Macroeconomics
Preliminary Examination
Fall 2014
Instructions: The preliminary exam has three questions. Choose two (2) questions. 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.
Question I
Models with Infinitely Lived Consumers
Consider an economy with two infinitely lived consumers. There is one good in each
period. Consumer i,i = 1,2, has the utility function
∞
∑ β ln ( c ) ,
t
i
t
t=0
where β , 0 < β < 1, is the common discount factor. Each of the consumers is endowed
with a sequence of goods:
w10 ,w11 ,w12 ,w13 ,... = ( 5,1,5,1,...)
(
)
( w ,w ,w ,w ,...) = (1, 4,1, 4,...) .
2
0
2
1
2
2
2
3
There is no production or storage.
(a) Describe an Arrow-Debreu market structure for this economy, explaining when
markets are open, who trades with whom, and so on. Define the Arrow-Debreu
equilibrium.
(b) Describe a sequential market structures for this economy, explaining when markets
are open, who trades with whom, and so on. Define a sequential markets equilibrium
for this economy.
(c) Carefully state a proposition or propositions that establish the essential equivalence of
the equilibrium concept in part (a) with that in part (b). Be sure to specify the
relationships between the objects in the Arrow-Debreu equilibrium and those in the
sequential markets equilibrium. (You are not asked to prove this proposition or
propositions.)
(d) Calculate the Arrow-Debreu equilibrium. (This equilibrium is unique, but you do not
have to prove this fact.) Use the answers to parts (c) and (d) to calculate the
sequential markets equilibrium.
Question II
TDCE and Ramsey Problem
Consider an infinite horizon setting in which there is one representative consumer, one
representative firm, and the government that finances a given stream of expenditure
∞
. Preferences of the representative consumer are given by
{gt }t=0
∞
∑ β u ( c ,l ) ,
t
t
t
t=0
where ct is consumption and lt is leisure in period t. The representative consumer is
endowed with an initial stock of capital k0 and decides how much to work, nt , consume,
ct , and invest, xt , in each period. The total amount of time is normalized to 1. The
(
)
representative firm operates a CRS production technology Ft ktf ,ntf . The feasibility
constraint for the firm is
ct + xt + gt = Ft ( ktf ,ntf ) .
There is no technological change. Investment is done at the household level, and the
standard law of motion for capital is assumed to hold. Capital depreciates at rate δ .
(a) Define a Tax Distorted Competitive Equilibrium (TDCE) in which the only taxes
present are taxes on consumption τ ct and labor income τ nt .
(b) Derive the conditions that fully characterize the TDCE defined in part (a). Make the
appropriate assumptions.
(c) Suppose τ ct → τ c* , τ nt → τ n* , and gt → g* . What should c* ,n* , k * , x * satisfy?
Characterize the steady state.
(d) Set up the Ramsey problem.
Question III
TDCE in the Neoclassical Growth Model with Human Capital and Optimal
Taxation
Consider the following Planner’s Problem:
∞
max ∑ β t
t=0
ct1−σ
1− σ
subject to
ct + xkt + xnt ≤ Aktα zt1−α
kt+1 ≤ (1− δ k ) kt + xkt
ht+1 ≤ (1− δ h ) ht + xht
zt ≤ nt ht
0 ≤ lt + nt ≤ 1
ct ,nt ,lt , kt ,ht ≥ 0
h0 , k0 given
Assume that the government has a fixed sequence of expenditures that it must finance,
gt , and that it can use taxes on capital and labor income, τ kt and τ zt .
(a) Define a Tax Distorted Competitive Equilibrium (TDCE) for this economy.
(b) Derive the conditions that fully characterize the TDCE defined in part (a).
(c) What is the Ramsey problem here for a benevolent government? In particular,
carefully drive and explain the implementability constraint for this environment.
τ
(d) Assume that δ k = δ h . What is kt in this case?
τ zt