Econ 4721: Money and Banking, Fall 2008
Homework 1 Solution
1
Problem 1. OLG
1.1
Problem
Consider an overlapping generations model in which consumers live for two periods.
The number of
people born in each generation is constant,
N
t
=
N
t
+1
=
N
.
In each period, young consumers are
endowed with
y
1
= 30 and old consumers are endowed with
y
2
= 0 units of the single consumption good.
Each member of the generations born in period 1 and later have the following utility function:
u
(
c
1
,t
, c
2
,t
+1
) = ln [
c
1
,t
] +
β
ln [
c
2
,t
+1
]
,
with
β
= 0
.
5.
Members of the initial old generation only live for one period and have utility
u
(
c
0
,
1
) = ln [
c
0
,
1
]. They
have no endowment.
(a) Define a
feasible consumption allocation
for this economy. Illustrate the set of feasible allocations
on a graph.
(b) Define a
Pareto efficient stationary allocation
for this economy.
(c) Solve for the Pareto efficient stationary allocation.
(d) Define a
competitive equilibrium without money
for this economy. Will there be any trades
between individuals in the economy? Explain.
(e) Solve for the consumption allocation in the competitive equilibrium without money. Is the allocation
the same as in (c)?
Which consumption allocation does each generation prefer (i.e.
compare the
competitive equilibrium allocation and the Pareto optimal allocation)?
(f) Now suppose that each member of the initial old generation is endowed with
m
0
units of fiat money.
(Therefore, the stock of fiat money in the economy
M
=
N
·
m
0
.) Define a
competitive equilibrium
with money
for this economy.
(g) Solve for the consumption allocation and the demand for real balances (
v
t
m
t
) in the competitive
equilibrium with money as a function of the
rate of return of fiat money
(
v
t
+1
v
t
).
(h) Find the rate of return of fiat money and the consumption allocation in the stationary competitive
equilibrium with money.
(i) Suppose that
m
0
= 10, that is, each initial old person is endowed with 10 units of money. Find the
demand for fiat money
m
t
and the value of fiat money
v
t
in every period.
1
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1.2
Solution
(a) An allocation is a pair of consumption quantities (
c
1
,t
, c
2
,t
+1
) for every guy born at time
t
≥
1, plus
c
0
,
1
for the initial old. A feasible allocation is an allocation that has the following property: for all
t
≥
1:
N
t
c
1
,t
+
N
t

1
c
2
,t
≤
N
t
y
(1)
Any version of the above equation that exploits the fact that
N
t
=
N
for all
t
or that
y
= 30 would
also be a correct answer. Presumably everyone should be able to draw this on a diagram.
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 Fall '08
 Staff
 Economics, Inflation, Monetary economics, Quantity Theory of Money

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