Ramsey Policies in a Small Open Economy with Sticky Prices and Capital Stéphane Aurayy

Beatriz de Blasz

Aurélien Eyquemx

April 15, 2010

Abstract In this paper we study jointly optimal …scal and monetary policies in a small open economy framework with capital, sticky prices and trade in both consumption and capital goods. We …rst consider the case of distortionary taxes on labor and capital, and no public debt. As in a closed economy set–up, in the steady state, the optimal in‡ation rate is zero, as well as the tax rate on capital earnings. The dynamic properties of optimal monetary and …scal policies in an open economy are qualitatively the same as those of a closed economy: the tax rate on capital income remains constant over the business cycle, while both the nominal interest rate and the tax rate on labor income move, although very smoothly, respectively to minimize the distortions implied by nominal rigidities and to balance the government budget. Adding non state-contingent public debt, leaves taxe almost constant over the business cycle as debt acts as a shock absorber, displaying the traditional near random walk properties. Keywords: small open economy, sticky prices, optimal monetary and …scal policies. JEL Class.: E52, E62, E63, F41.

We would like to thank Begoña Domínguez and participants at the Simposio de Análisis Económico 2009 for their helpful comments. Beatriz de Blas acknowledges …nancial support from ECO2008-04073 project of the Spanish MEC. y Universités Lille Nord de France (ULCO), CNRS–MESHS (USR 3185), GREDI, Université de Sherbrooke and CIRPÉE, Canada. Email: [email protected]. z Universidad Autónoma de Madrid, Departamento de Análisis Económico: T. e H. Económica, Campus de Cantoblanco, 28049 Madrid, Spain. Email: [email protected]. x GATE, UMR 5824, Université de Lyon and Ecole Normale Supérieure de Lyon, France and GREDI, Université de Sherbrooke, Canada. Email: [email protected].

1

Introduction

It is now well known that it is optimal to use unexpected variations in prices in a ‡exible prices set-up to smooth taxes. However, when prices are sticky, a trade o¤ appears between using unexpected in‡ation to smooth taxes and the fact that adjusting the price level is costly (Schmitt-Grohe & Uribe (2004a)). When an open-economy framework is considered, new issues come into play, such as whether to stabilize or not ‡uctuations on the exchange rate, and its implications on in‡ation and welfare (Benigno & de Paoli (2009)). The literature of optimal taxation following Lucas & Stokey (1983) has established that distorting taxes should be very smooth over time and states of nature, implying that capital taxes should be close to zero and that labor taxes should be roughly constant (e.g. Chari & Kehoe (1999)). Another result is that tax rates are an increasing function of the elasticity of input supply, i.e. the tax rate on labor increases when the elasticity of labor supply increases (see Chamley (1986)). Schmitt-Grohe & Uribe (2004a) …nd that the major results about optimal …scal policy are mostly unchanged in a closed economy when sticky prices are introduced. However, this is not the case in an open economy with nominal rigidities (Benigno & de Paoli (2009)). These authors argue that the open-economy dimension introduces shocks to the terms of trade that directly a¤ect labor supply decisions, although only for (implausibly) high levels of openness. Regarding monetary policy, the literature underlines that the optimal policy in a competitive environment (with ‡exible prices) is to follow the Friedman rule, i.e. to set the nominal interest rate to zero at each period. However, the Friedman rule is found to be suboptimal when sticky prices are introduced (see for instance Woodford (2003)). In this case, the optimal monetary policy rule should aim at stabilizing the in‡ation rate to minimize the distortions associated to nominal rigidities. To our knowledge little work has been done on the interaction between monetary and 1

…scal policy in an open-economy environment. This may look surprising because the e¤ectiveness and proper conduct of national macroeconomic policies should clearly depend on international linkages between national economies. Furthermore, previous literature omits capital as an input factor, while introducing this variable seems to be crucial for the results of optimal …scal policy. In this paper, we study the dynamic properties of setting taxes and monetary policy optimally in a small open economy with capital and Calvo sticky prices, where households trade both consumption and capital goods. The set-up is a standard new Keynesian DSGE model in which the government levies distortionary taxes on inputs (capital and labor) to provide individuals with an exogenously determined amount of public good, and does not have access to subsidies to undo the distortions introduced by imperfect competition. Our approach solves the dual Ramsey problem in the context of a small open economy with capital, extending the work of Schmitt-Grohe & Uribe (2004a). We study the optimal taxation system both in the Pareto-ine¢ cient steady state and around this steady state (dynamics). There are several ways in which the open dimension, combined with the presence of nominal rigidities and monopolistic competition may change some of the traditional results about optimal taxation. First, monopolistic competition distorts steady state allocations. In this case, taxes on inputs (labor and capital) as well as the in‡ation tax may help attaining the …rst-best. Second, as nominal rigidities imply departures from the Friedman rule, and as the dynamics of capital income are closely related to the path chosen for the nominal interest rate, the tax smoothing result on capital income taxes may not apply in this environment. Finally, the open-economy dimension may clearly a¤ect both the optimal level of tax rates and their dynamics over business cycles. Considering a closed economy version of our model, the results are as follows. In accordance with Chamley (1986), the tax rate on capital income should be zero in the steady state and should be kept nearly constant over the cycle, even in a sticky price environment. Optimal monetary policy implies large departures from the Friedman rule and should be aimed at stabilizing the in‡ation of production prices (as in Benigno & Woodford (2005)). 2

Fiscal and monetary policy instruments exhibit large persistence, re‡ecting the need to smooth the implied distortions on labor supply and consumption. When addressing the problem in a small open economy, we …nd that trade openness has no impact on the optimal steady state of the economy, as long as we impose symmetry between the small open economy and the rest of the world. Furthermore, the volatility of policy instruments is a¤ected only to the extent that the open-economy dimension implies changes in the volatility of tax bases (labor and capital income taxes). As a consequence, closed-economy results are fully preserved, whatever the degree of trade openness or the nature of goods traded (consumption or capital goods). Our results are robust to a sensitivity analysis with respect to the key parameters. Finally, an extension of the model with nominal non state-contingent public debt is discussed. The steady state properties of the Ramsey equilibrium in this case are almost isomorphic to the case of balanced-budget policies. In terms of the dynamics, tax rates now remain almost constant over the cycle and debt acts as a shock absorber. As shown by Schmitt-Grohe & Uribe (2004a), due to price stickiness, the central planner does not try to in‡ate production prices to reduce the amount of debt implied by (almost) not adjusting taxes and debt displays a near random walk behavior, just as in the case of state contingent public debt. The paper is structured as follows. The model and its general equilibrium conditions are presented in Section 2. Section 3 describes the parameterization used along the paper. Section 4 reports the results on optimal monetary and …scal policy both in the steady state and dynamically. Section 5 proceeds with a sensitivity analysis and presents an extension with public debt. The paper concludes with Section 6.

3

2 2.1

The model Households

The model is composed of two areas: the domestic economy and the rest of the world, of size n and 1

n, respectively. Both areas are symmetric, except for the size. In what

follows below, we denote rest of the world variables by an asterisk. Each country is populated by a unit mass of households j with in…nite life. In each area, the representative household maximizes a welfare index 0

(j) = E0

1 X

t

(1)

u (ct (j) ; `t (j))

t=0

subject to the budget constraint Et fqt;t+1 bt+1 (j)g+pc;t ct (j)+pk;t kt (j) = bt (j)+

t

(j)+(1

`;t ) wt `t

(j)+pk;t rk;t kt

1

(j) ; (2)

and subject to the appropriate transversality condition. In Equation (1), the parameter

is the subjective discount factor, ct (j) is the consump-

tion bundle chosen by the representative agent, `t (j) is the quantity of labor competitively supplied. In Equation (2), rk;t = 1 + (1

k;t )

zt pk;t

is the net (real) return on cap-

ital accumulation; wt and zt are the nominal wage and return on physical capital rental; Rn t (j) = 0 t (i; j)di refers to pro…ts paid by domestic …rms (operating on monopolistic

competition markets), indexed by i, to the representative domestic household. The variable bt (j) is a portfolio of state contingent assets held in period t

1, which pays in units

of domestic aggregate consumption, qt;t+1 denotes the stochastic discount factor for oneperiod ahead nominal payments attached to the portfolio. Finally, pc;t ; pk;t and pt denote the consumption goods, capital goods and produced goods price indices, respectively. The representative household chooses ct (j), `t (j), kt (j), and bt+1 (j) to maximize utility

4

(1) subject to the budget constraint (2). First order conditions imply u`;t wt (1 = 0; `;t ) uc;t pc;t pc;t uc;t+1 = qt;t+1 ; uc;t pc;t+1 pk;t+1 qt;t+1 rk;t+1 = 1: pk;t

(3) (4) (5)

Equation (3) is the standard labor supply function, describing the intratemporal trade o¤ between consumption and leisure. Equation (4) is the Euler equation relating the intertemporal choice of consumption as a function of in‡ation and the return on the …nancial portfolio. Denoting rt =

1 Et fqt;t+1 g

as the gross return on a risk-less one-period bond, and

taking conditional expectations on both sides of (4), the standard Euler equation writes r t Et

uc;t+1 uc;t

pc;t pc;t+1

= 1:

Finally, taking conditional expectations on both sides of Equation (5) shows that net returns on disposable assets should be equal in equilibrium Et

pk;t+1 rk;t+1 pk;t

= rt :

We follow Benigno & de Paoli (2009) and assume that the aggregate consumption of the representative household is a composite of consumption of goods produced at home (h); and goods produced in the rest of the world (f ) according to h 1 ct (j) = ' ch;t (j) where ' = 1

(1

n)

1

+ (1

1

') cf;t (j)

1

i

1

;

refers to the relative weight of home and foreign goods, which

is a function of the size of the domestic economy, n, and , a measure of trade openness (see Corsetti (2006), and Goldberg & Tille (2008)). Symmetrically, the consumption of a representative household in the rest of the world is h 1 ct (j) = ' ch;t (j)

1

+ (1

where ' = n . 5

1

' ) cf;t (j)

1

i

1

;

We assume that the law of one price holds at the producer level, then the companion consumption price indexes are given by 1

pc;t = ' (pt )1 + (1 ') ("t pt )1 h 1 + (1 ' ) (pt )1 pc;t = ' "t 1 pt

1

; i11

;

where "t denotes the nominal exchange rate, expressed as the price of foreign currency in terms of the domestic currency. In these expressions,

1 is the elasticity of substitution between domestic and foreign

goods. Standard Dixit & Stiglitz (1977) consumption subindexes are given by

ch;t (j) =

ch;t (j) =

"

"

1 n 1 n

1

Z

n

ch;t (i; j)

1

0

1

Z

n

ch;t (i; j)

0

1

#

1

, cf;t (j) =

di

#

1

, cf;t (j) =

di

"

1

1 1

"

n

1

cf;t (i; j)

1

n

1

1 1

Z

n

Z

1

cf;t (i; j)

n

1

#

1

;

di

#

di

1

;

where ch;t (j) (cf;t (j), respectively) is the consumption of …nal goods produced at home (in the rest of the world) by the representative, and

> 1 is the elasticity of substitution

across domestic varieties of …nal goods. Accordingly, optimal demands of domestic varieties can be expressed as ch;t (i; j) =

' pt (i) n pt

pt pc;t

ct (j) ; ch;t (i; j) =

' n

pt (i) pt

pt "t pc;t

ct (j) :

Households accumulate physical capital according to xt (j) = kt (j)

(1

) kt

1

(j) ;

where xt (j) denotes the investment of household j. The variable xt (j) is a bundle of domestic and foreign capital goods with a symmetric structure with respect to consumption goods, i.e. xt (j) =

h

1

xh;t (j)

1

+ (1

6

1

) xf;t (j)

1

i

1

;

where

= 1

(1

is a function of the size of the small open economy, n, and

n)

, the degree of openness on capital goods markets. Symmetrically, the bundle of a representative household in the rest of the world is xt (j) = where

h

1

1

xh;t (j)

i

1

1

+ (1

) xf;t (j)

1

;

=n .

Again, given that the law of one price holds, companion price indexes are pk;t = h pk;t =

(pt )1

+ (1

"t 1 p t

1

1

) ("t pt )1

+ (1

1

; i11

) (pt )1

:

Standard Dixit & Stiglitz (1977) subindexes are

xh;t (j) =

xh;t (j) =

"

"

1 n 1 n

1

Z

n

xh;t (i; j)

1

0

1

Z

0

n

xh;t (i; j)

1

#

1

, xf;t (j) =

di

#

di

1

, xf;t (j) =

" "

1

1 1

n

1

n

1

xf;t (i; j)

1

n

1

1

Z Z

1

xf;t (i; j)

n

Accordingly, optimal demands of domestic capital goods varieties are # " pt (i) pt pt (i) pt xh;t (i; j) = xt (j) ; xh;t (i; j) = n pt pk;t n pt "t pk;t

1

#

1

di

#

di

; 1

:

xt (j) :

Finally, let us de…ne terms of trade as st =

"t p t : pt

(6)

This de…nition is meant to be consistent with the de…nition of the real exchange rate, qt =

"t pc;t . pc;t

Given this notational convention, an increase of st denotes a deterioration

of terms of trade, implying an increase of the competitiveness of goods produced in the domestic economy.

7

2.2

Risk-sharing

Under the assumption of complete international markets of state contingent assets, a relation similar to Equation (4) holds in the rest of the world pc;t pc;t+1

uc ;t+1 uc ;t

"t "t+1

= qt;t+1 ;

which combined with Equation (4) gives the following risk-sharing condition uc ;t "t pc;t = = qt : uc;t pc;t

(7)

Equation (7) indicates that relative marginal utilities are related to the real exchange rate up to a constant that depends on initial conditions on relative net foreign asset position. Assuming symmetric initial conditions simply amounts to set

= 1, which is consistent

with the symmetric steady state around which we study the dynamic properties of the model.

2.3

Government

Public spending aggregates are composed only of domestic varieties gt =

"

1 n

1

Z

n

gt (i)

1

#

1

di

0

;

where gt+1 = 1 with

g;t+1

g

g+

g gt

+

g;t+1 ;

being an iid innovation with constant variance.

The government has access to distorting taxes on inputs (labor and capital) to …nance the exogenous stream of public spending. We only allow for balanced-budget policies, therefore the budget constraint of authorities in the domestic economy, expressed in nominal terms, is `;t wt `t

where `t =

Rn 0

`t (j) dj; and kt

1

=

+ Rn 0

k;t

(zt

kt

1

pk;t ) kt

(j) dj. 8

1

= pt gt

2.4

Firms

Firms, indexed by i, produce varieties yt (i) using domestic labor, lt (i); and physical capital, kt 1 (i); according to the following production function: yt (i) = at kt

1

(i) lt (i)1

:

The total factor productivity, at , evolves according to at+1 = (1 where

a;t+1

a) a

+

a at

+

a;t+1 ;

is an iid innovation with constant variance. Input prices are related by the

following e¢ ciency condition: wt lt (i) = (1

) zt kt

1

(i) ;

the nominal marginal cost being zt wt1

mct (i) = mct =

)1

(1

at 1 ;

which is the same for all …rms. Production prices are governed by Calvo (1983) pricing contracts. Only a fraction 1 of randomly selected domestic …rms is allowed to set new prices each period. The corresponding optimal price set by a …rm allowed to reset is:

pt (i) =

1 P

(

v=0

1

1 P

)v Et f

t+v yt+v

)v Et f

(

v=0

(i) mct+v g

;

t+v yt+v (i)g

where yt (i) is the aggregate demand addressed to …rm i yt (i) = ch;t (i) + ch;t (i) + xh;t (i) + xh;t (i) + gt (i) : Rn R1 Rn In this expression, ch;t (i) = 0 ch;t (i; j) dj, ch;t (i) = n ch;t (i; j) dj; xh;t (i) = 0 xh;t (i; j) dj R1 and xh;t (i) = n xh;t (i; j) dj. In addition, 1 is the steady state mark-up indicating the

distortion caused by monopolistic competition in …nal goods markets. 9

Aggregating among …rms and assuming that Calvo producers set the same price when free to reset, the aggregate production price index is pt = (1

) pt (i)

1

1

+ p1t

:

1

1

Production price in‡ation thus evolves according to1 1 t

+ (1

1

v1;t 1 v2;t

)

= 1;

where v1;t

1+ t+1

Et v1;t+1

1 c;t

= uc;t yt

mct ; pt

and v2;t

1 t+1 c;t

Et v2;t+1

Finally, the dispersion of production prices,

t

2.5

=

t 1 t

t

+ (1

= uc;t yt : h i R1 = 0 ptp(i) di t v1;t 1 v2;t

)

1; is given by :

Equilibrium

We focus on the case of a small open economy, i.e. n ! 0, in deriving the optimal monetary and …scal policy. One implication is that the aggregate consumer price index and the aggregate capital goods price index can respectively be expressed as ) (pt )1

pc;t = pt ; and pc;t = (1

) (pt )1

pk;t = pt ; and pk;t = (1 Let us de…ne aggregate output as yt = thus clears according to

h

1 n

1

Rn 0

yt = (1

) (1

) + st1

+ (1

) (1

) + s1t

See Appendix A for the derivation of these conditions.

10

1

1

("t pt )1

+ ("t pt )1

yt (i)

+ gt 1

+

1

i di

1

1 1

;

(8)

:

(9)

1 1

. The …nal goods market

ct + st ct xt + st xt

where xt and ct are exogenous processes of similar persistence and driven by the same iid innovations of constant variance,

c ;t+1

ct+1 = (1

c

)c +

c

ct +

xt+1 = (1

c

)x +

c

xt +

c ;t+1 ; c ;t+1 :

Finally, the labor market clearing condition is Z n Z n lt (i)di = `t (j)dj = `t 0

0

the capital market clearing condition is Z n Z kt (i)di = 0

n

kt (j)dj = kt

0

and the aggregate production is given by yt = Let fRt g1 t=0 = frt;

n;t ;

1 k;t gt=0

1 t

at kt 1 `1t

:

be a sequence of monetary and …scal policies.

1 Let fSt g1 t=0 = fat ; gt ; ct ; xt gt=0 be a sequence of exogenous shocks. 1 Let fQt g1 t=0 = fyt ; nt ; ct ; kt gt=0 be a sequence of quantities.

Let fPt g1 t=0 = f t ; pt ; pc;t ; pk;t ; wt ; zt ; mct ; st ; v1;t ; v2;t ;

1 t gt=0

be a sequence of prices.

De…nition A competitive equilibrium is de…ned as an allocation fQt g1 t=0 and a sequence

1 1 of prices fPt g1 t=0 such that, (i) for a given sequence of prices fPt gt=0 , shocks fSt gt=0 and

1 a given policy fRt g1 t=0 , the sequence fQt gt=0 satis…es households …rst-order conditions

(including the transversality conditions) and maximizes …rms pro…ts; (ii) for a given se1 1 quence of quantities fPt g1 t=0 , shocks fSt gt=0 and a given policy fRt gt=0 , the sequence

fPt g1 t=0 clears goods markets, …nancial markets and factors markets.

3

Parameterization

Before turning to the analysis of the optimal monetary and …scal policies, we assign numerical values to our deep parameters to proceed with the numerical simulations. 11

Utility function. We adopt the following functional form of the utility function: u (ct (j) ; nt (j)) =

ct (j)1 1

`t (j)1+ : 1+

Preferences. We consider a quarterly set-up. The subjective discount factor, ; is thus equal to 0:988; consistent with an average annual real interest rate of 5% in the steady state. The inverse of the intertemporal elasticity of consumption is

= 2 as in Corsetti

et al. (2008). We set the inverse of the elasticity of labor supply with respect to wages, , equal to 1. This value …ts the range proposed by Canzoneri et al. (2007). The elasticity of substitution between domestic and foreign goods is

= 1:5 as in Backus & Kehoe

(1994). The elasticity of substitution across domestic varieties of …nal goods determines the average mark-up and is set to

= 7, in accordance with Rotemberg & Woodford

(1997). Technology. The share of capital income in the GDP is

= 0:36. The depreciation rate,

, is assumed to be 10% annually. The degree of price stickiness, i.e. the Calvo parameter, is

= 0:75, implying an average duration of 4 quarters for prices.

Shocks. The characteristics of productivity, public expenditure and world demand innovations are to

a

=

g

( a) = =

c

(

c

) = 0:007 and

= 0:01. The persistence of shocks is set

= 0:9. We also calibrate the steady state value public expenditure to

represent 25% of the GDP, i.e. Openness.

g

=

g y

= 0:25.

Finally, the degree of trade openness is

economy (our benchmark), and

+

=

= 0 in the case of a closed

= 0:3 in the case of an open economy. This value

is in between the interval of parameters suggested by Galí & Monacelli (2005) and Pappa & Vassilatos (2007). In the baseline case, we set

= 0:3, implying

case of trade openness on the market of capital goods,

4

= 0. However, the

> 0, will also be investigated:

Optimal monetary and …scal policy

In this section, we present the Ramsey problem, and comment the implied results both in the steady state and around the steady state when shocks hit the economy. 12

4.1

The Ramsey problem

Equilibrium conditions summarized by Eqs. (12)–(25) (detailed in Appendix B) allow to determine all variables but

k;t

and rt . In particular, notice that

`;t

is determined

residually to balance the government budget at each period. We thus solve the dual form of the optimal monetary and …scal policy problem, that consists in …nding the sequence f

1 k;t ; rt gt=0

associated with the competitive equilibrium described above, that maximizes

individuals’welfare. The Ramsey problem thus writes M axE0 k;t ;rt

1 X t=0

t

Z

n

u (ct (j) ; `t (j)) dj

0

subject to Eqs. (12)–(25). This problem is known to result in time-inconsistent policies, as authorities may choose their policy instruments after agents have formed their expectations about forward key variables, such as the in‡ation rate, and therefore may take advantage of this situation in period zero. As a consequence, from period one onwards, authorities have an incentive to change their optimal policy since agents now take into account prior commitments when forming their expectations. Therefore, to avoid such inconsistency issues, we adopt the timeless perspective (see Woodford (2000)) and assume that the optimization problem is constrained by some former prior commitment, that is consistent with the optimal commitment chosen for period one onwards. We thus solve our dynamic Ramsey program and analyze its dynamic properties when the economy is closed or open respectively.2 Formally, this approach amounts to assume away optimality conditions in period 0.

4.2

The steady state

Table 1 describes the steady state implied by the Ramsey policy in the baseline case when the economy is open. 2 We solve the Ramsey problem analytically (in levels) using Matlab’s symbolic toolbox, and use Dynare’s second-order approximation algorithm (based on the method developed by Schmitt-Grohe & Uribe (2004b)) to simulate the model numerically.

13

Table 1: Steady state under Ramsey policy y c n ! n

2:2232 1:2057 0:6757 1:8048 0:4557

k k mc p

r

18:4678 0:0371 0:0000 0:8571 1:0121

1:0000 1:0000 1:0000 1:0000 1:0000

k c

s

q a g c x

1:0000 1:0000 0:5558 1:2057 0:4617

The …rst result is that the Ramsey steady state implies a zero in‡ation tax and a zero tax rate on capital income. Our results nest those of Chamley (1986) and Judd (1985) (similar optimal capital income tax rate) as well as those of Benigno & Woodford (2005) (similar optimal in‡ation tax). This leads to conclude that opening the economy does not alter the steady state results obtained in a closed economy. Interestingly, after setting k

= 0 and

= 1, the steady state tax rate on labor income boils down to `

=

(

1)(1

)

:

Notice that this expression does not depend either on the intertemporal elasticity of substitution of consumption ( ) or labor ( ), but depends positively on the steady state level of (exogenous) government expenditure ( ). It also depends negatively on the labor share ( ) (the higher the labor share, the higher the corresponding tax base, the lower the optimal tax rate on labor income), and negatively on the degree of monopolistic competition ( ). The intuition is the following. In this steady state, the in‡ation rate is zero, so that the distortions associated to monopolistic competition a¤ect the steady state through the level of the real marginal cost. A low level of competition generates important distortions by reducing the steady state value of the real wage inducing a higher tax rate on labor income to make provision of the exogenously given level of public spending in GDP, . A high level of competition, on the other hand, increases the real wage in the equilibrium and lowers the steady state labor supply and increases the steady state level of consumption.

14

4.3

Impulse response functions

In this section, we analyze the optimal response of policy instruments, as well as other key macroeconomic variables, under the Ramsey policy after each type of shock in the case of an open economy. We also report the impulse response functions (henceforth IRFs) of the same variables in a closed economy for comparison purposes. Trade in consumption goods only ( = 0:3)

4.3.1

Productivity shocks.

Figure 1 plots the IRFs of the variables of interest after a unit

productivity innovation. Figure 1: IRFs after a unit productivity shock under Ramsey policies ( = 0:3). Consumption

Output 0.6

2 1

0.4

0.5

% dev.

% dev.

% dev.

3

Hours worked

0.4 0.3

0.2 0

0.2 10

20

30

40

10

Labour income tax rate

40

10

10

10

20

30

40

10

Nominal interest rate

20

30

40

10

0

-0.04 α=0.3 α=γ=0

-0.06

-0.01

-0.08

-0.02 20 30 Quarters

40

30

40

0.8

-0.02

% dev.

% dev.

0.01

20

Terms of trade

0

10

0.8

PPI inf lation

0.02

40

0.6

5

-0.6

30

1 % dev.

-0.4

20

Real wages (PPI)

15 % dev.

% dev.

30

Capital

-0.2

% dev.

20

10

20 30 Quarters

0.7 0.6 0.5

40

10

20 30 Quarters

40

An increase in productivity produces the standard e¤ects on most macroeconomic variables. The marginal cost of production drops, driving …rms to increase their inputs 15

demands and in‡ation to fall. In this new Keynesian framework, both output and consumption increase as well as hours and the stock of capital, just as in the standard RBC set-up, driven by the increase of real wages and real capital revenues (not shown in the …gure). In the event of an increase in productivity, the Ramsey planner faces a tension between smoothing taxes to reduce the volatility of labor and reducing them because individuals may want to work more due to higher productivity. Which e¤ect is stronger will depend on preferences, mainly on the elasticity of labor supply. Given the parameterization employed in this model, the second e¤ect dominates, and labor income taxes fall considerably for more than 20 quarters. The tax rate on capital remains nearly constant (not reported). Since there is no public debt in the baseline model, taxes on labor income serve as a mechanism to balance the budget each period. Notice that the optimal monetary policy, re‡ected by movements in the nominal interest rate, is tight during the …rst 10 quarters and exhibits a moderate expansionary stance for the next 30 periods. This increase of the nominal interest rate shows the non-optimality of the Friedman rule in this case, also depicted in recent literature (see e.g. Faia (2008)), and closely follows the response of in‡ation. When we consider the open economy, the increase in domestic productivity drops the terms of trade (i.e. there is a real depreciation) and domestic goods become more competitive. The deterioration in terms of trade for domestic consumers reduces the response of consumption now that foreign goods become relatively more expensive. Home prices must fall by more in the open economy case to clear the market. The nominal interest rate falls on impact to increase afterwards. This a¤ects labor supply which rises as in the closed economy case, but becomes more persistent, and translates into a more persistent labor income tax rate too. This increase in labor supply, stronger than in the closed economy case, allows consumption to be smoother. The open dimension of the economy does not signi…cantly alter the dynamics of the optimal policy, except for the reaction of the nominal interest rate which becomes more persistent and volatile. In contrast to the 16

closed economy case, in‡ation becomes more volatile under the optimal policy when the economy is open to smooth the path for consumption, despite the cost of adjusting prices. Public spending shocks.

Figure 2 plots the same IRFs after a unit public spending

innovation. Figure 2: IRFs after a unit public spending shock under Ramsey policies ( = 0:3). Output

Consumption

Hours worked

-0.1

-0.2

0

-0.4 -0.6

% dev.

% dev.

% dev.

-0.15 -0.2 -0.25 -0.3 10

20

30

40

30

40

10

0.2

30

40

0.6 % dev.

-4

0.4 0.2 0

-6

0.1

-0.2 20

30

40

10

Nominal interest rate

20

30

40

10

PPI inf lation 0.08

0.04

α=0.3 α=γ=0

% dev.

0.06

0.02 0

20

30

40

Terms of trade -0.25 % dev.

10

% dev.

20

Real wages (PPI)

-2 % dev.

% dev.

20 Capital

0.5 0.3

-0.2 -0.3

10

Labour income tax rate

0.4

-0.1

0.04 0.02

-0.3 -0.35

0 10

20 30 Quarters

40

10

20 30 Quarters

40

10

20 30 Quarters

40

The shock produces an increase of public demand for …nal goods, however not met by an increase in output. With lump-sum taxes, private consumption would fall but hours would increase, which would make it possible for …rms to increase their production and would lead to a smoother reaction of in‡ation. Since lump-sum taxes have been ruled out and since the Chamley (1986) result on capital income taxation applies, i.e. taxes on capital income remain ‡at over the cycle (not reported), the government …nances the increase of spending by substantially increasing the tax rate on labor income. As a consequence, 17

hours drop, as well as output, which increases the response of the in‡ation rate. Note that as shown in Schmitt-Grohe & Uribe (2004a), in the presence of price stickiness, the Ramsey planner will resort to the use of taxes on labor income spread out over 40 quarters to minimize its negative e¤ects. This is also shown in the second-order moments below, which report high persistence of policy instruments. The optimal monetary policy consists of rising the nominal interest rate but moderately to limit the depressing e¤ect of the public spending shock on output. The increase in domestic demand drives in‡ation up, which in the open economy translates into an improvement in terms of trade, which damages the competitiveness of domestic …rms. As a consequence, the traditional “real exchange rate crowding out” e¤ect after a public spending shock applies in this model, since output drops more sharply when the economy is open than in the case of a closed economy. World demand shocks. Finally, Figure 3 plots the IRFs after a unit world demand innovation. Within our framework, the macroeconomic e¤ects of a world demand shock are the same as a terms of trade shock. As a consequence, when the economy is closed, all macroeconomic variables remain ‡at. In an open economy, the risk-sharing condition leads the terms of trade to rise (a real exchange rate appreciation), triggering an expenditure switching e¤ect and boosting the level of domestic private consumption. Notice, however, that the increase of private consumption occurs through the increase of imported …nal goods, at the expense of domestically produced goods, implying a depressing e¤ect on domestic global demand. The shock also modi…es households’ labor supply decisions, both through the channel of private consumption and through what Benigno & de Paoli (2009) call the terms of trade spillover. On the one hand, the real appreciation makes its possible to buy more expensive goods abroad, with a negative impact on domestic labor supply. On the other 18

Figure 3: IRFs after a unit world demand shock under Ramsey policies ( = 0:3). Output

Consumption

Hours worked 0

0.4

-0.1

0.3

% dev.

% dev.

% dev.

0

0.2

-0.05

-0.1

0.1 -0.2 10

20

30

0

40

10

Labour income tax rate

30

40

10

Capital

20

30

40

Real wages (PPI)

3

0.03

0.2

0.01 0

2

% dev.

% dev.

0.02

1

0.15 0.1 0.05

-0.01 10

20

30

0

40

-3 Nominal interest rate x 10

10

20

30

0

40

-3 x 10 PPI inf lation

4 2 0 -2 -4 -6 -8

10

20

30

40

Terms of trade 0 % dev.

α=0.3 α=γ=0

10 % dev.

% dev.

% dev.

20

5 0

10

20 30 Quarters

40

-0.5 -1 -1.5

10

20 30 Quarters

19

40

10

20 30 Quarters

40

hand, the real appreciation undermines the domestic purchasing power of households, pushing them to increase their labor supply. In our setting, the …rst e¤ect dominates and hours drop, driving the real wage up, as well as the marginal production cost of …rms. A moderate in‡ationary stance thus arises and the optimal response of the nominal interest rate consists in a restrictive policy during 5 to 7 quarters, followed by a long lasting expansionary policy over the next 25–30 quarters. Here again, it is noticeable that the path of the monetary policy instrument is very smooth. Since in this case, the optimal tax rate on capital income is almost zero all the time (not reported), the tax rate on labor income adjusts to balance government’s budget. Consequently, the tax rate on labor income increases during 20 quarters and then falls under its steady state level for another 20 quarters. This can be explained as follows. The combined e¤ect of the world demand shock on labor income is positive for 20 quarters (the increase of wages is dominated by the drop of hours), and then negative for the next 20 quarters (the increase of real wages more than compensates the drop of hours). 4.3.2

Trade in consumption and capital goods ( =

= 0:15)

As productivity and public spending shocks deliver qualitatively and quantitatively similar patterns when

=

= 0:15, we only report the dynamics after a world demand shock.

World demand shocks. when

=

Figure 4 plots the IRFs after a unit world demand innovation

= 0:15.

In this case again, the risk-sharing condition leads the terms of trade to rise (a real exchange rate to appreciation), implying an expenditure switching e¤ect depressing the trade balance, which should lead to a fall in output. However, because capital goods are cheaper, a supply driven increase in output more than compensates the consequences of the expenditure switching e¤ect on output after the …rst 2 quarters. This evolution is very clear when looking at the dynamics of the PPI in‡ation rate, which is driven by the marginal cost. Indeed, the latter increases in the very short run and drops because of the cheaper capital goods bought abroad. All in all, trade in capital goods changes 20

Figure 4: IRFs after a unit world demand shock under Ramsey policies ( = Output

Consumption

Hours worked

0.2

0.15

0.05

% dev.

% dev.

% dev.

0.15 0.1

0

= 0:15).

0.1

0 -0.02

0.05 10

20

30

0

40

-0.04 10

Labour income tax rate

20

30

40

10

Capital

20

30

40

Real wages (PPI)

0

-0.02

4

% dev.

% dev.

% dev.

0.2 -0.01

2

0.1 0.05

-0.03 20

30

0

40

10

20

30

0

40

-3 x 10 PPI inf lation

Nominal interest rate 4

0.04 % dev.

0.03 0.02 0.01

20

30

40

Terms of trade

α=γ=0.15 α=γ=0

2

10

0 % dev.

10

% dev.

0.15

0

-0.5 -1 -1.5

-2

0 10

20 30 Quarters

40

10

20 30 Quarters

21

40

10

20 30 Quarters

40

the adjustment pattern of output, hours worked, as well as policy instruments after a world demand shock. Indeed, the optimal response of the nominal interest rate consists in a restrictive policy during 10 quarters, followed by a long lasting expansionary policy over the next 30 quarters. The optimal tax rate on capital income is almost zero all the time (not reported) and the tax rate on labor income drops over the whole to balance the budget, since wages increase substantially.

4.4

Simulations

In this section, we report key moments generated by the model under the Ramsey policy for the benchmark parameterization and compare the results to those in the literature on optimal monetary and …scal policy. Table 2 reports the standard deviations and autocorrelations of key macroeconomic variables and policy instruments both in the closed and the open economy set-up. Table 2: Business cycles moments Standard deviations I II III

I

Autocorrelations II III

Output 1:0000 1:0000 1:0000 0:6962 0:7690 Consumption 0:2080 0:1701 0:1689 0:7579 0:7228 Hours worked 0:1801 0:1938 0:1770 0:7050 0:8708 Labor income taxes 0:2684 0:2488 0:2591 0:7244 0:8247 Capital stock 3:0271 2:9934 3:3022 0:9586 0:9631 Real wages (PPI) 0:3871 0:3437 0:3710 0:7369 0:7366 Nom. int. rate 0:0149 0:0176 0:0211 0:7160 0:6057 PPI in‡ation 0:0171 0:0305 0:0236 0:5544 0:5114 Terms of trade 0:5546 0:6563 0:7028 Capital income taxes 0:0001 0:0000 0:0000 0:7228 0:9040 I: = = 0, II: = 0:3 and = 0, III: = = 0:15 Note: standard deviations are expressed relative to the standard deviation

0:7251 0:7647 0:7783 0:7632 0:9596 0:7449 0:6296 0:5419 0:7032 0:7952 of output.

As observed in the dynamic analysis above, all policy instruments are very smooth, and show high autocorrelation as well as little volatility. Correlations are not reported, but as for optimal monetary policy, the nominal interest rate is weakly procyclical (between 0:7 and 0:4 for closed and open economy respectively) and shows a positive correlation 22

with in‡ation (between 0:41 and 0:43 for closed and open economy respectively). This suggests that both the output and in‡ation are relevant monetary policy targets (as in all studies studying the optimal monetary policy alone when prices are sticky). Finally, notice that the open dimension does not a¤ect optimal policies dramatically. It tends to smooth the paths of consumption, real wages and reduces the volatility of labor income taxes. However, the e¤ect of openness is ambiguous depending on the structure of trade. When only consumption goods are traded, hours are more volatile while the capital stock is less volatile. When both consumption and capital goods are traded in similar proportions, openness tends to reduce the volatility of hours while the volatility of capital increases. Openness also a¤ects the wedge between real factor returns received by households and paid by …rms, characterizing the so-called terms of trade spillover, as already discussed in the previous subsection. In terms of persistence, it is noticeable that openness increases the persistence of …scal instruments and tends to lower the persistence of monetary policy. Overall, the persistence of real variables is increased while that of nominal variables is lowered by opening the economy. Opening the economy, higher volatility of the nominal interest rate, triggered by higher volatility of the PPI in‡ation rate, is thus associated with a lower persistence of the policy instrument. For …scal instruments, as terms of trade spillovers tend to lower the volatility of tax bases, the volatility of tax rates falls along with an increase in persistence. Therefore the optimal policy consists in making policy changes smoother over the business cycle when policy instruments become less volatile. Conversely, when policy instruments need to be more volatile, the optimal policy consists in less persistent policy changes.

5

Sensitivity analysis and extension with public debt

In this section, we proceed to a sensitivity analysis by varying key parameters and discuss the implications of adding public debt in the economy.

23

5.1

Sensitivity analysis

As the values of some parameters are widely debated in the literature, we perform a sensitivity analysis with respect to them. For instance, in the macroeconomic literature, the elasticity of substitution between domestic and foreign goods, , is set between 1 and 2.5, to match the volatility of the trade balance (see Backus et al. (1992)). Similarly, Canzoneri et al. (2007) report that estimates of the Frisch elasticity, 1= , vary between 0.3 and 0.05. In a sense, our baseline calibration is quite conservative. We thus choose to address the robustness of our results to an increase in the value of risk-aversion parameter,

. Finally, the

, is usually calibrated to values between 1 and 6. As we set

= 2 in the baseline calibration, we check the robustness of our results for higher values of . Table 3 reports the standard deviations implied by the model for di¤erent values of , , and . This sensitivity analysis shows that changes in the elasticity of substitution between domestic and foreign goods, , has little e¤ect on the overall volatility in the economy. This is true whatever the structure of trade. Upward changes in the risk-aversion parameter, , lead consumption to be less volatile and therefore require more volatile terms of trade ‡uctuations to meet the risk-sharing condition. As a consequence, the terms of trade spillovers on real wages and hours are stronger, leading these variables to be more volatile. This e¤ect transmits to production costs, and in‡ation, as well as the nominal interest rate, are both more volatile. This pattern remains valid for di¤erent trade structures. The mechanism related to the terms of trade spillover is made very clear when looking at the pattern in the closed economy. In this environment, the volatility of hours does not increase, and the PPI in‡ation rate becomes less volatile when risk-aversion increases. Finally, upward changes in the intertemporal elasticity of substitution of labor,

, lower

the volatility of hours worked. The reduction in the elasticity of labor supply requires more volatile real wages, which in turn implies that consumption is more volatile. Due to the risk-sharing condition, as consumption is more volatile, terms of trade become less volatile, although the e¤ect remains of moderate magnitude. This sensitivity analysis 24

Table 3: Standard deviations –sensitivity analysis =2

= 2:5

Output Consumption Hours worked Labor income taxes Capital stock Real wages (PPI) Nom. int. rate PPI in‡ation Terms of trade Capital income taxes

1.0000 0.2080 0.1801 0.2684 3.0271 0.3871 0.0149 0.0171 – 0.0001

1.0000 0.2080 0.1801 0.2684 3.0271 0.3871 0.0149 0.0171 – 0.0001

Output Consumption Hours worked Labor income taxes Capital stock Real wages (PPI) Nom. int. rate PPI in‡ation Terms of trade Capital income taxes

1.0000 0.1707 0.2069 0.2443 3.0363 0.3370 0.0196 0.0355 0.5138 0.0001

1.0000 0.1736 0.2300 0.2455 3.0864 0.3358 0.0226 0.0416 0.4864 0.0000

Output 1.0000 1.0000 Consumption 0.1600 0.1559 Hours worked 0.1807 0.1878 Labor income taxes 0.2522 0.2480 Capital stock 3.4278 3.5955 Real wages (PPI) 0.3609 0.5583 Nom. int. rate 0.0237 0.0270 PPI in‡ation 0.0303 0.0370 Terms of trade 0.6180 0.5875 Capital income taxes 0.0000 0.0000 I: = = 0, II: = 0:3 and = 0, III: Note: standard deviations are expressed

=3

=5

=3

I 1.0000 1.0000 0.1153 0.2265 0.1550 0.1057 0.3139 0.2734 3.2975 2.8941 0.5247 0.5163 0.0170 0.0121 0.0150 0.0062 – – 0.0007 0.0000 II 1.0000 1.0000 1.0000 0.1643 0.1562 0.1996 0.2013 0.2530 0.1136 0.2611 0.2699 0.2530 3.1111 3.2717 2.9561 0.3863 0.4451 0.4459 0.0195 0.0224 0.0111 0.0346 0.0384 0.0124 0.8307 1.3486 0.5497 0.0000 0.0000 0.0000 III 1.0000 1.0000 1.0000 0.1394 0.1191 0.1738 0.1659 0.1709 0.1095 0.2750 0.2946 0.2646 3.6931 4.7922 3.2548 0.4230 0.5053 0.4826 0.0299 0.0517 0.0181 0.0276 0.0366 0.0076 1.0036 1.7544 0.6499 0.0000 0.0000 0.0000 = = 0:15 relative to the standard deviation 1.0000 0.1642 0.1685 0.2905 3.1477 0.4494 0.0159 0.0158 – 0.0001

25

=5 1.0000 0.2345 0.0748 0.2715 2.8567 0.5583 0.0112 0.0046 – 0.0001 1.0000 0.2131 0.0805 0.2521 2.9612 0.4860 0.0107 0.0075 0.5464 0.0000 1.0000 0.1774 0.0803 0.2655 3.2605 0.5319 0.0178 0.0043 0.6450 0.0000 of output.

shows that our main conclusions are unchanged. Most results are qualitatively robust and the quantitative e¤ect of changes in key parameter values are quite small.

5.2

An extension with public debt

In this subsection, we study the implications of considering non state-contingent public debt in our set-up. Adding nominal non state-contingent public debt provides the planner with an extra instrument, and therefore may a¤ect the behavior of the optimal policy (see for instance Schmitt-Grohe & Uribe (2004a)). The problem remains very similar to the balanced-budget version. Two key relations are a¤ected. First, the budget constraint of agent j becomes, Et fqt;t+1 bt+1 (j)g + bgt+1 (j) + pc;t ct (j) + pk;t kt (j) = rtg bgt (j) + bt (j) +

t

(j) + (1

`;t ) wt `t

(j) + pk;t rk;t kt

1

(j)

where bgt (j) is the holding of nominal public bonds in period t 1 paying a nominal interest rate rtg between period t

1 and t. In addition to the standard …rst-order conditions

described earlier, an additional condition, related to the optimal choice of public bonds, arises. Combining the latter with previous …rst-order conditions, we get a no-arbitrage relation, according to which, in equilibrium, the interest rate paid by the government must be equal to the nominal interest rate on risk-less bonds, i.e. g Et rt+1 = rt :

(10)

Second, the nominal budget constraint of the government becomes bgt+1 = rtg bgt + pt gt where bgt =

Rn 0

`;t wt `t

k;t

(zt

pk;t ) kt

1

(11)

bgt (j) dj.

With two additional variables (bgt and rtg ) and only one additional equilibrium condition (Eq. (10)), the Ramsey problem gains a degree of freedom and now consists in …nding

26

the sequence f

k;t ;

1 `;t ; rt gt=0

that maximizes individuals’welfare. The Ramsey problem

thus writes M ax E0

k;t ; `;t ;rt

1 X t=0

t

Z

n

u (ct (j) ; `t (j)) dj

0

subject to Eqs. (12) to (24) plus Eqs. (10) and (11). Concerning the calibration, we choose to impose a certain level of debt over quarterly GDP in the steady state, i.e.

bg y

= 2:4, corresponding to a 60% debt to annual GDP. This

…gure is broadly consistent with the situation of most the OECD countries. Given these changes, Table 4 reports the key moments both when the economy is closed and open. Table 4: Business cycles moments with public debt Means

Standard deviations Autocorrelations I II III I II III Output 2.1483 1.0000 1.0000 1.0000 0.6597 0.6971 0.6738 Consumption 1.1651 0.2002 0.1924 0.1609 0.7549 0.7167 0.7691 Hours worked 0.6530 0.1180 0.1112 0.1138 0.5401 0.6772 0.5789 Labor income taxes 0.5089 0.1701 0.1000 0.0978 -0.1644 0.0349 0.0026 Capital stock 17.8461 2.7613 2.7902 3.2211 0.9523 0.9540 0.9521 Real wages (PPI) 1.8048 1.0556 0.5307 0.7486 -0.0014 0.7541 0.2592 Nom. int. rate 1.0121 0.0134 0.0140 0.0231 0.4370 0.7053 0.6022 PPI in‡ation 1.0000 0.0276 0.0154 0.0125 -0.3622 -0.0437 -0.4500 Terms of trade 1.0000 – 0.6528 0.7680 – 0.6997 0.6966 Capital income taxes 0.0003 0.0002 0.0000 0.0000 0.1785 0.7728 0.2428 Public debt 5.1560 1.2665 1.1181 1.1635 0.8729 0.9613 0.9292 I: = = 0, II: = 0:3 and = 0, III: = = 0:15 Note: standard deviations are expressed relative to the standard deviation of output.

First, in terms of steady state, the stock of public debt is now positive, meaning that taxes should be higher to sustain the debt payment in the long run. Since the zero capital income taxation result still holds, the tax rate on labor income increases in comparison of the case without public debt. Other variables are broadly una¤ected. Second, in terms of volatility, when the economy is closed, consumption exhibits similar levels of volatility when considering public debt. Hours and labor income taxes are much less volatile while real wages and PPI in‡ation volatilities increase signi…cantly. Now when the economy 27

is open, we observe similar patterns in changes in volatilities induced by allowing for public debt in both cases (II and III in Table 4). Consumption becomes more volatile, while the volatility of hours and labor income taxes shrinks dramatically. Notice that at the same time the volatility of the nominal interest rate goes up. The volatility of real wages increases although not as much as when the economy is closed. Consequently, the dynamics of the PPI in‡ation rate is less volatile in the open economy. Whether trade occurs on the consumption or capital goods markets does not qualitatively change the e¤ects of introducing public debt. In terms of persistence, only policy instruments are a¤ected. As underlined by Schmitt-Grohe & Uribe (2004a), price stickiness induce that Ramsey equilibria exhibit near random walk properties even when debt is nominal and non state-contingent. When prices are sticky, the central planner …nds it costly to in‡ate the price level so that debt does not reach its steady state level and acts as a shock absorber. In our setting, we observe the same kind of adjustment pattern. When allowing for nominal non state-contingent debt, taxes move very little and exhibit almost zero persistence. Public debt is very volatile with a serial correlation close to unity, acting as a shock absorber and transmitting the near random walk properties to many variables in the economy.

6

Conclusion

In this paper, we analyze the joint determination of optimal …scal and monetary policies in a small open economy with sticky prices and capital. Solving for the optimal Ramsey policy, we show that our model nests the standard results about optimal …scal policy (both in the steady state and over the cycle) and optimal monetary policy in a closed-economy set-up. More precisely, we …nd that the optimal tax rate on capital income and the optimal in‡ation tax are zero in the steady state. The tax rate on capital income remains constant over the cycle, while both the nominal interest rate and the tax rate on labor income move, respectively to minimize the distortions implied by nominal rigidities and to bal28

ance the government budget. At the same time, movements of policy instruments are quite persistent to smooth the response of the economy to shocks and minimize the implied distortions (for example on labor supply decisions). Opening the economy does not fundamentally alter the results about optimal monetary and …scal policy, although the so-called terms of trade spillovers arise because of the wedge between factor income (wages and capital rental) received by households and paid by …rms. A simple extension with nominal non state-contingent public debt shows that policy instruments become less volatile and that debt acts as a shock absorber, exhibiting near random walk properties independently of the structure of trade.

References Backus, D. K. & Kehoe, P. J. (1994), ‘Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?’, American Economic Review 84(1), 84–103. Backus, D. K., Kehoe, P. J. & Kydland, F. E. (1992), ‘International Real Business Cycles’, Journal of Political Economy 100(4), 745–775. Benigno, G. & de Paoli, B. (2009), ‘On the International Dimension of Fiscal Policy’, CEPR Discussion Paper No 7232 . Benigno, P. & Woodford, M. (2005), ‘Optimal Taxation in an RBC Model: A LinearQuadratic Approach’, NBER Working Papers No 11029 . Calvo, G. (1983), ‘Staggered Prices in a Utility–maximizing Framework’, Journal of Monetary Economics 12(3), 383–398. Canzoneri, M., Cumby, R. & Diba, B. (2007), ‘The Cost of Nominal Inertia in NNS Models’, Journal of Money, Credit and Banking 39(7), 1563–1586. Chamley, C. (1986), ‘Optimal Taxation of Capital Income in General Equilibrium with In…nite Lives’, Econometrica 54(3), 607–622. Chari, V. V. & Kehoe, P. J. (1999), Optimal Fiscal and Monetary Policy, in J. B. Taylor & M. Woodford, eds, ‘Handbook of Macroeconomics’, Vol. 1, Elsevier, chapter 26, pp. 1671–1745. Corsetti, G. (2006), ‘Openness and the Case for Flexible Exchange Rates’, Research in Economics 60(1), 1–21.

29

Corsetti, G., Dedola, L. & Leduc, S. (2008), ‘High Exchange-rate Volatility and Low Pass-through’, Journal of Monetary Economics 55(6), 1113–1128. Dixit, A. & Stiglitz, J. (1977), ‘Monopolistic Competition and Optimum Product Diversity’, American Economic Review 67(3), 297–308. Faia, E. (2008), ‘Ramsey Monetary Policy with Capital Accumulation and Nominal Rigidities’, Macroeconomic Dynamics 12(S1), 90–99. Galí, J. & Monacelli, T. (2005), ‘Monetary Policy and Exchange Rate Volatility in a Small Open Economy’, Review of Economic Studies 72(3), 707–734. Goldberg, L. & Tille, C. (2008), ‘Macroeconomic Interdependence and the International Role of the Dollar’, NBER Working Paper No 13820 . Judd, K. L. (1985), ‘Redistributive Taxation in a Simple Perfect Foresight Model’, Journal of Public Economics 28(1), 59–83. Lucas, R. & Stokey, N. (1983), ‘Optimal Fiscal and Monetary Policy in an Economy without Capital’, Journal of Monetary Economics 12(1), 55–93. Pappa, E. & Vassilatos, V. (2007), ‘The Unbearable Tightness of Being in a Monetary Union: Fiscal Restrictions and Regional Stability’, European Economic Review 51(6), 1492–1513. Rotemberg, J. & Woodford, M. (1997), An Optimization–based Econometric Framework for the Evaluation of Monetary Policy, in B. S. Bernanke & J. J. Rotemberg, eds, ‘NBER Macroeconomics Annual’. Schmitt-Grohe, S. & Uribe, M. (2004a), ‘Optimal Fiscal and Monetary Policy under Sticky Prices’, Journal of Economic Theory 114(2), 198–230. Schmitt-Grohe, S. & Uribe, M. (2004b), ‘Solving Dynamic General Equilibrium Models Using a Second–order Approximation to the Policy Function’, Journal of Economic Dynamics and Control 28(4), 755–775. Woodford, M. (2000), ‘Pitfalls of Forward-Looking Monetary Policy’, American Economic Review (Papers and Proceedings) 90(1), 100–104. Woodford, M. (2003), Interest and Price, Princeton University Press.

30

A

Recursive formulation of pricing conditions

Putting price setting behavior in a recursive problem, we start from pt (i) = pt p1;t =

1 X

)v Et f

(

v=0

p2;t =

1 X

Using yt (i) =

pt (i) pt

i

yt and p1;t =

=

t

1 X

(i) mct+v g ;

t+v yt+v

(i)g :

uc;t : pc;t

uc;t+v p yt+v mct+v ; pc;t+v t+v

)v Et

(

v=0

p2;t =

t+v yt+v

)v Et f

(

v=0

h

p1;t ; 1 pt p2;t

1 X

uc;t+v p yt+v ; pc;t+v t+v

)v Et

(

v=0

and using a recursive transformation: pt yt mct ; pc;t p Et fp2;t+1 g = uc;t t yt : pc;t

Et fp1;t+1 g = uc;t

p1;t p2;t Finally, de…ning v1;t =

p1;t pc;t p2;t pc;t , and v = ; 2;t pt p1+ t

we get pt = pt

v1;t ; 1 v2;t

where v1;t v2;t

Et v1;t+1

1+ t+1

Et v2;t+1

1 c;t+1

1 t+1 c;t+1

31

mct ; pt = uc;t yt :

= uc;t yt

B

Summary of equilibrium conditions

Equilibrium conditions are expressed in real terms. We therefore de…ne the real wage as ! t = wptt , and real capital rental as t = pztt . Labour supply 1 un;t 1 = (1 ) + s1t (12) n;t ) ! t (1 uc;t Euler equation uc;t+1 (13) 1 = r t Et c;t+1 uc;t Arbitrage between physical and …nancial assets rt = Et

k;t+1

1 + (1

k;t+1 )

(14)

t+1

Risk-sharing condition uc ;t = uc;t Consumer prices index in‡ation

c;t

=

+ (1

(1

t

1

1

) st

1

st st 1

1

st st 1

1

v1;t 1 v2;t

1

)+

!

(15)

1 1

(16)

Capital goods prices index in‡ation

k;t

=

(1

t

)+

!

1 1

(17)

Production prices in‡ation 1 t

v1;t =

+ (1

) 1+ t+1

Et v1;t+1

v2;t =

Et v2;t+1

1 c;t

1 t+1 c;t

=1

(18)

t + uc;t y t mc pt

(19)

+ uc;t y t

(20)

v1;t 1 v2;t

(21)

Production prices dispersion t

=

t 1 t

+ (1

)

Marginal cost and inputs e¢ ciency

mct pt

=

t

! 1t

(1 ! t nt = (1 32

at 1 )1 ) t kt 1

(22) (23)

Goods markets clearing condition yt = (1 + (1

) + st1

) (1 ) (1

) + s1t

1

1

ct + st ct (24)

xt + st xt + gt

Government budget constraint `;t ! t `t

+

k;t

t

1

) + s1t

(1

1

kt

1

= gt

Government budget constraint with nominal non state-contingent public debt bgt+1 pt+1

t+1

= rtg

bgt + pt gt pt

`;t wt `t

33

k;t

(zt

pk;t ) kt

1

(25)