Macroeconomic Qualifying Exam-303 Module
Claremont Graduate University
September, 2006
Lamar
Answer all 3 questions. All parts are equally weighted.
1. Consider a closed endowment economy, with a single nonstorable
tradable good. Consider consumers who live for two periods, their utility
1
function is logarithmic U1 (C1,t , C2,t +1 ) = log C1,t + β1 log C2,t +1 , where β1 =
.
1 + ρ1
Agents receive endowments of this tradable good in each period denoted
by e1 and e2 respectively. In the first period of their lives, consumers
decide how much to save or borrow. The consumers can borrow and lend
at an interest factor Rt+1 = (1 + rt+1).
a) Write down the intertemporal budget constraint faced by the
consumer.
b) Solve the maximization problem and determine the consumer’s
consumption in the first period.
c) Determine when a consumer is going to be a saver or borrower as
a function of the interest rate rt+1 and his own discount rate ρ1.
Now assume there is a second type of consumer in the economy who
only differs in the discount factor, with a utility function
1
U1 (C1,t , C2,t +1 ) = log C1,t + β 2 log C2,t +1 , where β 2 =
, where ρ1 ≥ ρ2.
1 + ρ2
There is the same number of consumers of each type in the economy.
d) Determine the equilibrium interest rate.
e) Verify, using the condition you derived in part c) and the interest
rate you derived in part d) that indeed one type of consumer is
going to be a borrower and the other type a lender.
2. Consider an economy described by an overlapping generations model.
Agents work in the first period of their life earning a wage wt, and decide
how much to consume and save at the an interest factor Rt+1. In the
second period individuals consume all their savings. The individual’s utility
c11,−t θ
c12−,tθ+1
+β
function is U (c1,t , c 2,t +1 ) =
, β Є (0,1), θ > 0. The production
1−θ
1−θ
function is given by y = k α . Assume markets are competitive, no
depreciation, and no population growth.
a) Solve for saving.
b) What is needed (value of θ) to have constant saving. For that value
what form does the utility function take?
From now on assume that saving are constant.
c) Write down the capital market equilibrium condition, derive the
phase portrait, and find steady-state level of capital per capita.
d) How is the steady-state level of capital per capita affected by a tax
on labor income, where τwt is the tax rate. Ignore any implications
the tax may have for government revenue. Show your results.
3. Consider a standard optimal growth model with population growth.
a) Write down and solve the planning problem.
b) Write down the market problem and show that is equivalent to the
planning problem you solved in a). Under what assumptions the
two problems are equivalent.
c) Derive the phase portrait, showing vector fields, carefully
determining the relative positions of the KK and CC curves.
d) Assume you are the central planner, given an initial ko, what would
you like to do and why?