Essays on Macroeconomics and Contract Theory ARCHIVES by MASSACHUSETTS INSTITUTE OF TECHNOLOGY Juan Passadore B.A. Economics, Universidad de Montevideo (2006) OCT 15 2015 Submitted to the Department of Economics LIBRARIES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2015 @ 2015 Juan Passadore. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part.
redacted A u th o r ...... Signature ...... -- j ------Department of Economics August 15, 2015 redacted C ertified by ...... Signature . . .C,, - --- - Robert M. Townsend Elizabeth & James Killian/Professor of Economics ..I -1 Thesis Supervisor Certified by...... Signature redacted.. -- ~~ Ivan Werning Robert Solow Professor of Economics Thesis Supervisor
Accepted by ...... Signature redacted- ...... Ricardo J. Caballero nternational Professor of Economics Chairman, Departmental Committee on Graduate Studies
Essays on Macroeconomics and Contract Theory by Juan Passadore
Submitted to the Department of Economics on August 15, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Abstract
This thesis studies how contracting frictions affect the outcomes that the public sector or individual agents can achieve. The focus is on situations where the government or the agent lacks commitment on its future actions. Chapter 1, joint work with Juan Xandri, proposes a method to deal with equilibrium multiplicity in dynamic policy games. In order to do so, we characterize outcomes that are consistent with a subgame perfect equilibrium conditional on the observed history. We focus on a model of sovereign debt, although our methodology applies to other settings, such as models of capital taxation or monetary policy. As a starting point, we show that the Eaton and Gersovitz (1981) model features multiple equilibria-indeed, multiple Markov equilibria-when debt is sufficiently constrained. We focus on predictions for bond yields or prices. We show that the highest bond price is independent of the history, while the lowest is strictly positive and does depend on past play. We show that previous period play is a sufficient statistic for the set of bond prices. The lower bound on bond prices rises when the government avoids default under duress. Chapter 2, joint work with Yu Xu, studies debt policy of emerging economies accounting for credit and liquidity risk. To account for credit risk we study an incomplete markets model with limited commitment and exogenous costs of default following the quantitative literature of sovereign debt. To account for liquidity risk, we introduce search frictions in the market for sovereign bonds. In our model, default and liquidity will be jointly determined. This permits us to structurally decompose spreads into a credit and liquidity component. To evaluate the quantitative performance of the model we perform a calibration exercise using data for Argentina. We find that introducing liquidity risk does not harm the overall performance of the model in matching key moments of the data (mean debt to GDP, mean sovereign spread and volatility of sovereign spread). At the same time, the model endogenously generates bid ask spreads, that can match those for Argentinean bonds in the period of analysis. Regarding the structural decomposition, we find that the liquidity component can explain up to 50 percent of the sovereign spread during bad times; when the sovereign is not close to default, the liquidity component is negligible. Finally, regarding business cycle properties, the model matches key moments in the data. Chapter 3, studies the implications of reputation on equilibrium multiplicity in a model of sovereign debt. These models can exhibit multiple equilibria. In the worst equilibrium the government is in autarky. However, in reality, we do not observe the autarkic solution. Motivated by an apparent disconnection between theory and reality, I characterize a lower bound on the utility that the government can obtain for any positive probability that the government is from a commitment type that always repays debts. Chapter 4, joint with Ignacio Presno, studies the optimal risk sharing contract between a risk neutral money lender and an agent that faces Knightian uncertainty about the distribution of her endowment and cannot commit on future transfers. We find that in the optimal contract model uncertainty contributes to increase consumption of the agent over time independently of which shocks have been realized. This differs qualitatively from the case without Knigthian uncertainty.
Thesis Supervisor: Robert M. Townsend
3 Title: Elizabeth & James Killian Professor of Economics
Thesis Supervisor: Ivan Werning Title: Robert Solow Professor of Economics
4 Acknowledgments
My time at MIT has been a tremendously rewarding experience both in academic and personal terms. I owe special thanks to my advisors Robert Townsend and Ivan Werning. Both have been outstanding mentors and role models; along countless discussions during these years I have not only learned about economics, but more importantly, how to approach problems with an open and critical mind.
I would also like to thank George Marios Angeletos and Alp Simsek; both have been incredibly generous with their time. This dissertation has benefited from many discussions with them and from their insightful comments. I am also grateful to Arnaud Costinot for helpful comments, advice and support. Special thanks to my Professors at Universidad de Montevideo, Fernando Borraz, Marcelo Caffera, and Juan Dubra, and at Universidad Di Tella, Leandro Arozamena, Rodolfo Manuelli, Emilio Espino, and Martin Sola, whose encouragement and mentoring as an undergraduate student has been fundamental for my graduate years.
I have been fortunate to share this experience with an amazing group of friends and colleagues. I would like to thank Juan Pablo Xandri, Luis Zermeno, Sofia Barrera, Sebastian Di Tella, Mercedes Politi, Jose Montiel, Nicolas Caramp, Andres Sarto, Maria Fazzolari, Felipe Severino, Joaquin Blaum, Dejanir Silva, Giovanni Reggiani, Natalia Rigol, Joey Neggers, Marco Tabellini, Yu Xu, Miikka Rokkanen, Benjamin Feigenberg, Dan Rees, and Annalisa Scognamiglio, for making this journey both productive and fun. I would also want to thank an outstanding group of classmates and colleagues at the department who have been a constant source of inspiration.
The love from my family and friends back home has been fundamental during this journey. I thank my mother for her affection, unconditional support and constant encouragement. My brother has been a friend that provided wise words when needed. I would also like to thank Celia, Pedro, Martha, Maria and Jose, for always being there in spite of the distance. Last, but certainly not least, I would like to thank to an incredible group of friends back home for reminding me where I come from and where I am going.
5 To my father.
6 Contents
Acknowledgements 5
1 Robust Conditional Predictions in Dynamic Games: An Application to Sovereign Debt 11
1.1 Introduction ...... 11
1.2 A Model of Sovereign Debt ...... 15
1.2.1 Dynamic Game: Notation and Definitions ...... 16
1.3 Multiple Equilibria in Sovereign Debt Markets ...... 18
1.3.1 Lowest Equilibrium Price and Worst Equilibrium ...... 19
1.3.2 Highest Equilibrium Price and Best Equilibrium ...... 20
1.3.3 M ultiplicity ...... 22
1.3.4 Equilibrium Consistency: Focus on Outcomes ...... 23
1.4 Equilibrium Consistent Outcomes ...... 24
1.4.1 Equilibrium Consistency: Definitions ...... 24
1.4.2 Equilibrium Consistency: Characterization ...... 25
1.4.3 Equilibrium Consistent Prices ...... 27
1.4.4 Interpretation: Robust Bayesian Analysis ...... 31
1.5 Extensions: Excusable Defaults and Savings ...... 32
1.5.1 Excusable Defaults ...... 32
1.5.2 Best SPE ...... 34
1.5.3 Excusable Defaults and Savings ...... 35
1.6 Sunspots ...... 35
7 1.6.1 Equilibrium consistent distributions ...... 36
1.6.2 Expectations of Equilibrium Consistent Distributions . 38
1.6.3 Probability of Crises ...... 39
1.7 Discrete Income ...... 41
1.8 Conclusion and Discussion ...... 44
1.9 Appendix A ...... 45
1.10 Appendix B: Characterization of U (b, q S ...... 51
1.11 Appendix C: Computing U (b, q) ...... 54
1.12 Appendix D: Sunspot Proofs ...... 58
1.13 Appendix E: A connection to Robust II ayesian Analysis .. . 62
1.13.1 Robust Bayesian Analysis ...... 62
1.13.2 Main Result ...... 64
1.13.3 Further Results ...... 65
1.13.4 Proofs ...... 66
2 Illiquidity in Sovereign Debt Markets 70
9 . 1 In+rodr1t-in .mal ...... 70
2.2 Model .nvestors...... 73
2.2.1 Small Open Economy ...... 73
2.2.2 Investors ...... 74
2.2.3 Intermediaries ...... 74
2.2.4 Tim ing ...... 75
2.2.5 Decision Problem of the Government 76
2.2.6 Valuations of Debt: Before Default 77
2.2.7 Valuations of Debt: After Default 79
2.2.8 Equilibrium ...... 81
2.2.9 Numerical Algorithm ...... 81
2.3 Numerical Results ...... 86
8 2.3.1 Bond Prices, Bid-Ask Spreads, Decomposition ...... 86
2.3.2 Business Cycle Properties ...... 89
2.3.3 State dependent time in autarky? ...... 90
2.4 Conclusion ...... 93
2.5 Numerical Appendix ...... 93
3 Reputation and Debt Capacity 96
3.1 Introduction ...... 96
3.2 A Sovereign Borrower ...... 98
3.2.1 Setup ...... 98
3.2.2 Equilibrium ...... 100
3.3 Government Uncertainty ...... 101
3.3.1 Preliminaries ...... 101
3.3.2 Beliefs and Prices ...... 102
3.3.3 A Bound On Welfare ...... 104
3.4 Conclusion ...... 105
3.5 Appendix: Omitted Lemmas and Proofs ...... 106
4 Ambiguity, Insurance, and Lack of Commitment 108
4.1 Introduction ...... 108
4.2 Setting ...... 110
4.2.1 Agent ...... 110
4.2.2 Money Lender ...... 112
4.3 Equilibrium Allocations ...... 112
4.4 Efficient Allocations ...... 113
4.4.1 Pareto Program ...... 113
4.4.2 Benchmarks: Commitment and Expected Utility ...... 114
4.4.3 Characterizing Efficient allocations ...... 114
4.5 Conclusion ...... 117
4.6 Appendix: Omitted Lemmas and Proofs ...... 117
9 References 123
10 Chapter 1
Robust Conditional Predictions in Dynamic Games: An Application to Sovereign Debt
1.1 Introduction
Following Kydland and Prescott (1977) and Calvo (1978) the literature on optimal government policy with- out commitment has formalized these situations by employing dynamic game theory, finding interesting appli- cations for capital taxation (e.g. Chari and Kehoe, 1990, Phelan and Stacchetti, 2001, Farhi, Sleet, Werning, and Yeltekin 2012), monetary policy (e.g. Ireland, 1997, Chang, 1998, Sleet, 2001) and sovereign debt (e.g. Calvo, 1988, Eaton and Gersovitz, 1981, Chari and Kehoe, 1993a, Cole and Kehoe, 2000). This research has helped us understand the distortions introduced by the lack of commitment and the extent to which governments can rely on a reputation for credibility to achieve better outcomes.
One of the challenges in applying dynamic policy games is that these settings typically feature a wide range of equilibria with different predictions over outcomes. For example, for "good" equilibria the government may achieve, or come close to achieving, the optimum with commitment, while there are "bad" equilibria where this is far from the case, and the government may be playing the repeated static best response. In studying dynamic policy games, which of these equilibria should we employ? One approach is to impose refinements, such as various renegotiation proofness notions, that select an equilibrium or significantly reduce the set of equilibria. Unfortunately, no consensus has emerged on the appropriate refinements.
Our goal is to overcome the challenge multiplicity raises by providing predictions in dynamic policy games that are not sensitive to any equilibrium selection. The approach we offer involves making predictions
11 for future play that depend on past play. The key idea is that, even when little can be said about the unconditional path of play, quite a bit can be said once we condition on past observations. To the best of our knowledge, this simple idea has not been exploited as a way of deriving robust implications from the theory. Formally, we introduce and study a concept which we term "equilibrium consistent outcomes": outcomes of the game in a particular period that are consistent with a subgame perfect equilibrium, conditional on the observed history.
Although it will be clear that the notions we propose and results we derive are general and apply to any dynamic policy game, we develop them for a specific application, using a model of sovereign debt along the lines of Eaton and Gersovitz (1981). This model constitutes a workhorse in international economics. In the model, a small open economy faces a stochastic stream of income. To smooth consumption, a benevolent government can borrow from international debt markets, but lacks commitment to repay. If it defaults on its debt, the only punishment is permanent exclusion from financial markets; it can never borrow again.1
Given that our approach tries to overcome the challenges of multiplicity, as a starting point we first ensure that there is multiplicity in the first place. We show that in the standard Eaton and Gersovitz (1981) model, restrictions on debt, often adopted in the quantitative sovereign-debt literature, imply the existence of multiple equilibria. Our multiplicity relies on the existence of autarky as another Markov equilibrium. This result may be of independent interest, since it implies that rollover crises are possible in this setting. The quantitative literature on sovereign debt following Eaton and Gersovitz (1981) features defaults on the equilibrium path, but to shocks to fundamentals. 2 Another strand literature studies self-fulfilling debt crises following the models in Calvo (1988) and Cole and Kehoe (2000). Our results suggest that crises, defined as episodes where the interest rates are very high but not due to fundamentals, are a robust feature in models of sovereign debt.
Given multiplicity, our main result provides a characterization of equilibrium consistent outcomes in any period (debt prices, debt issuance, and default decisions). Aided by this characterization, we obtain bounds for equilibrium consistent debt prices that are history dependent. The highest equilibrium consistent price is the best Markov equilibria and, thus, independent of past play. The lowest equilibrium consistent price is strictly positive and depends on past play. In our baseline case, due to the recursive nature of equilibria, only the previous period play matters and acts as a sufficient statistic for the set of equilibrium consistent prices.
In our sovereign debt application, equilibrium consistent debt prices improve whenever the government avoids default under duress. In particular, if the country just repaid a high amount of debt, or did so under harsh economic conditions, for example, when output was low, the lowest equilibrium consistent price is higher.
'The key features are lack of commitment, a time inconsistency problem, and an infinite horizon that creates reputational concerns in the sense of trigger-strategy equilibria. These features are shared by all dynamic policy games, such as applications to capital taxation and monetary policy. 2 A recent exception is Stangebye (2014a) that studies the role of non fundamental shocks in sovereign crises in a model as in Eaton and Gersovitz (1981).
12 The choice to repay under these conditions reveals an optimistic outlook for bond prices that narrows down the set of possible equilibria for the continuation game. This result captures the idea that reputation is built for the long run by short-run sacrifices.
We apply our results to study the probability of a rollover debt crises. As we argued above, rollover debt crises may occur on the equilibrium path for any fundamentals. However, the probability of a rollover crisis, after a certain history, may be constrained. We derive these constraints, showing that rollover crises are less likely if the borrower has recently made sacrifices to repay. This result may be contrasted with Cole and Kehoe (2000). In their setting the potential for rollover crises induces the government to lower debt below a threshold that rules rollover crises out. Thus, the government's efforts have no effect in the short run, but payoff in the long run. In our model, an outside observer will witness that rollover crises are less likely immediately after an effort to repay.
In the Eaton and Gersovitz (1981) setting, all defaults are assumed to be punished by autarky. This may be seen as a feature of the game, or as a restriction to focus on the subset of equilibria with this property. In both cases, it may be viewed as somewhat ad hoc. In Section 1.5, we relax these assumptions. This captures an important alternative set of models from the literature of sovereign debt, when defaults are excusable, in the sense of Grossman and Huyck (1989). We have also assumed that the government can only borrow, but not save; in the context of excusable defaults, we also relax this assumption. The worst subgame perfect equilibrium remains autarky. We provide a characterization of equilibrium consistent outcomes and prices. As one might expect, the same principle applies: equilibrium consistent debt prices improve as the amount of debt just repaid increases, or if the conditions under which debt was repaid are less favorable. Repaying under duress rules out negative outlooks on prices.
The fact that the last period is a sufficient statistic may seem surprising. This result is a direct expression of robustness: the expected payoff rationalizing a decision may have been realized for histories that have not occurred. When income is continuous, any particular history has probability zero, so the realized expected payoff rationalizing past behavior can always been expected for those realizations that did not materialize. 3
To better understand the role of robustness in determining equilibrium consistent outcomes, we then move to the case where there are no randomization devices available and output is discrete. In a two period example, we provide a simple argument for why we obtain the sufficiency result in the continuous income case and we characterize how history impacts on the set of equilibrium consistent prices. We show that history can be summarized in a unique variable, a minimum value of utility that makes previous decisions rational. Intuitively, the restrictions that past decisions impose on current prices (and policies) can be decomposed into two terms. The first term is the value that the government receives in the realized history times the probability of that history. The second term is the value of the best equilibrium, off the equilibrium path. 3 This intuition was first introduced by Gul and Pearce (1996) to show that Forward induction has much less predictive power as a solution concept if there are correlating devices.
13 The link between current prices and past decisions depends on the strength (probability) and the length (discounting) of the link between current and past periods.
We relate equilibrium consistent outcomes to Robust Bayesian Analysis. In Bayesian Analysis, the econome- trician has a prior over fundamental parameters and derives a posterior after observing the data; in Robust Bayesian analysis the econometrician is uncertain about the prior and derives a set of posteriors from the set of possible priors. Given equilibrium multiplicity, in a dynamic policy game, the Bayesian econometrician has a prior over outcomes. In many contexts, it may be hard to form a prior; thus, uncertainty about the prior is a natural assumption. We show that the set of equilibrium consistent outcomes is the support of the posterior for a Bayesian that only assume that the data was generated by some subgame perfect equilibrium and is agnostic with respect to the prior over equilibrium outcomes.
Literature Review. Our paper relates to the literature on credible government policies; the seminal contributions in that literature are Chari and Kehoe (1990) and Stokey (1991).4 These two papers adapt the techniques developed in Abreu (1988) to dynamic policy games. Although these papers provide a full characterization of equilibria, they do not attempt to derive robust predictions across these equilibria, as we do here.
The two papers more closely related to our work are Angeletos and Pavan (2013) and Bergemann and Morris (2013). The first paper obtains predictions that hold across every equilibrium in a global game with an endogenous information structure. The second paper obtains restrictions over moments of observable en- dogenous variables that hold across every possible information structure in a class of coordination games. Our paper relates to them in that we obtain predictions that hold across all equilibria. Our results are weaker than Angeletos and Pavan (2013) because our predictions are regarding the equilibrium set. But it is also true that our problem has the additional challenge of being a (repeated) dynamic complete information game. The latter is precisely the root of weaker predictions.
The literature of sovereign debt5 has evolved in several directions. One direction, the quantitative literature on sovereign debt, focuses on a model where asset markets are incomplete and there is limited commitment for repayment, following Eaton and Gersovitz (1981), to study the quantitative properties of spreads, debt capacity, and business cycles. The aim of this strand of the literature is to account for the observed behavior of the data. The seminal contributions in this literature are Aguiar and Gopinath (2006) and Arellano (2008) which study economies with short term debt. Long term debt was introduced by Hatchondo and Martinez (2009), Arellano and Ramanarayanan (2012), Chatterjee and Eyigungor (2012). The quantitative literature of sovereign debt has already been successful in explaining the most salient features in the data.' Our paper
4 Atkeson (1991) extends the approach to the case with a public state variable. Phelan and Stacchetti (2001) and Chang (1998) extend the approach to study models where individual agents hold stocks (capital and money respectively). 5 For a review see Eaton and Fernandez (1995b), Aguiar and Amador (2013a). 60ther examples in this literature are Yue (2010), Bai and Zhang (2012), Pouzo and Presno (2011), Borri and Verdelhan (2009), D Erasmo (2008), Bianchi, Hatchondo, and Martinez (2012).
14 shares with this literature the focus on a model along the lines of Eaton and Gersovitz (1981) but rather than characterizing a particular equilibrium, it tries to study predictions regarding the set of equilibria.
Another direction of the literature focuses in equilibrium multiplicity, and in particular, in self fulfilling debt crises. The seminal contribution is Calvo (1988). Cole and Kehoe (2000) introduce self-fulfilling debt crises in a full-fledged dynamic model where the equilibrium selection mechanism is a sunspot that is realized simultaneously with output. Lorenzoni and Werning (2013) study equilibrium multiplicity in a dynamic version of Calvo (1988). Our paper studies multiplicity but in the Eaton and Gersovitz (1981) setting; the crucial difference between the setting in Calvo (1988) and the one Eaton and Gersovitz (1981) is that in the latter the government issues debt (with commitment) and then the price is realized. This implies that equilibrium multiplicity is coming from the multiplicity of beliefs regarding continuation equilibria. Stangebye (2014a) also studies multiplicity in a setting as in Eaton and Gersovitz (1981), but focuses on a Markov equilibrium.
Another strand of the literature studies the risk sharing agreement between international debt holders and a sovereign with some primitive contracting frictions. Worrall (1990b) studies an economy with limited commitment. Atkeson (1991) studies an economy with limited commitment and moral hazard and finds that capital outflows during bad times are a feature of the optimal contract. The model we use to discuss excusable defaults is closely related to these two models. 7
Outline. The paper is structured as follows. Section 1.2 introduces the model. Section 1.3 studies equi- librium multiplicity in our model of sovereign borrowing. Section 1.4 characterizes equilibrium consistent outcomes. Section 1.5 characterizes equilibrium consistent outcomes in a model with savings and excus- able defaults. Section 1.6 discusses the characterization of equilibrium consistent outcomes when there are correlating devices available after debt is issued and when income is discrete. Section 1.8 concludes.
1.2 A Model of Sovereign Debt
Our model of sovereign debt follows Eaton and Gersovitz (1981). Time is discrete and denoted by t E {0, 1,2, .... }. A small open economy receives a stochastic stream of income denoted by yt. Income follows a Markov process with c.d.f. denoted by F(ytel I yt). The government is benevolent and seeks to maximize the utility of the households. It does so by selling bonds in the international bond market. The household evaluates consumption streams according to
EE #tu(ct) 1t=0 7 Hopenhayn and Werning (2008) study optimal financial contracts when there is private information regarding the outside option and limited commitment to repayment and find that there is default along the equilibrium path. Aguiar and Amador (2013b) exploit this approach to study the optimal repayment of sovereign debt when there are bonds of different maturities.
15 where 13 < 1 and u is increasing and strictly concave. The sovereign issues short term debt at a price qt. The budget constraint is ct yt - bt + qtbt+1.
Following Chatterjee and Eyigungor (2012) we assume that the government cannot save
bt+1 > 0.
This helps focus our discussion on debt and may implicitly capture political economy constraints that make it difficult for governments to save, as modeled by Amador (2013).
There is limited enforcement of debt. Therefore, the government will repay only if it is more convenient to do so. We assume that the only fallout of default is that the government will remain in autarky forever after. We also do not introduce exogenous costs of default. As we will show below, our assumptions are sufficient for autarky to be an equilibrium. If the government cannot save, and there are no output costs of default, if the government expects a zero bond price for its debt now and in every future period, then it will default its debt. To guarantee multiplicity we need to introduce conditions to guarantee that best Markov equilibrium, the one usually studied in the literature of sovereign debt, has a positive price of debt. In Section 1.5 we characterize subgame perfect equilibrium and equilibrium consistent outcomes when the government can save and when defaults do not need to be punished.
Lenders. There is a competitive fringe of risk neutral investors that discount the future at rate r > 0. This implies that the price of the bond is given by
1 -i 1 + r where Jt if the default probability on bonds bt+1 issued at date t.
Timing. The sequence of events within a period is as follows. In period t, the government enters with bt bonds that it needs to repay. Then income yt is realized. The government then has the option to default dt c {0, 1}. If the government does not default, the government runs an auction of face value bt+1. Then, the price of the bond qt is realized. Finally, consumption takes place, and is given by ct = yt - bt + qtbt+i. If the government decides to default, consumption is equal to income, ct = yt. The same is true if the government has ever defaulted in the past. We adopt the convention that if d= 1 then d, = 1 for all t' > t.
1.2.1 Dynamic Game: Notation and Definitions
In this section we describe the basic notation for the dynamic game setup.
16 Histories. An income history is a vector yt = (Yo, y1, ... , yt) of all income realizations up to time t. A history is a vector ht = (ho, hi, ... , ht_ 1), where ht = (yt, dt, bt+i, qt) is the description of all realized values of income and actions, and h = h'h" is the append operator. A partial history is an initial history ht concatenated with a part of ht. For example, h = (ht, yt, dt, bt+1) is a history where we have observed ht, output yt has been realized, the government decisions (dt, bt+1) have been made, but market price qt has not yet been realized. We will denote these histories h = ht+1. The set of all partial histories (initial and partial) is denoted by W, and 7ig C W are those where the government has to make a decision; i.e., h = (ht, yt). 1 Likewise, Hm C '7 is the set of partial histories where investors set prices; i.e., ht+1 = (h', yt, dt, bt+i).
Outcomes. An outcome path is a sequence of measurable functions8
x = (dt (y') , bt+1 (yt) , qt (yt))tEN
The set of all outcomes is denoted by X. To make explicit that the default, bond policies and prices are the ones associated with the path x, sometimes we will write
(dtx (y') , bx+1 (yt) , qx V)
An outcome xt (the evaluation of a path at a particular period) is a description of the government's policy function and market pricing function at time t where the functions in Xt are dt : Y -+ {0, 1}, bt+i : Y -+ R+, and qt : Y -+ R+. Our focus will be on a shifted outcome, xt_ = (qt-, dt (.), bt+l (-)). The reason to do this is that the prices in qt-i1 will only be a function of the next period default decision.
Strategies. A strategy profile is a complete description of the behavior of both the government and the market, for any possible history. Formally, a strategy profile is defined as a pair of measurable functions a = (ag, q,), where ag : Wg -+ {0, 1} x R+ and qm : W, -+ R+. The government decision will usually be written as
01g (h', yt) (e gd(ht, yt) 7bo'g 1 (ht, yt)) so that dtg (-) and b'g (-) are the default decision and bond issuance decision for strategy ag. Eg is the set of all strategies for the government, and Em is the set of market pricing strategies. E = Eg x Em is the set of all strategy profiles. Given a history ht, we define the continuation strategy induced by ht as
h (h') = a (hths).
8 For our baseline case, where after default the government is permanently in autarky, the functions have the restriction that bond issues and prices are not defined after a default has been observed: bt+s+1 (yty-) = qt+. (ytys) = 0 for all y- and yt such that dt (yt) = 1.
17 Every strategy profile a generates an outcome path x := x (o).9 Given a set S C E of strategy profiles, we
denote x (S) = UGesx (a) for the set of outcome paths of profiles a E S.
Payoffs. For any strategy profile a E E, we define the continuation at h' E Wg
V(a I ht) = Et /3 [du(y, - bs + qsbs+1 ) + (1 - d,)u(y))]
where (y,, d , b8 +i, q.) are on the path x = x(0a0.10
Definition 1. A strategy profile a = (ag, qm) constitutes a subgame perfect equilibrium (SPE) if and only
if, for all partial histories ht C lHg
V(- I ht) ; V(o-,qm I ht) for all E9E 3g, (1.1)
and for all histories ht+1 = (ht, yt, dt, bt+i) E ,,
= (I -d"e(ht'+, yt+1)dF(yt+1 I yt). (1.2) qm (ht+') 1 1 + f
That is, the strategy of the government is optimal given the pricing strategy of the lenders qm (.), and
likewise qm,, (-) is consistent with the default policy generated by -g. The set of all subgame perfect equilibria is denoted as E C E.
1.3 Multiple Equilibria in Sovereign Debt Markets
This section characterizes the best and worst equilibrium prices in the dynamic game laid out in the previous section and discusses the scope for multiplicity of equilibria. For any history ht+1 we consider the highest and lowest prices (h_+) :=max qm 1 aEE (ht+ )
q(ht+1) minqm (ht+l).
The best and worst equilibria turn out to be Markov equilibria and we find conditions for multiplicity. The worst SPE price is zero and the best SPE price is the one of the Markov equilibrium that is characterized in the literature of sovereign debt as in Arellano (2008) and Aguiar and Gopinath (2006). Thus, our analysis may be of independent interest, providing conditions under which there are multiple Markov equilibria in a
9 It can be defined recursively as follows: at t = 0 jointly define (do (yo), bi (yo), q1 (yo)) = (dg (yo), b"' (yo), qm(yo, b' (yo)) and h' = (yo, do(yo),bi(yo),q1(yo)). For t > 0, we define (dt(yt),bt+ 1(yt),qt(yt)) = (d9(ht ,yt),b7yg(ht,yt),qm(ht,yt)) and t ht+1 = (h , y, d(yt),bt+ 1(yt),qt(yt)) 0 1 The utility of a strategy profile that specifies negative consumption is -oo.
18 sovereign debt model along the lines of Eaton and Gersovitz (1981).11 The importance of this result is that it opens up the possibility of confidence crises in models as in Eaton and Gersovitz (1981). Thus, confidence crises are not necessarily a special feature of the timing in Calvo (1988) and Cole and Kehoe (2000) but a robust feature in most models of sovereign debt.
The lowest price q(ht'+) will be attained by a fixed strategy for all histories ht+l. It will deliver the utility level of autarky for the government. Thus, the lowest price is associated with the worst equilibrium, in terms of welfare. Likewise, the highest price i(ht+1) is associated with a, different, fixed strategy for all histories (the maximum is attained by the same a for all ht+1 ) and delivers the highest equilibrium level of utility for the government. Thus, the highest price is associated with the best equilibrium in terms of welfare.
1.3.1 Lowest Equilibrium Price and Worst Equilibrium
We start by showing that, after any history ht_+', the lowest subgame perfect equilibrium price is equal to zero.
Proposition 2. Denote by B the set of assets for the government. Under our assumption of B > 0, the lowest SPE price is equal to zero q(htl+) = q(yt, bt+) = 0 and associated with a Markov equilibrium that achieves the worst level of welfare.
Whenever the government confronts a price of zero for its bonds in the present period and expects to face the same in all future periods, it is best to default. There is no benefit from repaying. The proof is simple. We need to show that defaulting after every history is a subgame perfect equilibrium. Because the game is continuous at infinity, we need to show that there are no profitable one shot deviations when the government plays that strategy. Note first that, if the government is playing a strategy of always defaulting, it is effectively in autarky. In a history ht+' with income yt and debt bt, the payoff of such a strategy is
u(yt)+ EYU(y').
Note also that, a one shot deviation involving repayment today has associated utility of
u(yt - bt) + Eflytu(y'). 1 -#
Thus, as long as bt+l is non-negative, a one shot deviation of repayment is not profitable. So, autarky is an SPE with an associated price of debt equal to zero.
1 1Our result complements the results in Auclert and Rognlie (2014); their paper shows uniqueness in the Eaton and Gersovitz (1981) when the government can save.
19 The equilibrium does not require conditioning on the past history, i.e. it is a Markov equilibrium. Notice, as well, that we have not yet introduced sunspots. Thus, multiplicity does not require sunspots. Sunspots may act as a coordinating device to select a particular continuation equilibrium. We introduce sunspots in Section 1.6.
Things are different when the government is allowed to save before default and the punishment is autarky, including exclusion from saving. Under this combination of assumptions, the government might want to repay small amounts of debt to maintain the option to save in the future. As a result, autarky is no longer an equilibrium and a unique Markov equilibrium prevails, as shown by Auclert and Rognlie (2014).
A similar result holds when there are output costs of default. The sufficient condition for multiplicity will be that for the government is dominant to default on any amount of debt that it is allowed to hold, for all b E B. With default costs, the value of defaulting is lower. Thus, we need to increase the static gain of defaulting for any history. A sufficient condition would then be that B > 0. The lower bound on debt will be increasing in the magnitude of the output costs of default.
1.3.2 Highest Equilibrium Price and Best Equilibrium
We now characterize the best subgame perfect equilibrium and show that it is the Markov equilibrium studied by the literature of sovereign debt. To find the worst equilibrium price, it was sufficient to use the definition of equilibrium and the one shot deviation principle. To find the best equilibrium price it will be necessary to find a characterization of equilibrium prices. Denote by W(yt, bt+ 1) the highest expected equilibrium payoff if the government enter period t + 1 with bonds bt+l and income in t was yt. The next lemma provides a characterization of equilibrium outcomes.
Lemma 3. xt = (qti 1,d (.), bt+1 (-)) is a subgame perfect equilibrium outcome at history ht if and only in the following conditions hold: a. Price is consistent q- 1 + (1 dt(yt)dF(yt I yt-1)), (1.3) b. IC government
(1 - d(yt)) [u(yt - bt +q(yt, bt+1 )bt+ 1 ) + /3W(yt, bt+1)] + d(yt)Vd(yt) > Vd(yt). (1.4)
The proof is omitted. Condition (1.3) states that the price qt-1 needs to be consistent with the default policy dt(-). Condition (1.4) states that a policy dt (-) , bt+1 (-) is implementable in an SPE if it is incentive compatible given that following the policy is rewarded with the best equilibrium and a deviation is punished
20 with the worst equilibrium. The argument in the proof follows Abreu (1988). These two conditions are necessary and sufficient for an outcome to be part of an SPE."
Markov Equilibrium. We now characterize the Markov equilibrium that is usually studied in the litera- ture of sovereign debt. The value of a government that has the option to default is given by
W(y_, b) = EVF_ [max {nd%(b, y), VD(y)}] . (1.5)
This is the expected value of the maximum between not defaulting Vnd(b, y) and the value of defaulting VD(y). The value of not defaulting is given by
Vnd(b, y) = maxu(y - b + q(y, b')b') + /W(y, b'). (1.6) b';>0
That is, the government repays debt, obtains a capital inflow (outflow), and from the budget constraint consumption is given by y - b + q(y, b')b'; next period has the option to default b' bonds. The value of defaulting is
Vd(y) = u(y) + EyU(y) (-7(1.7) 1-0
and is just the value of consuming income forever. These value functions define a default set
D(b) = {y E Y : Vnd(b, y) < Vd(y)} . (1.8)
A Markov Equilibrium (with state b, y) is a: set of policy functions (c(y, b), d(y, b), b'(y, b)), a bond price function q(b') and a default set D(b) such that: c(y, b) satisfies the resource constraint; taking as given q(y, b') the government bond policy maximizes Vnd; the bond price q(y, b') is consistent with the default set
1 - f,(,) dF(y' I y) f (1.9) q(y, b') = 1 +
The next proposition states that the best Markov equilibrium is the best subgame perfect equilibrium.
Proposition 4. The best subgame perfect equilibrium is the best Markov equilibrium.
From lemma 3, the value of the best equilibrium is the expectation with respect to yt, given yt-1, and is given by
max (1 - dt) [u(yt - bt + q(yt, bt+1)bt+ 1 ) + #W(yt, bt+1 )] + dtVd(yt). dt ,bt+ 12 Note that at any history (even on those inconsistent with equilibria) SPE policies are a function of only one state: the debt that the government has to pay at time t (bt). There are two reasons for this. First, the stock of debt summarizes the physical environment. Second, the value of the worst equilibrium only depends on the realized income.
21 Note that this is equal to the left hand side of (1.5). The key assumption for the best subgame perfect equilibrium to be the best Markov equilibrium is that the government is punished with permanent autarky after a default. We shall relax this assumption in Section 1.5, where we consider the possibility of excusable defaults. Excusable defaults allow for greater risk sharing, which improves welfare.
1.3.3 Multiplicity
Given that the worst equilibrium is autarky, a sufficient condition for multiplicity of Markov equilibria will be any condition that guarantees that the best Markov equilibria has positive debt capacity, a standard situation in quantitative sovereign debt models. In general some debt can be sustained as long as there is enough of a desire to smooth consumption. This will motivate the government to avoid default, at least for small debt levels. The following proposition provides a simple sufficient condition for this to be the case.
Define Vnd(b, y; B, 1 ) as the value function when the government faces the risk free interest rate q = + and some borrowing limit B as in a standard Bewley incomplete market model. The government has the option to default. This value is not an upper bound on the possible values of the borrower because default introduces state contingency and might be valuable. Our next proposition, however, establishes conditions under which default does not take place.
Proposition 5. Suppose that for all b G [0, B] and all y G Y, the
1 Vnd(b, y; B, ) >.; u(y) + /3Eg, 1,Vd (yI). (1.10)
Then there exist multiple Markov equilibria.
If the government is confronted with q = for b < B condition (1.10) ensures that it will not want to default after any history. This justifies the risk free rate for b < B. A SPE can implicitly enforce the borrowing limit b < B by triggering to autarky and setting qt = 0 if ever bt+1 > B. Since the debt issuance policy is optimal given the risk free rate, we have constructed an equilibrium. This proves there is at least one SPE sustaining strictly positive debt and prices. The best equilibrium dominates this one and is Markov, as shown earlier, so it follows that there exists at least one strictly positive Markov equilibrium. Finally, note that we only require checking this condition (1.10) for small values of B. However, the existence result then extends an SPE over the entire B = [0, oo). Indeed, it is useful to consider small B and take the limit, this then requires checking only a local condition. The following example illustrates this condition.
Example. Suppose there are two income shocks YL and YH that follow a Markov chain (a special case is the i.i.d. case). Denote by Ai the probability of transitioning from state i to state j =, i. We will construct an equilibrium where debt is risk free, the government goes into debt B and stays there as long as income is
22 low, and repays debt and remains debt free when income is high. Conditional on not defaulting, this bang bang solution is optimal for small enough B. To investigate whether default is avoided, we must compute the values
VBL = u(yL + (R - 1)B)+8 (ALVBH + (1 - AL)VBL)
VBH = u(yH - RB) + # (AHVOL + (1 - \H)VOH)
VOL = u(yL + B) +0 (ALVBH + (1 - AL)VBL)
VOH = u(yH) +1 (AHVOL + (1 - AH)VOH) where R = 1 + r. Write the solution to this system as a function of B. To guarantee that the government does not default in any state, we need to check that VBL(B) > vaut, VBH(B) > ul, VOL(B) > j"a' and
VOH(B) > Vaul (some of these conditions can be shown to be redundant).
Lemma 6. A sufficient condition for VBL Vaut,VBH Vaut, VOL VL'u. VH . 'S7to hold for some
B > 0 is V'sL(0) > 0,1V'H (0) > 0. When AH AL = 1 this simplifies to /u'(yL) > Ru'(yH) -
Note that the simple condition with AH = AL = 1 is met whenever u is sufficiently concave or if 0 is sufficiently close to 1. These conditions ensure that the value from consumption smoothing is high enough to sustain debt.
1.3.4 Equilibrium Consistency: Focus on Outcomes
The following example provides further intuition how a the best Markov equilibrium operates, and at the same time helps us to make the point why we focus on predictions about the set of equilibria. A standard property in sovereign debt models is that higher debt implies a higher default probability. That is, if b1 5 b2, then 3(bi) 5 6(b 2 ). Actually, this is a property of Markov equilibrium. If the government wants to default with lower debt bl, then it will also want to default with higher debt,
Vd(y) > u(y - b1 + q(y, b')b') + #7W(y, b')
> u(y - b 2 + q(y, b')b') + 3W(y, b').
A key step is that the continuation value W(y, b') does not depend on b. However, if we consider all subgame perfect equilibrium, this property will not necessarily hold. Different levels of debt, b1 versus b2, may be associated with different continuation equilibria and therefore, different default probabilities.
23 1.4 Equilibrium Consistent Outcomes
This section contains the main result of the paper, a characterization of equilibrium consistent outcomes. We work with the baseline case where income is a continuous random variable. After this characterization we turn to our attention to the implications for bond prices.
1.4.1 Equilibrium Consistency: Definitions
We first define the notion of consistent histories.
Definition 7. (Consistency) A history h is consistent with (or generated by) an outcome path x if and only if d= dx (y'), b,+ 1 = b 1 (y') and q, = qx (y') for all s < 1 (h) (where 1 (h) is the length of the history).
If a history h is consistent with an outcome path x we denote it as h E N (x). Intuitively, consistency of a history with an outcome means that, given the path of exogenous variables, the endogenous observed variables coincide with the ones that are generated by the outcome.
Definition 8. A history h is consistent with strategy profile o +-> h E W (X (0)).13
If a history h is consistent with a strategy a we denote it as h E N (a). Intuitively, a history is consistent with a strategy if the history is consistent with the outcome that is generated by the strategy. Given a set
S C E of strategy profiles, we use x (S) = UUESX (a) to denote the set of outcome paths of profiles a E S. The inverse operator for N (-) are respectively X (.) for the outcomes consistent with history h. We use E (h) to denote the strategy profiles consistent with h. For a given set of strategy profiles S C E, we write
R (S) = UIEs N (a) as the set of S-consistent histories. When S = E, we call R (S) the set of equilibrium consistent histories. The set of equilibria consistent with history h is defined as Slh := S n E (h).'
Definition 9. (S- consistent outcomes) An outcome path x = (dt (), bt+1 (), qt (-))t. is S- consistent with history h' - ] e E S n E (ht) such that x = x (a). If S =S we say x is equilibrium consistent with history ht, and we denote it as "x E x
The expected autarky continuation is
Vd(y) J u(y')dF (y' I y)
13 Remember that each strategy a generates an outcome path x := x (o). It can be defined recursively as follows: at t = 0 jointly define (do (yo), bi (yo) , q1 (yo)) (do" (yo) , b6 (yo) , qm (yo, b'g (yo))) and h' = (yo, do (yo), bi (yo), qi (yo)). For t > 0, we define (dt (yt) , bt+1 (yt) , qt (yt)) m (do" (ht, yt) , bo' (ht, yt) , qm (ht, yt)) and ht+1 = (ht, yt, dt (yt) , bt+1 (y') , qt (yt))- 14 This notation is useful to precisely formulate questions such as: "Is the observed history the outcome of some subgame perfect equilibria?" In our notation "h E 'H (SPE)".
24 and the autarky utility (conditional on defaulting) is simply
V, M =U (Y) + ov,(y). (11
The continuation utility (conditional on not defaulting) of a choice b' given bonds (b, y) is
rd (b, y, b') = u (y - b + b' (y, b') b') + OW (y, b') , (1.12) where -q(b') is the bond price schedule under the best continuation equilibrium (the Markov equilibrium that we just characterized), if yt = y and the bonds to be paid tomorrow are bt+1 = b'. Recall that
--- d -nd I - V (b, y) = maxV b y, . (1.13) b;>O
1.4.2 Equilibrium Consistency: Characterization
Suppose that we have observed so far ht = (ht-1, y1,dt-1, bt) an equilibrium consistent history (where price at time t has not yet been realized), and we want to characterize the set of shifted outcomes xt_ = (qt-i, dt (-) , bt+1 (-)) consistent with this history1 5 . Theorem 10 provides a full characterization of the set of equilibrium consistent outcomes xt- (e,1t ), showing that past history only matters through the opportunity cost of not defaulting at t - 1, u (yt-1) - u (Cet-1).
Proposition 10 (Equilibrium Consistent Outcomes). Suppose ht = (ht-1, yt-1, dt-1, bt) is an equilibrium
consistent history, with no default so far. Then Xt = (qt-1, d (-) , bt+1 (-)) is equilibrium consistent with ht if and only in the following conditions hold:
a. Price is consistent q- = (1 dt(yt)dF(yt I yt_1)), (1.14)
b. IC government
(1 - d(yt)) [u(yt - bt + q(yt, bt+1)bt+1) + /W(yt, bt+1)] + d(yt)Vd(yt) ;> V(yt), (1.15)
c. Promise keeping
,8 [f Vn (bt, yt, bt+1(yt)) dF(yt I yt_1) + Vd (yt) dF(yt I yt_1) > d.=O dt=1 5 " An outcome in period t was given by xt = (d' (-), bx4 (-), qx (-)); the policies and prices of period t. xt_ has the policies of period t but the prices of period t - 1. The focus in xt- as opposed to xt simplifies the characterization of equilibrium consistent outcomes.
25 [u (yt-1) - u (yt-1 - bt-1 + qt-ibt)] + ,V(yt-1). (1.16)
Proof. See Appendix.
If conditions (a) through (c) hold, we write simply
(qt_1, dt (-) , bt+l (-)) E ECC (bt_1, yt-1, bt) , where ECO stands for "equilibrium consistent outcomes".
As we mentioned in Section 1.3, conditions (1.14) and (1.15) in Proposition 10 provide a characterization of the set of SPE outcomes. Condition (1.14) states that the price qt-1 needs to be consistent with the default policy dt(-). Condition (1.15) states that a policy dt (.), bt+l (-) is implementable in an SPE if it is incentive compatible given that following the policy is rewarded with the best equilibrium and a deviation is punished with the worst equilibrium. The argument in the proof follows Abreu (1988)16.
Equilibrium consistent outcomes are characterized by an additional condition, (1.16), which is the main contribution of this paper. This condition characterizes how past observed history (if assumed to be generated by an equilibrium strategy profile) introduces restrictions on the set of equilibrium consistent policies. In our setting, condition (1.16) will guarantee that the government's decision at t -1 of not defaulting was optimal. That is, on the path of some SPE profile &, the incentive compatibility constraint from government's utility maximization in t - 1 is
u (ct_1) +6V (& ht) > u (yt-1) +/V(yt-1), (1.17) where V (& I ht) is the continuation value of the equilibrium, as defined before. One interpretation of (1.17) is that the net present value (with respect to autarky) that the government must expect from not defaulting, must be greater (for the choice to have been done optimally) than the opportunity cost of not defaulting: u (yt-1) - u (ct-1). This must be true for any SPE profile that could have generated ht_.
The intuition for why (1.16) is necessary for equilibrium consistency is as follows. Notice that the previous inequality also holds for the case the continuation equilibrium is actually the best continuation equilibrium. Therefore, for any equilibrium consistent policy (d (.), b' (-)) it has to be the case that
V ( ht ) =/Yt:dt(yt)=1 Vd(yt)dF(yt I yt-_)+
t I ht+l)] dF(yt I yt_1) :d(yt)=O [u (yt - bt + b' (y) ^m (h , yt, dt, bt+1 (yt))) + #V (a
16This is the argument in Chari and Kehoe (1990) and Stokey (1991)
26 I<)fVd (yt) dF(yt I yt-1) + (be, yt, bt+1) dF(yt I yt_1). (1.18) 1yt:dt (yt)=1 yt:d(yt)=O Equations (1.17) and (1.18) imply
[u (yt_1) - u (yi-1 - bt- 1 + qg-lbt)] + 3d (yt-1)
< 3 (bt, yt, bt+1(yt)) dF(yt I yt-1) + Vd(yt)dF(yt I yt-1). (1.19) Lfdt=O Zde=1 .I This is condition (1.16). So if the policies do not satisfy (1.16), they are not part of an SPE that generated the history ht_; in other words, there is no SPE consistent with ht_ with policies (dt (.), bt+ 1 (-)) for period t.
We also show that this condition is sufficient, so if (dt (-) , bt+1 (-)) satisfy conditions (1.14), (1.15), and (1.16), we can always find at least one SPE profile -that would generate Xt- on its equilibrium path. Even after a long history the sufficient statistics to forecast the outcome xt_ are
(bt-_1, bt , yt-_1).
Thus, effectively ECO(ht_) = ECO(bt1, yi_1, bt).
This result may seem surprising, but it is where robustness of the analyst (uncertainty about the equilibrium selection) is expressed. Because income y is a continuous random variable, any promises (in terms of expected utility) that rationalized past choices are "forgotten" each period; the reason is that the outside observer needs to take into account that promises could have been be realized in states that did not occur.
Finally, notice that even though the outside observer is using just a small fraction of the history, the set of equilibrium consistent outcomes exhibits history dependence beyond that of the set of SPE. The set of equilibrium consistent outcomes is a function variables (bt- 1, yt-, be). Thus, there is a role for past actions to signal future behavior. In contrast the set of subgame perfect equilibria after any history only depends on the Markovian states yt-1, bt.1
1.4.3 Equilibrium Consistent Prices
Aided with the characterization of equilibrium consistent outcomes in Proposition 10, we will characterize the set of equilibrium debt prices that are consistent with the observed history ht = (ht-1, yt-1, dt-1, be). The highest equilibrium consistent price solves
(ht) max ~(4,dt (-), bt+1(-)) 17 Notice that this role contrasts the dependence of the quantitative literature for sovereign debt that follows Eaton and Gersovitz (1981) as in Arellano (2008) and Aguiar and Gopinath (2006) where the fact that a country has just repaid a high quantity of debt, does not affect the future prices that will obtain.
27 subject to
S( ) , bt+1 (-)) E ECO(bt 1, yt 1 , b)
The lowest equilibrium consistent price solves
(ht min q (1.20) -(q,dt (-),bt+ (-) subject to (,dt - bt+1 (-))E 0C(bt-_1, yt-_1, bt) .
Highest Equilibrium Consistent Price. The highest equilibrium consistent price is the one of the Markov Equilibrium that we characterized in Section 1.2. Note that the expected value of the incentive compatibility constraint (1.15), is the value of the option to default W(y, b'), in the Markov Equilibrium. The promise-keeping will be generically not binding for the best equilibrium (given that the country did not default). For these two reasons, the best equilibrium consistent price is the one obtained with the default policy and bond policy that maximize the value of the option. Thus,
7 (h'_) = q (yt-1, bt) . (1.21)
Lowest Equilibrium Consistent Price. Our focus will be on the characterization of the lowest equilib- rium consistent price. Note that the lowest SPE price is zero. This follows because default is implementable after any history if we do not take into account the promise keeping constraint (1.16). On the contrary, we will show that lowest equilibrium consistent price is positive, for every equilibrium history. Furthermore, because the set of equilibrium consistent outcomes after history ht depends only on (bt_1, yt-1, bt), it holds that
q (h') q (bt- 1, yt_1, bt) . (1.22)
From (1.21) and (1.22), the set of equilibrium consistent prices will be
qt E [q (bti-1, yt1, bt), q (yt_1, bt)] . (1.23)
Proposition 11 establishes the main result of this subsection: a full characterization of q (b, y, b') (we drop time subscripts) as a solution to a convex minimization program, which can be reduced to a one equation/one variable problem.
Proposition 11. Suppose (b, y, b') are such that Vnd (b, y, b') > Vd (y) (i.e., not defaulting was feasible under the best continuation equilibrium). Then there exists a constant -y (b, y, ') > 0, such that
, 1 - f d (y') dF (y'I y) q(byb) =r
28 where
d (y') = 0 -> Vnd(b', y) Vd (y) + y for all y' E Y; y is the minimum solution to the equation:
f (And) = U (y) - U y - b + 1 -P( |yAdF)bl (1.24) /aAnd>, 1+ r where And v (b', y') -V (y') and F (A) its conditional cdf. If dF (-) is absolutely continuous, then -y is the unique solution to equation 1.24.
The proof is in the appendix. We provide a sketch of the argument. First, note that, by choosing the bond policy of the best equilibrium, all of the constraints imposed by equilibrium consistency are relaxed because the value of no default increases. So, finding the lowest ECO price will amount to finding the default policy that yields the lowest price and is consistent with equilibrium. Second, notice that the promise keeping constraint needs to be binding in the optimum. If not, the minimization problem has as its only constraint the incentive compatibility constraint, and the minimum price is zero (with a policy of default in every state). But, if the price is zero, the promise keeping constraint will not be satisfied. Third, notice that the incentive compatibility constraint will not be binding. Intuitively, imposing default is not costly in terms of incentives, and for the lowest equilibrium consistent price, we want to impose default in as many states as possible.
From these observations, note that the trade-off of the default policy of the lowest price will be: imposing defaults in more states will lower the price at the expense of a tighter promise keeping constraint. This condition pins down the states where the government defaults; as many defaults as possible, but not so many that no default in the previous period was not worth the effort. This, implies that the policy is pinned down by _d(y') =0 <= d 7nd(b' Iy') Vd (yI)+ where -y is a constant to be determined. This constant solves a single equation: is the minimum value such that the promise keeping holds with equality, with the optimal bond policy, evaluated at the best continuation Sf 1- F(7 1yl). ~ d AndF(~d y = u(y) - u(y - b+ .1 (1.25) and>- I+ r Remark 12. Note that the best equilibrium default policy at t
d(yt) = 0 <-vn V (bt, yt) Vd(yt).
29 On the contrary, the default policy of the lowest equilibrium consistent price is
d(yt) = 0 - (bt, y) > Vd(yt) + y where y is the constant that solves (1.25) and depends on (bt_1, yt-i1, be). The default policy is shifted to create more defaults, to lower the price, but not so many that the promise-keeping was not satisfied (i.e., we cannot rationalize previous choices).
Remark 13. Notice that by focusing on equilibrium consistent outcomes uncovers a novel tension that is not present in SPE. At a particular history h'_, implementing default is not costly because it is always as good as The next Proposition describes how the set of equilibrium consistent prices changes with the history of play. Proposition 14. Let q (b, y, b') be the lowest ECO (b, y, b') price. It holds that 1. q (b, y, b') is decreasing in b', 2. q (b, y, b') is increasing in b, 3. For every equilibrium (b, y, b'), -b + b'q (b, y, b) < 0, 4. If income is i.i.d., q is decreasing in y, and so is the set Q = [q (b, y,b') ,; (y,b')] First, note that as in the best equilibrium, the lowest equilibrium consistent price is decreasing in the amount of debt issued b'. The intuition is that higher amounts of debt issued imply a more relaxed promise keeping constraint. In other words, the past choices of the government can be rationalized with a lower price. A similar intuition holds for b; if the country just repaid a high amount of debt (i.e., made an effort for repaying), past choices are rationalized by higher prices. Second, note that if there is a positive capital inflow with the lowest equilibrium consistent price, it implies that u (y) - u (y - b + b'q (b, y, b')) < 0. Intuitively, the country is not making any effort in repaying the debt. Therefore, it need not be the case that the country was expecting high prices for debt in the next period. Mathematically, when there is a positive capital outflow with the lowest equilibrium consistent price, y is infinite. This implies that 1-F () = q (b, y, b') = 0, which contradicts a positive capital inflow. 1+* _ 6 b) 0 ostv aia Finally, because there are no capital inflows with the lowest equilibrium consistent price, repaying debt at this price will become more costly as income is lower; this due to the concavity of the utility function. 18 18 This observation is used in the literature of sovereign debt. For example, to show that default occurs in bad times, as in Arellano (2008), or to show monotonicity of bond policies with respect to debt, as in Chatterjee and Eyigungor (2012). 30 Mathematically, because of concavity, u(y) - u (y - b+ b'q (b, y, b')), is19 increasing as income decreases, and therefore, the promise keeping constraint tightens as income de- creases. Note that, in the non i.i.d. case, this property will not hold, because, even though the burden of repayment is higher, the value of repayment in terms of the continuation value can be increasing. 1.4.4 Interpretation: Robust Bayesian Analysis In this subsection we provide an interpretation of the set of equilibrium consistent outcomes relating to robust Bayesian analysis. Appendix 1.13 provides a more formal connection. In Bayesian analysis, the econometrician has a prior over the set of fundamental parameters E;20 here will be denoted by 7r(6). In addition, because of equilibrium multiplicity, she also has a prior p(x) over the set of outcomes X.21 Using data (in our case, in the form of a history) and these priors, she obtains a posterior. Suppose that she is interested in the (posterior) mean of a particular statistic T(x, 0). Conditional on the data, her prediction is EP'M [T(x, 0) 1 data]. There are many situations where the econometrician will not want to favor one equilibrium against another one; that is, there is uncertainty with respect to the prior p E P. Then, there is a whole range of posterior means of the statistic that is given (for a fixed 0, or a degenerate prior over 0) by [min EP [T (6, x) I data], max EP [T (0, x) I data]]. Our focus in on priors over equilibrium outcomes, but we are be agnostic about the particular prior (i.e. equilibrium selection). We characterized the set of equilibrium consistent (with history h) debt prices min qg, max qtl xEx(Eih) xEx(Ch) This interval characterizes the support of the posterior over prices, when the only assumption is that the observed history is part of an SPE. 19 The change in this expression will depend on the sign of u (y) - u y - b + b' I (-Y) that is positive due to the result of no capital inflows with the lowest equilibrium consistent price. 20 In our model, discount factor, parameters of the utility function, volatility of output, etc 2 1 For example, Eaton and Gersovitz (1981) select a Markov equilibrium. Chatterjee and Eyigungor (2012) choose an equi- librium with arbitrary probability of crises in their study of optimal maturity of debt. These would be examples of degenerate priors; in other words, there is a particular equilibrium selection. 31 1.5 Extensions: Excusable Defaults and Savings In this section we discuss how we characterize equilibrium consistent outcomes in a common different setting for the literature of sovereign debt: we do not restrict that a default needs to trigger a punishment and, we allow for savings. This will break the connection between the best SPE and the Markov equilibrium that we characterized in Section 1.2. However, autarky will still remain as the worst equilibrium. Given the best SPE values and prices, characterizing equilibrium consistent outcomes will follow the case in Section 1.4. 1.5.1 Excusable Defaults The setting where we do not impose that defaults need to be punished with financial exclusion is similar to the one in Atkeson (1991) and Worrall (1990b). 2 2 For the moment, assume that the government cannot save. Denote by WE (y_, b) the expected value of the best equilibrium if the government starts with bonds b. The following proposition characterizes equilibrium consistent outcomes in this case. Proposition 15 (ECO, excusable defaults). Suppose h' = (ht-, yt_, dt_1 , bt) is an equilibrium con- sistent history. Then xt_ = (qt-1,d (.), bt+ (.)) is an equilibrium consistent outcome at ht if and only if the following conditions hold: a. Price is consistent qt-1 r (1 dt(yt)dF(yt I yt-1)), (1.26) b. IC government u(yt - bt(1 - d(yt)) +E (yt,bt+1(yt))bt+1(yt)) -/W- E(yt,bt+1(yt)) Vd(yt), (1.27) c. Equilibrium consistency Of [u(yt - bt(1 - d(yt)) + qE(yt ,t+(yt))bt+1(yt)) + WE (yt, btl(yt))] dF(yt) > u(yt-1) - u (yt-1 - bt_ 1 (1 - d(yt.1)) + qt-libt+1(yt)) + /3W. (1.28) If conditions (a) through (c) hold, we write simply (qt_-1, dt (-) , bt+1I (-)) E IEC(DE (bt - 1 yt -1, bt) 2 2 1n our case, we restrict the contract to be one where the face value can be chosen, but can either be defaulted or repaid in full. 32 where ECO stands for "equilibrium consistent outcomes" and the subscript E stands for the case of excusable defaults. As in Section 1.4, conditions (1.26) and (1.27) characterize the set of SPE policies. The first condition (1.26) is again that the price has to be consistent with the default policy. The second condition (1.27) is the incentive compatibility for the government. The difference between (1.27) and the incentive compatibility of Proposition 10 that was given by (1 - d(yt)) [u(yt - bt + i(yt, bt+1)bt+1 ) + )W(yt, bt+1 )] + d(yt)Vd(yt) ;> Vd(yt) comes from the fact that defaults are not required to be punished. On the equilibrium path, defaults are excusable in the sense of Grossman and Huyck (1989);23 off the equilibrium path they are punished with autarky, the worst equilibrium. The intuition of condition (1.27) is similar to the incentive compatibility in Proposition 10 in Section 1.4. If in a history ht_ = (ht-1, yt-1, dt-1, bt) a default decision and bond issue decision wants to be implemented, it must be the case that it is weakly better than any deviation. Following Abreu (1988), any SPE can be implemented with strategies that impose the worst punishment in case of deviation; and, as in Proposition 10, we reward following the policy with the best equilibrium. This implies that dt, bt+1 is implementable if u(yt - bt(1 - dt) +(yt, bt+1)bt+1) + fW(ht_, dt, bt+1) > max u(yt - bt (1 - d) + q(yt, b')b') + 81W(h_, d, b') (1.29) d,b' ~ where W, W denote the best and worst continuation equilibria. The value of the best equilibrium is WE(yt, bt+1 ). Because the worst equilibrium is autarky with a price of debt equal to zero (q(yt, 6') = 0), the right hand side of (1.29) is equal to Vd(yt). Condition (1.27) follows. Again, conditions (1.26) and (1.27) are necessary and sufficient to characterize SPE outcomes. Equilibrium consistent outcomes are characterized by an additional condition (1.28). The right hand side of (1.28) is the opportunity cost from not taking the best deviation last period. The left hand side specifies the expected value of the policy under the best equilibrium. The reason why conditions (1.26) - (1.28) are necessary and sufficient is the same as before. The lowest equilibrium consistent price will solve qE (h_+') min q (q,dt(-),bt+ (-)) 23 The reason why defaults are part of the equilibrium path is that they introduce stay contingency for the country and are also expected by the borrowers, so they will make zero profits on average. 33 where (,dt .) bt+ 1 (-)) E ECOE (bt-1, yt-1, bt) The intuition of the solution to this program is similar the intuition that we had before. The bond policy will be the one of the best equilibrium, and the default policy will be tilted towards more defaults, but not so many that the previous choices cannot be rationalized. Again, the highest equilibrium consistent price will be qE, the best subgame perfect equilibrium price. Comparative statistics similar to the ones of Corollary 14 also hold. If the government issues more debt bt+i, the lowest equilibrium consistent price decreases. If the government has not defaulted debt, d(yt_1) = 0, an increase in bt increases qE. Also, after any history, the government cannot receive positive capital inflow in every continuation equilibrium. In other words, of the government receives a positive capital inflow, the lowest equilibrium consistent price is zero. 1.5.2 Best SPE Note that the characterization of equilibrium consistent outcomes will use as input the best equilibrium price qE(bt) and the value function of the best equilibrium WE(yt, bt+1(yt)).14 In this subsection we discuss how we characterize them. Best Equilibrium Price Taking as given WE(yt, bt+1(yt)), the price function qE solves the following functional equation 7 E(yt-1, bt) max q dt (yt),bt+ (yt) u(yt - bt(1 - d(yt)) + q(yt, bt+1(yt))bt+1(yt)) + WE (yt,bt+(yt)) Vd(yt) 1 dt(yt)dF(yt yt-1)) q = I + r (1- I A solution to the operator is guaranteed due to the monotonicity of the operator and because the set of continuous and weakly decreasing functions endowed with the sup norm is a complete metric space. Best Equilibrium Value Notice that we just obtained the best price taking as given the best equilibrium value for debt. Suppose now that we know the best price. The best equilibrium will be the equilibrium 2 4 Note that these ones will not be the ones of the Markov equilibrium that we characterized in Section 1.2. The reason is that now, the government is allowed to default, on the equilibrium path, without a punishment. A Markov equilibrium with states b, y would imply that the government will default every debt that it has acquired. Therefore, there has a to be a price keeping constraint. An alternative approach is one as in Atkeson (1991) or Worrall (1990b) that uses instead of b, y as a state variable, the funds that the government has after repayment, in our notation y - (1 - d(y))b. With this state variable, an approach as in Abreu, Pearce, and Stacchetti (1990) can be used to obtain the best equilibrium value and the policies. We develop an alternative approach where we iterate directly in the set of equilibrium payoffs and prices. This follows Abreu, Pearce, and Stacchetti (1990), but it is still not the same. 34 with highest expected value that meets the incentive compatibility and the price consistency constraint. It is given by -E F E'yyt] W E(yt_1, bt) = EY, lti ,_ l (yt_1, yt, bt)] where W (yt-1, yt, bt) solves WE' (yt-1, bt, yt) = max {U(yt - b(1 - dt(yt)) + E(yt, bt+1(yt))bt+1(yt)) dt(yt),bt+1(Vt) +#WE (yt, bt+1(yt))} subject to -(yt- = 1 (1 dt(yt)dF(yt I yt_1)) (1.30) Note that, constraint (1.30), is the one that makes sure that the amount lent, will be defaulted with the best equilibrium default rule. 1.5.3 Excusable Defaults and Savings The most general characterization of SPE allows the government to save, and does not impose any exogenous punishment if it defaults. We can show that the worst equilibrium price for debt is zero; autarky is the worst SPE. Subgame perfect equilibrium outcomes are characterized by qt-1 = 1+r (1 - f dt(yt)dF(yt I yt-1)) u(yt - bt(1 - d(yt)) +iqE(yt,bt+1(yt))bt+1(yt)) + wES(yt ,bt+1(yt)) naxu(yt - bt(1 - d) + q(yt,b')b') + 1 WES(ytb/)- d,b' Then, the worst SPE price for the case of savings and excusable defaults will be zero. So the characterization of equilibrium consistent outcomes is analogous to the one in Proposition 15 without the restriction that b>0. 1.6 Sunspots We are now interested in adding a sunspot variable. Adding a sunspot that is realized together with output adds nothing to the analysis. Effectively, output could already acting as a random coordination device. Thus, the interesting question is to add a sunspot variable after the bond issuance, but before the price is determined. As we shall see, conditional on any single realization, the set of equilibrium consistent outcomes 35 then coincides with the set of subgame perfect equilibria. Despite this we can obtain relevant history dependent predictions. In this section we do three things. First, we characterize what we term are equilibrium consistent distribu- tions. Those are distributions over prices that consistent with a subgame perfect equilibrium given history. Second, aided with this characterization we obtain bounds of the expectation over prices that hold across all equilibrium. This provides a way to obtain set identification of the set of structural parameters in our particular application. Finally, we provide an intuitive application of our results, and we find a bound on the probability of a non-fundamental debt crises; by crisis we mean an event where the realized price falls below a given threshold q, which we treat as a parameter. 1.6.1 Equilibrium consistent distributions Denote the sunspot by (t, realized after the bond issue of the government but before the price qt; i.e, a sunspot is realized after h'.. Without loss of generality25 we will assume (t ~ Uniform (0, 11 i.i.d. over time. If we assume that the game is on the equilibrium path of some subgame perfect equilibrium, then the government strategy before the realization of the sunspot was optimal; that is f[u (yt - bt + qt ((t) bt+1) + 13v ((t)] d~t ;> Vd (yt). The government ex-ante preferred to pay the debt and issue bonds bt+1 than to default, where q ((t) and v are the market price and continuation equilibrium value conditional on the realization of the sunspot (. The main difference in the characterization of equilibrium consistent distributions, is that now we cannot rely on the best continuation price, because it might not be realized. Best continuation. Define the maximum continuation value function T (b, q) given bonds b and price q as (bq)= max V(o-|bo=b) crESPE(b) subject to E(1 - d(yo)) 1 + r q This gives the best possible continuation value if we start at bonds b and we restrict prices to be equal to q. The following proposition provides a method to compute the function u (b, q) and show that is non-decreasing and concave in q. 2 5 This is because of robustness: we will try to map all equilibria that can be contingent on the randomizing device, and hence as long as the random variable remains absolutely continuous, any time dependence in (t can be replicated by time dependence on the equilibrium itself. 36 Proposition 16. For all q E [0, q(b)] the maximum continuation value T (b, q) solves F (b,q)= max f 5(y)Vd(y)+ [1-6(y)n (b, y) dF (y) subject to q= Jr (1-J6(y)dF(y)) (1.31) Furthermore, is F (b, q) non-decreasing and concave in q. Proof. See Appendix 1.11. 0 The fact that the function is non-decreasing in q follows from the fact that better prices are associated with better continuation equilibrium, as well as higher contemporaneous consumption (since bt+i > 0). This follows from the fact that defaults are punished but when the government does not default, it obtains the best continuation equilibrium (under the strategy associated with value U (bt+ 1 , qt)). Concavity, follows from the the fact that U(b, q) solves a linear programming problem. We use both properties to obtain sharper characterizations of the set of equilibrium consistent distributions and to obtain testable predictions. Main result. For a given equilibrium a at history h = (ht, yt, dt, bt+ 1) the equilibrium price distribution is defined by Pr (q E A) = Pr ((t : q' (h, (t) c A) Let Q (h) be the family of price distributions from history consistent equilibria. The following Proposition provides a characterization of this family. Proposition 17. Suppose h = (ht, yt, dt, bt+1) is equilibrium consistent. Then, a distributionP E A (R+) is an equilibrium consistent price distribution; i.e. P E Q (h) if and only if Supp (P) C [0,q(bt+1)] and { u (yt - bt + qbt+1) + 31v (bt+1, q)}I dP (q) > Vd (yt) (1.32) and hence Q (h) = Q (bt, yt, bt+1). Proof. See Appendix 1.12. Condition 1.32 parallels equation (1.16) in Proposition 10. There are some differences. We are now char- acterizing distribution over prices consistent with a decision of not defaulting dt = 0 and issuing debt bt+i. Note that we are taking an expectation with respect to q: the government does not know what particular price will be realized after it chooses a particular policy. The following two corollaries provide intuition regarding the set of equilibrium consistent distributions Q (bt, yt, bt+1)- 37 Corollary 18. The set of equilibrium price distributions Q (bt, yt, bt+1) is non-increasing(in set order sense) with respect to bt; if income is i.i.d, it is non-decreasing in yt. Proof. See Appendix 1.12. 0 The intuition of this comparative statistics is again coming from the revealed preference argument. If the government repaid a higher amount of debt, then the distribution of prices that they could be expecting needs to shift towards higher prices. If the set does not change, then there will be some distribution that will be inconsistent with equilibrium because it will violate the promise keeping constraint. Let P' [> P denote the relationship "Q' first order stochastically dominates Q". The next corollary provides a partial ordering in Q. Corollary 19. Suppose that P E Q (bt, yt, bt+1 ) and P' E A ([, (bt+ 1 )]) If P' > P ( i.e. it first order stochastically dominates P), then P' C Q (bt, yt, bt+ 1). Proof. See Appendix 1.12. E Once that a distribution is consistent with equilibrium, any distribution that first order stochastically domi- nates it will be an equilibrium consistent distribution. Intuitively, higher prices give both higher consumption and higher continuation equilibrium values for the government, since both are weakly increasing in the real- izations of debt price qt. 1.6.2 Expectations of Equilibrium Consistent Distributions The main application of the analysis in this section if to obtain bounds over expectations equilibrium out- comes. In particular we will focus on bounds over equilibrium consistent prices. The set of equilibrium consistent expected prices is given by E (bt, yt, bt+i) = {a E R+ : a = Ep (q) for some P c Q (bt, yt, bt+1)} where Ep (q) f qdP. The following Proposition shows that in fact, the set of expected values is identical to the set of equilibrium consistent prices when there are no sunspots. t Proposition 20. Suppose history h = (h , yt, dt = 0, bt+1 ) is equilibrium consistent. Then the set of expected prices is equal to the set of prices without sunspots Q (bt, yt, bt+ 1 ); i.e. E (bt, yt, bt+1) = [q (bt, yt, bt+1), i(bt+1)] 38 Moreover, if bt+1 > 0 then the minimum expected value is achieved uniquely at the Dirac distribution P that assigns probability one to q = q (bt, yt, bt+1). Proof. See Appendix 1.12 The result comes from the concavity of the value function U (bt+1, 4) and the fact that q (-) is the minimum price that satisfies: u (yt - bt + qbt+1) + rv (bt+1,, J = Vd (yt) (1-33) We showed concavity in Proposition 16. The equality at q = q (-) follows from the strict monotonicity in q of the left hand side expression: if the inequality was strict, then we can find a lower equilibrium consistent price, which contradicts the definition of q (-). Therefore, the integrand in 1.32 is bigger than Vd (yt) only when q > q (bt, yt, bt+ 1 ). Concavity of U (b, q) and Jensen's inequality then imply that for any distribution P E Q (bt, yt, bt+1) : u (yt - bt + Ep (q) bt+ 1 ) + 8v- (bt+1 , Ep (q)) > J {u (yt - bt + qbt+1) + OF (bt+1 , q)} dP (q) > V (yt) and therefore Ep (qt) > q (bt, yt, bt+ 1 ). The previous Proposition actually provides testable implications for the model. In particular, it yields a necessary and sufficient moment condition for equilibrium consistency at histories h = (ht, yt, dt, bt+1 ), Eqt {u (yt - bt + bt+1) + O (bt+1, qt) - Vd (yt) I h} 0 (1.34) The bounds that we just derived yields moment inequalities that are easier to check Eqt {qt I h} E [q (bt, yt, bt+1) , -i (bt+)] (1.35) Aided with these bounds we can perform estimation of the structural set of parameters as in Chernozhukov, Hong, and Tar. (2007). 1.6.3 Probability of Crises Our goal will be to infer the maximum probability (across equilibria) that the government assigns to the market setting a price q(() = 4; i.e., a financial crises. Formally, max Prp (q <) (1.36) PEQ(bt,yt,bt+i) 39 where Prp (q < 4) := fo dP (q). These bounds are independent of the nature of the sunspots (i.e. the distribution of sunspots, its dimensionality, and so on), in the same way as the set of correlated equilibria does not depend on the actual correlating devices. 26 Furthermore this bound will yield a necessary condition for a distribution to be an element in Q (bt, yj, bt+1). Upper bound on Pr (q = 0) To construct the maximum equilibrium consistent probability that qt = 0, we make the promise keeping constraint be as relaxed as possible. We do this by considering continuation equi- libria with two properties: first, assign the best continuation equilibria if q : 0 (i.e, under price -j (yt, bt+1)). Second, note that autarky is the best continuation equilibria feasible with q = 0; if the government receives a price of zero, in equilibrium, it will default with probability one in the continuation equilibrium". Let po = (= 0). The IC constraint is now PO [u (yt - bt + bt+1 x 0) + OVd(yt)] + (1 - PO) rV (bt, yt, bt+1 )] Vd (Yt). Then And (bt, yt, bt+1) < And (bt, yt, bt+i) + u (yt) - u (yt - bt) where And (-) denotes the maximum utility difference between not defaulting and defaulting (under the best equilibrium) And r~-d d(t A (bt, yt, b t+1) V (bt, yt, bt+i) Vd(yt) Thus, the probability of q = 0 is bounded away from 1 from an ex-ante perspective (i.e. before the sunspot is realized, but after the government decision). So we obtain a history dependent bound on the probability of a financial crises. Upper bound for general 4 < q (bt, yt, bt+1). Let p = P (a). Using the same strategy as before, to get the less tight the incentive compatibility constraint for the government we need to 1. for ( : q (() > 4, we consider equilibria that assign the best continuation equilibria, 2. maximize equilibrium utility for q : q < . Thus And (bt, yt, bt+1 ) r _ ) Vd (yt) - [u (yt - bt + 4bt+1 ) + #W7(bt, 4)] + And (bt, yt, bt+j) 2 6 As long as out interest is in characterizing all correlated equilibria. *The default decision in equilibrium needs to be consistent with the price: a price of zero is only consistent with default in every state of nature. And we assume that after default the government is in autarky forever. 40 Note that this is not an innocuous constraint only when the right hand side is less than 1. This happens only when U (yt - bt + dbt+1) + OU (bt, 4) ! Vd (yt). This constraint holds when and this holds if 4 q (bt, yt, bt+1) The last inequality comes from the charac- terization of q (bt, yt, bti). The following Proposition summarizes the results of this section: -nd Proposition 21. Take an equilibrium consistent history h = (ht, yt, dt, bt+j) and let And =- V (bt, yt, bt+)- Vd (yt). For any 4 < q (bt, yt, bt+i) A nd A nd- [u (yt - bt + 4bt+1) + / (bt, 4) - u(yt) - #Vd] For any 4 q (bt, yt, bt+i), P (4) = 1. In Proposition 21 we find the ex-ante probability (before (t is realized) of observing qt = 4 is less than P (4) < -nd 1 for any equilibrium consistent outcome. Note that if the income realization is such that V (bt, yt) = Vd (yt) (i.e. under the best continuation equilibrium, the government was indifferent between defaulting or not, and still did not default), then P (4) = 0 for any 4 < q (bt, yt, bt+1) = -4(yt, bt+ 1 ), which implies that at such income levels, even with these kind of correlating devices available, only q = -q(yt, bt+i) is the equilibrium consistent price. We also show that any price q E [q (-) , ij (-)] could be observed- with probability 1, since they are part of the path of a pure strategy SPE profile. When adding sunspots, any price in [0, ij (-)] can be observed ex-post, and since the econometrician has no information about the realization of the sunspot (or the particular equilibrium selection and use of the correlating device) any price is feasible ex ante. However, before more information is realized, the econometrician can place bounds on how likely different prices are, which can not be 1, so that the government incentive constraint is satisfied. Aided with the characterization of Proposition 21 we find a restriction satisfied by equilibrium consistent dis- tributions: they stochastically dominate P, in the first order sense. Note that it is a cumulative distribution function on q: it is a non-increasing, right-continuous function with range [0, 1], hence implicitly defining a probability measure over debt prices Corollary 22. The distribution _P (-) is the maximum lower bound (in the FOSD sense) of the set equilibrium consistent distributions; i.e. for every P E Q (bt, yt, bt+1) we have P > P, and if P' is some other lower bound, then P' t> P. Moreover, _P $ Q (bt, yt, bt+1) 1.7 Discrete Income Now, we study the case where there are no sunspots at all. For simplicity in the notation, focus in the case where income is i.i.d. Our main result in the previous section was a characterization of equilibrium consistent 41 outcomes. In this result, the fact that the last opportunity cost is a sufficient statistic for equilibrium consistent outcomes (and prices as a consequence) is somewhat surprising: the outside observer is only using observations from the last period to make an inference, even though she has a whole history of data available. However, as we will see below in a simple two period example, this result is a direct expression of Robustness: the econometrician needs to take into account that the expected payoff that rationalized a particular decision, could have been realized only in histories that did not occur. Suppose that we observe (VO, bo, qo, bi, I, b2 ). Denote h' = go, bo, qo, bl. We will show that 1(go, bo,1 go, bi, gi, Ib2) = q,1(bi, *yl,b2) when the probability p(yi) = 0, capturing the main feature of the case with continuous income, that any realization has zero probability. In order to do this, note that, because -o, bo, qo, b1 is an equilibrium history, there is a continuation value function such that p(y1 )Vo(yi, bi) 1 [u(go) - u(go - bo + qobi)] + W (1.37) where Vo(y1i, bi) is the continuation value function of a continuation equilibrium strategy after history VO, bo, qo, bi, y1i. Note also that V1, b2 and the decision not to default are part of an SPE. This implies that the following constraint has to hold - bi + q,(h0, gl, b2 )b2 ) +# p(y2)V(y2ib2) > u(91) + #_ (1.38) Note, also, it has to be the case that Vo(V1, bi) = ('Y1 - bi + q1(h1,11i, b2)b2) + 13 P(y2i)V1(Y2i, b2) (1.39) i That is, on the equilibrium path, the promised continuation needs to coincide with the continuation that we observe in history (go, bo, qo, bi, , b2 ). Finally, it also needs to be that the case that V1(y 2 i, b2) E [W, W(y 2 ,b 2 )] (1.40) V1 (y2i, b2) E [3K, W(y2i, b2)] (1.41) where we abuse notation slightly for the continuation value sets. Now, the lowest equilibrium consistent price solves min q (1.42) sV(1i,b1). jV1(ivt,2)}th r subject to (1.37), (1.38), (1.39), (1.40), and (1.41). Our objective is to show that if pAgi) = 0, the constraint 42 (1.37) is not binding, and therefore, the solution will not depend on ( O, bo, qo). To solve (1.42), we want to relax the constraint (1.37) as much as possible. So we pick the continuation value function ,bi) for yl Y= Vo(y1j, bi) = Vo(p 17 W(y 1 j, b1 ) for y1i 54V where V (p,bi) is free at the moment. Because the histories ("o,bo, qo, bi, for yli # V) are not realized, it could have been the case that the best continuation followed. The outside observer cannot neglect this possibility. Then, by adding and subtracting p(V 1 )V(Vj, bi), we can rewrite the left hand side of (1.37) as p(y1p)Vo(y1j, bi) = P(i) [Vo(ji, bi) - V(TI, bi)] + p(yi)W(yii, bi) (1.43) Plugging (1.43) in (1.37) p(Yi) [Vo(-j1 , bi) - V(9i, b 1)] + p(y1)W(y1j, bi) [u(-yo) - u(Vo - bo + qobi)] + W (1.44) where V(Fl, bi) is the value of not default in the best equilibrium when bonds are b1 and income is 71. So, when income is continuous, p(Yi) = 0. So, the constraint will not be binding if u(Vo - bo + qobi) + / Zp(yii)W(yz, b1) > u(FO) + ,3W holds. And this holds because Vo, bo, qo, b1 is an SPE history where the government did not default. If income is discrete, then bi, pI, b2 will not be sufficient statistics to summarize history. The intuition is that the future policies affect previous decisions, because the particular realized history does not have probability zero. Define oco = u(-o) - u(VO - bo + qobi) This is the opportunity cost of not defaulting. Rearrange (1.44), such that 1 1- Vo~pi, bi oco + W - W(bi)] + V(-yi, bi) If this constraint binds, the lowest equilibrium consistent price is , (9q,bo,go,b1,b2) with full history dependence. Whether it will bind or not, depends on the following. First, it depends on the past opportunity cost: if 43 in the past, the government passed on default under very harsh circumstances, then the continuation value needs to be higher. Second, it depends on the strength of the link between current and past decision. If the government discounts more the future, or the history is less likely, then the constraint is less likely to be binding. 1.8 Conclusion and Discussion Dynamic policy games have been extensively studied in macroeconomic theory to increase our understanding on how the outcomes that a government can achieve are restricted by its lack of commitment. One of the challenges in studying dynamic policy games is equilibrium multiplicity. Our paper acknowledges equilibrium multiplicity, and for this reason focuses on obtaining predictions that hold across all equilibria. To do this, we conceptually introduced and characterized equilibrium consistent outcomes. We did so under different settings, and we found that the assumption that a history was generated by the path of a subgame perfect equilibrium puts restrictions on current policies, and therefore on observables. In addition, we found intuitive conditions under which past decisions place restrictions on future policies; if the past decision occurred far away in time or in a history where the current history had low probability of occurrence, then it is less likely that a particular past decision influences current policies. In the extreme case that every particular history has probability zero, the restrictions of past decisions in current outcomes die out after one period. At first glance, this is surprising; but as we showed in the paper, this a direct consequence of robustness. As we discussed in the text, equilibrium consistency is a general principle. Even though we focus on a model of sovereign debt that follows Eaton and Gersovitz (1981), our results generalize to other dynamic policy games. An example is the model of capital taxation as in Chari and Kehoe (1990). In that model, the entrepreneur invests and supplies labor, then the government taxes capital, and finally, the entrepreneur receives a payoff. The worst subgame perfect equilibrium is one where the government taxes all the capital. Note that, if the government has been consistently abstaining from taxing capital, then as outside observers we can rule out that the government will tax all capital. Past behavior, and the sole assumption of equilibrium, is giving information to the outside observer about future outcomes. We think equilibrium consistency might have applications beyond policy games. The reason is that the sole assumption of equilibrium yields testable predictions. For example, the literature of risk sharing studies barriers to insurance and tries to test among different economics environments. Two environments that have received a lot of attention are Limited Commitment and Hidden Income. To test these two environments, a property of the efficient allocation with limited commitment is exploited: lagged consumption is a sufficient statistic of current consumption. If this hypothesis is rejected, then hidden income is favored in the data. However, the test is rejecting two hypotheses at the same time: efficiency and limited commitment. Our approach could, in principle, be suitable for a test that is tractable and robust to equilibrium multiplicity. 44 Over the course of the paper, we have been silent with respect to optimal policy. An avenue of future research is to relate equilibrium consistent outcomes and forward reasoning in dynamic games. Our conjecture is that, the set of equilibrium consistent outcomes will be intimately related with the set of outcomes if there is common knowledge of strong certainty of rationality. The reason is that, in the model of sovereign debt that we studied, the outside observer and the lenders have the same information set. Even in the motivating example, equilibrium consistent outcomes and outcomes when the solution concept is strong certainty of rationality are the same. In that case, our results have a different interpretation: the government is choosing the history to manage the expectations of the public. 1.9 Appendix A Proof. (Lemma 6 ) Note that we can rewrite the system of Bellman equations as A.v(B) = u(B) Thus, a condition in primitives is v'(O) = A-lu'(O) > 0 For the special case where A = 1, note that 3 VBH = 2 (u(yH - RB) + u(YL + B)) BH= 1~2- VOL = u(yL + B) +#VBH Then, v'H(0) > 0 implies that voL(0) > 0. A sufficient condition is 1u'(yL) > Ru'(yH). The intuition is that, the government is credit constrained in the low state, with no debt, and is willing to trade-off and have lower consumption in the high state. Proof. (Proposition 10). (Necessity, ==>) If (d (.) , b' (-)) is SPE - consistent, there exists an SPE profile & such that h' E 7C(&) and d (yt) = d& (ht, yt) and b' (y) = b I (ht, yt, d = 0) That is, there exists a SPE that generated the history ht_, specifies the contingent policy d (V), b' (-) in period t, and satisfies conditions (1.14) to (1.16). Because &is an SPE, using the results of Abreu, Pearce, and Stacchetti (1990) we know that if d (y) = 0 at hit = (ht, qt- 1 ) then u (yt - bt + b' (yt) qg (ht, dt = 0, b' (yt))) + 3W (a ht+l) ;> u (yt) + 13V(yt) (1.45) 45 By definition of best continuation values and prices W (& I h'+1 ) W (yt, b' (yt)) and q. (ht, dt = 0, b' (yt)) 5 ; (yt, b' (yt)) (1.46) Because b' (yt) 0 (no savings assumption), and u (.) is strictly increasing, we can plug in (1.46) into (1.45) to conclude that u (yt - b+ b' (yt)4 (yt, b' (yt))) + -W (yt, b' (yt)) u (y - bt + b' (yt) gn& (h', dt = 0, b' (yt))) + OW (& I ht+1) Proving condition (1.15). Further, since & generated the observed history, past prices must be consistent with policy (d (-) b' (-)). Formally: qt- = qi (htl,yt_1, dt_1, b) 1 + r* d& (ht, yt) dF(yt I yt-1)) yt_1)) = ( - d(yt)dF(yt I 1+ r y~tEY proving also condition (1.14). Condition (1.16) is the same as condition (1.15) but at t - 1, using the usual promise keeping accounting. Formally, if & is SPE and ht E R (&) then the government's default and bond issue decision at t - 1 was optimal given the observed expected prices u(yt- - bt- 1 + btqti) + 8W (& I ht) > u (yt-1) + /V3 (yt- 1 ) =ct-1 Using the recursive formulation of W (.) we get the following inequality: W (& I ht) = [u(yt - bt + b'(yt)q&(ht , yt, dt = 0, b'(yt))) + W (& ht+l)] dF(yt I yt_1) J yt:d(yt )=O nYtl +f [u (yt) + /Vd (yt)] dF(yt I yt-1) y~t:d(yt )=1 [u (yt - bt + b'(yt)q(b'(yt))) + W(b'(yt))] dF(yt I yt-1) Syt:d(yt)=0 + f [ (Vt) + 3Vd (yt)] dF(yt Iyt-1) yt:d(yt)=1 From the previous two inequalities, we show (1.16). (Sufficiency, +--) We need to construct a strategy profile o- c SPE such that ht_ E 71 (0-) and d (-) = d' (ht, -) and b'(-) = b-+ 1 (ht,). Given that ht E 1H (SPE), we know there exists some SPE profile a = (&g, 4m,) that generated ht . Let 7 (b, y) be the best continuation SPE (associated with the best price (-) ) when yt = y and bt+1 = b. Let o"t be the strategy profile for autarky (associated with qm = 0 for all 46 continuation histories). Also, let ht+l (yt) = (ht, yt, d (yt) , b' (yt) , q (yt, b' (yt))) be the continuation history at yt = y and the policy (d (-) , b' (-)) if the government faces the best possible prices. Define (h3, y,) -< ht as the histories that precede ht and are not equal to ht. That is, if we truncate ht to period s, we obtain h'. Denote (h', y,) $ ht as the histories that do not precede ht. The symbol -< denotes, histories that precede and can be equal. Construct the following strategy profile a = (ag, q0) : &g (h-, ys) for all (h', y) - ht aaut (Ys) for all s < tand (h", y) $ ht t ag (hs, ys)= dt (ht, yt) = d (yt) and bt+1 (ht yt) bY(yt) for (h ,yt) for all yt Dg (bs+ ,ys) (hs, ys) for all hs >- ht+1 (yt) oaut (yS) for all s > t, hS $ ht+1 (yt) and qm (hs, ys, d, bs+i) for all (hs, ys) -< ht for all s < t and (h', y,) $ht qm (h,7Ys, ds, b+,+) = 0 V (ys, b'(y)) for all h' >- (ht, y, d (yt) , ' (yO) 0 for all h8/>- (ht, yt, d (yt) , b' (yt)) By construction h'_ E R (a). This is because, o = &g for histories (hs, Ys) d ht. Also, the strategy a, prescribes the policy (d (-) , b' (-)) on the equilibrium path. Now we need to show that the constructed strategy profile is indeed an SPE. For this, we will use the one deviation principle. See that for all histories with s > t the continuation profile is an SPE (by construction); it prescribes the best continuation equilibrium, that is a SPE by definition. Now, we need to show that at ht this is indeed an equilibrium. This comes from the second constraint, the incentive compatibility constraint (1 - d(yt)) [u(yt - b + q(yt, bt+l((yt)))bt+1(yt)) + /W(yt, bt+1((yt)))] +d(yt)Vd(yt) ;> Vd(yt) Note also that the default policy at t - 1 was consistent with a (and is an equilibrium) and that qtI is consistent with the policy (d (-) , b' (-)). The promise keeping constraint (1.16) translates into the exact incentive compatibility constraint for profile a, showing that the default decision at t -1 was indeed optimal given profile a. The "price keeping" (1.14) constraint also implies that qt-i1 was consistent with policy (d (-) , b' (-)). The final step in sufficiency is to show that, s < t - 1 (that is hs --< ht ). Note that, because y is absolutely continuous, the particular y that is realized, has zero probability. So, the expected value of this new strategy is the same t W(& I hs) = W(a I h ) 47 for all h' -< h' with s < t - 1; the probability of the realization of ht, is zero. All this together implies that o- is indeed an SPE and generates history h_ on the equilibrium path, proving the desired result. i Proof. (Proposition 11) By Proposition 10, we can rewrite program (1.20) as, q(b, y, b') = min q q,d(.)E{,1} ,b"(-) subject to 1 - f d(y')dF(y' I y) q =1 + r (1 - d (y')) (7nd (b', y', b" (y')) - Vd (y')) > 0 (1.48) and 13 [d (y') Vd (y') + (1 - d (y')) Vnd (b', y', b" (y'))] dF (y' I y) - 3Vd(y) u (y) - u (y - b + b'q) (1.49) First, note that we can relax the constraint (1.48) and (1.49) by choosing b" (y') = argmax Vnd (b', y'I b>0 Second, define the set R (b') = {y' E Y : Vnd (b', y') Vd (y')} to be the set of income levels for which the government does not default, under the best continuation equilibrium. Note that, if y' V R (b'), it implies that no default is not equilibrium feasible for any continuation equilibrium (it comes from the fact that (1.48) is a necessary condition for no default). The minimization problem can now be written as q (b, y, b')= min q q,d(-)G{O,1 } subject to 1 - f d (y') dF (y' I y) 1+ r (1 - d (y')) [Vd (b', y') - Vd (y')] 0 for all y' E R (b') (1.50) d (y') = 1 for all y' R (b') (1.51) 3f [d (y') Vd (y') + (1 - d (y')) Vnd (b', y')] dF (y') - OVd(y) u (y) - u (y - b + b'q) As a preliminary step, we need to show that this problem has a non-empty feasible set. For that, choose the default rule that makes all constraints be less binding: i.e. d (Y') = 0 < Vnd (b', y') Vd ('/). This corresponds to the best equilibrium policy. If this policy is not feasible, then the feasible set is empty. Under this default policy, the one of the best equilibrium, the price q is equal to the best equilibrium price 48 q = j(y, b'). The feasible set is non-empty if and only if t3 [d (y') Vd (y') + (1 - d (y')) nd (b', y')] dF (y' I Y) - IVd(y) u (y) - u (y - b + b'4 (y, b')) u (y -- b + b'4 (y, b')) +,8W (b') ;> U (y) + BVd(y) = Vnd (b, y, b') Vd (y) where W (y, b') is the value of the option of defaulting b' bonds; this is the initial assumption of this propo- sition. Also, note that Vd(y) = [d (y') Vd (y') + (1 - d (y'))Vd (y')] dF (y' Iy) So, we can rewrite the promise keeping constraint as t3 (1 - d (y/)) [ind (b', y') - Vd (y')] dF (y') > u (y) - u (y - b + b'q) (1.52) We focus on a relaxed version of the problem. We will allow the default rule to be d (y') C [0, 1] for all y'. Given the state variables (b, y, b') the relaxed problem is a convex minimization program in the space (q, d (-)) C 0, x D (Y), where D (Y) = {d : Y -+ [0, 1] such that d (y') = 1 for all y' R (b')} is a convex set of default functions. Also, include the constraint for prices 1 - f d (y') dF (y' I y) 1+ r The intuition for this last constraint is that d (y') = 1 has to be feasible in the relaxed problem. The Lagrangian 1- d (y') dF (y' Iy) L(q,6(-))=q (-q+ 1-d I+r A (u (y) - u (y - b+ b'q) - 3 (1 - d (y')) nd (b', y') - Vd (y')] dF (y' Iy) The optimal default rule d (-) must minimize the Lagrangian L given the multipliers (p, A) (where p, A > 0). Notice that for y' E R (b') any d E [0, 1] is incentive constraint feasible, and 491C r +49d+ AO f7ynd[V-nd (b'(b', y')') _ V d (y')] dF (y' Iy) So, because it is a linear programming program, the solution is in the corners (and if it is not in the corners, it has the same value in the interior), then the values of y' such that the country does not default are given 49 by d (y') = 0 <-> A1A > (1.53) 3 (1 + r Note that A > 0 in the optimum. Suppose not; then d (y') = 1 for all y' C Y satisfies the IC and the price constraint. Then, the minimum price is 1 ~ 1 + r So, the minimizer will be zero, q = 0. But, this will not meet the promise keeping constraint. Formally, 3J V (y') dF (y' I y) - #Vd(y) - u (y) + u (y - b) = =,#g(Vd(y) - Vd(y)) + (y - b) - u (y) = u (y - b) - u (y) < 0 This implies A > 0. Note that, A > 0 implies that q (b, y, b') > 0. Define AO (1+ r) From (1.53) d (y') = 0 4-> And > - 4 Vn (b', y') > Vd (y) + 7 as we wanted to show. Aided with this characterization, from the promise keeping constraint we have an equation for -y as a function of the states [7nd (b/ y)_ Vd (y')] dF (y' I y) = u (y) - u (y - b + b'q) (1.54) JV-d(bI'y')>:Vd(y)+ y where Pr (Vnd (b', y') > Vd (y') + Y) (1.55) q= : 1 + r Define And(,') -- Vnd (b', y') - Vd (y') So, qft_ (A~d(y') -) 1+ r where - is the probability distribution of And(y'). The last step in the proof involves showing that the solution is well defined. Define the function G (-y) =/3 I n> AnddF (And I y) - u (y) + y - b+b (-I , ">' 1+ r First, note that G is weakly decreasing in y, that G (0) > 0 (from the assumption Vnd (b', y') - Vd (y') > 0) and lim-OO G (7) = u (y - b) - u (y) < 0. Second, note that G is right continuous in y. These two 50 observations imply that we can find a minimum -y : G (-y) > 0. If income is an absolutely continuous random variable, then G (-) is strictly decreasing and continuous, implying the existence of a unique y such that G (y) = 0. This determines the solution to the price minimization problem. 1.10 Appendix B: Characterization of U (b, q) Define the equilibrium value correspondence as vEJE001u = (c" (h'))} J E SPE (b) S (b)= (v, q) E R 2 : lo, { q = 1 (1 - f do (yo) dF (yo)) The set & (b) has the values and prices that can be obtained in a subgame perfect equilibrium. We need to find a policy that keeps the promise for prices, for one period. Enforceability. Take a bounded, compact valued correspondence W : R+ -4 R 2 . We will drop the dependence on d, and we will bear in mind that after default the government is not in the market. Definition 23. Given b > 0, a government strategy (d (.), b' (-)) is enforceable in W (b) if we can find a pair of functions v (y) and q (y) such that 1. (v (y), q (y)) C W (b'(y)) for all y E Y 2. For all y E Y, the policy (d (y) , b' (y)) solves the problem Vv(') q) (b, y) = max (1 u y - b + q (y) ] + 3v (y)} + d{u (y) + 3yd dE{0,1},b;>0(1dIII We will refer to the pair (v(), q (.)) as the enforcing values of policy (d (y) , b' (y)) and we will write (d (),b'(-)) E (W) (b). Definition 24. Given a correspondence W : R+ -: R2,we define the generating correspondence B (W) R+ -: R2 as B (W) (b) = (, q) E R2 : 3 (d(-),b'(-))EE(W)(b) : q= 1 (1-fd(y)) Definition 25. A correspondence W (-) is self-generating if for all b > 0 we have W (b) 9 B (W) (b). Theorem 26. Any bounded, self-generating correspondence gives equilibrium values: i.e. if W (b) C B (W) (b) for all b > 0, then W (b) C & (b). 51 Proof. The proof follows Abreu, Pearce, and Stacchetti (1990) and is constructive; we provide a sketch of the argument. Take any pair (v1, q-1) E W (b). We need to construct a subgame perfect equilibrium strategy profile o- c SPE (b). Since W (b) 9 B (W) (b) we know we can find functions (do (yo), b1 (yo)) and values (vo (yo) , qo (yo)) E W (b) for any b > 0 such that (do (yo) , b, (yo)) is in the argmax of vo(-),qo() (-) and V-1 Eo I Vvo(.),qo() (y, b) and -1= 1 r r1do(yo)dF(yo) Define o-g (ho) = (do (yo), fbi (yo)) and o-U (h) =qo where h_ = (bo, q-1). Because (vo (yo) , qo (yo)) E W (bi(yo)) and W is self-generating, we know that for any realization of yo, we can find policy functions (di (yi) , b 2 (yi)) and values (vi (yi) , qi (yl, b 2 (Yi))) E B (W) (b2 (yi)) such that (di (yi) , b2 (Y1)) is in the argmax of Vi (),qi( (.) and VO (Yo) = E (Vvi ()i (.)) , Orm (h'_) = qi (yi,b2) 1 (l fdi (yi)) Also define o, (h2) = (di (yi) , b2 (y1 )) is clear to see that strategy profiles om and o-r defined for all histories of type h' and h2 satisfy the first constraints of being a subgame perfect equilibrium. Doing it recursively for all finite t, we can then prove by induction (same as APS original proof) that this profile forms a SPE with initial values (vo, qo) as we stated. The finiteness of the value function is guaranteed because the set W is bounded. There are no one shot deviations by construction. E Theorem 27. The correspondence E (b) is the biggest correspondence (in the set order) that is a fixed point of B. That is, V (-) satisfies: B (E) (b) = E (b) (1.56) for all b > 0, and if another operator W (-) also satisfies condition 1.56, then W (b) 9 E (b) for all b > 0. Proof. Is sufficient to show that E (b) is itself self-generating. As in APS, we start with any strategy profile oU = (-g, um) and the values associated with it (vo, qo) with initial debt b. From the definition of SPE, we know 52 that the policy d, (yi) = dag (h', yi) and b' (yi) = b7' (h', yi) is implementable with functions q (yi,b) = q", (yi, d (yi) , b' (yi)) andy (yi, ) = V (o I h 2 (yi,)), whereh 2 (y, ) (hl, y, di (yi) , b' (yi) , q Moreover, because a is an SPE strategy profile, it means it also is a subgame perfect equilibria for the con- tinuation game starting with initial bonds b = 6, and hence (v (yi, 6) , q (yi, ) E V . This then means that (vo, qo) E B (V) (b), and hence V (-) is a self-generating correspondence. 0 Bang Bang Property. Now we are going to relate the APS characterization with the characterization in the main text .First, notice that the singleton set {(V, q)} = {(M , 0)} (corresponding to the autarky subgame perfect equilibria) is itself self-generating, and hence an equilibrium value. Let (v, q) = (V (b) , - (b)) denote the expected utility and debt price associated with the best equilibrium. Proposition 28. Let (d (-) .b' (-)) be an enforceable policy on V (b) (i.e. they are part of a subgame perfect equilibrium). Then, it can be enforced by the following continuation value functions: V (b' (y)) if d (y) = 0 and =b'(y) (1.57) Vd otherwise and q (YI il(b'(y)) if d (y) = 1 and = b' (y)W 0 otherwise Proof. Notice that the functions v (-) , q (-) satisfy the restriction (v (y, d) , q (y, E (d) for all 6. Since (d (-) , b' (-)) are enforceable, there exist functions (0 (.), (-)) such that for all y : d (y) = 0 we have u [y - b + 4 (y, b'(y)) b' (y)] + fj (y, b' (y)) ! u [y - b + 4 (y, ' +,3f) (y, b)(1.59) for all 6 > 0. Now, because the left hand side argument is an equilibrium value (since it is generated by an equilibrium policy), its value must be less than the best equilibrium value for the government, characterized by q = iq (b' (y)) and v = ?d(b'(y)) (that is, the best equilibrium from tomorrow on, starting at a debt value of 6 = b' (y). This means that Vnd (b, y, b' (y)) = u [y - b + ?J (y, b' (y)) b' (y)] + /V (b' (y)) > > u [y - b + 4 (y, b' (y)) b' (y)] + &b (y, b' (y)) .(1.60) 53 On the other side, we also have that autarky is the worst equilibrium value (since it coincides with the min-max payoff) which implies u [y - b + 4 (y,)b + & (y,b) u (y)+ Vd for allb O. (1.61) Combining 1.59 with the inequalities given in 1.60 and 1.61 we get u [y - b + -j (y, b' (y)) b' (y)] + /V (b' (y)) u (y) + IVd (1.62) which is the enforceability constraint (conditional on not defaulting) of the proposed functions (v, q) in equations 1.57 and 1.58. To finish the proof, we need to show that if it is indeed optimal to choose d (y) = 0 under the functions (iD (-) , 4 (-)), then it will also be so under functions (v (-) , q (.)). This is readily given by condition 1.62, since punishment of defaulting coincides with the value of deviating from bond issue rule b = b' (y). Hence, (v (-) , q (-)) also enforce (d (-) , b' (-)). L This proposition greatly simplifies the characterization of implementable policies. Remember the definitions of the objects vnd (b, y, b') =-u (y - b + ii(b') b') +#rV(b') as the expected lifetime utility under the best continuation equilibrium for any choice of debt b', and Vd (y) u (y)+_3 yd as the expected lifetime utility of autarky. Corollary 29. A policy (d ( Y),b' (-)) is enforceable on E (b) if and only if d (y) = 0 implies Vnd (b, y, b' (y)) > Vd (y). 1.11 Appendix C: Computing U (b, q) The function U (b, q) gives the highest expected utility that a government can obtain if they raised debt at price q and issued b bonds28 . This is the Pareto frontier in the set of equilibrium values. We now discuss how we compute U(b, q), which can be redefined using the equilibrium correspondence: U (b, q) := max {v : 34 > 0 such that (v, 4) E 8 (b) and 4 < q} (1.63) 2 8 Because this is the best equilibrium given a price 4 it does not depend on the amount of debt repaid; we are not characterizing equilibrium consistent outcomes. 54 Note that we focus in a relaxes version, where we replace the equality 4 = q by the inequality 4 5 q. We will show that this constraint is binding. The next three Lemmas characterize -U (b, q) and prove Proposition 16. Lemma 30 (Characterization of u). For all q E [0, ij (b)) the maximum continuation value ' (b, q) solves U(b,q)= max ] Ij (y) V (y)+ [1-6(y)]Vn (b, y)jI dF (y) (1.64) J(-)E[0,11 i subject to qs r 6 (y) dF (y)) (1.65) where the constraint 1.65 is always binding for all q > 0. Proof. Take an enforceable policy (5 (-), b' (-)) such that -1, (1 - f 3 (y) dF (y)) = q. By definition, there must exist functions (0 (y, b') , 4 (y, b')) E E (b') such that for all y (3 (y) , b' (y)) E argmax 5V (y) + (1 - 6) {u [y - b + 4 (y, b') b'] + 3,b (y, b')} (5,b') with the right hand side value (at the optimum) being the ex ante value of the policy. We show in Proposition 28 that (1) any enforceable policy can also be enforced by the "bang-bang values" 4 (y, b') = 4 (b' (y)) if b' = b' (y) b (y, b') = V (b' (y)) if b' = b' (y) and vd otherwise 0 otherwise and (2) the continuation value is maximized at this values, since 6 (Y) Vd (Y) + [1 - 6 (y)]{Ju[y - b + 4 (y, b' (y)) b'(y)] + Of) (y, b'(y))} < 5 (y) Vd (y) + [1 - 6 (y)] {u [y - b +4 (b' (y)) b'(y)] + /V [b'(y)]} = 5 (y) Vd (y) + [1 - 6 (y)] Vnd (b, y, b' (y)). (1.66) by def. Therefore, an enforceable policy (3 (.) , b' (-)) policy can generate (conditional on y) a value given by equation 1.66. Therefore, we can write the problem of finding the biggest continuation value consistent with a default price less than q as U(b, q) = max {6 (y) Vd (Y) + [ - 6 (y)] Vnd (b, y, b' (y))} dF (y) subject to the incentive constraint: Vnd (b, y, b'(y)) ;> Vd (y) for ally : 3 (y) = 0 55 and that its associated price is less than q: I 1 - f (y) dF (y)) Finally, notice that b'(y) only enters the problem through the term Vnd (b, y, b' (y)), and that making this object as large as possible makes both (1) the objective function bigger and (2) the constraints less binding (since it only enters through the incentive compatibility constraint). Therefore, we choose b' (y) to solve Vnd (b, y) = max Vd (b, y, b' (y)) b' >0 showing then the desired result. Finally, note that T (b, q) is weakly increasing in q, and that if we remove the price constraint, then the agent would choose the default rule to get price i4 (b) (the one associated with the best equilibrium), so for q < q(b) this constraint must be binding. E Remark 31. See that this is a linear programming problem in 6 (-), which we will see is easy to solve. If tractable, this Lemma will help us mapping the boundaries of the equilibrium correspondence S (b) for any given q. The following proposition solves the programming problem shown in Lemma 30, reducing it to solving a problem in one equation in one unknown. Lemma 32. Given(b, q) there exist a constant 7 =Y(b, q) such that T (b, q) = [ (y) Vd (y) + (1 - S (y)) Vnd (b, y)] dF (y) where 6(y0 < V/d(by);> Vd (y)+ for ally E Y and -y is the (maximum) solution to the single variable equation: 1 Pr y : vnd (b, y) > Vd (y) + } = q I1+ r I= Moreover, -y is also the Lagrange multiplier of constraint 1.65 in program 1.66, so that -) y (b, q). Proof. Using the Lagrangian in the relaxed program of letting 6 (y) E [0, 1] for all output levels for which no-default is feasible; i.e. for all y E D (b) {y Vnd (b, y) > Vd(y) . The Lagrangian (without the corner conditions for 6) is Sf [6 (y)V (y) + (1 -n (y))vd (b, y)] dF (y) + + fp (y) [1 - 6 (y)] [V-n (b, y) - Vd (y)] dF (y) + 56 +A (q(1+r) -I+ J (y)dF(y)) so that at a y : Vnd (y) > Vd (y) S=Vnd(b, y) + Vd (Y) + A) dF (y) =A(y) = [6d y d y) 1 otherwise Defining -y -- A we get the desired result, using the binding property of constraint for prices. E Lemma 33 (Concavity of 'U). The function U(b, q) = max {v : 3 < q such that (v, q) E & (b)} is concave in q. Proof. From Lemma 30 we know that the feasible set of the program in that Lemma is convex, having a linear objective function and an affine restriction. Take qo, qI E [0, ?(b)] and A G [0, 1]. We need to show that -u(b, Aqo+ (1- A) q1 ) ;> AU(b, qo) + (1- A) U(b, q1 ) Let G [6 (-)] = f [6 (y) Vd (y) + (1 - S (y)) Vn (b, y)j dF (y) be the objective function of the maximization in 1.64. Let So (y) be one of the solutions for the program when q = qo, and likewise 6 (y) be one of the solutions of the relaxed program when q = qI. Define 6 x (y) = A5o (y) + (1 - A) 61 (y) Clearly this is not a feasible default policy as it is, since 6 \ may be in (0, 1), but it is feasible in the relaxed program of Lemma 30. Note that it is feasible when q = q, := Aqo + (1 - A) qi, since (1- f 6,(y) dF (y)) = A -- f o (y) dF (y)) +. 1+r 1+ r + (1-A) (if so (y)dF(y)) Therefore, the optimal continuation value at q = q, must be greater than the objective function evaluated at 6,\. The reason is that the optimum will be at a corner even in the relaxed problem. Then 6 U(b, qx) G [6 (-)] AG [ o (-)] + (1 - A) G [61 ()J AU(b, qo) + (1- A) U (b, q1 ) (a) (b) using in (a) the fact that G [6 (-)] is an affine function in 6 (-) and in (b) the fact that both 6 o (-) and J1 (-) are the optimizers at qO and qi respectively. This concludes the proof. 57 1.12 Appendix D: Sunspot Proofs Proof of Proposition 17. Necessity (=->): Suppose there is an equilibrium strategy a such that h E R (o). This implies that the government decided optimally not to default at period t; i.e. [u (yt - bt + q' (h, () bt+i) + /V' (h, ()] d( ;> u (yt) +/d (1.67) Since o- is a SPE, we have that for all sunspot realizations ( E [0, 1] we must have (V' (h, () , q' (h, ()) E E (bt+1 ) using the self-generation characterization of (b). This further implies two things: 1. q' (h, () E [0, -q(bt+1)] (i.e. it delivers equilibrium prices) 2. V' (h, UT) (bt+ 1 , qo (h, ()) (because T is the maximum possible continuation value with price q q 0 (h,C) ) The price distribution given by o can be defined by a measure P over measurable sets A C R+ as S1 P (A) = 1 {q (h, () E A} d( = Pr {( : q" (h,C) E A} Note that numeral (1) shows that Supp (P) C [0, 4 (bt+ 1 )]. Changing integration variables in 1.67 and using the definitions above and properties (1) and (2), we get 1 (bt+ , 4)] dP (4) > J [u (yt - bt + bt+ 1 )+ 1 (yt) +)#Vd fn[u (yt - bt + q' (h, () bt+1) + OV' (h, ()] d( ;> U proving the desired result. Sufficiency (<-). Suppose that P is an equilibrium consistent distribution with cdf Fp. Let 0'* (b, q) E argmax V" (ho) s.t. qO < q aESPE(bt+i) i.e. it is a strategy that achieves the continuation value T (b, q). As we showed before, the constraint in this problem will be binding. Because h' is equilibrium consistent, we know there exist an equilibrium profile & such that h E W (&). For histories h' successors of histories ht+1 = (ht, dt, bt+ 1, (t, 4t) we define the profile 58 o as h od (h) if dt = 1, bt+l # bt+1 or dt V [0,4(bt+1)] a (h') ={.*(+, Ia* (bt+l I 4t) (h' ~ ht+1) otherwise and for histories h' = (ht , dt = 0, bt+, () let q" (ht, dt,bt+1, (t) =F ' ((t) where Fp (q) = P (4) is the cumulative distribution function of distribution P and F' ( )= inf {x E R: F (q) (} its inverse. It will be optimal to not default at t (if we follow strategy a for all successor nodes) if 1 f [u (yt - bt + Fi (() bt+1) + 8V' (bt+ 1 , ()] d( u (yt) + d. 0(a) f[u (yt - bt + dbt+1) +,3-v (bt+1, 4)] dP (4) ! U (yt) + # d (1.68) 7 using the classical result that FjA (() =d P if ( - Uniform [0, 1] and the fact that V (h') = V (o* (h')) = U (bt+ 1, qt) from the definition of a. Conditions 1.67 is satisfied, and Supp (P) ; [0, i (bt+ 1 )] imply that, if the government follows profile a, then h is also on the path of o, and a is indeed a Nash equilibrium at such histories (because both ad and o* (bt+1, 4) are subgame perfect profiles). Finally, for histories h' ) ht define a (h') = 6 (h'). Therefore, a (h') is itself a subgame perfect equilibrium profile (since it is a Nash equilibrium at every possible history) and generates h = (ht, dt = 0, bt+i) on its path. Proof of Corollary 18. This comes from the fact that the function U(P) = J{u (yt - bt + 4bt+1 ) + /3v (bt+l, 4)} dP (q) is strictly increasing in yt and strictly decreasing in bt, and the set can be rewritten as Q (bt, yt, bt+) =P E A ([0, ) U (P) Vd (yt) Proof of Corollary 19. The function H (q) := u (yt - bt + qbt+i) + #Uv (bt+1, q) is strictly increasing in q. Therefore, if P' > P and P E Q (bt, yt, bt+1 ) then f H (q) dP' > f H (q) dP > Vd (yt). Using Proposition 17 together with assumption (1) gives the result. l 59 It also has a greatest element, _. 1 if -4(bt+1) E A P(q CA) = 10 otherwise i.e. P is the Dirac measure over the best price q = Jq (bt+1). It also has an infimum, with respect to the first order stochastic dominance, given by the Lebesgue-Stjeljes measure associated with the cdf P (-) we characterize in section 3 below. However, this infimum distribution is not an equilibrium distribution. Proof of Proposition 20. We already know that max E (bt, yt, bt+j) = ?i (bt+1) since the Dirac distribution P over q = q (bt+1) is equilibrium feasible. In the same way, we also know that the Dirac distribution P that puts probability 1 to q = q (bt, yt, bt+ 1) is also equilibrium consistent; it corresponds to a case where both investors and the government ignore the realization of the correlated device, and the characterization of q () is exactly the only price that satisfies u (yt - bt + q (bt, yt, bt+1) bt+1) + 3v (bt+ 1 , q (bt, ye, bt+)) = Vd (y) and hence satisfies the conditions of Proposition 17. Lemma 33 shows that T (b, q) is a concave function in q, which together with the fact that u is strictly concave and b' > 0 implies that the function H (q) := u (yt - bt + qbt+1 ) + O (bt+1, q) is strictly concave in q. For any distribution P E Q (bt, yt, bt+ 1), let Ep (q) = f 4dP (4). Jensen's inequality then implies that u(yt - bt + Ep (q) bt+ 1 ) + 13-v (bt+1, Ep (q)) > [u(yt - bt + qbt+1 ) + rv (bt+1,4)] dP (4) > (1) > Vd(yt) (2) with strict inequality in (1) if P is not a Dirac distribution. Then, the definition of q(bt, yt, bt+ 1) implies that for any distribution P E Q (bt, yt, bt+1) we have Ep (q) > q(bt, yt, bt+i) and therefore the minimum expected value is exactly q (bt, yt, bt+1), which is achieved uniquely at the Dirac distribution P (because of strict concavity of u (-)). Finally, knowing that E is naturally a convex set, we 60 then get that E (bt, yt, bt+i) = min fddP() , max 4dP (4) 1PEQ(bt,yt,bt+1) f PEQ(bt,y,,bt+1) f = [q(bt,yt,bt+1) ,q(bt,yt,bt+i)] as we wanted to show. L Proof of Proposition 21. Upper bound for general 4 < q (bt, yt, bt+i). Here we replicate the same strategy: let p = Pr (( : q (C) 4). Using the same strategy as before, to get the less binding incentive compatibility constraint for the government we need to maximize equilibrium utility for ( : q (() 4 for ( : q (C) > 4, we consider equilibria that assign the best continuation equilibria (to make the incentive constraint of the government as flexible as possible). For (2) we just follow the same thing we did for the case where 4 = 0 and consider the continuation equilibria where q (() = - (bt+1) and v (C) = V (bt+1 ) (the fact that this corresponds to an actual equilibria is easy to check). For (1), we see that focusing on equilibria that have support q (() E {, - (bt+1 )} make the government incentive constraint as flexible as possible, since utility of the government is increasing in q and moreover, u (b, 4) (the biggest continuation utility consistent with q 4) is also increasing in q as we saw before. Therefore, if p is the maximum such probability, we must have p [u (yt - bt + 4bt+1 ) + /5u (bt, 4)] + (1 - p) V" (bt, yt, bt+i) > Vd (Yt) - Ad (bt, yt, bt+i) Vd (yt) - [u (yt - bt + dbt+1) + /8v (bt, 4)] + And (bt, yt, bt+l) See that this is not an innocuous constraint only when the right hand side is less than 1. This happens only when u (yt - bt + 4bt+1) + #Wv(bt, 4) ! Vd (yt) As we argued 4 > q (bt, yt, bt+i) where the last inequality comes from the characterization of q (bt, yt, bt+1). Proof of Corollary 22. P as defined in equation 1.36 cannot be an equilibrium consistent price: this im- plies that the Lebesgue-Stjeljes measure associated with P (-) has the property that Supp (P) = [0, q (bt, yt, bt+)] 61 and P (q = 0) = po > 0, which implies that - (.) ) + (bt+ , q (-)) J{u (yt - bt + dbt+ 1 ) + / (bt+ 1, 4)} dP (d) < u (yt bt + q bt+1 u 1 = Vd(yt) where the last equation comes from the definition of q () and the function H (4) =- u (yt - bt + qbt+1) + I5v (bt+1, 4) is strictly increasing in 4. L 1.13 Appendix E: A connection to Robust Bayesian Analysis We make a formal connection between equilibrium consistent outcomes and Robust Bayesian analysis. The main result is that, if the econometrician assumes the data generating process stems from a SPE of the game, then the set of equilibrium consistent outcomes essentially comprises the set of predictions a Bayesian econometrician can make, for any equilibrium Bayesian model (any prior over equilibrium outcomes). 1.13.1 Robust Bayesian Analysis Based in the principles of Robust Bayesian statistics (see Berger. et al. (1994)), we study the inferences that can be drawn from the observed data (a particular history h), which are not sensitive to the particular modeling assumptions (e.g., prior distribution chosen), across a given class of statistical models. Given that equilibrium multiplicity is a well-known problem of infinite horizon dynamic games, an econometrician must specify not only the physical environment for the economy, but also the equilibrium (or family of equilibria) on which they will focus their attention. Formally, the econometrician will try to draw inferences over: 1. Fundamental parameters 0 c E. These are parameters that fully describe the physical environment of the economy (examples are: The process for output F (yt I yt-1), the utility function u (ct), discount factor 13 E (0, 1), and international interest rate r. 2. Endogenous parameters a E A. These are parameters that given a physical description of the economy parametrize the stochastic process for the endogenous variables x (a) = (dt, bt+1, qt). These parameters comes from the equilibrium refinement (single valued or set valued) chosen by the econo- metrician. For example, in the Eaton and Gersovitz (1981) setting it amounts to the following. The process for income {yt}tEN can be an AR(1) process log yt = plog yt- 1 + Et 62 where yt is output, and et ~i.i.d N (0, o2). The utility function is u (c) = c 1 Y/ (1 - y) with y > 0. Hence, the fundamental parameters in this economy are The econometrician assumes that agents behave according to a particular rule, that relates exogenous vari- ables with endogenous variables. The literature of sovereign debt focuses in the best perfect Markov equilib- rium (with the restriction that after default there is a period in autarky). A special case is the equilibrium we covered in Section 1.2. So, xo (a) = best Markov equilibrium. Bayesian vs Frequentist. In a frequentist approach, parameters (0, a) are estimated (by calibration or some other statistical procedure) to best fit the observed historic data. In this section we will focus on the Bayesian approach where the econometrician (or outside observer) has a prior distribution for the parameters (0, a) and given data obtains a posterior. Our aim is to study inferences of Bayesian statistical models that hold any prior with support over equilibrium outcomes. Definition 34. A conditional model (me)Oee is a family of triples mo = {Ao, (a -+ xo (a) c X), Qo c A (Ao)} where Ao = (A0, EO) is the (measurable) space of process parametersa E AO; a -+ xe (a) is the mapping that assigns to every parameter a a particular stochastic process xo (a) for the variables (dt, bt+i, qt)teN given an exogenous process for yt; and Qo E A (Ac) is the Eo-measurable prior over a E Ae. w.l.o.g. we restrict attention to models where Qe is a full-support probability measure; i.e., supp (Qe) = Ac. 29 A conditional model me is parametric if Ae C Rkm (i.e., it has a finite-dimensional parameter space). Definition 35. A Bayesian model (or specification) is a pair m= {(mo)Oep ()} where (mo)oE is a conditional model and p (0) C A (9) is a prior over fundamental parameters. For the rest of this section, we will study Bayesian models conditional on a known fundamental parameter 0, fixing the physical environment (and hence dropping the dependence on 9). Once we condition on a particular value of the fundamental parameters, there is only uncertainty about the process followed by endogenous variables x. Given 9, one can map a particular model solely in terms of the probability distribution it implies over outcomes. Namely, given a conditional model m = {A, a -+ x (a), Q E A (A)}, we can define 2 9 1f this is not the case, the econometrician can work with an equivalent model, setting the parameter space to A0 = supp (Qo). 63 the implied measure over outcomes as Qm (B C X) = Q {a c A: x (a) E B} We will refer to Qm(.) as m's associated prior. Definition 36. m (with associated prior Qn) is a conditional equilibrium model if Qm (X E X (E8h)) 1 i.e., m assigns probability 1 to the process coming from a subgame perfect equilibrium profile. The class of conditional equilibrium models is written as Mg. Also, given an equilibrium consistent history h, we write M (h) {m : Q m (x (Elh)) = 1 and Qm (X (h)) > 0} i.e., the family of equilibrium models that assign positive probability to history h.30 1.13.2 Main Result In the following proposition we will study the inferences a Bayesian econometrician makes conditional on a given fundamental parameter 0. It states the main result of this section, showing that the set of equilibrium consistent outcome paths x (Elh) is essentially the union of all paths that have positive probability conditional on the observed history h, across all Bayesian equilibrium models. Proposition 37. Given an equilibrium consistent history h e W- (S) 1. The set of equilibrium consistent outcome paths satisfies: x (EIh) = {xcX::nmE M(h) andaC supp (Q(. I h)) such thatx=x(a)} (1.69) 2. For any measurable function T : X -+ R U fT(x(a))dQ(a I h)= chT (x(61h)) (1.70) mEME (h) Proof. See Subsection 1.13.4. 3 0 A more general definition for which the results of the next section hold is to ask that for X (h) to be in the support of QM . 64 Restrictions on support. First, note that (1.69) states that the outcomes that are equilibrium consistent after history h are the outcomes such that, there is a equilibrium conditional model that puts positive support on the parameters that maps into that outcome given the history. So, it formalizes the relation between a conditional equilibrium model and the set of equilibrium consistent outcomes given a history. Bounds on statistics. Second, note that (1.70) can be rewritten it in terms of the associated prior over outcomes Q U JT(x)dQm(x Ih)C inf T(x>, sup T(x) mE.ME(h) f E46jh) xEx(Ejh) with equality if ch T (x (61h)) is a closed set. Bayesian statisticians worry about the effect that the choice of the prior has for their inferences. To overcome this sensitivity, they choose a statistic T and report the interval of possible expected values of T under the posteriors in a family of priors f E F. For the case where T (x (91h)) is a compact set, we have that the set of all posterior expectations (conditional on h and 0) is identical to the interval [1 (h) , T (h)], where T(h) and T (h) are, respectively, the minimum and maximum values of the set {T (x) : x E X (Elh) }. The most important application of Proposition 37 is when we take yt and Tyt (x) = qf (yt). In this case, condition 1.70 helps us characterize the set of all expected values of bond prices qt across all equilibrium Bayesian models as: U f q (y) dQ (a I h)= , it mEMs (h) This is the interval characterized in Section 1.4. 1.13.3 Further Results In this section we study models that are based on small perturbations on equilibrium profiles. Our focus will be on "e- equilibrium models". Definition 38. Model m is an c-equilibrium model if Qm (x E x (9)) > 1 - e. For a given (non-equilibrium) model m, we define Q- (B C X) =Qm (B n x (E)) - QM (X (E)) as the equilibrium conditional prior. We will show that when e -+ 0, the posterior moments calculated with e- equilibrium models converge to the posterior means under their equilibrium conditional priors, and hence converge to elements in ch T (x (E1 h)). Proposition 39. Take an equilibrium history h and a family of models (m ),c(0,) (with a common parameter space) with associatedpriors (QE),E(o,1) such that 65 1. m, is an c-equilibrium model for all e E (0, 1) 2. There exist p > 0 such that for all e, Q, (X (h)) > p Then, for any bounded and measurable function T : X -+ R we have T(x) dQ, (x I h) - T (x) dQ-c (x I h) < ET T (1.71) where T = sup T (x) and T = inf T (x). This implies that as e -+ 0 T (x) dQE (x | h) - T (x) dQE (x I h) -*0 Proof. See Subsection 1.13.4. 0 Notice that for all e > 0, the prior Qf (-) is an equilibrium prior, since by construction it assigns probability one to the set of equilibrium consistent outcomes. Proposition 37 then implies that fT (x)dQ- (x I h) E ch T (x (Eh)) 1.13.4 Proofs Proof. (Proposition 37) Step 1. Showing the first statement (1). We first show if x E X (Elh) i.e. if x is equilibrium consistent at history h, we can construct an equilibrium model m and a in the conditional support such that x = x (a). We construct it as follows: the possible values for the parameter a are A = 1 }. The mapping is such that x (a = 1) = x. The measure Q is simply Q (a = 1) = 1. Since x E X (elh) we know there is an equilibrium o- that is consistent with x after h. Hence mx is an equilibrium model. Also, according to our model Pr (x E X (h)) = 1, and hence dQ (x I h) = Pr (x I h) = 1 > 0, finishing the proof. For the converse, take an equilibrium model m E .M such that a E supp (Q (- I h)) such that x = x (a). We will show that x E x (EIh). Using Bayes rule, the posterior distribution Q (a I h) after observing the history h dQ(ce)dQWd if h E W (x (a)) dQ (a I h) = - 0 if h N(x (a)) The prior was putting probability zero over non equilibrium outcomes, so the posterior has to be zero. This implies that a E supp (Q (- I h)) 4#=> h C N (x (a)) = W (oa) for some o-a C 8 1h (since m is an equilibrium model). Therefore x = x (cra) E X (Elh) finishing the proof. 66 Step 2. For (2), first define T := infxEx(,I) T (x) and T := supxEX(Eh) T (x). Take any equilibrium model m. Fix the history h. The expected value of T (-) under Q (- I h) is: EQ {T (xa) I h} = f T (x (a)) dQ (a I h) = T (x (a))dQ (a h) using in the second equality the definition of support, that was restricted without loss generality. Using equality 1.69 we know that for all a c supp (Q (- I h)) we have x (a) E X (Elh) and hence T < T (x (a)) ; T for all a C supp (Q (- I h)) Each of the inequalities are strict unless T E T (x (eh)) and T E T (x (61h)) respectively, showing that EQ {T (x (a)) I h} E [T, ] . We now need to show that it holds for every value in the convex hull. For any A E ch T (x (&1h)) there exist an equilibrium model mA such that EQO {T (x (a)) I h} = A. First, suppose A E (_T,7). If A E T (x (SIh)), we can specify model m as in the proof of (1) creating a model that assigns prob. 1 to x : T (x) = A. If not, we know there exist equilibrium outcomes x 1 , x2 C x (91h) and a number -y c (0, 1) such that A = yT (xi) + (1 - y) T (x 2 ) In this case, define mA with A = {1, 2}, with mapping a = 1 - X, and a = 2 -+ X2 and measure Q a = with prob. -y a = 2 with prob. 1 - -y is easy to check that EQA {T (x (a)) I h} = A. To finish the proof, we need to show the existence of such models on the cases when T C ch T (x (Elh)) and T E ch T (x (EIh)). In those cases, the construction from when A C (_T, T) applies. l Proof. (of Proposition 39) By Bayes rule: X (h)) QE (B|I h) Q(B n Q, (X (h)) which obviously implies that QE (X (h) I h) = 1 . Thus, to calculate EQ {T I h}, we can just integrate over X (h) C X to calculate the integral: T (x) dQ, (x I h) JT (x) dQE (x I h) = J T (x) dQ, (x I h) + IX(h)(xx()) As previously defined, x (eIh) = X (h) n x (E) is the set of equilibrium consistent outcomes with h, and denote 67 x (- E61h) := X (h) n (X ~ x (8)) as the outcomes consistent with h and not consistent with any subgame perfect strategy profile. Using these new definitions together with Bayes rule formula for QE (- h) we get (x) fT (x) dQ, (x I h) = i T (x) dQ (x) + T (x) dQ, (1.72) Jz~s,,)Q. (X (h)) fx(~Slh) Q, (X (h)) We now study the equilibrium conditional measure Q- (.). Applying Bayes rule and the definition of Q we get Q (B I h) :=. Q (B n X (h)) _ Q, (B n X (h) n x (8)) /Q, (x (8)) Q (X (h)) +-~--~ QE (X (h) n x (8)) /Q, (x (8)) by def. Q. (B n x (Elh)) QE (x (S)) and hence _Q, Qe(B n x (-61) I h) = .(B n x (Elh)) (x (-1h)) (B I h) (1.73) Q, (X (h)) Q,(X (h)) by def. It will be also useful to define the non-equilibrium conditional measure Q (B n (x ~ x ())) Q, (X ~ x (V)) for which we get, using Bayes rule: Q, (B n x(~ 86h) I h) = (X (- Eh)) Q (B I h) (1.74) Q (X (h)) Thus we can rewrite the conditional measure dQ, (x I h) as QE(X dQ (x Q, (X(h))~~ OEt h) if x E x (1h) dQ, (x I h) = if x E x (~ - (1.75) I (X (h))dQE - (x I h) 1h) 0 elsewhere Using 1.75, we then rewrite 1.72 as T (x) dQ,(xQE (, | h) Q (X(h))X (h)) T (x) dQ- (x I h)+ +Q, (X (~ E61h)) (x I h) Qe (X (h)) ST (x) dQ~' so that I T(x) dQ, (x h) - f T (x) dQ' (x I h) = 68 QE (x (Elh)) - QE (X (h)) dQ6 (x I h)+ QE (X (h)) fT(x) T(x) dQ~ (x I h) = Q, (X (hJ)) J Q, (X (--61h)) ( T (x) dQ~S (x I h) - T (x) dQ-6 (x I h) ) (1.76) IQ,(X (h)) (fJ\ EI4 using in the last equation the fact that Q, (X (h)) = Q (x (h)) +QE (x (-1 h)). See that since x (~ &Ih) C X - x (9), then Q, (x (- Elh)) 5 QE (X ~ X(1)) =- QE (x ()) < 6 using in the last inequality the fact that mE is an c-equilibrium model for all c E (0, 1). Also, because T is bounded, we get that =T-T Taking absolute values on both sides of 1.76, we get fT (x) dQ, (x I h) - fT (x) dQ6 (x I _QE (X (- -Elh)) h) < Q.E(X (h)) J(x) dQ (x I h) -f T (x) dQ-"(x T-T p for all E E (0, 1), proving the desired result. 69 Chapter 2 Illiquidity in Sovereign Debt Markets 2.1 Introduction The quantitative literature of sovereign debt studies business cycles in economies with endogenous spreads due to the risk of default. This literature has mainly focused on credit risk as the factor explaining spreads and debt capacity in sovereign nations.' However, the recent financial crisis in the US and the sovereign crisis in Europe have highlighted that there is substantial liquidity risk associated with sovereign lending. 2 Sovereign bonds are mostly traded in over-the-counter markets, where an investor who wants to sell a bond must search for a trading counterparty. While searching for a counterparty, this seller might incur in losses, and for this reason, investors need to be compensated to hold less liquid assets; this implies a higher risk premium. In this paper we study debt policy of emerging economies taking into account both credit and liquidity risk. To account for credit risk, we will study an incomplete markets model with limited commitment and ex- ogenous costs of default, as in Aguiar and Gopinath (2006), Arellano (2008), and Chatterjee and, Eyigungor (2012); default arises endogenously because of the relative costs and benefits of default. At the same time, we introduce search frictions in the market for Sovereign bonds, as in Duffie, Garleanu, and Pedersen (2005). At any given point in time an investor can receive a liquidity shock; in our model this means that the in- vestor now has a higher discount rate on payoffs until he sells the asset. Due to search friction, it takes time for the investor to find a counterparty. The time until he sells depends on the probability of finding a trading counterparty. The fact that some investors are liquidity constrained introduces a wedge between the 'This literature follows the setting in Eaton and Gersovitz (1981). See for example Arellano (2008), Aguiar and Gopinath (2006), Chatterjee and Eyigungor (2012), Hatchondo and Martinez (2009), Arellano and Ramanarayanan (2012), Yue (2010), Borri and Verdelhan (2009), Pouzo and Presno (2012), Bianchi, Hatchondo, and Martinez (2012); Aguiar and Amador (2013b) Section 6 provides a review of the literature. 2 Liquidity risk in sovereign debt markets has been recently documented by Pelizzon, Subrahmanyam, Tomio, and Uno (2013) and Bai, Julliard, and Yuan (2012). 70 valuations of the liquidity constrained and unconstrained investors. Intermediaries will exploit these wedges and bid ask spreads will emerge endogenously. In the model credit and liquidity risk are jointly determined. Because of the liquidity risk, the sovereign pays higher spreads today which affects the default decision. Therefore, liquidity affects default risk. At the same time, higher default risk feeds back into worse liquidity conditions because investors anticipate that the liquidity conditions will be worse during default. Therefore, default affects credit risk. As a consequence of this joint determination, the model enables us to quantify the relative contribution of credit and liquidity risk in sovereign spreads. To illustrate quantitatively the ability of the model to match key moments in the data, structurally decompose credit spreads, and resemble business cycles, we calibrate our model using data for Argentina. We find that introducing liquidity concerns does not harm the overall performance of the model in matching key moments of the data (mean debt to GDP, mean sovereign spread and volatility of sovereign spread). At the same time, the model endogenously generates liquidity spreads, that can match the ones for Argentinean bonds in the period of analysis. Regarding the structural decomposition, we find that the liquidity component can explain up to 50 percent of the sovereign spread during bad times; when the sovereign is not close to default, the liquidity component of spreads is negligible. Finally, regarding business cycle properties, the model matches key moments in the data. Literature Review. We build on the setting of the quantitative models of sovereign debt as in Aguiar and Gopinath (2006) and Arellano (2008); these two papers, extend the Eaton and Gersovitz (1981) framework of en- dogenous default to study business cycles in economies with risk of default. These early quantitative implementations study economies with short-term debt and no recovery on default. In our setting both long-term debt and recovery are crucial for the joint determination of credit and liquidity risk. Long- term debt was introduced by Hatchondo and Martinez (2009) and Arellano and Ramanarayanan (2012). Chatterjee and Eyigungor (2012) introduce randomization to guarantee convergence of the numerical al- gorithm and show the existence of an equilibrium pricing function. We follow this approach to modeling long-term debt. Endogenous recovery of defaulted debt was introduced by Yue (2010) by explicitly modeling the bargaining process between the sovereign and investors in the debt restructuring process. In our model recovery is exogenous.3 We build on the setting of over-the-counter markets first studied by Duffie, Garleanu, and Pedersen (2005).4 This framework was extended by Lagos and Rocheteau (2009) to allow for arbitrary asset holdings for in- 3 We abstract from this bargaining process because it is not crucial for our model. However, exogenous recovery implies the sovereign is willing to issue debt at low prices because he is certain that he will only repay a fraction; this behavior implies high mean and volatility of spreads. To rule out this behavior we introduce a reduced form cost of defaulting that depends on the level of debt. In a setting as in Yue (2010) this cost arises endogenously. 4 Another approach for secondary market frictions would be the adverse selection approach started by Kyle (1985), Glosten and Milgrom (1985), Back and Baruch (2004) Duffie and Lando (2001). 71 vestors. Lagos and Rocheteau (2007) studies the entry of dealers into the market. 5 Our paper structures the debt market as in Duffie, Garleanu, and Pedersen (2005) but to keep the model numerically tractable we follow He and Mildbrandt (2013) and we do not keep track of the asset holdings of high and low valuation investors. Our paper is closely related to He and Mildbrandt (2013), which extends the models of corporate default as in Leland and Toft (1996) by introducing an over-the-counter market as in Duffie, Garleanu, and Pedersen (2005). This, uncovers a joint determination of liquidity and credit risk. We are also closely related to Chen, Cui, He, and Milbradt (2014). Our paper extends the model of sovereign debt with long-term debt in- struments as in Chatterjee and Eyigungor (2012) to account for liquidity frictions as in Duffie, Garleanu, and Pedersen (2005). Despite the similarities, there is one crucial qualitative difference between the sovereign and corpo- rate settings. The value of default in our model is endogenously determined whereas in the corporate setting this value is fixed (does not depend future liquidity conditions) and is zero in most cases. Recent studies show that liquidity is a factor explaining sovereign spreads. Pelizzon, Subrahmanyam, Tomio, and Uno (2013) study market micro-structure using tick by tick data and document the strong non-linear relationship between changes in Italian sovereign risk and liquidity in the secondary bond market. Bai, Julliard, and Yuan (2012) find that most of the spread variations before the European sovereign debt crisis were due to liquidity and that most of the spreads were explained by credit risk in the onset of the crisis. Beber, Brandt, and Kavajecz (2009), on the contrary, show that for the Euro area, the majority of the spread is explained by credit risk.6 The evidence showing that liquidity is a factor explaining the spread of corporate bonds is more established. Longstaff, Mithal, and Neis (2005) use data of credit default swaps to measure the size of the default and non default component of credit spreads. They find that most of the spread is due to default risk and that the non default component is explained mostly by measures of bond illiquidity. Bao, Pan, and Wang (2011) show that there is a strong link between illiquidity and bond prices. Edwards, Harris, and Piwowar (2007) study transaction costs in OTC markets and find that transaction costs decrease significantly with transparency, trade size, and bond rating, and increase with maturity. Friewald, Jankowitsch, and Subrahmanyam (2012) liquidity effects account for approximately 14 per cent of the explained market-wide corporate yield spread changes. Chen, Lesmond, and Wei (2007) also find that liquidity is priced into corporate debt for a wide range of liquidity measures after controlling for common bond-specific, firm-specific, and macroeconomic variables. Layout. The paper is structured as follows. Section 2.2 describes the model environment and defines the equilibrium. Section 2.3 describes the calibration for Argentina and the numerical results. Section 2.4 5 There has been an extensive literature following Duffie, Garleanu, and Pedersen (2005). Some examples are Lagos, Rocheteau, and Weill (2011) which studies crises in over-the-counter markets; Afonso and Lagos (2014) which stud- ies high frequency trading in the market for federal funds; Atkenson, Eisfeldt, and Weill (2013) which studies the decisions of financial intermediaries to enter and exit an over-the-counter market. 6 Ashcraft and Duffie (2007) find evidence of trading frictions in the pricing of overnight loans in the federal funds market. Fleming (2002) finds evidence of liquidity effects in treasury markets. 72 concludes. 2.2 Model 2.2.1 Small Open Economy Time is discrete and denoted by t E {0, 1, 2,.}. The small open economy receives a stochastic stream of income denoted by yt. Income follows a first order Markov process P (yt+i = y/ I yt = y) = F(y', y) > 0. The government is benevolent and wants to maximize the utility of the households. To do this it trades bonds in the international bond market smoothing the households consumption. The household evaluates consumption streams according to o E 7#au(ct) 1t=0. The sovereign issues long-term debt7 . To simplify the maturity structure of debt, we follow Chatterjee and Eyigungor (2012)8. Each unit of outstanding debt will mature with probability m. If the unit does not mature, it pays a coupon z. The advantage of this formulation of debt is that it is memory-less; whether debt was issued 1 or n periods before, the probability that this debt will mature next period will be m. Therefore, the relevant state variable to measure the obligations of the government due in next period is the face value of debt. There is limited enforcement of debt. Therefore, the government will repay debts only if it is more convenient to do so. There are two consequences of default. First, the government looses access to the international credit market so it is effectively in autarky. It regains access next period with probability 0(b) 9 . Once the government regains access the face value of debt will be f b. Second, during default output is lower and given by y - <(y). There are two markets for debt. In the primary market, the government can sell bonds at a price qt. The price of debt will depend on next periods bond position and current income. Our convention is that bt+i > 0 denotes debt and bt+1 < 0 denotes savings. In the case of borrowing, after paying debt that matured this period mbt, and the coupon on outstanding debt (1 - m)zbt, the country increases its debt position to bt+1. The capital inflow that the country receives today is given by qt[bt+1 - (1 - m)bt]. The budget constraint for the economy is then Ct = yt - [m + (1 - m) z] bt + qt[bt+1 - (1 - m)bt] In the secondary market, government debt can be reselled. 7 We assume that there is a single type of bond in this economy. 8Long-term debt was introduced by Hatchondo and Martinez (2009) and Arellano and Ramanarayanan (2012); these two papers model long-term debt as consols. The approach is analogous. 9 The probability of re-entering the market will be an increasing function of the amount of debt the government has at the moment it defaults. This assumption is consistent with models of endogenous renegotiation as Yue (2010) and is important only for the quantitative performance of the model. We discuss this in detail in the calibration section. 73 2.2.2 Investors There are two types of investors (high valuation and low valuation) and two markets (primary and secondary). High valuation investors are risk neutral and discount payoffs at the rate ru. They are the only type of investors in the primary market buying debt from the government. An investor with high valuation receives an idiosyncratic liquidity shock that is un-insurable; with probability ( the investor will become liquidity constrained and his discount factor will now be rc, with rc > ru. Once a high valuation investor receives a liquidity shock he becomes a liquidity constrained (low valuation) investor and is a natural seller of the asset; he values the asset less than the high valuation investors. The liquidity constrained investor will sell the bond in the secondary market. As in Duffie, Garleanu, and Pedersen (2005), there is a search friction: a low valuation investor will meet a counter-party with probability A. Once a low valuation investor meets an intermediary and sells, she exits the market. We will denote the valuations of the international investors by qHDI qND for debt before default and qH, qL for debt in default for the high and low valuation investors respectively. 2.2.3 Intermediaries This section follows He and Mildbrandt (2013). There is a continuum of intermediaries (broker dealers) in perfect Bertrand competition holding no stock as in Duffie, Garleanu, and Pedersen (2007). The interme- diary buys from high valuation investors (H) and resells immediately to low valuation 'investors (L). The intermediaries contact low valuation investors with probability A. We will assume that there is a big mass of high valuation investors ready to buy in the primary market or in the secondary market. There is Nash bargaining between the intermediary and the investors. We assume that the bargaining power of the high and low valuation investors zero and a respectively. 10 Ask Price. The surplus for an intermediary that is trading with the high valuation investors is given by SH = A - M where A is the asking price at which they are buying from high valuation investors and M is the price at which the intermediary buys in the inter-dealer market. This surplus is zero because of Bertrand competition, the assumption that there is a high mass of high valuation investors, and the zero inventory restriction. Therefore, SH = 0 and this implies A = M. The surplus of the high valuation investors is (qf - A) - q', where qH denotes the valuation of the high valuation investor that has no bonds (where i E {D, ND}). Because they 10 This is natural because we assume that there are more high valuation investors than low valuation investors. These investors are ready to jump in and buy the bonds. The assumption that the number of type H investors is much higher than the number of type L investors is for tractability since it allows us to avoid keeping track of the distribution of asset holdings. 74 have no bargaining power, they have a surplus of zero. Also, qffO = 0, since the value of not having the asset is the claim on any future surplus; because this surplus is zero, the price is zero. Then A=M= q[ (2.1) Bid Price. Trading between the intermediary and the low valuation investor determines the selling price. The surplus for an intermediary trading with the low valuation investors is given by SL = M - B = q7 - B The surplus of the low valuation investors is given by (B - qfL) - q?. Because the low valuation investors exit the market once they sell, qjrO = 0. The total surplus (investors plus intermediary) is then qfy - qf. The bid price is such that the intermediary gets (1 - a) (from Nash bargaining) of the total surplus and is given by B = q, + a (qy - gi ) (2.2) Bid-Ask Spread. From (2.1) and (2.2) the bid ask spread will be A - B =(1-a)(q - q) 2.2.4 Timing In this subsection we spell out the timing of the model. Before Default. In period t, if the government is not in default, it starts the period with bt bonds outstanding. For these bonds the government will have to pay a coupon and pay principal as they mature. The total amount due in period t is [m + (1 - n) z] bt. Then, income yt is realized. After income is realized, the government decides whether to default or not dt E {0, 1}. If the government does not default, it issues [bt+1 - (1 - rn)btl debt in the primary market to the high valuation investors at a price qjD(yt, bt+1) . If the government decides to default, consumption this period is ct = yt - # (ye). The investors who started the period as low valuation investors will find an intermediary with probability A and will sell at a price qs2 e(yt, bt+1). Then, with probability ( the high valuation investors will receive a liquidity shock, so their effective discount rate will increase to rc from ru. If the government decides to default in period t, it will re-access the debt market in period t + 1 with debt f x bt. During Default. In period t, if the government is in default, it starts the period with current defaulted debt bt. Income yt is realized and consumption ct is given by yt - # (ye). Investors who started the period 75 qND (yt, bt+1) q"'(yt1 b t+1 ) Liq. shock I I I bt Yt dt E to, 1} y - O(y) qgiffe(yt, bt+ 1) Liq. shock 0(b) I I I I Figure 2-1: The figure summarizes the timing before and after default in period t. The government enters the period with bonds bt. Then income yt is realized and the government chooses whether to default. The upper branch depicts what happens when the government does not default. In this case, it issues debt in the primary market to the high valuation investors. The new face value of debt after the issue is bt+i. Then, liquidity constrained investors can sell their debt positions if they meet an intermediary. Finally the liquidity shock is realized. The lower branch depicts what happens in the case that the government defaults. In this case, consumption is equal to yt - (yt). Thus, liquidity constrained investors can sell their debt positions if they meet an intermediary. After this, the liquidity shock is realized. Finally, the government will re-access the debt market next period with probability 0(b). If the government is in default, the timing is depicted by the lower branch of the figure. as low valuation investors will find an intermediary with probability A and will sell at a price qSale(yt, bt+1). With probability ( the high valuation investors receive a liquidity shock, so their effective discount rate will be rc. With probability 0 the government will re access the international debt market in t + 1 with outstanding debt f x bt. Figure 2-1 summarizes the timing. 2.2.5 Decision Problem of the Government We represent the infinite horizon decision problem of the government as a recursive dynamic programming problem. The model has one endogenous state variable b and one exogenous state variable y. We focus on a Markov equilibrium with state variables (b, y). Value of the Option. The value of a government that has the option to default VND is the maximum between the values of defaulting on its debt and repayment. At a particular state (b, y) this value is given by VND(b, y) = max {VD (b,y),VC(b,y) {D,C} where VD (b, y) and VC (b, y) are the values of defaulting and repaying respectively. 76 Value of a Government in Default. The value of a government that defaults on its debt is VD(b, y) = u(y - 0(y)) + /Ey, [0(b)VND(f x b,y') + (1 - 0(b))VD(by)] The first term measures the flow utility: because the government defaults, the household consumes y - 0(y) instead of y. In the next period, with probability 6(b) the government will regain access to the international debt market with an outstanding debt of b. With probability (1 - 0(b)) it will remain in default. Value of Repayment. The value of a government that chooses to repay its debt is given by VC(b, y) = max {(1 - #)u(c) + #Ey, [VND(b', y)] where consumption is given by the budget constraint c = y - [m + (1 - m)z b + qgD(y, b') [b' - (1 - m)b] Default Sets. The default policy can be characterized by default and repayment sets. Let D(b) be the income levels such that the government prefers to default on its debt D(b) = {y C Y: VC(b, y) < VD(b,y)} When the borrower repays its debt, the policy function for debt issue is given by b' = b'(b, y) 2.2.6 Valuations of Debt: Before Default In this section we define the valuations of the high and low valuation investors before default. Suppose that the government has not decided to default in the state (b, y). High Valuation. The value of debt for the high valuation investors if the government wants to issue b - (1 - m)b so that total debt increases to b' is qgD(b', y) solves the following functional equation [z + (by) + (1 - D q b)y) = Ey, {D((b,((I -- d(b, y +))M(1 -m) ru +b ,y qD(b', y') + (1 - ()qg(b', y') +d(b', y') - 1 + ru (2.3) 77 The payoffs for the investor are as follows. If the government does not default on its debt in the next period, d (b', y') = 0, the investors will receive the fraction m of the debt that is maturing and the coupon on the remaining fraction (1 - m) given by z(1 - m). With probability ( in the next period they will receive a liquidity shock, so their remaining debt (1 - m) will have a value qND(b", y') for them. With probability (1 - () they will receive no liquidity shock and will value debt at qND(b", y'). Note that b" is the optimal policy for the government in the next period in the event they do not default. Should default occur, the government cannot borrow but keeps the defaulted debt b'. If the government does default on its debt in the next period, d (b', y') = 1, the investors will receive neither principal nor coupon payment; the debt will be valued qD' (b', y') and qD(b', y') if they receive a liquidity shock and if they do not, respectively. Note that, since these investors are not currently liquidity constrained, they discount at the rate ru. Liquidity Constrained. The price of debt for a liquidity constrained investor solves the following func- tional equation: L1/ / / m + (1 - rn) [z + (1 - A)qLD (b",y/) + 1qSale (by')] qD (byL E (- d(b , y')) NN ND (', y = Y1 + rL (1 - AD)qD (b', y) + AD q~alebI +d(b', y) D +rL / } (2.4) If the government does not default on its debt in the next period, d (y', b') = 0, investors will receive the fraction of debt m that is maturing and the coupon on the remaining fraction of debt (1 - n) given by z(1 - m). With probability A in the next period they will find an intermediary to trade their debt and will sell it at a price q~ale (y', b"). Otherwise, the investor will keep the debt and his valuation for it will be given by qND(y', b"). Again b" is the optimal policy for the government in the next period. The sale price is the outcome from the bargaining with the intermediary and is given by Sale / qND (b' y) = (1 - a)qND (b, y) + aqND(by) If the government does default on its debt next period, d (y', b') = 1, the investors will receive neither debt nor coupon payment; the debt will be valued qD(y', b") and q' (y', b") if they receive a liquidity shock and if they do not, respectively. The sale price in this case is Salerl L qD U(b', y) = (1 - a)q(b', y) + aq (b', y) Note that we assume that the probability of finding a counterparty to trade is lower when the investor is liquidity constrained. 78 High Valuation bt+l H ru t,ND zxbt Sovereign Intermediary M x bt qSale z x b tNID Low Valuation L rc qt,N1D Figure 2-2: The figure details the bond market if the sovereign is not in default and does not default in period t. It starts by issuing debt bt+ 1. This debt is bought by the high valuation investors in the primary market. After that, with probability A the low valuation investors will meet an intermediary. They will sell their bonds at the price q,ND--Sale aqt'NDH + (- ND. After selling their bonds they exit the market. The low valuation investors that do not meet an intermediary will try to sell their bonds next period. Then, with probability C, the high valuation investors will receive a liquidity shock. They will have the opportunity to sell the bond next period in the secondary market. Both the high and low valuation investors will receive the debt service m x bt and the coupon z x bt. 2.2.7 Valuations of Debt: After Default Suppose that the government decides to default or enters the period without market access, with current outstanding debt b, and income realization the income realization is y. High Valuation. The value of debt for the high valuation investors when the government is in default solves the following functional equation H 1 ~b qD(b, y) = 1+- ru E(b)E [(qH(b, y') + (1 - ()q' (b, y')] + (b)fq D(y, f x b) (2.5) With probability (1 - 0(b)) the default does not get resolved. Therefore, the value of the debt next period will be qD(y', b) and q' (y', b) if they receive or not the liquidity shock, respectively. With probability 0(b) default gets resolved and the investors receive a fraction f for every dollar of debt they have. They value this debt at qgD(y, f x b) given by (2.3). 79 High Valuation t,D rU Sovereign intermediary Sale bt I1D Low Valuation q%,D Figure 2-3: The figure details the bond market if the government is in default or defaults in period t. There is no debt issue or debt service. The sovereign has an outstanding balance of debt bt. The low valuation investors will meet an intermediary with probability A. They will sell their bonds at the price Sale H qt.D - aqtD + (1 - D)q'. After they sell the bond they exit the market. The low valuation investors that do not meet an intermediary will try to sell next period. Then, with probability (, the high valuation investors will receive a liquidity shock. They will have the opportunity to sell next period in the secondary market. Finally, with probability 0, the government resolves the default and re-accesses the next period with face value of debt bt x f. Liquidity Constrained. The value of debt for the low valuation investors when the government is in default solves the following functional equation qL (y, b) - ) EY [ADqD (b,y') + (1 AD)q(b Y'] 0(b)fqND (y, f X b) (2.6) 1+ rc With probability (1 - 0(b)) the default does not get resolved. With probability AD the liquidity constrained investors find an intermediary and they will sell the defaulted bond at a price q$ale(y', b) given by Sale~b)=( qD (by)=(1-aD)q(b,y) + aDq(b, y) With probability (1 - AD) they do not find an intermediary so they keep the unit of debt which they value it at qf(b, y'). With probability 0(b) the default gets resolved, they collect f for every unit of debt they had. Their valuation for this debt is qND(f x b, y) given by equation (2.4). 80 2.2.8 Equilibrium We focus in a Markov equilibrium with state variables (b, y). Definition 40. An equilibrium is a set of policy functions for consumption c(b, y), default d(b, y), and debt b'(b, y) such that: taking as given the bond valuation qND, the policy function for consumption c(b, y), debt issue b'(b, y) and the default set D(b), solve the borrowers optimization problem; the bond valuation functions qND (, y), qND (by )>7 D b ) '(,y satisfy (2.3) (2.4), (2.5) and (2.6) when default d(b', y') is consistent with D(b'). 2.2.9 Numerical Algorithm We follow a discrete state space method to solve for the equilibrium. As is discussed in Chatterjee and Eyigungor (2012) grid based methods11 have poor convergence properties when there is long-term debt. To overcome this problem, they propose a randomization procedure. We follow the prescription in Chatterjee and Eyigungor (2012) and compute a "slightly" perturbed version of the model described in this section. The details are given in the Numerical Appendix.Calibration We calibrate the model developed in the previous section to the case of Argentina. We choose to work with Argentina for two reasons. First, it makes the comparison with previous studies that focused on this case easy.1 Second, they had a recent episode of default with secondary market trading of debt. We will focus on the period of 1993:I and 2001:IV when Argentina had a fixed exchange rate with the dollar and was borrowing in international debt markets with the bonds traded in the secondary market. Preferences, Output. The utility function is CRRA u(c) =l-f. The endowment process follows ln yt = pln yt_1 + Ut with p E (0, 1) and ut ~ N(0, or). Default Costs. Following Chatterjee and Eyigungor (2012) the loss in terms of output during default is given by' 5 0(y) = max {0, doy + diy 2} 1 1An alternative would be to solve the model using Chebyshev polynomials as in Hatchondo and Martinez (2009). Hatchondo, Martinez, and Sapriza (2010) report the performance of the discrete state space techniques. 12 Examples are Chatterjee and Eyigungor (2012), Hatchondo and Martinez (2009) and Arellano (2008). 13 This is also the period analyzed in Arellano (2008), Chatterjee and Eyigungor (2012) and Hatchondo and Martinez (2009). 14 1n the Appendix we provide the details of the iid random shock to income Et ~ U[0, tmax] that is introduced to guarantee convergence of the numerical procedure. 15 As is explained in Chatterjee and Eyigungor (2012) this function nests several cases in the literature. In particular, when - do < 0 and dl > 0 the cost is zero when 0 < y -5 d, and rises more than proportional with output y > - djo. Alternatively, 81 The convexity of output costs is crucial to obtain spreads with, simultaneously, a high mean and a low volatility.16 We also introduce a functional form for the probability of reentering the international market after default. The functional form is 1 if b < 0 0(b)= 01b if 0 < b < 1.4 (2.7) 1 if 1.4 < b So, the expected time in autarky after a default is a linear function of defaulted debt, with a minimum of 8 quarters and a maximum of 40 quarters. We introduce a state dependent probability of reentry to associate default costs with the amount of debt that is defaulted. One of the differences in our paper from Chatterjee and Eyigungor (2012), Arellano (2008), Hatchondo and Martinez (2009), and Arellano and Ramanarayanan (2012) is that we introduce recovery in the case of default. Recovery is crucial for our mechanism. At the same time, it affects the incentives to borrow before default. In particular, the government finds it appeal- ing to borrow at high interest rates prior to default, because it effectively knows that will repay only a fraction of the face value. Therefore, in settings such as Chatterjee and Eyigungor (2012), Arellano (2008) , Hatchondo and Martinez (2009), and Arellano and Ramanarayanan (2012) introducing recovery increases the volatility of spreads because the government is borrowing at high interest rates prior to default. To disincentivize this behavior, we introduce a reduced form cost of default that depends on debt.1 7 We discuss the role of the debt dependent costs of default in the numerical results section. Parameters. With these functional forms, the model has 9 parameters that are standard in the literature of long-term debt: /, -y are preference parameters; Py, -,Y are the parameters for the process of output; Emax is the width of the support of the randomization variable; m, z rate at which debt matures and coupon rate; do, d, output costs parameters. Our paper introduces an over-the-counter market and endogenous time in autarky after default. So, we introduce additional parameters: ru, rc are the discount factor of the unconstrained and the constrained investor; C is the probability of receiving a liquidity shock; AND, AD are the probabilities of meeting a dealer in the case of default and not default; aND, aD are the bargaining powers of the intermediaries; f is the recovery rate the time in autarky function; 01 is the parameters in the probability of re-access. when do > 0 and di = 0 the cost is a linear function of output. The case studied in Arellano (2008) features consumption in default that is given by mean output if output is over the mean and equal to output if output is less than the mean. This implies a cost function 0'(y) = max{y - E(y), 0}, which closely resembles the case of do > 0 and di = 0. 16 The intuition is that during good times the probability of default is low because the costs of default are high and therefore spreads are high. Borrower's impatience implies that debt is built up during good times. However, during bad times, spreads increase quickly because the default costs decrease and the option of defaulting is more attractive. 17 This reduced form cost of default is qualitatively similar to what we would obtain with explicit micro-foundations, as in Yue (2010) and Bai and Zhang (2012). These two papers explicitly micro-found recovery after default by modeling the bargaining process between the government and the international lenders. Once the government defaults, it bargains with the international investors over a surplus generated by repayment. If the government re-accesses the debt market and agrees to repay a fraction of debt it is better off because they can smooth consumption; on the other hand, the international investors recover some of the principal they lent. The actual fraction that the government repays depends on the relative bargaining powers and the outside option. Once they agree, the government starts repaying debt and re-accesses the market when it payed all of the debt. 82 Preferences. Risk aversion -y is set to 2 and this is a standard value in the RBC literature and sovereign debt literature. Endowment. The parameters for output are estimated from linearly detrended data adjusted for season- ality of the real GDP of Argentina. The data is quarterly and the period is 1980:I and 2001:IV; the source is Neumeyer and Perri (2005). The estimated values are p. = 0.929 and or = 0.027. The width for the ran- domization parameter is set to be 6 max = 0.01 in the baseline model. In the computations, we approximate the AR(1) process with Rouwenhorst (1995) in 15 states for output. Discount rate. The discount rate of unconstrained investors is 1 percent, to match the risk free real quarterly return of the 3 month treasury bill in the period of study. Maturity. Regarding the parameters of debt maturity we match the average maturity and coupon informa- tion in Broner, Lorenzoni, and Schmukler (2013) as used in Chatterjee and Eyigungor (2012). The maturity m = is chosen to match the median maturity of Argentina's bonds that is equal to 20 quarters reported in Chatterjee and Eyigungor (2012). The coupon rate is set to z = 0.03 implying a coupon rate of 12 per- cent close to the 11 percent value weighted coupon rate for Argentina reported in Chatterjee and Eyigungor (2012). Recovery. We fix recovery f in 30 percent of face value following the target in Yue (2010). Market Reentry. We set (arbitrary) bmax - bmin at 1.4. This implies that if the country defaults with 140 percent of debt to GDP it will reenter with the lowest probability. In terms of the probabilities of reentering the financial market, there is a wide range of values used in the literature 18 ; we obtain Omax, 0 min from them. Beim and Calomiris (2001), report that for the 1982 default episode, Argentina spent until 1993 in a default state. For the 2001 default episode, Benjamin and Wright (2009) report that Argentina was in default starting in 2001 until 2005 when it settled with most of its bondholders. Chatterjee and Eyigungor (2012) fix 0 = 0.0385 and this implies an average exclusion period of 26 quarters or 6.5 years. We choose 0 max - Omin 0(b) = Omax - Omin b bmax - bmin such that the expected time in autarky ranges from 2 to 10 years. 8 1 As is discussed in Chatterjee and Eyigungor (2012), the definition of market access matters for these computations. Dias, Richmond, and Wang (2012) define a country as having normal market access whenever the country receives net re- source transfers of 1 percent of the GDP. Using this measure half of defaulting countries do not regain access until 7 years after a default. Gelos, Sahay, and Sandleris (2011) measure the period without market access as the period up until the country issues public and publicly guaranteed bonds or syndicated loans. Using this measure, the exclusion after the default in 1982 lasted only 4 years. 83 Matching Moments. So far, the parameters that remain to be calibrated are E = [3, do, di, AND, AD, ND, aD,rc ( where esd = (,8, do, dl) and E)"c = (AND, AD, ckND, O!D, rc, ). In the sovereign setting, as opposed to the corporate setting, calibrating eot" is challenging because of data availability. This is particularly the case for Argentina. He and Mildbrandt (2013) use turnover data to calibrate AD, aND and data from intermediaries 19 profits to calibrate aND, aD. Then, with data from bid ask spreads, rc, ( could be calibrated. So, E)"' is identified. For Argentinean bonds, there is no data on turnover and intermediaries profits; but, there is data on bid ask prices. So, we will rely on this for the calibration. We set the bargaining power aND, aD of the investors to zero; all of the gains from trade go to the intermediary. Second, the cost of a liquidity shock, given that an investor receives a liquidity shock, is pinned down by rc (discount factor of the constrained investor) and AND, AD (probabilities of meeting an intermediary). It can be shown that increasing rc is analogous to decreasing AND, AD. We fix rc = 0.2 arbitrarily and will use AND, AD to match moments. The probability of receiving a liquidity shock will be fixed in 0.25. So, the set of parameters that we will use to match moments are 0'= [13, do, di, AND, AD] These parameters will be chosen to match 5 moments: average debt to GDP ratio, mean and volatility of spreads, mean and volatility of bid-ask spreads [JE L] ,IF, [sprdt] ,o (sprdt) , E [bid - askt] ,o'(bid - askt) Target Yield and Bid Ask Spreads. For the target mean and volatility of spreads we use the series in Neumeyer and Perri (2005). Over the period 1993:1 and 2001:IV the mean and standard deviation of spreads was 0.0815 and 0.0443, respectively. The internal rate of return of bonds issued in the primary market is computed as qH(y, b/) = [m + (1 - m)z] / [m + r (y, b')]. The spread is then computed as (1+rH(y, b')) 4 - 1 minus (1 + rf) - 1. We will match this with the analogs in the data. We will use 100 basis points as target 20 bid ask spread. The model counter-parties are computed according to (qH - qL) /1 (qH _ qL 19 The data comes from Feldhutter (2011). 2 0 We are working in a more detailed analysis of the bid-ask spreads for Argentinean bonds. The three biggest issues (in terms of face value) defaulted in 2001 were issued in 2001, so there is only one year of data. For these biggest issues, the average bid ask spread was 95 basis points. This is comparable to the bid ask spreads of Corporate and Sovereign bonds of similar credit quality. Pelizzon, Subrahmanyam, Tomio, and Uno (2013) find that the bid ask spreads for European bonds have a median of 43 basis points and can rise up to 125 basis points (period June 2011 to November 2012). Chen, Cui, He, and Milbradt (2014) report bid ask spreads of 50 basis points during normal times for junk bonds and 218 during bad times. 84 Parameter Description Value 13 Sovereign's discount rate 0.954 -y Sovereign's risk aversion 2 Py Persistence of output 0.9485 o-, Volatility of output 0.0271 Emax Width of randomization parameter 0.01 m Rate at which debt matures 0.05 z Coupon rate 0.03 1 - 0 (b) Probability of reentry bmax - bmin = 1.4 do, di Output costs for default is {0, doy + diy2 } do= -0.18819, dl = 0.24558 ru Discount rate for unconstrained investors 0.01 rc Discount rate for constrained investors 0.02 Probability of getting a liquidity shock 0.25 A, AD Probability of meeting a market maker A = 0.8, AD = 0 O, aD Bargaining power 1 f Recovery rate for sovereign bonds 0.3 Table 2.1: Calibrated parameters Target Debt Capacity. For the target debt capacity we use the same average debt level as in Chatterjee and Eyigungor (2012) equal to 70 percent of the GDP. As is explained in Chatterjee and Eyigungor (2012), the database of World Bank development finance does not take into account coupon payments as debt because they only measure obligations at the face value. Therefore, the model analog of debt as reported in this database is just the face value of current obligations b. Summary of Parameter Calibration. The final parameter values can be found in Table .21 Model Moments. The results from our baseline calibration are summarized in Table 2.2. The last column lists the baseline results from Chatterjee and Eyigungor (2012) for comparison. Our baseline model generates mean debt to gdp of 54% which is below the empirical target of 100%. Despite having recovery, which intuitively should lessen default incentives and lead to better borrowing terms ex-ante, debt levels are lower in our setting. Part of this is due to liquidity effects, which raise the cost of borrowing: our model's sovereign spreads have a mean of 0.0975 with a volatility of 0.0519 both of which are higher than Chatterjee and Eyigungor (2012). Our model's mean bid-ask spread22 is 0.0136 in line with the data for Argentinean bonds. Note that the Bid-Ask spread is a quantitatively important component of borrowing costs. 2 1 In order to facilitate comparisons we have purposefully restricted most of our parameters to be equal to the ones in Chatterjee and Eyigungor (2012). In particular, the default costs are the same as in their paper. The only differences in the calibration are in (a) additional liquidity parameters, and (b) our specification of reentry probabilities which depend on the level of defaulted debt (see equation (2.7)). 22 Bid-ask spreads within the model are computed as (qH - qL) (qH + qL). According to our market structure assumptions, market makers buy at price qL and sell at price qH. The mid-quote is then - (q + qH). 85 Moment Data Model CE (2012), Table 3 mean debt to gdp 1.0 0.54 0.7 mean sovereign spread 0.0815 0.0975 0.0815 vol. sovereign spread 0.0443 0.0519 0.0443 mean bid-ask spread 0.0100 0.0136 - vol. bid-ask spread 0.002 0.0014 - Table 2.2: Model moments. 2.3 Numerical Results For the parameter values calibrated in Section 2.2.9, we structurally decompose credit spreads, we study business cycles, and we discuss our modeling assumptions. 2.3.1 Bond Prices, Bid-Ask Spreads, Decomposition Figure 2-4 plots model implied bond prices and bid-ask spreads as a function of output y and debt choice b'. First, panels A and C plot bond prices in the primary market qH (b', y) during the credit access and autarky regimes respectively. Note that standard comparative statics apply for bond prices; they are increasing in output and decreasing in debt. Also, they are always positive (event in autarky); this follows because the model features positive recovery upon default. Note that prices are much lower during autarky since recovery is set at 30% in the baseline calibration. Second, panel B plots bid-ask spreads during the credit access regime. Note that bid-ask spreads are small and flat when output is sufficiently high and default is not a concern, and rise as output falls and default becomes more likely. This is because prices are forward looking and take into account the possibility of worsening liquidity conditions for defaulted bonds. Finally, panel D plots bid-ask spreads for defaulted bonds. Because during default there are no bond issues, the state variable is the amount of debt in default b (and not debt choice b'). Note that bid-ask spreads are an increasing function of the level of debt in default. This is due to our assumption that the probability of reentry decreases for higher levels of defaulted debt and is a standard feature in sovereign debt models with renegotiation (see for example, Yue (2010), Bi (2008), and Benjamin and Wright (2009)). As in the credit access regime, bid-ask spreads are also higher when output is lower; this is due to default concerns after re-accessing credit markets. Liquidity Feedback. In Figure 2-4 we can observe that the liquidity-credit feedback loop highlighted in He and Mildbrandt (2013) for corporate bonds is also present in the sovereign setting. For example, from panel B, is clear that bid-ask spreads increase as the country nears default. On one hand, wider liquidity spreads traduce in higher ex-ante borrowing costs for the country. This in turn leads to increased debt rollover costs and increases default incentives. On the other hand, higher default risk implies that worse liquidity conditions are forecasted in the event of a default, because bid-ask spreads are higher during default. 86 A. Bond prices, credit access. B. Bid-ask spread, credit access 1.25-. Mo 0.75 50 11 1.2 0.5 1 0.5 1 bV 0 0.8 y b' 0 0.8 y C. Bond prices, autarky. D. Bid-ask spread, autarky. 0.4. 90,- 85. - 0.2 - -c 80'. 01- 75- -1.2 1.2 0.5 1 0.5 1 b' o 0-8 y b 0 0-8 y Figure 2-4: Bond prices and bid-ask spreads. This figure rlots bond prices at issue qH and bid-ask spreads which are defined as --. Panels A and B contains I lots during the credit access regime while Panels C and D contain plots for the autarky regime. These effects are nonlinear, in particular the feedback mechanism is stronger when output is low and/or when debt levels are high. Sovereign Spread Decomposition. One of the advantages of a model where liquidity and credit risk are jointly determined is that we can decompose spreads in credit and liquidity components. To investigate this in more detail, we follow He and Mildbrandt (2013) and decompose the total sovereign spread into default and liquidity components CS = CSD+CSL (2.8) 87 4A. spreads, total B. Default component C. Liquidity component 1511 10 0.8 5 0n 0.5 5 - - 0 0.6 - 0 0 0.4 .- 0 1 12 1 121 1.2 0 0.8 0 0.8 0 0.8 b' y b' y b y Figure 2-5: Sovereign spread decomposition. This figure decomposes total sovereign spreads CS into a default component CSDEF and a liquidity component CSLIQ- Panel A plots total spreads for bonds at the time of issue as a function of current output y and the amount of debt post issue b' Panels B and C respectively plot the default component and the liquidity component; both are expressed as a fraction of the total spread. where CS is the sovereign spread at the time of issuance2 3, CSD is the default component of the spread, and CSL is the liquidity component of the spread. The default component of the sovereign spread CSD is computed as follows. Take an individual investor without liquidity concerns operating in a marketplace that otherwise has liquidity concerns as a whole. That is, the bond price associated with CSD is still computed using equilibrium default and debt policies that take into account liquidity spreads, but discounting is done by an investor who faces no liquidity problems. The interpretation is that while there are liquidity concerns for the overall market (and the planner takes this into account in choosing debt and default policies), individual investors are heterogeneous and in particular there may be some investors without liquidity concerns who discount at the risk free rate. The liquidity component is just the residual CSL = CS - CSD- The above decomposition is plotted in Figure 2-5. Panel A plots the total sovereign spread CS as a function of current output y and debt choice b'. Panels B and C plot respectively, the default component CSD and the liquidity component CSL as a fraction of the total spread CS. Panel A highlights standard comparative statistics results for sovereign bonds: sovereign spreads increase during bad times (when output y is low) and when debt levels are high. Panels B and C show that when default risk is low (i.e. when output is high and/or debt levels are low) default risk is the predominant component, while the liquidity component becomes first order as overall default risk increases. For example, we see in Panel C that the fraction of the total sovereign spreads attributable to liquidity rises from around 0 to 50% as we move from right to left (i.e. from the low default risk region to the in default region). Even though the bonds are in default in the latter region, they nevertheless influence sovereign bond prices far away from default due to forward looking investors. These results are consistent with the feedback mechanism highlighted in He and Mildbrandt (2013). 2 3 More precisely, the annualized credit spread depends on current output y and the choice of debt b' so that CS = CS (b', y). For bonds at issue it is computed as CS(b', y) = 4 [m+(1-m)' +qND IV) - 1 - . I 'ND (b' ,y) r 88 Variable Data Model CE (2012), Table 4 0()/0(y) 1.09 1.01 1.11 a (') /u(y) 0.17 0.26 0.2 corr(c, y) 0.98 0.96 0.99 corr NX, y) -0.88 0.1 -0.44 corr(r - rf, y) -0.79 -0.46 -0.65 Debt service 0.053 0.044 0.055 Default frequency 0.125 0.023 0.068 Mean exclusion 4 years for 2001 2.9 6.5 (exog.) time (years) ARG default Table 2.3: Business cycle properties. Discussion. In the corporate setting (as in He and Mildbrandt (2013)) spreads can be decomposed in four terms CS = CSD,D + CSD,L + CSL,D + CSL,L (2.9) where CSD,D is a pure default component (default policies of a world without liquidity frictions), CSD,L is a liquidity induced credit component (how liquidity is changing default policies), CSL,D is a default induced liquidity component (calculated as a residual) and CSL,L is a pure liquidity component (abstracting from default risk, and only taking into account liquidity risk). All of these components are positive. Within the sovereign default setting there are additional complications that are not present within the corporate default setting: debt policy and the autarky continuation value are endogenous. A fixed debt policy is standard in a corporate setting; the rational is that the bond issue might have a covenant that restricts further issues, and this covenant is enforceable in a court. This assumption is usually for simplicity. Furthermore, in the corporate setting, the "autarky continuation value" for equity-holders corresponds to bankruptcy the value of which is usually exogenously fixed at zero (they optimally liquidate the firm when it has no value for them). On the contrary, in the sovereign debt setting debt policy responds to changes in liquidity conditions (because of changes in spreads) and the autarky continuation is endogenously determined. So, an increase in the liquidity friction might imply a decrease in credit risk, due to a different default policy; there is not guarantee that all the terms in (2.9) are positive.2 4 2.3.2 Business Cycle Properties The model's business cycle properties are summarized in Table 2.3. The second column lists the empirical moments in the data, while the last column lists the results from Chatterjee and Eyigungor (2012), for comparison. Qualitatively, the model performs well. As in the data, consumption is as volatile as output and nearly perfectly correlated with output. The volatility of the current account relative to output volatility is 0.26 in the model which is close to its empirical counterpart of 0.17. Sovereign spreads have a correlation 24 1n fact, in our simulations some of these terms are negative 89 of -0.46 with output. Also, qualitatively the model does a good job of capturing counter-cyclical sovereign credit risk although quantitatively the correlation still falls short of the empirical moment of -0.79. The model generates debt service (as a fraction of output) of 4.4% and a default frequency of 2.3%. While these two numbers are qualitatively correct, they are too low quantitatively. The reason being that the debt level is too low in the baseline model. As previously mentioned, if we allow for additional flexibility in choosing default costs so as to increase debt levels, then the quantitative performance of the model along these dimensions will also improve. Finally, note that the baseline model generates a positive correlation of 0.1 between the current account and output. This is an undesirable quality of the baseline model. However, the reason for this positive correlation is the functional form for exclusion times 0 (b) that for the current calibration is steep.25 Since defaulting with a high level of debt entails longer exclusion times, the optimal debt policy involves paying down debt during good times so that debt levels will on average be lower when defaults occur. For this reason, the current account and output are positively correlated in the model. Quantitatively, improvements can be made by decreasing the slope of 0 (b) so that the incentives for implementing the above mentioned debt reduction policies are not as strong. 2.3.3 State dependent time in autarky? To correctly capture the volatility of sovereign spreads, in our setting with recovery, it is crucial to impose additional costs of default that depend on the amount of debt defaulted. In particular, as we discussed before, we choose expected exclusion times that are increasing in the amount of defaulted debt (that is 0'(b) < 0). We now provide an intuition on why this helps in keeping volatility of spreads low. First consider a setting in which the reentry probability is constant and independent of the amount of debt in default. In this setting, Figure 2-6 compares default and debt policies for a model without recovery (i.e. f = 0) and one with 10 percent recovery (i.e. f = 0.1). Panels A and C respectively plot default policies for a model without recovery and one with recovery. Default occurs in the northwest region. The default threshold is "fuzzy" due to the randomization component e (see the appendix for details). Panels B and D respectively plot debt policy in the continuation region for the model without and with recovery; the plot fixed the randomization component at E = 0. In Panel B, we see that debt policy in the model without recovery is continuous. In contrast, we see in Panel D that debt policy in a model without recovery contains 26 a jump in the region where both output y and debt b is low. This jump involves the sovereign issuing debt at very low prices. The intuition for this jump in debt policy is as follows. In a setting with recovery, bond prices are always positive (see Figure 2-4), and as a result, the sovereign always has the option to issue additional debt to smooth consumption even if this means having to issue at extremely low prices. The discontinuity region 2 5 As we previously hinted, having this feature is nevertheless very important. 2 6 Note that this jump does violate the theorem of the maximum. This is because the numerical algorithm is discretized so that the choice set is not continuous. See app:numerical appendix for details for the numerical algorithm. 90 A. Default thresholds, no recovery. B. Debt policy, no recovery. 1.2 1 0.8 0.6 -. ....------.... 0.5 .....------0.4 - 0 1y 0.2 1 0 .141.6 0 0.8 1 1.2 1.4 1.6 b 0 0;8 y C.Default thresholds, 10% recovery. D Debt policy 10% recovery. 1.2. 1 0.8 0.6 i...... -- 0.4 ------1 1.6 0 0.5. 1.2 0.8 1 1.2 .4 1.6. b 0 0.8 Figure 2-6: Default and debt policy with constant reentry probabilities. This figure compares default and debt policies for a setting without recovery (f = 0) and one with a recovery rate of f = 10%. Reentry probabilities are constant and do not depend on the amount of defaulted debt. Panels A and C respectively plot default policies for a model without recovery and one with recovery. Default occurs in the northwest region. The default threshold is "fuzzy" due to the randomization component e (see the appendix for details). Panels B and D respectively plot debt policy in the continuation region for the model without and with recovery. The plot fixed the randomization component at e = 0. is one where the sovereign is precisely doing that. In fact, the level of debt is high enough such that in simulations default almost always occurs in the next period. This turns out to be optimal for the sovereign since upon reentering credit markets the country will only be responsible for the recovered amount of debt which is very small in comparison; in conjunction with delayed repayment, this high debt issuance strategy becomes attractive. In terms of the model, note that the benefit of issuing Ab > 0 units of debt is an increase consumption of Ac = Ab x qND (b + Ab, y). When bond prices are extremely low, a large amount of debt Ab must be issued in order to increase consumption by a small amount. This implies that Ab > Ac. Should default subsequently occur the country goes into autarky with additional debt Ab, the cost of which is having to repay an additional f Ab units of debt with certainty after regaining credit access (where f < 1 is the recovery rate). For low recovery rates (which is the case empirically), the marginal cost of default within the region where debt policy jumps is too low to discourage such behavior in the first place. The effect of discounting 91 B. Debt policy, 0(b) A. Default threshold, 0(b) 1.4 1.2 ...... 1 .5 - : 11 0 .8 -----.-.-.- 0.6...- - - .. ...- 0 .5 - - - - - 0 .4 - - -- 0 0.2- 1 0 0.5 1 . 0.8 0.9 1 1.1 1.2 1.3 yb 0 0.8 Figure 2-7: Default and debt policy with 0 (b). This figure plots default and debt policies for the baseline model which contains positive recovery rates and whose reentry probabilities are decreasing in the amount of defaulted debt (i.e. 0'(b) < 0). Panel A plots the default policy. Default occurs in the northwest region. The default threshold is "fuzzy" due to the randomization component e (see the appendix for details). Panel B plots debt policy in the continuation region (more precisely, the plot fixed the randomization component at E = 0). only further decreases marginal default costs within this reason. So, the sovereign has incentives to issue debt right before default at very high rates making spreads volatile. In simulations, the volatility of sovereign spreads are often many times larger than that of the mean of sovereign spreads. Furthermore, this is entirely attributable to the "jump" region for debt policy; in fact, if we truncate the period immediately prior to default in simulations (and hence ignore the "jump" part in debt policy), then the volatility of sovereign spreads will once again look reasonable. Second, Panel B plots debt policies when there is no recovery. There are no such jumps present when there is no recovery. Such policies of issuing a lot of debt at extremely high yields is obviously not feasible when bond prices are zero (and yields are infinite). As a result, the model without recovery can generate reasonable volatilities for sovereign spreads even when reentry probabilities are constant and independent of the amount of defaulted debt. Since having positive recovery is crucial for generating a feedback loop between liquidity risk and sovereign credit risk (as well as an important feature empirically), additional costs are required to rule out the above mentioned jumps in debt policy. When reentry probabilities depend on the amount of defaulted debt, issuing a lot debt at extremely high yields no longer becomes attractive (even when recovery rates are positive and bond prices are always positive). Such a debt policy implies high levels of debt upon default which is costly 92 due to extended exclusion times. To see this, note that defaulting after issuing Ab units of debt will decrease chance of regaining credit access by 0 (b) - 0 (b + Ab) ~-' (b) Ab or equivalently the average time spent in autarky increases by '(JbTA periods approximately. This additional cost is able to rule out jumps in debt policy.27 This is illustrated in Figure 2-7 which plots default and debt policies for the baseline model in which reentry probabilities decrease with the amount of debt in default. Notice that there are no longer any jumps regions in debt policy. By implication, this allows the model implied volatility of sovereign spreads to become reasonable. 2.4 Conclusion We studied debt policy of emerging economies taking into account credit and liquidity risk. To account for credit risk, we followed the quantitative literature of sovereign debt in studying an incomplete markets model with limited commitment and exogenous costs of default. To account for liquidity risk, we introduced search frictions in the market for sovereign bonds. By introducing liquidity risk in an otherwise standard model of sovereign debt, default and liquidity risk are now jointly determined. To illustrate quantitatively the ability of the model to match key moments in the data, structurally decompose credit spreads and resemble business cycles, we calibrated our model using data for Argentina. We found that introducing liquidity concerns does not harm the overall performance of the model in matching key moments of the data: mean debt to GDP, mean sovereign spread and volatility of sovereign spread. At the same time, the model generated endogenously liquidity spreads, which can match the ones for Argentinean bonds in the period of analysis. We also found that the liquidity component can explain up to 50 percent of the sovereign spread during bad times and the model matched key business cycle fluctuations data. 2.5 Numerical Appendix It is well known that numerical convergence is often a problem in discrete time sovereign debt models with long-term debt. To get around this problem, we adopt the randomization methods introduced in Chatterjee and Eyigungor (2012). Should the government choose to repay its debt, total output is given by yt + et where Et ~ Unif (0, max) is a small noise component that is iid across time. As shown in Chatterjee and Eyigungor (2012), this noise component et guarantees the existence of a solution of the pricing function equation. Qualitatively, it does not otherwise alter the model. A government that chooses to repay its debt will obtain VC (b, y, e) = max {(1 - /) u (c) + #Ey/, y [VND (b', y')] (2.10) 2 7 Another way of addressing this issue is to have upward adjustment costs for debt levels in order to rule out sudden large increases in debt issuance. 93 where the budget constraint is now given by c = y+E-b[m+(1-m)z+q'D(yb)[b'-(1-m)b] (2.11) which contains the randomization component e. Debt choice is denoted as b' (b, y, e). We impose that Et -= 0 during the autarky regime. The value to defaulting remains the same and is given by V' (b, y) = (I - 0) u (y - <0 (y)) +,#Ey, [OVND (f x b, y') + (I - 0) V,y ] (2.12) Note that the lack of choice variables in autarky means that randomization is not necessary for overall numerical convergence. The default decision is given by d (b, y, e) = 1JVC(b,y,e)>VD (b,y)} (2.13) and contains the randomization component. The continuation values are now adjusted as follows VND (b, y) = E, [max {VD (b, y), Vc (b, y e)}] (2.14) in order to take into account the randomization component. Finally, bond prices are adjusted accordingly so as to take into account the additional randomization variable: 1-d(b',y',E') z + qNi (bq(lLy) 1 I + Md+ ) D(b (b', y)', y') qP D (b', y) = i dEY,, (1 + (1 - () qD (b' (b', y', E') , y') II (2.15) 1_ru D,(,,,,) z + (1 - D () y/)D m++1 (- -) ) q) D (b', y) =Eye 1+rc [ +Aqse (b'D (b', y',e') ,y) I] (2.16) + (1 - AD) qD (b', y' A~q;ale I x b, y) (b, 1 - (b) E [q(b, y') + (1 - ()qE(b, y')] + 0 (b) fqHD(f (2.17) 6 q 1 (b,y E [AqSale(by) + (1 - AD)qL (b, y')] +0 (b) qTD(f x b,y) (2.18) + rc alL q VJD (b, y) (1 - a)qND(b, y) + aqND (b, Y) (2.19) q Da e(b, y) = (1 - aD)qD(b, y) + aDqD(b, y) (2.20) The rest of the numerical scheme is standard and follows the routine outlined in Chatterjee and Eyigungor (2012). We summarize the scheme in 4 steps: 1. Start by discretizing the state space. This involves choosing grids {yi}i' and {bj}Nb for output and debt. The grid points and transition probabilities for output is chosen in accordance with the 94 Rouwenhorst (1995) method. In the baseline model the number of states for output is chosen to be Ny = 15. The grid points for debt values are uniformly distributed over the range [0, bmal] with the upper limit bma, chosen large enough so as never to be binding in simulations. The baseline calibration has bma, = 1.4 and Nb = 41. In addition, the width for the randomization parameter is set to be emax = 0.01 in the baseline calibration. 2. Next perform value function iteration. Given bond prices, update value functions VC and VD. The debt and default policies b' (-) and d (-) are constructed using the algorithm outlined in Chatterjee and Eyigungor (2012). Where necessary, linear interpolation is used to obtain terms involving f x b. 3. Given debt and default policies, bond prices are then updated. 4. The above steps are iterated until both value functions and bond prices converge. 95 Chapter 3 Reputation and Debt Capacity 3.1 Introduction Models of sovereign borrowing are prone to multiple equilibria (see for example Calvo (1988), Cole and Kehoe (2000), Lorenzoni and Werning (2013), Passadore and Xandri (2015)). In some of these equilibria the gov- ernment has debt capacity, while in others, the government is effectively in autarky. The idea for a bad equilibrium is simple: if investors believe that the government will default on debts, and offer a price of zero, then the government has no incentive for repayment. Theorethically, given this equilibrium multiplicity, there is no reason why we should favor one equilibrium over another one. In reality, however, we rarely observe that countries are forever in autarky. In fact, quite the contrary; most countries are in some way connected to the international debt markets. For example, Cruces and Trebesch (2013) show that even countries that repeatedly default in their debts are only punished for a short period of time. Motivated by an apparent disconnect between theory and reality, I provide an argument on why we do not observe autarky as a solution, even though it is an equilibrium in the setting that I study. The benchmark setting follows Passadore and Xandri (2015). A small open economy borrows in the inter- national debt market to smooth consumption. The government lacks commitment. There is a fringe of risk neutral international investors that buy debt. The economy is variation of the setting in Eaton and Gersovitz (1981) with the additional assumption of no savings. In this game, it can be shown' that autarky is an equi- librium in addition to other equilibria where the government has positive debt capacity. I then introduce a perturbation of the benchmark model. There is an arbitrarily small probability that the government is a commitment type that always repays debt. I show that in this case the government can secure itself a payoff that is higher than autarky, across all equilibria. The idea, that is based on the 'See Passadore and Xandri (2015) for conditions to guarantee that there is a Markov equilibrium with positive debt capacity. 96 work of Fudenberg and Levine (1989) and Celentani and Pesendorfer (1996), is the following: if the rational government mimics the behavior of the commitment type, as the length of the history that is consistent with the commitment type increases, the probability that the international investors place on those actions also needs to increase. As this probability increases, the price of debt also increases. The rational government will then wait until good prices are back to reenter the market. Because this is a ex ante feasible strategy for the government, it can secure itself a payoff that is better than autarky. Literature Review. The literature on sovereign debt studies business cycles in economies where the government lacks commitment to repayment following the work of Eaton and Gersovitz (1981). Arellano (2008) and Aguiar and Gopinath (2006) are the first quantitative implementations. Hatchondo and Martinez (2009) and Chatterjee and Eyigungor (2012) introduce long term debt. The baseline model in this paper follows this tradition, but as in Passadore and Xandri (2015) and Chatterjee and Eyigungor (2012), I assume that the government cannot save to guarantee equilibrium multiplicity. The first papers to study reputation with commitment types are Fudenberg and Levine (1989) and Kreps and Wilson (1982). Fudenberg and Levine (1989) shows that if there is any probability of a commitment type that plays an action in a game the long lived player can secure itself a payoff by repeatedly playing that action. Kreps and Wilson (1982) examine Selten's chain store game and show that for arbitrarily small amounts of private information, the payoff of the entrant can be bounded away from the lowest equilibrium payoff. Celentani and Pesendorfer (1996) study a dynamic version of a setting that is similar Fudenberg and Levine (1989). The setting in this paper follows Fudenberg and Levine (1989) by introducing commitment types but as in Celentani and Pesendorfer (1996) the game is dynamic. Equilibrium multiplicity in sovereign debt markets has been widely studied. Calvo (1988) is the first paper to point out that expectations matter for debt repayment; the setting is static. Cole and Kehoe (2000) build on this setting and shows that in a dynamic stochastic model of sovereign debt there are situations under which there are multiple Markov equilibria. Lorenzoni and Werning (2013) study a dynamic version of Calvo (1988) and highlight the role of the market structure for multiplicity and the fact that sovereign crises do not need to occur immediately but build over time. This paper shares with these papers the focus on multiplicity, but studies a model as in Eaton and Gersovitz (1981). Another difference is that I focus on a bound across all equilibria. Layout. Section 3.2 presents the perfect information benchmark. Section 3.3 presents the main results in the paper. Section 3.4 concludes. 97 3.2 A Sovereign Borrower 3.2.1 Setup The perfect information setting is analogous to Passadore and Xandri (2015). Time is discrete and denoted by t E {0, 1, 2, .... }. A small open economy receives a stochastic stream of income denoted by yt. Income is iid. The government is benevolent and wants to maximize the utility of the households. To do this it trades bonds in the international bond market smoothing the households consumption. The household evaluates consumption streams according to 00 E #tu(ct) ,t=O . where 3 < 1 and u' > 0, u" < 0. The budget constraint of the economy is Ct = yt - (1 - dt)bt + qtbt+1 (3.1) Following Chatterjee and Eyigungor (2012) and Passadore and Xandri (2015) I assume that bt+1 > 0; the government cannot save.2 The government lacks commitment. Therefore, it will repay debts only if it is more convenient to do so. If the government defaults, the only cost is that it looses permanently access to the international credit market. 3 As shown in Passadore and Xandri (2015), the assumptions of no savings and no direct costs of default imply that autarky is an equilibrium in this setting. If there is positive debt capacity in the best equilibrium, then equilibrium multiplicity is guaranteed. Creditors. Investors are risk neutral and discount future payoffs at the rate r. They choose loans to maximize the expected profits 1 -5 4 = -qtbt+1 + 1 bt+ 1 + r where it is the endogenous probability of default in t + 1. The outside option of the investors is zero. Individual rationality and absence of arbitrage opportunities imply that the price of the bond is given by 1 - ir ge=1 +r 2 Amador (2013) provides a rational for this assumption: different political groups fighting for power might be introducing spending pressures and as a consequence the government does not save in equilibrium. 3I do not assume any further cost of default; I discuss at the end of this Section how direct costs of default imply that further restrictions on debt holdings have to be imposed to guarantee multiplicity. 98 So, the price of the bond is linked one to one to the perceived probability of default. For example, if the investors think that the government will default with probability one, then the price of debt will be zero. Timing. Every period t the government enters the debt market with an outstanding debt bt. Then income yt is realized. The government then has the option to default dt E {0, 1} on its debt bt. If the government decides to repay debt, it runs an auction with commitment and face value bt+i. After this the price of the bond qt is realized. Finally, consumption takes place; itis given by the budget constraint ct = yt - bt + qtbt+i- If the government decides to default on its debt bt consumption is equal to income, ct = yt. Consumption will be also equal to income if the government has ever defaulted in the past. I adopt the convention that if dt = 1 then dt, = 1 for all t' > t. Repeated Games Setting. A stage outcome is a tuple ht = (yt, dt, bt+i, qt, ct) where yt is the realization of the endowment; dt is the default decision; bt+i is the amount of debt that the government issues in period t; qt is the price of bonds; and ct is consumption by the households. Denote histories as ht := (ho,...., ht_ 1 ). A partial history will be a history plus some a subset of the stage outcome. For example, after the government announces a default and debt issue decision, the history is given by (ht , yt, dt, bt+i); and, this is the history where the market is choosing a price qt. The set of histories where the agent i is playing is denoted by Wi = U'= ot. The set of all partial histories will be denoted as W := Wg x 71m- Strategies. A strategy for the government will be a function og : $g -+ {0, 1} x R+ that specifies a default and debt issue decision at the beginning of each period. The government decision will usually be written as Og (ht, yt) = (d"g (ht, yt) , b'ig1 (ht, yt)) so that dt" () and b'- 1 (-) are the default decision and bond issuance decision for strategy o9 . A strategy for the market will be a function qm : Wm -+ [0, 1] that specifies a price for a given history. Eg is the set of all strategies for the government, and Em is the set of market pricing strategies. E = E9 x Em is the set of all strategy profiles. Payoffs. The payoff of a strategy ag for the government given a strategy qn for the market is given by V(ag, q, I h') = iE ,0 [dsu(y. - bs + qsbs+1) + (1 - d,)u(y,|)] I will use the convention that if consumption were to be negative given the strategy of the market, utility is minus infinity for the government; for example, if the government repays debts that are higher than income and the realized price is zero. Implicitly I assume that there is an infinitely costly technology such that the 99 government can obtain funds, so that the budget constraint is satisfied, consumption is zero, but utility is minus infinity. 3.2.2 Equilibrium Definition 41. A strategy profile o- = (og, q0) is a subgame perfect equilibrium (SPE) if and only if, for all partial histories h' E NHg V(o- h') > V(oiqm I ht) for all o E Eg, (3.2) and for all histories ht+1 = (ht, yt, dt, bt+i) E Jim qm (h+1) 1 (1 - d 9 (ht+, t+i)dF(yt+l I yt). (3.3) The set of all equilibrium strategies is denoted by E*. Aided with the definition of equilibrium strategies, I define the best and worst equilibrium prices and levels of welfare (ht+1) :=max qm (ht+1 ) ,V(ht 1 ) := max V (h+ 1 ) M OEE* - CEE* q(ht+1 ) min q. (ht+') ,V(ht') := min V (h+ 1 ) . M EE* M OEr,* Passadore and Xandri (2015) show that the best and worst equilibria are Markov, and that the worst equi- librium in terms of welfare is autarky, with a zero price of debt. The next proposition states these results. Proposition 42. Passadore and Xandri (2015). Denote by B the set of assets for the government. Under the assumption that the are no savings, i.e. B > 0, the lowest SPE price is equal to zero q(ht_+1 ) = q(bt+i) = 0 and lowest equilibrium value in terms of welfare is autarky E(h_+1 ) = Vaut(yt) The idea behind the proof this proposition is simple. Because the game is continuous at infinity it is sufficient to show that there are no profitable one shot deviations. If the government is defaulting after every history, then because investors price the bond according to the default rule of the government, the price of debt is zero. Furthermore, if for any history the price of debt is zero, by defaulting on its debts the government obtains a payoff of u(yt) + EyIytu(y'); 1-0 100 if the government deviates for one period and repays debt the payoff is u(yi - bt) + EY' YtU(y'). 1-# Because bt+i is non-negative, a one shot deviation of repayment is not profitable. Passadore and Xandri (2015) provide conditions to guarantee that in the best equilibrium there is a positive price of debt. These conditions are sufficient then to guarantee multiplicity and imply that consumption smoothing is valuable. Another approach to obtain a positive price for debt in the best equilibrium, the one followed by the quantitative literature of sovereign debt is to introduce direct costs of default. Note that in the latter case, the value of defaulting will be lower but the value of repaying stays the same. In order to make the deviations less appealing is sufficient to impose a lower bound on the amount of debt. 3.3 Government Uncertainty In the previous section I showed that in the worst equilibrium the government is effectively in autarky. In this Section I show that if there is a small probability that the government is a commitment type that always repays debts, the utility that the government obtains in any equilibrium is higher than the utility of autarky. 3.3.1 Preliminaries Suppose now that the government can be of two types 9 = { 9 c, 0,}; types are permanent. The government of type E) always repays its debts, dt = 0 for all t, and respects a debt ceiling b(y) = ao + aly. There is some prior probability p* > 0 that the government is of type O, . The government of type 0r, as in Section 3.2, repays debt only if it is more convenient to do so. I now define formally the set of committed governments 9 . Different commitment types will have different debt policies. Recall that Y is bounded. Denote by Y = {(yt)o : yt E Y} the set of all sequences of income realizations. An element in Y' is denoted as y. Define b(y) : b_}'O : b < b(yt), yt = y(t)} as the debt sequences such that given a sequence of output y they respect the debt ceiling. Denote by g = UyEYo6(y) the set of all policies that respect the debt ceiling. There will be one commitment type associated with each element in 9'. Denote as bi(y) an element in b(y). Index all of the elements in b(y) by i E I(y). Define 02(y) as the type that has debt policy bi(y). The set 9 is then defined as 9 0' := UyryoEc (y) where Ec(y) = UieI(y)Oi(y). 101 Debt Levels and Updating. Suppose that international investors observe a history h' = (...., yt, dt = 0, bt b(yt)). I will focus on the case where the investors will not make updates regarding debt levels as long as they meet the fiscal rule. Assuming that there is no updating is useful for two reasons. First, it simplifies the analysis; investors only update their beliefs when there is a default, and not with respect to debt levels. Second, even though investors are not sensible to debt levels, they are not totally insensible. One of the possible problems of using a commitment type that always repays debt is that debt might have always a positive price; that is limbI,,o q(b') > 0. Because of the fiscal rule, when the investors observe that b' -+ oo, or more precisely b > I(y), they will update their beliefs to the 0' type. 3.3.2 Beliefs and Prices I now characterize bounds on beliefs following Fudenberg and Levine (1989) and Celentani and Pesendorfer (1996). Recall that an outcome in period t is ht = (yt, dt, bt+i, qt) and a history is ht = (ho, hl, .... .ht- 1). I will focus on the histories h'+1 = (ht, yt, dt, bt+i) because these are the ones that the investors will use to price the bond. Define 3t := P ((dt+(yt+1) = O)y,,l I h+') as the probability that no default is played in every state of nature yt+i. Denote by h* the histories consistent with type 0'. The next Lemma characterizes an upper bound for the number of periods that the belief /3t can be lower than T. Define n(/3t < r) = #{t : #< < } as the number of periods such that the probability of no default 1t is below a certain threshold -k. Lemma 43. Suppose that p* > 0. Take any 0 < T < 1 and any strategy a for the 0r government. If a- is such that P(h* I Ec ) = 1, then P [n (#3t r) > _| h'] =0 log r Proof. See Appendix. E If investors consistently observe that the government is repaying debt, a strategy that is consistent with the commitment type, they need to eventually place a higher probability in the commitment type due to Bayesian updating. Furthermore, because of a higher probability of the commitment type, investors also place a higher probability in the action of the commitment type. Formally, after a history h'+1 , from Bayes rule, the posterior probability on the commitment type is given by 1 lp(PC ht+Ic ht)P(ht+ I ec, h) m P(h'm+1 I ht) 102 The second term in the numerator will be equal to one, because the commitment type will always repay debt. Note that the denominator is ft, because ht and hm+1 are histories consistent with no default so far, and in the next period. Thus, 0 _ P(E c I ht) P(Egc I ht+') 1 h)MP(h M P(ht+1 I htm Then, as long as 3t is below fr, we can bound the rate of increase in P(Ec I h+1) by -. Given the initial prior p* on the commitment type, there ae at most log periods such that the probability posterior on the commitment type can be less than ir. And finally note that 8t is weakly higher than this probability, so I also obtain a bound over &l. Aided with the evolution of beliefs, I can characterize the evolution of prices. Note that the price of debt in period t is given by 1 IE(l - d(yt+ ) I ht+ ) 1 - P ({dt+1 (yt+1 ) = I},| h+U) qt(ht' 1 M I+r 1+ r In the previous Lemma I found a bound on the number of periods such that the probability of no default given htm, P ({dt+1(yt+i) = 0 II hm+1), is below a constant ir. A direct application of Lemma 43 gives us a bound on the number of periods that the price can be below a particular value 7. Corollary 44. Suppose that p* > 0. Take any 0 < iTr < 1. Take any strategy for rationalgovernment o-. If o- is such that P(h* I E ) = 1. Then IP[n(qt <) > k(p*,) h*.= 0 Proof. See Appendix. The intuition is simple. As the number of periods with histories consistent with ec increases, the belief that (dt+l (yt+i) = 0 ),+, will be the policy also increases. And I can then use the bound on beliefs of Lemma 43, that implies a bound over prices. Formally, from the definition of ft, 1-#3t qt - Therefore, qt Bt +l <*+ q(1 +r). So, defining it = q(1 + r), and I obtain that P[n(qt )> 1 logA h* =0. log q(1 + r) 103 For convenience I just define p* k(Ip*,q) = log log q(1 + r) The bound is useful because it given a minimal price at which the debt that government issues will be priced. So, a government that is currently in autarky, after repaying some small amounts of debt, can guarantee it self a minimal price for debt. 3.3.3 A Bound On Welfare In this subsection I compute a lower bound in the utility that a government can achieve if there is a small prior probability I* over E). I start by defining some objects that will be useful for the bound. The value of a country that has currently bonds b, income y, confronts a price for debt q, follows a Markov strategy with state (b, y), and has the option to default is the maximum of the values of not defaulting and defaulting respectively, and is given by V0 (b, y, q) = max{V(b, y, q), Va"'(y)}. (3.4) The value of defaulting for this country is given by Vaut(y) = u(y) + /3EY,Vaut(YI). (3.5) This is the value of consuming its income and staying in autarky forever. The value of repaying debt, in case that the country confronts a price of q is Vc(b, y, q) = max u(y - b + qb') + 3#EY, V 0(b', y, q). (3.6) b' This is just the static payoff of issuing debt, plus the continuation value of having the option to default this debt next period. Note that the price q is a parameter and is not endogenously determined as in Section 3.2. Define V(o- I y, 0, [*) = min V(u- Iy, 0, A*) ar1-E*,1 as the minimum value that the government obtains in any Nash Equilibrium of the economy with private information, if it starts with current income y, zero debt, and prior probability p* over ec. The next proposition characterizes a bound on utility on this case. Proposition 45. For a given prior p* > 0, it holds that V(o I y, 0, t*) ;> max u(Y) + (1 Eu(y) + 6k(*,'!)EY Vo(0, y'i) bounded away from autarky for IL* > 0. 104 Proof. See Appendix. 0 The intuition of the proof is as follows. First, note that a country that has no reputation can borrow small amounts of debt and repay them. Note that the country is willing to do so, even though the interest rate might be very high, because it can always borrow an arbitrarily small amount, and the value functions are continuous. Second, note that as the investors observe that the country borrows and repays to them, investors update their beliefs. Third, note that at some point, the country will choose to enter the debt market following a Markov strategy, with a price that is weakly higher than q, for every level of debt that respects the ceiling. When the country will choose to enter the market will depend on the trade off between the number of periods the country will have to wait until it obtains a price of q (that is given by k(p*, q)), and the value at which it will reenter V'(0, y', q). The country will choose q optimally. The payoff of this strategy is given by [ (1-IF(*'3 max u(y) + U(y) + jk(y*,q)EY, V(0, Y', )] 4 1 - 1 On the first period it obtains utility arbitrarily close to income. For k(p*, ,) periods the country is obtaining also utility that is arbitrarily close to autarky, even though prices started to improve. After k(p*, q) periods, the country is following a Markov strategy, and obtaining at least V1(0, y', q). Note that the strategy played by the country need not to be optimal to provide a bound, just feasible; by following that strategy the sovereign can guarantee itself the payoff that I just characterized. Comments. First, note that as p* -+ 0 it holds that V(- I y, 0, t) -+ Va"(y). This is also the case for Fudenberg and Levine (1989) where if the prior probability is close to zero the payoff will not be "close" to the Stackelbergs payoff and will be close to the worst static equilibrium. Second, note that there are equilibria that for any p* > 0, there will be a utility that is higher than autarky. The equilibria could constructed in a similar way as cooperation in the Centipede Game. Examples are Kreps and Wilson (1982) and Kreps, Milgrom, Roberts, and Wilson (1982). 3.4 Conclusion Sovereign debt markets are prone to equilibrium multiplicity. In the worst equilibrium the government is effectively in autarky. Even though there are periods that resemble the autarkic solution, we do not observe that sovereign countries are consistently in that state. In this paper I provided an argument for why we do not observe the autarkic solution. Following the tradition of Fudenberg and Levine (1989) and Celentani and Pesendorfer (1996) I characterized a bound on the utility that the government can obtain if there is any initial probability that the government is a commitment type that always repays debt. 105 3.5 Appendix: Omitted Lemmas and Proofs Proof. The result is a direct application of Fudenberg and Levine (1992) and Celentani and Pesendorfer (1996). Recall that hm+1 = (ht, yt, dt, bt+i) and h' = (ht-1, yt_1, dt-1, b). By assumption, as long as bt+i bt+i, there is no updating. Suppose that both histories ht+1 and h, are consistent with a commitment type in ec. Bayes law implies that 1 - P(e'c Iht )P(ht- I EY, ht) P( c I ht+') ( c M) (3.7) M P(hi,+1 htm Note first that P(h,+1 I ht.) = IP(dt(yt) 0 1 ht) (3.8) because the market is not obtaining information from the price that they charge and there is no updating with respect to the debt levels. Note also that P(h+1 I e, h,) = JP(dt(yt) = 0 1ec, ht) (3.9) = 1 where we used the assumption of iid income. Therefore, from plugging (3.8) and (3.9) in (3.7) we obtain that P(Oc I ht+1 ) P(ec I hM) M 1P(d(yt) = 0 1 ht) Because P(dt(yt) = 0 1 ht) E (0, 1), we know that P(Ec I ht+l) is non decreasing. But also note that, when P ({dt(yt) = 0}, I ht) = i -r, P(9C I ht+1 ) increases at least by a factor of 1/-r. Given that P(WC) = > 0 we conclude that P n(t 5 fr) > | h-* =0 log7r J because the probability that dt = 0 is played is as least as high as the probability of the commitment types. 0 Proof. The next Lemma is a preliminary result that is needed for the proof. It states that is better for the country to get higher prices for debt. Lemma 46. The value function V0 (b, y, q) is weakly increasing in q. Proof. The proof of this lemma comes from the fact that the value function VC(b, y, q) is weakly increasing in q and from the fact that there is not reentry to the market after a default. If there was reentry, then it would be not obvious the result, because both of the value functions would be increasing. E 106 Consider the strategy of playing d = 0 borrowing an initial arbitrarily small amount of debt, repaying debt and interest rate for k(p*, q) periods, and then switching to a Markov equilibrium. Choose initial debt at c. For the first periods the government will be approximately obtaining the utility of autarky. At period k(p*, q), the price of debt will be at least q with probability one. Also, the amount of debt will be at most (1 k(* , E. This strategy has a payoff of at least '6 X -1 y,) (1 - , Ok~ ,gEyt,.,,V" , y(9,g, +u(y) + 1-6 y 0 where q* = . Because c is arbitrary, and V is continuous in the amount of debt lim V, C X7/1 - , yk(l,,), 1) -4 V" (0, yk(p*,4) We can always find a value of E , yk(A,q,), > Vau (yep,qg)) q* So, we just showed that the strategy has the same payoff as autarky for the first k(p*, q) periods and higher than autarky in the tail. Note that we can choose q optimally. Note that k(,i*, q) is increasing in q because d log /* _- + log p* dq log [q(1+ r)] (log [q(1 + r)])2 and because y is iid lEyk(*q) V(, yk(p*,q), ) = EyV' ( 0, y, j) As we said before, we can show that EYV'(0, y, q) is increasing weakly in q. Therefore, there is an optimal price that this bound is defining. 0 107 Chapter 4 Ambiguity, Insurance, and Lack of Commitment 4.1 Introduction There is substantial evidence of imperfect insurance in developing (e.g. Rosenzweig (1988), Townsend (1994), Udry (1994), Kinnan (2011), Morten (2013), among others) as well as in developed economies. To ex- plain this lack of perfect insurance one of the frictions that has been studied (e.g. Thomas and Worrall (1988a), Kocherlakota (1996), Ligon, Thomas, and Worrall (2002)) and received support in the data (e.g. Ligon, Thomas, and Worrall (2002), Karaivanov and Townsend (2014)) is lack of commitment of the agent regarding future transfers. At the same time, there is also substantial evidence of deviations from expected utility; one of the deviations that has received support in the data (e.g. Bryan (2010), Barham, Chavas, Fitz, Salas, (2014), Engle Warnick, Escobal, and Laszlo (2011)) is ambiguity aversion. Motivated by these two observations we study the design of a risk sharing agreement between a risk neutral money lender and an agent that is ambiguity averse and cannot commit to future transfers. To model the risk sharing agreement we follow Thomas and Worrall (1988a). The agent receives a stochastic stream of income every period. Because the agent dislikes risk and uncertainty she can enter a risk sharing agreement with a money lender to smooth his consumption. In this agreement she transfers his income every period and receives a transfer in exchange. The main friction is that the agent cannot commit to future transfers: she will transfer his endowment only if it is more convenient for him to do so. At the same time, the agent is uncertain about the process of his endowment; we model this by endowing the agent with multiplier preferences as in Hansen and Sargent (2008). Each period, the agent thinks that nature will distort, outcomes so she thinks that bad outcomes are more likely. 108 Consider first the setting in Thomas and Worrall (1988a) where the agent trusts the process of his income but cannot commit to future transfers. Current consumption and the promises of future consumption need to be such that the agent prefers staying in the agreement than walking out and consuming his endowment forever. The agent receives increases on consumption each time she receives a sufficiently high endowment shock; just enough that she stays in the contract. The principal finds optimal to defer the consumption increases because increasing promises of future consumption reduces the temptation of walking out in the future, without actually having to hand in consumption. When the agent receives the highest possible endowment shock, she receives full insurance. When the agent in uncertain about the process of her endowment, the contract changes qualitatively. Con- sumption not only increases every time the agent receives a high endowment shock but also increases over time. The intuition of this result is as follows. The fact that the agent does not trust his endowment process implies that she thinks that bad states are more likely. At the same time, there is monotonicity in terms of how much consumption the agent receives with respect to the endowment shock. The states such that consumption would not increase are the low states. But, because the agent distrusts his model, now consumption needs to increase even in the low endowment realization states. The main contributions of this paper are two. First, we show that introducing ambiguity aversion does not affect tractability of the contracting problem. This, implies that the setting could be readily applied to study quantitatively, for example, what is the price that agents are willing to pay to eliminate ambiguity. Second, that deviations from expected utility change qualitatively the path of efficient allocations over time. Literature Review. This paper studies the optimal risk sharing agreement between a risk neutral prin- cipal that can commit to transfers and an ambiguity averse agent that cannot commit to them. The risk sharing environment is analogous to Thomas and Worrall (1988a). The two sided limited commitment case was studied by Kocherlakota (1996). The latter has been applied to understand, for example, consumption insurance Ligon, Thomas, and Worrall (2002), international business cycles Kehoe and Perri (2002b), and sovereign debt Kletzer and Wright (2000). Our main deviation from this literature is to introduce model uncertainty. To model ambiguity aversion we endow the agent with multiplier preferences following Hansen and Sargent (2008). Multiplier preferences have been applied to study optimal monetary policy Orlik and Presno (2011), sovereign default Pouzo and Presno (2011), and optimal taxation Karantounias (2013). To the best of our knowledge there is almost no work in contracting with multiplier preferences. This paper belongs to a recent but growing literature of contracting with ambiguity averse agents. Most of this literature focuses on max-min objectives. Carroll (2013) studied a principal agent problem where the principal is uncertain about the technology of the agent. This paper shows that if the principal has a max min criterion to rank alternatives, the optimal contract is linear. Starting with the work of Bergemann and Morris 109 (2 005a) there is also substantial literature on mechanism design when there is uncertainty regarding higher order beliefs. 4.2 Setting Environment Time is discrete and indexed by t E {0, 1, 2...}. Let S = {si, s2 , ... , SN} be the set of states of the world. Denote st as the history of exogenous states (so, 8s, .... st). The probability that each state occurs is 7r(si) > 0 and the realization of states is i.i.d. over time. The probability of a particular history of states is 7r(st) = lh= lr(sh). There are two agents, one risk neutral money lender and one ambiguity averse agent. The agent receives an endowment of the unique final good every period; this realization is public information. The agent cannot commit to future transfers or store the good for consumption in other periods. Timing. The timing is as follows. At each period t, when the realized state is st, the agent receives an endowment y(st). Then she decides whether she transfers his endowment or not rt C {0, 1}. If she does (rt = 1), she receives a transfer from the money lender ct. A stage outcome is defined as ht = (yj, Tt, ct). A history is a concatenation of outcomes ht = (ho .... , ht- 1). The set of all histories is H. The set of all partial histories is H'. Strategies. A strategy for the agent is T(ht ,y) HT x V - {0, 1}, a function that assigns for every history a decision to transfer his endowment or not. For the money lender, a strategy is c(ht, yt, rt) : Ht x Y x {0, 1} -+ R+ a function that assigns for every history a transfer to the agent. Note that strategies induce outcomes in the usual way. A contract is a sequence {c(ht, yt,t)}"0 that specifies the consumption of the agent for every possible history when she is playing. 4.2.1 Agent We now define the utility of the agent at each particular history. In order to do so we first introduce probability distortions. Second, we show how the agent ranks consumption streams. Distortions via Martingales The agent is uncertain about the model. She has reference probability measure 7r(ht) but she thinks that nature is going to distort it to F(ht)l. To model these distortions we 1 Note that the measure 7r(ht) is induced from the measure over exogenous states and a particular strategy. In particular, given the measure over exogenous states and a pair of strategies, a measure is induced over histories. We abuse notation slightly and use 7r(ht) for the probability of the history instead of 7r'AUM (ht). Also, the agent in not uncertain regarding the mapping from histories to consumption. she is only uncertain about the exogenous state. 110 introduce a change of measure variable M(ht) = F(ht)/7r(ht). For a generic random variable X(ht) the expectation with respect to the distorted probability measure is given by IEX(ht ) = E ~(ht)X(ht) = E7r(ht)M(ht)X(h') = EM(ht)X(ht) The increment in M(ht) is given by m(ht+l Iht) = M(ht+1)/M(ht). Note that m(-) is a martingale (i.e., Etm(ht+l ht) = 1) and it measures the distortion on conditional probabilities t 1 t m(ht+lI ht) = i(ht+l)/7r(ht+l) _ f(h + h ) 7-(ht)/7r(ht) 7r(ht+l|ht) Evaluating Consumption Paths The agent evaluates contracts with Multiplier Preferences introduced by Hansen and Sargent (2008). We assume that the agents make decisions sequentially, period by period. t Fix a contract c(h ) = {c(ht)}&0 . Denote, c(ht) as instantaneous consumption in history h', c+(ht) = {c(hs)} 1+I as the sequence of consumption that succeeds history h', and (c(ht), c+(ht)) the sequence of consumption in which, at ht, c(ht) is consumed, and after that, consumption is given by c+(ht). The utility of a consumption stream (c(ht), c+(ht)) is defined, for every history ht, as + O3m(ht+l ht)]} (4.1) V(c(ht),c+(ht))=- mint E{m(ht) [u(c(ht)) +#V(c(ht),c+(ht)) {m(h )} subject to 7r(ht)m(ht) 1 ht where u : R+ -+ R is increasing, concave and twice continuously differentiable. The parameter 0 > 6 > 0 penalizes distortions by the nature; when 6 -+ +oo the agent fully trusts his model. In terms of the timing, we assume that at the start of each period, nature will move and change the outcome of the agent. The next proposition solves the inner minimization problem of the agent. Proposition 47. The value function in (4.1) is a fixed point of the following functional equation V(c(ht),c+(ht)) = -Olog (Sir(st+)e )c (4.2) The proof is in the appendix. The formulation in (4.2) will be useful to setup the contracting program recursively in the next section. The proof follows from explicitly solving the inner minimization problem. This proposition also shows the relationship between multiplier preferences and solving risk sensitive operators; that is, one can solve risk sensitive operators as solving a two person zero sum game. The risk sensitive operator introduces risk aversion in the continuation values, as well as additional risk aversion in consumption. 111 Finally, note that when 0 -> oo, equation (4.2) converges to expected utility; this is the case when the minimizing agent is extremely penalized by distortions. 4.2.2 Money Lender The money lender is risk neutral and can access a technology for savings with gross risk free return R. We assume for simplicity that R = 3-1. She has full commitment on the transfers she promises to the agent and wants to maximize the benefits of the risk sharing agreement. We assume that at t = 0 she makes a take it or leave it offer to the agent. The expected utility of the money lender is 00 11({c(h')}) = ZII(ht )3I(ht ) (4.3) t=o where I(ht) is the income in history h'. The expected profits generated by the contract for the money lender are given by Z' 0 w(ht)/3(y(st) - c(ht)), equation (4.3). 4.3 Equilibrium Allocations The solution concept is subgame perfect equilibrium. A subgame perfect equilibrium is a pair of strategies (r(-), c(-)) such that V(c(ht), c+(ht) | h') 2 V(c(ht), c+(ht) I ht) for every history h' and 11 (-r(-), c(-) I ho) > 0 2 H (T(.), a(-) I h ) . The next proposition characterizes the worst equilibrium payoff and provides necessary and sufficient conditions for an equilibrium. Lemma 48. The worst equilibrium is autarky forever with and associated payoff of V(y(ht), y+(h')) . A a contract and transfer sequence (T(-), c(-)) is a subgame perfect equilibrium if and only if V(c(h'), c+ (ht)) > V(y(h'), y+ (h')) The worst equilibrium is autarky forever with an associated payoff of V(y(ht), y+(ht)). Note that, if the money lender is given no transfer, and the agent is not giving any transfer, there is no incentive for anyone of them to deviate: if the agent deviates and gives a part of his endowment, she will be worse off. Because there are no profitable one shot deviations, autarky is a SPE. Also, this is the worst SPE, because the money lender cannot impose a higher punishment on the agent than not transferring income. The final part, that any SPE can be implemented with trigger strategies that specify the worst equilibrium after a deviation, follows Abreu (1988); the proof is omitted. 2 The one shot deviation principle holds because utility is continuous at infinity. This is a consequence of discounting. Note also that we are requiring that the money lender only is optimizing conditional on history ho. The reason for this is that it has full commitment. 112 4.4 Efficient Allocations 4.4.1 Pareto Program We now state the contracting problem of the money lender. The money lender chooses the best consumption path for the agent that is a subgame perfect equilibrium. The program is 00 max E ir(ht )3 t(y(st) - c(h')) t {c(h )} t=O subject to V(c(h'), c+(h')) > V(y(ht), y+(ht)) (4.4) where 4.4 holds for every history ht and V(c(hO), c+(hO)) Uo. To obtain a recursive formulation we use as a state variable the promised utility3 under the contract if state si is realized; this, is denoted by wi = V(c(ht+l), c+(ht+l)). Define vaut as Vaut = V(y(ht+l), y+(ht+l)). To simplify notation, denote consumption, endowment, and promised utility in each state s E S as C., Ys, WS respectively. The recursive formulation of the money lender problem (P1) is given by N P(v) = max E r, [(y, - c,) +3P(w,)] C3,Ws=1 subject to -Olog (Nr exp (U(CS)+/ 8 ) V (4.5) S=1 u(c8 ) + /3wS U(Ys) +3vaut (4.6) Cs E [cin , cmaxI (4.7) wS C [Vaut,V max] (4.8) The first restriction, (4.5), is coming from the promised utilities: the agent enters the period with a past promise by the money lender to receiving utility v. Because she is ambiguity averse her utility is distorted so the ex ante utility of {c8 ,WS}SES is given by the left hand side of (4.5). The second restriction, equation (4.6), is whether the agent wants to stay in the risk sharing agreement or prefers to consume his income 3 This follows from Spear and Srivastava (1987), Thomas and Worrall (1988a) and Phelan and Townsend (1991). A different approach, developed by Marcet and Marimon (2011), uses the Lagrange multipliers in the incentive constraints to keep track of promises. 113 and remain in autarky forever. Since this decision is taken knowing which particular state has been realized, utility is not distorted. The last two conditions just state that consumption and continuation utilities are chosen from equilibrium sets. 4.4.2 Benchmarks: Commitment and Expected Utility We now discuss two benchmarks: full commitment and no ambiguity aversion. Not surprisingly, an agent that has full commitment will be insured fully. The intuition is that there is no need to provide incentives; the principal just needs to provide enough for the agent to participate. The optimal contract in this case solves N P(v) = max 7r, [(y. - cS) + qP(w,)] subject to N --(U(C,) + OW.) -0 log ( 7r exp { > V 8=1 In the appendix we show that a solution to this program in fact prescribes a constant stream of consumption. When there is no ambiguity aversion by agent, but the agent lacks of commitment, (4.5) will collapse to the expected utility, and the problem of the money (P2) lender is then N P(v) = max E 7r, [(y8 - c') + 3P(ws)] {cs Ws I=1 subject to N E r, (u(cs) + 3w) > V s=1 u(c) + 3w, u(y) + /3vA For this case the characterization is the same as in Thomas and Worrall (1988a). In this case, consumption is constant over time for low realizations of the endowment and increases when a good shock is realized; once the agent has received the highest endowment shock, he will receive full insurance. For a detailed treatment of this program see, for example, Sargent and Ljungqvist (2004). 4.4.3 Characterizing Efficient allocations For every state s E S, the money lender will optimally choose how much to assign to current consumption and continuation value. From the first order conditions of P1, and the envelope condition, we get that u'(c8 )P'(w5 ) = -1 (4.9) 114 This is the rule that the money lender uses to distribute total utility between current consumption and future consumption (continuation value). The intuition is that it equalizes the marginal rate of substitution of the agent and the money lender between current and future consumption. The marginal rate of substitution between c, and w, for the household is u'(c5 )/3 ; the one of the money lender is - (P'(w8 ))'. This condition holds for every state. Note that, because of concavity of u and P, it implies that the money lender is assigning a higher continuation value, it has also to assign higher consumption. Define w := f(c), where f(-) is the implicit function defined by (4.9). This will be useful for the statement of the main proposition. Define also, exp _U(C.)+Ow. m(c, w, s) = e 0 ,= r, exp( u(c)+w This comes again from the first order conditions, specifies how much total utility changes when total utility in state s changes, and is the the probability distortion. Note that for all s, m(c, w, s) E (0, 1), and it adds up to one. Proposition 49, the main result in the paper, characterizes the evolution of consumption across histories ht, that are summarized by the promised utility v. As we will show, it is related one to one to consumption in the last period. Proposition 49. Denote last period consumption as c-. If {c5 , w,} solves P1, then: (1) In the states where the participation constraint is not binding, consumption is given by u'(c) = m(c, w, s)u'(c) where m(c, w, s) is the probability distortion in those states. In the states where the participationconstraint is binding, consumption is given by u(c) + #f(c) = u(y5 ) + vaut; (2) For every c- there is a threshold y(c_), such that the participationconstraint is binding if and only if y > y(c_); (3) Consumption is strictly increasing over time (c. > c- for all s) (4) Once sN is realized, the agent receives full insurance. That is, receives a constant consumption stream c such that u(c) + 8f (c) = u(y(sN)) + fvaut; (5) There is some T, such after these periods of time, the agent receives full insurance. The proof is in the appendix. We provide a sketch of the proof and the intuition in what follows. The first order conditions for P1 are given by c, : -r + pirau'(c 5)m(c, w, s) + Au'(c5 ) = 0 (4.10) W : r8/P'(w,) + pirm(c, w, s),3 + AS/ = 0 (4.11) P'(v) = -p (4.12) where A, and p are the Lagrange multipliers of (4.6) and (4.5), respectively. By increasing current con- sumption in state s, the static payoff of the money lender decreases, but it also increases the payoff of the 115 agent, in expected terms, and in state s. By changing the continuation value in state s, the money lender will have lower profits in the future, and is also changing expected utility and the utility if state s hits. The last condition is just the envelope condition. Rearranging these constraints we find that u'(c_) = m(c w, s)u'(c") (4.13) This is a modified Euler equation. In states that the participation constraint is not binding, A. = 0, the consumption of the agent is distorted by the term m(c, w, s), that is equal to the probability distortion in the optimal contract. So, in these states the agent is getting a compensation for the fact that she distrusts the model. The money lender takes this into account to choose how much consumption to allocate in state s: by increasing consumption is state s, will imply that marginal utility goes down, and also, that the agent thinks that this state is less likely. Note that in the case that 0 -+ oo, the distortion is zero, and consumption is constant. For the states where the participation constraint is binding, the agent receives enough so that it does not walk out of the contract c, + 3f (cS) = u(ys) + /vaut Given this constant consumption c, the agent will have no incentive to walk out of the contract. Note that the continuation value is pinned down by 4.9; and again, given the state, the relationship between current and future consumption is the same as in the case with 0 -+ o. Second, Proposition 49, also shows that there is a threshold value of c_, y(c_), such that the participation constraint is binding if and only if y ;> y(c_). The intuition is as follows. Define total utility in the optimal contract as h, = u(c8 ) + Ow,. It can be shown that in the optimal contract total utility is distributed on each state according to hs = max{h*, u}; (4.14) where h* is total utility when the participation constraint is not binding. The idea is that, when the participation constraint is binding, the agent receives just enough utility to stay on the contract; and when it does not bind, the agent receives more. Because h* depends on c_, or more explicitly on v, condition 4.14 already defines the threshold. The participation constraint will be binding in the sates such that u8 > h*(v).' Third, the fact that total consumption is increasing over time is more subtle. The expected value of m(h, s) is equal to one. Given the denominator, m decreases as h increases. So, at h*, m(h*, h, s), defined as exp (- m(h*, h, s) = 0 = 1 r exp (- ) and is greater than one. So, from the modified Euler equation (4.13) we conclude that consumption needs 4 Note that we still need to find h*. The only way is to guess and verify the number of constraints that will be binding. 116 to increase over time; bad states have more weight now, so higher consumption will be allocated in those states. Fourth, note that once the highest shock has been realized the agent receives full insurance. Mechanically, this follows from a guess and verify. Suppose that the best shock hits. Then the consumption and the continuation value are given by 3 3 U(CN) , f (cN) = U(y(SN)) / Vaut But, if consumption stays constant forever at this value, it will meet all of the participation constraints for every state, and also, it will meet the expected utility constraint. At this point, consumption stabilizes; as in the case where 0 -- oo. Finally, note that after a long enough string of periods, full insurance will be achieved, even though the highest shock has not been realized. This differs from the case without ambiguity aversion. The idea is as follows. We can bound how much consumption (and the continuation value) increases for every history, because we can bound the value of m away from 1. So, eventually, there is T high enough such that consumption will reach the maximum level CN- 4.5 Conclusion The main result in this paper is to characterize the evolution of consumption in the efficient arrangement between a risk neutral money lender that can commit to transfers and an ambiguity averse agent. In this setting, ambiguity aversion changes qualitatively the contract: the agent needs to be compensated for the fact that she distrust his model. Consumption increases every period, and eventually stabilizes. We think that this work shows that introducing ambiguity aversion does not affect tractability of the contracting problem; the analysis above can be readily applied to study quantitatively what is the price that agents are willing to pay to eliminate ambiguity. 4.6 Appendix: Omitted Lemmas and Proofs Proof. Now, we show the value function in (4.1) is a fixed point of the following functional equation V(c(st),c+(s')) = -Olog (Zlr(st+)e-+c (4.15) (8t+1 117 where, as we said before, (c(st+l), c+(si+l)) denotes the consumption stream with c(st+1) delivered in history st+1 and the rest according to the contingent plan c+(st+1). For this, note that V(c(st), c+(st))= mi E m(st+l) {u(c(st+1)) + 30 log m(st+l) + /V(c(st+l), c+(St+1)) } r(s t+1 ) m(S+1 t+l) (4.16) subject to r(st+)M(st+ 1 (4.17) st+1 The first order conditions yield t ZSt+i (st(l)exp (u(c(st+'))+fV(C(s +'),c+(s')) Plugging the solution into 4.16, yields u(c(se ))+13V(c(st+1),C+(.t+1)) V(c(st),c+(st)) = -0 log ( r(st+)e-u + + st+ 1 Proof. First, we show that when there is full commitment by the agent, there is full insurance. The con- tracting problem when there is full commitment is given by N P(v) =max E r, [(ys- c.) + qP(w,)] I SIW =1 subject to -0 log ( 7r, exp - (u(c,) + Ow') > V 8=1 Define exp {-u(c,,)+3w. m(c, w, s) e p ( u(cs s)+ =1 7r. exp (- ) where m(c, w, s) is the distortion in the one period ahead probability of state s. The first order conditions for this problem are cS: -7r + p7r u'(c8 )m(c, w, s) = 0 W, : 7r,#P'(wi) + p7rsm(c, w, s)/8 = 0 P'(v) = - p 118 For all s C S, P'(w') = P'(v)m(c, w, s) So, w, is constant. This implies that c. is also constant. 0 Lemma 50. P(v) that solves P1 is decreasing, strictly concave, and twice continuously differentiable. Lemma 51. Define utility under the contract and utility if the agent consumes his endowment stays in autarky as h, = u(c8 ) +8w, and u, _=U(Ys) +vaut respectively. Let h* be total utility when the participation constraint is not binding. The solution to P1 is h, = max{h*,u 8 }. Proof. To characterize the contract we divide Program P1 in two different problems. The first problem is an inter-temporal cost minimization problem in which for a given state and total utility the money lender chooses how to allocate consumption and continuation utility. The second problem will be a static optimization problem in which the money lender decides how much utility to allocate to each state given the solution of the inter-temporal problem. Step 1. Let the inter-temporal problem be P(h) = max -c + qP(w) (4.18) C,w subject to u(c) + 3w > h. (4.19) That is, minimize the inter-temporal cost of providing the agent utility h. A necessary condition for an optimum is U'(c)P'(w) = -1. (4.20) Since P and u are strictly concave, (4.20) defines an implicit function w = f(c), with f' > 0. This condition equalizes the marginal rates of substitution between consumption and continuation values for the agent and the money lender, for a given state. From the solution to (4.20) we can define for each state, total utility as hs = u(c8 ) + /3w, = u(c8 ) + ,f(c,). (4.21) Step 2. Now, we solve the static optimization problem of allocating total utility over each state s E S. Define U" = u(ys) + 3vaut. (4.22) Then, using the solution to the inter temporal problem, we can rewrite P1 as N P(v) = max ir (ys + P(h8 )] {h.} 1 119 subject to hs})LI -C log (7ir, exp{ (4.23) ns ? us. (4.24) The first order conditions for this problem are h: 7rP'(h.) - ex - =0 :7r. exp V: P'(V) = -A For the states with rn = 0, h, = h* > u8 , and for the ones with 7 > 0, h, = us. Then, a solution to this problem satisfies hS = max{h*, us}. Proof. For part 1, define exp ( u(c )+Ow8) m(c, w, s) = X: 7ir, exp ( u(cs)+8wt) where m(c, w, s) is the distortion in the one period ahead probability of state s. The first order conditions for P1 are given by c, : - 7r, + pgru'(cr)m(c,w,s)+Au'(c) 0 (4.25) w: 7r5 3P'(w5 ) + pwram(c, w, s)/3 + A8/3 0 (4.26) P'(v) = -A (4.27) For the states in which, the participation constraint is not binding, A, = 0. Then, for s such that the participation constraint is not binding, from 4.26 and 4.27 we have that P'(wS) = P'(v)m(c, w, s) = P'(v)rm(h, s) (4.28) and this implies that u'(c-) = m(c, w, s)u'(c) . For the states where the participation constraint is binding, u(c8 ) + /6w = u(cS) + Of(c') = u(y.) + /VaUt For part 3, note that n(h, s) is a martingale with mean one, and it has to be over and below one with 120 positive probability. Because h, = max{h*, U,}, the maximum value is over states where h, = h*. And, this maximum value needs to be greater than one. Then, from 4.28, P'(w) < P'(v), so, w. > v. Because, f(.) is strictly increasing c, >f(v) = c-. For part 4, we guess and verify. This will be the optimal contract following from the uniqueness of the optimal contract. Let S {s* E St Vi t, and si = SN for some i < t}. The optimal contract for any history st E S~t is w, = f(c) = v and c, = c for all s E S. This implies that, m(c, w, s) = 1, for all s E S . Choose A, = 0 for all s. First order conditions hold and these are necessary and sufficient for an optimum. where f is defined in the proof of Lemma and is weakly increasing. Finally, for part 5, we characterize the dynamics of consumption after histories that exclude any state sn for n < N - 1, i.e. st E &n = {st E S' : maxy(si) y(sn) for all si E st} i This follows from the fact that h, = max{h*I uS} showed in lemma 51. Second, recall that v < wi (st) < wi(st). So, P'(wi(st)) < K = P'(wl(st)) < P'(v) by strict concavity of P. After some history st, the first order conditions imply that P'(w8 ) = P'(v)m8 = P'(v) + (m - 1)P'(v) < P'(v)+(Km-1)K where the inequality follows from the definition of bounds Km and K. Third, the optimal contract prescribes that WN(s) > wi(s') ,Vsi , s. Strict concavity of P, and again lemma 51, imply that 0 > P"(w) > K mini P'(w8 ) - P'(v) (Km - 1)K ws-v Ws-v Therefore, (Km - 1)K (we -v) > k.-- Let E K > 0. Then, w. - v > E Vst E S,,. This result implies that for any history s E SE,, the 121 increments on the continuation value for any state st+1 < SN are at least E. Therefore, for histories s' long enough, wi(st) (for i < n) would eventually surpass u,. 0 122 References ABREU, D. (1988): "On the Theory of Infinitely Repeated Games with Discounting," Econometrica, 56(2), pp. 383-396. ABREU, D., D. PEARCE, AND E. STACCHETTI (1990): "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring," Econometrica, 58(5), pp. 1041-1063. AFONSO, G., AND R. LAGOS (2014): "Trade dynamics in the market for federal funds," Discussion paper, National Bureau of Economic Research. AGUIAR, M., AND M. AMADOR (2013a): "Sovereign debt," chapterprepared for the Handbook of International Economics, 4. (2013b): "Sovereign Debt: A Review," Working Paper 19388, National Bureau of Economic Re- search. AGUIAR, M., AND G. GOPINATH (2006): "Defaultable debt, interest rates and the current account," Journal of InternationalEconomics, 69(1), 64 - 83. AMADOR, M. (2013): "Sovereign Debt and the Tragedy of the Commons," Manuscript. ANGELETOS, G.-M., AND A. PAVAN (2013): "Selection-free predictions in global games with endogenous information and multiple equilibria," Theoretical Economics, 8(3), 883-938. ARELLANO, C. (2008): "Default Risk and Income Fluctuations in Emerging Economies," The American Economic Review, 98(3), 690-712. ARELLANO, C., AND A. RAMANARAYANAN (2012): "Default and the Maturity Structure in Sovereign Bonds," Journal of Political Economy, 120(2), pp. 187-232. ASHCRAFT, A. B., AND D. DUFFIE (2007): "Systemic Illiquidity in the Federal Funds Market," American Economic Review, 97(2), 221-225. ATKENSON, A., A. EISFELDT, AND R. 0. WEILL (2013): 'The Market for OTC Derivatives," Working Paper. ATKESON, A. (1991): "International Lending with Moral Hazard and Risk of Repudiation," Econometrica, 59(4), pp. 1069-1089. AUCLERT, A., AND M. ROGNLIE (2014): "Unique Equilibrium in the Eaton-Gersovitz Model of Sovereign Debt," . BACK, K., AND S. BARUCH (2004): "Information in Securities Markets: Kyle Meets Glosten and Milgrom," Econometrica, 72(2), 433-465. 123 BAI, J., C. JULLIARD, AND K. YUAN (2012): "Eurozone Sovereign Bond Crisis: Liquidity or Fundamental Contagion," Working Paper. BAI, Y., AND J. ZHANG (2012): "Duration of sovereign debt renegotiation," Journal of International Eco- nomics, 86(2), 252 - 268. BAO, J., J. PAN, AND J. WANG (2011): 'The Illiquidity of Corporate Bonds," The Journal of Finance, 66(3), 911-946. BARHAM, B. L., J.-P. CHAVAS, D. FITz, V. R. SALAS, AND L. SCHECHTER (2014): "The roles of risk and ambiguity in technology adoption," Journal of Economic Behavior & Organization, 97, 204-218. BEBER, A., M. W. BRANDT, AND K. A. KAVAJECZ (2009): "Flight-to-Quality or Flight-to-Liquidity? Evidence from the Euro-Area Bond Market," Review of Financial Studies, 22(3), 925-957. BEIM, D., AND C. CALOMIRIs (2001): "Emerging Financial Markets," McGraw-Hill. BENJAMIN, D., AND M. WRIGHT (2009): "Recovery Before Redemption: A Theory of Sovereign Debt Renegotiation," Working Paper. BERGEMANN, D., AND S. MORRIS (2005a): "Robust mechanism design," Econometrica, 73(6), 1771-1813. (2013): "Robust predictions in games with incomplete information," Econonetrica, 81(4), 1251- 1308. BERGER, J. 0., ET AL. (1994): "An overview of robust Bayesian analysis," Test, 3(1), 5-124. BI, R. (2008): "Beneficial Delays in Debt Restructuring Negotiations," IMF Working Papers, pp. 1-29. BIANCHI, J., J. C. HATCHONDO, AND L. MARTINEZ (2012): "International reserves and rollover risk," Discussion paper, National Bureau of Economic Research. BORRI, N., AND A. VERDELHAN (2009): "Sovereign risk premia,". BRONER, F. A., G. LORENZONI, AND S. L. SCHMUKLER (2013): "WHY DO EMERGING ECONOMIES BORROW SHORT TERM?," Journal of the European Economic Association, 11, 67-100. BRYAN, G. (2010): "Ambiguity and insurance," Unpublished manuscript. CALVO, G. A. (1978): "On the time consistency of optimal policy in a monetary economy," Econometrica: Journal of the Econometric Society, pp. 1411-1428. CALVO, G. A. (1988): "Servicing the Public Debt: The Role of Expectations," The American Economic Rcvicw, 78(4), pp. 647-661. CARROLL, G. (2013): "Robustness and linear contracts," American Economic Review. 124 CELENTANI, M., AND W. PESENDORFER (1996): "Reputation in dynamic games," Journal of Economic Theory, 70(1), 109-132. CHANG, R.. (1998): "Credible Monetary Policy in an Infinite Horizon Model: Recursive Approaches," Journal of Economic Theory, 81(2), 431 - 461. CHARI, V., AND P. J. KEHOE (1993a): "Sustainable Plans and Debt," Journal of Economic Theory, 61(2), 230 - 261. CHARI, V. V., AND P. J. KEHOE (1990): "Sustainable Plans," Journal of Political Economy, 98(4), pp. 783-802. CHATTERJEE, S., AND B. EYIGUNGOR (2012): "Maturity, Indebtedness, and Default Risk," American Eco- nomic Review, pp. 2674-2699. CHEN, H., R. Cui, Z. HE, AND K. MILBRADT (2014): "Quantifying liquidity and default risks of corporate bonds over the business cycle," Discussion paper, National Bureau of Economic Research. CHEN, L., D. LESMOND, AND J. WEI (2007): "Corporate Yield Spreads and Bond Liquidity," The Journal of Finance, 62(1), 119-149. CHERNOZHUKOV, V., H. HONG, AND E. TAMER (2007): "Estimation and confidence regions for parameter sets in econometric models," Econometrica, 75(5), 1243-1284. COLE, H. L., AND T. J. KEHOE (2000): "Self-Fulfilling Debt Crises," The Review of Economic Studies, 67(1), pp. 91-116. CRUCES, J. J., AND C. TREBESCH (2013): "Sovereign Defaults: The Price of Haircuts," American Economic Journal: Macroeconomics, 5(3), 85-117. D ERASMO, P. (2008): "Government reputation and debt repayment in emerging economies," Manuscript, University of Texas at Austin. DIAS, D., C. RICHMOND, AND T. WANG (2012): "Duration of Capital Market Exclusion: An Empirical Investigation.," UIUC Working Paper. DUFFIE, D., N. GARLEANU, AND L. H. PEDERSEN (2005): "Over-the-Counter Markets," Econometrica, 73(6), 1815-1847. (2007): "Valuation in Over-the-Counter Markets," Review of Financial Studies, 20(6), 1865-1900. DUFFIE, D., AND D. LANDO (2001): "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, 69(3), 633-664. EATON, J., AND R. FERNANDEZ (1995): "Sovereign Debt," National Bureau of Economic Research. 125 EATON, J., AND M. GERSOVITz (1981): "Debt with Potential Repudiation: Theoretical and Empirical Analysis," The Review of Economic Studies, 48(2), pp. 289-309. EDWARDS, A., L. HARRIS, AND M. PIWOWAR (2007): "Corporate Bond Market Transaction Costs and Transparency," The Journal of Finance, 62(3), 1421-1451. ENGLE WARNICK, J. C., J. ESCOBAL, AND S. C. LASZLO (2011): "Ambiguity aversion and portfolio choice in small-scale Peruvian farming," The BE Journal of Economic Analysis & Policy, 11(1). FARHI, E., C. SLEET, I. WERNING, AND S. YELTEKIN (2012): "Non-linear Capital Taxation Without Commitment," The Review of Economic Studies, 79(4), 1469-1493. FELDHUTTER, P. (2011): "The Same Bond at Different Prices: Identifying Search Frictions and Selling Pressures," Review of FinancialStudies. FLEMING, M. J. (2002): "Are Larger Treasury Issues More Liquid? Evidence from Bill Reopenings," Journal of Money, Credit and Banking, 34(3), pp. 707-735. FRIEWALD, N., R. JANKOWITSCH, AND M. G. SUBRAHMANYAM (2012): "Illiquidity or credit deterioration: A study of liquidity in the {US} corporate bond market during financial crises," Journal of Financial Economics, 105(1), 18 - 36. FUDENBERG, D., AND D. K. LEVINE (1989): "Reputation and equilibrium selection in games with a patient player," Econornetrica: Journal of the Econometric Society, pp. 759-778. (1992): "Maintaining a reputation when strategies are imperfectly observed," The Review of Eco- nomic Studies, 59(3), 561-579. GELOS, R. G., R. SAHAY, AND G. SANDLERIS (2011): "Sovereign borrowing by developing countries: What determines market access?," Journal of InternationalEconomics, 83(2), 243 - 254. GLOSTEN, L. R., AND P. R. MILGROM (1985): "Bid, ask and transaction prices in a specialist market with heterogeneously informed traders," Journal of FinancialEconomics, 14(1), 71 - 100. GROSSMAN, H. I., AND J. B. V. HUYCK (1989): "Sovereign Debt as a Contingent Claim: Excusable Default, Repudiation, and Reputation," Working Paper 1673, National Bureau of Economic Research. GUL, F., AND D. G. PEARCE (1996): "Forward induction and public randomization," journal of economic theory, 70(1), 43-64. HANSEN, L. P., AND T. J. SARGENT (2008): "Robustness," Princeton University Press. HATCHONDO, J. C., AND L. MARTINEZ (2009): "Long-duration bonds and sovereign defaults," Journal of InternationalEconomics, 79(1), 117 - 125. 126 1111---l- ...... -- -- 1- 1-.1------.1-1-1-1 1- --.- w - - , - I - , -- :,- , - - , , , - -- - ".------1- " - ,-1. 1-1--l- HATCHONDO, J. C., L. MARTINEZ, AND H. SAPRIZA (2010): "Quantitative properties of sovereign default models: solution methods matter," Review of Economic Dynamics, 13(4), 919-933. HE, Z., AND K. MILDBRANDT (2013): "Endogenous Liquidity and Defaultable Bonds," Econornetrica, forth- coming. HOPENHAYN, H., AND 1. WERNING (2008): "Equilibrium Default," Working paper. IRELAND, P. N. (1997): "Sustainable monetary policies," Journal of Economic Dynamics and Control, 22(1), 87-108. KARAIVANOV, A., AND R. M. TOWNSEND (2014): "Dynamic financial constraints: Distinguishing mecha- nism design from exogenously incomplete regimes," Econometrica, 82(3), 887-959. KARANTOUNIAS, A. G. (2013): "Managing pessimistic expectations and fiscal policy," Theoretical Eco- nomics, 8(1), 193-231. KEHOE, P. J., AND F. PERRI (2002): "International Business Cycles with Endogenous Incomplete Markets," Econometrica, 70(3), 907-928. KINNAN, C. (2011): "Distinguishing barriers to insurance in Thai villages," Mimeo. KLETZER, K. M., AND B. D. WRIGHT (2000): "Sovereign debt as intertemporal barter," American econo-mic review, pp. 621-639. KOCHERLAKOTA, N. R. (1996): "Implications of efficient risk sharing without commitment," The Review of Economic Studies, 63(4), 595-609. KREPs, D. M., P. MILGROM, J. ROBERTS, AND R. WILSON (1982): "Rational cooperation in the finitely repeated prisoners' dilemma," Journal of Economic theory, 27(2), 245-252. KREPS, D. M., AND R.. WILSON (1982): "Reputation and imperfect information," Journal of economic theory, 27(2), 253-279. KYDLAND, F. E., AND E. C. PRESCOTT (1977): "Rules Rather than Discretion: The Inconsistency of Optimal Plans," Journal of Political Economy, 85(3), pp. 473-492. KYLE, A. S. (1985): "Continuous Auctions and Insider Trading," Econometrica, 53(6), pp. 1315-1335. LAGOS, R., AND G. ROCHETEAU (2007): "Search in Asset Markets: Market Structure, Liquidity, and Welfare," The American Economic Review, 97(2), pp. 198-202. (2009): "Liquidity in Asset Markets With Search Frictions," Econometrica, 77(2), 403-426. LAGOS, R., G. ROCHETEAU, AND P.-O. WEILL (2011): "Crises and liquidity in over-the-counter markets," Journal of Economic Theory, 146(6), 2169 - 2205. 127 LELAND, H., AND K. TOFT (1996): "Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads," The Journal of Finance, 51(3), 987-1019. LIGON, E., J. P. THOMAS, AND T. WORRALL (2002): "Informal insurance arrangements with limited commitment: Theory and evidence from village economies," The Review of Economic Studies, 69(1), 209-244. LONGSTAFF, F., S. MITHAL, AND E. NEIS (2005): "Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market," The Journal of Finance, 60(5), 2213-2253. LORENZONI, G., AND I. WERNING (2013): "Slow Moving Debt Crises," Working Paper 19228, National Bureau of Economic Research. MARCET, A., AND R.. MARIMON (2011): "Recursive Contracts," Mimeo. MORTEN, M. (2013): "Temporary migration and endogenous risk sharing in village india," Unpublished Manuscript. NEUMEYER, P. A., AND F. PERRI (2005): "Business cycles in emerging economies: the role of interest rates," Journal of Monetary Economics, 52(2), 345 - 380. ORLIK, A., AND 1. PRESNO (2011): "On credible Monetary Policies with Model Uncertainity," Mimeo. PASSADORE, J., AND J. XANDRI (2015): "Robust Conditional Predictions in Dynamic Games: An Applica- tion to Sovereign Debt," MIT Working Paper. PELIZZON, L., M. SUBRAHMANYAM, D. ToMio, AND J. UNO (2013): "The Microstructure of the European Sovereign Bond Market: A Study of Euro-zone Crisis," Working Paper. PHELAN, C., AND E. STACCHETTI (2001): "Sequential Equilibria in a Ramsey Tax Model," Econometrica, 69(6), 1491-1518. PHELAN, C., AND R. M. TOWNSEND (1991): "Computing Multi-Period, Information-Constrained Optima," The Review of Economic Studies, 58(5), pp. 853-881. Pouzo, D., AND I. PRESNO (2011): "Sovereign Default Risk and Uncertainty Premia," Mimeo. (2012): "Sovereign default risk and uncertainty premia," Discussion paper, Working Papers, Federal Reserve Bank of Boston. ROSENZWEIG, M. R. (1988): "Risk, implicit contracts and the .family in rural areas of low-income countries," The Economic Journal, pp. 1148-1170. ROUWENHORST, G. (1995): "Asset Pricing Implications of Equilibrium Business Cycle Models," T. Cooley (ed.), Frontiers of Business Cycle Research. 128 SARGENT, T., AND L. LJUNGQVIST (2004): "Recursive Macroeconomic Theory," SLEET, C. (2001): "On credible monetary policy and private government information," Journal of Economic Theory, 99(1), 338-376. SPEAR, S. E., AND S. SRIVASTAVA (1987): "On Repeated Moral Hazard with Discounting," The Review of Economic Studies, 54(4), pp. 599-617. STANGEBYE, Z. (2014a): "Dynamic Panics: Theory and Application to the Eurozone," Job Market Paper. STOKEY, N. L. (1991): "Credible public policy," Journal of Economic Dynamics and Control, 15(4), 627 -- 656. THOMAS, J., AND T. WORRALL (1988a): "Self-Enforcing Wage Contracts," The Review of Economic Studies, 55(4), pp. 541-553. TOWNSEND, R. M. (1994): "Risk and insurance in village India," Econometrica: Journal of the Econometric Society, pp. 539-591. UDRY, C. (1994): "Risk and insurance in a rural credit market: An empirical investigation in northern Nigeria," The Review of Economic Studies, 61(3), 495-526. WORRALL, T. (1990): "Debt with potential repudiation," European Economic Review, 34(5), 1099 - 1109. YUE, V. Z. (2010): "Sovereign default and debt renegotiation," Journal of InternationalEconomics, 80(2), 176 - 187. 129