Econ 208 Marek Kapicka Lecture 15 Financial Intermediation

Econ 208

Marek KapickaLecture 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 ProblemTiming

Time Two assets:

Liquid, short-term (short) asset unit of consumption in period t can be converted

to unit of consumption in period Illiquid, long-term (long) asset

unit of consumption in period can be converted into units of consumption in period

Long asset yields more in the long run, but nothing in the short run!

A Liquidity ProblemPreferences

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

An Example of Early Consumers

A Liquidity ProblemPreferences

Probability of being early: Preferences of a consumer:

expected utility

Trade-off: investing in long asset yield higher return but does not insure against the risk of being an early consumer

𝜃𝑈 (𝐶1)+(1−𝜃 )𝑈 (𝐶1+𝐶2)

A Liquidity Problem

1. Autarkic Solution2. Market Solution3. Efficient Solution4. Banking Solution

1. Autarkic Solution

The consumer has initial wealth Invests fraction in the short

(liquid) asset

Chooses to maximize

𝜃𝑈 (𝜆 )+ (1−𝜃 )𝑈 (𝜆+(1−𝜆 ) 𝐹 )

1. Autarkic SolutionThe Budget Constraint

1

1

𝐶1

𝐶2

𝐹

1. Autarkic Solution

If the utility is logarithmic, the solution is

If increases, increases If increases, decreases

𝜆=min [𝜃

1−1𝐹

¿,1]¿

A Liquidity Problem


2. A Market SolutionMarket 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 SolutionMarket vs. Autarky

1

1

𝐶1

𝐶1+𝐶2

𝐹

Autarkic choices

Market Equilibrium

A Liquidity Problem


3. The Efficient SolutionWhat is efficiency?

Pareto Efficiency: What would a social planner, not bound by markets, do?

Social planner: Choose feasible consumption Choose the amount and the society

invests in illiquid (long) and liquid (short) assets𝑥+𝑦=1

3. The Efficient SolutionSocial planner’s problem

Social planner: Maximize the expected utility

Subject to

WLOG assume that late consumers only consume in period 2

𝜃𝑈 (𝐶1)+(1−𝜃 )𝑈 (𝐶2)

3. The Efficient SolutionSocial Planner’s problem

Social planner: Maximize the expected utility

First order condition

max𝑥𝜃𝑈 ( 1−𝑥𝜃 )+ (1−𝜃 )𝑈 ( 𝐹𝑋

1−𝜃)

𝑈 ′ (𝐶1)=𝐹𝑈 ′(𝐶2)

3. The Efficient SolutionCase 1: Too little liquidity in the market solution

1

1

𝐶1

𝐶2

𝐹

Market Equilibrium

𝐶1∗

𝐶2∗

Efficient Solution

3. The Efficient SolutionCase 2: Too much liquidity in the market solution

1

1

𝐶1

𝐶2

𝐹

Market Equilibrium

𝐶1∗

𝐶2∗

Efficient Solution

3. The Efficient SolutionCase 3: The right amount of liquidity in the market solution

1=𝐶1∗

1

𝐶1

𝐶2

𝐹=𝐶2∗

Market Equilibrium = Efficient solution

3. The Efficient SolutionWhat next?

In general, the market solution is not efficient

How to get efficiency? Can banking improve on the market

solution?

A Liquidity Problem


5. Banking SolutionA 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 . This inequality holds in the efficient

allocation

5. Banking Solution

A bank Collects depositors’ investments at

time 0 Invests in a portfolio Offers to pay consumers (A deposit

contract) Free entry into the banking sector

Banks maximize investors’ expected utility

5. Banking SolutionEquilibrium without runs

Later on, we’ll see that banks are prone to runs, but ignore it for now

The bank maximizes the expected utility

Subject to

𝜃𝑈 (𝐶1)+(1−𝜃 )𝑈 (𝐶2)


Maximize the expected utility

First order condition

Identical to the social planner’s problem

The (good) equilibrium is efficient!

max𝑥𝜃𝑈 ( 1−𝑥𝜃 )+ (1−𝜃 )𝑈 ( 𝐹𝑥

1−𝜃)

𝑈 ′ (𝐶1)=𝐹𝑈 ′(𝐶2)


To make the problem interesting, we assume that

We also assume that the illiquid asset can be liquidated in period 1 to yield

𝑈 (𝐶 )=𝐶1−𝜎

1−𝜎,𝜎>1


1

1

𝐶1

𝐶2

𝐹

𝐶1∗

𝐶2∗

Equilibrium without runs

5. Banking SolutionEquilibrium with runs

Assume that the bank operates under a sequential service constraint: Everyone who comes to the bank in

period 1 is paid , until bank resources are depleted

The liquidated value of all the bank’s assets is𝑆= 𝑓𝑥+𝑦 ≤ 𝑥+𝑦=1


Suppose that everyone decides to withdraw in period 1

Since

1. Not everyone in can be paid in period 1

2. Those who wait until period 2 will get nothing

The bank will become insolvent

𝐶1>1≥𝑆


A payoff matrix: late consumer (rows) vs every other late consumer (columns):

Note: the run/run payoff is the expected payoff

There are two equilibria: No run/No run (good equilibrium) Run/Run (bad equilibrium)

Run No Run

Run

No Run

Documents

Econ 208 Marek Kapicka Lecture 15 Financial Intermediation