Self-enforcing Debt, Reputation, and the Role of Interest Rates

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Interest Rates

Victor Filipe Martins da Rocha, Yiannis Vailakis

To cite this version:

Victor Filipe Martins da Rocha, Yiannis Vailakis. Self-enforcing Debt, Reputation, and the Role of Interest Rates. 2014. �hal-01097114�

V. Filipe Martins-da-Rochaa and Yiannis Vailakisb

How domestic costs of default do interact with the threat of exclusion from credit markets to determine interest rates and sovereign debt sustainability? In this paper, we address this question in the context of a stochastic general equilibrium model with lack of commitment and self-enforcing debt in which default has two consequences: loss of access to international borrowing and output costs. In contrast toBulow and Rogoff (1989), we show that part of the ability to borrow is merely attributed to the threat of credit exclusion, or equivalently, to the loss of the sovereign’s reputation. Apart from the limit case–analyzed byHellwig and Lorenzoni(2009)–where output costs are absent, equilibrium interest rates are always higher than growth rates, implying that the way “reputation for repayment” supports debt does not depend on whether debt limits allow agents to exactly roll over existing debt period by period.

1. Introduction

Unlike domestic loans, sovereign loans are not usually secured by any form of col- lateral or enforced by a supranational legal system. This leads to the question of why sovereign governments ever repay their debts, or, equivalently, why investors accept to lend to sovereigns. One of the most advocated reason is the well-known reputational argument (first formalized byEaton and Gersovitz(1981)): sovereign states who repudi- ate their debts may tarnish their reputation, be denied access to credit, and, therefore, loose part of their ability to smooth consumption. However, the exclusion from credit markets is not the only consequence of default. Recent evidence from emerging markets suggests that default episodes are usually accompanied by direct costs on the sovereign economy. For instance, when the country cannot discriminate between debts owned by foreigner creditors and debts owned by its own creditors, default may be damaging for the domestic financial system. Similarly, default may trigger trade sanctions that alter the sovereign’s international trade pattern. To the extent that trade or banking is essen- tial for production, such costs may cause a decline in the domestic output.¹ A question then arises: How domestic costs of default do interact with the threat of exclusion from credit markets to determine interest rates and borrowing levels?

This paper addresses this question in the context of a stochastic general equilibrium model with lack of commitment and self-enforcing debt. It is assumed that default con- sequences consist of two components: loss of access to international borrowing and direct

1We would like to thank Manuel Amador, Yves Balasko, Gaetano Bloise, Jan Eeckhout, Christian Hellwig, Timothy Kehoe, Patrick Kehoe, Felix Kubler, Thomas Mariotti, Herakles Polemarchakis, Karl Shell, Stephen Spear and Neil Wallace for comments and suggestions.

aCNRS, U. Paris-Dauphine and FGV/EESP,[email protected]

bU. of Glasgow,[email protected]

1See among othersCohen and Sachs(1986),Cohen(1992) andTomz and Wright(2007,2013).

output costs modeled as the loss of an exogenous fraction of income.² FollowingAlvarez and Jermann(2000) andHellwig and Lorenzoni(2009), agents have access to a complete set of one period contingent bonds and face debt limits that are compatible with maximal risk sharing subject to debt repayment being individual rational. We fully characterise self-enforcing debt limits and obtain two striking implications: (1) unless output costs are zero, interest rates are always higher than agents’ growth rates; (2) part of the ability to borrow–captured by equilibrium self-enforcing debt limits–must necessarily reflect the threat of credit exclusion, or equivalently, the loss of a country’s reputation.

The first implication of our results stands in contrast with the limit case where there is no output drop. Assuming that credit exclusion is the only consequence of default,Hellwig and Lorenzoni(2009) observed that the lack of commitment implies that the debt limits typically bind and interest rates may be lower than agents’ growth rates. Their key finding is that, with sufficiently low interest rates, debt limits allow agents to roll-over existing debt period by period. This turns out to be a necessary and sufficient condition for debt repayment. The implied connection of repayment incentives to the ability to roll over debt makes possible to interpret their characterisation result as “rational bubbles” on equilibrium debt limits. Exploiting this characterization they subsequently constructed an example where positive debt levels are sustained at equilibrium. Our first result shows that the presence of “rational bubbles” on equilibrium debt limits is not robust to a realistic strengthening of the default punishment. Indeed, we show that, no matter how small is the output drop parameter, interest rates must be higher than growth rates.

This, in turn, prevents agents to roll-over their debt limits.

The second implication of our results stands in contrast toBulow and Rogoff(1989)’s analysis of repayment incentives. They argued that the threat of enjoining from ever borrowing in the future in no way enhances a country’s ability to borrow.³ More pre- cisely, when output drop costs are present, they showed that self-enforcing debt limits coincide with the present value of output losses and claimed that these debt limits reflect solely the loss in utility due to threat of output drop. In other words, according to their interpretation, “reputation debt”–defined as the maximum amount of debt supported by the mere threat of credit exclusion–is never sustained. We instead argue that, even if the

“bubble mechanism” exhibited byHellwig and Lorenzoni (2009) is not robust to output costs, their reputation debt sustainability result is. Formally, we show that, when default induces both exclusion from asset markets and output losses, then part of the ability to

2We have chosen to represent output loss as an exogenous fraction of income only for the sake of simplifying the analysis. In that respect, we followCole and Kehoe(2000),Aguiar and Gopinath(2006), Tomz and Wright (2007), Arellano (2008), Abraham and Carceles-Poveda (2010) andBai and Zhang (2010,2012).

3Bulow and Rogoff (1989) led to a vast literature studying alternative mechanisms to answer why countries repay their debts. It includes, among others,Cole and Kehoe(1995),Cole and Kehoe(1998), Kletzer and Wright (2000), Kehoe and Perri (2002), Gul and Pesendorfer (2004), Krueger and Uhlig (2006) andAmador(2012). We refer toAguiar and Amador(2014),Wright(2011) andTomz and Wright (2013) for a thorough discussion of this literature.

borrow may reflect the threat of loss of access to international credit markets.

To achieve our goals, we first characterize not-too-tight debt limits when there is output drop. The novel feature of our characterization result, which has no analogue in the absence of output costs, is its implication about the level of interests rates. We do not impose a priori any restriction on interest rates. We instead show that no matter how small is the output drop parameter, interest rates must be higher than agents’ growth rates if debt is self-enforcing. This, in turn, allows us to use the characterization result in Hellwig and Lorenzoni (2009) to prove that any process of self-enforcing debt limits is the sum of the present value of output losses and a “bubble” component. Taking into account market clearing conditions, we derive that, at equilibrium, self-enforcing debt limits coincide with the present value of future output losses (i.e., equilibrium debt limits do not have a bubble component).

The fact that with direct output costs low interest rates are not compatible with self- enforcing debt has perhaps some value, but it says nothing on how each punishment (exclusion versus output drop) affects agents’ ability to borrow. We would like to go further and understand whether some amount of borrowing can be attributed solely to the threat of credit exclusion as in the limit case (with no output drop) analyzed by Hellwig and Lorenzoni(2009).Bulow and Rogoff(1989) argued that the present value of future output losses represents exactly the amount of resources an agent is ready to give up in order to avoid loosing output in the future. According to this interpretation, our characterization result would mean that the threat of credit exclusion does not provide by itself enough incentives for repayment when default also entails some loss of output. We argue that disentangling repayment incentives in this way is questionable. Actually, it is true that with full commitment, the present value of future output losses does represent the utility loss in terms of current resources. This is because debt limits are set to avoid Ponzi schemes and never bind. However, in our environment without commitment, the self-enforcing debt limits typically bind and the present value of future output losses may not anymore represent the debt level sustained exclusively by the threat of output drop.

To support our claim we modify the stationary example of Hellwig and Lorenzoni (2009) by introducing output drop and analyze the asymptotic behavior of equilibria when the output drop parameter converges to zero. According to Bulow and Rogoff (1989)’s way of disentangling repayment incentives, when the output drop parameter vanishes, we should expect that the debt level–if it reflects only the output costs–also van- ishes. Interestingly enough, our analysis shows that this is not the case: the equilibrium self-enforcing debt limits of the modified economy do not vanish, and more importantly, they converge to the positive debt limits of the economy inHellwig and Lorenzoni(2009) where exclusion is the only consequence of default. Therefore, by a continuity argument, it sounds reasonable to consider that a fraction of self-enforcing debt limits must reflect the loss of consumption smoothing (and risk-sharing) due to the exclusion from credit markets.

Our analysis motivates the study of an alternative way of disentangling repayment

incentives which is not in contradiction with the aforementioned asymptotic result. We address this issue by proposing a new definition for the part of the self-enforcing debt limits that reflects the loss of utility merely due to the output drop. We show that our definition delivers some intuitive properties. More precisely, the debt level attributed to output costs never exceeds the present value of output losses and, more importantly, it converges to zero when the drop parameter vanishes.⁴ In addition, since the overall debt in our modified example converges to the reputation debt sustained in the original example ofHellwig and Lorenzoni(2009), we obtain the following two results: (a) credit exclusion is strong enough to sustain positive levels of debt in an environment with output costs and high interest rates; (b) the reputation debt sustained through a bubble in an equilibrium with low interest rates can be approximated by the reputation debt sustained in an equilibrium with high interest rates. We interpret this as a robustness property of the positive result in Hellwig and Lorenzoni (2009). Indeed, even if bubbles are not robust to a small (but realistic) change in the default punishment, sustained reputation debt levels are.

The paper is organized as follows: Section 2 describes the stochastic dynamic econ- omy with lack of commitment. Section 3 characterizes self-enforcing debt when default consequences consist of both exclusion from international borrowing and direct output costs. Section4contains our main findings. It illustrates the implications of our charac- terization result for reputation debt sustainability in the presence of output costs and proposes a new way to disentangle repayment incentives. Section5concludes. The proofs of the main results are detailed in Section 6 while technical proofs are presented in an appendix.

2. The Model

Here we present an infinite horizon general equilibrium model with lack of commitment and self-enforcing debt limits, along the lines of Bulow and Rogoff (1989) and Hellwig and Lorenzoni(2009). Time and uncertainty are both discrete and there is a single non- storable consumption good. The economy consists of a finite set I of infinitely lived agents sharing risks in an environment where debtors cannot commit to their promises.

2.1. Uncertainty

We use an event tree Σ to describe time, uncertainty and the revelation of information over an infinite horizon. There is a unique initial date-0 event s⁰ ∈ Σ and for each date t ∈ {0,1,2, . . .} there is a finite set S^t ⊂ Σ of date-t events s^t. Each s^t has a unique predecessorσ(s^t) inS^t−1 and a finite number of successorss^t+1inS^t+1 for which σ(s^t+1) = s^t. We use the notation s^t+1 s^t to specify that s^t+1 is a successor of s^t.

4Recall that the present value of output losses reflects the overall ability to borrow due to both default punishments.

Events^t+τ is said to follow event s^t, also denoted s^t+τ s^t, if σ^(τ)(s^t+τ) = s^t. The set S^t+τ(s^t) := {s^t+τ ∈ S^t+τ : s^t+τ s^t} denotes the collection of all date-(t+τ) events following s^t. Abusing notation, we letS^t(s^t) :={s^t}. The subtree of all events starting from s^t is then

Σ(s^t) := [

τ≥0

S^t+τ(s^t).

We use the notation s^τ s^t when s^τ s^t or s^τ = s^t. In particular, we have Σ(s^t) = {s^τ ∈Σ : s^τ s^t}.

2.2. Endowments and Preferences

Agents’ endowments are subject to random shocks. We denote by yⁱ = (yⁱ(s^t))_s^t∈Σ

agenti’s process of positive endowmentsyⁱ(s^t)>0 of the consumption good contingent to event s^t. Preferences over (non-negative) consumption processes c = (c(s^t))_s^t∈Σ are represented by the lifetime expected and discounted utility functional

U(c) :=X

t≥0

β^t X

s^t∈S^t

π(s^t)u(c(s^t))

where β ∈(0,1) is the discount factor, π(s^t) is the unconditional probability of s^t and u : R+ → [−∞,∞) is a Bernoulli function assumed to be strictly increasing, concave, continuous on R+, differentiable on (0,∞), bounded from above and satisfying Inada’s condition at the origin.⁵

Given a date-t events^t, we denote by U(c|s^t) the lifetime continuation utility condi- tional to events^t, defined by

U(c|s^t) :=u(c(s^t)) +X

τ≥1

β^τ X

s^t+τs^t

π(s^t+τ|s^t)u(c(s^t+τ))

where π(s^t+τ|s^t) := π(s^t+τ)/π(s^t) is the conditional probability of s^t+τ given s^t. We assume thatU(yⁱ|s⁰)>−∞ for every agent i.⁶ Since the bernoulli function is bounded from above, we then get that U(yⁱ|s^t)>−∞ for every events^t.

A collection (cⁱ)i∈I of consumption processes is called an allocation. It is said to be resource feasible if P

i∈Icⁱ=P

i∈Iyⁱ.

5The functionuis said to satisfy the Inada’s condition at the origin if limε→0[u(ε)−u(0)]/ε=∞. This property is automatically satisfied ifu(0) =−∞. We assume that agents’ preferences are homogenous.

This is only for the sake of simplicity. All arguments can be adapted to handle the heterogenous case where the preference parameters (β, π, u) differ among agents.

6This assumption is automatically satisfied if eitheru(0)>−∞or the allocation (yⁱ)i∈Iis uniformly bounded away from zero, in the sense that there existsε >0 such thatyⁱ(s^t)≥ε for each agentiand events^t.

2.3. Markets

At every date-t event s^t, agents can issue and trade a complete set of one-period contingent bonds, which promise to pay one unit of the consumption good contingent on the realization of any successor event s^t+1 s^t. Let q(s^t+1) > 0 denote the price at event s^t of the s^t+1-contingent bond. Agent i’s holding of this bond is aⁱ(s^t+1). The amount of state-contingent debt agent i can issue is observable and subject to state- contingent (non-negative) upper bounds (or debt limits) Dⁱ = (Dⁱ(s^t))_s^t_∈Σ. Given an initial financial claim aⁱ(s⁰), we denote by Bⁱ(Dⁱ, aⁱ(s⁰)) the budget set of an agent who never defaults.⁷ It consists of all pairs (cⁱ, aⁱ) of consumption and bond holdings satisfying the following constraints: for every event s^t,

(2.1) cⁱ(s^t) + X

s^t+1s^t

q(s^t+1)aⁱ(s^t+1)≤yⁱ(s^t) +aⁱ(s^t)

and

(2.2) aⁱ(s^t+1)≥ −Dⁱ(s^t+1).

Given some initial claim b∈Rat events^τ, we denote byJⁱ(Dⁱ, b|s^τ) the largest contin- uation utility defined by

Jⁱ(Dⁱ, b|s^τ) := sup{U(cⁱ|s^τ) : (cⁱ, aⁱ)∈Bⁱ(Dⁱ, b|s^τ)}

where Bⁱ(Dⁱ, b|s^τ) is the set of all plans (cⁱ, aⁱ) satisfying aⁱ(s^τ) = b, together with Equations (2.1) and (2.2) for every successor events^ts^τ. The demand set contingent to events^τ, denoted bydⁱ(Dⁱ, b|s^τ), consists of all optimal consumption and bond holdings plans in the budget setBⁱ(Dⁱ, b|s^τ). Formally,dⁱ(Dⁱ, b|s^τ) := argmax{U(cⁱ|s^τ) : (cⁱ, aⁱ)∈ Bⁱ(Dⁱ, b|s^τ)}.

2.4. Default Costs

We consider an environment where there is no commitment. Agentimight not honor his debt obligations and decide to default if it is optimal for him. His decision depends on the consequences of default. Following Bulow and Rogoff (1989) (see also Hellwig and Lorenzoni(2009)), we assume that a defaulting agent starts with neither assets nor liabilities, is excluded from future credit but retains the ability to purchase bonds, and suffers a drop in output.⁸

7The initial financial claimaⁱ(s⁰) can be interpreted as an initial transfer (by a social planner) or as the consequence of (un-modeled) past transactions.

8Bulow and Rogoff (1989) used the terminology “direct sanctions” for the output drop (see their Section III.B).

It has been extensively documented in the sovereign debt literature that output falls during sovereign default.⁹ Disruption of international trade or domestic financial sys- tems can lead to output drops if either trade or banking credit is essential for produc- tion. Mendoza and Yue (2012) and Gennaioli et al. (2014) propose explicit models of the mechanisms by which sovereign default causes an efficiency loss in production. We follow the tradition in the sovereign debt literature (see for instance Cohen and Sachs (1986),Cole and Kehoe(2000),Aguiar and Gopinath(2006),Arellano(2008),Abraham and Carceles-Poveda (2010), and Bai and Zhang (2010,2012)) and model the negative implications on output by the loss of an exogenous fractionλ∈[0,1) of income. Formally, we denote byBⁱ_λ(D, b|s^t) and dⁱ_λ(D, b|s^t) the budget and demand sets corresponding to Bⁱ(D, b|s^t) and dⁱ(D, b|s^t) where the endowmentyⁱ(s^τ) at every successor events^τ s^t is replaced by (1−λ)yⁱ(s^τ). Given these notations, countryi’s default option at event s^t is

V_λⁱ(s^t) :=J_λⁱ(0,0|s^t)

where J_λⁱ(0,0|s^t) := sup{U(cⁱ|s^t) : (cⁱ, aⁱ) ∈ B_λⁱ(0,0|s^t)} is the largest continuation utility at event s^t when agent i starts with zero financial claim, cannot borrow forever and endows income (1−λ)yⁱ(s^τ) at every s^τ s^t. The fraction λ ∈[0,1) is called the output drop parameter.¹⁰

2.5. Self-enforcing Debt Limits

We should now incorporate the fact that agents have the option to default. Since borrowers issue contingent bonds, lenders have no incentives to provide credit contingent to some event if they anticipate that the borrower will default.¹¹The maximum amount of debtDⁱ(s^t) at any events^ts⁰ should reflect this property. If agenti’s initial financial claim at events^tcorresponds to the maximum debt−Dⁱ(s^t), then he prefers to repay his debt if, and only if,Jⁱ(Dⁱ,−Dⁱ(s^t)|s^t)≥V_λⁱ(s^t). When a process of bounds satisfies the above inequality for every events^ts⁰, it is called self-enforcing.¹² Competition among lenders naturally leads to consider the largest self-enforcing boundDⁱ_λ(s^t) defined by the equation

(2.3) Jⁱ(D_λⁱ,−Dⁱ_λ(s^t)|s^t) =V_λⁱ(s^t).

We followAlvarez and Jermann (2000) and refer to such debt limits as not-too-tight.

9We refer toCohen and Sachs(1986),Cohen(1992) andTomz and Wright(2007,2013) for details.

10We assume that the output loss fraction is homogenous among agents. This is only for the sake of simplicity. All the arguments follow almost verbatim in the heterogenous case.

11Since the default punishment is independent of the default level, an agent either fully repays his debt or defaults totally. There is no partial default.

12Indeed, since the functionJⁱ(Dⁱ,·|s^t) is increasing, for any bond holdingaⁱ(s^t) satisfying the re- strictionaⁱ(s^t)≥ −Dⁱ(s^t), agentiprefers honoring his obligation than defaulting onaⁱ(s^t).

2.6. Competitive Equilibrium

Fix an allocation (aⁱ(s⁰))i∈I of initial financial claims that satisfies market clear- ing, i.e., P

i∈Iaⁱ(s⁰) = 0. A competitive equilibrium with self-enforcing debt is a list (q,(cⁱ, aⁱ, Dⁱ_λ)i∈I) which consists of state-contingent bond prices q, a resource feasible consumption allocation (cⁱ)i∈I, a market clearing allocation of bond holdings (aⁱ)i∈I

and an allocation of debt limits (D_λⁱ)i∈I such that (cⁱ, aⁱ) belongs to the demand set dⁱ(D_λⁱ, aⁱ(s⁰)|s⁰) and debt limits are not-too-tight.¹³

2.7. Natural Debt Limits

Given state-contingent bond prices q = (q(s^t))_s^t_s⁰ we denote by p(s^t) the associ- ated date-0 price of consumption at s^t defined recursively by p(s⁰) = 1 and p(s^t+1) = q(s^t+1)p(s^t) for everys^t+1 s^t. We use PV(q;x|s^t) to denote the present value at date-t events^t of a processx restricted to the subtree Σ(s^t) and defined by

PV(q;x|s^t) := 1 p(s^t)

X

s^t+τ∈Σ(s^t)

p(s^t+τ)x(s^t+τ).

When there is no ambiguity regarding asset prices, we use the simpler notation PV(x|s^t).

We say that interest rates are higher than agent i’s growth rates when the present value PV(yⁱ|s⁰) of his future endowments is finite, and lower (or equal) when it is infinite.¹⁴ When interest rates are higher than agent i’s growth rates, his natural debt limits are then defined byNⁱ(s^t) := PV(yⁱ|s^t).

3. Characterizing Self-enforcing Debt

Bulow and Rogoff(1989) analyzed initially a special case of the model where default entails solely credit exclusion (i.e.,λ= 0). They proved a striking characterization result:

self-enforcing debt is never sustained if interest rates are higher than growth rates and debt limits are tighter than natural debt limits.¹⁵

13Since bonds are in zero net supply, market clearing meansP

i∈Iaⁱ= 0.

14The choice of this terminology is driven by the following particular case. Assume that interest rates and bounds on growth rates are time and state invariant, i.e.,q(s^t+1) =π(s^t+1|s^t)(1 +r)⁻¹ and yⁱ(s^t+1) = (1 +gⁱ(s^t+1))yⁱ(s^t) for everys^t+1s^t, where the stochastic growth rate is such thatmⁱgⁱ≤ gⁱ(s^t+1) ≤Mⁱgⁱ for some parameter gⁱ with 0< mⁱ ≤Mⁱ <∞. Then, the present value of agent i’s endowments is finite if, and only if,r > gⁱ, and infinite if, and only if,r≤gⁱ.

15The model studied byBulow and Rogoff (1989) is slightly different than the one presented here.

They analyzed repayment incentives of a small open economy borrowing from competitive, risk neutral foreign investors. The sovereign country trades at the initial event a complete set of state-contingent contracts that specify the net transfers to foreign investors in all future periods and events. Contracts are restricted to be compatible with repayment incentives and to allow investors to break even in present value terms (this environment is in the spirit ofKehoe and Levine (1993), but with a different default option). We show inMartins-da-Rocha and Vailakis(2014) that the First BR Theorem is a corollary of Theorem 1 inBulow and Rogoff(1989).

Theorem3.1 (First BR Theorem) Assume that the output drop parameter is zero (i.e., λ= 0), and that interest rates are higher than agent i’s growth rates. IfDⁱ₀ is a process of not-too-tight debt limits tighter than natural debt limits,¹⁶ then for every events^t, we have D₀ⁱ(s^t) = 0.

Bulow and Rogoff(1989) also studied the case where there is a loss in production after default. They found that self-enforcing debt can be sustained but up to a level that does not exceed the present value of the loss in output. Formally, their second characterization result is as follows.¹⁷

Theorem3.2 (Second BR Theorem) Assume that output drop is positive after default (i.e., λ > 0), and that interest rates are higher than agent i’s growth rates. If D_λⁱ is a process of not-too-tight debt limits tighter than natural debt limits,¹⁸ then for every event s^t, we have D_λⁱ(s^t) = PV(λyⁱ|s^t).

The First and Second BR Theorem should be interpreted as partial equilibrium re- sults. Indeed, in the context of a general equilibrium model, interest rates are determined endogenously to clear bond markets. Therefore, assuming a priori that interest rates are higher than growth rates is an ad-hoc assumption on endogenous variables. In an envi- ronment with full commitment, there is no loss of generality in assuming that equilibrium interest rates are higher than growth rates. Indeed, in such an environment, debt limits are only set to avoid Ponzi schemes and do not bind. In contrast,Hellwig and Lorenzoni (2009) observed that without commitment, debt limits typically bind and interest rates are not necessarily higher than growth rates. They asked subsequently whether the First BR Theorem is valid without imposing ad-hoc restrictions on interest rates.¹⁹Working in this direction, they provided the following characterization of not-too-tight debt limits.

Theorem 3.3 (HL Theorem) Assume that the output drop parameter is zero (i.e., λ= 0). If D₀ⁱ is a process of not-too-tight debt limits, then D₀ⁱ allows for exact roll-over, in the sense that

(3.1) D₀ⁱ(s^t) = X

s^t+1s^t

q(s^t+1)Dⁱ₀(s^t+1), for alls^t∈Σ.

16That is,D0ⁱ≤Nⁱ andJⁱ(Dⁱ0,−Dⁱ0(s^t)|s^t) =V0ⁱ(s^t) for everys^ts⁰.

17Again, the model inBulow and Rogoff(1989) is not exactly the same as ours. However, we show in Martins-da-Rocha and Vailakis(2014) that the Second BR Theorem follows as a corollary of Theorem 2 inBulow and Rogoff(1989).

18That is,D_λⁱ ≤NⁱandJⁱ(D_λⁱ,−Dⁱ_λ(s^t)|s^t) =V_λⁱ(s^t) for everys^ts⁰.

19When interest rates are lower than agent i’s growth rates, his natural debt limits are infinite, therefore imposing that not-too-tight debt limits are tighter than natural debt limits is an innocuous assumption.

HL Theorem is a complete characterization of not-too-tight debt limits since any process of debt limits allowing for exact roll-over is not-too-tight.²⁰Hellwig and Lorenzoni (2009) build on this result to construct a simple stationary economy with two agents where, at equilibrium, debt limits are positive and interest rates are lower than each agent’s growth rates (zero non-contingent interest rate). That is, their example implies that the First BR Theorem does not have a general equilibrium counterpart.

The first contribution of this paper is to explore whether the Second BR Theorem holds true in a general equilibrium setting. To achieve our goal, we extend the HL Theorem and characterize not-too-tight debt limits allowing for the possibility of interest rates to be lower than growth rates, but without assuming that the output drop parameter is zero. We obtain the following characterization result.

Theorem 3.4 Assume that output drop is positive after default (i.e., λ > 0). If there exists a process Dⁱ_λ of not-too-tight debt limits, then interest rates are necessarily higher than agenti’s growth rates and there exists a non-negative processMⁱ allowing for exact roll-over such that Dⁱ_λ= PV(λyⁱ) +Mⁱ.

Some remarks are in order. As expected, more severe punishments, by reducing the risk of default, enlarge insurance opportunities. However, the striking observation concerns with the implications of Theorem 3.4 about interest rates: no matter how small is the output drop parameter, interest rates must be higher than agent i’s growth rates. This is because the not-too-tight debt limit Dⁱ_λ(s^t) must be larger that the present value PV(λyⁱ|s^t) of the future output losses.

The detailed proof of Theorem3.4is postponed to Section6.1. Here we only comment on the key steps. The novel step, which has no counterpart when the output drop param- eter is zero, amounts to show that the not-too-tight debt limitDⁱ_λ(s^t) is necessarily larger than the present value PV(λyⁱ|s^t) of output drop. The natural approach to prove this is to show that D_λⁱ(s^t)≥λyⁱ(s^t) + ˜D_λⁱ(s^t) where ˜D_λⁱ(s^t) :=P

s^t+1s^tq(s^t+1)Dⁱ_λ(s^t+1), and then use a standard iteration argument. Given the definition of the not-too-tight debt limitDⁱ_λ(s^t), this is equivalent to proving that

(3.2) Jⁱ(D_λⁱ,−λyⁱ(s^t)−D˜ⁱ_λ(s^t)|s^t)≥V_λⁱ(s^t).

Observe that we always have

(3.3) V_λⁱ(s^t)≥u((1−λ)yⁱ(s^t)) +β X

s^t+1s^t

π(s^t+1|s^t)V_λⁱ(s^t+1).

If we had an equality in (3.3), then the inequality (3.2) would be straightforward.²¹ In- deed, consuming (1−λ)yⁱ(s^t) and borrowing up to the debt limitsDⁱ_λ(s^t+1) at event s^t

20This follows from a simple translation invariance of the budget restrictions (2.1) and (2.2).

21In the simpler environment where saving is not possible after default (as it is the case inAlvarez and Jermann(2000)) we always have an equality in (3.3).

leads to the right hand side continuation utility in (3.3), and satisfies the solvency con- straint at events^t in the budget set defining the left hand side of (3.2). However, in our environment where a country can save after default, (3.3) need not be satisfied with an equality. Dealing with this problem constitutes the technical contribution in the proof of Theorem3.4.

The second step of the proof consists in showing that the difference between the debt limit and the present value of output drop involves a bubble (in the sense that it allows for exact roll-over). The proof of this step is based on a weaker but technically less involved version of the HL Theorem we propose (see Section 6.2), for which interest rates are known to be higher than the agents’ growth rates.

We know that the First BR Theorem is not valid in a general equilibrium environ- ment since, due to Hellwig and Lorenzoni (2009)’s example, positive levels of debt can be sustained at equilibrium. Interestingly enough, we show below that the Second BR Theorem does extend to a general equilibrium setting.

Corollary 3.1 Assume that output drop is positive after default (i.e., λ > 0). At any competitive equilibrium with self-enforcing debt (q,(cⁱ, aⁱ, Dⁱ_λ)i∈I), interest rates are higher than any agent’s growth rates and the debt limits coincide with the present value of future output losses, i.e., Dⁱ_λ(s^t) = PV(λyⁱ|s^t) for all s^t∈Σ.

If we assume that equilibrium debt limits are tighter than natural debt limits then the above result is a simple corollary of Theorem3.4. Indeed, we know that equilibrium interest rates must be higher than growth rates. Moreover, a non-negative and non-zero process allowing for exact roll-over cannot be bounded from above by the natural debt limits (see the proof of Proposition 3 in Hellwig and Lorenzoni (2009)). However, not- too-tight debt limits, like prices, are endogenously determined. Therefore, we cannot assume a priori that they are tighter than the natural debt limits. We should instead prove this is necessarily the case at equilibrium. One may argue (see for instance the discussion following Proposition 3 in Hellwig and Lorenzoni (2009)) that if debt limits are not tighter than the natural debt limits, then the aggregate stock of debt, and thus the savings of some lender, will eventually exceed the value of aggregate endowments. This way of reasoning is not as simple as it may appear since the debt limits are associated to savings of some lender only when they bind. To complete the argument and remove the bubble component from the equilibrium self-enforcing debt limits we prove the necessity of a “market transversality” condition.

Proof of Corollary 3.1: Let (q,(cⁱ, aⁱ, D_λⁱ)i∈I) be a competitive equilibrium with self-enforcing debt. It follows from Theorem3.4that interest rates are higher than every agent’s growth rates, i.e., PV(yⁱ|s⁰)<∞for eachi. Since consumption markets clear, we haveP

i∈Icⁱ=P

i∈Iyⁱ which implies that each consumption processcⁱhas finite present

value. Applying Lemma6.3, we deduce the following market transversality condition²²

t→∞lim X

s^t∈S^t

p(s^t)[aⁱ(s^t) +D_λⁱ(s^t)] = 0.

Market clearing of bond markets then implies (3.4) lim

t→∞

X

s^t∈S^t

X

i∈I

p(s^t)Dⁱ_λ(s^t) = 0.

SinceDⁱ_λ = PV(λyⁱ) +Mⁱ whereMⁱ is non-negative and satisfies exact roll-over (Theo- rem 3.4), we have

(3.5) X

s^t∈S^t

X

i∈I

p(s^t)D_λⁱ(s^t) =X

i∈I

Mⁱ(s⁰) +λ X

s^t∈S^t

X

i∈I

p(s^t) PV(yⁱ|s^t).

Combining Equations (3.4) and (3.5), and using the fact that PV(yⁱ|s⁰)<∞, we deduce thatP

i∈IMⁱ(s⁰) = 0. Non-negativity of eachMⁱ(s⁰) implies thatMⁱ(s⁰) = 0 for everyi.

SinceMⁱ is non-negative and allows for exact roll-over, we get the desired result:Mⁱ = 0

for everyi. Q.E.D.

Corollary 3.1 reveals that with output drop, some risk sharing is always possible at equilibrium. The fact that this result is obtained without assuming a priori that interest rates are higher than growth rates is an improvement to Bulow and Rogoff (1989) and an extension of Hellwig and Lorenzoni (2009)’s analysis. However, the result itself says nothing on how each punishment (exclusion versus output drop) affects agents’ ability to borrow. In particular, it does not reveal whether some amount of borrowing is attributed to the threat of credit exclusion. To answer this question we need to study and disentangle borrowers’ repayment incentives. This is the subject of the following section.

4. Repayment Incentives

Following the terminology introduced inBulow and Rogoff (1989), we call reputation debt the maximum amount of debt supported by the mere threat of credit exclusion in case of default. When credit exclusion is the only punishment for default (i.e., λ = 0), the not-too-tight debt limitsDⁱ₀ defined by the equations

Jⁱ(D₀ⁱ,−Dⁱ₀(s^t)|s^t) =Jⁱ(0,0|s^t), for all s^t∈Σ

represent exactly the reputation debt. In that respect, the First BR Theorem can be interpreted as a negative result: no reputation debt is sustainable if interest rates are higher than growth rates and debt limits are not allowed to exceed the natural debt

22Our market transversality condition differs from the standard transversality condition. Indeed, due to the lack of commitment, debt limits may bind for some agent iin which case we do not necessarily have thatp(s^t) =β^tπ(s^t)u⁰(cⁱ(s^t))/u⁰(cⁱ(s⁰)).

limits. When the output drop parameter is positive (i.e., λ >0), the not-too-tight debt limitsDⁱ_λ defined by the equations

Jⁱ(D_λⁱ,−Dⁱ_λ(s^t)|s^t) =J_λⁱ(0,0|s^t), for all s^t∈Σ

represent instead the maximum debt level sustained by the threat of two punishments:

the credit exclusion and the loss of the amount λyⁱ of future endowments.

Bulow and Rogoff (1989) propose to disentangle the two sources of repayment in- centives. They identify the present value PV(λyⁱ|s^t) as the level of debt merely due to the output drop costs. The reputation debt is then defined as the difference D_λⁱ(s^t)− PV(λyⁱ|s^t). Given this terminology, the Second BR Theorem can also be interpreted as a negative result: if interest rates are higher than growth rates and, in addition to the credit exclusion, there is some loss of output after default, then debt can be sus- tained (D_λⁱ(s^t) >0) but only on the basis of the output drop sanction (since D_λⁱ(s^t) = PV(λyⁱ|s^t)). In other words, with interest rates being higher than agents’ growth rates, even if there is a loss of output after default, reputation debt cannot be sustained.

When there are no direct sanctions (i.e.,λ= 0), the HL Theorem connects borrowers’

repayment incentives to the ability to roll-over their debt. This feature can be interpreted as a rational bubble on debt limits which is compatible with equilibrium when the present value of the agents’ endowments is infinite, or equivalently when interest rates are lower than growth rates. But with output drop (i.e., λ > 0), Corollary 3.1 says that rolling over debt is impossible since equilibrium interest rates must necessarily be higher than agents’ growth rates. Moreover, the debt limit D_λⁱ(s^t) coincides with the present value PV(λyⁱ|s^t), implying that reputation debt as defined byBulow and Rogoff(1989) cannot be sustained even in a general equilibrium setting.

If we accept Bulow and Rogoff (1989)’s disentanglement of repayments incentives, Corollary 3.1suggests that sustaining reputation debt in Hellwig and Lorenzoni (2009) is not robust to a more realistic modeling of the default consequences. This is because we obtain the following discontinuity result: reputation debt can never be sustained when λ >0, but it does in the limit case whereλ= 0. However, the discontinuity may not arise, if a part of the sustained debt PV(λyⁱ|s^t) does actually reflect the threat of exclusion from credit markets. In this case, Corollary 3.1 would suggest three things. First, sustaining reputation debt is robust to a realistic strengthening of the default punishment. Second, reputation debt cannot be attributed to the “rational bubble mechanism” of Hellwig and Lorenzoni(2009). Third, the disentanglement of repayments incentives proposed by Bulow and Rogoff (1989) is not the appropriate one. The next session is devoted to the investigation of those issues. Working in this direction, we modify the example inHellwig and Lorenzoni(2009) by introducing output drop (λ >0), and we analyze the asymptotic behavior of equilibria whenλconverges to 0.

4.1. Adding Output Drop inHellwig and Lorenzoni(2009)

Denote by E^hl the economy considered in Hellwig and Lorenzoni (2009). There are two agentsI ={a, b}and uncertainty is captured by a Markov process with state space Z = {z^a, z^b}. The state zⁱ corresponds to the situation where agent i’s endowment is

¯

e ∈ (1/2,1) and agent j’s endowment, with j 6= i, is e := 1−e <¯ ¯e. The transition probabilities are symmetric where α ∈(0,1) is the probability of switching states, i.e., Prob(zⁱ|z^j) = α for i 6= j. The Bernoulli function u : [0,∞) → R is strictly concave, strictly increasing, continuously differentiable and bounded; and the discount factor β belongs to (0,1). The parameters (u, β, α,e) are chosen such that there exists ¯¯ c > 0 satisfying

(4.1) ¯c <¯e and 1−β(1−α) =βαu⁰(1−c)¯ u⁰(¯c) .

The corresponding event tree Σ can be defined as follows. The initial event iss⁰ :=z^a and a date-t event is a history of state realizations s^t = (s⁰, s₁, . . . , s_t) with s_r ∈ Z for each 1 ≤ r ≤ t. The transition probabilities are defined by π(s^t+1|s^t) = α if s_t+1 6= s_t andπ(s^t+1|s^t) = 1−αifst+1=st. The endowment processyⁱis defined byyⁱ(s^t+1) := ¯e if s_t+1 = zⁱ and yⁱ(s^t+1) := e if s_t+1 6= zⁱ. Since the initial state is z^a, agent a begins with the high endowment ¯ewhile agent b’s initial endowment is e.

4.2. Credit Exclusion without Output Drop

We now recall the equilibrium with positive levels of reputation debt proposed by Hellwig and Lorenzoni(2009). The state-contingent bond prices q are defined by

q(s^t+1) :=







q^c:=βαu⁰(1−¯c)

u⁰(¯c) if s_t+16=s_t q^nc:=β(1−α) if s_t+1=s_t.

The consumptioncⁱ satisfies cⁱ(s^t+1) := ¯cif s_t+1 =zⁱ and cⁱ(s^t+1) := 1−¯c ifs_t+1 6=zⁱ. Bond holdings aⁱ are given byaⁱ(s^t+1) :=−ω ifst+1=zⁱ and aⁱ(s^t+1) :=ω ifst+16=zⁱ, whereωis defined by the equation (1−q^nc+q^c)ω= ¯e−¯c. Initial asset positions (aⁱ(s⁰))i∈I

area^a(s⁰) :=−ω and a^b(s⁰) :=ω.

Hellwig and Lorenzoni(2009) proved that (q,(cⁱ, aⁱ, D₀ⁱ)i∈I) is a competitive equilib- rium with self-enforcing debt where the not-too-tight debt limits (D₀ⁱ)i∈I are given by D₀ⁱ(s^t) :=ω. Observe that the state-contingent bond prices are such that q^c+q^nc = 1, that is, the non-contingent interest rate is zero. In particular, interest rates are lower than each agent’s growth rates. Trivially, Dⁱ₀ allows for exact roll-over and is therefore not-too-tight.

4.3. Credit Exclusion with Output Drop

We propose to modify the above example by incorporating a positive output drop after default. For any small enough output drop parameter λ >0, we show that there exists a symmetric Markov equilibrium where the two equilibrium consumption levels ¯c_λ and c_λ := 1−¯c_λ are defined as follows.²³

Lemma4.1 There exists ¯λ >0 small enough such that for everyλ∈(0,λ), there exists¯

¯

c_λ ∈(1/2,c) satisfying¯

(4.2) [1−q^nc+q^c(¯c_λ)]λw(¯c_λ) = ¯e−¯c_λ where for anyx∈(0,c),¯

q^c(x) :=βαu⁰(1−x)

u⁰(x) and w(x) := (1−q^nc)¯e+q^c(x)e (1−q^nc)²−(q^c(x))². Moreover, we have

(4.3) lim

λ→0¯c_λ = ¯c.

We denote by E_λ^hl the economy where default entails output drop with parameter λ∈(0,λ) and where agents’ initial asset holdings are defined by¯ a^a_λ(s⁰) :=−λw(¯c_λ) and a^b_λ(s⁰) :=λw(¯c_λ). All the other primitives remain the same as inE^hl.²⁴ We now describe a symmetric Markovian equilibrium of the economy E_λ^hl.

Letq_λbe the price process defined byq_λ(s^t+1) :=q^c(¯c_λ) ifst+1 6=standq_λ(s^t+1) :=q^nc if s_t+1 = s_t. We claim that interest rates are higher than each agent’s growth rates.

Indeed, since endowments are uniformly bounded from above, it is sufficient to show that

q^c(¯c_λ) +q^nc<1.

This follows from the fact that x 7→ q^c(x) is increasing, ¯c_λ < c¯ and q^c(¯c) +q^nc = 1 (by construction of ¯c). Observe that the present value of future endowments yⁱ at any events^t computed with pricesqλ is given by

PV(q_λ, yⁱ|s^t) =







w(¯c_λ) ifs_t=zⁱ w(¯c_λ)− ¯e−e

1−q^nc+q^c(¯c_λ) ifst6=zⁱ. (4.4)

Letcⁱ_λ be the consumption process defined bycⁱ_λ(s^t+1) := ¯cλ ifst+1=zⁱ and cⁱ_λ(s^t+1) :=

1−¯c_λifst+1 6=zⁱ; and letaⁱ_λbe the bond holdings process defined byaⁱ_λ(s^t+1) :=−λw(¯c_λ) ifs_t+1 =zⁱ and aⁱ_λ(s^t+1) :=λw(¯c_λ) if s_t+1 6=zⁱ.

23The proof of Lemma4.1is postponed to Section 6.3.1.

24We modify the initial endowments in order to get existence of a symmetric Markovian equilibrium and simplify the asymptotic analysis. In particular, we will show thataⁱ_λ(s⁰)−→aⁱ(s⁰) whenλ→0.

We can show that (q_λ,(cⁱ_λ, aⁱ_λ, D_λⁱ)i∈I) is a competitive equilibrium of the economyE_λ^hl where the not-too-tight debt limits (D_λⁱ)i∈I are given by

D_λⁱ(s^t) :=λPV(q_λ;yⁱ|s^t) = PV(q_λ;λyⁱ|s^t), for all s^t∈Σ.

We postpone the detailed arguments to Section 6.3.2. The important issue we want to analyze is the asymptotic behavior of the equilibrium (q_λ,(cⁱ_λ, aⁱ_λ, Dⁱ_λ)i∈I) when λ→0.

Proposition 4.1 When the output drop parameter converges to zero, i.e., λ → 0, the competitive equilibrium (q_λ,(cⁱ_λ, aⁱ_λ, D_λⁱ)i∈I) of the economy E_λ^hl converges (for the product topology) to the equilibrium (q,(cⁱ, aⁱ, Dⁱ₀)i∈I) of the economyE^hl. In particular,

λ→0limD_λⁱ(s^t) = lim

λ→0PV(qλ;λyⁱ|s^t) =ω=D₀ⁱ(s^t)>0.

Proof of Proposition 4.1: The convergence of (q_λ) and (cⁱ_λ) follows from (4.3). Re- call that ¯c_λ satisfies

[1−q^nc+q^c(¯cλ)]λw(¯cλ) = ¯e−¯cλ. Passing to the limit, we get

λ→0limλw(¯c_λ) = e¯−c¯

1−q^nc+q^c =ω.

This implies that (aⁱ_λ) converges to aⁱ.

Recall that D_λⁱ(s^t) = λPV(qλ;yⁱ|s^t) where PV(qλ;yⁱ|s^t) is specified in (4.4). If the current state is the high endowment, i.e., s_t=zⁱ, then

D_λⁱ(s^t) =λw(¯c_λ)−−−−→

λ→0 ω.

If the current state is the low endowment, i.e.,s_t6=zⁱ, then D_λⁱ(s^t) =λw(¯cλ)− λ(¯e−e)

1−q^nc+q^c(¯c_λ) −−−−→

λ→0 ω.

That is, when the output drop λ converges to 0, the non-contingent interest rates also converge to zero (q^nc+q^c(¯c_λ) −→ q^nc+q^c = 1) and the present value PV(q_λ;yⁱ|s^t) of future endowments converges to infinite. The decrease in the output drop is compensated by the increase in the present value of income in a way that the debt limitD_λⁱ(s^t), being

the product of the two effects, converges to ω. Q.E.D.

This result is intriguing. Given the interpretation proposed by Bulow and Rogoff (1989), the debt level D_λⁱ(s^t) should only reflect the threat of loss of output. When λvanishes, the output dropλyⁱ vanishes and the outside optionJ_λⁱ(0,0|s^t) converges to Jⁱ(0,0|s^t). It is then intuitive to expect that the debt limitDⁱ_λ(s^t) converges to zero too.

However, this level of debt does not vanish, and more importantly, it converges to the