Macroeconomics Qualifying Exam-303 Module
Claremont Graduate University
May, 2007
Lamar
Answer 2 of 3 questions. All parts are equally weighted.
1. Consider a household living for two periods with preferences given by
1
1
U (c1 , c2 ) = − e ( −γc1 ) + β (− )e ( −γc2 ) , where . The agent receives an endowment in
γ
γ
the two periods (e1, e2). Savings are rewarded at the interest rate r. The discount
1
, where ρ is the discount rate.
factor β can be written as β =
1+ ρ
a) Write down the dynamic optimization problem of the household and derive
the optimality condition.
b) Using the approximation ln(1+x)≈x solve for the optimal consumption
choices c1 and c2 as a function of the parameters.
c) Under what condition is c2 larger than c1? Explain your answer.
d) Now suppose the government introduces a tax τ on positive interest
income, that is, lenders receive interest net of taxes equal to (1- τ)r on
their savings. Borrowers instead face the usual interest rate r. Consider
the optimization problem of a household with preferences U (c1,c2) and
endowment (e1, e2). Graph the budget constraint of the household in
(c1,c2) space and show how it changes after the imposition of the interest
income tax τ.
e) How the introduction of the interest income tax τ affects the optimal
consumption choices? Does your answer depend on whether the
household is a net borrower or net lender?
2. Consider an OLG model where agents live two periods, receive income y
when young and are retired when old. Leisure is not valued. Population
grows geometrically at rate n > -1, Nt+1 = (1+n)Nt . Consider a Pay-As-You-Go
social security scheme, where each period the government taxes the young
(τ) so as to make transfers to the old (σ).
a) Set up the utility maximization problem faced by the consumer.
Write down the government budget constraint at time t. What is the
value of the transfer σ in per youngster terms?
b) Find the consumer’s saving function when preferences are
represented by
U (C1,t , C 2,t +1 ) = (1 − β ) ln(C1,t ) + β ln(C 2,t +1 ) with 0 < β < 1
c) Analyze the effect of an increase in τ on the savings of each young
consumer, and on the welfare of each generation. Recall that each
generation’s welfare depends on consumption. Does an increase in
τ increases welfare?
d) Now suppose a fully-funded social security system where the
government taxes the young in period t, invests the proceeds of the
tax, and then makes transfers to the old in period t+1. Find the
consumer’s saving in this case.
e) Analyze the effect of an increase in τ on the savings of each young
consumer, and on the welfare of each generation. Recall that each
generation’s welfare depends on consumption.
3. Consider an OLG model with productive capital and government debt.
Population grows geometrically at the rate n > -1, Nt+1 = (1+n)Nt .
Households solve the following maximization problem:
Maxc1 ,c2 (1 − β ) ln c1 + β ln c2
s.t.
c1,t = wt − st
c2,t = Rt +1 st
The net return on savings is R = 1 + r –δ, where δ ε (0,1) is the
depreciation rate on capital. Production takes place with a Cobb-Douglas
technology Y = K α N 1−α . The government rolls over its debt in perpetuity,
and has the following budget constraint Bt +1 = Bt Rt , where B is the
aggregate stock of government debt which has the same return as
physical capital.
a) Derive consumer and firm optimality conditions.
b) Write down and interpret the asset market equilibrium condition in
per capita terms.
c) How many steady states are in this model? Identify each one. Draw
the phase portrait and determine the dynamics of the system.
d) Discuss the stability of each steady state equilibrium. What is the
dynamic path of the economy when debt is zero. How the
introduction of debt affects economic growth?
e) What is the effect of government debt on interest rates? Explain.
f) Using a phase portrait analyze the effect on the economy of a very
high ratio of debt relative to capital.