Federal Reserve Bank of MinneapolisResearch Department Sta� Report

September 2016

Optimal Austerity*

Juan Carlos ConesaStony Brook University

Timothy J. Kehoe University of Minnesota,Federal Reserve Bank of Minneapolis,and National Bureau of Economic Research

Kim J. RuhlPennsylvania State University

ABSTRACTWe extend a model of self-ful�lling debt crises to allow the government to optimally choosethe tax rate. We consider di�erent assumptions regarding when the government chooses thetax rate and whether the government can commit to the chosen rate. When the governmentcan commit to a tax rate before international lenders choose whether to lend, it optimallyruns down its debt and increases taxes � austerity � even when the country is not in dangerof a debt crisis. This preemptive austerity deters panics and supports more debt comparedto a model with no commitment.

JEL Codes: E6, F34, H2, H3Keywords: Debt crisis; �scal policy; commitment; Eurozone

*We thank Morten Ravn for helpful discussions. All of the data used in this paper are available at

www.econ.umn.edu/ tkehoe. The views expressed herein are those of the authors and not necessarily those

of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

1 Introduction

Europe's deep and long-lived recession has left policy makers, burdened with potentially un-

sustainable debt levels, struggling to �nd the appropriate response. One group of countries,

exempli�ed by Greece, has chosen to continue to run budget de�cits, borrow from abroad,

and hope that conditions will improve � in the language of Conesa and Kehoe (2015),

these countries are �gambling for redemption.� Another set of countries, notably the United

Kingdom, has implemented austerity policies, increasing taxes and decrease expenditures,

to decrease the outstanding debt. In this paper, we ask when is austerity optimal, and when

should a country gamble for redemption?

Our point of departure is the Conesa and Kehoe (2015) model of gambling for redemp-

tion, in which stochastic output generates a need for consumption smoothing. Governments

choose the amount of debt to issue, but cannot commit to repay this debt. As in Cole and

Kehoe (1996; 2000), self-ful�lling debt crises are possible: International lenders, in response

to the realization of a sunspot, may panic and choose not to roll over the debt, even when,

in the absence of the sunspot, lenders would have purchased the debt.

We extend the Conesa and Kehoe (2015) model to allow the government to optimally

chose the tax rate. This gives governments a choice between implementing austerity and

gambling for redemption. We study the government's decision problem under four scenarios

that di�er in the government's ability to commit to a tax rate. When a government can

commit to a tax rate, it may be able to stave o� a marginal crisis by committing to higher

tax rates.

We �nd that in models without commitment, austerity is never optimal outside of the

crisis zone � the set of debt levels for which a self-fulling crisis occurs conditional on the

sunspot. In these models, during a recession, there is no reason to implement austerity before

a crisis. Without commitment, a higher tax rate has no e�ect on the lenders' behavior. In

this model, it is optimal to keep tax rates low, smooth consumption, and increase debt.

When the government can commit to a tax rate, however, it may choose to implement

austerity for levels of debt outside of the crisis zone. When debt levels are near the lower

threshold of the crisis zone, a government with commitment will increase the tax rate and

decrease the debt outstanding. The ability to commit to a higher tax rate allows the model

to sustain higher debt levels.

A natural question arises: How do di�erences in the government's ability to commit

to �scal policy in�uence household welfare when the government cannot commit to repay

its debt? In our model, the answer depends crucially on the level of debt the government

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inherits. For very low or very high levels of debt, commitment does not the matter: Either

the government will always repay its debt or it will always default on its debt.

In the intermediate range of debt levels, particularly when debt is close to the lower

threshold that de�nes the crisis zone, the model without commitment delivers higher wel-

fare. In the model with commitment, when debt levels are near the lower threshold of the

crisis zone, the government will commit to higher tax rates: austerity. In a model without

commitment, the best the government can do is to gamble for redemption, increasing its

debt level and delivering higher welfare to households. For higher levels of debt within the

crisis zone, the model with commitment delivers higher welfare. When the government can

commit to tax rates, it can sustain higher debt levels, including levels of debt for which the

government without commitment would have defaulted.

The commitment needed to support better outcomes at higher debt levels is likely to be

di�cult for many countries to sustain � particular when governments need to implement

austerity outside of a crisis. Extra-national institutions, such as the International Monetary

Fund or the European Stability Mechanism, play an important role in this regard, although

a lack of consistency across these extra-national institutions may weaken their ability enforce

commitment (Corsetti, Erce, and Uy, 2016).

In section 2, we lay out a model of self-ful�lling debt crises, in which the government

optimally chooses the tax rate subject to di�erent assumptions regarding its ability to com-

mitment to the tax rate. In section 3 we characterize the equilibrium of the model and in

section 4 we show how di�erent assumptions about the government's ability to commit to

tax rates change the conditions under which austerity is optimal. Section 5 concludes.

2 Model

Our model closely follows Conesa and Kehoe (2015). We model a small open economy

populated by a representative household and a benevolent government. The government

chooses the labor income tax rate � under various timing protocols � and borrows from a

continuum of risk-neutral international lenders to smooth household consumption. Aggregate

productivity is stochastic, which generates the need for consumption smoothing. As in Cole

and Kehoe (1996; 2000), the government may default on the debt owed to the international

lenders and the lenders may panic (conditional on a sunspot) and refuse to lend to the

government.

In each period, the value of an exogenous sunspot variable, ζ, is realized. This sunspot

variable is uniformly and independently distributed on the interval [0, 1]. The sunspot pro-

2

vides a coordination device for the lenders' expectations and allows us to construct an equi-

librium with an arbitrary crisis probability.

2.1 Technology

Output y is produced using labor supplied by the household, `,

y(a, z) = θ(a, z)`, (1)

where aggregate productivity is θ(a, z) = A1−aZ1−z θ, with A < 1 and Z < 1. The produc-

tivity level depends on the business cycle and the default history of the government. The

economy may be in normal times (a = 1) or in a recession (a = 0). When the economy is in

a recession, productivity falls. If the government defaults (z = 0), productivity permanently

falls; z = 0, forever.

Before period 0, the economy is in normal times and the government has not defaulted,

(a = 1, z = 1). In period 0, the economy unexpectedly enters a recession and output falls.

In every period that follows, the private sector may return to normal times with probability

0 < p < 1. When the economy eventually returns to normal times (a = 1) it stays in normal

times forever.

2.2 Households

The representative household supplies labor and consumes private (c) and public (g) goods

to maximize utility,

u(c, `, g) = (1/ρ) log[µcρ + (1 − µ)(1 − `)ρ]1/ρ + γ log(g − g) (2)

subject to the budget constraint,

c(a, z) = (1 − τ)θ(a, z)`, (3)

where τ is the government-levied tax rate. The parameter ρ governs the labor supply elastic-

ity. To ensure that taxes will have a negative impact on labor supply, we require 0 < ρ < 1.

The parameters γ and g govern the importance of public consumption. Notice that we do

not allow for private savings, so the household's problem is static.

2.3 Government

As in Cole and Kehoe (1996; 2000), the government collects tax revenue from the household,

issues debt (B), and chooses whether to default on existing debt or to repay in order to

3

maximize household welfare. We model multi-period debt as in Hatchondo and Martinez

(2009) and Chatterjee and Eyigungor (2012). The fraction δ of the existing stock of debt

comes due in each period. The government's budget constraint is

g + zδB = τθ(a, z)`(a, z, τ) + q(B′, s)(B′ − (1 − δ)B), (4)

where q(B′, s) is the price of new debt. The aggregate state of the economy is s = (B, a, z−1, ζ, τ),

where z−1 = 1 if the government has defaulted in the past.

In contrast to earlier research on self-ful�lling debt crises, the government chooses the

tax rate. To understand the role of �exibility and commitment in setting the tax rate, we

consider several scenarios, which di�er in when the government sets the tax rate. We discuss

these timing assumptions below.

2.4 Lenders

A unit continuum of risk-neutral international lenders are each endowed with quantity w of

the consumption good. The lenders observe the sunspot variable before choosing how much

debt to purchase. If ζ > 1 − π, a panic occurs and the lenders will choose not to purchase

any government debt.

When a panic does not occur, each lender chooses an amount to lend, b, so solve

J(b, B′, s) = max x+ βEJ(b′, B′′, s′) (5)

x+ q(B′, s)b′ =w + z(B′, s, q(B′, s))b,

with the additional constraint b ≥ −D, which is large enough to rule out Ponzi schemes, but

otherwise does not bind. We assume that lenders have deep pockets: w is large enough to

rule out corner solutions. The �rst-order condition from this problem is

q(B′, s) = βEz(B′(s′), s′, q(B′(s′), s)). (6)

Notice that bond prices only deviate from the risk free price to adjust for expected default.

When there is no risk of default, Ez(B′(s′), s′, q(B′(s′), s)) = 1, the bond price is β.

2.5 Timing

Within a period, the general sequence of events is: 1) The random variables a and ζ are

realized and the government chooses B′, the amount of debt to sell. 2) Each of the continuum

of lenders chooses b′, the amount of debt to purchase. 3) The government chooses whether

to default, household choose labor e�ort, and production takes place.

4

We consider four variations on this sequence of events. Each variation introduces the

government's choice over the tax rate in a di�erent way.

Timing 0 Once and for all, the government commits to a tax rate in period 0. The gov-

ernment chooses τ after observing a0. This tax rate cannot be changed in subsequent

periods.

Timing 1 In each period, the government commits to a tax rate after the realization of the

recession (a), but before the realization of the sunspot and the possible panic. This

tax rate cannot be changed within the period, but can be changed in the next period.

Timing 2 In each period, the government commits to a tax rate after the realizations of

both the recession and the sunspot, but before the lenders choose the amount of debt

to purchase. This tax rate cannot be changed within the period, but can be changed

in the next period.

Timing 3 The government can not commit to a tax rate. This implies that the government

chooses the tax rate after the recession and sunspot have been realized and the lenders

have chosen the amount of debt to purchase.

Timing zero and timing three are at the extremes: either the government commits to a once

and forever tax rate, or the government cannot commit at all. In the intermediate cases,

the government has some �exibility to respond to changes in the economy and, since it can

commit to a tax rate, some ability to in�uence a lender's behavior.

3 Equilibrium

The government's debt policy can be characterized by a pair of threshold values for normal

times and pair of threshold values for recessions. If the debt level is below the lower threshold,

B ≤ b(a), the government will always repay its debt, even if the lenders choose not to rollover

the debt. We refer to [0, b(a)] as the safe zone. If the debt level is above the upper threshold,

B > B(a), the government will always default on its debt.

If the debt level falls between the thresholds, b(a) < B ≤ B(a), the country is in the

crisis zone. In the crisis zone, when a bad realization of the sunspot occurs, ζ > 1 − π, the

lenders will not purchase any debt and the government's optimal policy is to default. If the

sunspot realization is good, ζ < 1 − π, lenders will purchase positive amounts of debt and

the government's optimal policy is to repay.

5

Generally, the thresholds decrease when the economy transitions to recession. We focus

on the case in which

b(0) < b(1) < B(0) < B(1). (7)

Outside of default or a crisis, bond prices are determined by the �rst-order condition from

lender's problem. Given the characterization of default, equilibrium bond prices in normal

times are

q(B′, (B, 1, 1, ζ)) =

β(δ + (1 − δ)q′(·)) if B′ ≤ b(1)

β(1 − π)(δ + (1 − δ)q′(·)) if b(1) < B′ ≤ B(1)

0 if B(1) < B′

(8)

and in recessions

q(B′, (B, 0, 1, ζ)) =

β(δ + (1 − δ)q(·)) if B′ ≤ b(0)

β(p+ (1 − p)(1 − π))(δ + (1 − δ)q′(·)) if b(0) < B′ ≤ b(1)

β(1 − π)(δ + (1 − δ)q′(·)) if b(1) < B′ ≤ B(0)

βp(1 − π)(δ + (1 − δ)q′(·)) if B(0) < B′ ≤ B(1)

0 if B(1) < B′.

(9)

If default has occurred in the past (z−1 = 1) or if the economy is in crisis (B > b(a) and

ζ > 1 − π) lenders do not purchase government debt and q(B′, s) = 0.

4 Optimal austerity

We parameterize the model so that it generates outcomes that are consistent with the recent

European experience. When choosing parameters, we consider the outcomes of the model

without commitment (timing 3) in normal times. Our benchmark country has a debt-output

ratio of about 80 percent. We think that these characteristics � no commitment, normal

economic output, and an 80 percent debt-output ratio � capture the situation that many

European countries faced in the early 2000s. Our parameter choices are summarized in

table I.

The model period is one year. We choose β, the international lender's discount factor,

so that the yield on a safe bond is two percent. The real interest rate on debt in the crisis

zone is determined by π, the probability of a panic. We choose π so that the spread on risky

debt is about three percent, similar to the spreads over German debt that we observe for

6

Table I � Parameters

Value Target/assumption

A 0.95 productivity loss in recession = 5%

Z 0.95 default penalty

p 0.20 expected recovery = 5 years

β 0.98 safe bond yield = 2% (annual)

π 0.03 real interest rate in crisis zone = 5% (annual)

δ 1/6 average debt maturity = 6 years

γ 0.12 government revenue/output = 30%

µ 0.08 share of time devoted to work = 0.33

ρ 0.5 labor supply elasticity = 0.7

g 5.0 sacrosanct share of government expenditure = 0.50

countries like Italy and Spain. We set the share of the outstanding debt that comes due each

period, δ, so that the average debt maturity is six years.

We set p, the probability of exiting a recession, so that an average recession lasts �ve

years, and we assume that productivity falls by �ve percent during recessions. During a

recession, the decrease in productivity leads to a decrease in hours worked of �ve percent.

This implies that output falls by 10 percent during a recession, similar to the deep recession

that many European countries experienced in 2008.

When a government defaults, productivity permanently falls by �ve percent. Along

with the accompanying fall in labor e�ort, output decreases by about 10 percent following

default. Models that generate endogenous output loss after a default, such as Mendoza and

Yue (2012) and Sosa-Padilla (2012), �nd output loses between 6 and 12 percent of output.

We take the labor supply elasticity to be 0.7. The utility function (2) parameter µ is

set so that the representative household devotes 33 percent of its time endowment to work.

The parameter γ governs the relative weight of public consumption in the household's utility

function. We choose γ so that government revenues as a share of output is 30 percent in

normal times. Lastly, we assume that 50 percent of government expenditure is necessary,

which pins down g.

4.1 No commitment

We begin with timing three, in which commitment is not feasible. In �gure 1, we plot

the government's policy function. In panel A we plot the policy function in normal times.

7

Figure 1 � Debt policy function with no commitment

(a) Normal times

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

b(1) B(1)

(b) Recession

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

b(0) b(1) B(0) B(1)

Government debt is constant in the safe zone and decreases in the crisis zone. In panel B

we plot the policy function when the economy falls into a recession. The thresholds shift

left to b(0) and B(0). If the government's initial debt position is below b(0), it is optimal

to increase borrowing to smooth consumption. In this case, the government is gambling for

redemption � increasing its debt level while it waits for the recession to end.

How does the government choose taxes when it cannot commit to a tax rate? In �gure 2,

we plot the tax policy function. Consider �rst the tax policy in normal times. When the

debt level is in the safe region, tax rates are relatively �at: The government is holding debt

constant in this region. If the debt level is in the crisis zone, however, austerity is optimal.

The government increases the tax rate to decrease its debt level and move out of the crisis

zone. If debt is greater than the upper threshold, the government defaults on its debt and

lowers the tax rate.

When the economy enters a recession with a debt level that remains in the safe zone,

it lowers the tax rate. The lower tax rate and increased borrowing smooth household con-

sumption. When the economy enters a recession with debt level in the crisis zone it increases

the tax rate and decreases its borrowing to move out of the crisis zone. Notice that for debt

levels between ¯b(0) and b(1), the economy was initially in the safe zone, but is thrust into

the crisis zone when the recession occurs.

In timing three we considered the case in which the government cannot commit to a tax

rate. In the other extreme, in timing zero, the government must set a once-and-for-all tax

rate that it cannot change.

8

Figure 2 � Tax policy function

(a) No commitment (timing three)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80

OptimalTax Recession

Optimal TaxNormal Times

b(0) b(1) B(0) B(1)

(b) Once and for all tax rate (timing zero)

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.40

0 10 20 30 40 50 60 70 80 90

Optimal TaxNormal Times

OptimalTax Recession

4.2 Commitment within the period

In timing zero the government committed to a once and for all tax rate, and in timing three

the government could not commit to any tax rate. We now consider the intermediate case

� timing two � in which the government observes the recession shock and commits to a

tax rate for the rest of the period. The government must make its choice before the sunspot

is revealed.

In �gure 3, we plot the debt policy functions for normal times and recessions. In nor-

mal times, the lower threshold in this case is identical to the lower threshold in the no-

commitment case in �gure 1.

In normal times, the safe region is now divided into two sub-regions. For very low levels of

debt, the government rolls over its debt. As the debt level increases, the government chooses

to decrease the debt level, even though it is in the safe region. As can be seen in �gure 4,

this is accompanied by higher tax rates � the government �nds austerity optimal in the safe

zone as a way to deter panics. As the debt level approaches the lower threshold, austerity

becomes increasingly severe, with tax rates rising by more than 20 percentage points. This

stands in stark contrast to the optimal policy when the government cannot commit (�gure 2).

Without commitment, it is never optimal to implement austerity in the safe zone because

lenders would not be persuaded by higher taxes that could be changed later.

Since the government can commit to higher tax rates in order to deter panics, this model

supports larger debt levels in the crisis zone. The upper threshold is signi�cantly larger than

in the no-commitment case.

9

Figure 3 � Debt policy function with within-period commitment

(a) Normal times

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

b(1) B(1)

(b) Recession

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

b(0) b(1) B(0) B(1)

Figure 4 � Tax policy function with within-period commitment

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80

OptimalTax Recession

Optimal TaxNormal Times

b(0) b(1) B(0) B(1)

4.3 Welfare

What are the consequences of changing the degree of �exibility the government has in setting

tax policy? In �gure 5, we plot household welfare in the model under timing two (committing

to a tax after the sunspot and recession are known) and timing three (no commitment).

In both timing scenarios, welfare is higher in normal times than in recession times. The

relative importance of commitment is similar in normal times and recessions, so we focus

our discussion on the plots associated with a recession.

There are small welfare di�erences for very low or very high levels of debt. As we showed

10

Figure 5 � Welfare

20.5

21.0

21.5

22.0

22.5

23.0

23.5

0 10 20 30 40 50 60 70 80 90debt

Timing 3, normal times

Timing 2,normal times

Timing 2, recession

Timing 3, recessions

in section 4.2, the government in timing two replicates the behavior the government without

commitment for low levels of debt. For high enough levels of debt, the government always

defaults. If default occurs after the government has committed to a tax rate (timing two),

the government resets the tax rate to the level that is optimal when borrowing is no longer

possible.

When the debt level is near the lower threshold the model in which governments cannot

commit delivers higher welfare. In this region, the government with commitment will raise

taxes and reduce its debt level to stave o� a marginal crisis. The government without

commitment, in contrast, will gamble for redemption, continuing to borrow in hopes that

the recession will end.

For debt levels deeper into the crisis zone, the government with commitment is able to de-

liver more welfare than the government without commitment, because the crisis zone without

commitment is smaller than with commitment. As debt levels rise, the government without

commitment approaches unsustainable debt levels before the government with commitment,

generating relatively smaller household utility. As the debt level crosses the upper threshold

in the model without commitment, welfare in the no-commitment model falls to its default

value, while the government with commitment is able to sustain larger levels of debt and

welfare. This is the trade o� that commitment brings: The ability to tie the government's

hands is costly at low levels of debt, where austerity is necessary, but is bene�cial at high

levels of debt, which can only be sustained by the credible promise of higher tax rates.

11

5 Conclusions

When is austerity optimal? We develop a model in which governments borrow from inter-

national investors to smooth household consumption using debt that is subject to roll over

risk. During a recession, if the government cannot commit to a tax rate, it is only optimal to

implement austerity when it is in danger of a debt crisis. If a government can commit to a

tax rate, it will implement austerity before it is in danger of a crisis, foregoing consumption

smoothing to avoid fundamentals that may lead to a debt crisis.

12

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