.
BE SURE TO STATE THEM CLEARLY.
Part II
Question I
Ramsey Taxation
Suppose that an infinite horizon setting in which there is a representative
consumer and a representative firm as in the standard single sector growth
model. The utility function of the representative consumer is given by
∞
∑ β u ( c ,l , g ) ,
t
t
t
t
t=0
where gt is the amount of government goods and services produced in each
period.
The feasibility constraint for the firm is
ct + xt + gt = F ( kt ,nt ) .
There is no technological change.
Investment is done at the household level, and the standard law of motion
for capital is assumed to hold.
Suppose that the government has at its disposal only labor and capital
income taxes for financing expenditures, but can freely borrow and lend
(i.e., it faces a present value budget constraint). Assume that the consumer
takes gt , τ nt , and τ kt as given when making its decisions.
(a) Set up and define a TDCE (Tax Distorted Competitive Equilibrium) for
each fixed sequence ( gt , τ nt , τ kt ) .
(b) Assuming interiority of the equilibrium, give the FOCs that characterize
the equilibrium you defined in (a).
(c) Set up the Ramsey problem for this economy.
(d) If the government acts benevolently in choosing ( gt , τ nt , τ kt ) , will it be
true that τ kt → 0 ? That is, does the Chamley-Judd characterization of the
asymptotic behavior of Ramsey tax systems extend to this setting in
which gt enters the utility function? Prove it. Assume that in the Ramsey
allocation, all quantities converge to constant levels, ct → c∞ , etc.
Question II
Overlapping Generations
Consider an overlapping generations economy in which there is one good
in each period and each generation, except the initial one, lives for two
periods. The representative consumer in generation t,t = 1,2,... , has the
utility function
t
log ctt + log ct+1
(
)
t
= ( 3,1) . The representative consumer in
and the endowment wtt ,wt+1
generation 0 lives only in period 1, has the utility function
log c10
and has the endowment w10 = 1. There is no fiat money.
(a) Define an Arrow-Debreu equilibrium for this economy. Calculate the
unique Arrow-Debreu equilibrium.
(b) Define a sequential markets equilibrium for this economy. Calculate the
unique sequential markets equilibrium.
(c) Define a Pareto efficient allocation. Is the equilibrium allocation Pareto
efficient? Prove it.
Suppose now that the representative consumer in generation t,t = 1,2,... ,
is endowed with one unit of labor when young and none when old. The
representative consumer in generation 0 is endowed also with k 1 units of
capital. The resource constraint is
ctt−1 + ctt + kt+1 − (1− δ ) kt = θ ktα nt1−α ,
where θ > 0,0 < α < 1, and 0 < δ ≤ 1 .
(d) Define an Arrow-Debreu equilibrium for this economy. Characterize the
Arrow-Debreu equilibrium. (Make all necessary assumptions.)
Question III
Dynamic Programming
Consider the optimal growth problem
∞
max ∑ β t log ct
t=0
s.t. ct + kt+1 ≤ θ ktα
ct , kt ≥ 0
k0 = k 0 .
Here 0 < α < 1,0 < β < 1, and θ > 0.
(a) Write down the Euler conditions and the transversality condition for this
problem.
(b) Calculate the steady state values for c and k.
(c) Write down the functional equation that defines the value function for
this problem. Guess that the value function has the form a0 + a1 log k.
Calculate the policy function.
Consider now an overlapping generations economy in which there is one
good in each period and each generation, except the initial one, lives for two
periods. The representative consumer in generation t,t = 0,1,... , has the
utility function
t
log ctt + log ct+1
and is endowed with one unit of labor when young and none when old. The
representative consumer in generation -1 lives only in period 0, has the
utility function
log c0−1
and is endowed with k 0 units of capital and amount b 0 of nominal
government assets. The resource constraint is
ctt−1 + ctt + kt+1 − (1− δ ) kt = θ ktα ,
where θ > 0 , 0 < α < 1, and 0 < δ ≤ 1.
(d) Define a sequential markets equilibrium. Calculate the two steady states.
Make all necessary assumptions. (Hint: one steady state is when r b = 0 ,
where r b is the interest rate on bonds, and another one is when b = 0. )