A Primer on Banking
From Diamond, D. and P. Dybvig (1983) “Bank Runs, Deposit Insurance,
and Liquidity”,
Journal of Political Economy
, Vol. 91, p. 40119.
1
Environment
•
Three periods
t
= 0
,
1
,
2
.
•
Unit measure of exante (i.e.
t
= 0) identical agents
•
All agents have 1 unit of the good at
t
= 0 and prefer to consume either
at
t
= 1 or
t
= 2
.
•
Storage technologies
1. Productive Storage technology: 1 unit of goods invested at
t
= 0 yields
R >
1 units at
t
= 2
.
If the storage is interrupted at
t
= 1
,
the salvage
value is the initial investment.
t
= 0
t
= 1
t
= 2
−
1
1
R
2. Pillow Storage technology: 1 unit of goods invested at
t
= 1 yields 1 unit
at
t
= 2
.
t
= 1
t
= 2
−
1
1
Storage in this technology is unobservable.
•
Agents face an iid preference shock (
θ
) which is realized at
t
= 1 and
determines their type:
1. Early consumers:
prob
(
θ
= 1) =
π
with preferences
u
(
c
1
)
.
That is,
they only want to consume at
t
= 1
.
2. Late consumers
prob
(
θ
= 2) = (1
−
π
) with preferences
u
(
c
2
)
.
That
is, they only want to consume at
t
= 2
.
•
Given that these shocks are iid and there is a continuum of agents,
π
and
(1
−
π
) also denote the population fractions of early and late consumers
in the economy.
•
Assume
u
0
(
c
)
>
0
, u
0
(0) =
∞
,
u
0
(
∞
) = 0
,
and CRRA
>
1. Consumption is
unobservable.
2
Autarkic Allocation
•
At
t
= 1
,
choose
W
= 1 if
θ
= 1 and
W
= 0 if
θ
= 2
.
This generates
c
A
1
= 1 and
c
A
2
=
R.
1
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3
Planner’s problem when type is observable
(First Best)
max
(
c
1
,c
2
)
∈
R
+
,
(
S,W
)
∈
[0
,
1]
πu
(
c
1
) + (1
−
π
)
u
(
c
2
)
s.t.S
+
πc
1
=
W
(1
−
π
)
c
2
=
R
(1
−
W
) +
S
where the two constraints are resource feasibility at
t
= 1 and
t
= 2 respectively
and the objective function should not be considered expected utility but the
sum of utilities for each agent in the economy.
•
Since the short return between
t
= 1 and
t
= 2 (i.e. 1) is dominated by the
return to the long asset (i.e.
R
), it is strictly better not to liquidate more
than you need to cover expenditure by early consumers. Hence
S
= 0
.
In
this case the problem can be reduced to
max
W
πu
μ
W
π
¶
+ (1
−
π
)
u
μ
R
(1
−
W
)
1
−
π
¶
yields
u
0
(
c
∗
1
)
=
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 Spring '07
 CORBAE
 Game Theory, Bank run, late consumers

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