ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
ABM and banking networks
Lecture 3: Some motivating economics models
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
M. R. Grasselli
The Bernanke,
Gertler and
Gilchrist
Model (1996)
McMaster University
February, 2012
Liquidity preferences
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
An asset is illiquid if its liquidation value at an earlier time
is less than the present value of its future payoff.
For example, an asset can pay 1 ≤ r1 ≤ r2 at dates
t = 0, 1, 2.
The lower the ratio r1 /r2 the less liquid is the asset.
At time t = 0, consumers don’t know in which future date
they will consume.
The consumer’s expected utility is
ωU(r1 ) + (1 − ω)U(r2 ),
where ω is the probability of being an early consumer
(type 1).
Sufficiently risk-averse consumers prefer the liquid asset.
Example: Diamond (2007)
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Let A = (r1 = 1, r2 = 2) represent an illiquid asset and
B = (r1 = 1.28, r2 = 1.813) a liquid one.
Assume investors with power utility u(c) = 1 − c −1 and
ω = 1/4.
The expected utility from holding the illiquid asset is
1
3
E [u(c)] = u(1) + u(2) = 0.375
4
4
By comparison, the expected utility from holding the liquid
asset is
1
3
E [u(c)] = u(1.28) + u(1.813) = 0.391
4
4
Observe, however, that risk-neutral investors would prefer
the illiquid asset, since:
E [A] = 1.75 > 1.68 = E [B]
Liquidity risk sharing
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Consider an economy with dates t = 0, 1, 2, a liquid asset
(numeraire) (1, 1) and an illiquid asset (technology)
(r , R), with r ≤ 1 and R > 1.
Suppose that consumer’s preferences are given by
u(c1 )
with probability ω
U(c1 , c2 ) =
(1)
u(c1 + c2 ) with probability 1 − ω
Denoting by cki the consumption of agents of type i at
time k, the optimal risk sharing for publicly observed
preferences is
c12 = c21 = 0
(2)
u 0 (c11 ) = Ru 0 (c22 )
ωc11 + (1 − ω)
c22
R
(3)
=1
(4)
Private information and incentives
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
However, liquidity preferences are private unverifiable
information !
Fortunately, it follows from R > 1 that the optimal risk
sharing solution satisfy the self-selection condition
1 < c11 < c22 < R.
This implies that there exists a contract that achieves
(2)–(4) as a Nash equilibrium.
The key insight of Diamond and Dybvig (1983) is that
such a contract can take the form of a demand deposit
offered by a bank.
(5)
A model for banks
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
Suppose now that a bank offers a fixed claim r1 per unit
deposited at time 0.
Assume that withdrawers are served sequentially in
random order until bank runs out of assets.
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Denoting by fj the fraction of withdrawers before j and by
f their total fraction, the payoffs per unit deposited are
V1 (fj , r1 ) = r1 1{fj <r −1 }
1
V2 (f , r1 ) = [R(1 − r1 f )/(1 − f )]+
Setting r1 = c11 , a good equilibrium corresponds to f = ω,
since this leads to V2 = c22 > c11 = V1 .
However, it is clear that f = 1 (run) is also an equilibrium
leading to V1 ≤ c11 and V2 = 0 < c22 .
Example revisited: Diamond (2007)
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Let the illiquid asset be A = (1, 2), u(c) = 1 − c −1 and
ω = 1/4
√
Then the marginal utility condition becomes c22 = Rc11 .
Substituting into the budget constraint (4) gives
√
R
1
√ = 1.28,
c1 =
c22 = 1.813.
1−ω+ω R
Suppose the bank offers the liquid asset B = (1.28, 1.813)
to 100 depositors each with $1 at 0 and invests in A.
If f = 1/4, the bank needs to pay 25 × 1.28 = 32 at t = 1.
At t = 2 the remaining depositors receive 68×2
75 = 1.813.
Therefore a forecast fˆ = 1/4 is a Nash equilibrium.
However, the forecast fˆ = 1 is another Nash equilibrium.
A model for interbank loans
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Consider an economy with 4 banks (regions) A, B, C , D.
There is a continuum of agents with unit endowment at
time 0 and liquidity preferences given according to (1).
The probability ω of being an early consumer varies from
one region to another conditional on two states S1 and S2
with equal probabilities:
Table: Regional Liquidity Shocks
S1
S2
A
ωH
ωL
B
ωL
ωH
C
ωH
ωL
D
ωL
ωH
Each bank can invest in a liquid asset (1, 1) and an illiquid
asset (r < 1, R > 1) and promises consumption (c1 , c2 ).
The central planner solution
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
The central planner solution consists of the best
allocation (x, y ) of per capita amounts invested in the
illiquid and liquid assets maximizing the consumer’s
expected utility.
This is easily seen to be given by
γc1 = y , (1 − γ)c2 = Rx,
ωH + ωL
where γ =
is the fraction of early consumers.
2
Once liquidity is revealed, the central planner moves
resources around.
For example, in state S1 , A and C have excess demand
(ωH − γ)c1 at t = 1, which equals the excess supply
(γ − ωL )c1 from B and D.
At t = 2 the flow is reversed, since the excess supply
(ωH − γ)c2 from A and C equals the excess demand
(γ − ωL )c2 from B and D.
Optimal interbank loans
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
In the absence of a central planner, interbank loans can
overcome the maldistribution of liquidity.
Suppose that the network is completely connected (i.e
links between all banks).
To achieve the optimal allocation, it is enough for banks
to exchange deposits zi = (ωH − γ)/2 at time t = 0.
At t = 1, a bank with high liquidity demand satisfies
ωH − γ
3(ωH − γ)c1
ωH +
c1 = y +
,
2
2
which reduces to γc1 = y .
At t = 2, the same bank satisfies
[(1 − ωH ) + (ωH − γ)]c2 = Rx,
which reduces to (1 − γ)c2 = Rx.
Shocks and stability
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
Allen and Gale (2000) then analyze the effects of small
shocks to interbank markets with networks of the form:
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
They show that the complete network absorbs shocks
better than the incomplete one.
Their analytic model is difficult to generalize to arbitrary
(asymmetric) networks.
Financial Accelerator
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Firms need external financing to engage in profitable
investment opportunities.
Their ability to borrow depends on the market value of
their net worth.
If an initial shock induces a fall in asset prices, it
deteriorates the balance sheets of the firms and their
ability to borrow declines.
Tightening financing conditions limit investment, which in
turn reduces output.
Decreased economic activity further cuts the asset prices
down, which leads to deteriorating balance sheets,
tightening financing conditions and declining economic
activity.
A simple theoretical model
ABM and
banking
networks
Lecture 3:
Some
motivating
economics
models
M. R. Grasselli
The Diamond
and Dybvig
Model (1983)
The Allen and
Gale Model
(2000)
The Bernanke,
Gertler and
Gilchrist
Model (1996)
Consider a firm with cash holdings C and illiquid assets A.
To produce output Y the firm uses inputs X financed with
borrowed funds B.
Suppose that the interest rate is zero and that A can be
sold with a price of P per unit after the production, and
the price of X is normalized to 1.
Thus X = C + B.
Suppose now that it is costly for the lender to seize firms
output Y in case of default, but ownership of A can be
transferred.
Then B ≤ PA, which implies X ≤ C + PA.
Thus, an initial decline in the asset prices limits borrowing
and leads to decreased economic activity, which feeds back
to a fall in asset demand and further fall is asset prices
further, causing a vicious cycle.