1

Avoiding costly default with state-contingent contracts:

issues of market power and moral hazard

Marcus Miller and Lei Zhang

Houblon Norman Fellows at the Bank of England1*

December, 2013

Abstract

To reduce the severity of sovereign debt problems, the CIEPR report on Revisiting Sovereign

Bankruptcy advocates statutory change. The focus of recent proposals from economists at the

Bank of England and the Bank of Canada, however, is on promoting contractual innovations,

the issuance of state-contingent securities in particular.

In this paper, where Stone-Geary preferences are used to capture ‗inability to pay‘, we

discuss how the maturity-extension clauses of ‗sovereign cocos‘ can support the competitive

equilibrium for solvent but illiquid debtors. But if the status quo is advantageous to powerful

creditors, is this not an innovation they will resist?

Secondly, we analyse the proposal for GDP bonds in terms of ―completing the market‖ by

adding Arrow securities. When debtor moral hazard is introduced, however, a puzzle

emerges: how to price such securities if ‗hidden actions‘ by the debtor can change state

probabilities?

*The views expressed here are those of the authors and do not represent those of the Bank or the Monetary

Policy Committee.

1 For comments and discussion, we thank Ken Binmore, Martin Brooke, Oliver Bush, Alex Pienkowski and

Dania Thomas, but remain responsible for the views expressed and errors made; and we are grateful to

Efthymia Mantellou for research assistance, funded by ESRC/CAGE.

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―If debt contracts embodied more risk-sharing between debtors and creditors, or were

written in ways that made countries less vulnerable to rollover crises or exchange rate

movements, debt crises would be both less frequent and less severe.

Sturzenegger and Zettelmeyer (2006)

1 Introduction

According to Revisiting Sovereign Bankruptcy, co-written by lawyers and economists and

issued by Brookings, there are three reasons why sovereign debt problems have become more

salient over the last decade. First because they are no longer confined to emerging markets,

but involve North Atlantic economies too – especially in the Euro area; second because the

use of debt swaps may be jeopardised by the recent US court ruling giving ‗holdout creditors‘

the right to interfere with payments to those that have accepted a debt swap; and finally

because distorted incentives and/or policy mistakes lead to ‗overborrowing‘ ex ante. Such

factors, they conclude, ‗create a much stronger case for an orderly sovereign bankruptcy

regime than ten years ago‘ CIEPR (2013).

But what if – to strike a more optimistic note – crises are a spur for financial innovation? This

is the perspective taken in Brooke et al. (2013) ―Sovereign default and State-Contingent

Debt,‖ a Financial Stability Paper No. 27, co-authored by economists at the Bank of England

and Bank of Canada which studies contractual changes that may help avoid costly default.

They are quick to concede that experience of EMEs shows evidence of the high costs of

sovereign default, including substantial deadweight losses. In fact, they cite evidence that‘

sovereign debt crises in EMEs have led to median output losses of at least 5% in levels

terms‘, De Paoli et al. (2009); and, since these periods of low growth seem last for about a

decade, it would appear that the output cost of crisis cumulate to about one fifth of one

year‘s GDP.

They also claim, however, that the contractual nature of sovereign debt and of procedures for

crisis resolution have evolved in response: this is shown in Table 1, where the last line refers

to innovations proposed by the authors.

Date Event Subsequent Development

1980s Latin American Debt Crisis Baker Plan for liquidity provision/

Brady swap for debt relief

Late 1990s East Asian Financial Crisis Collective Action Clauses

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2001/2002 Argentine default Pari Passu principle

2008/9 North Atlantic Financial

Crisis

Proposal for state-contingent debt:

Cocos and GDP bonds

Table 1 Debt crises and changes in contracts and procedures

It should be noted, however, that the paper by Brooke et al. focuses on how to handle

liquidity shocks, rather than how to achieve debt write downs for insolvent sovereigns, which

is the key feature of the Brookings study.

In what follows we begin with liquidity issues which can affect a sovereign debtor who is

solvent but nevertheless exposed to balance sheet pressures which, in the absence of external

support, may lead to default and substantial deadweight losses. After highlighting the two

proposals for state-contingent debt in Brooke et al. (2013), a ‗toy‘ model of an endowment

economy with some initial short term debt is used to illustrate the risk of default from a

liquidity shock; and how Cocos and may help to avoid them. Issues of market power,

coordination failure and asymmetric information are then discussed.

What if the liquidity shock panic was driven by the risk of an adverse shift in future

fundamentals that might lead to future default? In this case Arrow Securities can be used by

the debtor to avoid default, essentially by buying insurance against the adverse shock. But

pricing problems arise if there is moral hazard.

2 Highlights from the paper by Brooke et al. (2013)

2.1 Sovereign Cocos

The proposal is for bonds which ‗automatically extend in repayment maturity when a

country receives official sector emergency liquidity assistance. (Italics added) Activation

of the maturity extension would not require approval by the existing bondholders. If the

entire debt stock of a country were to contain these clauses, the entire amortisation profile of

the sovereign would shift into the future when a crisis occurs and official sector assistance

was provided. The details of this automatic private sector bail-in would be defined ex ante in

the bond‘s legal documentation.‘

While acknowledging that that state-contingent rollovers had been advocated earlier by

others – by Buiter and Sibert (1999) and, in the context of Euro area bonds, by Weber et al.

(2011) – the key innovative feature they propose is to use the provision of official liquidity

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support as the trigger for ‗bailing-in‘ private creditors. A possible boiler plate is provided, as

follows.

Feature

Design

Trigger for maturity extension:

when the sovereign receives emergency

liquidity from the official sector – for

example when the sovereign draws upon

credit from the IMF or the ESM.

Length of maturity extension:

should match that of typical official sector

support programmes – around three years for

an IMF programme.

Bonds covered: all sovereign and sovereign-guaranteed debt

(bonds and loans) would include this clause.

Treasury bills with an original maturity of

one year or less would be excluded

Coupon payments: will continue at their original level and

frequency. ‗Amortising bonds‘ would have

the principal (but not coupon) payments

postponed

Number of maturity extensions:

Only one per coco. But any sovereign cocos

issued after the trigger event would be

unaffected – these could

be triggered in the normal way

Table 2 Proposed design features of a sovereign coco

It is argued that this may reduce creditor moral hazard as creditors ‗could no longer anticipate

full repayment by the official sector in times of crisis‘. Without an IMF/ESM ‗put‘, creditors

will have to share more of the downside risk and should, for that reason, take more care in

lending.

2.2 GDP bonds

The second proposal involving state-contingent instruments in the paper by the Central

bankers is to recommend issuing GDP bonds in place of plain vanilla bonds, effectively a

pre-emptive debt equity swap.

‗While sovereign cocos are primarily designed to tackle liquidity crises, GDP-linked bonds

help reduce the likelihood of solvency crises. And both are state-contingent instruments,

which can be defined in bond contracts at issuance. [The contracts to include] the following

features: first, the bond‘s principal would be directly indexed to nominal GDP; and second,

the coupon on this bond is paid as a fixed proportion of this principal, and therefore also

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varies with nominal GDP. GDP-linked bonds are not a new idea. Shiller (1993 and 2003)

argues that these bonds would allow households and companies to take an ‗equity stake‘ in a

country‘s economic performance, helping risk diversification and hedging.

Barro (1995) focuses on the benefits to the government, in particular, the ability to use GDP-

linked bonds as a means to smooth taxes through time. Others, including Chamon and Mauro

(2005) and Ruban, Poon and Vonatsos (2008) demonstrate how GDP-linked bonds can

reduce the credit risk on sovereign debt.

GDP-linked bonds can reduce the likelihood of sovereign default through two related means.

First, they reduce the size of increases in sovereign debt related to contractions in GDP.

Second, GDP-linked bonds can raise the maximum sustainable debt level of the sovereign,

providing countries with more ‗fiscal space‘ in times of crisis, Barr et al. (2012).

3. A ‘toy’ model of debt and default

As do the authors of FSP No 27, we assume that debtors are greatly concerned to avoid

default because of the deadweight and other costs of so doing. Yue (2010) discusses recent

evidence of how the balance of power shifts towards the creditors in such circumstances.

From an anthropological perspective, Graeber (2010) stresses that severe treatment of

defaulting debtors has been highly persistent for millennia2.

We use a highly-stylised model of a sovereign who is solvent but has considerable short term

debt exposure to illustrate some key aspects of these proposals. Stone-Geary utility functions

– where a minimum level of consumption is required for subsistence – are used to explain

default triggered by ‗inability to pay‘. This formulation yields a sharp contrast between

competitive equilibrium and one where a creditor with market power can take advantage of

the debtor‘s lack of a viable outside option.

Specification

Consider an endowment economy which lasts for two periods with two players (a creditor and a

sovereign debtor). There is no uncertainty in period one, and uncertainty in period two is captured by

2 Widespread evidence of ‗fair‘ responses in the ‗ultimatum game‘ might lead one to expect creditors

to treat defaulting debtors with more compassion; but default may be seen as a serious violation of

cooperative behaviour which is evolutionarily efficient, Binmore (2010).

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two possible states: High or Low with probability of and . Assume that the debtor has

inherited a given level of one period debt of which needs to be serviced (amortised or rolled over at

interest) in period 1.

Endowments

The endowment structure is shown in the following table:

Period t=1 t=2

State and state

probability

Creditor

Debtor

Table 3. Endowment structure

Endowment is not storable and .

Stone-Geary Preferences

Preferences of the two players are assumed to be identical with period utility of and a discount

factor of . Each player has a subsistence level of consumption at where . The life-time

utility of a player is given by

where , , , and subscripts denote time period and superscripts the

states. We assume that consumption at any time and state is at least the level of subsistence,

Unless otherwise stated, we assume

(i) Perfect information.

(ii) Initial debt is large enough to trigger current period default if there is no rollover, i.e.,

. (So the debtor‘s initial endowment is insufficient to cover subsistence and paying

off the debt in period one)

(iii) The debt is small enough so that the debtor is solvent in two periods (to be specified

later).

Each agent chooses a consumption profile to maximise expected utility subject their endowments, the

rate of interest and Arrow prices, if applicable. . (Here we assume away any strategic interaction

between players, so decisions are made taking the interest rate as given.)

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In Section 3.1 we start with the case of no uncertainty, and in Section 3.2, we examine a case where

the debtor can share risks by issuing state-contingent securities, whose prices are determined in

competitive markets.( In the Annex we look at a two period bond.)

To simplify the presentation, we scale all endowment and consumption relative to the subsistence

level of consumption. Measured in this way the endowment structure becomes:

Period t=1 t=2

Player/State H L

Creditor 1 1 1

Debtor 1 1

Table 4. Rescaled endowments.

The rescaled consumption is defined as

Now we can rewrite utility function in terms of ―discretionary consumption‖

with subsistence constraints ; and the high initial debt assumption, .

3.1 Competitive equilibrium without uncertainty

To set the scene, we start with the simple case where the debtor is illiquid but solvent, i.e.

, there is no uncertainty in the debtor‘s endowment in period 2 ( , and

both agents are ‗price-takers‘.

Both debtor and creditor take the interest rate as given and

(1)

subject to

, or , as appropriate, and (2, 3)

implying an Euler equation for each player

(4, 5)

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So the two budget constraints (2, 3) and the Euler equations (4, 5) determine the consumption

profile for the two agents given the interest rate; which in competitive equilibrium is

determined by the market clearing in period 1:

. (6)

3.2 Liquidity shocks and sovereign cocos – a graphical treatment

In this section we look at the case with log utility. As shown in Figure 1, where the

dimensions of the Edgeworth box measure the total endowments in each period (namely

), the Stone-Geary utility functions only apply to consumption above subsistence, so

the logarithmic utility curves are measured from the origins indicated by and , with

reference to the debtor and creditor respectively. In the absence of any initial debt, the ―no-

trade equilibrium‖ would be at A in the middle of the diagram, where the slope of the budget

line (tangent to the indifference curves for both parties) is the gross interest rate which

with constant endowments is simply

(7)

As the agents have parallel linear Engle curves, redistribution will not affect the interest rate,

it will merely redistribute consumption from debtor to creditor, giving stationary

consumption for the creditor at the level

(8)

and for the debtor at the level

. (9)

As the debtor is solvent, i.e. , the rollover implied by these choices of

consumption improve both player‘s welfare.

As can be seen from Figure 1, where the initial endowment is indicated at (as the debt is

all short term), the debtor is illiquid and cannot pay off the entire debt in period 1 as

this would take consumption below subsistence. This poses no problem for market

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equilibrium3, however, as the creditor effectively rolls over the short-term debt by choice, as

shown by the competitive equilibrium at where the budget line has been shifted left to

and the debtor amortises a little over half the debt in period 1. In fact, as can be seen

from the consumption choices (8) and (9), the debtor amortises the debt by two equal

payments of ), so the amount of the debt rollover in period 1 is

. (10)

Figure 1. A liquidity shock that triggers default.

But what if this voluntary rolling over ceases – for reasons of market panic perhaps or doubts

about the solvency of the debtor – i.e. there is a liquidity or ‗balance sheet‘ shock which has

no impact on the endowments but shifts the equilibrium abruptly to E? The debtor will be

forced to default and face the costs that incurs.

3 As Milton Friedman would doubtless have predicted, given his views on how the anonymity of

markets prevents discrimination.

E

1

L

L

R

s s 2

b

a

Default

OB

C

OC

Subsistence

for Borrower

Subsistence

for Creditor

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If, in the face of this shock, the sovereign were to apply and qualify for official emergency

liquidity4 then the ―sovereign cocos‖ proposed by Brooke et al. (2013) would effectively

reverse the liquidity shock and shift the equilibrium back to . Action taken by the official

sector resembles that of a central bank which provides liquidity in the case of a bank run; but

here it is the creditors that provide liquidity to the sovereign. In the terminology of Corsetti et

al. (2006), one could say that official lending effectively acts as a catalyst for private lending.

3.3 Creditor Market Power

There are, of course, important issues to be borne in mind often missing from market

equilibrium models; issues involving market power or market dislocation, for example, and

asymmetric information. Take the case of market power where the creditor has a monopoly in

the supply of credit.

In the circumstances described, with a solvent but illiquid debtor, it is obvious that the ‗take

it or leave it‘ offer from a maximizing creditor would be , where the debt is rolled over but

only at the cost of the debtor losing all discretionary consumption in both periods. It might of

course be that the option of outright default could put a limit on creditor power. In other

words, the benefits the creditor can extract from the ‗take it or leave it‘ offer would be limited

to the costs of a financial crisis to the debtor. But for convenience we proceed on the

assumption that the debtor sees outright default as a more costly option.

What if the creditor acts as a market monopolist offering rollover credit at interest rates

higher than the competitive market rate? Given Stone-Geary preferences, we find that a

monopoly creditor will also achieve the same result! To find the monopoly outcome we first

determine the debtor‘s offer curve by solving

s.t. (11)

and the Euler condition with log utility.

Solving for and as functions of the interest rate set by the creditor:

(12)

4 In amount which, for convenience, we set at zero.

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(13)

By substitution we obtain the offer curve explicitly as

(14)

This is shown in Figure 2 starting from the origin ( ; passing through the

competitive equilibrium where and

(15)

and tending asymptotically to as .

Figure 2. The Debtor’s Offer Curve and Monopoly Equilibrium

Given this offer curve above, the monopoly creditor will choose the origin in the figure above

by setting . This is obvious when one inserts the offer curve into the earlier

Figure 1 where the indifference curve of the creditor is shown passing through . What is

striking is how, instead of proceeding from the initial endowment to the competitive

equilibrium as would be the case for the creditor, the offer curve for the debtor starts from

, before heading to . This simply reflects the fact that a sole supplier facing an illiquid

Oligopoly Outcomes

Debtor’s Endowment

Competitive

Equilibrium

Monopoly

Equilibrium

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debtor with no outside option can raise the cost of credit until it reduces the ―continuation

value‖ of the debtor to subsistence.

Assuming imperfect competition, with Cournot Oligopoly in the supply of credit will yield

outcomes on the offer curve tending towards as a number of oligopolist increases without

limit (Miller et al., 2013). Such oligopoly outcomes are of course less likely to trigger the

option of outright default by the debtor.

One implication of such strategic behaviour is that the cost of credit will rise in liquidity

crisis with a corresponding transfer of resources to the creditor.5 Another is that tough

creditors will not see reducing the dead-weight cost of crisis as being in their interest: it will

give the debtor a more attractive outside option. By the same token, such creditors will

presumably resist the inclusion of rollover clauses in sovereign debt!

Domestic credit markets provide analogous evidence of market power being used in this way.

Recently, for example, a British bank has been accused of market manipulation for cutting off

credit to various SMEs so as to drive them into the arms of its restructuring unit, which

purchased their collateral assets at knock down prices (Tomlinson, 2013). There are similar

accounts of predatory behaviour by banks in the US Great Depression.6

Creditor coordination and moral hazard

What if the liquidity shock has a negative impact on the net worth of the borrower, reducing

its capacity to pay? In circumstances where the supply of credit is not coordinated but

delivered by many individual agents, each acting to minimise its losses, this could in

principle result in a ‗debt run‘ leading to a self-fulfilling solvency crisis, as discussed in

Sachs (1984). The wide-spread use of sovereign cocos would presumably avoid coordination

problems of this sort. 7

What of incentive effects of rollovers on debtor behaviour, effects that may not be directly

observable or contractible? There is enormous literature dealing with debtor and creditor

moral hazard and we do not propose to add to it. It is worth noting that institutional features

may be important in terms of revealing hidden actions giving appropriate incentives.

5 Could this be an element of what Hausmann and Sturzenegger (2007a, b) refer to as ―dark matter‖ – the

capitalized value of return privileges obtained by powerful creditors? 6 See also Allen and Gale‘s (2006) discussion of ―cash-in-the-market‖ pricing, and multiple equilibria in asset

markets with limited participation. 7 See Sturzenegger and Zettelmeyer (2006) for discussion of an alternative contingent loan scheme proposed

by Chamon (2004).

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As regards debtor moral hazard, for example, some argue that automatic rollovers without

conditionality give local elites the opportunity to arrange things so that they receive non-

contractible benefits while the creditors take a haircut, Ghosal and Miller (2003). But Corsetti

et al. (2006), in their model of ‗the IMF‘s catalytic approach‘ argue that the time provided by

a rollover may allow a benevolent government to implement policies that enhance its

capacity to pay its creditors. The different predictions here depend largely on the involvement

of agent with powers of discovery and conditionality. The potential debtor moral hazard

effect of the ‗sovereign coco‘ is presumably offset because it is emergency lending by the

IMF and/or ESM which acts as a trigger for maturity extension.

3.4 Risk of Insolvency: GDP Bonds as Arrow Securities

In this section, we turn to case of exogenous endowment shocks with known probability. We

analyse the proposal for GDP bonds in terms of ―completing the market‖ by adding Arrow

securities to secure efficient risk sharing. This offers an interpretation of the proposed GDP

bonds.

Let p and q be the Arrow prices for date 2 High and Low state consumption measured in date

1 consumption. The creditor‘s problem is then

(16)

subject to

, (17)

(18)

Which has two first order conditions

(19)

(20)

With log utility, one can show that

(21)

(22)

. (23)

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The debtor‘s problem is the same as that of the creditor‘s, except the budget constraint being

replaced by

, (24)

Solvency requires

(25)

The demand functions of the debtor under log utility are then

(25)

(26)

.

(27)

Using market-clearing conditions in period 2

one obtains the Arrow prices as

(26)

(27)

Given (26) and (27), all consumption allocation can be backed out.

What is the rollover in this case?

Using (10), we get the amount of debt that is rolled over is

(28)

in exchange for period 2 repayment, in the high state, of

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and in the low state, of

The key points can be illustrated using a crudely adjusted version of the earlier figure, where

the endowment shock appears as a reduction of the size of the Edgeworth box , see Figure 2

(where , for simplicity, only discretionary consumption is plotted).

Given the risk of default in the lower state and the consequences that may follow, the debtor

could ensure no default by saving sufficiently in period 1, i.e. by reducing period 1

consumption to the point shown as . (This form of self-insurance is hardly optimal,

however, as it would involve great consumption variance in period 2 – some more savings in

period 1 would prove worthwhile. )

Figure 3. Avoiding default by precautionary saving; or by issuing Arrow Securities

Δ

b

E

Default

Δ exogenous endowment shock

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The obvious alternative to self-insurance is a risk-sharing contract where the creditor

provides insurance against the adverse shock, as discussed above. In terms of Figure 2

required savings by the debtor will diminish as indicated at the point which lies to the right

of , with state contingent repayments in the second period are indicated by , .

In outline, such risk-sharing contracts seem to correspond to the issuance of GDP, bonds as

proposed by Brooke et al. (2013) for sovereign debtors.

3.5 Risk-sharing by Arrow securities: discussion

With Arrow securities and GDP bonds the debtor effectively buys insurance cover against

bad outcomes. As with insurance contracts in general, however, moral hazard issues will

arise: might the debtor not behave differently in the presence of such insurance?

For this purpose it is tempting to extend the analysis of the previous section by allowing for

costly effort on the part of the debtor to reduce the probability of the bad outcome. This is

relatively straightforward for non-contingent, plain vanilla long term debt.

Take the case where the debtor can choose either High or Low effort. With High effort, the

probability of High state is larger than that resulted from Low effort, . But exerting

High effort costs the debtor in terms of its utility (where the utility cost of Low effort is

normalized to zero). Does the issuance of a long term debt affect the debtor‘s incentive to put

in effort?

Since the debt contract is not state contingent, creditor‘s behavour will depend only on the

interest rate and not on state probability. So whether High or Low effort is an equilibrium

will depend on debtor‘s utility with respect to these two effort levels.

Applying the envelope theorem to debtor‘s utility function (gross of effort cost), one can

show that

where and are two positive Lagrange multipliers (because budget constraints of the

debtor are binding). It can be shown from the market clearing condition that <0, i.e.,

increasing effort reduces the equilibrium interest rate. As ,

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and , so . Hence, excluding the effort cost, increasing the

effort level improves debtor‘s utility.

Consequently whether the High effort level will be an equilibrium outcome or not simply

depends on whether it is efficient, i.e.,

So presence of longer term debt contract does not per se create debtor moral hazard even if

effort level is not observable.

Before proceeding further to analyse the case of state-contingent Arrow securities, it is

crucial to note that one is dealing with an environment in which, by definition, the action of a

large agent changes state probabilities. One might well expect a loss of Pareto efficiency in

such a case – a sort of ―pecuniary externality‖ perhaps where state prices are affected by the

action (or inaction) of the debtor? In fact, the problem may be more serious. Magill, Quinzii

and Rochet (2012), in an illustration with discrete choice of effort which affects state

probability, demonstrate that Arrow prices do not exist.

They begin with a criticism of the state-space formulation of the Arrow-Debreu paradigm as

follows:

The Arrow-Debreu model is however of far more limited applicability than the

literature […] would have us believe, and this for two reasons. The first is that the

Arrow-Debreu (AD) model is based on assumptions about the way productive

uncertainty within firms can be described and the nature of the markets on which firms

are supposed to operate which are difficult, if not impossible, to match to anything we

observe, or could even imagine observing, in the real world. The conditions that would

need to be satisfied are the following:

(1) it must be possible to make a complete enumeration of primitive causes (states of

nature) which, when combined with the actions (investment) of the firms, serve to

explain the firms' different possible outcomes;

(2) these primitive causes must:

(i) have probabilities which are exogenous and independent of the actions of the

firms;

(ii) be known and understood, not only by the manager of each firm, but by all

agents in the economy;

(iii) be sufficiently simple to describe and verify to form the basis for contracts

traded on markets. In the standard Arrow-Debreu model these contracts are

(promises to make) forward delivery of goods contingent on the primitive

causes (states of nature).

Secondly, even if all these conditions could be satisfied, on which Arrow (1971) raised

serious doubts, he also argued that the technical conditions required to obtain existence

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of an equilibrium may be violated by the unavoidable presence of indivisibilities or non

convexities. Magill et al. (2012, p. 2)

They go on to consider a case of discrete action where an industrial product can be good or

bad depending, in discrete fashion, on the non-verifiable effort of the producers.

For example consider the problem faced by an automobile company like Toyota which

needs to design and implement the production of a new model or improve on the design

and production of an existing model. It can hire engineers to design the various

components of the car, test the prototypes, and set up a factory to produce and assemble

all the components. At the end of the period of design and production, cars get

produced which are either ―good‖ (no flaws) or ―bad‖ (have flaws in the functioning of

some parts leading for example to unintended acceleration). […] The contingencies

which condition the outcome of the production process are numerous and difficult to

describe. Furthermore, whatever the difficulties involved in their enumeration, they are

very much ―internal‖ to the firm, and are hence unlikely to be observable and verifiable

by outsiders. However the precise description of these contingencies is an essential

ingredient of the Arrow-Debreu model since it assumes that prices are based on these

contingencies. Magill et al. (2012, p. 14)

The authors then proceed to prove that in such a case Arrow-Debreu prices do not exist:

In the informal discussion preceding the description of the AD model we expressed

doubts about the realism of the market structure based on states of nature. We now

show that even if we accepted the strong assumption that such markets can be put in

place, it would not suffice to solve the inefficiency, since this economy has no Arrow-

Debreu equilibrium. Magill et al. (2012, p. 14)

As yet, we have not been able to solve for the Arrow prices even in our toy model when it is

expanded to introduce unobserved effort on the part of the debtor. Is it simply a matter of

trying harder; or is it mission impossible even in this primitive setting? While output

increases for high effort in any given state (and likewise for low effort), there is no simple

correlation between the unobserved effort and output: and with discrete choice of effort it

seems to satisfy the conditions for which this non-existence theorem is proved. If so, the

impossibility result should apply here too!

If it is correct to interpret GDP bonds as Arrow securities, how can they be priced? Maybe

the interpretation of GDP bonds as Arrow-Debreu securities is a blind alley that leads

nowhere? Is there an alternative approach that will work in these circumstances?

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4. Conclusion

We should emphasize, in conclusion, that the two reports we have mentioned deal with

different aspects of sovereign debt problems. That by the Central Bankers, on which we have

focussed, is explicitly limited to handling problems facing a solvent debtor - by providing

maturity extensions in the face of liquidity shock and by providing insurance against shocks

to net wealth. The CIEPR report, on the other hand, is more concerned with sovereigns who

have ‗over-borrowed‘ and need a debt write-down for sustainability; and the focus is on the

institutional changes required to achieve this - by aggregating across creditors and protecting

the debtor‘s assets from seizure by vultures, for example.

Although the paper by Brooke et al. (2013) emphasises the positive benefits of contractual

evolution, it is nevertheless true that institutions play an important part, certainly for

sovereign cocos where the institutional trigger which plays a vital role. This may also be true

of GDP bonds, if the problems of pricing go deeper than GDP revisions, as we discuss

above. Could it be that institutional monitoring of policy effort is a necessary condition for

proper pricing of contracts like this that include elements of macroeconomic insurance?

References

Allen, F. and D. Gale (2007), Understanding Financial Crises, Oxford: Oxford University

Press.

Barr, D., O. Bush and A. Pienkowski (2012), ―GDP-Linked Bonds and Sovereign Default‖,

Presentation at Universidad Nacional de Cordoba.

Barro, R. (1995), ―Optimal debt management‖, NBER Working Paper No. 5327.

Binmore, K. (2010), Natural Justice, Oxford: Oxford University Press.

Brooke, M., A. Pienkowski, R. Mendes, and E. Santor (2013), ―Sovereign Default and State-

contingent Debt‖, Bank of England Financial Stability Paper No. 27.

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22

Annex

A1. Rollover to a debt contract with uncertainty

Let the market determined gross real interest rate be , the creditor‘s problem is as follows

(A1)

Subject to

(A2)

(A3)

where the rollover measured in terms of period one consumption is

(A4)

The first order condition for optimality to (A1)-(A2) is

(A5)

Note that (A2) and (A5) determine creditor‘s demand functions. Using log utility, these result

in

. (A6)

Similarly, the problem faced by the debtor is

(A7)

Subject to

(A8)

, (A9)

. (A10)

23

Assume

(A11)

so the debtor is solvent ex ante.

The first order condition to (A7)—(A9) is

With log utility, one can solve for period 1 consumption of the debtor:

(A12)