· online appendix to fiscal consolidation in an open economy with sovereign premia and without...
TRANSCRIPT
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).
4 International Journal of Central Banking December 2017
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
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 5
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)
6 International Journal of Central Banking December 2017
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)
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 7
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.
8 International Journal of Central Banking December 2017
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)
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 9
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):
10 International Journal of Central Banking December 2017
∂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)
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 11
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)
12 International Journal of Central Banking December 2017
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
.
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 13
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)
14 International Journal of Central Banking December 2017
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)
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 15
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
)
16 International Journal of Central Banking December 2017
+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
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 17
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.
18 International Journal of Central Banking December 2017
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
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 19
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)
20 International Journal of Central Banking December 2017
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.
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 21
Table A2. Steady-State Solution of the ReformedEconomy with f = 0 when the Residual Fiscal
Instrument Is sg
Steady-StateVariables Description Solution
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
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6002k
yH Physical Capital as Share ofGDP
4.2291
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.3240
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
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
22 International Journal of Central Banking December 2017
Table A3. Steady-State Solution of the ReformedEconomy with f = 0 when the Residual Fiscal
Instrument Is τ c
Steady-StateVariables Description Solution
u Period Utility 0.9227rk Real Return to Physical
Capital0.0749
w Real Wage Rate 1.2442n Hours Worked 0.3403
yH Output 0.8237TT Terms of Trade 1.01τ c Capital Tax Rate 0.1593
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6086k
yH Physical Capital as Share ofGDP
4.2291
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.3240
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
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
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 23
Table A4. Steady-State Solution of the ReformedEconomy with f = 0 when the Residual Fiscal
Instrument Is τk
Steady-StateVariables Description Solution
u Period Utility 0.9311rk Real Return to Physical
Capital0.0729
w Real Wage Rate 1.2649n Hours Worked 0.3393
yH Output 0.8348TT Terms of Trade 1.01τk Capital Tax Rate 0.2930
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6040k
yH Physical Capital as Share ofGDP
4.3448
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.3240
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
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.
24 International Journal of Central Banking December 2017
Table A5. Steady-State Solution of the ReformedEconomy with f = 0 when the Residual Fiscal
Instrument Is τn
Steady-StateVariables Description Solution
u Period Utility 0.9290rk Real Return to Physical
Capital0.0749
w Real Wage Rate 1.2442n Hours Worked 0.3435
yH Output 0.8314TT Terms of Trade 1.01τn Capital Tax Rate 0.4018
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6086k
yH Physical Capital as Share ofGDP
4.2291
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.3240
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
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.
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 25
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
26 International Journal of Central Banking December 2017
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).
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 27
Figure A2. IRFs of Consumption(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).
28 International Journal of Central Banking December 2017
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
˜f ≡ (1−λ)dTT1−ν−TT ν∗fh
yH Total Foreign Debt asShare of GDP
0.2194 0.2109
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 29
Table A10. Reformed Steady-State Solution
Steady-StateVariables Description Solution
u Period Utility 0.9326rk Real Return to Physical
Capital0.0727
w Real Wage Rate 1.2673n Hours Worked 0.3394
yH Output 0.8367TT Terms of Trade 1.01τk Capital Tax Rate 0.2908
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6035k
yH Physical Capital as Share ofGDP
4.3583
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.2880
TT1−νdyH Total Public Debt as Share of
GDP0.8
˜f ≡ (1−λ)dTT1−ν−TT ν∗fh
yH Total Foreign Debt as Share ofGDP
0
Table A11. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)
Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output
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.
30 International Journal of Central Banking December 2017
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
kyH Physical Capital as
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
˜f ≡ (1−λ)dTT1−ν−TT ν∗fh
yH Total Foreign Debt asShare of GDP
0.2122 0.2109
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 31
Table A13. Reformed Steady-State Solution
Steady-StateVariables Description Solution
u Period Utility 0.9296rk Real Return to Physical
Capital0.0731
w Real Wage Rate 1.2625n Hours Worked 0.3391
yH Output 0.8328TT Terms of Trade 1.01τk Capital Tax Rate 0.2951
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share of GDP 0.6045k
yH Physical Capital as Share ofGDP
4.3314
TT ν∗ fh
yH Private Foreign Assets as Shareof GDP
0.3600
TT1−νdyH Total Public Debt as Share of
GDP1
˜f ≡ (1−λ)dTT1−ν−TT ν∗fh
yH Total Foreign Debt as Share ofGDP
0
Table A14. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)
Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output
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.
32 International Journal of Central Banking December 2017
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
Steady-StateVariables Description Solution
u Period Utility 0.9315yH Output 0.835228TT Terms of Trade 1.00615
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share ofGDP
0.6029
kyH Physical Capital as
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
yH Total Foreign Debt asShare of GDP
0.1
Table A16. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)
Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output
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.
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 33
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
Steady-StateVariables Description Solution
u Period Utility 0.9320yH Output 0.8358TT Terms of Trade 1.002
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share ofGDP
0.6017
kyH Physical Capital as
Share of GDP4.3397
TT ν∗ fh
yH Private Foreign Assets asShare of GDP
0.1050
TT 1−νdyH Total Public Debt as
Share of GDP0.9
f ≡ (1−λ)dTT 1−ν−TT ν∗fh
yH Total Foreign Debt asShare of GDP
0.2109
Table A18. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)
Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output
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.
34 International Journal of Central Banking December 2017
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.
Table A19. Optimal Reaction to Debt and Output withDebt Consolidation (optimal fiscal policy mix)
Fiscal Optimal Reaction Optimal ReactionInstruments to Debt to Output
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.
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 35
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.
36 International Journal of Central Banking December 2017
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
f ≡ (1−λ)dTT 1−ν−TT ν∗fh
yH = 0, it follows fh = (1 − λ) dTT ν−1yH .
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 37
Table A21. Ramsey Steady-State Solution (unrestricted)
Steady-StateVariables Description Solution
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
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share ofGDP
0.5327
kyH Physical Capital as
Share of GDP6.1284
TT ν∗ fh
yH Private Foreign Assets asShare of GDP
0.3240
TT 1−νdyH Total Public Debt as
Share of GDP0.9
f ≡ (1−λ)dTT 1−ν−TT ν∗fh
yH Total Foreign Debt asShare of GDP
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
38 International Journal of Central Banking December 2017
Table A22. Ramsey Steady-State Solution(restricted τn
t = 0)
Steady-StateVariables Description Solution
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
Q − Q∗ Interest Rate Premium 0TT 1−ν c
yH Consumption as Share ofGDP
0.5443
kyH Physical Capital as
Share of GDP5.8387
TT ν∗ fh
yH Private Foreign Assets asShare of GDP
0.3240
TT 1−νdyH Total Public Debt as
Share of GDP0.9
f ≡ (1−λ)dTT 1−ν−TT ν∗fh
yH Total Foreign Debt asShare of GDP
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).
Vol. 13 No. 4 Fiscal Consolidation in an Open Economy 39
Figure A3. IRFs under Flexible Exchange Rates
Note: IRFs are in levels and converge to the reformed steady state, while thesolid horizontal line indicates the point of departure (status quo value).
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.
40 International Journal of Central Banking December 2017
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.