Optimal Fiscal Consolidation underFrictional Financial Markets∗

Dejanir H. SilvaUIUC †

January 2020

Abstract

This paper studies optimal fiscal policy in a currency union subject to capital flow shocks. Iconsider an economy with two main ingredients: i) sticky prices and ii) financially constrainedinternational arbitrageurs. Given capital outflows and high external debt, the fiscal authority facesa trade-off between stimulating the economy or paying off the external debt to reduce sovereignspreads. I derive three main results. First, I show that it is not optimal to use government spendingto stimulate the economy, regardless of the severity of a recession or the degree of financial frictions,as long as tax instruments are available. Second, in the empirically relevant case, it is optimal to limitthe extent of a recession by having an increasing path of value-added taxes. Third, it is optimal toraise and front-load the overall level of taxes, i.e. the sum of value-added, payroll, and personalincome taxes. Therefore, it is optimal to engage in a fiscal consolidation as government debt fallscompared with a passive fiscal policy.

JEL Codes: F45, F33, E62Keywords: Currency union; Fiscal consolidation; Optimal policy; Capital flows

∗This paper is a revised version of a chapter of my PhD dissertation at MIT. I am extremely grateful to my advisors RobertTownsend, Iván Werning, and Alp Simsek for invaluable guidance and support. I also thank Arnaud Costinot and JuanPassadore for valuable comments.†University of Illinois at Urbana-Champaign. Email: dejanir@illinois.edu.

1

1 Introduction

The European periphery has received a large inflow of capital since the adoption of a common cur-rency. Such inflows have compressed domestic interest rates, leading to a boom financed by externaldebt. When these flows reversed, countries were forced to roll their debts over in adverse conditions,leading to a sharp increase in spreads, as can be seen in figure 1. A severe recession followed, resultingin a significant increase in unemployment. Given the lack of an independent national monetary author-ity, fiscal policy became the main tool available to policymakers. The combination of weak economicperformance with higher interest costs led, however, to a rapid increase in the levels of governmentdebt, reducing fiscal capacity. Policymakers then faced an important trade-off: either to engage in fiscalstimulus, improving the performance of the economy but worsening the fiscal situation, or to performa fiscal consolidation, reducing spreads but potentially aggravating the recession.

In this paper, I study the optimal resolution of this trade-off. I consider the optimal response offiscal policy to capital flow reversals in the context of a currency union model with two main ingre-dients: (i) nominal rigidities and (ii) frictional financial markets. In the presence of sticky prices, anincrease in sovereign spreads will cause a recession. Fiscal policy will then have a role to play inpotentially reducing the inefficiencies created by the lack of demand. I will refer to this role as themacroeconomic-stabilization motive for intervention. The second main ingredient of the model is a fi-nancial friction as in Gabaix and Maggiori (2015), where international borrowing is intermediated byfinancially constrained arbitrageurs. An important implication of this friction is that domestic interestrates will respond to capital flows and the country’s external borrowing. In particular, the governmentis able to affect sovereign spreads by reducing the amount of external debt. I will refer to this effect asthe debt-management motive for intervention.

Introducing financial frictions allow us to capture the effects of movements in capital flows. Themodel is able to generate the main features of the boom-bust cycle experienced by the southern Eu-ropean countries. In the model, capital inflows generate a compression in spreads, followed by anappreciation of the real exchange rate and an economic boom financed by external debt. As the move-ment of capital reverts in direction, the need to finance the country’s stock of external debt leads to anincrease in spreads, followed by a deep recession, a reversion of the current account, and an internaldevaluation, consistent with the experience observed during the European sovereign debt crisis.

To respond to the increase in spreads and the subsequent recession, the government has access toa rich set of conventional fiscal instruments, including government spending and the level of value-added, personal income, and payroll taxes. I derive three main lessons from the optimal policy prob-lem, related to government spending, taxes, and public debt. The first lesson regards the (sub)optimalityof stimulus spending. The second main result is that economies with high enough degree of home biasshould respond to higher spreads by having value-added taxes (VATs) to increase over time. The thirdlesson is that it is optimal to raise and front-load the overall level of taxes, i.e. the sum of value-added,payroll, and personal income taxes. By raising and front-loading taxes, the social planner is able toreduce government debt relative to what occurs under a passive policy that keeps the overall tax levelconstant, engaging then in a form of optimal fiscal consolidation. I will consider now each of these resultsin detail.

2

Figure 1: The cost of capital flows reversals: spreads, unemployment, and government debt

2000 2005 2010 20150

2

4

6

8

10Sovereign spreads

year

%

PortugalIrelandItalySpain

2000 2005 2010 20150

10

20

30Unemployment

year

%

2000 2005 2010 20150

20

40

60

80

100

120

140Debt-to-GDP ratio

year

%

Source: Eurostat. Sovereign spreads denote the difference on 10-year nominal rates between each country and Germany.

Let’s consider first the determination of the optimal level of stimulus spending. In the absence offrictions, the optimal provision of public goods is determined by the equalization of the marginal ben-efit of public spending and the marginal cost of producing the additional good. I will define stimulusspending as deviations from this frictionless benchmark. I show that it is never optimal to deviate fromthe frictionless benchmark as long as tax instruments are available, regardless of the degree of pric-ing or financial frictions. This result contrasts with the old Keynesian idea that even purely wastefulspending could be beneficial for a depressed economy with constrained monetary policy.1 The differ-ence lies in the availability of instruments to the planner. Stimulus spending is suboptimal not becauseof its inability to attenuate a recession, but because spending is in fact a dominated instrument. Arecession corresponds to an inefficient use of labor in the economy or, in the language of Chari et al.(2007), a labor wedge. If the planner wants to reduce the inefficiencies of a recession, she should use aninstrument that directly affects the labor wedge, such as the payroll or personal income tax, insteadof distorting the allocation of public goods to achieve this goal. This applies even if the set of instru-ments is not enough to achieve the first-best, as a simple payroll tax is enough. The logic is similar tothe Principle of Targeting, commonly used in international trade.2 In the absence of tax instruments, Ifind that the planner distorts the allocation of public goods to stabilize the economy.3 As discussed inSection 4, however, countries at the European periphery did in fact change their tax rates following thesovereign debt crisis, suggesting they are not constrained in the use of these instruments. This justifiesthe focus on the optimal design of tax instruments in response to the increase in spreads.

The second result regards the optimal timing of VATs in response to the increase in spreads. An im-

1Gali and Monacelli (2008), for instance, say that "each country’s fiscal authority plays a dual role, trading-off between theprovision of an efficient level of public goods and the stabilization of domestic inflation and output gap", consistent with this view.

2The Principle of Targeting was developed in Bhagwati and Ramaswami (1963). See Dixit (1985) for a discussion.3My result, therefore, complements the one by Werning (2011) who finds a minor role for stimulus spending in an

economy facing the zero lower bound (ZLB). Adding a tax instrument eliminates any residual role for stimulus spending.

3

portant observation is that aggregate demand depends on the change, instead of the level, in the VAT. Ifthe VAT is increasing over time, then households have an incentive to shift consumption to the present,when consumption is relatively cheap. In contrast, if the VAT is decreasing over time, then householdshave an incentive to shift consumption to the future. The two motives for intervention will push theVAT in opposite directions. The debt-management motive will induce the planner to adopt a decreas-ing path for the VAT. In this way households have an incentive to reduce consumption, pay off externaldebt, and reduce the size of sovereign spreads. The intervention is correcting a pecuniary externality, ashouseholds do not internalize the fact that they may reduce interest rates by paying off external debt.In contrast, because the labor wedge is particularly large on impact, the macroeconomic-stabilizationmotive pushes the planner to adopt an increasing path for the VAT, so households consume more inthe present, when demand is particularly lacking. Therefore, the prediction about the path of the VATis, in general, ambiguous.

It turns out that the degree of openness of an economy is a key determinant of which motive forintervention dominates. In relatively open economies, the debt-management motive dominates be-cause, as we reduce the degree of home bias, households consume fewer domestic goods, reducing theimportance of domestic distortions. In contrast, the macroeconomic-stabilization motive dominatesin relatively closed economies. As households consume fewer foreign goods, they should react lessintensively to the cost of obtaining foreign goods, i.e. the interest rate at which the country can bor-row abroad. In the limit, as the economy barely trades with the rest of the world, it is optimal to usethe VAT to offset the increase on spreads, completely avoiding the recession. An important questionis which motive dominates empirically. In Section 5, I derive a bound on the threshold determiningthe direction of the intervention and show that in the empirically relevant case the macro stabilizationmotive dominates.

The third result regards the overall level of taxes and the path of government debt. The discus-sion above implies that the overall level of taxes is not determined by the desire to control aggregatedemand, as demand can be influenced by the change in the VAT. The disconnection between demandand tax level illustrates how it is possible to use fiscal policy to provide stimulus without necessarilyhaving an adverse impact on government finances. The overall tax level will instead play a role indetermining the behavior of domestic inflation and the terms of trade. Under flexible prices, the termsof trade depreciate on impact in response to the increase in foreign spreads. Given that countries sharea common currency, this requires a coordinated discrete cut in prices by domestic producers. Understicky prices, the price level cannot jump in our continuous-time setting, so the internal devaluationhappens slowly over time. The planner can approximate the flexible price equilibrium by sharplycutting taxes and accelerating the internal devaluation process. Even though the planner can (approx-imately) replicate the flexible prices allocation, I show that this will in general not be optimal since thislogic ignores the cost of deflation when prices are set in a staggered manner. If the standard cost ofinflation (or deflation) is high enough, then it is optimal to raise taxes instead. In the limit where thecost of inflation becomes arbitrarily high, it is optimal to use taxes to completely avoid the deflation inproducer prices. In contrast, if the elasticity of output to foreign demand is relatively high, then depre-ciating the terms of trade has a large impact on output, and it would be optimal to cut taxes instead.Therefore, the prediction regarding the level of taxes is ambiguous.

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The question is then which effect dominates empirically. The cost of inflation depends on theelasticity of substitution between varieties, corresponding to the highest level of disaggregation. Theelasticity of output to the terms of trade depends on how substitutable bundles of goods from differentcountries are, corresponding to a higher level of aggregation. As discussed in Section 6, the empiri-cal literature systematically estimates higher elasticity of substitution for more disaggregated goods.Therefore, in the empirically relevant case the cost of inflation dominates, and it is optimal to raiseand front-load taxes. An important implication of this result is that government debt under an optimalpolicy is lower than it is in an economy that keeps the level of taxes constant. Therefore, the countryengages in an optimal fiscal consolidation as the government increase taxes and reduces public debt inresponse to the shock.

Finally, I compare the timing and magnitudes of tax changes under the optimal policy with theactual tax changes in Portugal, Italy, Greece, and Spain. The magnitudes predicted by the model are inline with the experience of these Southern European countries. Regarding the timing of tax changes, Ifound the behavior of Greece and Spain roughly consistent with the behavior predicted by the theory,and to a lesser extent the behavior of Portugal and Italy.

Literature. The analysis of fiscal policy in currency unions can be traced back to Kenen (1969).Modern contributions focusing on optimal policy can be found in Beetsma and Jensen (2005), Gali andMonacelli (2008), and Ferrero (2009). Importantly, they consider only a limited set of fiscal instruments.Beetsma and Jensen (2005) and Gali and Monacelli (2008) study the role of government spending, butabstract from the choice of distortionary taxes, which explains why they find an active role for stimulusspending. Ferrero (2009) considers a single tax instrument, but abstracts from optimal governmentspending. Correia et al. (2013) studies optimal fiscal policy under a richer set of tax instruments, butin the context of a closed economy in a liquidity trap, and find that it is optimal to keep the overalllevel of taxes constant. In contrast, I show that in an open economy with a common currency it isoptimal to raise and front-load taxes, leading to a fiscal consolidation. Another strand of the literaturefocus on fiscal devaluations. Correia (2011) and Farhi et al. (2013) show how to mimic an exchangerate devaluation using fiscal instruments. In contrast, I focus on the determination of the conditionsunder which a fiscal devaluation would be optimal. I find that, in the empirically relevant case, thecosts of the internal devaluation actually exceed the benefits of reducing the recession, which impliesit is optimal to engage in a fiscal appreciation. The assumption of staggered price setting is crucial forthis result. Therefore, my results differ from the ones in Adao et al. (2009), who find that the exchangerate regime is irrelevant, since they are able to implement any flexible price allocation with a constantexchange rate and constant producer prices. The difference is due to the fact that, while I consider a richset of tax instruments, it is not rich enough to render the exchange rate regime irrelevant. In particular,while I consider a VAT, which does not affect the terms of trade due to the border adjustment, theyconsider a consumption tax without a border adjustment.4 Hence, they are able to control the termsof trade independently of the level of producer prices, while in our model terms of trade adjustmentsrequire changes in producer prices. Given that in practice the operation of value-added taxes involvesa border adjustment, I consider this to be the relevant case.

4See Barbiero et al. (2019) for a positive analysis of border adjustment. Chari et al. (2018) shows that it is optimal to havea VAT with border adjustment in the context of a cooperative Ramsey problem.

5

Another important distinction with the previously mentioned literature is the focus on frictionalfinancial markets and endogenously determined spreads. As in Neumeyer and Perri (2005) and Uribeand Yue (2006), equilibrium interest rates are affected by both foreign and domestic conditions in oureconomy. In contrast to their work, the financial friction leads to a debt-elastic interest rate, similar tothe one in Schmitt-Grohé and Uribe (2003). Interest rates also depend on domestic conditions in thesovereign debt literature, as in Arellano (2008). Bianchi et al. (2018) studies optimal fiscal policy withdefault risk in a model with downward nominal wage rigidity as in Na et al. (2018), but they focuson the case of limited instruments, so they find an active role for stimulus spending. Finally, a relatedbody of literature considers the impact of alternative policies in a similar environment, such as capitalcontrols in Farhi and Werning (2012), or foreign exchange interventions in Fanelli and Straub (2019)and Cavallino (2019).

Organization. The paper is organized as follows. In Section 2, I describe the environment andthe equilibrium conditions. In Section 3, I discuss the impact of capital flow reversals. In Section 4,I present the Ramsey problem and discuss the suboptimality of stimulus spending. In Section 5, Idiscuss the optimal path for the VAT and I derive the optimal fiscal consolidation result in Section 6.The appendix contains proofs and extensions.5

2 A Currency Union with Imperfect Financial Markets

The economy consists of a continuum of countries, indexed by i ∈ [0, 1], sharing a common currency.The economy is populated by households, firms, international financial intermediaries, and nationalgovernments. Preferences and technology are symmetric across countries. I will focus attention on aspecific country which will be called "Home," with index i = H.

For ease of exposition, I will consider a perfect foresight equilibrium. Following the practice in thesticky prices literature, most of the analysis will focus on a first-order approximation of the equilibriumconditions. Hence, due to the linearization, explicitly introducing uncertainty would not change themain results, given the certainty-equivalence property.6

2.1 Environment

Households

Preferences are given by ˆ ∞

0e−ρt

[C1−σ

t1− σ

+ χ log Gt −N1+φ

t1 + φ

]dt.

Households derive utility from an aggregate of consumption goods Ct, an aggregate of governmentpurchases Gt, and there is a disutility from supplying labor Nt. Aggregate consumption Ct is a Cobb-

5In the appendix, I consider two extensions: (i) an economy with a terms-of-trade-manipulation motive, as in Costinotet al. (2011), and (ii) an economy with downward nominal wage rigidity as in Schmitt-Grohé and Uribe (2016).

6For a discussion of the certainty-equivalence property of linearized models, see, e.g., Schmitt-Grohé and Uribe (2004).

6

Douglas composite of domestic and foreign goods:

Ct =

(CH,t

1− α

)1−α (CF,t

α

)α

,

where CF,t is a composite of foreign goods

CF,t =

(ˆ 1

0C

γ−1γ

i,t di

) γγ−1

, Ci,t =

(ˆ 1

0Ci,t(j)

ε−1ε dj

) εε−1

.

Ci,t is an index of goods produced in country i, and the bundle of domestic goods is obtained for i = H.The parameter α controls the degree of home bias. As α goes to zero, there is extreme home bias

and the economy barely trades with the rest of the world. As α goes to one, there is no home bias, andhouseholds in all countries will consume the same basket of goods. The parameter ε represents theprice elasticity of goods within a given country and γ represents the elasticity of substitution betweengoods from different countries. Both elasticities are greater than one, ε > 1 and γ > 1.

The per-period budget constraint is given by

Bt = itBt + (1− τlt )WtNt + Πt + Tt −

[ˆ 1

0Pc

H,t(j)CH,t(j)dj +ˆ 1

0

ˆ 1

0Pc,∗

i,t (j)Ci,t(j)djdi

],

where it represents the nominal interest rate, Bt the value of domestic nominal bonds, Wt the nominalwage, and Πt =

´ 10 ΠH,t(j)dj aggregate nominal profits. Pc

H,t(j) and P∗,ci,t (j) denote, respectively, theprices consumers pay on domestic and foreign goods.7

Importantly, households have access only to domestic bonds, so the return on households’ assetsis given by the domestic nominal interest rate.8 Households face a labor income tax τl

t and receivelump-sum transfers from the government Tt.

Households are subject to the usual No-Ponzi condition:

limt→∞

e−´ t

0 isdsBt ≥ 0.

The demand for home and foreign bundles are given by the standard CES expressions

CH,t = (1− α)

(PcH,t

Pct

)−1

Ct, CF,t = α

(PcF,t

Pct

)−1

Ct.

and the demands for the bundles of individual countries and the goods of individual firms are

Ci,t =

(P∗,ci,t

PcF,t

)−γ

CF,t, Ci,t(j) =

(P∗,ci,t (j)

P∗,ci,t

)−ε

Ci,t.

7Consumer and producer prices will differ in this economy because of the presence of taxes.8In appendix A.3, I allow households to trade in international financial markets subject to trading costs and show that

the results do not change compared with the simpler case of no international financial market participation.

7

where Pct = (Pc

H,t)1−α(Pc

F,t)α is the consumer price index, Pc

H,t =(´ 1

0 PcH,t(j)1−εdj

) 11−ε

is the price of

domestic goods, PcF,t =

(´ 10 (P∗,ci,t )

1−γdi) 1

1−γis the price of the bundle of imported goods, and P∗,ci,t =(´ 1

0 P∗,ci,t (j)1−εdj) 1

1−εis the price of goods imported from country i.

Firms

Each differentiated good is produced using labor as the only input:

Yt(j) = AtNt(j).

Profits of domestic firms are given by

ΠH,t(j) = (1− τvt )Pc

H,t(j)(CH,t(j) + Gt(j)) + (1− τxt )P∗,cH,t(j)C∗H,t(j)− (1 + τ

pt )WtNt,

where τvt denotes a value-added tax (VAT), τx

t an export tax, and τpt a payroll tax. Gt(j) is government

demand for good j.Notice the border adjustment involved in the operation of the VAT. Domestic firms pay the tax on

domestic sales, but they are reimbursed for the VAT on foreign sales. Domestic firms must instead payτx

t on exports. In contrast, imports pay the VAT, so foreign firms receive the price (1− τvt )P∗,ci,t (j) on

their sales to the home country. For simplicity, I assume foreign countries do not impose an export tax.Firms are subject to Calvo pricing, i.e. the periods during which firms are allowed to reset their

prices are determined by a Poison arrival with intensity ρδ. Importantly, we need to specify howtaxes affect consumer and producer prices between the reset periods. Let PH,t(j) = (1 − τv

t )PcH,t(j)

and P∗H,t(j) = (1− τxt )P∗,cH,t(j) denote the prices producers receive on domestic and foreign sales, re-

spectively. I follow Barbiero et al. (2019) in assuming that producer prices for domestic and foreignconsumers are preset, so there is full pass-through of the VAT and export taxes to consumers.9 Firmswill choose the domestic producer price PH,t(j) and the price for foreign consumers is determined bythe law of one price: P∗H,t(j) = PH,t(j).

The problem of the firm is given by:

maxPH,t(j)

ˆ ∞

0e−´ s

0 (it+z+ρδ)dz[

PH,t(j)Yt+s|t − (1 + τpt )Wt+s

Yt+s|tAt+s

]ds,

subject to demand Yt+s|t =(

PH,t(j)PH,t+s

)−ε(Ct+s + Gt+s + C∗H,t+s).

Notice that between reset periods, consumer prices are given by

PcH,t+s(j) =

PH,t(j)1− τv

t+s, P∗,cH,t+s(j) =

PH,t(j)1− τx

t+s.

9In Barbiero et al. (2019), full pass-through for foreign consumers is obtained in the case of producer currency pricing(PCP).

8

International Financial Intermediaries and Gross Capital Flows

In contrast to households, international financial intermediaries can trade financial assets issued byother countries. Intermediaries have no capital and finance their positions in the domestic bonds BI

t

by borrowing internationally. Intermediaries rebate profits to foreign households. I follow Gabaix andMaggiori (2015) and assume that intermediaries can divert a fraction Γ|BI

t | of the borrowed funds, sothe total amount of diverted funds is Γ(BI

t )2. Arbitrageurs will have an incentive to repay only if the

equilibrium profits exceed the amount that can be diverted. The problem of financial intermediaries isgiven by

V It = max

BIt

(it − i∗t )BIt ,

subject toV I

t ≥ Γ(BIt )

2.

In equilibrium, the constraint will be binding as long as the spread is non-zero, so the demand fordomestic bonds from financial intermediaries is

BIt =

1Γ(it − i∗t ).

As in Gabaix and Maggiori (2015) and Cavallino (2019), I introduce noise traders with exogenousdemand for domestic bonds BN

t . Such exogenous demand can be interpreted as a form of liquiditytrading or as resulting from hedging needs.10 Notice that, even though only the total positions ofintermediaries and noise trades BI

t + BNt are needed to determine the net asset position of the home

country, the gross positions will matter for the determination of sovereign spreads because of thefinancial friction.

Government, external debt, and market clearing

Government consumption is an aggregate of domestically produced goods:

Gt =

(ˆ 1

0Gt(j)

ε−1ε dj

) εε−1

.

Government purchases individual goods to minimize costs, given the aggregate amount Gt. Thegovernment flow budget constraint is given by

Dgt = itD

gt + Pc

H,tGt + Tt − τvt Pc

H,t(CH,t + Gt)− τvt Pc

F,tCF,t − (τlt + τ

pt )WtNt − τx

t P∗,cH,tC∗H,t,

where Dgt denotes government debt.

10See, for instance, Dávila and Parlatore (2019) for a discussion of alternative microfoundations.

9

The government is also subject to a No-Ponzi condition:

limt→∞

e−´ t

0 isdsDgt ≤ 0.

Combining the flow budget constraint for households and the government, we obtain the law ofmotion of external debt Et = Dg

t − Bt:

Et = itEt + P∗t CF,t − P∗,cH,tC∗H,t.

Finally, the market-clearing conditions for domestic goods and labor are given by

YH,t(j) = CH,t(j) + Gt(j) + C∗H,t(j), Nt =

ˆ 1

0Nt(j)dj,

and the market-clearing condition for domestic bonds is given by

Et = BIt + BN

t ,

and analogous conditions hold for foreign countries.

2.2 Equilibrium characterization

Let us next consider the characterization of the equilibrium behavior. For simplicity, I will focus on thecase of a symmetric rest of the world, where Pi,t(j) = P∗t for all j ∈ (0, 1) in country i 6= H, and assumezero foreign inflation, P∗t = 1. I will also assume productivity is constant, At = 1.0 for all t ≥ 0.

Terms of Trade, Real Exchange Rate, and Aggregate Demand

Following Gali and Monacelli (2008), I define the terms of trade faced by domestic and foreign house-holds, respectively, as the ratio of the relevant import to export prices,

St ≡Pc

F,t

PcH,t

=P∗t

PH,t, St ≡ (1− τx

t )P∗t

PH,t,

where we used the fact that foreigners face price PH,t1−τx

tfor domestic goods.

Notice that, due to the border adjustment, the VAT does not directly affect the terms of trade fordomestic or foreign households. In contrast, the export tax distorts the terms of trade that are relevantto foreign consumers.

As all countries share the same currency, the nominal exchange rate is fixed and equal to one. Thereal exchange rate is then given by the ratio of domestic and foreign consumer price indices (CPI):

Qt =P∗tPc

t= (1− τv

t )S1−αt .

Hence, a depreciation of the real exchange rate is associated with a reduction in the VAT tax or an

10

increase in the terms of trade. This expression enable us to focus on the determination of the terms oftrade and to interpret movements in St as movements in the real exchange rate, given a path for τv

t .The terms of trade will play an important role in determining the level of aggregate demand, as

given by Yt = CH,t + Gt + C∗H,t, or more explicitly

Yt = (1− α)Sαt Ct + Gt + αSγ

t C∗t , (1)

using demand CH,t = (1− α)(

PcH,tPc

t

)−1Ct and C∗H,t = α

(PH,t

(1−τxt )P∗t

)−γC∗t .

The first term represents demand by domestic agents for domestic output, the second term rep-resents government demand and the third term corresponds to exports. This equation captures theintratemporal effects of movements in the real exchange rate. An increase in the terms of trade, a de-preciation of the real exchange rate, will induce domestic and foreign households to shift consumptiontowards home goods, an expenditure-switching effect.

Consumption and labor supply

The solution to the consumer problem involves the transversality condition limt→∞ e−´ t

0 isdsBt = 0, anintratemporal condition,

Cσt Nφ

t = (1− τvt )(1− τl

t )Wt

Pt, (2)

and an intertemporal condition,Ct

Ct= σ−1 (it − πt − ˙τv

t − ρ)

, (3)

where πt ≡ PtPt

and ˙τvt is the time derivative of − log(1− τv

t ).Notice that the levels of labor income and VATs affect the labor-supply condition, while only

changes in the VAT affect the Euler equation directly. Here we see the intertemporal effects of move-ments in the real exchange rate, as a real appreciation triggered by an increase in either producer pricesor VAT taxes will effectively reduce the real interest rate.

Pricing condition

Aggregate demand for labor is given byNt = ∆tYt, (4)

where

∆t ≡ˆ 1

0

(PH,t(j)

PH,t

)−ε

dj. (5)

The term ∆t captures the effect of price dispersion on aggregate labor demand (and production).Price dispersion creates misallocation of inputs, reducing the aggregate productivity of the economy.

Under sticky prices, the optimal price-setting condition is given byˆ ∞

0e−´ s

0 (it+z+ρδ)dzPt+sYt+s|t

[PH,t(j)

Pt+s− ε

ε− 1(1 + τ

pt+s)

(1− τvt+s)(1− τl

t+s)Cσ

t+sNφt+s

]ds = 0. (6)

11

The above equation indicates that a firm will choose prices as a weighted average of all futuremarginal costs. The three previous conditions are standard in the sticky prices literature and are de-rived explicitly in Appendix B.1.

In the limit ρδ → ∞, the expression above collapses to the flexible price supply condition:

S−αt =

ε

ε− 1(1 + τ

pt )

(1− τvt )(1− τl

t )Cσ

t Yφt , (7)

where I used the fact that S−αt =

PH,tPt

and PH,t = PH,t(j) under flexible prices.

Sovereign spreads and external debt

Combining demand for domestic bonds from intermediaries with the market-clearing condition forbonds, we obtain the following condition for the interest rate:

it = ρ + (i∗t − ρ)− ΓBNt︸ ︷︷ ︸

exogeneous component≡ψt

+ ΓEt︸︷︷︸endogenous component

. (8)

The domestic interest rate has two components. First, there is an exogenous component ψt thatdepends on the foreign interest rate i∗t and the amount of (gross) capital inflows to the country BN

t .Second, there is an endogenous component that depends on the level of external debt.

Plugging the above equation into the law of motion of external debt, we obtain

Et = (ρ + ψt)Et + ΓE2t + αSα−1

t Ct − αSγ−1t C∗t , (9)

for a given E0, where αSγ−1t C∗t − αSα−1

t Ct corresponds to the countries’ net exports.In Appendix B.2, I show that the government’s intertemporal budget constraint can be written as

Dg0 =

ˆ ∞

0e−´ t

0 isds

[τv

t1− τv

tSα−1

t Ct +τl

t + τpt

(1− τvt )(1− τl

t )Sα−1

t Cσt N1+φ

t + τxt αSγ−1

t C∗t − S−1t Gt − Tt

]dt.

(10)This completes the description of the equilibrium. An equilibrium is a sequence of allocations

(Ct, C∗t , Nt, Yt, Et), prices (it, Wt, [PH,t(j)]j∈[0,1], St, St), and government policies (Gt, τvt , τx

t , τpt , τl

t , Tt, Dgt )

such that conditions (1)-(10) are satisfied.

2.3 Log-linearization and calibration

To obtain a sharper characterization of the equilibrium behavior, it will be useful to log-linearize themodel around the steady state with zero inflation and zero capital flow shocks, ψt = 0. AppendixB.3 derives the steady state and log-linear approximation. Log-deviations from the steady state aredenoted by lower case letters, e.g. ct ≡ log(Ct/C) and st ≡ log(St/S), where a bar over a variable de-notes the corresponding value in the steady state. For a given specification of fiscal policy, equilibrium

12

is determined by five conditions. Aggregate output,

yt = ςc(αst + ct) + ςggt + ςxγst, (11)

where ςc, ςg, and ςx denote the share of private domestic consumption, public consumption, and ex-ports in the steady state GDP.

The Euler equation,ct = σ−1(ψt + Γet − (1− α)πH,t − ˙τv

t ), (12)

where τvt ≡ − log 1−τv

t1−τv , et ≡ Et

Y, and Γ ≡ ΓY.11

The law of motion of external debt,

et = ρet + ςm ((α− 1)st + ct)− ςx(γ− 1)st, (13)

given e0, where ςx and ςm denote exports and imports relatives to GDP in the steady state.Finally, the dynamics of domestic inflation and the terms of trade,

πH,t = ρπH,t − κ[φyt + σct + αst + τv

t + τpt + τl

t

], (14)

st = −πH,t, (15)

given s0, where κ is the Phillips curve parameter as defined in the appendix.Calibration. The preference parameters are obtained from Gali and Monacelli (2008): φ = 3, ρ =

0.04, and σ = 1.5. The parameter controlling home bias is set at α = 0.31, the elasticity of substitutionbetween varieties is ε = 6, and foreign elasticity of demand is γ = 2.0, consistent with the evidencediscussed in sections 5 and 6. Average price duration is set at seven months, according to the evidencereported in Klenow and Kryvtsov (2008). The share of government spending on total demand is setat ςg = 0.19, the average for Portugal, Italy, Greece, and Spain. Debt-to-GDP ratio is calibrated at 0.77to match the average for the same group of countries in 2007, prior to the spike in spreads and rise indebt in the recent European sovereign debt crisis. The steady state VAT is set at τv = 0.23, the averagefor the European countries we are considering, and τp = 0.47, capturing both the average payroll taxand income taxation. The financial friction parameter Γ is set at 0.105 to match the average impact ofhigher external debt on spreads during the sovereign debt crisis.12

3 Capital Flow Reversals and the Boom/Bust Cycle

In this section, I consider the positive implications of capital flows reversals and show that the model isable to capture the main features of the data. In the following sections, I focus on the optimal responseof fiscal policy to such capital flow reversals.

11As in Schmitt-Grohé and Uribe (2003), the steady-state level of net external debt is determined, since interest rates aredebt-elastic. In the absence of capital flow shocks, we have E = 0 in the steady state.

12Our measure of net external debt corresponds to the outstanding amount of non-contingent liabilities owed to non-residents by residents.

13

Figure 2: Sovereign yields, gross capital flows, and net external debt

2000 2002 2004 2006 2008

3.5

4.0

4.5

5.0

5.5

10 year yield

%

PortugalItalyGreeceSpainGermany

2000 2002 2004 2006 20080

5

10

15

20

25

30

35Gross Capital Inflows

% (

Tre

nd G

DP

)

2000 2002 2004 2006 2008 2010

40

60

80

100

Net External Debt

% G

DP

Source: Eurostat and Broner et al. (2013).

3.1 Capital inflows and the boom

Countries at the periphery of Europe received a large inflow of capital following the introductionof the common currency, as can be seen from figure 2. Gross capital flows, measured by total capitalinflow from foreign agents relative to GDP, averaged almost 16% over the 2000-2007 period. Moreover,foreign interest rates fell substantially during this period, as the 10-year yield on German bonds fell by250 b.p. during the 2000s. In response to these foreign conditions, peripheral countries accumulated asubstantial amount of debt, where the net external debt of Portugal, Greece, and Spain increased by anaverage of 35% of GDP over the period spanning the adoption of the Euro by Greece until the financialcrisis in the US.

Despite being stylized, the model is able to capture the main features of this period. I considera reduction in ψt = i∗t − ρ − ΓBN

t , corresponding to the drop in German interest rates and the grossinflows described above. We are focusing on reductions in long-term interest rates, and I thereforefocus on the case where the change in ψt is considered permanent by the agents. Qualitatively, theresults are similar for the case of transitory, but persistent, ψt. As a benchmark, I assume that fiscalpolicy does not respond to the shock, i.e., gt = Tt = τx

t = τvt + τ

pt = 0.13

Figure 3 shows the equilibrium dynamics in response to the shock. Consider first the behaviorunder sticky prices. In response to the capital inflows and the reduction in the foreign interest rate,households increase consumption in the short run, causing a boom and a reduction in net exports. Theincrease in consumption is financed by an increase in external debt. The domestic boom leads to anincrease in inflation and an appreciation of the terms of trade.

Figure 4 shows that all these features are observed in the data in the years preceding the crisis. Netexports over GDP fell by over 2% on average for Portugal, Italy, Greece, and Spain from 2002 through2008. Growth in these countries expanded relative to growth in Germany for several years before

13The initial value of τv0 is pinned down by the intertemporal budget constraint.

14

Figure 3: Effect of capital inflows: reduction in ψt

0 1 2 3 4 5 6

-0.02

0.00

0.02

0.04

Output

time0 1 2 3 4 5 6

-0.08

-0.06

-0.04

-0.02

0.00

Terms of trade

time

0 1 2 3 4 5 6

-0.10

-0.08

-0.06

-0.04

-0.02

Net exports

time0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

External debt

time

Sticky pricesFlex prices

slowing down.14 The real exchange rate appreciates sharply, as inflation in the peripheral countriesaccelerates in relative terms.

Notice that the pricing friction is important in generating the right response of output. Underflexible prices, the price level jumps in response to the shock, generating an appreciation of the realexchange rate and a large decline in net exports, which ends up depressing output in the model. Understicky prices, the internal appreciation happens slowly, muting the response of net exports.

3.2 Capital outflows and the bust

Following the financial crisis in the US, and especially after the sovereign debt crisis, there was areversal in capital flows and peripheral countries were forced to roll over the higher levels of debtincurred during the boom under adverse conditions. We will capture this phenomenon by assumingthat after T∗ = 6.0 years of low interest rates caused by negative values of ψt, unexpectedly ψt = 0 forall t ≥ T∗. We then compute the transition back to the steady state starting from the level of externaldebt and terms of trade obtained at the end of the boom phase.

Figure 5 shows the dynamics after the change in external conditions given e0 > 0 and s0 < 0.15

Sovereign spreads it − i∗t = Γet will now follow the evolution of external debt. Given the calibratedvalue of Γ = 0.105, spreads increase on impact by 325 basis points, in line with the average 350 bpincrease in the 10-year yield of Portugal, Italy, and Spain from 2009 through 2011. In response to thisincrease in spreads, the model generates a sharp recession and a reversal in the external balance. Given

14Greece is the exception, with a very pronounced boom in the years following the adoption of the common currency andan equally pronounced drop in growth since the beginning of the sovereign debt crisis.

15Period 0 in the plot corresponds to the date the news about ψt = 0 arrives.

15

Figure 4: Net exports, cumulative growth, and real exchange rate

2002 2004 2006 2008 2010 2012 2014 2016

-12

-10

-8

-6

-4

-2

0

2

Net exports

% G

DP

2002 2004 2006 2008 2010 2012 2014 20160.00

0.05

0.10

0.15

0.20

0.25

Cumulative growth differential

%

2002 2004 2006 2008 2010 2012 2014 2016

85

90

95

100Real exchange rate

PortugalItalyGreeceSpain

Source: Eurostat.

the recession, the peripheral countries undergo internal devaluation of the real exchange rate. As seenin figure 4, we also observed in the data a sharp reversal in net exports, a reduction in economic growth,and a depreciation of the real exchange rate. Hence, the model is able to capture the main features ofthe data in the period since the crisis. Sticky prices are again an important feature, as prices wouldimmediately drop under flexible prices, depreciating the real exchange rate and causing a boom led byexternal demand, instead of the large recession observed in the data. Moreover, the positive responseof the economy under flexible prices implies that the output gap, measured by the difference betweenoutput with sticky prices and output with flexible prices, was even larger than the actual measureddrop in output.

The outcomes shown in figure 5 are obtained under the assumption of no response of fiscal policy.However, countries could either use fiscal stimulus to fight the recession or pay off the debt at a fasterpace to reduce spreads. In the following sections, we study the optimal policy mix in this environment.

4 The Suboptimality of Stimulus Spending

I will consider now the optimal fiscal policy under the two extremes of price flexibility: fully flexibleand fully rigid prices. This will keep the analysis simple even in the context of the non-linear Ramseyproblem. The more complex case of intermediary degree of price stickiness is considered in Section 6.

4.1 The Ramsey Problem

Implementability. It will be convenient to adopt a primal approach, where the social planner choosesthe equilibrium quantities (Yt, Gt, Ct, St, St, Et) instead of choosing taxes directly. Under flexible prices,the planner must satisfy two conditions: the market-clearing condition for goods (1) and the evolutionof external debt (9). These conditions are sufficient for determining a competitive equilibrium in thefollowing sense. Given (Yt, Gt, Ct, St, St, Et), we can use the labor-supply condition (2) to pin down the

16

Figure 5: Effects of high initial external debt: transition given E0 > 0

0 1 2 3 4 5 6

-0.04

-0.02

0.00

0.02

Output

time0 1 2 3 4 5 6

0.00

0.02

0.04

0.06

0.08

Terms of trade

time

0 1 2 3 4 5 6

0.02

0.04

0.06

0.08

Net exports

time0 1 2 3 4 5 6

0.10

0.15

0.20

0.25

0.30

External debt

time

Sticky pricesFlex prices

real wage and (8) to determine the nominal interest rate. From the Euler equation (3), we obtain therate of change of the VAT and from the pricing condition (7) the total level of taxes.16 Importantly, theplanner cannot use lump-sum taxes to finance the budget.17 Instead, the government’s intertemporalbudget constraint pins down the initial value of the VAT. Notice that by considering a richer set ofinstruments, where the planner can choose spending and tax policy at the same time, neither the Eulernor the pricing condition appears as an explicit constraint on the Ramsey problem. This observationwill have important implications for the design of spending policy.

Therefore, the Ramsey problem is given by

maxCt,St,St,Yt,Gt,Et

ˆ ∞

0e−ρt

[1

1− σC1−σ

t + χ log Gt −1

1 + φY1+φ

t

]dt (16)

subject to

Yt = (1− α)Sαt Ct + Gt + αSγ

t C∗t ,

Et = (ρ + ψt)Et + ΓE2t + αSα−1

t Ct − αSγ−1t C∗t ,

the No-Ponzi condition on Et, and St = S if prices are fully rigid, given E0.The problem takes the perspective of the domestic country, so the optimal policy can be imple-

16The solution to the Ramsey problem does not uniquely determine τpt and τl

t , so for concreteness I will assume that τlt is

kept at the steady-state value and solve for τpt .

17The only role of the lump-sum transfers is to rebate the revenue of the export tax and of the labor income tax, which arekept at their steady-state level, so we can isolate the budgetary impact of the VAT and payroll tax. Eliminating the lump-sumwould not significantly change our results.

17

mented without international cooperation. This contrasts with previous work on fiscal policy in cur-rency unions (see Ferrero (2009) and Gali and Monacelli (2008)) where the perspective of the wholeunion was taken, implicitly assuming coordination of both monetary and fiscal policy.

Optimality conditions

The first-order conditions for output is given by

λt = Yφt , (17)

where λt, the Lagrange multiplier on the resource constraint, captures the marginal cost (in utilityterms) of increasing domestic production.

The optimality condition for consumption is given by

C−σt = (1− α)Sαλt + αSα−1

t µt, (18)

where µt is the co-state on the external solvency condition and it captures the social cost of obtainingforeign goods.

The left-hand side of (18) gives the marginal benefit of increasing consumption. The right-hand sidegives the corresponding marginal cost. To increase Ct by one unit, one must increase consumption ofdomestic goods by (1− α)Sα

t at social cost λt, and consumption of foreign goods by αSα−1t at social cost

µt.The co-state µt can be expressed as

µt = e−´ t

0 (ψs+2ΓEs)dsµ0. (19)

The term µt acts as an intertemporal price for foreign goods. Importantly, the social discount rateis ψt + 2ΓEt instead of ψt + ΓEt under the laissez-faire economy. In the language of Chari et al. (2007),this difference corresponds to an intertemporal wedge. The behavior of the intertemporal wedge willplay a key role in determining the dynamics of VATs.

The first-order conditions for St and St are given by

St =µt

λt, St =

γ− 1γ

µt

λt, (20)

under the assumption of flexible prices.The first condition implies that the terms of trade faced by domestic households should equal the

relative social cost of foreign and domestic goods. An important implication is that the labor wedge,defined as the gap between the marginal product (in utility terms) and the marginal disutility of labor,

18

is equal to zero under flexible prices:18

ωLt ≡

S−αt C−σ

tλt

− 1 = 0. (21)

The behavior of the labor wedge will be tightly connected to the total level of taxes. The secondcondition in (20) introduces a wedge in the terms of trade faced by foreign consumers. The wedgewill align the social cost with the marginal revenue in foreign sales, so the planner effectively actsas a monopolist in foreign markets. This international monopolist effect is behind the standard optimaltrade tariff in international trade (see, e.g., Dixit and Norman (1980)) and it will pin down the optimalexport tax in our economy. Given the presence of the international monopolist effect, it is important tohave an export tax to obtain a clean trade-off between the financial and pricing frictions. Otherwise,domestic instruments would react in part to try to compensate for the missing export tax, even in aneconomy with no financial friction or macroeconomic stabilization problems. In Appendix A.1, I solvethe constrained optimal policy problem where the planner does not have access to foreign trade taxesand show that our main conclusions hold in this case as well.

A benchmark economy

One can use conditions (17)-(20) to determine the optimal policy. As a benchmark, let’s consider firstthe case of no frictions on financial intermediaries, Γ = 0, and fully flexible prices, ρδ → ∞. The nextlemma characterizes optimal policy in this economy. All proofs can be found in the appendix.

Lemma 1 (Optimal policy in the benchmark economy). Suppose prices are fully flexible and there is nofinancial friction, Γ = 0. The optimal policy is given by

i. Domestic taxes1 + τ

pt

(1− τvt )(1− τl

t )

ε

ε− 1= 1, τv

t = 0.

ii. Export taxes1

1− τxt=

γ

γ− 1.

iii. Government spendingGt =

χ

Yφt

.

The three conditions are standard. The condition on domestic taxes implements the result derivedabove of zero labor wedge under flexible prices. It corresponds to the standard correction of domesticmonopoly power in optimal policy problems with sticky prices. The second condition comes from theabsence of an intertemporal wedge when Γ = 0. In this case, it is optimal to keep taxes constant toavoid distorting consumption allocation over time. The export tax effectively implements a constant

18The marginal product of labor is At = 1, given our normalization of productivity. The additional unit of domestic goodsis worth S−α

t units of the consumption bundle, so the utility of the marginal product is S−αt C−σ

t . The marginal disutility of

labor is Yφt = λt, while taking the ratio of the two expressions gives S−α

t C−σt

λt.

19

markup on foreign sales, a consequence of the CES demand structure and the international monopo-list effect. The final condition describes the optimal provision of public goods, as it implies that themarginal benefit of increasing government spending must equal the marginal cost of producing thecorresponding goods, i.e. χ

Gt= λt. In particular, if χ = 0, which means that government spending is

purely wasteful, then it is optimal to set Gt = 0.

4.2 Zero Stimulus Spending and the Pecking Order of Fiscal Instruments

Let’s consider now optimal government spending in the context of the general economy with financialfrictions and sticky prices. The next proposition characterizess optimal government spending. Inparticular, we will be interested in the amount of stimulus spending, defined as deviations from theoptimality condition in the benchmark economy:19

GSt ≡ Gt −

χ

Yφt

. (22)

If GSt = 0, then government spending is determined as in the frictionless economy and there is no

stimulus spending. If GSt > 0, then it is optimal to spend more than what would be expected based

purely on a static cost-benefit analysis. The next proposition characterizes the conditions under whichit is optimal to use stimulus spending.

Proposition 1 (Pecking order of fiscal instruments). Suppose Γ ≥ 0, ρδ ≥ 0, and let gSt ≡ gt + φyt denote

the log-linear approximation of GSt .

i. Full set of instruments. If the planner has access to the full set of tax instruments, then

gSt = 0.

ii. No tax instruments. If the planner has no access to tax instruments, then

gSt = κφγt

where γt is the Lagrange multiplier on the pricing condition.

Proposition 1 implies that the government should not engage in wasteful forms of spending, ordeviate from a static cost-benefit analysis, when it has the option of adjusting taxes. In particular, thisrule does not depend on which extreme of the price flexibility spectrum characterizes the economy orhow frictional international financial markets are.20

It is not optimal to distort the optimal provision of public goods to stimulate the economy in ourenvironment mainly because spending is a dominated instrument. In the presence of an inefficient labor

19See Werning (2011) for a similar definition of stimulus spending in a closed economy without tax instruments.20I show in the appendix that this result holds even more generally. In Appendix A.2, I consider the case of downward

nominal wage rigidity. In Appendix C.4, I consider the case of non-separable private and public consumption, and the casewhere spending can affect production directly. In all cases, it is not optimal to deviate from the condition for the optimalprovision of pubic goods in a frictionless economy, given appropriate tax instruments.

20

wedge, it is better to use a tax instrument that acts directly at the relevant margin, instead of distortingother economic decisions. This is essentially an application of the Principle of Targeting, commonlyused in the analysis of trade or environmental policy.21 In this case, it is optimal to use the payrollor personal income tax to affect the labor wedge directly, keeping the government spending decisionundistorted. Therefore, regardless of the multiplier effect of government spending or how deep therecession is, it is not optimal to use stimulus spending.

Consider now the case of no fiscal instruments. In this case, the Euler equation and the pricingcondition become constraints on the optimal policy problem. In the absence of tax instruments thatenable the planner to shift demand intertemporally, it may be optimal to distort the allocation of publicgoods to generate inflation and affect the real rate faced by households. This explains why the literaturehas previously identified how the use of government spending for stimulus may be optimal.22

To the extent that the degree of stimulus spending depends on the multiplier of the pricing condi-tion, we can see how stimulus is the result of a missing instrument problem, as a sufficient conditionfor γt = 0 is for the planner to have access to a payroll tax or a personal income tax. According to dataon the cyclical behavior of tax rates reported by Vegh and Vuletin (2015), countries at the peripheryof Europe did change their tax rates after the beginning of the sovereign crisis. For instance, Spainincreased the highest marginal personal income tax rate in 2011 and 2012 but had reduced it back tothe pre-crisis level by 2016, and it increased the VAT rate twice after 2009. Portugal, Italy, and Greecealso enacted changes in the VAT or the personal income tax. This suggests that these countries werenot constrained in their choice of tax instruments.

Therefore, from this result we obtain a pecking order of fiscal instruments. First, the governmentshould use taxes, in this case value-added, payroll, or personal income taxes, when trying to stim-ulate the economy. Only if these instruments are absent, and provided government spending has amultiplier effect, the government should use spending to stimulate the economy.

5 Optimal VAT Dynamics and Demand Management

In this section I show how the timing of the VAT is determined by the balance of two forces, a debt-management motive, by which the planner pays off foreign debt at a faster pace to reduce sovereignspreads, and a macroeconomic-stabilization motive, by which the planner induces more spending to stim-ulate the economy. I will consider the equilibrium dynamics under the optimal policy, and then discussthese two motives for intervention in turn.

5.1 Optimal Equilibrium Allocation

Let’s consider the behavior of the solution to the optimal policy problem (16). I will characterize thedynamics using a first-order approximation around an optimal steady state, i.e. fiscal policy in steadystate will be determined by the solution of the Ramsey problem in the absence of disturbances. Given

21See Dixit (1985) for a discussion and Kopczuk (2003) for a general treatment.22In environments without tax instruments, Ferrero (2009) and Gali and Monacelli (2008) found that it is optimal to use

stimulus spending in an open-economy case. Werning (2011) found a similar result in the case of a liquidity trap.

21

τp and τv, the remaining taxes and spending (τl , τx, G) satisfy the same conditions as in the benchmarkeconomy.

First, consider the dynamics of external debt and the co-state variable µt:

et = e−νote0, µt =2Γνo

e−νote0,

where νo > 0, νo is increasing in Γ, and νo → 0 as α→ 0.External debt et, and consequently sovereign spreads Γet, decay at rate νo, which is increasing with

the degree of financial frictions Γ. Hence, a higher value of Γ implies higher spreads on impact, but theyare also less persistent, as the planner has a stronger incentive to quickly reduce the debt. Moreover,debt decays at a slower pace for relatively closed economies, as net exports represent a smaller shareof GDP and it becomes harder to repay a given amount of external debt.

The co-state µt captures the social cost of foreign goods, which increases on impact and decays overtime at rate νo. Moreover, µ0 is increasing in Γ, so the initial social cost of foreign goods is higher whenthe financial friction is more severe. The evolution of µt has implications for the equilibrium dynamics,as shown in proposition 2.

Proposition 2 (Optimal allocation). Suppose prices are either fully flexible or fully rigid, E0 > 0, and ψt = 0for t ≥ 0. The equilibrium dynamics under the optimal policy is, up to first-order, given by

i. Flexible prices:

ct = −χcf µt, st = χs

f µt, nxt = χnxf µt, yt = χ

yf µt,

where χcf , χs

f , χnxf are positive constants independent of Γ, and χ

yf is positive if σ ≥ 1.

ii. Rigid prices:

ct = −χcr µt, ωL

t = χLr µt, nxt = χnx

r µt, yt = −χyr µt,

where χcr , χL

r , χnxr and χ

yr are positive constants independent of Γ.

First, notice that the financial friction affects the equilibrium dynamics only through µt. Hence,as we increase Γ, consumption drops by a greater extent on impact, but it recovers at a faster pace.Under flexible prices, the terms of trade will depreciate. The combination of the reduction in domesticconsumption with a more depreciated terms of trade will increase net exports. The impact on outputis ambiguous. If the response of net exports is strong enough, a sufficient condition being σ ≥ 1, thenoutput will actually increase in response to higher sovereign spreads. This counterfactual predictionhighlights the importance of allowing for some degree of price rigidity. In contrast to the case of flexibleprices, output unambiguously falls under rigid prices. Without the response of the terms of trade, theeconomy enters a recession. This will lead to an inefficient use of labor, captured by the positive laborwedge ωL

t . The dynamics of the labor wedge will play an important role in explaining the optimal taxdesign.

22

5.2 Debt-Management Motive

We now consider the behavior of taxes that are necessary to implement the optimal allocation. Toisolate the debt-management motive, consider first the case of flexible prices, so there is no need formacro stabilization. Combining the expression for the co-state (19) and the Euler equation (3) as wellas the labor wedge (21) with the pricing condition (7), we obtain the optimal tax conditions

τvt + τ

pt + τl

t = 0, ˙τvt = −Γet,

where τpt ≡ log 1+τ

pt

1+τp and τlt ≡ − log 1−τl

t1−τl .

As in the benchmark economy, the labor wedge is equal to zero. This implies that the sum of thepayroll, personal income, and VATs will be constant. Because the optimal policy only pins down thesum τ

pt + τl

t , but not the level of theses taxes separately, I will assume that τlt = τl for all t ≥ 0, and

τl is chosen to have a zero labor wedge in the steady state. Therefore, from this point onwards, I willconsider only variations in τ

pt and τv

t .The presence of the financial friction changes the dynamics of the VAT. Given positive external

debt et > 0, it is optimal to have a declining rate on the VAT. By setting taxes temporarily high, theplanner induces households to postpone consumption, paying off external debt at a faster pace. Thisenables the planner to correct a pecuniary externality, as households do not internalize that, by payingoff external debt, they are (collectively) able to reduce the interest rate the country faces. I will refer tothis incentive to pay off debt faster than in the laissez-faire economy as the debt-management motive.

An important aspect of the optimal policy under flexible prices is that the payroll tax should offsetany changes in the VAT. Otherwise, the period of temporarily high taxes would induce a distortion inthe use of labor, a positive labor wedge. As shown by Farhi et al. (2013), this combination of an increasein the VAT and an offsetting payroll tax corresponds to a fiscal appreciation, i.e., it mimics the effects ofan appreciation of the nominal exchange rate. I will revisit the issue of optimal fiscal devaluationsunder sticky prices in section 6.

5.3 Macroeconomic-Stabilization Motive

We now consider the case of fully rigid prices. In this case, there is no pricing condition to pin downthe level of payroll tax, allowing us to focus on the determination of the VAT.

Proposition 3 gives the main result of this section, a characterization of the optimal VAT dynamics.

Proposition 3 (Optimal VAT dynamics). Suppose prices are fully rigid, E0 > 0, and ψt = 0 for t ≥ 0. Theoptimal policy satisfies

i. Optimal VAT dynamics:˙τvt = −1− α

α˙ωL

t︸ ︷︷ ︸macro-stabilization

− Γet︸︷︷︸debt-management

.

ii. Role of openness:˙τvt = χv

r Γet, where χvr ∈ [−1, 1],

23

and χvr > 0 iff α < α∗, for a threshold α∗ satisfying 0.5 < α∗ < 1.

iii. Severity vs. persistence trade-off:

y0

yp0< 1,

νr

νp> 1, if α < α∗,

and the inequalities are reversed if α > α∗.

There are two forces affecting the optimal VAT dynamics. First, we have the debt-managementmotive, which is present even in the flexible prices economy. This motive pushes the planner to havea declining VAT, so households have an incentive to postpone consumption and pay off external debtat a faster pace. The macroeconomic-stabilization motive is specific to an economy with pricing frictions.Because the labor wedge is declining over time, the planner has an incentive to adopt an increasingVAT, as lower taxes in the present induce households to consume more exactly when the economy isparticularly depressed. Since these two forces push in opposite directions, in general, the predictionregarding the path of VATs is ambiguous.

It turns out that the relative importance of these motives for intervention depends crucially on thedegree of openness of the economy. Consider, for instance, an economy that imports the vast majorityof its consumption and exports nearly all of its production, i.e. α ≈ 1. In this case, demand for domesticgoods is not affected by the increase in interest rates and yt = 0. The planner then simply aligns theprivate and social costs of foreign goods by setting ˙τv

t = −Γet, as in the flexible price economy. Supposenow that the economy barely trades with the rest of the world, α ≈ 0. In this case, households consumeonly domestic goods, so they should not react to the intertemporal price of foreign goods. The plannerthen sets ˙τv

t = Γet, completely offsetting the effects of higher interest rates, so ct = 0. For intermediarycases, 0 < α < 1, which force will dominate depends on how close to each extreme α is.

An important question then is which force dominates empirically. The macroeconomic-stabilizationmotive dominates for α < α∗, as in the closed-economy limit discussed above. The exact value of α∗

depends, however, on the details of the economy, as the value of the EIS, the importance of govern-ment spending, or the Frisch elasticity. Regardless of these details, we are able to derive a lower boundon the value of the threshold, α∗ ≥ 0.5. Hence, we will be able to determine whether the macro stabi-lization motive dominates by simply checking whether α < 0.5. The parameter α equals ςx

ςc+ςx, where

ςx is the share of exports in GDP in the steady state and ςc represents the absorption of domestic goodsby domestic agents, which can be measured as the sum of total consumption and investment minusimports.

Table 1 shows the estimated value of α over a ten-year interval before the start of the turmoil infinancial markets since the financial crisis in the US. The average for Portugal, Italy, Greece, and Spain(denoted by "PIGS" in the table) is around 30%, well below the lower bound for α∗ of 0.5. Therefore,in the empirically relevant case, the macroeconomic-stabilization motive dominates and the plannershould choose an increasing path of value-added taxes.

This will have implications for the severity and persistence of the recession. When the macroeconomic-stabilization motive dominates, the planner shifts demand to the present, reducing the severity of therecession. By slowing the pace, however, the country pays off external debt, the interest rate stays

24

Table 1: Share of foreign demand in private absorption

Portugal Italy Greece Spain PIGSςx

ςc+ςx35.7% 31.1% 26.5% 31.8% 31.3%

Source: Eurostat. Average value for the period 1998 through 2008.

high for a longer period of time, increasing the persistence of the recession. Therefore, α effectivelydetermines how the recession will play out over time.

On the effectiveness of VAT changes

The previous discussion showed how a planner can use the VAT path to achieve its goals by managingthe timing of aggregate demand. Hence, an important question is whether such tax announcementshave the desired effect on demand as expected. The recent work by D’Acunto et al. (2016) examinesexactly this question. The authors study the effects of an announcement in Germany that the gov-ernment would increase the VAT two years in advance. The authors document that the policy wassuccessful in increasing inflation expectations. This is consistent with recent studies estimating theimpact of VAT changes on inflation.23 Moreover, they show that there is a positive association betweeninflation expectations and purchases of durable goods, especially during the announcement period ofthe new tax. This evidence is consistent with the prediction in the model that changes in the VAT affectthe level of aggregate demand.

6 Front-Loading of Taxes and Optimal Fiscal Consolidation

We have so far considered either the case of flexible prices or we have assumed no price adjustmentat all. Under flexible prices, the economy was able to achieve a coordinated, discrete cut in prices, en-abling the real exchange to adjust immediately to the shock. As in the classical argument of Friedman(1953), it may be hard in practice to achieve this coordinated reduction in prices, and the actual processof adjustment may take a long time. To capture this sluggish adjustment, I will assume next that pricesare sticky and study optimal fiscal policy in this context.

6.1 Optimal Policy under Sticky Prices

Under sticky prices, the optimal policy can be obtained by solving the linear-quadratic problem:

min[ct,st,gt,yt,et,πH,t]

12

ˆ ∞

0e−ρt

[σ(ςc + ςx)c2

t + (γςx + αςc)s2t + ςgg2

t + φy2t + 2Γe2

t +ε

κπ2

H,t

]dt,

23See, e.g., Gautier et al. (2013), Benedek et al. (2015), and Karadi and Reiff (2019).

25

subject to

yt = ςcct + ςggt + (γςx + αςc)st,

et = ρet − [(γςx + (1− α)ςm)st − ςmct] ,

st = −πH,t,

where s0 = 0 and e0 > 0 are given.The cost function captures the effects of deviations from the steady state. There are two important

distinctions with respect to the flexible prices problem. First, the terms of trade becomes a state variableinstead of a control variable. The price level cannot jump on impact, instead it starts at the steady states0 = 0, and the adjustment comes from an internal devaluation that happens over time.24 The seconddistinction is that this process of internal devaluation is costly. Generating deflation (or inflation) iscostly because of the staggered nature of the price adjustment, which creates price dispersion andmisallocation of resources. This is the standard cost of inflation in New Keynesian models (see, e.g.Galí (2015)); however it will have important implications for the design of tax policies in an openeconomy.

The constraints on this problem are the log-linearized resource constraint, the evolution of externaldebt, and st = −πH,t, which captures the fact that an internal devaluation comes through deflation. Ialso impose the condition that the planner cannot change the export tax from its steady-state value, sost = st. The Euler equation (3) and the New Keynesian Phillips curve (14) do not enter as constraintson this problem, as they are used to pin down the behavior of the payroll tax and the VAT, for a givensolution to the linear-quadratic Ramsey problem above.

Optimality conditions

The optimality conditions for output and government spending are given by

λt = φyt, gt = −λt. (23)

The Lagrange multiplier on the resource λt captures again the marginal cost of producing domesticgoods. The optimality condition for government spending extends our previous result of no stimulusspending to the case of sticky prices. Even though the economy could be potentially in a recession,the government should not deviate from a static cost-benefit analysis when deciding on a level ofspending.

The optimality condition for consumption is

− σct = (1− α)λt + αµt. (24)

This corresponds to the log-linear version of (18) and it captures the fact that the marginal costof consumption has both a domestic component, λt, and a foreign component, µt. The social cost of

24I consider the impact of alternative initial conditions for st in subsection 6.4 .

26

foreign goods evolves according toµt = −2Γet, (25)

which again differs from the corresponding private cost.The optimality conditions so far coincide with the ones we derived before. The new element comes

from the dynamics of inflation,

πH,t = −κ

ε

ςx

α(γ + 1− α)

ˆ ∞

te−ρ(s−t)ωL

s ds, (26)

which is obtained by combining the optimality conditions for inflation and for the co-state for st.Hence, if the economy is expected to be in a recession in the future, i.e. the labor wedge is on

average positive, then the economy will experience deflation and the terms of trade will depreciateover time.

Equilibrium dynamics

In contrast to the optimal policy problem with flexible or rigid prices studied so far, there are two statevariables in the Ramsey problem under sticky prices: external debt and terms of trade. Characterizingthe equilibrium dynamics requires solving a four-dimensional system of ODEs, involving the law ofmotion of the two state variables and the corresponding co-states, given in (25) and (26).25 In theappendix, I provide closed-form expressions for the solution. I focus on the case in which there is aunique bounded solution and the eigenvalues of the dynamic system are real-valued, as this will bethe case in our calibration. A sufficient condition for obtaining real-valued eigenvalues is that Γ issufficiently small, so the following analysis assumes that Γ is small.

Proposition 4 (Dynamics of state variables). Suppose E0 > 0, s0 = 0, and ψt = 0 for t ≥ 0. For Γsufficiently small, we obtain the state/co-state dynamics

i. State variables:et = −χe

1e−ν1t + χe2e−ν2t, st = −χs (e−ν1t − e−ν2t) ,

where ν1 > ν2 > 0, χe2 > χe

1 > 0, and χs > 0.

ii. Co-state variables:

µt = −χµ1 e−ν1t + χ

µ2 e−ν2t, πH,t = −χπ

1 e−ν1t + χπ2 e−ν2t,

where 0 < χµ1 < χ

µ2 and χπ

1 > χπ2 > 0.

From proposition 4, we are able to obtain a complete characterization of the state variables. I showin the appendix that the following properties hold under the optimal policy. Terms of trade exhibithump-shaped dynamics, as it is initially increasing and eventually decreasing. Therefore, there isdeflation on impact, as the economy enters in a recession, and eventually inflation as the economy

25Formally, the co-state associated with st is equal to εκ πH,t instead of πH,t.

27

recovers. External debt is eventually decreasing and converges back to the steady state.26 The socialcost of foreign goods µt jumps on impact, it is always positive, and it is eventually decreasing. Giventhe behavior of the state and co-state variables, we can determine the evolution of the control variables.

Proposition 5 (Equilibrium dynamics). Suppose E0 > 0, s0 = 0, and ψt = 0 for t ≥ 0. For Γ sufficientlysmall, we obtain

i. Control variables:

ct = −χcµµt − χc

sst, yt = −χyµµt + χ

ys st, nxt = χnx

µ µt + χnxs st,

where all coefficients are positive.

ii. Labor wedge:

ωL0 > 0, ˙ωL

0 < 0,ˆ ∞

0e−ρtωL

t dt > 0, limt→0

ωLt = 0.

Proposition 5 gives the dynamics of the control variables and of the labor wedge. Consumptionand output fall on impact while net exports increase, as µ0 > 0 and s0 = 0. The labor wedge is positiveon impact, capturing the inefficiencies created by the recession. As the social cost of foreign goodsdrops, consumption recovers, alleviating the recession and reducing the labor wedge. Similarly, theinternal devaluation of the terms of trade caused by the deflation will stimulate net exports, helpingto reduce the labor wedge as well. Another important property of the labor wedge is that it is positiveon average, as it will be relevant to pinning down the average level of taxes.

Figure 6 shows the equilibrium dynamics under the optimal and passive fiscal policies. Passivefiscal policy is defined by the conditions gS

t = τxt = τ

pt + τv

t = ˙τvt = 0, so the government does

not directly affect any wedge and adjusts taxes only to satisfy its intertemporal budget constraint.The planner is able to substantially stabilize output through the proper use of tax policy. The mainsource of adjustment is consumption, insofar as net exports do not respond as much to the shockunder the optimal policy. The reason for net exports being less responsive is a combination of higherconsumption and a slower process of internal devaluation. That the terms of trade appreciate initiallymore sharply under the optimal policy is tightly connected to the behavior of taxes and its connectionwith the labor wedge. To see this fact, notice that the New Keynesian Phillips Curve can be written asfollows:

πH,t = ρ− κ[τ

pt + τv − ωL

t

].

Hence, front-loading taxes tends to alleviate the deflationary pressure caused by the positive laborwedge. Back-loading taxes would have the opposite effect and accelerate the internal devaluation. Wenow turn to the determination of the optimal path of taxes.

26Whether external debt is initially decreasing depends on parameters. By reducing the extent of the recession, the plannerlimits the initial response of net exports. If net exports are insufficient to pay for the interest on the stock of debt, the countryinitially accumulates debt faster than it is able to repay it.

28

Figure 6: Dynamics under optimal and passive fiscal policy

0 1 2 3 4 5 6

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

Output

time0 1 2 3 4 5 6

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Terms of trade

time

0 1 2 3 4 5 6

0.02

0.03

0.04

0.05

0.06

0.07

Net exports

time0 1 2 3 4 5 6

0.05

0.10

0.15

0.20

0.25

0.30

External debt

time

Passive policyOptimal policy

6.2 Optimal Front-Loading of Taxes

The next proposition characterizes the optimal behavior of taxes.

Proposition 6 (Optimal taxes: sticky prices). Suppose E0 > 0, and ψt = 0 for t ≥ 0. The optimal policysatisfies

i. Optimal VAT dynamics:˙τvt = −1− α

α˙ωL

t︸ ︷︷ ︸macro stabilization

− Γet︸︷︷︸debt-management

.

ii. Optimal payroll tax:

τpt + τv

t =

[1− ςxγ + ςcα

εα

]ωL

t .

A sufficient condition for the term in brackets to be positive is ε > γ + 1.

The expression for the evolution of the VAT is the same as in the rigid prices case. It captures thetrade-off between the macroeconomic-stabilization motive, which runs in the direction of having anincreasing VAT, and the debt-management motive, which pushes VATs in the opposite direction. Aswe have seen, the first effect dominates in the empirically relevant case.

The second expression is new. It determines the overall level of taxes, captured by the sum of thepayroll and VATs, as a function of the labor wedge. As the labor wedge is high on impact and positiveon average, then it is optimal to raise and front-load taxes, provided the term in brackets is positive.

29

The fact that it may be optimal to raise taxes in response to a recession may sound surprising. Stickyprices impose the constraint on the planner that the price level cannot jump, so the adjustment in pricesmust happen over time. Remember that in the absence of this constraint, i.e. in the flexible prices case,it is optimal to have the price level jump down, depreciating the terms of trade on impact. One couldimagine the planner cutting taxes to achieve the desired internal devaluation. This, however, ignoresthe fact that deflation is costly when prices are set in a staggered manner. The planner will cut taxeson impact if the response of output to a depreciation, captured by the average elasticity ςxγ + ςcα,is large relative to the cost of inflation, parameterized by ε. If ε is sufficiently large, then the cost ofdeflation is larger than the benefit of depreciating the real exchange rate, so the planner increases taxesto attenuate the extent of the deflation.

An important implication of the above result is that it is not optimal to use the tax instruments toreplicate the allocation under flexible prices (or a flexible exchange rate). Even though it is feasible forthe planner to induce a fiscal devaluation, cutting the VAT and increasing the payroll tax by the sameamount, this is in general not optimal. While the only role of the payroll tax in a fiscal devaluation is toundo the labor distortions caused by the change in the VAT, here the planner uses the payroll tax in anactive manner to face the trade-off between increasing the demand by depreciating the real exchangerate or limiting the depreciation to avoid the cost of deflation.

Evidence on elasticities of substitution. It remains to determine which case is the empirically rel-evant one: front-loading or back-loading of taxes. The answer depends essentially on the comparisonof ε, the elasticity of substitution between differentiated individual goods, and γ, the elasticity of sub-stitution between bundles of foreign goods. The standard calibration in macro and trade is ε = 6.0,see e.g. Gali and Monacelli (2005) and Costinot and Rodríguez-Clare (2014), but higher values can befound in the literature, for example ε = 8.3 for Eaton and Kortum (2002). There is less consensus onthe exact value of γ, but the literature systematically estimates smaller elasticities for higher levels ofaggregation. For instance, Broda and Weinstein (2006) estimate an average elasticity of 12 for ten-digitgoods and of 4 for three-digit goods. Hobijn and Nechio (2018) use variation in the VAT to estimateupper level elasticities of substitution and find results in the range of 1 to 3 depending on the level ofaggregation. In the calibration I set γ = 2, the mid-point of this interval. Therefore, in the empiricallyrelevant case ε > γ + 1 and it is optimal to front-load taxes.

6.3 Optimal Fiscal Consolidation

The previous results indicate that, in the empirically relevant case, it is optimal to front-load taxes, asthe short-run level of taxes is higher than the long-run level, i.e. τv

0 + τp0 > limt→(τv

t + τpt ). Moreover,

it is optimal to raise the average level of taxes. These two results have important implications for howthe government is financed, in particular for the path of government debt.

Figure 7 shows the path of taxes, public debt, and the labor wedge under optimal and passive fiscalpolicy, as deviations from the steady state. Recall that passive fiscal policy is defined by ˙τv

t = τvt + τ

pt =

0, i.e., the fiscal authority adjusts the mix of VAT and payroll taxes in period 0 (without changing thetotal level of taxes) and keep taxes constant after the initial change. Given our calibration, the VATincreases (and the payroll decreases) under the passive policy. The fiscal authority must rely to a

30

Figure 7: Taxes and public debt under optimal and passive fiscal policy

0 1 2 3 4 5 6

-0.02

0.00

0.02

0.04

0.06

Value-added tax

time0 1 2 3 4 5 6

-0.06

-0.03

0.00

0.03

0.06

Payroll tax

time

Passive policyOptimal policy

0 1 2 3 4 5 60.00

0.05

0.10

0.15

0.20

0.25

Labor wedge

time0 1 2 3 4 5 6

0.00

0.05

0.10

0.15

Government debt

time

greater extent on the VAT after the shock because the VAT has a larger tax base than the payroll tax,so this change increases total tax revenue, compensating in part for the effects the recession has ongovernment finances.

While the sum of the VAT and payroll taxes is kept constant under the passive policy, it increaseson impact and declines over time under the optimal policy. The initial increase in taxes is large, as theincrease in the payroll tax of 6% more than compensates for the 2% decline in the VAT. The level oftaxes declines relatively quickly though, as τv

t + τpt ≈ 2.4% after a year of the shock and is about 0.8%

after two years. Hence, as discussed above, taxes are front-loaded under the optimal policy.This tax path is able to substantially mitigate the inefficiencies created by the recession, as the labor

wedge is much larger on impact under the passive policy. This results in part from the increasing pathof VATs, which induces households to consume more when demand is lacking. The increase in thepayroll tax also contributes to the reduction in the labor wedge, as it limits deflation and reduces thereal interest rate faced by households.

The combination of front-loaded and high taxes leads to a smaller increase in government debt un-der the optimal policy. The impact on the level of debt can be substantial, as the increase in governmentdebt is 30% larger under the passive policy after a decade compared with what it is under the optimalpolicy. Moreover, notice that the initial drop in the VAT is not dictated by an optimality condition fromthe Ramsey problem, but it is chosen to satisfy the intertemporal budget constraint of the government.The fact that τv

0 < 0 implies that the effects of higher taxes on average more than compensate for theloss in revenues caused by the recession and the increase in financing costs, as the present discountedvalue of primary surpluses would exceed the value of the debt if τv

0 = 0. The planner could alter-natively keep the initial value of the VAT at the steady-state level and rebate the additional revenue

31

Figure 8: Terms of trade, output, and net exports: the role of initial terms of trade

0 2 4 6 8

-0.06

-0.03

0.00

0.03

0.06

Terms of Trade

time0 2 4 6 8

-0.020

-0.015

-0.010

-0.005

0.000

0.005

Output

time0 2 4 6 8

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

Net exports

time

through a lump-sum transfer in the future. This would reduce the level of government debt by evenmore than in the implementation using the initial VAT. Therefore, the optimal policy is not only ableto limit the recession, but it also limits the expansion of public debt.

6.4 Optimal Response to Terms of Trade Appreciation

We have focused so far on the impact of having to finance the debt accumulated during the boom yearsunder adverse conditions. However, countries have to manage not only higher debt during the bust,but also more appreciated terms of trade as a result of substantial capital inflows, as we have seen infigure 4. In the presence of sticky prices, initial conditions matter.

The role of initial conditions

Figure 8 shows the impact of the initial appreciation of the terms of trade on the evolution of equilib-rium variables under the optimal policy. Having more sharply appreciated terms of trade accentuatesthe adverse effects of the crisis. Net exports play an important role in supporting the economy whendomestic demand falls. Since an appreciated terms of trade drag down net exports, output falls bymore on impact when it starts with domestic products being relatively more expensive than in thesteady state.

One important feature of the equilibrium under the optimal policy is that, even though the reces-sion is more severe on impact, the economy reaches roughly the same point after two years. Evenwhen the terms of trade starts 8% below the steady-state level, output reaches essentially the samelevel at year two of an economy that started with s0 = 0. This reflects how policy responds to varyinginitial conditions.

Figure 9 shows the response of taxes and the evolution of government debt for alternative valuesof s0. Starting with the terms of trade more appreciated calls for more aggressive intervention. Theincrease in the VAT and the payroll tax is more strongly accentuated when the real exchange rate beginsat a more appreciated level, as a higher level of taxes is needed to fight the deflationary pressure present

32

Figure 9: Optimal tax instruments and government debt: the role of initial terms of trade

0.0 2.5 5.0 7.5 10.0

-0.10

-0.05

0.00

0.05

Value-added tax

time0.0 2.5 5.0 7.5 10.0

0.000

0.025

0.050

0.075

0.100

VAT + Payroll tax

time0.0 2.5 5.0 7.5 10.0

0.00

0.03

0.06

0.09

0.12

Government debt

time

Table 2: Change in taxes and initial terms of trade

Portugal Italy Greece Spain PIGS

VAT 3.0% 2.0% 5.0% 5.0% 3.8%

PIT 6.0% 0.0% 5.0% 9.0% 5.0%

s0 -4.3% -4.7% -12.9% -8.8% -7.7%

Source: Vegh and Vuletin (2015) and Eurostat.

in a more depressed economy. The additional revenue from taxes will more than compensate for theloss in revenues caused by the recession, which ends up reflected in the lower value of the initial VAT.Despite the weaker economy and the larger initial cut in the VAT, the path of government debt remainsfor several years roughly the same as in the economy with no initial appreciation.

The above analysis suggests that countries like Greece and Spain that entered the crisis after a moreintensive process of internal appreciation should have increased taxes more sharply than countries likePortugal and Italy that had experienced a more modest appreciation. This seems to be in line with theobserved tax changes. Table 2 shows the changes in VATs and the highest marginal personal incometax (PIT) rate, which performs a role similar to that of the payroll tax in the model, for Portugal, Italy,Greece, and Spain, as well as the average for the four countries, denoted by "PIGS." Since 2009, theaverage increase in the VAT plus PIT for Portugal and Italy was 5.5 percentage points. In contrast,the average increase in the VAT plus PIT for Greece and Spain was 12.0 percentage points. Hence, weobserve a stronger policy reaction for countries that started with more appreciated real exchange rates.

The magnitudes implied by the model are also in line with the observed tax changes. We measurethe initial terms of trade as the cumulative inflation differential with Germany since 2002. The averagevalue of s0 for the four countries was roughly 8% and the average tax change was 8.8%. The totalincrease in taxes generated by the model was 9.6%, a difference of less than one percent. The predictionof the model also lines up with the evidence if we condition on the specific initial conditions. Given theaverage level of s0 for Portugal and Italy compared with Greece and Spain, the predicted total increase

33

in taxes was 8.0% and 11.2%, respectively. The actual average tax changes for Portugal and Italy was5.5%, while the average for Greece and Spain was 12%, again differing by less than one percent fromthe predicted change.27

While the observed magnitude of tax changes was roughly in line with those derived under theoptimal policy, the evidence is mixed regarding the timing of tax changes. All four countries adoptedan increasing path for the VAT, in most cases gradual. Portugal adopted an increasing path for thepersonal income tax, while Italy did not change the personal income tax, in contrast with the initialincrease followed by a reduction of the PIT required by the theory. The behavior of Greece and Spainwas closer to the theoretical prediction. Greece kept the PIT at a higher level from 2010 through 2012and reduced it from 2013 through 2015. Spain increased the PIT sharply in 2011 and 2012 and reducedit in 2015 and 2016. This indicates that the observed tax policy was not at odds with the predictions ofan optimal policy problem.

7 Conclusion

In this paper, I study optimal fiscal policy in a currency union subject to financial frictions. Sovereignspreads are endogenous and depend on the countries’ net external debt and on capital flows. I derivethree main lessons regarding fiscal policy. First, it is not optimal to use government spending to stim-ulate the economy, regardless of whether a country is in a deep recession or of the magnitude of thefiscal multiplier. As long as the planner has access to tax instruments, such as a payroll or a personalincome tax, it should use these instruments instead of distorting the provision of public goods. Thesecond lesson regards the timing of VAT changes. The choice of the VAT will reflect the trade-off be-tween two motives for intervention: a debt-management motive, where the planner has an incentive toadopt a declining value-added tax to induce households to postpone consumption and reduce debt,and a macroeconomic-stabilization motive, where the planner has an incentive to enact an increasing VATto induce households to consume more in the present and stimulate the economy. The key determi-nant of which force is more important quantitatively is the degree of home bias and I find that in theempirically relevant case the macro stabilization motive dominates. Finally, I study the determinationof the total level of taxes, including the VAT and payroll taxes. In the empirically relevant case, it isoptimal to raise and front-load taxes, as the planner uses taxes to limit the extent of deflation caused bythe recession. The combination of higher and front-loaded taxes leads to a reduction in governmentdebt compared with what would occur under a passive fiscal policy, i.e. it is optimal to engage in aform of fiscal consolidation.

Fiscal policy is especially important for a member of a currency union, as the country loses theability to use monetary policy to stabilize its economy. Hence, it is important to analyze how fiscalpolicy should be designed in such an environment, in particular given current debates over the relativeimportance of fiscal stimulus and fiscal consolidation. A formal analysis of the trade-off enabled us tofind ways of achieving the desired level of stimulus without sacrificing the soundness of governmentfinances.

27Italy seems to be the exception, with an actual tax change that was much smaller than that predicted by the theory.

34

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36

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37

A Extensions

I consider now two extensions of the basic model. First, I consider an economy without access to exporttaxes. This will create a new motive for intervention, a terms-of-trade-manipulation motive, a frictioncommonly used in the study of small open economies. The second extension introduces downwardnominal wage rigidity, a friction that has received supporting evidence as being important in the recentEuropean experience.

A.1 Terms of trade manipulation

So far in the analysis, I have assumed that countries have access to an export tax in the steady state.This enabled us to isolate the role of the debt-management and macroeconomic-stabilization motives.However, countries may be bound by trade agreements and unable to intervene in foreign trade. In theabsence of an instrument that can be used to directly affect foreign trade, the planner has an incentiveto distort the use of domestic instruments to perform in part the functions of the export tax.

Consider the Ramsey problem under flexible prices with the additional constraint that τxt = 0 for

all t ≥ 0. We show in appendix F.1 that the labor wedge will be positive in this economy, even underfull price flexibility:

ωLt =

αSγt C∗t

(1− α)Sαt Ct + (γ− 1)Sγ

t C∗t. (27)

Recall that, when the export tax is available, the planner wants to effectively charge a mark-upon foreign sales, acting as a monopolist on international markets.28 In the limit case α → 1, the laborwedge is 1+ωL

t = γγ−1 , i.e., the planner uses domestic taxes to obtain the appropriate wedge on foreign

sales. Because the country sells all production abroad in this case, it creates no domestic distortion. Inthe limit case α → 0, then ωL

t = 0 and it is optimal to use taxes to eliminate the domestic distortionsinstead. In the intermediary case 0 < α < 1, the planner does not completely eliminate the domesticdistortion to charge higher prices to foreign consumers.

Importantly, the relevance of this international monopolist effect depends on the relative impor-tance of foreign demand. The planner will have a stronger incentive to keep a higher labor wedge andmore appreciated terms of trade in periods where foreign demand is larger. As the relative importanceof foreign demand varies with shocks to the economy, the planner has an incentive to manipulate theterms of trade in response to shocks. I refer to this effect as the terms-of-trade-manipulation motive.29

The next proposition characterizes the optimal policy in this case.

Proposition 7 (Optimal taxes: terms of trade manipulation). Suppose E0 > 0, ψt = 0 for t ≥ 0, prices areflexible, and τx

t = 0 for t ≥ 0. The optimal policy satisfies

i. Optimal VAT dynamics:˙τvt = − γ− α

γ + 1− α

1− α

α˙ωL

t︸ ︷︷ ︸terms of trade manipulation

− Γet︸︷︷︸debt-management

.

28See Broda et al. (2008) for evidence regarding how market power affects the choice of tariffs.29Models featuring this same motive for intervention were used to study, e.g., capital controls (Costinot et al. (2011)) and

foreign exchange intervention (Fanelli and Straub (2019)).

38

ii. Optimal payroll tax:τ

pt + τv

t = χτωωL

t ,

where χτω > 0 if σ ≥ 1.

Proposition 7 shows that the main features of optimal policy in the case of sticky prices are presenteven under flexible prices if we consider the case of a planner who does not have access to an instru-ment to intervene in foreign transactions. The labor wedge increases on impact, as domestic demandshrinks and foreign demand expands. In response, it is optimal to front-load taxes, just as in thesticky prices case. Moreover, there is a trade-off between the terms of trade manipulation motive,which pushes the planner to shift demand to the present in response to the high labor wedge, andthe debt-management motive, which pushes in the opposite direction. In a sense, the terms-of-trade-manipulation motive strengthens the result regarding the optimality of raising and front-loading taxes,as it holds now even under flexible prices.

A.2 Downward Nominal Wage Rigidities

We now consider an economy that is subject to downward nominal wage rigidities.30 Our main resultwill be that the optimality of raising and front-loading taxes is maintained even under downwardnominal wage rigidity. Moreover, we will find that fiscal policy is even more powerful in managingthis friction than with staggered price stickiness, as the planner is able to obtain the same utility as inthe case without any price or wage rigidity. This extension will allow us to account for the increasein unemployment observed in economies that are exposed to an increase in sovereign spreads in therecent European debt crisis. The inability of wages to go down will limit the market-clearing processin the labor market, creating the possibility of unemployment.

Assume that prices are completely flexible, but nominal wages are downwardly rigid. We willassume that e0 > 0, so interest rates will be temporarily high. In order to obtain a downward-slopinglabor demand curve, it is useful to consider the extension with decreasing returns to scale, i.e. theproduction function is given by

Yt = AtN1ϕ

t . (28)

The log-linearized wage and pricing equations are given by

wt − pt = σct + φnt + τvt + τl

t , (29)

pH,t = (ϕ− 1)yt + wt + τpt . (30)

Wages are downwardly rigid wt ≥ 0, i.e., wages are allowed to go up, but not to go down. If theconstraint is binding, then there will be unemployment, i.e. labor supply will exceed labor demand.

30For empirical evidence of downward nominal wage rigidities in the European case, see Babecky et al. (2010). SeeSchmitt-Grohé and Uribe (2016) for a discussion of the impact of downward nominal rigidities under fixed exchange rates ina related model.

39

We can rewrite the equations above as labor demand and labor supply equations:

nst =

1φ

[wt + (1− α)st − σct − τv

t − τlt

],

ndt =

ϕ

ϕ− 1[−wt − st − τ

pt]

.

using pt = −(1− α)st.Unemployment is then given by

ut = nst − nd

t =1φ

[−σθt − τv

t − τlt

]− ϕ

ϕ− 1[−st − τ

pt]

, (31)

where I used the fact that wt = 0 and θt ≡ ct − 1−ασ st.

Consider first the case of passive fiscal policy. The decline in consumption in response to the shockinduces a positive income effect on the labor supply such that, when wages are not allowed to decline,will lead to unemployment. The next proposition characterizes the unemployment and full employ-ment periods induced by the shock.

Proposition 8 (Dynamics under downward wage rigidity). Suppose that fiscal policy is passive, wagesmust satisfy wt ≥ 0, prices are fully flexible, and ϕ = 1. Then,

i. State and co-state dynamics:

et = e−νpte0, µpt =

Γνp

e−νpte0, (32)

where νp > 0 and increasing in Γ.

ii. Equilibrium dynamicsct = −χc

pµpt , yt = −χ

ypµ

pt , ut = χu

pµpt , (33)

and wt = st = 0, where χcp, χ

yp, and χu

p are positive constants.

As in the sticky-prices economy, output declines on impact. The new aspect is that unemploymentincreases with the shock and slowly reverts back to the steady state. We next consider the optimalpolicy in this environment. I will assume that the planner does not have access to the export tax. Animportant distinction with the sticky-prices case is that it will no longer be possible to normalize eitherthe VAT or the payroll tax to zero, as both taxes will be necessary for the implementation of the optimalpolicy.

Proposition 9. (Optimal fiscal policy) Suppose that fiscal policy is passive, wages must satisfy wt ≥ 0, pricesare fully flexible, and ϕ = 1.

• Optimal government spending:

gt = −φyt.

40

• Optimal taxation:

τvt + τc

t + τlt = χτµt, ˙τv

t = − γ− α

γ + 1− α

1− α

α˙ωL

t − Γet, τpt = −χτp

µt,

where the coefficients are all positive.

As before, it is not optimal to deviate from the optimal provision of public goods. By the Principleof Targeting, it is better to use instruments that directly affect labor demand and labor supply whenan economy is facing unemployment instead of indirectly by changing government spending. Theoptimal allocation involves no unemployment, as the planner shifts labor demand upward by reducingthe payroll tax. As in the case of sticky prices, it is optimal to raise and front-load taxes, confirming therobustness of this result.

A.3 Trading on International Financial Markets

Consider an extension where domestic households have access to foreign bonds that are subject totrading frictions. Domestic households can buy domestic bonds Bd

t and foreign bonds B∗t . Householdsare unconstrained in their choice of domestic bonds, but households must pay a quadratic trading costχb2 (B∗t )

2 to buy amount B∗t of foreign bonds.The law of motion of the household’s wealth is given by

Bdt + B∗t = itBd

t −χb

2(B∗t )

2 + i∗t B∗t + (1− τlt )WtNt + Πt + Tt −

(Pc

H,tCH,t + P∗t CF,t)

. (34)

Let Bt ≡ Bdt + B∗t , then

Bt = itBt − (it − i∗t )B∗t −χb

2(B∗t )

2 + (1− τlt )WtNt + Πt + Tt −

(Pc

H,tCH,t + P∗t CF,t)

. (35)

The household will maximize the profit from trading foreign bonds subject to the trading friction

B∗t = − 1χb

(it − i∗t ). (36)

The law of motion of household’s wealth is then given by

Bt = itBt +(it − i∗t )

2

2χb+ (1− τl

t )WtNt + Πt + Tt −(

PcH,tCH,t + P∗t CF,t

). (37)

The net external debt of the home country is still given by Et = Dgt − Bt. The law of motion of

external debt is given by

Et = itEt −(it − i∗t )

2

2χb+ P∗t CF,t − P∗,cH,tC

∗H,t. (38)

Notice that the market clearing for domestic bonds is given by

Bdt − Dg

t + BIt + BN

t = 0⇒ BIt = Et + B∗t − BN

t . (39)

41

Using the demand for bonds from intermediaries, we obtain

it − i∗t = Γ(Et + B∗t − BNt )⇒ it − i∗t =

χb

χb + ΓΓ(Et − BN

t ). (40)

The law of motion of external debt is then given by

Et =

(i∗t −

χbΓχb + Γ

BNt

)Et +

χbΓχb + Γ

E2t −

12χb

(χb

χb + ΓΓ(Et − BN

t )

)2

+ P∗t CF,t − P∗,cH,tC∗H,t. (41)

Suppose BNt = 0 and i∗t = ρ, then

Et = ρEt + ΓE2t + P∗t CF,t − P∗,cH,tC

∗H,t. (42)

where Γ ≡(

1− 12

Γχb+Γ

)χb Γ

χb+Γ < Γ.The constraint above assumes the same form as in the case of no participation on the part of do-

mestic households in foreign markets, but with a smaller value for the financial friction, as Γ < Γ.Moreover, notice that Γ → Γ as χb → ∞ and Γ → 0 as χb → 0. Hence, by varying the tradingcost χb, we obtain the extremes of perfect financial markets and no international trading by domestichouseholds.

Insofar as the dynamics of net external debt is essentially the same as in the case of no internationalparticipation, we will then obtain the same results for the optimal policy in the case of internationaltrading by households.

We now consider the case where BNt 6= 0,

Et =(ρ + ψt

)Et + ΓE2

t −1

2χb

(χbΓ

χb + Γ

)2

(BNt )2 + P∗t CF,t − P∗,cH,tC

∗H,t, (43)

where ψt ≡ i∗t − ρ−(

χbχb+Γ

)2ΓBN

t .

Log-linearizing the above expression around the steady state with BNt = 0, we obtain

et = ρet + ςm((α− 1)st + ct)− ςx(γ− 1)st (44)

which corresponds to equation (13).The Euler equation is given by

ct = σ−1(ψt + Γet − (1− α)πH,t − ˙τvt ) (45)

where Γ = χbχb+Γ ΓY and ψt ≡ i∗t − ρ− χb

χb+Γ ΓBNt , which corresponds to (3).

Hence, for properly defined parameters, we obtain the same equations as in the case of no partic-ipation in international financial markets. Therefore, our results apply to the case where householdscan trade in international financial markets subject to trading frictions.

42

B Derivations for section 2

B.1 Derivation of the pricing condition under sticky prices

The problem of the firm can be written as

maxPH,t(j)

ˆ ∞

0e−´ s

0 (it+z+ρδ)dzPt+sYt+s

[PH,t(j)

Pt+s

(PH,t(j)PH,t+s

)−ε

− (1 + τpt )

Wt+s

Pt+s

(PH,t(j)PH,t+s

)−ε]

ds, (46)

where I used the normalization of productivity.The first-order condition is given by

ˆ ∞

0e−´ s

0 (it+z+ρδ)dzPt+sYt+s

[1− ε

Pt+s

(PH,t(j)PH,t+s

)−ε

+ ε(1 + τpt )

Wt+s

Pt+s

(PH,t(j)PH,t+s

)−ε 1PH,t(j)

]ds = 0, (47)

rearranging the expression above,

ˆ ∞

0e−´ s

0 (it+z+ρδ)dzPt+sYt+s|t

[PH,t(j)

Pt+s− ε

ε− 1(1 + τ

pt+s)

(1− τvt+s)(1− τl

t+s)Cσ

t+sNφt+s

]ds = 0. (48)

Price dispersion and aggregate productivity. Demand for labor from firm j is

Nt(j) =(

PH,t(j)PH,t

)−ε

Yt ⇒ˆ 1

0Nt(j)dj =

ˆ 1

0

(PH,t(j)

PH,t

)−ε

djYt. (49)

From the market-clearing condition, labor is given by

Nt = ∆tYt, (50)

where ∆t ≡(

PH,t(j)PH,t

)−ε.

B.2 Derivation of the government’s intertemporal budget constraint

The flow budget constraint can be written as

Dgt = itD

gt +(1− τv

t )PH,t

1− τvt

Gt +Tt−τv

t1− τv

t(PH,tCH,t + PF,tCF,t)−

τlt + τ

pt

(1− τlt )(1− τv

t )Cσ

t Yφt −

τxt

1− τxt

PH,tC∗H,t,

(51)using the relationship between consumer and producer prices and the labor-supply condition.

Using the expression for demand for domestic and foreign goods, we obtain

Dgt = itD

gt + S−1

t Gt + Tt −τv

t1− τv

tSα−1

t Ct −τl

t + τpt

(1− τlt )(1− τv

t )Cσ

t Yφt −

τxt

1− τxt

PH,tC∗H,t, (52)

using Pt = Sα−1t , PH,t = S−1

t , and PH,t1−τx

t= S−1

t given the normalization P∗t = 1.Integrating the above expression and rearranging it gives the expression in the text.

43

B.3 Derivation of the log-linearized conditions

Steady state

Let’s consider a steady state with zero inflation and zero capital flow shocks, ψt = 0. From the Eulerequation (3), given zero inflation and constant value-added taxes, we obtain

E = 0. (53)

From the law of motion of external debt, we obtain

Sα−1C = Sγ−1

C∗, (54)

and S = (1− τx)S.In the steady state, the pricing condition (6) boils down to

PH(j)P

=ε

ε− 11 + τp

(1− τv)(1− τp)CσY. (55)

Since prices are equalized for all firms, we can write PH(j)P

= S−α. Combining the above equations,

we obtainS−α−σ(γ−α)

=ε

ε− 11 + τp

(1− τv)(1− τp)(1− τx)σ(γ−1)(C∗)σY. (56)

The market-clearing condition for goods can be written as

Y = (1− α)SαC + G + α(1− τx)γSγC∗. (57)

Combining the above equations, we can solve for S:

S =

[ε

ε− 1(1 + τp)(1− τx)σ(γ−1)

(1− τv)(1− τp)

((1− α)(1− τx)γ−1 + G + α(1− τx)γ

)(C∗)1+σ

]− 1α+σ(γ−α)+γ

. (58)

Given fiscal policy, the equations above determine (C, S, Y).

Log linearization

Let’s now derive the log-linearized conditions around the steady state.Euler equation. Let ct ≡ log(Ct/C). From (3), we then obtain

ct = σ−1(it − πt − ˙τvt − ρ). (59)

From (8), we obtain

it = ρ + ψt + ΓYEt

Y= ρ + ψt + Γet, (60)

where Γ ≡ ΓY, et ≡ EtY

, and in the steady state ψt = 0.

44

The price level, before the VAT, is given by Pt = (P∗t )αP1−α

H,t , so taking logs and differentiating withrespect to time, we obtain

πt = απ∗t + (1− α)πH,t, (61)

where π∗t ≡P∗tP∗t

and πH,t ≡ PH,tPH,t

.Using the above equations and assuming π∗t = 0, we obtain

ct = σ−1(ψt + Γet − (1− α)πH,t − ˙τvt ). (62)

External debt. External debt evolves according to

et = (ρ + ψt + Γet)et +P∗t CF,t

Y−

P∗,cH,tC∗H,t

Y.

The value of exports is given by

P∗,cH,tC∗H,t

Y=

PH,tC∗H(1− τx

t )Y= P∗ςxe−st+γst+c∗t , (63)

using C∗H,t = α(

PH,t(1−τx

t )P∗t

)−γC∗t and the definition ςx ≡ C∗H

Y.

The value of imports is given by

P∗t CF,t

Y= ςme(α−1)st+ct , (64)

using CF,t = α

(P∗t

P1−αH,t (P∗t )α

)−1

Ct and ςm ≡ P∗CFY

.

Combining the above expressions, we obtain

et = (ρ + ψt + Γet)et + ςme(α−1)st+ct − ςxe(γ−1)st (65)

≈ (ρ + ψt + Γet)et + ςm((α− 1)st + ct)− ςx(γ− 1)st, (66)

using st = st − τxt and P∗ = 1.

Aggregate output. From (1), we obtain

eyt =(1− α)SαC

Yeαst+ct +

GY

egt +αS

γC∗

Yeγst , (67)

and linearizing the expression above, we obtain

yt = ςc(αst + ct) + ςggt + ςxγst. (68)

New Keynesian Phillips Curve. Up to the first order, we have pH,t =´ 1

0 pH,t(j)dj and, applying

45

an exact law of large numbers, we obtain

pH,t = ρδ

ˆ t

−∞e−ρδ(t−s)psds⇒ πH,t = ρδ(pt − pH,t). (69)

The optimal price-setting condition is given by

ˆ ∞

0e−(ρ+ρδ)s

C−σt+s

Pt+s(1 + τct+s)

[(1− τv

t+s)Pt(j)−ε

P−εH,t+s

Yt+s −ε

ε− 1Wt+s

Pt(j)−ε−1

P−εH,t+s

(Yt+s

At+s

)]ds = 0. (70)

Linearizing the expression above,ˆ ∞

te−(ρ+ρδ)(s−t)

[−τv

s − ε(pt − pH,s) + yt − (τcs + τl

s + σcs + αst + (1 + φ)(ys − as)) + (ε + 1)(pt − pH,s)]

ds = 0,

(71)

which when rearranged is

pt = (ρ + ρδ)

ˆ ∞

te−(ρ+ρδ)(s−t)

[τv

s + τcs + τl

s + σcs + αss + φys + pH,s

]ds. (72)

Taking the derivative of the above expression above with respect to t and using the fact that πH,t =

ρδ(pt − pH,t), we obtain

πH,t = ρπH,t − κ[τv

t + τct + τl

t + σct + αst + φyt

], (73)

where κ ≡ ρδ(ρ + ρδ).Terms of trade dynamics. Taking logs of St =

P∗tPH,t

and differentiating with respect to time,

st = π∗t − πH,t. (74)

Using π∗t = 0, gives st = −πH,t.

C Derivations and proofs for section 4

C.1 The non-linear Ramsey problem

The Ramsey problem under flexible prices can be written as

maxCt,St,St,Yt,Gt,Et

ˆ ∞

0e−ρt

[1

1− σC1−σ

t + χ log Gt −1

1 + φY1+φ

t

]dt, (75)

subject to

Yt = (1− α)Sαt Ct + Gt + αSγ

t C∗t , (76)

Et = (ρ + ψt)Et + ΓE2t + αSα−1

t Ct − αSγ−1t C∗t , (77)

given E0, and we add the constraint St = S in the case of fully rigid prices.

46

The Hamiltonian for this problem is

Ht =1

1− σC1−σ

t + χ log Gt −1

1 + φY1+φ

t − µt

[(ρ + ψt)Et + ΓE2

t + αSα−1t Ct − αSγ−1

t C∗t]

− λt[(1− α)Sα

t Ct + Gt + αSγt C∗t −Yt

].

The first-order condition with respect to output is given by

λt = Yφt . (78)

The first-order condition with respect to government spending is given by

χ

Gt= λt. (79)

The optimality condition for consumption is

C−σt = (1− α)Sα

t λt + αSα−1t µt. (80)

The optimality condition for St is given by

(γ− 1)µt = γλtSt. (81)

The evolution of the co-state µt is given by

µt = ρµt −(ρ + ψt + 2ΓEt

)µt. (82)

If prices are fully flexible, then we have the additional condition

µt = λtSt. (83)

Assuming either flexible or rigid prices, we can write (Yt, Ct, Gt, St, St) as a function of (λt, µt).Plugging the expressions for the controls into the resource constraint gives an implicit equation defin-ing λt = λ(µt). Hence, we obtain all the controls given µt and the solution can be obtained by solvingthe dynamic system in (Et, µt).

C.2 Proof of lemma 1

Domestic taxes. Plugging the optimality condition for St into the first-order condition for consumption,we obtain

C−σt = λtSα

t . (84)

Plugging the above expression into the pricing condition (7), we obtain

1 + τpt

1− τvt

ε

ε− 1= 1. (85)

47

Differentiating the optimality condition for consumption with respect to time, we obtain

− σCt

Ct=

λt

λt+ α

St

St⇒ ψt + ΓEt − ˙τv

t = − µt

µt, (86)

using the Euler equation (3) and the condition λt =µtSt

.Rearranging the above expression and using the law of motion of µt gives

˙τvt = −ΓEt, (87)

which gives τct = 0 in the case Γ = 0.

Export tax. Using St = (1− τxt )St and the expression for the terms of trade faced by domestic and

foreign consumers, we obtain the expression for the export tax.

11− τx

t=

γ

γ− 1. (88)

C.3 Proof of proposition 1

Proof. Full set of instruments. We now assume that the government has access to the full set of fiscalinstruments. For the case of rigid prices, ρδ = 0, or fully flexible prices, ρδ → ∞, the Ramsey problemis characterized in C.1. The optimality condition for government spending is

λt =χ

Gt⇒ Gt =

χ

λt, (89)

so stimulus spending is equal to zero.The problem for intermediary degree of sticky prices is defined in E.2. The optimality condition for

government spending is given bygt = −λt, (90)

where λt = φyt, up to first order, stimulus spending is again equal to zero.No tax instruments. The optimal policy problem under sticky prices can be written as

min[ct,st,gt,yt,et,πH,t]

12

ˆ ∞

0e−ρt

[σ(ςc + ςx)c2

t + ςx(γ + 1− α)s2t + ςgg2

t + φy2t + 2Γ

(et +

ψt

2Γ

)2

+ε

κπ2

H,t

]dt,

subject to

yt = ςcct + ςggt + (ςcα + ςxγ)st,

et = ρet − ςx [(γ + 1− α)st − ct] ,

st = −πH,t

πH,t = ρπH,t − κ [φyt + σct + αst]

ct = σ−1 (ψt + Γet − (1− α)πH,t)

48

where s0 = 0 and e0 > 0 are given.The first-order condition with respect to government spending is given by

ςggt + φytςg − κφςgγt = 0, (91)

where γt is the multiplier associated with the New Keynesian Phillips curveRearranging the expression above, we obtain

gt = −φyt + κφγt ⇒ gSt = κφγt. (92)

Hence, stimulus spending is non-zero when the multiplier γt is non-zero.

C.4 Generalizations of the no stimulus spending result

Suppose now there are two types of government spending, consumption spending and productivespending, denoted by Gc

t and Gpt , respectively. A household’s utility is now given by

ˆ ∞

0e−ρt

[u(Ct, Gc

t )−N1+φ

t1 + φ

]dt. (93)

The production function is given by

Yt = F(Nt, Gpt ), (94)

where Gpt is used as an intermediary input, and F is increasing in both arguments and concave.

It will be useful to define the inverse of the production function with respect to the first argument,i.e. F(Yt, Gp

t ), satisfying Yt = F(F(Yt, Gpt ), Gp

t ) for all values of Yt.The market-clearing condition is now given by

Yt = (1− α)Sαt Ct + Gc

t + Gpt + αSγ

t C∗t . (95)

The Ramsey problem under flexible prices can be written as

maxCt,St,St,Yt,Gt,Et

ˆ ∞

0e−ρt

[u(Ct, Gc

t )−1

1 + φF(Yt, Gp

t )1+φ

]dt, (96)

subject to

Yt = (1− α)Sαt Ct + Gc

t + Gpt + αSγ

t C∗t , (97)

Et = (ρ + ψt)Et + ΓE2t + αSα−1

t Ct − αSγ−1t C∗t , (98)

given E0, and we add the constraint St = S in the case of fully rigid prices.

49

The optimality condition with respect to the two types of spending are

uG(Ct, Gct ) = λt, −F(Yt, Gp

t )φ FG(Yt, Gp

t ) = λt, (99)

where λt = F(Yt, Gpt )

φ FY(Yt, Gpt ).

The above conditions are independent of Γ or whether prices are flexible or fully rigid. Hence, it isnot optimal to deviate from the optimal condition for the provision of public goods in the frictionlesseconomy.

D Derivations and proofs for section 5

D.1 Proof of proposition 2

Proof. Equilibrium dynamics. Log-linearizing the optimality conditions, we obtain

yt = ςc(αst + ct) + ςggt + ςxγst

λt = φyt,

gt = −λt,

−σct = (1− α)(αst + λt) + α(µt − (1− α)st),

˙µt = −ψt − 2Γet,

et = ρet + ςm(ct − (1− α)st)− ςxγst.

where st = st = µt − λt if prices are flexible or st = st = 0 if prices are rigid.Consumption is given by

ct = −σ−1 ((1− α)λt + αµt)

. (100)

Let 1 f lex denote an indicator function that equals one if prices are flexible. If 1 f lex = 1, we willobtain the expression for the case of flexible prices and if 1 f lex = 0 we will obtain the expression forthe case of rigid prices. Plugging the expression for consumption, terms of trade, and governmentspending into the resource constraint, we obtain λt = χλµt, where31

χλ = α(γ + 1− α)1 f lex − (ξ − α)

Ω−1 + (1− α)(ξ − α) + α(γ + 1− α)1 f lex, (101)

where ςk ≡ φςk for k ∈ c, g, x, Ω ≡ ςc+ςx1+ςg

, and ξ = α + 1−ασ .

If prices are flexible, then terms of trade are given by st = st = χsµt, where

χs =Ω−1 + ξ − α

Ω−1 + (1− α)(ξ − α) + α(γ + 1− α), (102)

31In the derivations, we repeatedly use the fact that ςc = (1− α)(ςc + ςx) and ςx = α(ςc + ςx) in the steady state.

50

Consumption is given by ct = −χcµt

χc = σ−1αΩ−1 + (γ + 1− α)1 f lex

Ω−1 + (1− α)(ξ − α) + α(γ + 1− α)1 f lex, (103)

Net exports is given by nxt = χnxµt

χnx = ςxσ−1α(Ω−1 + (γ + 1− α)1 f lex) + (γ + 1− α)(Ω−1 + ξ − α)

Ω−1 + (1− α)(ξ − α) + α(γ + 1− α)1 f lex. (104)

If prices are rigid, the labor wedge is given by ωLt = −σct − λt = χLµt

χL = α1 + (ξ − α)Ω

1 + (1− α)(ξ − α)Ω. (105)

D.2 Derivation of state and co-state dynamics

Optimal policy. External debt and co-state µt evolve according to

et = ρet − χnxµt,

˙µt = −ψt − 2Γet.

The system can be written as[et

˙µt

]=

[ρ −χnx

−2Γ 0

] [et

µt

]+

[0−ψt

]. (106)

The eigenvalues of the system are

ν =ρ +

√ρ2 + 8Γχnx

2, ν =

ρ−√

ρ2 + 8Γχnx

2. (107)

The matrix of eigenvectors and the inverse are given by

V =

[− ν

2Γ − ν2Γ

1 1

], V−1 =

2Γν− ν

[−1 − ν

2Γ

1 ν2Γ

]. (108)

Let zt = [et, µt]′, ut = [0,−ψt]′, A denote the matrix of coefficients. Define the transformed variablesZt = V−1zt and Ut = V−1ut,

Zt =1

ν− ν

[−2Γet + |ν|µt

2Γet + νµt

], Ut = −

[ |ν|ν−ν

νν−ν

]ψt. (109)

51

One can solve for Z1,t forward and Z2,t backwards to obtain

Z1,t = −ˆ ∞

te−ν(s−t)U1,sds, Z2,t = evtZ2,0 +

ˆ t

0ev(t−s)U2,sds, (110)

transforming back to the original coordinates

µ0 = −2Γν

e0 +

ˆ ∞

0e−νtψtdt (111)

The initial value of Z2,t is given by

Z2,0 = −2Γv

e0 +ν

ν− ν

ˆ ∞

0e−νsψsds. (112)

External debt is given by

et =v

2Γ

ˆ ∞

te−v(s−t)U1,sds− ν

2Γ

[evtZ2,0 +

ˆ t

0ev(t−s)U2,sds

]=

vν

ν− ν

[1− e−(v−ν)t

2Γ

ˆ ∞

te−v(s−t)ψsds +

eνt

2Γ

ˆ t

0(e−νs − e−νs)ψsds

]+ eνte0,

and the co-state is given by

µt = −ˆ ∞

te−v(s−t)U1,sds + evtZ2,0 +

ˆ t

0ev(t−s)U2,sds

=

[e−(v−ν)tν− ν

ν− ν

ˆ ∞

te−v(s−t)ψsds− νeνt

ν− ν

ˆ t

0(e−νs − e−νs)ψsds

]− 2Γ

νeνte0.

If ψt = 0 for all t ≥ 0, then

et = eνte0, µt = −2Γν

eνte0. (113)

Setting νo ≡ −ν, we obtain the expression in the text.Passive policy. Consider next the evolution of equilibrium variables under the passive policy for

flexible and rigid prices.Flexible prices. From the pricing and Euler conditions, we have

− σct − αst = λt, σct = ψt + Γet + (1− α)st. (114)

Combining the above conditions, we obtain

st + λt = −(ψt + Γet). (115)

Define µpt ≡ st + λt. Then,

ct = −σ−1 ((1− α)λt + αµpt)

. (116)

52

This implies that the equilibrium variables can be written as

ct = −χcf µ

pt , st = χs

f µpt , nxt = χnx

f µpt , yt = χ

yf µ

pt . (117)

External debt and µpt evolve according to the dynamic system

et = ρet − χnxp µ

pt , ˙µp

t = −ψt − Γet.

The solution to the system above is

et = eνpte0, µt = −Γνp eνpte0. (118)

where νp =ρ−√

ρ2+4Γχnx2 and |νp| < |ν|.

Rigid prices. From the Euler equation and the resource constraint, we obtain

σct = ψt + Γet, λt = (1− α)Ωct. (119)

Define µpt ≡ −σct, then

χpc = σ−1, χ

pλ = (1− α)Ωχ

pc , χnx

p = ςxχpc . (120)

Similarly, the coefficients under the optimal policy satisfy

χcr = σ−1 α

1 + (1− α)(ξ − α)Ω, χλ

r = (1− α)Ωχrc, χnx

r = ςxχrc. (121)

The evolution of et and µpt is given by

et = eνpte0, µt = −Γνp eνpte0, (122)

where νp =ρ−√

ρ2+4Γχnxp

2 .

D.3 Proof of proposition 3

Proof. Optimal VAT dynamics. From the Euler equation for consumption, we can solve for ˙τvt

˙τvt = ψt + Γet − σct

= −(1− α)(µt − λt)− Γet

= −1− α

α˙ωL

t − Γet,

using the optimality condition for consumption and the fact that ωLt = α(µt − λt).

53

Role of openness. We can write the expression for the evolution of the VAT as follows,

˙τvt =

[2(1− α)

1 + (ξ − α)Ω1 + (1− α)(ξ − α)Ω

− 1]

Γet

= χvr Γet,

whereχv

r ≡ 1− 2α

1 + (1− α)(ξ − α)Ω. (123)

The second term in the above expression is monotonically increasing in α, and it is equal to zerofor α = 0 and equal to 2 for α = 1. Therefore, χv

r ∈ [−1, 1]. Moreover, there exists 0.5 < α∗ < 1 suchthat χv

r > 0 if and only if α < α∗. Hence, relatively closed economies will choose an increasing path ofVATs.

Severity vs. persistence trade-off. First, notice that |νp| < |νr| if

2α

1 + (1− α)(ξ − α)Ω> 1, (124)

which requires α > α∗.The ratio of initial consumption under the optimal policy and the passive policy is given by

cr0

cp0= 2

α

1 + (1− α)(ξ − α)

|νp||νr| . (125)

Notice that the following function f (x) = (√

ρ2 + x− ρ)/x is decreasing,

f ′(x) =(ρ2 + x)−

12

2x−√

ρ2 + x− ρ

x2 =x

2x2√

ρ2 + x− 2(ρ2 + x)− 2ρ

√ρ2 + x

2x2√

ρ2 + x= − x− 2ρ

√ρ2 + x + 2ρ2

2x2√

ρ2 + x< 0,

(126)because the numerator is convex in x, zero at x = 0, and the global minimum is at x = 0.

We can write the ratio of initial consumptions as

cr0

cp0=

f (4Γχnxp )

f(

4Γχnxp × 2 α

1+(1−α)(ξ−α)

) . (127)

If α > α∗, then cr0

cp0> 1, i.e., consumption falls by more under the optimal policy than under the

passive policy. A similar inequality holds for output.

54

E Derivations and proofs for section 6

E.1 An approximate welfare criterion under flexible prices

Utility from consumption can be written as

C1−σt

1− σ=

C1−σ

1− σe(1−σ)ct ≈ C1−σ

1− σ+ N1+φ

(ςc + ςx)

[ct +

1− σ

2c2

t

], (128)

where I used the fact that CSα

Y = ςc + ςx and

C−σ= YφSα ⇒ C1−σ

= N1+φ CSα

Y. (129)

The log-linear version of the Backus-Smith condition holds exactly:

ct = θt +(1− α)

σst. (130)

The utility from government spending can be written as

χ log Gt = χ log G + N1+φςggt, (131)

where I used the fact that χ = N1+φςg.

The disutility from labor is given by

N1+φt

1 + φ=

N1+φ

1 + φe(1+φ)yt ≈ N1+φ

1 + φ+ N1+φ

[yt +

1 + φ

2y2

t

]. (132)

The aggregate demand condition can be expressed as:

eyt = ςceθt+ξst + ςgegt + ςxeγst , (133)

up to second order, we get

yt +12

y2t = ςc(θt + ξst) + γςx st + ςggt +

12[ςc(θt + ξst)

2 + ςgg2t + ςx(γst)

2] .

Let ut ≡ C1−σt

1−σ + χ log Gt− 11+φ N1+φ

t , u ≡ C1−σ

1−σ + χ log G− 11+φ N1+φ, and ut ≡ ut−u

N1+φ . The term ut canbe written as

ut = (ςc + ςx)

[ct +

1− σ

2c2

t

]+ ςggt −

[yt +

1 + φ

2y2

t

]= −1

2[ςc(θt + ξst)

2 + ςx(γst)2 + ςgg2

t + φy2t]− 1

2(σ− 1)(ςc + ςx)c2

t + ςx(θt + (ξ − 1)st − γst),

where I am ignoring terms independent of policy.We can use the external solvency constraint to eliminate the linear terms. First, taking a second-

55

order approximation, we obtain

et = ρet + ψtet + ΓYe2t + ςm(ct − (1− α)st +

12(ct − (1− α)st)

2)− ςxγ

γ− 1((γ− 1)st +

12((γ− 1)st)

2).

Integrating the expression above and rearranging, we obtain

ˆ ∞

0e−ρtςx [(θt + (ξ − 1)st)− γst] dt =

12

ˆ ∞

0e−ρt

[ςxγ(γ− 1)s2

t − ςx(ct − (1− α)st)2 − 2Γ

(et +

ψt

2ΓY

)2]

.

(134)Let U =

´e−ρtutdt be our measure of welfare. This can be written as

U = −ˆ ∞

0

e−ρt

2

[ςc(θt + ξst)

2 + ςxγs2t + ςgg2

t + φy2t + (σ− 1)(ςc + ςx)c2

t + ςx(θt + (ξ − 1)st)2 + 2Γ

(et +

ψt

2ΓY

)2]

,

expanding the quadratic terms and grouping the terms involving θt,

U = −ˆ ∞

0

e−ρt

2[σ(ςc + ςx)c2

t + ςxγs2t + ςx(1− α)s2

t + ςgg2t + φy2

t + 2ΓYe2t]

.

assuming ψt = 0.

Derivation of the loss function under sticky prices

Taking a second-order approximation of the expression for the domestic price index

P1−εH,t =

ˆ 1

0PH,t(j)1−εdj⇒

ˆ 1

0(ph,t(j)− pH,t)dj =

ε− 12

ˆ 1

0(ph,t(j)− pH,t)

2dj. (135)

Approximating the expression for the price dispersion (5) up to second order, we obtain

∆t = −ε

ˆ 1

0(pH,t(j)− pH,t)dj +

ε2

2

ˆ 1

0(pH,t(j)− pH,t)

2dj

=ε

2

ˆ 1

0(pH,t(j)− pH,t)

2dj.

Hence, up to first order, ∆t = 0, so nt = yt − at. Applying an exact law of large numbers:

∆t =ε

2ρδ

ˆ t

−∞e−ρδ(t−s)(ps − pH,s)

2ds⇒ ˙∆t =ε

2$δ(pt − pH,t)

2 − ρδ∆t. (136)

Subtracting ρ∆t from both sides, multiplying by e−ρt, and integrating the expression above, weobtain ˆ ∞

0e−ρt∆tdt =

ε

2κ

ˆ ∞

0e−ρtπ2

H,tdt, (137)

where κ ≡ ρδ(ρ + ρδ) and using the fact that ∆0 = 0 and πH,t = ρδ(pt − pH,t).

56

Taking a second-order approximation of the labor disutility gives

N1+φt

1 + φ=

N1+φ

1 + φe(1+φ)∆t+(1+φ)yt ≈ N1+φ

1 + φ+ N1+φ

[∆t + yt +

1 + φ

2y2

t

]. (138)

Hence, the (per-period) loss function under sticky prices corresponds to that under flexible pricesplus the term ε

κ π2t .

E.2 Optimal policy under sticky prices

The optimal policy problem under sticky prices can be written as

min[ct,st,gt,yt,et,πH,t]

12

ˆ ∞

0e−ρt

[σ(ςc + ςx)c2

t + ςx(γ + 1− α)s2t + ςgg2

t + φy2t + 2Γe2

t +ε

κπ2

H,t

]dt,

subject to

yt = ςcct + ςggt + (ςcα + ςxγ)st,

et = ρet − ςx [(γ + 1− α)st − ct] ,

st = −πH,t,

where s0 = 0 and e0 > 0 are given.Let ut denote the term in brackets in the objective function. We can write the Hamiltonian for this

problem as follows

H = −12

ut − λt[ςcct + ςggt + (ςcα + ςxγ)st − yt

]− µt [ρet − ςx [(γ + 1− α)st − ct]] + γtπH,t.

The first-order conditions for the control variables are

−σct = (1− α)λt + αµt,

gt = −λt,

φyt = λt,ε

κπH,t = γt.

The co-states µt and γt evolve according to

µt = ρµt − µtρ− 2Γet,

γt = ργt − ςx(γ + 1− α)st − (ςcα + ςxγ)λt + µtςx(γ + 1− α).

Rearranging the expression above, we obtain

πH,t = ρπH,t +κ

εςx(γ + 1− α) [µt − λt − st] . (139)

57

We can write the expression above in terms of the labor wedge. The labor wedge is given by

ωLt = −σct − λt − αst

= α(µt − λt − st).

Domestic inflation then evolves according to

πH,t = ρπH,t +κ

ε

ςx

α(γ + 1− α)ωL

t . (140)

E.3 Proof of proposition 5

Proof. Optimal policy. Combining the optimality conditions for consumption, output, and the market-clearing condition gives

λt = α(γ + 1− α)st − (ξ − α)µt

Ω−1 + (1− α)(ξ − α), (141)

which allows us to write yt = −χyµµt + χ

ys st, for χy > 0 and χ

ys > 0.

Consumption is given by

ct = −σ−1αµt + (1− α)(γ + 1− α)Ωst

1 + (1− α)(ξ − α)Ω= −χc

µµt − χcsst. (142)

Net exports are given by

nxt = ςxσ−1αµt + (γ + 1− α) [1 + (ξ − α)Ω] st

1 + (1− α)(ξ − α)Ω

= χnxµ µt + χnx

s st.

The labor wedge is given by

ωLt = α

[1 + (ξ − α)Ω

1 + (1− α)(ξ − α)Ωµt −

1 + (αγ + (1− α)ξ)Ω1 + (1− α)(ξ − α)Ω

st

]= χL

µµt − χLs st. (143)

Combining the above expression with the law of motion of inflation,

πH,t = ρπH,t +κ

εςx(γ + 1− α)

[1 + (ξ − α)Ω

1 + (1− α)(ξ − α)Ωµt −

1 + (αγ + (1− α)ξ)Ω1 + (1− α)(ξ − α)Ω

st

]= ρπH,t + µπ

µ µt − µπs st.

Passive policy. Under the passive policy, the Euler and New Keynesian Phillips Curve equationsare given by

σct = Γet + (1− α)st,

πH,t = ρπH,t − κ [φyt + σct + αst] .

58

Define µpt ≡ (1− α)st − σct and λ

pt = φyt. Net exports can be written as

nxt = ςx

[(γ + 1− ξ)st + σ−1µ

pt

]= χ

µ,pnx µ

pt + χ

s,pnx st,

where χµ,pnx > χ

µnx and χ

s,pnx < χs

nx.The multiplier λt is given by

λt = −(ξ − α)

Ω−1 µpt +

αγ + (1− α)ξ

Ω−1 st. (144)

Inflation evolves according to

πH,t = ρπH,t + κ[µ

pt − λt − st

]= ρπH,t + κ

[(1 + (ξ − α)Ω)µ

pt − [1 + (αγ + (1− α)ξ)Ω] λt

]= ρπH,t + χ

µ,pπ µt − χ

s,pπ µt,

where χµ,pπ > χ

µπ and χ

s,pπ > χs

π for ε > γ + 1.

E.4 Proof of proposition 4

Proof. Optimal policy. The evolution of the system can be characterized by the dynamic systemµt

πH,t

et

st

=

0 0 −2Γ 0

χπµ ρ 0 −χπ

s

−χnxµ 0 ρ −χnx

s

0 −1 0 0

µt

πH,t

et

st

.

The matrix has four eigenvalues,

ν1 =ρ +

√ρ2 + 2χπ

s + 4Γχnxµ + 2∆

2, ν2 =

ρ +√

ρ2 + 2χπs + 4Γχnx

µ − 2∆

2, (145)

and

ν1 =ρ−

√ρ2 + 2χπ

s + 4Γχnxµ + 2∆

2, ν2 =

ρ−√

ρ2 + 2χπs + 4Γχnx

µ − 2∆

2, (146)

where∆ =

√(χπ

s + 2Γχnxµ )2 − 8Γ(χnx

s χπµ + χnx

µ χπs ). (147)

We will assume that ∆ is real-valued, i.e., the following inequality holds

(χπs + 2Γχnx

µ )2 > 8Γ(χnxs χπ

µ + χnxµ χπ

s ), (148)

such that we have two positive eigenvalues and two negative eigenvalues and the dynamic system

59

has a unique solution. Notice that the above condition will be satisfied for sufficiently small Γ.Let zt = [µt, πH,t, et, st]′, then the general solution to the system above is

zt = c1eν1tV1 + c2eν2tV2 + c1eν1tV1 + c2eν2tV2, (149)

where c1, c2, c1, c2 are coefficients to be determined.Insofar as we are interested in bounded solutions, we must have c1 = c2 = 0. The relevant eigen-

vectors are the ones associated with the negative eigenvalues, given by

V1 =[

χπs −∆−2Γχnx

µ

2χπµ

|ν1|χnx

µ χπs +χnx

s χπµ

χπµ (ρ+|ν|1)

− χnxµ |ν1|χπ

µ1]′

, (150)

andV2 =

[χπ

s +∆−2Γχnxµ

2χπµ

|ν2|χnx

µ χπs +χnx

s χπµ

χπµ (ρ+|ν|2)

− χnxµ |ν2|χπ

µ1]′

. (151)

Let Vµj , j = 1, 2, denote the first element of the eigenvector, corresponding to the equation for µ.

Notice that ∆ < χπs for small Γ,32 so Vµ

1 > 0 for Γ small, and then Vµ2 > Vµ

1 > 0. Analogously,let Ve

j denote the third element of the eigenvector, corresponding to the equation for e. Notice that|vj|+ ρ = νj and

|v1|(|v1|+ ρ) =2χπ

s + 4Γχnxµ + 2∆

4→Γ→0 χπ

s ,

|v2|(|v2|+ ρ) =2χπ

s + 4Γχnxµ − 2∆

4→Γ→0 0.

From the above results, we have that

limΓ→0

Ve1 =

χnxs χπ

µ

χπµ (ρ + |ν|1)

, limΓ→0

Ve2 =

χnxµ χπ

s + χnxs χπ

µ

χπµ (ρ + |ν|2)

. (152)

Therefore, Ve2 > Ve

1 > 0 for sufficiently small Γ. The remaining undetermined coefficients can beexpressed as

c1 = −c2, c1 =e0

Ve1 −Ve

2. (153)

State variables. The terms of trade are given by

st = −χs(

e−|ν1|t − e−|ν2|t)

, (154)

where χs ≡ e0Ve

2−Ve1> 0.

The above expression is zero at t = 0, and the derivative with respect to time, which equals theinflation rate with a minus sign, is

st = χs

(|ν1|e−|ν1|t − |ν2|e−|ν2|t

), (155)

32We can write ∆ as follows ∆ =√(χπ

s )2 − 4Γχπs χnx

µ − 8Γχnxs χπ

µ + 4Γ2(χnxµ )2, because the quadratic term in Γ is negligible

for small Γ, so ∆ < χπs .

60

which is positive at zero and negative for sufficiently large t. Hence, the terms of trade is zero at t = 0,it is initially increasing and eventually decreasing.

External debt is given by

et = χs

(Ve

2e−|ν2|t −Ve1e−|ν1|t

)= −χe

1e−|ν1|t + χe2e−|ν2|t, (156)

where χe1 < χe

2.The derivative of the external debt is

et = χs

(−|ν2|Ve

2e−|ν2|t + |ν1|Ve1e−|ν1|t

), (157)

which is eventually negative, since the positive term decays faster than the negative one.Co-state variables. The social cost of foreign goods is given by

µt = χs

(χπ

s + ∆− 2Γχnxµ

2χπµ

e−|ν2|t −χπ

s − ∆− 2Γχnxµ

2χπµ

e−|ν1|t)

= −χµ1 e−|ν1|t + χ

µ2 e−|ν2|t, (158)

where χµ1 < χ

µ2 and χ

µ1 > 0 for sufficiently small Γ. Moreover, µt > 0 for all t, because

µt = e−|ν2|t(

χµ2 − χ

µ1 e−(|ν1|−|ν2|)t

)e−|ν2|t > 0, (159)

and µt is eventually decreasing,

µt = χµ1 |ν1|e−|ν1|t − |ν2|χ

µ2 e−|ν2|t, (160)

because the positive term decays faster than the negative one.The labor wedge can be written as

ωLt ∝ χπ

µ χs

(−

χπs − ∆− 2Γχnx

µ

2χπµ

e−|ν1|t +χπ

s + ∆− 2Γχnxµ

2χπµ

e−|ν2|t)+ χπ

s χs(

e−|ν1|t − e−|ν2|t)

= χs

(χπ

s + 2Γχnxµ + ∆

2e−|ν1|t −

χπs + 2Γχnx

µ − ∆2

e−|ν2|t)

.

The expected discounted value of the labor wedge is

ˆ ∞

0e−ρtωL

t dt =αεχs

κςx(γ + 1− α)

(χπ

s + 2Γχnxµ + ∆

2(ρ + |ν1|)−

χπs + 2Γχnx

µ − ∆2(ρ + |ν2|)

), (161)

which is positive for sufficiently small Γ.

61

Passive policy. The system under the passive policy is given byµt

πH,t

et

st

=

0 0 −Γ 0

χµ,pπ ρ 0 −χ

s,pπ

−χµ,pnx 0 ρ −χ

s,pnx

0 −1 0 0

µt

πH,t

et

st

.

The matrix has four eigenvalues,

νp1 =

ρ +√

ρ2 + 2χs,pπ + 2Γχ

µ,pnx + 2∆

2, ν

p2 =

ρ +√

ρ2 + 2χs,pπ + 2Γχ

µ,pnx − 2∆

2, (162)

and

νp1 =

ρ−√

ρ2 + 2χs,pπ + 2Γχ

µ,pnx + 2∆

2, ν

p2 =

ρ−√

ρ2 + 2χs,pπ + 2Γχ

µ,pnx − 2∆

2, (163)

where∆ =

√(χ

s,pπ + Γχ

µ,pnx )2 − 4Γ(χs,p

nx χµ,pπ + χ

µ,pnx χ

s,pπ ). (164)

Notice that, since χs,pπ > χs

π, then ∆p > ∆, and

χsπ + 2Γχ

µnx =

κ

ε

ςx(γ + 1− α)α

α(1 + (1− α)(ξ − α)Ω)χs

π +2α

1 + (1− α)(ξ − α)ΩΓχ

µ,pnx < χ

s,pπ + Γχ

µ,pnx , (165)

so |νp1 | > |ν1|.

The eigenvectors are given by

Vp1 =

[χπ

s −∆−2Γχnxµ

2χπµ

|ν1|χnx

µ χπs +χnx

s χπµ

χπµ (ρ+|ν|1)

− χnxµ |ν1|χπ

µ1]′

, (166)

andVp

2 =[

χπs +∆−2Γχnx

µ

2χπµ

|ν2|χnx

µ χπs +χnx

s χπµ

χπµ (ρ+|ν|2)

− χnxµ |ν2|χπ

µ1]′

. (167)

E.5 Proof of proposition 6

Proof. VAT dynamics. Differentiating the expression for consumption, we obain:

− σct = (1− α)λt + αµt ⇒ ˙τvt = (1− α)(λt + st − µt)− Γet. (168)

Using the expression for the labor wedge, we obtain

˙τvt = −1− α

α˙ωL

t − Γet. (169)

62

Payroll tax. The New Keynesian Phillips curve is given by

πH,t = ρπH,t − κ[φyt + σct + αst + τv

t + τpt]

. (170)

We can write the expression above as

πH,t = ρπH,t − κ[α(λt + st − µt) + τv

t + τpt]

= ρπH,t − κ[τ

pt + τv

t − ωLt

].

Combining the expression above with the dynamics of πH,t under the optimal policy, we obtain

τpt + τv

t =

[1− ςc + ςx

ε(γ + 1− α)

]ωL

t , (171)

where a sufficient condition for the term in brackets to be positive is ε > γ + 1.

E.6 Government budget constraint

The government’s flow budget constraint can be written as

Dgt = itD

gt −

[τv

t1− τv

tSα−1

t Ct +τl

t + τpt

(1− τvt )(1− τl

t )Sα−1

t Cσt N1+φ

t + τxt αSγ−1

t C∗t − S−1t Gt − Tt

]. (172)

In the steady state, we obtain

Dg

Y=

τv

1− τvt(ςc + ςx) +

τp

1 + τp (1− ε−1)− ςg −TY

. (173)

The above expression can be written as

Dgt = (ρ + ψt + ΓEt)Dg

t −Y

[(eτv

t

1− τv − 1)(ςc + ςx)e−(1−α)st+ct +

(1 + τp)eτpt − 1 + τl

(1− τv)(1− τl)eτv

t −(1−α)st+σct+(1+φ)yt

+τx

1− τx ςxe(γ−1)st − ςge−st+gt − Tt

].

We assume that the revenue from the export tax and the cost of the labor subsidy are rebated backto households in a lump-sum, which is kept at the steady-state level.

Log-linearizing the above expression

dgt = ρdg

t + (ψt + Γet)dg −

[ςc + ςx

1− τv τvt +

τv(ςc + ςx)

1− τv (ct − (1− α)st) + (1− ε−1)τpt

+(1− ε−1)τp

1 + τp (τvt − (1− α)st + σct + (1 + φ)yt)− ςg(−st + gt)

],

63

which can be rearranged to obtain,

dgt = ρdg

t −[(

ςc + ςx

1− τv − (1− ε−1)

)τv

t + (1− ε−1)(τpt + τv

t ) +

(τv(ςc + ςx)

1− τv +τp(1− ε−1)

1 + τp σ

)ct

−[(

τv(ςc + ςx)

1− τv +τp(1− ε−1)

1 + τp

)(1− α)− ςg

]st +

(τp(1− ε−1)

1 + τp (1 + φ) + ςgφ

)yt − (ψt + Γet)d

g]

.

Integrating the expression above, we obtain(ςc + ςx

1− τv − (1− ε−1)

)τv

0ρ

=

ˆ ∞

0e−ρt

[−(

ςc + ςx

1− τv − (1− ε−1)

) ˆ t

0˙τvs ds− (1− ε−1)(τ

pt + τv

t ) + (ψt + Γet)dg

−(

τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp σ

)ct +

[(τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp

)(1− α)− ςg

]st

−(

τp(1− ε−1)

1 + τp (1 + φ) + ςgφ

)yt

]dt.

Hence, provided that the VAT and the payroll tax have different tax bases, i.e. ∆base ≡ ςc+ςx1−τv − (1−

ε−1) 6= 0, the government’s intertemporal budget constraint can be satisfied by adjusting the initialvalue of the VAT.

Optimal policy. We now characterize the value of the initial VAT in the case of the optimal policy.First, notice that we can write the term involving the derivative of the VAT as follows,

ˆ ∞

0

ˆ t

0e−ρt ˙τv

s dsdt =ˆ ∞

0

ˆ ∞

se−ρt ˙τv

s dtds =1ρ

ˆ ∞

0e−ρs ˙τv

s ds, (174)

and1ρ

ˆ ∞

0e−ρt ˙τv

t dt = −1− α

α

(− ωL

0ρ

+

ˆ ∞

0e−ρtωL

t dt)− Γ

ρ

ˆ ∞

0e−ρtetdt. (175)

We can then write the initial VAT as follows,

∆baseτv0

ρ= −∆base

(1− α)ωL0

αρ+

ˆ ∞

0e−ρt

[χd

ωL ωLt + χd

e et + χdc ct + χd

s st + χdyyt

]dt,

where ∆base ≡ ςc+ςx1−τv − (1− ε−1) and

χdωL ≡ ∆base

1− α

α− (1− ε−1)

[1− ςc + ςx

ε(γ + 1− α)

],

χde ≡

∆base

ρΓ + d

gΓ,

χdc ≡ −

(τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp σ

),

χds ≡

(τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp

)(1− α)− ςg,

χdy ≡ −

(τp(1− ε−1)

1 + τp (1 + φ) + ςgφ

).

64

Given τv0 , we can compute the value of τv

t as follows:

τvt = τv

0 +

ˆ t

0τv

s ds

= τv0 −

1− α

α(ωL

t − ωL0 )− Γ

(χe

11− e−ν1t

ν1− χe

21− e−ν2t

ν2

).

Passive policy. We now consider the case of passive fiscal policy, i.e. ˙τvt = τv

t + τpt = 0. The initial

value of VAT satisfies

∆baseτv0

ρ=

ˆ ∞

0e−ρt

[χ

d,pe et + χ

d,pc ct + χ

d,ps st + χ

d,py yt

]dt,

where ∆base ≡ ςc+ςx1−τv − (1− ε−1) and

χd,pe ≡ d

gΓ,

χd,pc ≡ −

(τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp σ

),

χd,ps ≡

(τv(ςc + ςx)

1− τv +(1− ε−1)τp

1 + τp

)(1− α)− ςg,

χd,py ≡ −

(τp(1− ε−1)

1− τv (1 + φ) + ςgφ

).

Debt dynamics. Government debt evolves according to

dgt = ρdg

t + χdc ct + χd

s st + χdyyt + d

gΓet − ∆baseτ

vt − (1− ε−1)(τ

pt + τv

t ).

Integrating from 0 to t and using the fact that debt starts at the steady state, then

dgt =

ˆ t

0eρ(t−z)

[χd

c cz + χds sz + χd

yyz + dgΓez − ∆baseτ

vz − (1− ε−1)(τ

pz + τv

z )]

dz. (176)

Expanding the term involving τv,

ˆ t

0e−ρzτv

z dz =

ˆ t

0e−ρz

[τv

0 +1− α

αωL

0 −1− α

αωL

z −ˆ z

0Γesds

]dz

=

ˆ t

0e−ρz

[τv

0 +1− α

αωL

0 −1− α

αωL

z −Γρ

ez

]dz +

Γρ

e−ρtˆ t

0ezdz.

We can then write the expression for government debt as follows:

dgt =

ˆ t

0eρ(t−z)

[χd

ωL ωLt + χd

e et + χdc cz + χd

s sz + χdyyz

]dz− ∆base(eρt − 1)

ρ

(τv

0 +1− α

αωL

0

)− ∆baseΓ

ρ

ˆ t

0ezdz.

(177)

65

F Derivation and proofs for the extensions.

F.1 The case without export taxes

Consider the Ramsey problem under flexible prices with the additional constraint St = St. The first-order condition for St is now given by

µt = λtSt(1− α)Sα

t Ct + γSγt C∗t

(1− α)Sαt Ct + (γ− 1)Sγ

t C∗t. (178)

Plugging the above expression into the optimality condition for consumption,

C−σt = Sα

t λt(1− α)Sα

t Ct + (γ− (1− α))Sγt C∗t

(1− α)Sαt Ct + (γ− 1)Sγ

t C∗t. (179)

In the zero external debt steady state, we obtain

C−σ = Sαλγ

γ− α, µ = λS

γ + 1− α

γ− α. (180)

Linearizing the two above optimality conditions, we obtain

µt = λt + st +1− α

γ + 1− αst −

1− α

(γ− α)(γ + 1− α)ct, −σct = αst + λt +

(1− α)α

γst −

α(1− α)

γ(γ− α)ct.

(181)Define θt ≡ ct − (ξ − α)st, where ξ = α + 1−α

σ . We can write the above equations as

µt = λt + st +1− α

(γ− α)(γ + 1− α)((γ− ξ)st − θt), −σθt = st + λt +

α(1− α)

γ(γ− α)((γ− ξ)st − θt).

(182)The market-clearing condition for goods can be written as

λt = Ω [(1− α)θt + (αγ + (1− α)ξ)st] , (183)

where Ω ≡ ςc+ςx1+ςg

.Combining the above expressions, we obtain

st = −σ(γ− α)2 + α(γ− ξ) + γ(γ− α)(ξ − α)Ω

γ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)θt, (184)

where the coefficient multiplying θ is negative provided γ > ξ.The multiplier λt is given by

λt = σΩ

(ξ − α)− (αγ + (1− α)ξ)1− α(ξ−α)

γ(γ−α)+ Ω(ξ − α)

1 + Ω(αγ + (1− α)ξ) + α(1−α)γ(γ−α)

(γ− ξ)

θt

= −σαγ(γ− ξ)(γ + 1− α)Ω

γ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)θt,

66

where the coefficient is again negative if γ > ξ.Consumption is given by

ct = θt + (ξ − α)st

=α(γ− α)(γ + 1− α) (1 + γΩ)

γ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)θt.

Finally, we can write θt as a function of µt

(γ− α)(γ + 1− α)µt = (γ− α)(γ + 1− α)λt + [(γ− α)(γ + 1− α) + (1− α)(γ− ξ)]st − (ξ − α)σθt.

Plugging the expressions for λt and st, we rearrange it to obtain

θt = −σ−1 (γ− α)(γ + 1− α) [γ(γ− α) (1 + (αγ + (1− α)ξ)Ω) + α(1− α)(γ− ξ)]

D0 + γ(γ− α) [α(γ− ξ)(γ + 1− α)2 + ((γ− α)(γ + 1− α) + γ)(ξ − α)]Ωµt,

where

D0 ≡ ((γ− α)(γ + 1− α) + (1− α)(γ− ξ))((γ− α)2 + α(γ− ξ)

)+ (ξ − α) (γ(γ− α) + α(1− α)(γ− ξ)) .

The law of motion of external debt is given by

et = ρet − ςx((γ− ξ)st − θt)

= ρet − χtte µt,

where χtte > 0 given the above results.

A derivation similar to the one with the full set of instruments gives

et = eνttte0, µt = −2Γνtt eνttte0. (185)

where νtt is the negative eigenvalue of the system using χtte instead of χe.

F.2 Proof of proposition 7

Proof. We now derive the optimal taxes in this case. From the Euler equation, we obtain

ct = σ−1(ψt + Γet + (1− α)st − τvt )⇒ θt = σ−1(ψt + Γet − τv

t ). (186)

We can write the optimality conditions for consumption and terms of trade as

µt = λt + st +γ

α(γ + 1− α)ωL

t , −σct = αst + λt + ωLt , (187)

67

where

ωLt =

α(1− α)

γ(γ− α)((γ− ξ)st − θt).

Combining the expressions above, we obtain

− σct = −(1− α)st + µt −(γ− α)(1− α)

α(γ + 1− α)ωL

t . (188)

Differentiating the expression above and using the Euler equation, we obtain

ψt + Γet − ˙τvt = − ˙µt +

(γ− α)(1− α)

α(γ + 1− α)˙ωL

t , (189)

which can be rearrange to give

τvt = − (γ− α)(1− α)

α(γ + 1− α)˙ωL

t − Γet. (190)

More explicitly, we obtain

τvt =

[(γ− α)2(1− α) ((1− α)γ + αξ − α + γ(ξ − α)Ω)

D0 + γ(γ− α) [α(γ− ξ)(γ + 1− α)2 + ((γ− α)(γ + 1− α) + γ)(ξ − α)]Ω− 1]

Γet. (191)

One can show that the term in brackets is always negative and is equal to −1 when α = 1. Againthe debt-management motive is strongest at α = 1.

From the expression for the labor wedge, we obtain the level of taxes

τvt + τ

pt =

α(γ− α)(γ + 1− α) [(1− α)γ + αξ − α + γ(ξ − α)Ω]

D0 + γ(γ− α) [α(γ− ξ)(γ + 1− α)2 + ((γ− α)(γ + 1− α) + γ)(ξ − α)]Ωµt. (192)

Notice that the labor wedge is equal to zero for α = 0 and for α = 1− α. In the first case, there is nogain of distorting domestic production to obtain higher foreign revenue, given the negligible level offoreign demand. In the second case, it is optimal to keep a constant mark-up, so taxes do not deviatefrom the steady-state level.

F.3 The economy with downward nominal wage rigidity

Passive Policy

Consider first the equilibrium under the passive policy. We can conjecture that the economy starts withpositive unemployment given a positive level of external debt. Equilibrium is then determined by

yt = ςcθt + (ςcξ + ςxγ)st,

−st = (ϕ− 1)yt,

68

solving the system

yt =ςcθt

1 + (ςcξ + ςxγ)(ϕ− 1); st = −

ςc(ϕ− 1)θt

1 + (ςcξ + ςxγ)(ϕ− 1). (193)

Unemployment is given by

ut = −[

σ

φ+

ϕςc

1 + (ςcξ + ςxγ)(ϕ− 1)

]θt. (194)

Net exports is given bynxt = ςx [(γ− ξ)st − θt] . (195)

External debt evolves according to

et = ρet − ςx [(γ− ξ)st − θt] . (196)

In the special case ϕ = 1, then st = 0 if unemployment is positive. I will guess and verify thatunemployment is always positive. Under this assumption, et and θt satisfy the system[

et

θt

]=

[ρ ςx

σ−1Γ 0

] [et

θt

],

using the Euler equation for consumption.The eigenvalues of the system above are

ν =ρ +

√ρ2 + 4ςxσ−1Γ

2, ν =

ρ−√

ρ2 + 4ςxσ−1Γ2

, (197)

and the eigenvectors are

V =

[σνΓ

1

], V =

[σνΓ

1

]. (198)

The unique bounded solution to the system above is

et = eνte0, θt =Γe0

σνeνt. (199)

As θt < 0 for all t ≥ 0, unemployment is always positive. Hence, on impact output and consump-tion will decrease, while unemployment will increase. All variables revert back to the steady state atrate |ν|.

Optimal policy

Consider now the optimal policy. Given enough instruments, the downward nominal wage rigiditydoes not impose any additional restriction on the problem under flexible prices. In the case where

69

γ = ξ, the value of θt is the same as above and θt = −σ−1µt. Wages and labor supply satisfy

wt = −st − τpt , φyt = −st − σθt − τv

t − τlt − τ

pt . (200)

Imposing wt = 0, we obtain

τpt = −χsµt, τv

t + τlt + τ

pt =

[1− σ−1χs

θ − χλθ

]σχθ

µµt, (201)

where χsθ and χλ

θ are obtained from the solution to the problem with flexible prices and no export tax

χsθ = σ

(γ− α)2 + α(γ− ξ) + γ(γ− α)(ξ − α)Ωγ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)

,

χλθ = σ

αγ(γ− ξ)(γ + 1− α)Ωγ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)

.

Notice that 1− σ−1χs − σ−1χλ is given by

1− σ−1χs − σ−1χλ = α(1− α)(γ− α) + α(ξ − α) + γ(ξ − α)Ω

γ(γ− α) [1 + (αγ + (1− α)ξ)Ω] + α(1− α)(γ− ξ)> 0. (202)

It is therefore optimal to raise and front-load taxes.

70