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Homework 4
Answer Key
Problem 1. Bank Runs
Consider an economy in which consumers live for three periods (called period 1, 2, and 3). Each consumer
has an endowment of 20 goods when young, and nothing when middle-aged and when old. Utility from
consumption depends on what ”type” a consumer is, as follows:
u
(
c
1
,c
2
,c
3
) =
log [
c
2
]
for type 1 — impatient
log [
c
2
+
c
3
]
for type 2 — patient
There are 200 consumers, and half will become each type. However, consumers only learn their type
in period 2. Assume no one else can observe the types (an agent knows her type only).
There are two assets available in the economy, money and capital, oﬀering the following rates of
return:
One-Period Return
Two-Periods Return
Fiat Money
1
1
Capital
0
.
9
1
.
5
(a) Suppose a consumer, on her own, saves half her goods in each asset in period 1. After ﬁnding out
her type, she consumes all the savings in the appropriate period. Calculate the consumption the
consumer would receive in either period.
(b) Suppose a competitive bank oﬀers demand deposit contracts to all consumers, specifying interest
rates
r
1
and
r
2
for deposits withdrawn in periods 2 and 3, respectively. Which assets, and in what
amounts, would the bank need to hold in order to oﬀer
r
1
= 1 and
r
2
= 1
.
5?
(c) Calculate the consumption a consumer would receive in each period from depositing with a bank.
(d) Suppose a bank run occurs in this economy (all type 2 people pretend to be type 1 people and
withdraw early). How many people, in total, would the bank be able to pay in period 2 at the
promised rate of return before the bank runs out of assets? How many additional goods would the
bank need in order to pay back all its depositors?
(e) Suppose that in the period after you made your deposit at the bank you turn out to be a type 2
person and you learn that all the other type 2 people are about to pretend to be type 1 people so
that they can withdraw early. Is it in your self-interest to also try to withdraw early?
1

(f) Suppose that there are two cities,
A
and
B
, each with consumers and a bank as described above.
There are no bank runs, but in each city the true distribution of consumer types turns out to be
diﬀerent from what people initially expected. Speciﬁcally, in city
A
only 25% of the people learn
they are type 2 and in city
B
75% of the people learn they are type 2 (instead of 50% in each city).
The banks in each city, however, expected that half of their customers would be of each type, so

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