The Puzzling Behavior of Spreads during Covid∗
Stelios Fourakis

Loukas Karabarbounis

Johns Hopkins University

University of Minnesota and FRB of Minneapolis

January 2024
Abstract

Advanced economies borrowed substantially during the Covid recession to fundtheir fiscal policy. The Covid recession differed from the Great Recession in thatsovereign debt markets remained calm and spreads barely responded. We study
the experience of Greece, the most extreme manifestation of the puzzling behaviorof spreads during Covid. We develop a small open economy model with long-term
debt and default, which we augment with official lenders, heterogeneous households
and sectors, and Covid constraints on labor supply and consumption demand. The
model is quantitatively consistent with the observed boom-bust cycle of Greece
before Covid and salient observations on macro aggregates, government debt, andthe sovereign spread during Covid. The spread is stable despite a rise in externalborrowing during Covid, because lockdowns were perceived as transitory and thebailouts of the 2010s had tilted the composition of debt at the beginning of Covid
away from defaultable private debt. The ECB’s policy of purchasing debt in secondary markets during Covid did not stabilize spreads so much, but allowed thegovernment to provide transfers that reduced inequality.
JEL Classifications: E20, E58, E60, F34, F44.
Keywords: Sovereign Debt, Lockdowns, Official Lending, Inequality.

∗
Contact information: sfourakis@jhu.edu and loukas@umn.edu. The views in this paper do not necessarilyreflect the views of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

1

Introduction

According to the International Monetary Fund (IMF), GDP of advanced economies contracted
by 4.2 percent in 2020.1 To combat the Covid recession, governments borrowed significantly tofinance fiscal stimulus programs. As a result, between 2019 and 2020, government debt relative
to GDP of advanced economies increased by almost 20 percentage points. The changes in
GDP and government debt during Covid were as significant as those observed during the GreatRecession, when advanced economies’ GDP fell by 3.4 percent and their debt-to-GDP ratio
increased by roughly 20 percentage points. However, in contrast to the sovereign debt crisesof the Great Recession in the periphery of the eurozone, sovereign debt markets in advanced
economies during Covid did not experience turbulence. The goal of this paper is to understand
why this time was different. Why did sovereign spreads not increase during Covid, despite
countries engaging in fiscal expansions that were as significant as past expansions that wereaccompanied by turbulence in sovereign debt markets?
To answer our question, we study the experience of Greece. Figure 1 shows the evolutionof the Greek sovereign spread and debt-to-GDP ratio since 2000. After joining the euro area,
Greece was borrowing at a spread of less than one percentage point over the interest rate
paid by the German government. Greece’s government debt-to-GDP ratio started increasing
significantly in 2007 and reached almost 200 percent in 2011. Its sovereign spread started
increasing in late 2008 and reached roughly 20 percentage points in 2011. The Covid stimulus
in 2020 elevated Greece’s debt-to-GDP ratio to an all-time high, but its spread barely responded.
Greece is an ideal case study for several reasons. Like other advanced economies, Greece
experienced a significant recession in 2020, with GDP falling by 9 percent, and expanded
its fiscal policy dramatically, leading to a roughly 25 percentage points rise in its debt-toGDP ratio. Furthermore, shortly before Covid, Greece experienced a sovereign debt crisis
during a macroeconomic contraction of unprecedented magnitude (Gourinchas, Philippon, andVayanos, 2016; Chodorow-Reich, Karabarbounis, and Kekre, 2023). While the puzzling patternof stable spreads despite rising debt levels is not unique to Greece, Greece is the most extreme
manifestation of the puzzle. It had the highest external debt-to-GDP ratio in the world andexperienced a sovereign debt crisis just a decade before Covid. Yet, Greece was able to borrow
massively again during the Covid recession, without an increase in its borrowing costs.
To give a sense of the magnitude of the missing Greek spread during Covid, we examine
the statistical relationship between spreads, debt, and other variables in samples of advanced
1

The data for the tabulations in this paragraph are from the October 2023 version of the World Economic
Outlook (https://shorturl.at/efp78).

1

30

225

10

Spread (p.p.)

20

200
175
150

0

125

Debt-to-GDP (percent)

100

2000

2005

2010

Debt-to-GDP

2015

2020
SpreadFigure 1: Greek Debt-to-GDP Ratio and Sovereign Spread

economies before Covid. The predictions using samples that include the Greek sovereign debt
crisis of the 2010s generate an upper bound on the missing spread, because they project fromthe previous Greek experience, which features the strongest statistical association between the
spread and debt and the largest persistent depression of the past 60 years among middle- andhigh-income economies. Using only the experiences of Ireland, Italy, Portugal, and Spain, we
instead obtain a lower bound. Pooling across samples and specifications, we find a missing
spread of around 4 percentage points, with a lower bound of around 2 percentage points and an
upper bound of around 7 percentage points. Using these estimates, we can pose our researchquestion more precisely: Why did the historical relationships that predict a roughly 4 percentage
points increase in the Greek spread break down during Covid?
We begin by developing a model that aims to account for the experience of Greece both
before and during Covid. We augment a standard small open economy model of long-term
debt and default with four elements that allow us to accommodate several different forces that
could plausibly account for the stability of the spread.2 First, we introduce official lenders thatprovide bailout loans to the government. These loans have a longer maturity and a lower coupon
rate than the debt held by private investors and are risk-free. The government anticipates that
official lenders may provide loans during debt crises, and this expectation affects its incentives
to accumulate debt and the price of the debt that it issues. We use the words “loans” and“bailouts” interchangeably, and include in them both the loans that were extended to Greece
2

These elements aside, the model is similar to Chatterjee and Eyigungor (2012). This is also the baselinemodel described in Aguiar, Chatterjee, Cole, and Stangeby (2016) without rollover risk and risk-averse foreign
investors.

2

in the 2010s as part of the economic assistance programs and the direct purchases of Greeksovereign debt by the European Central Bank (ECB) in secondary markets beginning in 2020
through the Pandemic Emergency Purchase Programme (PEPP).
Second, in addition to internationally traded goods, we model a non-traded sector that produces domestically consumed goods, because Covid lockdowns had a disproportionate impact
on some types of services that are not traded. Third, motivated by the significant transfersthat Greek households received both in the boom of the 2000s and during Covid, we model two
types of households and introduce a motive for the government to redistribute income. The first
type is optimizing households, which have access to an internationally traded risk-free bondthat allows them to smooth their consumption over time. The second type is hand-to-mouthhouseholds, which cannot access asset markets, have lower labor income than the optimizing
households, and are disproportionally affected by lockdowns because of their limited ability towork from home. Finally, in our production economy, the private sector responds to government
policies by changing its savings and labor supply, which introduces a meaningful trade-off for
the government when choosing the optimal level of redistribution.
We quantify the model and show that it does a good job of generating the observed boomof the 2000s, the bust of the 2010s, and the dynamics of the debt-to-GDP ratio and the spread
both in the 2000s and in the 2010s. In doing so, we depart from the typical strategy forquantifying sovereign default models, which is to compare averaged model outcomes to averaged
statistics in the data over a long sample or close to default episodes.3 Instead, we map, ina detailed way, exogenous and endogenous model variables to their data counterparts, feed
the measured exogenous variables into the model, and evaluate the model’s performance interms of matching the time series of the endogenous variables in the whole sample. While thisprocess is more elaborate than typical strategies followed by the literature, our approach has two
advantages. First, by feeding in the exogenous shocks as measured in the data before solving
the model, we remove degrees of freedom from the researcher when it comes to generatingboom-bust cycles and sovereign debt crises. Second, we assess the model’s performance morecomprehensively than it is done in the literature, as we compare production, consumption,
private assets, transfers, sovereign debt, and spreads in the model to the observed time series
throughout our sample period.
Covid lockdowns take the form of two constraints, one limiting the non-traded goods thatcan be consumed in 2020 and the other limiting the labor that hand-to-mouth households can
3

For example, see the quantification of typical models described in Aguiar, Chatterjee, Cole, and Stangeby(2016).

3

provide in 2020. In our open economy model, the supply-side constraint is important for generating the drop in production, and the demand-side constraint is important for generating thedrop in consumption. We discipline our quantitative exercises by requiring that the parametersof the economy before Covid be held constant when we assess the performance of the model
during Covid. Despite this discipline, we find that the model matches very closely the increasein transfers and government debt in 2020 and 2021, while at the same time generating stablespreads in both years. The model also matches the observed increase in private savings in 2020
and the observed rebound of consumption and output in 2021.
We illustrate that the constraint on labor supply is mostly responsible for the substantial
increase in transfers and government debt during Covid. The mechanism is that lockdowns disproportionally affect hand-to-mouth households in 2020, so the government provides insuranceby transferring resources to them. However, some transfers spill over to optimizing households. Since optimizing households face a binding constraint on consuming non-traded goods
and anticipate that the rise in transfers is transitory, they save the additional resources. When
lockdowns are lifted in 2021, optimizing households increase their consumption substantially,
and the government continues to provide higher transfers to hand-to-mouth households, which
reduce inequality.
Having developed a model that generates a significant recession, a significant increase inboth government borrowing and private savings, and stable spreads during Covid, we run ahorse race to assess which of the competing explanations accounts for the missing spread during
Covid. Quantitatively, the most important factor for the low spread is the belief that lockdowns
would not persist for longer. To be precise, by “belief” we mean the expectation of how
lockdowns would evolve after 2020 and not whether lockdowns actually occurred after 2020. The
incidence of lockdowns also matters for spreads, but their expected duration is quantitatively
more important. The mechanism by which a higher expected duration of lockdowns increases
the spread is subtle and specific to the nature of the Covid shock. If the 2020 recession had beencaused by a productivity shock, the expected duration of the shock would not have played an
important role for the spread. The difference between lockdowns and productivity shocks is that
lockdowns are redistributive in nature, because in the absence of a policy intervention, they hit
hand-to-mouth households hardest. Had lockdowns been perceived as more persistent, foreign
investors would have anticipated more redistribution by the government in the future, which
would have pushed the economy closer to default with a higher spread and lower equilibriumborrowing.
The second most important factor in generating a low spread during Covid is the old bailouts
4that Greece received during the 2010s. By old bailouts, we mean the official loans provided to
Greece under the economic assistance programs. Whereas in the 2000s, less than 20 percent of
Greek government debt was from official lenders, that proportion rose to more than 60 percent
by the end of 2019. In a counterfactual world in which these loans had not been extended in
the 2010s, Greece would have entered Covid with a debt composition that was tilted toward
defaultable private debt, and the spread would have increased more during Covid. Similarly,
as part of the economic adjustment programs in the 2010s, Greece wrote down less than halfof its private debt. If prevailing policies had extracted more debt relief from private investors,
expectations for future haircuts in 2020 would have been higher, and the spread would haveincreased by significantly more during Covid.
By contrast, we find a smaller role for the PEPP. The ECB’s purchases of Greek debt insecondary markets were roughly 10 percent of GDP in both 2020 and 2021. In their absence,
spreads would have increased, but by less than the lower bound predicted by the historical
relationship between the spread and debt. This occurs because removing ECB purchases leads
the government, in equilibrium, to borrow significantly less in order to moderate the rise of the
spread. We conclude that ECB purchases did not stabilize spreads so much, but they allowed
the government to provide increased transfers to households, which reduced inequality.
We find a negligible role for moral hazard in keeping spreads stable, because debt crises andsignificant extensions of loans are rare events. Changes in the maturity of private debt issuance,changes in the risk-free rate, and private sector responses also play a limited role in stabilizing
the spread. Finally, we study interactions among various forces. While the ECB purchases in2020 are not quantitatively important on their own for stabilizing the spread, removing thesepurchases and increasing the expected duration of lockdowns raises the spread to levels closer
to the ones observed during the sovereign debt crisis of the 2010s. A policy implication of
our findings is that ECB policies during Covid mostly ameliorated inequality, but would haveaverted a sovereign debt crisis had investors perceived lockdowns as more persistent.
Our paper contributes to several strands of literature. We extend the research frontier ofsovereign default models in the spirit of Eaton and Gersovitz (1981), Arellano (2008), andChatterjee and Eyigungor (2012) to include heterogeneous households and sectors, private savings, redistributive motives for the government, official lenders, and Covid lockdowns.4 While
4

Gordon and Guerron-Quintana (2018) also consider a production economy by including capital in theirmodel. They study a social planner’s problem that can be decentralized under suitable lump-sum transfers to arepresentative household. A difference with their paper is that private savings constrain the government in ourmodel, because transfers also accrue to households that do not have access to private savings. Arellano, Bai,and Mihalache (2018) extend the model with capital accumulation to a two-sector environment that accounts
for the observation that non-traded production declines more than traded production during sovereign debt

5

most of these elements separate us from papers such as Bocola, Bornstein, and Dovis (2019)
and Bocola and Dovis (2019), we share with them our focus on the experience of advanced
economies, and in particular the eurozone, rather than the experiences of emerging markets,
which originally motivated the sovereign default literature. As discussed earlier, a contributionof our paper is to further develop the quantitative implementation of sovereign debt models by
more thoroughly mapping our model to the data and by more comprehensively assessing themodel’s performance.
An emerging literature tries to understand how official lending impacts sovereign borrowing
in private markets. Callegari, Marimon, Wicht, and Zavalloni (2023) and Liu, Marimon, andWicht (2023) consider how a financial stability fund can improve allocations by offering a longterm contingent contract to a country that cannot commit to repaying its debt. We are focusedon the quantitative effects of and the fiscal policy responses to bailouts, rather than on theiroptimal design, and thus we take bailout policies as given from the data without modelingthe problem faced by the official lender. In a standard sovereign default model, Fink and
Scholl (2016) demonstrate that bailouts reduce default risk in the short term at the cost ofincreasing the long-run probability of default as countries accumulate more debt. The authors
also demonstrate that conditioning loans on some modest level of fiscal discipline lowers thelong-run probability of default. Our model is similar to theirs in that government incentives toborrow are affected by the possibility of official loans. However, we abstract from conditionality
in fiscal policy because PEPP did not impose conditions on fiscal policy.
Our model predicts an increase in the spread by 0.5 percentage point in 2020 in the absenceof the purchases of sovereign debt by the ECB. In the absence of the 2010s loans, the model generates a roughly 2 to 2.5 percentage points increase in the spread between 2012 and 2018. Ourestimates come from a structural model with equilibrium adjustments and specific mechanisms,
and thus are not directly comparable to empirical estimates of the effects of asset purchaseson sovereign spreads that do not include equilibrium adjustments but include other channels
by which purchases affect spreads. Despite these differences, we document that empirical estimates of the effects of asset purchases on spreads (Trebesch and Zettelmeyer, 2018; Rostagno,
Altavilla, Carboni, Lemke, Motto, and Saint-Guilhem, 2021) are quantitatively consistent with
our model-based counterfactuals of the effects of official lending on the Greek spread.
Our paper contributes to the study of fiscal policy during recessions in the presence ofsovereign risk.5 Bianchi, Ottonello, and Presno (2023) study a model with nominal rigidity andcrises. Sosa-Padilla (2018) studies the interaction between sovereign debt crises and banking crises in a closed
economy model of default, private capital accumulation, and distortionary labor taxation.
5
In an early contribution, Cuadra, Sanchez, and Sapriza (2010) develop a model with endogenous fiscal policy

6uninsurable unemployment risk for hand-to-mouth households. Despite these elements, which
tend to make countercyclical fiscal policy optimal, the authors demonstrate that such a policy
may be undesirable because it increases the probability of a debt crisis during a recession. In
our model, transfers to poor households decline substantially in the recession of the 2010s,
because the government is excluded from international financial markets and does not haveenough resources to sustain transfers. However, transfers increase substantially during Covid.
Consistent with their insights, we illustrate that the optimality of expanding fiscal policy ina recession is state dependent. Had Greece entered the Covid period with a larger stockof private debt and a lower stock of official loans, transfers would have been substantiallysmaller. Further, in our model, the ECB’s purchases of sovereign debt provide a buffer that the
government uses to expand its fiscal policy during Covid. Another difference from their paper
is that the redistributive nature of the Covid recession is key for how fiscal policy responds
to it, and that response is qualitatively different from the response with respect to standard
productivity shocks.62Some Observations on Sovereign Spreads and Debts

We begin by presenting some observations on the relationship between sovereign spreads andgovernment debt. We restrict our analysis to the group of advanced economies, which is therelevant comparison group for Greece. In Figure 2 we present the cross-sectional relationshipbetween the five-year CDS spread and the change in the debt-to-GDP ratio.7 The left panels
show this relationship for the sample before Covid, covering the years between 2003 and 2019.
and sovereign risk, and establish that public spending is procyclical and tax rates are countercyclical. Arellano
and Bai (2017) study the Greek crisis and show that a rise in tax revenues does not prevent sovereign default
because they are used to finance public consumption.
6
Arellano, Bai, and Mihalache (2023) study debt relief programs in emerging markets during Covid, in aframework that integrates epidemiological dynamics into sovereign default models. Unlike advanced economies,
emerging markets had less fiscal space to transfer resources to households and thus the authors find substantial
benefits from relief programs that eased financial constraints. We instead find that the ECB’s intervention had
a modest impact on spreads but a larger impact on the consumption of poor households. Motivated by theEuropean sovereign debt crisis, D’Erasmo and Mendoza (2021) study the distributional motive of a government
to not repay its debt in a model in which households have uninsurable risk and hold government debt. We
abstract from such a motive, because for the case of Greece, most government debt is held externally. Instead,
we focus on redistribution by way of government transfers, which was the primary policy instrument used toaddress the impact of Covid.
7
For this figure, we use the fiscal space database from the World Bank (Kose, Kurlat, Ohnsorge, and Sugawara, 2022). The CDS spread data come from Bloomberg and J.P. Morgan. We use CDS spreads relative toGermany. We exclude countries for which less than 10 percent of government debt is held externally, a criterion
which drops Switzerland and Japan from the sample. To improve the visibility of the figure, we also exclude
observations with spreads higher than 100 percentage points, a criterion that drops Greece in 2012 from thesample.

7

8
6
4
0

2

CDS Spread (p.p.)

8
6
4
2

0

5

10

15

20

25

-15 -10

-5

0

5

10

15

20

(a) Sample 2003-2019(b) Sample 2020-2022

-5

0

5

10

15

20

25

3
2
1
0

4
3
2
1
0

-15 -10

4

Change in Debt-to-GDP (p.p)

Change in CDS Spread (p.p.)

Change in Debt-to-GDP (p.p)

25-1

CDS Spread (p.p.)

0

-5

-1

Change in CDS Spread (p.p.)-15 -10-15 -10

-5

0

5

10

15

20

Change in Debt-to-GDP (p.p)

Change in Debt-to-GDP (p.p)(c) Sample 2003-2019(d) Sample 2020-2022

25

Figure 2: Cross Section of CDS Spreads and Debt-to-GDP Ratios
Each dot represents the average CDS spread for a bin of countries that consists of 2.5 percentof observations, where observations are sorted according to their annual change in the debt-toGDP ratio in the horizontal axis. The right panels show the same relationship for the Covid
sample, covering the years between 2020 and 2022. Each bin of countries consists of 5 percentof observations, as we have fewer observations in this shorter sample. In both figures we fit aquadratic relationship.
Beginning with the top left panel, we observe that the CDS spread is uncorrelated withchanges in the debt-to-GDP ratio for negative changes in the debt-to-GDP ratio. The spreadincreases as the debt-to-GDP ratio becomes positive, reaching almost 8 percentage points forcountries that experience a 20 percentage points change in their debt-to-GDP ratio.8 The
bottom left panel confirms that the same convex relationship arises between the annual change
8

Greece during its sovereign debt crisis is a member of this bin. Other countries with relatively high spreads
and changes in their debt-to-GDP ratios are Ireland, Italy, Portugal, and Spain during the euro debt crisis andCyprus in 2015.

8

8
6
0

2

4

Spread (p.p.)

8
6
4
0

2

Spread (p.p.)

2018

2019

2020

Observed
CS: Changes (Pre-20)
TS: GIIPS (99-11)

2021

2022

2018

CS: Levels (Pre-20)
TS: IIPS (99-11)
Greece (99-11)

2019

2020

Observed
CS: Changes (Pre-20)
TS: GIIPS (99-11)(a) Predictor: Debt/GDP

2021

2022

CS: Levels (Pre-20)
TS: IIPS (99-11)
Greece (99-11)(b) Predictors: Debt/GDP and GDP

Figure 3: Actual vs Predicted Spread for Greecein the spread and the annual change in the debt-to-GDP ratio, but the maximum change inthe spread is now 4 percentage points.
In the right panels, for the Covid period, we observe that the range of values for the change
in the debt-to-GDP ratio is similar to the one in the left panels for the period before Covid.
Several governments during Covid increased their debt ratio by around 20 percentage points
relative to GDP. Despite the significant changes in the debt-to-GDP ratio during Covid, spreads
are low. No advanced economy in the world experienced a spread higher than 2 percentage
points, and the spread is uncorrelated with the debt-to-GDP ratio.
Figure 3 quantifies the puzzling behavior of the spread for the case of Greece. The solid
blue line presents the observed Greek sovereign spread between 2018 and 2021. The other lines
predict the Greek spread from 2020 onward, using the historical behavior of sovereign spreads
before 2020. The lines “CS: Levels (Pre-20)” and “CS: Changes (Pre-20)” in the left panel of
Figure 3 predict the Greek spread using the linear relationship between the level or the change
in the spread and the change in the debt-to-GDP ratio for the observations that underlie thebinscatters of the left panels of Figure 2. The lines “CS: Levels (Pre-20)” and “CS: Changes(Pre-20)” in the right panel of Figure 3 predict the Greek spread using the same relationship but
also add the change of log GDP as a predictor. The other three lines use time series variation
before the euro debt crisis, for a sample that only includes only Ireland, Italy, Portugal, andSpain (“TS: IIPS (99-11)”), a sample that adds Greece to these four countries (“TS: GIIPS (9911)”), and a sample that covers only Greece (“Greece: 99-11”). As the figures show, predicting
9

solely on the experience of other euro countries with turbulence in their sovereign debt markets
produces a lower prediction of the increase in the spread. By contrast, including the Greek
experience before the 2012 default significantly raises the predicted spread.9
To summarize, from the historical relationship between spreads and debt-to-GDP ratios,
we predict that the increase in Greece’s debt-to-GDP ratio during Covid should have been
associated with a spread that rises to roughly 5.5 percentage points, when averaged acrossestimation methods.10 The observed spread is, however, well below 2 percentage points after
the second half of 2020. The remainder of the paper attempts to understand why the historical
relationship broke down during Covid.

3

Model

We first describe the economy for the period between 2002 and 2019, which includes the boomin the 2000s and the sovereign debt crisis in the 2010s, but not the Covid recession. We use thisperiod to parameterize the model and assess the plausibility of its mechanisms. Later, we extend
the environment to model the economic impacts of lockdowns and to perform counterfactuals
during Covid.

3.1

Population

Time is discrete and the horizon is infinite, t = 0, 1, 2, .... The domestic economy is small
and populated by a measure one of households, a representative firm in each sector, and agovernment.11 There are two types of households, j = {o, h}. The key difference between
them is that households o optimize their consumption intertemporally, while households hconsume hand-to-mouth their after-tax-and-transfer income. There are two sectors i = {T, N },with sector T producing internationally traded goods and sector N producing domesticallyconsumed non-traded goods. In addition to these domestic participants, there is a continuumof international investors that hold and price government private debt, as well as official lenders
9

The data for the time series predictions in Figure 3 come from the ECB’s Interest Rate Statistics and are
the same data underlying the series shown previously in Figure 1. For the time series regressions with more than
one country, we include time and country fixed effects in a regression of the spread on the debt-to-GDP ratio
and log GDP. We have confirmed the robustness of our results to including alternative or additional controls
such as GDP growth, the trade balance, the primary deficit, and an index of political risk.
10
Our range of estimates is consistent with the estimates reported by Cruces and Trebesch (2013). The
authors estimate how spreads comove with the debt-to-GDP ratio for a larger set of countries that includes
emerging markets and for a longer sample. Using their estimated coefficient on debt-to-GDP, we would predictthat the Greek spread rises to around 4.5 percentage points during Covid.
11
The economy is small in the sense that it takes as given the price of traded goods and the interest rate on
savings. Traded goods serve as the numeraire, with a world price equal to one.

10

that may extend loans to the government.

3.2

Timing of Events

Vector st collects the state variables at the beginning of the period. The government moves
first and decides whether to restructure the debt held by foreign investors. Conditional on
not restructuring its debt, it chooses how much new debt to issue and how many resources
to transfer to households. If the government decides to restructure its debt, it may receive aloan from official lenders. After the government makes its decisions, households choose theirconsumption, labor, and assets, firms choose their output and inputs, foreign investors pricethe government debt, and labor and non-traded goods markets clear. Private decisions dependon the vector xt , which includes st and all other variables realized after the government makes
its decisions.

3.3

Households

A measure γ of households are type o, “optimizing,” and use savings to smooth their consumption intertemporally. They choose their consumption of traded goods cT , consumption of
non-traded goods cN , labor supply `, and assets a to maximize the expected sum of discounted
utility flows,

E0

∞
X

βct

t=0

 α 
gt
χoc(.)1−σ − 1
o 1+1/ε
−
(` )
,
ḡ
1−σ
1 + 1/ε t(1)where parameter βc ∈ (0, 1) is households’ discount factor, parameter σ ≥ 0 is the coefficientof relative risk aversion, parameter χo > 0 is the disutility of work, and parameter ε ≥ 0 is the
Frisch elasticity of labor supply.
The factor (gt /ḡ)α that multiplies the bracket is the utility effect of exogenous government
consumption g, normalized by some constant level ḡ. Parameter α allows us to consider in a
flexible way how government consumption affects the private marginal utility of consumption.
When α = 0, private and government consumption are separable. Government consumption
increases the private marginal utility of consumption when α > 0, and the opposite when α < 0.
The same factor (gt /ḡ)α multiplies both the utility of consumption and the disutility of labor,
so that government consumption does not mechanically generate a labor wedge in the work
decision.12
12

For equilibrium allocations and prices, the only relevant consideration is whether government consumption
is a complement or a substitute to private consumption, and not whether government consumption increases
or decreases the level of utility. To consider the welfare effects of government consumption, we could add a11

The consumption aggregator in the utility function is
φ
 1
 φ−1
φ−1
φ−1
1
φ
cot = ωc (coT t ) φ + (1 − ωc ) φ (coN t ) φ,(2)

where ωc ∈ (0, 1) is the weight on traded goods and φ ≥ 0 is the elasticity of substitution
between traded and non-traded goods. The budget constraint of optimizing households is
coT t + pN t coN t + aot+1 = (1 − τt )wt θo `ot + (1 + r)aot + Tto ,(3)

where pN t is the price of non-traded goods relative to traded goods, τt is the exogenous tax rateon labor income, wt is the wage per efficiency unit of labor, parameter θo > 0 is the production
efficiency of optimizing households, parameter r > 0 is the risk-free return on savings, and Tto
is the government’s transfers to optimizing households. Optimizing households cannot borrow:aot+1 ≥ 0.13
A measure 1 − γ of households are type h, “hand-to-mouth,” without access to assets. They
choose consumption and labor supply to maximize the expected sum of discounted utility flows,

E0

∞
X
t=0

βct

 α 
gt
c(.)1−σ − 1
χhh 1+1/ε
−
(` )
,
ḡ
1−σ
1 + 1/ε t(4)subject to the sequence of budget constraints
chT t + pN t chN t = (1 − τt )wt θh `ht + Tth ,(5)

where Tth are government’s transfers to hand-to-mouth households. To introduce inequality
considerations into the model, we allow hand-to-mouth households to have different production
efficiency θh > 0 and disutility of work χh > 0 from those of optimizing households. We
motivate this modeling choice with the observation that the Greek government transferredsignificant resources to poorer households during our sample period.

3.4

Firms

Firms are perfectly competitive and operate linear technologies that transform labor into output. In the traded sector, the profit maximization problem of the representative firm is
max ΠT t = yT t − wt `T t ,`T t ≥0subject to yT t = zT t `T t ,(6)separable term in preferences that depends only on government consumption in order to distinguish between
complementarities and level effects of government consumption.
13
The asset a is a risk-free international bond and does not represent claims to the government’s bonds. Mostof Greek debt is held externally, and so we simplify the model by assuming that domestic households do nothold government debt. For example, in 2019, 89 percent of Greek government liabilities were held externally.

12where yT t is traded production, `T t is total efficiency units of labor employed in the traded sector, and zT t is exogenous traded productivity. In the non-traded sector, the profit maximization
problem of the representative firm is
max ΠN t = pN t yN t − wt `N t ,`N t ≥0subject to yN t = zN t `N t ,(7)where yN t is non-traded production, `N t is total efficiency units of labor employed in the nontraded sector, and zN t is exogenous non-traded productivity.
The linearity in production simplifies the computation of the model, because it allows us
to pin down the wage per efficiency unit of labor and the relative price of non-traded goods asa function of productivities when both sectors operate in equilibrium. In this case, we obtainwt = zT t and pN t = zT t /zN t .

3.5

Government

The government purchases gT traded goods and gN non-traded goods that provide utility tohouseholds.14 It transfers T o resources to optimizing households and T h resources to handto-mouth households. The government raises revenue by taxing labor income at an exogenous
rate of τ . To cover any gap between spending and tax revenues, the government borrows
internationally.
One type of external financing is long-term debt b issued to competitive, risk-neutral foreign
investors at a price of q. While in good credit standing, the maturity rate of this debt is λp > 0
and its coupon rate is κp > 0. While in bad credit standing, the maturity rate of this debt
is λd > 0 and its coupon rate is κd > 0. Another type of external financing is loans ffrom official lenders. The maturity rate of loans is λg > 0 and their coupon rate is κg = r.
A crucial difference between b and f is that the former is defaultable, whereas the latter isnot. We think this is a reasonable assumption, because research shows that official loans frommultilateral organizations have higher seniority than debts held by private lenders.15 Thus, f isrisk-free and its actuarially fair price equals one. Changes in the flow of loans can be triggered
endogenously, by the government’s decision to restructure, or exogenously, by the passage oftime.
In each period, the first decision of the government is whether to restructure its debt,
ηt+1 = {0, η̄}. Restructuring debt is associated with a haircut η̄ ∈ [0, 1] on bt+1 and new
maturity rate λd and coupon rate κd for the remaining debt. Restructuring may trigger a
14

The government consumption aggregator g is Cobb-Douglas in gT and gN , with traded share parameter ωg .
See Schlegl, Trebesch, and Wright (2019) for evidence on the seniority of the IMF and the World Bank andTrebesch and Zettelmeyer (2018) for evidence on the seniority of the ECB.
15

13

flow of official loans equal to δ̂t . Official loans δ̂t |(ηt+1 , δt ) are a random variable, with theirdistribution depending on both the government’s restructuring decision ηt+1 and the loans δtthat are announced in the beginning of the period. Finally, restructuring debt is associatedwith a utility cost C(ηt+1 , st ) for the government and exclusion from issuing debt for at least
the current period.
We distinguish between restructuring ηt+1 , which is the control variable of the government,
and the credit standing of the government at the beginning of each period dt , which is the statevariable. The two variables are linked by the law of motion0
if ηt+1 = 0 and dt = 0,
dt+1 = 0
if ηt+1 = η̄ or dt = 1, with probability ψ,




1
if ηt+1 = η̄ or dt = 1, with probability 1 − ψ.

(8)

If the government enters a period in good credit standing (dt = 0) and does not restructure its
debt (ηt+1 = 0), then it always enters next period in good credit standing (dt+1 = 0). If thegovernment enters a period in bad credit standing (dt = 1) or restructures its debt (ηt+1 = η̄),then with probability ψ ∈ [0, 1] it regains access to markets in the next period (dt+1 = 0) andwith probability 1 − ψ it enters the next period in bad credit standing (dt+1 = 1).
We now describe the maximization problem of a government that begins the period in good
credit standing and chooses not to restructure its debt (dt = ηt+1 = 0). At this point, the flowof loans δ̂t has been realized, so the value of the government isno
V n (st , ηt+1 , δ̂t ) = max ζUto + (1 − ζ)Uth + βg Et V (st+1 ) ,bt+1 ,Tt(9)where the parameter βg ∈ (0, 1) is the government’s discount factor and the parameter ζ ∈ [0, 1]
is the weight that the government places on the utility of optimizing households, which may
differ from their share γ in the population. The continuation value V (st+1 ) denotes the valueof the government at the beginning of the next period and is defined below.
The government is subject to the budget constraint,gT t + pN t gN t + Tt + (λp + κp )bt + (λg + κg )ft = τt yt + qt (bt+1 − (1 − λp )bt ) + δ̂t − it ,(10)

where yt = wt `t = yT t + pN t yN t is aggregate output in terms of traded goods and it ≥ 0 is an
issuance cost introduced for technical reasons (see Appendix B).
Total transfers are
Tt = γTto + (1 − γ)Tth = γξTth + (1 − γ)Tth ,

(11)
14

where the parameter ξ ≥ 0 controls the progressivity of the transfer system. When ξ = 1, thetransfer to every optimizing household equals the transfer to every hand-to-mouth household.
In our quantitative implementation, we have ξ < 1, which means that the transfer system
is progressive, as each hand-to-mouth household receives more transfers than each optimizing
household. We do not allow the government to tax in a lump-sum way optimizing households,
because that would make the decision to redistribute uninteresting.
Finally, the law of motion for official loans is given by
ft+1 = (1 − λg )ft + δ̂t .

(12)

In choosing debt bt+1 and transfers Tt , the government internalizes the private sector equilibrium
and the price of debt q(st , ηt+1 , δ̂t , bt+1 , Tt ) that ensue after its choices.
Next, we describe the maximization problem of a government that either begins the period
in bad credit standing (dt = 1) or decides to restructure its debt (ηt+1 = η̄). At this point, theflow of loans δ̂t has been realized, so the value of the government isnoV d (st , ηt+1 , δ̂t ) = max ζUto + (1 − ζ)Uth − C(ηt+1 , st ) + βg Et V (st+1 ) .
Tt

(13)

Similar to Bianchi, Ottonello, and Presno (2023), we specify the utility cost of default as
C = µ + µT log(zT t ) + µN log(zN t ) if ηt+1 > 0 and zero otherwise. The government is subject
to the budget constraint,gT t + pN t gN t + Tt + (λd (ηt+1 , δ̂t ) + κd (ηt+1 , δ̂t ))bt + (λg + κg )ft = τt yt + δ̂t ,(14)

the law of motion for private debt,
bt+1 = (1 − ηt+1 )(1 − λd (ηt+1 , δ̂t ))bt ,(15)

and the law of motion for official loans in equation (12). We allow the new maturity rateλd (ηt+1 , δ̂t ) and coupon rate κd (ηt+1 , δ̂t ) to depend on the country’s restructuring decision andbailout state.16 Again, the government internalizes the private sector equilibrium and the priceof debt q(st , ηt+1 , δ̂t , bt+1 , Tt ) that ensue after its choices.
Finally, the government chooses whether to restructure its debt to maximize the value
V (st ) = max
dt V̄ d (st , ηt+1 )ηt+1 ∈{0,η̄}



d
n
+ (1 − dt ) I{ηt+1 > 0}V̄ (st , ηt+1 ) + I{ηt+1 = 0}V̄ (st , ηt+1 ).
16
Dvorkin, Sanchez, Sapriza, and Yurdagul (2021) show that restructuring events are often associated with asignificant increase in a country’s debt maturity. Modeling λd and κd as a function of the restructuring decisionand bailout state allows us to mimic the profile of payments and path of the debt during the 2010s.

15

The flow of official loans δ̂t is not known yet when the government contemplates restructuring
its debt. Thus, the expected value under good credit standing isZdV̄ (st , ηt+1 ) = V d (st , ηt+1 , δ̂t )dF (δ̂t |ηt+1 , δt ),(16)

and the expected value under bad credit standing isZnV̄ (st , ηt+1 ) = V n (st , ηt+1 , δ̂t )dF (δ̂t |ηt+1 , δt ),(17)

where F (δ̂t |ηt+1 , δt ) is the conditional distribution function of official loans.17

3.6

Foreign Investors

Foreign investors who hold the government’s debt are risk-neutral and competitive. Therefore,
the price of debt is actuarially fair and satisfies
"
h
i
1
Et (1 − dt+1 )I{ηt+2 (st+1 ) = 0} λp + κp + (1 − λp )q(xt+1 )q(xt ) =1+rh+dt+1 + (1 − dt+1 )I{ηt+2 (st+1 ) > 0} λd (ηt+2 , δ̂t+1 ) + κd (ηt+2 , δ̂t+1 )#i+ (1 − ηt+2 )(1 − λd (ηt+2 , δ̂t+1 ))q(xt+1 ) ,(18)

where r is the same risk-free rate at which optimizing households save.
Changes in the price of debt reflect only changes in the expected payoff of debt, which
includes the return of principal, coupon payment, the probability of default, and the haircut
rate conditional on default. We abstract from the risk premium for two reasons. First, inAppendix A, we demonstrate that, under the null hypothesis that our model generates thedata, the predicted sovereign spread tracks very closely the observed spread in the run-up
to the Greek debt crisis of the 2010s. The prediction of the spread is based only on statevariables of the model specific to Greece and not on creditor-side variables that determine
the risk premium. We document two significant deviations of the observed spread from thepredicted spread, one in 2012 and one in 2015, which we could interpret as an increase in therisk premium. However, model parameters are not estimated to match spread levels during
17
The timing assumption, in which the government first decides whether to restructure and then issues new
debt, excludes the possibility of self-fulfilling rollover crises. We view this as a reasonable assumption. Bocola
and Dovis (2019) document that only about one-seventh of the rise of Italian spread between 2008 and 2012reflects non-fundamental risk. The deterioration of fundamentals in Greece was much larger than the one in
Italy. In the run-up to the crisis, we do not observe a lengthening of the maturity structure of Greek bonds,
which would be suggestive of increased rollover risk.

16

these years. Second, with respect to the Covid period, one would need to argue that the risk
premium decreased by 4 percentage points in order to rationalize the puzzle of the missing
spread, which we find implausible. We note that the variation of spreads in the cross-sectional
results presented in Section 2 absorbs common time-varying risk premia. In the time series
results with more than one country in the sample, we additionally absorb fixed differences inrisk premia across countries.

3.7

Equilibrium

The exogenous state variables, zT , zN , gT , gN , τ, δ̂, δ, include productivity, government consumption, the tax rate, and official loans. We specify the stochastic processes that govern these statevariables in our quantitative implementation.
Given the exogenous state variables and their perceived stochastic processes, an equilibrium consists of allocations cji (x), `j (x), `i (x), a0 (x) for j = {o, h} and i = {T, N }, prices
w(x), pN (x), q(x), and government policies η 0 (s), b0 (s, δ̂), T (s, δ̂) such that (1) households maximize their values; (2) firms maximize their profits; (3) the government maximizes its value;(4) the non-traded goods market clears: cN + gN = yN ; and (5) the price of debt q(xt )satisfies the functional equation (18). The state vector at the beginning of the period is
s = (zT , zN , gT , gN , τ, δ, d, f, b, a), and the state vector after the government has made its decision is x = (s, η 0 , δ̂, b0 , T ). Appendix C describes the computation of the equilibrium.18

4

Quantification

We begin by measuring model objects in the data. We continue by parameterizing the model sothat model-generated outcomes match their data counterparts between 2002 and 2019, including
the Greek boom in the 2000s and the sovereign debt crisis in the 2010s. Later, we extend themodel to the Covid period and perform counterfactuals to evaluate the economic forces that
determine spreads and other macroeconomic outcomes during that period.
18

Given w and pN , the market clearing condition in the non-traded sector pins down equilibrium labor `N .
There is no equilibrium condition in the traded goods sector, because production that exceeds consumption isexported and production that falls short of consumption is imported. Net exports nxt = yT t − cT t − gT t andthe current account equals net exports plus net income and transfers from abroad (see Appendix B). For the
traded sector to operate in equilibrium, `T t = γθo `ot + (1 − γ)θh `ht − `N t > 0, we need the disutility of work to
be sufficiently low or production efficiency to be sufficiently high. These conditions are always satisfied in ourquantitative results, so we do not consider any further the corner solution with zero traded production.

17

4.1

Measurement

Our measurement procedure disaggregates economic activity between traded and non-traded
goods, maps economic concepts in the model to statistics in the national accounts, and gathers
key statistics of government’s fiscal and borrowing policies. The primary data source for macro
variables is the System of National Accounts (SNA), available from the OECD. For government
statistics, we additionally use data from the ECB’s Interest Rate Statistics, the Greek Government’s Quarterly Public Debt Bulletin (QPDB), and the World Bank’s Quarterly Public Sector
Debt Statistics (QPSD). Our sample period covers annual observations between 2002 and 2021.
We begin by allocating industry data on value added, compensation to employees, othertaxes on production less subsidies, and hours worked between the traded and the non-tradedsector. The traded sector consists of agriculture, mining, manufacturing, and transportation,and the non-traded sector consists of the remaining industries (see Appendix Table A.1). All
values are divided by the price of traded goods, defined as the Paasche index of the prices of
the industries that compose the traded sector. All quantities are divided by population. Hours
worked include hours from both employees and the self-employed. We normalize units in thedata in 2019, so that y19 = pN 19 = `j19 = zi19 = 1.
There are three deviations between aggregate economic activity in our model and aggregate
economic activity recorded by the national accounts. On the production side, we do not consider
taxes on production less subsidies; on the expenditure side, we do not consider investment;
and on the income side, we do not consider capital income earned from domestic production.
The key step that allows us to map model objects to statistics in the national accounts is toscale statistics in the national accounts by the labor share of income. Appendix A details themeasurement of the labor share and how we perform these adjustments.
We split labor income between the wage and the labor input in order to estimate theproductivity process. We use data on self-employed worker’s hours to impute their labor income
and add this income to compensation of employees from the national accounts to estimate totallabor income. The unadjusted wage equals this total labor income divided by total hours,
where the totals include labor income and hours of the self-employed. We then use data onlabor services from KLEMS and derive the wage w adjusted for compositional changes acrossdemographic and educational groups. Traded productivity zT = w and non-traded productivity
zN = zT /pN , where pN is the Paasche index of the prices of the industries that compose thenon-traded sector. To make sectoral productivities consistent with sectoral outputs, we definelabor inputs residually as `N = pN yN /zN and `T = yT /zT .

18

0 = 1Moving to the government sector, we trigger the sovereign debt crisis in 2011, so η11and d12 = 1. Greece regained access to international markets in 2019, so d19 = 0. For thetax rate τ , we scale government revenues from all sources with the labor share and divide theresulting revenues with output y. For transfers T , we scale with the labor share the sum of SNA
item D.6 (“social contributions and benefits”), item D.39 (“other subsidies on production”),
and item D.99 (“other capital transfers”). Items D.39 and D.99 include paycheck protections
and similar transfers to producers and households during Covid.19
We calculate maturity and coupon rates of Greek debt using data on the maturity of new issuances and the residual maturity of the total debt stock from the Greek QPDB and Bloomberg.
We augment these data with data on the terms of the 2012 restructuring and the time series ofthe stock of debt securities from 2011 to 2019 in order to better match the profile of payments
after the default. For private debt during good credit standing, we find that λp = 0.10 andκp = 0.05. Official loans have a significantly longer duration than private debt, λg = 0.04, anda significantly lower coupon rate, κg = 0.02. To match the changes in private debt during thedefault episode, we calculate λd = 0.07 between 2011 and 2014 and λd = 0.05 between 2015
and 2018. In order to match the profile of debt service payments over the same period, we
set the coupon rates to κd = 0.05 in 2011, κd = 0.10 between 2012 and 2014, and κd = 0.03
for years beginning in 2015. Given maturities and coupon rates, we obtain the price of debt
using the formula qt = (λp + κp )/(λp + r + spreadt ), where r = 0.02 is the risk-free rate andthe spread is the difference between the interest rate on Greece’s 10-year government debt andthat of Germany, as recorded by the ECB Interest Rate Statistics.
We use the government budget constraint to back out the total issuance of debt and loans:issuancet = gT t + pN t gN t + Tt − τt yt +(λp + κp )bt + (λg + κg )ft{z}|+buybackt ,(19)

Maturing Stocks and Interest Payments

where all variables in the right-hand side are observed. Maturing stocks and loans data comefrom the ECB and government interest payments from the SNA. We also add the market valueof the buyback of debt that foreign investors held in 2012 to the total issuance.
We allocate the flow of total issuance in equation (19) between revenues from issuing private
debt, qt (bt+1 −(1−λp )bt ), and flows of official loans δ̂t , using data on the maturity rates, default
dates, and purchases of Greek sovereign debt by the ECB under the PEPP (see Appendix A for
19

While the SNA explicitly advises against allocating Covid transfers to item D.99 (https://shorturl.
at/CEJT3), we find that the Hellenic Statistical Authority (ELSTAT) allocates paycheck protections for selfemployed to item D.99. Item D.39 increased from 3.1 billion euros in 2019 to 6.9 billion euros in 2020 and to
9.6 billion euros in 2021. Item D.99 increased from 1.5 billion euros in 2019 to 7.5 billion euros in 2020 and to
5.3 billion euros in 2021. These changes equal roughly 6 percentage points of 2019 GDP.

19

details). Given the allocation of issuances between debt and loans, we build the stocks of debt b
and loans f from their laws of motion and the calculated maturities and price of debt.20 Finally,
we obtain private savings at as output minus domestic absorption, issuances of government debt
and loans, and flows of debt service. The initial value of a02 is set so that wealth over output
in 2002 matches the value of the external wealth-to-GDP ratio that is reported for Greece by
the External Wealth of Nations (Lane and Milesi-Ferretti, 2018).
To summarize, our measurement process develops a dataset of variables that are internallyconsistent with the concepts of the model. In Appendix A, we confirm that data variables
under the lens of our model align closely with variables in other datasets that one might haveobtained without referring to a model. For example, we show that our time series of private
debt-to-output b/y has the same level and is highly correlated with external measures of thedebt-to-GDP ratio that do not need to conform to our model concepts.
Figure 4 presents the evolution of the exogenous driving forces between 2002 and 2019. In
the first panel, we see that productivity in both sectors is booming until roughly 2009, falls dramatically until 2015, and then remains depressed until 2019. The second panel shows a similar
pattern for government consumption, where most of government consumption is concentrated
on non-traded goods. In the third panel, we see that official loans reached roughly 30 percent
in 2011 and remained elevated for many years after 2011. The last panel plots the increase in
the tax rate from roughly 40 percent before the crisis to 48 percent after 2011.21

4.2

Parameterization

We begin by specifying stochastic processes for the driving forces. The productivity process for
sector j = {T, N } islog zjt = (1 − ρj )z̄jt + ρj log zjt−1 + zjt ,(20)

where z̄jt is the permanent component of productivity and zjt is the innovation to the transitory
component of productivity. The innovations of the two sectors are jointly normally distributed,
20

We choose 2011 as the year of debt restructuring because 2011 is the first year without significant private debt
flows. However, Greece received some official loans in 2010. Our model does not accommodate these, becausethe δ̂t variable becomes positive for the first time in 2011 when ηt+1 becomes positive. To accommodate the 2010flows of official loans, we augment the law of motion for official loans in equation (12) to ft+1 = (1−λg )ft + δ̂t + δ̃t ,where δ̃t is another flow of official loans that allows us to model the 2010 loans. We assume that δ̃t is realized
at the beginning of the period, before the government and the private sector make their decisions, that δ̃t is
independent over time, and that a positive value of δ̃t is unexpectedly realized in 2010. In the third panel of
Figure 4, we plot the sum of both flows, δ̂t + δ̃t .
21
Our average tax rate in the sample is 0.45, which is equal to the average combined tax wedge from consumption and labor income taxes calculated for Greece by Chodorow-Reich, Karabarbounis, and Kekre (2023).
The authors use a more elaborate procedure to calculate tax rates, because they have investment and capitalin their model. The correlation between our series and theirs is 0.94.

20

g_N
g_T

Government Consumption

z_N
z_T

0.20

Productivity

1.3

0.15

1.2

0.10

1.1

0.05

1.0

0.00 2002 2005 2008 2011 2014 2017 20202002 2005 2008 2011 2014 2017 2020

0.500

0.3

0.475

Bailout

Tax Rate

0.2

0.450
0.425

0.1
0.0

0.400
2002 2005 2008 2011 2014 2017 20202002 2005 2008 2011 2014 2017 2020

Figure 4: Exogenous Driving Processeszt ∼ N (0, Σz ). We assume that households, the government, and foreigners always anticipate aconstant permanent component, Ez̄jt = z̄jt−1 , but that in 2011 there is a one-time unanticipatedchange in the permanent component.
The top panel of Table 1 shows our estimates of the parameters of the productivity processes.
The processes are relatively persistent, with autoregressive coefficients of ρN = 0.86 and ρT =0.84. As evidenced in the first panel of Figure 4, mean productivity falls dramatically after 2010.
These drops lead us to estimate one-time, unanticipated changes in the permanent componentof productivity βz̄11N = −0.29 for the non-traded sector and βz̄11T = −0.42 for the traded sector.
The innovations to productivity have a correlation of σzN zT = 0.45.22
To economize on state variables for the computation of the model, we specify government
consumption as a function of productivities:
log gjt = ḡjt + βgj N log zN t + βgj T log zT t + βgj N T log zN t log zT t + gjt , gjt ∼ N (0, σg2j ), (21)where ḡjt is the permanent component of government consumption and gjt ∼ N (0, σg2j ) is theinnovation to the transitory component of government consumption. As we did for productivity,
22
For each sector j = {T, N }, we run the regression log zjt = (1−ρj )z̄jt +(1−ρj )βz̄11j I(t ≥ 2011)+ρj log zjt−1 +zjt , where I(t ≥ 2011) is a dummy variable indicating the regime after 2011. We estimate elements of the Σzmatrix using the in-sample variances and covariances of the residuals of these regressions.

21

Table 1: Parameter Values: Stochastic Processes

Productivity

Government

Taxes
Official Flows
Lockdowns

Parameter Value
ρN
0.86
z̄T
0.32
σ zN
0.03
βgN N
0.95
βgT T
1.34
σgN
0.04
ḡT
−3.77
20
0.10
βḡN
τL
0.40
δL
0.35
π`
0.50

Parameter Value
ρT
0.84
11
βz̄N
−0.29
σzT
0.04
βgN T
0.34
βgN N T
1.72
σgT
0.04
11
βḡN
0.15
20
0.10
βḡT
τH
0.48
δH
0.65

Parameter Value
z̄N
0.23
11
βz̄T
−0.42
σ zN zT
0.45
βg T N−0.05
βg T N T
1.72
ḡN
−1.98
11
βḡT
0.15

πδ

0.25

Table 2: Parameter Values: Without Solving the Model
Parameter
Value
Explanationωc
0.30
expenditure share of traded goods, private sector
ωg0.14
expenditure share of traded goods, government
φ0.44
substitution elasticity, private, traded and non-traded goods
γ
0.40
share of optimizing households
r
0.02
risk-free rate
κp
0.04
coupon rate, government debt in good credit standing
λp
0.10
maturity rate, government debt in good credit standingκd (.)[0.03, 0.10] coupon rate, government debt in bad credit standing
λd (.)[0.05, 0.07] maturity rate, government debt in bad credit standing
κg
0.02
coupon rate, government loans
λg
0.04
maturity rate, government loansη̄
0.47
haircut on government debt
ψ
0.14
reentry probability¯
d0.80
issuance cost parameterν1
0.00
taste shock parameterν2
0.25
taste shock parameter22

we assume that households, the government, and foreigners always anticipate a constant permanent component, Eḡjt = ḡjt−1 . We introduce unanticipated changes in the permanent component of government consumption in 2011 and during Covid, because government consumption
is a function of productivity. The middle panel of Table 1 summarizes our estimates.23
The tax rate takes two values, τt ∈ {τL , τH }. The perceived tax rate process is Eτt = τt−1 ,so that households, the government, and foreigners always expect the tax rate to equal its value
in the previous year. We introduce a one-time, unanticipated change in the tax rate in 2011,
consistent with the significant increase in the tax rate shown in Figure 4. The third panel ofTable 1 shows our estimates of τL = 0.40 for the period before 2011 and τH = 0.48 for theperiod after 2011.
Next, we specify the variables that characterize flows of official loans: δ, which is announced
at the beginning of the period, and δ̂, which is realized after the government makes its decisions.
We specify these variables as multiples of the government’s spending on goods and debt service,δg [gT + pN gN + (λk + κk )b + (λg + κg )f ] for k ∈ {p, d}. The multiple δg takes three values,
δg = {0, δL , δH }. The second-to-last panel of Table 1 shows our estimates of δL = 0.35 andδH = 0.65. We feed into the model δg = δH between 2011 and 2014 and δg = δL between 2015
and 2018.
The variables characterizing the flows of official loans are linked through two transition
matrices. The first transition matrix maps the multiple at beginning of the period (the δg of δ)to the multiple realized after the government makes its decision (the δg of δ̂). With some abuseof notation,
1 − πδ̂ πδ̂ /2 πδ̂ /2,δ̂|(δ, η 0 = η̄) = 
0
1
0


0
0
1

δ̂|(δ, η 0 = 0) = 1.

(22)

The first row of the transition matrix says that when the government begins the period witha multiple of zero for δ and chooses to restructure its debt (η 0 = η̄), with probability 1 − πδ̂
it transitions to a δ̂ of zero, and with equal probability πδ̂ /2 it transitions to a δ̂ characterizedby either a low multiple δL or a high multiple δH . The parameter πδ̂ governs the moral
hazard induced by the possibility of bailouts, because starting from good credit standing thegovernment places a higher probability of receiving a loan when πδ̂ is higher. We estimate πδ̂23
For every sector j = {T, N }, we run regression (21) and add a time dummy that indicates the regime between
2011 and 2019 and another time dummy that indicates the regime between 2020 and 2021. The estimatedcoefficients on these dummies are βḡ11j and βḡ20j . We estimate σg2j using the in-sample variances of the residuals ofthese regressions. Innovations to the transitory components of government consumption are uncorrelated acrosssectors, because government consumptions are correlated through their dependence on sectoral productivities.

23

jointly with other parameters when we solve the model. The second and third rows of thetransition matrix show that, starting from a low or high multiple always leads to the same
low or high multiple when the government restructures its debt (η 0 = η̄). Similarly, when thegovernment does not restructure its debt (η 0 = 0), official loans always remain in the same
multiple, δ̂|(δ, η 0 = 0) = 1.
The second transition matrix specifies how the realized values of official loans δ̂ transits tonext period’s announced loans δ 0 :


1
0
0


.
δ 0 |δ̂ = 
π
1
−
π
0
δ
 δ

0
πδ
1 − πδ

(23)

In the first row, starting with a zero realized multiple for official loans always leads to a zeromultiple at the beginning of the next period. The parameter πδ governs the transience of official
loans, because it equals the probability that the multiple decreases by one step starting froma positive multiple. In the second-to-last panel of Table 1, we estimate this probability tobe πδ = 0.25, so that official loans are quite persistent over time. Our estimate reflects theevolution of official loans in the third panel of Figure 4, which shows that Greece received loans
for most years in the 2010s.
Table 2 reports values of parameters that we set before solving the model. We set theweight parameters in the consumption aggregators to match expenditure shares of traded goods:ωc = 0.30 for households and ωg = 0.14 for the government. We follow Stockman and Tesar(1995) and set the elasticity of substitution between traded and non-traded goods to φ = 0.44.
We set the fraction of optimizing households to γ = 0.40, close to the value that Martin andPhilippon (2017) estimate for Greece in their study of the boom-bust cycle in the eurozone.
The risk-free rate on private savings is r = 0.02. We calculate maturity and coupon rates
for government debt using data from the ECB and the Greek QPDB as described before in
our measurement section. Using the Greek sovereign debt crisis, we calculate a haircut rateη̄ = 0.47 and a reentry probability ψ = 0.14.
Table 3 presents values of the remaining parameters. The first panel displays parameters
estimated with data between 2002 and 2019, before Covid. We estimate these parameters byrequiring that endogenous variables of the model match their analogs in the data along certain
dimensions (see Appendix B for details on the process). Parameters affect all model variables
simultaneously, but here we informally discuss features of the data that most intuitively identify
the parameters. In the first column, the production efficiency parameters θo , θh are estimatedso that the model matches the inequality in pre-tax-and-transfer Greek income reported by24

Table 3: Parameter Values: Solving the Model

θ
θh
χo
χh
σ
ε
ξ

Value
1.79
0.36
0.84
0.46
1.16
0.56
0.87

Explanation
productivity, oproductivity, hdisutility, odisutility, hrisk aversion
Frisch elasticity
progressivity of transfers

c̄N

0.71

Covid constraint, consumption

o

βc
βg
µ
µN
µT
ζ
α
πδ̂
`¯

Value
0.97
0.96
0.05
0.11
0.09
0.24
0.90
0.74
0.66

Explanation
discount factor, households
discount factor, government
cost of default, constantcost of default, slope in log(zN )cost of default, slope in log(zT )government weight on ocomplementarity (g, c)
moral hazard
Covid constraint, labor

the Luxembourg Income Study. In order to match data on inequality in post-tax-and-transfers
income reported by the Luxembourg Income Study, our model requires a progressivity of transfers parameter ξ = 0.87.24 The disutilities of labor are chosen to normalize labor supply ofboth types of households to one in 2019. We find that the model does a good job in terms ofaccounting for the time series of output, labor supply, and consumption, with a coefficient ofrelative risk aversion σ = 1.16 and a Frisch elasticity of labor supply ε = 0.56. These values
fall comfortably in the range of values used in dynamic general equilibrium models and models
of sovereign debt.
Moving to the second column, we estimate the discount factor of households so that households discount at a rate which is lower than the risk-free rate, βc = 0.97. This allows the model
to match the decline in private savings a during the boom of the 2000s. The government isslightly more impatient than households, with a discount factor of βg = 0.96. This discountfactor is a key determinant of the speed of sovereign debt accumulation. Our estimate of βg
differs from much of the sovereign default literature in that we do not require a high degreeof impatience for the government, partly because the unusually large fluctuations in productivity generate a significant sovereign spread during the crisis. The constant in the default
cost function µ controls the amount of sustainable debt at the average values of productivity.
We find that default costs are substantial and that they increase with productivity, so thatparameters µN and µT are positive. The high cost of default allows the model to sustain a high24

In the Luxembourg Income Study, we observe that pre-tax-and-transfer income per capita at the top 40
percent of the distribution is roughly 5 times the pre-tax-and-transfer income per capita at the bottom 60 percentof the distribution. The ratio of disposable income, which we equate with the ratio of post-tax-and-transfer
income, is around 2.5.

25

debt-to-GDP ratio.25
The government places a lower weight ζ = 0.24 on the welfare of the optimizing households
relative to their share γ = 0.4 in the population. This feature of the model allows it to generate
the high level of transfers that we observe in the data. The complementarity between private
and government consumption, α = 0.90, allows the model to generate an increasing path ofboth government borrowing and transfers during the boom of the 2000s, because the government transfers more resources to households to increase their consumption when government
consumption is high. Finally, the relatively high value of the moral hazard parameter, πδ̂ = 0.74,
also allows the model to sustain high levels of government debt, because the government expects
a disbursal of loans in case of default with high probability.

4.3

Time Series of Model Outcomes

Figure 5 compares time series of the endogenous variables generated by the model with theiranalogs in the data. We focus on the period between 2002 and 2019 and discuss the Covid
period when we introduce our extended model. To generate these time series, we feed into
the model’s policy functions the paths of productivity, government consumption, tax rate, andofficial loans displayed previously in Figure 4 (see Appendix C for more details).
In the first panel, we see that the model does a good job of matching the boom in output
in the 2000s. The model matches almost exactly the more than 30 log points decline in output
between the late 2000s and 2016. After 2016, the model generates around half of the output
recovery. The dynamics of output follow closely the dynamics of productivity and government
spending, as discussed previously in Figure 4.26 Deviations of output in the data from output
in the model arise because of variations in labor supply. Since these deviations are generally
small, we conclude that labor supply in the model behaves quite reasonably over the sample
period.
In the second panel, we see that the model also does a good job of matching the consumption
boom in the 2000s and the dramatic decline in the first part of the 2010s. Further, as the next
two panels show, the model is generally consistent with the observed sectoral comovement ofconsumption. The model’s success in matching the consumption dynamics is also reflected in25

To get a sense of magnitudes, from the perspective of the government the utility cost of default is equivalent
to a one-time consumption drop between 55 and 80 percent, depending on the state of the economy, for bothtypes of households. Converting this to a annualized flow using the government’s discount factor, yields apermanent consumption loss between 2 and 3 percent.
26
We acknowledge that productivity may be responding to more primitive shocks. For instance, ChodorowReich, Karabarbounis, and Kekre (2023) demonstrate how tax, foreign demand, and financial shocks reduced
utilization during the Greek bust.

26

1.1

Data
Model

1.0

Output

1.1

0.9

1.0

0.8

0.9

0.7

2002 2005 2008 2011 2014 2017 2020

0.40

Data
Model

0.30

0.6

0.25
0.20

0.5
2002 2005 2008 2011 2014 2017 2020

0.4

2002 2005 2008 2011 2014 2017 2020
0.40

Data
Model

Data
Model

0.35

AssetsTransfers

0.3

0.30

0.2

0.25

0.1
0.0

Data
Model

0.35

0.7

0.20

2002 2005 2008 2011 2014 2017 20202002 2005 2008 2011 2014 2017 2020

20

Data
Model

Data
Model

15

Debt

Spread

1.5

10

1.2

5

0.9
0.6

2002 2005 2008 2011 2014 2017 2020

Tradable Consumption

Nontradeable Consumption

0.8

1.8

Data
Model

Consumption

1.2

02002 2005 2008 2011 2014 2017 20202002 2005 2008 2011 2014 2017 2020

Figure 5: Time Series of Endogenous Variables

27

the path of private assets in the fifth panel. Private assets decline substantially in the boom,
because impatient optimizing households deaccumulate assets. Private assets recover in theearly part of the 2010s, because transfers from government increase or remain high and thesetransfers are perceived by optimizing households as transitory. The model and the data differ
with respect to the magnitude of the recovery of assets, with the model accounting for only
about half of the recovery of assets.
In terms of fiscal policy, the model matches closely the level of sovereign debt. It also matches
the rise of debt during the boom period. The model is generally successful in accounting forthe dynamics of transfers, with transfers rising during the boom period, peaking around theheight of the crisis, and then falling precipitously as the economy goes into the recession.
The model generates a restructuring decision for the government in 2011.27 Additionally,
the model successfully accounts for the time series of the spread.28 It accounts for the low
spread before 2009, the spike of the spread between 2010 and 2012, and the decline in the
spread after the restructuring. With respect to the spread, the two most important deviationsof the model from the data are in 2012 and in 2015. However, as we show in Appendix A,
these deviations may be attributable to a rise in the risk premium that we did not model.
Subtracting our estimate of the risk premium in 2012 allows the model to match the dataalmost exactly in terms of the spread. The risk premium plausibly increased in 2015, following
political uncertainty and the referendum on exiting the euro. However, these specific events are
not relevant for the parameters of the model that matter for the counterfactuals during Covid,
because Greece had a stable government in this period.
Before discussing the Covid period, we pause to contrast our quantitative strategy to alternatives in the sovereign default literature. The most common approach is to choose parameterssuch that moments from a long-run simulation of the model match their data counterparts (see,for example, Chatterjee and Eyigungor, 2012). Another common approach is to produce similar long-run simulations, extract some number of years leading to a default, and then requirethat moments calculated using those samples match their data counterparts (see, for example,
Arellano, 2008). These approaches are not appropriate for Greece, because Greece’s experience27

We add taste shocks to the government’s problem in equation (13) to compute in a robust way the model’s
solution (see Appendix B for details). We classify a year as a year with restructuring when there is a larger than
50 percent probability that the government restructures across the different realizations of the taste shocks.
λ +κ
28
The spread is calculated with the standard convention, spreadt = pqt p − λp − r, where r is the risk-freerate, κp is the coupon rate, λp is the maturity rate, and qt is the price of debt in equation (18). In 2010, thespread is calculated using the price qt after the change to the regime with lower productivity and higher taxes.
In 2011, the spread is calculated after default, but before the flow of official loans δ̂t is realized. In 2012, thespread is calculated before the haircut η̄ on debt is applied. See Appendix C for more details.

28

from 2002 to 2019 is a outlier relative to other default episodes on both macroeconomic andgovernment borrowing dimensions. While the unique nature of the crisis makes it a very informative sample to estimate key model parameters, this uniqueness also means that long-run
average outcomes are not the appropriate model analogue for the 18 years of data. Following
the standard practice, we show in Appendix Table A.3 that the long-run model moments under
our parameterization deviate in important dimensions from averaged sample moments.

5

The Covid Period

We begin by extending the model to consider the economic impacts of lockdowns during theCovid period. We demonstrate the success of the model in accounting for salient observations on
the dynamics of macro variables, external borrowing, and the spread during the Covid recession.
Finally, we perform various counterfactuals to assess the economic forces that determine thespread and other macroeconomic outcomes.

5.1

Extension of the Model during Covid

The economic environment during the Covid period is the same as the one in the period
before Covid, except for the lockdowns. We model lockdowns using two constraints. The first
constraint applies to the consumption of non-traded goods:

I(lockt = 1) (cN t − c̄N ) ≤ 0,

(24)

where c̄N is a parameter that governs the tightness of the constraint and lockt is an exogenous
variable that takes the value of one during lockdowns. We motivate constraints on consumptionof non-traded goods with the observation that lockdowns affected mostly service industries, suchas accommodation, education, and entertainment, while stay-at-home orders did not prevent
households from consuming traded goods, such as those produced by agriculture and manufacturing, to such a large extent.
The second constraint applies to the labor supply of the hand-to-mouth households:
I(lockt = 1) `ht − `¯ ≤ 0,

(25)

where `¯ is a parameter that governs the tightness of the constraint. We motivate this constraint
with the observation that higher-income households mostly retained their jobs because they
could be performed from home, while lower-income households faced significant earning losses
because jobs in the service sector could not be performed as easily from home.
29

Our modeling of lockdowns is consistent with previous narratives of the economic impactsof the pandemic. For instance, Stantcheva (2022) summarizes results from multiple researchteams and countries that show the disproportionate impact of lockdowns on non-traded sectors
which were deemed non-essential and had more difficulty substituting from in-person to onlinetransactions. Lockdowns also had a disproportionate impact on lower-income households, which
had fewer opportunities for remote work and experienced a significantly larger probability ofjob loss.
The last panel of Table 3 shows the values of the two constants that parameterize theconstraints. We pick c̄N and `¯ such that the model matches the decline in output and nontraded consumption in 2020 relative to 2019. The perceived persistence of lockdowns isπ` = Prob(lockt+1 = 1|lockt = 1) = 0.5,

Prob(lockt+1 = 0|lockt = 0) = 1.

(26)

We set the persistence π` to 0.5, which implies a two-year expected duration of lockdowns.
The duration is two years based on some scenarios considered by the International Monetary
Fund (2020) that projected potential lockdowns also in 2021. This parameter turns out tobe crucial in understanding the behavior of the spread during Covid, and thus we discuss
other parameterizations in our counterfactuals. Once lockdowns are lifted, everyone expects
lockdowns to be lifted forever.
Finally, we augment the process for official loans in equation (23) to model ECB’s purchasesof sovereign debt under the PEPP. In equation (27), the ECB extends loans to the government
at the beginning of the period in the first year of lockdowns without being prompted by debt
restructuring,



0

1

0




δt |(δ̂t−1 , lockt = 1, lockt−1 = 0) = 
0

1

0


.


0 π δ 1 − πδ

(27)

When lockt−1 = 1 or lockt = 0, the beginning of the period loans δ follow the process inequation (23). The realized loans δ̂ after the government makes its debt restructuring decision
follow the process in equation (22).29
29

In the stock of official loans f , we model identically both the loans that were extended to Greece in the2010s under the economic assistance programs and the ECB’s purchases of Greek sovereign debt in secondary
markets after 2020 through the PEPP. Appendix B shows the equivalence of loans and purchases of sovereign
debt under the PEPP when the government realizes that profits from bond purchases are rebated.

30

Table 4: Performance of Model during Covid
Year
2020

2021

5.2

Statistic (×100)
log y20 − log y19
log c20 − log c19(b21 − b20 )/y19(T20 − T19 )/y19(a21 − a20 )/y19
spread20 − spread19
log y21 − log y19
log c21 − log c19(b22 − b20 )/y19(T21 − T19 )/y19(a22 − a20 )/y19
spread21 − spread19

Data
−8.1
−4.1
12.0
6.1
4.4
0.1
−4.0
1.4
21.0
5.5
5.4
0.0

¯ c̄N
Model, `,
−8.1
−10.5
12.0
9.0
3.9
0.4
−2.9
2.7
17.3
3.7
−5.8
0.0

Model, `¯ Model, c̄N
−6.9
1.3
2.4
−8.9
12.3
3.0
10.0
3.6
−5.2
1.9
0.4
0.0
−2.2
−3.3
1.8
3.5
17.6
9.7
4.0
5.2
−13.8
−7.4
0.0
0.0

Performance of the Model during Covid

We introduce lockdowns unexpectedly in 2020. We choose to introduce lockdowns only in 2020
because Greece lifted the lockdowns before the middle of 2021, and we discuss below the casein which lockdowns extend to 2021. We solve for the endogenous variables of the model using¯ and π` . The results of this exercise are disciplined by keeping all
the calibrated values of c̄N , `,other parameters constant at their values shown in Tables 1, 2, and 3. These parameter values
were informed only by observations through 2019.30
Table 4 presents changes in key variables in 2020 relative to 2019 (first panel) and in 2021
relative to 2019 (second panel). The first column presents statistics in the data and the second
column presents the corresponding statistics in the model when we introduce both the constraint
on labor supply `¯ and the constraint on non-traded consumption c̄N . The only statistics targeted
during our calibration are the changes in output and non-traded consumption between 2019and 2020.31
The model matches almost perfectly the increase in government debt in both 2020 and30

We start the model from the state variables (a, b, f ) observed in the data at the beginning of 2020. We proceed similarly to our analysis of the period before Covid and feed in productivity and government consumption
shocks during Covid. Productivities increase somewhat in 2020 relative to 2019, but in 2021 they revert backto their 2019 values. Government consumption increases substantially in 2020 and 2021. However, these shocks
tend to stimulate economic activity, and so we focus our discussion on the effects of lockdowns.
31
We define consumption expenditure in terms of traded goods similarly to output, c ≡ cT + pN cN , whereci = γcoi + (1 − γ)chi for each sector i = {T, N }.

312021. Further, it matches closely the increase in transfers in 2020 and 2021 and the increase in
private savings in 2020. Finally, the model is successful in accounting for the stability of thesovereign spread. In 2020, the spread increases by 0.1 percentage point in the data, whereas in
the model it increases by 0.4 percentage point. As argued in Section 2, these are small changes
when compared with the historical changes in spreads during high government borrowing andrecessions. The model matches perfectly the stability of the spread in 2021 relative to 2019.32
The economics of lockdowns are that the government borrows substantially in order totransfer resources to hand-to-mouth households, which, because of the constraint on their labor
supply, suffer a sharp drop in their labor income that they cannot smooth. However, some of thetransfers spill over to the optimizing households. Optimizing households also face a constraint
on non-traded consumption and realize that increased transfers are transitory, so they save theextra resources. When lockdowns are lifted in 2021, optimizing households reduce their savingsand begin to consume significantly more. The government keeps transfers elevated because it is
impatient and the ECB purchases under PEPP increase its fiscal space. The consumption boomof optimizing households and expansionary fiscal policy drive a faster recovery of consumption
relative to the recovery of output. Indeed, as Table 4 shows both for the data and for themodel, output is still depressed in 2021 relative to 2019, whereas consumption is higher in 2021
than in 2019.
The last two columns of the table present model statistics when we introduce separately
the constraint on labor supply `¯ or the constraint on non-traded consumption c̄N . The modelwith the constraint on labor supply generates almost the entire drop in output, but misses thedrop in consumption. Conversely, the model with the constraint on non-traded consumption
generates almost the entire drop in consumption, but misses the drop in output. For the behavior of government debt and transfers, the constraint on labor supply is the crucial constraint.
In that model, the government borrows substantially to transfer resources to hand-to-mouthhouseholds, which are the only households hit by the labor supply constraint. On the other
hand, in the model with only a constraint on non-traded consumption, the government does
not borrow as much, because hand-to-mouth households and optimizing households are equallyaffected by lockdowns.
These results offer insights into the debate about the nature of the Covid shock. Most
32

The spread calculation in the data is somewhat sensitive to the 2019 value, because 2019 is the year inwhich Greece reenters the market and the spread falls by more than 2 percentage points between the beginningof the year and the end of the year. For this reason, we use the spread at the end of 2019 as the baseline valueof the spread before the Covid recession, and we difference out the spreads in 2020 and 2021 relative to this2019 spread.

32

of the literature has focused on disentangling demand versus supply origins of the recession
using closed-economy frameworks (Baqaee and Farhi, 2022; Guerrieri, Lorenzoni, Straub, andWerning, 2022). In an open economy, however, consumption is potentially disconnected fromproduction because households and the government can trade in international markets to financetheir expenditures. Although constraints on both demand and supply are important for thedrop in macroeconomic aggregates, we find that constraints on supply are more importantin terms of understanding the fiscal policy responses following the lockdowns. The difference
between the two types of constraints is that demand constraints do not increase the value ofproviding transfers as much as constraints on the supply side do in the open economy.

5.3

Counterfactuals during Covid

We are now ready to answer the question that motivated this paper. We developed a modelthat generates a significant increase in government borrowing alongside a significant recession
in 2020. Yet, spreads remain relatively constant. The behavior of government debt and spread
during the Covid recession contrasts sharply with their behavior in other time periods. So, why
was this case during Covid, despite a significant recession and accommodative fiscal policies?
Table 5 performs a series of counterfactual exercises to assess the economic forces affecting
spreads and other macroeconomic outcomes. The first panel of the table presents the changes
in output, consumption, debt, transfers, relative consumption, and spreads in 2020 in the data
and the model. Each row in the other panels presents a different counterfactual in which wechange values of certain parameters or shut off mechanisms. The entries display changes inmodel variables under the counterfactual relative to the baseline model. For example, the firstentry says that in the absence of ECB’s policies during Covid, output would have been 0.6
percentage point higher than in the baseline model, which features these policies. Thus, output
would have declined by 7.5 percent in the counterfactual instead of 8.1 percent in the baseline.
5.3.1

Pandemic Emergency Purchases

The first row assesses the role of the PEPP, under which the ECB purchased sovereign debt
equal to 10 percent of output in both 2020 and 2021. We find that these purchases allowed
the Greek government to fund substantially more transfers than it would have otherwise. Increased transfers lowered inequality substantially, as, in the absence of increased transfers, theconsumption of optimizing households would have increased by almost 17 percentage points
more than the consumption of hand-to-mouth households. Despite the importance of ECB
purchases for increasing transfers, spreads would have increased by only 0.5 percentage point
33

Table 5: Counterfactuals

y

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.

(relative to 2019 values)
Data 2020
Model 2020
(relative to 2020 model)δ̂t≥20 = 0πδ̂ = 0.9πδ̂ = 0.1
λp = 0.04
δ̂t<20 = 0q̄bnew = 0.71(q̄b + f )η̄ = 0.73
η̄ = 0.22
ε = 0.1
a0 = ar = 0.03π` = 0.85
persistent z shocks
permanent z shocksδ̂t<20 = 0, δ̂t≥20 = 0π` = 0.85, δ̂t≥20 = 0(relative to 2021 model)
lock21 = 1
lock21 = 1, π` = 0.85
lock21 = 1, π` = 0.85, δ̂t≥20 = 0(100 × ∆ log)cbT

−8.1 −4.1 12.0
−8.1 −10.5 12.0

co /ch(∆ p.p.)Spread6.1
9.0 −19.5

0.1
0.4

3.5 −7.3 16.6
0.2 0.2 −0.3
−0.7 −0.5
1.0
2.5 −0.4
0.8−5.8 −1.4
3.3−5.5 −6.2 14.1
−4.7 −7.3 16.5
6.9 8.8 −16.7−2.6 −3.9 31.9
−2.2 −4.1 10.4
−0.7 −0.7
0.6−7.1 −10.3 24.6
−3.7 −3.6 14.9
−6.0 −6.6 18.0
−2.9 −10.1 24.1
−0.6 −17.3 45.4

0.5
0.0
−0.2
0.4
0.7
2.3
1.8
−0.6
0.7
0.0
−0.3
2.4
−0.4
−0.2
1.9
5.1

−9.6 −16.2 6.6 1.7 −7.1
−9.2 −22.1 −3.9 −5.0
9.8
−8.5 −32.4 −1.0 −15.8 44.6

1.0
3.7
12.9

0.6 −6.3
0.0 0.1
0.1 −0.4−0.1 −0.3−0.5 −1.0−0.1 −5.2
0.4 −6.2−1.0 7.1
0.0 −1.4−3.7 −2.5
2.7 −1.2
0.1 −8.8
0.1 13.0
0.1 9.5
0.0 −8.5
0.5 −15.3

in the absence of the purchases.
To understand this result, panel (a) of Figure 6 presents the relationship between the spreadand the debt issuance. The blue solid curve presents the menu of spreads and debt issuancethat the government faces under the baseline model. The spread is increasing in debt issuance,
because issuing more debt moves the economy closer to the restructuring region and the price ofdebt q in equation (18) falls. The star in the blue solid curve indicates the optimal choice of thegovernment among all combinations of spreads and borrowing. Removing the ECB purchases
shifts the menu of spreads and debt issuances upward to the red dashed curve. If debt issuance
34

2.5

3.5
Baseline
3.0
No Loans Before 2020
2.5
2.0
1.5
1.0
0.5
0.0 0.1
0.0
0.1

Baseline
No ECB Purchases

Spread (p.p.)

Spread (p.p.)

2.0
1.5
1.0
0.5
0.0 0.7

0.8

0.9

b21/y19

1.0

1.1(b21 b20)/y19(a) No ECB Purchases

57Baseline6= 0.73
5
4
3
2
1
0 0.7
0.8

Spread (p.p.)

Spread (p.p.)

3
2
1
0

0.0

0.1

0.2

0.3

(b21 b20)/y19(c) Different composition

2.5

2.5

Spread (p.p.)

Spread (p.p.)

1.5

1.0

1.0

0.5

0.5

0.0 0.7

0.8

0.9

b21/y19

1.0

0.0

1.1

0.7

(e) Low ε

0.8

0.9

b21/y19

1.0(f) No savings

5

Baseliner = 0.03

Baseline= 0.85Spread (p.p.)

Spread (p.p.)

4
3

1.0

2

0.5
0.0

1.0

Baseline
a0 = a

2.0

1.5

1.5

0.9

b21/y19(d) Higher η̄Baseline
= 0.1

2.0

2.0

0.3

(b) No old bailouts

Baseline
Liabilities 71% Private

4

0.2

1

0.7

0.8

0.9

b21/y1901.0(g) Higher r

0.7

0.8

0.9

b21/y19(h) Higher π`

Figure 6: Menu of Spreads and Debt Issuance35

1.0

is held constant, removing the ECB purchases would increase the spread. The flatness of thecurve around the optimal choice of the government determines how much of the equilibrium
adjustment is borne by either spreads or debt issuance. To offset the lower purchases from the
ECB, the government is willing to borrow 3.5 more percentage points of output from private
investors by paying a spread that is 0.5 percentage point higher than that in the baseline modelwith the purchases. But the government does not borrow enough to compensate for the entire
loss of purchases, because doing so would move the economy to the steeper part of the spreadfunction. Had the Greek government borrowed an additional 7 percentage points to offset themissing PEPP resources, the spread would have increased by around 0.9 percentage point in
our model relative to the baseline.
How do our model estimates of the effects of the ECB’s sovereign debt purchases comparewith empirical estimates? The model predicts an increase in the spread by 0.5 percentage pointin 2020 in the absence of the purchases by the ECB. Without the loans in the 2010s, the model
generates a roughly 2 to 2.5 percentage points increase in the spread between 2012 and 2018.
Trebesch and Zettelmeyer (2018) examine how ECB purchases under the Securities Market
Program affected the cross-section of Greek bond yields in 2010. In comparing their estimateswith ours, we note that their effects may incorporate channels other than reductions in default
probability and that ECB purchases were concentrated on lower-maturity and higher-yieldbonds. The authors estimate that ECB purchases reduced the yield of the purchased bonds
by between 0.8 percentage point and 1.9 percentage points and that the drop in yields did not
spill over to non-targeted bonds. Combining an event-study with a Bayesian VAR methodology
between 2013 and 2020, Rostagno, Altavilla, Carboni, Lemke, Motto, and Saint-Guilhem (2021)
estimate that the ECB’s Asset Purchase Program (APP) before the Covid period reduced 10year yields of Germany, France, Spain, and Italy by around 0.5 percentage point in 2014. They
also estimate that the incremental effect of the Pandemic Emergency Purchase Program wasroughly 0.5 percentage point, which is identical to our model-based estimate. According to theauthors, the cumulative effect of all quantitative easing interventions on the four country yields
was around 2 percentage points, which is consistent with our estimate of the cumulative effectsof the 2010s loans on the Greek spread.
5.3.2

Moral Hazard and Maturity Length

The second and third rows of the second panel of Table 5 assess how changes in the level of moral
hazard might have prevented spreads from rising. In our baseline calibration, the probability ofreceiving a bailout is πδ̂ = 0.74. Increasing the probability to 0.9 produces negligible changes
36

on all variables. Lowering the probability to 0.1 decreases the spread slightly. The effects ofchanging the parameter πδ̂ are small because bailouts affect outcomes only when the government
chooses to restructure, and restructuring happens in less than 1 percent of years in the long-run
equilibrium of our model.
Did spreads remain low because of changes in the maturity structure of debt? Recall thatin our baseline parameterization we have λp = 0.10, which implies an average maturity of 10
years. In row 4 of the table, we lengthen the maturity of private debt to 25 years to matchthe maturity of official loans. Lengthening the maturity increases the spread by 0.4 percentage
point, because the government has increased incentive to dilute the debt and foreign investors
perceive it as more risky. Conversely, shortening the maturity of debt would have lowered the
spread. To the extent that policy interventions allowed Greece to borrow at maturities longer
than the ones it would have existed without these interventions, we conclude that lengthening
maturities do not explain the puzzle of the missing spread.
5.3.3

Old Bailouts

The next rows of Table 5 quantify the role of the bailouts that Greece received between 2010
to 2018. In row 5, labelled δ̂t<20 = 0, we remove the entire history of bailouts in the 2010s andbegin the model in 2020 with only private debt and no official loans. In this case, the spreadincreases by 0.7 percentage point more than in the baseline, which exceeds the effect of the
PEPP on the spread. Unlike removing the PEPP purchases, removing the old loans causes thegovernment to cut back significantly on borrowing. We visualize the government’s choice toreduce debt issuance in panel (b) of Figure 6, which shows that the schedule of spreads shifts up
and becomes steeper and more convex. To understand the difference between the old loans and
PEPP purchases, we note that removing the old loans makes all debt private at beginning of
Covid. This reduces the country’s fiscal space and implies a substantial increase in the spread,
unless the government reduces its borrowing.
Row 6, labelled q̄bnew = 0.71(q̄b + f ), uses a different approach to assess the role of the oldbailouts. In this counterfactual, we convert Greece’s total liabilities at the beginning of 2020
from the observed split of 66 percent official and 34 percent private to the counterfactual splitof 29 percent official and 71 percent private. We use a 29 percent share of official loans to matchthe average share of official loans to countries between 1995 and 2019, as reported by the WorldBank’s QPSD. In this case, the spread rises by 2.3 percentage points, which is close to the lower
bound of our estimate of the missing spread. The logic is that as we tilt the composition of
liabilities toward private debt, the gain of the government from defaulting rises, and investors
37

demand a higher compensation to hold its bonds. Panel (c) of Figure 6 shows that the menuof spreads shifts up and becomes substantially steeper. As a result, the government chooses toborrow less during Covid, which reduces consumption and increases inequality.
Rows 7 and 8 of Table 5 consider the effects of changing the extent of debt relief underrestructuring. Recall that in our baseline parameterization, the haircut rate is η̄ = 0.47. Aninterpretation of the bailout policies of the 2010s is that they allowed for only limited debt
relief in order to protect foreign investors from losses, in exchange for longer-maturity official
loans that reduced the welfare losses of default for Greece. To assess the role of debt relief
within this coordinated set of policies, in row 7 we raise the haircut rate to η̄ = 0.73. A
higher haircut increases the losses of lenders in the event of a restructuring and increases the
incentive of the government to restructure, with both mechanisms causing foreign investors
to demand higher compensation in order to break even. As we see in panel (d) of Figure 6,
these mechanisms lead to a higher and steeper schedule of spreads. In equilibrium, the spreadincreases by 1.8 percentage points, and the government reduces significantly the amount ofborrowing and transfers.33 To the extent that old intervention policies limited haircuts upondefault, we conclude that these policies played an important role for the stability of the spread
during Covid. In row 8, we instead reduce the haircut rate to η̄ = 0.22. To the extent that
Covid intervention policies signaled to governments and foreign investors more debt relief upondefault, we conclude that these interventions do not explain the puzzle of missing spread.
To summarize, rows 5 to 8 of Table 5 demonstrate the importance of old bailout policies forthe stability of the spread during Covid. The quantitative effects vary quite a bit, depending onwhether one conceptualizes the old policies as removing all loans, or changing the compositionof debt while keeping total debt constant, or limiting debt relief. However, in all cases thesepolicies generate changes in the spread that are at least as large as the changes in the spread
generated by the ECB purchases under PEPP during Covid. Further, the largest estimatesof how old loan policies affect the spread during Covid coincide with the lower bound of ourestimate of the missing spread.
5.3.4

Private Responses

In row 9 of Table 5, we consider the role of labor supply responses. In our baseline, the valueof the Frisch elasticity was ε = 0.56, whereas now we make labor close to inelastic by setting
33
Our estimate of the effect of haircut size on the spread is consistent with the empirical estimates of Cruces
and Trebesch (2013), who estimate the effect of past restructuring events on future interest rate spreads. We
interpret these estimates as the effects of different restructuring regimes in our setting, which are perceived tobe permanent. The authors’ estimates that are most consistent with our modeling of the restructuring process,
imply that a 26 percentage points increase in η̄ is associated with a 1.5 percentage points increase in the spread.

38ε = 0.1. In this case, redistributive transfers become substantially less distortionary, which
increases the willingness of the government to borrow and provide transfers. In anticipationof government’s willingness to redistribute more during Covid, foreign investors increase therequired compensation to hold debt. As seen from panel (e) of Figure 6, the schedule of spreads
that the government faces shifts up and becomes steeper. The resulting equilibrium featuresa spread that is higher by 0.7 percentage point, lower borrowing and transfers, and higher
inequality.
The recession in 2020 was unusual in that it featured a substantial increase in private savings,
a feature of the data that our model replicates quite well. A priori, a plausible explanation forthe low spread during Covid is that resources saved by a country’s citizens are accessible to thegovernment in order to repay external debts, so the relevant measure of indebtedness is net,
and not gross, indebtedness. In order to assess the role of this mechanism, in row 10 of Table
5 we assume that optimizing households cannot save or dissave and are unexpectedly forced tochoose private assets a21 = a20 . In panel (f) of Figure 6, we see that the schedule of spreads
barely changes. While the equilibrium spread does not change, equilibrium allocations bearsignificant adjustments. Wealth effects on labor supply are the key mechanism accounting forthis result. If the government maintains the same level of transfers under the baseline allocationand optimizing households are forced to consume these transfers, labor supply and output fallsignificantly. To avoid an even larger recession, the government cuts back on borrowing andtransfers.34
Did spreads remain low because of a decline in the safe rate during Covid? To assess thispossibility, in row 11 of Table 5, we increase the risk-free rate from 2 percent in the baseline to
3 percent. Recall that the government is relatively impatient, with a discount factor of 0.96.
Reducing the distance between government’s discount rate and the private sector’s discountrate generates lower gains from intertemporal trade, and so the government engages in lessborrowing. In panel (g) of Figure 6, we can see that the schedule of spreads shifts down relative
to the baseline. We conclude that the decline in risk-free rate does not explain the puzzle of themissing spread, because in a counterfactual world with a higher risk-free rate, both the spreadand borrowing are lower.
34

In the baseline model with endogenous savings, a 10 percentage points increase in government debt isassociated with a negligible drop in output. With constrained savings, a 10 percentage points increase ingovernment debt is associated with a 5 percent drop in output.

39

5.3.5

Beliefs about the Persistence of Lockdowns

Row 12 of Table 5 presents our key counterfactual, which concerns the belief that lockdowns
would not persist for a longer time. Recall that in our baseline parameterization, which matched
the responses of the spread and debt during Covid, the expected duration of lockdowns is two
years. We now raise the probability π` from 0.5 to 0.85, which matches the duration of Greek
depression between 2009 and 2016.35 As we can see in panel (h) of Figure 6, increasing theperceived persistence of lockdowns dramatically reduces the country’s fiscal space by shifting
the schedule of spreads upward. In the new equilibrium, the spread increases by 2.4 percentage
points. In addition, borrowing decreases by 7 percentage points, transfers decrease by 10
percent, consumption drops by 9 percent, and inequality increases by 25 percentage points.
The mechanism by which a higher expected duration of lockdowns increases the spread is
specific to the nature of the Covid shock. To illustrate this, in rows 13 and 14, we use either
persistent or permanent productivity shocks to engineer a recession of similar magnitude to
the one caused by lockdowns.36 As the table shows, in this case the spread does not change
significantly. Additionally, the government does not borrow as much, and transfers do notincrease as much as they do under the lockdowns.
Why are lockdowns different from productivity shocks? The answer lies in the distributional
nature of the lockdowns. In the absence of a policy intervention, lockdowns impact hand-tomouth households most negatively. In our model, lockdowns result in a 43 percent drop in
the pre-tax-and-transfer labor income of hand-to-mouth households, as opposed to a 2 percent
increase in the income of optimizing households.37 The heterogeneous impact of lockdowns
increases substantially the government’s incentive to borrow and transfer resources to hand-tomouth households. Had lockdowns been perceived as more persistent, foreign investors would
anticipate a higher redistributive motive from the government in the future. This expectation
would have pushed the economy closer to default, with a higher spread and lower equilibrium
35

Our counterfactual with π` = 0.85 is reasonable in light of historical rare disasters. Barro and Ursua (2012)assemble a sample of 40 countries that extends back to 1870. The authors report a mean output decline of 21
percent during a disaster and a 4-year mean duration of disasters. Approximately 16 percent of observations
have disasters that persist for 6 or more years.
36
We reduce both traded and non-traded productivities by a common proportion that is set to match the fall in
output from 2019 to 2020. For the case of persistent shocks, we introduce negative innovations to productivities
and assume that productivities are expected to follow their autoregressive process in equation (20). For thecase of permanent shocks, we lower the permanent components of productivities and adjust the innovations sothat we replicate the same decline in output.
37
Our estimate concerns the increase in inequality in the absence of policy responses. We do not compare thisestimate with observed inequality measures, because measured inequality includes substantial policy responses
that took place during Covid. For instance, paycheck protection programs increase the relative income oflower-income households relative to a counterfactual world without paycheck protection programs.

40

borrowing.
By contrast, productivity shocks do not affect the distribution of pre-tax-and-transfer labor
income across households as much. With persistent productivity shocks, the drop in labor
income is 10 percent for hand-to-mouth households and 7 percent for optimizing households.
With permanent productivity shocks, the drop in labor income is 7 percent for hand-to-mouthhouseholds and 8 percent for optimizing households. Thus, the incentive of the government
to redistribute resources during a recession caused by productivity shocks is muted, and thepersistence of the shock does not matter quantitatively for borrowing, transfers, and the spread.
5.3.6

Interactions

Next, we consider interactions between various mechanisms. In row 15 of Table 5, we remove
the ECB purchases under PEPP, as in row 1. At the same time we remove the old bailouts,
as in row 5. The spread increases by 1.9 percentage points, which exceeds the sum of theindividual effects by 0.7 percentage point. The interaction between the old and new loans is
the closest analog to the estimates reported by Rostagno, Altavilla, Carboni, Lemke, Motto,
and Saint-Guilhem (2021) on the cumulative effect of both APP and PEPP from 2014 to 2020.
According to these authors, the combined effects of these policies is to depress the ten-yearyields of Germany, France, Spain, and Italy by roughly 2 percentage points, which is close to
our estimate of 1.9 percentage points.
In row 16, we interact the ECB purchases under PEPP with a higher expected duration of
lockdowns, as in row 12. While the individual effects on the spread are 0.5 and 2.4 percentage
points respectively, introducing both changes increases the spread by more than 5 percentage
points. Considering the isolated effect of the PEPP, we conclude that ECB policies mostly
ameliorated inequality during Covid. However, at the beginning of Covid, there was considerable uncertainty about the evolution of the virus, the invention of a vaccine, and the durationand economic impact of lockdowns. The counterfactual, in which we interact the PEPP policy
with a higher expected duration of lockdowns, illustrates that these policies would have averted
a significant sovereign debt crisis had lockdowns been perceived as more persistent by investors
and governments.
5.3.7

Extended Lockdowns

Row 17 of Table 5 examines the case in which lockdowns extend to 2021. Compared with thebaseline model, which does not feature lockdowns in 2021, this case generates a substantial dropin output and consumption. However, the spread increases by only one percentage point. This41

result suggests that the incidence of lockdowns is not as important as the expected durationof lockdowns for the stability of the spread. To illustrate this more formally, in row 18 we
extend the lockdowns by one more year and increase π` to 0.85, as in the counterfactual in row
12. In this case, we find that the spread increases by 3.7 percentage points. The last row of
the table illustrates that the spread would reach levels observed in the crisis of the 2010s if, inaddition to the 2021 incidence and the higher expected duration of lockdowns, we remove the
ECB purchases under PEPP. In this case, the probability of default exceeds 70 percent. Again,
this result highlights that ECB policies would have averted a significant sovereign debt crisis
had lockdowns been perceived as more persistent.

6

Conclusion

The goal of this paper is to understand the puzzling behavior of sovereign spreads during Covid.
The puzzle is that advanced economies went through a major recession and expanded their fiscal
policies by borrowing significantly, yet spreads on sovereign debt barely responded. Studying
the most extreme manifestation of the puzzle, the experience of Greece, we proceeded in threesteps. First, we developed a small open economy model with long-term debt and default, which
we augmented it with official lenders, heterogeneous sectors and households, and constraints
on consumption and labor supply to mimic the Covid lockdowns. Second, we showed that aquantitative version of the model is consistent with the observed boom-bust cycle of Greece
before Covid and salient observations on the behavior of macro aggregates, government debt,
and the sovereign spread during Covid. In our final step, we performed a series of counterfactual
exercises to assess which forces account for the stability of the spread during Covid.
We reached several substantial conclusions. The government expanded fiscal policy byborrowing externally during Covid to ameliorate the uneven impact of lockdowns on the labor
supply of poorer and asset-constrained households. Despite this expansion, the spread did
not respond, because lockdowns were perceived as transitory and old bailouts allowed Greece
to enter Covid with a debt composition that was less tilted toward defaultable private debt.
By contrast, we find a smaller role for the new PEPP policies and for other margins suchas lengthening maturity of debt, moral hazard, and private responses to government policies.
While some of our lessons are specific to institutional details of the Greek sovereign debt
markets in the 2010s, we conjecture that our analyses provide insights on the stability ofthe spread in other countries that also expanded their fiscal policy by borrowing externally,
went through relatively shorted-lived recessions, and were subject to ECB interventions that42

purchased sovereign debt.
Methodologically, our paper contributes to the literature by improving upon the quantification of sovereign debt models and by extending these models to consider the distributional
effects of shocks and the effects of lockdowns. On the policy side, we highlight the importanceof the history of official lending for countries entering recessions and considering new debt issuances. We also conclude that ECB purchases of sovereign debt during Covid did not stabilize
spreads so much, but allowed the government to provide increased transfers to households,
which stabilized consumption and reduced inequality. A rationalization of the ECB policies
during Covid is that they would have stabilized sovereign debt markets substantially had the
lockdowns been perceived as more persistent by private investors.

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45

The Puzzling Behavior of Spreads during Covid
Online Appendix
Stelios Fourakis and Loukas Karabarbounis
Appendix A provides details for the measurement. Appendix B elaborates on some model
elements. Appendix C presents our algorithm for solving the model and details of the simulations.

A

Data

Risk premium and sovereign spread. In this section, we discuss why abstracting from the risk
premium for the case of Greece is appropriate. Our starting point is the equilibrium price ofan asset:
qt = Et [Mt,t+1 Qt+1 ] = Et [Mt,t+1 ]Et [Qt+1 ] + Covt (Mt,t+1 , Qt+1 ),(A.1)

where Mt,t+1 is the stochastic discount factor of the investor pricing the asset and Qt+1 is thepayoff of the asset in period t + 1. Our model assumes that (i) lenders are risk neutral and havea constant discount rate; and (ii) the price of the sovereign debt depends on fundamentals of
the domestic economy, the amount of sovereign debt, and the country’s status vis-a-vis private
and official lenders. In other words, we have a model of qt that assumes that Et [Mt,t+1 ] isconstant and the covariance term is 0,
qt =1
Et [Qt+1 ].
1+r

(A.2)

In the data there may be a nonzero risk premium for sovereign debt. To investigate this, ourapproach begins by predicting the model-consistent spread on the basis of the fundamentals of
the domestic economy, the amount of sovereign debt, and the country’s status vis-a-vis private
and official lenders. Under the null hypothesis that our model generates the data, the residual
between the observed spread and the predicted spread reflects the covariance term in equation(A.1).
We calculate the model-consistent bond price asqt =λp + κp,
λp + r + spreadt(A.3)

where spreadt is the observed spread, r = 0.02 is the risk-free rate, λp = 0.10 is the maturity
rate, and κp = 0.045 is the coupon rate. We project this bond price from 2001 to 2019 on a
1

20
15
10
0

5

Spread (p.p.)

2002

2005

2008

2011

Observed Spread

2014

2017

2020

Predicted Spread w/o Risk Premium

Figure A.1: Greek Observed and Predicted Sovereign Spreaddummy denoting whether the country has access to private debt markets, GDP growth, thedebt-to-GDP ratio, and a full set of second degree terms involving these measures. We use
monthly data to form the prediction.
In Figure A.1, we plot the observed spread and our prediction, both aggregated at theannual frequency as in the rest of our model. We interpret the difference between them as
the effects of the omitted creditor-side variables that determine the risk premium.1 It turns
out that the small set of state variables that predict spreads under our model fits the data
extremely well in sample, with the R-squared of the prediction being 93 percent. There arelarge residuals in 2012, which coincides with the peak of the eurozone debt crisis, and in 2015,
which coincides with political instability and the referendum of exit from the euro. We findthese deviations reassuring, because we would expect the risk premium to increase in precisely
these two circumstances.
In sum, abstracting from the risk premium for the case of Greece is appropriate for two
reasons. First, the predicted spread under our model that does not feature a risk premium
tracks extremely closely the observed spread in the run up to the Greek crisis. We use the runup to 2012 to estimate model parameters, which implies that the deviations in 2012 and 2015
are not relevant for the estimation of the parameters. Second, during Covid, one would needto argue that the risk premium fell dramatically in order to rationalize the puzzling behaviorof the Greek spread. As discussed in the main text, the missing Greek spread during Covid
1

Because the average residual is zero, the residuals represent only the relative size of the risk premium. In
order to derive a meaningful representation of its absolute magnitude, we assume that the average risk premium
until 2007 is zero and adjust the prediction accordingly.

2

is around 4 percentage points. In fact, if we use our prediction until 2019 to estimate out ofsample the risk premium in 2020, we would obtain a decline in the risk premium of around 6
percentage points. We find such a decline in the risk premium during Covid implausible.
Adjusting the national accounts to confront to the model. Table A.1 shows which industries areclassified as part of the traded sector and which are classified as part of the non-traded sector.
To understand the deviations of our model from the national income and product accounts,
we write output in the model as
y = pT yT + pN yN(A.4)= pT (cT + gT ) + pN (cN + gN ) + nx
= w(`T + `N ).
In the national accounts we haveY

= PT YT + PN YN + Taxes on Products(A.5)= PT (CT + XT + GT ) + PN (CN + XN + GN ) + NX
= W L + RK,
where we use uppercase letters for SNA concepts and lowercase letters for model concepts.
On the production side, our model does not have taxes on products, and the value added byindustry excludes other taxes on production. On the expenditure side, our model does nothave investment. On the income side, our model does not have capital income generated bydomestic capital.
Our adjustments begin by measuring the labor share of income for the aggregate economy,
Employees
Compensation × 1 + Total Hours−Hours
Hours Employees
s` =
.
(A.6)
GDP − Taxes on Production
The numerator of the labor share adds the labor income of the self-employed to the compensation of employees. The imputation of the labor income of the self-employed assumes that the
self-employed have the same labor share as the rest of the economy (see Karabarbounis (2023)
for more details).
The labor share of income in the non-traded sector is
N −Hours EmployeesN
CompensationN × 1 + Total Hours
Hours EmployeesNs`N =,Value AddedN − Other Taxes on ProductionNand the labor share of income in the traded sector is
T −Hours EmployeesT
CompensationT × 1 + Total Hours
Hours EmployeesTs`T =.
Value AddedT − Other Taxes on ProductionT
3

(A.7)

(A.8)

Table A.1: Traded and Non-Traded Sectors
Traded Sector

Non-Traded Sector

agriculture, forestry, and fishingmining and quarrying
manufacturing
land transport and pipelines transportwater transport
air transport

electricity, gas, steam
water supply, sewerage, waste
construction
wholesale and retail trade
warehousing
postal and courier
information and communication
financial and insurance
professional, scientific, and technicaladministrative and support services
public administration and defence
education
human health and social workarts, entertainment, and recreationaccommodation and food service
other services

We note a difference between GDP and the sum of value added across sectors. GDP includes
taxes on production less subsidies, which we allocate proportionally between labor and capitalincome. Value added at the industry level includes only other taxes on production, which area subcomponent of taxes on production less subsidies. Thus, for the sectoral labor shares,
we allocate other taxes on production less subsidies proportionally between labor and capitalincome.
We use the labor shares to map model concepts to their data counterparts. Output of
non-traded goods in the model,
pN yN = s`N × Value AddedN +Value AddedN× s` × Taxes on Products,
Aggregate Value Added(A.9)

equals the labor income generated by the non-traded sectors plus a share of the missing taxes
on production (called taxes on products) attributed to labor income. This share equals theshare of value added of non-traded goods before the adjustment is made. Similarly, we measure
output of traded goods in the model as
yT = s`T × Value AddedT +Value AddedT× s` × Taxes on Products.
Aggregate Value Added
4(A.10)

Our next step is to construct measures of government and private consumption. We scale
the corresponding measures in the data by the labor share of income, since our model does nothave domestic capital. For government consumption of non-traded goods, we estimate
pN gN = (6/7)s` × Government Consumption,(A.11)

where the factor 6/7 comes from the estimates of Chodorow-Reich, Karabarbounis, and Kekre(2023) for the fraction of government consumption that is accruing to non-traded goods. We
thus obtain government consumption of traded goods:
gT = (1/7)s` × Government Consumption.

(A.12)To measure non-traded goods, we use the market clearing condition,cN = yN − gN .

(A.13)

To adjust for the lack of investment from our model, we equate the sum of private consumption
by sector to the sum of private consumption and gross capital formation in the SNA, scaled
down by the labor income share. Given the measured consumption of non-traded goods, wethen obtain consumption of traded goods residually:pT cT = s` × (Hh Consumption + Gross Capital Formation) − pN cN .

(A.14)

These adjustments imply that net exports in the model equal net exports in the data, again
scaled by the labor income share:
nx = yT − cT − gT = s` × NX.

(A.15)

Allocating issuances between revenues from issuing private debt and flows of official loans. Given
issuances in equation (19) of the main text, default dates, and maturities, we allocate to private
debt




issuancet





0
qt (bt+1 − (1 − λp )bt ) =issuancetissuancet − PEPPt

if t < 2018 and dt+1 = 0,
if t = 2018,
if t = 2019,
if t ≥ 2020,

5(A.16)

and we allocate to official loans
issuancetissuancetδ̂t = ft+1 − (1 − λg )ft =


0





PEPPt

if t < 2018 and dt+1 = 1,
if t = 2018,(A.17)

if t = 2019,
if t ≥ 2020.

The variable PEPP equals the ECB’s estimate of purchases of Greek sovereign debt in secondary
markets during the Covid period.
In Figure A.2, we compare the adjusted data through the lens our model with the raw datafrom statistical agencies reported in national accounts and the ECB.

B

Model

Equivalence of loans and purchases. In addition to providing loans δ̂t , official lenders can purchase privately-held government debt from investors in secondary markets. We demonstratethat loans from official lenders and purchases of sovereign debt in secondary markets are equivalent for allocations when accrued profits Πg from the purchases of the sovereign debt are rebatedto the country.
Let vt denote the stock of these purchased debts at the beginning of period t and let q̄tdenote the average price at which it was purchased. The interest rate realized on investmentsin these debts is
λp + κp(λp + κp + (1 − λp )q̄t )vt=− λp = r̄t .
q̄t vtq̄t(A.18)

Since the official lender’s cost of capital is the risk-free rate r, total profits on its investmentsareΠgt = (r̄t − r)q̄t vt =
λp + κp− λp − r q̄t vt = (λp + κp )vt − (λp + r)q̄t vt .
q̄t

(A.19)

We assume that before the government chooses how much debt it intends to issue, the official
lender announces how much debt it intends to buy this period, or, equivalently, how much debt
it intends to hold, if possible, at the end of the period v̂t+1 . Let b̂t+1 denote the amount ofprivately-held debt right after the auction of any new issuances or buyback of any existing debt.
Privately-held bonds at the end of the period are
bt+1 = max{0, b̂t+1 − (v̂t+1 − (1 − λp )vt )}.

(A.20)
6

1.05

1.3

2005

2008

2011

2014

2017

2020

2002

2017

2020

Statistical Accounts

.9

.95
2011

2014

2017

2020

2002

2005

PWT

2008

2011

Model Concept

2014

2017

2020

Statistical Accounts

1.4
1.2
1
.6

.8

-.05

Govt. Debt to Output

1.6

0

KLEMS

-.1

2005

2008

2011

2014

2017

2020

2002

2005

2008

2011

Model Concept

2014

2017

2020

Statistical Accounts

-1

0

-2

.5

1

External Wealth to Output

1.5

-.5

Statistical Accounts

-1.5

-.15

Net Exports to Output

2014

.8
2008

Model Concept

Govt. Loans to Output

2011

.85

Private Absorption to Output

1.2
1.1
.9

2005

Model Concept

2002

2008

Model Concept

1

Productivity

2002

2005

Statistical Accounts

1.3

Model Concept

.85

.9

.9

.95

Labor Input

1

1.2
1.1
1

Output

2002

2002

2005

2008

2011

Model Concept

2014

2017

2020

2002

Statistical Accounts

2005

2008

2011

Model Concept

Figure A.2: Model versus Statistical Concepts

7

2014

2017

2020

Statistical Accounts

Officially-held bonds are
vt+1 = (1 − λp )vt + min{v̂t+1 − (1 − λp )vt , b̂t+1 }.

(A.21)

The average price at which debt holdings were purchased isq̄t+1 =

(1 − λp )q̄t vt + (vt+1 − (1 − λp )vt )qt,
vt+1(A.22)

which implies that the book value of the official lender’s debt holdings evolves asq̄t+1 vt+1 = (1 − λp )q̄t vt + (vt+1 − (1 − λp )vt )qt .

(A.23)The government budget constraint is
gT t +pN t gN t +Tt +(λp +κp )(bt +vt )+(λg +κg )ft = τt yt +qt (b̂t+1 −(1−λp )bt )−it +δ̂t +Πgt , (A.24)
where profits Πgt are rebated to the government. Substituting these profits from equation (A.19)
yieldsgT t + pN t gN t + Tt + (λp + κp )(bt + vt ) + (λg + κg )ft = τt yt + qt (b̂t+1 − (1 − λp )bt ) − it+ δ̂t + (λp + κp )vt − (λp + r)q̄t vt .
(A.25)
Next, we use that the risk-free price of official loans is equal to one (κg = r) and separateb̂t+1 into the portion that will be privately held at the end of the period, bt+1 , and the portion
that the official lender will purchase after the auction, vt+1 − (1 − λp )vt . Thus, we write thegovernment budget constraint as
gT t + pN t gN t + Tt + (λp + κp )bt + (λg + κg )(ft + q̄t vt ) = τt yt + qt (bt+1 − (1 − λp )bt ) − it+ δ̂t + qt (vt+1 − (1 − λp )vt )+ (λg − λp )q̄t vt .

(A.26)To see the equivalence between loans and repurchases, let us consider first the case of
λg = λp , in which the last term of equation (A.26) becomes zero. In this case, both the stocks
ft and q̄t vt and their increments δ̂t and qt (vt+1 − (1 − λp )vt ) are perfect substitutes. In the
more general case with λg 6= λp , the flow of official loans δ̂t can be adjusted appropriately tomaintain the equivalence.
External positions. Let mt be government’s borrowing position and ιt be the effective interest
rate on government’s liabilities. We define
mt+1 − (1 + ιt )mt ≡ qt (bt+1 − (1 − λp )bt ) + δ̂t − I(dt = ηt+1 = 0)(λp + κp )bt− (1 − I(dt = ηt+1 = 0))(λd + κd )bt − (λg + κg )ft . (A.27)
8

Using this definition on the budget constraints of households and the government and themarket clearing condition for non-traded goods leads tocT t + gT t = yT t + [(1 + r)at − at+1 ] − [(1 + ιt )mt − mt+1 ],(A.28)

where cT t = γcoT t + (1 − γ)chT t is aggregate consumption of traded goods and at = γaot isaggregate private savings. Net exports are nxt = yT t − cT t − gT t = yt − ct − gT t − pN t gN t andthe external wealth of the country is ewt = at − mt . Thus, the current account is
cat = ewt+1 − ewt = nxt + (rat − ιt mt ),(A.29)

where rat − ιt mt equals net income and transfers from abroad.
Issuance cost. In our model, we have long-term debt and non-zero recovery. Thus, we use
an issuance cost i(.) to reduce the government’s incentive to dilute to an extreme degree thedebt before defaulting. See Chatterjee and Eyigungor (2015) for details on the economics ofsuch behavior. The cost is continuous and, for each (bt , bt+1 ), depends on the one period ahead
default probability Et [I{ηt+2 (st+1 > 0}]:
it = i Et [I{ηt+2 (st+1 > 0}] qt (bt+1 − (1 − λp )bt ),wherei(x) =   1 1 + sin x−d¯ − 1 π , bt+1 − (1 − λp )bt > 0,
2
2
1−d¯

0,

bt+1 − (1 − λp )bt ≤ 0.

Along the equilibrium path, the issuance cost is almost always zero.
Taste shocks. In order to robustly compute the solution to our model, we augment the modelwith a set of taste shocks over the decisions available to the government at any point in time,
following Dvorkin, Sanchez, Sapriza, and Yurdagul (2021).2 Specifically, we assume that there
are Generalized Type I Extreme Value preference shocks associated with the set of choiceprofiles available to the government at the beginning of each period. When the government
enters the period in bad credit standing, these choice profiles are {(ηt+1 = 0), (ηt+1 = η̄)}.
2

The sovereign default literature has concluded that, even in simpler models than ours with long-term debt,
it is impossible to compute robustly pure strategy equilibria in a discrete state space representation. Motivatedby the fact that mixed strategy solutions are even harder to compute, Chatterjee and Eyigungor (2012) provethe existence of a pure strategy equilibrium and provide an algorithm to find equilibria by adding an i.i.d.
continuous random variable that affects choices in each period. Introducing this variable makes the probabilityof choosing each feasible level of debt in the next period continuous in the price function and the continuation
value function.

9

Table A.2: Moments Targeted
Statistic (×100)
log y09 − log y02
log y05 − log y04
log y15 − log y09
log b10 − log b03
log a10 − log a03
log a15 − log a10
log T09 − log T02
spread10
spread11

Data
26.5
8.2
−37.7
79.8
−32.5
22.0
14.9
6.3
13.1

Model
24.7
−0.2
−31.5
91.0
−40.4
13.3
3.5
7.4
13.9

When the government is in good credit standing, these choice profiles are {{(ηt+1 = 0, bt+1 =b0 )}b0 ∈B , (ηt+1 = η̄)}, where the set of points in the grid of bt+1 is B.
Calibration of parameters. We follow a two step process to calibrate the parameters shown in
Table 3. In the first step, we take as given that the model matches the path of the dynamic
model variables, a and b, and pick θo , θh , σ, , ξ so that the model produces time series of
o o

o o

o

o o

γ[(1−τ )wθ ` +ra+T ]
γwθ `
γwθ `
y, c, T, (1−γ)wθ
h `h , (1−γ)[(1−τ )wθ o `o +ra+T h ] that match their data analogs. The ratio (1−γ)wθ h `ho ooγ[(1−τ )wθ ` +ra+T ]is the pre-tax-and-transfer inequality and the ratio (1−γ)[(1−τ)wθo `o +ra+T h ] is the post-tax-and-transfer inequality, which we obtain from the Luxembourg Income Study (https://shorturl.
at/QY146). We also pick χo , χh to normalize labor supplies in 2019 to one. In the second step,
we calibrate the remaining 8 parameters, βc , βg , ζ, α, , µ, µN , µT , πδ̂ , so that the dynamic model
reproduces as closely as possible the targets shown in Table A.2.
Long-run moments. In Table A.3, we compare the long-run moments generated by the modelwith the sample averages in the data. We perform the comparison under two regimes. The firstregime is between 2002 and 2010, before the sovereign debt crisis. The second regime covers
the sovereign debt crisis after 2011.

C

Computation

Solution. Our strategy begins by solving for the optimal policy of households, conditional on
the government’s policy. Then, we solve for the optimal choices of the government, given the
optimal responses of households to these policies. We use a discrete state space approximation

10

Table A.3: Long-Run Moments
2002-2010 Regime
Data
Model
1.15
1.72
0.16
0.00
0.01
0.06
0.11
0.00
1.19
0.16
2.02
0.11
0.26
0.14
0.20
0.22
0.22
−0.07
0.83
0.56
0.83
0.60
0.83
0.91
0.69
0.52
0.87
−0.02
0.71
0.82

StatisticE[bt+1 /yt ]E[at+1 /yt ]E[ft+1 /yt ]E[I{ηt+1 > 0}]
100 × E[spreadt ]100 × σ[spreadt ]E[Tt /yt ]E[(gT t + pN t gN t )/yt ]ρ[spreadt , log(yt )]ρ[spreadt , bt+1 /yt ]ρ[log(ct ), log(yt )]ρ[log(gT t + pN t gN t ), log(yt )]ρ[log(ct ), Tt ]
ρ[log(gT t + pN t gN t ), Tt ]
ρ[log(gT t + pN t gN t ), log(ct )]2011-2019 Regime
Data
Model
0.75
0.55
0.29
0.00
1.21
0.38
0.00
0.00
8.68
2.84
5.57
3.84
0.32
0.29
0.21
0.21
0.41
−0.30
0.52
0.88
0.89
0.74
0.85
0.96
0.58
0.24
0.63
−0.25
0.85
0.87

of the stochastic processes.3 For the endogenous state variables b, a, and f , we use fixed, evenly
spaced grids for the beginning-of-period values. We allow end-of-period values a0 and f 0 to be
off grid and interpolate the associated values using cubic splines. We restrict b0 , a0 , and f 0 tolie within the bounds of their grids. Whenever the flow of loans implied by δ̂ would result inf 0 being above the maximum value in the grid, that flow is reduced so that the f 0 implied by
the law of motion for f is exactly equal to the maximum value in the grid for f .4
To describe our computational procedure, it is useful to collect state variables in the vectorŝ = (zN , zT , gN , gT , τ, d, δ, lock). The set of objects used to determine convergence consists ofthe continuation value of the government, the price of the debt, the probability of restructuringin the next period, and the expected marginal utility of traded consumption of optimizing
3

We use 13 points for zN and 11 points for zT conditional on zN . We use 1 point for gN and gT conditionalon both zN and zT , so government consumptions are set at their conditional means.
4
For our approximation, we use 41 points for b, 18 points for a, and 11 points for f . We use 151 points for0
b , which is chosen directly from the grid.

11households in the next period,E[V (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 )|ŝt , ηt+1 , δ̂t ],q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),Pr(ηt+2 (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 ) > 0|ŝt , ηt+1 , δ̂t ),hi
E βc (1 + r)UT (ŝt+1 , ft+1 , at+1 , bt+1 )|ŝt , ηt+1 , δ̂t .
Here, νt+1 is a vector of preference shocks associated with the choices available to the government, which we described in Appendix B.
Our solution algorithm proceeds in the following steps.
1. Computation before iteration. Before beginning the policy function and value function
iterations, we compute a number of key equilibrium outcomes conditional on ŝt and certain
endogenous variables. This step allows us to compute the full model in a tractable way
because it needs to be performed only once.
(a) Taking as given ŝt , we calculate the expected marginal utility of traded consumption
optimizing households in the next period (Ēt+1 ) and the government’s net revenue
from borrowing and debt service (N Bt ). When the government is in good credit
standing, N Bt is
qt (bt+1 − (1 − λp )bt ) + δ̂t − it − (λp + κp )bt − (λg + κg )ft .
When it is in bad standing, N Bt isδ̂t − (λd (ηt+1 , δ̂t ) + κd (ηt+1 , δ̂t ))bt − (λg + κg )ft .
The values of ŝt , Ēt+1 , and N Bt uniquely determine the equilibrium outcomes inthe domestic economy,{wt , pN t , Pt }, {cot , coT t , coN t , `ot }, {cht , chT t , chN t , `ht }, {Tt , Tth , Tto }, {yN t , yT t , `N t , `T t }as well as the net savings of the optimizing households:
N St = at+1 − (1 + r)at .
The equilibrium is determined from the following system of 19 equations that canbe solved for 19 unknowns. Beginning with the case of lock = 0, and dropping time12

subscripts to ease the notation, the system is:
w = zT αg1u0 (co ) = ĒḡPw = pN zN1 1−φ
P = ωc + (1 − ωc )p1−φN

P co + N S = (1 − τ )wθo `o + To

co = c(coT , coN )P ch = (1 − τ )wθh `h + T h

ch = c(chT , chN )u0 (co )(1 − τ )wθo
= v 0 (`o )
P

coT
ωc φ
=
p
o
cN
1 − ωc N

u0 (ch )(1 − τ )wθh
= v 0 (`h )
P

chT
ωc φ
=
p
h
1 − ωc N
cN

pN gN + gT + T = τ w(`T + `N ) + N B

T o = ξT h

T = (1 − γ)T h + γT oyT = zT `T

yN = zN `N

yN = gN + γcoN + (1 − γ)chN`T + `N = γθo `o + (1 − γ)θh `h .
When lock = 1, there are additional inequality constraints, and some of the above
equations may not hold. In particular, households may not be able to choose ratiosof traded to non-traded consumption as in the unconstrained CES demand system above and labor supply in an interior solution. When this is the case, we use
pN ciN +ciT instead of P ci in the budget constraints of the households and the binding
constraints replace optimality conditions in the system above.
(b) We also compute a second set of equilibrium values in which the value of N St is taken
as given, but the value of Ēt is solved for. This second set of values allows us tocompute period payoffs for the government directly from (ŝt , N St , N Bt ), regardlessof whether the Euler equation holds or not. We write the equilibrium payoff to
the government from these outcomes as Û (ŝ, N S, N B) and the equilibrium marginalutility of tradable consumption for optimizing households as Ê(ŝ, N S, N B).
Since these computations only need to be performed once, we use very fine grids for Ē,N S, and N B in order to increase the accuracy of the approximations.
2. Value/Policy Function Iteration. Each time we begin an iteration, we have in memory a13

guess for the set of objects used to determine convergenceE[V (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 )|ŝt , ηt+1 , δ̂t ],q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),Pr(ηt+2 (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 ) > 0|ŝt , ηt+1 , δ̂t ),hi
E βc (1 + r)UT (ŝt+1 , ft+1 , at+1 , bt+1 )|ŝt , ηt+1 , δ̂t .
Given this guess, we proceed as follows.
(a) First, we solve for the policy functions of the households, conditional on the government’s policy, using the endogenous grid method for policy function iteration, as
described by Carroll (2006). Specifically, for each ŝt , ηt+1 , δ̂t , ft , bt , bt+1 , at+1 , we use
our computed solutions to determine the implied N St , which immediately yields theat such that at+1 is optimal. In order to do this, we calculate ft+1 using the law of
motion for f , calculate N Bt using its definition and our guess for the price function,and take Ēt directly from our guess. This procedure yields the inverse policy function a? (a0 ) for every ŝt , ηt+1 , δ̂t , ft , bt , bt+1 . We then use linear interpolation to obtainthe values of a0? associated with points at ∈ A , i.e. a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ).
(b) Given the optimizing households’ conditional policy functions, we then step back to
the government’s decision problems, given ŝt , ηt+1 , δ̂t . When the government is in
bad credit standing, then it has no decisions to make, and its payoff is given by:
V d (ŝt , ηt+1 , δ̂t , ft , at , bt ) = Û (ŝt , N St , N Bt )+ βg E[V (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 )|ŝt , ηt+1 , δ̂t ],ft+1 = (1 − λg )ft + δ̂t ,bt+1 = (1 − ηt+1 )(1 − λd (ηt+1 , δ̂t ))bt ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),N St = at+1 − (1 + r)at ,N Bt = δ̂t − (λd (ηt+1 , δ̂t ) + κd (ηt+1 , δ̂t ))bt − (λg + κg )ft .

14

We can also calculate the payoff to lenders as
Qd (ŝt , ηt+1 , δ̂t , ft , at , bt ) = (λd (ηt+1 , δ̂t ) + κd (ηt+1 , δ̂t ))+ (1 − ηt+1 )(1 − λd (ηt+1 , δ̂t ))q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),ft+1 = (1 − λg )ft + δ̂t ,bt+1 = (1 − ηt+1 )(1 − λd (ηt+1 , δ̂t ))bt ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),and the marginal utility of traded consumption of optimizing households as
UTd (ŝt , ηt+1 , δ̂t , ft , at , bt ) = Ê(ŝt , N St , N Bt ),ft+1 = (1 − λg )ft + δ̂t ,bt+1 = (1 − ηt+1 )(1 − λd (ηt+1 , δ̂t ))bt ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),N St = at+1 − (1 + r)at ,N Bt = δ̂t − (λd (ηt+1 , δ̂t ) + κd (ηt+1 , δ̂t ))bt − (λg + κg )ft .
When the government is in good credit standing, then it chooses how much to borrowbt+1 , and its payoff is given by:
V n (ŝt , ηt+1 , δ̂t , ft , at , bt , νt ) = max Û (ŝt , N St , N Bt ) + νt (ηt+1 , bt+1 )bt+1 ∈B+ βg E[V (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 )|ŝt , ηt+1 , δ̂t ],ft+1 = (1 − λg )ft + δ̂t ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),N St = at+1 − (1 + r)at ,N Bt = q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 )(bt+1 − (1 − λp )bt ) + δ̂t− (λp + κp )bt − (λg + κg )ft − it .
With our assumptions about the distribution of the preference shocks, the set ofvalues of feasible choices bt+1 determine their choice probabilities. We write these
conditional choice probabilities as Prnc (bt+1 |ŝt , ηt+1 , δ̂t , ft , at , bt ) and the value of thegovernment’s objective function, excluding the contribution of the preference shocks,
for a given choice of bt+1 as V nc (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ).

15

The possible payoffs to lenders are:Qnc (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ) = λp + κp + (1 − λp )q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),ft+1 = (1 − λg )ft + δ̂t ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),and the marginal utility of traded consumption of optimizing households is:
UTnc (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ) = Ê(ŝt , N St , N Bt ),ft+1 = (1 − λg )ft + δ̂t ,at+1 = a0? (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),N St = at+1 − (1 + r)at ,N Bt = q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 )(bt+1 − (1 − λp )bt ) + δ̂t− (λp + κp )bt − (λg + κg )ft − it .
The expected payoff to lenders conditional on repayment is:
Qn (ŝt , ηt+1 , δ̂t , ft , at , bt ) =X

Prnc (bt+1 |ŝt , ηt+1 , δ̂t , ft , at , bt )Qnc (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ),bt+1 ∈B

and the expected marginal utility of traded consumption of optimizing households:
UTn (ŝt , ηt+1 , δ̂t , ft , at , bt ) =X

Prnc (bt+1 |ŝt , ηt+1 , δ̂t , ft , at , bt )UTnc (ŝt , ηt+1 , δ̂t , ft , at , bt , bt+1 ).

bt+1 ∈B(c) Next, we step back to the point in time when the government chooses ηt+1 , before δ̂tis realized. We calculate the expected value to the government of choosing ηt+1 = η̄asdV̄ (ŝt , η̄, ft , at , bt , νt ) =Z

V d (ŝt , η̄, δ̂t , ft , at , bt )dF (δ̂t |η̄, δt ) + νt (η̄).

Since δ̂t = δt with probability 1 whenever ηt+1 = 0, the expected value to thegovernment of choosing ηt+1 = 0 is simply
V̄ n (ŝt , 0, ft , at , bt , νt ) = max {V nc (ŝt , 0, δt , ft , at , bt , bt+1 ) + νt (0, bt+1 )},bt+1 ∈B

if the government enters the period in good standing, andV̄ d (ŝt , 0, ft , at , bt , νt ) = V d (ŝt , 0, δt , ft , at , bt ) + νt (0, (1 − λd (0, δt ))bt ),16

if the government enters the period in bad standing. When the government enters
the period in good standing, we use the set of values {{V nc (ŝt , 0, δt , ft , at , bt , bt+1 ) +νt (0, bt+1 )}bt+1 ∈B , V̄ d (ŝt , ft , at , bt , νt )} to calculate the probability that it chooses
to restructure. When the government enters the period in bad standing, we instead use the set of values {V̄ d (ŝt , η̄, ft , at , bt , νt ), V̄ d (ŝt , 0, ft , at , bt , νt )} to calculate
the probability that it chooses to restructure. Call these probabilities Pr(ηt+1 =η̄|ŝt , δt , ft , at , bt ). The two sets of values above also allow us to calculate the ex ante
expected value to the government, that is, before the preference shocks are realized,of entering the period in state (ŝt , ft , at , bt ), which is Eνt [V (ŝt , ft , at , bt , νt )].
Furthermore, we can calculate the ex ante payoff to private bondholders as
Z
Q(ŝt , ft , at , bt ) = Pr(ηt+1 = η̄|ŝt , δt , ft , at , bt ) Qd (ŝt , η̄, δ̂t , ft , at , bt )dF (δ̂t |η̄, δt )
+Pr(ηt+1 = 0|ŝt , δt , ft , at , bt ) (1 − dt )Qn (ŝt , 0, δ̂t , ft , at , bt )+ dt Qd (ŝt , 0, δ̂t , ft , at , bt ) ,and the ex ante expected marginal utility of traded consumption of optimizing households asZ

UT (ŝt , ft , at , bt ) = Pr(ηt+1 = η̄|ŝt , δt , ft , at , bt ) UTd (ŝt , η̄, δ̂t , ft , at , bt )dF (δ̂t |η̄, δt )
+Pr(ηt+1 = 0|ŝt , δt , ft , at , bt ) (1 − dt )UTn (ŝt , 0, δ̂t , ft , at , bt )+ dt UTd (ŝt , 0, δ̂t , ft , at , bt ) .
(d) Finally, given these beginning of period values, we use the transition probabilities forŝt+1 |ŝt , ηt+1 , δ̂t to update our guess for the set of objects that determine convergence:E[V (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 )|ŝt , ηt+1 , δ̂t ],q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),Pr(ηt+2 (ŝt+1 , ft+1 , at+1 , bt+1 , νt+1 ) > 0|ŝt , ηt+1 , δ̂t ),hi
E βc (1 + r)UT (ŝt+1 , ft+1 , at+1 , bt+1 )|ŝt , ηt+1 , δ̂t ,where the price of debt isq(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ) =1
E[Q(ŝt+1 , ft+1 , at+1 , bt+1 )|ŝt , ηt+1 , δ̂t ].
1+r

We calculate the distance in the sup norm between the updated objects and theguesses we started the iteration with. If the distance for each object is below a17specified tolerance, we conclude the iterative process. If not, then we calculate thenext guess for each of the relevant objects and return to step 2 of the process.
Denoting each object by F , we set Fnext = αF ∗ Fold + (1 − αF ) ∗ Fupdated , where αF
is a object-specific smoothing factor.

Simulation. We now describe how we use the model solution obtained with the above algorithm to produce the main results of the paper. As described in the main text, these results
are produced by feeding the measured sequence of exogenous variables into the model’s policy
functions. We refer to this process here as “simulation,” because preference shocks introduce
dispersion in model’s outcomes for a given set of exogenous driving forces. We use the averaged
model outcomes across preference shocks to calculate the model’s prediction.5 In our simulations, whenever a policy function or an initial value results in a b, a, or f taking a value off
their respective grids, we randomly assign the value to one of the two adjacent grid points thatbracket the value. The probability of reaching each of the two adjacent points is negatively
related to the distance between the value and the grid point.
For our simulations, we generally need to determine optimal policies at values of the exogenous variables zN , zT , gN , gT , δ, and δ̂ that are not necessarily on the grids used for thecomputation of the solution, because the time series of exogenous variables were measured without restricting their values to lie on their grids. The problem is that our solution algorithm did
not necessarily produce optimal policies for the particular values of the exogenous variables inthe data. Forward-looking variables, such as the price of debt or the expected marginal utilityof traded consumption, depend on the current value of any persistent exogenous state variable.
Since exogenous variables in the time series are not necessarily on the grids used by our solution
algorithm, our algorithm does not compute forward-looking variables at values that fall outsideof the grid.
In order to address this problem, for each vector of exogenous variables, we compute a
new set of forward-looking values that assume that, although the economy is off-grid in thecurrent period, it will be on-grid in all future periods. Since our algorithm does compute ongrid values for all relevant outcomes from the next period forward, the only additional object
that we need in order to calculate forward-looking values in the current period is the set oftransition probabilities associated with the current, off-grid, exogenous state. For δ̂ and δ, we
5
Whenever it might be necessary to draw a set of preference shocks in order to determine an optimal choice,
we do not generate those preference shocks directly, but rather sample from the implied distribution of choices.
This is without loss of generality, because we care only about the choices that arise because of the preference
shocks, and not about their realized values.

18

set transition probabilities to their values at the nearest grid point in absolute value. For theremaining states zN , zT , gN , gT , we use the transition probabilities implied by the estimatedparameters of the stochastic processes.
Even if there were no persistence in any exogenous variables, the exogenous variables would
still affect the payoffs from choices made today, so optimal policies at off-grid points would ingeneral differ from those our algorithm calculates for on-grid points. Here, we use the forwardlooking values obtained by taking expected values using the transition probabilities we describedabove. Only a single iteration is required to compute these values because we are off-grid onlyin the current period.
This procedure allows us to generate a full set of policy functions for off-grid values ofthe exogenous state variables.6 We use these policy functions, combined with the measured
sequences of exogenous variables described in Appendix A, to generate our various simulations.
Adjusting the price of debt during restructuring and exclusion. To make the spread producedby our model comparable to the spread measured in the data, we make two overall adjustments
throughout all simulations. First, because haircuts in the model are not actually applied until
the transition from one period to the next, we calculate the spread in years when the government
chooses to restructure as:
spreadt =λp + κp(1 − ηt+1 )q(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 )− λp − r.

(A.30)

This spread uses the model’s ex-dividend price of the debt during the year leading up the actual
application of the haircut and is therefore the model analogue of the data in such years.
Second, during periods of exclusion, in equation (A.30) we use the price of a marginal bondwith the same maturity λp and coupon κp as those during the periods in which the country
can access the international market. We call this price q̂(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ), and it iscalculated from the following fixed point problem:Q̂d (ŝt , ηt+1 , δ̂t , ft , at , bt ) = (λ̂d (ηt+1 , δ̂t ) + κ̂d (ηt+1 , δ̂t ))+ (1 − ηt+1 )(1 − λ̂d (ηt+1 , δ̂t ))q̂(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ),6In order to improve the accuracy of these simulations, we also solve for these optimal policies on finer grids
for bt , at , and ft than those used for the iterative solution. In particular, we use 61 points for bt , 27 points forat , and 18 points for ft . We leave the number of points for bt+1 fixed at 151.

19

"

1
E (1 − dt+1 )Qn (ŝt+1 , 0, δ̂t+1 , ft+1 , at+1 , bt+1 )1+rZ?
+ dt+1 I{ηt+2 = η̄} Qd (ŝt+1 , η̄, δ̂t+1 , ft+1 , at+1 , bt+1 )dF (δ̂t+1 |η̄, δt+1 )#q̂(ŝt , ηt+1 , δ̂t , ft+1 , at+1 , bt+1 ) =?
+ dt+1 I{ηt+2= 0}Q̂d (ŝt+1 , 0, δt+1 , ft+1 , at+1 , bt+1 )|ŝt , ηt+1 , δ̂t ,where (λ̂d , κ̂d )(ηt+1 , δ̂t ) = (λp , κp ) when δ̂t > 0 and both are 0 when δ̂t = 0.
In addition to these overall changes, we make three other adjustments to the conventionof the timing of the spread measurement to make the spread comparable to its data analog
between 2010 and 2012. First, in 2010, we assume that foreign investors expect Greece to movein 2011 to the new regime with lower productivity, lower government spending, and higher tax
rate, and so the price of debt reflects this new regime. Second, in 2011, we measure the spread
after η12 = η̄ has been chosen, but before the loan δ̂ is realized. This yields a expected value ofthe bond, inclusive of the dividend.7 Third, in 2012, we measure the spread before the haircut
is applied, since in the data the application of the haircut was effectively not complete until theend of the year. This means that in equation (A.30) we use the price of the debt from 2011.

Appendix References
Carroll, C. (2006): “The Method of Endogenous Gridpoints for Solving Dynamic Stochastic
Optimization Problems,” Economics Letters, 91(3), 312–320.
Chatterjee, S., and B. Eyigungor (2012): “Maturity, Indebtedness, and Default Risk,”
American Economic Review, 102(6), 2674–2699.
(2015): “A Seniority Arrangement for Sovereign Debt,” American Economic Review,105(12), 3740–65.
Chodorow-Reich, G., L. Karabarbounis, and R. Kekre (2023): “The Macroeconomicsof the Greek Depression,” American Economic Review, 113(9), 2411–2457.
Dvorkin, M., J. M. Sanchez, H. Sapriza, and E. Yurdagul (2021): “Sovereign Debt
Restructurings,” American Economic Journal: Macroeconomics, 13(2), 26–77.
Karabarbounis, L. (2023): “Perspectives on the Labor Share,” NBER Working Paper No.
31854.
7

We then obtain spreadt =(1−λp )(1−η̄)E[q(ŝt ,ηt+1 ,δ̂t ,ft+1 ,at+1 ,bt+1 )|ηt+1 ,δt ]−1(1−η̄)E[q(ŝt ,ηt+1 ,δ̂t ,ft+1 ,at+1 ,bt+1 )|ηt+1 ,δt ]−(λp +κp )20− r.