Macroeconomics
Preliminary Examination
Spring 2015
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 ,... = ( 2,2,2,2,...)
(
)
( 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.
(e) Suppose now that there is a third consumer. Consumer 3 has the same utility function
∞
∑ β ln ( c ),
t
3
t
t=0
and has the endowment sequence
( w ,w ,w ,w ,...) = ( 4,1, 4,1,...).
3
0
3
1
3
2
3
3
Define a sequential markets equilibrium.
Question II
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
Question III
Consider an overlapping generations economy in which the representative consumer born
in period t,t = 1,2,..., has the utility function over consumption of the single good in
periods t and t + 1
t
t
u ctt ,ct+1
= ctt + log ct+1
(
)
(
)
t
= ( w1 ,w2 ) . Suppose that the representative consumer in the
and endowments wtt ,wt+1
initial old generation has the utility function
u 0 ( c10 ) = log c10
and endowment w10 = w2 of the good in period 1 and endowment m of fiat money.
(a) Describe an Arrow-Debreu market structure for this economy, explaining when
markets are open, who trades with whom, and so on. Define an Arrow-Debreu
equilibrium for this economy.
(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) Suppose that m = 0. Calculate both the Arrow-Debreu equilibrium and the sequential
markets equilibrium.
(d) Define a Pareto efficient allocation. Suppose that w2 > 1. Is the equilibrium allocation
in part (c) Pareto efficient? Explain carefully why or why not.
(e) Suppose now that there are two types of consumers of equal measure in each
generation. The representative consumer of type 1 born in period t,t = 1,2,..., has the
utility function over consumption of the single good in periods t and t + 1
1t
u1 ( ct1t ,ct+1
) = ct1t + log ct+11t,
while the representative consumer of type 2 has the utility function
2t
u1 ( ct2t ,ct+1
) = log ct2t + ct+12t.
The
endowments
of
these
consumers
are
( w ,w ) = ( w ,w ),i = 1,2.
it
t
it
t+1
i
1
i
2
The
representative consumers of type 1 and 2 who live only in period 1 have utility
functions log c110 and c120 , endowments w110 = w12 and w120 = w22 of the good in period 1,
and endowments m1 and m 2 of fiat money. Define an Arrow-Debreu equilibrium for
this economy.