Macroeconomics
Preliminary Examination
Part II
Fall 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
2 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 0 < β < 1. Each of the consumers is endowed with a sequence of
goods:
(ω ,ω ,ω ,ω ,...) = ( 5,2,5,2,...)
(ω ,ω ,ω ,ω ,...) = ( 2,5,2,5,...)
1
0
1
1
1
2
1
3
2
0
2
1
2
2
2
3
In other words, ω t1 = 5 if t is even and ω t2 = 2 if t is odd, with the pattern
reversed for ω t2 . 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 an
Arrow-Debreu equilibrium.
(b) Define a sequential markets equilibrium 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.
i
i
(d) Prove that the equilibrium consumption sequences are constant, c t = c .
1
2
Explain how you would calculate the values of c and c .
(e) Define a Pareto efficient allocation. Is the equilibrium allocation Pareto
efficient? Explain.
3 Question II
Dynamic Programming in an Economy with Leisure
Consider the social planning problem of choosing sequences ct , xt ,lt , kt to
solve
∞
max ∑ β t ( ln ct + γ ln xt )
t=0
s.t.
ct + kt+1 ≤ θ ktα lt1−α
xt + lt ≤ 1
ct , xt ,lt , kt ≥ 0
k0 ≤ k 0.
(a) Write down the Bellman equation for this problem.
(b) Guessing that the value function V ( k ) has the form a0 + a1 ln k and that
the policy function for labor l ( k ) is constant, find analytic solutions for
the value function V ( k ) and the policy functions c ( k ) , x ( k ) ,l ( k ) , k ' ( k ) .
4 Question III
Overlapping Generations Models with Production
Consider an overlapping generations economy in which each generation
has a representative consumer who lives two periods and has preferences
represented by
t
ln(ctt ) + ln ct+1
.
( )
The consumer has endowment of one unit of time to work when young and
no endowment when old. Output is produced using capital and labor. The
resource feasibility constraint for this economy reads as
ctt−1 + ctt + kt+1 − (1− δ ) kt = θ ktα lt1−α .
There is an initial old consumer that is endowed with k 1 units of capital. He
has preferences represented by
( )
ln c10 .
(a) Define an Arrow-Debreu equilibrium for this economy.
(b) Define a sequential markets equilibrium for this economy.
t
(c) Derive the demand functions ctt (⋅) and ct+1
(⋅) as function of the wage and
the interest rate.
(d) Combine the profit maximization conditions for the firm, the demand
functions for consumers and the feasibility condition to reduce the
conditions for a sequential markets equilibrium for period t to a single
second order difference equation in kt−1 , kt , and kt+2 .
(e) Find the steady state kt = k * .
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