hellwig on bubbles and sovereign debt

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Econometrica, Vol. 77, No. 4 (July, 2009), 11371164

BUBBLES AND SELF-ENFORCING DEBT

BY CHRISTIAN HELLWIG AND GUIDO LORENZONI1

We characterize equilibria with endogenous debt constraints for a general equilib-rium economy with limited commitment in which the only consequence of default islosing the ability to borrow in future periods. First, we show that equilibrium debt lim-its must satisfy a simple condition that allows agents to exactly roll over existing debtperiod by period. Second, we provide an equivalence result, whereby the resulting setof equilibrium allocations with self-enforcing private debt is equivalent to the alloca-tions that are sustained with unbacked public debt or rational bubbles. In contrast tothe classic result by Bulow and Rogoff (1989a), positive levels of debt are sustainablein our environment because the interest rate is sufficiently low to provide repaymentincentives.

KEYWORDS: Self-enforcing debt, risk sharing, rational bubbles, sovereign debt.

1. INTRODUCTION

CONSIDER A WORLD with integrated capital markets in which countries canborrow and lend at the world interest rate. Suppose that the only punishmentfor a country that defaults on its debt is the denial of credit in all future periods.What is the maximum amount of borrowing that can be sustained in equilib-rium? In this paper, we address this question in the context of a general equilib-rium model with complete securities markets and endogenous debt constraints.Following Alvarez and Jermann (2000), debt limits are set at the largest pos-

sible levels such that repayment is always individually rational. We show that,under some conditions, positive levels of debt are sustainable in equilibrium,and we characterize the joint behavior of intertemporal prices and debt lim-its. In the process, we identify a novel connection between the sustainability ofdebt by reputation and the sustainability of rational asset price bubbles.

The key observation for our sustainability result is that the incentives fordebt repayment rely not only on the amount of credit to which agents haveaccess, but also on the interest rate at which borrowing and lending takes place.After default, agents are excluded from borrowing, but are allowed to save.

1We thank for useful comments a co-editor, three anonymous referees, Marios Angeletos,Andy Atkeson, V. V. Chari, Hal Cole, Veronica Guerrieri, Nobu Kiyotaki, Narayana Kocher-lakota, John Moore, Fabrizio Perri, Balzs Szentes, Aleh Tsyvinski, Ivn Werning, Mark Wright,and seminar audiences at the Columbia, European University Institute (Florence), Mannheim,Maryland, Max Planck Institute (Bonn), NYU, Notre Dame, Penn State, Pompeu Fabra, Stan-ford, Texas (Austin), UCLA, UCSB, the Federal Reserve Banks of Chicago, Dallas, and Min-neapolis, the 2002 SED Meetings (New York), the 2002 Stanford Summer Institute in Theo-retical Economics, and the 2003 Bundesbank/CFS/FIC Conference on Liquidity and FinancialInstability (Eltville, Germany) for feedback. Lorenzoni thanks the Research Department of theFederal Reserve Bank of Minneapolis for hospitality during part of this research. Hellwig grate-fully acknowledges financial support through the ESRC. All remaining errors and omissions areour own.

2009 The Econometric Society DOI: 10.3982/ECTA6754
http://www.econometricsociety.org/http://www.econometricsociety.org/http://dx.doi.org/10.3982/ECTA6754http://dx.doi.org/10.3982/ECTA6754http://www.econometricsociety.org/http://www.econometricsociety.org/


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1138 C. HELLWIG AND G. LORENZONI

Lower interest rates make both borrowing more appealing and saving afterdefault less appealing. We show that sustaining positive levels of debt requireslow interest rates, that is, intertemporal prices such that the present value of

the agents endowments is infinite.2

Our result stands in contrast to the classic result of Bulow and Rogoff (1989a;henceforth BR), who analyzed the repayment incentives of a small open econ-omy borrowing at a fixed world interest rate and showed that no debt can besustained in equilibrium if the only punishment for default is the denial of fu-ture credit.3 The crucial difference is that BR took the world interest rate asgiven and assumed that the net present value of the borrowers endowment,evaluated at the world interest rate, is finite. This rules out precisely the in-tertemporal prices that emerge in our general equilibrium setup.

The difference between BR and this paper can also be interpreted in termsof unilateral versus multilateral lack of commitment. BR assumed that, af-ter default, the borrowing country can save by entering a cash-in-advancecontract with some external agent (some other country or some internationalbank), paying up front in exchange for future state-contingent payments. BRimplicitly assumed that this agent has full commitment power for exogenousreasons, so he will never default on the cash-in-advance contract.4 In ourmodel, instead, there is multilateral lack of commitment: no agent can com-mit to future repayments. Therefore, one countrys ability to save after default,and hence its default incentives, rests on the other countries repayment incen-tives.5

To illustrate our results, we first present a simple stationary example where,

in equilibrium, positive borrowing arises and the interest rate is zero. Moregenerally, debt is self-enforcing as long as the real interest rate is less than orequal to the growth rate of debt limits, which in steady state is equal to thegrowth rate of the aggregate endowment. As is well known, this is the samecondition that makes bubbles sustainable in equilibrium (Tirole (1985)).

2This terminology follows Alvarez and Jermann (2000).3This result has sparked a vast literature that considers alternative ways to sustain positive

levels of debt. A nonexhaustive list of contributions includes Eaton and Gersovitz (1981), Bulowand Rogoff(1989b), Cole and Kehoe (1995, 1998), Kletzer and Wright (2000), Kehoe and Perri(2002), Amador (2003), and Gul and Pesendorfer (2004).

4The role of this assumption in BR was first pointed out by Eaton (1990) and Chari and Kehoe(1993a).

5Two papers that share with BR the assumptions of one-sided commitment and credit exclu-sion after default are Chari and Kehoe (1993b) and Krueger and Uhlig (2006), the first in thecontext of a model of government debt with distortive taxes and lack of government commit-ment; the second in the context of competitive insurance markets with one-sided commitment byinsurers but not households. Krueger and Uhlig (2006) showed that BRs assumption about creditexclusion emerges naturally from competition among insurers with one-sided commitment. Twopapers which, instead, assume multilateral lack of commitment are Kletzer and Wright (2000)and Kehoe and Perri (2002). In both papers, borrowing can arise in equilibrium, but the mecha-nism is quite different from ours: all transfers are publicly observable, and risk sharing and debtare sustained by a folk-theorem-type argument.


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BUBBLES AND SELF-ENFORCING DEBT 1139

In the rest of the paper, we give a complete characterization of the conditionsunder which debt is sustainable. Our first general result (Theorem 1) statesthat debt limits and intertemporal prices are consistent with self-enforcement

if and only if all agents are able to exactly refinance outstanding obligationsby issuing new claims. The result relies on arbitrage arguments that comparethe feasible consumption sequences with and without a default (as in BR), aswell as weak restrictions on preferences that guarantee the monotonicity of theagents optimal asset holdings in initial wealth.

Our second result then establishes conditions for the existence of an equilib-rium with self-enforcing debt and gives a characterization of sustainable equi-librium allocations by means of an equivalence result. Consider an alterna-tive environment with no private debt, but where the government issues state-contingent debt that is not backed by any fiscal revenue, that is, the government

must finance all existing claims by issuing new debt. If this unbacked publicdebt is valued in equilibrium, it is a rational bubble.6 Theorem 2 shows thatany equilibrium allocation of the economy with self-enforcing private debt canalso be sustained as an equilibrium allocation of the economy with unbackedpublic debt, and vice versa. Therefore, conditions that ensure the existence ofvalued unbacked public debt or, more generally, the existence of rational bub-bles, also ensure the sustainability of positive levels of private debt in our econ-omy. The possibility of rational bubbles in models with exogenous borrowingconstraints la Bewley (1980) has been recognized in Scheinkman and Weiss(1986), Kocherlakota (1992), and Santos and Woodford (1997). Our equiva-

lence result shows that self-enforcing private debt can play the same role as abubble and can take its place in facilitating intertemporal exchange.Our model is closely related to models with limited commitment and en-

dogenous borrowing constraints (Kehoe and Levine (1993), Kocherlakota(1996), Alvarez and Jermann (2000)). The central difference is our assump-tion about default consequences: while previous authors assumed that agentsare completely excluded from financial markets after default, our exclusiononly rules out future credit. From a methodological point of view, the majordifference is that under our punishment, the equilibrium utility after default isendogenous and depends on equilibrium prices. This prevents us from starting

from a social planners problem with exogenous participation constraints andthen decentralizing its solution, as in Alvarez and Jermann (2000).In Section 2, we describe our general model and define competitive equilib-

ria with self-enforcing private debt and unbacked public debt. In Section 3, weillustrate our main results with a simple example. In Section 4, we study repay-ment incentives for individual agents and characterize debt limits consistentwith no default (Theorem 1). In Section 5, we study the resulting general equi-librium implications (Theorem 2). The online Supplemental Material (Hellwig

6In a deterministic environment, unbacked public debt is equivalent to fiat money.


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and Lorenzoni (2009)) contains additional results and extensions for the exam-ple in Section 3.

2. THE MODEL

Uncertainty, Preferences and Endowments

Consider an infinite-horizon endowment economy with a single nonstorableconsumption good at each date t {0 1 2 }. For each t, there is a positivefinite set St of date-t events st. Each st has a unique predecessor (st) St1

and a positive, finite number of successors st+1 St+1 for which (st+1) = st.There exists a unique initial date-0 event s0. Event st+ is said to follow eventst (denoted st+ st) if ()(st+) = st. The set S(st) = {st+ : st+ st} {st}denotes the subtree of all events starting from st and S S(s0) denotes the

complete event tree.At date 0, nature draws a sequence {s0 s1 }, such that st1 = (st) for all

t. At date t, st is then publicly revealed. The unconditional probability of st isdenoted by (st), where (s t) > 0 for all st S. For st+ S(st), (s t+|st) =(st+)/(st) denotes the conditional probability ofst+ given st.

There is a finite number J of consumer types, each represented by a unitmeasure of agents and indexed by j. Each consumer type is characterized bya sequence of endowments of the consumption good, Yj {yj(st)}stS .

7 Pref-erences over consumption sequences C {c(s t)}stS are represented by thelifetime expected utility functional

U(C) =stS

t(st)u(c(st))(1)

where (0 1), and u() is strictly increasing, concave, bounded, twice differ-entiable, and satisfies standard Inada conditions.

Markets

At each date st, agents can issue and trade a complete set of contingent secu-rities, which promise to pay one unit of period t+ 1 consumption, contingent

on the realization of event st+1 st, in exchange for current consumption. Ifno agent ever defaults (as will be the case in equilibrium), securities issued bydifferent agents are perfect substitutes and trade at a common price.

If agents had the ability to fully commit to their promises, they would be ableto smooth all type-specific endowment fluctuations. In our model, however,agents cannot commit: at any date st, they can refuse to honor the securitiesthey have issued and default. Any default becomes common knowledge and

7Throughout the paper, for any variable x, x(st) denotes the realization of x at event st, Xdenotes the sequence {x(st)}stS , and X(s

t) denotes the subsequence {x(st+ )}st+ S(st).


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DEFINITION 2: For given {aj(s0)}j=1J withJ

j=1 aj(s0) = 0, a competitive

equilibrium with self-enforcing private debt consists of consumption profiles,asset holdings, and debt limits {Cj Aj j}j=1J, and state-contingent bond

prices Q such that (i) for each j, {Cj

Aj

} maximize (1), subject to (2) and(3), for given aj(s0), (ii) the debt limits j are not too tight for each j, and

(iii) markets clear:J

j=1 cj(st) =

Jj=1 y

j(st) andJ

j=1 aj(st) = 0 for all st S.

Our equilibrium definition follows exactly Alvarez and Jermann (2000) ex-cept for the default punishment, which only allows for the denial of futurecredit, instead of complete autarky. Conceptually, the debt limits are similar toprices in Walrasian markets, in that individuals optimize taking prices and debtlimits as given, but both are endogenously determined by the market equilib-rium to satisfy the market-clearing and self-enforcement conditions.

Unbacked Public Debt

For our equivalence result, we consider an alternative economy with un-backed public securities. As before, there are sequential markets with com-plete contingent securities. However, unlike before, agents can no longer issuethese claims. Claims are only supplied by a government, which rolls over a fixedinitial stock of claims d(s0) period by period by issuing new securities. The gov-ernment must satisfy its budget constraint d(s t)

st+1st q(s

t+1)d(st+1), for all

st S, that is, at st the amount of resources raised by issuing new claims for

all st+1 st must be sufficient to honor the previous periods commitments.This budget constraint captures the notion that these securities are not backedby any tax revenues or other government income. We assume that the govern-ments budget constraint is satisfied with equality each period:

d(st) =

st+1st

q(st+1)d(st+1) for all st S(6)

Given initial asset holdings aj(s0) 0, optimal consumption allocations andasset holdings maximize (1), subject to (2) and the nonnegativity constraint

aj(st) 0 for all st S. A competitive equilibrium with unbacked public debtis then defined as follows:

DEFINITION 3: For given initial asset positions aj(s0) 0 for all j and debt

supply d(s0) =J

j=1 aj(s0), a competitive equilibrium with unbacked public

debt consists of consumption and asset profiles {Cj Aj}j=1J, a debt supplyprofile D, and bond prices Q such that (i) for each j, {Cj Aj} are optimal given

aj(s0), (ii) D satisfies (6), and (iii) markets clear:J

j=1 cj(st) =

Jj=1 y

j(st) andJj=1 a

j(st) = d(s t) for all st S.


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3. AN EXAMPLE

In this section, we illustrate the main results of our paper by means of asimple example with two types of consumers. In each period, one type re-

ceives the high endowment e and the other receives the low endowment e,with e + e = 1. The types switch endowment with probability from one periodto the next. Formally, uncertainty is captured by the Markov process st, withstate space S= {s1 s2} and symmetric transition probabilities Pr[st+1 = s1|st =s2] = Pr[st+1 = s2|st = s1] = . The event s

t corresponds here to the sequence{s0 st} and the endowments y

j(st) only depend on the current realizationofst, with y

j(st) = e ifst = sj and = e ifst = sj.We construct a symmetric Markov equilibrium, in which consumption allo-

cations, asset holdings, debt limits, and prices of state-contingent bonds de-pend only on the current state st, consumption allocations and asset holdings

are symmetric across types and states, and the debt limit is binding in thehigh-endowment state for each agent. To focus on a stationary equilibrium,assume that the economy begins in state s0 = s1 and the initial asset positionsare a1(s0) = and a

2(s0) = , where is the debt limit for both agents.Proposition 1 shows that equilibria with positive debt levels can exist.

PROPOSITION 1: Let c be defined by 1 (1 ) = u(1 c)/u(c). Ifc < e, there exists a stationary equilibrium with self-enforcing private debt in whichthe following conditions are observed:

(i) State-contingent bond prices are q(st+1) = qc = 1 (1 ) if st+1 = stand q(st+1) = q

nc= (1 ) ifs

t+

1= s

t.10

(ii) Consumption allocations are cj(st) = c ifst = sj and = c ifst = sj, wherec= 1 c.

(iii) Asset holdings are aj(st) = if st = sj and = if st = sj, where =(e c)/(2qc).

(iv) Debt limits are j(st) = for all st S.

PROOF: Given the conjectured equilibrium prices, the proposed allocationssatisfy the consumers first-order conditions qcu

(c) = u(c), qnc = (1 ),and qcu

(c) u(c). In addition, budget constraints and market-clearingconditions are satisfied by construction, given our definition of . We there-

fore only need to check that debt limits are not too tight. Theorem 1 belowshows that a sequence of debt limits satisfies (5) if and only if (s t) =

st+1st q(st+1)(st+1) for all st S. With constant debt limit j(st) = ,

this reduces to 1 = qc + qnc, which the equilibrium prices satisfy by construc-tion. Q.E.D.

Notice that c is well defined and unique, given Inada conditions and strictconcavity. Moreover, c is defined independently ofe, so the condition c < e is

10The subscripts c and ncstand for change and no change.


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satisfied for an open set of parameters, under any smooth parametrization ofu().

Proposition 1 illustrates our first general result: positive levels of debt are

sustainable in equilibrium if interest rates are sufficiently low. In particular,the no-default condition requires that qc + qnc = 1, tying the risk-free inter-est rate to zero. To explain where this result comes from and why it relatesthe incentives for repayment to bond prices, let us suppose the prices qc andqnc = (1 ) are stationary, and restrict agents to stationary consumption al-locations ofch in high-endowment periods and cl in low-endowment periods.

11

For a consumer in the high-endowment state, the expected lifetime utility as-sociated to (ch cl) is

v(ch cl) =1

1 + 2(1 (1 ))u(ch) + u(cl)

A consumer who chooses constant asset positions of in high-endowmentstates and a in low-endowment states faces the budget constraints ch =e + qnc qca and cl = e + a qnca + qc. Substituting for a gives theintertemporal budget constraint

(1 qnc)ch + qccl = (1 qnc)e + qce + [q2c (1 qnc)

2](7)

If the consumer never defaults, his optimal allocation (ch cl) maximizesv(ch cl) subject to (7). After defaulting in a high-endowment period, the agent

maximizes the same objective function subject to the same constraint (7), butwith = 0. A default thus represents a parallel shift of the intertemporal bud-get constraint and is optimal as long as the shift is positive, which is the casewhenever 1 qnc > qc. On the other hand, not defaulting is strictly preferredwhen 1 qnc < qc, and the agent is exactly indifferent when qc = 1 qnc. Thisillustrates how repayment incentives depend on the interest rate: the higher isthe interest rate, the less appealing is the opportunity to borrow and the moreappealing is the option to lend after default.

Figure 1 illustrates this argument graphically and shows how the equilibriumallocation is determined. The figure depicts the space of stationary allocations

(ch cl), along with the indifference curves for the function v(ch cl). The solidstraight line going through (ee), with slope 1, depicts the aggregate resourceconstraint ch + cl = e + e = 1. Our equilibrium allocation (cc) corresponds tothe allocation that maximizes v(ch cl), subject to the resource constraint.

Consider now an allocation along the resource constraint to the upper leftof (cc), for example, (ca ca). The intertemporal budget constraint that sup-ports this allocation as an equilibrium (the dashed line through (ca ca)) must

11Since stationary allocations are optimal whenever state prices are stationary and satisfy qnc =(1 ), these restrictions are without loss of generality.


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FIGURE 1.Feasible stationary allocations and default incentives.

be tangent to the indifference curve going through (ca ca); its slope in turnequals the ratio of state prices (1 qnc)/qc. Since by construction the slope ofthe indifference curve at (cc) is 1, the state prices supporting (ca ca) satisfy(1 qnc)/qc > 1. However, our previous argument implies that the no-defaultcondition is violated. In the figure, the consumer can default in the high-endowment state, trade along the post-default budget constraint (the dashedline through (ee)), and reach the consumption bundle (cdh c

dl ), which gives

strictly higher utility than (ca ca).The same argument applies to all allocations to the upper left of (cc), but

notice that as the allocation moves closer to (cc), the two intertemporal bud-get constraints become flatter and the gap between them, and hence the ben-efit from defaulting, becomes smaller. At the allocation (cc), at which thesupporting state prices satisfy (1 qn)/qc = 1, the two intertemporal budgetconstraints exactly coincide with each other and with the aggregate resourceconstraint, implying that agents are indifferent between defaulting and not de-faulting. This corresponds to our equilibrium allocation with self-enforcing pri-vate debt.

If we move to the lower right of (cc), the allocations between (cc) and(ee) along the feasibility constraint require supporting state prices that sat-

isfy (1 qnc)/qc < 1. At these prices, the intertemporal budget constraint withno default is to the right of the one with default, and hence agents wouldstrictly prefer no default to default. Finally, the default and no-default bud-get constraints converge again at (ee), which corresponds to the autarkicequilibrium with zero borrowing and prices equal to qautc = u

(e)/u(e) andqautnc = (1 ). Moreover, all allocations to the right of(ee) or to the left of(ca ca) can be immediately ruled out since they are dominated by the autarkyallocation, which is always within the agents budget sets.

Our example allows for a simple comparison between our results and thoseobtained when default is punished by complete exclusion from financial mar-


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This theorem also shows that the budget set of an agent facing debt limits which are not too tight is identical to that of an agent facing zero debt limitswho starts with a higher initial asset position. Hence, optimal consumption and

asset profiles are the same for the two agents. This simplifies the equilibriumcharacterization, since, rather than computing the fixed point between the con-sumers optimization problem and the self-enforcement condition (8), we onlyneed to compute optimal consumption allocations for agents with zero debtlimits, and these are identical to the optimal consumption allocations withoutdefault. Equilibrium debt limits are then constructed so as to satisfy (8) andmarket clearing in the asset market.

Condition (8) states that debt can only be sustained if, instead of repaying,the borrower is able to infinitely roll over outstanding debt. This leads to a sim-ple comparison with BRs no-lending result, which follows almost immediately

from Theorem 1.

PROPOSITION 3Bulow and Rogoff: Suppose prices and endowments aresuch that

w(st)

st+ S(st)

y(st+)p(st+)/p(st) < for all st

Suppose the debt limits are not too tight and satisfy (s t) w(st) for all st.Then (s t) = 0 for all st.

PROOF: If w(st) < , it must be the case that limT

st+Tst p(st+T)

w(st+T)/p(st) = 0. Now, using the exact rollover condition and the condition(s t) w(st), we have

(s t) =

st+Tst

p(st+T)

p(st)(s t+T)

st+Tst

p(st+T)

p(st)w(st+T)

for all T, which implies

(s t) limT

st+Tst

p(st+T)

p(st)w(st+T) = 0

Q.E.D.

BR showed that positive levels of self-enforcing debt are ruled out if twoconditions hold: (i) endowments are finite-valued at the prevailing state prices,that is, interest rates are high, and (ii) the agents debt is bounded by thepresent value of his future endowments, the natural debt limit. Condition (i)is imposed as an exogenous restriction on state prices. Condition (ii) on theother hand is a standard restriction which is usually imposed to rule out Ponzi


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games.15 With high interest rates, if debt limits are not zero and are not tootight, they are eventually inconsistent with the natural debt limits. With low in-terest rates, however, natural debt limits are infinite and impose no restriction

on debt levels. While BR only focused on the first scenario, our characteriza-tion in Theorem 1 is sufficiently general to encompass both.16

To sustain positive debt levels, we must abandon condition (i) or (ii). From apartial equilibrium point of view, relaxing either one can lead to self-enforcingdebt. However, a general equilibrium argument shows that the interesting casearises when we dispose of condition (i). If we relax condition (ii), but maintainhigh interest rates, Theorem 1 implies that if there are positive levels of debt,the aggregate stock of debt, and thus the savings of some lender, will eventu-ally exceed the value of aggregate endowments. This clearly cannot happen ingeneral equilibrium. On the other hand, it is possible to construct economies

where, in general equilibrium, condition (i) fails to hold, as shown in the exam-ple in Section 3 and, more generally, in Section 5 below.

Self-Enforcement and Exact Rollover

Before we proceed to the proof of Theorem 1, we illustrate the relation be-tween self-enforcement and exact rollover through a series of figures. For this,we assume that endowment fluctuations are deterministic and that agents tradea single uncontingent bond.17 The agents budget constraint can then be rewrit-ten as ct = yt + (ptat pt+1at+1)/pt. For a given sequence of prices {pt}, wecan thus compare the consumption profiles resulting from different asset planssimply by comparing the period-by-period changes in the present value of assetholdings, ptat pt+1at+1. In the following figures, we plot the time paths {ptat}of the present values of asset profiles with and without defaults to evaluaterepayment incentives.

Figure 2 considers repayment incentives when debt limits allow for exactrollover. In a deterministic environment, this requires that ptt is constantover time; such debt limits are represented by the dotted line A. Line B rep-resents an arbitrary asset profile that is feasible for an agent who defaults atsome date t. Line C represents a parallel downward shift of the asset profile Bto an initial asset position of t. Notice that C generates the same consump-

tion sequence as B. Moreover, since profile B remains nonnegative, C always

15In the working paper version, Bulow and Rogoff (1988, p. 5) hinted at the idea that relax-ing this condition may lead to positive debt when they remarked that this assumption rules outPonzi-type reputational contracts.

16Our formulation of the BR result is slightly different from the original version of the no-lending result, which stated that any asset sequence that gives an agent no incentive to ever defaultmust always remain nonnegative. The formal equivalence between the two statements followsfrom the additional (almost immediate) observation that any sustainable asset position can bebounded below by a sequence of debt limits that are not too tight.

17We thus replace the dependence on st by a time subscript to simplify notation.


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FIGURE 2.Debt limits satisfying exact rollover.

remains above A, and is therefore feasible for an agent who starts with an asset

position oft and does not default. Hence, this agent must be weakly better offnot defaulting at date t. On the other hand, for any asset profile C that is fea-sible without default starting from an asset position oft, there is some assetprofile B that is feasible starting from a default at date t and gives the agent thesame consumption profile as C. If debt limits allow for exact rollover, the agentmust therefore be exactly indifferent between defaulting on an asset positionoft and not defaulting, that is, the self-enforcement condition is satisfied withequality.

Along similar lines, we can illustrate how repayment incentives are violatedwhen the present value of debt limits is shrinking over time. Figure 3 plots

the case of BR, in which the natural debt limits are finite and act as a lowerbound on the agents asset profile. In this case, their present value falls overtime (line A). Then, given any asset profile B that is consistent with these debtlimits and admits positive debt at some date t, there exists a date t t atwhich the present value of debt reaches a maximum. At that point, the agentcan default and replicate the same consumption profile as B, just using positiveasset holdings (line C), and can even improve upon the no-default profile bystrictly increasing consumption at date t (line D).

FIGURE 3.Shrinking debt limits.


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FIGURE 4.Expanding debt limits.

Likewise, if the present value of debt limits is expanding over time (Figure 4),

for every profile B that is feasible after a default at some date t, there exists aprofile C that is feasible without default starting from asset position t and im-plements the same consumption. Moreover, profile D remains feasible withoutdefault starting from asset position t, but delivers strictly higher consumptionat date t, where the debt limits are expanding. Hence, agents strictly prefernot to default when the present value of debt limits is expanding over time.

PROOF OF THEOREM 1: Proposition 4 establishes the sufficiency part ofTheorem 1 by generalizing the graphical argument of Figure 2.

PROPOSITION 4: Suppose that the debt limits allow for exact rollover. Then

V (a (st); st) = VD(a (st); st) for all st Sand any a (s t)(9)

PROOF: Starting from an arbitrary st, consider asset profiles {a(st+)}st+ S(st)and {a(st+)}st+ S(st) that satisfy a(s

t+) = a(st+) (s t+) for all st+ S(st),with a(st) = a (s t) and a(st) = a (s t). By construction, {a(st+)}st+ S(st)is feasible under no default if and only if {a(st+)}st+ S(st) is feasible after de-faulting at st. Moreover, the exact rollover condition (8) implies a(st+)

st++1st+ q(st++1)a(st++1) = a(st+)

st++1st+ q(s

t++1)a(st++1) for allst+ S(st) and, therefore, starting from a(st) = a, asset plan {a(st+)}st+ S(st)implements the same consumption allocation {c(s t+)}st+ S(st) as asset plan{a(st+)}st+ S(st) starting from a(s

t) = a j(st). But then C(a(st); st) =C(a (s t)O(st); st) and V (a (st); st) = VD(a (s

t); st). Q.E.D.

The self-enforcement condition (5) follows from setting a = (s t) in (9).Condition (9) further implies that an agent who defaults on his maximum grossamount of debt (s t), but keeps his own asset holdings a (s t) after adefault, is always exactly indifferent between defaulting and not defaulting.The assumption that agents start with a net financial position of zero afterdefault can therefore be relaxed without weakening repayment incentives.


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Proposition 5 establishes the necessity part of Theorem 1 by showing thatdebt limits are not too tight only if they allow for exact rollover. This wasalready suggested by the graphical arguments in Figures 24. However, this

graphical intuition is incomplete, since it only applies to sequences of debt lim-its whose present values are monotone increasing or decreasing. The argumentfor arbitrary nonmonotone sequences of debt limits turns out to be consider-ably more involved.

PROPOSITION 5: If the debt limits are not too tight, they allow for exactrollover.

The proof of this proposition is given in the Appendix: here we sketch thekey steps. Consider a sequence of debt limits which are not too tight. Startingfrom some arbitrary event st, we first construct a sequence of auxiliary debt

limits (st), as follows:

(st+) =

(s t+) ifa(st+) = (s t+)st++1st+

q(st++1) min{(s t++1) (st++1)}

otherwise

(10)

where A(st) denotes the optimal no-default asset profile of an agent startingfrom st with asset position a(st) = (s t). The first step of the proof (and themajor technical hurdle) consists of showing that this sequence is well defined

and finite-valued. This is complicated by the fact that present discounted val-ues need not be well defined in our environment since we cannot rely on anassumption of high interest rates. The characterization in turn makes use ofthe time separability, concavity, and boundedness of u().

Next, using arbitrage arguments, we show that (i) (st+) (s t+) and

(st+) = (s t+) whenever a(st+) = (s t+), and (ii) (st+) st++1st+ q(s

t++1)(st++1) for every st+ S(st). The first property states

that is a lower bound of and is equal to whenever the actual debt limit is

binding. The second property states that satisfies (ER) with a weak inequal-

ity, so that under , at any event, the maximum outstanding debt obligationsare weakly less than the funds that can be raised by exhausting debt limits onthe continuation events.

Property (i), together with the concavity ofu() then implies that the value ofthe no-default problem starting from asset holdings of a(st) = (s t) at st is thesame under the original debt limits, (st), as under the auxiliary debt limits,

(st) (since the latter only relaxes nonbinding debt limits). Since the origi-

nal debt limits are not too tight and (s t) (st), we thus obtain VD(0; st) =

V ((st); (st) st) = V ((st); (st) st) V ((st); (st) st). On the otherhand, using the same arbitrage argument as Proposition 4, property (ii) implies


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that C(0O(st); st) C((st) (st) st) and V ((st); (st) st) VD(0; st)

with strict inequality if(st+) >

st++1st+ q(st++1)(st++1) for some st+

S(st). Therefore, both inequalities must hold with equality, which requires

(s t) = (st) and that (st) satisfies the exact rollover condition as an equalityfor all st+ S(st).

To complete the proof, we show that (st+1) = (s t+1) for all st+1 st. Re-peating the same steps as above, we construct additional sequences of aux-

iliary debt limits (st+1), together with optimal asset holdings A(st+1), for

each st+1 st. Clearly, (st+1) satisfies (ER) and (s t+1) = (st+1). More-over, due to concavity and additive separability of U, optimal asset profiles

are monotone in initial asset holdings, so that a(st+) = (s t+) = (st+)

whenever a(st+) = (s t+) = (st+). Together with (10), this implies that

(s

t+

) =

(s

t+

) for all st+

S

(s

t+1

) and hence

(s

t+1

) =

(s

t+1

) = (s

t+1

),which completes our proof. Q.E.D.

REMARK 1: Proposition 5 is the only result where we use the assumptionsof additive time separability, concavity, and boundedness of u(). All otherresults rely purely on arbitrage arguments and therefore require only strictmonotonicity. The boundedness assumption is a strong restriction, but it is re-quired only for a partial equilibrium characterization. If one restricts attentionto debt limits such that optimal consumption allocations are bounded aboveby aggregate endowments (a condition that must hold in general equilibrium),

Proposition 5 holds under the following weaker restriction.

ASSUMPTION 1: For all C such that U(C) minj U(Yj) and c(s t) [0J

j=1 yj(st)] for all st S,

stS

t(st)c(st)u(c(st)) < .

This regularity condition bounds the rate at which individual and aggregateendowments can grow or decline, relative to the discount factor . When rela-

tive risk aversion is bounded, this assumption holds whenever U(J

j=1 Yj) and

minj U(Yj) are both finite.

5. GENERAL EQUILIBRIUM CHARACTERIZATION

We now turn to the question of whether there exist equilibria with positivelevels of self-enforcing debt and how they can be characterized. Theorem 2shows that a given consumption allocation and price vector constitute a com-petitive equilibrium with self-enforcing private debt if and only if the sameallocation and prices are an equilibrium of the corresponding economy withunbacked public debt. For the latter economy, there are known existence andcharacterization results (e.g., Santos and Woodford (1997)), which then extendimmediately to the economy with self-enforcing private debt.


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THEOREM 2: An allocation {Cj}j=1J and prices Q are sustainable as a com-petitive equilibrium (CE) with self-enforcing private debt if and only if {Cj}j=1Jand Q are also sustainable as a competitive equilibrium with unbacked public

debt.

PROOF: Step 1. If{Cj Aj j Q} is a CE with self-enforcing private debt for

initial asset positions {aj(s0)}j=1J, then {Cj Aj

j} is optimal given initial

asset holdings ofaj(s0) j(s0), state prices Q, and zero debt limits, J

j=1 j

satisfies (6), andJ

j=1(Aj

j) =

Jj=1

j, so that market clearing is sat-

isfied. {Cj Aj j J

j=1 j Q} therefore constitutes a CE with unbacked

public debt for initial asset holdings {aj(s0) j(s0)}j=1J.

Step 2. If {Cj

Aj

DQ}j=1J constitutes a CE with unbacked public debt,starting from initial asset positions {aj(s0)}j=1J, we can construct debt limits

j = aj(s0)/d(s0) D and asset holdings Aj = Aj + j. By construction, debt

limits j are not too tight, and the allocations {Cj Aj} are optimal given initial

asset holdings of aj(s0) = 0 and debt limits holdings of j. Moreover, sinceJj=1 A

j =J

j=1(Aj + j) =

Jj=1 A

j D = 0, these allocations clear markets

in the private debt economy. Therefore {Cj aj j p}j=1J is a CE with self-enforcing private debt, for initial asset positions of zero. Q.E.D.

The proof of Theorem 2 makes repeated use of the exact rollover property.First, with exact rollover, the feasible, and hence the optimal, consumption al-locations in the default and no-default problems exactly coincide for suitablychosen initial values of asset holdings. Second, the exact rollover property im-plies a mapping from debt limits j which are not too tight to a sequence ofpublic debt supply that satisfies the government rollover constraint (6), andvice versa. Given this mapping from debt limits to public debt circulation andthe equivalence of optimal consumption allocations, market-clearing condi-tions in the two economies turn out to be exactly equivalent.

Theorem 2 formally establishes the connection between sustaining repay-ment incentives for private debt and sustaining rational bubbles. In the privatedebt economy, the agents commitment and enforcement power is so limitedthat any contract that, at some date, requires a positive transfer of resources innet present value is not sustainable. Likewise, in the economy with unbackedpublic debt, the government does not have the power to use taxation to guar-antee its debt holders a positive net transfer of resources. In both cases, the re-sult is that the only sustainable allocations roll over existing debt forever. Theequivalence arises because neither side can credibly commit to future transfers,either via contract enforcement or via taxation.


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6. CONCLUDING REMARKS

In this paper, we have studied a general equilibrium economy with self-enforcing private debt, in which borrowers, after default, are excluded from

future credit, but retain the ability to save at market interest rates. For a par-tial equilibrium version of this model, in which a small open economy borrowsinternationally at fixed, positive interest rates, BR showed that debt cannot besustained by reputational mechanisms only: eventually, the country always hasan incentive to default. The BR result can be interpreted as follows: if thereare some agents who are able to commit to intertemporal transfers at highinterest rates, then the remaining agents, who are unable to commit, will ac-cumulate and decumulate the securities issued by the committed agents, butwill never become net borrowers. Krueger and Uhlig (2006) provided the ana-lytical foundations for this interpretation.

In contrast, we show that positive levels of debt can be sustained when noparty has commitment power. The key to our result is that interest rates ad-just downward to provide repayment incentives to all the potential borrowingparties. As a result, low interest rates emerge in equilibrium.

These results helps to clarify the BR result in two directions. From a for-mal point of view, they highlight the role played by the interest rate in theBR argument. From a more substantive point of view, they clarify the role ofmultilateral versus unilateral lack of commitment and the role of credit exclu-sion versus stronger forms of punishment for the sustainability of debt in gen-eral equilibrium. This analysis helps to bridge the gap between the negative

result of BR and the positive results obtained with stronger forms of punish-ment, in particular those in Kehoe and Levine (1993) and Alvarez and Jermann(2000).18

From an empirical point of view, the central implication of our model is thatequilibrium borrowing requires low interest rates. In principle, this predictioncould be used to test the hypothesis that international debt is sustained by ourreputational mechanism. In practice, this is tantamount to testing for rationalbubbles in international debt.19 One standard approach to such a test wouldbe to compare the rate of return on international debt with the growth rateof the aggregate stock of debt in circulation. An alternative approach focuses

18See the discussion in Section 3. Our analysis also relates to the model of private internationalcapital flows of Jeske (2006) and Wright (2006). In their model, the ability of agents to participatein domestic capital markets after defaulting on external debt has the same effects as the ability tosave in our model. See the discussion in Wright ( 2006) for formal details.

19See Ventura (2004) and Caballero and Krishnamurthy (2006) for related work on bubblesin international capital flows. Notice that rational bubbles are often ruled out on theoreticalgrounds if there is a tradable asset that pays an infinite stream of dividends. This argument doesnot necessarily apply in a world with sovereign borrowers and multilateral lack of commitment.Real assets are typically tied to a physical location where the dividend is generated. The countrycontrolling that location may choose to expropriate foreign nationals and bar them from accessingthe assets dividends in the event of a default.


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on net flows, taking a sample of countries who borrow on the internationalcapital markets and evaluating whether their debtor position is associated to anet outflow or to a net inflow of resources. This is analogous to the approach in

Abel, Mankiw, Summers, and Zeckhauser (1989), who focused on the net flowsbetween the corporate sector and consumers in a closed economy. Empiricalwork in either direction will face two significant challenges. First, as empha-sized in Gourinchas and Rey (2005), net financial flows between countries arethe outcome of gross flows that include instruments with very different ratesof return and risk profiles (debt, equity, direct investment, and so on). Sec-ond, especially when looking at emerging economies, default episodes do takeplace regularly in international financial markets.20 In this paper, we allow forfully state-contingent securities, so we do not treat explicitly gross holdings ofdifferent assets and do not take a stand on how to interpret actual defaultswithin the context of our model. To map the models predictions to observeddata on financial flows that include defaults, it would probably be necessary toextend the analysis by modelling explicitly gross positions in various financialinstruments and to formulate an explicit interpretation of default episodes.

Finally, our paper also has implications for the literature on inside and out-side money. In particular, in our setup unbacked public debt and self-enforcingprivate debt are analogous, respectively, to outside (fiat) money and insidemoney. The existing monetary literature discusses the circulation of fiat moneyand inside money largely separately from each other. The circulation of fiat

money requires that an intrinsically useless asset (a rational bubble) is tradedat a positive price. The circulation of inside money instead relies on having theproper reputational mechanisms in place to guarantee that outstanding claimsare honored. Although on the surface these seem to be conceptually distinctproblems, our analysis shows that they are closely related.21

20After disentangling the various gross positions and estimating rates of returns on the variousinstruments, Gourinchas and Rey (2005) found that the United States after 1988 became a netdebtor and, in spite of that, has been typically receiving a net positive inflow of resources from

the rest of the world. This finding is consistent with our story. Klingen, Weder, and Zettelmeyer(2004) looked at ex post realized returns for a broad sample of emerging markets over the last 30

years and argued that returns were very close to the returns on 10-year U.S. treasuriesa levelwhich could very well be consistent with our story. Eichengreen and Portes (1989) and Lindertand Morton (1989) found similar results based on historical evidence.

21See Cavalcanti, Erosa, and Temzelides (1999) and Berentsen, Camera, and Waller (2007)for matching models of private debt circulation under limited commitment. In Cavalcanti, Erosa,and Temzelides (1999), a fixed subset of agents is allowed to issue notes, which are sustained bythe loss of a noncompetitive note-issuing rent if outstanding notes are not redeemed on demand.Berentsen, Camera, and Waller (2007) studied a matching model with money and competitivesupply of bank credit, and showed among other things that with lack of commitment, such creditis sustainable only if the inflation rate is nonnegative.


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APPENDIX: PROOF OF PROPOSITION 5

Suppose that V ((st)(st); st) = VD(0; st) for all st S, and that (s t) ( st+) and (st+) N1(st)}

B(st) = {st+ st : a(st+) = (s t+) and (st+) N1(s

t)}

N(st) =

=0

N(st) B(st) =

=0

B(st)

N(st+; st) =N(st) S(st+)

B(st+; st) = B(st) S(st+) for all st+ N(st)

N(s

t

) denotes the set of histories s

t+

along which the debt limit was neverbinding between events st and st+, and N(st) denotes the union of all suchsets, including st. B(s

t) denotes the set of histories st+ at which the debt limitis binding for the first time after st and B(st) denotes the union of all such sets.N(st+; st) defines the subtree ofN(st), which starts at st+, and B(st+; st)denotes the set of events at which the debt limit is binding for the first timeafter st+. Finally, we define

(st+; st) =

st++kB(st+ ;st)

p(st++k)

p(st+)(s t++k)

and

w(st+; st)

st++kN(st+ ;st)

p(st++k)

p(st+)y(st++k)

as the present value of the first binding debt limits and of the endowments overN(st).

The proof of Proposition 5 then proceeds in five steps, which are stated asseparate lemmas. Lemma 1 establishes that under mild regularity conditions,


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w(st+; st) and (st+; st) are both finite-valued. Lemma 2 establishes the ex-

istence of a well defined, finite-valued sequence of auxiliary debt limits (st),which satisfies the recursive characterization:

(st+; st) =

st++1st+ q(s

t++1) min{(s t++1) (st++1; st)}

ifa(st+) > ( st+)

(s t+) ifa(st+) = (s t+)

(11)

for all st+ S(st). Lemma 3 establishes that (st) bounds (st) from belowand satisfies the (ER) condition as a weak inequality. Lemma 4 establishes that

(st) satisfies (ER) as an equality and that (st; st) = (s t). Finally, Lemma 5

shows that (st+1; st) = (s t+1) for all st+1 st.

LEMMA 1: Suppose either that u() is bounded or that c

(st+

) J

j=1 yj

(st+

)for all st+ S(st), and that Assumption 1 holds. Then w(st+; st) < and

(s t+) + w(st+; st) > (st+; st) > .

PROOF: Summing the agents budget constraint over st++k N(st+; st), weget

st++kN(st+ ;st)

p(st++k)

p(st+)c(st++k)

= a

(s

t+

) + w(s

t+

; s

t

)

(s

t+

; s

t

)

limK

st++KN(st+ ;st)

p(st++K)

p(st+)a(st++K)

Now, using the agents first-order condition for st++k N(st+; st), we have

st++kN(st+ ;st)

p(st++k)

p(st+)c(st++k)

=

st++kN(st+ ;st)

k (s

t++k)

(st+)

u(c(st++k))

u(c(st+)) c

(st++k

) <

either because u(c)c u(c) u(0) U ifu() is bounded or because of As-sumption 1 ifc(st+)

Jj=1 y

j(st+) holds for all st+ S(st). In addition, the

agents first-order conditions over st++k N(st+; st), along with the transver-sality condition, imply

limK

st++KN(st+ ;st)

p(st++K)

p(st+)a(st++K) = 0


24/28


But then it follows immediately that w(st+; st) < and (st+; st) > .Finally, if a(st) = y(st) +

st+1st q(s

t+1)(st+1), the only feasible alloca-tion without default yields c(s t) = 0 and a(st+1) = (s t+1) for all st+1 st,

which yields a lifetime expected utility ofu(0) +

st+1st (st+1

|st

)V ((st+1

)(st+1) st+1). If instead the agent defaults and sets d(st+1) = 0 for all st+1 st,his lifetime expected utility is u(y(st)) +

st+1st (s

t+1|st)VD(0 st+1), which

must be strictly preferred to no default if(st) satisfies (SE). But then (s t) >y(st) +

st+1st q(s

t+1)(st+1) for all st S, and summing this inequality over

st++k N(st+; st), we find w(st+; st) + (s t+) (st+; st) > 0. Q.E.D.

Lemma 2 establishes the existence of a finite-valued sequence of auxiliary

debt limits (st).

LEMMA 2: There exists a finite-valued sequence (st) which satisfies (11),for all st+ S(st) and for which limK

st++KN(st+ ;st) p(s

t++K)(st++K;st) = 0.

PROOF: Step 1 establishes the existence of such a solution for st+ N(st),together with the limit property. Step 2 extends the construction to S(st).

Step 1. Let (0) be defined by (0)(st+) = (st+; st) Y (st+; st) and define(K)(st) recursively by (K) = T(K1), where the operator T on sequencesB RN(s

t)is defined by

(TB)(st+) =

st++1N(st+ ;st)

q(st++1) min{(s t++1)b(st++1)}

+

st++1B(st+ ;st)

q(st++1)(st++1)

Since (s t+) (st+; st) w(st+; st) = (0)(st+), we have(1)(st+) (0)(st+) for all st+ N(st) and, therefore, (1) T(0). ButT is a monotone operator and, therefore, {(K)} is a nondecreasing sequence.Moreover, since (K)(st+) 0 for all Kand st+ N(st), {(K)} must converge

to a finite limit (st

) which satisfies (11). The limit property then follows im-mediately from 0 (st++K; st) (0)(st++K) = (st++K; st) w(st++K; st)

and the fact that limK

st++KN(st+ ;st) p(st++K)[(st++K; st) w(st++K;

st)] = 0.Step 2. Define B(1)(st) = B(st) and let

B(k)(st) =

=k

st+ st : a(st+) = (s t+) and

(st+) Nst+

for some st+ B(k1)(st)


25/28


26/28


st++1 N(st+; st) and (s t++1) = (st+; st+) for all st++1 B(st+; st) in(11) then implies (ER) for all st+ N(st). Solving (11) forward and using the

limit property from Lemma 2, we find (st+; st) = (st+; st).

(ii) Applying the same arbitrage argument as in part (i) at st+

B(st

), wehave (s t+) (st+; st+) and (st+; st+) =

s++1s+ q(s

t++1)(st++1;

st+). Moreover, by construction, (st+; st) = (s t+) and (st++1; st) =

(st++1; st+) for all st++1 st+, from which the result follows immediately.Q.E.D.

Lemma 4 uses these two properties to show that (s t) = (st; st) and that

(st) satisfies (ER) with equality for all st+ S(st).

LEMMA

4: For all s

t S, (s

t

)=

(s

t;s

t

)and

(s

t

)satisfies (ER)with equal-ity.

PROOF: Consider the consumer problem with borrowing constraints equal

to (st). Since the objective is strictly concave, (s t+) (st+; st) for all

st+ S(st), and (s t+) > (st+; st) only if a(st+) > (st+), (st) relaxesonly nonbinding constraints and, hence, A(st) is also optimal for the relaxed

problem with borrowing constraints (st), implying V ((st) (st) st) =V ((st)(st) st). Using the self-enforcement hypothesis and the monotonic-

ity of V (a (st) st) in a and (s t) (st; st), this implies VD(0 st) =

V ((st)(st) st) = V ((st) (st) st) V ((st; st) (st) st). On the otherhand, since (st) satisfies the exact rollover property as a weak inequality, the

same argument as Proposition 4 implies that V ((st; st) (st) st) VD(0 st),

where the inequality is strict whenever (st+; st) >

st++1st+ q(st++1)

(st++1; st) for some st+ S(st). Together these inequalities can hold

only as equalities, which requires that (s t) = (st; st), and (st+; st) =st++1st+ q(s

t++1)(st++1; st) for all st+ S(st). Q.E.D.

To complete the proof that (st) satisfies (ER), we thus need to show that

(st+1

) = (st+1

; st

) for all st+1

S(st

). Whenever st+1

B1(st

), that is, when-ever the debt limit is binding at st+1, this is true by construction. Our finallemma shows that this is also true whenever the debt limit is not binding.

LEMMA 5: For all st+1 N1(st), (s t+1) = (st+1; st).

PROOF: Since st was chosen arbitrarily, applying all the preceding argu-

ments to st+1 N1(st) implies that (s t+1) = (st+1; st+1), so it suffices to show

that (st+1; st) = (st+1; st+1). Now, from the monotonicity of {a(st+)} withrespect to the initial asset holdings, it follows that N(st+1) N(st+1; st) and


27/28


B(st+1) B(st+1; st) N(st+1; st), that is, debt limits must be binding for anagent starting from st+1 with assets (s t+1) whenever they are binding for anagent starting from st+1 with assets a(st+1) > ( st+1). But then, by the defini-

tion of (11), (st+

; st+1

) = (st+

; st

) for all st+

B(st+1

; st

), and both (st

)and (st+1) satisfy (ER) for all st+ N(st+1; st), from which it follows im-

mediately that (st+; st+1) = (st+; st) for all st+ N(st+1; st) and, hence,

(st+1; st) = (st+1; st+1). Q.E.D.

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Dept. of Economics, University of California at Los Angeles, Box 951477, LosAngeles, CA 90095-1477, U.S.A. and Centre for Economic Policy Research, Lon-don EC1V 0DG, U.K.; [email protected]

and

Dept. of Economics, Massachusetts Institute of Technology, E52-251C, 50Memorial Drive, Cambridge, MA 02142, U.S.A. and NBER; [email protected].

Manuscript received October, 2006; final revision received December, 2008.
http://www.econometricsociety.org/ecta/Supmat/6754_extensions.pdfhttp://www.econometricsociety.org/ecta/Supmat/6754_extensions.pdfhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:21/Jes2006&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehPer2002&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehPer2002&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehPer2002&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehPer2002&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehPer2002&rfe_id=urn:sici%2F0012-9682%28200907%2977%3A4%3C1137%3ABASD%3E2.0.CO%3B2-Dhttp://www.e-publications.org/srv/ecta/linkserver/openurl?rft_dat=bib:22/KehP

hellwig on bubbles and sovereign debt

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