A Model with Costly Enforcement, Slides of Algebra

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A Model with Costly Enforcement

Jes˙s Fern·ndez-Villaverde

University of Pennsylvania

December 25, 2012

A Model with Costly Enforcement

We keep the basic structure as before, except that now the Önancial market friction is the cost of enforcing contracts.

Structure:

(^1) Borrower may decide to renege on debt.

(^2) If that is the case, the lender can only recover the fraction ( 1 θ ) of the gross return Rtk + 1 pt qt kt where:

(^1 ^ θ ) Rtk + 1 <^ Rt

and the borrower keeps the rest, θ Rtk + 1 pt qt kt.

Costly Enforcement Model III

Advantage: much easier to handle than costly state veriÖcation model.

Disadvantage: no default in equilibrium, no spreads.

When to use each of them?

Can we move from Örms to banks?

An Application I: Bank Runs

2007-2010: run on investment funds instead of classical run on banks.

Suggests we may want to think again about runs on Önancial institutions.

Calling them or not a bank is somewhat irrelevant: any institution that engages in maturity transformation.

Gorton (2010)ís emphasis on runs on funds during the crisis.

An Application II: Bank Runs

Kiyotaki and Gertler (2012) incorporate Diamond and Dybvig (1983) into the dynamic macro model we saw in last lecture.

Main idea: maturity mismatch.

To keep the presentation simple, we will get rid of nominal rigidities.

Also, this will facilitate comparison with a neoclassical framework.

But, Örst, let us review Diamond and Dybvigís model.

A Review of Diamond and Dybvig: Agents

Continuum of agents.

Three-dates economy:

(^1) t = 0: each agents endowed with 1 unit of good. (^2) t = 1: early consumption, with probability π 1 and utility u (c 1 ). (^3) t = 2: late consumption, with probability π 1 = 1 π 2 and utility u (c 2 ).

Think about the need of consumption as a liquidity shock i.i.d. for each agent. Law of large numbers.

Consumption in other date does not yield utility.

Expected utility of agents:

π 1 u (^) (c 1 ) + π 2 u (^) (c 2 )

A Review of Diamond and Dybvig: E¢ cient Allocation

Social planner: perfect risk-pooling among agents. Invest I and store 1 I in such a way that no long-term project is liquidated too early. We solve

max π 1 u (c 1 ) + π 2 u (c 2 ) s.t. π 1 c 1 = 1 I π 2 c 2 = RI

Then: max π 1 u

1 I

π 1

• π 2 u

RI

π 2

Optimality condition:

u^0

1 I

π 1

= Ru^0

RI

π 2

A Review of Diamond and Dybvig: Autarky

Each agent invests I in the long-term project at t = 0 and stores 1 I. If liquidity shock at t = 1,

c 1 = 1 I + lI = 1 ( 1 l ) I  1

Otherwise c 2 = RI + 1 I = 1 + (R 1 ) I  R

(at least one of the two inequalities is strict). Expected utility:

π 1 u ( 1 ( 1 l ) I ) + π 2 u ( 1 + (R 1 ) I )

I is always ex post ine¢ cient: either too low or too high. Inferior to e¢ cient allocation.

A Review of Diamond and Dybvig: Financial Markets II

Expected utility: π 1 u ( 1 ) + π 2 u (R)

dominates autarky, but it is still not e¢ cient because liquidity is not properly allocated.

To see this, note that, in general

u^0 ( 1 ) = Ru^0 (R)

For instance, if u^0 ( 1 ) > Ru^0 (R), impatient consumers get more in the optimal allocation than in the equilibrium with Önancial markets (she needs to be insured against the liquidity risk better than what she can get on her own by storing all her endowment).

A Review of Diamond and Dybvig: Fractional Reserve

Banking I

A bank can o§er a contract to depositors: (c 1  , c 2  ).

It must be the case that c 2  > c 1  (otherwise, depositors will always cash-in at t = 1 regardless of the liquidity shock).

Let us suppose that agents withdraw funds when they want to consume.

Then, bank keeps reserves π 1 c 1  and invests in the long-term project 1 π 1 c 1 .

A Review of Diamond and Dybvig: Fractional Reserve

Banking II

Problem: what if the depositors show up at t = 1?

Bank run (self-fulling prophecy).

Sequential service constraint.

It is a Nash, regardless of the investors beliefs about the soundness of the portfolio of the bank.

Ine¢ cient allocation where the bank has to liquidate early the long-run project.

A Review of Diamond and Dybvig: Solutions

(^1) Narrow banking Wallace (1996):

1 Pay in all events: even worse than autarky.

2 Pay if liquidation: same than autarky.

(^3) Securitization: same than equilibrium with Önancial markets.

(^2) Suspension of Convertibility.

(^3) Equity: Jacklin (1986).

(^4) Deposit insurance.

Capital

Capital is hold by banks and households:

ktb + kth = k = 1

When capital ktb is hold by a bank at period t, it produces zt + 1 ktb of nondurable good at period t + 1. When capital kth is hold by a household at period t, it requires f

kth

to produce zt + 1 ktb of nondurable good at period t + 1. Interpretation as management cost. Assumption:

f

kth

α 2

kth

for kth  kht 2 ( 0 , 1 )

α k

h t

kth k^

h t 2

for kth > k

h t

Kink in management costs allows the household to absorb all the capital in case of a banking collapse.

Representative Household

Preferences: E 0

∞

t = 0

β t^ log cth

Endowment of nondurable goods zt w h^.

Deposits in a bank that pay Rt + 1 if no bank run. If bank run, a depositor receives either the full payment or nothing, depending on the timing of the withdrawal.

We assume that, ex ante, the household gives zero probability to bank run.

Hence, budget constraint:

cth + dt + qt kth + f

kth

= zt w h^ + Rt dt 1 + (qt + zt ) kth 1