· Online Appendix to Fiscal Consolidation in an Open Economy with Sovereign Premia and without...

Online Appendix to Fiscal Consolidation in anOpen Economy with Sovereign Premia andwithout Monetary Policy Independence∗

Apostolis Philippopoulos,a,b Petros Varthalitis,c andVanghelis Vassilatosa

aAthens University of Economics and BusinessbCESifo

cEconomic and Social Research Institute, Trinity College Dublin

Appendix 1. Households

This appendix presents and solves the problem of the household insome detail. There are i = 1, 2, . . . , N identical domestic householdsthat act competitively.

Household’s Problem

Each household i maximizes expected lifetime utility given by

E0

∞∑t=0

βtU (ci,t, ni,t, gt) , (1)

where ci,t is i’s consumption bundle, ni,t is i’s hours of work, gt is percapita public spending, 0 < β < 1 is the time preference rate, and E0is the rational expectations operator conditional on the informationset.

The consumption bundle, ci,t, is defined as

ci,t =

(cHi,t

)ν (cFi,t

)1−ν

νν(1 − ν)1−ν, (2)

∗Corresponding author: Apostolis Philippopoulos, Department of Econom-ics, Athens University of Economics and Business, Athens 10434, Greece. Tel:+30-210-8203357. E-mail: [email protected].

1

2 International Journal of Central Banking December 2017

where cHi,t and cF

i,t denote the composite domestic good and the com-posite foreign good, respectively, and ν is the degree of preferencefor the domestic good.

We define the two composites, cHi,t and cF

i,t, as

cHi,t =

[N∑

h=1

[cHi,t(h)]

φ−1φ

] φφ−1

(3)

cFi,t =

⎡⎣ N∑f=1

[cFi,t(f)]

φ−1φ

⎤⎦φ

φ−1

, (4)

where φ > 0 is the elasticity of substitution across goods producedin the domestic country.

The period budget constraint of each i expressed in real terms is(see, e.g., Benigno and Thoenissen 2008, for similar modeling)

(1 + τ ct )[PH

t

PtcHi,t +

PFt

PtcFi,t

]+

PHt

Ptxi,t + bi,t +

StP∗t

Ptfh

i,t

+φh

2

(StP

∗t

Ptfh

i,t − SP ∗

Pfh

)2

=(1 − τk

t

) [rkt

PHt

Ptki,t−1 + ωi,t

]+ (1 − τn

t ) wtni,t

+ Rt−1Pt−1

Ptbi,t−1 + Qt−1

StP∗t

Pt

P ∗t−1

P ∗t

fhi,t−1 − τ l

i,t, (5)

where Pt is the consumer price index (CPI), PHt is the price index of

home tradables, PFt is the price index of foreign tradables (expressed

in domestic currency), xi,t is i’s domestic investment, bi,t is i’s end-of-period real domestic government bonds, fh

i,t is i’s end-of-periodreal internationally traded assets denominated in foreign currency,rkt is the real return to inherited domestic capital, ki,t−1, ωi,t is

i’s real dividends received by domestic firms, wt is the real wagerate, Rt−1 ≥ 1 is the gross nominal return to domestic governmentbonds between t − 1 and t, Qt−1 ≥ 1 is the gross nominal returnto international assets between t − 1 and t, τ l

i,t are real lump-sumtaxes if positive (or transfers if negative) to each household, and

Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 3

τ ct , τk

t , τnt are tax rates on consumption, capital income, and labor

income, respectively. The parameter φh ≥ 0 captures adjustmentcosts related to private foreign assets, where variables without timesubscripts denote steady-state values.

Each household i also faces the budget constraints

Ptci,t = PHt cH

i,t + PFt cF

i,t (6)

PHt cH

i,t =N∑

h=1

PHt (h)cH

i,t(h) (7)

PFt cF

i,t =N∑

f=1

PFt (f)cF

i,t(f), (8)

where PHt (h) is the price of each variety h produced at home and

PFt (f) is the price of each variety f produced abroad (expressed in

domestic currency).Finally, the law of motion of physical capital for household i is

ki,t = (1 − δ)ki,t−1 + xi,t − ξ

2

(ki,t

ki,t−1− 1)2

ki,t−1, (9)

where 0 < δ < 1 is the depreciation rate of capital and ξ ≥ 0 is aparameter capturing adjustment costs related to physical capital.

For our numerical solutions, we use the usual functional form:

ui,t (ci,t, ni,t, mi,t, gt) =c1−σi,t

1 − σ− χn

n1+ηi,t

1 + η+ χg

g1−ζt

1 − ζ, (10)

where χn, χg, σ, η, ζ are preference parameters.

Household’s Optimality Conditions

Each household i acts competitively, taking prices and policy asgiven. Following the literature, to solve the household’s problem, wefollow a two-step procedure. We first suppose that the householddetermines its desired consumption of composite goods, cH

i,t and cFi,t,

and, in turn, chooses how to distribute its purchases of individualvarieties, cH

i,t(h) and cFi,t(f).


The first-order conditions of each i include the budget constraintswritten above and also

∂ui,t

∂ci,t

∂ci,t

∂cHi,t

Pt

PHt (1 + τ c

t )

= βEt∂ui,t+1

∂ci,t+1

∂ci,t+1

∂cHi,t+1

Pt+1

PHt+1

(1 + τ c

t+1

)RtPt

Pt+1(11)

∂ui,t

∂ci,t

∂ci,t

∂cHi,t

Pt

PHt (1 + τ c

t )StP

∗t

Pt

[1 + φh

(StP

∗t

Ptfh

t − SP ∗

Pfh

)]= βEt

∂ui,t+1

∂ci,t+1

∂ci,t+1

∂cHt+1

Pt+1

PHt+1

(1 + τ c

t+1

)QtSt+1P

∗t+1

Pt+1

P ∗t

P ∗t+1

(12)

∂ui,t

∂ci,t

∂ci,t

∂cHi,t

1(1 + τ c

t )

{1 + ξ

(ki,t

ki,t−1− 1)}

= βEt∂ui,t+1

∂ci,t+1

∂ci,t+1

∂cHi,t+1

1(1 + τ c

t+1

)×

⎧⎨⎩ (1 − δ) − ξ2

(ki,t+1ki,t

− 1)2

+

ξ(

ki,t+1ki,t

− 1)

ki,t+1ki,t

+(1 − τk

t+1)rkt+1

⎫⎬⎭ (13)

−∂ui,t

∂ni,t=

∂ui,t

∂ci,t

∂ci,t

∂cHi,t

Pt

PHt

(1 − τnt ) wt

(1 + τ ct )

(14)

cHi,t

cFi,t

=ν

1 − ν

PFt

PHt

(15)

cHi,t(h) =

[PH

t (h)PH

t

]−φ

cHi,t (16)

cFi,t(f) =

[PF

t (f)PF

t

]−φ

cFi,t. (17)

Equations (11)–(13) are, respectively, the Euler equations fordomestic bonds, foreign assets, and domestic capital, and (14) isthe optimality condition for work hours. Finally, (15) shows theoptimal allocation between domestic and foreign goods, while (16)and (17) show the optimal demand for each variety of domestic and


foreign goods, respectively; these conditions are used in the firm’soptimization problem below.

Implications for Price Bundles

Equations (15), (16), and (17), combined with the household’s bud-get constraints, imply that the three price indexes are

Pt = (PHt )ν(PF

t )1−ν (18)

PHt =

[N∑

h=1

[PHt (h)]1−φ

] 11−φ

(19)

PFt =

⎡⎣ N∑f=1

[PFt (f)]1−φ

⎤⎦1

1−φ

. (20)

Appendix 2. Firms

This appendix presents and solves the problem of the firm in somedetail. There are h = 1, 2, . . . , N domestic firms. Each firm h pro-duces a differentiated good of variety h under monopolistic compe-tition in its own product market facing Calvo-type nominal pricefixities.

Demand for the Firm’s Product

Demand for firm h’s product, yHt (h), comes from domestic house-

holds’ consumption and investment, CHt (h) and Xt(h), where

CHt (h) ≡

∑Ni=1 cH

i,t(h) and Xt(h) ≡∑N

i=1 xi,t(h), from the domes-tic government, Gt (h), and from foreign households’ consumption,CF∗

t (h) ≡∑N∗

i=1 cF∗i,t (h). Thus, the demand for each domestic firm’s

product is

yHt (h) = CH

t (h) + Xt(h) + Gt (h) + CF∗t (h). (21)

Using demand functions like those derived by solving the house-hold’s problem above, we have

cHi,t(h) =

[PH

t (h)PH

t

]−φ

cHi,t (22)


xi,t (h) =[PH

t (h)PH

t

]−φ

xi,t (23)

Gt (h) =[PH

t (h)PH

t

]−φ

Gt (24)

cF∗i,t (h) =

[PF∗

t (h)PF∗

t

]−φ

cF∗i,t , (25)

where, using the law of one price, we have in (25)

PF∗t (h)PF∗

t

=P H

t (h)St

P Ht

St

=PH

t (h)PH

t

. (26)

Since, at the economy level, aggregate demand is

Y Ht = CH

t + Xt + Gt + CF∗t , (27)

the above equations imply that the demand for each firm’s product is

yHt (h) =

[PH

t (h)PH

t

]−φ

Y Ht . (28)

The Firm’s Problem

The real profit of each firm h is (see, e.g., Benigno and Thoenissen2008 for similar modeling)

ωt(h) =PH

t (h)Pt

yHt (h) − PH

t

Ptrkt kt−1(h) − wtnt(h). (29)

The production technology is

yHt (h) = At[kt−1(h)]α[nt(h)]1−α, (30)

where At is an exogenous stochastic total factor productivity processwhose motion is defined below.

Profit maximization is also subject to the demand for its productas derived above:

yHt (h) =

[PH

t (h)PH

t

]−φ

Y Ht . (31)


In addition, firms choose their prices facing a Calvo-type nomi-nal fixity. In each period, firm h faces an exogenous probability θ ofnot being able to reset its price. A firm h, which is able to reset itsprice, chooses its price P#

t (h) to maximize the sum of discountedexpected nominal profits for the next k periods in which it may haveto keep its price fixed. This is modeled next.

Firm’s Optimality Conditions

To solve the firm’s problem, we follow a two-step procedure. We firstsolve a cost-minimization problem, where each firm h minimizes itscost by choosing factor inputs given technology and prices. The solu-tion will give a minimum nominal cost function, which is a functionof factor prices and output produced by the firm. In turn, given thiscost function, each firm, which is able to reset its price, solves amaximization problem by choosing its price.

The solution to the cost-minimization problem gives the inputdemand functions:

wt = mct(h)(1 − a)yt (h)nt (h)

(32)

PHt

Ptrkt = mct(h)a

yt (h)kt−1 (h)

, (33)

where mct(h) = Ψ′t(h)Pt

denotes the real marginal cost.Then, the firm chooses its price to maximize nominal profits

written as (see also, e.g., Rabanal 2009 for similar modeling)

Et

∞∑k=0

θkΞt,t+k

{P#

t (h) yHt+k (h) − Ψt+k

(yH

t+k (h))}

,

where Ξt,t+k is a discount factor taken as given by the firm,

yHt+k (h) =

[P#

t (h)P H

t+k

]−φ

Y Ht+k and Ψt(h) denotes the minimum nom-

inal cost function for producing yHt (h) at t so that Ψ′

t(h) is theassociated marginal cost.


The first-order condition gives

Et

∞∑k=0

θkΞt,t+k

[P#

t (h)PH

t+k

]−φ

Y Ht+k

{P#

t (h) − φ

φ − 1Ψ′

t+k(h)}

= 0.

(34)

Dividing by PHt , we have

Et

∞∑k=0

θk

⎡⎣Ξt,t+k

[P#

t (h)PH

t+k

]−φ

Y Ht+k

{P#

t (h)PH

t

− φ

φ − 1mct+k(h)

Pt+k

PHt

}⎤⎦= 0. (35)

Summing up, the behavior of each firm h is summarized by(32), (33), and (35). A recursive expression of the equation above ispresented below.

Notice, finally, that each firm h, which can reset its price in periodt, solves an identical problem, so P#

t (h) = P#t is independent of h,

and each firm h, which cannot reset its price, just sets its previousperiod price PH

t (h) = PHt−1 (h) . Thus, the evolution of the aggregate

price level is given by

(PH

t

)1−φ= θ

(PH

t−1)1−φ

+ (1 − θ)(P#

t

)1−φ

. (36)

Appendix 3. Government Budget Constraint

This appendix presents in some detail the government budget con-straint and the menu of fiscal policy instruments. Recall that wehave assumed a cashless economy for simplicity.

Then, the period budget constraint of the government written inaggregate and nominal quantities is

Bt + StFgt = Pt

φg

2

(StF

gt

Pt− SF g

P

)2

+ Rt−1Bt−1 + Qt−1StFgt−1

+ PHt Gt − τ c

t (PHt CH

t + PFt CF

t )

− τkt (rk

t PHt Kt−1 + PtΩt) − τn

t WtNt − T lt , (37)


where Bt is the end-of-period nominal domestic public debt and F gt

is the end-of-period nominal foreign public debt expressed in foreigncurrency so it is multiplied by the exchange rate, St. The parameterφg ≥ 0 captures adjustment costs related to public foreign debt. Therest of the notation is as above. Note that Bt =

∑Ni=1 Bi,t, CH

t =∑Ni=1 cH

i,t, CFt =

∑Ni=1 cF

i,t, Kt−1 =∑N

i=1 ki,t−1, Ωt ≡∑N

i=1 ωi,t,Nt ≡

∑Ni=1 ni,t, and T l

t denotes the nomimal value of lump-sumtaxes.

Let Dt = Bt + StFgt denote the total nominal public debt at the

end of the period. This can be held both by domestic private agents,λtDt, where in equilibrium Bt ≡ λtDt, and by foreign private agents,StF

gt ≡ (1 − λt) Dt, where 0 ≤ λt ≤ 1. Then, dividing by the price

level and the number of households, the budget constraint in realand per capita terms is

dt =φg

2[(1 − λt) dt − (1 − λ) d]2 + Rt−1

Pt−1

Ptλt−1dt−1

+ Qt−1StP

∗t

Pt

P ∗t−1

P ∗t

Pt−1

P ∗t−1St−1

(1 − λt−1) dt−1

+PH

t

Ptgt − τ c

t

(PH

t

PtcHt +

PFt

PtcFt

)− τk

t

(rkt

PHt

Ptkt−1 + ωt

)− τn

t wtnt − τ lt , (38)

where, as said in the main text, in each period, one of the policyinstruments (τ c

t , τkt , τn

t , gt, τ lt , λt, dt) follows residually to satisfy

the government budget constraint.

Appendix 4. Decentralized Equilibrium (DE)

This appendix presents in some detail the DE system. Following therelated literature, we work in steps.

Equilibrium Equations

The DE is summarized by the following equations (quantities are inper capita terms):


∂ut

∂ct

∂ct

∂cHt

1(1 + τ c

t )Pt

PHt

= βEt∂ut+1

∂ct+1

∂ct+1

∂cHt+1

1(1 + τ c

t+1

) Pt+1

PHt+1

RtPt

Pt+1

(39)

∂ut

∂ct

∂ct

∂cHt

1(1 + τ c

t )Pt

PHt

StP∗t

Pt

(1 + φp

(StP

∗t

Ptfh

t − SP ∗

Pfh

))= β

∂ut+1

∂ct+1

∂ct+1

∂cHt+1

1(1 + τ c

t+1

) Pt+1

PHt+1

QtSt+1P

∗t+1

Pt+1

P ∗t

P ∗t+1

(40)

∂ut

∂ct

∂ct

∂cHt

1(1 + τ c

t )

{1 + ξ

(kt

kt−1− 1)}

= βEt∂ut+1

∂ct+1

∂ct+1

∂cHt+1

1(1 + τ c

t+1

)×

⎧⎨⎩ (1 − δ) − ξ2

(kt+1kt

− 1)2

+ξ(

kt+1kt

− 1)

kt+1kt

+(1 − τk

t+1)rkt+1

⎫⎬⎭ (41)

−∂ut

∂nt=

∂ut

∂ct

∂ct

∂cHt

Pt

PHt

(1 − τnt ) wt

(1 + τ ct )

(42)

cHt

cFt

=ν

1 − ν

PFt

PHt

(43)

kt = (1 − δ)kt−1 + xt − ξ

2

(kt

kt−1− 1)2

kt−1 (44)

ct ≡(cHt

)ν (cFt

)1−ν

νν (1 − ν)1−ν (45)

wt = mct(1 − a)Atkat−1n

−at (46)

PHt

Ptrkt = mctaAtk

a−1t−1 n1−a

t (47)

ωt =PH

t

PtyH

t − PHt

Ptrkt kt−1 − wtnt (48)

Et

∞∑k=0

θk

⎡⎣Ξt,t+k

[P#

t

PHt+k

]−φ

yHt+k

{P#

t

Pt− φ

φ − 1mct+k

Pt+k

Pt

}⎤⎦ = 0

(49)


yHt =

1[˜P H

t

P Ht

]−φAtk

at−1n

1−at (50)

dt = Rt−1Pt−1

Ptλt−1dt−1 + Qt−1

StP∗t

Pt

P ∗t−1

P ∗t

Pt−1

P ∗t−1St−1

(1 − λt−1) dt−1

+PH

t

Ptgt − τ c

t

(PH

t

PtcHt +

PFt

PtcFt

)− τk

t

(rkt

PHt

Ptkt−1 + ωt

)− τn

t wtnt − τ lt +

φg

2[(1 − λt) dt − (1 − λ) d]2 (51)

yHt = cH

t + xt + gt + cF∗t (52)

− PHt

PtcF∗t +

PFt

PtcFt +

φg

2[(1 − λt) dt − (1 − λ) d]2

+φp

2

(StP

∗t

Ptfh

t − SP ∗

Pfh

)2

Qt−1StP

∗t

Pt

P ∗t−1

P ∗t

×

⎛⎝(1 − λt−1) dt−1St−1P ∗

t−1Pt−1

− fht−1

⎞⎠ = (1 − λt) dt − StP∗t

Ptfh

t (53)

(PH

t

)1−φ= θP 1−φ

t−1 + (1 − θ)(P#

t

)1−φ

(54)

Pt =(PH

t

)ν (PF

t

)1−ν(55)

PFt = StP

H∗t (56)

P ∗t = (PH∗

t )ν∗(

PHt

St

)1−ν∗

(57)(PH

t

)−φ

= θ(PH

t−1

)−φ

+ (1 − θ)(P#

t

)−φ

(58)

Qt = Q∗t + ψ

⎛⎝e

dtP H

tPt

Y Ht

−d

− 1

⎞⎠ (59)

lt ≡Rtλtdt + Qt

St+1St

(1 − λt)dt

P Ht

PtyH

t

, (60)


where f∗gt ≡ (1−λt)dt

StP ∗t

Pt

, Ξt,t+k ≡ βk c−σt+k

c−σt

Pt

Pt+k

τct

τct+k

, yHt =[∑N

h=1[yHt (h)]

φ−1φ

] φφ−1

, and PHt ≡

(∑Nh=1 [Pt (h)]−φ

)− 1φ

. Thus,(˜P H

t

P Ht

)−φ

is a measure of price dispersion.

We thus have twenty-two equations in twenty-two variables, {yHt ,

ct, cHt , cF

t , nt, xt, kt, fht , PF

t , Pt, PHt , P#

t , PHt , wt, mct, ωt, rk

t , Qt,dt, P ∗

t , Rt, lt}∞t=0. This is given the independently set monetary and

fiscal policy instruments, {St, τ ct , τk

t , τnt , gt, τ l

t , λt}∞t=0; the rest-

of-the-world variables, {cF∗t , Q∗

t , PH∗t }∞

t=0; technology, {At}∞t=0; and

initial conditions for the state variables.In what follows, we shall transform the above equilibrium con-

ditions. In particular, following the related literature, we rewritethem—first, by expressing price levels in inflation rates, secondly,by writing the firm’s optimality conditions in recursive form, and,thirdly, by introducing a new equation that helps us to computeexpected discounted lifetime utility. Finally, we will present the finaltransformed system that is solved numerically.

Variables Expressed in Ratios

We first express prices in rate form. We define seven new variables,which are the gross domestic CPI inflation rate, Πt ≡ Pt

Pt−1; the gross

foreign CPI inflation rate, Π∗t ≡ P ∗

t

P ∗t−1

; the gross domestic goods

inflation rate, ΠHt ≡ P H

t

P Ht−1

; the auxiliary variable, Θt ≡ P#t

P Ht

; the

price dispersion index, Δt ≡[

˜P Ht

P Ht

]−φ

; the gross rate of exchange

rate depreciation, εt ≡ St

St−1; and the terms of trade, TTt ≡ P F

t

P Ht

=StP

∗Ht

P Ht

.1 In what follows, we use Πt, Π∗t , Π

Ht , Θt, Δt, εt, TTt instead

of Pt, P∗t , PH

t , P#t , Pt, St, PF

t , respectively.Also, for convenience and comparison with the data, we express

fiscal and public finance variables as shares of nominal output,

1Thus, TTtTTt−1

=St

St−1

P ∗Ht

P ∗Ht−1

P Ht

P Ht−1

= εtΠ∗Ht

ΠHt

.


PHt yH

t . In particular, using the definitions above, real governmentspending, gt, can be written as gt = sg

t yHt , while real government

transfers, τ lt , can be written as τ l

t = slty

Ht TT ν−1

t , where sgt and sl

t

denote, respectively, the output shares of government spending andgovernment transfers.

Equation (49) Expressed in Recursive Form

We now replace equation (49), from the firm’s optimization prob-lem, with an equivalent equation in recursive form. In particular,following Schmitt-Grohe and Uribe (2007), we look for a recursiverepresentation of the form

Et

∞∑k=0

θkΞt,t+k

[P#

t

PHt+k

]−φ

yHt+k

{P#

t − φ

(φ − 1)mct+kPt+k

}= 0.

(61)

We define two auxiliary endogenous variables:

z1t ≡ Et

∞∑k=0

θkΞt,t+k

[P#

t

PHt+k

]−φ

yHt+k

P#t

Pt(62)

z2t ≡ Et

∞∑k=0

θkΞt,t+k

[P#

t

PHt+k

]−φ

yHt+kmct+k

Pt+k

Pt. (63)

Using these two auxiliary variables, z1t and z2

t , as well as equation(61), we come up with two new equations that enter the dynamicsystem and allow a recursive representation of (61).

Thus, in what follows, we replace equation (49) with

z1t =

φ

(φ − 1)z2t , (64)

where we also have the two new equations (and the two new endoge-nous variables, z1

t and z2t )

z1t = Θ1−φ

t ytTT ν−1t + βθEt

c−σt+1

c−σt

1 + τ ct

1 + τ ct+1

(Θt

Θt+1

)1−φ( 1ΠH

t+1

)1−φ

z1t+1

(65)


z2t = Θ−φ

t ytmct + βθEt

c−σt+1

c−σt

1 + τ ct

1 + τ ct+1

(Θt

Θt+1

)−φ( 1ΠH

t+1

)−φ

z2t+1.

(66)

Lifetime Utility Written as a First-Order Difference Equation

Since we want to compute social welfare, we follow Schmitt-Groheand Uribe (2007) by defining a new endogenous variable, Vt, whosemotion is given by

Vt =c1−σt

1 − σ− χn

n1+ηt

1 + η+ χg

(sg

t yHt

)1−ζ

1 − ζ+ βEtVt+1, (67)

where Vt is household’s expected discounted lifetime utility at time t.Thus, in what follows, we add equation (67) and the new vari-

able Vt to the equilibrium system. Note that, in the welfare numericalsolutions reported, we add a constant number, 2, to each period’sutility; this makes the welfare numbers easier to read.

Equilibrium Equations Transformed

Using the above, the transformed DE system is summarized by

Vt =c1−σt

1 − σ− χn

n1+ηt

1 + η+ χg

(sg

t yHt

)1−ζ

1 − ζ+ βEtVt+1 (68)

βEt c−σt+1

1(1 + τ c

t+1

)Rt1

Πt+1= c−σ

t

1(1 + τ c

t )(69)

βEtc−σt+1

1(1 + τ c

t+1

)QtTT v∗+ν−1t+1

1Π∗

t+1

= c−σt

1(1 + τ c

t )TT v∗+ν−1

t

[1 + φp

(TT ν∗+ν−1

t fht − TT ν∗+ν−1fh

)](70)

βc−σt+1TT ν−1

t+11(

1 + τ ct+1

) {1 − δ − ξ

2

(kt+1

kt− 1)2

+ ξ

(kt+1

kt− 1)

kt+1

kt+(1 − τk

t+1)rkt+1

}= c−σ

t TT ν−1t

1(1 + τ c

t )

[1 + ξ

(kt

kt−1− 1)]

(71)


χnnηt = c−σ

t

(1 − τnt ) wt

(1 + τ ct )

(72)

cHt

cFt

=ν

1 − νTTt (73)

kt = (1 − δ)kt−1 + xt − ξ

2

(kt

kt−1− 1)2

kt−1 (74)

ct ≡(cHt

)ν (cFt

)1−ν

(ν)ν (1 − ν)1−ν (75)

wt = mct(1 − a)Atkat−1n

−at (76)

1TT 1−v

t

rkt = mctaAtk

a−1t−1 n1−a

t (77)

ωt =1

TT 1−vt

yHt − 1

TT 1−vt

rkt kt−1 − wtnt (78)

z1t =

φ

(φ − 1)z2t (79)

yHt =

1Δt

Atkat−1n

1−at (80)

dt =φg

2[(1 − λt)dt − (1 − λ)d]2 + Rt−1

1Πt

λt−1dt−1

+ Qt−1TT v+v∗−1t

1Π∗

t

1TT v+v∗−1

t−1(1 − λt−1)dt−1 + TT ν−1

t sgt y

Ht

− τ ct

(1

TT 1−vt

cHt + TT v

t cFt

)− τk

t

(rkt

1TT 1−v

t

kt−1 + ωt

)− τn

t wtnt − TT ν−1t sl

tyHt (81)

yHt = cH

t + xt + sgt y

Ht + cF∗

t (82)

(1 − λt)dt − TT ν∗+ν−1t fh

t = −TT ν−1t cF∗

t + TT νt cF

t

+ Qt−1TT ν∗+ν−1t

1Π∗

t

(1

TT v+v∗−1t−1

(1 − λt−1)dt−1 − fht−1

)


+φp

2

(TT ν∗+ν−1

t fht − TT ν∗+ν−1fh

)2

+φg

2((1 − λt)dt − (1 − λ)d)2 (83)(

ΠHt

)1−φ= θ + (1 − θ)

(ΘtΠH

t

)1−φ(84)

Πt

ΠHt

=(

TTt

TTt−1

)1−ν

(85)

TTt

TTt−1=

εtΠH∗t

ΠHt

(86)

Π∗t

ΠH∗t

=(

TTt−1

TTt

)1−ν∗

(87)

Δt = θΔt−1(ΠH

t

)φ+ (1 − θ) (Θt)

−φ (88)

Qt = Q∗t + ψ

(e

(dt

T Tν−1t yH

t

−d)

− 1

)(89)

z1t = Θ1−φ

t ytTT ν−1t + βθEt

c−σt+1

c−σt

1 + τ ct

1 + τ ct+1

(Θt

Θt+1

)1−φ( 1ΠH

t+1

)1−φ

z1t+1

(90)

z2t = Θ−φ

t ytmct + βθEt

c−σt+1

c−σt

1 + τ ct

1 + τ ct+1

(Θt

Θt+1

)−φ( 1ΠH

t+1

)−φ

z2t+1

(91)

lt =Rtstdt + Qtεt+1 (1 − λt) dt

TT ν−1t yH

t

. (92)

We thus have twenty-five equations in twenty-five variables,{Vt, y

Ht , ct, cH

t , cFt , nt, xt, kt, f

ht , TTt, Πt, ΠH

t , Θt, Δt, wt, mct, ωt, rkt ,

Qt, dt, Π∗t , z

1t , z2

t , Rt, lt}∞t=0. This is given the independently set pol-

icy instruments, {εt, τct , τk

t , τnt , sg

t , slt, λt}∞

t=0; the rest-of-the-worldvariables, {cF∗

t , Q∗t , Π

H∗t }∞

t=0; technology, {At}∞t=0; and initial condi-

tions for the state variables.

Equations and Unknowns (Given Feedback Policy Coefficients)

We can now define the final equilibrium system. It consists of thetwenty-five equations of the transformed DE presented in the pre-vious subsection, the four policy rules in subsection 2.7 in the main


text, and the equation for domestic exports in subsection 2.8 in themain text. Thus, we have a system of thirty equations. By usingtwo auxilliary variables, we transform it to a first order,2 so weend up with thirty-two equations in thirty-two variables, {yH

t , ct,cHt , cF

t , xt, nt, fht , dt, kt, ωt, mct, Πt, ΠH

t , Π∗t , Θt, Δt, TTt, wt,

rkt , Qt, lt, z1

t , z2t , Vt, Rt; τ c

t , sgt , τk

t , τnt ; kleadt, TT lagt, cF∗

t }∞t=0.

This is given the exogenous variables, {Q∗t , Π

H∗t , At, εt, sl

t, λt}∞t=0,

and initial conditions for the state variables. The thirty-two endoge-nous variables are distinguished in twenty-four control variables,{yH

t , ct, cHt , cF

t , xt, nt, ωt, mct, Πt, ΠHt , Π∗

t , Θt, TTt, wt, rkt , z1

t , z2t , Vt,

kleadt, cF∗t , τ c

t , sgt , τk

t , τnt }∞

t=0, and eight state variables,{fh

t−1, dt−1, kt−1, Δt−1, Qt−1, Rt−1, lt−1, TT lagt}∞t=0. All this is given

the values of feedback policy coefficients in the policy rules, whichare chosen optimally in the computational part of the paper.

Status Quo Steady State

In the steady state of the above model economy, if εt = 1, our equa-tions imply ΠH

t = ΠH∗t = Π∗ = Π. If we also set ΠH∗ = 1 (this is

the exogenous component of foreign inflation), then ΠHt = ΠH∗

t =Π∗ = Π = 1; this in turn implies Δ = Θ = 1. Also, in order to solvethe model numerically, we need a value for the exogenous exports,cF∗; we assume cF∗ = 1.01cF , as it is the case in the data. Finally,since from the Euler for bonds, 1 = βR

Π , we residually get R = 1β .

(Note that these conditions will also hold in all steady-state solutionsthroughout the paper.)

The steady-state system is (in the labor supply condition, we useνccH TT 1−v = 1)

1 = β[1 − δ +(1 − τk

)rk] (93)

1 = βQ (94)

χnnη = c−σ νc

cHTT 1−v (1 − τn)

(1 + τ c)w (95)

2In particular, when we use the Schmitt-Grohe and Uribe (2004) MATLABroutines, we add two auxiliary endogenous variables, klead and TT lag, to reducethe dynamic system into a first-order one.


cH

cF=

ν

1 − νTT (96)

x = δk (97)

c =

(cH)ν (

cF)1−ν

(ν)ν (1 − ν)1−ν (98)

w = mc(1 − a)Akan−a (99)

rk = TT 1−vmcaAka−1n1−a (100)

ω =1

TT 1−v yH − 1TT 1−v rkk − wn (101)

yH = Akan1−a (102)

d = Qd + TT ν−1sgyH − τ c

(1

TT 1−v cH + TT vcF

)− τk

(rk 1

TT 1−v k + ω

)− τnwn − TT ν−1slyH (103)

yH = cH + x + sgyH + cF∗ (104)

(1 − λ)d − TT ν∗+ν−1fh = −TT ν−1cF∗ + TT νcF

+ QTT ν∗+ν−1(

1TT v+v∗−1 (1 − λ)d − fh

)(105)

Q = Q∗ + ψ(e( d

T T ν−1yH −d) − 1)

(106)

z1 =φ

(φ − 1)z2 (107)

z1 = TT ν−1yH + βθz1 (108)

z2 = yHmc + βθz2. (109)

We thus have seventeen equations in seventeen variables, c, cH ,cF , k, x, n, yH

t , TT , mc, ω, d, z1, z2, Q, fh, rk, w. The numericalsolution of this system is in table 2 in the main text.

Appendix 5. The Reformed Economy

This appendix presents the reformed economy as defined in subsec-tion 4.1 in the main text. The equations describing the reformed


economy are as in the previous appendix except that now we setQ = Q∗, so that the public debt ratio is determined by the no pre-mium condition, d

TT ν−1yH ≡ d or d ≡ dTT ν−1yH . Note also thatsince Q = Q∗, we set β = 1

Q∗ in the parameterization stage.In what follows, we present steady-state solutions of this econ-

omy. We will start with the case in which the country’s net foreigndebt position is unrestricted so that f ≡ (1−λ)TT 1−νd−TT ν∗

fh

yH isendogenously determined. In turn, we will study the case in whichwe set the country’s net foreign debt position equal to zero (meaningthat the trade is balanced) so that f = 0 in the new reformed steadystate.

Steady State of the Reformed Economy withUnrestricted Foreign Debt

Now Q = Q∗ so that d ≡ dTT ν−1yH . The steady-state system is

1 = β[1 − δ + (1 − τk)rk] (110)

χnnη = c−σ νc

cHTT 1−ν (1 − τn

t )(1 + τ c

t )w (111)

cH

cF=

ν

1 − νTT (112)

x = δk (113)

c =

(cH)ν (

cF)1−ν

(ν)ν (1 − ν)1−ν (114)

yH = Akan1−a (115)

yH = cH + x + sgyH + cF∗ (116)

rk = TT 1−vmcaAka−1n1−a (117)

w = mc(1 − a)Akan−a (118)

ω =1

TT 1−v yH − 1TT 1−v rkk − wn (119)

z1 =φ

(φ − 1)z2 (120)


Table A1. Steady-State Solution of the ReformedEconomy with Unrestricted f

Steady-StateVariables Description Solution

u Period Utility 0.9458yH Output 0.858759TT Terms of Trade 0.838459

Q − Q∗ Interest Rate Premium 0TT 1−ν c

yH Consumption as Share of GDP 0.5522k

yH Physical Capital as Share ofGDP

4.2291

TT ν∗fh

yH Private Foreign Assets as Shareof GDP

−4.5742

TT1−νdyH Total Public Debt as Share of

GDP0.9

˜f ≡ (1−λ)dTT1−ν−TT ν∗fh

yH Total Foreign Debt as Share ofGDP

4.8982

z1 = yTT ν−1 + βθz1 (121)

z2 = ymc + βθz2 (122)

d ≡ dTT ν−1yH (123)

d = Q∗d + TT ν−1sgyH − τ c

(1

TT 1−v cH + TT vcF

)− τk

(1

TT 1−v rkk + ω

)− τnwn − TT ν−1slyH

t (124)

(1 − λ)d − TT ν∗+ν−1fh = −TT ν−1cF∗ + TT νcF

+ QTT ν∗+ν−1(

1TT v+v∗−1 (1 − λ)d − fh

). (125)

We thus have sixteen equations in c, cH , cF , k, x, n, yHt , TT ,

mc, ω, d, z1, z2, rk, w, fh. The numerical solution of this system ispresented in table A1. Notice that, for the reasons explained in thetext, private foreign debt, −fh > 0, and hence the country’s netforeign debt-to-GDP ratio, f , are extremely high in this case.


Table A2. Steady-State Solution of the ReformedEconomy with f = 0 when the Residual Fiscal

Instrument Is sg


u Period Utility 0.9125rk Real Return to Physical

Capital0.0749

w Real Wage Rate 1.2442n Hours Worked 0.3403

yH Output 0.8237TT Terms of Trade 1.01sg Capital Tax Rate 0.2306




4.2291

TT ν∗ fh


0.3240


GDP0.9



0

Steady State of the Reformed Economy withRestricted Foreign Debt

Now Q = Q∗, so that d ≡ dTT ν−1yH as above, and, in addition,f ≡ (1−λ)TT 1−νd−TT ν∗

fh

yH = 0, so that (1 − λ) dTT 1−ν −TT ν∗fh = 0

or fh ≡ (1−λ)dTT 1−ν

TT ν∗ . In this case, the equation for the balance ofpayments (125) simplifies to

TTcF = cF∗. (126)

Therefore, using this equation in the place of (125), we have sixteenequations in c, cH , cF , k, x, n, yH

t , TT , mc, ω, d, z1, z2, rk, w, andone of the tax-spending policy instruments, sg, τ c, τk, τn. Note thatin turn fh follows residually from fh ≡ (1−λ)dTT 1−ν

TT ν∗ . The numeri-cal solution of this system under each public financing instrument



Instrument Is τ c



Capital0.0749


yH Output 0.8237TT Terms of Trade 1.01τ c Capital Tax Rate 0.1593




4.2291

TT ν∗ fh


0.3240


GDP0.9



0

is presented in tables A2–A5, while a summary of these solutionsappears in table 3 in the main text.

Restricted Changes in Fiscal Policy Instruments

We now repeat the main computations restricting the magnitude offeedback coefficients in the policy rules so that all tax-spending pol-icy instruments cannot change by more than 10 percentage pointsfrom their averages in the data (we report that the results also donot change when we allow for 5 percentage points only).

Restricted results for optimal feedback policy coefficients, theimplied statistics, and welfare over various time horizons arereported in tables A6, A7, and A8, which correspond to tables 4,5, and 6, respectively, in the main text.

Inspection of the new tables, and comparison to their counter-parts in the main text, implies that the main qualitative results do



Instrument Is τk



Capital0.0729


yH Output 0.8348TT Terms of Trade 1.01τk Capital Tax Rate 0.2930




4.3448

TT ν∗ fh


0.3240


GDP0.9



0

not change. Namely, debt consolidation is again preferable to non-debt consolidation after the first ten years. Also, although obviouslyfeedback policy coefficients are now smaller, the best fiscal policymix again implies that we should earmark public spending for thereduction of public debt and, at the same time, cut taxes to mitigatethe recessionary effects of debt consolidation. The only difference isthat now, since cuts in income taxes are restricted, we should alsocut consumption taxes.

Impulse Response Functions (IRFs) withOne Fiscal Instrument at a Time

Figures A1 and A2 compare debt consolidation results when the fis-cal authorities can use all instruments jointly to the case in whichthey are restricted to use one instrument at a time only. Results arealways with optimized rules. To save on space, we focus on resultsfor the public debt-to-GDP ratio and private consumption.



Instrument Is τn



Capital0.0749


yH Output 0.8314TT Terms of Trade 1.01τn Capital Tax Rate 0.4018




4.2291

TT ν∗ fh


0.3240


GDP0.9



0

Table A6. Optimal Reaction to Debt and Output withDebt Consolidation (restricted optimal fiscal policy mix)

Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output

sgt γg

l = 0.4 γgy = 0.0011

τ ct γc

l = 0.0005 γcy = 0.4865

τkt γk

l = 0.0026 γky = 0.9496

τnt γn

l = 0.0023 γny = 0.95

Note: V0 = 79.9336.


Tab

leA

7.Sta

tist

ics

Implied

by

Tab

leA

6

Fiv

eTen

Tw

enty

Ela

stic

ity

Ela

stic

ity

Per

iods

Per

iods

Per

iods

Dat

ato

Lia

bilitie

sto

Outp

ut

Min

/Max

Ave

rage

Ave

rage

Ave

rage

Ave

rage

sg t−

2.01

6%−

0.00

35%

0.13

10/0

.227

30.

1473

0.15

760.

1760

0.22

22τ

c t0.

0032

%1.

97%

0.13

52/0

.179

90.

1492

0.15

450.

1614

0.17

56τ

k t0.

0093

%2.

16%

0.21

43/0

.301

40.

2418

0.25

200.

2654

0.31

18τ

n t0.

0062

%1.

61%

0.34

23/0

.429

50.

3698

0.38

0.39

340.

421


Table A8. Welfare over Different Time Horizonswith, and without, Debt Consolidation(restricted optimal fiscal policy mix)

Two Four Ten Twenty ThirtyPeriods Periods Periods Periods Periods

With Consolid. 1.4862 3.0164 7.7221 15.3294 22.2939Without Consolid. (1.6251) (3.1791) (7.4481) (13.4127) (18.1904)Welfare Gain/Loss −0.0458 −0.0327 0.0267 0.1075 0.1693

Figure A1. IRFs of Public Debt to GDP(comparison of alternative fiscal policies)

Note: IRFs are in levels and converge to the reformed steady state, while thesolid horizontal line indicates the point of departure (status quo value).


Figure A2. IRFs of Consumption(comparison of alternative fiscal policies)



Appendix 6. Robustness to the Public DebtThreshold Parameter

The Case with a Lower Public Debt Threshold

We now assume d ≡ 0.8. This implies ψ ≡ 0.0319 to hit the data.Then, we have the solution shown in tables A9–A11.

Table A9. Status Quo Steady-State Solution

Steady-StateVariables Description Solution Data

u Period Utility 0.8217 —rk Real Return to

Physical Capital0.0908 —

w Real Wage Rate 1.1140 —n Hours Worked 0.3313 0.2183

yH Output 0.7124 —TT Terms of Trade 0.9945 —

Q − Q∗ Interest Rate Premium 0 0.011TT 1−ν c

yH Consumption as Shareof GDP

0.6334 0.5961

kyH Physical Capital as

Share of GDP3.4872 —

TT ν∗ fh

yH Private Foreign Assetsas Share of GDP

0.1749 0.1039

TT1−νdyH Total Public Debt as

Share of GDP1.0965 1.0828


yH Total Foreign Debt asShare of GDP

0.2194 0.2109


Table A10. Reformed Steady-State Solution



Capital0.0727






4.3583

TT ν∗ fh


0.2880


GDP0.8



0

Table A11. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)


sgt γg

l = 0.4338 γgy = 0

τ ct γc

l = 0 γcy = 0.0279

τkt γk

l = 0 γky = 2.2569

τnt γn

l = 0.0001 γny = 2.136

Note: V0 = 80.1038.


The Case with a Higher Public Debt Threshold

Next, we assume d ≡ 1. This implies ψ ≡ 0.108 to hit the data.Then, we have the solution shown in tables A12–A14.

Table A12. Status Quo Steady-State Solution

Steady-StateVariables Description Solution Data

u Period Utility 0.8217 —rk Real Return to

Physical Capital0.0908092 —

w Real Wage Rate 1.11373 —n Hours Worked 0.331273 0.2183

yH Output 0.712311 —TT Terms of Trade 0.995009 —

Q − Q∗ Interest Rate Premium 0.011 0.011TT 1−ν c

yH Consumption as Shareof GDP

0.6335 0.5961


Share of GDP3.4872 —

TT ν∗ fh

yH Private Foreign Assetsas Share of GDP

0.1827 0.1039

TT1−νdyH Total Public Debt as

Share of GDP1.0965 1.0828



0.2122 0.2109


Table A13. Reformed Steady-State Solution



Capital0.0731






4.3314

TT ν∗ fh


0.3600


GDP1



0



sgt γg

l = 0.4588 γgy = 0.0017

τ ct γc

l = 0 γcy = 0.0018

τkt γk

l = 0.0014 γky = 2.2569

τnt γn

l = 0.1306 γny = 1.5629

Note: V0 = 79.8482.


Appendix 7. Robustness to the Net Foreign Debt Positionin the Reformed Steady State

The Case with 0.1 Net Foreign Debt Ratio in the ReformedSteady State

When we set f ≡ (1−λ)TT 1−νd−TT ν∗fh

yH = 0.1 in the reformed steadystate, the results are as shown in tables A15 and A16.

Table A15. Steady-State Solution of theReformed Economy with f = 0.1




yH Consumption as Share ofGDP

0.6029


Share of GDP4.3425

TT ν∗ fh

yH Private Foreign Assets asShare of GDP

0.2240

TT 1−νdyH Total Public Debt as

Share of GDP0.9

f ≡ (1−λ)dTT 1−ν−TT ν∗fh


0.1



sgt γg

l = 0.6725 γgy = 0.0020

τ ct γc

l =0 γcy = 0.7435

τkt γk

l = 0.0011 γky = 2.2572

τnt γn

l = 0.0015 γny = 2.14

Note: V0 = 80.2538.


The Case with 0.2109 Net Foreign Debt Ratioin the Reformed Steady State

When we set f ≡ (1−λ)TT 1−νd−TT ν∗fh

yH = 0.2109 (as in the data) inthe reformed steady state, the results are as shown in tables A17and A18.

Table A17. Steady-State Solution of theReformed Economy with f = 0.2109





0.6017


Share of GDP4.3397

TT ν∗ fh


0.1050


Share of GDP0.9



0.2109



sgt γg

l = 0.5558 γgy = 0.0114

τ ct γc

l =0.0094 γcy = 1.3091

τkt γk

l = 0.0159 γky = 1.3759

τnt γn

l = 0 γny = 2.14

Note: V0 = 80.3931.


Appendix 8. Robustness to World Interest Rate Shocks

Using the new specification presented in the main text, the resultsare as shown in tables A19 and A20.



sgt γg

l = 0.2068 γgy = 0

τ ct γc

l = 0.0002 γcy = 0.2097

τkt γk

l = 0 γky = 2.2569

τnt γn

l = 0 γny = 2.1360

Note: V0 = 79.5457.


Tab

leA

20.

Wel

fare

over

Diff

eren

tT

ime

Hor

izon

sw

ith,an

dw

ithou

t,D

ebt

Con

solidat

ion

(optim

alfisc

alpol

icy

mix

)

Tw

oFou

rTen

Tw

enty

Thir

tyPer

iods

Per

iods

Per

iods

Per

iods

Per

iods

Wit

hC

onso

lid.

1.45

892.

8359

7.06

6014

.412

921

.902

6W

itho

utC

onso

lid.a

(1.5

236)

(2.9

533)

(6.8

521)

(12.

6868

)(1

7.86

57)

Wel

fare

Gai

n/Los

s−

0.02

16−

0.02

370.

0208

0.09

630.

1662

aT

heva

lues

ofth

eop

tim

ized

feed

back

pol

icy

coeffi

cien

tsw

hen

we

star

tfr

om,

and

retu

rnto

,th

esa

me

stat

usqu

ost

eady

stat

ear

eγ

g l=

0.01

09,γ

c l=

0.52

06,γ

k l=

0,γ

n l=

0.30

32,γ

g y=

0.01

29,γ

c y=

0,γ

k y=

2.25

64,an

dγ

n y=

0.02

36.


Appendix 9. Ramsey Policy Problem

The Ramsey government chooses the paths of policy instruments tomaximize the household’s expected discounted lifetime utility, Vt,subject to the equations of the DE. As shown in appendix 4 above,the DE system contains twenty-four equations (we do not includethe equation for Vt).

As said in the main text, we solve for a “timeless” Ram-sey equilibrium, meaning that the resulting equilibrium conditionsare time invariant (see also, e.g., Schmitt-Grohe and Uribe 2005).If we follow the dual approach to the Ramsey policy problem,the government chooses the paths of {yH

t , ct, cHt , cF

t , nt, xt, kt, fht ,

TTt, Πt, ΠHt , Θt, Δt, wt, mct, ωt, r

kt , Qt, dt, Π∗

t , z1t , z2

t , Rt, lt}∞t=0, which

are the twenty-four endogenous variables of the DE system, plusthe paths of the three tax rates, {τ c

t , τkt , τn

t }∞t=0. Notice that pub-

lic debt, {dt}∞t=0, is also among the choice variables. Also notice

that, in solving this optimization problem, we take as given therate of exchange rate depreciation, {εt}∞

t=0; public spending ongoods/services, {sg

t }∞t=0; lump-sum government transfers, {sl

t}∞t=0;

and the share of public debt issued for the domestic market, {λt}∞t=0.

We treat these variables as exogenous for different reasons: in a cur-rency union model, εt = 1 all the time; we take {sg

t }∞t=0 as given for

simplicity; we take {slt}∞

t=0 as given because, if the Ramsey govern-ment can choose a lump-sum policy instrument, then its problembecomes trivial; finally, we set λt exogenously and equal to its dataaverage value (as we have done so far anyway) because, if it is chosenby the Ramsey government, its optimality condition will be identi-cal to the household’s optimality condition for financial assets/debtin a Ramsey steady state without sovereign interest rate premia,so the Ramsey steady-state system will be indeterminate (in par-ticular, in the steady state, the government’s optimality conditionfor λt is reduced to 1 = βQ, which is also the household’s opti-mality condition for foreign assets/debt). For the very same reason(namely, the optimality conditions of the household and the gov-ernment with respect to fh

t are both reduced to 1 = βQ in theRamsey steady state without sovereign interest rate premia), weset fh

t so as to satisfy f = 0 in the reformed steady state (as wehave done so far anyway in the reformed steady state). Thus, from


yH = 0, it follows fh = (1 − λ) dTT ν−1yH .


Table A21. Ramsey Steady-State Solution (unrestricted)


u Period Utility 1.2215yH Output 1.56176TT Terms of Trade 1.01τ c “Optimal” Consumption

Tax Rate26.5726

τk “Optimal” Capital TaxRate

0.00272056

τn “Optimal” ConsumptionTax Rate

−26.4976



0.5327


Share of GDP6.1284

TT ν∗ fh


0.3240


Share of GDP0.9



0

Working in this way, and including Ramsey multipliers, we end upwith a well-defined system consisting of fifty-one equations in fifty-one endogenous variables (the system is available upon request fromthe authors). The numerical solution of this system in the steadystate is presented in table A21 (we do not report solutions for Ram-sey multipliers). In this solution, as said before, Q = Q∗ = R = 1

β ,and fh = (1 − λ) dTT ν−1yH . (Note that nominal variables, likeR, do not affect the real allocation, and hence utility, at steadystate.)

Notice, as also said in the text, and as is well established in theRamsey literature with optimally chosen consumption taxes, thatthe solution in table A21 is characterized by an extremely largeconsumption tax rate, τ c = 26.5726, and an equally large labor


Table A22. Ramsey Steady-State Solution(restricted τn

t = 0)


u Period Utility 1.0950yH Output 1.11622TT Terms of Trade 1.01τk “Optimal” Capital Tax

Rate0.0500465

τ c “Optimal” ConsumptionTax Rate

0.810199



0.5443


Share of GDP5.8387

TT ν∗ fh


0.3240


Share of GDP0.9



0

subsidy, τn = −26.4976 < 0. Hence, following the same literature,we rule out a subsidy to labor by setting τn

t = 0 at all t. This givesthe solution in table A22 (this table is also presented in the maintext, but we repeat it here for the reader’s convenience).

Appendix 10. Flexible Exchange Rates

The IRFs under flexible exchange rates are shown in figure A3 (thezero lower bound for the nominal interest rate is not violated).


Figure A3. IRFs under Flexible Exchange Rates


References

Benigno, G., and C. Thoenissen. 2008. “Consumption and RealExchange Rates with Incomplete Markets and Non-tradedGoods.” Journal of International Money and Finance 27 (6):926–48.

Rabanal, P. 2009. “Inflation Differentials between Spain and theEMU: A DSGE Perspective.” Journal of Money, Credit andBanking 41 (6): 1141–66.


Schmitt-Grohe, S., and M. Uribe. 2004. “Optimal Fiscal and Mon-etary Policy under Sticky Prices.” Journal of Economic Theory114 (2): 198–230.

———. 2005. “Optimal Fiscal and Monetary Policy in a Medium-Scale Macroeconomic Model.” In NBER MacroeconomicsAnnual, ed. M. Gertler and K. Rogoff, 385–425. Cambridge, MA:MIT Press.

———. 2007. “Optimal Simple and Implementable Monetary andFiscal Rules.” Journal of Monetary Economics 54 (6): 1702–25.

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