sovereign debt 1

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A Dynamic Model of Sovereign Debt and Defaults: Understanding the Incentives ofthe Borrower.Marcela Giraldo

Abstract:

This paper develops a non-stochastic sovereign debt dynamic model, with increasing cost of capital. It findsthat there are parameters for which there is a stable fixed borrowing level, and there are parameters for whichthe model depicts the case of serial defaulters. Levels of debt that tend towards the fixed point correspond tohigher levels of consumption than would be attainable under financial autarky. In the cases of unstable paths(those that eventually end in default), consumption decreases as debt increases, because an increasing amountof domestic production needs to be added to current borrowing in order to pay debt obligations.The model shows how, as the cost of capital decreases, the equilibrium level of borrowing increases, and

more levels of debt become sustainable in the long run. On the other hand, higher desired returns fromcreditors have the effect of lowering the stable borrowing level and the threshold for sustainable levels ofdebt. Very high levels of desired returns, R, may lead countries to become serial defaulters. This occursbecause the stable equilibrium point disappears and every initial borrowing amount ends in default. A highlevel of R is not the only thing that takes a country to a state of chronic default; short lengths of punishmentare far more important. In the benchmark model, at least 21 periods of punishment are necessary to avoidchronic defaults. High levels of impatience can also make a country a serial defaulter.


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IntroductionThis paper addresses the question of when default of sovereign debt is optimal using adynamic model of lending. It does so by developing a non-stochastic dynamic model withincreasing cost of capital for a small open economy. The paper builds on the modeldiscussed in Aguiar and Gopinath (2004) and Arellano (2008). To the knowledge of the

author, this is the first paper to use the results of a dynamic optimization model to analyzethe optimal path of the level of debt, the levels after which countries should consider defaultas their best strategy, and the factors that alter this path. In order to achieve that goal, thepaper focuses on the decision making of the borrower, and concludes with the implicationsfor a general equilibrium. The fact that the model is non-stochastic allows the finding of adeterministic optimal path of debt levels, for a given production capacity under financialautarky.

In this model default is punished by not allowing the defaulting country to accessinternational financial markets for a predetermined length of time, as is common in the

literature and now supported by the findings in Cruces and Trebesch (2013). There are alsoindirect costs to defaulting. If a country is an active participant of the financial markets, itcan achieve higher levels of output1. However, countries may default if the repayment ofdebt becomes too burdensome, and both production and consumption suffer to the pointthat it is preferable to be excluded for a period of time from the financial markets than torepay the debt. Driven by recent episodes of default, the output function was modeled sothat it may be lower with excessive debt levels than under financial autarky.

The model shows intervals of borrowing levels that lead to eventual defaults and howdifferent parameters affect their length. These are also intervals where creditors and debtorsshould consider debt restructuring / renegotiation. The borrowing equilibrium levels under

some parameter values are well below 25% of production. This is significant given theobservation from Reinhart and Rogoff (2009) that most defaults occur in countries withdebt levels below 50% of GDP (which means the equilibrium must be well below that level).

This model is also interesting in that the right combination of parameters can representthe case of a serial defaulter.

Aguiar and Gopinath (2004) and Arellano (2008) also study a model of debt that allowscountries to default. The parameters are assumed to be consistent with empiricalobservations from Argentina. Although their model includes shocks that can havepermanent or transitory effects, and the model presented here does not have uncertainty and

does not explicitly model shocks, the set up of the problem is very similar. All of thesemodels follow the classic framework in which lending is limited to one period bonds anddefault is punished with financial autarky. As in Aguiar and Gopinath, borrowers recognizethat the cost of the loans increases with each additional unit of debt. However, their paperneeds a high level of impatience in order to observe default. That is not necessary in this

1Arellano (2008), Cohen and Sachs (1986), and Cole and Kehoe (2000) also model sovereigndefaults as having a negative impact on output.


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paper. In fact, the benchmark model has ! ! !!!"and default exists for high borrowinglevels. It is important to keep in mind that the model in this paper is not calibrated, but thecharacteristics of the equilibrium are analyzed for different values of the parameters. Hence,the ultimate question and the numerical methodology between both papers are somewhatdifferent.

Other related papers in the literature include Gelos, Sahay and Sandleris (2011) and Yue(2010). Gelos et al, report several factors that seem to determine access to international creditmarkets. That paper shows that a countrys frequency of defaults is not relevant indetermining access, but the vulnerability to shocks and the perceived quality of the economicpolicies and institutions are significant. That result is very important because models in

which the history is not directly relevant, but only the state of the world today is, are mucheasier to solve. The authors also report that the average exclusion from the internationalcredit markets after a default declined from four years in the 1980s to two years in the 1990sand that access to financial markets seems to be more consistent with lower levels of debt to

GDP ratio. Both of these results are consistent with the model presented here. Yue (2010)is another paper with a similar model to Aguiar and Gopinath (2004) and Arellano (2008)but it allows for renegotiation of the debt after default.

In this paper, as is common in the literature, countries either pay their debt in full, or defaultand pay nothing. For a paper that looks at the implications of restructuring andrenegotiations in debt sustainability, see Kletzer and Wright (2000). In that paper, theauthors show that long-term debt relationships are sustainable, even without collateral, butthey are subject to constant renegotiations and threat of entry by competing lenders.

This paper is organized as follows. It first explains the main assumptions and sets up the

model. Since the model is solved using a numerical analysis, the paper then details themethodology and later reports the results. The result section is divided in two. First, aparticular set of parameters is used to explore the existence of an equilibrium. Secondly, the

values of the parameters are allowed to change and the paper reports the effects of thechanges in the parameters on the equilibrium points using bifurcation diagrams. To close,there is a brief discussion on the implications of the findings for the general equilibrium.

ModelConsider a country that can produce p under financial autarky. Actual production, I,depends on how much it borrows. This is not necessarily because the funds obtained are

used to invest, but it is a way to model spillover effects in the economy from having accessto financial markets. Excessive borrowing by the government will have the effect of reducingthe production of the country during the following period. The reduction may takeproduction below the financial autarky level for the extreme levels. The possible decrease inproduction below the autarky capacity with very high levels of borrowing is not crucial andthe general dynamics do not change if that did not occur.


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This can be modeled as follows. Let !!!! ! !!be the production at period t. Its domain isthe possible borrowing levels, L. The notation !!!!!!!!indicates the production in period tis a function of the amount borrowed in the previous period.

!!!!!!!! ! !! !!!!!!!! !! (1)

where !!!!! ! !, !!!! !!! ! !, for some M>0, !!!!!!!!is concave and, as the notationindicates, it depends on the borrowing level of the previous period.

Consumption comes from production and net capital inflows (amount borrowed todayminus the repayment of old debt), and all borrowing occurs in the form of one-periodbonds. So,

!! ! !!!!!!!!! !! ! !!!! !!!!!

where !!is the amount borrowed in period t, and !! ! !!! !! !!!. If a country canborrow up to 100% of its production in autarky, M=1. M can be thought of as depending onthe economic policies and institutions of a country that influence the governments ability toaccess credit markets. In the next section, M will be an important determinant of the

borrowing costs. !!!!is the price that the government promised to pay for the capital thatwas acquired last period.

In order to simplify notation and leave together all the terms that are composed only of statevariables, lety be the disposable income and define it as

!!!!!!!!!

!!!!!!!!!

!!!! !!!!! (2)

where

!! ! ! !!"#$!!!!

Cost(L) is the cost of capital, which is increasing. This is equivalent to the inverse of theprobability of repaying the loans that is common in similar models that include uncertaintyand shocks to income. Let the desired return of the creditor be a constant r, and R=1+r.

This value must be at least the return on risk free assets.

It follows that

!! ! !!!!!!!!! !!

2Countries are only considering borrowing. Lending is not directly modeled. The reader canthink of returns from lending as included in the production equation.


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The timing of decisions within one period is as follows. The government observes the debtobligations and then it decides how much to borrow. After that, it decides whether to pay ordefault on its debt from the previous period. So, in the same period the government mayborrow and use some of that money to pay its debt obligations. An equivalent set up wouldbe to have the country decide L on the previous period, and start period t by deciding

whether to default or not. No significant differences exist between both models.

As is common in the literature, countries either pay the debt in full, or go into default andpay nothing. If a country defaults, it does so after borrowing today. The consequences of thedecisions materialize starting in the following period. During the punishment period there isno borrowing, and consumption equals production.

The Value function for a government that has the option to default is

!!!!!! ! !"#!!! ! !!!!!!!where !!is the best that can be achieved if default is not considered, and !!is the value ofdefaulting. (1) and (2) are the transition equations.

If a country does not consider default, the problem it is solving is the following.

!!!!! ! !"#

!!!!!!"!! ! ! ! ! ! ! !!!!!

where !! ! !!!!.

If a country defaults, the value function will be

!!!!! ! !"#

!!!!!!"!! ! ! ! ! ! ! !!!!!! !! ! !!!

!

!!!

!! !!!! ! !!!!

where !! ! !!!!and N is the number of periods that the country is excluded from the

international credit market. Since !!depends on the borrowing level of the previous period,

!and Iare the state variables. The control variable is L.

Numerical Analysis and Solution

To solve the model, the Value Function Iteration method is used, and hence, the state-spaceis discretized. In order to do this, the following functional forms for the utility function andthe cost of capital were adopted:

! ! !

!!!!

! ! !


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!"#$!!!! !

!!!

!!!!!!

!"!! !!!!! !!!! !"!!"#$%! (3)

This functional form of the cost of capital guarantees that the cost of the loan increases with

each additional unit borrowed. The functional form is assuming that the markets will allowthe borrowing country to access up to a certain multiple of its autarkic production level ofcapital. At this threshold point, the cost of the loan will be infinity. Although anycontinuously increasing function would yield similar results, the assumption of the existenceof an M value allows the model to assess what happens if markets behave as if the rangeof acceptable debt levels increases or decreases (M increases or decreases, respectively), andhence, the cost of accessing capital varies.

Plugging (3) into (2):

!! !!!! ! ! ! !! !!!! !! ! !"#

!"!!!!!

!!!!!

Where !!!!!!!!will be assumed to take the following functional form

!!!!!!!! !!

!"!!!!!!" ! !!!!! ! !!! !!!!!

Figure 1 illustrates why this functional form for !! !!!! was selected. It shows the

functions x(1-x) and 0.2x. !!is the difference between both curves when M and p equal one.

!!is initially zero, then it increases up to a certain point, after which it decreases. Thefunction becomes negative for excessive levels of borrowing. This means that if a country is

an active participant of the financial markets, it can achieve higher levels of output.However, the output function may be lower with excessive debt levels than under financialautarky.


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Figure 1

Note that the control variable L(t) fully determines the state variable y(t+1) since there is nouncertainty in this model, andpdoes not change from period to period. Each level of debttoday has a unique corresponding disposable income tomorrow. Using this fact, the control

variable was discretized, and the corresponding state variable values estimated. Then, acomputer code was generated that would

1. Find !!and !!for each !!!!, given an initial V vector.

2. Use !!!!!! ! !"#!!!!!!!to generate a new V vector.3. Repeat points 1 and 2 until the difference between the old and the new V vector is lessthan some epsilon.

Using the last cycle of the iteration process, the !!that maximizes the value function for

each !!!!is saved, including whether default was chosen or not. This allows the author togenerate a sequence of optimal loan amounts, given an initial value. As discussed in the nextsection, several sequences will converge to a stable fixed point.

ResultsBenchmark Model

To evaluate the results of the model, the following parameters values where used: !=2,

!=0.95, R=1.001, M=1, p=1, N=50. Henceforth, this set of parameters is referred to as thebenchmark model. All the results in this section are evaluating the incentives of theborrowing country. The implications for the general equilibrium will be discussed in thefollowing section.

Figure 2 shows the value function is decreasing in !!!!. Note that this does not imply thatthe value function is maximized by not lending today, but there can be a positive L that


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maximizes the value function for each state. This maximum is the greatest when no lendingoccurred last period, in other words, when y=p=1.

Figure 2

Figure 3 shows the best response to each state. The 45 degree line is included to easilyidentify the fixed points of equilibrium. The red portion of the best response linecorresponds to periods when default is optimal. Since in the period following default no loanis possible, any fixed points on this portion of the graph will be ignored.

The best response function is increasing and is relatively flat (slope less than one) for small

values of !!!!. Since the best response in period t fully describes the state variable in periodt+1, a sequence can be generated for any initial point. Hence, if the 45-degree line interceptsthe best response function in this flat section, there is a stable fixed point, as is the case inthe benchmark model graphed in Figure 3. All numerical exercises confirmed convergenceto that point for initial points to the left of the second fixed point. The slope of the functionincreases abruptly before 0.5, and the response function crosses the 45-degree line again.

This second fixed point is unstable. The third abrupt change in the shape of the responsefunction occurs when defaulting becomes optimal. In this case, a country tries to borrow the

amount that maximizes !!, and then defaults in all its debt.

The shape of the best response function is consistent for different parameter combinations,although fixed points are not always present (see Figure 4 for an example.) Note that a case

such as the one in Figure 3 is a hopeful one. A country that finds itself borrowing anamount to the right of the unstable fixed point will start a new sequence in L=0 after itdefaults, and after the punishment period has lapsed. Given the shape of the responsefunction, the sequence that starts in L=0 converges to the stable fixed point. In thebenchmark model, this is a constant borrowing of 0.181, or 16.3% of total production. Infact, the stable fixed point has a basin of attraction that starts at zero, and has the unstablefixed point as its upper limit. All the points inside the basin of attraction are sustainable debtlevels, where as all the other points are unsustainable levels.


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Figure 3

Figure 4 corresponds to a case of chronic default. The stable and unstable equilibriadisappear, and every initial point has a sequence that ends in default. After a forcedpunishment time, the country restarts at zero. But zero will also end, eventually, in default.

Figure 4


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Figure 5shows a sequence that starts with L(1)=0.4, in the benchmark model. It clearlygradually converges to the fixed point.

Figure 5

Figure 6 shows the corresponding sequence of consumption. The low level of consumptionthat corresponds to lending above the equilibrium indicates that the country is borrowingonly to pay previous debt (some of its domestic income is also used to complete thepayment), and is also experiencing low levels of production. As the level of the debt movestowards the equilibrium, consumption settles above the level of autarky.

Figure 6

For the points with borrowing greater than in equilibrium, consumption decreases as Lincreases. In the cases of unstable paths (those that eventually end in default), this occurs


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because an increasing amount of domestic production needs to be added to L(t) in order tocollect enough to pay previous debts. This continues until it is preferable to default andconsume the amount of production attainable under financial autarky.

Bifurcation diagrams

In this section the model was run with varying parameters to evaluate the consequentialchanges in the stable equilibrium. In order to do that, the following steps where repeated fordifferent values of the parameter in question3:

1. Given the parameter value, the best response vector is found.2. The sequence of levels of borrowing is found starting at a low level. The length of the

sequence was generally 200 since this model exhibits rapid convergence.3. The last 10 points of the sequence where graphed for each value of the parameter. If

the graph only shows one point, it is because the sequence achieved a stableequilibrium and all 10 Ls were the same.

Figure 7 shows the bifurcation diagram for M. M was allowed to change between 0.1 and 2.All the other parameters stayed as in the benchmark model. As the figure shows, within thisrange, there is always a stable equilibrium. This equilibrium point increases linearly with M,and never surpasses 0.4. Note that if M starts to decrease for a country (or the costsincrease), the fixed stable point also decreases, which implies that as the amount of capitalthat a country has access to decreases, lower levels of borrowing start becomingunsustainable.

Figure 7

3Bifurcation diagrams such as these can be found often in Gian-Italo Bischis work. See, forexample, Bischi, Gatti and Gallegati (2004).


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Figure 8 is the bifurcation diagram for R, for values between 1 and 1.05. This diagram isdiscontinuous and decreasing. The discontinuity in the graph is due to the discretization of Lused to solve the problem numerically. As the grid becomes more detailed, the number ofsteps in this diagram increases. This happens because the slope of the continuous function isnegative, but close to zero. Higher desired returns from creditors (or return from a risk free

asset) have the effect of lowering the fixed and stable borrowing level, and the threshold forsustainable levels of debt, although by a very small amount.

Figure 8

Figure 9 shows the same bifurcation diagram as Figure 8, but for a longer interval of R. AfterR=1.119, stability is lost. This is a 11.9% return, per period. Periods can be thought of asmonths, since there are 50 punishment periods (a little over 4 years), which is similar to what

was observed in practice during the 1980s, as reported by Gelos, Sahay and Sandleris (2011).So, the desired rate of return must increase to excessive levels for the equilibrium todisappear.

After R=1.119, we observe cycles of less than 6 periods. This is a state of chronic defaults.After a period of punishment, the economy starts at L=0. Then the level of borrowing

increases in each period, until default is optimal again and the country enters a new period ofpunishment (where L is forced to zero again).


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Figure 9

High levels of R is not the only thing that takes a country to a state of chronic defaulting,short lengths of punishment are far more significant. In the benchmark model, at least 21periods of punishment are necessary to avoid chronic defaults. Gelos, Sahay and Sandleris(2011) report that during the 1990s, the average exclusion from international credit markets

was two years, slightly above the benchmark threshold for the existence of a stable fixedpoint. The bifurcation diagram for the parameter N is in Figure 10.

Figure 10


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The bifurcation diagram for the coefficient of relative risk aversion is a horizontal line.

Changes in !have no effect on the stable equilibrium.

Figure 11 shows the diagram for delta. Low values of delta, equivalent to high levels ofimpatience, eliminate the stable equilibrium. Not surprisingly, countries must have a

minimum level of concern for the future to make their participation in financial marketspossible. Delta must be greater than 0.92 in the benchmark model for a sustainableborrowing level to exist.

Figure 11

Table 1shows the results for different combinations of the values of the parameters. Thefirst column indicates the parameters that are different from the benchmark model. Thebasin of attraction is the interval from which any sequence that starts there will converge tothe stable fixed point.

Table 1

Parameters Smallest L of Default Stable Fixed Point Basin AttractionBenchmark 0.48 0.181 0 to 0.419M=3 1.47 0.542 0 to 1.209

M=3, R=1.05 1.44 0.504 0 to 1.117M=0.5 0.233 0.09 0 to 0.203R=1.05, N=5 0.42 dne NAR=1.05, N=10 0.446 dne NAM=3, N=10 1.42 dne NAM=3, N=25 1.46 0.542 0 to 1.124R=1.1 0.45 0.155 0 to 0.255

Unless otherwise noted: !=2, !=0.95, R=1.001, M=1, p=1, N=50.


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Borrowing levels outside the basin of attraction are levels at which a renegotiation shouldoccur since default is imminent. In practice, one knows that a region is outside of the basinof attraction, or close the upper limit, when very small changes in borrowing levels lead to

very different reactions (the best response function has a very steep slope.)

Notice that high levels of impatience (low levels of !) are not necessary to achieve default, asis the case in Aguiar and Gopinath (2006). There can be default with high !s also. Thedifference is that, in the latter case, default is escapable. In addition, there is no need for anincome shock to observe default. Changes in the parameters R and N can put a countrysborrowing levels outside the basin of attraction of the unique fixed and stable point, causingan eventual default.

Implications for a general equilibriumIf creditors can perfectly predict the behavior explained in the previous section, they wouldadjust the supply function and increase the price of their lending for amounts outside the

basin of attraction, perhaps to infinity. So that the upper bound of the basin of attractionwould also be a point of discontinuity in the supply for credit. In other words, the creditmarket would dry out before there is default.

Proposition 1: In the general equilibrium with perfect information, there is no lending outsideof the basin of attraction of the stable equilibrium.

Proof:If there is perfect information, the lender can find the best response function in graph 3 or4. Hence, for every L(t), the lender can find out L(t+1) and whether there will be default.Outside the basin of attraction, all sequences end in default. Then, the creditor knows there

is a period T, in which the borrower will not pay back the debt. In period T-1, the creditorwill not lend because it knows that debt will not be paid next period.Assume the lender will not lend in period T-i. Then, it will not lend in period T-(i+1)because it knows the borrower not repay. Recall that outside the basin of attraction,borrowing is increasing at a rapid pace. All funds borrowed are used to pay old debt, andsome of the domestic production must be added to the new loans to complete the payment.Hence, by not having access to funds in T-i, it will be forced into default.

The borrower himself has strategic incentives not to repay. If a borrower knows that it willbe forced into financial autarky in the next period, independent of its decision to repay ornot, all the incentives for repayment are eliminated. Hence, if there will be no lending in T-i,

there would be default in T-(i+1). Knowing this, the lender does not lend in that period.QED

If there is perfect information, default would only be observed if when an exogenous changein the parameters occurs, such that the level of debt of a country is suddenly outside of theinterval of sustainable debt. This is an example of a classic Dornbusch/Calvo-type suddenstop.In the literature, there are other ways to explain the often-observed events of sovereign


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default and restructuring. Theoretically, that is done either with of shocks to income4, orbecause the reaction of the markets is delayed. The latter might happen when information isnot perfect5or incentives are altered6. When either of these two explanations are part of thestory, this paper sheds light on how an unsustainable debt crisis may precede, or even cause,a decline in consumption and a preference for default.

ConclusionsThis paper develops a non-stochastic sovereign-debt dynamic model with increasing cost ofcapital. It finds that there are parameters for which there is a stable fixed point, and there areparameters under which the model pictures the case of serial defaulters. Levels of debt thattend towards the fixed point correspond to higher consumption than would be attainableunder financial autarky. In the cases of unstable paths (those that eventually end in default),consumption decreases as L increases because an increasing amount of domestic productionneeds to be added to L(t) to collect enough to pay previous debt levels. This continues untilit is preferable to default and consume the amount of production attainable under financial

autarky.The model shows how as the cost of capital decreases (equivalent to M increasing), theequilibrium level of borrowing increases, and more levels of debt become sustainable overthe long run. On the other hand, higher desired returns from creditors have the effect oflowering the stable borrowing level, and the threshold for sustainable levels of debt. Veryhigh levels of desired returns, R, may lead countries to become serial defaulters. This occursbecause the stable equilibrium point disappears and every initial borrowing amount ends ineventual default.High levels of R is not the only thing that takes a country to a state of chronic defaulting,short lengths of punishment are far more significant. In the benchmark model, at least 21periods of punishment are necessary to avoid chronic defaults.

Low levels of impatience are necessary for the stable equilibrium to exist. If a countrys !starts to decrease, it will become a serial defaulter. In sum, an exogenous change in someparameters can put the borrowing levels of some countries outside the stable region.

Bibliography

Aguiar, M., & Gopinath, G. (2006). Defaultable Debt, Interest Rates and the Current Account.

Journal of International Economics, 69(1), 64-83.

4Income shocks are usually included in default models, but their implications for the particularanalysis presented here are left for future research.5Reinhart and Rogoff (2011 and 2009) highlight that a possible reason for the imperfect informationis that public domestic debt data is often opaque. Complete information on the total governmentdebt is not always fully accessible to creditors.6Such as the effects of insurance.


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Arellano, C. (2008). Default Risk and Income Fluctuations in Emerging Economies. The AmericanEconomic Review, 98(3), 690-712.

Bischi, G. I., Gatti, D. D., & Gallegati, M. (2004). Financial Conditions, Strategic Interaction and

Complex Dynamics: a Game-Theoretic Model of Financially Driven Fluctuations.Journal of EconomicBehavior & Organization, 53, 145-171.

Cohen, D., & Sachs, J. (1986). Growth and External Debt under Risk of Debt Repudiation.EuropeanEconomic Review, 30(3), 529-560.

Cole, H. L., & Kehoe, T. J. (2000). Self-Fulfilling Debt Crises. Review of Economic Studies, 67(1), 91-116.

Cruces, J. J., & Trebesch, C. (2013). Sovereign Defaults: The Price of Haircuts.American EconomicJournal: Macroeconomics, 5(3), 85-117.

Gelos, G. R., Sahay, R., & Sandleris, G. (2011). Sovereign Borrowing by Developing Countries:What Determines Market Access?Journal of International Economics, 83, 243-254.

Kletzer, K. M., & Wright, B. D. (2000). Sovereign Debt as Intertemporal Barter. The AmericanEconomic Review, 90(3), 621-639.

Reinhart, C. M., & Rogoff, K. S. (2011). From Financial Crash to Debt Crisis.American EconomicReview, 101(5), 1676-1706.

Reinhart, C. M., & Rogoff, K. S. (2009). This Time is Different.Princeton, New Jersey: PrincetonUniversity Press.

Yue, V. Z. (2010). Sovereign Default and Debt Renegotiation.Journal of International Economics, 80,176-187.

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