Econ 809: Assignment 3
Spring 2006
David Andolfatto
1. Consider an economy consisting of 2-period-lived overlapping generations
(with an initial old generation that lives for one period only). Let us
restrict attention to stationary allocations. A representative young agent
has preferences given by:
h(l) + βc2 ,
where l denotes time devoted to home-production when young (which
generates h(l) units of home-production, with h0 > 0 > h00 ) and c2 denotes
future consumption (i.e., consumption when old). Each young agent is
endowed with one unit of divisible time that can be allocated either to
leisure or work (e), so that n + l = 1. The initial old generation has a zero
time endowment. Time spent at work generates output y1 = ωe, where
ω > 0 is an exogenous productivity parameter. Output is nonstorable.
Let Nt denote the population of young agents at date t and assume that
Nt = nNt−1 , with n > 0 (gross population growth rate).
(a) Characterize the Pareto optimal allocation (assuming that the initial
old receive the same consumption level as old agents of any generation). Describe the pattern of trades that are necessary to achieve
this allocation.
(b) Show the competitive equilibrium allocation is autarkic. Explain.
(c) Imagine now that the initial old are endowed with fiat money, M0 /N0
per agent. Assume that the government keeps the stock of money
fixed over time. Characterize the competitive monetary equilibrium
and compare it to the Pareto optimal allocation.
(d) Assume now that the government expands the stock of fiat at the
exogenous rate μ > 0, so that Mt = μMt−1 . Assume that new money
is distributed as a lump-sum transfer to old agents (e.g., as a social
security payment). Characterize the competitive monetary equilibrium. How does the economy’s real GDP depend on μ? Explain.
Is money ‘neutral’ in this economy? Is money ‘superneutral’ in this
economy? Explain.
2. Consider two economies, labelled A and B. Each economy has an identical
and fixed population size. The populations consist of 2-period-lived overlapping generations. A representative young agent in each economy has
preferences given by u(c1 , c2 ) = c2 . Each young agent has a nonstorable
endowment y > 0. The government in each economy manages their respective money supplies, MtA and MtB . Assume that M0A = M0B is distributed
evenly to the initial old in each economy. Individuals are free to hold any
currency they please.
1
(a) Assume, for the moment, that each government keeps its money supply constant. Show that the equilibrium nominal exchange rate is
indeterminate. Explain.
(b) Imagine now that both countries agree to fix the nominal exchange
rate at par. At the same time, imagine that the government in economy B begins to expand its money at some constant rate μB > 1,
using the new money to finance government purchases. Derive an
expression for the equilibrium (gross) rate of inflation Πt . Show that
limt→∞ Πt = μB .
(c) Explain why (in the previous question) there is inflation in economy
A even though the domestic money supply M A remains constant.
Explain how the government in economy B is effectively extracting
tax revenue from agents in economy A. (Use a diagram with indifference curves and budget constraints to show where this tax revenue
comes from in each country).
(d) What policy advice might one offer the government in economy A to
improve the welfare of its citizens? Explain.
2