Working Paper No.

343
Efficient frameworks for sovereign
borrowing
Gregor Irwin and Gregory Thwaites

March 2008
Working Paper No. 343
Efficient frameworks for sovereign borrowing
Gregor Irwin(1) and Gregory Thwaites(2)

Abstract

This paper presents a theoretical model of strategic default to assess how national and international
policymakers should seek to influence the cost of default and the distribution of bargaining power in the
event of a default. We find that, in the absence of restrictions on the parameter space, deadweight costs
of default should be driven to zero. Moreover, if the debtor is risk-averse, there is an optimal division
of bargaining power between the debtor and its creditors. Even with restrictions on the parameter space,
marginally lower deadweight costs, possibly in some combination with greater creditor bargaining
power, can always raise social welfare ex ante. However, once debt has been contracted, the debtor’s
trade-off between creditor bargaining power and deadweight costs changes fundamentally. In
equilibrium, the deadweight costs of default may therefore tend to be too high, and the allocation of
bargaining power inefficiently skewed towards the debtor. The challenge for policymakers is to find
credible, time-consistent combinations of policies that can both reduce deadweight costs and shift
bargaining power towards creditors.

Key words: Sovereign debt, default, restructuring.

JEL classification: F34, G15.

(1) International Finance Division, Bank of England. Email: gregor.irwin@bankofengland.co.uk

(2) Conjunctural Assessment and Projections Division, Bank of England.

The views expressed in this paper are those of the authors, and not necessarily those of the Bank of England. We are
grateful to Glenn Hoggarth, Adrian Penalver, Chris Salmon and Misa Tanaka for helpful comments. This paper was finalised
on 20 February 2008.

The Bank of England’s working paper series is externally refereed.

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ISSN 1749-9135 (on-line)
Contents

Summary 3

1 Introduction 5

2 The model 13

3 Unconstrained optimum 17

4 Constrained optima 20

5 Credibility and ex-post optimality 22

6 Conclusions and policy implications 24

Appendix 28

References 31

Working Paper No. 343 March 2008 2

Summary

There is no supranational authority that can enforce sovereign debt contracts. Consequently, the
decision by a government to default on its debts is often as much a question of willingness to pay
as it is of ability to pay. Debt restructurings, which change both the size and the timing of
payments made to creditors, are therefore brought about through negotiation between the parties
to the contract, rather than by court adjudication. When a sovereign decides whether to default it
has to weigh the benefit against the cost. The main benefit comes in the form of a reduced
repayment, which is often referred to in understated terms as a ‘haircut’ for creditors. The cost
comes in a number of different forms, such as loss of reputation, or loss of current and future
access to private capital markets. Moreover, sovereign default is often associated with costly
currency crises and banking crises. The multiple costs of default, and the partial extent of the
haircut that can be achieved in practice, both serve to limit the incentive that the sovereign has to
default and underpin the very existence of sovereign debt markets.

National and international policymakers have some control ex ante over the size and form of the
cost of default and the distribution of bargaining power in the event of a default. This paper
presents a theoretical model of strategic default to assess how policymakers should exercise their
control over these levers. We consider a world in which the sovereign issues fixed interest debt
to finance an investment with uncertain returns. After both the productivity of the investment
and the resulting income stream are known, the sovereign must decide whether to repay the debt
in full, or to seek a restructuring. If the sovereign takes the second option we assume it must pay
a deadweight cost, to reflect the loss of reputation and the economic disruption that ensues
following a default. It must then negotiate over the size of the haircut, which is ultimately
determined by the distribution of bargaining power. These factors – the deadweight cost and the
distribution of bargaining power following a default – are the two key dimensions by which the
‘framework for sovereign borrowing’ is characterised in our model. We assess the
welfare-maximising values for both these policy parameters.

We find that, if there are no restrictions on the distribution of the bargaining power, the
deadweight costs of default should be driven to zero. Both deadweight costs and the need to
settle with creditors can dissuade a debtor from defaulting. However, the latter is more efficient,
as resources denied to the debtor are reallocated to creditors rather than being destroyed.

Working Paper No. 343 March 2008 3

Assuming creditors are competitive and risk-neutral, this should benefit the debtor through lower
interest rates. If the debtor is risk-averse, then in the event of a restructuring the optimal
outcome requires bargaining power to be shared between the debtor and its creditors. This is
because shifting bargaining power to creditors has two conflicting effects on the debtor’s welfare.
On the one hand, by dissuading default and lowering interest rates, it allows a sovereign to
borrow more at a lower cost. But on the other, if creditors capture too much of the available
resources after a default, the risk-sharing benefit of default is diminished, as creditors receive
additional resources from the debtor when the latter needs them most. The optimal regime
should balance this tension.

In constrained policy settings we find that, whenever welfare can be raised by marginally
increasing the deadweight costs of default, welfare is also improved by shifting bargaining power
to creditors. It follows that for any given value of the deadweight cost, if bargaining power is
optimally allocated between the parties, it must be welfare-improving to reduce the deadweight
cost. Moreover, starting from any situation where the welfare impact of marginally raising the
deadweight cost is positive, there is always a step increase in the allocation of the bargaining
power to creditors which is sufficient to ensure that the impact of raising the deadweight cost
becomes negative. Taken together, these results mean that, so long as creditor bargaining power
can be increased, lower deadweight costs can always raise social welfare ex ante.

The analysis shows that, once debt has been contracted, the debtor’s trade-off between creditor
bargaining power and deadweight costs changes fundamentally. With the interest rate on debt
fixed, the incentives of the debtor change so that it no longer cares whether, after a default,
resources are transferred to creditors or are wasted in the form of deadweight costs. There is
therefore a need to design mechanisms that allow debtors to commit to the ex-ante optimal
combination of policy parameters.

In sum, these results suggest that domestic and international policymakers should pay careful
attention to the impact of their policies, not just on the deadweight costs of default, but also on
the allocation of bargaining power in the event of a restructuring. The final result, in particular,
suggests that in equilibrium the deadweight costs of default may tend to be too high, and the
allocation of bargaining power inefficiently skewed towards the debtor. A challenge for all
policymakers, therefore, is to find credible policies that can both reduce deadweight costs and
shift bargaining power towards creditors. In due course this should raise welfare.

Working Paper No. 343 March 2008 4

1 Introduction

There is no supranational authority that can enforce sovereign debt contracts. Courts in foreign
jurisdictions sometimes rule that a sovereign in default has a legal obligation to repay its debts,
but even so the principle of sovereign immunity severely curtails the ability of creditors to make
good on their claims when the sovereign is unwilling to co-operate. This sets sovereign debt
apart from corporate debt in two respects (Bedford et al (2005)). First, the decision to default (or
to seek a restructuring pre-default) is a question of willingness to pay as much as ability to pay.
Second, restructurings are brought about through negotiation between the debtor and its
creditors, albeit usually indirectly and informally, rather than by court adjudication.1

When a sovereign decides whether to default on its external debts it has to weigh the benefit
against the cost. The benefit comes in the form of a reduced repayment – the ‘haircut’ the
sovereign obtains on the debt it owes to its creditors. This is the outcome of the negotiation
process, which in turn depends on the relative bargaining strengths of the parties to the
negotiation. Sturzenegger and Zettelmeyer (2007) find a considerable variation in the haircut
achieved on external debt in recent restructurings, ranging from 13% on average in Uruguay in
2003, to around 50% in Russia in 1999 and 2000, and 74% in Argentina in 2005. The
deadweight cost of default can come in a number of different forms, such as loss of reputation, or
loss of current and future access to private capital markets. Moreover, sovereign default is often
associated with costly currency crises and banking crises. The overall impact is often a
significant and enduring output loss for the economy as a whole. De Paoli, Hoggarth, and
Saporta (2006) estimate the median output loss from recent crises to be 6.9% of GDP in each
crisis year, with the median length of crises eight years.2 The multiple costs of default, and the
partial extent of the haircut that can be achieved in practice, both serve to limit the incentive that
the sovereign has to default. Dooley (2000) has gone so far as to conjecture that deadweight
costs of default in particular are an essential component of sovereign debt markets: in their
absence countries might never choose to repay their debts and, foreseeing this, creditors would
never lend to them in the first place.

Domestic and international policymakers have some control ex ante over the size and form of the

1 Inmature domestic bankruptcy regimes corporate restructurings are often concluded in the ‘shadow of the law’, in order to preempt
predictable court rulings. By so doing the parties concerned can avoid unnecessary legal costs. In sovereign restructurings there is no
‘shadow of the law’.
2 This is the cumulative difference per year between potential and actual output.

Working Paper No. 343 March 2008 5

cost of default and the distribution of bargaining power in the event of a default. For example,
sovereigns can raise deadweight costs by issuing debt in forms that are hard to restructure
(Dooley (2000), Bolton and Jeanne (2005)), or difficult to default on selectively (Broner and
Ventura (2005)). Sovereigns can also increase their creditors’ bargaining power by placing some
resources under foreign jurisdictions, or by entering into international treaties which bolster
creditor rights.3 The official sector can also influence deadweight costs and the distribution of
bargaining power through ‘architectural’ innovations, such as encouraging the widespread
inclusion of collective-action clauses in sovereign bond contracts (Haldane et al (2005)). By
lending to a debtor in arrears the IMF can directly reduce the deadweight cost of default, but may
also shift bargaining power away from creditors to the sovereign, as the latter has a less urgent
need to re-access private international capital markets.

How should policymakers exercise their control over these levers? Should they seek to raise or
lower the cost of default? Should they attempt to shift bargaining power to or from the sovereign
in the event of a default? What impact will these measures have on the incentive to default in the
first place and on welfare? Weak incentives – due to a low deadweight cost of default and/or
high debtor bargaining power – will result in more frequent and larger defaults, leading creditors
to require high interest rates. But strong incentives may cause the debtor undue hardship in bad
times and result in insufficiently frequent defaults with suboptimally low haircuts. How can
domestic and international policymakers strike a balance between these competing effects? Is
Dooley right when he conjectures that deadweight costs are necessary to sustain sovereign debt
markets?

This paper presents a stylised theoretical model of strategic external debt default to address these
questions. We consider a world in which the sovereign issues plain vanilla debt to finance an
investment with uncertain returns. After the productivity of the investment is known the
sovereign must decide whether to repay the debt in full, or to seek a restructuring of the debt. If
the sovereign takes the second course of action it must pay a deadweight cost, equal to a fixed
proportion of the output from the investment, and negotiate over the size of the haircut, which is
ultimately determined by the distribution of bargaining strength between the sovereign and its
creditors, which we take to be exogenous. These factors – the deadweight costs and the
distribution of bargaining power following a default – are the two key dimensions by which the

3 For
example, the ‘Convention on the Settlement of Investment Disputes between States and Nationals of Other States’, which is
administered under the auspices of the World Bank’s ICSID.

Working Paper No. 343 March 2008 6

‘framework for sovereign borrowing’ is characterised in our model.4 We then assess the ex-ante
welfare-maximising values for these policy parameters, both with and without constraints on the
parameter space. The analysis has implications for both domestic and international
policymakers, insofar as they can influence these parameters ex ante.

The analysis produces five key results. The first main result is that, in the absence of restrictions
on the parameters space, deadweight costs of default should be driven to zero. It is always better
to tackle the enforcement problems of sovereign debt by giving creditors the power to bargain for
some claim on the debtor’s resources after default. The simple intuition for this result is as
follows. Both deadweight costs and the need to settle with creditors can dissuade a debtor from
defaulting. But an ex-post transfer is more efficient, because the resources denied to the debtor
go to creditors rather than being destroyed. Knowing that they will receive some payment in the
event of default, competitive creditors demand a lower interest rate ex ante. With lower
contractual interest rates to pay, debtors default less frequently, other things being equal, and
have higher consumption when they do repay. Put another way, creditors in a competitive market
never earn any surplus, so whatever framework maximises debtor surplus must also maximise
social welfare. By the same token, debtors pay all the expected deadweight costs, so for a given
level of lending it is optimal to reduce deadweight costs as far as possible. This result is germane
to Dooley’s conjecture that deadweight costs are necessary to support the existence of sovereign
external borrowing. Our simple framework demonstrates that sovereign borrowing can be
supported in equilibrium in the absence of default costs, providing creditors have sufficient
bargaining power post-default. Moreover, this is superior in welfare terms.

The second main result is that, if the debtor is risk-averse, then in the event of a restructuring
there is an interior optimum allocation of bargaining power between the debtor and its creditors.
This is because shifting bargaining power to creditors has two conflicting effects on the debtor’s
welfare. On the one hand, by dissuading default and lowering interest rates, it allows a sovereign
to borrow more at a lower cost. But on the other, if creditors capture too much of the available
resources after a default, the risk-sharing benefit of default is diminished, as creditors receive
additional resources from the debtor when the latter needs them most. The optimal regime
should balance this tension.

4 Inpractice, the distribution of bargaining power may be partly influenced by the extent to which creditors are able to threaten credibly to
take actions which would impose additional costs on the debtor. We do not explicitly model this interconnection. Instead we take the
distribution of bargaining power and the deadweight costs that are incurred in the event of a default as exogenously given, and consider
the effect that varying each has on welfare.

Working Paper No. 343 March 2008 7

In our specification, it is optimal to allocate creditors just enough bargaining power so they are
willing to extend the amount of lending that maximises the expected consumption of the debtor
(also equal to expected Gross National Product (GNP)). Returns to capital are diminishing, so
the level of investment that maximises expected consumption is finite. Giving debtors a
borrowing capacity beyond this level cannot raise expected consumption, and the required
increase in creditor bargaining power must reduce the debtor’s consumption when it defaults,
which is when the marginal utility from consumption is highest. On the other hand, reducing
borrowing capacity below the efficient level reduces the debtor’s consumption for every
realisation of productivity.

A related implication of this result is that ex post benchmarking a haircut against ‘ability to pay’
alone is an inappropriate test of the merits of a restructuring deal. Ex-ante efficiency, and in
particular the desire to share risk, requires that creditors should not be able to bargain for all that
a sovereign can repay in the event of a default.

The third and fourth results relate to constrained policy settings, when the policymaker does not
have complete discretion over the size of the deadweight costs and the degree of creditor
bargaining power. This analysis is important because in practice policymakers have only a
limited ability to influence the size of each. Moreover, the factors which influence each are
likely to be under the imperfect control of several parties. Consequently it is perhaps most useful
for policymakers to understand the impact on welfare of marginal changes to either the
deadweight cost or the allocation of bargaining power.

The third key result is that, for any arbitrary level of deadweight costs and creditor bargaining
power, whenever welfare can be raised by increasing the deadweight costs of default, welfare is
also improved by shifting bargaining power to creditors. Both measures discourage default, and
can therefore increase credit supply. But they can also harm welfare by punishing debtors in bad
states of the world. We show that whenever the former beneficial effect outweighs the latter
adverse one for deadweight costs, such that it improves welfare to raise them, this must also be
the case for increased creditor bargaining power. It then follows that for any given value of the
deadweight cost, if bargaining power is optimally allocated between the parties, it must be
welfare-improving to reduce the deadweight cost. Moreover, starting from any combination of
the policy parameters for which the welfare impact of marginally raising the deadweight cost is
positive, there is always a step increase in the allocation of the bargaining power to creditors

Working Paper No. 343 March 2008 8

which is sufficient to ensure that the impact of raising the deadweight cost becomes negative.
Taken together, these results mean that, so long as creditor bargaining power can be increased,
lower deadweight costs can always raise social welfare ex ante. Once again, this substantially
qualifies Dooley’s conjecture.

The fourth main result (obtained numerically) suggests that, if deadweight costs cannot be
reduced to zero, the optimal degree of creditor bargaining power is generally lower than in the
unconstrained case, and may even fall to zero. Intuitively, the two policy measures are imperfect
substitutes, in that both dissuade default, and both punish debtors in bad states of the world. So a
constrained policymaker should generally attempt to offset excessive deadweight costs by
reducing creditor bargaining power.

The final main result is that, once debt has been contracted, the debtor’s trade-off between
creditor bargaining power and deadweight costs changes fundamentally. With the interest rate on
outstanding debt fixed, the debtor no longer cares whether, after a default, resources are
transferred to creditors or are wasted in the form of deadweight costs. The ex-post optimal
combination of policy parameters diverges from the ex-ante optimal combination and the change
in debtor incentives therefore gives rise to a time-consistency problem. With this, there is a need
to design mechanisms that allow debtors to commit to resolving debt problems in ways that
transfer resources to creditors rather than wasting them.

Taken together these results suggest that domestic and international policymakers should pay
careful attention to the impact of their policies, not just on the deadweight costs of default, but
also on the allocation of bargaining power in the event of a restructuring. The final result in
particular suggests in equilibrium the deadweight costs of default may tend to be too high, and
the allocation of bargaining power inefficiently skewed towards the debtor. A challenge for all
policymakers, therefore, is to find credible, time-consistent combinations of polices that can both
reduce deadweight costs and shift bargaining power towards creditors. In due course this should
increase the amount of finance available to the debtor and raise welfare.

The remainder of the paper is structured as follows. The next subsection reviews the related
literature in more detail. Subsection 1.2 then illustrates how policymakers can influence the key
parameters in our model using the IMF’s lending-into-arrears policy as an example. Section 2
sets out the model. Section 3 characterises optimal policy ex ante, assuming the policymaker is

Working Paper No. 343 March 2008 9

unconstrained, while Section 4 considers the case where the policymaker faces exogenous
constraints on the parameter space. Section 5 highlights the tension between ex-ante and ex-post
optimality, and the time-consistency problem that follows from it. Section 6 concludes.

1.1 Related literature

There is a large literature on the question of why sovereign debtors repay external debts. In the
past, incentives to repay were often provided by the threat of force. Mitchener and Weidenmier
(2005) document the use of ‘supersanctions’ – gunboat diplomacy, direct fiscal control and asset
seizures – as a means of ensuring the repayment of sovereign debt during the gold standard era.
Most modern explanations are less direct and fall into one of two categories. The first suggests
that countries repay in order to avoid immediate economic costs of default, which might either be
an inevitable consequence (eg Bolton and Jeanne (2005)), or voluntarily imposed by creditors
and therefore a threat point in any negotiation (eg Bulow and Rogoff (1989)). The second
category proposes that countries repay in order to avoid reputational costs, which may
subsequently manifest themselves as poor financing terms when capital markets are re-accessed
(eg Grossman and van Huyck (1985) and Cole, Dow and English (1995)).

There is also a literature on the risk-shifting aspect of the option to default. Several papers (eg
van Wijnbergen (1990), Zame (1993), Zha (2001)) recognise that, when securities markets are
incomplete, the option to default in some states of the world can raise welfare by providing
partial insurance against bad states of the world.

But few papers address, even indirectly, the question of what environment domestic and
international policymakers should aim for. The strategic default model of sovereign borrowing
in Gai et al (2001) suggests that higher deadweight costs increase the amount of lending that is
incentive compatible, but reduce the amount that is efficient through its negative effect on the
expected productivity of investment. Dooley (2000) notes that sovereign borrowers may
endogenously choose debt structures that are costly to restructure as a means of committing to
repay their debts. Bolton and Jeanne (2005) find that a debtor may choose to issue debt with
high restructuring costs in order to commit not to borrow excessively at a later date. Cavallo and
Velasco (2006) find that a sovereign should only commit to limited creditor sanctions in the event
of a default, using a model in which the sovereign enters into what is essentially an insurance
contract with creditors, but in which payments by creditors are unpredictable and subject to

Working Paper No. 343 March 2008 10

sudden stops. Haldane et al (2005) examine the optimal threshold for collective-action clauses
in sovereign bond contracts, given the effect this has on both the likelihood of a default and the
restructuring outcome. They conclude that different debtors should choose different thresholds,
depending on their degree of risk aversion and their creditworthiness.

Two papers touch directly on the class of policy questions we are trying to address. Hamann
(2004) and Yue (2006) both compute and simulate infinite-horizon models of a small open
economy with exogenous stochastic output, able to issue plain vanilla debt only. In Hamann’s
model, the sovereign can choose to default at the cost of a perpetual reduction in output and
permanent exclusion from credit markets. He calibrates the model to Argentine data and finds
that increasing the deadweight cost from 10% to 35% of output reduces to zero the incidence of
default and raises welfare by around 7% of consumption. However, Hamann holds creditor
bargaining power fixed across these experiments. Yue examines the impact of shifting
post-default bargaining power from creditors to debtors. In a baseline parameterisation which
includes unavoidable deadweight costs of default, she finds that welfare is maximised by giving
debtors all of the bargaining power, such that a default implies complete repudiation of debt.
However, Yue does not examine the sensitivity of these results to the size of the deadweight cost
of default, which are held fixed.

Our paper does not attempt to match the data in the same way as these two papers do and our
framework is essentially static. However, unlike these previous studies, the model we use
incorporates both creditor bargaining power and variable deadweight costs of default. This
enables us to assess the extent to which policies that redistribute bargaining power or vary
deadweight costs are complements or substitutes and to identify both the globally-optimal and
the constrained-optimal combinations of these ‘policy parameters’. In so doing, our aim is to
provide domestic and international policymakers with a better understanding of the trade-offs
they face when assessing the merits of reforms that have been proposed for the framework for
sovereign borrowing.

1.2 IMF lending into arrears

In this subsection we illustrate how policymakers can influence the policy parameters in our
model using the IMF’s lending-into-arrears (LIA) policy as an example. The provision of
financing by the IMF to a sovereign debtor who is in arrears with private creditors can impact on

Working Paper No. 343 March 2008 11

the deadweight costs of default in a number of ways. In particular, financing can be used to
offset the effect on exporters, who might otherwise find they are denied access to trade credit.
Equally, IMF financing can be used to mitigate the impact of a default on the domestic banking
sector, for fear that this would otherwise result in a credit crunch and so push the economy
further into recession.

But IMF LIA also impacts on the distribution of bargaining power between a debtor in default
and its creditors. Before 1989 it was IMF policy not to lend to members in arrears with their
commercial bank creditors, who at that time provided the vast bulk of private lending to
emerging market sovereigns. However, this had the unintended consequence of providing
commercial banks with an effective veto over IMF lending decisions, hence giving a
considerable boost to their bargaining position during a restructuring.

In 1989 the IMF introduced a new LIA policy which enabled it to lend to members in arrears on
bank credit, subject to three conditions. The first condition was that the financing package be
considered essential for the successful implementation of the debtor’s adjustment program and
therefore to help limit the scale of economic dislocation. The second and third conditions were,
respectively, that restructuring negotiations had begun between the debtor and its creditors and
that it was expected that a package to restore external sustainability would be agreed within a
reasonable period.

In 1998, in order to reflect the changing nature of capital flows to emerging markets, the policy
was amended to also allow the IMF to lend to members in arrears on international sovereign
bonds. This raised concerns that not all creditors would be willing or able to enter into timely
negotiations with a debtor, in part because of difficulties in co-ordinating among a more
dispersed creditor base, but also because of fears that bondholders might have less of an interest
in maintaining an ongoing relationship with a debtor in difficulty and hence attach a lower
priority re-establishing the debtor’s good financial standing. As a result, in 1999 the third
condition was amended to instead require that the debtor be making ‘good-faith efforts’ to
negotiate with creditors.5

The history of the LIA policy illustrates how the IMF can, through its actions, impact on both the

5 In response to concerns that the good faith criterion was too vague, the policy was amended again in 2002 with the intention of

providing greater procedural clarity.

Working Paper No. 343 March 2008 12

deadweight cost of default and the distribution of bargaining power between debtors and creditors
in the event that a restructuring is required. The fundamental rationale for IMF lending to a
debtor in arrears is to reduce the deadweight cost or the ‘economic dislocation’ that can result.
But the development of the policy also illustrates how the Fund has been sensitive to the potential
impact of LIA on bargaining power and hence the outcome of restructuring negotiations. Prior
to 1989 the concern was that the IMF’s lending policy, and in particular its non-toleration of
arrears, gave the whip hand in a restructuring negotiation to commercial bank creditors. Since
then the evolution of the policy has seen the pendulum swing in the opposite direction, so that
now the provision of finance to debtors in arrears is subject to only a very loosely defined
requirement that the debtor engage in good-faith efforts to negotiate with creditors.

The model set out below helps to illustrate some of the trade-offs that must be faced when
assessing this and other policies. In particular it helps to identify the conditions under which IMF
financing that reduces deadweight costs is welfare-improving. It underscores the importance of
designing the LIA policy in such a way as to ensure that the combined effect on deadweight costs
and the distribution of bargaining power is beneficial. At the very least it illustrates that the
impact of the LIA policy on the balance of bargaining power should not be neglected.

2 The model

We model an economy that has access to a production technology with uncertain returns. It has
no resources of its own which it can use to invest in the project, so must borrow abroad to finance
it. The country chooses how much to borrow, at an interest rate determined endogenously in the
international capital market, so as to maximise the expected utility of consumption when the
returns of the project are realised.

Crucially, there is no external commitment technology that can force the country to repay its
debts. Once the returns on the project have been realised, the sovereign must choose whether to
repay or default. If it defaults, we assume the sovereign loses a fraction α of output as a
deadweight cost, and must relinquish a fraction β of the remainder to creditors in partial
settlement of its debts. We assume that the parameters α and β can be set with credibility, either
by the debtor itself or by some outside agency. We consider, in turn, the cases where the
parameter space is unrestricted and where it is subject to exogenous restrictions which reflect the
limited control that policymakers have over these parameters in practice.

Working Paper No. 343 March 2008 13

The proportionality of the deadweight cost to output can be justified as, other things being equal,
the absolute value of the deadweight cost is likely to be higher in larger economies.
Alternatively we could specify the deadweight cost as a function of the size of the default, but
this would add substantial algebraic complexity with little additional economic insight.

The parameter β is taken to represent the strength of creditor bargaining power. The assumption
that the settlement creditors can bargain for is an increasing function of GDP, net of the
deadweight cost, can be justified if we interpret creditors’ bargaining power as arising from the
credible threat of some economic damage, such as trade sanctions or the denial of future official
financing or capital market access. These costs are likely to be increasing in the debtor’s
economic size. The impulse to settle with one’s creditors may also come from the non-pecuniary
‘diplomatic’ costs of being seen to have treated creditors unfairly (see Sturzenegger and
Zettelmeyer (2007)). Clearly, these costs are hard to quantify, but it seems likely a government’s
willingness to pay to avoid them will depend on the resources it has available.

Once the decision to default or not has been made, the country consumes the project’s returns, net
of any deadweight cost and the payment it must make to creditors. The capital markets anticipate
this behaviour and demand an interest rate, r, which depends on the amount the country borrows,
the probability of a default, and the amount creditors expect to receive in the event of a default.
We assume creditors are competitive and risk-neutral, and therefore break even on average.

Perhaps the most fundamental assumption embedded in our model relates to the set of financial
contracts that are available. The debtor can only finance investment with plain vanilla debt, with
no de jure indexation to the productivity shock or any other state variable in the model. The
option to default, as we will see, gives rise to some de facto conditioning of payment flows on
other economic variables. But, particularly when the exercise of this option triggers a
deadweight cost, we are left a long way from a first-best world in which risk-neutral creditors
would assume all the risk of the project. However, the assumption that all debt is plain vanilla is
realistic, given the rarity of GDP-linked sovereign bonds observed in practice. We do not
attempt to model the underlying frictions that prevent such contracts from being written.

Working Paper No. 343 March 2008 14

2.1 Production

The output of the project, y, depends on the amount of resources invested, L, an exogenous
stochastic productivity shock, θ, and the deadweight cost, proportional to α, that is incurred in
the event of default: |
θ L λ if no default
y=
(1 − α) θ L λ if default
where θ ∼ U [0, 1] and λ ∈ (0, 1). The sovereign decides how much to borrow and invest before
the value of productivity θ is realised.6

This particular production function has been chosen for analytical convenience: it can be shown
that all of our main results go through with any well-behaved production function.7 A uniform
distribution is chosen for the productivity shock in order to make it possible to derive an
analytical expression for the interest rate in Subsection 2.3 below.8

2.2 Consumption and default

After the productivity shock, θ, is realised, the debtor decides whether or not to default, so as to
maximise consumption. The debtor consumes the project’s returns net of any deadweight cost
and less the payment it must make to creditors. In addition, the debtor consumes an exogenous
amount equal to c, which is interpreted as a non-stochastic endowment of goods that can be
consumed, but not used for investment, nor pledged to creditors. It plays a largely technical role
in the model.9 However, economically it seems reasonable to assume that the debtor has access
to some positive level of consumption even if it does not borrow anything. Using the
superscripts D and N D to denote ‘default’ and ‘no default’ respectively, consumption in each of
the states is given by
C N D (θ) = c + θ L λ − r L (1)
C D (θ) = c + (1 − α) (1 − β) θ L λ (2)

We define E
θ as the critical value of the productivity shock at which the debtor is indifferent

6 A more general framework might endogenise the productivity parameter θ. For example, productivity could be thought of as a function

of the debtor’s policy effort.

7 We require only that f (0) = 0, f (0) = ∞ and f (L) < 0.
8 The upper bound of the support of this distribution has no effect on the results: one could always rescale the production function to give
the same result. However, setting the lower bound of the support at zero, as opposed to some strictly positive number, may have some
effect on the results relating to optimal risk-sharing that we derive below.
9 It allows us to evaluate the CES utility function at L = 0 and θ = 0 when the coefficient of relative risk aversion γ ≥ 1.

Working Paper No. 343 March 2008 15

between default and full repayment, ie where C N D = C D . This yields
E r L 1−λ
θ= (3)
α + β − αβ
This expression captures the main factors at play in an intuitive fashion. The probability of
default (equal to E
θ, given the assumed distribution for θ) is increasing in the contractual interest
rate and the degree of indebtedness, and decreasing in the size of the deadweight cost of default
and the degree of creditor bargaining power.

2.3 Equilibrium interest rate

The sovereign’s payment to creditors in each of the states is given by

PND = rL (4)
P D = β (1 − α) θ L λ (5)
Creditors expect to break even ex ante. With the risk-free interest rate normalised to unity, and
given (4) and (5), this implies
= E
θ
b c
L = 1 −E
θ rL + β (1 − α) θ L λ dθ (6)
0
Substituting in the expression for E
θ gives a quadratic in r, the lower root of which10 is
√
1 − 1 − 2L 1−λ
r (L) = (7)
L 1−λ
(2α + β − αβ)
where = (8)
(α + β − αβ)2
This expression only is real-valued if L ≤ L, where
t u 1−λ 1
1
L= (9)
2
We can interpret L as an upper borrowing limit. It is straightforward to show that r (L) ≥ 1 and
d e
∂r (L) /∂ L > 0 for all L ∈ 0, L , both of which are intuitively appealing conditions. If L = L
then r = 2. Furthermore, if L = L and α = 0 then E
θ = 1, otherwise E
θ < 1.

2.4 Equilibrium debt level

Given the terms on which it can borrow, r (L), the debtor chooses borrowing, L, to maximise
expected utility. With a utility function, u (C), this is given by
= Eθ = 1
b Dc b c
E [U (L; α, β)] = u C dθ + u C N D dθ
0 E
θ

10 Both roots are a solution to the break-even condition, but only the lower root is a competitive equilibrium. If the prevailing interest rate

were given by the higher root, individual lenders would find it profitable to offer credit at an infinitesimally lower interest rate, and the
debtor would accept funds on these terms. This is not the case at the lower root.

Working Paper No. 343 March 2008 16

Using Leibniz’ rule, which allows us to eliminate terms in ∂E
θ/∂ L, given C D = C N D at E
θ, the
corresponding first-order condition is

= E
θ
∂ E [U (L; α, β)] b c
= (1 − α) (1 − β) θλL λ−1 u C D dθ (10)
∂L 0
= 1t t uu
λ−1 ∂r b c
+ θλL − r+L u C N D dθ = 0
E
θ ∂L

We define E
L (α, β) as the expected utility maximising amount of lending as function of the
policy parameters, α and β:
E
L (α, β) = arg max E [U (L; α, β)]

Note that when L = E

L raising L must increase C D and reduce C N D , if the first-order condition is
to be satisfied. A risk-averse debtor will be content to trade off a lower expected consumption if
it can shift consumption from the state where marginal utility is low (ie the no-default state) to
where it is high (the default state). Consequently we can infer that a risk-averse debtor will
necessarily choose to borrow an amount higher than that which maximises expected
consumption, if unconstrained by creditors.

3 Unconstrained optimum

We turn to the question of what values for the policy parameters, α and β, maximise the debtor’s
expected utility. By assumption, creditors are risk-neutral and always expect to break even, so
we can ignore their utility when determining social welfare.

In this section we solve for the optimal values of the policy parameters when the policymaker has
complete freedom to set both α and β over the set [0, 1] × [0, 1], with only weak restrictions on
the utility function.11 In the next section we consider the case where exogenous restrictions are
placed on the policy parameter space. Our strategy is to establish as much as we can analytically,
employing numerical methods to illustrate these results and to shed light on the remaining
questions.

11 In the risk-neutral case, we assume only that u (.) > 0. In the risk-averse case, we add the condition that u (.) < 0.

Working Paper No. 343 March 2008 17

3.1 Optimal setting for α

We first consider the optimal setting for α, before examining the optimal setting for β in the next
subsection. It is relatively straightforward to identify the optimal setting for α when there are no
constraints on the policy parameter space. Equations (1), (2) and (3) show that, taking r and L as
given, raising β discourages default in the same way that raising α does, by reducing the debtor’s
consumption in the event of a default. In particular note that α and β enter symmetrically into
both (2) and (3). By contrast, equations (4) and (5) show that, taking r and L as given, α and β
have opposing effects on the payment to creditors, specifically in the case of a default. We can
therefore conclude that discouraging default by raising β dominates doing so by increasing α, as
raising β entails an additional transfer of resources to creditors in the event of a default, which
will reduce r in equilibrium and in so doing benefit the debtor. Consequently, it is always
optimal for the debtor to trade a lower α for a higher β, and so welfare maximisation necessarily
requires α = 0, providing β is unrestricted. This is true regardless of how risk-averse is the
debtor.

3.2 Optimal setting for β

We first assume the sovereign debtor is risk-neutral, before considering the case where the debtor
is risk-averse. Define P (θ, L) as the ex-post payment from the debtor to creditors as a function
of realised productivity and the amount of lending. When there are no deadweight costs (α = 0)
we can express the consumption of the debtor as follows:

C|α=0 = c + θ L λ − P (θ, L)

Note that this is also equal to GNP in the debtor country: it is output, less factor payments
abroad. The condition that creditors expect to break even ex ante means E[P (θ, L)] = L and
so, once again assuming there are no deadweight costs, expected consumption/GNP is
Lλ
E [C]|α=0 = c + −L (11)
2
The first-order condition tells us that this is maximised when L = H
L where
t u 1−λ
1

H λ
L= (12)
2

Proposition 1 If the sovereign debtor is risk-neutral, welfare is maximised when α = 0 and

β ≥ λ.

Working Paper No. 343 March 2008 18

Proof. See Appendix.

The intuition behind this result is as follows.12 The creditors and the sovereign are both
risk-neutral, so expected consumption/GNP is a sufficient statistic for social welfare. If α = 0
this is maximised when L = H
L and this is feasible providing L ≥ H
L. Given α = 0 a comparison
of (9) and (12) informs us this requires β ≥ λ. If β = λ the sovereign will choose to borrow
L=H
L = L, and from (3), given (7), we can also infer that the sovereign will default with unit
probability. If β > λ the sovereign still chooses to borrow L = H
L, even though H
L < L, as this
remains the level of borrowing that maximises expected consumption (and hence expected
utility). From (11), given (12), expected consumption is independent of β when L is not
constrained: as β is increased beyond λ the default probability and consumption given default
both fall, but these effects are completely offset by a lower interest rate when the sovereign
chooses to repay. Consequently, the risk-neutral debtor is indifferent between any β ≥ λ.

Proposition 2 If the sovereign debtor is risk-averse, welfare is maximised when α = 0 and

β = λ.

Proof. See Appendix.

When the sovereign is risk-averse it does not just care about expected consumption; it also cares
about how consumption is distributed across different states of the world. Given α = 0, for
β ≤ λ it can be shown that the sovereign always exhausts its credit limit and defaults with unit
probability, so the debtor’s consumption is C D = θ (1 − β) L λ for all θ. It can be shown that the
effect that raising β has in reducing consumption (in the certain event of default) for a given L is
more than offset by the increase in the credit limit and the optimal choice of L. This is true for
every realisation of θ and so higher β will be preferred by the debtor.

For β > λ, the sovereign is able to borrow more than H

L if it chooses. By definition, borrowing
more than H
L reduces expected consumption. So, loosely speaking, the debtor can yield higher
utility only if this delivers a better ‘risk profile’ (ie more consumption in bad states of the world
and less in good states). However, it can be shown that the risk profile from any combination of
higher β and higher L is always inferior to β = λ with L = H
L, so increasing β above λ must

12 This result is related to the finding in Tanaka (2005) that IMF intervention to mitigate deadweight costs can be welfare-improving as

long as debtors are required to reach a ‘fair’ settlement with their creditors.

Working Paper No. 343 March 2008 19

reduce expected utility. Put another way, higher creditor bargaining power produces extra
borrowing capacity but, given the larger share that goes to creditors in the event of default,
consumption turns out to be lower in bad states of the world.

4 Constrained optima

It is implausible to regard domestic and international policymakers as having complete freedom

to set the size of the deadweight cost and creditor bargaining power. Some loss of output in the
event of default may be inevitable and there are likely to be limits to the amount of bargaining
power that can be credibly promised to creditors. It is therefore salient to analyse the case where
the policymaker’s choice of α or β is constrained. We begin with a general result that holds at
any point in the policy parameter space {α, β}.

Proposition 3
d E [U (C(L; α, β))] d E [U (C(L; α, β))]
If >0 then > 0.
dα dβ

Proof. See Appendix.

In words, Proposition 3 means that if marginally increasing the deadweight cost raises welfare,
marginally increasing creditor bargaining power will do so as well. This result can be explained
as follows. Suppose to begin with that the debtor is not credit constrained. Raising the
deadweight cost or increasing creditor bargaining power both reduce debtor welfare in the event
of a default. Either of these changes can therefore only raise overall welfare if they have an
offsetting positive effect on consumption when the borrower repays.13 The channel through
which this can occur is by lowering the interest rate. Raising creditor bargaining power always
reduces the interest rate; increasing the deadweight cost only does so when these are sufficiently
small in the first place, and creditor bargaining power is sufficiently weak. Whenever a higher
deadweight cost reduces the interest rate sufficiently to increase expected utility, it is necessarily
the case that giving bargaining power to creditors does so too. In the case in which the debtor is
credit constrained, the interest rate is unaffected by the deadweight cost or bargaining power, but

13 The Envelope Theorem allows us to ignore the effect of changes in the policy parameters on the optimal amount of lending. Leibniz’

rule allows us to ignore the direct effect on the default probability, as in the region of the default barrier the debtor is indifferent between
defaulting and not defaulting.

Working Paper No. 343 March 2008 20

the credit limit is. The direct effect of a higher credit limit is always to raise expected utility,
otherwise the constraint would not be binding. Raising the deadweight cost or increasing
creditor bargaining power both lift the credit limit, but raising increasing creditor bargaining
power does so more efficiently. Consequently, if the effect of a higher deadweight cost in raising
the credit limit offsets the adverse effect it has on consumption in the event of a default, this will
be true a fortiori for higher creditor bargaining power.

Corollary 4
d E [U (C(L; α, β))] d E [U (C(L; α, β))]
If ≤0 then < 0.
dβ dα

Taken together Proposition 3 and its Corollary have a number of implications. First, starting
from any point in the policy parameter space either it is welfare-improving to reduce the
deadweight cost in isolation, or there is some combination of lower deadweight cost and higher
creditor bargaining power that is welfare-improving. Second, for any value of the deadweight
cost, if creditor bargaining power is set optimally, so that the derivative of expected utility with
respect to β is zero, then it is always welfare-improving to reduce the deadweight cost. Third,
starting from any point in the policy parameter space at which derivative of expected utility with
respect to α is positive, there is always a step increase in β which is itself welfare-improving, but
which also in turn causes the sign of the derivative of expected utility with respect to α to become
negative.

4.1 Numerical results

The policy problem does not yield an analytic solution in the case where α and/or β are
constrained. To say any more about the nature of the constrained optima, we must assume a
functional form for utility and evaluate the solution numerically. We adopt a standard CRRA
utility function with parameter γ
c1−γ
u (c) =
1−γ
Expected utility is therefore given by
= 1d λ e1−γ = Eθ d e1−γ
θL − rL + c (1 − α) (1 − β) θ L λ + c
E [U (L)] = dθ + dθ
E
θ 1−γ 0 1−γ
for γ = 1. Our solution method is to analytically integrate the utility function, numerically
d e
evaluate it over the range 0, L , and search for the optimal value, L ∗ . This is repeated for a grid

Working Paper No. 343 March 2008 21

of points within the set {α, β} ∈ (0, 1) × (0, 1), and a range of values for {γ , λ, c}. Higher
values of maximised utility U (L ∗ (α, β) ; α, β) indicate superior policy settings.

Chart 1 shows the maximised value of utility as a function of α and β.14 This confirms our
earlier result that the global optimum is α = 0, β = λ. Moreover, it confirms that there is only a
particular set of circumstances for which the impact of a marginal decrease in α on expected
utility is negative: this is only the case when both α and β are sufficiently low. Taken together
with the analytical results of the previous subsection this suggests there are only restrictive
circumstances under which it becomes suboptimal to reduce deadweight costs. This is only the
case when both α and β are low and the policymaker is unable to increase β sufficiently in
tandem with a decrease in α.

Chart 1 also shows that the optimal value of β is typically decreasing in α, and may even be zero
for high enough α. Chart 2 plots the optimal choice of β as a function of α, as well as the values
of utility, the credit limit and the default probability associated with it. If deadweight costs
cannot be reduced to zero, it is optimal to give creditors less bargaining power than in the
unconstrained case; in extremis, creditors might optimally have no bargaining power at all. This
is because deadweight costs are a substitute for creditor bargaining power, albeit an inefficient
one, in that both measures dissuade default. Note that, as deadweight costs increase, it becomes
more expensive for the borrower to share risk with its creditors. Moreover, positive deadweight
costs reduce the expected productivity of investment. For these reasons, the default probability
falls.

5 Credibility and ex-post optimality

Up to this point we have assumed that the values of α and β, while potentially constrained, can
nevertheless be credibly fixed in advance by the policymaker. Yet once debt has been contracted,
the interest rate determined and the productivity realised, both the debtor and creditors will face
incentives to deviate from this.

As before, consumption is given by

d e d e
C = max C N D , C D = max θ L λ − r L , (1 − α) (1 − β) θ L λ

14 Utility is measured in units of certainty equivalent consumption. We set γ = 1.5, λ = 0.2 and c̄ = 0.01.

Working Paper No. 343 March 2008 22

Chart 1: Maximised utility function

0.2
Certainty equivalent consumption

0.15

0.1

0.05

0
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
alpha 0 beta

Charts 3 and 4 show consumption as a function of α and β, holding the interest rate and lending
constant, for two different realisations of the productivity parameter θ.15 Ex post, the debtor will
seek to maximise this quantity. The charts illustrate the obvious point that debtors will seek to
minimise both the deadweight cost and creditor bargaining power ex post. Note, however, that
the policy setting that yields the highest attainable utility ex post (at α = 0, β = 0) is also the
worst policy configuration ex ante. This can be seen by comparing Charts 3 and 4 with Chart 1 at
this particular combination of parameter values.

Ex post, the debtor is willing to trade the policy measures off against each other at a suboptimal
rate. Once debt has been contracted, the relative inefficiency of deadweight costs – the fact that
resources are wasted rather than transferred to creditors – is no longer internalised by the
debtor.16 The contour diagrams in Charts 5 and 6 illustrate this point.

Therefore, faced with an ex-post trade-off between deadweight costs and creditor bargaining
power, debtors will choose ‘too much’ of the former, relative to the ex-ante optimum. For

15 In E H
b c charts, the interest rate, lending and the policy parameters are set at the ex-ante optimal levels, ie α = 0, β = λ, L = L,
these
r =r HL .
16 This problem may be overcome to some extent if there is a repeated interaction between the debtor and creditors, as reputation
concerns may constrain the debtor’s actions. However, this effect is likely to be limited when the policymaker believes they are unlikely
to be in office for much longer. Such fears are often justified in crisis situations (Frankel (2005)).

Working Paper No. 343 March 2008 23

Chart 2: Optimal creditor bargaining power as a function of deadweight costs

3.5 0.25

Default probability (LHS)

3
Credit limit (LHS) 0.2
2.5 Optimal beta (RHS)
Utility (RHS)
0.15
2

1.5
0.1

1
0.05
0.5

0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Alpha

example, faced with a choice between a pre-default restructuring (which one might characterise
as relatively low α, high β) and a post-default restructuring (high α, low β), the debtor will be
biased towards the latter, relative to the ex-ante optimum. This suggests that, to the extent that
commitment problems prevent time-consistent choices of the policy parameters, we will observe
an excessive incidence of deadweight costs.

Furthermore, considering the effect of marginal changes in the policy parameters, Charts 3 and 4
illustrate that these tensions are more likely to surface when realised productivity is low. When
productivity is high, the debtor is locally indifferent to changes in the policy parameters over
more of their range: intuitively, high output makes default less tempting, so small changes in the
terms of default are less relevant.

6 Conclusions and policy implications

The model outlined above illustrates how sovereign borrowing can be supported in equilibrium
by a combination of deadweight costs of default and/or the ability of creditors to bargain for a
share of the debtor’s post-default resources. The model allows us to explore the welfare
implications of different combinations of deadweight costs and creditor bargaining power, which
are the key ‘policy parameters’ of our analysis. We find that welfare is globally maximised when
deadweight costs are zero and bargaining power is efficiently shared between the debtor and

Working Paper No. 343 March 2008 24

Chart 3: Ex-post consumption, θ = 0.5

Chart 4: Ex-post consumption, θ = 1

Working Paper No. 343 March 2008 25

Chart 5: Ex-ante isoutility contours

Chart 6: Ex-post isoutility contours

Working Paper No. 343 March 2008 26

creditors.

How should domestic and international policymakers respond to these conclusions? No

policymaker has complete control over the policy parameters. But sovereign debtors and the
IMF, in particular, exert some influence over them. The model illustrates how policymakers
should exercise this influence when they are partly constrained. Starting from any point in the
policy parameter space either it is welfare-improving to reduce the deadweight cost in isolation –
as is the case for most of the parameter space – or there is some combination of lower
deadweight costs and higher creditor bargaining power that is welfare-improving. Moreover, if
the welfare impact of a marginal reduction in deadweight costs is negative, there is always a step
increase in the allocation of bargaining power to creditors that in turn means this welfare impact
becomes positive.

The model also illustrates how difficult it might be to implement the optimal framework in
practice, as commitments by the debtor are likely to be time-inconsistent: in the event of a
default the debtor has an incentive to drive the policy parameters in the direction which is
welfare-minimising from an ex-ante perspective. This suggests that some form of commitment
technology is required so that the debtor can tie its hands. Alternatively, or in addition, the role
of those outside agencies which also exert control over the key policy parameters, such as the
IMF, becomes more important, to the extent that they are not subject to the same
time-inconsistency problem.

The structure of the IMF’s lending into arrears policy is potentially a key determinant of the
policy parameters. Post-default balance of payments support can directly reduce deadweight
costs of default, although how significant this impact is will depend on both the timing of the
support and the particular circumstances faced by the debtor. The analysis suggests that reducing
deadweight costs is either beneficial in its own right, or if not, it can be rendered beneficial if
there is an accompanying shift in bargaining power from the debtor to creditors. In the future, as
the LIA policy continues to evolve, the IMF should pay explicit attention to the impact the policy
has on the balance of bargaining power in the event of a restructuring.

Working Paper No. 343 March 2008 27

Appendix

Proof of Proposition 1

The debtor is risk-neutral, so maximising expected consumption will maximise expected utility.
Expected consumption is given by
= Eθ = 1
λ
E [C] = c + (1 − α) (1 − β) θ L dθ + (θ L λ − r L)dθ
0 E
θ
Given the equilibrium condition for the interest rate, expressed in equation (6), this simplifies to
r 2
s
E [C] = c + 1 − αE θ L λ /2 − L

Expected consumption is maximised when α = 0 and L = H

L. From (8) and (9) it follows that
when α = 0 then H
L ≤ L if and only if β ≥ λ. Consequently H
L is feasible and will be chosen
providing β ≥ λ. This completes the proof.

Proof of Proposition 2

The first step is to show that β = λ dominates any configuration with β < λ. We know from
Subsection 2.4 that if a risk-averse debtor is not credit constrained it will choose L > L̃.
However, when β < λ, a comparison of (12) and (9) shows that L̃ > L, and so the debtor must
be credit constrained. If the debtor is credit constrained then the default probability is unity and
for all θ consumption is then given by
t u 1−λλ
β
C D = θ (1 − β) +c (A-1)
2
which is necessarily increasing in β when β < λ, for all realisations of θ. It follows that higher
β is preferred by any agent with a non-decreasing utility function. This establishes that the
optimal β ≥ λ.

The next step is to show that β = λ dominates any configuration with β > λ. Suppose, first, that
the debtor is credit constrained. Then, once again, the default probability is unity and for all θ
consumption is then given by (A-1). This is necessarily decreasing in β when β > λ, for all
realisations of θ. It follows that lower β = λ is preferred by any agent with a non-decreasing
utility function. This establishes that the optimal β ≯ λ if the debtor is credit constrained.

Working Paper No. 343 March 2008 28

Suppose, instead, that the debtor is not credit constrained, and so E
L (β) ≤ L. We can evaluate
the impact on expected utility of an increase in β using the Envelope Theorem, which allows us
to ignore the effect of the change on the optimal amount of lending, and using Leibniz’ rule,
which allows us to eliminate terms in ∂Eθ/∂β, given C D = C N D at E
θ. This means
d b ce = Eθ =
d E U C(E L (β) ; β) λ
b D c ∂r 1 b N D c
= −L θu C (θ) dθ − L u C (θ) dθ (A-2)
dβ 0 ∂β Eθ
We want to show that (A-2) is negative when β > λ. Given the equilibrium interest rate, equation
(7), we can infer from the first-order condition (10) for L that

= t ut u = Eθ
∂r 1 b c 1−β λ b c
−L u C N D (θ) dθ = Lλ θu C D (θ) dθ (A-3)
∂β E
θ β 1−λ 0
t u = 1t u
λ λ λ b c
+ L θ − θ̂ u C N D (θ) dθ
1−λ E
θ β

If u(.) is strictly concave then the marginal utility from consumption must always be higher in
default states than in non-default states, given that C D > C N D for any of the values θ may take in
each of the states. Consequently, given (A-3), a sufficient condition for (A-2) to be negative is
that v t ut uw = Eθ t u= 1 t u
1−β λ λ λ
1− θdθ > θ − θ̂ dθ
β 1−λ 0 1 − λ Eθ β
After solving these integrals this sufficient condition simplifies to L > L̃, which we know will be
chosen by a debtor that is not credit constrained. Consequently, (A-2) must be negative when
β > λ. This establishes that the optimal β ≯ λ if the debtor is not credit constrained.

Taking these results together we have now established that the optimal β ≥ λ and that the optimal
β ≯ λ. Consequently, it must be the case that the optimal β = λ. This completes the proof.

Proof of Proposition 3

We first consider the case in which the debtor is not credit constrained. We can evaluate the
effect of a marginal increase in α or β on expected utility using the Envelope Theorem, which
allows us to ignore the effect of the changes on the optimal amount of lending, and Leibniz’ rule,
which allows us to eliminate terms in ∂E
θ/∂α and ∂E θ/∂β, given C D = C N D at E
θ. This means:
d b ce = =
d E U C(E L; α, β) λ
E
θ b D c ∂r 1 b N D c
= −(1−β)L θu C (θ) dθ − L u C (θ) dθ (A-4)
dα 0 ∂α Eθ

Working Paper No. 343 March 2008 29

 d b ce = Eθ =
d E U C(E L; α, β) λ
b D c ∂r 1 b N D c
= −(1−α)L θu C (θ) dθ − L u C (θ) dθ (A-5)
dβ 0 ∂β Eθ
Comparing (A-4) and (A-5), this part of the proof requires only that the following condition is
satisfied
∂r ∂r
(1 − α) > (1 − β)
∂α ∂β
Given equation (7) it can be shown that this condition holds for all values α and β may take.
This completes the proof for the case in which the credit constraint is not binding.

We now consider the case where the debtor is credit constrained. By substitution of (8) and (9)
into (7) we can see that the interest rate is now fixed at r = 2 and is therefore independent of α
and β. However, higher α or higher β will raise the credit limit and (at the margin) lead the
debtor to borrow more. Consequently, (A-4) and (A-5) must now be modified as follows
d b ce d b ce = Eθ
d E U C(L; α, β) ∂ E U C(L; α, β) ∂ L ∂ λ b c
= − (1 − β)L θu C D (θ) dθ
dα ∂L ∂ ∂α 0
d b ce d b ce = Eθ
d E U C(L; α, β) ∂ E U C(L; α, β) ∂ L ∂ λ b c
= − (1 − α)L θu C D (θ) dθ
dβ ∂L ∂ ∂β 0
Note that d b ce
∂ E U C(L; α, β) ∂ L
≤0
∂L ∂
Therefore, it is sufficient to show that
∂ ∂
(1 − α) > (1 − β)
∂α ∂β
From equation (8) it can be shown that this condition is satisfied for all values α and β may take.
This completes the proof.

Note that our specific assumption y = θ L λ is not necessary for this proof: the above result holds
for any well-behaved production function of the form y = F (L).

Working Paper No. 343 March 2008 30

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market-based approach and the role of the IMF’, Bank of England Financial Stability Review,
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Bolton, P and Jeanne, O (2005), ‘Structuring and restructuring sovereign debt: the role of
seniority’, NBER Working Paper no. 11071.

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