Macroeconomic Qualifying Exam-303 Module
Claremont Graduate University
May, 2006
Lamar
Answer 2 of 3 questions. All parts are equally weighted.
1. Consider a “small” endowment-open economy, with a single nonstorable
tradable good. Agents receive endowments of this tradable good in each
period. Since the economy is “small” consumers face an exogenous world
interest factor Rt+1, where Rt+1=(1+rt+1). Each country has perishable
endowments e1,e2. Population is normalized to 1. The utility function of
this economy is logarithmic U (C1,t , C 2,t +1 ) = log C1,t + β log C 2,t +1 .
a) Set up the consumer problem and find optimal savings.
b) Determine the optimal consumption levels.
The Current Account (CA) of a country is given by its saving, i.e.,CA1 = e1-C1,t.
c) Suppose the following supply shock, endowment in period 1 falls
and the endowment in period 2 rises in such way that the present
value of wealth is unchanged. Find the effect on the current
account balance CA1. Interpret your result.
Hint: Express the optimal consumption level C1 as a function of the present value
of wealth (W).
d) Suppose now other supply shock, the endowment in period 1 falls
with no change in the endowment in period 2. Find the effect on the
current account balance CA1. Interpret your result. Compare your
result with your finding in c).
e) Suppose there is an exogenous change in the world discount
interest factor. Find the effect of such a change on the Current
Account Balance.
2. Consider a standard Overlapping Generations Model with productive
capital where agents live two periods, working when young and retired
when old. Population growth is given by Nt+1 = (1+n)Nt, where Nt is the
number of youngsters at time t, and n > -1. Consumers maximize lifetime
utility, which is given by,
U (c1,t , c 2,t +1 ) =
c11,−t θ
1−θ
+β
c12−,tθ+1
1−θ
, β Є (0,1), θ > 0, θ ≠ 1.
Net production is given by a Cobb-Douglas production function,
Y − δK = F ( K , N ) − δK = K α N 1−α − δK .
a) Derive the agent optimality conditions. Find optimal savings.
b) Obtain firm’s first order conditions on their choice of labor and capital.
c) Write down the Capital Market Equilibrium Condition.
d) Assume now that θ =1, find steady states, and derive the phase portrait
including arrows of motion.
e) Show the stability and uniqueness properties of all steady states.
f) Now assume α >1, redo e) and sketch the phase portrait. Justify your
answer.
3. Consider an Overlapping Generations Model with production, and
population growth given by Nt+1 = (1+n)Nt, where Nt is the number of
youngsters at time t, and n > -1. Agents work when young and enjoy
retirement when old. Assume we have a Pay-As-You-Go (PAYG) social
security system.
Utility is separable and given by U (c1,t , c2,t +1 ) = U (c1,t ) + β U (c2,t +1 ) .
a) Write down the government budget constraint in per capita terms.
b) Derive first order conditions for the consumer.
c) From what you found in b) show graphically and analytically the effect of
taxes on savings and capital accumulation.
d) Now suppose there is a permanent reduction in the fertility rate n of the
population, starting next generation. Reducing the benefits to the old
generation is politically unfeasible; in order to keep the social security
benefits constant, the social security administration increases the
contributions by young people to match the benefits to be paid to the
old.
This question asks you to the effects over capital accumulation of a
reduction in the fertility rate of the population while keeping social
security payments. If your answer is ambiguous, at least try to identify
the effects at play.