Default and Fragility in the Payment System

Scott Freeman

Department of Economics, University of Texas at Austin

Paula Hernandez-Verme

Department of Economics, Texas A&M University

March, 2004

Outline

1. Motivation.

2. The environment.

3. The Planner’s Problem.

4. Trade and travel patterns.

5. Optimality and Fragility under Alternative Settlement Rules.

6. Conclusions and extensions.

1. Motivation

Payment System Arrangements to settle debt (obligations).

• CHIPS clears and settles over $1.2 trillion daily

with only $2.4 billion in pre-funding.

• Most consumption and investment purchases, and all financial transactions are conducted with debt settled by third parties (checks, electronic fund transfers), not with cash.

If the payments system is fragile, the entire market economy is fragile, i.e.: vulnerable to Pareto

dominated equilibria with low financial activity.

Net Debt = IOUs payable – IOUs receivable

Gross Debt = IOUs payable

Net settlement

• In a given period, must bring cash to clear net debt.

• Credit risk (default) and potential spillover.

• Fedwire, CHIPS.

Gross settlement

• In a given period, must bring cash to clear gross debt.

• Reserves.

Rules of Settlement of Debt

Unwinding (Strict)

Amount owed is independent of default

by others.

Debt Forgiveness

Amount owed depends on

default by others.

Requirements of Theoretical Modeling of the Payment System, Zhou (2000)

1. Consumption/Investment debt: the system design affects the allocation of real resources.

2. Treat consumption/investment debt as distinct from payment debt, which is created only for payment needs.

3. Incorporate settlement liquidity shortage.

Done

Previous analysis

(1)-(3)

Perfect enforcement

Exogenous default probabilities

or

Begun 4. Incorporate credit risk.

(1)-(3) are insufficient guides for settlement policy.

Without endogenous default choices, we cannot discuss the effect of settlement rules on:

• Default incentives.

• The stability of the payments system.

Camera and Li (2003):

• Strategic complementarity and multiplicity of equilibria.

• Descentralized payments system.

• Lack of study of local stability analysis of equilibria.

Kahn, McAndrews and Roberds (2000):

Closer to our attempt. But absence of:

• Fiat money as the payments instrument.

• Proposal of net settlement with debt forgiveness.

• Local stability analysis of equilibria.

When debtors have the option to default, which rule for the settlement of debt has equilibria that are:

Optimal (right long-run incentives)?

Unique and Stable (free from systemic risk)?

Our Question:

We present a model with the following features:

1. Exchange involving debt.

2. Debt cleared by third parties.

3. Debt settlement requires final payment using fiat money.

4. Debtors have a nontrivial default option.

5. Interdependence of default decisions.

2. The Environment

• Closed, endowment economy.

• 2 period-lived overlapping generations: young and old.

• Population is constant and time is discrete.

Outer Islands

• There are I outer islands.

• A continuum of households with unit mass in each of the outer islands.

• There are I different goods: each good is island-specific.

• Contracts cannot be enforced in the outer islands.

• A place where all contracts are enforced.

• Think about the civil and monetary authority being here.

• There may be a location-specific utility/disutility of going to the Central Island.

Central Island

Utility of living under the law in a place where contracts can be enforced.

Endowments:

• Each young household born in island i (i=1,2,…I) is endowed with w units of the island-i-specific good.

• Old households have no endowment of goods.

• There is a generation of initial old endowed with the constant money supply M.

Preferences:

Ex-ante identical households same utility function:

)()( 21 cvcuU

Consumption when young

Consumption when old

Location-specific utility, random

o

c Utility from going to the Central Island

Utility from going to outer islands

= Net utility = c - o

Young households born in island i wish to consume only the good specific to island (i+1).

Old households born in island i want to consume only the good specific to island (i+I/2) (Modulo I).

The good you have is not the good you want:

3. The Planner’s Problem

Maxccc *

221 ,ˆ,~, Household’s Expected Utility

s.t. Feasible Set

*

Social Optimum

• No rules of settlement.

• The only constraint is feasibility.

*

,,2

,,21

,,,2

,,21

,ˆ,~,

ˆ~ s.t.

ˆ~

*

*

*

**221

jijijiji

jijijijiji

ccc

dfcdfccw

dfcvdfcvcuMax

Or:Utility if stay

away from C.I.

Utility if travel to C.I.

Feasible set

The Social Optimum is characterized by:

222 ˆ~ ccc

1, 21

2

1

ccMRScv

cu

, ,0

0 ,0 0

*

-

*

*

If

If

1.

2.

3.

Optimality involves some default

Optimality involves no default

From 22 ˆ~ cvcv

• Golden Rule.

• c1 = consumption acquired with debt, c2 = consumption acquired with fiat money.

4. Trade and Travel Patterns

Each period has two parts

First part

Second part

Intra-generational trade:

- Young with young (outer islands).

- Old with old (Central Island).

Inter-generational trade:

Young with old.

First Part of the Period Second Part of the Period

Old

Young

Buyer i travels to island i+1 to purchase good

i+1. Issues IOUs.

Seller i stays in island i: waits for buyers

Household chooses whether to travel to

Central Island or not.

Household i sells remainder of good i to incoming old

households in exchange for fiat money.

Household i travels to another island to purchase a

consumption good.

Intra-generational trade Inter-generational trade.

• They constitute “consumption debt”: goods acquired for the promise of future payment (debt for goods).

• Promise must be repaid next period on the Central Island: only time when people will get together.

• The repayment of debt can only be enforced in the Central Island.

• Only fiat money is useful to old agents for their purchases in the outer islands. Therefore, old agents will require fiat money for the repayment of debt.

• r= gross real interest rate promised on the debt issued.

About the IOUs :

• At the beginning of the period, the old household j observes the realization of the random variable ,j.

• ,j = utility that old household j derives only if it chooses to travel to the Central Island during the first part of the period.

,j is i.i.d. across households and islands and it is stationary.

• f(,j ) is the pdf, with support on ,

Utility of Central Island Travel:

*j cut-off value

for household j

If j < *j, do not travel to Central

Island choose to default

If j *j, do travel to the central

island chooses not to default

Ex-ante preferences are identical, but they are different ex-post.

If old household goes to the Central Island:

• Carries fiat money from the previous period.

• Must repay its debt.

• Gets paid only by households who show up in the Central Island.

• Consumes goods.

• R = effective real interest rate paid on loans.

• Gets the utility j .

If old household doesn’t go to the Central Island:

• Carries fiat money from the previous period.

• Does not repay its debt.

• Does not get paid for the debt it accepted the previous period.

• Consumes goods.

2c

2~c

= utility of going to the C.I. – utility of not going to C.I.

5. Alternative Rules for the Settlement of Debt

We examine 3 alternative rules:

• A Flexible Net Settlement Rule (Debt Forgiveness).

• A Net Settlement Rule with Unwinding (Strict).

• A Gross Settlement Rule.

Net Settlement Gross Settlement

• IOUs receivable can be used to pay IOUs payable.

• Bring cash for anything extra: net debt.

• IOUs receivable cannot be used to pay IOUs payable.

• Bring enough cash to pay gross debt.

Net debt = IOUs payable – IOUs receivable

• Your gross debt is $100 payable, and you have $100 receivable.

• Only 80% of the people who owe you money show up at the Central Island.

Suppose:

Strict Net Settlement Net Settlement with Debt Forgiveness

• You pay $100.

• You get paid $80.

• You pay $80.

• You get paid $80.

A) A Flexible Net Settlement Rule

• Only net debt matters.

• π = fraction of old households traveling to the Central Island.

• Debt forgiveness: Old households who travel to the Central Island pay only a fraction π of their debt.

Net debt = IOUs receivable - (IOUs payable)

If <1 debt forgiveness kicks in: debt and IOUs receivable reduced by the same percentage.

Gross debt = (IOUs payable)

The resulting equilibrium is:• Unique.

• Optimal.

222 ˆ~ ccc

12

2

1

rcv

cu

1.

2.

3.

0,

0,00

*

*

*

Some default

No default

Equilibrium:

*j

*j

0

Cut-off value chosen by household j as a function of *

-j

Cut-off value chosen by the other households

Nash equilibrium

B) A Strict Net Settlement Rule

• Only net debt matters.

• Old households who travel to the Central Island pay the total value of their debt but receive only a fraction of what they are owed.

• π = fraction of old households traveling to the Central Island.

Gross debt = IOUs payable

The amount you owe is independent of the fraction of households who default.

Net debt = IOUs receivable - (IOUs payable)

System of 3 equations in 3 unknowns rp

s

t

tj ,,*

22* ˆ~ cvcvj

0ˆ21 cvrcu

22~1ˆ1 cvcvr

Cut-off value

Inter-temporal choice

Intra-temporal choice

Equilibria:

• Equilibria are complicated. No-uniqueness seems typical.

• Strategic Complementarities may be present: If you think that no one else will show up at the Central Island, you may not want to show up either. The more others default, the more you want to default.

Why?Because others’ default does reduce

what you receive but it does not reduce what you owe.

Due to strategic complementarities, the reaction function has a positive slope. If multiple equilibria:

• Equilibria that are Pareto-ranked.

• Could implement a policy rule to get to a Pareto superior allocation.

• Multiplier effects: interior solution.

(j)

1Locally

unstable

Locally stable

0 1 (-j)

Figure 1: Equilibria in the absence of nonpecuniary utilityStrict Net Settlement

optimal

Figure 2: Strategic complementarities and multiple equilibria when travel to the Central Island is always beneficial

Strict Net Settlement Rule

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1.0

-j)

(j)

Locally stable

Locally stable

Locally unstable

optimal

1.0 Optimal

0.8

Locally

unstable

(j)

0.6

Locally Locally stable

0.4 stable but not optimal

0.2

0.0 0.2 0.4 0.6 0.8 1.0 (-j)

Figure 3a: Equilibria when travel to the Central Island is costly for some,Strict Net Settlement

1.0

0.8

Optimal

allocation

(j)

0.6

Locally

0.4 stable

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Figure 3b: Equilibria when costs of travel to the Central Island aresufficiently high, Strict Net Settlement

(-j)

C) A Gross Settlement Rule

• IOUs receivable cannot be used to pay IOUs payable.

• Bring enough cash to pay gross debt additional constraint on real money balances.

22~1ˆ)1( cvcvr

rcvrcu 21 ˆ

22* ˆ~ cvcv Cut-off value

Inter-temporal choice

Intra-temporal choice

A Gross Settlement Rule is weakly Pareto inferior to a Strict Net Settlement rule.

• Equilibrium allocations resulting from an unconstrained gross settlement rule (=0) are identical to those resulting from a strict net settlement rule.

• For each equilibrium resulting from a constrained gross settlement rule (>0) there is a Pareto superior equilibrium resulting from a strict net settlement rule.

1.0 Optimal

0.8 Constrained Gross

Settlement

(j)

0.6

0.4 Strict Net

Settlement

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Figure 4: Equilibria under Strict Settlement Rule and Equilibriaunder a Constrained Gross Net Settlement

(-j)

6. Conclusions

A Flexible Net Settlement Rule

With Debt forgiveness

When debtors have the option to default, which rule for the settlement of debt has equilibria that are: Optimal (right long-run incentives)?

Unique and Stable (free from systemic risk)?

A Flexible Net Settlement Rule

With Debt forgiveness

Pareto Ranking of Settlement Rules

Order0

0

If If

1 Net settlement with debt forgiveness

Strict net settlement

Gross settlement (constrained)

Net settlement with debt forgiveness

Gross settlement (constrained)

Strict net settlement, other equilibria2

3

Strict net settlement with universal repayment

Gross settlement (unconstrained)

Gross settlement (unconstrained)