Download Sovereign debt issuance and selective default

Sovereign debt issuance and selective default
PRELIMANRY AND INCOMPLETE
∗
Wojtek Paczos†, Kirill Shakhnov‡
First Version: May 2013
This Version: February 2014
Abstract
While the empirical evidence shows that governments, in vast majority of cases, default selectively either
on home or foreign debt holdings, theory has not yet been able to account for this fact. We build a general
equilibrium incomplete markets model with government discretion and two types of investors, domestic and
foreign to study mechanisms and trade-offs of selective defaults. The model matches important data moments,
in particular frequencies of different types of defaults and debt levels. Secondly, the model is a useful tool for
analysing how government optimally finances itself using three sources of funds: taxation, domestic debt and
foreign debt. We thus provide a simple theory of sovereign debt issuance and debt composition. Thirdly, contrary
to recent theoretical findings, we show that when taxes are distortionary secondary markets are not sufficient to
prevent sovereign defaults.
Keywords: sovereign debt, selective default, debt composition, secondary markets
JEL Classication: F34, G15, H63, E43
∗ Authors
would like to thank Davide Debortoli, Harris Dellas, Antonia Diaz, Juan Dolado, Dirk Niepelt and Evi Pappa for many
useful comments and suggestions. We are especially grateful to Árpád Ábrahám, Russell Cooper and Ramon Marimon for all their
advice and constant support.
† European University Institute, [email protected]
‡ European University Institute, [email protected]
1
1
Introduction
Humanity has witnessed sovereign debt crises as long as the history of civilization1 . Sovereign default has been
studied extensively in the literature. However, mainly default on external debt has been considered, while attention
to domestic defaults has been neglected. Reinhart and Rogoff (2011)[12] have documented and categorized all
default events in the last 150 years. Based on their observations, there have been at least 58 de jure defaults
on domestic public debt2 . Also, out of 267 defaults in this period only 17 times did the governments default
simultaneously on both domestic and foreign debt3 . As the question why governments usually default selectively
on either foreign or domestic debt remains open, this paper is an attempt to bridge this gap.
We build incomplete markets model with limited commitment of government. Government has to cover its
expenditures and has three means of financing: it issues one-period defaultable bonds on international and domestic
markets and collects taxes. Tax collection is costly, as taxes are distortionary. Economy is subject to two shocks:
the output4 and the tax distortion shock5 . While the output shock provides incentives for government borrowing
on international markets, the tax distortion shock creates the wedge between domestic borrowing and taxation. To
show intuitive mechanism of selective default, we present the tractable two-period version of the model. Because
of uncertainties and default penalties, government is facing two trade-offs: between foreign repayment and default
and between domestic default and repayment through distortionary taxation. First one is common in the literature.
The government decides each period whether to transfer resources away from the economy as repayment of debt
to foreign investors or keep resources at home and suffer default penalties. When output is low ceteris paribus, it
is more costly for a risk averse borrower to respect the contract. Second trade-off is new and drives domestic debt
and default policies. This trade-off is drawn along different lines, as both repayment and default on domestic debt
is a transfer of resources within the economy. In the case of default on domestic debt government suffers default
penalties similar to foreign default penalties. When it decides to repay however, it needs to finance this repayment
1 The
2 It
first recorded incidence of sovereign default dates back to 377 B.C. in ancient Greece
is certainly an underestimated number due to the difficulty to detect pure domestic defaults. For example, the large-scale 1989
pure domestic default, which did not coincide with default on external debt, is relatively unknown outside Argentina. The most well
know pure domestic default happened in Russia 1998, which was one of the largest local currency debt default (US $39 billion). Besides,
this number does not include de facto default through inflation, the nationalization of pensions and other forms. See Reinhart and
Rogoff (2011)[12] for detailed empirical evidence
3 These numbers may differ with other studies that use same database as we divide defaults into three instead of two types: foreign,
domestic and total. Whenever a country is in default on both international and domestic markets we count this situation as one total
default, instead of one domestic and foreign.
4 The output shock summarizes all possible external shocks to net income of economy, such as term of trade, exchange rate and etc.
5 The tax distortion shock is a structural shock to how efficiently the available resources are used. For example, a way to think of
this is how successful a new implemented tax reform is. The recent evidence of Greece austerity measures has shown that it can be
a challenge to implement a new tax reform. Another way, following Pouzo (2010) [10], is simply to say that the degree of distortion
coming through taxation is unknown.
2
with tax collection, which is costly. Thus this trade-off draws distinction between tax-financed and debt-financed
expenditure policies. In this we provide simple theory of domestic debt, which we call ”betting for redemption”.
When taxation distortion is high, government prefers to accumulate debt rather than collect taxes in the hope that
in future tax collection will be less distortionary and it would gain ability to repay debt cheaply. Government is
thus accumulating domestic debt up to endogenous debt limit and if this possibility does not arrive it has no other
choice than to default.
Next we develop quantitative infinite horizon version of the model. We show that including two types of
debt allows to generate reasonable repayment with lower output costs and shorter market exclusion than previous
studies6 . Also, our model is able to generate reasonably high debt-to-GDP ratio of 25% with reasonably low default
probability of 3% 7 .
The main contribution of this paper is the quantitative framework to study sovereign debt issuance and selective
defaults. But our analysis also has some other, quite novel implications. Firstly, we provide simple theory of domestic
debt. Secondly, as in our model government is optimally borrowing on both domestic and foreign markets, this
paper can be seen as as a novel theory of debt composition. Lastly, we focus on studying how secondary sovereign
debt markets affect government’s incentives to default. With introduction of unconventional monetary policies after
the Great Recession, secondary sovereign debt markets attract an increasing interest among economists. Based on
two-period version of the model we can provide useful insights on the efficiency of these policies. Contrary to Broner,
Martin and Ventura (2010) [4] we show that well functioning secondary markets are not sufficient to solve sovereign
risk problem. We find equilibria where the repayment problem is amplified through the secondary markets.
The remainder of the paper is organized as follows. In the next section we revise empirical facts on domestic
and foreign public debt holdings and selective defaults. Section 3 offers literature review. Section 4 introduces the
model. Section 5 studies equilibrium in two-period model and shows intuitively main trade-offs. Section 6 presents
calibration and results from the infinite-horizon version of the model. In section 7 we study the role of the secondary
markets for creating repayment incentives. Last section concludes.
2
Facts
In this section we show empirical findings that motivate the positive contribution of this paper. Before we continue
with the discussion we need to set the scene with some definitions. There are three different definitions of domestic
and foreign debt. According to legal definition domestic debt is debt issued according to domestic law, regardless
6 Borensztein
and Panizza (2009)[3] survey empirically costs of sovereign defaults. One of their strong results is that costs are short
lived. Tomz (2007) on the other hand finds that sanctions are less important mechanism for generating repayment than reputation loss.
7 In Arellano (2008) average debt-to-GDP ratio is 6%, Chatterjee and Eyigungor (2012) [5] attain best result up to date of 70% but
with a high default probability of 7% by employing stochastic maturity and highly non-linear default cost.
3
of its currency, and regardless of who is holding claims. According to economic definition domestic debt is held
by residents, regardless of currency and the law under which it was issued. Finally, the last definition has it, that
domestic debt is debt denominated in home currency, regardless of law and nationality of bond holders. Let us
refer to it as currency definition.
With respect to these definitions first important point to raise, is that they do not necessarily overlap. Secondly,
under legal and currency definition, sovereign can easily default selectively, as foreign and domestic bonds are
effectively two different obligations. Still, it is the economic definition that creates clear differential incentives for
sovereign to default. For this reason we throughout the model we adopt economic definition. Another reason is the
consideration of the role of the secondary markets, which would cease to function if foreign and domestic bonds
were two different obligations.
1. Sovereign defaults usually happen in selective manner. The database collected by Reinhart and
Rogoff8 accompanying their seminal work This Time It’s Different[11] reveals interesting features of catalogued
sovereign default episodes between 1800 and 2010. Firstly, home debt, usually neglected in theoretical literature on
sovereign risk, plays important role in the build-up, during and after sovereign default on foreign holdings. This
argument is extensively developed in Reinhart and Rogoff (2011)[12]. Secondly, sovereign defaults happen on both
domestic and foreign debt holdings, usually in a selective fashion. Whereas foreign defaults are common, home
defaults are hardly rare. Out of 267 episodes of sovereign debt crisis identified across 70 countries in the last 210
years, 205 were pure foreign 26 were pure domestic and 36 times did the governments default on both domestic and
foreign. Only 17 times the default on home and foreign debt happened within the same year. In Figure 1 we plot
fraction of borrowers that remained in foreign, domestic of total default in a given year between 1800 and 2010.
This findings suggest that assumption of selective sovereign default fits reality better than usual assumptions: that
home debt is irrelevant and only foreign debt is defaulted or that sovereign cannot discriminate and can only default
on all types of debt outstanding
2. Governments have number of tools to discriminate among types of bondholders. How can
government default on foreign investors while repaying home investors or vice-versa? The assumption that two
types of bond-holders are indistinguishable, and therefore sovereign can only default on total debt outstanding,
underestimates the creativity of governments.
Among the tools that government use to discriminate against types of bondholders the most popular are capital
controls, exchange controls and freeze on deposits. In 1990 Brazil defaulted on its domestic debt but kept servicing
its foreign debt. All foreign exchange transactions were directed through Central Bank, multiple exchange rate
regime was introduce as well as freeze on local currency deposits.
In 1998 Russia defaulted on both foreign and local currency debt imposing capital and exchange rate controls.
8 www.reinhartandrogoff.com
4
Figure 1: Fraction of countries in different types of default
Source: Own calculations based on Reinhart and Rogoff (2009)
However in subsequent years Russia ”undefaulted” on its foreign obligations and kept servicing debts to foreign
investors. Moreover, also bonds held by domestic companies were repaid, therefore effectively Russia defaulted only
on domestic household sector holding of public debt. Default was accompanied by both foreign and local currency
deposit freeze.
Argentina 2001 default is often considered as model case for foreign default9 . In fact this episode is catalogued
as total default. First all resident-held bonds both domestic and foreign currency denominated were converted to
government-guaranteed loans, which were all later converted to peso at much lower rate than market exchange rate.
Also 60% of the debt defaulted in December 2001 was held by Argentines.
Examples of what could be considered as pure foreign default (in the peaceful times) include among most
recent cases: Bolivia 1989 (most of domestic debt was repurchased a year before default), Pakistan 1999 (stopped
payments on outstanding obligations to creditors in the U.K., Europe and the U.S. and put a freeze on foreign
currency deposits mostly owned by non-residents) and most probably Cyprus 2013 (freeze and partial expropriation
of deposits in the amount exceeding 100,000 euro which are mostly owned by non-residents).
3. Composition of bondholders matters. The empirical work on composition of bondholders is growing.
9 Many
sovereign default models are calibrated to mimic salient features of this default, e.g. Arellano 2008 ([2])
5
We draw on this literature, particularly on Andritzky (2012) [1] and Dell’Erba et. al. (2013)[14] to show that
composition of investors is correlated with interest rates and total level of debt to GDP. The latter contribution
finds that there is significant correlation between spreads and debt levels when majority of debt is denominated
in foreign currency (similarly in emerging economies and Eurozone countries). They also document, that financial
crises had more profound effects on economies that rely more on foreign borrowing. The former one finds strong
positive correlations between fraction of domestic debt in total debt and total debt-to-GDP ratio and negative
correlation between fraction of foreign debt and spreads in advanced economies. This paper contributes to this
literature by providing framework to study driving forces behind debt composition and its consequences on spreads,
total debt default incentives.
3
Literature
The paper combines the well developed literature on foreign default with a new fast growing one on domestic
default. Looking first at foreign default, the standard mechanism of foreign default can be described as follows:10
a benevolent government accumulates defaultable debt in order to smooth a country’s consumption over business
cycle. Interest rates reflect default probabilities which are endogenous to the borrower’s incentives to default.
Default happens along the equilibrium path in order to complete markets. The long enough sequence of negative
output shocks leads to the default. To generate a positive borrowing and repayment, it is necessary to impose
default costs. Two default costs are usually considered: temporary exclusion from international financial markets
and direct output loss.
While the mechanism and trade-offs behind foreign default are clear, domestic default literature is still only developing. There are two recent contributions that adhere to benevolent government assumption and study domestic
default in Ramsey setting. D’Erasmo and Mendoza (2012)[6] propose a heterogeneous agent model in which a utilitarian government relies on lump-sum taxes and defaultable bonds to finance stochastic government expenditures.
Since there exists the background distribution of individual income, the rich accumulates more government debt.
Default has a redistributive aspect, because it hurts mostly the rich, while repayment by taxation hurt mostly the
poor. Pouzo(2010)[10] on the other hand considers a representative agent model in which a government relies on
distortionary labour income taxes and defaultable bonds to finance its stochastic expenditures. The government
might default to mitigate the distortions.
Broner, Martin and Ventura(2010)[4] (henceforth BMV(2010)) turn their attention to the role of secondary
markets in solving sovereign default problem. They show that, even in the absence of default penalties, sovereign
risk does not destroy foreign asset trade, if foreign creditors can resell their assets to domestic investors on secondary
10 See
for example J. Eaton and M. Gersovitz (1981)[7] and C. Arellano(2008)[2].
6
markets. Our findings put this result into question.
An important contribution that studies how distribution of debt among domestic and foreign investors influences
government incentives to default is Cooper, Kempf and Peled (2008)[13]. Authors study the debt repayment game
between regional governments in a financial federation with central government. They find conditions (government
expenditure and fraction of debt being held by foreign investors being high enough) under which government has
incentives to default. However, the incentives to default in this model derive from the fact, that regional government
expects central government to step in and fully bail-out all obligations by imposing federation-wide tax. In our
work we abstract from bail-out considerations and derive endogenously fractions of public debt held by domestic
and foreign investors.
Finally, two contributions that study selective nature of sovereign default in different settings than ours are
Vasishta (2010)[15] and Erce (2012)[8]. The former is the two-period open economy model with endowments,
foreign and domestic borrowing and output shock. The incentive problem of the government is similar to the one
described in our paper, however, in equilibrium foreign default never happens, because it is strictly dominated by
total default. The latter is the three-period economy model with government, domestic and foreign investors and
banking sector. The model derives optimal thresholds for partial-defaults scenario for the government, which are
state-dependent, and therefore neither foreign nor domestic debt is senior. The focus of this paper is however, on
how selective nature of default affects the economy after the default episode (banks, domestic and foreign investors
make their decision after observing partial default decision), and, in turn, how the following actions would affect
government’s decision to default. In our model the future possibility of default also affects the economy in the time
preceding default-repayment decision.
4
4.1
The Model
Environment
We build incomplete markets model with government’s limited commitment. The economy consists of three types of
actors, domestic households, foreign investors and benevolent government, which acts as Ramsey planner. Let the
time be indexed as t = 0, 1, 2, ... . The economy is subject to exogenous stochastic stream of income yt ∈ Y, which
is a Markov process. At each time t the government has to cover exogenous stream of government expenditure
g. It can collect resources in three different ways: issue bonds in home market, issue bonds with foreign investors
or collect taxes. Taxes are lump-sum, but collecting taxes comes at a cost to the economy. We assume raising T
amount of taxes by government induces a loss of T (1 + τ ) resources to agents. We assume for τ has convex support
τ ∈ Φ and is a stochastic Markov process. This is a key element that will allow to break Ricardian equivalence in
our endowment economy and create a trade-off between two sources of domestic financing. If the debts outstanding
7
are being repaid with taxes government imposes distortions on economy, if they are repaid with issuance of new
debt government risks going into default and suffering penalties.
In each period t the government decides either to repay or default on outstanding foreign and domestic debt.
When government chooses to default the economy suffers from output penalties and is excluded from borrowing on
the market where default happened for random number of periods. We allow exclusion and output cost to differ
between types of default.
4.2
Households
Households are identical and risk averse. Their utility is given by:
∞
X
β t u(ct )
t=0
where β is the discount factor, c is consumption and u(c) is increasing and strictly concave. Households are allowed
to save using domestically issued government bonds bh . They take bond discount prices and taxes as given. They
face intratemporal budget constraint, which differs depending on government’s decision to default on either of two
bonds. If the governments repays both domestic and foreign debt households budget constraint is the following:
cr = y − T (1 + τ ) + bh − qh b0h
(1)
where bh is amount of domestic debt owed and repaid by government to households, b0h is new issue of government
domestic debt (household’s savings) and qh is domestic bond discount price. If the government defaults on foreign
debt, households are still allowed to save in domestic market but they suffer output cost:
cf d = y f d − T (1 + τ ) + bh − qh b0h
(2)
In case of domestic default government maintains foreign borrowing, but domestic debt market is closed:
chd = y hd − T (1 + τ )
(3)
Similarly in case of simultaneous domestic and foreign default, which we will refer to as total default:
ctd = y td − T (1 + τ )
4.3
(4)
Foreign Investors
Foreigners are risk neutral and deep pocket investors with access to international credit markets, where they can
save and borrow and constant interest rate r. When lending resources to government they account for possibility
of default and break even in expected terms, therefore their policy can be summarized in:
qf =
(1 − δ f )
1+r
8
where qf is a discount price of government bond issued with foreign investors, δ f d is probability of foreign default
is probability of total default.
4.4
Recursive equilibrium
We define recursive equilibrium in which domestic households, foreign investors and government act sequentially
and government acts discretionary. The aggregate state of the economy is given by two endogenous debts and two
exogenous processes for income and tax distortions S = (bh , bf , s) and s = (y, τ ). Every period government decides
whether to repay its both outstanding debts, default on domestic debt, default on foreign debt or default on both:
V 0 (bh , bf , s) = max{V r (bh , bf , s), V f d (bh , s), V hd (bf , s), V td (s)}
(5)
The government repayment decision is summarized by two default indicators df ∈ {0, 1} and dh ∈ {0, 1} where
di = 0 stand for repayment, df = 1 stands for foreign and dh = 1 for domestic default. If the government decides
to repay it solves the following problem:
n
0 0 0 0 o
r
V r (bf , bh , s) = max
u(c
)
+
βE
V (bf , bh , s
0
0
(6)
bf ,bh
subject to households budget constraint (1), households first order condition:
n
o
E (1 − d0h (S 0 ))u0 (c0 (S 0 ))
qh (b0f , b0h , s) = β
u0 (c(S))
(7)
where prices, default indicator are equilibrium objects defined over state space § = {bf , bh , y, τ } foreign bond price
schedule:
qf (b;f , b0f , s)
E 1 − d0f (S 0 )
=
1+r
(8)
and implementability constraint:
T = g + (1 − dh )bh + (1 − df )bf
(9)
given that df = 0 and dh = 0. If the government defaults on foreign debt (and keeps servicing its domestic
obligations) the economy suffers output cost and is allowed to return to international borrowing in the future with
probability θf . With probability 1 − θf country remains only on domestic bond market and government can still
decide to also default on domestic bonds (yielding total default).
(
)
fd 0 0
fd
fd
f 0
0
0
f
td 0
V (bh , s) = max
u(c ) + βE θ V (0, bh , s ) + (1 − θ )max V (bh , s ), V (s )
0
(10)
bh
subject to households budget constraint (2), households first order condition (7) and implementability constraint
(9) given dh = 0 and df = 1. Thirdly, if the government decides to default on domestic debt outstanding it remains
9
active on international markets, comes back to domestic borrowing with probability θh , can still default on foreign
debt and suffers domestic output penalty:
(
V
hd
(bf , s) = max
u(c ) + βE θh V 0 (b0f , 0, s0 ) + (1 − θh )max V hd (b0f , s0 ), V td (s0 )
0
)
hd
(11)
bf
subject to households budget constraint (3), foreign bond price schedule (8) and implementability constraint (9)
given dh = 1 and df = 0. Lastly, at any given time the government can decide to pursue total default. Economy
suffers output penalties for both domestic and foreign default and government comes back to international and
domestic borrowing with probabilities θf and θh respectively.
V td (s) = u(ctd ) + βE θf θh V 0 (0, 0, s) + θf (1 − θh )V hd (0, s0 ) + (1 − θf )θh V f d (0, s0 ) + (1 − θf )(1 − θh )V f d (s0 ) (12)
subject to households budget constraint (4) and implementability constraint (9) given dh = 1 and df = 1.
Now, that actions and optimization problems are defined for each actor in the economy we can define equilibrium:
Definition 1. Recursive equilibrium in this economy is (i) the set of prices for domestic bond qh (S) and foreign
bond qf (S), (ii) government debt policies b0h (S) and b0f (S) and (iii) government default schedules dh (S) and df (S)
such that:
1) Taking as given domestic bond price schedule dh (S)and government domestic debt b0h (S) and tax policies consumption c(S) satisfies households budget constraint and first order condition
2) Taking as given government foreign default schedule df prices qf (S) are consistent with foreign investors expected
zero profits
3) Taking as given prices qh (S) and qf (S) governments default schedules dh (S) and df (S) and debt policies b0h (S)
and b0f (S) solve government optimization problem
4) Government bond and tax policies and default schedules satisfy implementability constraint
5
Equilibrium in a two period model
The main driving forces of government optimal policy are two trade-offs. First is the trade off between transferring
resource away from economy in form of foreign debt repayment and loosing resources due to foreign default penalties.
Second one is between imposing distortion on the economy in form of tax collection and imposing loss of resources
in form of domestic default penalties. For expositional purposes we find it informative to study an equilibrium in
a version model reduced to two periods.
In order to isolate key mechanisms we strip the model down to only necessary ingredients:
• Government has to cover expenditures only in the first period g1 = g > 0. Government expenditures in the
second period are g2 = 0. This creates incentive to borrow due to consumption smoothing motive
10
• The economy is subject to two shocks only in the second period. Output can be high or low y2 = {yl , yh } as
well as tax distortions τ2 = {τl , τh }.
• In the first period output is high y1 = yh and taxes are non-distortive τ1 = τl = 0
• Economy starts with no debts outstanding bh0 = 0 and bf 0 = 0
• After domestic default the output of the economy is y(1 − δ h )
As before, domestic households are identical and risk averse and foreign investors are risk neutral.
5.1
Default schedule
We solve the model by backward induction starting in the second period and moving back to the first period. Given
government debt decision from the first period bh and bf in the second period government takes default decision
that maximize domestic households utility from consumption. As this is terminal period there is no demand for
government bonds in the second period, so the only source of income for government is taxation. In the second period
four scenarios may arise: repayment, foreign default, domestic default and total default. Combining households
budget constraints (1)-(4), implementability constraint (9) and our simplifying assumptions for two-period economy,
household consumption in each of four scenarios are respectively given by following equations. Notice that, in order
to repay bh amount of domestic bonds to households government need to raise bh (1 + τ ) taxes yielding a loss of τ bh
resources to domestic economy.
cr
= y2 − bf (1 + τ2 ) − bh τ2
(13)
cf d
= y2 (1 − δ f ) − bh τ2
(14)
chd
= y2 (1 − δ h ) − bf (1 + τ2 )
(15)
ctd
= y2 (1 − δ h )(1 − δ f )
(16)
A. Foreign debt limit
When deciding whether to default on foreign investors government compares household utilities under repayment
and under foreign default. It is immediate to see, that foreign debt will be repaid whenever:
bf
δf
≤
y
1 + τ2
(17)
where left-hand-side is foreign debt-to-gdp ratio and right-hand-side is a number defined by parameters of the
model. Whenever then inequality is the opposite foreign debt is defaulted.
Proposition 1. If taxation is costly there is finite limit to foreign debt sustainable in repayment equilibrium. If
taxation distortion is stochastic this limit can be broken and foreign default arises in equilibrium.
11
Proof. Follows directly from comparing (13) and (14).
B. Domestic debt limit
Similarly we can define domestic debt limit. Domestic debt will be repaid whenever:
bh
δh
≤
y
τ2
(18)
Proposition 2. If taxation is costless and home default induces small positive costs to the economy the any level
of domestic debt is repaid. If taxation is costly there is finite limit to domestic debt sustainable in repayment
equilibrium. If taxation distortion is stochastic this limit can be broken and domestic default arises in equilibrium.
Proof. Follows directly from comparing (13) and (15).
Inequalities (17) and (18) completely characterize government policy in the second period. Notice that whenever
both inequalities are reverse it also holds that ctd > cr , which is consistent with the definition of total default being
simultaneous default on both domestic and foreign debts outstanding.
C. Default policies in the second period
Having established debt limits in repayment equilibrium we want to find an equilibrium in which, depending on
realization of stochastic shocks, all four outcomes (repayment, foreign default, domestic default and total default)
can arise in the second period. Both of two stochastic processes in this economy assume two outcomes. Therefore
we impose equilibrium conditions that would map four possible realization of joint (y, τ ) stochastic processes into
four equilibrium outcomes. These conditions are:
1. After bad output shock y2 = yl independently on realization of tax distortion government defaults on foreign
debt
2. After good output shock y2 = yh independently on realization of tax distortion government repays foreign
debt
3. After bad tax distortion shock τ2 = τh independently on output realization government defaults on domestic
debt
4. After good tax distortion shock τ2 = τl independently on output realization government repays domestic debt
Mathematicaly these conditions can be summarized by four inequalities that follow from substituting realizations
of y and τ into (17) and (18):
yl δ f
yh δ f
< bf ≤
1 + τl
1 + τh
h
yh δ
yl δ h
< bh ≤
τh
τl
12
(19)
(20)
where inequalities in (19) correspond to conditions 1) and 2) respectively and inequalities in (20) correspod
to conditions 3) and 4) respectively. How these conditions translate into mapping between (y, τ ) outcomes and
repayment/default decisions can be easily understood by looking at Figure 2. Red (dotted line) represents limit
to domestic debt sustainable in repayment equilibrium, while blue (solid) line represents limit to foreign debt
sustainable in equilibrium. In the second period four situations may happen. Circles show situation when debt is
repaid, while crosses show defaults. Circles and crosses colors represent respective debt (blue-foreign, red-home).
A negative shock to output is shows as an increase in debt-to-GDP ratio. Figure shows four outcomes denoted
Figure 2: Debt Sustainability and Selective Defaults
by letters A to D. A) After good output shock and successful tax reform both debts lie below their respective
sustainability limits and therefore both are repaid. B) After bad output shock and unsuccessful tax reform foreign
debt (blue cross) lies above its sustainability limit and therefore is defaulted. Home debt (red circle) however is still
repaid, as it lies below its sustainability limit. C) After good output shock but unsuccessful tax reform situation is
reverse to B. D) After bad output shock and unsuccessful tax reform both debts break their respective sustainability
limits and therefore are defaulted
13
5.2
Debt policies in the first period
In this section and in the remainder of the paper we apply CRRA instantaneous utility function:
u(ci ) =
c1−σ
1−σ
(21)
We solve for optimal government domestic and foreign debt policies in the first period following these steps:
1. Assuming that (19) and (20) are satisfied in second period we can rewrite government problem as (26) - see
Appendix 1
2. Solution to the problem is then a set of two first-order conditions (27) and (28) and pricing rules (29) and
(30)
3. We pick set of parameters and solve (27)-(30) numerically (for parameter choices see Appendix 1)
4. We confirm that resulting policy functions bf , bh and equilibrium prices qf , qh satisfy conditions (19)-(20)
therefore expectations in (26) are consistent in equilibrium
5. We vary one parameter at a time within a range where (19)-(20) are satisfied to derive comparative statics
Comparative statics reveal that this simple two-period environment can account for empirically observed facts
that share of foreign investors is negatively correlated with interest rates and total debt of the economy. For wider
discussion of two-period model results and graphical exposition of comparative statics for different parameters
please refer to Appendix 2. Now that the trade-offs behind our model have been described in detail we can turn to
quantitative analysis of infinite-horizon version of the model.
6
6.1
Quantitative Analysis
Calibration
To solve the model numerically, we need to assume specific functional forms and assign parameters. Table 1
represents the parameters, which is selected directly from data. We assume the CRRA utility function with the
risk aversion coefficient σ is equal to two. The risk-free interest rate r is set to 1.7% percent, which is the average
quarterly interest rate of a five-year US treasury bond during this time period. These parameters are common values
used in real business cycle literature as well as default literature. We employ Argentinian as a common framework
for default literature. We calibrate the AR(1) stochastic process for output, based on the series of Argentinian
GDP.
14
log(yt ) = ρy log(yt−1 ) + ut
(22)
where ut ∼ N (0, y ).
The government faces two type of costs upon default. The output cost is assumed to be asymmetric as in
Arellano (2008).
ytdef = min{yt , γ ∗ y}
(23)
where y is the mean of the output process and γ takes one of three values g. The cost function implies that
default is more costly with high output realisation. The government expenditure is set to be average Argentinian
government expenditure 25% of GDP for the period 1993-2011. This number is not substantially different from the
cross country average 31% Since domestic default is not highly frequent event, there are only after 1950 21 domestic
default episodes vs. 101 foreign default episodes in Reinhart and Roggoff dataset, we calculate the median length
of foreign default 2.5 years and domestic default 4.6 years for the whole dataset. This estimate is slightly low in
comparison with the average exclusion period for Argentina 7.5 years.
11
Table 1: Parameters Selected Directly
Parameters
Values
Source
Risk-free interest rate
Rf = 1.7%
US 5-year bond quarterly yield
Risk aversion
σ=2
Standard in literature
Persistence of output
ρ y = 0.945
Argentinian data
Std. dev. of output
y = 0.025
Argentinian data
Government expenditure
g = 25%
Gov. exp. of Argentina as % of GDP
Prob. of Re-entry
Avg. exclusion length
to foreign market
θf = 0.35
from foreign market 2-6 years
to domestic market
θh = 0.25
from domestic market 4-9 years
Low distortion shock
τ l = 1%
Normalization
After choosing directly eight parameters, we are left with six parameters to be calibrated. Table 2 summarizes
parameters and moments to be matched. We report the range of moments, because the exact value depends on
time and county sample as well as whether we use mean or median. Due to limited availability of data on domestic
vs. foreign default, we have to employ different datasets. We use Reinhart and Rogoff data to calculate frequencies
of different types of default. Unfortunately, Reinhart and Rogoff do not report debt composition. Therefore to
11 Gelos,
Sahay and Sandleris (2011) measure exclusion as the years between default and the date of the next issuance of public and
publicly guaranteed bonds or syndicated loans.
15
calculate debt-to-GDP ratios, we employ Panizza dataset, who constructs his data based on legal definition, under
what legal jurisdiction debt is issued, which is consistent with Reinhart and Rogoff. Interest rate date on domestic
debt is extremely scarce. We know only one source is standardized sovereign debt database of LAC Debt Group,
which contains information about Latin American Countries debt holdings and interest rates from 2006 onwards.
standard deviation of interest spread on domestic debt is almost twice as high as standard deviation of interest
spread on foreign debt. Standard deviation of interest spread on foreign debt is compatible with literature.12
Table 2: Parameters Selected by Matching Moments
Parameters
Values
Moment
Value
High distortion
τh = 10%
Domestic Debt-to-GDP-ratio
10-100%
Prob. of Low distor. state
πl l = 0.8
Domestic default frequency
0.5-3%
Prob. of High distor. state
πh h = 0.8
Std. dev. of interest spread
0.04161
on domestic debt
Discount factor
β = 0.95282
Foreign default frequency
1-6%
Cost of foreign default
γf = 0.998
Foreign Debt-to-GDP-ratio
10-100%
Cost of domestic default
γh = 0.99
Std. dev. of interest spread
0.02497
on external debt
6.2
Simulation Results
In this section we analyze default policies, debt policies and equilibrium prices in the calibrated model. Next we
examine quantitative performance of the model against the data. We explain the algorithm of solving the model
numerically in Appendix 3. Both default and debt policies are four dimensional objects, as the state space for the
economy consists of two endogenous (domestic and foreign debt) and two exogenous (output and tax distortions)
states. For each variable of interest we compare policies for different levels of the same type of debt keeping the
value of second type of debt constant. Figures 3 and 4 plot repayment and default policies in debt-output space.
White color stands for repayment, light grey for foreign default, dark grey for domestic default and black for total
default. We can see that repayment-default trade-off for foreign debt is mostly driven by output process, while tax
distortions don’t matter. On the other hand default area for domestic debt is much bigger for high tax distortion
than for low tax distortion scenario. Also, as in both cases we set second type of debt to zero, we cannot observe
total default.
12 The
spread is calculated as the difference between the Argentinean interest rates reported by Neumeyer and Perri (2005)[9] and the
rate of a 3-month U.S. Treasury bill in the period from 1993Q1-2001Q4.
16
Figure 3: Default sets for foreign debt given bh=0
Figure 4: Default sets for domestic debt given bf=0
17
Figure 5: Foreign debt policies given bh=1.8
Most interesting findings of the model are revealed by Figures 6 and 6. Figure 19 plots debt policies for
foreign debt given that outstanding domestic debt is positive bh = 1.8. Foreign debt policies are similar to other
quantitative models of sovereign default. Country accumulates foreign debt when output is high due to low interest
rates. Interest are low, as a result of default set decreasing in y. Also, government accumulates more debt when
economy suffers from high tax distortions. This is explained by the fact, that government avoids using distortionary
taxation and instead finances its expenditures via both foreign and domestic (as we shall see) debt.
Figure 6 plots policies for domestic debt. When tax distortions are low and output is high government for any
level of debt outstanding finances its expenditures in full via taxation. This is the situation when raising taxes
comes at lowest cost for the economy. On the other hand, when distortions are still low, but output is also low,
government runs the policy of covering its expenditures via domestic debt, whenever there is positive amount of
foreign debt outstanding. More interestingly, when distortions are high and output is low government is prohibited
from borrowing abroad and tax collection is costly, therefore optimal way of financing expenditures is again via
domestic debt. When output is high however government uses domestic debt also to repay foreign debt, as tax
necessary to repay foreign debt are costly to collect.
To assess the performance of the model, we simulate 10,000 paths from the model, each with length 1000 and burn
first 100 simulation of each path. Then we compare the resulting business cycle statistics with the corresponding
18
Figure 6: Domestic debt policies bf=1.8
statistics from data. Table 6.2 shows that the results for the benchmark calibration are in line with the data. Our
model performs well in most of dimensions. The model replicate reasonably high debts levels and the same time
reasonably low defaults probabilities. It predicts that consumption is more volatile than output and net exports are
strongly countercyclical, and net exports are strongly countercyclical.13 . It is worth to stress once again, that two
shocks have the opposite effect on the economy. While tax distortion shock has a substantial impact on domestic
debt accumulation, it has a mild impact on the foreign debt accumulation. The opposite is true for output shock.
The model also predicts mildly procyclical behaviour of interest rate on foreign debt, it generate countercyclical
behaviour of interest rate on domestic debt. It contradicts with findings by Neumeyer and Perri (2005), who showed
that in a sample of emerging economies the interest rate of foreign debt are countercyclical, but mildly procyclical
in developed economies.
7
Secondary Markets
In this section we will use the two-period model setting from Section 5 to study how secondary markets affect and
create incentives for government repayment and default. We will introduce secondary markets in the second period.
13 See
Neumeyer and Perri (2005)[9]
19
Table 3: Cyclical properties
Mean
Std. dev
corr(x,y)
corr(x,tau)
Domestic default frequency
3.20%
Foreign default frequency
3.29%
Total default frequency
0.47%
Domestic Debt-to-GDP-ratio
12.4%
8%
0.11
0.62
Mean Foreign Debt-to-GDP-ratio
11.8%
18%
0.61
-0.04
Interest spread on domestic debt
19.87%
1.65
-0.36
0.38
Interest spread on external debt
16.90%
0.03
0.17
-0.05
Consumption
7.67
1.35
0.99
0.17
Output
10.00
1.19
1
0
Trade balance in % GDP
26.80 %
3.47 %
-0.93
0.29
Secondary markets open after nature selects output and taxation shock. Therefore all participants on the market
have perfect foresight of what the government is going to do (repay or default), if no trade in assets will take place
on the secondary markets. Our first findings are summarized in Proposition 1 and 2. To summarize, with costly
enforcement there exist finite sustainable debt limits to both foreign and domestic debt, which can be broken due to
stochastic nature of output and taxation distortion. There are four possible outcomes of the model in the moment
of opening of secondary markets, which are summarized in Figure 2. Both in situation A and D workings of the
secondary markets would not change the final outcome, as both debts are repayable on primary markets or both
are going to be defaulted. Therefore we only study situation B, as situation C is its mirror image. First let us
summarize what is happening in the economy in the moment of opening of the secondary markets and introduce
some notation:
• There are positive both foreign and domestic debts outstanding: bf and bh
• Each debt has its respective sustainability limit which we derive from (19) and (19) and denote in levels: B¯f
and B¯h
• After good shock to taxation and bad shock to output foreign debt has exceeded it sustainability limit bf > B¯f
but domestic debt lies below its sustainability limit bh < B¯h 14 .
14 We
assume sharp inequalities, so that neither debt is exactly at its sustainability limit. The rationale for that is that we will
introduce sharp distinction between situations in which investors internalize the aggregate effects of their actions and in which they do
not. When debt lies exactly at its sustainability limit even a small investor with no market power understands that his participation in
debt repurchasing would change repayment incentives for the government.
20
• As foreigners know they will defaulted upon they are willing to sell their claims in the secondary market. As
domestic investors know there is still some room for an increase in repayable domestic debt they are willing
to provide demand for those bonds. Bonds in the secondary market sell at discount price p.
• Additionally we assume that the government is not able to repay full debt if it is held entirely by domestic
agents bh + bf > B¯h
For the matter of consistency with the model we keep track of domestic debt outstanding. However, our analysis
is also valid for the case when bh = 0, as in BMV (2010) [4]. Therefore we can see this section as generalization of
their work in which we allow for costly enforcement15 As we shall see what matters for creating repayment incentives
through secondary markets is not the level of home or foreign debt outstanding, but the relative difference between
above-the-limit foreign holdings bf − B¯f and below-the-limit domestic accommodation possibilities B¯h − bh . The
first difference we will call defaultable foreign debt overhang, while the second domestic debt accommodation limit.
Foreign investors strategy-space is the quantity of bonds they offer in the secondary market sf = {q f }
[0, bf ]. Domestic investors strategy space is quantity demanded in the secondary market sh = {q h }
qf ∈
q h ∈ [0, y2 ].
Discount price clears the market pq f = q h . We start with defining conditions for which BMV (2010) [4] equilibrium
holds.
Proposition 3. If domestic investors can accommodate all of defaultable foreign debt overhang, that is if (B¯h −bh ) >
(bf − B¯f ):
a. There exists a set of Nash equilibria in pure strategies {sf , sh } = {q ∗ , q ∗ } where each foreign investor supplies
and each domestic investor demands the same quantity q ∗ ∈ [
bf −B¯h B¯h −bh
]
bh ,
bf
of her bond holdings and the market
clearing discount price is p = 1.
b. As the result both debt fall under their respective sustainability limits and are repaid by government.
c. All equilibria with quantity strategies in the interior of this set are stable. The two equilibria with strategies on
the boundary of this set are not stable.
Proof. See Appendix part 4.
Note that there also exists a set of intuitive equilibria in mixed strategies which deliver the same allocation
f
{s , sh } = {{π q (bf ), (1 − π q )(0)} , {π q (bf ), (1 − π q )(0)}}: each foreign investor mixes between supplying her total
debt holdings bh to the secondary market with the same probability π q ∈ [
bf −B¯h B¯h −bh
]
bh ,
bf
and supplying zero with
q
probability (1 − π ) and home investors do the same. Bonds trade at discount price p = 1.
This result affects the workings of the primary market, as it turns risky foreign debt into riskless. Therefore
interest rate on the primary market is r = 0. Important to notice, is that necessary condition for the existence
15 In
BMV (2010) government wants to default on foreign debt due its discretionary behaviour under no penalties upon default δ f = 0.
In our analysis it is due to governments discretionary behaviour with δ f > 0 and unfortunate output shock.
21
of this class of equilibria is that domestic economy can fully accommodate defaultable foreign debt overhang in
repayment equilibrium. If this is not the case, then existence of equilibria and the allocation that they deliver
depends on the market power of investors, as we shall see next.
Proposition 4. If both domestic and foreign investors have zero market power (they do not internalize their
actions on aggregates and cannot coordinate) and defaultable foreign debt overhang is greater than domestic debt
accommodation limit there exists no Nash equilibrium in pure strategies.
Proof. See Appendix part 4.
Proposition 4 shows that under certain circumstances secondary markets do not help create incentives for repayment of government debt, when those incentives are absent on the primary market. Moreover, as the equilibrium
does not exist, the outcome of the secondary markets trade is uncertain. This result may shed some light on why,
in turbulent times, secondary markets may cease to function. Russia in 1998 defaulted effectively on its obligations
towards households but repaid its obligations to firms. Why there was no significant re-trade of bonds between
households and firms in the secondary markets remains an open question, but this proposition may provide some
intuition.
We investigate this result further in altering the assumptions that neither domestic nor foreign investors have
market power and cannot coordinate their actions. We formalize the idea of investor having market power by lifting
an assumption of each investor being zero-measure. Instead we assume that the measure of each investor is > 0,
so that the economy is populated by
1
number of investors. We still maintain assumption that defaultable foreign
debt overhang is larger than domestic debt accommodation limit.
Proposition 5. If domestic investors have market power (are able to coordinate) and defaultable foreign debt
overhang is greater than domestic debt accommodation limit B¯f − bf > bh − B¯h :
a. There exists a Nash equilibrium in pure strategies where {sh , sf } = {q h , q h} where q h = bh − B¯h .
b. Discount price that clears the market is any p from the range [0, 1].
c. Foreign debt is defaulted.
d. Equilibrium is not stable.
Proof. See Appendix part 4.
Note that equilibrium in Proposition 5 holds for foreign investors both with and without market power. What
is interesting, is that although this equilibrium delivers improvement upon no trade in secondary markets scenario,
the outcome is yet, as in Proposition 4, uncertain. What this equilibrium can establish, is that trade on secondary
markets will take place, and this may positively influence trade on primary markets as well (decrease in the interest
rate on international borrowing in the first period). However, as the result is ambiguous, the exact transmission
22
from secondary to primary markets is impossible to quantify. Lastly, let us study the reverse situation, in which it
is foreign investors who have market power, and domestic are all zero-measure.
Proposition 6. If foreign investors have market power (are able to coordinate), domestic investors do not have
market power (are unable to coordinate) and defaultable foreign debt overhang is greater than domestic debt accommodation limit B¯f − bf > bh − B¯h :
a. There exists a Nash equilibrium in pure strategies where {sh , sf } = {q f , q f } where qf = B¯f − bf .
b. Foreign investors are able to re-trade their defaultable debt overhang to home investors and are repaid by the
government. Domestic investors exceed domestic debt accommodation limit and are defaulted by the government.
c. The price on the secondary market is p = 0.
Proof. See Appendix part 4.
Proposition 6 shows an interesting result. Under the mix of unfavourable circumstances for domestic investors,
which is low accommodation limit relative to foreign debt overhang and no market power on their side, contrary
to foreign investors side, the selective default result that would occur on primary market absent any trade on
secondary market is reversed. Now, instead of defaulting on its foreign obligations government defaults on domestic
debt holdings and foreign obligations are repaid.
The aim of this section is to show that the workings of the secondary markets for government bonds in the
situation where either domestic or foreign debt is going to be defaulted, are ambiguous. This section by no means
exhaust the topic. What this section proves is that strengthening the role and efficiency of the secondary markets
is not a remedy that can automatically heal the sovereign risk problem. We find that the equilibria are dependent
on underlying conditions, such as investors market power and relative size of demand and supply of bonds. Clearly
more research both empirical and theoretical on the workings of the secondary markets during sovereign risk crisis.
8
Conclusions
The conclusions of the paper are two-fold. On the positive side we develop a model of sovereign debt issuance
on intenational and domestic markets and selective defaults. Our model is capable of replicating business cycle
statistics and we show that including two types of investors brings the model closer to the data, especially in terms
of debt-to-GDP ratio and direct output penalties upon default. Our model is useful tool to study how fraction of
investors in public debt arise engogenously in equilibrium and how debt composition is correlated with spreads and
total debt. On a positive side we also provide a simple theory of domestic public debt.
On the normative side we analyze how debt is being retraded on secondary markets when taxation is distortionary
and government is about to default. We characterize conditions under which defaultable debt overhang is fully
23
retraded on secondary markets and both domestic and foreign debt are repaid. However, we show that if these
conitions are not met other type of equilibria may arise. Depending on the relative demand and supply for bonds
and market power of we find equilibria in which default incetives are reversed - instead of defaulting on foreign debt
goverment defaults on domestic debt after trade on secondary markets or
References
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[4] Fernando Broner, Alberto Martin, and Jaume Ventura. Sovereign risk and secondary markets. American
Economic Review, 100(4):1523–55, September 2010.
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24
[13] Hubert Kempf Russell Cooper and Dan Peled. Is it is or is it ain’t my obligation? regional debt in a financial
federation. International Economic Review, 2008.
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in emerging and advanced economies. Oxford Review of Economic Policy, 2013.
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International Economics, 2010.
Appendix 1. Two period model
A. Derivations
Output and tax distortion processes are given by:


yh = y with prob. π y
y2 =

yl with prob. 1 − π y


τl
with prob. π τ
τ2 =

τh > τl with prob. 1 − π τ
(24)
(25)
Assuming that (19) and (20) are satisfied in second period we can rewrite government problem as
u (y − qh bh − (g − qh bh − qf bf ) (1 + τ1 ))
max
bf ,bh
+βπ y π τ
u (y − bf (1 + τl ) − τl bh )
u y 1 − δ h − bf (1 + τh )
u yl 1 − δ f − τl bh
!
u yl 1 − δ f 1 − δ h
+βπ y (1 − π τ )
+β (1 − π y ) π τ
+β (1 − π y ) (1 − π τ )
(26)
First order conditions of the problem yield:
(bf :)
(bh :)
βπ τ (1 + τ l )u0 (cH,τ
2
(1 + τ1 )qf u0 (c1 ) =
+
β(1 − π τ )(1 + τ h )u0 (cH,τ
2
τ1 qh u0 (c1 ) =
βπ τ τ l u0 (cH,τ
2
y
+
β
25
1 − π τ l 0 L,τ
π τ u (c2
πy
l
,f r,hr
h
l
l
)
,f r,hd
)
,f r,hr
)
,f d,hr
)
(27)
(28)
And equilibrium prices are:
1 − πy
piy
(29)
π y π τ u0 (cr2 ) + (1 − piy )piτ u0 (cf2 d )
u0 (c1 )
(30)
qf =
qh = β ∗
B. Parametrization
Parameter
Value
y
1
Output today / High output tomorrow
σ
1
Risk aversion of Home agents
Πy
0.72
[0.5 , 1]
Probability of high output
Πτ
0.8
[0.5 , 1]
Probability of Tax Reform
g
0.7
[0.5 , 0.8]
Gov. expenditure
yL
0.5
[0.1 0.7]
Low output
τ1
0.1
[0 , 0.2]
Tax distortions today
τ2H
0.15
[0.1 , 0.2]
Tax distortions tomorrow (high)
τ2L
0.05
[0 , 0.15]
Tax distortions tomorrow (low)
f
0.65
[0.42 , 0.87]
Output cost of Foreign default
δh
0.05
[0 , 0.15]
Output cost of Home default
r
0.00
δ
Range
Description
Risk-free interest rate
Appendix 3. Solution algorithm
1. Guess price schedules p0f and p0h
2. Calculate consumption in autarchy caut and value of permanent autarchy V aut
3. Guess four value functions V 00 , V 0f d , V 0hd and V 0td using V aut
4. Calculate optimal policies bf and bh in repayment given V 00 as continuation value and prices
5. Calculate value of repayment V r given optimal policies and continuation value
6. Repeat steps 4 and 5 for foreign default and domestic default to obtain V 1f d and V 1hd
7. Calculate value of total default V 1td given V 1f d and V 1hd and V 00
8. Derive optimal default policies d comparing four value functions V r , V 1f d V 1hd V 1td at each grid point
{bf , bh , y, τ }
26
9. Derive new value function V 10 as maximum of four value functions used in previous step at each grid point
10. Substitute V 00 = V 10
11. Repeat steps 3-9 until convergence in value function
12. Given optimal default policies d calculate prices of foreign and domestic debt qf1 and qh1 at each grid point
using pricing rules (7) and (8)
13. Update prices qf0 = αf qf0 + (1 − αf )qf1 and qh0 = αh qh0 + (1 − αh )qh1
14. Repeat steps 1-13 until convergence in prices
Appendix 4. Proofs
Proof of Proposition 3. a and b. There is no profitable deviation for any of the participants as bonds are retraded
at face value and both investors are repaid.
c. Consider first strategy q ∗ =
bf −B¯h
bh .
If any single foreign investors makes mistake and instead supplies q ∗ − amount in the secondary market, then foreign debt holdings after secondary markets close are B¯h + and exceed
sustainability limit. As the result foreign debt is defaulted. Similarly when q ∗ =
B¯h −bh
bf
if any foreign investor
makes mistake and instead supplies q ∗ + amount, then domestic debt holdings after secondary markets close are
B¯h + and exceed domestic debt sustainable limit. Any -deviation from q ∗ in the interior of this set does not
destroy repayment equilibrium.
Proof of Proposition 4. a. Let’s first consider pair of strategies {sf , sh } = {qa , qa } where qa = B¯h − bh , that is
exactly what domestic economy can accommodate in the repayment equilibrium. Therefore under this strategies
discount price is p = 1. However, as foreign debt is still defaultable each foreign investor has profitable deviation
to supply more bonds on the secondary market. As they do the market clearing price goes down. As the price
goes down domestic investors have more incentives to profitably deviate by increasing their demand, as they don’t
internalize their deviation will affect aggregate outcome. Therefore {sf , sh } = {qa , qa } is not an equilibrium.
b. As foreign investors have profitable deviation to increase bond supply let us consider pair of strategies {sf , sh } =
{qb , qa }, where qb = bf − B¯f and qa is the same as in point a. The market clearing price is p =
qa
qb .
As domestic
investors assume more bonds than is sustainable in repayment equilibrium they have profitable deviation to decrease
their demand in the secondary market.
c. The argument in point a. can be extended for any pair of strategies {sf , sh } = {q f , q h }
q f ∈ (0, qb ), q h ∈ (0, qa ).
d. The argument in point b. can be extended for any pair of strategies {sf , sh } = {q f , q h }
q f ∈ [qb , bf ], q h ∈ (0, bf ].
e. What is left to see is whether there could be an equilibrium where foreign investors supply less than foreign debt
27
overhang and domestic investors demand more than domestic debt accommodation limit {sf , sh } = {q f , q h }
qf ∈
[0, qb ), q h ∈ [qa , bf ]. Clearly, as long as p > 0 both parties have profitable deviations: foreign investors increase their
supply and domestic investors decrease their demand. In the limiting case we have q f = qb , q h = qa which is not an
equilibrium as proven in point b.
Thus no pair of strategies constitutes an equilibrium. QED
Proof of Proposition 5. a. Each domestic investor internalizes that any δ > 0 increase in demand deviation from
her strategy would result in the total domestic debt being equal B¯f + δ and therefore exceeding sustainability
limit. Therefore any increase in demand by domestic investors is not a profitable deviation. Any δ < 0 deviation
in her strategy would be weakly pay-off reducing as the the bonds are purchased on the secondary market at the
discount price p ∈ [0, 1] and are later repaid by government at face value. Therefore any decrease in demand is not
a profitable deviation for domestic investors.
For foreign investors any decrease in supply would be weakly reducing pay-off because the trade on secondary
market is the only source of revenue, as their debt holdings will be fully defaulted on the primary market. On the
other hand, any increase in supply cannot be met by domestic investors demand, as they can coordinate and would
never demand more than bh − B¯h
b. On the secondary market trade is demand-determined, therefore trade at any price p ∈ [0, 1] constitutes improvement upon no-trade scenario for both parties.
c. Any mistake in the form decrease in demand on the side of domestic investors will be met by an increase in
supply by foreign investors and will lead to domestic default as shown in point a.
Proof of Proposition 6. The proof rests on the logic presented in Proof of Proposition 4, where no pair of strategies
constitutes an equilibrium, as each investor has profitable deviation. Now, as foreign investors are able to coordinate,
none of them has profitable deviation of supplying anything less than exactly her part of defaultable foreign debt
overhang (B¯f − bf ), as supplying even δ > 0 less would result in foreign debt being defaulted. Also, as price is
p = 0 none of them has profitable deviation to supply anything more, as trading even δ > 0 would mean giving
away for free repayable bonds.
Domestic investors in this equilibrium are defaulted upon. Therefore each of them is indifferent among demanding
any amount of bonds on the secondary market as long as the price is p = 0.
28