Banking as Insurance
•
Banks as suppliers of liquidity
•
(What is liquidity?) Insurance problem
•
Demand-deposit contracts
•
Bank runs
•
The case for suspension of convertibility and deposit insurance
•
How markets and contracts interact
1
The Diamond and Dybvig (1983) model
1.1
•
The environment
A continuum of consumers with preferences
u (c1 , c2 ) = u (c1 + θc2 )
• θ ∈ {0, 1}
•
If
θ=1
is an idiosyncratic shock, realized at
the consumer is late or patient, otherwise he is early or impatient
• π ≡ Pr (θ = 0).
By LLN,
π
is also the fraction of early consumers in the population
•
Each consumer has an endowment of
•
Two possible technologies:
t = 1.
Storage: transforms
x
goods at
t
1
at
into
1
t=0
x
1 As
goods at
t+1
you know, this requires abusing the law of large numbers. See Judd (1985) on why this is a problem and
Uhlig (1996) on how to x it.
1
Econ 661
•
Fall 2019
t = 0 into Ry > y goods at t = 2. The
investment can be terminated early (liquidated) at t = 1. If z ≤ y units are liquidated,
then the investment yields λz goods at t = 1 and R (y − z) goods at t = 2. Assume
λ ≤ 1.
Long-term investment: transforms
Note: original paper has
of technology is
1.2
•
λ = 1,
y
goods at
so the storage technology is redundant
not the main issue.
⇒ the ex-ante choice
First-best planner problem
Assume (it's easy to prove) that the solution is symmetrical, that it involves early consumers
consuming at
t=1
and late consumers consuming at
t=2
and that no long-term projects
are liquidated
max πu (c1 ) + (1 − π) u (c2 )
c1 ,c2 ,x,y
s.t. πc1 ≤ x
(1 − π) c2 ≤ Ry
x+y =1
or equivalently
max πu (c1 ) + (1 − π) u (c2 )
c1 ,c2 ,x,y
s.t. πc1 + (1 − π)
•
c2
≤1
R
FOC:
πu0 (c1 ) − µπ = 0
1
(1 − π) u0 (c2 ) − µ (1 − π) = 0
R
so
•
MRS = MRT
•
What if types are not observable?
u0 (c1 )
=R
u0 (c2 )
(1)
Planner can set up a mechanism:
(early or late) and you get a consumption bundle in return.
•
The rst best allocation is incentive compatible!
2
R>1
implies
c2 > c1
declare your type
Econ 661
1.3
Fall 2019
Complete markets allocation
•
Markets for consumption at
•
Assume (it's easy to prove) again that in equilibrium the agent only buys positive amounts
t = 1 and t = 2 contingent on realization of θ for each individual.
of the following contingent claims: c1 if impatient (at price
p1 ) and c2
if patient (at price
p2 )
max πu (c1 ) + (1 − π) u (c2 )
c1 ,c2
s.t. p1 c1 + p2 c2 ≤ 1
with FOC:
•
1 − π p1
u0 (c1 )
=
0
u (c2 )
π p2
t = 0 goods into π1 units of the contingent
units of the contingent claim c2 if patient . Competition
By using the LLN, rms can transform one unit of
R
claim c1 if impatient, or into
1−π
implies
p1 = π
1−π
p2 =
R
•
Therefore
u0 (c1 )
=R
u0 (c2 )
and we recover the rst best allocation.
•
(There's nothing special about this, it just illustrates the Second Welfare Theorem)
•
Anything other than this which arises in the model must be a result of some form of market
incompleteness
1.4
Incomplete markets
•
Suppose there are no insurance markets
•
The only market is at
t = 1,
where agents can trade
equivalently, ongoing projects)
•
Let the price of
t=2
goods at
t=1
be
p.
3
t=1
goods against
t=2
goods (or,
Econ 661
•
Fall 2019
The consumer's problem is
max πu (c1 ) + (1 − π) u (c2 )
c1 ,c2 ,x,y
s.t. c1 ≤ x + pRy
x
c2 ≤ Ry +
p
x+y =1
•
In any equilibrium where consumers invest in both technologies, it must be that
p=
•
If
p>
1
R
1
, then it is always preferable to invest in the long-term technology at
R
t=1
all early consumers will be trying to sell at
and nobody would buy
1
, then it is always preferable to invest in storage at t = 0, but then all late consumers
R
1
cannot be the equilibrium price.
would be willing to pay 1 for t = 2 goods, so p <
R
•
If
•
Therefore the consumer's problem reduces to
p<
max πu (c1 ) + (1 − π) u (c2 )
c1 ,c2
s.t. c1 ≤ 1
c2 ≤ R
•
The allocation
c1 = 1 c2 = R
does not in general coincide with the complete markets allocation
•
t = 0, but then
Allocations coincide for the special case of log preferences:
General optimality condition:
Special case of log
so the allocation
c1 = 1, c2 = R
u0 (c1 )
=R
u0 (c2 )
c2
=R
c1
satises optimality
4
Econ 661
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Fall 2019
Assume that the consumer is more risk averse than log, i.e.
u00 (c) c
− 0
u (c)
| {z }
> 1 ∀c
coecient of relative risk aversion
•
Then
cF1 B > 1
and
cF2 B < R.
0
Proof:
0
Z
R
Ru (R) = u (1) +
1
= u0 (1) +
Z
∂cu0 (c)
dc
∂c
R
[cu00 (c) + u0 (c)] dc
1
00
Z R
cu (c)
0
0
u (c)
= u (1) +
+ 1 dc
u0 (c)
1
< u0 (1)
1.5
A bank
•
Consumers get together and create a bank
•
They each deposit their endowment with the bank, which invests the rst best amount in
each technology
•
The bank sets up the following contract: each consumer can ask for
wait until
t=2
cF1 B
at
t = 1,
or can
and get a pro-rata share of whatever is left.
•
Demand deposit
•
The bank commits to satisfying sequential service: satisfy consumer's withdrawals in the
order in which they arrive (which, assume, is random)
•
If the stored goods are not enough to satisfy a consumer who demands payment, then the
bank must liquidate the long-term investment until that, too, runs out.
•
Consumers then play a game, where the actions are withdraw or wait (or you could allow
for partial withdrawal, it doesn't matter)
•
Let
fj
be the number of depositors who arrived in line before consumer
j
withdraw and
f
be the total number of consumers that will eventually ask to withdraw
5
and asked to
Econ 661
•
Fall 2019
The payos for an impatient consumer are:
Withdraw

c F B
if
0
otherwise
1
•
Wait
cF1 B fj < xF B + λy F B
The payos for a patient consumer are:
2
Withdraw

c F B
if
0
otherwise
1
•
0
Wait
cF1 B fj < xF B + λy F B
n
o
1 FB
B
1−πcF
1 −(f −π) λ c1
max R
,0
1−f
Derivation:
If
The rst best allocation has
invested in the
long-term technology.
consumers who
f
consumers withdraw, this requires
f cF1 B
goods at
t=1
πcF1 B invested in storage and 1 − πcF1 B
FB
Therefore (f − π) c1v of the t = 1 goods from
withdraw have to come from liquidating long-term projects
•
This requires liquidating
Each produces
These will be split among the
R
1
λ
goods at
(f − π) cF1 B
t = 2,
for a total of
1−f
1−πcF1 B −(f − π) λ1 cF1 B
R 1 − πcF1 B − (f − π) λ1 cF1 B
projects, so only
remain
depositors who did not withdraw
The symmetric straegy prole withdraw i impatient is a Nash equilibrium, with payos
equal to the rst best allocation.
Impatient consumers don't want to deviate because they don't care about future consumption
•
Patient consumers don't want to deviate because
cF2 B > cF1 B
The symmetric strategy prole withdraw no matter what is also a Nash equilibrium
Given that
cF1 B > 1 ≥ xF B + λy F B ,
if everyone tries to withdraw the money will run
out
Therefore anyone who waits will obtain zero
This justies withdrawing
This equilibrium produces a very bad allocation!
2 These
payos are accurate for the case where f ≥ π, which will be true in equilibrium because all impatient
consumers withdraw. To be fully correct you have to take into account that if f < π you need to use storage
between t = 1 and t = 2
6
Econ 661
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Fall 2019
Note on strategic complementarity:
Best response is a step function of
UW ithdraw − UW ait
f
is not globally increasing in
f
Goldstein and Pauzner (2005) show how to solve for a unique equilibrium if one assumes
incomplete information as in Morris and Shin (1998)
1.6
Suspension of convertibility
•
One variant of the contract can rule out the bad equilibrium
•
The contract states that you can withdraw
cF1 B
at
t = 1 as long as less than π other consumers
have withdrawn before you. After that, you are forced to wait
•
Now the payos for an impatient consumer are:
Withdraw

c F B
if
0
otherwise
1
•
fj < π
Wait
0
3
and the payos for a patient consumer are:
Withdraw

c F B
if
0
otherwise
1
fj < π
Wait
cF2 B
•
So now waiting is dominant for patient consumers, and runs will not take place.
•
There won't be suspension of convertibility in equilibrium
•
Problem: you need to know
1.7
•
π
exactly in order to use these contracts!
Remarks
Liquidity transformation as a form of insurance, not provided by narrow banking (Wallace
1996)
•
We'll come back to the idea of narrow banking later on.
What is the reason for the sequential-service constraint? What does it mean?
3 Again,
you would need to take into account what happens if f < π
7
Econ 661
Fall 2019
Wallace (1988) camping trip economy. Minnesota approach to microfoundations
Also Green and Lin (2003), Ennis and Keister (2009)
•
Why have a bank rather than a direct mechanism?
•
Deposit insurance as a way to undo the sequential service constraint
•
2
Can work even with stochastic
Taxation can be conditioned on aggregate withdrawals
π
Chicken models and the why doesn't the government just do everything argument.
The Jacklin (1987) critique
•
Jacklin (1987) makes three related points:
1. Under the Diamond and Dybvig (1983) assumptions, you don't really need a bank,
equity is good enough
2. The Diamond and Dybvig (1983) assumptions are very special.
Under more general
preferences:
(a) You can do more with an bank than with equity
(b) A bank may or may not be able to achieve the rst best
3. If there is a market, a bank that is useful cannot survive
2.1
Who needs a bank?
•
Suppose instead of a bank consumers set up a rm
•
The rm invests the rst-best amount and issues shares. Each consumer gets one share
•
It declares (and commits to) the following dividend policy: a dividend of
paid at
•
t=1
and a dividend of
Consumers can trade shares in the rm at
dividends) at a price of
•
d2 = (1 −
π) cF2 B will be paid at
p
t = 1
goods per share.
Supply of shares (impatient consumers sell):
S=π
8
d1 = πcF1 B
will be
t=2
(they trade ex-dividend - after paying
Econ 661
•
Fall 2019
Demand for shares (patient consumers perhaps buy):
D=
•

 (1−π)d1
if
0
otherwise
p
p < d2
Market clearing:
(1 − π) d1
p
(1 − π) d1
= (1 − π) cF1 B
p=
π
π=
•
This is sometimes known as cash-in-the-market pricing
Buyers are at a corner solution. They would want to buy more but have no more money.
Consumption attained by early consumers
c1 = d1 + p = cF1 B
•
Consumption attained by late consumers
d1
c2 = d 2 1 +
p
FB
= (1 − π) c2
1+
πcF1 B
(1 − π) cF1 B
= cF2 B
•
2.2
•
No need for demand deposits, no risk of bank runs!
General preferences
Diamond and Dybvig (1983) assume a very special form of preferences. Consider the more
general formulation:
(
u (c1 , c2 , θ) =
where
uE
and
uL
uE (c1 , c2 )
uL (c1 , c2 )
with prob
are smooth functions
Original version is
uE = u (c1 ), uL = u (c1 + c2 )
9
θ
1−θ
with prob
Econ 661
Fall 2019
To keep the spirit of the original model, you may assume that
uL1 (c1 , c2 )
uE
1 (c1 , c2 )
>
uE
uL2 (c1 , c2 )
2 (c1 , c2 )
so that early and late are still meaningful terms, but the argument doesn't depend
on this.
•
Planner problem:
max
E L L
cE
1 ,c2 ,c1 ,c2
E
L
cL1 , cL2
θuE cE
1 , c2 + (1 − θ) u
s.t.
E
c2
cL2
E
L
θ c1 +
+ (1 − θ) c1 +
R
R
•
Before we assumed (it was obvious) that the planner went to a corner, so he was only choosing
two numbers
•
First order conditions:
L
uE
1 (·) = u1 (·)
L
uE
2 (·) = u2 (·)
uE
1
=R
E
u2
•
May or may not be incentive compatible
•
Edgeworth box illustration:
10
Econ 661
Fall 2019
early
consumer
c2
late
consumer
c1
11
Diamond & Dybvig
Allocation
Econ 661
Fall 2019
early
consumer
mirror
image
c2
late
consumer
interior
allocation
c1
12
Econ 661
Fall 2019
early
consumer
c2
late
consumer
mirror
image
c1
•
General deposit contracts: give the depositor the following options at
Bundle
E
E : cE
1 , c2
Bundle
L: cL1 , cL2
cE1 , cE2 , cL1 , cL2
t=1
or
are any four numbers.
Original version, assuming corner solution, constrains this to
Contracts that make sense are those that respect resouce constraints and incentive
cE
2 = 0
and
compatibility
(They might be vulnerable to runs, but we don't worry about this here)
13
cL1 = 0
Econ 661
•
Fall 2019
If social optimum is incentive compatible, implementable via deposits
Proof: trivial
Otherwise, social optimum not implementable and need to think about incentive-constrained
or second-best optimum
•
What can you do with equity contracts? Notice the following about the FB allocation in the
original model
The price of shares is
p = (1 − π) cF1 B
Shares pay a dividend of
d2 = (1 − π) cF2 B
Eectively, the price of
t=2
godds at
t=1
q=
is:
cF B
p
= 1F B
d2
c2
Patient households have MRS of
Both types go to a corner; neither equates price to MRS. The FOCs for the planner in
1;
impatient household have and MRS of
∞.
the smooth case DO NOT hold
•
The consumption bundles of each type cost the same at the equilibrium prices
This NOT a general property
In general, equity contracts can implement the rst best if and only if two allocations cost
the same at the equilibrium price
•
2.3
More restrictive conditions that for implementation via deposits
In general case, deposits are useful relative to equity
Free riding and trading restrictions
•
Go back to original model
•
Suppose that, after a bank/rm is set up, at
trade
t=2
goods against
t=1
t = 1
a market opens where consumers can
goods
•
What is the price?
•
The bank/rm oers consumers either
cF1 B
at
14
t=1
or
cF2 B
at
t=2
Econ 661
•
Therefore the price of
•
•
Fall 2019
t=2
goods in terms of
t=1
goods must be
B
cF
1
.
F
B
c2
q=
Otherwise, either withdrawing or not-withdrawing is dominant
Consider a free-riding strategy:
Don't join the bank/rm
Invest in the long term technology
If you are impatient, sell your long term project at
If you are patient, hold on to your project and consume at
t=1
t=2
This delivers
c1 = Rq = R
cF1 B
> cF1 B
cF2 B
c2 = R > cF2 B
so the bank\rm would unravel!
•
The same issue would arise in the equity implementation of the planner's allocation
•
In general, opening a market of any kind at
t = 1 means that you cannot attain with deposits
any more than you can attain with equity.
•
Pecuniary externality:
Investing in the bank rather than in long-term projects increases
rate between periods 1 and 2) by making
t=1
q
(lowers the interest
goods more abundant
This is good for insurance purposes (it makes up for the missing insurance market)
Without restrictions, individuals do not internalize the externality so they invest in
long-term projects rather than in the bank
•
Restrictions on trading are needed to sustain these insurance-like arrangements. Are they
reasonable?
•
This relates to the microfoundations of the sequential service constraint
In the camping economy of Wallace (1988) a
15
t=1
market is impossible
Econ 661
3
Fall 2019
The Holmström and Tirole (1998) Model
3.1
Environment
•
Three periods:
•
Consumers
0, 1, 2
Utility
u = c0 + c1 + c2
•
Large endowments at each of the three periods
Entrepreneurs
Utility
u = c0 + c1 + c2
Endowment
Can carry out a project of scale
Output is realized at
Pr (y) = pH
A
or
at
pL
t=0
t=2
I
and can be
yI
or
0
depending on entrepreneur's eort
Lack of eort gives utility
At
B
t = 1 the project needs an extra amount of money ρI
in order to continue (e.g. some
machines broke down). This is the assumption then adopted by Lorenzoni (2008).
ρ ∼ F (ρ)
You can call
If the entrepreneur does not invest
ρ
a liquidity shock
ρ
then the project terminates and pays
private benet of shirking also goes away if the project is terminated
•
Assumption:
Z
Z
max {pH y − ρ, 0} dF (ρ) − 1 > 0 >
•
max {pL y + B − ρ, 0} dF (ρ) − 1
Timing:
1. Investment takes place
2.
ρ
is realized
3. Either
ρ
is reinvested or the project is terminated
16
0.
The
Econ 661
Fall 2019
4. Entrepreneur chooses eort
5. Output is realized
•
The fact the eort is chosen after the reinvestment decision is really important. It means that
even at that stage there is a limit to how much the entrepreneur can pledge to consumers
3.2
•
Second-best allocation
Contracts between entrepreneur and consumers specify:
The size of the investment
When the project will be terminated
I
(equivalently the loan amount
λ (ρ) ∈ {0, 1}
I − A)
(later we allow for fractional liqui-
dation)
The dividends paid to each party
de , dc
if the project succeeds, possibly depending on
the size of the liquidity shock
•
Solve
Z
max
I pH de (ρ) λ (ρ) dF (ρ)
Z
s.t. I [pH dc (ρ) − ρ] λ (ρ) dF (ρ) ≥ I − A
I,de (ρ),dc (ρ),λ(ρ)
de (ρ) ∆p ≥ B
dc (ρ) + de (ρ) = y
•
Note that the implicit assumption is that the consumer agrees to pay for both the gap in the
entrepreneur's initial funds and the liquidity shock (in those cases where they agree that the
shock will be met)
•
Since entrepreneurs gets surplus, objective is equivalent to just maximizing surplus
Z
I
•
It's immediate (from linearity) that the optimum will be given by a cuto
shock is met i
•
[pH y − ρ] λ (ρ) dF (ρ) − 1
ρ < ρ̂
and the project is terminated otherwise
Dene
ρ 1 = pH y
• ρ1
is the expected value of the project (per unit of investment)
17
ρ̂
such that the
Econ 661
Fall 2019
•
In the rst-best allocation, the cuto would be
•
Dene
ρ0 = pH
• ρ0
ρ1
B
y−
∆p
is the maximum divident the entrepreneur can promise the consumers while still satisfying
the IR constraint
•
Why is this a useful thing to compute?
At
If they don't meet the liquidity shock, both parties lose everything
The entrepreneur can credibly promise
t = 1,
than ρ0
the parties will always nd some way to meet the liquidity shock if it is less
suade them to put up
ρ0
ρ0
to the consumers, so he will be able to per-
in order not to lose everything
•
Moral hazard model is just one way to justiy nonpledgeability
•
What matters for the rest of the analysis is just the values of
•
(At MIT people knew this model in its various incarnations as the ρ0 -ρ1 model ')
•
Holmström and Tirole (2011) is a book-length version of this paper, where
ρ0
and
ρ1
ρ0
and
ρ1
are
taken as primitives of the model
•
Write the program as
ρ̂
Z
max I
I,ρ̂
Z
s.t. I
[ρ1 − ρ] dF (ρ) − 1
0
ρ̂
[ρ0 − ρ] dF (ρ) ≥ I − A
0
•
Assume that
Z
ρ0
[ρ0 − ρ] dF (ρ) < 1
0
Economically, this means that maximum pledgeable income after meeting liquidity
shocks is not enough to pay for invesment
Otherwise investment would go to
The entrepreneur will need to put in his own money to be able to invest at all
∞
18
Econ 661
•
Fall 2019
Let
Z
m (ρ̂) ≡
ρ̂
[ρ1 − ρ] dF (ρ) − 1
0
k (ρ̂) ≡
• m (ρ̂)
• k (ρ̂)
1
1−
R ρ̂
0
1
=
[ρ0 − ρ] dF (ρ)
1+
0
ρdF (ρ) − ρ0 F (ρ̂)
is the marginal net social return on investment. It is maximized at
ρ̂ = ρ1
is the equity multiplier. It measures how much investment can take place for each
unit of the entrepreneur's wealth. It is maximized at
•
R ρ̂
ρ̂ = ρ0
The problem reduces to
max U (ρ̂) = m (ρ̂) k (ρ̂) A
ρ̂
R ρ̂
[ρ1 − ρ] dF (ρ) − 1
0
=
A
R ρ̂
1 + 0 ρdF (ρ) − ρ0 F (ρ̂)
=
ρ1 −
R ρ̂
0
•
R ρ̂
0
ρdF (ρ)+1
F (ρ̂)
ρdF (ρ)+1
F (ρ̂)
ρ∗ = arg min
ρ̂
•
(2)
− ρ0
Therefore the optimal cutof solves
R ρ̂
•
A
0
ρdF (ρ) + 1
F (ρ̂)
Interpretation: minimize the cost per unit of investment that reaches period
Numerator: Total cost
Denominator: Number of projects that reach
What is the tradeo ?
19
t=2
(3)
2
Econ 661
•
Fall 2019
FOC:
∗
∗
∗
∗
ρ f (ρ ) F (ρ ) − f (ρ )
R
ρ∗
0
ρdF (ρ) + 1
=0
[F (ρ∗ )]2
R ρ∗
∗
ρf (ρ) dρ + 1
F (ρ∗ )
R ρ∗
ρ∗ F (ρ∗ ) − 0 F (ρ) dρ + 1
=
F (ρ∗ )
0
ρ =
Z
ρ∗
F (ρ) dρ = 1
(4)
ρ1 − ρ∗
A
ρ∗ − ρ0
(5)
0
•
Therefore:
U=
(This follows from integration by parts in equation (4):
∗
Z
∗
ρ F (ρ ) −
ρ∗
ρf (ρ) dρ = 1
0
ρ∗ =
1+
R ρ∗
ρf (ρ) dρ
F (ρ∗ )
0
and replacing in (2)
•
3.3
•
What happens if there is a mean-preserving spread in
Lower
Higher utility
Why?
F (ρ)?
ρ∗
Implementation
Pure loan doesn't work:
Lend
At
Lenders are willing to be diluted by the new lenders who provide
I −A
t = 1,
at
t=0
in exchange for some promise at
entrepreneur needs
t=2
ρ
ρ
because otherwise
they will lose everything
(They will of course require compensation ex-ante for this possibility)
New lenders could be the same people as old lenders or dierent ones
20
Econ 661
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At
So whenever
t = 1,
the most that the entrepreneur can pledge to new lenders is
ρ ∈ (ρ0 , ρ∗ ),
ρ0
the second-best-optimal thing to do would be to continue the
project, but we will need to terminate it!
•
Initial loan plus line of credit
Lend
(For simplicity, suppose that
at
I −A
t = 1.
at
t=0
and grant an irrevocable line of credit for
ρ
ρ∗ I
is observable, or that the entrepreneur cannot consume
Otherwise you have to worry about entrepreneurs claiming they've had a bad
liquidity shock and running away with they money)
In exchange for this, the entrepreneur promises
Notice that the consumers need to be able to commit to the line of credit
For
at
•
ρ ∈ (ρ0 , ρ∗ ), the extra money they put up at t = 1 is less than what they will recover
t = 2, so they are making an expected loss!
That's the point of a line of credit!
Material adverse change clauses
Would the parties want to renegotiate? To lower credit line? To higher credit line?
Large initial loan plus liquid assets
Lend
Contractually limit investment to exactly
ρI
I − A + ρ∗ I
I
and agree that the entrepreneur will keep
in liquid assets
(Requires that such liquid assets exist!)
When the liquidity shock comes, the entrepreneur meets it with the liquid assets
The entrepreneur promises
Notice that the entrepreneur would in general not want to choose the right combination
ρ0 I + (ρ∗ − ρ) I
of investment/liquid assets at
•
to the lender.
∗
•
ρ0 I
t = 0.
to the consumers
The investment level has to be enforceable
Contrast this to the Diamond and Dybvig (1983) version of liquidity shocks
Why not rely on the market at
D&D: there is no market
H&T: the market will say no
t = 1?
What if consumers cannot commit (and there is no storage technology)?
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No commitment
∗
3.4
•
⇒
no credit line
Similar issue in Lorenzoni (2008)
No storage
Liquid assets cannot be claims on consumers, because consumers cannot commit
They cannot be stored goods because goods are not storable
Firms can acquire claims on other rms
⇒
what are the liquid assets? Do we have enough of them?
Idiosyncratic ρ shocks
Paper argues that intermediary is necessary: direct stakes in rms are not enough. This is
actually wrong (problem set)
•
Similar issue in D&D: actually, equity can be enough!
The total amount of resources needed at
t=1
to implement the second best allocation is
ρ∗
Z
D=I
ρdF (ρ)
0
•
An intermediary is set up at
•
The intermediary obtains
•
(There are many rms and the law of large numbers applies: there is no aggregate risk)
•
The intermediary also oers a credit line to rms for an amount of
•
In exchange, the intermediary gets a claim to
t=0
I −A
from investors in exchange for shares and lends it to rms
ρ0 I
at
t=2
ρ∗ I
from each rm (but the rm of
course will not pay if the project is terminated)
•
The intermediary can commit to the credit line, but this doesn't solve the problem, because
the intermediary does not have an endowment at
t = 1:
it must persuade consumers to give
up some endowment.
•
At
•
It can sell its claims on
•
Is this enough?
t=0
the intermediary has to nd
t=2
D
to honour the credit lines.
ouptut to consumers, obtaining up to
∗
F (ρ∗ ) ρ0 I .
ρ∗
Z
F (ρ ) ρ0 I − I
ρdF (ρ) = I − A > 0
0
so yes, by selling its claims on rms to consumers the intermediary can satisfy the credit line
and have some left over, which it will then use to repay the original investment
22
I −A
Econ 661
Fall 2019
•
There is enough inside liquidity to implement the second-best allocation
•
Liquidity here is used to mean something like credible promises of goods at a specic date
3.5
Aggregate ρ shocks
•
Now suppose that all rms have the same
•
Now whenever
ρ
shock
ρ ∈ (ρ0 , ρ∗ ) the value of the intermediary's portfolio (assuming that the second
best allocation is implemented) will be
ρ0 I
•
But in order to face the liquidity shock it needs to raise
ρI > ρ0 I
which is not possible even by selling all its holdings to consumers
•
Therefore this doesn't work
•
The economy does not generate sucient inside liquidity
3.6
Government bonds
•
Suppose that, although consumers can't commit, the government can commit for them!
•
i.e. the government can issue debt and credibly promise to tax consumers in order to pay
back its debt
•
Now government debt plays the role of a storage technology!
•
Outside liquidity
•
We can go back to implementing the second-best allocation
•
Is this a chicken model?
3.7
Liquidity premium on bonds
•
Suppose that shocks are aggregate / there is just one rm
•
There is a limited supply of government bonds, so that in equilibrium they trade at a price
q>1
•
(i.e. you pay
q
at
t=0
to get
1
at
t = 1)
How does that change the optimal decision of rms?
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Econ 661
•
Fall 2019
Invest
Meet liquidity shock until some cuto
In order to do so, hold
(because you can always raise
I
(ρ̂ − ρ0 ) I
ρ0
ρ̂
government bonds
at
t=1
by diluting the original investors)
Program is:
Z
ρ̂
[ρ1 − ρ] dF (ρ) − 1 − (q − 1) (ρ̂ − ρ0 )
Z ρ̂
ρdF (ρ) + (q − 1) (ρ̂ − ρ0 ) I
s.t. F (ρ̂) ρ0 I ≥ I − A + I
max
I
I,de ,dc ,λ(ρ)
0
0
•
Now
1
k (ρ̂, q) =
R ρ̂
1 + 0 ρdF (ρ) − ρ0 F (ρ̂) + (q − 1) (ρ̂ − ρ0 )
Z ρ̂
m (ρ̂, q) =
[ρ1 − ρ] dF (ρ) − 1 − (q − 1) (ρ̂ − ρ0 )
0
•
And, again, the problem can be rewritten as
max k (ρ̂, q) m (ρ̂, q) A
ρ̂
•
As before, the optimal cuto will satisfy
∗
ρ = arg min
3.8
1+
R ρ̂
0
ρdF (ρ) + (q − 1) (ρ̂ − ρ0 )
F (ρ̂)
The Jacklin (1987) critique rears its ugly head
•
Suppose all rms are doing the optimal policy of holding bonds
•
A single rms can deviate and choose to buy shares in other rms instead of bonds
•
(The expected return on shares is
•
What is the value of shares at
1>
t=1
1
, so they are better than bonds in that sense)
q
when
ρ ∈ (ρ0 , ρ∗ )?
(ρ∗ − ρ) I
(i.e. the value of excess bonds that are not needed because the shock is only
24
ρ)
Econ 661
•
Fall 2019
By buying enough shares in other rms the rm can implement the
q = 1-second-best
allocation
•
And investors will be willing to lend it enough money to buy these shares at
they are pledgeable and have a return of
3.9
•
t=0
because
1
Partial liquidation
Is a simple cuto still the correct solution of the problem?
termination. We buy
z
Suppose we could do partial
bonds and use these to either partially or fully meet the liquidity
shock
Z
[ρ1 − ρ0 ] λ (ρ, z) dF (ρ)
max I
Z
s.t. I
0
•
I,,λ(ρ,z),z
∞
∞
0
[ρ0 − ρ] λ (ρ, z) dF (ρ) ≥ I − A + I (q − 1) z
z
λ (ρ, z) ≤ min 1,
ρ − ρ0
Notice here we express the objective as the entrepreneur's surplus rather than total surplus.
Because IR binds, this is equivalent. The paper switches back and forth in order to be harder
to follow.
•
Let
•
The entrepreneur will choose the maximum possible continuation (i.e.
δ
be the multiplier on the investor's IR constraint.
n
o
z
λ (ρ, z) ≤ min 1, ρ−ρ
)
0
i
ρ1 − ρ0 + δ [ρ0 − ρ] > 0
⇔ρ<
•
The choice of
z
is governed by
Z
ρ̄
ρ0+z
•
ρ1 − ρ0 (1 − δ)
≡ ρ̄
δ
∂λ (ρ, z)
· [ρ1 − ρ0 + δ (ρ0 − ρ)] dF (ρ) − δ (q − 1) = 0
∂z
q > 1 implies that partial liquidation will indeed be used!
∂λ(ρ,z)
use partial liquidation, we would have
= 0 a.s., which
∂z
(6)
Note that
(if we had enough bonds
to not
is not optimal according
to (6)
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Econ 661
3.10
•
Fall 2019
The demand for liquidity
Go back to the case without partial liquidation (the same holds in the case with partial
liquidation)
•
We had that
∗
ρ = arg min
1+
R ρ̂
0
ρ̂
ρdF (ρ) + (q − 1) (ρ̂ − ρ0 )
F (ρ̂)
•
Demand for bonds will be decreasing in
•
At some
•
At
•
Shape of demand curve for bonds
•
Evidence from Krishnamurthy and Vissing-Jorgensen (2012)
q=1
q max
q
liquidity will be so expensive that projects will just be infeasible
consumers become willing to buy government bonds.
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Econ 661
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References
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Journal of
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Goldstein, I. and Pauzner, A.: 2005, Demand-deposit contracts and the probability of bank runs,
The Journal of Finance 60(3), 12931327.
Green, E. J. and Lin, P.: 2003, Implementing ecient allocations in a model of nancial intermediation,
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Holmström, B. and Tirole, J.: 1998, Private and public supply of liquidity,
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Holmström, B. and Tirole, J.: 2011,
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Journal
Krishnamurthy, A. and Vissing-Jorgensen, A.: 2012, The aggregate demand for treasury debt,
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Lorenzoni, G.: 2008, Inecient credit booms,
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Uhlig, H.: 1996, A law of large numbers for large economies,
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Federal Reserve Bank of