Private capital flows, capital controls, and default risk

Journal of International Economics 69 (2006) 120–149

www.elsevier.com/locate/econbase

Private capital flows, capital controls,

and default risk

Mark L.J. Wright *

Department of Economics, Stanford University, 579 Serra Mall, Stanford CA 94305-6072, United States

Received 20 August 2004; received in revised form 18 April 2005; accepted 10 May 2005

Abstract

What has been the effect of the shift in emerging market capital flows toward private sector

borrowers? Are emerging market capital flows more efficient? If not, can controls on capital flows

improve welfare? This paper shows that the answers depend on the form of default risk. When private

loans are enforceable, but there is the risk that the government will default on behalf of all residents,

private lending is inefficient and capital controls are potentially Pareto-improving. However, when

private agents may individually default, capital flow subsidies are potentially Pareto-improving.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Capital flows; Default; Borrowing constraints; Capital controls; Capital subsidies

JEL classification: F21; F34; F41; O19

1. Introduction

Capital flows to emerging markets have shifted markedly away from public sector

towards private sector borrowers. In the past decade, the private sector share of the stock

of emerging market debt rose more than fivefold from around five percent in 1990 to more

than twenty-five percent of the total by the end of the century. Private non-debt capital

flows have also increased rapidly, with gross foreign direct investment as a share of all

private capital inflows more than doubling over the same period.

0022-1996/$ -

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see front matter D 2005 Elsevier B.V. All rights reserved.

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ress: [email protected].

M.L.J. Wright / Journal of International Economics 69 (2006) 120–149 121

The rise in private capital flows has been accompanied by a change in attitudes towards

capital controls. In the decade of the 1990s, a number of countries, including Brazil and

Chile, introduced controls to limit short-term inflows on capital, while others, most

notably Spain, Malaysia and Thailand, introduced controls to limit capital outflows. And

while many of these actual experiences with controls were justified as temporary measures

in the face of a crisis, there have been increasing calls for controls to be used both as a tool

for macroeconomic management (for example, Obstfeld and Taylor, 2004), and to correct

other financial market imperfections.1

Motivated by these developments, in this paper we present a theory of private capital

flows in the presence of different forms of default risk and ask the normative question:

Does default risk provide a justification for government intervention in private

international capital markets? We first show that the risk of default by both individual

borrowers, as well as by national governments on their behalf, makes private financial

markets imperfect and introduces the possibility of default crises even in the absence of

sovereign debt. We then study the regulation of capital flows to prevent crises by

mitigating individual and national incentives to default on foreign borrowing, and show

that the optimal role of capital controls, taxes and subsidies depends upon the precise form

of default risk.

When there is national default risk, the government of a developing country has the

power to enforce private contracts between individual residents and foreigners, but can

default on the nation’s entire borrowings through various means such as the nationalization

of industries, or by closing the border to private debt repayments via exchange and capital

controls. National default risk thus introduces a separation between the decision to borrow

and the decision to default which creates the possibility of inefficient capital flows. In

particular, as private agents do not take into account the effect of their own private

borrowing decisions on the default decision of the government, they tend to overborrow.

In the presence of national default risk, the optimal form of government intervention

depends upon the degree of sophistication of financial markets. If financial markets are

bsimpleQ, in the sense that there is no price-discrimination, government intervention in the

form of capital controls (or taxes) is necessary to implement the constrained efficient level

of capital flows. That is, appropriate taxes on borrowing in normal (that is, non bcrisisQ)periods constrain private borrowing to the point that the government does not find it

optimal to default: default crises never occur. If, however, private capital markets are

bsophisticatedQ, in the sense of being able to price discriminate by offering non-linear

contracts, the constrained efficient level of capital flows can be attained without

government intervention.

We then go on to add the risk of resident default on international, but not domestic,

borrowing. In this case, even if financial markets are bsophisticatedQ, private capital flowsare inefficiently low in equilibrium. The reason lies in the asymmetry of enforcement;

because contracts between domestic residents are enforced, if the resident of a country

1 Ariyoshi et al. (2000) provide an excellent review of country experiences with capital controls, and the various

arguments in favor of their use. The International Monetary Funds changing views on capital controls are

summarized in the IMF Survey of May 19, 2003 (for the public reaction, see The Economist bA Slightly

Circuitous RouteQ and bA Place for Capital ControlsQ, May 1st, 2003).

M.L.J. Wright / Journal of International Economics 69 (2006) 120–149122

defaults on international debts and is excluded from direct access to international markets,

they are still able to access international capital markets using other residents as

intermediaries. This reduction in the penalty for default severely constrains the level of

capital flows that can be supported by private financial markets and leads to the potential

for beneficial government intervention. However, with resident default, efficiency is

achieved by capital flow subsidies: when private agents are tempted to default it is optimal

for the government to subsidize debt repayments in order to lower the benefits of default.

Such a policy acts like tax on default, in that a defaulter who is excluded from international

borrowing loses future access to the subsidy. However, unlike a tax on default, it is not

necessary for the government to know the identity of the defaulter.

Both forms of default risk are important in practice. Historically national default risk

has been a consequence of the doctrine of sovereign immunity which prohibited legal

action against a sovereign state. Although this doctrine has been eased over time, the

process of attaching assets remains difficult as many debtor countries have little in the way

of foreign assets. Moreover, national default risk extends beyond the borrowings of the

government itself to include default on private contracts either directly, by nationalizing

private industries or taxing foreign remittances, or indirectly by imposing capital controls

and other measures that prevent residents from honoring their obligations. Resident default

risk, on the other hand, results from the failure of domestic contract enforcement

institutions to enforce foreign claims on domestic residents. In some cases this is an

explicit feature of the law: for example, the 11th Amendment to the U.S. Constitution

prevents legal action against a U.S. State by non-residents. In other cases, this is as a result

of judicial corruption or partiality.2 Even in developed countries the ability to use domestic

courts to force fulfillment of a private contract can be difficult, as illustrated by the 1976–

77 Japan–Australia sugar dispute in which a group of Japanese sugar refiners refused to

honor contracts after world sugar prices fell dramatically (see, for example, March, 1979).

The theory we develop builds on a substantial literature on trade in financial assets in

the presence of enforcement frictions. One strand of this literature studies international

debt in the presence of default risk, and has been surveyed by Eaton and Fernandez (1995).

For the most part, this literature focuses on public sector lending and assumes that a

country is associated with a representative agent that makes all borrowing and default

decisions on behalf of its residents.3 In contrast, we examine borrowing by private agents

in the face of default risk, and show how the allocations obtained in this earlier literature

might result out of the interaction of private agents in a competitive equilibrium. Another

strand of this literature looks at the effects of enforcement frictions in closed economy debt

markets, as in Alvarez and Jermann (2000) and Kehoe and Levine (1993). Where the

current paper differs is in drawing a distinction between the borrowing decision and the

default decision (when private agents borrow subject to national repudiation risk), and in

studying the interaction of enforcement frictions in international markets and the lack

2 Judicial partiality is commonly cited as an important component of corruption in emerging markets. The

World Bank’s World Development Report of 1997 emphasized a lack of predictability in court rulings as a

constraint on foreign investment. Al-Kilani (1999) argues that discrimination on the basis of nationality is

common in middle eastern legal proceedings.3 Exceptions include Cole and English (1991, 1992), and Chang (1995).


thereof in domestic markets (which allows residents who have defaulted to continue

accessing international markets using other residents as intermediaries).

In two recent papers, Jeske (2001) and Kehoe and Perri (2002) study borrowing by

private agents in environments subject to default risk and argue that efficient

allocations require the imposition of capital controls. Jeske was the first to study the

problem of resident default risk, and argued that prohibitive capital controls, combined

with a centralized mechanism for apportioning international borrowing among

residents, can obtain the constrained efficient level of borrowing. In this paper, we

build on the analysis of Jeske and show that with resident default risk, constrained

efficient capital flows can be attained by capital flow subsidies without the need for

centralized borrowing. Kehoe and Perri study an economy with national repudiation

risk and show that private agents can be induced to borrow efficiently by government

taxes on capital flows. By contrast, we analyze environments with both national and

resident default risk and show that, if financial markets are sophisticated, capital

controls are either not necessary or are dominated by capital flow subsidies in terms of

efficiency.

The rest of this paper is organized as follows. Section 2 begins with a simple

example that illustrates the different forms of default risk and necessity for different

forms of government intervention. Section 3 outlines the general modeling environment

with national default risk, characterizes the constrained efficient allocations and studies

their implementation both with capital controls, and without controls when financial

markets are sophisticated. Section 4 then adds the potential for resident default risk and

establishes the necessity of capital subsidies. Section 5 discusses some alternative

enforcement arrangements that turn out to be equivalent to the environment of Section

4, while Section 6 concludes. A Technical appendix collects proofs of all results in the

text.

2. An expository example

We begin by developing the themes of this paper in the context of a simple example of

international capital flows. We first study the need for, and optimal form of, capital market

intervention in the presence of national default risk, before considering how the form of

intervention changes with resident default risk.

Consider a world economy consisting of two countries, each of which is inhabited by

a unit-measure of residents with identical logarithmic utility functions. In both countries,

output of the single non-storeable good alternates between high 1+y and low 1�y

levels, for some ya (0,1). Time lasts forever, with the bodd countryQ receiving the high

endowment in all odd numbered periods, and the beven countryQ receiving the high

endowment in even numbered periods, so that the world endowment is fixed at two in

all periods. Agents in both countries discount the future using the common discount

factor b.In this world, a first-best outcome is characterized by both countries perfectly

smoothing consumption. However, the presence of default risk – either national or resident

– will, in general, constrain trade so that the first-best is not attained.


2.1. National default risk

When nations, but not individual residents, can default we have national default risk. In

particular, suppose that each country is represented by a benevolent government that

borrows and makes default decisions, and that if the government of a country were to

default, the country can be punished by exclusion from financial markets. In this case,

capital flows between countries must be nationally self-enforcing, and so must deliver a

level of welfare for the country at least as large as can be attained in autarky. In the

symmetric case (the general version is studied in Section 3), it is easy to see that the

constrained efficient level of consumption alternates between 1+xN and 1�xN, where the

N stands for national default, and where xN is the smallest self-enforcing deviation from

consumption smoothing given by

xNuminzz0flog 1þ zð Þ þ blog 1� zð Þzlog 1þ yð Þ þ blog 1� yð Þg:

Note that if residents are sufficiently patient, or bzbFIu� log(1+y)/log(1�y), the

solution to this problem is xN=0 and full consumption smoothing is achievable. By

contrast, if residents are sufficiently impatient, or bVbAu (1�y)/(1+y), the expression

log (1+ z)+blog(1� z) reaches a maximum at z =y and there is no trade in equilibrium so

that the world is in autarky. In what follows we focus exclusively on the intermediate case

in which some, but not full, consumption smoothing takes place and ask whether or not

government intervention is required so that the borrowing decisions of private individuals

attain this constrained efficient outcome. The answer turns out to depend on both the

degree of sophistication of international financial markets, and whether or not residents

can also individually default (so that there is also resident default risk).

2.2. Efficient capital controls with national default risk

We first assume that both domestic and international capital markets are anonymous so

that no-one knows whether they are trading with another resident of the same country or

with a foreigner. In this case, all bonds must trade at the same price. However, this does

not support the constrained efficient outcome. To see this, suppose that world interest rates

are determined by the marginal rate of substitution of the country that has high output

today (and hence saves), so that the price of a bond is given by qN=b(1+xN)/(1�xN)Nb.As qNNb(1�xN)/(1+xN), residents in the country that has low output today would like to

borrow more than is efficient. The intuition is straightforward: because residents are small,

they do not take into account the effect of their borrowing decisions on the likelihood that

the government defaults and overborrow relative to the constrained efficient allocation.

In this case, government intervention to reduce borrowing is necessary to attain

efficient capital flows. In particular, if the government of the country whose residents are

borrowing today taxes borrowings at the rate

sN ¼ 1��1� xN

1þ xN

�2

N0;

and rebates the proceeds in lump-sum fashion, efficient capital flows can be attained. In

this case, for every claim sold today against one unit of consumption tomorrow, agents in


the country with low output receive (1�sN)qN units of consumption today, so that interest

rates are increased. As long as the proceeds are rebated to residents in lump sum fashion,

this satisfies each agent’s budget constraint, while the tax ensures that agents choose to

borrow the constrained efficient amount.

2.3. Sophisticated financial markets and national default risk

Now suppose that financial markets are sophisticated so that they can discriminate

between residents of different countries and price non-linearly. This places a greater

burden on financial markets, requiring that total borrowings are observable to all market

participants. However, in this case, efficiency can be attained without any government

intervention.

If bonds trade at price qN and bond holdings are stationary, then the efficient allocation

implies bond holdings of f N=( y�xN)/(1+qN), so that countries borrow when output is

low. Then, if financial markets were to offer each resident a non-linear price schedule

consisting of a linear price of qN up to a borrowing constraint of f N=( y�xN)/(1+qN)b0

(so that asset holdings must be greater than this negative amount), a competitive

equilibrium with borrowing constraints attains the efficient allocation.

Note that in this simple example, all agents are identical and so face the same individual

borrowing constraints. More generally, when agents differ, borrowing constraints must

also differ, sometimes in quite complicated ways. A focus of Section 3.3 will be on

characterizing these borrowing constraints in a more complicated setting.

2.4. Resident default risk

Now consider the case in which enforcement institutions discriminate in favor of

domestic residents so that there is also resident default risk, in addition to national default

risk. Financial markets are assumed to be sophisticated and treat contracts traded between

domestic residents differently from contracts traded internationally. In the event of a

default, a resident is excluded from international financial markets. However, because

domestic contracts are enforced, they can continue to access international capital markets

using other residents as intermediaries. This drastically reduces the amount of borrowing

that can be supported in equilibrium.

To characterize the stationary allocations that result in this case, note that if the value (at

domestic prices pR) of a resident’s expected series of repayments on their debts is negative,

then that resident would benefit from defaulting on their debts. An equilibrium in this

world therefore requires that the discounted value of these repayments is never negative,

and equals zero whenever an agent is borrowing constrained. If we focus on stationary

allocations of bonds trading at price qR, letting f R denote bond holdings that pay off when

output is high, and f RV bonds that pay off when output is low, then this requires

f R � qRf RV þ qRð f RV � qf RÞ1� pRqR

¼ xR � yþ qRðy� xRÞ1� pRqR

¼ 0;

where 1+xR is the equilibrium consumption in this world when output is high. There are

two solutions to this equation. The first is autarky, or xR=y, while the second requires


qR=1, in which case we get xR=(1�b)/(1+b)zxN. Both are worse than the constrained

efficient outcome, opening up the possibility of beneficial government intervention.

Outside autarky, international interest rates are zero so that international debt is a

bubble asset. Domestic interest rates are positive, which provides enough surplus from the

access to international financial markets to deter default. Exploitation of the apparent

arbitrage opportunity is prevented by the existence of the debt constraints. Note that debt is

also a bubble asset in the model of private liquidity in Hellwig and Lorenzoni (2003), and

that similar allocations are found in the turnpike monetary model of Townsend (1980). We

explore the relationship between all three of these frameworks in Section 5.

2.5. Efficient capital flow subsidies with resident default risk

With resident default risk, capital flow subsidies can implement the constrained

efficient allocation. To see how, note that residents of countries that are unconstrained at

time t must face domestic interest rates determined by world bond prices pU=qN, while a

country that is constrained has domestic interest rates determined by pC =b(1�xN)/

(1+xN)bpU. When unconstrained, agents buy and sell international claims at qN; when

constrained, we let the after tax (or subsidy) price be given by (1�sR)qN. We show that

the efficient allocation can be sustained by appropriately choosing a negative sR, orsubsidy.

Consider an agent facing interest rates (after taxes and subsidies) pt and recursively

define the sequence of time-zero domestic prices by Pt=ptPt�1 for some P0 given. Then

the discounted value of the stream of future repayments by a resident, starting from some

period r, is given by

Prfr � Prqrþ1 frþ1 þ Prþ1 frþ1 � Prþ1qrþ2 frþ2 þ Prþ2 frþ2 � N ::

If the agent is constrained in period r, then this must equal zero. Also, as the agent is not

constrained in period r +1, Pr+1=PrpU=Prq

N, while by symmetry the agent is constrained

in period two so that all the terms starting with Pr+2 fr+2 onwards sum to zero. Hence the

no-default condition in this case reduces to

0 ¼ Prfr � PrðqNÞ2ð1� sRÞfrþ2:

But then by symmetry we have (1�sR)= ( qN)�2, which is greater than one. This

implies a borrowing subsidy: because the after subsidy price of a bond is greater than

one in periods in which the consumer is selling bonds, for a given amount of repayment

tomorrow the consumer is able to borrow a greater amount today. That is, as long as the

subsidy is set appropriately, we can support the constrained efficient level of capital

flows.

3. National default risk

The previous example illustrated that the necessity for, and optimal form of, capital

controls on private capital flows depended on the form of default risk and the

sophistication of international financial markets. The following sections establish these


results in greater generality, and develop some features that emerge once the assumptions

of symmetry are relaxed.

3.1. Constrained efficient allocations

Consider a world economy populated by a finite number of countries denoted

m =1, . . . , M. Each country m is populated by N types of agents. We will use

subscripts to denote types of individuals and superscripts to denote countries, so that

knm N0 denotes the measure of agents of type n in country m. Time is discrete and in

each period t=0, 1, 2 . . . information about current and future endowments is

indexed by the state ht, an element of the finite set H. Information about states forms

a Markov chain, and the transition probability from h to hV is given by p(hV|h) with

the initial state h0 given. We let htu (h0, h1, . . . , ht)aHt denote a history of states

up to date t. The notation hs|ht for sN t refers to a history hs that continues ht in the

sense that hs=(ht, ht+1, ht+2, . . . , hs). The probability of observing history ht is

denoted by p(ht), and that of observing history hs conditional on having been in ht is

p(hs|ht).There is one non-storable good available for consumption in each history. A state h

specifies a vector of endowments for each type of agents in each country. We denote by

enm(ht) the endowment of a type n agent in country m after history ht and by cn

m(ht) thecorresponding consumption level. It will be convenient to define, for all countries m and

for all histories ht, the aggregate endowment of a country, m, as em htð Þ ¼PN

n¼1 kmn emn htð Þ

and the aggregate consumption level of country m as cm htð Þ ¼PN

n¼1 kmn cmn htð Þ.Individuals

have preferences ordered by a time additively separable function

Xlt¼0

btXht

pðhtÞUmn ðcðh

tÞÞ;

where ba (0,1) is the discount factor of an agent, which is assumed the same over all

types, while the function Unm is the period utility function which is strictly increasing,

strictly concave, and twice continuously differentiable.

In the presence of national default risk, capital flows between countries must be such

that they are nationally self-enforcing: an allocation must deliver the residents of a nation

sufficient utility to deter their national government from defaulting on their behalf. It is

assumed that each national government is benevolent with respect to its own citizens and

acts to maximize a weighted average of their lifetime utilities where /nmN0 is the weight

assigned to agents of type n by the government of country m, and where for convenience

we normalizeP

n /mn ¼ 1 and use /mu{/n

m} to denote the vector of weights for

residents in country m. We allow governments to make arbitrary lump-sum transfers

between its citizens.

In this environment, a consumption allocation c ={cnm(ht)}n,m,h t,t is resource feasible if

it is non-negative and satisfies the resource constraint

XMm¼1

XNn¼1

kmn cmn ðh

tÞVXMm¼1

XNn¼1

kmn emn ðh

tÞ; ð1Þ


for all t and all ht. A consumption allocation satisfies the sequence of national

participation constraints if it satisfies

XNn¼1

/mn

XsNt

bs�tXhsjht

pðhsjhtÞUmn ðcmn ðhtÞÞzDmðht; /mÞ; ð2Þ

for all m, t and for all ht. The constraint (2) states that the continuation of an allocation for

a country m must, after each history, deliver at least as much weighted lifetime utility as

default to autarky for that country as a whole. We denote the latter by Dm(ht, /m), which is

calculated by maximizing the {/nm}-weighted average of residents’ utilities subject to the

constraint that cm(ht) not exceed em(ht). For future reference, we let Dnm(ht, /m) denote

the lifetime utility of an agent of type n in country m that is in international autarky.

In this environment, an allocation is constrained efficient if it satisfies resource and

incentive feasibility, and there is no other allocation satisfying these constraints that is

strictly Pareto-preferred. To characterize the set of constrained efficient allocations, let

um N0 denote the Pareto weight attached to the welfare of country m by the international

social planner (so that um/nm is the welfare weight attached to a citizen of type n living in

country m). We can characterize the set of efficient allocations by solving the following

convex planning problem:

maxc

XMm¼1

umXNn¼1

/mn

Xlt¼0

btXht

pðhtÞUmn ðcmn ðhtÞÞ; ðPPÞ

subject to the sequence of resource and incentive feasibility constraints in (1) and (2).

The strong separability embodied in this problem implies that the solution to the

planning problem in (PP) can be found in two stages. In the second stage, aggregate

country consumption is taken as given at cm(ht) and is allocated between residents of a

country to maximize the weighted average of their utilities defined as Vm(cm(ht), /m). The

solution to the second-stage problem equates marginal rates of substitution across residents

within a country. In the first stage, the planner allocates consumption between countries to

maxc

XMm¼1

umXlt¼0

btXht

pðhtÞVmðcmðhtÞ; /mÞ;

subject to

XMm¼1

cmðhtÞVXMm¼1

emðhtÞ;

for all t, andXsNt

bs�tXhsjht

pðhsjhtÞVmðcmðhtÞ; /mÞzDmðht; /mÞ;

for all m and all t.

This result is useful because it allows us to solve a standard limited commitment

problem at an international level to determine the efficient level of intertemporal trade


between countries before later allocating these goods between residents of a country.4 In

what follows, we make use of some well known properties of the solution to the first stage

problem which we note here without proof. First, there exist two values for the discount

factor, bA and bFCI corresponding to autarky and full consumption insurance, with

bAbbFCI, such that if countries are sufficiently impatient (b bbA) autarky is the only

feasible allocation, while if countries are sufficiently patient (b NbFCI), full consumption

insurance is attainable. Second, if b NbA, in two country models, the ergodic set for

consumption displays strictly less variability than the autarky allocation. Proofs of these

results can be found in, for example, Alvarez and Jermann (2000), Kocherlakota (1996)

and Kletzer and Wright (2000), while further references can be found in Eaton and

Fernandez (1995).

For the rest of this paper we take the constrained efficient allocations as given and ask

whether different market environments are capable of obtaining these allocations. We

focus on the case in which residents are sufficiently patient so as to admit some

international capital flows to all countries. That is, we assume that for all m there exists a

ht such that Vm(cm(ht), /m)NVm(em(ht), /m).

3.2. Implementation with capital controls

Assume that capital markets are anonymous so that no-one knows whether they are

trading with another resident of the same country or with a foreigner. In this case, residents

of all countries must be able to trade at the same prices. International financial markets

trade a full set of state contingent one-period international Arrow securities, with prices

denoted by q(ht, ht+1) for securities purchased after history ht for payment in period t+1

after observing event ht+1, and we denote by bnm(ht, ht+1) the holdings of these securities

by a type n agent in country m. Domestic courts act to enforce all contracts entered into by

domestic agents. However (as long as b bbFCI) marginal rates of substitution are not

equalized across residents in different countries at the constrained efficient allocation.

Hence, if all agents are able to trade at these same prices, they would not all choose the

constrained efficient allocation. In particular, agents would like to borrow more against

income in future states of the world in which output is higher than is allowed in the

constrained efficient allocation. Capital controls serve the role of constraining this

borrowing.

In order to ensure that agents in unconstrained countries borrow and save appropriately,

it must be the case that equilibrium prices equal their marginal rate of substitution. This

leads to our candidate equilibrium price sequence for bonds

qðht; htþ1Þumaxn;m

�bUmV

n ðcmn ððht; htþ1ÞÞÞUmV

n ðcmn ðhtÞÞpðhtþ1jhtÞ

�; ð3Þ

4 However, in general the solution of the first-stage problem requires knowledge of the appropriate resident

welfare weights /mu{/nm} which determine the form of the period country welfare functions Vm(cm(h t), /m) as

well as the default values Dm(ht, /m). An exception is the special case in which preferences are homothetic and

identical within a country.


for all dates t and histories ht. In order to induce agents in constrained countries to borrow

appropriately, we introduce capital controls, sm(ht, ht+1), in the form of taxes on Arrow

securities such that

qmðht; htþ1Þuð1� smðht; htþ1ÞÞqðht; htþ1Þ ¼ bUmV

n ðcmn ððht; htþ1ÞÞÞUmV

n ðcmn ðhtÞÞpðhtþ1jhtÞ;

where qmVq are after tax domestic prices. These controls are independent of the agents’

type n by the fact that marginal rates of substitution are equalized within a country. Using

these prices we can define, for each country, the after tax time zero Arrow–Debreu price of

a unit of consumption at time t after realization of history ht as

Qmðhtjh0Þ ¼ qmðh0; h1Þqmðh1; h2Þ: : :qmðht�1; htÞ: ð4Þ

Before tax world prices, Q(ht|h0), are defined analogously.

The problem of a resident of type n of country m in this implementation is to choose

sequences for consumption and security holdings to maximize

Xlt¼0

btXht

pðhtÞUmn ðcðhtÞÞ;

subject to

cðhtÞ þXhtþ1

ð1� smðht; htþ1ÞÞqðht; htþ1Þbmn ðht; htþ1Þ

¼ emn ðhtÞ þ bmn ðht�1; htÞ þ Tmn ðhtÞ;

for all ht =(ht�1, ht) for all t, and bnm(ht, ht+1)z�B�nm(ht, ht+1), for all ht for all t, with b

(h0) given. Here, B�nm(ht, ht+1) is the natural borrowing constraint defined as the value of

the agent’s endowment and lump-sum transfers

�B

mn ðht; htþ1Þ ¼ �

Xsz1

Xhtþs

QmðhsjhtÞ½emn ðhtþsÞ þ Tm

n ðhtþsÞ�;

which is sufficient to ensure that the budget set is compact and never binds if the period-

utility function obeys the Inada conditions, while Tnm(ht) is a lump-sum transfer from the

government. The following proposition shows that if we construct capital controls in this

way and rebate the proceeds in lump-sum fashion, we can support the constrained efficient

allocation as a competitive equilibrium with capital controls.

Proposition 1. Let {cnm(ht)} be an allocation that solves the planning problem above with

some international capital flows. Define Arrow and Arrow–Debreu prices by (3) and (4).

Then there exists sequences of capital controls {sm(ht, ht+1)} and lump-sum transfers

{Tnm(ht, ht+1)}, an initial wealth allocation defined by {bn

m(h0)}, and a sequence of asset

holdings {bnm(ht,ht+1)} such that the plan {cn

m(ht), bnm(ht, ht+1)} is a competitive

equilibrium with capital controls.


3.3. Decentralization with solvency constraints

In the previous subsection, capital controls, in the form of taxes on foreign borrowing,

were introduced to limit borrowing by private agents to levels that ensured national

governments had no incentive to default. Capital controls were necessary because

constrained efficiency requires the marginal rates of substitution of agents in different

countries to differ when their governments are tempted to default. As long as financial

markets must price assets linearly, they cannot discriminate between agents in different

countries, and government intervention is necessary.

In practice, non-linear pricing is common in markets that, in all other respects, look

competitive. A number of authors have made use of this fact to consider asset prices in

economies where trade is limited by information (Kocherlakota and Pistaferri (2004)) and

enforcement (Alvarez and Jermann (2000)) frictions. In this subsection, we show that with

a simple form of non-linear pricing – linear pricing up to a solvency constraint –

international financial markets can attain the constrained efficient allocation. That is, the

constrained efficient allocation can be decentralized, and no government intervention is

necessary.

As in the previous subsection, it is assumed that there exists a full set of state contingent

one-period international Arrow securities with prices set by the marginal rate of

substitution of agents in unconstrained countries. The problem of a resident of type n of

country m in this decentralization is to choose sequences for consumption and securities

holdings to maximize

Xlt¼0

btXht

pðhtÞUmn ðcðh

tÞÞ;

subject to

cðhtÞ þXhtþ1

qðht; htþ1Þbmn ðht; htþ1Þ ¼ emn ðhtÞ þ bmn ðht�1; htÞ;

for all ht =(ht�1, ht) for all t, and

bmn ðht; htþ1Þz��Bm

n ðht; htþ1Þ; ð5Þ

for all ht for all t, with bnm(h0) given. Here, all agents face the same prices, but now each

agent is constrained by B�nm(ht, ht+1), a borrowing constraint that is set so as to prevent

agents from accumulating so much debt that their government is tempted to default on

their behalf. Following Alvarez and Jermann (2000), the constraint in (5) will be referred

to as a solvency constraint and is type, country and history dependent. This leads to our

definition of a competitive equilibrium with solvency constraints.

Definition 1. A competitive equilibrium with solvency constraints is a sequence of

solvency constraints B�nm(ht, ht+1) for all countries m, types n, dates t and histories ht,

a price system { q(ht, ht+1)} for all countries m, dates t, and histories ht, and an

allocation {cnm(ht), bn

m(ht, ht+1)} such that for all countries m and types n, the

allocation solves the resident’s problem given the price system, solvency constraints,


and initial asset holdings, and markets clear. That is, for all dates t and all histories ht,goods markets clear

XMm¼1

XNn¼1

kmn cmn ðhtÞV

XMm¼1

XNn¼1

kmn emn ðhtÞ;

and for all ht+1 the markets for securities clear

XMm¼1

XNn¼1

kmn bmn ðh

t; htþ1Þ ¼ 0:

We will exhibit a sequence of solvency constraints for every agent that prevent default

by not allowing agents to accumulate more debt than the country as a whole would be

willing to repay. The set of such solvency constraints is in general very large, and so we

aim to set these constraints to be as minimally constraining as possible. In the closed

economy model of Alvarez and Jermann, indeterminacy in the solvency constraints arose

because there were many possible levels to set the solvency constraint in periods in which

it does not bind. Alvarez and Jermann introduce the terminology bnot too tightQ to capture

their solution which involves setting the solvency constraints so that an agent on its

borrowing constraint would be indifferent to default in all states.

In addition to this form of indeterminacy, in the present context of international lending

there is an extra indeterminacy introduced by the fact that the borrowing of each type of

agent needs to be constrained so that aggregate borrowing is consistent with repayment

incentives for the national government. That is, the separation of borrowing and default

decisions means that there needs to be solvency constraints for each type of agent,

although there is only one participation constraint for the country. We pin down the

solvency constraints across agents within a country by, first, requiring that if all agents are

on their borrowing constraints, the country would be indifferent to default, and second,

that agents would have no incentive to trade away from these borrowing constraints. The

latter is imposed by requiring that marginal rates of substitution are equalized (or

equivalently that marginal values of wealth are proportional).5

To make these requirements precise, letWnm(b, ht) be the value to a resident of type n in

country m who begins at history ht with bond holdings b. The functional equation for this

agent defines Wnm(b, ht) as

maxc; bV; f Vf g

�U cð Þ þ b

Xhtþ1

Wmn ðeVðh

t; htþ1Þ þ bVðht; htþ1Þ; ðht; htþ1ÞÞpðhtþ1jhtÞ�;

subject to

cþXhtþ1

qðht; htþ1Þbðht; htþ1Þ ¼ emn ðhtÞ þ b;

5 When the national participation constraint binds, the weighted sum of resident utilities of repayment equals

the weighted sum of resident utilities in default. This does not, in general, imply that the individual resident’s

utility levels are equal to their level in default.


bðht; htþ1Þz��B m

n ðht; htþ1Þ:

This leads to the following definition.

Definition 2. An equilibrium with solvency constraints that are minimally constraining is

such that the solvency constraints satisfy, for all dates t and all histories ht

XNn¼1

/mn W

mn ð �

�B

m

n ðht�1; htÞ; htÞ ¼ DmðhtÞ;

and such that for all n, nV

/mn

BWmn ð �

�B

m

n ðht�1; htÞ; htÞBb

¼ /mnV

BWmnV ð �

�B

m

nVðht�1; htÞ; htÞBb

:

When solvency constraints are minimally constraining, they will bind simultaneously

for all agents if and only if the corresponding participation constraint in the planner’s

problem binds. When the solvency constraints bind, the marginal rates of substitution of

all agents are not equalized across countries, although they are equalized within countries.

Agents in constrained countries will accumulate assets according to their solvency

constraints.

Using these results, we are able to state a version of the second welfare theorem for our

economy.

Proposition 2. Let {cnm(ht)} be an allocation that solves the planning problem above with

some international capital flows, and define Arrow and Arrow–Debreu prices by (3) and

(4). Then there exists a sequence of solvency constraints {B�nm(ht, ht+1)}, an initial wealth

allocation defined by {bnm(h0)}, and a sequence of asset holdings {bn

m(ht, ht+1)} such that

the plan {cnm(ht), bn

m(ht, ht+1)} is a competitive equilibrium for the solvency constraints

and initial wealth allocation. Moreover, the sequence of solvency constraints can be

chosen so that they are minimally constraining.

The equilibrium concept used here has two desirable features. First, individual

residents retain access to international financial markets, and it is to these markets that

the burden of ensuring that default does not occur is devolved. Second, the

decentralization does not rule out the possibility of default: default remains feasible.

However, default does not occur because the solvency constraints leave agents with no

incentive to default.

There are also two obvious criticisms of this approach. First, and in contrast to the

closed economy model, the individual solvency constraints may require that some agents

save a minimum amount in bonds. To see this, note that complete markets within a country

serve to ensure that marginal rates of substitution are equalized within a country. Hence,

when one agent is borrowing constrained, all agents must be constrained. In general, the

asset positions of agents will depend upon their initial wealth levels, and at any moment in


time some agents within a country may be creditors even when a nation as a whole is a

debtor.

Second, there is a sense in which the burden of ruling out default that has been

placed upon financial markets is very large. As countries must have no incentive to

default, in setting these solvency constraints it is necessary for financial markets to

have knowledge of the preferences of a nation’s government, which in turn depends

upon the preferences of its residents and the vector of national Pareto weights.

Similarly, financial markets must have knowledge of the aggregate endowment process

of a nation, as it is this process that determines the worst that can happen to a country

in default.

In summary, we have shown that when domestic courts enforce contracts

symmetrically between both residents and foreigners, it is possible to decentralize the

constrained efficient allocations in an economy in which individuals borrow subject to

their own solvency constraint. The requirements for financial markets, in terms of

information and the ability to discriminate between agents, are large, but this obviates

the need for government intervention. This contrasts with the previous subsection in

which financial markets were limited in their sophistication, but the same information

burden is borne by the government in setting its capital controls. In the next section we

consider the alternative case in which domestic courts do not enforce contracts with

foreigners and residents can individually default and show that even if financial markets

are sophisticated, government intervention in the form of capital subsidies is necessary

to attain efficiency.

4. Resident default risk

4.1. Competitive equilibrium

In this section we examine the level of capital flows that can be attained in a

competitive equilibrium in which there is both the risk of national default, as well as

the risk of default by a nation’s residents. This resident default risk results from the

failure of domestic contract enforcement institutions to enforce foreign claims on

domestic residents, and results in practice from both explicit partiality in the legal

system as well as implicit partiality arising from judicial corruption. Resident default

risk was first studied by Jeske (2001) whom we follow in assuming that the domestic

court system perfectly enforces contracts between domestic residents, but does not

enforce contracts with foreigners. The only punishment that can be levied against an

agent in default on their foreign borrowings is exclusion from international financial

markets.

In this economy, debts must be distinguished according to whether they are between

two residents, or between a resident and a foreigner. That is, given M countries there are

M+1 sets of state contingent securities. The first M sets of these securities are country

specific trading only within country m while the last set are securities traded on world

markets. We denote the prices of these one-period securities as q(ht, ht+1) for international

securities purchased after history ht for payment in period t+1 after observing event ht+1,


and, analogously for securities in country m, by pm(ht, ht+1). Corresponding period-zero

dated security prices are given recursively by

PmðhsÞ ¼ Pmðhs�1Þpmðhs�1; hsÞ; for all m; and QðhsÞ ¼ Qðhs�1Þqðhs�1; hsÞ:

Holdings of foreign securities are denoted f(ht, ht+1) and of domestic securities in country

m, by bm(ht, ht+1).

As residents are small relative to the market, a resident that defaults does so assuming

that prices in domestic markets stay unchanged. Given a domestic price vector pm the

problem of a resident that defaults at a given point in time s after history hs with domestic

bond holdings b is represented as

Dmn ðb; hs; fpmgÞ ¼ max

c; bVf g

Xlt¼s

bt�sXht jhs

pðhtjhsÞUmn ðcðh

tÞÞ

subject to

cþXhtþ1

pmðht; htþ1Þbðht; htþ1Þ ¼ eðhtÞ þ b; bðht; htþ1Þz��B;

for all ht that continue hs with b given, and where in the last constraint B�

is set sufficiently

large so that it never binds, and ensures compactness of the budget set. Note that the

default values Dnm here differ from those calculated in Section 3.

The value Dnm(b, ht, {pm}) plays an important role in determining the set of self-

enforcing allocations. Specifically, the resident’s problem (RP) is to choose sequences for

consumption and domestic and international securities holdings to maximize

Xlt¼0

btXht

pðhtÞUmn ðcðhtÞÞ; ðRPÞ

subject to

cðhtÞ þXhtþ1

qðht; htþ1Þf ðht; htþ1Þ þXhtþ1

pmðht; htþ1Þbðht; htþ1Þ

VeðhtÞ þ f ðht�1; htÞ þ bðht�1; htÞ; ð6Þ

for all ht =(ht�1, ht) for all t,

Xlt¼s

bt�sXht jhs

pðhtjhsÞUmn ðcðhtÞÞzDm

n ðbðhs�1; hsÞ; hs; fpmgÞ; ð7Þ

for all hs, for all s, and

bðht; htþ1Þz��B; f ðht; htþ1Þz��F ð8Þ

for all ht for all t, and b (h0) given. Here, (6) is the usual flow budget constraint, and (7) is

the participation constraint which guarantees that a resident cannot choose a path for

consumption that would leave them preferring to default after any history hs. The final


constraints (8) ensure compactness of the budget set. Note that we have followed Kehoe

and Levine (1993) in imposing the participation constraint (7) on the consumption set of a

resident. This makes default infeasible for the resident, which is much stronger than in the

previous section where it was feasible, but never optimal, for an agent to default.

In this framework, we allow the domestic and international state contingent securities to

trade at different prices. This is despite the fact that such securities have the same payoffs

and are free of default risk (domestic securities by virtue of domestic enforcement

mechanisms, and international by virtue of the unfeasibility of default implied by

constraint (7)). Indeed, it will emerge that, unless the interest rate on domestic borrowing

is higher than that on international borrowing, there is no effective penalty for default.

Arbitrage opportunities are ruled out by the participation constraint (7).

This leads to the definition of an equilibrium in this economy. To distinguish it from the

equilibrium with solvency constraints above, we follow Kehoe and Levine (1993) and

refer to it as a debt constrained equilibrium.

Definition 3. A debt constrained equilibrium is an allocation {cnm(ht), bn

m(ht, ht+1), fnm(ht,

ht+1)}, and a price system { q(ht, ht+1), pm(ht, ht+1)} for all countries m, such that for all

countries m and types n, the allocation solves the resident’s problem given the price

system and initial asset holdings, and markets clear:

XMm¼1

XNn¼1

kmn cmn ðh

tÞVXMm¼1

XNn¼1

kmn emn ðh

tÞ; for all dates t and all histories ht

XNn¼1

kmn bmn ðh

t; htþ1Þ ¼ 0; for all htþ1 and all m;

XMm¼1

XNn¼1

kmn fmn ðh

t; htþ1Þ ¼ 0: for all htþ1:

The characterization of this equilibrium is inhibited by the fact that the consumption set

of a resident is not convex. This follows from the fact that Dnm(b, hs, { pm}) is typically

strictly concave in b for all hs and { pm}. Jeske (2001) shows how this problem can be

surmounted. For a resident of type n in country m, the transfer they make to foreigners

after any history hs is equal to

nxmn ðhsÞ ¼

Xhsþ1

qðhs; hsþ1Þf mn ðhs; hsþ1Þ � f mn ðh

sÞ;

which we denote by nxnm(hs) as the net exports of a resident of type n.

Clearly, if the value of the future stream of such transfers, evaluated at domestic prices,

was positive, a resident would be better off by defaulting on their debts. That is, a

necessary condition for repayment is that foreign bond holdings satisfyXhszht

PmðhsÞnxmn ðhsÞV0; ð9Þ

for all histories ht, where the notation hszht is used to denote that the summation is over

all histories that continue ht. Note that this constraint defines a convex subset of the set of


affordable allocations. Jeske (2001, Propositions 3 and 4) showed that the solution to an

auxiliary resident’s problem, which is simply problem (RP) modified in that the sequence

of participation constraints (7) is replaced by the sequence in (9), is also the solution to the

original problem.

As domestic markets are complete and domestic contracts are enforceable, in

equilibrium we have

pmðht�1; htÞ ¼ bpðhtjht�1ÞUmV

n ðcðht�1; htÞÞUmV

n ðcðht�1ÞÞ; ð10Þ

for all n. That is, all residents of a country have the same marginal rate of substitution. This

also implies that, in any state of the world ht, either all residents are on their participation

constraints (7) or none are constrained. The intuition is that, if some residents were

constrained while others were not, the unconstrained residents could borrow from

international markets and lend to the constrained residents. In this sense, we can refer to a

country as a whole as being either constrained or unconstrained. Further, if a country m is

unconstrained, it must be that the international bond price equals the marginal rate of

substitution of all residents in that country. If the residents of a country are constrained,

they would like to borrow more, and their marginal rates of substitution must be lower

than this price. This implies that

qðht�1; htÞ ¼ maxm¼1; N ; M

pmðht�1; htÞ: ð11Þ

That is, domestic interest rates are always at least as high as international interest rates.

It is this fact that acts as a deterrent to default; a resident who defaults would only have

access to a market in which interest rates are higher in the periods in which an individual

wants to borrow.

So far we have concentrated solely on the resident’s decision to default. A natural

question that arises is whether it is possible for residents to voluntarily enter into

agreements that their government would like to repudiate? As agents cannot

coordinate in default, is it possible that they could receive allocations worse than

autarky in some states of the world? The following proposition shows that this is

never the case.

Proposition 3. In the debt constrained equilibrium, the {/nm}-weighted average of

residents’ utilities, in the continuation of the allocation after any history ht, is always no

less than Dm(ht, /m).

The debt constrained allocations are resource feasible and, by the result of Proposition

3, satisfy the national participation constraints. Hence they are no better than the

constrained efficient allocation. It is also straightforward to produce examples, such as the

one examined in Section 2.4, in which the debt constrained equilibrium is strictly worse

than the constrained efficient equilibrium. The intuition is straightforward: because

domestic debt contracts are enforced, an agent in default who is excluded from direct

access to international capital markets continues to have indirect access to international

capital markets using other residents as intermediaries. This contrasts with the case of

national default when all agents are excluded from international financial markets. Thus,


the punishment to default has been severely reduced, and only smaller levels of capital

flows can be supported in equilibrium.

The fact that private financial markets do not achieve the constrained efficient level

of capital flows opens up the possibility for Pareto-improving government intervention.

The obvious remedy is for the government to reform the domestic law enforcement

system to allow domestic residents to commit to honoring contracts with foreigners.

However, in practice such reform may not be possible. Current efforts at reducing

corruption in the developing world, including removing partiality towards domestic

residents in judicial systems, have run into a number of difficulties. However, while

national governments are often unable to effect reform of domestic institutions, they

typically retain control over their borders, and hence also over flows of capital. We show

in the next subsection that capital flow subsidies can be used to produce a Pareto

improvement.

4.2. Efficient borrowing subsidies

Consider the constrained efficient allocation {cnm(ht)}n,m,h t,t constructed in Section 3.1.

Given this allocation, we can construct sequences of domestic and international bond

prices using the formulae (10) and (11). By the results above, we know that this allocation

evaluated at these prices will in general violate the sequence of participation constraints

(9). Moreover, the equality of marginal rates of substitution over residents of a country

implies that this constraint will be violated for all residents of a country at the same time.

Knowledge of the allocation also allows us to calculate the contribution of each type to net

exports, as nxnm(ht)=en

m(ht)�cnm(ht), which given an international price for bonds in this

new equilibrium q(ht, ht+1) (which will in general differ from the q calculated from Eq.

(11)), implies asset holdings of

nxmn ðhtÞ ¼

Xhtþ1

qðht; htþ1Þf mn ðht; htþ1Þ � f mn ðh

tÞ:

Our approach will be to introduce a subsidy on international lending so that the

rents gained from access to international markets are larger, and the penalties from

being excluded are greater. Attaching the subsidy to new lending is notationally

simpler, but equivalent to, a policy of subsidizing a resident’s interest payments on past

borrowing.

In order to introduce such a subsidy, it will be necessary to distinguish between before-

tax and after-tax international bond prices. Specifically, denote by qm(ht, ht+1), the price ofan international state contingent security, after taxes and subsidies, in country m. This will

be determined by the international price q(ht, ht+1) and any tax (or subsidy if negative)

sm(ht, ht+1), according to

qmðht; htþ1Þ ¼ ð1þ smðht; htþ1ÞÞqðht; htþ1Þ;

where for the moment we are allowing this subsidy to be country specific. A government

policy is then a sequence of taxes and subsidies on foreign lending sm(ht), and a sequence

of lump-sum taxes Tnm(ht), which for the moment we allow to be resident specific.


Our task is, given a consumption allocation, to find a government policy and a

sequence of before-tax prices { q(ht, ht+1)} such that the implied securities holdings can be

supported in a debt constrained equilibrium. This leads to the following definition.

Definition 4. A debt constrained equilibrium with taxes is an allocation {cnm(ht), bn

m(ht,ht+1), fn

m(ht, ht+1)}, a government policy {sm(ht), Tnm(ht)}, and a price system for

international bonds { q(ht, ht+1)}, and domestic bonds { pm(ht, ht+1)} for all countries m,

such that for all countries m and types n, the allocation solves the resident’s problem given

the price system, government policy and initial asset holdings, each government’s budget

balances,

Xlt¼0

Xht

XNn¼1

PðhtÞ½smðhtÞqðht; htþ1Þf mn ðht; htþ1Þ � Tm

n ðhtÞ� ¼ 0;

and markets clear:

XMm¼1

XNn¼1

kmn cmn ðh

tÞVXMm¼1

XNn¼1

kmn emn ðh

tÞ; for all dates t and all histories ht

XNn¼1

kmn bmn ðh

t; htþ1Þ ¼ 0; for all htþ1 and all m;

XMm¼1

XNn¼1

kmn fmn ðh

t; htþ1Þ ¼ 0: for all htþ1:

The following proposition establishes that with appropriately designed borrowing

subsidies, the constrained efficient level of capital flows can be achieved. The intuition is

straightforward: by subsidizing borrowing, the government of a country increases the

penalty to default, which supports more lending in the market equilibrium.

Proposition 4. Let {cnm(ht)} be a constrained efficient consumption allocation with some

international capital flows, and define Arrow–Debreu prices from (3) and (4). Then this

consumption allocation can be supported as a debt constrained equilibrium with taxes.

Moreover, the taxes form a borrowing subsidy.

5. Alternative default penalties

In the previous section, international capital flows were constrained by the fact that

residents in default could not be completely excluded from access to financial markets.

Although unable to access them directly, they were able to use other domestic residents as

intermediaries who could access world markets on their behalf. The fact that agents in

default retain some access to international capital markets after default is reminiscent of

the environment introduced by Bulow and Rogoff (1989) in which agents continued to

have access to international savings after default. Unlike that partial equilibrium model,

but in common with the recent general equilibrium version of that model introduced by

Hellwig and Lorenzoni (2003), borrowing can be supported with resident default risk,


despite the reduced default penalty. Indeed, in the simple example presented in Section 2,

the allocations were the same as the ones produced in the private liquidity equilibria of

Hellwig and Lorenzoni.

In this section, we probe the connections between the two frameworks and establish an

equivalence result between them. The equivalency result is limited only by the fact that in

the model of this paper, there is an extra dimension of heterogeneity among residents of a

nation. Consequently, we show first that every private liquidity equilibrium is also a debt

constrained equilibrium with resident default risk. We then go on to show that for every

debt constrained equilibrium with resident default risk, there exists a set of representative

agents such that the equilibrium allocations are attained in the corresponding private

liquidity equilibrium.

The intuition for this equivalence is straightforward. Although in our model a resident

in default is excluded from all international financial markets, when they want to save

domestically they get the international interest rate and so it is as though they could save

internationally. However, when they want to borrow, they face a domestic interest rate that

is much higher than the world interest rate. Thus it is as if they were excluded from

borrowing internationally.

To understand this equivalence, it is first necessary to describe the private liquidity

equilibrium framework of Hellwig and Lorenzoni, as applied to our stochastic

international lending environment. Consider a group of countries each populated by a

unit measure of residents with identical period utility functions Vm. Take a sequence of

prices { q(ht, ht + 1)}, and postulate a sequence of borrowing constraints for each country mB�m(ht, ht+1). Let U(ht) denote the sequence of borrowing constraints corresponding to

histories that continue ht. The problem of a country at any point ht starting with

international bond holdings of f(ht) is given by

WmPLðf ðh

tÞ; UðhtÞÞ ¼ maxc

Xls¼t

bs�tXhsjht

pðhsjhtÞVmðcðhsÞÞ

subject to

cðhsÞ þXhsþ1

qðhs; hsþ1Þf ðhs; hsþ1ÞVemðhsÞ þ f ðhsÞ;

f ðhs; hsþ1Þz�B mðhs; hsþ1Þ:

It is assumed that, following a default, a country is excluded from future borrowing in

international markets but retains access to savings. That is, they solve

WmD ðh

tÞ ¼ maxc

Xls¼t

bs�tXhsjht

pðhsjhtÞVmðcðhsÞÞ;

subject to

cðhsÞ þXhsþ1

qðhs; hsþ1Þf ðhs; hsþ1ÞVemðhsÞ þ f ðhsÞ;

f ðhs; hsþ1Þz0:

with f (ht)=0 given. The borrowing constraints B�m(ht) are then set such that


WmPLð�B

mðhtÞ;UðhtÞÞ ¼ WmD ðh

tÞ:

This leads to the definition of a private liquidity equilibrium.

Definition 5. A private liquidity equilibrium is an allocation {cm(ht), f m(ht)}m=1M , a

sequence of borrowing limits {B�m(ht)}, and a price system { q(ht)} such that the

allocation solves the consumers problem given prices and borrowing constraints, the

sequence of implied value functions satisfies the no-default condition

WmPLðf mðh

tÞ;UðhtÞÞzWmD ðh

tÞ;

and goods and asset markets clear

XMm¼1

cmðhtÞVXMm¼1

emðhtÞ; for all dates t and all histories ht

XMm¼1

f mðht; htþ1Þ ¼ 0: for all htþ1:

The following proposition establishes the first half of our equivalence result.

Proposition 5. Let {cm(ht), fm(ht)}m=1M , {B

�m(ht)}, and {q(ht)} be a private liquidity

equilibrium, and suppose that for each m there exists a constant fm such that for all t and

all ht

jVmðcmðhtÞÞjVfmVmVðcmðhtÞÞcmðhtÞ: ð12Þ

Then there exists a sequence of domestic prices {pm(ht)} and a sequence of domestic bond

holdings {bm(ht)}m=1M identically equal to zero, such that the allocation {cm(ht), bm(ht),

f m(ht)}m=1M , and the price system {q(ht), pm(ht)} constitutes a debt constrained

equilibrium.

To show the converse, it is important to note that the debt constrained equilibria of

Section 4 have an extra dimension – the presence of resident heterogeneity – over private

liquidity equilibria. However, as contracts are enforced between residents of a country,

national borrowing decisions can be represented by an aggregate agent with preferences

Vm. The following proposition establishes that for every debt constrained equilibrium there

exists a private liquidity equilibrium with appropriately defined representative agents that

attains the same aggregate consumption and international asset flows.

Proposition 6. Let {cnm(ht), bn

m(ht, ht+1), fnm(ht, ht+1)}, and {q(ht, ht+1), p

m(ht, ht+1)} bea debt constrained equilibrium. Then there exists a representative agent world economy

and a sequence of borrowing constraints {B�m(ht)} such that the world allocation fcm

ðhtÞ; f mðht; htþ1Þ ¼P

n fmn ðht; htþ1Þg and borrowing constraints {B

�m(ht)} with world

prices {q(ht, ht+1)} is a private liquidity equilibrium.

This equivalence result is especially useful when combined with Proposition 6 of

Hellwig and Lorenzoni which establishes an equivalence result between self-enforcing


private debt equilibria, and a class of monetary economies that includes Townsend’s

(1980) turnpike model. In particular, this allows us to apply equilibrium existence results

from the monetary economics literature to our model of international capital flows with

resident default risk.

6. Conclusion

The shift in emerging market capital flows away from sovereign towards private

borrowers has led to much speculation about changes in the pattern of future capital flows,

their efficiency, and the likelihood of future debt repudiations. Some authors have used

this shift to justify the imposition of capital controls. In this paper, we have shown that

both the potential for Pareto-improving government intervention, as well as the specific

form that intervention should take, depend on the precise details of the enforcement

institutions of a country, the extent to which they discriminate between domestic residents

and foreigners in the enforcement of contracts, and the sophistication of financial markets.

If enforcement is symmetric and financial markets are sophisticated, private capital flows

are constrained efficient and no government intervention is necessary. If, however,

enforcement favors domestic residents, private capital flows will tend to be inefficiently

low, but it is a subsidy on capital flows that promotes efficiency by increasing the benefit

to accessing international markets and thus increasing the penalty associated with

exclusion from these markets.

A subsidy on capital flows may also be preferable to alternative forms of government

intervention on the basis of the information required by the government: unlike policies

that directly punish default, the government need not know the identity of the borrower

or whether they have defaulted. Nonetheless, the information requirements for a subsidy

are significant. In the present perfect information environment, it is impossible to

definitively investigate the superiority of a particular scheme in this regard. To

investigate the optimal regulation of capital flows in a private information environment

will require further work on the decentralization of optimal allocations with imperfect

information.

Acknowledgements

I thank, without implicating, Fernando Alvarez, Christine Groeger, Christian Hellwig,

Ken Kletzer, Narayana Kocherlakota, Robert Martin, two anonymous referees, the co-

editor Mick Devereux, and numerous seminar participants for comments.

Appendix A. Technical appendix

Proof of Proposition 1. The proof is constructive. Given an allocation {cnm(ht)}, compute

sequences of international Arrow {q(ht, ht+1)} and after-tax country Arrow–Debreu

{Qm(ht|h0)} prices, and taxes {sm(ht, ht+1)} as described in the text. Then compute the


bond holdings of an agent of type n in country m after history ht as the value of the

difference between consumption and endowments from that history onwards

bmn ðhtÞ ¼ ½cmn ðh

tÞ � emn ðhtÞ� þ

Xsz1

Xhtþs

QðhsjhtÞ½cmn ðhtþsÞ � emn ðh

tþsÞ�:

As there is some international trade in capital, this sum is well defined by Proposition

4.10 of Alvarez and Jermann (2000) which guarantees the finite value of the world

allocation at these prices. Define the sequence of lump-sum transfers by

Tmn ðh

tÞ ¼ �Xhtþ1

smðht; htþ1Þqðht; htþ1Þbmn ðht; htþ1Þ:

By construction, the sequences {cnm(ht), bn

m(ht, ht+1)} satisfy the agent’s flow budget

constraints.

That {cnm(ht), bn

m(ht, ht+1)} attains the maximum in the agent’s problem for after tax

prices {(1�sm(ht, ht+1)) q(ht, ht+1)} can be verified by checking the necessary and

sufficient Euler and transversality conditions for the agent’s problem. From the definition

of Arrow prices in (3), we have that

UmVn ðcmn ðh

tÞÞð1� smðht; htþ1ÞÞqðht; htþ1Þ ¼ bUmVn ðcmn ðh

t; htþ1ÞÞpðhtþ1jhtÞ;

which verifies that the Euler equation is satisfied. To see that the transversality condition

holds, note that

limTYl

XhT

bTUmVn ðcmn ðh

T ÞÞ½bmn ðhT Þ þ �B m

n ðhT Þ�pðhT jh0Þ

V limTYl

XhT

bTUmVn ðcmn ðh

T ÞÞ� Xl

s¼1

XhsþT

QmðhsþT jhT Þcmn ðhsþT Þ

�pðhT jh0Þ

V limTYl

XhT

bTUmVn ðcmn ðh

T ÞÞ� Xl

s¼1

XhsþT

QðhsþT jhT ÞeðhsþT Þ�pðhT jh0Þ

VUmVn ðcmn ðh0ÞÞ lim

TYl

XhT

QðhT jh0ÞXls¼1

XhsþT

QðhsþT jhT ÞeðhsþT Þ ¼ 0;

where the first inequality follows by construction of bnm(hT) and the borrowing constraint

B�

nm(hT), the second by feasibility and the fact that after tax prices are below international

prices, the third from the definition of the Arrow–Debreu and Arrow prices, and the last

equality from the assumption that the value of the endowment is finite. 5

Proof of Proposition 2. The proof is constructive, and adapts that of Alvarez and Jermann

(2000) to our framework. Given an allocation {cnm(ht)}, define the implied sequences of

Arrow {q(ht, ht+1)} and international Arrow–Debreu {Q(ht|h0)} prices, and the bond

holdings of an agent of type n in country m after history ht, as in the proof of the previous

proposition (noting that lump-sum tax proceeds are now zero).

As an intermediate step, construct a first candidate sequence of solvency constraints

B̃nm(ht, ht+1). For all histories ht such that the participation constraint of country m

binds, set e B̃nm(ht, ht+1)=�bnm(ht, ht+1) for all agents in that country. For all other


histories, set them equal to the natural borrowing constraints, B�

nm(ht|h0), defined

above.

That {cnm(ht), bn

m(ht, ht+1)} attains the maximum in the agent’s problem for prices

{q(ht, ht+1)} and solvency constraints {B̃nm(ht, ht+1)} can be verified by checking the

necessary and sufficient Euler and transversality conditions for the agent’s problem. From

the definition of Arrow prices in (3), we have that

UmVn ðcmn ðh

tÞÞqðht; htþ1ÞzbUmVn ðcmn ðh

t; htþ1ÞÞpðhtþ1jhtÞ;

with equality if �bnm(ht, ht+1)N� B̃nm(ht, ht+1) which verifies that the Euler equation is

satisfied. To see that the transversality condition holds, note that

limTYl

XhT

bTUmVn ðcmn ðh

T ÞÞ½bmn ðhT Þ þ B̃Bm

n ðhT Þ�pðhT jh0Þ

V limTYl

XhT

bTUmVn ðcmn ðhT ÞÞ

� Xls¼1

XhsþT

QðhsþT jhT Þcmn ðhsþT Þ�pðhT jh0Þ

V limTYl

XhT

bTUmVn ðcmn ðh

T ÞÞ� Xl

s¼1

XhsþT

QðhsþT jhT ÞeðhsþT Þ�pðhT jh0Þ

VUmVn ðcmn ðh0ÞÞ lim

TYl

XhT

QðhT jh0ÞXls¼1

XhsþT

QðhsþT jhT ÞeðhsþT Þ ¼ 0;

where the first inequality follows by construction of bnm(hT) and the borrowing constraint

B�

nm(hT), the second by feasibility, the third from the definition of the Arrow–Debreu and

Arrow prices, and the last equality from the assumption that the value of the endowment is

finite.

Finally, to show that we can set these solvency constraints so that they are minimally

constraining, define the value functions {Wnm} that solve the agent’s problem for all n and

m. As the plan {cnm(ht), bn

m(ht, ht+1)} satisfies the Euler and transversality conditions, it

constitutes a solution for these value functions as long as the initial conditions are given by

{bnm(h0)}. For all ht such that the participation constraint of country m binds, define our

final sequence of solvency constraints {B�

nm(ht, ht+1)} such that B

�nm(ht, ht+1)=B̃n

m(ht, ht+1).For al l other ht, define them such that

XNn¼1

/mn W

mn ð �

�B

m

n ðht�1; htÞ; htÞ ¼ DmðhtÞ;

and

/mn

BWmn ð�

�B

m

n ðht�1; htÞ; htÞ

Bb¼ /m

nV

BWmnV ð�

�B

m

nVðht�1; htÞ; htÞ

Bb:

Using the envelope condition, it is easy to verify that this latter condition is also

satisfied at states at which the participation constraint binds. 5

Proof of Proposition 3. We show the result for t =0; the argument is analogous after any

other history. As the government is able to set transfer to maximize the weighted average


of residents’ utilities, the weighted average of residents’ utilities in country m can be

written as the solution to the problem

maxcm

XNn¼1

/mn

Xlt¼0

btXht

pðhtÞUmn ðcmn ðhtÞÞ;

subject to the date zero budget constraint

Xlt¼0

Xht

PmðhtÞXNn¼1

cmn ðhtÞVXlt¼0

Xht

PmðhtÞXNn¼1½emn ðhtÞ � nxmn ðhtÞ�;

given sequences of prices and net exports. By the result at (9) the autarky allocation is

affordable at these prices, but as it was not chosen it cannot yield a greater value for {/nm}-

weighted resident utility. 5

Proof of Proposition 4. The proof is constructive. Given an allocation, and a first estimate

of prices from (10) and (11), we can check whether at the efficient allocation a resident’s

participation constraint is satisfied. Let H be the (countable) set of all possible histories ht,and let NVm be that subset of H, such that for all htaNVm no country m resident’s

participation constraints are violated at ht. For all htaNVm set the after tax international

prices qm(ht, ht+1)=q(ht, ht+1) so that for all m, sm(ht)=0.

Given the efficient allocation, we can form net exports for a country after any history as

nxm(ht)=em(ht)�cm(ht). For all htaH\NVm, we can use these values to construct an

estimate of the extent to which a country as a whole is violating its participation constraint

(noting that if some residents of a country have violated their constraints, then all residents

of that country have, and we can without loss focus on country aggregates)

�Xhszht

PmðhsÞnxmðhsÞ:

This is negative (as the constraint is violated) and finite by assumption (as

Pm(ht)VQ(ht)). Note that this implies that the amount by which the constraints are

violated is bounded above by zero, and below by the finite present value of the country’s

endowment.

In equilibrium, at the world pre-tax bond price, we have

nxmðhtÞ ¼Xhtþ1

qðht; htþ1Þf mðht; htþ1Þ � f mðhtÞ:

Our aim is to find a sequence of values for extra surplus Sm(ht)z0 for each country and

each htaH\NB for which its participation constraints are violated, such that when the

surplus is added, these participation constraints are then satisfied. In equilibrium, we will

distribute this surplus in terms of a subsidy on lending, so that

SmðhtÞ ¼ � sðhtÞXhtþ1

qðht; htþ1Þ f mðht; htþ1Þ:

Note that as the country is borrowing, the non-negativity of S implies the non-

negativity of s, and hence a subsidy. This is a countable sequence of affine equations in a


countable number of unknowns which we can index by j. It is convenient to view this as a

mapping from the closed e-ball in ll into itself, where e is defined as the largest possible

value of the aggregate endowment. Then, as the Arrow–Debreu prices are absolutely

summable, the mapping is a bijection and there is a unique solution for the surplus

sequence.

Given initial asset holdings, we can then solve for future asset holdings by iterating on

the following. Given f m(ht), we can find f m(ht, ht+1), for countries whose constraints bindat ht, from

nxmðhtÞ � SmðhtÞ ¼ ð1þ smðhtÞÞXhtþ1

qðht; htþ1Þf mðht; htþ1Þ � f mðhtÞ

¼Xhtþ1

qmðht; htþ1Þf mðht; htþ1Þ � f mðhtÞ;

where qm(ht, ht +1) is the level of after-tax international bond prices in country m. Given

estimates of bond holdings for countries whose constraints are binding at ht, we can

construct the aggregate holdings for countries whose constraints are not binding from the

market clearing condition. If the set of such countries at ht is given by M(ht), then as these

countries set no subsidy, and trade at q(ht, ht+1), we can use

XmaM htð Þ

nxm htð Þ ¼X

maM htð Þ

� Xhtþ1

q ht; htþ1ð Þf m ht; htþ1ð Þ � f m htð Þ�;

to get an estimate of the q(ht, ht+1) (and hence s(ht)) where we have noted that s(ht) iscommon over all assets at this history, and all countries whose constraints bind.

The precise sequence of lump-sum taxes across residents is indeterminate. However, its

present value, along with initial domestic bond transfers, can be determined from

resident’s lifetime budget constraints. 5

Proof of Proposition 5. Given the private liquidity allocation, define the domestic price

sequence by

pmðht�1; htÞ ¼ bpðhtjht�1ÞVmVðcmðht�1; htÞÞVmVðcmðht�1ÞÞ

;

and the domestic bond sequence to be identically zero. This allocation is resource feasible

for the debt constrained economy. All that remains is to show that the allocation is optimal

for residents. We first show that this allocation lies within the budget set of the residents of

the debt constrained economy, and then show that the allocation is optimal within that

budget set.

To compare budget sets, consider the default problem from the private liquidity

economy. By the convexity of the problem, an allocation attains the solution to the default


problem at time t, if and only if there exists a sequence of non-negative multipliers kPL and

jPL such that

bs�tpðhsjhtÞVmVðcðhsÞÞ ¼ kPLðhsjhtÞ

� kPLðhsjhtÞqðhs; hsþ1Þ þ kPLððhs; hsþ1ÞjhtÞ þ jPLðhsjhtÞ ¼ 0

where jPL=0 if f N0. This can be rearranged to get

kPLððhs; hsþ1ÞjhtÞkPLðhsjhtÞ

¼ bpðhsþ1jhsÞVmVðcðhs; hsþ1ÞÞVmVðcðhsÞÞ ¼ pmðhs; hsþ1Þ

¼ qðhs; hsþ1Þ if f ðhs; hsþ1ÞN0:

Similarly, the solution to the default problem in the debt constrained economy is

associated with non-negative multipliers kDC such that

bs�tpðhsjhtÞVmVðcðhsÞÞ ¼ kDCðhsjhtÞ

� kDCðhsjhtÞpmðhs; hsþ1Þ þ kDCððhs; hsþ1ÞjhtÞ ¼ 0;

which can be rearranged to get

kJ;Dððhs; hsþ1ÞjhtÞkJ;DðhsjhtÞ

¼ bpðhsþ1jhsÞVmVðcðhs; hsþ1ÞÞVmVðcðhsÞÞ ¼ pmðhs; hsþ1Þ:

But f(hs, hs+1)N0 if pm(hs, hs+1)=q(hs, hs+1), while f(hs, hs+1)=0 if pm(hs, hs +1)Nq(h

s,

hs+1).Hence, the budget set for the debt constrained default problem is a superset of the

budget set for the private liquidity default problem, and so WDm(ht)VDn

m(b, ht, { pm}).However, the first order conditions are identical up to the scaling of the k multipliers.

Hence the solution of the private liquidity default problem also attains the maximum of the

debt constrained default problem, and we have WDm(ht)=Dn

m(b, ht, { pm}). But then

WmPLðf mðh

tÞ; UðhtÞÞzWmPLð�B

mðhtÞ; UðhtÞÞ ¼ WmD ðh

tÞ ¼ Dmn ðb; ht; fpmgÞ;

from the no default condition and the definition of the borrowing constraints, and the

allocation is feasible for the residents problem in the debt constrained economy.

To show optimality, note that at the solution to the residents problem from the private

liquidity problem there exist non-negative multipliers kPL(ht) and lPL(ht) such that

btpðhtÞVmVðcmðhtÞÞ ¼ kPLðhtÞ; kPLðhtÞqðht; htþ1Þ � kPLðht; htþ1Þ ¼ lPLðhtÞ;

where lPL(ht)N0 if f(ht)N0. To verify optimality for the debt constrained economy

we exhibit the necessary multipliers to show that the allocation solves the residents


auxiliary problem. Define kDC(ht)=kPL(ht) and noting that the participation constraint

does not bind in period zero, we can define lDC(ht) recursively from

lPLðhtÞ ¼ PmðhtÞ" X

hsVðht ; htþ1ÞlDCðhsÞ � qðht; htþ1Þ

XhsVht

lDCðhsÞ#:

But then the allocation satisfies the first order necessary and sufficient conditions for

an optimum in the debt constrained economy, the problem for which is well-defined

under assumption (12) by the Proof of Proposition 4.6 in Alvarez and Jermann

(2000). 5

Proof of Proposition 6. Define the representative agents for each country with preferences

Vm as in Section 3. We have to show that there exists a sequence of borrowing constraints

such that the allocation and borrowing constraints at these prices constitute a private

liquidity equilibrium. By Lemma 2 of Hellwig and Lorenzoni (2003) the suitable sequence

of borrowing constraints exists as long as the continuation utilities are always above the

default level.

To establish that the continuation utilities are above default, note that by definition of

the debt constrained equilibrium, the continuation utilities of every individual agent are

larger than what that agent can get by defaulting Dnm. Therefore, the value to the

government of an economy is at least as large as the weighted sum of the default values for

each resident. The value to the aggregate economy in default is given by

Dmð0; hs; fpmgÞ ¼ maxc; bVf g

Xn

/mn

Xlt¼s

bt�sXht jhs

pðhtjhsÞUmn ðcmn ðh

tÞÞ;

subject toXhszht

PmðhsÞ½emðhsÞ � cmðhsÞ�V0;

for all ht continuing hs. This clearly returns a value at least as large as the weighted sum of

the default values of each residentP

n /mn D

mn . To show that it is exactly as large, note that

the solution for the individual consumer’s problem involves setting

pmðht�1; htÞ ¼ bpðhtjht�1ÞUmV

n ðcmn ðht�1; htÞÞUmV

n ðcmn ðht�1ÞÞ;

for all n. But this also solves the national problem. Hence we have that the value to the

government of a country is at least as large as the default level Dm(0, hs, { pm}).Hence the allocation is feasible for the residents problem for the private liquidity

economy. But by the Proof of Proposition 5 the residents’ budget set from the debt

constrained economy is a superset of the residents budget set for the private liquidity

economy, and hence the allocation must attain the optimum for the private liquidity

economy. 5


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Documents

Private capital flows, capital controls, and default risk