sovereign default: which shocks matter?

Review of Economic Dynamics 14 (2011) 553576

Contents lists available at ScienceDirect

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overeign default: Which shocks matter?

ernardo Guimaraes a,b,,1

ao Paulo School of Economics/FGV, Brazilondon School of Economics, United Kingdom

r t i c l e i n f o a b s t r a c t

ticle history:ceived 24 July 2009vised 8 October 2010ailable online 15 October 2010

L classication:4

ywords:vereign debtfaultorld interest ratestput shocks

This paper analyses a small open economy that wants to borrow from abroad, cannotcommit to repay debt but faces costs if it decides to default. The model generates analyticalexpressions for the impact of shocks on the incentive compatible level of debt. Debtreduction generated by severe output shocks is no more than a couple of percentage points.In contrast, shocks to world interest rates can substantially affect the incentive compatiblelevel of debt.

2010 Elsevier Inc. All rights reserved.

Introduction

What generates sovereign default? Which shocks are behind the episodes of debt crises we observe? The answer toe question is crucial to policy design. If we want to write contingent contracts,2 build and operate a sovereign debtstructuring mechanism,3 or an international lender of last resort,4 we need to know which economic variables are subjectshocks that signicantly increase incentives for default or could take a country to bankruptcy.I investigate this question by studying a small open economy that wants to borrow from abroad, cannot commit to repaybt but faces costs if it decides to default. There is an incentive compatible level of sovereign debt beyond which greaterbt triggers default and it uctuates with economic conditions. The renegotiation process is costless and restores debtits incentive compatible level. The model yields analytical expressions that allows us to quantify the impact of differentocks on the incentive compatible level of debt.One important result from this framework is that shocks to domestic productivity (or output) do not generate sizablectuations in the incentive compatible level of debt. Following a severe output shock, debt relief of a couple of percentageints would restore incentive compatibility. This is consistent with some empirical evidence (Tomz and Wright, 2007),

Address for correspondence: Sao Paulo School of Economics/FGV, Brazil.E-mail address: [email protected] thank Giancarlo Corsetti, Nicola Gennaioli, Alberto Martin, Silvana Tenreyro, Alwyn Young, the anonymous referees and, especially, Francesco Caselli

d the associate editor for helpful comments, several seminar participants for their inputs, and Zsoa Barany and Nathan Foley-Fisher for excellent researchsistance nanced by STICERD.Borensztein and Mauro (2004) argue for GDP-indexed debt. Kletzer et al. (1992) defend indexing debt payments to commodities prices.Krueger (2002) and Bolton and Jeanne (2007) argue for a sovereign debt restructuring mechanism.Fischer (1999) argues for an international lender of last resort.

94-2025/$ see front matter 2010 Elsevier Inc. All rights reserved.i:10.1016/j.red.2010.10.002

554 B. Guimaraes / Review of Economic Dynamics 14 (2011) 553576

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Sz11though most of the recent quantitative models on debt and default focus on output shocks.5 This result suggests that theerature has been focusing on the wrong type of shocks.On the other hand, shocks to risk-free world real interest rates are much more important. Debt relief in response to

asonable uctuations in world interest rates is an order of magnitude higher than that generated by shocks to output.hile a change in interest rates from 1% to 2% doubles the cost of servicing debt, a 1% fall in output reduces by just 1% thest of not repaying. The large effect of shocks to world interest rates on debt default is present in some empirical workribe and Yue, 2006).Using data from the Latin American debt crisis of the 1980s, I compare the models prediction of debt reduction withe observed debt relief. The increase in world interest rates at the beginning of the 1980s can solely account for over halfthe debt forgiveness obtained by the main Latin American countries through the Brady agreements.The model builds on the literature of endogenous sovereign debt and default (Eaton and Gersovitz, 1981; Arellano, 2008).e of the key assumptions in the model is that if a country repudiated its debt, it would be excluded from capital marketsd incur an output loss.6 The assumption of such costs is a simple way of modeling the costs that could be imposed oncountry that defaults. Debt repudiation might inhibit foreign direct investment and undermine a countrys capacity totain benecial deals in multi-lateral organisations such as the WTO. In addition, creditors can threaten countries thatight repudiate debt with sanctions such as the loss of access to short-term trade credit and seizure of assets.7

In reality, however, observed punishment for default is arguably tame and temporary. But that is, at least in part, becausebtor and creditors renegotiate the debt and, sooner or later, a new agreement is reached. Although we rarely observeplicit contingent contracts, premium rates on borrowing and occasional debt reduction are observed, so contracts arentingent de facto even if not written as such.8 The possibility of renegotiation makes contracts contingent.9

Here, I assume that the borrower can costlessly call for debt renegotiation, but lenders have all the bargaining power ine renegotiation stage. In equilibrium, the assumption on bargaining power implies that debt renegotiation simply restoresbt to its incentive compatible level. Costs, delays and further debt reductions associated with the renegotiation processight play an important role, but abstracting from them allows us to better understand the effects of different shocks one incentive compatible level of debt.10

I begin with an endowment economy, where borrowers impatience drives debt decisions. Then I move to a model withoduction, where borrowing occurs due to differences in the marginal productivity of capital. Models with endogenouscision of debt repayment and capital accumulation are not very tractable analytically. However, for some limiting cases,an derive simple and intuitive analytical solutions for the level of debt and debt reduction (haircut). The method consiststaking rst order approximations of the Bellman equations. I also solve the model numerically for non-limiting cases andow that the analytical solutions are good approximations.The results on debt reduction for both economies are very similar. Debt reduction predicted by the model dependssitively on the magnitude of the shocks and the persistence of states. Importantly, the output cost of defaulting, a variablehich is dicult to measure, has no signicant effect on debt relief as a fraction of the outstanding debt. So the conclusionsthe quantitative effects of shocks do not rest on any particular value of this variable.There are two key differences between the similar results for both economies. First, in the economy with capital, dif-

rences in the consumption-savings decisions for different states of the economy could inuence decisions on debt andfault. However, I show they produce no rst order effects on the incentive compatible level of debt, as they have onlycond order effects on the value functions. Second, growth prospects inuence the level of debt and debt relief, but thosefects are not large. The amount of capital inuences debt relief in both directions, rendering the result ambiguous.11

erefore, the conclusions about the impact of shocks on the incentive compatible level of debt are basically the same inth economies.Section 2 studies an endowment economy, Section 3 adds capital accumulation to the model, Section 4 contrasts the

sults with data from the Latin American debt forgiveness under the Brady Plan, Section 5 concludes. Most proofs andmerical exercises are in Appendices A and B.

For example, Arellano (2008), Aguiar and Gopinath (2006) and Yue (2010).The output cost of debt repudiation, as modeled here, is present in Cohen and Sachs (1986), Bulow and Rogoff (1989) and Arellano (2008), to name a.For a discussion of such costs, see Bulow and Rogoff (1989), English (1996), Sturzenegger and Zettelmeyer (2006) and Tomz (2007).Sovereign debt was analysed as an (implicitly) contingent claim by Grossman and Van Huyck (1988), Atkeson (1991) and Calvo and Kaminsky (1991)ong others. Grossman and Van Huyck show that an equilibrium in which excusable default is allowed without sanctions can be sustained. Alfaro andnczuks (2005) quantitative analysis builds on Grossman and Van Huyck. Calvo and Kaminsky (1991) take the optimal contract approach to study whethere small default premium paid by Latin American countries in the 1970s would be compatible with large debt reductions in the 1980s.Kocherlakota (1996) and Alvarez and Jermann (2000) analyse models where agents can trade contingent contracts but cannot commit to repay debt.Renegotiation in models of sovereign debt is studied by Bulow and Rogoff (1989), Fernandez and Rosenthal (1990) and Yue (2010). Kovrijnykh and

entes (2007) study the return from debt overhang to the credit market.In the numerical examples, debt relief depends negatively on the level of capital.

B. Guimaraes / Review of Economic Dynamics 14 (2011) 553576 555

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mJedethEndowment economy

In this section I consider a discrete time model of an open endowment economy that can borrow from abroad, butnnot commit to repay its debts. First I describe a generic two-state economy, then I consider the two special cases inhich output or world interest rates uctuate and nally contrast debt reductions following each type of shock. The focusthe paper is sovereign debt, so all results in this paper are derived from the point of a view of a domestic planner.12

The economy is populated by a continuum of innitely lived agents whose preferences are aggregated to form the usualpresentative agent utility function:

t=0

tu(ct)

here (0,1) is the subjective time discount factor, ct is consumption at time t and u(.) is the felicity function thattises the Inada conditions.I model debt default costs as an instantaneous permanent fall in output and loss of access to capital markets. Thermanent fall specically captures the loss that a country suffers by taking an antagonistic position towards the rest of theorld and never repaying a cent of its debts. I denote by the fraction of output lost due to default, so output is given by:

yt ={yt, if it has never defaulted(1 )yt, if it has ever defaulted

In the model this is out-of-equilibrium behaviour, which corresponds to never observing such action in reality. For thisason, it is dicult to obtain an estimate of , but the main results of this paper do not depend on the value of .The country can issue only one-period debt (dt ). The price of debt is denoted qt .There is a continuum of risk-neutral lenders that, in equilibrium, lend to the country as long as the expected return oneir assets is not lower than the risk-free interest rate in international markets, r . The price of a bond that delivers oneit of the good next period with certainty, (1+ r)1, is denoted q . There is a maximum amount of debt the country canntract that prevents it from running Ponzi schemes but it is never reached in equilibrium.The economys ow budget constraint is then given by:

ct ={yt dt + qtdt+1, if it has never defaulted(1 )yt, if it has ever defaulted

In a stochastic world, the incentive compatible level of debt uctuates. If debt goes above its incentive compatible level,e debtor prefers not to repay it, but then both creditors and debtors have incentives to renegotiate. I make the extremesumption that the country can costlessly renegotiate its debt there are no delays and no punishment.There is a stochastic state variable s. At the beginning of the period, its value is revealed and the country makes decisionsout debt and default. The country can:

Repay its debt d; Default: in which case the output loss is incurred and the country loses access to international capital markets; Renegotiate: the debt becomes ad, where a is the outcome of the bargaining process, as specied below. There is nopunishment.

Using the ow budget constraint and the expressions for output, and denoting by dF the face value of debt to be repayedthe following period, the value functions associated with each option are:

Vpay(d, s) =maxdF

{u(y d + q(dF , s)dF )+ E[V (dF , s)]}

Vdef (s) = u((1 )y)+ Vdef (s)

Vreneg(d, s) =maxdF

{u(y ad+ q(dF , s)dF )+ E[V (dF , s)]}= Vpay(ad, s)

here dF is the face value of debt and q(dF , s) is its unit price. The country chooses the maximum of them:

V (d, s) =max{Vpay(d, s), Vpay(ad, s), Vdef (s)}

In the standard Ramsey model, the central planner solution and the decentralised equilibrium are the same. However, that is not true without com-itment to repay debt. The distinction between the central planner solution and the decentralised allocation is analysed by Kehoe and Perri (2004) andske (2006). Kehoe and Perri (2004) show that the central planner solution can be decentralised if the central government is in charge of deciding aboutfaulting or not and taxes capital income to counteract an externality of capital accumulation. The logic behind Jeskes argument is similar and he ndsat capital controls may be welfare improving.


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16The key difference between this and Yue (2010) is that the renegotiation process is costless and instantaneous. Followinge (2010), a is determined by a Nash bargaining game, where the outside option for the country is autarky and the outsidetion for the lenders is walking away with no repayment at all. Thus the surpluses for borrower and lenders are:

B(ad, s) = Vpay(ad, s) Vdef (s)L(ad, s) = ad

The level of debt after renegotiation, (s)d, is given by:

(s)d = argmaxad

([B(ad, s)

] [L(ad, s)

]1 )bject to

B(ad, s) 0 and L(ad, s) 0

As a only appears multiplying d, the solution for (s)d does not depend on d.I assume = 0: lenders have all the bargaining power in the renegotiation stage, so the borrower gets no surplusm the bargaining process. Most of the literature following Eaton and Gersovitz (1981) assumes that the borrower fullypays its debt provided it is incentive compatible to do so. As noted by Fernandez and Rosenthal (1990), that is equivalentassuming that all bargaining power lies with the creditors (ex-post). Here, by assuming = 0, I am adding costlessnegotiation and maintaining that assumption: creditors are able to extract the maximum incentive compatible paymentsm the borrower up to the face value of debt, dF .The recovery rate (s) is given by:

B((s)d, s

)= 0 Vpay((s)d, s)= Vdef (s)For = 0, (s)d is the incentive compatible level of debt, beyond which greater debt would make the country prefer tofault.13 Hence, the country is always indifferent between defaulting and renegotiating, and it is assumed it chooses thetter.14

As renegotiation is costless, at state s lenders know they will receive at most (s)d, and borrowers will always repayeir debt up to (s)d. If d > (s)d for some state s, there is no cost for borrowers to convert their debt d into (s)d. Thessibility of costless renegotiation with = 0 makes debt contingent in a particular way: debt can be costlessly reduced toincentive compatible level.Higher values of would lead to larger reductions in debt. Owing to its bargaining power, the borrower would obtainrt of what lenders could lose in the renegotiation process (the full debt, d). In the extreme case of = 1, the borrowerould have all the bargaining power and (s) would be 0.15

For simplicity, there are 2 possible states, st {h, l}, and the probability of switching states is . Pr(st = st1) = ,(st = st1) = 1 . 0.5. Let V h(d) and V l(d) be value functions in the high and low states, respectively, and denotedh and dl the incentive compatible level of debt in each state such that:

V hpay(dh)= V hdef , V lpay(dl)= V ldef

Assume WLOG dh > dl . If the face value of debt dF is smaller than dl , debt is not contingent and there is no debtduction in the following period. But if the country chooses dF such that dl < dF dh , debt repayments are de-factontingent.Consider dF (dl,dh]. If s = h then dF will be repayed; if s = l, repayment will be dl and debt reduction will be

d = dF dl . Denote the expected repayment by d . If s = h, the expected repayment d = (1 )dF + dl . If s = l, thepected repayment d = (1 )dl + dF . Thus if s = h, repayments in the following period are given by dF = d + d (if= h) and dl = d (1 )d (if s = h).16As lenders are risk neutral and competitive, equilibrium bond prices q(dF , s) are such that q(dF , s)dF = qd . Consider these s = h and s = l. Then, debt will be renegotiated and reduced. The haircut corresponds to the proportional fall in thevel of debt, d/dF . That is the key variable of the model. The spread over treasuries is the rate paid above the risk-freete if there is no renegotiation, so it is the difference between the face value of debt dF and the expected repayment d ,vided by dF . Thus the spread over treasuries is d/dF the probability of renegotiation ( ) times the haircut.Two stochastic versions of the model are considered: stochastic r and stochastic y.

Vpay is decreasing in the level of debt, and Vdef is independent of the level of debt. So (s)d is the maximum value of debt such that Vpay((s)d, s)ef (s).Alternatively, I am assuming is larger than 0, but very small.As renegotiation is costless, if = 1 there would be no borrowing in equilibrium. A model with = 1 requires some costs for renegotiating debt.Likewise, if s = l, repayments in the following period are dl = d d (if s = l) and dF = d + (1 )d (if s = h).


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diThroughout the paper, I make assumptions to ensure that the country exhausts its borrowing possibilities: it borrows= dh , making debt as high as possible in each possible state. In this section I will assume that is very low, as is

pically assumed in the recent quantitative models of sovereign default.17 In Section 3, the assumption of a very low ismoved and replaced by the assumption of a high marginal productivity of capital. As in general dF dh , the results in thisper can be seen as upper bounds for debt relief following a negative shock, under the assumption that the renegotiationocess generates no further reductions in debt.

1. Stochastic world interest rates

Here, I analyse the contingent debt contract for an economy with xed per-period endowment, but uctuations in worldterest rates, r , leading to uctuations in the price of risk-free debt, q . The price of a riskless bond in internationalarkets is qh in the high state and ql in the low state, qh > ql .In equilibrium bond prices are such that q(dF , s)dF = qhd in the high state and q(dF , s)dF = qld in the low state. If< dl , the value functions conditional on repayment are:

V hpay(d) =maxdF

{u(y d + qhdF )+ [(1 )V hpay(dF )+ V lpay(dF )]}

V lpay(d) =maxdF

{u(y d + qldF )+ [(1 )V lpay(dF )+ V hpay(dF )]}

If dF (dl,dh], the value functions conditional on repayment are:

V hpay(d) =maxdF

{u(y d + qh[(1 )dF + dl])+ [(1 )V hpay(dF )+ V lpay(dl)]}

V lpay(d) =maxdF

{u(y d + ql[(1 )dl + dF ])+ [(1 )V lpay(dl)+ V hpay(dF )]}

The value functions in the following period are those associated with repayment and the level of debt is adjusted to itscentive compatible level.In case of default, the value function in both states is:

Vdef = u((1 )y)+ Vdef = u((1 )y)1

It makes no difference whether foreign interest rates are low or high if the country is excluded from internationalancial markets.I assume < ql , which implies that the country will choose to borrow and, if possible, increase current consumption atpense of future consumption. I focus on the steady state of the model, where consumption might depend on the state,t does not depend on time.

oposition 1. If < ql , an equilibrium with cht = ch and clt = cl for all t and some constants ch and cl such that ch cl implies thate country always borrows as much as it can, dF = dh in all periods and states.

oof. See Appendix A. As the country is borrowing as much as it can, the countrys debt is equal to its incentive compatible level in each future

ate: debt payment is dh if s = h and dl if s = l which is equivalent to what they would be in a world with completentingent contracts but no commitment to repay.As the value function in case of default does not depend on the state, V hpay(d

h) = Vdef = V lpay(dl). Consequently, con-mption is given by:

ch = cl = (1 )yDenote q = (qh + ql)/2. The following proposition establishes the value of d:

oposition 2. If dF = dh in all periods and states, d is given by:dh dl

dh= q

h ql1 ql + 2 q ,

dh dldl

= qh ql

1 qh + 2 q (1)

Arellano (2008) uses = 0.82/year, Aguiar and Gopinath (2006) and Yue (2010) use lower values. Such extreme impatience is often interpreted as thescount rate of the policy maker, different from the population.


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Moof. As dF = dh and V hpay(dh) = Vdef = V lpay(dl), the value functions in the high and low states are:

V hpay(dh)= u(y dh + qh[(1 )dh + dl])+ Vdef

V lpay(dl)= u(y dl + ql[(1 )dl + dh])+ Vdef

V hpay(dh) = V lpay(dl) implies:

y dh + qh[(1 )dh + dl]= y dl + ql[(1 )dl + dh]hich yields the claim. Debt relief depends on: (i) the magnitude of interest rate uctuations and (ii) the persistence of the interest rate process.the i.i.d. case, = 0.5, debt relief when the state switches from h to l is approximately equal to (qh ql). In the othertreme, as 0, debt reduction is much higher: (qh ql)/(1 ql), as it has to compensate for all the expected futuress brought on by the fall in q . Hence, higher persistence implies higher difference between dh and dl .The output cost has no effect on d/d. It is important to determine the level of d, but it does not inuence the ratiotween the incentive compatible levels of debt in both states.

2. Stochastic endowment

In this section, I x the world interest rates at r and allow y to uctuate between yh in the high state and yl in thew state, yh > yl .If dF (dl,dh], the value functions conditional on repayment are:

V hpay(d) =maxdF

{u(yh d + q[(1 )dF + dl])+ [(1 )V h(dF )+ V l(dl)]}

V lpay(d) =maxdF

{u(yl d + q[(1 )dl + dF ])+ [(1 )V l(dl)+ V h(dF )]}

Should the country go into default, the value functions are:

V hdef = u((1 )yh)+ [(1 )V hdef + V ldef ]

V ldef = u((1 )yl)+ [(1 )V ldef + V hdef ]

I assume < ql . An argument identical to Proposition 1 shows that in the steady state of the model, the countryhausts its borrowing possibilities, choosing dF = dh in all states and periods.The following proposition establishes the value of d:

oposition 3. If dF = dh in all periods and states, d is given by:dh dl

d= (1 q

)1 q(1 2)

yh yly

(2)

here d = (dh + dl)/2 and y = (yh + yl)/2.

oof. Consider the case y = yh . If the borrowing constraint is binding and debt is at its maximum level, then:

V hpay(dh)= u(yh dh + q[dl + (1 )dh])+ [(1 )V hdef + V ldef ]

aking V hpay(dh) = V hdef , we get:

dh q[dl + (1 )dh]= yh (3)Analogously:

dl q[dh + (1 )dl]= yl (4)Subtracting (4) from (3), we get:

(dh dl)[1 q(1 2)]= (yh yl) (5)


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chHothSumming (4) and (3) and manipulating, we get:

= dy

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Substituting the value of in (5), we get the claim. As before, larger uctuations and more persistent states imply higher debt relief and has no effect on d/d.

3. Contrasting stochastic q and stochastic y

In order to contrast debt relief in the cases of stochastic interest rates and endowments, we need to contrast the numer-ors of Eqs. (1) and (2), as the denominators are virtually the same. The key distinction is that the numerator of Eq. (1)the difference between interest rates in both states, while the numerator of Eq. (2) is the relative change in endowmentultiplied by (1 q), the present discounted interest rate. A reasonable range for the numerator of Eq. (1) (stochasticterest rates) is between 2% and 4%. On the other hand, a reasonable range for endowment uctuations is from 2% to, say,, which in combination with a range of average real interest rates from 1% to 3% gives a range for the numerator of Eq. (2)tochastic technology) of 0.02% to 0.2%. This is one or two orders of magnitude below what we get from uctuations inorld interest rates.Even permanent uctuations in output would not generate sizable debt relief. The effect of a permanent output fall onbt relief can be found by taking the limit 0. In that case, we get

dh dld

= yh yly

a permanent 2% fall in output leads to a fall of 2% in the incentive compatible level of debt.Fluctuations in q alter the cost of servicing debt. Shocks to y change the present value of output losses due to default.th these changes affect the incentive compatible level of debt. However, there is a great distinction in quantitative effectsshocks to y and q .Shocks to y affect the economy regardless of the decision on default. Defaulting on debt does not solve the problem,es not increase the endowment. Low output increases the incentives for default only because the punishment for defaultassumed to be proportional to output. Therefore, in order to generate a fall in the incentive compatible level of debt of, the expected present value of output (considering the present and all future periods) needs to decline by x%. But thatnnot be more than a couple of percentage points.18

On the other hand, if the country decides not to repay and stops interacting with world nancial markets, q has nofect in the economy. In this sense, default solves the problem of high interest rates. Importantly, while percentageanges in output are relatively small, we observe very large percentage changes in real interest rates (a switch from 1% tois an increase of 100%), which lead to large impacts on the incentive compatible level of debt. As an illustration, if it is

centive compatible to repay $1 a year, at a permanent interest rate of 1% this implies the country can sustain a debt of00. If interest rates are constant and equal to 2%, that implies an incentive compatible level of debt of $50. How large ise debt reduction generated by observed shocks to world interest rates? Section 4 provides the answer in the case of therge interest rate hikes of the early 1980s.If is small, debt repayments are a small proportion of the present value of output. Thus shocks to world interest

tes will not have large effects on Vpay . However, the incentive compatible level of debt might be signicantly affectedcause the effect of world interest rate shocks on Vdef is zero. In contrast, shocks to the present level of output may havelarge effect on Vpay , but because they affect the country regardless of decisions on debt, they do not play a key role intermining the incentives for defaulting. Therefore, the results of this paper point to the importance of shocks that haverge impacts in the economy only (or mostly) if the country chooses to keep interacting with world nancial markets orly (or mostly) if the country chooses to default on its debt. By affecting only Vpay or only Vdef , those shocks providerger incentives to switch to the other value function.In the model of sovereign default and debt renegotiation in Yue (2010), the recovery rate depends on the bargainingwer of the borrower, [0,1]. As in this paper, if = 1, the haircut is 100%, by construction. This paper corresponds toe case = 0, the haircut is given solely by the fall in the incentive compatible level of debt, which is close to 0 in the casean output shock. Yue (2010) uses = 0.72, which means that the borrower gets 72% of the total surplus and, indeed, thetained haircut is 73%. That is not inconsistent with a very small fall in the incentive compatible level of debt.In a setup with renegotiation, a key determinant of the recovery rate is the bargaining power . If is high, any shockat triggers a renegotiation process can lead to large debt reductions. But if output shocks could trigger a renegotiation

In Aguiar and Gopinath (2006) and Yue (2010), the loss of output in autarky is multiplicative, exactly like here. Productivity shocks may lead to largeranges in the incentive compatible level of debt if the costs of defaulting are smaller at the low state. That is an implicit assumption in Arellano (2008).wever, if that is the case, uctuations in the cost of defaulting are playing the main role not uctuations in output and a better understanding ofe costs of defaulting is needed before we can accept productivity shocks as important drivers of default.


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bovaocess, so would any shock that generates a fall of a few percentage points in the incentive compatible level of debt cluding mild changes in world interest rates.

Debt and default in a growth model

In this section, I introduce capital accumulation in the model. The domestic country borrows because its marginal productcapital is higher, not because of risk sharing or impatience. The assumption of a low is replaced with = q . In these of stochastic world interest rates, q uctuates around .The main conclusion is that the impact of output and interest-rate shocks on the incentive compatible level of debt arery similar to their impact in the endowment economy.The model follows the structure of the previous section, but with capital, kt , depreciating at rate and output a functionlely of the level of capital, as labour is normalised to 1:

yt ={At . f (kt), if it has never defaultedAt(1 ). f (kt), if it has ever defaulted

The economys ow budget constraint is then given by:

ct + kt+1 ={At . f (kt) + (1 )kt dt + qtdt+1, if it has never defaultedAt(1 ). f (kt) + (1 )kt , if it has ever defaulted

. Deterministic model

The deterministic version of the model with capital accumulation serves two purposes: it sheds light on the relationshiptween growth and capital ows, and helps to understand the results of the stochastic model.19

The value functions of the deterministic model are given by:

V (k,d) =max{Vpay(k,d), Vdef (k, )}d:

Vpay(k,d) =maxk,d

{u(A f (k) + (1 )k k d + qd)+ V (k,d)}

Vdef (k, ) =maxk{u((1 )A f (k) + (1 )k k)+ Vdef (k)}

I assume that decisions about k and d are made simultaneously and lenders can observe k before taking their lendingcisions (or condition their decisions on k). As noted by Cohen and Sachs (1986), the country would otherwise have ancentive to borrow d but then invest less, consume more and default on its debt.20The following results hold in an equilibrium with no uncertainty:

1. q = q , a constant. As there is no uncertainty, q = q if the country will repay and q = 0 otherwise. The choice d = 0 isstrictly better than any choice d such that q = 0 because that yields the same amount of consumption today and moreproduction next period (by avoiding the A f (k) output loss). The no-default condition is Vpay(k,d) Vdef (k, ).

. dmax(k, ) is the maximum level of incentive compatible debt and is increasing in . Since by differentiating the valuefunction we obtain that Vpay is decreasing in d and Vdef is decreasing in , Vpay(k,dmax) = Vdef (k, ). Thus, an increasein implies an increase in dmax(k, ).

. If k is below the steady state level of capital, k , then d = dmax. In the steady state, k = k = k and the marginalproductivity of capital, mpk = A f (k) , equals the marginal cost of renting an extra unit of capital, r . In thiscase, the country has no incentive to change the level of its debt, because capital is at the optimal level and smoothconsumption can be achieved by always choosing d = d. In contrast, if k < k , mpk > r and d cannot be smaller thandmax, otherwise the no-default condition would not bind, so the country could borrow an extra unit at r , invest it andobtain a greater return than r next period.

The study of external debt and default is closely related to the question of why capital does not ow from rich to poor countries. One proposedplanation is that the risk of default prevents larger capital inows in emerging economies (Reinhart and Rogoff, 2004 and Reinhart et al., 2003). Alternativeplanations emphasise differences in productivity (Lucas, 1990) and question whether the marginal productivity of capital is really higher in poor countriesaselli and Feyrer, 2007). Yet, most of the recent work on debt and default building on Eaton and Gersovitz (1981) focuses on risk sharing. Cohen andchs (1986) present a growth model in which debt is repaid only if it is incentive compatible to do so, but assume a linear production function andve no uncertainty. They also analyse a numerical example with decreasing returns to capital which is essentially the deterministic model of this section.arcet and Marimon (1992) and Kehoe and Perri (2004) also study economies with capital accumulation.

To see this, note that in the optimal plan Vpay(k,d) = Vdef (k, ) and u(c) = Vpayk (k,d). But Vpayk (k,d) >Vdefk (k

), so if the country has alreadyrrowed d and hasnt committed to k , a marginal decrease in k leads to an increase in todays utility that is bigger than the decrease in tomorrowslue. This type of moral hazard problem is studied by Atkeson (1991).


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eq4. If = 0, no debt can be sustained. If = 0, any positive discounted stream of repayment is worse than defaulting, sothe maximum incentive compatible positive discounted stream of repayment is zero.21

5. If = 1, we obtain full commitment. If = 1 with no default, in the steady state, the country obtains consumptionequal to A f (k) k rd which is positive because d k , A f (k) = r and f (k) > k f (k). Hence the rstbest, full commitment equilibrium is incentive compatible.

Using these results, I now derive the path of debt in the neighbourhood of = 0. I will detail two observations that areed to derive it.First, consider kp and d such that Vpay(kp,d) = Vdef (kp, ) and kp k . Then there exists some k kp and d such thate country is indifferent between repaying and choosing (kp,d) and defaulting and choosing (kd). The value functionsill be equivalent in this case and are given by:

Vpay(k,d) =maxkp ,d

{u(y + (1 )k kp d + qd

)+ Vpay(kp,d)}Vdef (k, ) =max

kd

{u((1 )y + (1 )k kd

)+ Vdef (kd, )}Second, from result 4, we know that the value functions when = d = d = 0 are identical and have a common optimal

vel of capital:

V0(k) =maxk{u(y + (1 )k k)+ Vdef (k,0)}

=maxk{u(y + (1 )k k)+ Vpay(k,0)}

Then, from result 3, we always have d = dmax and therefore the no default condition will always bind: Vpay(k,dmax) =def (k, ). Using the rst observation, we rewrite these value functions in the form above. Then by taking a linear ap-oximation of Vpay(k,d) and Vdef (k, ) around V0(k) and manipulating the linearised expressions, we can nd d and at equate Vpay(k,d) and Vdef (k, ) without solving for the value functions. The expression for d turns out to be a goodproximation for its true value if is not more than a few percentage points. This is because when 0, the optimaloice of capital is independent of the decision about defaulting which allows us to get the analytical results. As movesay from 0, that is no longer true, however the impact on the value function of reoptimising the level of capital due to aor 2% fall in productivity is very small, and so is its impact on the maximum incentive compatible level of debt.The results do not depend on the functional forms of utility or production, which have only second order effects.

oposition 4. In this economy, for a very small :

1. In steady state:

d = y

1 q (6)

2. For yt < y:

dt+1yt

= 1 q +

dt+1(1 q)yt

and:

dt = q.dt+1 + (1 q) yt1 q

oof. See Appendix A. Part (1) of the proposition shows that in the steady state, the country keeps repaying its debt if the interest payment,

(1 q), is not greater than the output loss, y , due to default. The debt as proportion of GDP is, to a rst orderproximation, equal to /(1q). Positive debt with no uncertainty arises in equilibrium to nance convergence. If = 1%d q = 0.98 (r 2%), the debt-GDP ratio is 50%.The level of debt is proportional to the output loss and inversely proportional to the risk-free interest rate. Note that aange in interest rate from 1% to 2% has the same impact on d as a 50% decrease in GDP. The intuition for the different

This result is due to the absence of uncertainty in the model. It has already been shown in the literature that, with uncertainty, there may be debt inuilibrium even in the absence of output costs (Eaton and Gersovitz, 1981).


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wchpacts of A and q on the incentive compatible level of debt (Eq. (6)) is similar to the reasons for their distinct effects ofocks in Section 2 (Eqs. (1) and (2)).Part (2) of the proposition shows that for yt < y , the condition for default reduces to a comparison between output

sses and resources paid to foreign agents in the present period. But the increase in debt is endogenously determined bynsidering that the country will be indifferent in the next period between repaying and defaulting so, ultimately, debtperiod t is obtained by backward induction from the steady state level of debt.For yt < y , the absolute level of debt is increasing over time because the present value of the output loss due to defaultincreasing as output rises to its steady state level. The debt-GDP ratio is decreasing over time, because positive capitalows generate greater incentive for the country to repay, and capital inows are decreasing over time.The proposition also shows that in equilibrium the country must experience net outows of resources on the path ofnvergence. Debt is increasing (nancial account is in surplus) but the increase is smaller than the interest paid on itsbt. So, even though the current account is in decit, the country is a net exporter of goods.In Appendix B, I present a numerical example that conrms the analytical expression is a good approximation for the

sults and illustrates convergence in this economy.

2. Stochastic world interest rates

Here, I analyse an economy with xed technology, A, and uctuations in world interest rates, r , which lead to uctua-ns in the price of risk-free debt, q . Except for capital accumulation and the assumption that qh and ql are close to ,e model is identical to that of Section 2.1.As in Section 2, renegotiation restores debt to its incentive compatible level. Denote by dhand dl the incentive compatible

vel of debt in the high and low states, respectively, such that V hpay(k,dh) = V hdef (k) and V lpay(dl,k) = V ldef (k). If the face

lue of debt dF dl , the value functions are:

V hpay(k,d) =maxk,dF

{u(ch)+ [(1 )V hpay(k,dF )+ V lpay(k,dF )]}

V lpay(k,d) =maxk,dF

{u(cl)+ [(1 )V lpay(k,dF )+ V hpay(k,dF )]}

here ci = A f (k) + (1 )k k d + qidF for i {l,h}.If dF (dl,dh], then dl is repayed if s = l and dF is repayed if s = h. The value functions associated with repayment are:


{u(ch)+ [(1 )V hpay(k,dF )+ V lpay(k,dl)]}


{u(cl)+ [(1 )V lpay(k,dl)+ V hpay(k,dF )]}

here ch = A f (k) + (1 )k k d + qh[(1 )dF + dl] and cl = A f (k) + (1 )k k d + ql[(1 )dl + dF ].In the event of default, the value function in both states is:

Vdef (k, ) =maxk{u((1 )A f (k) + (1 )k k)+ Vdef (k, )}

Dene mpk = A f (k) and ri = 1/qi 1. If the marginal productivity of capital mpk is larger that the interest ratesd qh and ql are arbitrarily close to , then the country borrows as much as it can, the face value of debt dF is dh . Thext proposition formalises that.

oposition 5. If mpk > ri , qh ql is arbitrarily small, and qh and ql are arbitrarily close to , then dF = dh.

oof. See Appendix A. If the countrys borrowing constraint is binding, d = dh dl . In order to expand its borrowing possibilities, a country

ith a high marginal productivity of capital chooses to make debt in each of the future states as high as possible, respectinge incentive compatibility constraints. Debt repayment in each state, dh and dl , are such that V hpay(k,d

h) = V lpay(k,dl) =def (k).When the borrowing constraint is binding, the problem of the country is analogous to choosing debt in each state dh and or to choosing the expected debt repayment d and debt reduction d subject to incentive compatibility constraints.

2.1. The value of dFor analytical convenience, I consider that dq = qh ql is suciently small and work with linear approximations,

hich implies we are not considering the effects on the value function of reoptimising the choice of k when the countryanges state.


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stIn addition, I temporarily consider an alternative process for q that I denote by the -process, as opposed to the -ocess that we described above. At time t = 0, q0 = q ; from time t = 1 on, there is a constant probability at each periodat q permanently goes to q. So, for t > 0:

if qt1 = q , Pr(qt = q ) = and Pr(qt = q) = 1 ; if qt1 = q, qt = q.

The value function at (k,d) if qt = q is:

V (k,d,q

)= maxk,d,d

{u(c) + [(1 )V det(k,d)+ V (k,d ,q )]}

here c = A f (k)+ (1 )k k d+ q (d(1 )+ d ) and V det is the value function in the model with no uncertainty.The -process and the -process are related using the following lemma:

mma 6. Dene q = (qh + ql)/2 and denote by V h(k,d,qh) the value function for the -process. Then V h(k,d,qh) = (k,d,qh) if = 1 2 .

oof. See Appendix A. Compare the following two cases when q follows the -process: (1) q = q and debt is d0 and (2) q = q and debt is. We want to nd the values of d0 and d0 that make the country indifferent between both cases in order to determined using Proposition 5. By taking a linear approximation of V (k,d ,q ) around V det(k,d0) and using the indifferencendition that V (k,d ,q ) = V det(k,d0), we get the following lemma:

mma 7. Indifference between both states, V (k,d ,q ) = V det(k,d0), implies:

u(c0)(d0 d0

)(7)

=t=0

()tu(ct)(q q)dt+1 +

t=0()t

[u(ct)q + V (kt+1,dt+1)

d

](dt+1 dt+1

)

here dt+1 is debt contracted at time t if qt = q and dt+1 is debt contracted at time t if qt = q.

oof. See Appendix A. Suppose that q > q. The rst line in the above expression shows the utility cost of having higher debt. The second lineows the utility benet of borrowing at a lower rate, taking into account the probability of cheaper borrowing in futureriods, plus the benet of being able to borrow more due to lower interest rates. If the borrowing constraint is binding,en

u(ct)q > V (kt+1,dt+1)d

hich means that the benet of borrowing an extra unit this period is greater than the cost of holding an extra unit of debtxt period.From Lemma 7, we can write:

d0 d0d0

>

t=0

()tu(ct)u(c0)

(q q)dt+1

d0

It is convenient to consider Eq. (7) as we approach the steady state of the deterministic economy. Formally, rst I takee limit of small shocks (qh ql 0+) and then I consider the limit of small differences in the marginal product ofpital (mpk r 0+).In the limit of small shocks, as k approaches k , the borrowing constraint stops binding, ct and dt approach their steady

ate, constant, values, and we get:

d0 d0d0

= q q

1 (8)

And that leads to the following proposition:


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woposition 8. Consider a deterministic steady state around, k, d and q, such that qh and ql are close to q = and q = (qh +ql)/2.en, a linear approximation around the steady state will satisfy V (k,dh,qh) = V (k,dl,ql) when:

dh dld

= qh ql

1 q(1 2) (9)

here d = (dh + dl)/2.

oof. See Appendix A. Eq. (9) is very similar to Eq. (1), and for small uctuations of q , they are exactly the same. There are two differencestween the case with capital accumulation and the endowment economy. One is that, with capital accumulation, thetimal decision on consumption and savings depends on the state, which inuences the value function. But because of thevelope theorem, around the point of maximum, those differences in the choice of k have only second order effects on thelue function. It follows that they have no rst order impact on the incentive compatible level of debt.The second difference is that growth prospects have an inuence on the incentive compatible level of debt. Proposition 8ows that in the steady state, debt relief for economies with and without capital coincide. But Lemma 7 shows that,tside the steady state, d/d depends not only on the magnitude of interest rate uctuations and the persistence of theterest rate process, but also on the current level of capital and its marginal productivity. The lower is the level of capital,e greater are the marginal productivity of capital and the difference between u(c)q and V (k,d)

d , which contribute tocrease d: a switch to the low state that prevents the country from borrowing is more punitive when capital is lower.lower level of capital also implies lower consumption and, therefore, higher marginal utilities, so present consumptionmore important and higher costs of borrowing in the future are less relevant, which induces a decrease in d. Lastly,higher ratio between future and present debt increases the importance of future costs of borrowing, which induces ancrease in d. Thus the overall effect cannot be deduced from the formula. In the numerical examples, d is slightlycreasing in k, implying the effect of the borrowing constraint predominates.The analysis has focused on the two-state case, but the same insights apply if we consider more general processes. Thext proposition considers the case of an auto-regressive process for q .

oposition 9. Suppose that q follows an AR(1) process:

qt+1 q = (qt q

)+ t+1d Var(t) is arbitrarily small. If the economy is close to its steady state (k k), V (k,d1,q1) = V (k,d2,q2) for any {q1,q2}se to q and {d1,d2} when:

d1 d2d

= q1 q21

here d is the level of debt in the deterministic model when q = q.

oof. See Appendix A. 3. Stochastic technology

In this section, I consider xing the world interest rates at r and allowing for uctuations in A. Productivity is Ah ine high state and Al in the low state, Ah > Al . It is assumed q = . If the face value of debt dF dl , there is no debtduction and the value functions are:


{u(ch)+ [(1 )V hpay(k,dF )+ V lpay(k,dF )]}


{u(cl)+ [(1 )V lpay(k,dF )+ V hpay(k,dF )]}

here ci = Ai f (k) + (1 )k k d + qdF for i {l,h}. If dF (dl,dh], then dl is repayed if s = l and dF is repayed if= h. The value functions associated with repayment are:


{u(ch)+ [(1 )V hpay(k,dF )+ V lpay(k,dl)]}


{u(cl)+ [(1 )V lpay(k,dl)+ V hpay(k,dF )]}

here ch = Ah f (k) + (1 )k k d + q[(1 )dF + dl] and cl = Al f (k) + (1 )k k d + q[(1 )dl + dF ].


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loamIn case of default, the value functions are:

V hdef (k, ) =maxk{u((1 )Ah f (k) + (1 )k k)+ [(1 )V hdef (k, )+ V ldef (k, )]}

V ldef (k, ) =maxk{u((1 )Al f (k) + (1 )k k)+ [(1 )V ldef (k, )+ V hdef (k, )]}

As before, if the marginal productivity of capital is higher than the interest rates (mpk > r) and uctuations (Ah Al) areall, the country borrows up to a debt limit in each possible future state that ensures it remains incentive compatible forto repay. The proof is analogous to the proof of Proposition 5. The face value of debt dF is equal to dh , and V hpay(k

,dh) =hdef (k

, ) and V lpay(k,dl) = V ldef (k, ).As before, we need to obtain an expression for d. The analogy to Proposition 8 for the case of stochastic technology

quires the additional assumption that is arbitrarily small and yields the following result:

oposition 10. Consider a deterministic steady state, {k, d, A}, such that Ah and Al are close to A = (Ah + Al)/2. Then, for arbitrarilyall , a linear approximation around the steady state will satisfy V hpay(k

,dh) = V hdef (k, ) and V lpay(k,dl) = V ldef (k, ) when:

dh dld

= (1 q)

1 q(1 2)Ah Al

A(10)

here d = (dh + dl)/2.

oof. See Appendix A. Eq. (10) is exactly the same as Eq. (2).

The Latin American debt crisis of the 1980s

In this section, I contrast the predictions of the model with data from the Latin American debt crisis of the 1980s. Thesults show that the interest rate shock at the beginning of the 1980s can account for a large part of the observed debtlief.

1. Observed debt relief

External shocks were important factors in the Latin American debt crisis of the 1980s. As noted by Diaz-Alejandro984), countries with different policies and distinct economies ended up in similar crises in the beginning of the 1980s,cing problems that in 1979 would have been considered unlikely.One key external shock was the increase in US real interest rates, shown in Fig. 1 (from Dotsey et al., 2003). In contrastthe 1970s when real interest rates were around 0%, in the 1980s they were around 4%. Such large increase in US realterest rates makes the Latin American crisis a convenient case to evaluate the models predictions.22

In the beginning of the 1980s, the prices of Latin American bonds in secondary markets collapsed, capital ows to thoseonomies dried or reverted and the fast process of economic growth of the 1970s stopped. After countless IMF missions,veral debt reschedules and some attempts of debt renegotiation (including the Baker Plan), came the Brady agreements,arting in 1989. In the period between 1989 and 1994, most of the main Latin American countries got some debt relief.ble 1 shows the debt relief following the Brady Plan agreements as a percentage of the outstanding long term debt in theain Latin American countries. With the exception of Venezuela, at 20%, the other four countries are around 30%.23

As the Brady agreements did not cover all forms of external debt, the gures in Table 1 should be seen as upper boundsline, 1995).In the model, renegotiation is costless, instantaneous and the borrower extracts no surplus from the process. In reality,bt relief came ten years after the shock, and what happened in those ten years had some inuence on the nal agreement.espite the delay, it is worth comparing the debt relief prescribed by the model and the Brady agreements because the latterere in fact the relief solution to the crisis.

There are other cases in which interest rate increases in the US contributed to crises in other countries. For example, the sharp increase in US interesttes in 1994 is sometimes mentioned as one of the factors that almost led Mexico to default in December 1994 (see Calvo et al., 1996).As noted by Cline (1995), the initial approach for dealing with the problem of debt overhang was aimed both at reducing debt and providing new

ans, but for practical purposes the Brady Plan has been all forgiveness and no new money (Cline, 1995, p. 236). Indeed, according to the model, if theount of debt exceeds its incentive compatible level, new money will not be made available.


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HFig. 1. US real interest rates.

Table 1Debt relief Brady plan agreements,Cline (1995)

Venezuela 20%Brazil 28%Argentina 29%Mexico 30%Uruguay 31%

2. Debt relief according to the model

If the borrowing constraints of the Latin American countries were binding in 1979, then the interest rate rise at theginning of the 1980s would bring debt, d, above its incentive compatible level. In this section, I compute the debtduction prescribed by the model and compare it to the data.Consider the model calibrated to represent the 1970s and 1980s. Suppose that world interest rates may be either 0% ora year and that each state lasts for an average of 10 years: qh = 1.00, ql = 1.041, = 1.021 and = 0.10. Eq. (9)

elds the debt reduction given a switch from the high state to the low:

d

d= 1.00 1.04

1

1 1.021(1 2 0.10) = 0.178

That implies a spread over treasury of d/d = 1.8% when the state is high but debt relief of 18% when the stateitches to low and interest rates jump from 0% to 4%. Data from 19731980 show a spread over treasury of 1.4% for Brazild Argentina and 1.1% for Mexico (Calvo and Kaminsky, 1991).d/d is not signicantly affected by the length of debt contracts. If we work with a period of 5 years instead of onear, qh = 1.00, ql = 1.045, = 1.025 and = 0.50, we still obtain (dh dl)/d = 0.178.The result holds for any > 0 and is robust to other interest rates processes. Using the auto-regressive process and

suming a half-life of 3 years for the interest rate increase, the AR-1 coecient, , would be 0.79. A jump in real interesttes from 1% a year to 6% a year would then imply even greater debt relief: d/d = 22.5%.In sum, the decrease in the level of debt predicted by the model in response to an interest rate increase of the magnitudeserved in the data exceeds half the debt relief of the Brady agreements. In contrast, a negative productivity shock wouldt generate results of similar magnitude. A huge 10% reduction in productivity, assuming an average persistence of 10ars ( = 0.10) and world interest rates of 2% a year would imply debt reduction slightly below 1% according to Eq. (10).Numerical results, presented in Appendix B, show that lower marginal productivity of capital (combined with low ad-

stment costs for capital) reduces the amount of debt relief prescribed by the model, but conrm they are quantitativelyportant.

Concluding remarks

Recent quantitative models of sovereign debt aim at explaining the recent debt crisis in Argentina using output shocks.owever, according to the model in this paper, output uctuations do not have a sizable effect on the incentive compatible


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anshobvel of debt nothing remotely close to the observed debt reduction of 71%.24 On the other hand, uctuations in worldterest rates can have a strong impact on the incentive compatible level of sovereign debt.Given the costs of renegotiating debt, should countries issue debt contingent on world real interest rates? In the casethe Latin American debt crisis of the 1980s, such contracts could have avoided 10 years of costly bargaining. However,hile the real interest rate shock of 1980 can be seen as a policy decision, other movements on world real interest ratesight be correlated with variables that affect the incentive compatible level of debt, in the opposite direction. For example,e nancial crises of 2008/2009 led to lower real interest rates but capital outows from emerging economies. Indeed,ley-Fisher and Guimaraes (2009) nd that policy-induced increases in US rates raise default risk in emerging markets, bute overall correlation between default risk and US real interest rates is actually not positive.Incentives for defaulting on debt depend not on how much a shock affects the country, but on the differential effects of aock if the country repays and if the country defaults. Output shocks affect a country regardless of its decisions on default.contrast, world interest rate shocks affect the country only if it keeps interacting with international nancial markets, sove a much stronger effect on the incentive compatible level of debt. Other shocks that have an impact on internationalancial markets could lead to similar effects for instance, by affecting investors risk appetite. Fluctuations that affecte country only if it defaults, such as shocks that inuence the costs of defaulting, could also have large effects on thecentive compatible level of debt.25

ppendix A. Proofs

While proving the propositions of the model with capital accumulation, I often use the following method and I refer toas a rst order Taylor approximation. Consider the following Value function:

V (x, y) =maxa,b,c

{u(x, y,a,b, c) + V (x(a,b, c), y(a,b, c))}

So the values of a,b, c are chosen, which determine the next periods x, y (denoted as a convention by x, y). Take thenction that is to be maximised:

f (x, y,a,b, c) = u(x, y,a,b, c) + V (x(a,b, c), y(a,b, c))Denote by a , b and c the maximising values of V (x, y) and by a , b and c the maximising values of V (x, y). Then

(x, y) = f (x, y,a,b, c) and V (x, y) = f (x, y, a, b, c). Now, if (x, y) and (x, y) are suciently close, we can take theproximation of the maximand function with respect to the variables to be chosen:

f(x, y,a,b, c

) f (x, y, a, b, c)+ z=x,y,a,b,c

f (x, y, a, b, c)z

(z z)

= V (x, y) +

z=x,y,a,b,c

(u(x, y, a, b, c)

z+ V (x

, y)z

)(z z)

If there is a binding constraint on the possible values of a variable, then its maximised value will be determined by thenstraint. Otherwise, the envelope theorem applies and the derivative of f with respect to that variable will be zero.This method is different from the general rst order Taylor approximation: the original function V (x, y) is not a functionthe variables with respect to which the approximation is done. This is why the maximand function has to be dened.

1. Proposition 1

oof. Suppose qF < ql . Then the rst order condition with respect to dF would have to hold with equality in both states:

V h

dF= 0 (qh (1 ))u(ch)= u(cl)

V l

dF= 0 (ql (1 ))u(cl)= u(ch)

qh > implies qh (1 ) > , so the rst order condition implies u(ch) < u(cl). But similarly, ql > impliesl (1 ) > , so the second rst order condition implies u(cl) < u(ch). Contradiction.

The negative correlation between probability of default and output does not tell us about causality, because a high probability of default has a negativepact on output of emerging economies through its impact on interest rates (Neumeyer and Perri, 2005).Foley-Fisher (2008) applies the method developed in this paper to study the effect of terms of trade shocks on the incentive compatible level of debt,

d contrasts the results to data on debt relief for the highly indebted poor countries. In his model, countries in default are subject to trade sanctions, soocks to the terms of trade have a large effect on the cost of defaulting. His results show that terms of trade shocks can explain a signicant part of theserved debt relief.


st

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arre

V

pu

(d

inNow suppose qF (ql,qh). Again, the rst order condition with respect to dF would have to hold with equality in bothates:

V h

dF= 0 u(ch)qh(1 ) = (1 )u(ch)

V l

dF= 0 u(cl)ql = u(ch)

hich contradicts the assumptions that < ql and cl ch . 2. Proposition 4

oof. The value functions in case of repayment and in case of default are maximised at (kp,dp) and (kd) respectively,hich means that:

Vpay(k,d) = u(y + (1 )k kp d + qdp

)+ Vpay(kp,dp)Vdef (k, ) = u

(y + (1 )k kd y

)+ Vdef (kd, )When d = d = 0 and = 0, the value functions are identical in the two cases, V0(k). It is maximised by choosing k = ko .Consider (k,d, ) such that the country is indifferent between repaying and choosing (kp,d) or defaulting and choosing

d), which means that Vpay(k,d) = Vdef (k, ). Approximate the functions that are to be maximised,

fpay(d,k,d

)= u(y + (1 )k k d + qd)+ Vpay(k,d)fdef

(k,

)= u(y + (1 )k k y)+ Vdef (k, )ound V0(k). The rst order Taylor approximation of these functions around V0(k) with respect to (k,d,d) or (k, )spectively yields

fpay(d,k,d

)= Vpay(k,0) + fpay(0,ko,0)d

(d 0) + fpay(0,ko,0)

d(d 0)

+ fpay(0,ko,0)

k(k ko

)+ Ok,p= V0(k) + u(co)

(d + qd)+ Vpay(ko,0)d

d + Ok,p

fdef(k,

)= Vdef (k,0) + fdef (ko,0)

( 0) + fdef (ko,0)

k(k ko

)+ Ok,d= V0(k) u(co) y + Vdef (k

o,0)

+ Ok,d

Where I used that since Vpayk =

Vdefk = 0 due to the Envelope condition for the unconstrained maximisation of Vpay and

def with respect to k , their evaluationVpay(k,0)

k =Vdef (k,0)

k = 0 as well. The optimal consumption with no borrowing, nonishment is denoted by co = y+ (1 )kko . Furthermore, lim(d,d,k)(0,0,ko)

Ok,pd,d,k2 = 0 and lim( ,k)(0,ko)

Ok,p ,k2 = 0.

Note that Vpay(k,d) = Vdef (k, ) fpay(d,kp,d) = fdef (kd, ). Using the rst order Taylor expansions at points,kp,d) and (kd, ), we get:

u(co)(d + qd + y)+ (Vpay(ko,0)

dd Vdef (k

o,0)

)+ (Ok,p Ok,d) = 0

The last part is to show that Vpay(ko,0)

d d Vdef (ko,0)

is approximately zero.If k < k then the country borrows the maximum level of incentive compatible debt. This debt level makes the country

different between repaying and defaulting at (kp,d):

Vpay(kp,d

)= Vdef (kp, )First order Taylor approximations of Vpay(kp,d) and Vdef (kp, ) around Vo(ko) with respect to only k , d and yield:


u

A.

Prho

w

w

fo

A.

LeorVpay(kp,d

)= V0(kp)+ Vpay(ko,0)k .(kp ko

)+ Vpay(ko,0)d

.d + Ok(d2)

Vdef(kp,

)= V0(kp)+ Vdef (ko,0)k .(kp ko

)+ Vdef (ko,0)

. + Ok( 2)

Vpay(kp,d) = Vdef (kp, ) implies:

Vpay(kp,0)d

.d Vdef (kp,0)

. 0

Using this last equation the difference of the original Taylor expansions simplies to:

u(co)(d + qd + y)+ (Ok,p Ok,d) 0

(co) = 0, and (Ok,p Ok,d) is very small near (d,d,k) = (0,0,kp) and ( ,k) = (0,kd) imply that

d + qd + y = 0 d = qd + yWhich yields the second part of the claim. In steady state, d = d , and we get the rst part of the claim.

3. Proposition 5

oof. First, consider s = h and dF is such that dl < dF < dh . The rst order condition with respect to dF would have told with equality. The rst order conditions with respect to k and dF are:

u(cht)= (A f (k)+ 1 )[(1 )u(cht+1)+ u(clt+1)]

u(cht)qh(1 ) = (1 )u(cht+1)

Combining both yields:

1= qh(A f (k)+ 1 )(1 [1 u(clt+1)u(cht+1)

])

As clt+1 and cht+1 are similar, that implies:

1

qh 1 A f (k)

hich contradicts the assumption that mpk > rh .If s = l and dF is such that dl < dF < dh , a similar procedure leads to

1= ql(A f (k)+ 1 )(1 (1 )[1 u(clt+1)u(cht+1)

])

hich also contradicts the assumption that mpk > rh .Last, if dF < dl , combining the rst order conditions with respect to dF and k yields:

1

qi 1= A f (k)

r i {l,h}, which is a contradiction. 4. Lemma 6

Before proving Lemma 6, we need an auxiliary result:

mma 11. Consider the model with 2 states, h and l, and probability of changing state equal to . Dene q = (qh + ql)/2. In a rstder approximation,

V (k,d, q) = [V (k,d,qh)+ V (k,d,ql)] 2


Pr

Pr1

Nefu

w

re

Asbe

Si

so

No

th

If

an

If

anoof. Comes directly from a Taylor expansion of V (k, d,qh) and V (k, d,ql) around V (k, d, q). We are ready to prove Lemma 6.

oof. First, we need to show that, close to the deterministic steady state (k = k,d = d), V (k, d,qh) = V (k, d,qh) if = 2 .

V (k,d,qh

)= maxk,d,d

{u(A f (k) + (1 )k k d + qhd)

+ [(1 )V det(k,d d)+ V (k,d + (1 )d,qh)]}ar the deterministic steady state, choosing the optimal (d,k) instead of (d, k) has only second order effect on the valuenction V . As a rst order approximation, we can write:

V (k,d,qh

)= u(ch)+ [(1 )V det(k,d d) + V (k,d + (1 )d,qh)]V det(k,d) = u(c) + V det(k,d)

here ch = A f (k) k d(1 qh) and c = A f (k) k d(1 q).Taking a rst order Taylor approximation of f (k,d,qh,d) = V (k,d,qh) around V det(k,d) (qh = q and d = 0) with

spect to q,d, we get:

V (k,d,qh

) u(c) + u(c)(qh q)d + (V det(k,d) + (1 )V det(k,d)d

()(d 0))

+ (

V (k,d,q)

d(1 )(d 0) + V

(k,d,q)

q

(qh q))

= u(c) + u(c)(qh q)d + V det(k,d) + (qh q) V (k,d,q)q

a simple rst order Taylor approximation, V (k,d,q)q (q

h q) = V (k,d,qh) V (k,d,q), so the above approximation canwritten as:

V (k,d,qh

)= u(c) + u(c)(qh q)d + V det(k,d) + [V (k,d,qh) V (k,d,q)]nce V (k,d,q) = V det(k,d) = u(c) + V det(k,d),

V (k,d,qh

) V det(k,d) = u(c)(qh q)d + [V (k,d,qh) V det(k,d)]

V (k,d,qh

) V det(k,d) = u(c)(qh q)d1 (11)

w, note that, as a rst order approximation:

V(k, d,qh

) V det(k, d) = u(c)(qh q)d + [(1 )V (k, d,qh)+ V (k, d,ql) V det(k, d)]= u(c)(qh q)d + (1 2)[V (k, d,qh) V det(k, d)]

e last equality follows from Lemma 11. Then:

V(k, d,qh

) V det(k, d) = u(c)(qh q)d1 (1 2) (12)

= 1 2 , Eqs. (11) and (12) imply that V (k, d,qh) = V (k, d,qh).Now, we complete the proof by induction. Away from the steady state, we have:

V (k,d,qh

)= u(A f (k) + (1 )k k d + qhd)+ [V (k,d,qh)+ (1 )V det(k,d)]d

V(k,d,qh

)= u(A f (k) + (1 )k k d + qhd)+ [(1 )V (k,d,qh)+ V det(k,d)]V (k,d,qh) = V (k,d,qh), then, using Lemma 11, we can write:

V(k,d,qh

)= u(A f (k) + (1 )k k d + qhd)+ [V (k,d,qh)+ (1 )V det(k,d)]d thus V (k,d,qh) = V (k,d,qh).


A.

Prva

ag

anV

w

Th

A5. Lemma 7

oof. Subscripts denote time (k0 is capital at time 0). The superscript for t > 0 means that at time t 1, when theriable (k or d) has been chosen, q = q .

V (k0,d

0,q

)= max

k1,d1,d

{u(A f (ko) + (1 )ko k1 d0 + qd1

)+ [(1 )V det(k1,d1 d)+ V (k1,d1 + (1 )d,q )]}

ain dene the function to be maximised:

f(k0,d

0,q

,k1,d1,d

)= u(A f (ko) + (1 )ko k1 d0 + qd1)+ [(1 )V det(k1,d1 d)+ V (k1,d1 + (1 )d,q )]

d take a Taylor approximation with respect to k1,d0,d1,q,d around q = q, d = d0, d = 0 that is when f () =det(k0,d0).

V (k0,d

0,q

)= f (k0,d0,q ,k1,d1,d) u(A f (ko) + (1 )ko k1 d0 + qd1)+ [(1 )V det(k1,d1) + V (k1,d1,q)]

+ f (k0,d0,q,k1,d1,0)k1

(k1 k1

)+ f (k0,d0,q,k1,d1,0)d0

(d0 d0

)+ f (k0,d0,q,k1,d1,0)

d1

(d1 d1

)+ f (k0,d0,q,k1,d1,0)q

(q q)

+ f (k0,d0,q,k1,d1,0)d

(d 0)= V det(k0,d0) +

[V (k1,d1,q) V det(k1,d1)

]+ u(c0)[(d0 d0)+ q(d1 d1)+ d1(q q)]+ V

det(k1,d1)

d1

{(1 )[(d1 d1) (d 0)]}

+ V (k1,d1,q)

d1

{[(d1 d1

)+ (1 )(d 0)]}+ V (k1,d1,q)q

(q q)

here I used that V det(k0,d0) = u(A f (ko) + (1 )ko k1 d0 + qd1) + V det(k1,d1).Note that as a rst order Taylor approximation:

V (k1,d1,q

)= V (k1,d1,q) + [(1 )V

det(k1,d1)

q+ V

(k1,d1,q)

q

](q q)

= V (k1,d1,q) + V (k1,d1,q)

q

(q q)

en q = q, V det(k1,d1) = V (k1,d1,q) and V det(k1,d1)d1 =V (k1,d1,q)

d1. Using these last equations I get:

V (k0,d

0,q

) V det(k0,d0) = u(c0)[(d0 d0)+ q(d1 d1)+ d1(q q)]+ V

(k1,d1,q)

d1

(d1 d1

)+ [V (k1,d1,q ) V det(k1,d1)]nalogously, for t > 0 and starting with equal initial debts at both states, we get:

V (kt ,dt,q

) V det(kt ,dt) = u(ct)[q(dt+1 dt+1)+ dt+1(q q)]+ V

(kt+1,dt+1,q)dt+1

(dt+1 dt+1

)+ [V (kt+1,dt+1,q ) V det(kt+1,dt+1)]


Re

Im

A.

Pr

An

Us

A.

PrcoV

w

Socursive substitution leads to

V (k0,d

0,q

) V det(k0,d0) = u(c0)(d0 d0)+

t=0

()t[u(ct)

[q(dt+1 dt+1

)+ dt+1(q q)]

+ Vdet(kt+1,dt+1)

dt+1(dt+1 dt+1

)]

posing V (k0,d0,q

) = V det(k0,d0) we get the claim. 6. Proposition 8

oof. Using Lemma 6 and Eq. (8):

V(k,dh,qh

)= V (k, d, q) V (k,dh,qh

)= V (k, d, q) dh d

d= q

h q1 (1 2)

alogously,

V(k,dl,ql

)= V (k, d, q) dl d

d= q

l q1 (1 2)

ing both equations, we get the claim. 7. Proposition 9

oof. Consider a Taylor approximation of V AR(k,di,qi) around the deterministic steady state (V det(k,d,q)). The borrowingnstraint is not binding, so choosing the optimal (d,k) instead of (k,d) has only second order effects on the value functionAR . As a rst order approximation, we can write:

V AR(k,di,qi

)= E(

t=0tu

(A f (kt) + (1 )kt kt+1 dit + qidit+1

))

t=0

tu(c) + E(

t=0tu(c)d

(qit q

)) u(c)(di0 d)

here c = A f (k) k (1 q)d.Looking at the middle part of this expression,

E

( t=0

tu(c)d(qit q

))

= u(c)d[(qi0 q)+ E(( (qi0 q)+ 1)+ 2( ( (qi0 q)+ 1)+ 2)+ )]= u(c)d

[(qi0 q

) t=0

( )t + E(

t=1tt

1

1

)]

=t=0

( )tu(c)d(qi0 q

)= u(c)d(qi0 q) 11 :

V AR(k,di,qi

) t=0

tu(c) + u(c)d(qi0 q) 11 u(c)(di0 d

)


N

A.

Pr

w

w

w

wde

wow imposing V AR(k,d1,q1) = V AR(k,d2,q2), we get:t=0

tu(c) + u(c)d(q1 q) 11 u

(c)(d1 d)

=t=0

tu(c) + u(c)d(q2 q) 11 u

(c)(d2 d)

(d2 d) (d1 d)= 11 d

((q2 q) (q1 q))

d2 d1 = 11 d

(q2 q1)

d2 d1d

= q2 q11

8. Proposition 10

oof. Consider that productivity follows the -process, so that A0 = A and for t > 0:

if At1 = A , Pr(A = A ) = and Pr(A = A) = 1 ; if At1 = A, At = A.

The value function at (k,d) if At = A is:

V (k,d, A

)= maxk,d,d

{u(c) + [(1 )V det(k,d)+ V (k,d , A )]}

here c = A f (k)+ (1 )k k d+ q(d(1 )+ d ) and V det is the value function in the model with no uncertainty.A Taylor approximation of V (k,d, A ) around the deterministic steady state (V det(k,d)) yields:

V pay(k,d, A

)= u(c) + u(c)(A A) f (k) + (1 )V det(k,d) + V (k,d, A )here c = A f (k) k (1 q)d.So:

V pay(k,d, A

) V det(k,d) = u(c)(A A) f (k) + [V pay(k,d, A ) V det(k,d)]hich yields:

V pay(k,d, A

)= V det(k,d) + u(c)(A A) f (k)1

If d is close to d, V pay(k,d , A ) can be written as:

V pay(k,d , A

)= V det(k,d) + u(c)(A A) f (k)1 u

(c)(d d) (13)

Out of the equilibrium path, the value function conditional on default is:

V def(k, , A

)=maxk{u(c) + [(1 )V detdef (k, )+ V def (k, , A )]}

here c = (1 )A f (k) + (1 )k k and V detdef is the value function in the model with no uncertainty if the countrycides to default.A Taylor approximation of V def (k, , A

) around the deterministic steady state (V detdef (k, )) yields:

V def(k, , A

)= u(cd) + u(cd)(A A)(1 ) f (k) + (1 )V detdef (k, ) + V def (k, , A )here cd = (1 ) A f (k) k, which yields:

V def(k, , A

)= V detdef (k, ) + u(cd)(1 )(A A) f (k)1 (14)


an

Var

w

at

Ap

B.1

an

ansi

thstslunou60

InthofsitoinUsing an argument similar to Lemma 6, if = 12 and uctuations of technology are small, V h(k,d, Ah) = V (k,d, Ah)d an argument similar to Proposition 5 shows that if the country is constrained, d and d are such that V pay(k

,d ) =

def (k, ) and V detpay(k,d) = V detdef (k, ). We want to know the values of d and d that make such equalities hold when we

e close to the deterministic steady state.Using Eqs. (13) and (14), V detpay(k,d) = V detdef (k, ) imply:

V pay(k,d , A

)= V def (k, , A ) u(cd)(1 )(A A) f (k)1 + u

(c)(A A) f (k)1 u

(c)(d d)

= V def(k, , A

)+ [ u(c) + u(c) u(cd)](A A) f (k)1 u

(c)(d d)

From Proposition 4:

V detpay(k,d) = V detdef (k, ) d = A f (k)

1 qhich implies cd = c.So V pay(k,d

, A ) = V def (k, , A ) if:

(1 q)dA f (k)

u(c) (A A) f (k)1 = u

(c)(d d)

(1 q)d1

(A A)A

= d d

Using = 1 2 and substituting (A ,d ) for (Ah,dh) and (Al,dl) we get 2 equations that relate debt and productivityeach of the two states. Combining both equations, we get Eq. (10). pendix B. Numerical examples

. Deterministic model

In the numerical examples of this paper, specic utility and production functional forms are assumed as follows:

u(c) = c1

1 , f (k) = k

I calibrate the specic parameters as follows: one period corresponds to one year. A = 1, = 0.36, = 1.021, = 3d = 0.10. The price of a riskless bond, q , equals . The output loss in terms of default, = 0.01.The numerical solution is obtained through value function iteration. The state space is discretised using grids for debtd capital but the planner can choose any point in the grid. From Fig. 2, the numbers obtained in this solution are verymilar to those from the analytical formulae using the path of yt given by the numerical example.Fig. 2 also shows the behaviour of capital in this economy, = 0.01, compared to the closed-economy case, = 0, ande full-commitment open-economy case, = 1. Without the possibility of default, the level of capital jumps to its steadyate level and the marginal productivity of capital equals r in one period. The possibility of default makes convergenceower. Due to the initial capital inow, the level of capital is higher in this economy than in the closed economy casetil they converge. However, the closed economy slowly catches up, as the open economy will be experiencing net capitaltows (trade balance surpluses) during the whole history, as shown at Fig. 2. Debt stabilises at 51% of GDP but reaches% of GDP at earlier stages.A usual intuition is that nancially open economies should converge faster to their steady states (Barro et al., 1995).contrast, the equilibrium from this model shows that an indebted open economy would take more time to convergean a closed economy with the same level of capital. After the initial capital inow, the country experiences net outowsresources i.e. a positive trade balance. In addition, a closed economy that opens to capital ows would not converge

gnicantly faster but, on the way towards the steady state, would have higher output than if it remained closed. In orderexperience faster convergence, emerging economies need trade decits and, as Proposition 4 shows, that does not occurequilibrium.


B.

obch

k

twFig. 2. Deterministic model.

Fig. 3. Debt relief.

2. Stochastic model

In this section, the accuracy of the analytical approximations used in Section 4.2 is checked using numerical simulations.I want to obtain the values of d/d that make V h(k,dh) = V l(k,dl) at every state (k,d). The numerical solution istained through value function iteration. The state space is discretised using grids for debt and capital but the planner canoose any point in the grid. At the beginning of each iteration, d is calculated to make V h(k,dh) = V l(k,dl).I use the same stylisation of the 1970s and 1980s to calibrate the model: one period correspond to one year, = 0.36,

= 1.021, = 0.01, = 3 and = 0.10. A = 1, and q uctuates around : ql = 1.041 and qh = 1.00. I constrain k to lie in some interval adjustment costs for capital are zero in that interval and innity outside it. Fig. 3 showsd/d as a function of the marginal productivity of capital if the borrowing constraint is binding and the state is high ino situations: (a) k k (0.10k,0.10k) and (b) k k (0.05k,0.10k).The main results are as follows:

For mpk > 0.04 = rl , the linear approximation works well: d/d is around 0.17 and gradually increasing in mpk. Thepossibility of borrowing an additional unit in the high state is worth slightly more to countries with high mpk.


For mpk below rl = 4%, d/d is considerably smaller. For lower values of mpk, when the state shifts to low, interestrates are higher than mpk, so the country sells capital and could even end up buying high-interest-rate foreign bonds.This might sound unrealistic because adjustment costs for capital would prevent such rapid capital movement. Indeed,the debt reduction d/d for lower values of mpk are sensitive to the assumptions on adjustment costs. Fig. 3 showsthat when mpk is at its lowest value, k = k, then d/d is around 0.10 with adjustment costs given by (a) and around0.11 in case (b).

References

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JesKeKl

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Sovereign default: Which shocks matter?IntroductionEndowment economyStochastic world interest ratesStochastic endowmentContrasting stochastic q* and stochastic y

Debt and default in a growth modelDeterministic modelStochastic world interest ratesThe value of d

Stochastic technology

The Latin American debt crisis of the 1980'sObserved debt reliefDebt relief according to the model

Concluding remarksProofsProposition 1Proposition 4Proposition 5Lemma 6Lemma 7Proposition 8Proposition 9Proposition 10

Numerical examplesDeterministic modelStochastic model

References

sovereign default: which shocks matter?

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