Econ 4721: Money and Banking, Fall 2008
Homework 2 Answer Key
1
Problem 1. Inflation
Consider an overlapping generations model in which consumers live for two periods.
The number of
people born in each generation grows in each period, according to
N
t
=
nN
t

1
, where
n
= 1
.
2. In each
period, young consumers are endowed with
y
= 60 and old consumers are endowed with 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
) = log
c
1
,t
+
β
log
c
2
,t
+1
(1)
with
β
= 0
.
5.
Members of the initial old generation only live for one period and have utility
u
(
c
0
,
1
) = log
c
0
,
1
.
The government expands the money supply by a factor of
z
each period,
M
t
=
zM
t

1
. Assume that
z
= 1
.
5. The money created each period is used to finance a lumpsum subsidy of
a
t
+1
goods to each old
person.
(a) Solve for the (stationary) Pareto efficient allocation. The answer should be two numbers
(
c
P O
1
, c
P O
2
)
.
(b) Write the government’s budget constraint in period
t
+ 1.
(c) Define a competitive equilibrium with money for this economy.
(d) Solve for the rate of return of money (
v
t
+1
/v
t
) and the growth rate of the price level (
p
t
+1
/p
t
) in a
stationary equilibrium. The answer should be two numbers.
(e) Solve for the consumption allocation (
c
*
1
, c
*
2
) and a lumpsum subsidy
a
*
in a stationary equilibrium.
The answer should be three numbers.
(f) Verify that agents prefer the Pareto Optimal allocation to the competitive equilibrium allocation
with inflation.
(g) Illustrate the Pareto Optimal allocation
(
c
P O
1
, c
P O
2
)
and the competitive equilibrium allocation (
c
*
1
, c
*
2
)
on the (
c
1
, c
2
) plane. Your graph should also include the feasibility line, the lifetime budget con
straint, and their indifference curves.
1.1
Solution
To find the Pareto efficient allocation, we set up the following problem, explicitly incorporating the fact
its solution will be stationary:
max
c
1
,c
2
log
c
1
+
β
log
c
2
(2)
1
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s.t.
N
t
c
1
+
N
t

1
c
2
=
N
t
y
(3)
Using the fact that
N
t

1
=
1
n
N
t
, and getting rid of
N
t
terms, we rewrite constraint as:
c
1
+
1
n
c
2
=
y
(4)
You can solve this problem in multiple ways. I use the quick solution formula I’ve discussed in class
at some point:
c
P O
1
=
1
1 +
β
y
= 40
(5)
c
P O
2
=
β
1 +
β
y
1
n
= 24
(6)
So your answer for part (a) is
(
c
P O
1
, c
P O
2
)
= (40
,
24).
The government budget constraint is very simple:
v
t
+1
(
M
t
+1

M
t
) =
a
t
+1
N
t
(7)
where the left hand side (LHS) is the seignorage revenue, and the RHS is the total value of lumpsum
rebates that are given to the people.
We define the monetary equilibrium for this economy as follows:
Definition 1
A competitive equilibrium with money for this economy would be an allocation
(
c
*
1
,t
, c
*
2
,t
+1
)
for every person born at time
t
≥
1
,
c
*
0
,
1
for the initial old, prices of money
v
*
t
for all
t
≥
1
and government
transfers
a
*
t
for all
t
≥
1
such that:
1. [1.]
2. For every
t
≥
1
, every person born at
t
, taking
v
*
t
,
v
*
t
+1
and
a
*
t
+1
as given, chooses
(
c
*
1
,t
, c
*
2
,t
+1
)
to
solve:
max
c
1
,t
,c
2
,t
+1
,m
t
log
c
1
,t
+
β
log
c
2
,t
+1
(8)
s.t.
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 Fall '08
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