Macroeconomics
Preliminary Examination
Part II
Spring 2013
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
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
= (ω 1 , ω 2 ) . 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 = ω 2 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 market are open, who trades with whom, and so on. Define a
sequential markets equilibrium for this economy.
(c) Derive the aggregate excess demand function z0 ( p1 , m ) for the initial old
generation.
(d) Derive the aggregate excess demand functions y ( pt , pt+1 ) of the
generation of young agents and z ( pt , pt+1 ) of the old generation.
Demonstrate that they are homogeneous of degree zero in prices and
satisfy Walras’ law
pt y ( pt , pt+1 ) + pt+1z ( pt , pt+1 ) = 0 .
(e) Suppose that m = 0 (no outside money). Calculate the sequential markets
equilibrium.
Question II
TDCE in the Neoclassical Growth Model with Human Capital and
Optimal Taxation
Consider the two (human and physical) capital version of the endogenous
growth model, where the representative consumer solves:
∞
max ∑ β t ln ( ct )
t=0
subject to
∞
∑ p [c + x
t
t=0
t
∞
kt
+ xht ] ≤ ∑ pt ⎡⎣(1− τ kt ) rt kt + (1− τ nt ) wt nt ht ⎤⎦
t=0
kt+1 ≤ (1− δ ) kt + xkt , ∀t = 0,1,...,∞
ht+1 ≤ (1− δ ) ht + xht , ∀t = 0,1,...,∞
0 ≤ nt ≤ 1,
h0 , k0 given,
∀t = 0,1,...,∞
where all the variables have their standard interpretation. The effective labor
supply is given by nt ht , and τ kt and τ nt denote tax rates on capital and labor
incomes, respectively. Not that the consumer will choose to supply labor
inelastically nt = 1 . Also note that human and physical capital depreciate at
the same rate.
1−α
Firms maximize profits under the production function Aktα ( nt ht ) and
market clearing is given by
1−α
ct + xkt + xht + gt = Aktα ( nt ht ) .
Assume that all equilibrium quantities are strictly interior.
(a) Define a TDCE (Tax Distorted Competitive Equilibrium) for this
economy. Characterize the competitive equilibrium.
(b) What is the equilibrium value for the ratio ht to kt in the absence of taxes
and government spending? How is this ratio affected by the presence of
capital and labor income taxes?
(c) If the tax rates are fixed across time, show how the growth rate of γ c ,
given by
ct+1
,
ct
depends on these tax rates. (Hint: find γ c as a function of τ k and τ h .)
γc =
(d) Assuming p0 = 1, show that for arbitrary sequence of taxes, in interior
equilibria, only the initial values of h and k enter the consumer’s budget
constraint.
Question III
A Deterministic Optimal Growth Model
Consider an economy with a representative consumer with the utility
function
T
∑ β ln ( c ),
t
t
t=0
where 0 < β < 1.
Output is produced using capital as input and is given by
f ( kt ) = ktα , 0 < α < 1,
where kt is used to represent capital available at the beginning of period t .
This is capital that was accumulated in the previous period t − 1. The law of
motion for the capital stock is given by:
kt+1 = (1− δ ) kt + it ,
where it is the amount of investment expenditure in a period toward building
up the capital stock for the following period. Assume that kT +1 = 0 and
k0 > 0 is given. Capital fully depreciates at the fixed rate δ = 1.
The resource constraint for the social planner is given by:
ct + it = f ( kt ) .
(a) Define the social planner’s problem for this economy.
(b) Write down the Euler equation as a function of kt−1 , kt , and kt+1 .
k
(c) Let zt = αt , t = 0,1,...,T . Using the change of variable verify that the
kt−1
first-order difference equation (from part (b)) in zt can be written as
zt+1 = 1+ αβ −
αβ
.
zt
(d) Using the boundary condition zT +1 = 0, show that the paths for capital
and consumption are given by
1− (αβ )
T −t
kt+1 = αβ
1− (αβ )
T −t+1
ktα , t = 0,1,...,T ,
and
ct =
respectively.
1− αβ
α
t = 0,1,...,T ,
T −t+1 kt ,
1− (αβ )