royal holloway, university of london€¦ · sovereign debt and structural reforms andreas müllery...
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Sovereign Debt and Structural Reforms�
Andreas Müllery Kjetil Storeslettenz Fabrizio Zilibottix
December 19, 2015
Abstract
Motivated by the European debt crisis, we construct a tractable theory of sovereign debt andstructural reforms under limited commitment. A sovereign country which has fallen into a recessionof an uncertain duration issues one-period debt to smooth consumption. It can also implementstructural policy reforms to speed up recovery from the recession. The sovereign can renege on itsobligations by su¤ering an observable stochastic default cost, in which case creditors o¤er a take-it-or-leave-it debt haircut to avert default. We characterize the competitive Markov equilibriumand compare it to the constrained e¢ cient allocation. The competitive equilibrium features large�uctuations in consumption, debt, and reform e¤ort. During the recession, consumption fallswhenever debt is honored. A large debt destroys incentives for structural reforms since some ofthe gains from reform accrue to the lenders. The constrained optimum yields step-wise increasingconsumption and step-wise decreasing reform e¤ort. Markets for state-contingent debt alone cannotrestore e¢ ciency. The optimum can be interpreted as a �exible assistance program providing budgetsupport during recession followed by a debt increase after recovery. Restrictions are imposed ondebt issuance and reform e¤ort. The terms of the program are improved each time the country cancredibly threaten to leave.JEL Codes: E62, F33, F34, F53, H12, H63Keywords: Austerity, Debt overhang, Default, European debt crisis, Fiscal policy, Great Re-
cession, Limited commitment, Markov equilibrium, Moral hazard, Renegotiation, Risk premia, Risksharing, Sovereign debt, Structural reforms.
�We would like to thank George-Marios Angeletos, Cristina Arellano, Marco Bassetto, Tobias Broer, Fernando Broner,Patrick Kehoe, Enrique Mendoza, Juan-Pablo Nicolini, Ugo Panizza, Aleh Tsyvinski, Jaume Ventura, ChristopherWinter, and seminar participants at Annual Meeting of the Swiss Society of Economics and Statistics, Cà FoscariUniversity of Venice, Cambridge, CEMFI, CREi, European University Institute, Goethe University Frankfurt, GraduateInstitute of Geneva, Humboldt University, Istanbul School of Central Banking, NORMAC, Oxford, Swiss National Bank,Universitat Autonoma Barcelona, University College London, University of Konstanz, University of Oslo, University ofOxford, University of Pennsylvania, University of Padua, University of Toronto, and Yale University. We acknowledgesupport from the European Research Council (ERC Advanced Grant IPCDP-324085).
yUniversity of Oslo, Department of Economics, [email protected] of Oslo, Department of Economics, [email protected] of Zurich, Department of Economics, [email protected].
1 Introduction
The recent European debt crisis has revamped the interest in understanding the dynamics of sovereigndebt during economic downturns. Economic theory o¤ers two simple policy prescriptions for countriessu¤ering a temporary decline in output. First, they should borrow on international markets to smoothconsumption. Second, they should undertake reforms �possibly painful ones �to speed up economicrecovery. However, the fact that there is limited enforcement of sovereign debt may have major e¤ectson the simple policy prescriptions. On the one hand, risk sharing may be hampered by rising defaultpremia associated with the risk of debt renegotiations.1 On the other hand, a large debt can reducethe borrower�s incentive to undertake economic reforms to boost economic growth since some of thegains from growth would accrue to the lenders.
This paper proposes a theory of sovereign debt to address the following questions. First, what doesthe optimal dynamic contract between a planner and a sovereign, temporarily impoverished, countryprescribe in an environment where the country cannot commit to honoring its debt and costly reformscan speed up economic recovery? Second, does the market equilibrium attain the e¢ cient level ofstructural reforms and consumption smoothing? If not, what policies and institutional arrangementscan improve e¢ ciency?
The theory rests on three building blocks. The �rst is that sovereign debt is subject to limitedenforcement, and that countries can renege on their obligations subject to real costs, as in e.g. Aguiarand Gopinath (2006), Arellano (2008) and Yue (2010). Di¤erent from this literature, we assume thesize of default costs to be stochastic, re�ecting exogenous changes in the domestic and internationalsituation. The second building block is that whenever creditors face a credible default threat, theycan make a take-it-or-leave-it renegotiation o¤er to the indebted country. This approach conformswith the empirical observations that unordered defaults are rare events, and that there is great het-erogeneity in the terms at which debt is renegotiated, as documented by Tomz and Wright (2007) andSturzenegger and Zettelmeyer (2008). The third building block is the possibility for the governmentof the indebted country to make "structural" policy reforms that speed up recovery from an existingrecession. Examples of such reforms include labor and product market deregulation, and the establish-ment of �scal capacity that allows the government to raise tax revenue e¢ ciently (see, e.g., Ilzkovitzand Dierx 2011). While these reforms are bene�cial in the long run, they may entail short-run costsfor citizens at large, governments or special-interest groups (see, e.g., Blanchard and Giavazzi 2003,and Boeri 2005).
More formally, we construct a dynamic model of sovereign debt. Productivity is subject to aggre-gate shocks following a two-state Markov process. A benevolent local government can issue sovereigndebt to smooth consumption.2 The sovereign country starts in a recession of an unknown duration.The probability that the recession ends is endogenous, and hinges on the government�s reform e¤ort.Debt issuance is subject to a limited-commitment problem: the government can, ex-post, repudiateits debt, based on the publicly observable realization of a stochastic default cost. When this realiza-tion is su¢ ciently low relative to the outstanding debt, the default threat is credible. In this case, a
1Greece, the hardest hit country in Europe, saw its debt-GDP ratio soar from 107% in 2008 to 170% in 2011, atwhich point creditors had to agree to a debt haircut implying a 53% loss on its face value. Meanwhile, internationalorganizations stepped in to provide �nancial assistance and access to new loans, asking in exchange �scal austerity and acommitment to economic reforms. A new crisis with a request from the Greek government of a haircut on the outstandingdebt was barely averted in the summer of 2015.
2 In the benchmark model, sovereign debt is a non-state-contingent bond. Our results do not hinge on this assumption,and we show in an extension that if the government could issue state-contingent debt, i.e., securities whose pay-o¤ iscontingent to the realization of the aggregate productivity, the main results would remain unchanged.
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syndicate of creditors makes a take-it-or-leave-it debt haircut o¤er, as in Bulow and Rogo¤ (1989). Inequilibrium, there is no outright default, but debt renegotiations. Haircuts are more frequent duringrecession, and more frequent the larger is the outstanding sovereign debt. Consumption increases aftera renegotiation, in line with the empirical evidence that economic conditions improve in the aftermathof debt relief, as documented in Reinhart and Trebesch (2016).
We �rst characterize the laissez-faire equilibrium. During recessions, the government would liketo issue debt in order to smooth consumption. However, as debt accumulates, the probability ofrenegotiation increases, implying a rising risk premium and consumption volatility. The reform e¤ortexhibits a non-monotonic pattern: it is increasing with debt at low levels of debt because of thedisciplining e¤ect of recession. However, for su¢ ciently high debt levels the relationship is �ippedbecause of a debt overhang problem: very high debt levels deter useful reforms because most of thegains from the reforms accrue to foreign lenders in the form of capital gains on the outstanding debt.The moral hazard problem exacerbates the country�s inability to achieve consumption smoothing: athigh debt levels, creditors expect little reform e¤ort, are pessimistic about the economic outlook, andrequest an even higher risk premium.
Next, to establish a normative benchmark, we characterize the optimal dynamic contract betweena planner without enforcement power and a country that has fallen into recession. In contrast tothe laissez-faire equilibrium, the constrained optimal allocation features non-decreasing consumptionand non-increasing reform e¤ort during the recession. More precisely, consumption and e¤ort remainconstant whenever the country�s participation constraint is not binding. However, when the constraintis binding the planner increases the country�s promised utility and consumption, and reduces therequired reform e¤ort.
Having characterized the constrained e¢ cient allocation, we consider its implementation in a de-centralized environment. We �rst show that, in the absence of aggregate productivity shocks, thelaissez-faire equilibrium attains the constrained-e¢ cient allocation. However, the equilibrium is notconstrained-e¢ cient when the economy is in a recession. In the laissez-faire equilibrium, consumptionand reform e¤ort �uctuate too much, and reform e¤ort is ine¢ cient. The ine¢ ciency cannot be over-come by allowing the government to issue state-contingent debt. In standard models, state-contingentdebt would provide insurance against the continuation of a recession �i.e., Arrow securities paying o¤conditional on the aggregate state, recession or normal times. However, the better insured the coun-try is, the more severe the moral hazard problem becomes. For instance, full insurance would removeincentives to exert reform e¤ort. Since this is priced into the debt, the social value of state-contingentdebt is limited. In a calibrated version of the model, we show that state-contingent debt yields verysmall welfare gains relative to the equilibrium with non-state contingent debt.
The constrained optimal allocation can be attained through the intervention of an independentinstitution (e.g., the IMF) that monitors the country�s �scal policy and the structural reform program.During the recession the optimal program entails a persistent budget support through extending loanson favorable terms, combined with a larger reform e¤ort than the borrower would choose on its own.Upon recovery from the recession, the sovereign is settled with a (large) debt on market terms. Acommon objection to schemes implying deferred repayment is that the country may refuse to repayits loans when the economy recovers. In our theory, this risks is factored in as part of the contract.The optimal program has the interesting feature that, whenever the country can credibly threaten todefault, the international institution should relent and improve the terms of the agreement for thedebtor by granting her higher consumption and a lower reform e¤ort. In other words, the austerityprogram is relaxed over time, whenever necessary to avert the breakdown of the program.
We provide a quantitative evaluation of the theory with the aid of a calibrated version of the model.
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The model matches realistic debt-to-GDP ratios, as well as default premia, renegotiation frequencies,and recovery rates. We regard this as a contribution in itself. In the existing quantitative literature,it is di¢ cult to sustain high debt levels, contrary both to the observation that many countries havemanaged to �nance debt-GDP ratios above 100%, and to the estimates of a recent study by Collard,Habib, and Rochet (2015) showing that OECD countries can sustain debt-GDP ratios even in excessof 200%. We �nd that an assistance program implementing the constrained optimum yields largewelfare gains, equivalent to a transfer of 45% of the initial GDP with a zero expected cost for theinstitution running the assistance program.
1.1 Literature review
Our paper relates to several streams of the literature on sovereign debt. By focusing on Markovequilibria, we abstract from reputational mechanisms, being close in the spirit to the direct-punishmentapproach proposed by Bulow and Rogo¤ (1989).3 Our work is related to the more recent quantitativemodels of sovereign default such as Aguiar and Gopinath (2006) and Arellano (2008). Yue (2010)considers, as we do, the possibility of renegotiation, although in her model renegotiation is costly andis determined by Nash bargaining between creditors and debtors - with no stochastic shocks to outsideoptions. In her model, ex-post ine¢ cient restructuring helps ex-ante discipline and provides incentivesto honor the debt.4 This literature does not consider the e¢ cient allocation nor economic reforms.Moreover, we pursue an analytical characterization of the properties of the model, whereas the mainfocus of this literature has been quantitative.
In terms of reform e¤ort, our paper is related to Krugman (1988), Atkeson (1991) and Jeanne(2009). Krugman (1988) showed that when a borrower has a large debt, productive investments mightnot be undertaken (the �debt overhang�). His is a static model where debt is exogenous. Atkeson(1991) studies an environment in which an in�nitely-lived borrower faces a sequence of two-periodlived lenders. The borrower can use funds to invest in productive future capacity or to consume thefunds. However, the lenders cannot observe the allocation to investment or consumption. Our paperdi¤ers from Atkeson�s in three key respects. First, in our model we focus on Markov equilibria wherethe borrower cannot commit the reform e¤ort, but the lender can observe it. This seems a plausibleabstraction in the context of, for example, the European debt crisis. Second, in our theory structuralreforms a¤ect the future stochastic process of income, while his model investments only a¤ect nextperiod�s income. Third, in our model all agents (and the planner) have an in�nite horizon. Theresults are di¤erent. Atkeson (1991) shows that the optimal contract involves capital out�ow from theborrower during the worst aggregate state. Our model predicts instead that in a recession the borrowerkeeps accumulating debt and renegotiates it periodically. Moreover, in our model the constrainedoptimal allocation (though not necessarily the market equilibrium) has non-decreasing consumption.Jeanne (2009) studies an economy where the government takes a policy action that a¤ects the returnto foreign investors (e.g., the enforcement of creditor�s right) but this can be reversed within a timehorizon that is shorter than that at which investors must commit their resources.
Hopenhayn and Werning (2008) study the optimal corporate debt contract between a bank and a
3The distinction between the reputation approach and the punishment approach as the two main conceptual frame-works in the literature on sovereign debt crisis has been introduced recently by Bulow and Rogo¤ (2015).Pioneer contributions to the analysis of debt repudiation based on reputational mechanisms such as the threat of future
exclusion from credit markets include Eaton and Gersovitz (1981), Grossman and Van Huyck (1988), and Fernandez andRosenthal (1989).
4Other papers focusing on the restructuring of sovereign debt include Asonuma and Trebesch (2016), Benjamin andWright (2009), Bolton and Jeanne (2007), Dovis (2014), and Mendoza and Yue (2012).
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risk-neutral borrowing �rm. As we do, they assume that the borrower has a stochastic default cost.Di¤erent from us, they focus on the case when this outside option is not observable to the lender andshow that this implies that default can occur in equilibrium. They do not study reform e¤ort nor dothey analyze the case of sovereign debt issued by a country in recession.
Another recent paper complementary to ours is Conesa and Kehoe (2015). In their theory, undersome circumstances, the government of the indebted country may opt to �gamble for redemption.�Namely, it runs an irresponsible �scal policy that sends the economy into the default zone if therecovery does not happen soon enough. The source and the mechanism of the crisis are di¤erentfrom ours. Their model is based on the framework of Cole and Kehoe (1996, 2000) inducing multipleequilibria and sunspots.
Our paper is related also to the literature on endogenous incomplete markets due to limited en-forcement or limited political commitment. This includes Alvarez and Jermann (2000) and Kehoeand Perri (2002). The analysis of constrained e¢ ciency is related to the literature on competitiverisk sharing contracts with limited commitment, including Thomas and Worrall (1988), Kocherlakota(1996), and Krueger and Uhlig (2006). An application of this methodology to the optimal design ofa Financial Stability Fund is provided by Abraham, Carceles-Poveda, and Marimon (2014). In ourmodel all debt is held by foreign lenders. Recent papers by Broner, Martin, and Ventura (2010),Broner and Ventura (2011), and Brutti and Sauré (2016) study the implications for the incentives todefault of having part of the government debt held by domestic residents. Song et al. (2012) andMüller et al. (2016) focus, as we do, on Markov equilibria to study the politico-economic determina-tion of debt in open economies where governments are committed to honor their debt. An excellentreview of the sovereign debt literature is provided by Aguiar and Amador (2014).
In the large empirical literature, our paper is related to the �nding of Tomz and Wright (2007).Using a dataset for the period 1820�2004, they �nd a negative but weak relationship between economicoutput in the borrowing country and default on loans from private foreign creditors. While countriesdefault more often during recessions, there are many cases of default in good times and many instancesin which countries have maintained debt service during times of very bad macroeconomic conditions.They argue that these �ndings are at odds with the existing theories of international debt. Our theoryis consistent with the pattern they document. In our model, due to the stochastic default cost, countriesmay default during booms (though this is less likely, consistent with the data) and can conversely failto renegotiate their debt during very bad times. Their �ndings are reinforced by Sturzenegger andZettelmeyer (2008) who document that even within a relatively short period (1998-2005) there are verylarge di¤erences between average investor losses across di¤erent episodes of debt restructuring. Theobservation of such a large variability in outcomes is in line with our theory, insofar as the bargainingoutcome hinges on an outside option that is subject to stochastic shocks. Borensztein and Panizza(2009) evaluate empirically the costs that may result from an international sovereign default, includingreputation costs, international trade exclusion, costs to the domestic economy through the �nancialsystem, and political costs to the authorities. They �nd that the economic costs are generally short-lived. Finally, the relationship between consumption and renegotiations is in line with the evidencedocumented by Reinhart and Trebesch (2016), as discussed above. For a thorough review of theevidence, see also Panizza et al. (2009).
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2 The model environment
The model economy is a small open endowment economy populated by an in�nitely-lived representativeagent. The endowments follow a two-state Markov switching process, with realizations w 2 fw; �wg,where 0 < w < �w. We label the two endowment states, respectively, recession and normal times.Normal times is assumed to be an absorbing state. If the economy starts in a recession, it switches tonormal times with probability p and remains in the recession with probability 1� p. The assumptionthat normal times is an absorbing state aids tractability and allows us to focus sharply on the salientaspects of a recovery from a deep economic crisis. A benevolent government can implement a costlyreform policy to increase the probability of a recovery. In our notation, p is both the reform e¤ortand the probability that the recession ends.
The preferences of the representative agent are described by the following expected utility function:
E0X
�t�u (ct)� �Ifdefault in tg �X (pt)
�:
The utility function u is twice continuously di¤erentiable and satis�es limc!0 u(c) = �1, u0 (c) > 0, and u00 (c) < 0. I 2 f0; 1g is an indicator switching on when the economy is in a default stateand � is a stochastic default cost assumed to be i.i.d. over time and to be drawn from the p.d.f.f (�) with an associated c.d.f. F (�) :We assume that F (�) is continuously di¤erentiable everywhere,and denote its support by @ � [0; �max] � R+, where �max < 1. The assumption that shocks areindependent is inessential, but aids tractability. X is the cost of reform, assumed to be an increasingconvex function of the probability of exiting recession, p 2 [p; �p] � [0; 1]. X is assumed to be twicecontinuously di¤erentiable, with the properties that X
�p�= 0; X 0 (p) > 0 and X 00 (p) > 0. In normal
times, X = 0.To establish a benchmark, we characterize the optimal allocation under full insurance and full
enforcement. The economy is assumed to start with an outstanding obligation of b given an implicitgross rate of return of R = ��1. The �rst best entails perfect insurance: the country enjoys a constantstream of consumption and exerts a constant reform e¤ort during recession (during normal times,there is no e¤ort). The level of b lowers consumption and increases reform e¤ort in recession.
Proposition 1 Let WFB (b; w) ; cFB (b; w) and pFB (b) denote, respectively, the discounted utility,consumption and e¤ort as a function of the outstanding obligation b, with w 2 fw; �wg denoting theinitial state of productivity. Then, for an economy starting in normal state:
cFB (b; �w) = �w � (1� �) b; WFB (b; �w) =u�cFB (b; �w)
�1� � :
For an economy starting in recession:
cFB (b; w) =(1� �)w + �pFB (b) �w1� � (1� pFB (b)) � (1� �) b;
WFB (b; w) =u�cFB (b; w)
�1� � �
X�pFB (b)
�1� � (1� pFB (b))
where pFB (b) is the reform e¤ort exerted for as long as the economy stays in recession. pFB (b) is theunique solution for pFB satisfying the following condition:
�
1� � (1� pFB)
0B@ ( �w � w)� u0�cFB (b; w)
�| {z }increase in output if econ. recovers
+ X�pFB
�| {z }saved e¤ort cost if econ. recovers
1CA = X 0 �pFB� : (1)
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Moreover, when e¤ort is interior, cFB (b; w) and pFB (b) are, respectively, decreasing and increasingfunctions of b.
3 Laissez-faire equilibrium
In the laissez-faire equilibrium, the benevolent government can issue a one-period bond (sovereigndebt) to smooth consumption. The bond, b, is a claim to one unit of the next-period consumptiongood, which sells today at the price Q (b; w). Bonds are purchased by a representative risk neutralforeign creditor who has access to an international risk-free portfolio paying the world interest rate R.For simplicity, we focus on the case in which �R = 1, although our main insights would carry over tothe case in which �R < 1. After issuing debt, the country decides its reform e¤ort.
The key assumptions are that (i) the country cannot commit to repay its sovereign debt, and(ii) the reform e¤ort is not contractible. At the beginning of each period, the government observesthe realization of the default cost �; and decides whether to repay the debt that reaches maturityor to announce default on all its debt. The cost � is publicly observed, and captures in a reducedform a variety of shocks including both taste shocks (e.g., the sentiments of the public opinion aboutdefaulting on foreign debt) and institutional shocks (e.g., the election of a new prime minister, a newcentral bank governor taking o¢ ce, the attitude of foreign governments, etc.).5 If a country defaults,no debt is reimbursed.6
When the government announces its intention to default, a syndicate of creditors can make atake-it-or-leave-it renegotiation o¤er that we assume to be binding for all creditors. By accepting therenegotiation o¤er, the government averts the default cost. In equilibrium, a haircut is o¤ered onlyif the default threat is credible, i.e., if the realization of � is su¢ ciently low to make the countryprefer default to full repayment. When they o¤er renegotiation, creditors make the debtor indi¤erentbetween an outright default and the proposed haircut.
In summary, the timing is as follows: The government enters the period with the pledged debt b,observes the realization of w and �, and then decides whether to announce default. If the threat iscredible, the creditors o¤er a haircut. Next, the country decides whether to accept or decline the o¤er.Then, the government issues new debt subject to the period budget constraint Q� b0 = b+ c�w. Fortechnical reasons we also impose that debt is bounded, b 2 [b;~b] where b 2 (�1; 0] and ~b = �w= (R� 1)is the natural borrowing constraint. In equilibrium, these bounds will never be binding. If the countrycould commit to honor its debt, it would sell bonds at the price Q = 1=R. However, due to the riskof default or renegotiation, it sells at a discount, Q � 1=R. Next, consumption is realized, and �nallythe government decides its reform e¤ort.
5Alternatively, � could be given a politico-economic interpretation, as re�ecting special interests of the groups inpower. For instance, the government may care about the cost of default to its constituency rather than to the populationat large. In the welfare analysis, we stick to the interpretation of a benevolent government and abstract from politico-economic factors, although the model could be extended in this direction.
6For simplicity, we assume that � captures all costs associated with default. In an earlier version of this paper, weassumed that the government could not issue new debt in the default period, but were allowed to start issuing bondsalready in the following period. The results are unchanged. One could even consider richer post-default dynamics, suchas prolonged or stochastic exclusion from debt markets. Since outright default does not occur in equilibrium, the detailsof the post-default dynamics are immaterial.
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3.1 De�nition of Markov equilibrium
In the characterization of the laissez-faire equilibrium, we restrict attention to Markov-perfect equilib-ria where the set of equilibrium functions only depend on the pay-o¤ relevant state variables, b and �.This rules out that the government�s decisions can be a¤ected by the desire to establish or maintaina reputation.
De�nition 1 A Markov-perfect equilibrium is a set of value functions fV;Wg, a threshold renegoti-ation function �, an equilibrium debt price function Q, a set of optimal decision rules fB; B; C;g,equilibrium law of motions of debt B and of the aggregate state w; such that, conditional on the state
vector (b; �; w) 2�[b;~b]� [0; �max]� fw; �wg
�, the sovereign and the international creditors maximize
utility, and markets clear. More formally:
� The value function V satis�es
V (b; �; w) = max fW (b; w) ;W (0; w)� �g ; (2)
where W (b; w) is the value function conditional on the debt level b being honored,
W (b; w) = maxb02[b;~b]
u�Q�b0; w
�� b0 + w � b
�+ Z
�b0; w
�;
and where Z is de�ned as
Z�b0; w
�= max
p2[p;�p]
��X (p) + �
�p� E
�V�b0; �0; �w
��+ (1� p)� E
�V�b0; �0; w
���; (3)
Z�b0; �w
�= �E
�V�b0; �0; �w
��(4)
and E�V�x; �0; w
��=R@ V (x; �;w) dF (�).
� The threshold renegotiation function � satis�es
� (b; w) =W (0; w)�W (b; w) : (5)
� The debt price function satis�es the following arbitrage conditions:
Q (b; �w) = Q̂ (b; �w) (6)
Q (b; w) = (b)� Q̂ (b; �w) + [1�(b)]� Q̂ (b; w) (7)
where Q̂ (b; w) is the bond price conditional on next period being in state w,
Q̂ (b; w) � 1
R(1� F (� (b; w))) + 1
R
1
b
Z �(b;w)
0b̂ (�;w)� f (�) d�; (8)
and where b̂ (�;w) is the new debt after a renegotiation given a realization �. b̂ is implicitly
de�ned by the condition W�b̂ (�;w) ; w
�=W (0; w)� �:
� The set of optimal decision rules comprises:
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1. A take-it-or-leave-it debt renegotiation o¤er:
B (b; �; w) =�b̂ (�;w) if � � � (b; w) ;b if � > � (b; w) :
(9)
2. An optimal debt accumulation and an associated consumption decision rule:
B (B (b; �; w) ; w) = arg maxb02[b;~b]
�u�Q�b0; w
�� b0 + w � B (b; �; w)
�+ Z
�b0; w
�; (10)
C (B (b; �; w) ; w) = Q (B (B (b; �; w) ; w) ; w)�B (B (b; �; w) ; w) + w � B (b; �; w) : (11)
3. An optimal e¤ort decision rule:
�b0�= arg max
p2[p;�p]
��X (p) + �
�p� E
�V�b0; �0; �w
��+ (1� p)� E
�V�b0; �0; w
���: (12)
� The equilibrium law of motion of debt is b0 = B (B (b; �; w) ; w) :
� The probability that the recession ends is p = (b0).
V and W denote the value functions of the benevolent government. Equation (2) states that when� > � (b; w) there is no renegotiation, otherwise the government renegotiates its debt. Since, ex-post,creditors have all the bargaining power, the discounted utility accruing to the sovereign equals thevalue that she would get under outright default. Thus,
V (b; �; w) =
�W (b; w) if b � b̂ (�;w) ;
W (0; w)� � if b > b̂ (�;w) :
Consider, next, the equilibrium debt price function. Since creditors are risk neutral, the expectedrate of return on the sovereign debt must equal the risk-free rate of return. Then, the arbitrageconditions (6)�(7) ensure market clearing in the bond market and pin down the equilibrium bondprice in normal times and recession, respectively. The function Q̂ de�ned in equation (8) yields thebond price after the state w has realized but before knowing �:With probability 1�F (� (b; w)) debtis honored, where � (b; w) denotes the threshold default shock realization such that, conditional onthe debt b; the government cannot credibly threaten to default for all � � � (b; w). With probabilityF (� (b; w)), debt is renegotiated to a level that depends on the realization of �: This level is givenby b̂ (�;w) which, recall, denotes the renegotiated debt level that keeps the government indi¤erentbetween accepting the creditors� o¤er and defaulting. In the rest of the paper, we use the morecompact notation EV (b; w) � E [V (b; �; w)] and EV (b0; w) � E
�V�b0; �0; w
��:
Consider, �nally, the set of decision rules. (9) stipulates that creditors always extract the entiresurplus at the renegotiation stage. Equations (10)-(11) yield the optimal consumption-saving decisionssubject to a resource constraint. Equation (12) yields the optimal e¤ort decision. Note that the e¤ortexerted depends on b0, since e¤ort is chosen after the new debt is issued.
3.2 Existence of a laissez-faire Markov equilibrium
The crux for proving the existence of a Markov equilibrium lies in establishing the existence of thevalue functionW: Once this is done, all the equilibrium functions (V;�; b̂; Q; Z;B; B;) can be derivedfrom the set of de�nitions above. Before proving the existence of W; we establish an intuitive propertylinking b̂ and �:
8
Lemma 1 Suppose a value functionW (b; w) exists and is strictly decreasing in b. Then, b̂ (� (b; w) ; w) =b: Moreover, � (b; w) is strictly increasing in b, hence, b̂ (�; �w) = ���1 (�) and b̂ (�;w) = ��1 (�), where�� (b) � � (b; �w) ; and � (b) � � (b; w) :
The lemma follows from the de�nitions of b̂ and �. On the one hand, b̂ (�;w) is the debt level that,conditional on �, makes the debtor indi¤erent between honoring and defaulting. On the other hand,� (b; w) is the realization of � that, conditional on b, makes the debtor indi¤erent between honoringand defaulting.
The next proposition establishes the existence of a Markov equilibrium. The proof establishes thatthe value function W is a �xed-point of a monotone mapping following Theorem 17.7 in Stokey andLucas (1989).
Proposition 2 There exists a Markov equilibrium, i.e., a set of equilibrium functions (V;W;�; Q;B; B; C;)satisfying De�nition 1. The value functions V and W are continuous and non-increasing in b. Theequilibrium functions �; Q and are also continuous in b: The bond revenue, Q(x;w)x, is non-decreasing in x: The policy function B (b; w) is non-decreasing in b.
Proposition 2 establishes the existence but not the uniqueness of the Markov equilibrium. Thecorollary of Theorem 17.7 in Stokey and Lucas (1989) provides a strategy to verify numerically whetheran equilibrium is unique. We state this as Corollary 1 in the appendix.
The next proposition establishes local di¤erentiability properties of the value function W; and thatthe �rst-order conditions are necessary. Due to the possibility that debt is renegotiated, the valuefunctions are not necessarily concave. Nevertheless, we establish that the equilibrium functions aredi¤erentiable at all debt levels that can result from an optimal choice given some initial debt level.We de�ne formally the set of such debt levels as B(w) = fx 2 [b;~b] j B (b; w) = x; for some b 2 [b;~b]g.7
Proposition 3 The equilibrium functions W (b; w), Z(b; w); �(b; w), Q(b; w), (w) are di¤erentiablefor all b 2 B(w). Moreover, for any b0 2 B(w); the �rst-order condition (@=@b0)u (Q(b0; w)b0 + w � b)+(@=@b0)Z (b0; w) = 0 and the envelope condition @W (b0; w)=@b0 = �u0 (C (b0; w)) hold true.
The proof is an application of the envelope theorem of Clausen and Strub (2013) that appliesto problems including endogenous functions such as default probabilities and interest rates (see alsoArellano et al. 2014).
3.3 Equilibrium in normal times
In this section, we characterize the equilibrium when the economy is in normal times (w = �w), inwhich case there is no aggregate uncertainty. The next lemma establishes properties of the thresholdand of the debt revenue function, Q (x; �w)x at the optimal interior debt choice.
Lemma 2 The debt revenue function Q(b0; �w)b0 is concave in b0 2 [b;~b] and strictly increasing anddi¤erentiable for all b0 2 B(w) such that b < �b. Moreover, @ (Q (b0; �w) b0) =@b0 = R�1 (1� F (� (b0; �w))).
7We prove later that during normal times the equilibrium functions are di¤erentiable everywhere. However, this isnot true in recession. In this case Proposition 3 shows that di¤erentiability still holds for all b that can be attained asan optimal choice.
9
An immediate implication of the lemma is that if we de�ne �b to be the lowest debt inducingrenegotiation almost surely (i.e., such that limb0!�b F (�(b
0; �w)) = 1), then, �b is also the top of the La¤ercurve, i.e., the endogenous debt limit. More formally, �b = argmaxb2[b;~b] fQ (b; �w) bg < ~b: Although theborrower could issue debt exceeding �b, the marginal debt revenue would be zero for b > �b since thisdebt would never be honored.
We now study consumption and debt dynamics. We introduce a de�nition that will be usefulthroughout the paper.
De�nition 2 A Conditional Euler Equation (CEE) describes the (expected) marginal rate of substitu-tion between current and next-period consumption in all states of nature �0 that induce the governmentto honor its debt next period.
Next, we characterize formally the CEE. The government solves the consumption-saving problemgiven by (10). The �rst-order condition and the envelope theorem yield the following proposition.
Proposition 4 If the realization of �0 induces no renegotiation, then the following CEE holds true:
�Ru0 (c0jH; �w)u0 (c)
= 1; (13)
where c = C (B (b; �; �w) ; �w) is current consumption and c0jH; �w = C (b0; �w) = C (B (B (b; �; �w) ; �w) ; �w)is next-period consumption. Since �R = 1; then b0 = B (b; �w) = b, and consumption remains constant.Moreover, for all b < �b, the value function W (b; �w) is strictly decreasing, strictly concave and twicecontinuously di¤erentiable in b, and consumption C(b; �w) is strictly falling in b.
Although the CEE (13) resembles a standard Euler equation under full commitment, the similarityis deceiving: R is not the ex-post interest rate when debt is fully honored; the realized interest rate isin fact higher due to the default premium.
When debt is renegotiated, consumption increases discretely, hence u0 (ct) =u0 (ct+1) > �R. This isnot surprising, since the country bene�ts from a reduction in debt repayment.8 Thus, consumptionand debt are, respectively, increasing and decreasing step functions over time: they remain constantin every period in which the country honors its debt, while changing discretely upon every episodeof renegotiation. Figure 1 illustrates a simulation of the consumption and debt dynamics. Note thatthe sequence of renegotiations eventually brings the debt to a su¢ ciently low level where the riskof renegotiation vanishes. This consumption path is di¤erent from the �rst-best allocation whereconsumption and debt are constant for ever. Interestingly, in the long run, consumption is higher inthe market equilibrium with the risk of repudiation than in the �rst best allocation.
It is straightforward to generalize the results to the case of �R < 1 under the assumption thatutility features constant relative risk aversion. In this case, when the debt is honored debt wouldincrease and consumption would fall. After each episode of renegotiation the economy would startagain accumulating debt. In a world comprising economies with di¤erent �; e.g., some with �R = 1and some with �R < 1; economies with low � would experience recurrent debt crises.
8The prediction that whenever debt is renegotiated consumption increases permanently is extreme, and hinges on theassumptions that �R = 1 and that � is i.i.d. with a known distribution. In Section 6 we extend the model to a settingwhere there is uncertainty about the true distribution of � and the market learns about this distribution by observingthe sequence of ��s. In this case, a low realization of � has two opposing e¤ects on consumption: on the one hand, alow � triggers debt renegotiation which on its own would increase consumption; on the other hand, a low � a¤ects thebeliefs about the distribution of �, inducing the market to regard the country as less creditworthy (namely, the countrydraws from a distribution where low � is more likely). This tends to increase the default premium on bonds and to lowerconsumption.
10
Time1 5 10 15 20 25 30
Deb
t
0
0.2
0.4
0.6
0.8
1
1.2
Con
sum
ptio
n
0.92
0.94
0.96
0.98
1
1.02DebtConsumption
Figure 1: Simulation of debt and consumption for a particular sequence of ��s during normal times.
3.4 Equilibrium under recession
When the economy is in recession the government chooses, sequentially, whether to honor the currentdebt, how much new debt to issue, and how much reform e¤ort to exert. In this section, we assumethat the government cannot issue state-contingent debt, i.e., securities whose payment is contingenton the aggregate state of the economy. In Section 5.1 below we relax this restriction.
A natural property of the laissez-faire equilibrium allocation is that C (b; w) < C (b; �w) for allb � �b: conditional on honoring a giving debt level, consumption is higher in normal times than in arecession. Although we could �nd no numerical counterexample to this property, it is di¢ cult to proveit in general because the equilibrium functions for consumption, e¤ort and debt price are determinedsimultaneously. However, we can provide a su¢ cient condition in terms of the primitive parametersof the model.
Proposition 5 The following conditions are su¢ cient to ensure that C (b; w) < C (b; �w) for all b 2�0;�b�: (i) �w � w > �
1�� �w; and (ii) F [(u ( �w)� u ((1� �) ( �w � w))) = (1� �)] = 0.
Conditional on the consumption ranking, it is straightforward to show, using De�nition 1, that�(b; w) > �(b; �w), b̂(�;w) < b̂(�; �w), Q(b; w) < Q(b; �w) and W (b; w) < W (b; �w): Note, in particular,that the price of the bond increases if the recession ends. The reason is that the probability ofrenegotiation decreases as the economy switches into normal times. The property that �(b; w) >�(b; �w) implies that one can partition the state space into three regions:
- if b < b�; the country honors the debt with a positive probability, irrespective of the aggregatestate (the probability of renegotiation being higher if the recession continues than if it ends);9
- if b 2�b�;�b
�; the country renegotiates with probability one if the recession continues, while it
honors the debt with a positive probability if the recession ends;9b� is implicitly determined by the equation W
�b�; w
�=W (0; w)� �max.
11
- if b > �b; the country renegotiates its debt with probability one, irrespective of the aggregatestate.
Note that the risk of repudiation introduces some elements of state contingency, since debt isrepaid with di¤erent probabilities under recession and normal times.
3.4.1 Reform e¤ort in equilibrium
We denote by (b0) the equilibrium policy function for e¤ort, i.e., the probability that the recessionends next period, as a function of the newly-issued debt. More formally, the �rst-order condition from(12) yields:
X 0 � �b0�� = � �Z 1
0V�b0; �0; �w
�dF (�)�
Z 1
0V�b0; �0; w
�dF (�)
�: (14)
The continuity and monotonicity of X; together with the continuity of the value functions andthe continuity of the c.d.f. F ensure that is a continuous function. Intuitively, the government hasan incentive to do some reform e¤ort because a recovery will increase expected utility. The gain isthe di¤erence in expected utility between recovery and recession, and the larger this di¤erence, thelarger will the reform e¤ort be. However, e¤ort is not provided e¢ ciently. To see why, recall that thebond price increases upon economic recovery. Thus, the creditors reap part of the gain from economicrecovery, whereas the country bears the full burden of the e¤ort cost.
We can prove that e¤ort is ine¢ ciently provided with the aid of a simple one-period deviationargument. Consider an equilibrium e¤ort choice path consistent with (14) �corresponding to the caseof non-contractible e¤ort. Next, suppose that, only in the initial period, the country could contracte¤ort before issuing new debt. As it turns out, the country would choose a higher reform e¤ort in the�rst period than in the laissez-faire equilibrium. We state this result as a lemma.
Lemma 3 Suppose that b0 > 0 and that the borrower could, in the initial period, commit to an e¤ortlevel upon issuing new debt, then the reform e¤ort would be strictly larger than in the case in whiche¤ort is never contractible.
If the government could commit to reform, its reform e¤ort would be monotone increasing in thedebt level, since a high debt increases the hardship of a recession. However, under moral hazard, theequilibrium reform e¤ort exhibits a non-monotonic behavior. More precisely, (b) is increasing atlow levels of debt, and decreasing in a range of high debt levels, including the entire region
�b�;�b
�.
Proposition 6 establishes this result more formally.
Proposition 6 There exist three ranges, [0; b1] � [0; b�]; [b2;�b] � [b�;�b]; and��b;1
�such that:
1. If b 2 [0; b1) ; 0 (b) > 0;
2. If b 2�b2;�b
�; 0 (b) < 0;
3. If b 2��b;1
�; 0 (b) = 0.
The following argument establishes the result. Consider a low (possibly negative) debt rangewhere the probability of renegotiation is zero. In this range, there is no moral hazard. Thus, a higherdebt level has a disciplining e¤ect, i.e., it strengthens the incentive for economic reforms: due to the
12
concavity of the utility function, the discounted gain of leaving the recession is an increasing functionof debt.
As one moves to a larger initial debt, however, moral hazard becomes more severe, since thereform e¤ort decreases the probability of default, and shifts some of the gains to the creditors. Thisis reminiscent of the debt overhang e¤ect in Krugman (1988). The debt overhang dominates over thedisciplining e¤ect in the region [b�;�b]. In this range, debt has a stark state contingency. If the economyremains in recession, debt is renegotiated for sure, so the continuation utility
R10 V
�b0; �0; w
�dF (�)
is independent of b. If the recession ends, the continuation utility is decreasing in b. Therefore, in thisregion the value of reform e¤ort necessarily declines in b. By continuity, the same argument appliesto a range of debt below b�. Finally, when b > �b; debt is always renegotiated, and thus the gain fromleaving the recession is independent of b.10
Note, �nally, that the debt-overhang e¤ect hinges on the presence of some renegotiation risk andan associated premium on debt. If the borrower instead could commit to repay the debt, the priceof debt would be 1=R regardless of the aggregate state, so an economic recovery would not yield anybene�ts to the lenders. Consequently, there would be no moral hazard in the e¤ort choice and thee¤ort function would be monotone increasing in debt.
3.4.2 Debt issuance and consumption dynamics
In this section we characterize the dynamics of consumption and debt. We proceed in two steps. First,we derive the properties of the CEE. Then we summarize its characterization in a formal proposition.
The government solves the problem (10) for w = w. Using the �rst-order condition together withthe envelope theorem yields the following CEE:
E
�MUt+1MUt
jdebt is honored at t+ 1�
(15)
= 1 +0 (bt+1)
Pr (debt is honored at t+ 1)RhQ (bt+1; �w)� Q̂ (bt+1; w)
ibt+1:
Equation (15) is the analogue of (13). There are two di¤erences. First, the left-hand side has theexpected ratio between the marginal utilities, due to the uncertainty about the future aggregate state.Second, there is a new term on the right-hand side capturing the e¤ect of debt on reform e¤ort.
For expositional purposes, consider �rst the case in which the probability that the recession endsis exogenous, i.e., 0 (:) = 0. In this case, the CEE requires that the expected marginal utility beconstant. For this to be true, conditional on debt being honored, consumption growth must be positiveif the recession ends, and negative if the recession continues. The lack of consumption insurance stemsfrom the incompleteness of �nancial markets, and would disappear if the government could issue state-contingent bonds. However, this conclusion does not carry over to the economy with moral hazard,as we show in Section 5.1 below.
Consider, next, the general case. Moral hazard introduces a new strategic motive. By changingthe level of newly-issued debt, the government manipulates its own ex-post incentive to make reforms.The sign of this strategic e¤ect is ambiguous, and hinges on the sign of 0 (see Proposition 6). Whenthe outstanding debt is low, 0 > 0. Then, more debt strengthens the ex-post incentive to reform,thereby increasing the price of the newly-issued debt. The right-hand side of (15) is in this case larger
10 In a variety of numerical simulations, we have always found to be hump-shaped with a unique peak (see Figure3), although in general this depends on the distribution F (�).
13
than unity, and the CEE implies a lower consumption fall (hence, higher debt accumulation) thanin the absence of moral hazard. In contrast, in the region of high initial debt, 0 < 0. In this case,it is optimal to issue less debt than in the absence of moral hazard. The reason is that the marketanticipates that a larger debt reduces the reform e¤ort. In response, the government restrains itsdebt issuance strategically in order to mitigate the ensuing fall in the debt price. Thus, when therecession continues, a highly indebted country will obtain less consumption insurance when the reformis endogenous than when p is exogenous.
We summarize the results in a formal proposition.
Proposition 7 If the economy starts in a recession and the realization of �0 induces no renegotia-tion, the optimal debt level, b0 = B (B (b; �; w) ; w) ; induces a consumption sequence that satis�es thefollowing CEE:
�R
0BBBBBBBBBBBB@
�1�
�b0����1� F
���b0���| {z }
prob. of repayment and continuing recession
Pr�Hjb0
�| {z }unconditional prob. of repayment
�u0�c0jH;w
�u (c)| {z }
MRS if rec. cont.
+
�b0���1� F
����b0���| {z }
prob. of repayment and end of recession
Pr�Hjb0
�| {z }unconditional prob. of repayment
� u0 (c0jH; �w)u (c)| {z }
MRS if rec. ends
1CCCCCCCCCCCCA= 1+
0 (b0)
Pr (Hjb0)�R�Q�b0; �w
�� Q̂
�b0; w
��b0| {z }
gain to lenders if recession ends
(16)where c = C (B (b; �; w) ; w) is current consumption, c0jH;w = C (b0; w) = C (B (B (b; �; w) ; w) ; w) isnext-period consumption conditional on w and no renegotiation, and Pr (Hjb0) is the unconditionalprobability that the debt b0 be honored, i.e., Pr (Hjb0) = [1�(b0)] �
�1� F
��� (b0)
��+ (b0) �
(1� F (� (b0))) :
We end this section by noting that the top of the La¤er curve of debt corresponds to a lower debtlevel in recession than during normal times.
Lemma 4 Let �b = argmaxb fQ (b; �w) bg and �bR = argmaxb fQ (b; w) bg. Then, �bR � �b; with equalityholding only if (b) = p (i.e., if the probability of staying in a recession is exogenous).
The reason why the top of the La¤er curve under recession is located strictly to the left of �b isthat the reform e¤ort is decreasing in debt (i.e., 0 < 0) for b close to �b, as established in Proposition6. This implies that for b close to but smaller than �b, bond revenue is strictly decreasing in b. Byreducing the newly-issued debt, the borrower increases the subsequent reform e¤ort, which in turnincreases the current bond price and debt revenue.
3.4.3 Taking stock
The previous sections have established the main properties of the laissez-faire equilibrium. The �rstproperty is that moral hazard induces an ine¢ cient provision of reform e¤ort in equilibrium, especiallyfor high debt levels. Figure 2 shows the e¤ort function (b) in a calibrated economy. Note that thereform e¤ort plunges for high debt levels.11 The hump-shaped e¤ort function contrasts sharply with11This prediction is consistent with the casual observation that in the recent European debt crisis structural reforms
have met stronger opposition in highly indebted countries. Countries with moderate initial debt levels, such as forinstance Spain, have arguably been more prone to enact structural reforms than has Greece.
14
Debt issued, b'0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Ref
orm
Effo
rt,(b
')
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
Figure 2: Reform e¤ort function (b). The parameter values correspond to the calibration in Section7 with � �xed to 0:5.
the optimum e¤ort in Proposition 1. In the �rst best, reform e¤ort is monotone increasing in theinitial debt level, and remains constant over time. The second property is that the possibility ofrenegotiating a non-state-contingent debt may improve risk sharing. This is per se welfare-enhancingbut it exacerbates the moral hazard in reform e¤ort.
The third property is that in periods when debt is fully honored, the equilibrium features positivedebt accumulation if the economy remains in recession, and constant debt when the economy returnsto normal times. An implication of the �rst and third property is that, as the recession persists, thereform e¤ort initially increases, but then, for high debt levels, it declines over time. Figure 3 illustratesa time path for debt and consumption (left panel) and of the corresponding reform e¤ort (right panel)for a particular simulated sequence of ��s. The volatility in consumption and e¤ort contrast sharplywith the optimal allocation of Proposition 1 where consumption and reform e¤ort are constant overtime.
The fourth property concerns post-renegotiation debt dynamics. Debt accumulation resumes im-mediately after the haircut, while consumption increases upon debt relief and start falling againthereafter. This prediction is broadly consistent with the empirical evidence that economic conditionsof debtors improve following a debt relief, as documented in Reinhart and Trebesch (2016). It isalso consistent with the recent debt dynamics of Greece �after the 2011 debt relief, the debt-GDPratio fell from 171% to 157%, but subsequently it increased back to 177%. Interestingly, the theorypredicts that for highly indebted countries a large haircut may enhance the reform e¤ort, contrary tothe common view that pardoning debt would have perverse e¤ects on incentives.
For simplicity we have assumed that the government can only issue one-period non-state-contingentdebt. Issuing debt at multiple maturities could in principle allow the borrower to obtain some addi-tional insurance. In a world without moral hazard, this could complete the markets (cf. Angeletos2002). However, as we show in Section 5.1 below, in our model even an economy with a full set
15
Time1 5 T=10 15
Deb
t
0.4
0.6
0.8
1
1.2
Con
sum
ptio
n
0.5
0.6
0.7
0.8
0.9
1
DebtConsumption
Time1 5 T=10
Ref
orm
Effo
rt
0.13
0.14
0.15
0.16
0.17
0.18
Figure 3: Simulation of debt, consumption and e¤ort for a particular sequence of ��s in the competitiveequilibrium. In this particular siumulation the recession ends at time T = 10.
of state-contingent assets would fail to attain e¢ ciency due to the moral hazard problem associatedwith structural reforms. This mitigates the concern about the loss of generality associated with theassumption that there is only one-period debt.12
4 Constrained Pareto optimum
In this section, we study the constrained e¢ cient allocation in an environment where the plannercannot overrule the limited commitment constraint. We characterize the optimal dynamic contract,subject to limited commitment: the country can quit the contract, su¤er the default cost, and resortto market �nancing. The problem is formulated as a one-sided commitment with lack of enforcement,following Ljungqvist and Sargent (2012) and based on a promised-utility approach in the vein of Spearand Srivastava (1987), Thomas and Worrall (1988 and 1990) and Kocherlakota (1996). Allowing two-sided limited commitment, where even the planner could terminate the contract, would yield identicalresults, since in our environment the planner has never an incentive to quit.
We assume that the planner has full information: she observes the realization of � (as does themarket) and can control e¤ort. This is a useful benchmark for two reasons. First, it simpli�es theanalysis and makes for sharper results. It is well-known that solving models involving both incompleteinformation and limited commitment is involved. Second, from an applied perspective, an internationalagency might observe reforms but have limited instruments to prevent sovereign debt renegotiation.While in this section we empower the planner with the ability to actually enforce the desired e¤ort
12 In addition, we conjecture that if we extened the model to allow the borrower to issue debt at multiple maturities,it would only issue one-period debt in steady state in order to limit the extent of moral hazard. Aguiar and Amador(2013) reach a similar conclusion in a di¤erent model. From an empirical standpoint, Broner, Lorenzoni, and Schmukler(2013) document that in emerging markets governments issue mostly short term debt.
16
level, in Section 4.2 below we relax this assumption and consider an environment where e¤ort mustbe self-enforcing.
We denote by � the utility promised to the risk-averse agent in the beginning of the period, beforethe realization of �. � is the key state variable of the problem. We denote by �!� and !� the promisedcontinuation utilities conditional on the realization � and on the aggregate state �w and w, respectively.P (�) and �P (�) denote the expected present value of pro�ts accruing to the principal conditional ondelivering the promised utility � in the most cost-e¤ective way in recession and in normal times,respectively. The planning problem is evaluated after the uncertainty about the aggregate state hasbeen resolved (i.e., the economy is either in recession or in normal times in the current period), butbefore the realization of � is known.
In Proposition 14 in the online appendix we prove, following the strategy in Thomas and Wor-rall (1990), that the functional equations de�ned in equations (17) and (22) below are contractionmappings, that the pro�t functions P (�) and �P (�) are decreasing, strictly concave and continuouslydi¤erentiable, and that the associated maximands are unique.
4.1 Constrained e¢ ciency
In normal times, the optimal value �P (�) satis�es the following functional equation:
�P (�) = maxfc�;�!�g�2@
Z@
��w � c� + � �P (�!�)
�dF (�) ; (17)
where the maximization is subject to the constraintsZ@[u (c�) + ��!�] dF (�) � �; (18)
u (c�) + ��!� � �� � �; � 2 @; (19)
c� 2 [0; �w]; �; �!� 2 [�� � E [�] ; ��]:
Here, �� is the value of the outside option for the agent. We return to the interpretation of �� below.(18) is a promise-keeping constraint, whereas (19) is a participation constraint (PC).
The application of recursive methods allows us to establish the following proposition.
Proposition 8 Assume the economy is in normal times. (I) For all realizations � such that the PCof the agent, (19), is binding, �!� > � and the solution for (c�; �!�) is determined by the followingconditions:
u0 (c�) = �1
�P 0 (�!�); (20)
u (c�) + ��!� = �� � �: (21)
The solution is not history-dependent, i.e., the initial promise, �; does not matter. (II) For all real-izations � such that the PC of the agent, (19), is not binding, �!� = � and c� = c (�) : The solution ishistory-dependent.
The e¢ cient allocation has standard properties. Whenever the agent�s PC is not binding, consump-tion and promised utility remain constant over time. Whenever the PC binds, the planner increasesthe agent�s consumption and promised utility in order to meet her PC.
17
Next, we consider an economy in recession. The principal�s pro�t obeys the following functionalequation:
P (�) = maxfc�;p�;�!�;!�g�2@
Z@
�w � c� + �
�(1� p�)P
�!��+ p� �P (�!�)
��dF (�) ; (22)
where the maximization is subject to the constraintsZ@
�u (c�)�X (p�) + �
�(1� p�)!� + p��!�
��dF (�) � �; (23)
u (c�)�X (p�) + ��(1� p�)!� + p��!�
�� � � �; � 2 @; (24)
c� 2 [0; �w]; p� 2 [p; �p]; �; !� 2 [� � E [�] ; �]; �!� 2 [�� � E [�] ; ��];
and where � is the outside option for the agent of breaking the contract when the economy is inrecession. Note that there are two separate promised utilities, !� and �!�; associated with the twopossible realizations of the aggregate state in the next period. The following proposition can beestablished.
Proposition 9 Assume the economy is in recession. (I) For all realizations � such that the PC ofthe agent, (19), is binding, the solution for
�c�; p�; !�; �!�
�is determined by the following conditions:
u0 (c�) = � 1
P 0�!�� ; (25)
� � � = u (c�)�X (p�) + ��(1� p�)!� + p��!�
�; (26)
P 0�!��= P 0 (�!�) ; (27)
X 0 (p�) = ��u0 (c�)
��P (�!�)� P
�!���+��!� � !�
��: (28)
The solution is not history-dependent, i.e., the promised utility � does not matter.(II) For all realizations � such that the PC of the agent, (19), is not binding, !� = �; �!� = �! (�) ;c� = c (�) ; and p� = p (�) : The solution is history-dependent. The reform e¤ort is decreasing in thepromised utility level. (III) For all � 2 @; �!� > !�.
When the agent�s PC is slack, consumption, reform e¤ort, and promised utilities remain constant.Every time the PC binds, the planner increases the promised utilities, and grants the agent an increasein consumption and a reduction in the reform e¤ort. The agent will, when entering the contract,be o¤ered lower consumption and be required to exercise higher e¤ort, compared to the �rst bestallocation. The conditions the agent faces improve subsequently every time the PC binds. Note that,if we compare two countries entering the contract with di¤erent initial promised utilities, the countrywith a lower promised utility earns a lower consumption and is asked to exercise higher e¤ort. Thus,the country with the lower promised utility is expected to recover faster from the recession.
As the recession ends, the promised utility increases (cf. part III of Proposition 9) and e¤ort goesto zero. Consumption may either remain constant or increase depending on whether the PC binds.Interestingly, the set of states such that the PC binds expands. Namely, there are realizations of � suchthat consumption rises only if the recession ends. In contrast, for su¢ ciently large ��s, the agent�s PCis binding irrespective of whether the recession continues or ends. In this case, consumption remainsconstant. In other words, because of limited commitment, the agent is o¤ered partial insurance againstthe continuation of the recession.
18
Time1 5 T=10 15
Con
sum
ptio
n
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
Ref
orm
Effo
rt
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
ConsumptionReform Effort
Time1 5 T=10 15
Pro
mis
edU
tility
, Rec
essi
on
22.15
22.05
21.95
21.85
21.75
21.65
Pro
mis
edU
tility
, Nor
mal
Tim
es
1.85
1.845
1.84
1.835
1.83
RecessionNormal Times
Figure 4: Simulation of consumption, e¤ort, and promised utilities for a particular sequence of ��s inthe constrained optimum. In this simulation the recession ends at time T = 10.
Figure 4 illustrates an example of the time path of consumption and e¤ort (left panel) and of thecorresponding promised utilities (right panel) in the constrained e¢ cient allocation when the economystarts in a recession.
4.2 Self-enforcing reform e¤ort
Thus far we have assumed that the planner can dictate the reform e¤ort during recession as long as theagent stays within the contract.13 In this section, we consider an alternative environment where theplanner can verify reform e¤ort only at the end of each period. More formally, the allocation is identicalto the solution to the planning problem (22)�(24) subject to the additional incentive constraint (IC)stipulating that, for all � 2 @,
�X (p�) + ��(1� p�)!� + p��!�
�� �� � �; (29)
where � � 0 is an exogenous direct punishment that the planner can in�ict on the agent when shedoes not exert the prescribed reform e¤ort (note that in this case the country pays no default cost).
If the IC (29) is slack, the allocation of Proposition 9 is not susceptible to pro�table deviations,and the solution is as in Proposition 9. Note that, if the IC constraint is not binding at t, it will neverbind in future, since the allocation of Proposition 9 entails a non-decreasing promised utility and anon-increasing e¤ort path. The following Lemma establishes properties of the constrained allocationwhen the IC is binding.
13Assuming that reforms are observable seems natural to us. It is possible, for instance, to verify whether a countryintroduces labor market reforms, removes product market regulation, cuts employment in the public sector, or passeslegislative measures to curb tax evasion (e.g., by intensifying tax audits and enforcing penalties). Nevertheless, it maybe di¢ cult to prevent deviations such as delays, lack of implementation, or weak enforcement of agreed-upon reforms.In other words, the agent might accept the consumption decided by the planner but not exert the agreed e¤ort.
19
Lemma 5 When the IC is binding, e¤ort and promised utilities are constant at the levels (p�; !�; �!�),where the triplet is uniquely determined by the equations
� � ~� = �X (p�) + � ((1� p�)!� + p��!�) ; (30)�P 0 (�!�) = P 0 (!�) ; (31)
X 0 (p�) = � (�!� � !�)� �
P 0 (!�)
��P (�!�)� P (!�)
�; (32)
where ~� � � � (1� �) �; and the pro�t functions �P and P are de�ned as in Section 4.
Equation (30) yields the IC when it holds with equality. Equations (31) and (32) then followfrom the FOCs (see equations (66)-(68) in the appendix). These two conditions hold true regardlessof whether or not the IC constraint is binding.14 Although the pro�t functions of the problem withan IC constraint generally di¤er from those of the problem without IC constraint, we prove that thepro�t functions coincide when evaluated at the promised utilities !� and �!�.
The following proposition characterizes the equilibrium dynamics.
Proposition 10 Suppose that the economy starts in a recession, and is endowed with the initialpromised utility �.
1. If � � !�, then the IC is never binding, and the constrained optimal allocation,�c�; p�; !�; �!�
�;
is identical to that in Proposition 9.
2. If � < !�; then there exist two thresholds, �� and ~�(�); where �� = ~�(!�) (expressions in theproof in the appendix) such that:
(a) If � < ��, the PC is binding while the IC is not binding. The solution is not history-dependent and is determined as in Proposition 9 (in particular, !� > !
� and p� < p�).
(b) If � 2 [��; ~�(�)], both the PC and the IC are binding. E¤ort and promised utilities areequal to (p�; !�; �!�) as given by Lemma 5. Consumption is determined by equations (24)and (29) which yield:
c�� = u�1�~� � �
�: (33)
Consumption and e¤ort are lower than in the allocation of Proposition 9.
(c) If � > ~�(�), the IC is binding, while the PC is not binding. E¤ort and promised utilities areequal to (p�; !�; �!�) : The consumption level is determined by the promise-keeping constraint(23). In particular, consumption is constant across � and given by:
c�~�(�) = u�1�~� � ~� (�)
�: (34)
For given � and �, consumption and e¤ort are lower than in the allocation of Proposition9.
14To see why the solution to (29)�(32) is unique, note that the concavity and monotonicity of P and �P imply thatEquation (31) determines a positive relationship between !� and �!�. Thus, Equation (32) yields an implicit decreasingrelationship between p� and !�, while (29) yields an implicit increasing relationship between p� and !�.
20
Time1 5 T=10 15
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sum
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n
0.64
0.66
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0.74
0.76
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orm
Effo
rt
0.13
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Time1 5 T=10 15
Pro
mis
edU
tility
, Rec
essi
on
22.15
22.05
21.95
21.85
21.75
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Pro
mis
edU
tility
, Nor
mal
Tim
es
1.85
1.845
1.84
1.835
1.83
RecessionNormal Times
Figure 5: Simulation of consumption, e¤ort, and promised utilities for a particular sequence of ��swhere the IC is initially binding. Solid lines refer to an economy with an IC constraint. Dashed linesrefer to the economy without an IC constraint. In this particular siumulation the recession ends attime T = 10.
Consider an economy where, initially, � < !�. If the �rst realization of � is su¢ ciently low (case2.a of Proposition 10), the binding PC induces an e¤ort level so low that the IC is not binding.Therefore, the allocation is not history-dependent, and the characterization of Proposition 9 applies.If the �rst realization of � is larger than ��, the IC is binding, and Lemma 5 implies that e¤ort andpromised utility are equal to (p�; !�; �!�), i.e. the maximum e¤ort and the minimum promised futureutilities consistent with the IC. If � 2 [��; ~�(�)] (case 2.b), consumption is pinned down jointly bythe PC and the promise-keeping constraint. In this case, consumption is decreasing in �. Finally, if� > ~�(�) (case 2.c) the PC imposes no constraint, and the initial consumption is determined only bythe promise-keeping constraints. When the IC is binding (cases 2.b and 2.c), both consumption ande¤ort are lower than in the second-best solution of Proposition 9. Intuitively, the planner cannot sete¤ort at the e¢ cient level due to the IC, and adjusts optimally to the constraint by reducing currentconsumption and e¤ort, and increasing promised utilities relative to the case of no IC constraint.Thus, the contract provides less consumption insurance but more e¤ort smoothing. After one periodthe equilibrium is characterized as in the constrained optimum of Proposition 9.
Figure 5 is the analogue of Figure 4 in an economy in which the IC is binding in the initial period,i.e., � < !�. It shows simulated paths of consumption, e¤ort and promised utility in the constrainedoptimal allocation for two otherwise identical economies where one economy (solid lines) is subjectto the IC constraint, while the other economy (dashed lines) has no such constraint. The initialpromised utility � (not displayed) is lower than !� implying that the IC is binding. In the �rst period,consumption and e¤ort are lower in the economy with an IC constraint. In contrast, promised utility ishigher. In other words, the planner provides less insurance by making consumption and e¤ort initiallylower, but growing at a higher speed. As of the second period, the dynamics of both economies arethe same as in Figure 4.
21
4.3 Comparison between constrained optimum and laissez-faire equilibrium
In this section, we compare the constrained optimum with the laissez-faire equilibrium.15 To thisaim, we assume that the borrower�s outside options corresponds to outright default followed by apermanent reversion to the Markov equilibrium allocation characterized in Section 3. More formally,we let �� = W (0; �w) and � = W (0; w) : We view this as a natural assumption in our environment.The purpose of our analysis is to identify market ine¢ ciencies that could be ameliorated by variousinstitutional arrangements that we discuss below. Our assumption implies that debtors can alwaysquit any institutional arrangement and revert to the market equilibrium.16
The assumption above allows us to derive an important equivalence result: in normal times, theconstrained e¢ cient allocation of Proposition 8 is identical to the laissez-faire equilibrium. To establishthis result we return, �rst, to the laissez-faire equilibrium. Let
�� (b) =�1� F
��� (b)
��b+
Z ��(b)
0b̂ (�; �w) dF (�) (35)
denote the expected value for the creditors of an outstanding debt b before the current-period un-certainty is resolved. Note that �� (b) yields the expected debt repayment, which is lower than theface value of debt, since in some states of nature debt is renegotiated. To prove the equivalence, wepostulate that �� (b) = �P (�) ; and show that in this case � = EV (b; �w) :17 If the equilibrium were notconstrained e¢ cient, the planner could do better, and we would �nd that � > EV (b; �w) :
Proposition 11 Assume that the economy is in normal times and that �� = W (0; �w). The laissez-faire equilibrium is constrained Pareto e¢ cient, namely, �� (b) = �P (�), � = EV (b; �w) :
Intuitively, renegotiation provides the market economy with su¢ ciently many state contingenciesto attain second-best e¢ ciency. This result hinges on two features of the renegotiation protocol. First,renegotiation averts any real loss associated with unordered default. Second, creditors have all thebargaining power in the renegotiation game.18
The e¢ ciency result of Proposition 11 does not carry over to recessions. In the laissez-faire equilib-rium, consumption is falling (and debt accumulates) when the country honors its debt. In contrast, theplanner would insure the agent�s consumption by keeping it constant whenever the PC is not binding.Therefore, the market underprovides insurance. The dynamics of the reform e¤ort also are sharplydi¤erent. In the constrained e¢ cient allocation, e¤ort is a monotone decreasing function of promisedutility which is in turn step-wise increasing over time (cf. Figure 4). In contrast, in the laissez-faireequilibrium the reform e¤ort is hump-shaped in debt. Since debt increases over time (unless it isrenegotiated), e¤ort is also hump-shaped over time conditional on no renegotiation.
15We ignore the incentive constraint discussed in section 4.2, since, as we show, this a¤ects at most the allocation inthe initial period.16 In principle, one could consider more sophisticated punishment strategies. These would a¤ect the initial value of the
outside option without changing the qualitative properties of the constrained optimum allocation.17Recall that EV (b; �w) =
R@ V (b; �; �w) dF (�) denotes the discounted utility accruing to a country with the debt level
b in the competitive equilibrium.18We view this as a useful benchmark. In reality, renegotiations may entail costs associated with legal proceeds and
lawsuits, trade retaliation, temporary market exclusion, etc. Also, creditors may not have the full ex-post bargainingpower at the renegotiation stage as in Yue (2010). This would reduce the amount of loans creditors can recover. In allthese cases, the competitive equilibrium would fail to implement the second best.
22
5 Decentralization
In this section, we discuss policies and institutions that decentralize the constrained e¢ cient allocation.
5.1 Laissez-faire (Markov) equilibrium with state-contingent debt
The analysis of the laissez-faire equilibrium in Section 3 was carried out under the assumption thatthe government can issue only a non-contingent asset. In this section, we extend the analysis to allowstate-contingent debt. We show that a laissez-faire equilibrium with state-contingent debt wouldattain constrained e¢ ciency if and only if there were no moral hazard. However, when the reforme¤ort is endogenous, the combination of moral hazard and limited commitment curtails the insurancethat markets can provide. Consequently, the Markov equilibrium with state-contingent debt is notconstrained e¢ cient. In the quantitative analysis of Section 7 below, we show that markets for state-contingent debt yield only small quantitative welfare gains relative to the benchmark laissez-faireeconomy.
Let bw and b �w denote two securities paying one unit of output if the economy is in a recession or innormal times, respectively. We label these securities recession-contingent debt and recovery-contingentdebt, respectively, and denote by Qw
�b0w; b
0�w
�and Q �w
�b0w; b
0�w
�their corresponding prices. The budget
constraint in a recession is given by:
Qw�b0w; b
0�w
�� b0w +Q �w
�b0w; b
0�w
�� b0w = bw + c� w:
Under limited commitment, the price of each security depends on the two outstanding debt levels,as both a¤ect the reform e¤ort and the probability of renegotiation.19 The value function of thebenevolent government can be written as:
V�bw; �; w
�= max
fb0w;b0�wg�u�Qw�b0w; b
0�w
�� b0w +Q �w
�b0w; b
0�w
�� b0�w + w � B (b; �; w)
�(36)
�X��b0w; b
0�w
��+ �
�1�
�b0w; b
0�w
��EV
�b0w; w
�+ �
�b0w; b
0�w
�EV
�b0�w; �w
�:
Mirroring the analysis in the case of non-state-contingent debt, we proceed in two steps. First, wecharacterize the optimal reform e¤ort. This is determined by the di¤erence between the discountedutility conditional on the recession ending and continuing, respectively (cf. equation (14)):
X 0 � �b0w; b0�w�� = � �Z 1
0V�b0�w; �
0; �w�dF (�)�
Z 1
0V�b0w; �
0; w�dF (�)
�:
Note that the incentive to reform would vanish under full insurance.Next, we characterize consumption and debt issuance. To this aim, consider �rst the equilibrium
asset prices. The prices of the recession- and recovery-contingent debt are given by, respectively:
Qw�b0w; b
0�w
�=
1��b0w; b
0�w
�R
�1� F
���b0w���
+1
b0w
Z �(b0w)
0
���1 (�) dF (�)
�!; (37)
Q �w
�b0w; b
0�w
�=
�b0w; b
0�w
�R
�1� F
����b0�w���
+1
b0�w
Z ��(b0�w)
0
����1 (�) dF (�)
�!: (38)
19Note that these assets are not Arrow-Debreu assets since their payo¤s are not conditional on the realization of �.An alternative approach would have been to follow Alvarez and Jermann (2000) and issue an Arrow-Debreu asset foreach state (w; �) and let the default-driven participation constraint serve as an endogenous borrowing constraint.
23
The next proposition characterizes the CEE with state-contingent debt.
Proposition 12 Assume that there exist markets for two securities delivering one unit of output ifthe economy is in recession and in normal times, respectively, and subject to the risk of renegotiation.Suppose that the economy is initially in recession. The following CEEs are satis�ed in the laissez-faireequilibrium:(I) If the recession continues,
u0�c0jH;w
�u0 (c)| {z }
MRS if rec. continues
= 1 +@
@b0w�b0w; b
0�w
�| {z }
>0
�R��
�b0w; b
0�w
��1� F
���b0w��� �
1��b0w; b
0�w
��| {z }>0
: (39)
(II) If the recession ends,
u0 (c0jH; �w)u0 (c)| {z }
MRS if rec. ends
= 1 +@
@b0�w�b0w; b
0�w
�| {z }
<0
�R��
�b0w; b
0�w
��1� F
��� (b0�w)
���b0w; b
0�w
�| {z };>0
(40)
where
��b0w; b
0�w
��Q �w
�b0w; b
0�w
�� b0�w
�b0w; b
0�w
� �Qw�b0w; b
0�w
�� b0w
1��b0w; b
0�w
� � 0: (41)
Moreover,
c = Qw�b0w; b
0�w
�� b0w +Q �w
�b0w; b
0�w
�� b0�w + w � B (b; �; w) ;
c0jH;w = Qw�B �w
�b0w�; B �w
�b0w���Bw
�b0w�+Q �w
�Bw�b0w�; B �w
�b0w���B �w
�b0w�+ w � b0w;
c0jH; �w = Q�B�b0w; �w
�; �w��B
�b0w; �w
�+ �w � b0�w;
where Bw�bw�and B �w
�bw�denote the optimal level of newly-issued recession- and recovery-contingent
debt when the recession continues, and debt is honored.
If the probability that the recession ends were exogenous, consumption would be independent ofthe realization of the aggregate state. In this case, the CEEs imply constant consumption c0jH;w =c0jH; �w = c where, recall, c0jH;w is consumption conditional on debt being honored in the next period.The solution has the same properties as the constrained Pareto optimum without moral hazard:consumption is constant when debt is honored, and increases discretely when it is renegotiated. Thenext proposition establishes formally that the two allocations are equivalent. To this aim, de�ne��bw�to be the valuation of debt conditional on staying in recession but before the realization of �.
Proposition 13 If the probability that the recession ends is independent of the reform e¤ort (i.e., = p), then the laissez-faire equilibrium with state-contingent debt is constrained Pareto e¢ cient,namely, �
�bw�= P (�), � = EV
�bw; w
�.
An immediate implication of Proposition 13 is that if e¤ort could be contracted before issuingthe new debt, then the laissez-faire equilibrium with state-contingent debt would implement the con-strained optimal allocation. This equivalence breaks down if there is moral hazard. In this case,the consumption and e¤ort dynamics of the laissez-faire equilibrium are qualitatively di¤erent from
24
those of the constrained optimum. In the laissez-faire equilibrium, consumption falls (and recession-contingent debt increases) whenever the economy remains in recession and debt is honored, as shownby equation (39). On the contrary, consumption increases whenever the recession ends, as shownby equation (40). Therefore, di¤erent from the e¢ cient allocation (which, recall, features constantconsumption when the outside option is not binding), the laissez-faire equilibrium features imperfectinsurance, even conditional on honoring the debt.
The intuition is as follows. By issuing more recession-contingent debt, the country strengthens itsincentive to make reforms, since @=@b0w > 0. This induces the government to issue more recession-contingent debt than in the absence of moral hazard. This e¤ect is stronger the larger is �
�b0w; b
0�w
�which can be interpreted as the net expected gain accruing to the lenders from a marginal increasein the probability that the recession ends. On the contrary, issuing more recovery-contingent debtweakens the incentives to do reform. As a result, consumption increases if the recession ends and fallsif the recession continues (and debt is honored). This result highlights the trade-o¤ between insuranceand incentives: the country must give up insurance in order to gain credibility about its willingnessto do reforms.
The behavior of e¤ort is also di¤erent between the laissez-faire equilibrium with state-contingentdebt and the constrained e¢ cient allocation. In the planning allocation, e¤ort is constant wheneverthe outside option is not binding. In contrast, in the equilibrium changes in debt in�uence the reforme¤ort: this is increasing in the newly-issued recession-contingent debt and decreasing in the newly-issued recovery-contingent debt. In summary, the Markov equilibrium with state-contingent debt isine¢ cient and provides less smoothing of consumption and reform e¤ort than does the planner.
5.2 Reputational mechanisms
We have focused so far on Markov equilibria abstracting from reputational mechanisms. We �nd thisrestriction reasonable in our environment, as the decentralized nature of the international credit marketmakes it di¢ cult for creditors to coordinate expectations on appropriate punishment schemes. Nev-ertheless it is interesting to investigate under which conditions non-Markov equilibria can implementthe constrained e¢ cient allocation. In this section, we show that a simple reputational equilibriumdecentralizes the constrained optimum if markets for state-contingent debt exist and the discount fac-tor is su¢ ciently high. A thorough investigation of the non-Markov equilibria is beyond the scope ofour analysis. Instead, we focus on an implementation where the threat point is the in�nite reversionto the Markov equilibrium, in the spirit of Barro and Gordon (1983).
The candidate constrained e¢ cient equilibrium has the following features. The agent exerts thesecond best e¤ort level given by the planner solution above, and the market prices (state-contingent)debt accordingly. The agent receives full consumption insurance if it honors its debt. Debt may berenegotiated along the equilibrium path. However, if the agent does not exert the required e¤ort level,the reputational equilibrium reverts to the Markov equilibrium studied above. We prove in the onlineappendix that in order for such an equilibrium to be sustained the following condition must hold, forall bw and � :
X�pC��X
�(b0�w; b
0w)�< �
� �pC �(b0�w; b0w)
���EV (b0�w; �
0; �w)� EV (b0w; �0; w)�
+(1� pC)��EV C(b0w; �
0; w)� EV (b0w; �0; w)� �
; (42)
where pC , b0�w, and b0w are the optimal e¤ort and debt issuances given an initial debt bw and a realization
�; and EV C denotes the expected utility along the constrained e¢ cient candidate equilibrium. Theleft-hand side is the value of the deviation in terms of the reduction in the e¤ort cost in the current
25
period. The right-hand side is the cost (punishment) for abandoning the e¢ cient allocation. Thiscan be decomposed into two parts. The �rst is the reduction in the probability that the recessionends associated with the deviation times the expected gain of leaving the recession in the Markovequilibrium. The second is welfare loss of switching to the Markov equilibrium in recession. Thiscondition is more likely to be satis�ed when � is large. On the contrary the condition necessarily failswhen � is su¢ ciently small.
5.3 Interpreting the optimal plan as an austerity program
In this section, we discuss an institutional interpretation of the constrained optimum allocation. Con-sider a stand-by program run by an international institution (e.g., the IMF) which, like the planner,can monitor the reform, but cannot get around the limited commitment problem, i.e., the indebtedcountry can walk away unilaterally. We show that the planner allocation can be interpreted as acombination of transfers (or loans), repayment schedules, reform program and renegotiation strategy.This program has two key features. First, the country cannot run an independent �scal policy, i.e., itis not allowed to issue additional debt in the market. Second, the program is subject to renegotiation.More precisely, whenever the country credibly threatens to abandon the program, the internationalinstitution should sweeten the deal by increasing the transfers, reducing the required e¤ort, and reduc-ing the debt the country owes when the recession ends. When no credible threat of default is on thetable, consumption and reform e¤ort should be held constant as long as the recession lasts. When therecession ends, the international institution receives a payment from the country, �nanced by issuingdebt in the market.
Let � denote the present discounted utility guaranteed to the country when the program is �rstagreed upon. Let c� (�) and p� (�) be the consumption and reform e¤ort associated with the promisedutility in the planning problem. Upon entering the program, the country receives a transfer equal toT (�) + b0; where T (�) = c� (�) � w (note that T (�) could be negative). In the subsequent periods,the country is guaranteed the transfer �ow T (�) so long as the recession lasts and there is no crediblerequest of renegotiating the terms of the austerity program. In other words, the international insti-tution �rst bails out the country from its obligations to creditors, and then becomes the sole residualclaimant of the country�s sovereign debt. The country is also asked to exert a reform e¤ort p� (�). Ifthe country faces a low realization of � and threatens to leave the program, the institution improvesthe terms of the program so as to match the country�s outside option. Thereafter, consumption ande¤ort are held constant at new higher and lower levels, respectively, as in the planner�s allocation.And so on, for as long as the recession continues.
As soon as the recession ends, the country owes a debt bN to the international institution, deter-mined by the equation
Q (bN ; �w)� bN = c� (�N )� �w + bN :
Here �N is the expected utility granted to the country after the most recent round of renegotiation.After receiving this payment, the international institution terminates the program and lets the country�nance its debt in the market.
This program resembles an austerity program, in the sense that the country is prevented fromrunning an independent �scal policy and reform program. In particular, the country would like toissue extra debt after entering the stand-by agreement, so austerity is a binding constraint. In addition,the country would like to shirk on the reform e¤ort prescribed by the agreement. Thus, the governmentwould like to (temporarily) deviate from the optimal plan, and promises about future transfers is anessential feature of the program.
26
A distinctive feature of the assistance program is that the international institution sets "harsh"entry conditions in anticipation of future renegotiations. How harsh these conditions are depends on�: In turn, � may re�ect a political decision about how many (if any) own resources the internationalinstitution wishes to commit to rescuing the indebted country. A natural benchmark is to set � suchthat the international institution makes zero pro�ts (and zero losses) in expectation. Whether, ex-post, the international institution makes net gains or losses hinges on the duration of the recessionand on the realized sequence of ��s.
Another result that has important policy implications is that there would be no welfare gain ifthe international institution committed never to accept any renegotiation. On the contrary, such apolicy would lead to welfare losses because, on the one hand, there would be ine¢ cient default inequilibrium; on the other hand, the country could not expect future improvements, and thereforewould not accept a very low initial consumption, or a very high reform e¤ort. If the internationalinstitution�s expected pro�t were zero in both programs, the country would receive a lower expectedutility from the alternative (no renegotiation) program.
In summary, our theory prescribes a pragmatic approach to debt renegotiation. Credible threats ofdefault should be appeased by reducing the debt and softening the austerity program. Such approachis often criticized for creating bad incentives. In our model, such appeasement is precisely the optimalpolicy under the reasonable assumption that penalties on sovereign countries for breaking an agreementare limited.
6 Extension: learning
In our theory, renegotiation is unambiguously good for the borrower. On the one hand, consumptionalways increases upon renegotiation, in line with the empirical evidence documented by Reinhartand Trebesch (2016). On the other, renegotiations do not a¤ect the terms at which the country canborrow in future. In particular, conditional on the debt level, the risk premium is independent ofthe country�s credit history. In this section, we sketch an extension where bond prices depend onthe frequency of previous renegotiations. We assume that there is imperfect information about thedistribution from which countries draw their realizations of �: In particular, there are two types ofcountries, creditworthy (CW) and not creditworthy (NC), that draw from di¤erent distributions.20
In particular, FNC (�) � FCW (�), with strict inequality holding for some �; implying that the NCcountry is more likely to have lower realizations of the default cost. We assume that priors are commonknowledge, and denote by � the belief that the borrower is CW. Beliefs are updated according to Bayes�rule:
�0 =fCW (�)
fCW (�)� � + fNC (�)� (1� �)� � � (�; �) :
Moreover, we de�neF (�j�) � �FCW (�) + (1� �)FNC (�) ;
and restrict attention to laissez-faire equilibria during normal times (w = �w). For the ease of exposi-tion, normal times variables will be indicated with a bar on top for the rest of this section.
20One could assume that the distributions have a common support in order to rule out perfectly revealing realizations.However, this is not essential.
27
In the new environment, the price of debt depends on the prior about the country�s type, i.e.,�Q (b0; �). No arbitrage implies the following bond price:
�Q�b0; �
�=1
R
(1� F (�� (b0; �) j� (�� (b0; �) ; �)))+
�b0R ��(b0;�)0 b̂ (�;� (�; �)) dFCW (�) +
1��b0R ��(b0;�)0 b̂ (�;� (�; �)) dFNC (�)
!where �� (b0; �) denotes the threshold �0 such that debt will be honored next period if and only if�0 � �� (b0; �) : More formally, �� is the unique �xed point of the following equation
�� = ���b0;� (��; �)
�:
The function �� takes into account that the realization of �0 will itself alter next-period beliefs, whichin turn a¤ect the country�s incentive to renegotiate.21 The bond price is falling in b0 and increasingin �:
Consider, next, the consumption-savings decision. The CEE yields:22
1� F�����B (b; �) ; �
�j� (��; �)
�= �
Z 1
��(b;�)
u0��C����0; �
�; �B (b; �)
��u0��C (b; �)
� dFCW��0�
+(1� �)Z 1
��(b;�)
u0��C����0; �
�; �B (b; �)
��u0��C (b; �)
� dFNC��0�:
If next-period consumption conditional on honoring the debt did not depend on �0; then the CEE wouldboil down to equation (13). However, in this extension, the realized consumption growth depends on�0 because creditors learn over time about the borrowers�types. For example, take two realization of�0; say �0h and �
0l; such that �
0h > �
0l; neither inducing renegotiation. Here, consumption will be larger
under �0h because the larger realization has a stronger positive e¤ect on the belief that the country isCW. This improves the terms of borrowing, and hence consumption. Note that renegotiation mightbe associated with a fall in consumption � for example if the realized � is just below �� (b0; �) thee¤ect of a very small renegotiation is more than o¤set by that of Bayesian updating. Conversely, acountry experiencing a sequence of large ��s which induces it to honor debt for a long time will enjoyan increasing consumption.
In summary, this simple extension shows that our theory can incorporate learning e¤ects throughwhich countries prone to renegotiation are punished by the market with high interest rates.
7 Quantitative Analysis
In this section, we study the quantitative properties of the model. To this end, we calibrate the modeleconomy to match salient facts on default premia, investor losses, and debt-to-GDP ratios.23 Ourmain purpose here is to evaluate the welfare gain of going from the laissez-faire equilibrium to theconstrained optimal allocation. We also assess the quantitative importance of limited commitment,non-contractible e¤ort, and incomplete �nancial markets.21Note that some functions must be rede�ned to take into account their dependence on public beliefs. Apart from�Q (b0; �) ; de�ned in the text, b̂ (�; �) is the renegotiated debt given � and �: Moreover, �� (b0; �) denotes the thresholdthat makes the country indi¤erent between honoring the debt level b0 and defaulting, conditional on the realized belief�:22We defer the derivation to the online appendix.23The model is solved by discretizing the state space. The benchmark calibration uses 5,000 points for debt and
600 points for �: The numerical approximation is remarkably accurate in the sense that the Euler equation errors arenegligible. See the online appendix for further details.
28
7.1 Calibration
A model period corresponds to one year. We normalize the GDP during normal times to �w = 1 andassume that the recession causes a drop in income of 38%, i.e., w = 0:62 � �w. This corresponds tothe fall of GDP per capita for Greece between 2007 and 2013, relative to trend.24 Since we focuson the return on government debt, the annual real gross interest rate is set to R = 1:02, implying� = 1=R = 0:98. The utility function is assumed to be CRRA with a relative risk aversion of 2.
We assume an isoelastic e¤ort cost function; X(p) = �1+1='(p)
1+1=', where � regulates the averagelevel of e¤ort and ' regulates the elasticity of reform e¤ort to changes in the return to reforms. Weset the two parameters, ' and �; so as to match two points on the equilibrium e¤ort function (b). Inparticular, we assume that the e¤ort at the debt limit is
��b�= 10%, so that a country at the debt
limit would choose an e¤ort inducing an expected duration of the recession of one decade (we haveGreece in mind). Moreover, we assume that the maximum e¤ort is maxb(b) = 20%, inducing anexpected recession duration of �ve years (we have Iceland and Ireland in mind). This implies setting' = 20:68 and � = 23:70.
Finally, we determine the distribution of the default cost �, assumed to have bounded support�0; ���. We calibrate the distribution f (�) so that the model matches key moments of quantities and
prices of sovereign debt. One common problem in the quantitative literature on sovereign debt isthat those models fail to match observed values of debt-to-GDP ratios under realistic parametrization(Arellano 2008; Yue 2010). This is not a problem in our model. In fact, the maximum defaultcost realization �� is set so that the debt limit during normal times is �b= �w = 180%. Moreover, thedistribution f is chosen so that the model matches an average default premium of 4% for a countrywhich has a debt-output ratio of 100% during recession.25 This was the average debt and averagedefault premium for Greece, Ireland, Italy, Portugal, and Spain (GIIPS) during 2008-2012. To matchthis target we assume that � is distributed according to a Beta distribution with c.d.f. given byF (�; ��; �) = �(�=��; �)=�(1; �), where �(x; �) =
R x0 (1 � t)
��1dt. By setting � = 0:139 the target ismet.
7.2 Welfare Comparison
We use the calibrated economy to evaluate the welfare gains of di¤erent policy arrangements. Thewelfare gains are measured as the equivalent variation in terms of the market value of an initialdebt to output reduction in the market economy, namely, the market value of a reduction in initialdebt required to make the borrower indi¤erent between staying in the market arrangement (with thereduction in debt) and moving to an alternative allocation.
We assume that the economy starts in a recession and has an initial debt-output ratio of b0=y0 =100%. Table 1 reports the welfare gains of the policy arrangements relative to the market allocation.The results for the benchmark calibration with a relative risk aversion of two, are listed in the middlecolumn of the table. Given the initial b0, the welfare gain of going from the market allocation to
24GDP per capita of Greece fell from 18,924 to 14,551 Euro between 2007 and 2013 (Eurostat). The annualized growthrate between 1997 and 2007 was 3.8%. The fall in output between 2007 and 2013 relative to trend is therefore 38%.25Our calibration yields simulation results that are also in line with the �ndings of Reinhart and Trebesch (2016)
who document an average debt relief of 40% of external government debt across both the 1930s and the 1980s/1990s.Moreover, the average debt relief is reported to be 21% of GDP for advanced economies in the 1930s, and 16% of GDPfor emerging market economies in the 1980s/1990s. Even though we do not target these moments of the data in thebenchmark calibration, our simulations yield an average face value debt relief of 38% corresponding to a reduction worth29% of recession GDP at market value, which is within the ballpark of their estimates.
29
RRA1 2 3
Market Equilibrium 0% 0% 0%First-best 76.7% 143.2% 198.1%Second-best 21.2% 44.6% 64.1%State-contingent 1.2% 5.0% 5.4%Grexit 6.3% -5.7% -19.5%
Table 1: Welfare equivalent transfer (market value, % of GDP)
the �rst best is equivalent to a one-time transfer of 143% of GDP. Similarly, the gain of going to theconstrained optimum is equivalent to a one-time transfer of 45% of GDP. The large welfare di¤erencebetween the �rst- and the second-best allocation shows the large losses associated with the limitedenforcement.
Allowing for state-contingent debt, on the other hand, yields a gain equivalent to a mere 5% one-time transfer, substantially less than the second-best gains. As discussed above, this illustrates thatthe trade-o¤ between moral hazard and insurance renders access to full insurance not very useful.Moreover, it shows that the large gains of the constrained optimum must be originating from theplanner�s ability to mitigate the moral hazard problem.
To understand the role of risk aversion, we vary the relative risk aversion from 1 to 3 (see Table1).26 As expected, the welfare gains relative to the market allocation are increasing in the degree ofrisk aversion.
7.3 Consumption volatility and recession duration
In this section we report some important moments of our simulation results. Recall that for eachconsidered policy arrangement, we start the simulation at the initial debt level (or a level of promisedutility) corresponding to a zero pro�t intervention in the market equilibrium at a debt-output ratio of100%.
RRA1 2 3
Market Equilibrium 0.0209 0.0190 0.0162First-best 0 0 0Second-best 0.0080 0.0066 0.0055State-contingent 0.0187 0.0152 0.0132Grexit 0.0312 0.0338 0.0351
Table 2: Consumption volatility over the �rst 10 years
Table 2 shows the variance of log consumption over the �rst ten simulation periods. The constrainedoptimum (second best) yields substantially smoother consumption than the market allocation. Theconsumption volatility is not much a¤ected by the degree of relative risk aversion.
26When changing the relative risk aversion, the internally calibrated parameters, ', �, ��, and �, are recalibrated sothe model meets the stated empirical moments.
30
Table 3 reports the expected duration (the cross-sectional mean duration resulting from a largenumber of stochastic simulations) of the recession which is inversely related to the average level ofreform e¤ort over the recession period. For example, in the benchmark calibration the �rst-best alloca-tion implies an expected recession of 16:1 years, which corresponds to a constant recovery probability(reform e¤ort level) of 6:2%.
RRA1 2 3
Market Equilibrium 6.2 6.7 7.2First-best 15.1 16.1 16.9Second-best 7.0 6.7 6.3State-contingent 7.4 8.2 8.5Grexit 8.0 6.6 5.7
Table 3: Expected duration of the recession (in years)
Interestingly, the expected duration of the recession is roughly equal for the market equilibriumand the constrained optimum (where the ranking depends on the relative risk aversion). In contrast,the allocation with state-contingent debt yields signi�cantly lower reform e¤ort. This illustrates amain point of our paper: access to better insurance magni�es the moral hazard problem.
In the last row of Table 1 we consider a simple version of an �austerity program�with the followingfeatures: (1) the external institution guarantees the debt of the borrower (so the market price is always1=R), (2) the borrower cannot issue additional debt in the private market (i.e., ��scal austerity�), (3)the institution commits to terminate the arrangement and induce an outright default ("Grexit") if theborrower attempts to renegotiate the debt obligations (in which case the institution steps in an honorsthe issued guarantees) or increase the debt level, (4) once the recession is over, no new guarantees willbe issued, and (5) the initial debt is increased to a level such that the institution has zero expectedpro�ts (i.e., the austerity program breaks even in expectation). However, there are no conditionsimposed on the reform e¤ort.
One the one hand, this program could potentially be good because it mitigates the moral hazardprogram. To see this, note that with a debt guarantee the bond price does not change when therecession ends, so more of the gains from reform e¤ort will accrue to the borrower. On the otherhand, the program will induce ine¢ cient default. For the benchmark calibration, we �nd that such anausterity program yields a loss equivalent to a one-time tax of 5:7%, so the borrower would prefer themarket to this austerity program. However, when the risk aversion is su¢ ciently low, the austerityprogram is actually better than the market allocation. With a lower risk aversion, the insurance valueis smaller and the improvements in the moral hazard problem outweighs the ine¢ ciencies.
8 Conclusions
This paper presents a theory of sovereign debt dynamics under limited commitment. A sovereign coun-try issues debt to smooth consumption during a recession whose duration is uncertain and endogenous.The expected duration of the recession depends on the intensity of (costly) structural reforms. Bothelements � the risk of repudiation and the need for structural reforms �are salient features of therecent European debt crisis.
31
The laissez-faire equilibrium features repeated debt renegotiations. Renegotiations are more likelyto occur during recessions and when the country has accumulated a high level of debt. As a recessiondrags on, the country has an incentive to go deeper into debt. A higher level of debt in turn maydeter rather than stimulate economic reforms.
The theory bears normative predictions that are relevant for the management of the Europeancrisis. The market equilibrium is ine¢ cient for two reasons. On the one hand, the government ofthe sovereign country underinvests in structural reforms. The intuitive reason is that the short-runcost of reforms is borne entirely by the country, while future bene�ts of reforms accrue in part to thecreditors in the form of an ex-post increased price of debt, due to a reduction in the probability ofrenegotiation. On the other hand, the limited commitment to honor debt induces high risk premiaand excess consumption volatility. A well-designed intervention by an international institution canimprove welfare, as long as the institution can monitor the reform process. While we assume, fortractability, that the international institution can monitor reforms perfectly, our results carry overto a more realistic scenario where reforms are only imperfectly monitored. The optimal policy alsoentails an assistance program whereby an international organization provides the country with aconstant transfer �ow, deferring the repayment of debt to the time when the recession ends. Theoptimal contract takes into account that this payment is itself subject to renegotiation risk.
A second implication is that, when the government of the indebted country credibly threatensto renege on an existing agreement, concessions should be made to avoid an outright repudiation.Contrary to a common perception among policy makers, a rigid commitment to enforce the terms ofthe original agreement is not optimal. Rather, the optimal policy entails the possibility of multiplerenegotiations, which are re�ected in the terms of the initial agreement.
To retain tractability, we make important assumptions that we plan to relax in future research.First, in our theory the default cost follows an exogenous stochastic process. In a richer model, thiswould be part of the equilibrium dynamics. Strategic delegation is a potentially important extension.In the case of Greece, voters may have an incentive to elect a radical government with the aim ofdelegating the negotiation power to an agent that has or is perceived to have a lower default costthan voters do (cf. Rogo¤ 1985). In our current model, however, the stochastic process governing thecreditor�s outside option is exogenous, and is outside of the control of the government and creditors.
Second, again for simplicity, we assume that renegotiation is costless, that creditors can perfectlycoordinate and that they have full bargaining power in the renegotiation game. Each of these assump-tions could be relaxed. For instance, in reality the process of negotiation may entail costs. Moreover,as in the recent contention between Argentina and the so-called vulture funds, some creditors mayhold out and refuse to accept a restructuring plan signed by a syndicate of lenders. Finally, thecountry may retain some bargaining power in the renegotiation. All these extensions would introduceinteresting additional dimensions, and invalidate some of the strong e¢ ciency results (for instance,the result that the laissez-faire economy attains the constrained optimum in the absence of income�uctuations). However, we are con�dent that the gist of the results is robust to these extensions.
Finally, by focusing on a representative agent, we abstract from con�icts of interest betweendi¤erent groups of agents within the country. Studying the political economy of sovereign debt wouldbe an interesting extension. We leave the exploration of these and other avenues to future work.
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A Appendix: Proofs of lemmas, propositions, and corollaries
Proof of Proposition 1. The case of normal times is straightforward. Consider an economystarting in a recession. Perfect insurance implies constant consumption and e¤ort. The e¢ cientsolution maximizes the discounted utility, W = u (c) = (1� �)�X (p) = (1� � (1� p)) with respect toc and p, subject to the following intertemporal budget constraint:
1
1� � (1� p) (w � c) +�
1� �p
1� � (1� p) ( �w � c) = b: (43)
Note that since insurance is provided in actuarially fair markets the budget constraint holds in expectedterms. Writing the Lagrangean, and di¤erentiating it with respect to c yields u0 (c) = �: Di¤erentiatingthe Lagrangean with respect to p and simplifying terms yields
X 0 (p)
�1� ��
+ p
��X (p) = ( �w � w)� u0 (c) ; (44)
which is identical to equation (1) in the text. Equation (43) de�nes a positively sloped locus in theplane (p; c) ; while equation (44) de�nes a negatively sloped locus in the same plane. Unless the solutionfor e¤ort is a corner, the two equations pin down a unique interior solution for p and c. Consider thecomparative statics with respect to b (note that b only features in equation (43)). An increase in byields a decrease in c and an increase in p. This concludes the proof.
Proof of Lemma 1. The �rst part follows from the de�nitions of � and b̂. For the second part, notethat � (b; w) =W (0; w)�W (b; w) such that if W (b; w) is strictly decreasing in b, then � (b; w) mustbe strictly increasing in b. Thus, the inverse function of � (b; w) exits and is given by the expressionsstated in Lemma 1.
35
Proof of Proposition 2. We prove the existence of the Markov equilibrium by showing that thevalue function W is a �xed-point of a monotone mapping following Stokey and Lucas (1989, Theorem17.7).
Let �(w) be the space of bounded, continuous, and decreasing functions de�ned over [b;~b]. More-over, let d1 denote the supremum norm such that (�; d1) is a complete metric space. Let z 2 �(w)and be a real constant representing the outside option under outright default. T (z; ) is similar tothe Bellman equation of the Markov equilibrium in the text, but di¤ers in that the value of outrightdefault in the recession state is exogenously given by � �. We �rst establish the existence of anequilibrium for an exogenous ; and then extend the argument to an endogenous outside option as inthe Markov equilibrium.
De�ne the following mapping:
T (z; ;w) (b) = maxb02[b;~b]
u(Q(b0; z; )b0 � b+ w) + Z(b0; z; ; w); (45)
where � (x; z; ) = � z(x); and z�b̂ (�; z; )
�= � �: In addition, let T 0(z; ) = z, Tn(z; ) =
T (Tn�1(z; ); ), n = 1; 2; : : : , and zn � Tn(z; ). Moreover, when w = �w,
Z(x; z; ; �w) = �E�max
�z (x) ; � �0
�; and
Q(x; z; ; �w)x = R�1Ehmin
nx; b̂(�; z; )
oi:
Correspondingly, when w = w
Z(x; z; ; w) = �X((x; z; )) + ��(x; z; )E
�max
�W (x; �w) ;W (0; �w)� �0
�+(1�(x; z; ))E
�max
�z (x) ; � �0
� �;
X 0((x; z; )) = �
�[1� F (� (x; �w))]W (x; �w)� [1� F (�(x; z; ))] z(x)
�;
Q(x; z; ; w)x = R�1E
24 (x; z; )minnx; b̂ (�; �w)
o+(1�(x; z; ))min
nx; b̂ (�; z; )
o 35 :Note that the mapping during recession, T (z; ;w) ; takes as given the existence of the equilibriumfunctionsW (b; �w) ; � (b; �w) and b̂ (�; �w) : This is legitimate because one can prove existence recursively,�rst for normal times and then in recession.
We de�ne upper and lower bounds for the value functions. More formally, WMIN � u(w)=(1 ��) � �max, and WMAX � u( �w)=(1 � �). It is straightforward to see that WMIN and WMAX are,respectively, lower and upper bounds to the present utility the country can attain in equilibrium.
We establish �rst that the operator T (z; ;w) (b) is a uniformly continuous, bounded and decreas-ing (in b) mapping of the function space � into itself. Continuity follows by the Theorem of theMaximum. Boundedness follows from the fact that utility is bounded because consumption, reforme¤ort, the support of the default cost, and the elements of � are bounded. Finally, to establish thatthe mapping T is decreasing in b note that, for any � > 0; T (z; ;w) (b+�) < T (z; ;w) (b) since
T (z; ;w) (b+�) = maxb02[b;~b]
u�Q(b0; z; ; w)b0 + w � (b+�)
�+ Z
�b0; z; ; w
�= u (Q(B (b+�; z; ; w) ; z; ; w) �B (b+�; z; ; w) + w � (b+�)) + Z (B (b+�; z; ; w) ; z; ; w)< u (Q (B (b+�; z; ; w) ; z; ; w) �B (b+�; z; ; w) + w � b) + Z (B (b+�; z; ; w) ; z; ; w)� u (Q (B (b; z; ; w) ; z; ; w) �B (b; z; ; w) + w � b) + Z (B (b; z; ; w) ; z; ; w) = T (z; ;w) (b) ;
36
where B(b; z; ; w) = argmaxb02[b;~b] u(Q(b0; z; ; w)b0 � b+ w) + Z(b0; z; ; w):
Next, we establish that the mapping T (z; ;w) is monotone in z, that its �xed-point z� (z; ;w) (b) =limn!+1 Tn (z; ;w) (b) exists, and that z� 2 �. To this aim, consider z; z+ 2 � with z (b) < z+ (b),8b 2 [b;~b]. Since z and z+ are decreasing in b; implying that z+
�b̂(�; z+; )
�= �� > z
�b̂(�; z+; )
�,
it follows immediately that b̂(�; z; ) < b̂(�; z+; ). Consequently, � (b; z; ) � � (b; z+; ) for allb 2 [b;~b], which in turn implies that Q(b; z; ; w) � Q(b; z+; ; w). Consider �rst the case of w = �w:We establish that z+ > z , T (z+; ; �w) (b) > T (z; ; �w) (b) ; since
T�z+; ; �w
�(b) = max
b02[b;~b]u�Q(b0; z+; ; �w)b0 � b+ �w
�+ Z
�b0; z+; ; �w
�= u
�Q(B
�b; z+; ; �w
�; z+; ; �w) �B
�b; z+; ; �w
�� b+ �w
�+ Z
�B�b; z+; ; �w
�; z+; ; �w
�� u
�Q(B (b; z; ; �w) ; z+; ; �w) �B (b; z; ; �w)� b+ �w
�+ Z
�B (b; z; ; �w) ; z+; ; �w
�> u (Q(B (b; z; ; �w) ; z; ; �w) �B (b; z; ; �w)� b+ �w) + Z (B (b; z; ; �w) ; z; ; �w)
= maxb02[b;~b]
u(Q�b0; z; ; �w
�b0 + �w � b) + Z
�b0; z; ; �w
�= T (z; ; �w) (b) :
where the �rst inequality follows from the fact that B (b; z; ; �w) yields a lower utility relative to theoptimal B (b; z+; ; �w). The second inequality follows from the fact that Q (b; z; ; �w) � Q (b; z+; ; �w)for all b 2 [b;~b].
Consider, next, the case of w = �w: Let E �V (b) � E [V (b; �; �w)] : We establish that z+ > z ,T (z+; ;w) (b) > T (z; ;w) (b) ; since
T�z+; ;w
�(b) = max
b02[b;~b]u�Q(b0; z+; )b0 � b+ w
�+ Z
�b0; z+;
�= u
�Q(B
�b; z+;
�; z+; ) �B
�b; z+;
�� b+ w
��X
��B�b; z+;
�; z+;
��+
�(B (b; z+; ) ; z+; )�E �V (B (b; z+; ))+
(1�(B (b; z+; ) ; z+; ))�E [max f � �; z+ (B (b; z+; ))g]
�� u
�Q(B (b; z; ) ; z+; ) �B (b; z; )� b+ w
��X ( (B (b; z; ) ; z; ))
+
�(B (b; z; ) ; z; )�E �V (B (b; z; ))+
(1�(B (b; z; ) ; z; ))�E [max f � �; z+ (B (b; z; ))g]
�> u (Q (B (b; z; ) ; z; ) �B (b; z; )� b+ w)�X ( (B (b; z; ) ; z; ))
+
�(B (b; z; ) ; z; )�E �V (B (b; z; ))+
(1�(B (b; z; ) ; z; ))�E [max f � �; z (B (b; z; ))g]
�= max
b02[b;~b]u(Q(b0; z; )b0 � b+ w) + Z(b0; z; ) = T (z; ) (b; w) ;
where the same logic as above applies.We have established that T (z; ;w) is a monotone mapping with the sup norm. This mapping is
an equicontinuous family (each function in � is uniformly continuous and the continuity is uniformfor all functions in �). Then, Stokey and Lucas (1989, Theorem 17.7) ensures that the �xed point ofT (z; ;w) exists, is an element of � and is given by z�(z; ;w) = limn!+1 Tn(z; ;w).
Thus far, we have proven the existence of at least one �xed point of the mapping T for anyexogenous outside option, 2 [WMIN ;WMAX ] : We now use a di¤erent �xed point argument toshow that, conditional on an initial z; there exists a unique �xed point that the outside optionsin normal times and recession are, respectively, �z ( �w) and
�z (w) with the following properties:
z� (z; �z ( �w) ; �w) (0) = �z ( �w) = WMAX and z� (z; �z (w) ; w) (0) = �z (w) 2 (WMIN ;WMAX) : To
37
see why, note that, by the Theorem of the Maximum, z� (z; ;w) (b) = limn!+1 Tn (z; ;w) (b) iscontinuous in . Moreover, z� is bounded since z� (z; ;w) 2 [WMIN ;WMAX ]. Thus, the Brouwer�xed-point theorem ensures that there exists a z 2 [WMIN ;WMAX ] such that z� (z; z; w) (0) = z. Since z� (z; z; w) (0) = z � � (0; z�; z), this is equivalent to say that, at each �xed point,� (0; z�; �z (w)) = 0: To prove uniqueness, we note then that �
�0; z� ;
�is monotone increasing
for all 2 [WMIN ;WMAX ] ; as the set of (potential) states of nature in which the outside optionis attractive expands when increases. Therefore, there exists a unique �xed point �z (w) suchthat � (0; z�; �z (w)) = 0. In particular in normal times �z ( �w) = W (0; �w) = WMAX : In recession, �z (w) =W (0; w) 2 (WMIN ;WMAX).
The results proven thus far allow us to claim the existence of an equilibrium value function Wsuch that W (b; w) = T (W ;W (0; w) ; w) (b) : The de�nition of the remaining equilibrium functionsfollow from De�nition 1. This establishes the existence of a Markov equilibrium. The continuity ofthe value function W (b; w) in b follows from the Theorem of the Maximum, and implies that alsothe equilibrium functions �; Q and are continuous in b: It is also straightforward to show that Wis strictly decreasing in b and, hence, that � (b; w) = W (0; w) �W (b; w) is strictly increasing in b:Finally, we claim that the bond revenue, Q(b0; w)b0, is (weakly) monotone increasing in b0: This followsfrom equations (6)�(8), the fact that � increasing in b; and from Lemma 1.
Next, we prove that B is monotone decreasing in b by applying Topkis�Theorem: To this aim,de�ne B(b; w) = argmaxb02[b;~b]O(b
0; b; w) where
O(b0; b; w) = u(Q(b0; w)b0 � b+ w) + �Z(b0; w): (46)
We �rst establish that the objective function O(b0; b; w) is supermodular in (b0; b), i.e., if b0H > b0L and
bH > bL, then, O(b0H ; bH ; w)�O(b0L; bH ; w) � O(b0H ; bL; w)�O(b0L; bL; w). To this aim, note that
u(Q(b0H ; w)b0H � bH + w)� u(Q(b0L; w)b0L � bH + w) (47)
� u(Q(b0H ; w)b0H � bL + w)� u(Q(b0L; w)b0L � bL + w):
This inequality follows from the concavity of the utility function and the fact thatQ (b; w) b is increasingwith b; implying that, for � > 0;
u0(Q(x;w)x� b+ w)� u0(Q(x+�; w) (x+�)� b+ w) � 0:
Rearranging terms in (47), and adding and subtracting continuation values on both sides of theinequality yields
u(Q(b0H ; w)b0H � bH + w)� u(Q(b0H ; w)b0H � bL + w)
+�Z(b0H ; w)� �Z(b0H ; w)� u(Q(b0L; w)b
0L � bH + w)� u(Q(b0L; w)b0L � bL + w)
+�Z(b0L; w)� �Z(b0L; w)
that is equivalent to O(b0H ; bH ; w)�O(b0H ; bL; w) � O(b0L; bH ; w)�O(b0L; bL; w): This establishes thatO(b0; b; w) is supermodular in (b0; b): Topkis�Theorem implies then that B(bH ; w) � B(bL; w). Thisconcludes the proof of Proposition 2.
Corollary 1 Let the operator T be de�ned as in equation (45). If, for w 2 fw; �wg; W (b; w) =limn!1 Tn (WMIN ;W (b; w) ; w) (b) = limn!1 Tn (WMAX ;W (b; w) ; w) (b), then the Markov equilib-rium is unique (i.e., there exists a unique equilibrium value function W (b; w) satisfying De�nition1).
38
Proof of Corollary 1. The proof follows immediately from the Corollary to Theorem 17.7 in Stokeyand Lucas (1989).
Proof of Proposition 3. The proof is an application of the generalized envelope theorem in Clausenand Strub (2013) which allows for discrete choices (i.e., repayment or renegotiation) and non-concavevalue functions. Consider the program W (b; w) = maxb02[b;~b]O(b
0; b; w) where O is de�ned in equation(46). Theorem 1 in Clausen and Strub (2013) ensures that if we can �nd a di¤erentiable lower supportfunction (DLSF) for O; then O is di¤erentiable for all b0 2 B(w):
We start by proving the lemma for the case of w = w. The strategy of the proof involves �ndingDLSF for the equilibrium functions Q(b0; w)b0 and Z(b0; w). To this aim, we follow the strategy ofBenveniste and Scheinkman (1979), and consider the value function of a pseudo-borrower that choosesdebt issuance b0 = B (x;w) instead of the optimal b0 = B (b; w),
fW (b; x; w) � u (Q (B (x;w) ; w)B (x;w)� b+ w) + Z (B (x;w) ; w) :
Note that fW is di¤erentiable and strictly decreasing in b: Since debt issuance is chosen suboptimally,it must be the case that fW (b; x; w) � W (b; w) with equality holding at x = b. Furthermore, letthe pseudo-borrower set the default threshold at the level e� (b; x; w) = W (0; w) �fW (b; x; w), wheree� (b; x; w) � � (b; w). Thus, the pseudo-borrower will �nd it optimal to renegotiate for a range of �larger than �(b; w). Note that e� (b; x; w) is di¤erentiable and strictly increasing in b: Thus, the inversefunction exists and is such that e��1x;w (�) � b̂(�;w) (where we de�ne e�x;w (b) � e� (b; x; w)).
Consider �rst the case in which w = w. Let
eO �b0; b; x; w� = u� eQ(b0; x; w)b0 � b+ w�+ eZ(b0; x; w);
where the pseudo bond revenue function is given by
eQ �b0; x; w� b0 =1
Re(b0; x) h1� F (e� �b0; x; �w�)i b0 + Z e�(b0; �w)
0
e��1x; �w(�)dF (�)!
+1
R(1� e(b0; x)) h1� F (e� �b0; x; w�i b0 + Z e�(b0;w)
0
e��1x;w(�)dF (�)!;
and the continuation value is given by
eZ(b0; x; w) = �X(e(b0; x)) + �24 e(b0; x)EmaxnfW (b0; x; �w) ;fW (0; x; �w)� �0o+(1� e(b0; x))EmaxnfW (b0; x; w) ;fW (0; x; w)� �0o
35 ;having de�ned e as
e(b0; x) = �X 0��10@�24 Emax
nfW (b0; x; �w);fW (0; x; �w)� �0o�Emax
nfW (b0; x; w);fW (0; x; w)� �0o351A :
Note that eQ; eZ and e are di¤erentiable in b0 since we established above thatfW and e� are di¤erentiable.Then, eO is a DLSF for O such that eO(b0; b; x; w) � O(b0; b; w) with equality (only) at b0 = x. Thus,
Theorem 1 in Clausen and Strub (2013) ensures that the objective function O (b0; b; w) is di¤erentiable
39
in b0 at b0 = B (b; w) and that @O(b0; b; w)=@b0 = @ eO(b0; b; B (b; w) ; w)=@b0 = 0. In this case, a standard�rst-order condition yields
@u (Q(b0; w)b0 � b+ w)@b0
+@Z(b0; w)
@b0= 0:
Moreover, Lemma 3 in Clausen and Strub (2013) ensures that the functions W (b; w), Z(b; w),�(b; w), Q(b; w), and (b) are di¤erentiable and that a standard envelope condition applies, namely,
@Z(b0; w)
@b0= �
"(b0) [1� F (�(b0; �w))] @W (b0; �w)
@b0
+(1�(b0)) [1� F (�(b0; w))] @W (b0;w)@b0
#;
@W (b; w)
@b= �u0 (Q(B(b; w); w)B(b; w)� b+ w) < 0:
The proof for the case of w = �w follows the same strategy and is therefore omitted.
Proof of Lemma 2. In normal times, the bond revenue function can be written as
Q(b0; �w)b0 = R�1Eminnb0; b̂(�; �w)
o.
Note that the minimum function in the expectation operator is concave in b0. Since the sum of concave
functions is still concave, then also Enmin b0; b̂(�; �w)
omust be concave in b0. This implies that the
marginal bond revenue is falling in b0. Di¤erentiating Q(b0; �w)b0 with respect to b0 2 [b;~b] yields:
@ (Q(b0; �w)b0)
@b0= Q
�b0; �w
�+@
@b0
R�1
�1� F
���b0; �w
���+ (Rb0)�1
Z �(b0; �w)
0
���1 (�)� f (�) d�!� b0
= Q�b0; �w
��R�1b0f
���b0; �w
��� @� (b
0; �w)
@b0
+b0(Rb0)�1 ���1���b0; �w
��f���b0; �w
���@� (b
0; �w)
@b0� 1
R
1
b0
Z �(b0; �w)
0
����1 (�)� f (�) d�
�| {z }
=Q(b0; �w)� 1R(1�F (�(b0; �w)))
= R�1�1� F
���b0; �w
���:
Proof of Proposition 4. The �rst-order condition of (10) for w = �w yields:
@
@b0�b0 �Q
�b0; �w
�� u0 (c) + @
@b0�EV
�b0; �w
�= 0;
The function V has a kink at b0 = b̂ (�; �w) : Consider, �rst, the range of realizations � > �� (b) implyingthat b0 < b̂ (�; �w). Di¤erentiating V in this range yields:
@
@bV (b; �; �w) = �u0 [Q (B (b; �w) ; �w)�B (b; �w) + �w � b] :
40
Next, consider the region of renegotiation, � < �� (b) ; implying that b > b̂ (�; �w) : In this case,@@bV (b; �; w) = 0. Using the results above one obtains:
@
@bEV (b; �w) =
Z �(b; �w)
0
@
@bV (b; �; �w) dF (�) +
Z 1
�(b; �w)
@
@bV (b; �; �w) dF (�)
= �Z 1
�(b; �w)u0 [Q (B (b; �w) ; �w)�B (b; �w) + �w � b] dF (�)
= � [1� F (� (b; �w))]� u0 [Q (B (b; �w) ; �w)�B (b; �w) + �w � b] (48)
Plugging this expression back into the FOC, and leading the expression by one period, yields
0 =@
@b0�b0 �Q
�b0; �w
�� u0 (c)� �
�1� F
���b0; �w
���� u0
�c0jH; �w
�:
Finally, recall that @@b fb�Q (b; �w)g =
1R (1� F (� (b; �w))) ; as established in the proof of Lemma
2. Thus, for F (� (b; �w)) < 1; the above equation is equivalent to the CEE (13) in the proposition.Although the �rst-order condition is also satis�ed at b0 = �b; it is possible to show that this is never anoptimal solution as long as b < �b (details in the online appendix).
Consider, next, the properties of the equilibrium functions C and W: Using the de�nition of Cin equation (11) and Lemma 2, standard algebra shows that C is continuously di¤erentiable anddecreasing, with derivative @C(b; �w)=@b = R�1 [1� F (�(b; �w))] � 1 < 0: Since B(b; �w) = b; thenB maps the complete domain of b into itself. Proposition 3 implies then that value function isdi¤erentiable everywhere, and that the envelope condition @W (b; �w)=@b = �u0(C(b; �w)) applies. Dif-ferentiating this condition with respect to b yields @2W (b; �w)= (@b)2 = �u00(C(b; �w))@C(b; �w)=@b =u00(C(b; �w))
�1�R�1 [1� F (�(b; �w))]
�< 0: This establishes that the value function W is twice con-
tinuously di¤erentiable and strictly concave. This concludes the proof of the proposition.
Proof of Proposition 5. We start by claiming that cFB (b; w) > C (b; w), where cFB (b; w) wasde�ned in Proposition 1 and C (b; w) denotes consumption in the laissez-faire equilibrium when debtis honored. To see why, note that @=@b
�WFB (b; w)
� @=@b fW (b; w)g. This follows from observing
that the di¤erence between WFB (b; w) and WFB (b+�; w) ; where � > 0; merely re�ects the utilityloss from a permanent reduction in consumption cFB (b; w) and (in recession) a permanent increasein e¤ort pFB (b). In contrast, in the market equilibrium a larger debt induces a higher volatility ofconsumption (recall that � (b; w) is monotone increasing in b �see Lemma 1 �so higher debt increasesthe probability of renegotiation) and (in recession) e¤ort. Since @=@b
�WFB (b; w)
= �u0
�cFB (b; w)
�and @=@b fW (b; w)g = �u0 (C (b; w)) ; then the claim that cFB (b; w) > C (b; w) follows.
Let �min � [u ( �w)� u ((1� �) ( �w � w))] = (1� �) and bPV � w + �1�� �w. The assumption in the
Proposition implies that F (�min) = 0 and bPV < �w: Consider �rst the range b � bPV : In this range,W (b; �w) = WFB (b; �w) : To see this, note that W (bPV ; �w) = u ( �w � (1� �) bPV ) = (1� �) = �min:Since by assumption F (�min) = 0, no renegotiation is possible for b � bPV , so the claim follows. Thetwo claims above and Proposition 1 imply that C (b; �w) = cFB (b; �w) > cFB (b; w) � C (b; w) in therange b � bPV : Consider next the range b 2 (bPV ;�b]: In this range, C (b; w) � 0 since debt exceeds themaximum present value of future income. In contrast, C (b; �w) > 0 since bPV < �w: We have thereforeestablished that C (b; �w) > C (b; w) for all b � �b:
Proof of Lemma 3. If in the initial period (but not later) the country can contract on e¤ort whileissuing new debt, the problem becomes
maxb0;p�
�u (c)�X (p�) + �p� � EV
�b0; �w
�+ � (1� p�)� EV
�b0; w
�:
41
Note that the next-period value function V is the same as in the benchmark problem with non-contractible e¤ort, since we are considering a one-period deviation. The �rst-order condition withrespect to p yields
0 =d
dp��Q�b0; w
�b0� u0 (c)�X 0 (p�) + �
�EV
�b0; �w
�� EV
�b0; w
��) X 0 (p�) =
hQ�b0; �w
�� Q̂
�b0; w
�ib0 � u0 (c)
+�
�Z 1
0V�b0; �0; �w
�dF (�)�
Z 1
0V�b0; �0; w
�dF (�)
�> �
�Z 1
0V�b0; �0; �w
�dF (�)�
Z 1
0V�b0; �0; w
�dF (�)
�; (49)
where the last equation follows from the facts that Q (b0; �w) > Q̂ (b0; w) ; and that
d
dp
�Q�b0; w
�b0=
d
dp
nhpQ�b0; �w
�+ (1� p) Q̂
�b0; w
�ib0o
=hQ�b0; �w
�� Q̂
�b0; w
�ib0:
The right-hand side of the inequality in equation (49) is the optimal e¤ort in the benchmark case withnon-contractible e¤ort, given in equation (14). This establishes the lemma.
Proof of Proposition 6. Consider, �rst, the range b 2 [0; b1). Di¤erentiate equation (14) withrespect to b0,
X00 ��b0��0�b0�= �
�Z 1
0
@
@b0V�b0; �0; �w
�dF (�)�
Z 1
0
@
@b0V�b0; �0; w
�dF (�)
�= ��
�1� F
����b0���
� u0�Q�B�b0; �w
�; �w��B
�b0; �w
�+ �w � b0
�(50)
+��1� F
���b0���
��Q�B�b0; w
���B
�b0; w
�+ w � b0
�Taking the limit of equation (50) as b0 ! 0 yields
X00( (0))0 (0) = � [1� F (� (0))]� [Q (B (0; w))�B (0; w) + w � 0]
���1� F
��� (0)
��� u0 [Q (B (0; �w) ; �w)�B (0; �w) + �w � 0]
= ��u0 (Q (B (0; w))�B (0; w) + w � 0)� u0 ( �w)
�> 0; (51)
where the last equation uses the facts that �� (0) = � (0) = F (0) = 0 and that during normal timesc = �w if b = 0. Note that during recession, the annualized present value of income is strictly smallerthan �w. Therefore, it can never be optimal to choose consumption during recession larger than orequal to �w when b = 0. Since the marginal utility of consumption is larger in a recession thanduring normal times, the right-hand side of equation (51) is strictly positive. Since X 00 > 0, thenlimb!0
0 (b) = 0 (0) > 0. By continuity, it follows then that 0 (b) will be positive for a range of bclose to b = 0, so there must exist a b1 > 0 such that 0 (b) > 0 for all b 2 [0; b1).Consider, next, the range b 2
�b�;�b
�, in which case F (� (b)) = 1 and F
��� (b)
�< 1. This implies that
equation (50) can be written as
X00( (b))0 (b) = ��
�1� F
��� (b)
��� u0 [Q (B (b; �w) ; �w)�B (b; �w) + �w � b] < 0;
42
which establishes that 0 (b) < 0 for all b 2�b�;�b
�and with strict inequality also for b = b�. By
continuity, it follows then that there exists a b2 < b� such that 0 (b) < 0 for all b 2�b2;�b
�. Finally,
in the range where b � �b, F (� (b)) = F��� (b)
�= 1 so the right-hand side of equation (50) becomes
zero, implying that 0 (b) = 0.
Proof of Lemma 4. Di¤erentiating the bond revenue with respect to b yields
d
dbfQ (b; w) bg =
d
db
npbQ (b; �w) + (1� p) bQ̂ (b; w)
o+0 (b)�
�Q (b; �w)� Q̂ (b; w)
�b
= (b)� 1
R
�1� F
��� (b)
��+ (1�(b))� 1
R(1� F (� (b))) (52)
+0 (b)��Q (b; �w)� Q̂ (b; w)
�b;
where the second equality can be derived as following:
d
db
npbQ (b; �w) + (1� p) bQ̂ (b; w)
o= p
1
R
�1� F
��� (b)
��+ (1� p) Q̂ (b; w)
+ (1� p)"
� bRf (� (b))� �
0 (b)�1R1b
R �(b)0
���1 (�)� f (�) d�
�+ 1
Rbf (� (b))�0 (b)
#= p
1
R
�1� F
��� (b)
��+ (1� p) 1
R(1� F (� (b)))
Consider, �rst, the case in which (b) is constant, (b) = p: In this case, debt revenue is increasingfor all b < �b; since, then, p=R�
�1� F
��� (b)
��+(1� p) =R�(1� F (� (b))) > 0. Moreover, it reaches
a maximum at b = �b (recall that F��� (b)
�< F (� (b)) for all b < �b). This establishes that, if is
constant, then �bR = �b:Consider, next, the general case. Proposition 6 implies that, in the range where b 2 [b2;�b]; 0 (b) <
0: Since Q (b; �w) > Q̂ (b; w), then, in a left neighborhood of �b; 0 (b)�hQ (b; �w)� Q̂ (b; w)
ib < 0: This
means that, starting from �b; one can increase the debt revenue by reducing debt, i.e., �bR < �b:
Proof of Proposition 7. The procedure is analogous to the derivation of the CEE in normal times.The �rst-order condition of (10) for w = w yields
0 =d
db0�Q�b0; w
�� b0
� u0 (c) + �
�1�
�b0�� ddb0EV
�b0; w
�+ �
�b0� ddb0EV
�b0; �w
�;
where a term has been cancelled by the envelope theorem. Using the same argument as in the proofof Proposition 4 we can write:
d
dbEV (b; w) = � [1� F (� (b))]� u0 [Q (B (b; w) ; w)�B (b; w) + w � b] :
Plugging this back into the FOC (after leading the expression by one period) yields the CEE
0 = u0 (c)�n0�b0��R
hQ�b0; �w
�� Q̂
�b0; w
�ib0+
+ �b0���1� F
����b0���
+�1�
�b0����1� F
���b0���
��R��1�
�b0����1� F
���b0���
u0�c0jH;w
�+
�b0���1� F
����b0���
u0�c0jH; �w
��;
43
where the equality follows from Lemma 4. Rearranging terms yields equation (16).
Proof of Proposition 8. We write the Lagrangian,
�� =
Z@
��w � c� + � �P (�!�)
�dF (�) + ��
�Z@[u (c�) + ��!�] dF (�)� �
�+
Z@��� [u (c�) + ��!� � �� + �] dF (�) ;
with the associated multipliers �� and ���. The �rst-order conditions yield
f (�) = u0 (c�)���f (�) + ���
�; (53)
��� + ��f (�) = � �P 0 (�!�) f (�) : (54)
The envelope condition yields� �P 0 (�) = �� (55)
The two �rst-order conditions and the envelope condition jointly imply that
u0 (c�) = � 1�P 0 (�!�)
(56)
�P 0 (�!�) = �P 0 (�)����f (�)
: (57)
Note that (56) is equivalent to (20) in the text. Consider, next, two cases, namely, when the PC isbinding and when it is not binding.
When the PC is binding, ��� > 0: (57) implies then that �!� > �: Then, (56) and (21) determinejointly the solution for (c�; �!�) : When the PC is not binding, ��� = 0: (57) implies then that �!� = �and c� = c (�) :
What remains to be shown is that the �rst-order conditions are su¢ cient. The proof followsThomas and Worrall (1990, Proof of Proposition 1). The details of this proof are in Proposition 14 inthe online appendix.
Proof of Proposition 9. We write the Lagrangian,
� =
Z@
�w � c� + �
�(1� p�)P
�!��+ p� �P (�!�)
��dF (�)
+�
�Z@
�u (c�)�X (p�) + �
�(1� p�)!� + p��!�
��dF (�)� �
�+
Z@���u (c�)�X (p�) + �
�(1� p�)!� + p��!�
�� � + �
�dF (�) ;
where � and �� denote the multipliers in recession. The �rst-order conditions yield:
f (�) = u0 (c�)�� f (�) + ��
�(58)
�� + � f (�) = �P 0�!��f (�) (59)
�� + � f (�) = � �P 0 (�!�) f (�) (60)
���P (�!�)� P
�!���f (�) =
��� + � f (�)
� �X 0 (p�)� �
��!� � !�
��(61)
44
The envelope condition yields:�P 0 (�) = �� (62)
Combining the �rst-order conditions and the envelope condition yields:
u0 (c�) = � 1
P 0�!�� (63)
P 0�!��= P 0 (�)�
��f (�)
P 0�!��= �P 0 (�!�)
X 0 (p�) = ��u0 (c�)
��P (�!�)� P
�!���+��!� � !�
��We distinguish two cases, namely, when the PC is binding and when it is not binding.(I) When the PC is binding and the recession continues, �� > 0; !� > �; and
u (c�)�X (p�) + ��(1� p�)!� + p��!�
�= � � �
Then, (25), (27), (28) and (26) determine jointly the solution for�c�; p�; !�; �!�
�. In this case, there
is no history dependence, i.e., � does not matter.(II) When the PC is not binding, �� = 0: Then, !� = � and c� = c (�) : The solution is history
dependent. Moreover, (28) implies that
� ��u0 (c (�))
��P (�! (�))� P (�)
�+ [�! (�)� �]
= X 0 (p (�)) ; (64)
namely, the planner requires constant e¤ort over the set of states for which the constraint is notbinding: p� = p (�). Di¤erentiating the left-hand side yields
u00 (c (�)) c0 (�)���P (�! (�))� P (�)
�| {z }<0
+�u0 (c (�))P 0 (�) + 1
� ��!0 (�)� 1
�= u00 (c (�)) c0 (�)�
��P (�! (�))� P (�)
�< 0
since, recall, (25) implies that �P 0 (�) = �1=u0 (c (�)). This implies that the right-hand side must alsobe decreasing in �. Since X is concave and increasing, this implies in turn that p (�) must be increasingin �.
(III) Finally, we prove that !� < �!�. To this aim, note that � > �� since it is more expensive todeliver a given promised utility during recession than in normal times. Hence, the respective envelopeconditions, (55) and (62), imply that, for any x, P 0 (x) < �P 0 (x). Next, note that since both P (x) and�P (x) are decreasing concave functions, then P 0 (x1) = �P 0 (x2) , x1 < x2. Therefore, the conditionP 0�!��= �P 0 (�!�) (see equation (63)) implies that !� < �!�.
Proof of Lemma 5. The Lagrangian of the planner�s problem reads as
� =
Z@
�w � c� + �
�(1� p�)P
�!��+ p� �P (�!�)
��dF (�)
+�
�Z@
�u (c�)�X (p�) + �
�(1� p�)!� + p��!�
��dF (�)� �
�+
Z@���u (c�)�X (p�) + �
�(1� p�)!� + p��!�
�� � + �
�dF (�)
+
Z@ �
�� � ~� �X (p�) + �
�(1� p�)!� + p��!�
��dF (�) ;
45
where the Lagrange multipliers of the PC and IC must be non-negative for all �, �� � 0; � � 0. The�rst-order conditions yield:
f (�) = u0 (c�)�� f (�) + ��
�; (65)
�� + � f (�) + � = �P 0�!��f (�) ; (66)
�� + � f (�) + � = � �P 0 (�!�) f (�) ; (67)
���P (�!�)� P
�!���f (�) =
��� + � f (�) + �
� �X 0 (p�)� �
��!� � !�
��; (68)
while the envelope condition yields P 0 (�) = �.The �rst-order conditions (66)�(68) imply equations (31)-(32) in the text. Since P and �P are
monotonic and concave, equation (31) implies a positive relationship between !� and �!�. Equation(32) yields then a negative relationship between p� and !�. Consider, next, the IC constraint. Whenthe IC constraint is binding, equations (30), (31), and (32) pin down a unique solution for p�; !� and�!�; denoted by (p�; !�; �!�).
Proof of Proposition 10. This proof builds on the proof of Lemma 5. Suppose � > !� (case1). Then the IC is not binding in the initial period. Moreover, by Proposition 9, the promisedutility is non-decreasing over time. Thus, the IC will never bind in the future, and can be ignoredaltogether. Suppose, next, that � � !� (case 2). We �rst determine the upper bound on �; denotedby ��, such that the PC is binding while the IC is not binding. Let
�c�; p�; !�; �!�
�denote the solution
characterized in Proposition 9 when the IC is not binding and (c��; p�; !�; �!�) the solution characterized
in Proposition 10 when the IC is binding. Note that c�� is de�ned in (33). At the threshold realization��, the two allocations must be equivalent, i.e.,
(c�� ; p�� ; !�� ; �!��) = (c��� ; p
�; !�; �!�):
Given a promised utility of !�, the promise-keeping constraint implies:
!� = !�� =
Z ��
0
�u(c�)�X(p�) + �
�(1� p�)!� + p��!�
��dF (�)
+
Z �max
��
�u(c��)�X(p��) + �
�(1� p��)!�� + p�� �!��
��dF (�)
=
Z ��
0(v � �) dF (�) + (v � ��) [1� F (��)] = v �
Z ��
0�dF (�)� �� [1� F (��)] : (69)
Since !� is decreasing in �; then �� is unique. Moreover, if � < ��; then !� > !
�: In this case, thesolution is not history-dependent and is determined as in Proposition 9 (case 2.a). If, to the opposite,� � ��; then !� = !�: Two subcases must be distinguished here. First, if � � �� and � = !�, then themultipliers of both the IC and PC must be zero, �� = � = 0; because the planner keeps the triplet(p�; !�; �!�) constant. More formally, the envelope condition together with equation (66) implies that
P 0 (�) = P 0�!��+�� + �f (�)
:
Thus, � = !� = !�; and both the multiplier of the IC and that of the PC must be zero. In otherwords, as long as the IC was binding in the previous period, and continues to bind in the currentperiod, the planner keeps consumption, e¤ort and promised utilities constant.
46
Second, if � � �� and � < !�, then the planner must adjust promised utility, !� = !�, to satisfythe IC. In this case, the multiplier of the IC must be strictly positive, � > 0.
For the determination of consumption, two separate cases must be distinguished. In the �rst case(2.b), � � ~�(�) (where the expression for ~�(�) is given below), so that both the IC and the PC bind.In this case, �� > 0, and the IC and the PC determine jointly the consumption level, whose level isgiven by c�� as de�ned in equation (33). In the second case (2.c), � > ~�(�), so that the PC does notbind. In this region, �� = 0, and the consumption level provided by the planner is pinned down bythe promise-keeping constraint (23), and given by equation (34).
Next, we determine the unique threshold, ~�(�); that sets apart case (2.b) from case (2.c) and showthat ~�(�) � ��. Because the PC holds with equality at the threshold realization ~�(�), the consumptionlevel for all realizations of � > ~�(�) where the PC is not binding must be given by
c�~�(�) = u�1�~� � ~�(�)
�:
This condition, together with the promise-keeping constraint, fully characterizes the threshold ~�(�):
� =
Z ��
0
�u(c�)�X(p�) + �
�(1� p�)!� + p��!�
��dF (�)
+
Z ~�(�)
��
�u(c��)�X(p�) + � [(1� p�)!� + p��!�]
�dF (�)
+
Z �max
~�(�)
hu(c�~�(�))�X(p��) + �
�(1� p��)!�� + p�� �!��
�idF (�)
=
Z ��
0(v � �) dF (�) +
Z ~�(�)
��(v � �) dF (�) +
Z �max
~�(�)
hu(c�~�(�)) + Z(b0)
idF (�)
= F (~�(�))v �Z ~�(�)
0�dF (�) +
�v � ~�(�)
� h1� F (~�(�))
i= v �
Z ~�(�)
0�dF (�)� ~�(�)
h1� F (~�(�))
i:(70)
If � = !� then equations (69) and (70) imply that �� = ~�(�). The right-hand side of (70) is falling in~�. It follows that if � < !�, then ~�(�) � ~�(!�) = ��.
Proof of Proposition 11. We prove the proposition by deriving a contradiction. To this aim,suppose that, for �� (b) = �P (�) ; the planner can deliver more utility to the agent than can the laissez-faire equilibrium. Namely, � > EV (b; �w). Then, since �P is a decreasing strictly concave function,we must have that �P (EV (b; �w)) > �P (�) and �P 0 (EV (b; �w)) > �P 0 (�) : We show that this inequality,along with the set of optimality conditions, induces a contradiction.
First, recall, that equation (6) implies that �� (b) = RQ (b; �w) b: Thus,
�P (EV (b; �w)) > �P (�) = RQ (b; �w) b; (71)
where EV (b; �w) is decreasing in b. Di¤erentiating the two sides of the inequality (71) with respect tob yields
�P 0 (EV (b; �w))� d
dbEV (b; �w) >
d
db[Q (b; �w) b]�R = 1� F
��� (b)
�; (72)
where the right-hand side equality follows from the proof of Lemma 2. Next, equation (48) impliesthat
d
dbEV (b; �w) = �
�1� F
��� (b)
��� u0 [C (b; �w)] ;
47
where C (b; �w) = Q (B (b; �w) ; �w) � B (b; �w) + �w � b is the consumption level in the laissez-faire equi-librium when the debt b is honored. Plugging the expression of d
dbEV (b; �w) into (72), and simplifyingterms, yields
u0 (C (b; �w)) > � 1�P 0�EV
��b; �w
�� : (73)
Next, note that C (b; �w) = c (�) : Equation (73) yields u0 (c (�)) > �1= �P 0�EV
��b; �w
��; while (56)
yields that u0 (c (�)) = �1= �P 0 (�) : Thus, the two conditions jointly imply that �P 0 (�) > �P 0�EV
��b; �w
��which in turn implies that � < EV
��b; �w
�, since �P is decreasing and concave. This contradicts the
assumption that � > EV��b; �w
�:
The analysis thus far implies that � � EV��b; �w
�:We can also rule out that � < EV (b; �w), because
it would contradict that the allocation chosen by the planner is constrained e¢ cient. Therefore,� = EV (b; �w) :
Proof of Proposition 12. We proceed in two steps: �rst, we derive the CEEs (step A), and thenwe show that �
�b0w; b
0�w
�> 0 (step B).
Step A: The �rst-order conditions with respect to b0�w and b0w in problem (36) yields
0 = u0 (c)� d
db0�wREV
�b0w; b
0�w
�+ �
�b0w; b
0�w
� d
db0�wEV
�b0�w; �w
�;
0 = u0 (c)� d
db0wREV
�b0w; b
0�w
�+ �
�1�
�b0w; b
0�w
�� d
db0wEV
�b0w; w
�;
where REV�b0w; b
0�w
�� b0w�Qw
�b0w; b
0�w
�+ b0�w�Q �w
�b0w; b
0�w
�is the bond revenue and c is de�ned in the
Proposition. Note that both equations have been simpli�ed using an envelope condition. The valuefunction has a kink at bw = b̂ (�;w) : Consider, �rst, the range of realizations � 2 [� (b) ;1) ; implyingthat bw < b̂ (�;w). Di¤erentiating the value function yields:
d
dbV (b; �; �w) = �u0 [Q (B (b; �w) ; �w)�B (b; �w) + �w � b] ;
d
dbV (b; �; w) = �u0
�Qw�Bw (b) ; B �w (b)
��Bw (b) +Q �w
�Bw (b) ; B �w (b)
��B �w (b) + w � b
�;
where Bw and B �w denote the optimal issuance of the two assets, respectively. Next, consider the rangeof realizations � < � (b) ; implying that bw � b̂ (�;w). In this case, ddbV (b; �; w) = 0.
In analogy with equation (48), we obtain:
d
dbwEV
�bw; w
�= �
�1� F
���bw���
� u0�c0jH;w
�: (74)
Plugging (48) and (74) into the respective �rst-order conditions, and leading by one period, yields
u0 (c)� d
db0�wREV
�b0w; b
0�w
�= �
�b0w; b
0�w
���1� F
����b0�w���
� u0�c0jH; �w
�;
u0 (c)� d
db0wREV
�b0w; b
0�w
�= �
�1�
�b0w; b
0�w
����1� F
����b0�w���
� u0�c0jH;w
�:
48
The marginal revenues from issuing recession-contingent debt is given by:
d
db0wREV
�b0w; b
0�w
�=
1��b0w; b
0�w
�R
�1� F
���b0w���
�@(b0w;b0�w)
@b0w
1��b0w; b
0�w
�Qw �b0w; b0�w�� b0w + b0�w � @
@b0wQ �w
�b0w; b
0�w
�=
1��b0w; b
0�w
�R
�1� F
���b0w���
+@�b0w; b
0�w
�@b0w
���b0w; b
0�w
�; (75)
where, note, @@b0wQ �w
�b0w; b
0�w
�=
@(b0w;b0�w)@b0w
Q �w(b0w;b0�w)(b0w;b0�w)
follows from applying standard di¤erentiation to the
de�nition of Q �w
�b0w; b
0�w
�in equation (38). Applying the same methodology to the recovery-contingent
debt, we obtain:
d
db0�wREV
�b0w; b
0�w
�=�b0w; b
0�w
�R
�1� F
����b0�w���
+@�b0w; b
0�w
�@b0�w
���b0w; b
0�w
�:
The CEEs conditional on the recession continuing and ending, respectively, are then:
�u0�c0jH;w
�u0 (c)
=1
R+
@
@b0w�b0w; b
0�w
��
��b0w; b
0�w
��1� F
���b0w��� �
1��b0w; b
0�w
���u0 (c0jH; �w)u0 (c)
=1
R+
@
@b0�w�b0w; b
0�w
��
��b0w; b
0�w
��1� F
��� (b0�w)
���b0w; b
0�w
� :Setting �R = 1 yields equations (39)-(40).
Step B: Next, we prove that, in equilibrium, ��b0w; b
0�w
�> 0. To prove the claim, it is useful to
de�ne the two functions
� �w�b0�w��
Q �w
�b0w; b
0�w
�� b0�w
�b0w; b
0�w
� =1
R
�1� F
����b0�w���
b0�w +
Z ��(b0�w)
0
���1 (�)� f (�) d�!;
�w�b0w��
Qw�b0w; b
0�w
�� b0w
1��b0w; b
0�w
� =1
R
�1� F
���b0w���
b0w +
Z �(b0w)
0��1 (�)� f (�) d�
!;
where, recall, � (x) > �� (x) and F (� (x)) > F��� (x)
�. Note that �
�b0w; b
0�w
�= � �w (b
0�w) � �w
�b0w�;
where both � �w and �w are increasing functions in the relevant range, i.e., b0�w � �b and b0w � �b. Weproceed in two steps. First, we show that �
�b0w; b
0�w
�� 0 ) b0�w > b
0w (step B1). Next, we show that
b0�w > b0w ) ��b0w; b
0�w
�> 0 (step B2). Steps B1 and B2 establish jointly a contradiction ruling out
that ��b0w; b
0�w
�� 0 (step B3).
Step B1: Suppose that ��b0w; b
0�w
�� 0. Then, the CEEs (39)�(40) and the assumption that
u00 < 0, imply thatc0jH; �w � c � c0jH;w: (76)
Suppose, to derive a contradiction, that b0w � b0�w. Recall that, if the recession ends and debt is honored,debt remains constant, i.e., b
00= B (b0�w) = b
0�w. Moreover, Q (b
0�w; �w) = Q �w
�b0w; b
0�w
�=�b0w; b
0�w
�. Thus,
49
c0jH; �w = Q (b0�w; �w) b0�w + �w � b0�w = � �w (b0�w) + �w � b0�w.
c0jH; �w = �b0w; b
0�w
�� �w�b0�w�+�1�
�b0w; b
0�w
��� �w�b0�w�+ �w � b0�w
� �b0w; b
0�w
�� �w�b0�w�+�1�
�b0w; b
0�w
��� �w�b0w�+ �w � b0w
� �b0w; b
0�w
�� �w�b0�w�+�1�
�b0w; b
0�w
���w�b0w�+ �w � b0w
> �b0w; b
0�w
�� �w�b0�w�+�1�
�b0w; b
0�w
���w�b0w�+ w � b0w = c0jH;w:
The �rst inequality follows from the assumption that b0w � b0�w and the fact that (1� p) � �w (x)�x < 0 forany p 2 [0; 1], which is due to the fact that � �w (x) � x=R < x for any x. The second inequality followsfrom the fact that � �w
�b0w�� �w
�b0w�, see equation (77) below. The last inequality follows from the
maintained assumption that �w > w. We have therefore proven that if b0w � b0�w then c0jH; �w > c0jH;w,which contradicts (76) and, hence, implies that �
�b0w; b
0�w
�� 0. We conclude from Step B1 that
��b0w; b
0�w
�� 0) b0�w > b
0w.
Step B2: Suppose that b0w = b0�w = x: Then, for any x:
�(x; x) = � �w (x)� �w (x) =1
R
Z �(x)
��(x)
��x� ��1 (�)
�� f (�) d�
�| {z }
>0
(77)
+1
R
Z ��(x)
0
�����1 (�)� ��1 (�)
�� f (�) d�
�| {z }
>0
> 0:
Since � �w (x) is an increasing function for x � �b, equation (77) implies that � �w (b0�w) > �w�b0w�, for all
b0w < b0�w � �b. We conclude from Step B2 that b0�w > b
0w ) �
�b0w; b
0�w
�> 0.
Step B3: Putting together the conclusions of Step B1 and Step B2, we derive a contradiction:��b0w; b
0�w
�� 0) b0�w > b
0w ) �
�b0w; b
0�w
�> 0. Therefore, we must have that �
�b0w; b
0�w
�> 0:
Proof of Proposition 13. The strategy of the proof is the same as that of Proposition 11. We provethe proposition by deriving a contradiction. To this aim, suppose that, for �
�bw�= P (�) ; the planner
can deliver more utility than the agent gets in the laissez-faire equilibrium. Namely, � > EV�bw; w
�.
Then, since P is a decreasing strictly concave function, we must have that P�EV
�bw; w
��> P (�)
and P 0�EV
�bw; w
��> P 0 (�) : Note that, absent moral hazard, the price of recession-contingent
debt is independent of the amount of recovery-contingent debt. It is therefore legitimate to de�ne~Qw�bw�� Qw
�bw; b �w
�:
First, the same argument invoked in the proof of Proposition 11 implies that��bw�= R
1�p~Qw�bw�bw:
Hence,
P�EV
�bw; w
��> P (�) =
R
1� p~Qw�bw�bw: (78)
where EV�bw; w
�is decreasing in bw. Di¤erentiating the two sides of the inequality (78) with respect
to bw yields:
P 0�EV
�bw; w
��� d
dbwEV
�bw; w
�(79)
>R
1� pd
dbw
�~Qw�bw�bw
�= �
�1� F
���bw���;
50
where the right-hand side equality follows from equation (75). Next, equation (48) implies that
d
dbEV (b; �w) = �
�1� F
��� (b)
��� u0 (C (b; �w)) ;
where C�bw; w
�is the consumption level assuming that the recession-contingent debt bw is honored.
Plugging in the expression of ddbwEV
�bw; �w
�allows us to simplify (79) as follows:
u0�C�bw; w
��> � 1
P 0�EV
�bw; w
�� : (80)
Next, note that C�bw; w
�= c (�) : Equation (80) yields u0 (c (�)) > � 1
�P 0(EV (b �w; �w)); while (63) yields
that u0 (c (�)) = � 1P 0(�)
: Thus, the two conditions jointly imply that � 1P 0(�)
> � 1P 0(EV (b �w;w))
which inturn implies that � < EV (b �w; w), since P is decreasing and concave. This contradicts the assumptionthat � > EV (b �w; w) :
The analysis thus far establishes that � � EV (b �w; w) : We can also rule out that � < EV (b �w; w)because it would contradict that the allocation chosen by the planner is constrained e¢ cient. There-fore, � = EV (b �w; w).
Proof of results in Section 6. The price of state-contingent debt along the equilibrium pathyields:
QCw�b0w; p
�=
1� pR
Q̂Cw�b0w�; QC�w
�b0�w; p
�=p
RQ̂C�w
�b0�w�;
Q̂Cw�b0w�=
1
R
�1� F
���Cw�b0w���
+1
b0w
Z ��C(b0w)
0
���C�1w (�) dF (�)
�:
The corresponding value function is:
V C�bw; �; w
�= max
fb0w;b0�w;pg
�u
�1� pR
Q̂Cw�b0w�� b0w +
p
RQ̂C�w
�b0�w�� b0�w + w � B (b; �; w)
��X (p) + � [1� p]EV
�b0w; w
�+ �pEV
�b0�w; �w
�:
The e¢ cient e¤ort is pinned down by the following condition:
X 0 (p) = u0 (c)��Q̂Cw
�b0w�� b0w � Q̂C�w
�b0�w�� b0�w
�+ � [1� p]EV C
�b0w; w
�+ �pEV
�b0�w; �w
�:
The CEEs yieldu0�c0jH;w
�u0 (c)
= 1 ;u0 (c0jH; �w)u0 (c)
= 1:
Consider, next a deviation such that the agent exercises the ex-post optimal e¤ort level conditionalon the debt issued. In this case, the agent is punished by a permanent reversion to the Markovequilibrium. The value of such deviation is
V DEV�bw; �; w
�= u(c)�X
�(b0�w; b
0w)�+�
�(b0�w; b
0w)EV (b
0�w; �
0; �w) + (1�(b0�w; b0w))EV (b0w; �0; w)�:
The incentive compatibility constraint can then be expressed as
�X�pC�+ �
�pC � EV (b0�w; �0; �w) + (1� pC)� EV C(b0w; �0; w)
�> �X
�(b0�w; b
0w)�+ �
�(b0�w; b
0w)EV (b
0�w; �
0; �w) + (1�(b0�w; b0w))EV (b0w; �0; w)�;
that simpli�es to equation (42) in the text.
51
Online Appendix of�Sovereign Debt and Structural Reforms�
Andreas Müller, Kjetil Storesletten and Fabrizio Zilibotti
B Additional proofs and details on numerical algorithm
This online appendix contains technical analysis and additional proofs related to Sections 3.4, 4 and
6. It als contains details of the numerical algorithm in Section 7.
B.1 Details of the proof of Proposition 4
We have established in Lemma 3 that an interior solution necessarily satis�es the FOC
[1� F (�(B(b; �w); �w))]u0(Q(B(b; �w); �w)B(b; �w)� b+ �w)
= [1� F (�(B(b; �w); �w))]u0(Q(B(B(b; �w); �w))B(B(b; �w); �w)�B(b; �w) + �w):
Both, constant debt accumulation, B(b; �w) = b and maximial debt accumulation B(b; �w) = �b areobvious solution candidates. Note however, that B(b; �w) = �b can only be a global maximum whenthe outstanding debt level is at the maximum, b = �b (where the two solution candidates coincide),because otherwise the objective is strictly falling in the left neigborhood of �b since
u0(Q(�b; �w)�b� b+ �w) < u0(Q(�b; �w)�b� �b+ �w); b < �b:
Thus, we are therefore left to show that B(b; �w) = b is the unique solution that satis�es the FOC. Forthe ease of exposition, let us rewrite the FOC as�
1� F (�(b0; �w))�u0(Q(b0; �w)b0 � b+ �w)
=�1� F (�(b0; �w))
�u0(Q(b00; �w)b00 � b0 + �w):
Suppose there existed a solution candidate where the current debt accumulation b0 was strictly reduced,
b � b0 > 0. Because the marginal bond revenue is falling and smaller than R�1 this leads to a
smaller reduction of today�s consumption relative the increase in next period�s consumption for a
given b00. Therefore, b00 has to be lowered even further to equalize consumption intertemporally,
b0 � b00 > b � b0 > 0. This argument can be expanded to further periods such that the equilibrium
would feature accelerated asset accumulation and ever falling consumption which contradicts the
requirement that it is a global maximum. Suppose to the contrary that current debt accumulation
b0 was strictly increased, b0 � b > 0. Then, by the same argument as before, b00 has to be increased
even further to equalize consumption intertemporally, b00 � b0 > b0 � b > 0, and the equilibrium would
1
feature accelerated debt accumulation. This implies that the economy will hit the upper bound on
debt accumulation �b for some outstanding debt level below the maximum, b < �b. However, we have
already shown that this cannot be optimal. Thus, B(b; �w) = b is the unique maximizer of the objective
function.
B.2 Formal properties of the constrained e¢ cient allocation (Section 4)
Proposition 14 There exists unique pro�t functions �P and P that solve the programs (17) and
(22), respectively. Moreover, �P and P are continuously di¤erentiable and strictly concave. Given
the promised utility � and the realization �, (i) if w = �w there exists a unique optimal pair of promised
utility and consumption f�!� (�) ; c� (�)g ; (ii) if w = w there exists a unique optimal 4-tuple of promisedutilities, consumption and e¤ort,
�!� (�) ; �!� (�) ; p� (�) ; c� (�)
. The �rst-order conditions in Propo-
sitions 8 and 9 are necessary and su¢ cient when the solution is interior.
The proof strategy follows Thomas and Worrall (1990, Proof of Proposition 1), i.e., we show that
the problem is a contraction mapping to establish the uniqueness and strict concavity of �P and P .
The di¤erentiabilty of �P and P follows from an application of Lemma 1 in Benveniste and Scheinkman
(1979). Finally, we prove that �P and P pin down uniquely promised utilities, e¤ort and consumption.
The arguments used to prove Proposition 14 in normal times and recession are mirror image of
each other, except that the recession case is complicated by the presence of an e¤ort choice. For
this reason, we prove the results when w = w (assuming the properties of �P follow the proposition),
omitting the simpler proof for the case in which w = �w [more precisely, the arguments are extended
by setting X(p�) = 0 and p� = 1].
We prove the results in the form of three lemmas and one corollary.De�ne, �rst, the mapping T (x)(�) as the right-hand side of the planner�s functional equation
T (x)(�) = max(fc�;p�;�!�;!�g�2@)2�(�)
Z@
�w � c� + �
�p� �P (�!�)
+(1� p�)x(!�)
��dF (�)
where maximization is constrained by the set �(�) de�ned byZ@
�u(c�)�X(p�) + �
�p��!� + (1� p�)!�
��dF (�) � �
u(c�)�X(p�) + ��p��!� + (1� p�)!�
�� � � �; 8� 2 @;
c� 2 [0; �w]; p� 2 [p; �p]; �; !� 2 [� � E [�] ; �]; �!� 2 [�� � E [�] ; ��]:
Recall that � and �� are the values of the outside option during recession and normal times, respectively.
We take as given the uniqueness, strict concavity, and di¤erentiability of the pro�t function in normal
times, �P . Moreover, let the pro�t in normal times be bounded between �PMIN = 0 and PMAX =
�w=(1� �).
2
Lemma 6 T (x) maps concave functions into strictly concave functions.
Proof. Let � 0 6= � 00 2 [� � E [�] ; �], � 2 (0; 1), �o = �� 0 + (1� �)� 00, P k(�) = T (P k�1)(�), and P k�1be concave. Then,
P k�1(��0 + (1� �)� 00) � �P k�1(� 0) + (1� �)P k�1(� 00):
We follow the strategy of Thomas and Worrall (1990, Proof of Proposition 1), i.e., we construct a
feasible but (weakly) suboptimal contract,nco�; p
o�; �!
o�; !
o�
o�2@, such that even the pro�t generated by
the suboptimal contract P ok(��0+(1��)� 00) � P k(�� 0+(1��)� 00) dominates the linear combination of
maximal pro�ts �P k(�0)+(1��)P k(� 00). De�ne the weights �; �� 2 (0; 1) and the 4-tuple (co�; po�; !o�; �!o�)
such that
� � � [1� p�(� 0)]�(1� p�(� 0)) + (1� �)(1� p�(� 00))
� � 1� p�(�0)
1� po�(�o)
�� � �p�(�0)
�p�(� 0) + (1� �)p�(� 00)� � p�(�
0)
po�(�o)
!o�(�o) = �!�(�
0) + (1� �)!�(� 00)�!o�(�
o) = ���!�(�0) + (1� ��)�!�(� 00)
co�(�o) = u�1
��u(c�(�
0)) + (1� �)u(c�(� 00))�:
Hence,
(1� po�(�o))!o�(�o) = ��1� p�(� 0)
�!�(�
0) + (1� �)(1� p�(� 00))!�(� 00)po�(�
o)�!o�(�o) = �p�(�
0)�!�(�0) + (1� �)p�(� 00)�!�(� 00)
By construction the suboptimal allocation satis�es
co� 2 [0; �w]; p0� 2 [p; �p]; !o� 2 [� � E [�] ; �]; �!o� 2 [�� � E [�] ; ��];
and, given the promised-utility �o, is also consistent with the promise-keeping constraintZ@
�u�co�(�
o)��X(po�(�o)) + �
�(1� po�(�o))!o�(�o) + po�(�o)�!o�(�o)
��dF (�)
=
Z@
24 �u (c�(� 0)) + (1� �) (c�(� 00))�X (�p�(� 0) + (1� �)p�(� 00))+���(1� p�(� 0))!�(� 0) + (1� �)(1� p�(� 00))!�(� 00)
�+� [�p�(�
0)�!�(�0) + (1� �)p�(� 00))�!�(� 00)]
35 dF (�)>
Z@
24 �u (c�(� 0)) + (1� �) (c�(� 00))� [�X(p�(� 0) + (1� �)X(p�(� 00))]+���(1� p�(� 0))!�(� 0) + (1� �)(1� p�(� 00))!�(� 00)
�+� [�p�(�
0)�!�(�0) + (1� �)p�(� 00))�!�(� 00)]
35 dF (�)= �� 0 + (1� �)� 00 = �o:
The fact that X (�p�(� 0) + (1� �)p�(� 00)) < �X(p�(� 0) + (1� �)X(p�(� 00)) follows from the convexity
3
of X. Moreover, the participation constraint for any � yields
u�co�(�
o)��X(po�(�o)) + �
�(1� po�(�o))!o�(�o) + po�(�o)�!o�(�o)
�=
24 �u (c�(� 0)) + (1� �) (c�(� 00))�X(�p�(� 0) + (1� �)p�(� 00))+���(1� p�(� 0))!�(� 0) + (1� �)(1� p�(� 00))!�(� 00)
�+� [�p�(�
0)�!�(�0) + (1� �)p�(� 00))�!�(� 00)]
35>
24 �u (c�(� 0)) + (1� �) (c�(� 00))� [�X(p�(� 0) + (1� �)X(p�(� 00))]+���(1� p�(� 0))!�(� 0) + (1� �)(1� p�(� 00))!�(� 00)
�+� [�p�(�
0)�!�(�0) + (1� �)p�(� 00))�!�(� 00)]
35= �
�u�c�(�
0)��X(p�(� 0) + �(1� p�(� 0))!�(� 0) + �p�(� 0)�!�(� 0)
�+(1� �)
�c�(�
00)�X(p�(� 00)) + �(1� p�(� 00))!�(� 00) + �p�(� 00))�!�(� 00)�
� � (� � �) + (1� �) (� � �) = � � �;
where we used again the strict convexity of the cost function X. Thus, we have proven that the
suboptimal allocationnco�; p
o�; !
o�; �!
o�
o�2@
is feasible. Namely, it satis�es the participation constraints
and delivers at least the promised utility �o. The pro�t function evaluated at the optimal contract�c�; p�; !�; �!�
�2@ then implies the following inequality,
�P k(�0) + (1� �)P k(� 00)
= �T (P k�1)(�0) + (1� �)T (P k�1)(� 00)
=
Z@
24 w � [�c�(� 0) + (1� �)c�(� 00)]+���(1� p�(� 0))P k�1(!�(� 0)) + (1� �)(1� p�(� 00))P k�1(!�(� 00))
����p�(�
0) �P (�!�(�0)) + (1� �)p�(� 00) �P (�!�(� 00))
�35 dF (�)
=
Z@
24 w � [�c�(� 0) + (1� �)c�(� 00)]+�(1� po�(�o))
��P k�1(!�(�
0)) + (1� �)P k�1(!�(� 00))�
�po�(�o)��� �P (�!�(�
0)) + (1� ��) �P (�!�(� 00))�
35 dF (�)<
Z@
24 w � u�1 (�u(c�(� 0)) + (1� �)u(c�(� 00)))+�(1� po�(�o))P k�1(�!�(� 0) + (1� �)!�(� 00))
�po�(�o) �P (���!�(�
0) + (1� ��)�!�(� 00))
35 dF (�)=
Z@
�w � co�(�o) + �
�po�(�
o) �P (�!o�(�o)) + (1� po�(�o))P k�1(!o�(�o))
��dF (�)
� P ok(�o) � P k(�o) = P k(�� 0 + (1� �)� 00):
The �rst inequality follws from the strict concavity of the utility function and the pro�t function in
normal times, along with the assumed concavity of P k�1. The second inequality, P k(�o) � P ok(�
o),
follows from the fact that the optimal allocation delivers (weakly) larger pro�ts than the suboptimal
one. We conclude that P k(��0+(1� �)� 00) > �P k(� 0)+(1� �)P k(� 00), i.e., P k is strictly concave. This
concludes the proof of the lemma.
Let denote the space of continuous functions de�ned over the interval [��E [�] ; �] and boundedbetween PMIN = �( �w�w)=(1� �) and PMAX = �w=(1� �). Moreover, let d1 denote the supremum
norm, such that (; d1) is a complete metric space.
4
Lemma 7 The mapping T (x) is an operator on the complete metric space (; d1), T (x) is a con-
traction mapping with a unique �xed-point P 2 .
Proof. By the Theorem of the Maximum T (x)(�) is continuous in �. Moreover, T (x)(�) is boundedbetween PMIN and PMAX since even choosing zero consumption for any realization of � would inducepro�ts not exceeding PMAX
w + �
Z@
�p� �P (�!�) + (1� p�)x(!�)
�dF (�) < �w + �=(1� �) �w
= �w=(1� �) = PMAX ,
and choosing the maximal consumption of �w for any � would induce pro�ts no lower than PMIN ,
�( �w � w) + �Z@x(!�)dF (�) � PMIN :
Thus, T (x)(�) is indeed an operator on (; d1):
According to Blackwell�s su¢ cient conditions T is a contraction mapping (see Lucas and Stokey
(1989, Theorem 3.3) if: (i) T is monotone, (ii) T discounts.
1. Monotonicity: Let x; y 2 with x(�) � y(�), 8� 2 [� � E [�] ; �]. Then
T (x)(�) = max(fc�;p�;�!�;!�g�2@)2�(�)
Z@
�w � c� + �
�p� �P (�!�)
+(1� p�)x(!�)
��dF (�)
� max(fc�;p�;�!�;!�g�2@)2�(�)
Z@
�w � c� + �
�p� �P (�!�)
+(1� p�)y(!�)
��dF (�)
= T (y)(�):
2. Discounting: Let x 2 and a � be a real constant. Then
T (x+ a)(�) = max(fc�;p�;�!�;!�g�2@)2�(�)
Z@
�w � c� + �
�p� �P (�!�)
+(1� p�)�x(!�) + a
� �� dF (�)� T (x)(�) + �a
< T (x)(�) + a;
since � < 1:
Thus, T is indeed a contraction mapping and according to Banach�s �xed-point theorem (see Lucasand Stokey (1989, Theorem 3.2)) there exists a unique �xed-point P 2 satisfying the stationaryfunctional equation,
P (�) = T (P )(�):
5
Corollary 2 The pro�t function P (�) is strictly concave in � 2 [� � E [�] ; �].
This follows immediately from Lucas and Stokey (1989, Corollary 1). Since the unique �xed-point
of T is the limit of applying the operator n times Tn(x)(�) starting from any (and, in particular the
concave ones) element x in , and the operator T maps concave into strictly concave functions the
�xed-point P must be strictly concave.
Lemma 8 The pro�t function P (�) is continuously di¤erentiable in � 2 [� � E [�] ; �].
Proof. The proof is an application of Benveniste and Scheinkman (1979, Lemma 1). Recall that �Pand P are strictly concave. Consider the pseudo pro�t function
eP (~�; �) �Z ~�(~�)
0
�w � ~c�(~�) + �
�p�(�) �P (�!�(�)) + (1� p�(�))P (!�(�))
��dF (�) (81)
+
Z 1
~�(~�)
hw � ~c~�(~�)(~�) + �
�p�(�) �P (�!�(�)) + (1� p�(�))P (!�(�))
�idF (�)
where the triplet (p�(�); !�(�); �!�(�)) is the same as in the optimal contract given an initial promise�, and the consumption function, ~c�(~�), is de�ned implicitly by the condition
~� =
Z ~�(~�)
0
�u (~c�(~�))�X(p�(�)) + �
�(1� p�(�))!�(�) + p�(�)�!�(�)
��dF (�)
+
Z 1
~�(~�)
hu�~c~�(~�)(~�)
��X(p�(�)) + �
�(1� p�(�))!�(�) + p�(�)�!�(�)
�idF (�): (82)
Note that for ~� = �, equation (82) is equivalent to the promise-keeping constraint. Moreover, for allstates � � ~�(~�) such that the (pseudo-)participation constraint is binding,
u (~c�(~�))�X(p�(�)) + ��(1� p�(�))!�(�)+p�(�)�!�(�)
�= � � �: (83)
Otherwise, when � > ~�(~�) then consumption and promised utility are history dependent, implyingthat
u�~c~�(~�)(~�)
��X(p�(�)) + �
�(1� p�(�))!�(�)+p�(�)�!�(�)
�= � � ~�(~�): (84)
Substituting in the right hand-side of (83) and (84), respectively, in the �rst and second line of (82),pins down the threshold ~� (~�) that that separates states in which the participation constraint is bindingfrom states with in which it is not binding:
~� = � �Z ~�(~�)
0
�dF (�)��1� F
�~� (~�)
��� ~� (~�) :
Di¤erentiating equation (83) with respect to ~� shows that, for � � ~�(~�); u0 (~c�(~�)) ~c0�(~�) = 0:
Di¤erentiating equation (82) shows that the consumption function, ~c~�(~�)(~�); is also continuously dif-
ferentiable when � > ~�(�). In particular,
1 = (1� F (~�(�)))u0�~c~�(~�)(~�)
�~c0~�(~�)(~�): (85)
6
Recall that the function eP has the properties that eP (~�; �) � P (�) with eP (�; �) = P (�). Thus,Lemma 1 in Benveniste and Scheinkman (1979) implies that the pro�t function P (�) is continuouslydi¤erentiable in � 2 [� � E [�] ; �], with derivative
P 0(�) = eP ~�(�; �) = �(1� F (~�(�)))~c0~�(�)(�)= �1=u0 [(c(�))] < 0:
The value of eP ~� follows from the di¤erentiation of (81) using standard methods. The last equality
follows from (85) and from the fact that ~c~�(�)(�) = c(�). This establishes that the pro�t function P (�)
is continuously di¤erentiable, concluding the proof.
We can now establish that the constrained allocation is unique.
Lemma 9 The constrained-optimal allocation is characterized by a unique 4-tuple of state-contingent
promised utilities, consumption and e¤ort levels,�!� (�) ; �!� (�) ; p� (�) ; c� (�)
.
Proof. Lemma 6 implies that there cannot be two optimal contracts with distinct promised-utilities.
Suppose not, so that there exists a 4-tuple of promised utilitiesn!0�; !
00�; �!
0�; �!
00�
osuch that either
!0�(�) = !00�(�) or �!0�(�) 6= �!00�(�) (or both). Then, from the strict concavity of P and �P ; it would
be possible to construct a feasible allocation that dominates the continuation pro�t implied by the
proposed optimal allocations, i.e., either P (�!0� + (1 � �)!00�) > �P (!0�) + (1 � �)P (!00�), or �P (���!0� +(1 � ��)�!00�) > �� �P (�!0�) + (1 � ��) �P (�!00�) (or both). This contradicts the assumption that the proposedallocations are optimal, establishing that the optimal contract pins down a unique pair of promised
utilities,n!�; �!
0�
o:
Finally, we show that a unique pair of promised utilities pins down uniquely e¤ort and consumption.More formally, the �rst order conditions imply that
X 0 (p�) = ���P 0
�!���1 � �P (�!�)� P �!���+ ��!� � !��� ;
�P 0�!���1
= u0(c�);
implying that, given � and �, e¤ort and consumption are uniquely determined.
B.2.1 Self-enforcing reform e¤ort
The proof of the pro�t function�s strict concavity and di¤erentiability when reform e¤ort is self-enforcing is by-and-large a corollary of the case without the additional incentive constraint. We knowfrom Proposition 10 that for � > !� the additional incentive constraint is never relevant, thus strictconcavity and the di¤erentiability of the pro�t function follows immediately from the recession case
7
discussed above. On the other hand, if � � !�, than the pro�t function evaluated at the optimalcontract reads as
P (�) =
Z ��
0
�w � c� + �
�(1� p�)P (!�) + p� �P (�!�)
��dF (�)
+
Z ~�(�)
��
�w � c�� + �
�(1� p�)P (!�) + p� �P (�!�)
��dF (�)
+
Z �max
~�(�)
hw � c�~�(�) + �
�(1� p�)P (!�) + p� �P (�!�)
�idF (�):
Note that the promised-utility � only enters the last two terms such that the �rst derivative of thepro�t function is given by (we will prove di¤erentiability of the pro�t function below)
P 0(�) =hw � c�~�(�) + �
�(1� p�)P (!�) + p� �P (�!�)
�if(~�(�))~�
0(�)
�Z �max
~�(�)
dc�~�(�)
d~�(�)~�0(�)dF (�)
�hw � c�~�(�) + �
�(1� p�)P (!�) + p� �P (�!�)
�if(~�(�))~�
0(�)
= �Z �max
~�(�)
dc�~�(�)d�
dF (�) < 0;
where equation (34) implies that dc�~�(�)=d� = �u0(~� � ~�(�))�1~�0(�) > 0 as the threshold ~�(�) is
decreasing in the promised utility. The negative second derivative follows immediately
P 00(�) = �"Z �max
~�(�)
d2c�~�(�)d�2
dF (�) +dc�~�(�)d�
f(~�(�))(�~�0(�))#< 0;
because
d2c�~�(�)=d�2 =
h�u00(~� � ~�(�))~�0(�)2 + u0(~� � ~�(�))~�00(�)
i�hu0(~� � ~�(�))�1~�0(�)
i�2> 0:
The positive sign of the second derivative is based on the fact that ~�00(�) > 0. This can be veri�ed
from totally di¤erentiating equation (70) with respect to �
1 = �h~�(�)f(~�(�))~�
0(�)i�h~�0(�)h1� F (~�(�))
i� ~�(�)f(~�(�))~�0(�)
i= �~�0(�)
h1� F (~�(�))
i) ~�
0(�) = �
h1� F (~�(�))
i�1< 0:
~�00(�) = �f(~�(�))~�0(�)=
h1� F (~�(�))
i2> 0:
Finally, as ~�(�) is continuously di¤erentiable, so is consumption, c�~�(�); and the pro�t function, P (�).
This concludes the argument.
8
B.3 Formal properties of the equilibrium with learning (Section 6)
Let the value functions of the government be denoted by �V (b; �; �) and �W (b; �). Since outrightdefault is never observed in equilibrium, the value function simpli�es to
�V (b; �; �) = maxb0[b;~b]
�u��Q�b0; �
�� b0 + �w � B (b; �; �w; �)
�+ � � EV
�b0; �
�: (86)
where EV (b0; �) � E �V�b0; �0;�
��0; �
��:
The function �� is such that ��(b; �) = �W (0; �)� �W (b; �). Given a debt issuance of b0 and a currentprior of �; debt will be honored next period if �0 � �� (b0; �) where �� is the unique �xed point of thefollowing equation
�� = ���b0;� (��; �)
�:
The probability of renegotiation is
E�F����b0; �0
�j�0�; �= �FCW
����b0;� (��; �)
��+ (1� �)FNC
����b0;� (��; �)
��= �FCW
����b0; �
��+ (1� �)FNC
����b0; �
��= F
����b0; �
�;� (��; �)
�;
and an arbitrage argument then implies the following bond price
�Q�b0; �
�=1
R
1� F (�� (b0; �) j� (��; �))+
�b0R ��(b0;�)0 b̂ (�;� (�; �)) dFCW (�) +
1��b0R ��(b0;�)0 b̂ (�;� (�; �)) dFNC (�)
!:
In what follows, we assume that the relevant equilibrium functions are di¤erentiable in b. Then,di¤erentiating b� �Q (b; �), with respect to b yields
d
db
�b� �Q (b; �)
= �Q (b; �) + b� d
db�Q�b0; �
�= �Q (b; �)� b
R
�@F (�� (b; �) j� (��; �))
@�
@�� (b; �)
@b
�+
+b
R
�
bb̂ (�� (b; �) ;� (�� (b; �) ; �)) fCW (�
� (b; �))@�� (b; �)
@b
� 1R
�
b
Z ��(b;�)
0b̂ (�;� (�; �)) dFCW (�)
+b
R
1� �bb̂ (�� (b; �) ;� (�� (b; �) ; �)) fNC (�
� (b; �))@�� (b; �)
@b
� 1R
1� �b
Z ��(b;�)
0b̂ (�;� (�; �)) dFNC (�) ;
9
such that
d
db
�b� �Q (b; �)
= �Q (b; �)� b
R
�@F (�� (b; �) j� (��; �))
@�
@�� (b; �)
@b
�| {z }
A+B
+
+�b
RfCW (�
� (b; �))@�� (b; �)
@b| {z }A
+ (1� �) bRfNC (�
� (b; �))@�� (b; �)
@b| {z }B
� �Q (b; �) + 1
R(1� F (�� (b; �) j� (��; �)))
=1
R(1� F (�� (b; �) j� (��; �))) :
Next, consider the consumption-savings decision. The �rst-order condition of (86) reads
d
db0��Q�b0; �
�b0� u0
��Q�b0; �
�� b0 + �w � B (b; �; �w; �)
�+d
db0�EV
�b0; �
�= 0:
The value function has a kink at b = b̂ (�; �) : Consider, �rst, the range where b < b̂ (�; �) :Di¤erentiating the value function yields
d
db�V (b; �; �) = �u0
��Q��B (b; �) ; �
�� �B (b; �) + �w � b
�;
where �B denotes the optimal issuance of new bonds. Next, consider the region of renegotiation,
b > b̂ (�; �) : In this case, ddb�V (b; �; �) = 0.
Using the results above one obtains
d
dbEV (b;� (�; �)) = �
Z@
d
db�V (b; �;� (�; �)) dFCW (�) + (1� �)
Z@
d
db�V (b; �;� (�; �)) dFNC (�)
= �
R ��(b;�)0
ddb�V (b; �;� (�; �)) dFCW (�)
+R1��(b;�)
ddb�V (b; �;� (�; �)) dFCW (�)
!
+(1� �) R ��(b;�)
0ddb�V (b; �;� (�; �)) dFNC (�)
+R1��(b;�)
ddb�V (b; �;� (�; �)) dFNC (�)
!
= �
Z 1
��(b;�)
d
db�V (b; �;� (�; �)) dFCW (�)
+ (1� �)Z 1
��(b;�)
d
db�V (b; �;� (�; �)) dFNC (�)
=
Z 1
��(b;�)
d
db�V (b; �;� (�; �)) dF (�j�)
= �Z 1
��(b;�)u0��Q��B (b;� (�; �)) ;� (�; �)
�� �B (b;� (�; �)) + �w � b
�dF (�j�) :
10
Plugging this expression back into the FOC, and leading the expression by one period, yields
0 =1
R
�1� F
����b0; �
�j� (��; �)
��� u0
��Q�b0; �
�� b0 + �w � B (b; �; �w; �)
���Z 1
��(b;�)u0��Q��B�b0;� (�; �)
�;� (�; �)
�� �B
�b0;� (�; �)
�+ �w � b
�dF (�j�)
thus
�R =�1� F
����b0; �
�j� (��; �)
�� Z 1
��(b;�)
u0��C�� (�; �) ; �B (b; �)
��u0��C (�; b)
� dF (�j�)!�1
;
where the �rst step uses the fact that ddb0��Q (b0; �) b0
= 1
R (1� F (�� (b0; �) j� (��; �))) ; as shown
above. Since �R = 1; then
1� F�����B (b; �) ; �
�j� (��; �)
�=
Z 1
��(b;�)
u0��C�� (�; �) ; �B (b; �)
��u0��C (�; b)
� dF (�j�)!
= �
Z 1
��(b;�)
u0��C�� (�; �) ; �B (b; �)
��u0��C (�; b)
� dFCW (�) + (1� �)Z 1
��(b;�)
u0��C�� (�; �) ; �B (b; �)
��u0��C (�; b)
� dFNC (�) :
11
B.4 Numerical Algorithm
B.4.1 Laissez-faire Equilibrium
We solve for the laissez-faire equilibrium described in Section 3 with an augmented value function
iteration algorithm. Let b = (b1; b2; :::; bN ) denote the equally spaced and inreasingly ordered grid for
sovereign debt. Let � = (�1; �2; :::�S) be the increasingly ordered grid for the default cost, where the
location of any grid point, �s, is chosen such that the cumulative weighted sum, eF (�s) �Psk=1 1=S,
approximates the CDF of the default cost shock F (�s). We choose N = 50000 and S = 600 to get a
solution with high accuracy.27
1. Guess the default threshold, �0(b; w) 2 �, for both aggregate states w and guess the reforme¤ort, 0(b), over the debt grid b. Compute the associated bond revenue,
Q0(b; �w)b = Q̂0(b; �w)b
Q0(b; w)b = 0(b)Q̂0(b; �w)b+ (1�0(b))Q̂0(b; w)b;
where the discounted recovery values are given by
Q̂0(b; w)b = R�1
0@(1� eF (�0(b; w)))b+ X�s2�1;:::;�0(b;w)
b̂0(�s; w)=S
1Aand b̂0(�;w) 2 b is the inverse function of �0(b; w).
2. Guess the value functions conditional on honoring the debt, W0;0(b; w). For any given debt level,bn � b̂0(�S ; �w), on the debt grid update the value function in normal times according to
Wi+1;0(bn; �w) = maxb02(b1;:::;b̂0(�S ; �w))
u�Q0(b
0; �w)b0 + �w � bn�
+�
24(1� eF (�0(b0; �w)))Wi;0(b0; �w) +
X�s2�1;:::;�0(b0; �w)
Wi;0
�b̂0(�s; �w); �w
�=S
35 ;until convergence. For the remaining grid points, bn � b̂0(�S ; �w), setWi+1;0(bn; �w) =Wi+1;0(b̂0(�S ; �w); �w).In recession, for any given debt level, bn � b̂0(�S ; w), on the debt grid, update the value function
27Since we are using value function iteration to solve for the decentralized Markov equilibrium, we can use the condi-tional Euler equation to evaluate the accuracy of the solution in terms of consumption. The mean Euler equation errorin normal times (recession) is 1:16� 10�16 (2:31� 10�4). Thus, in recession, there is a $2:31 mistake for each $100000.Note that we approximate and evaluate the accuracy of the solution globally over the full debt grid. Moreover, theconditional Euler equation in recession involves the derivative of an equilibrium function (reform e¤ort) and non-smoothdebt accumulation. Thus, we consider the accuracy of the solution to be good. When increasing the number of gridpoints to N = 100000 and S = 10000, the Euler equation errors can be further reduced to 5:91 � 1020 in normal timesand 9:54� 10�5 in recession at the usual cost of computational time.
12
according to
Wi+1;0(bn; w) = maxb02(b1;:::;b̂0(�S ; �w))
u�Q0(b
0; w)b0 + w � bn�
+��1�0(b0)
� " (1� eF (�0(b0; w)))Wi;0(b0; w)
+P�s2�1;:::;�0(b0;w)Wi;0
�b̂0(�s; w); w
�=S
#
+�0(b0)
"(1� eF (�0(b0; �w)))W1;0(b0; �w)
+P�s2�1;:::;�0(b0; �w)W1;0
�b̂0(�s; �w); �w
�=S
#;
until convergence. For the remaining grid points, bn � b̂0(�S ; w), setWi+1;0(bn; w) =Wi+1;0(b̂0(�S ; w); w).
W1;0(b0; w) denotes the converged value function conditional on the guess for the threshold and
the reform e¤ort.
3. Update the default threshold and the reform e¤ort according to
�j+1(b; w) =W1;j (0; w)�W1;j (b; w) ;
and Equation (14). Go back to step 1 and iterate until convergence.
B.4.2 Constrained Pareto Optimum
We solve for the second-best allocation with an augmented function iteration algorithm. Consider thesame grid � = (�1; �2; :::�S) for the default cost that we used above. Let �w = (�w(�1); :::; �w(�S))denote the grid for promised utility, where
� �w(�s) = �� �sXk=1
�k=S � �s�1� eF (�s)�
�w(�s) = � �sXk=1
�k=S � �s�1� eF (�s)� :
Note that given a promised utility, �w(�s), the default cost realization �s = ~�(�w(�s)) corresponds
to the state s where the participation constraint of the debtor starts binding. It turns out to be
convenient to set the promised utility for a continued recession, �! �w = � �w, and the promised utility for
a continued recession, !w = �w.
1. Guess the reform e¤ort, pw;0(�w), over the grid �w.
2. Guess the future consumption, c0w;0;0(�w), and the promised utilitiy, �!w;0;0(�w), over the grids
� �w and �w.
3. Compute current consumption from the Euler equations (which holds for all states s where theparticipation constraint is not strictly binding)
cw;0;0(�w) =�u0��1 �
u0�c0w;0;0(�w)
��R�;
13
and the initial promised utility, ~�w;0 (�w), implicitly de�ned by
�� � ~�(~� �w;0;0 (� �w)) = u(c �w;0;0(� �w)) + �� �w
� � ~�(~�w;0;0��w�) = u(cw;0;0(�w))�X(pw;0(�w)) + �
�pw;0(�w)�!w;0;0(�w) + (1� pw;0(�w))�w
�;
4. Update the guess for the future consumption function by interpolating �w on the pairs (~�w;0;0; cw;0;0)
to yield c0w;1;0(�w) Update the guess for promised utility by interpolating c0w;1;0(�w) on the pairs�
cw;0;0; ~�w;0;0�to yield �!w;1;0(�w). Go back to step 3 and iterate until convergence. Let cw;1;0(�w)
and �!w;1;0(�w) denote the converged functions given the guess on the reform e¤ort.
5. Guess the pro�t functions, �P0(� �w) and P 0(�w). Update the pro�t function in normal timesaccording to
�Pi+1;0(~� �w;1;0 (� �w)) =�1� eF (~�(~� �w;1;0 (� �w)))� � �w � c �w;1;0(� �w) +R�1 �Pi;0(� �w)�+
X�s2�1;:::;~�(~� �w;1;0(� �w))
��w � c �w;1;0(� �w(�s)) +R�1 �Pi;0(� �w(�s))
�=S;
until convergence, �P1;0(� �w). In recession, update according to
P i+1;0(~�w;1;0��w�) =
�1� eF (~�(~�w;1;0 ��w�))�
24 w � cw;1;0(�w)
+R�1�pw;0(�w) �Pi;0(�!w;1;0(�w))+(1� pw;0(�w))P i;0(�w)
� 35+
X�s2�1;:::;~�(~�w;1;0(�w))
24 w � cw;1;0(�w(�s))
+R�1�pw;0(�w(�s) �Pi;0(�!w;1;0(�w(�s)))+(1� pw;0(�w(�s)))P i;0(�w(�s))
� 35 =S:
6. Update the reform e¤ort function according to
pw;j+1(�w) =�X 0��1 � u0(cw;1;j(�w))R�1 � �P1;j(�!w;1;j(�w))� P1;j(�w)�
+���!w;1;j(�w)� �w
� �:
Go back to step 3 and iterate until convergence.
14