The Diamond – Dybvig Model
These are the underlying baseline assumptions of the model
I.
II.
III.
there are three time periods 𝑡 = 0, 𝑡 = 1, 𝑡 = 2
there are 𝑁 individuals in the economy
𝜃 represents the probability of consuming in 𝑡 = 1;
(1 − 𝜃) represents the probability of consuming in 𝑡 = 2
(with 0 ≤ 𝜃 ≤ 1 the larger 𝜃 the more impatient the consumer,
if 𝜃 = 1 then the consumer is very impatient and will only consume in 𝑡 = 1 and not consume
in 𝑡 = 2,
if 𝜃 = 0 then the consumer is very patient and will only consume in 𝑡 = 2 and not consume in
𝑡 = 1)
IV.
the bank invests in an asset 𝑥
Utility: here we assume the following functional form for the utility functions
!
𝑈 𝑐! = 1 − !
!
(1)
𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑡 = 1,2,3
(any other following functional form for the utility functions can alternatively be chosen)
The overall expected utility for the whole 𝑡 = 1,2,3 is then
!
𝑈 𝑐! , 𝑐! = 𝜃𝑢 𝑐! + 1 − 𝜃 𝑢 𝑐! = 𝜃 1 − !
!
!
+ 1 − 𝜃 (1 − ! )
!
(2)
The world without banks
In a world without banks the following payoffs would be resulting:
in 𝑡 = 0 :
customers save/invest their money
in 𝑡 = 1:
withdraws would provide the individual with a payoff of 1 (so that 𝑐! =1)
in 𝑡 = 2:
withdraws would provide the individual with a payoff of 1 + 𝑟 where r is the real
interest rate (𝑐! = 1 + 𝑟)
The overall utility for the consumer would be/ (2) becomes:
1
1
1
𝑈 𝑐! , 𝑐! = 𝜃𝑢 𝑐! + 1 − 𝜃 𝑢 𝑐! = 𝜃 1 −
+ 1−𝜃 1−
=𝜃 1− + 1−𝜃
𝑐!
𝑐!
1
= 1−𝜃
!
1 − !!!
1−
(3)
1
1+𝑟
and the ratio of the consumption/income
!!
!!
(4)
=1+𝑟
The world with banks
In a world with banks the following payoffs would be resulting:
in 𝑡 = 0 :
customers deposit their income in the bank
in 𝑡 = 1:
withdraws would provide the individual with a payoff of 𝑦!
in 𝑡 = 2:
withdraws would provide the individual with a payoff of 𝑦!
In 𝑡 = 1, there are 𝑁𝜃 consumers who prefer to withdraw their money from the bank so that the
bank needs to withdraw sufficient funds from the investment to cover the funds 𝑦! demanded by the
consumers
(5)
𝑁𝜃𝑦! = 𝑁𝑥
In 𝑡 = 2, the remaining 1 − 𝜃 𝑁 consumers will paid the income 𝑦! from the remainder of the asset
that has been invested in by the bank (1 − 𝑥)
𝑁 1 − 𝜃 𝑦! = 1 − 𝑥 1 + 𝑟 𝑁
(6)
here 1 + 𝑟 represents the return on the asset 𝑥 for the bank.
(In both (5) and (6) the left hand side represents the total amount the bank has to pay to all the
consumers demanding their funds back in 𝑡 = 1 and 𝑡 = 2 respectively, whereas the right hand side
represents the funds that the bank needs to liquidate from the investment 𝑥 to cover the total
amount demanded by the consumers).
Competition amongst banks drives the profits for the banks to 0, so that the bank identifies the
optimal amounts of 𝑦! and 𝑦! to maximise the expected utility of the consumers.
We can simplify the consumers’ utility functions in one variable by defining 𝑦! as a function of 𝑦!
with (5) and (6):
from (5)
𝑁𝜃𝑦! = 𝑁𝑥
↔
𝜃𝑦! = 𝑥
(7)
from (6)
𝑁 1 − 𝜃 𝑦! = 1 − 𝑥 1 + 𝑟 𝑁
↔
1 − 𝜃 𝑦! = (1 − 𝑥)(1 + 𝑟)
(8)
Substitute (7) for 𝑥 into (8)
(9)
1 − 𝜃 𝑦! = 1 − 𝑥 1 + 𝑟 = (1 − 𝜃𝑦! )(1 + 𝑟)
↔ 𝑦! = (1 − 𝜃𝑦! )
!
↔
!
!!
= (!!!!
(!!!)
(10)
!!!
!!!
(11)
! ) (!!!)
The consumer’s utility function (2) becomes
!
𝑈 𝑦! , 𝑦! = 𝜃𝑢 𝑦! + 1 − 𝜃 𝑢 𝑦! = 𝜃 1 − !
!
!
!
substitute (11) for
!!
(12)
+ 1 − 𝜃 (1 − ! )
!
into (12)
!
𝑈 𝑦! = 𝜃 1 − !
!
!
+ 1 − 𝜃 (1 − (!!!!
!)
!!!
(!!!)
(13)
)
so that the expected utility can be maximised:
!
max!.!.!.!! 𝜃 1 − !
!
+ 1+𝜃
!
1−!
!
!
!
! !!! !
!
!!! !!!!! !
+ (1 − 𝜃)(1 − (!!!)(!!!! ))
!
𝐹. 𝑂. 𝐶. 𝑤. 𝑟. 𝑡. 𝑦! : 1
1
1 + 𝑟 1 − 𝜃𝑦! × 0 − 1 − 𝜃
0=𝜃 !−(
𝑦!
1 + 𝑟 1 − 𝜃𝑦!
= 𝜃 !! −
!!!
=𝜃 1−!
!
!
−𝜃 1 + 𝑟
!
=𝜃
(14)
1
𝜃 1−𝜃 ! 1+𝑟
−
𝑦!!
1 + 𝑟 1 − 𝜃𝑦!
!
(15)
from which follows
!
! !!! !
!
!!! !!!!! !
𝜃 !! =
(16)
divide by 𝜃
!
!!!
expand (17) by
!!!
!!!
=
!!! !
(17)
!!! !!!!! !
(= 1)
!
!!!
=
(!!!) !!! !
!!!
!
!!!!!
!
= (1 + 𝑟)
!!! !
!!! ! !!!!! !
1a detailed description of the differentiation can be found at the end of this document
(18)
where the last fraction of (18) is equivalent to
!
!!!
=
(!!!) !!! !
!!! ! !!!!! !
!
!!!
(see (11)), so that (18) can be expressed as
!
(19)
= (1 + 𝑟) ! !
!
which can be reorganised as
!!!
!!!
= (1 + 𝑟)
(20)
=
(21)
so that the ratio of incomes is
!!
!!
(1 + 𝑟)
and also
𝑦!! = (1 + 𝑟)𝑦!! so that 1 + 𝑟 > 𝑦! > 𝑦! > 1
With
!!
!!
=
(1 + 𝑟) (21) <
!!
!!
(22)
= 1 + 𝑟 (22), the banks allow for a better consumption smoothing.
The differentiation of the expected utility under banks
The above derivative from the second term of the function comes from
𝑓 𝑥
𝑔 𝑥
𝑤𝑖𝑡ℎ 𝑖𝑡𝑠 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒
𝑔 𝑥 × 𝑓 ! 𝑥 − 𝑔! 𝑥 × 𝑓 𝑥
𝑔 𝑥
!
with
𝑓 𝑥 = 1−𝜃
!
𝑎𝑛𝑑 𝑓′(𝑥)!.!.!.!! = 0
and
𝑔 𝑥 = (1 + 𝑟)(1 − 𝜃𝑦! ) 𝑎𝑛𝑑 𝑔′(𝑥)!.!.!.!! = −𝜃 1 + 𝑟
Alternatively, you can also solve this as follows:
𝐴𝑠 𝜃 𝑎𝑛𝑑 𝑟 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠, 1 − 𝜃 𝑎𝑛𝑑 1 + 𝑟 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
𝑇ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑖𝑠 0 (𝑒. 𝑔. 𝑓 5 = 5 𝑡ℎ𝑒𝑛 𝑓 ! 5
= 0, 𝑎𝑛𝑑 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑠 𝑗𝑢𝑠𝑡 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑒. 𝑔. 𝑓 𝑥
= 5𝑥 𝑡ℎ𝑒𝑛 𝑓 ! 𝑥 = 5 .
𝐷𝑒𝑛𝑜𝑡𝑒 1 − 𝜃 ! 𝑎𝑠 𝐴 𝑎𝑛𝑑 1 + 𝑟 𝑎𝑠 𝐵 𝑎𝑛𝑑 𝜃 𝑎𝑠 𝐶 𝑎𝑛𝑑 1 + 𝜃 𝑎𝑠 𝐷,
𝑡ℎ𝑒𝑛 𝑤𝑒 𝑐𝑎𝑛 𝑟𝑒𝑤𝑟𝑖𝑡𝑒
1
1
1
1−𝜃
+ 1−𝜃 1−
=𝜃 1−
+ 1−𝜃 1−
𝑦!
𝑦!
𝑦!
1 + 𝑟 1 − 𝜃𝑦!
1
1−𝜃
=𝜃 1−
+ 1+𝜃 1−
𝑦!
1 + 𝑟 1 − 𝜃𝑦!
1
𝐴
1
𝐴
=𝐶 1−
+𝐷−
=𝐶 1−
+ 𝐷 − 1 − 𝐶𝑦! !!
𝑦!
𝐵 1 − 𝐶𝑦!
𝑦!
𝐵
max!.!.!.!! 𝜃 1 −
The derivative will be
𝐹. 𝑂. 𝐶. 𝑤. 𝑟. 𝑡. 𝑦! :
0=𝐶
1
𝐴
! + 0 − 𝐵 −1 1 − 𝐶𝑦!
𝑦!
!!
(−𝐶)
the derivative of the second term comes from the chain rule where
𝑓 𝑔 𝑥
𝑤𝑖𝑡ℎ 𝑓 𝑔 𝑥
𝑤𝑖𝑡ℎ 𝑖𝑡𝑠 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑓 ! 𝑔 𝑥 𝑔! 𝑥
= 1 1 − 𝐶𝑦!
!!
𝑠𝑜 𝑡ℎ𝑎𝑡 𝑓 ! 𝑔 𝑥
= −1 1 − 𝐶𝑦!
!!
and
𝑔 𝑥 = 1 − 𝐶𝑦! 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑔! 𝑥 = −𝐶
replace back for A, B, C, D
1
𝐴
1
1−𝜃 !
1
!!
0 = 𝐶 ! + 0 − −1 1 − 𝐶𝑦!
−𝐶 = 𝜃 ! +
𝐵
1 + 𝑟 1 − 𝜃𝑦!
𝑦!
𝑦!
!
1
1−𝜃
=𝜃 !−𝜃
1 + 𝑟 1 − 𝜃𝑦! !
𝑦!
!
−𝜃
Both options lead to exactly the same result, you also do not explicitly substitute for the constants
since that is not particularly pretty (I did this here explicitly to highlight the difference between
constants and the variables).