Download Econ 101

Econ 208
Marek Kapicka
Lecture 15
Financial Intermediation
Announcements

PS5 will be posted today, due next
Thursday before the section (3pm)


Give them directly to Xintong, or to her
mailbox
Read “Zero sum debate” – the
Economist article about capital taxation
Why Financial Crises?

Key insight: Banks are here to transform
illiquid assets to liquid liabilities



Depositors prefer to withdraw deposits easily
(preference for liquidity)
Borrowers need time to repay the loans
Tension between both sides of the
balance sheet:

If everyone wants to withdraw deposits,
there is not enough resources
A Liquidity Problem




How to choose between liquid and illiquid
assets?
Liquid assets: can be converted into
immediate consumption without any costs
Illiquid assets: it is costly to convert them
into immediate consumption
People have preference for liquidity: they
are unsure when they need to consume
A Liquidity Problem
Timing


Time 𝑡 = 0,1,2
Two assets:

Liquid, short-term (short) asset


Illiquid, long-term (long) asset


1 unit of consumption in period t can be converted to 1
unit of consumption in period 𝑡 + 1, 𝑡 = 0,1
1 unit of consumption in period 0 can be converted into
𝐹 > 1 units of consumption in period 2
Long asset yields more in the long run, but
nothing in the short run!
A Liquidity Problem
Preferences

Liquidity preference: Two types of
consumers:



Early consumers: only want to consume in
period 1
Late consumers: are indifferent about the
timing of consumption
The consumer learns about his type at
the beginning of period 1
An Example of Early Consumers
A Liquidity Problem
Preferences


Probability of being early: 𝜃
Preferences of a consumer: expected
utility
𝜃𝑈 𝐶1 + 1 − 𝜃 𝑈(𝐶1 + 𝐶2 )

Trade-off: investing in long asset yield
higher return but does not insure
against the risk of being an early
consumer
A Liquidity Problem
1.
2.
3.
4.
Autarkic Solution
Market Solution
Efficient Solution
Banking Solution
1. Autarkic Solution


The consumer has initial wealth 𝑊 = 1
Invests fraction 𝜆 in the short (liquid)
asset
𝐶1 = 𝜆
C2 = 𝜆 + 1 − 𝜆 𝐹

Chooses 𝜆 to maximize
𝜃𝑈 𝜆 + 1 − 𝜃 𝑈(𝜆 + 1 − 𝜆 𝐹)
1. Autarkic Solution
The Budget Constraint
𝐶2
𝐹
1
1
𝐶1
1. Autarkic Solution

If the utility is logarithmic, the solution
is
𝜃
𝜆 = min [
1−1


, 1]
𝐹
If 𝜃 increases, 𝜆 increases
If 𝐹 increases, 𝜆 decreases
A Liquidity Problem
1.
2.
3.
4.
Autarkic Solution
Market Solution
Efficient Solution
Banking Solution
2. A Market Solution
Market vs. Autarky


In a market, early consumer are
allowed to sell long assets and buy
short assets
We don’t have time to go through this,
but one can show:


Market can achieve more risk sharing than
autarky
We will see that with banks we can do
even better than that
2. A Market Solution
Market vs. Autarky
𝐶1 + 𝐶2
Market Equilibrium
𝐹
Autarkic choices
1
1
𝐶1
A Liquidity Problem
1.
2.
3.
4.
Autarkic Solution
Market Solution
Efficient Solution
Banking Solution
3. The Efficient Solution
What is efficiency?


Pareto Efficiency: What would a social
planner, not bound by markets, do?
Social planner:


Choose feasible consumption 𝐶1 , 𝐶2
Choose the amount 𝑥 and 𝑦 the society
invests in illiquid (long) and liquid (short)
assets
𝑥+𝑦 =1
3. The Efficient Solution
Social planner’s problem

Social planner:

Maximize the expected utility
𝜃𝑈 𝐶1 + 1 − 𝜃 𝑈(𝐶2 )
Subject to
𝑦
𝐶1 = 𝜃
𝐹𝑥
𝐶2 =
1−𝜃
𝑥+𝑦 =1

WLOG assume that late consumers only
consume in period 2
3. The Efficient Solution
Social Planner’s problem

Social planner:

Maximize the expected utility
1−𝑥
𝐹𝑋
max 𝜃𝑈
+ 1 − 𝜃 𝑈(
)
𝑥
𝜃
1−𝜃

First order condition
𝑈 ′ (𝐶1 ) = 𝐹𝑈′(𝐶2 )
3. The Efficient Solution
Case 1: Too little liquidity in the market solution
𝐶2
Market Equilibrium
Efficient Solution
𝐹
𝐶2∗
1
1 𝐶1∗
𝐶1
3. The Efficient Solution
Case 2: Too much liquidity in the market solution
Efficient Solution
Market Equilibrium
𝐶2
𝐶2∗
𝐹
1
𝐶1∗
1
𝐶1
3. The Efficient Solution
Case 3: The right amount of liquidity in the
market solution
𝐶2
Market Equilibrium
= Efficient solution
𝐹 = 𝐶2∗
1
1 = 𝐶1∗
𝐶1
3. The Efficient Solution
What next?


In general, the market solution is not
efficient
How to get efficiency?

Can banking improve on the market
solution?
A Liquidity Problem
1.
2.
3.
4.
Autarkic Solution
Market Solution
Efficient Solution
Banking Solution
5. Banking Solution
A note on Information Structure

It is reasonable to assume that agent’s
type is private information



Only the agent knows if he is early or late
No one else cannot observe it
The (late) agents will not want to
misrepresent their type if 𝐶1 ≤ 𝐶2 .

This inequality holds in the efficient
allocation
5. Banking Solution

A bank




Collects depositors’ investments at time 0
Invests in a portfolio (𝑥, 𝑦)
Offers to pay consumers (𝐶1 , 𝐶2 ) (A deposit
contract)
Free entry into the banking sector

Banks maximize investors’ expected utility
5. Banking Solution
Equilibrium without runs


Later on, we’ll see that banks are prone
to runs, but ignore it for now
The bank maximizes the expected utility
𝜃𝑈 𝐶1 + 1 − 𝜃 𝑈(𝐶2 )

Subject to
𝐶1 =
𝑦
𝜃
𝐹𝑥
𝐶2 =
1−𝜃
𝑥+𝑦 =1
5. Banking Solution
Equilibrium without runs

Maximize the expected utility
1−𝑥
𝐹𝑥
max 𝜃𝑈
+ 1 − 𝜃 𝑈(
)
𝑥
𝜃
1−𝜃

First order condition
𝑈 ′ (𝐶1 ) = 𝐹𝑈′(𝐶2 )


Identical to the social planner’s problem
The (good) equilibrium is efficient!
5. Banking Solution
Equilibrium without runs

To make the problem interesting, we
assume that
𝐶 1−𝜎
𝑈 𝐶 =
,𝜎 > 1
1−𝜎

We also assume that the illiquid asset
can be liquidated in period 1 to yield
𝑓≤1
5. Banking Solution
Equilibrium without runs
𝐶2
Equilibrium without
runs
𝐹
𝐶2∗
1
1 𝐶1∗
𝐶1
5. Banking Solution
Equilibrium with runs

Assume that the bank operates under a
sequential service constraint:


Everyone who comes to the bank in period
1 is paid 𝐶1 , until bank resources are
depleted
The liquidated value of all the bank’s
assets is
𝑆 = 𝑓𝑥 + 𝑦 ≤ 𝑥 + 𝑦 = 1
5. Banking Solution
Equilibrium with runs


Suppose that everyone decides to
withdraw in period 1
Since
𝐶1 > 1 ≥ 𝑆
1.
2.

Not everyone in can be paid in period 1
Those who wait until period 2 will get
nothing
The bank will become insolvent
5. Banking Solution
Equilibrium with runs

A payoff matrix: late consumer (rows) vs
every other late consumer (columns):
Run
Run
No Run
No Run
(𝑓𝑥 + 𝑦, 𝑓𝑥 + 𝑦)
(𝐶1 , 𝐶2 )
(0, 𝑓𝑥 + 𝑦)
(𝐶1 , 𝐶2 )
Note: the run/run payoff is the expected payoff

There are two equilibria:


No run/No run (good equilibrium)
Run/Run (bad equilibrium)