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European Economic Review

European Economic Review 53 (2009) 689–714

0014-29

doi:10.1

� CorE-m1 W2 A

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journal homepage: www.elsevier.com/locate/eer

To react or not? Technology shocks, fiscal policy and welfare inthe EU-3

Jim Malley a, Apostolis Philippopoulos a,b,�, Ulrich Woitek c

a University of Glasgow and CESifo, UKb Athens University of Economics and Business, Greecec University of Zurich and CESifo, Switzerland

a r t i c l e i n f o

Article history:

Received 1 March 2007

Accepted 30 January 2009Available online 19 March 2009

Keywords:

Fiscal policy

Business cycles

Welfare

JEL classification:

E6

H5

21/$ - see front matter & 2009 Elsevier B.V. A

016/j.euroecorev.2009.01.005

responding author at: Department of Econom

ail address: aphil@aueb.gr (A. Philippopoulos

e will use the terms active, feedback, state-c

lso note that policymakers may even follow p

y associated with political distortions. It is no

a b s t r a c t

This paper develops a dynamic stochastic general equilibrium (DSGE) model to examine

the quantitative macroeconomic implications of counter-cyclical fiscal policy for France,

Germany and the UK. The model incorporates real wage rigidity and consumption habits,

as the particular market failures justifying policy intervention. We subject the model to

productivity shocks and allow policy instruments to react to the output gap and the

debt-to-output ratio. A welfare analysis reveals that the most effective instrument-target

combination is to use public consumption to stabilize the output gap. Moreover, welfare

gains from counter-cyclical fiscal policy are much stronger in the presence of wage

rigidities compared with consumption habits. Finally, since active policy and automatic

stabilizers are substitutes, it is possible that relatively undistorted economies may be in

need of countercyclical fiscal action due to inadequate automatic stabilizers.

& 2009 Elsevier B.V. All rights reserved.

The empirical evidence points to the adoption of increasingly countercyclical polices by governments in OECDcountries over the postwar period. (Galı́, 2005)

1. Introduction

Despite the relative neglect of fiscal compared to monetary policy since the 1970s, there has been somewhat of a revivalin the interests of European policymakers and academics for counter-cyclical fiscal stabilization policy (see, e.g. Andréset al., 2008; Andrés and Doménech, 2006, the papers in the CESifo Economic Studies, 2005 volume and the theoreticalpapers on feedback fiscal policy, e.g. Aloi et al., 2003; Guo and Lansing, 1998 and Leeper, 1991). While most theorists seemto prefer the use of automatic stabilizers to active policy rules,1 in practice, as pointed out by e.g. Galı́ (2005) above,policymakers do change their fiscal policies when the economic fundamentals change.

In Europe, this is obviously related to the fact that monetary policy is no longer an option for individual countries, butalso to the recent sustained slow growth in several European economies. At the same time, due to the conditions of theMaastricht Treaty (MT) and the Stability and Growth Pact (SGP), there have also been recommendations in Europe tocorrect public finance imbalances by adding, for instance, the public debt-to-GDP ratio, or the budget deficit-to-GDP ratio,to the set of fundamentals that fiscal authorities should respond to.2

ll rights reserved.

ics, Athens University of Economics and Business, Athens 10434, Greece. Tel.: +30 210 8203357.

).

ontingent and state-dependent interchangeably.

rocyclical policy which further undermines the role of automatic stabilizers. Procyclical policies are

t the aim of this paper to analyze the effects of these policies.

www.sciencedirect.com/science/journal/eerwww.elsevier.com/locate/eerdx.doi.org/10.1016/j.euroecorev.2009.01.005mailto:aphil@aueb.gr

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This renewed interest in counter-cyclical fiscal stabilization policy begs a number of important questions regardingwhich, if any, of the potential fiscal instruments available to policymakers will be able to deliver the desired degree ofmacroeconomic stability. More importantly, will active relative to passive policy be welfare improving?3

With this background in mind, we construct a dynamic stochastic general equilibrium (DSGE) model to examine thequantitative macroeconomic implications, and the resulting welfare implications, of counter-cyclical state-contingent fiscalpolicies for France, Germany and the UK. These policies imply that the fiscal authorities adjust their policy instruments tothe economic situation, in a manner that is beyond the role played by automatic stabilizers. In an attempt to more closelyreplicate the output dynamics that appear in macroeconomic data, our model includes private and public capitaladjustment costs, habit persistence in consumption and real wage rigidity. The latter is particularly relevant for Europe andis one of the market failures justifying policy intervention in our setup.4

Our policy instruments include the three major items of public spending (public consumption, investment andtransfers) as shares of output and the two main sources of tax revenues (income and consumption tax rates). Thegovernment’s allocative role in our setup is the provision of public consumption services that augment household’s utilityand public investment that enhances public capital entering the firm’s production function. When modelling feedbackpolicy, we allow policy instruments to respond to the cyclical state of the economy as measured by the output gap and thedeviation of the public debt-to-GDP ratio from a 60% long-run level (which is the reference level of the SGP). When suchcounter-cyclical reaction is switched off, policymakers follow passive policy which in our setup is defined as automaticstabilization.

We next subject the model to a temporary stochastic productivity shock to evaluate the quantitative welfare implicationsof active versus passive policy. To conduct our general equilibrium welfare analysis, we first follow the linear-quadraticmethod introduced by Rotemberg and Woodford (1997). This approach requires that welfare be calculated using inputsapproximated in the area of the optimal (i.e. social planner’s) steady-state. However, we also use a more general measure, inwhich these inputs are approximated in the area of the non-optimal steady-state (i.e. a long-run that includes market andpolicy distortions). Using these two measures, we quantify welfare under active and passive policy and solve for acompensating consumption supplement that can make welfare equal in the two policy regimes (see, also e.g. Lucas, 1990).

Our main findings are as follows. First, the dependence of tax rates on the output and debt volatility introduces localindeterminacy (i.e. expectations-driven outcomes), at least for a large range of parameter values (see also e.g. Schmitt-Grohé and Uribe, 1997 for the US).5 Second, reaction to public debt can also lead to local indeterminacy. This most likelyoccurs since reaction to public debt requires ceteris paribusa stronger counter-cyclical reaction than reaction to the outputgap.6 Third, using public investment for stabilization yields negligible welfare benefits, probably because its output share isvery small in the data (e.g. 2.9%, 3.0% and 1.1% for Germany, France and the UK, respectively). Fourth, the most effectiveinstrument-target combination is to use public consumption spending, as a share of output, to stabilize the output gap.7

Fifth, welfare gains from counter-cyclical fiscal policy are much stronger in the presence of wage rigidities compared withconsumption habits. Sixth, the welfare gains from active policy are much greater when we work in the area of the distortedversus the socially optimum steady-state. This happens because there are more non-internalized distortions in the formercase, and this naturally increases the scope for active policy. Finally, the largest welfare gains from active policy accrue toGermany followed by France and the UK. This order follows the ranking of (market and policy) distortions across the threecountries. However, the relative distribution of welfare gains may change when we take automatic stabilizers into account.For instance, although the UK is the least distorted economy in our setup, in some cases, it benefits most from counter-cyclical fiscal action because of inadequate automatic stabilizers.

The rest of the paper is organized as follows. Sections 2 and 3 set out the theoretical and parameterized models. Section4 studies the model’s fit relative to the data. The welfare analysis is contained in Section 5 and Section 6 concludes thepaper. Details pertaining to the steady-state, the linearized system, and welfare analysis under the two criteria, arecontained in Appendices A–D, respectively.8

2. The theoretical model

The DSGE model we develop below is populated by a large, but constant, number of identical infinitely lived privateagents (households and firms). The government imposes distorting taxes and issues bonds to finance public consumption,

3 Here we focus on fiscal policy rules, which are in the spirit of Taylor rules in the context of monetary policy.4 Indeed, it is widely believed that real wage rigidity is one of the most important distortions in most European economies (see, e.g. Smets and

Wouters, 2003 and Blanchard and Galı́, 2007).5 Also, tax rates, especially income tax rates, change infrequently via reforms and political debates (see also e.g. King and Rebelo, 1999, p. 974). For

these reasons, tax rates are not state-contingent in our setup. Also, since transfers do not affect the real allocation in our setup, their share in output will

remain constant.6 This explanation is consistent with the result that the possibility of indeterminacy increases with the intensity of countercyclicality (see e.g.

Schmitt-Grohé and Uribe, 1997).7 Note that this is the policy instrument that most economists seem to prefer, at least for fiscal consolidation reasons. Also note that our finding

regarding the relative (in)effectiveness of public investment versus consumption maintains when absolute instead of proportional deviations in public

consumption and public investment are employed.8 Note that further details of the derivations in Appendices A–D, plus model solution and Watson’s measure are available upon request.

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investment and transfers. The government’s policy instruments can follow feedback rules which depend on bothexogenous factors and the endogenous state of the economy. We allow for the well-known persistence of macroeconomicvariables by including capital adjustment costs, habit persistence in consumption and real wage rigidity.

2.1. Households

The preferences of each household, indexed by the superscript h, are given by the intertemporal utility function:

E0X1t¼0

btUht ¼ E0X1t¼0

btUðCht � xCt�1; Lht ;G

c

t Þ (1)

where E0 is the expectations operator; Cht is private consumption of household h at time t; Ct�1 is average (per capita)

lagged-once private consumption; Lht is the leisure time of household h at t; Gc

t is average consumption services provided bythe government at t; 0pxo1 is a habit persistence parameter; and 0obo1 is the rate of time preference. In other words,utility depends positively on consumption relative to an external habit variable, where the latter is assumed to beproportional to average past consumption (see, e.g. Smets and Wouters, 2003; Christiano et al., 2005), leisure time andpublic consumption services.

Instantaneous utility, Uht ; is increasing in its three arguments, concave and satisfies the Inada conditions. We employ thewidely used form:

Uht ¼ ½ðCht � xCt�1Þ

m1 ðLht Þm2 ðGct Þ

1�m1�m2 �1�s1� s (2)

where 0om1, m2, ð1� m1 � m2Þo1 are the weights given to relative consumption, leisure time and public consumptionservices, respectively, and s41 is the degree of risk aversion.

Each household can save in the form of capital, Iht , and government bonds, Dht . It receives interest income, r

kt K

ht , from

accumulated capital and rbt Bht from accumulated bonds, where r

kt and r

bt denote the gross returns to capital and bonds,

respectively. The household divides its time endowment between leisure and work so that Lht þ Hht ¼ 1 at each t.

Labor augmenting technological progress at time t is Zt ¼ zt , where zX1 is a constant growth rate. Given Zt , each householdreceives labor income, wtZtH

ht , per unit of effective time worked. Finally, the household receives dividends paid by

firms, Pht , and average lump-sum transfers paid by the government, Gtr

t . Accordingly, the budget constraint of each household is

ð1þ tct ÞCht þ I

ht þ D

ht ¼ ð1� t

yt Þðr

kt K

ht þwtZtH

ht þP

ht þ r

bt B

ht Þ þ G

tr

t (3)

where 0otyt o1 and 0otcto1 are, respectively, the tax rates on income and private consumption at t. Private holdings ofgovernment bonds and capital grow according to the following evolution equations:

Bhtþ1 ¼ Bht þ D

ht (4)

and

Khtþ1 ¼ ð1� dpÞKht þ I

ht �

ap

2

IhtKht� I

h

Kh

!2Kht (5)

where 0 pdpp1 is a constant depreciation rate, apX0 captures internal adjustment costs on gross private investment, andIh=Kh denotes the ratio of household’s investment to capital in the long run (where this ratio will be stationary in equilibrium;see Section 2.5). Note that, in common with the RBC literature, adjustment costs will be zero in the long run.

Households act competitively by taking market prices, policy variables and economy-wide variables as given. Thus, eachhousehold h chooses fCht , H

ht , L

ht , I

ht , D

ht , K

htþ1, B

htþ1g1t¼0 to maximize (1) subject to (2)–(5), the restriction L

ht þ H

ht ¼ 1 and

initial conditions for Kh0 and Bh0.

9 The first-order conditions include the constraints (3)–(5), the optimality condition forlabor supply, (6a), and the Euler-equations for private capital and government bonds ((6a)–(6b)):

wt ¼�ð@Uht =@H

ht Þð1þ tct Þ

ð@Uht =@Cht Þð1� t

yt Þzt� MRSt (6a)

@Uht =@Cht

1þ tct¼ bEt Zktþ1

@Uhtþ1=@Chtþ1

1þ tctþ1

!" #(6b)

@Uht =@Cht

1þ tct¼ bEt Zbtþ1

@Uhtþ1=@Chtþ1

1þ tctþ1

!" #(6c)

9 We assume that each household takes average lagged-once consumption, Ct�1 ; as given. Thus, this is an ‘‘external habit’’ model (see the discussion

in e.g. Campbell, 1999, p. 1284 for different models of habit formation). Also see Appendix C, for the social planner’s solution that internalizes all variables.

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J. Malley et al. / European Economic Review 53 (2009) 689–714692

where we define

eIht � IhtKht� I

h

Kh

Zktþ1 � ð1� apeIht Þ ð1� tytþ1Þrktþ1 þ

1� dp þ apeIhtþ1 Ihtþ1Khtþ1

!� a

p

2ðeIhtþ1Þ2

1� apeIhtþ1

8>>>>>>>:

9>>>>=>>>>;Zbtþ1 � 1þ ð1� t

ytþ1Þr

btþ1

2.2. Firms

Each firm, indexed by the superscript f , produces an homogeneous final product, Yft , by using private capital, Kft , private

labor, Hft , and average (per firm) public capital, Kg

t .10 The production function of each firm takes the form:

Yft ¼ AtðKft Þa1 ðZtHft Þ

a2 ðKgt Þ1�a1�a2 (7a)

where At is exogenous stochastic productivity (whose motion is specified below) and 0oa1, a2; (1� a1 � a2Þo1 are theproductivity of private capital, labor and public capital, respectively. We follow e.g. Lansing (1998) by assuming CRS in thethree factors.

Firms act competitively by taking market prices, policy variables and economy-wide variables as given. Thus, each firm fchooses Kft and H

ft to maximize

Pft � ð1� tst ÞYft � rkt K

ft �wtZtH

ft (7b)

where �1otsto0 denotes an output subsidy and vice versa for 0otsto1.11 The returns to capital and labor are given by

rkt ¼a1ð1� tst ÞY

ft

Kft(7c)

wt ¼a2ð1� tst ÞY

ft

ZtHft

(7d)

2.3. Wage setting

To avoid further complicating the model but to also help it replicate the stylized facts in Europe regarding inertia inwage adjustment, we follow the setup employed in Blanchard and Galı́ (2007). In particular, we assume that

wt � ðwt�1ÞZðMRStÞ1�Z (8)

where 0pZp0 measures the degree of wage sluggishness and MRSt is given by (6a). The basic idea behind this partial adjustmentmodel is that real wages respond only sluggishly to current conditions in the labor market. As pointed out by Blanchard and Galı́(2007), ‘‘this is a parsimonious way of modeling the slow adjustment of wages to labor market conditions, as found in a variety ofmodels of real wage rigidities, without taking a stand on what is the ‘right’ model’’. In other words, although ad hoc, thisspecification can be consistent with a number of possible sources of rigidity in European labor markets, e.g. institutional, legal andsociopolitical rigidities and safety nets, etc. Blanchard and Katz (1997) review the literature and provide empirical evidence thatwages depend strongly on lagged wages. Finally, notice that this modeling has the following implications: (i) if Z ¼ 0; the standardneoclassical model obtains; (ii) in the steady-state, i.e. when wt ¼ wt�1 ¼ w, it follows that again w ¼ MRS:

2.4. Government

In per capita terms, the within-period government budget constraint is (as above, variables with an upper bar denote percapita quantities):

Gc

t þ Gi

t þ Gtr

t þ ð1þ rbt ÞBt ¼ Btþ1 þ t

yt ðr

kt Kt þwtZtHt þPt þ r

bt BtÞ þ t

stYt þ t

ct Ct (9a)

10 The constant number of firms equals the constant number of households.11 This tax/subsidy will only be used to conduct our Rotemberg–Woodford type of welfare analysis (see Section 5.2). In all other parts, this is set to be

zero. Having this in mind, it will suffice to use a flat rate over time, i.e. tst � ts0 at all t.

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J. Malley et al. / European Economic Review 53 (2009) 689–714 693

where Gc

t , Gi

t , and Gtr

t are, respectively, per capita government consumption, government investment and governmenttransfers at t, and Btþ1 is the end-of-period per capita stock of bonds issued by the government.

12

Government investment, Gi

t , is used to augment the stock of public capital whose motion is given by (in per capita terms)

Kg

tþ1 ¼ ð1� dgÞKgt þ G

i

t �ag

2

Gi

t

Kg

t

� Gi

Kg

!2K

g

t (9b)

where 0pdgp1 is a constant depreciation rate, agX0 captures adjustment costs on gross public investment, and Gi=Kg

denotes the ratio of per capita public investment to public capital in the long run (where this ratio will be stationary inequilibrium; see Section 2.5).

2.5. Decentralized competitive equilibrium

Given the paths of technology and labor augmenting technical progress fAt , Ztg1t¼0, economic policy instruments fGct , G

it ,

Gtrt , tyt , tct , tstg1t¼0 and initial conditions for the state variables ðB0, K0, K

g0, w�1, C�1Þ, a symmetric decentralized competitive

equilibrium (DCE) is defined to be a sequence of allocations fYt ;Ct ; It ;Ht ;Ktþ1;Kgtþ1;Btþ1g1t¼0 and prices frkt ; rbt ;wtg1t¼0 such

that: (i) households maximize utility; (ii) firms maximize profits; (iii) all markets clear; for instance, in the capital, laborand bond markets, Kht ¼ K

ft ¼ Kt , H

ht ¼ H

ft ¼ Ht and B

ht ¼ Bt , respectively; and (iv) the government budget constraint given

by (9a) and the motion equation for public capital in (9b) are satisfied. Note that equilibrium quantities will be denotedwithout the superscripts h and f .

We next transform the relevant quantities to stationary variables by defining xt � Xt=zt , where Xt � ðYt , Ct , It , Kt ; Kgt , Bt ,Gct , G

it , G

trt Þ. Accordingly, small letters will denote quantities in per capita and efficiency units. An exception is ht � Ht ; which

is per capita work time. Also note that we define at � At . Our DCE derived in the previous sub-sections can now be rewrittenin stationary form as follows:

yt ¼ atðktÞa1 ðhtÞa2 ðkgt Þ

1�a1�a2 (10a)

yt ¼ ct þ it þ gct þ git (10b)

zktþ1 ¼ ð1� dpÞkt þ it �ap

2ðeitÞ2kt (10c)

wt ¼ ðwt�1ÞZm2ð1þ tct Þ ct �

xzct�1

� �m1ð1� t

yt Þð1� htÞ

0BB@1CCA

1�Z

(10d)

z ct �xzct�1

� �m1ð1�sÞ�1ð1� htÞm2ð1�sÞðgct Þ

ð1�m1�m2Þð1�sÞ

ð1þ tct Þ

¼ ebEt zktþ1 ctþ1 �xzct

� �m1ð1�sÞ�1ð1� htþ1Þm2ð1�sÞðgctþ1Þ

ð1�m1�m2Þð1�sÞ

ð1þ tctþ1Þ

0BBB@1CCCA

2666437775 (10e)

z ct �xzct�1

� �m1ð1�sÞ�1ð1� htÞm2ð1�sÞðgct Þ

ð1�m1�m2Þð1�sÞ

ð1þ tct Þ

¼ ebEt zbtþ1 ctþ1 �xzct

� �m1ð1�sÞ�1ð1� htþ1Þm2ð1�sÞðgctþ1Þ

ð1�m1�m2Þð1�sÞ

ð1þ tctþ1Þ

0BBB@1CCCA

2666437775 (10f)

zkgtþ1 ¼ ð1� dgÞkgt þ g

it �

ag

2ðegitÞ2kgt (10g)

gct þ git þ g

trt þ ½1þ ð1� t

yt Þrbt �bt ¼ zbtþ1 þ t

styt þ t

yt ð1� tstÞyt þ t

ct ct (10h)

12 Since lump-sum transfers/taxes do not affect real allocations, they will be set to zero. Only in the long run, when the public debt-to-output ratio is

set at a given value (e.g. the reference value of 60% as implied by the SGP), lump-sum transfers/taxes will be non-zero playing the role of the residual

variable that satisfies the government budget constraint.

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J. Malley et al. / European Economic Review 53 (2009) 689–714694

where the stationary returns to capital and labor are:

rkt ¼a1ð1� tst Þyt

kt(10i)

wt ¼a2ð1� tst Þyt

ht(10j)

and where we define

eb � bzð1�m2Þð1�sÞeit � it

kt� i

k; egit � gitkgt � g

i

kg

zktþ1 � ½1� apeit� ð1� tytþ1Þrktþ1 þ 1� d

p þ apeitþ1 itþ1ktþ1� �

� ap

2ðeitþ1Þ2

1� apeitþ18>>>:

9>>=>>;zbtþ1 � 1þ ð1� t

ytþ1Þr

btþ1

In other words, the stationary DCE is defined by the above system of 10 nonlinear stochastic difference equations in fyt ,ct , it , ht , ktþ1, k

gtþ1, r

bt , btþ1;wt , r

kt g1t¼0 for given paths of technology, fatg1t¼0 and the independent policy instruments, fgct , git , gtrt ,

tyt , tct , tstg1t¼0, whose evolution is explained in the next sub-section.

2.6. The motion of technology and fiscal policy instruments

Following most of the RBC literature, we assume that the stochastic process determining at is an exponential first-orderMarkov process

at ¼ að1�raÞ

0 arat�1e

et (11)

where a040 is a constant, 0orao1 is the autoregressive parameter and et�iidð0;s2Þ are random shocks to productivity.Also following e.g. Guo and Lansing (1998), Aloi et al. (2003), Andrés and Doménech (2006), Schmitt-Grohé and Uribe

(2007) and Andrés et al. (2008), the policy instrument rates ðgct=yt ; git=yt ; gtrt =yt ; tyt ; tct , tstÞ follow feedback rules. These rules

consist of an exogenous, or non-state contingent, part and an endogenous, or state-contingent, part. Concerning the latter,we assume that the fiscal authorities can react to deviations of output, yt ; and the public debt-to-output ratio, bt=yt ; fromtheir long-run values, y and b=y, respectively. The output gap, yt=y, is the most common indicator of economic activity,while the public debt-to-output ratio relative to its long-run value, ðbt=ytÞ=ðb=yÞ, is an indicator of public finances.

13 Thevalue of b=y will be assumed to be 60% which is the reference value implicitly required by the SGP. Note that our feedbackpolicy rules are also in accordance with Tanzi’s (2005) suggestion that ‘‘counter-cyclical fiscal policy should not beabandoned in depressions and it could be tried in milder slowdowns when the fiscal accounts of a country are in goodinitial conditions’’. Note finally that, despite the recent loosening of the SGP, national policymakers continue to take intoaccount the fiscal accounts and imbalances.14

For the reasons discussed so far (see the Introduction and footnotes 5, 11 and 12), we assume that gtrt =yt , tyt , tct and tst do

not contain a state-contingent part. We also assume that the non-state-contingent part of all instruments, except tyt ; is a

13 When we undertake the welfare analysis below, we also explore the implications of using the output growth gap instead of the output gap

following Walsh (2003). While we found that the rank ordering of welfare gains across countries generally remains unchanged, the quantitative size of

these gains is uniformly smaller when employing the output growth gap. Hence we do not consider its use further.14 The CESifo DICE Report, 2/2004, pp. 85–86, points to other possible fiscal rules, not considered in this study but which are used or being considered

in various OECD countries. For instance, there can also be ceilings on different items of government spending, or golden rules stating that net government

borrowing should not exceed net public investment. Also, in the EU countries, there is the additional 3% of GDP ceiling on net government borrowing.

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J. Malley et al. / European Economic Review 53 (2009) 689–714 695

constant. This implies the following specific policy rules:

gctyt¼ gc0

yty

� �ggy0 yt=bt

y=b

� �ggb0 yt�1

y

� �ggy1 yt�1=bt�1

y=b

� �ggb1

(12a)

gityt¼ gi0

yty

� �giy0 yt=bt

y=b

� �gib0 yt�1

y

� �giy1 yt�1=bt�1

y=b

� �gib1

(12b)

gtrtyt¼ gtr0 (12c)

tyt ¼ ðty0Þð1�rtÞðtyt�1Þ

rt (12d)

tct ¼ tc0 (12e)

tst ¼ ts0 (12f)

where the constants gc0, gi0, g

tr0 represent the steady-state shares of each component of public spending to output, and the

constants ty0, tc0, ts0 are the steady-state values of the three tax rates.

15 The parameters ggy0, ggb0; g

iy0, gib0, g

gy1; g

gb1, g

iy1, gib1 are

feedback policy coefficients, which can be positive, negative or zero depending on whether the policy instrument rate isused procyclically, countercyclically or acyclically.16

Eqs. (10a)–(10j), jointly with Eqs. (11) and (12a)–(12f), summarize the stationary DCE. Its long-run solution andlinearized version are in Appendices A and B, respectively.17

3. The parameterized model

The model’s structural parameters relating to preferences, production and capital accumulation are next calibratedusing annual data for France, Germany and the United Kingdom from 1970 to 2005. The individual country data areobtained from the OECD, IMF and ECFIN. The OECD databases include: (i) Main Economic Indicators (MEI); (ii) EconomicOutlook (EO); and (iii) International Sectoral Database (ISDB). The IMF data are from the International Financial Statistics(IFS) database. Effective tax rates were obtained from ECFIN.

3.1. Calibrated parameters

The parameters of our model and their calibrated values are listed in Table 1. Average labour’s share, a2, is obtaineddirectly from the ISDB dataset. An approximate value across our three-countries of 0.6 has also been used by Smets andWouters (2003) for the Euro Area aggregate. Private and public capital’s shares, a1 and (1� a1 � a2Þ, respectively, areobtained by decomposing the implied aggregate capital share into private and public shares using average private andpublic investment shares from the EO database. Our implied values for the productivity of public capital, 0.053, 0.046, 0.015for Germany, France and the UK, respectively, are similar in magnitude to those found in, for example, Lansing (1998) forthe US.

Since we do not have data pertaining to capital adjustment costs and given that our specification of these costs is basedon Canova and De Nicoló (2002), we follow their study and Cooper and Haltiwanger (2002) and set ap ¼ 2 for all countries.Note that in the absence of data, we adopt the convention of fixing the same parameter value across countries in order notto bias the subsequent results and analysis. In other words, when the appropriate data exists, we let it define cross-countrydifferences in economic structure.

Given the relationship between the gross real rate of interest and the discount rate, i.e. ð1þ rÞ ¼ 1=b, we use MEI data onex post real interest rates to imply the values of b reported in Table 1. In light of the lack of reliable depreciation rate data,following Smets and Wouters (2003), we set the private one to 10% per annum.18 We also applied the same rate to publiccapital. In all cases, this contributed to producing reasonable private and public capital to output ratios, e.g. private: 2.12,2.56, 2.47 and public: 0.31, 0.31, 0.10 for Germany, France, and the UK, respectively.

Following Kydland (1995, Chapter 5, p. 134), we use data on average hours worked, H, to imply the leisureweight in utility, m2 ¼ 1� h, where h ¼ H=ð7 � 14 � 52Þ is the average share of the total time endowment allocated to workand H is obtained from the EO database. The normalization of H to obtain h follows Jorgenson (1995) and Correia et al.

15 As said above, the flat over time output subsidy, ts0, is zero, except in the Rotemberg–Woodford type of welfare analysis conducted in Section 5.3.The other policy constants are defined in Section 3.2.

16 Given the inherent delays in the conduct of fiscal policy in practice and the presence of habits in our model, we have also added a one-period lag to

the standard setup adopted in the papers cited above. We will examine the welfare implications of the feedback rules both without and with the lag

below.17 We focus on two market distortions (persistence in wages and consumption—both of which are not internalized by private agents) to justify active

policy. Obviously, we exclude many other distortions. For instance, we do not study non-market clearing, capital market imperfections, market power,

nominal rigidities, rule-of-thumb agents, etc. We also assume homogeneous agents and hence solve for symmetric equilibria. As a result, our policy

instruments do not include unemployment benefits, redistributive taxes in the form of progressive taxation, etc.18 Note that for Germany, Heer and Maussner (2005, p. 66) use an annualized depreciation rate of 0.0433. As a robustness check, we also use their rate

when calculating welfare and find that our results do not significantly change.

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Table 1Parameter values.

Parameter Ger Fra UK Definition

0oa1o1 0.359 0.378 0.384 Productivity of private capital0oa2o1 0.588 0.576 0.601 Productivity of laborapX0 2.000 2.000 2.000 Adjustment costs on private capitalagX0 2.000 2.000 2.000 Adjustment costs on public capital0obo1 0.963 0.968 0.976 Rate of time preference0pdgp1 0.100 0.100 0.100 Depreciation rate on public capital0pdpp1 0.100 0.100 0.100 Depreciation rate on private capital0pZp1 0.978 0.959 0.910 Degree of real wage rigidity0ogc0o1 0.200 0.149 0.110 Public consumption to output ratio0ogi0o1 0.029 0.030 0.011 Public investment to output ratio0om1o1 0.308 0.314 0.341 Consumption weight in utility0om2o1 0.592 0.586 0.559 Leisure weight in utilitysX1 2.000 2.000 2.000 Degree of risk aversion0pxo1 0.700 0.700 0.700 Habit persistence parameter0ptc0o1 0.200 0.149 0.110 Indirect tax rate0pty0o1 0.313 0.303 0.287 Direct tax ratezX1 1.016 1.010 1.021 Labor augmenting tech progress

J. Malley et al. / European Economic Review 53 (2009) 689–714696

(1995).19 Our implied leisure weights reported in Table 1 are similar in magnitude to those found in other calibrationstudies, see e.g. Cooley and Prescott (1995) for the US. Given the lack of relevant data, we follow the study by Baier andGlomm (2001) setting the relative weight of public consumption services at ð1� m1 � m2Þ ¼ 0:1 for all countries. Alsofollowing the literature, we set the habit persistence parameter x equal to 0.7 for all countries. For example, this value is themidpoint of the range reported by Smets and Wouters (2003) for the Euro Area and Christiano et al. (2005) for the US andequal to the value reported by Batini et al. (2003) for the UK. The gross rate of labour productivity, z, is calculated usingISDB data.

Finally, we obtain the wage persistence parameter, Z, via maximum likelihood estimation of (B.4) (see Appendix B)using real compensation data from the MEI and IFS databases and the data mentioned above for ct, ht , tyt and tct .

20 Toestimate (B.4), we take log deviations from an HP trend for each of the series and condition on the relevant calibratedparameters described above. All estimates of Z reported in Table 1 above are significant at the 1% level. Our estimates of Zwhich range from 0.91 to 0.98 appear in line with others used in DSGE exercises. For example, Blanchard and Galı́ (2007)set the persistence parameter of lagged once real wage equal to 0.9 and Christoffel and Linzert (2005) set it between 0.9and 0.97.

3.2. Parameters for technology and the policy rules

In the subsequent analysis, we will consider shocks to productivity across countries. Given that the resulting volatility inmacroeconomic aggregates depends on the size of the shock and the degree of persistence in the process drivingtechnology, we normalize these parameters across countries. To understand the differential effects on volatility, we fix ra at0.95 and shock the standard deviation of technology, sa, by 1%.

Unless otherwise defined, the values of gc0, gi0, t

y0 and t

c0 in (12a)–(12f) are given in Table 1.

21 Data for the public spendingas shares of output for each country are from EO. Data for income and consumption tax rates for each country are from theECFIN paper by Martinez-Mongay (2000). The income rates reported in Table 1 are the weighted average of the effective taxrates on gross capital and employed labour.

We now turn to the reaction coefficients in the fiscal policy instruments (see Eqs. (12a), (12b)). Based on the empiricalfinding of Galı́ and Perotti (2005) and Claeys (2006) and the lack of other robust estimates, we will employ a range runningfrom 0 to 0.2 for each country. Finally the AR parameters for the effective tax rates, rt in (12d) were obtained using theECFIN data. The estimates of rt for France, Germany and the UK, respectively, are: 0.944, 0.74 and 0.71 and are allsignificant at the 1% level.

19 The assumption is that 10 h per day are necessary for physical needs ‘‘physiological time’’ and therefore do not count in the total hours available for

the labor-leisure choice, i.e. the remaining 14 h.20 The data for income and consumption tax rates, tyt and tct ; are from Martinez-Mongay (2000).21 To be more precise: (i) the constants gc0, g

i0, t

y0 are given by the data averages for all t; (ii) t

s0 is constant and is set either to zero, or to the value

needed to undo a long-run distortion in the Rotemberg–Woodford welfare measure (in Section 5.2); (iii) tc0 is set either to the data average or to the valueneeded to undo a long-run distortion in the Rotemberg–Woodford welfare measure; and (iv) we set gtr0 ¼ 0 along the transition path (in the long run,where public debt is set to be 0.6 of output, gtr0 becomes residually endogenous to satisfy the government budget constraint (see e.g. (A.10) in the

Appendix).

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4. Model evaluation

We next undertake an empirical assessment of the implications of our above calibration and key modelling choices (i.e.to include habits in consumption, wage rigidity and adjustment costs in private and public capital). To this end, wecompare both the autocovariance and impulse response functions of our model to the data using the Watson (1993) andChristiano et al. (2005) methods, respectively.22 Both measures are based on a comparison of an estimated VAR with our, ineffect, calibrated VAR. The baseline RBC model (with government) is defined as a special case of our model without privateand public capital adjustment costs, real wage rigidity and habits (i.e. Z ¼ ap ¼ ag ¼ x ¼ 0). To understand the implicationsin terms of model fit, we move from this baseline to our full DCE model by adding back in the mechanisms which generatepersistence and thus should have the effect of capturing some of the inertia observed in the actual data.

The data VAR is estimated using annual data from 1970 to 2005.23 To make the VAR model as parsimonious as possible,we estimate a system consisting of output, private investment, private consumption, and wages. Government expenditure(consumption and investment) is calculated using the accounting identity byt ¼ ocbct þoibit þogbgt, and hours can beobtained from the relation bwt ¼ byt � bht . This is possible since we can still obtain the spectral density matrix for the dataincluding output via the following transformation:

U ¼

1og �

oiog �

ocog 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 0 0 �1

0BBBBBBBBB@

1CCCCCCCCCA; Fg;y;i;c;w;hðoÞ ¼ UFy;i;c;wðoÞU0

where oc , oi, and og are the average output weights of the three expenditure components calculated from the data. For theimpulse-response matrix W at lag t, an equivalent transformation gives

Wg;y;i;c;w;hðtÞ ¼ UWy;i;c;wðtÞU0

4.1. Watson measure of fit

The Watson (1993) method allows us to compare characteristics of the actual data corresponding to our model with thedata generated by our model. Watson points out, in the context of models which are calibrated, that it is important not toview the economic model as the data generating process, but rather as an approximation to it. The essence of Watson’sapproach is to determine the size of the stochastic error necessary to reconcile the model-generated covariances with thesample covariances.

To briefly illustrate the main points of the approach, consider an ðn� 1Þ vector of stationary variables xt explained by aneconomic model, and its empirical counterpart yt , with covariance generating functions GxðzÞ and GyðzÞ, respectively.Watson’s method poses the following question, ‘‘How much error would have to be added to xt so that the autocovariancesof xt þ ut are equal to the autocovariances of yt ’’ (see Watson, 1993, p. 1015). This setup implies that the difference betweenthe model and the data can be expressed as ut ¼ yt � xt or GuðzÞ ¼ GyðzÞ þ GxðzÞ � GyxðzÞ � GxyðzÞ where GuðzÞ is thecovariance generating function for the difference between the model and the data.

To obtain the various covariance functions requires that we (i) estimate GyðzÞ from the data; (ii) calculate GxðzÞ from themodel; and (iii) choose GxyðzÞ to minimize the variance of ut , subject to the constraint that GxyðzÞ is positive semi-definite.24

With these calculations in place, we can then derive Watson’s RMSAE. More specifically we compute the ratio of theautospectrum of uj to the autospectrum of yj

25:

RjðoÞ ¼R p�p Guðexpð�ioÞÞjj doR pp Gyðexpð�ioÞÞjj do

(13)

This measure is conceptually similar to the unexplained variance of a standard regression. Although it is not boundedbetween zero and unity, smaller values do imply a better model approximation to the data that larger ones. Finally, notethat in all the results reported below, for any variable j, we use the entire range of the spectrum, i.e. ½�p;p�.26

22 Note that for this exercise, the values of the feedback parameters in various policy rules are set to zero. Once we establish our preferred model

based on the data, we proceed in the next section to examine the welfare implications of the parameterized feedback policy rules.23 As in Watson, the data (in logs) are pre-filtered using the Hodrick and Prescott (1997) filter (smoothing weight: 100). In order to mimic the

characteristics of a Baxter and King (1999) filtered series, Ravn and Uhlig (2002) propose a smaller weight m ¼ 6:25. Since we apply the sametransformation to both the model and the data spectrum, the choice of the smoothing weight is not important.

24 In other words, the spectrum, FxyðoÞ, is positive semi-definite.25 The spectrum is given as FðoÞ ¼ ð1=2pÞGðexpð�ioÞÞ (see, Hamilton, 1994, Sections 3.6, 10.3).26 Note that this measure can also be calculated at any frequency o or between desired ranges ½o1;o2�. Since all variables are equally important when

calculating the ‘‘goodness of fit’’ measure, we weight them equally (see, e.g. Watson, 1993, p. 1018).

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To assess the statistical significance of the fit measure, we generate 1000 replications for each model (non-parametricbootstrap). Based on these replications, we obtain the empirical distribution for the fit measure given in (13), whichwe then compare with the performance of the baseline model ( Z ¼ x ¼ ag ¼ ap ¼ 0). To calculate the model spectrumincluding output, consumption, investment, government expenditure and hours of work from the full calibratedmodel spectrum (12 variables), we use a (5� 12) selection matrix E with ones in the appropriate position such thatFxðoÞ ¼ EF12�12ðoÞE0.27

4.2. Christiano et al. (2005) measure of fit

A complementary exercise, which can shed further light into the issue of model fit, can be carried out using the methodsdeveloped in Christiano et al. (2005). In contrast to the emphasis on autocovariances as carried out above, we nowconcentrate on our model’s impulse responses relative to those of a similarly identified VAR model of the data. Let ŵj be avector of impulse responses for lags 1; . . . ;5 of variable j to an identified technology shock, and wðjÞj the correspondingmodel responses dependent on the calibrated parameter vector j. Christiano et al. propose the weighted sum of squareddeviations:

L ¼ ðŵj �wðjÞjÞV�1ðŵj �wðjÞjÞ

0 (14)

which is minimized with respect to the parameter vector j. We use it to compare the four versions of the model(adjustment cost, habits, wage rigidity, full model) with the baseline RBC model. The matrix V has the variances of the dataimpulse-responses on the diagonal. In our case, they are calculated from 1000 non-parametric bootstrap replications of theestimated VAR. The model impulse responses are calculated in the same way as the model spectrum, using a (5� 12)selection matrix E with ones in the appropriate position.

4.3. Results: autocovariance functions

The results reported in Tables 2a–c are in percent differences from the baseline RBC model. Accordingly, a positive valuedenotes the percent improvement in fit across models and vice versa for negative values. The numbers in parentheses arethe bootstrap standard errors.28

The sign of the results for the French case indicate (excluding adjustment costs for all variables expect hours) animprovement in fit for all variables and all models. Note that the full model fit is uniformly better than the base with thelargest gains accruing to bht and bgt followed by byt, bct bit and bwt .

In the German case, adjustment costs again do not improve model fit relative to the base but as in the French case, thereis a statistically significant improvement in fit for all other variables across the remaining models (except hours in thehabits model). Comparing the full model fit in Table 2b with that in Tables 2a and 2c reveals that the improvement in modelfit for the full model is, on average, quantitatively the largest for Germany followed by France and the UK, respectively.

Finally, the UK case appears to follow a similar pattern to the French and German results. In particular, the adjustmentcosts model leads to a significant deterioration in fit for all variables exceptbit and bct (which are not significant, nsÞ and to aworse fit for hours in the habits model. Otherwise, the remaining results show a statistically significant improvement in fitfor all variables and models.

Overall the results in Table 2 provide an interesting picture and some empirical support for several of our key modellingchoices. First, the presence of both real wage rigidity and habits (except for hours) improves model fit for all countries bothin the marginal and full model cases. Second, despite adjustment costs not performing well when considered in isolation,they do not adversely affect the full model results.

4.4. Results: impulse response functions

To obtain the measures of fit reported in Table 3, we examine the percent difference between L (see Eq. (14)) for each ofour four model specifications and the L for the baseline RBC model. Since lower values of L imply a better fit between themodel and data impulse responses than higher ones, negative values in Table 3 imply the model under consideration fitsthe data better than the baseline RBC model. It is worth noting that the ACF based measure of fit, calculated in Table 2, isbroadly analogous to an overall measure of fit in standard regression analysis and the impulse responses based measurereported below is akin to partial measures of fit. Hence the two measures should be interpreted as complements and notsubstitutes.

The broad message across all three countries is that the marginal models for adjustment costs (except for bgt in France, bctand bwt in Germany and the UK) and habits (except forbit in the UK) do not improve the model fit for the impulse response

27 We follow Uhlig (1999) and filter the model spectrum using the power transfer function of the HP-filter with the same smoothing weight as the

data.28 In Table 2 all results are statistically significant at the 1% level unless otherwise indicated by ns.

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Table 2(a) Model fit France (ACFs); (b) model fit Germany (ACFs); and (c) model fit UK (ACFs).

Adj. costs Habits Wage rigidity Full model

(a)byt �0.076 0.049 2.138 0.855(0.006) (0.004) (0.172) (0.069)bit �0.209 0.239 1.641 0.776(0.024) (0.028) (0.190) (0.090)bct �0.033 0.025 1.943 0.817(0.003) (0.002) (0.186) (0.078)bgt �0.095 0.061 2.670 1.067(0.001) (0.001) (0.029) (0.012)bht 0.109 �0.032 5.188 1.947(0.003) (0.008) (1.353) (0.509)bwt �0.096 0.109 0.538 0.167(0.001) (0.001) (0.007) (0.002)

(b)byt �0.064 0.058 5.119 2.225(0.007) (0.006) (0.522) (0.227)bit �0.250 0.308 3.748 1.852(0.020) (0.025) (0.306) (0.151)bct �0.021 0.029 4.545 2.009(0.003) (0.005) (0.695) (0.307)bgt �0.076 0.069 6.108 2.654(0.005) (0.004) (0.375) (0.163)bht �0.005 �0.009 13.323 5.461(0.002) (0.003) (3.662) (1.501)bwt �0.091 0.105 0.766 0.293(0.003) (0.004) (0.026) (0.010)

(c)byt �0.206 0.047 0.619 0.218(0.070) (0.008) (0.110) (0.039)bit �0.381 0.223 0.663 0.285ð1:468Þns (0.038) (0.112) (0.048)bct �0.089 0.021 0.431 0.171ð0:844Þns (0.005) (0.099) (0.039)bgt �0.330 0.075 0.990 0.349(0.014) (0.002) (0.024) (0.009)bht �0.006 �0.007 1.264 0.414ð0:758Þns (0.003) (0.403) (0.133)bwt �0.287 0.105 0.278 0.057(0.041) (0.004) (0.011) (0.002)

J. Malley et al. / European Economic Review 53 (2009) 689–714 699

based measure. However, the sticky real wage and full models appear to fit better than the base RBC model for the majorityof variables for each country.

If we concentrate on the full model in Tables 2 and 3, the preponderance of evidence is supportive of our key modelingchoices. Accordingly, based on these findings, the stylized facts regarding the key sources of persistence for France,Germany and the UK and the general modelling practice in the DSGE literature, we retain each source of rigidity in theanalysis which follows.

4.4.1. Model versus empirical impulse response functions

To shed further light on the properties of the model relative to the data over the business cycle range, we next presentthe full model’s responses to a one-percent shock to technology (see Fig. 1). The sub-plots in Fig. 1 are constructed using thesame set of variables contained in Tables (2(a)–3(c)) and include plots of the 95% confidence regions for both the model anddata VAR impulse response functions.29 These regions are useful for establishing whether there is a statistically significantdifference between the impulse response functions of the model and the data. The light and dark gray regions in the figuresrepresent the model and the data, respectively. If the regions do not overlap, as shown by white space between the grayregions, then we can conclude that there is a significant difference between the model and the data.

29 Note that 1000 simulations were employed to obtain each confidence region. We use asymmetric 95% confidence intervals based on Davidson and

MacKinnon (1993, p. 766).

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Table 3(a) Model fit France (IRs); (b) model fit Germany (IRs); and (c) model fit UK (IRs).

Adj. costs Habits Wage rigidity Full model

(a)byt 0.12 0.01 �0.09 �0.09bit 0.71 0.17 �0.04 0.06bct 0.45 0.16 �0.21 �0.07bgt �0.14 0.00 1.49 1.49bht 0.15 0.05 �0.17 �0.21bct 2.10 0.40 �0.63 �0.75(b)byt 0.04 0.00 �0.18 �0.17bit 0.03 0.16 0.04 0.34bct �0.07 0.13 �0.26 �0.11bgt 0.04 0.01 0.89 1.15bht 0.02 0.06 �0.55 �0.56bwt �0.10 0.21 �0.86 �0.76(c)byt 0.02 0.00 �0.16 �0.13bit 0.10 �0.01 �0.28 �0.20bct �0.04 0.07 �0.12 �0.06bgt 0.01 0.00 0.41 0.22bht 0.02 0.03 �0.34 �0.26bwt �0.11 0.21 0.02 0.06

1 5 10-0.02

0

0.02

0.04

0.06

FRA

it

1 5 10-0.02

-0.01

0

0.01

0.02

0.03yt

1 5 10-0.04

-0.02

0

0.02

0.04

0.06ct

1 5 10-0.01

0

0.01

0.02

0.03gt

1 5 10-0.01

0

0.01

0.02

0.03ht

1 5 10-0.04

-0.02

0

0.02

0.04wt

it yt ct gt ht wt

it yt ct gt ht wt

1 5 10-0.02

0

0.02

0.04

0.06

GE

R

1 5 10-0.01

0

0.01

0.02

0.03

0.04

1 5 10-0.02-0.01

00.010.020.030.04

1 5 10-0.01

0

0.01

0.02

0.03

0.04

1 5 10-0.01

0

0.01

0.02

0.03

0.04

1 5 10-0.02

-0.01

0

0.01

0.02

0.03

1 5 10-0.02

0

0.02

0.04

0.06

lag

GB

R

1 5 10-0.04

-0.02

0

0.02

0.04

lag1 5 10

-0.04

-0.02

0

0.02

0.04

0.06

lag1 5 10

-0.04

-0.02

0

0.02

0.04

lag1 5 10

-0.01

0

0.01

0.02

0.03

lag1 5 10

-0.04

-0.02

0

0.02

0.04

0.06

lag

Fig. 1. Impulse-responses.

J. Malley et al. / European Economic Review 53 (2009) 689–714700

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J. Malley et al. / European Economic Review 53 (2009) 689–714 701

The results reported in Fig. 1 generally suggest that the data VAR contains relatively more cyclical responses to thetechnology shock whereas the model’s responses are relatively more persistent. However, despite these differences, thereare only two instances where the two sets of impulse response functions significantly differ. These occur for the wage ratein France at lags 1–2 and also in the UK at lag 1. Thus the model impulse responses do not appear to be inconsistent withthe data as depicted by the VAR model.

5. Welfare and fiscal stabilization

We now turn to a formal welfare ranking of the two policy regimes (i.e. passive versus active) to determine whetherpolicymakers should act or not and, if so, by how much and to which state variable. To carry out this analysis, we first solvefor a compensating consumption supplement that can make welfare equal in the two policy regimes (see, e.g. Lucas, 1990).The value of this supplement will provide us with a measure of the welfare difference between the two policy regimesexpressed as a share of private consumption. To calculate this requires a measure of aggregate welfare (defined as expecteddiscounted lifetime utility) under passive and active policy. We will follow two approaches to obtain these measures, bothbased on the popular linear-quadratic local tradition. Then, with these measures in place, we will subject the model to aseries of stochastic productivity shocks (i.e. 1000 per experiment) to understand the quantitative welfare implications ofmoving from passive to active fiscal policy. In light of the necessary condition for fiscal solvency, we will evaluate oneinstrument and one target per experiment.

5.1. Compensating consumption supplement

Let Wp denote the expected discounted lifetime utility that the household enjoys under passive policy, and Wa denotethe same under active policy other things equal. When variables are re-expressed in stationary form (see our notation inSection 2.5), and since Zt ¼ zt , Eqs. (1) and (2) imply

Wj � E0X1t¼0

btUh;jt ¼ E0X1t¼0

ebt cjt �xzcjt�1

� �m1ð1�sÞð1� hjtÞ

m2ð1�sÞ

1� s ðgc;jt Þð1�m1�m2Þð1�sÞ

2666437775 (15)

where the superscripts j ¼ p; a denote outcomes under the passive and active policy regime, respectively, and 0oeb �bzð1�m2Þð1�sÞo1.

Working as in e.g. Lucas (1990), we assume a compensating consumption supplement at each date t under the activepolicy regime that is proportional by z to private consumption under the passive reference regime and makes Wa ¼Wp.Thus, z is such that

Wa ¼ E0X1t¼0

ebt ð1þ zÞm1ð1�sÞ cpt �xzcpt�1

� �m1ð1�sÞð1� hpt Þ

m2ð1�sÞ

1� s ðgc;pt Þð1�m1�m2Þð1�sÞ

2666437775

The relationship between welfare in the active and passive cases can then be given by

Wa

Wp¼ ð1þ zÞm1ð1�sÞ

or

z ¼ Wa

Wp

� �1=m1ð1�sÞ� 1 (16)

Therefore, if z40, there is a welfare gain of moving from passive to active policy and vice versa for zo0. But to obtain anestimate of z, we first need to obtain estimates of Wa and Wp: To do so, we will use two measures of welfare.

5.2. A first welfare measure (socially optimal steady-state)

We start by applying the welfare measure introduced by Rotemberg and Woodford (1997). The advantage of thisapproach is that it addresses a well-recognized problem that a second-order approximation of the within-period utilityfunction may not be consistent with a first-order approximation to the equilibrium solution of the endogenous variablesdue to the presence of linear (deviation) terms in the former (see e.g. Rotemberg and Woodford, 1997; Woodford, 2003,pp. 383–387).

To derive a purely quadratic approximation to utility (see Eq. (C.14) in the Appendix), the method requires that the long-run equilibrium, around which we approximate, is socially optimal. That is, in the long-run, the economy reproduces thereal allocations of the social planner. This is achieved by deriving a set of policy rules for the long-run DCE economy which

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J. Malley et al. / European Economic Review 53 (2009) 689–714702

mimics the allocations of the social planner’s.30 In our setup, this is achieved in the steady-state by: (i) using an outputsubsidy to offset the distortion resulting from the income tax (see Eq. (C.11a)); (ii) using a consumption tax rate to offsetthe distortion from not internalizing habit persistence in the decentralized solution (see Eq. (C.11b));31 and (iii) settingpublic consumption and public investment, as shares of output, to their socially optimal values (see Eqs. (C.10j) and(C.10k)). We then use the times-paths of private consumption, leisure and public consumption, as derived by solvingthe first-order approximate system in Appendix B,32 and substitute them into (C.14) to obtain the expected discountedlifetime utility.

5.3. A second welfare measure (distorted steady-state)

We also obtain estimates of Wa and Wp; and hence z, by working in the area of the distorted long-run DCE. Theadvantage of using this measure of welfare, in contrast to the one described above, is that a distorted steady-state is a morerealistic point around which to approximate welfare. Moreover, given that the effects of distortions on welfare increasewith the size of distortions, we expect the welfare effects of active versus passive policy to be relatively higher when usingthis method.33

To obtain this measure of welfare (see Eq. (D.1) in the Appendix), so that it is comparable with (C.14), we derive asecond-order approximation to utility around the steady-state of the DCE derived in Appendix A. We then use the times-paths of private consumption, leisure and public consumption, as derived by solving the first-order approximate system inAppendix B,34 and substitute them into (D.1) to obtain the expected discounted lifetime utility.

5.4. Welfare results

Before we apply the above two welfare measures, recall that the policy instruments that respond to innovations in thestate of the economy are public consumption and public investment as shares of output (see Eqs. (12a)–(12b)). Moreover,these two shares are linked to the output gap and the public debt-to-output target. The detailed results reported belowconcentrate on the former policy instrument and target only, since public investment yielded negligible welfare resultsquantitatively35 and the debt target led to local indeterminacy36 across a large range of reaction coefficients considered.37

5.4.1. Welfare and size of policy reaction (current output gaps)

Using Eqs. (C.14), which is around the socially optimal steady-state, and (D.1), which is around the distorted steady-state, we first report the welfare implications of active versus passive policy in Figs. 2 and 3, respectively (usinggovernment consumption as a share of output to respond to the current output gap).

Fig. 2 suggests that the welfare gains from counter-cyclical fiscal action are highest for Germany followed by France andthen by the UK Moreover, the welfare gains for each country increase monotonically in the range we consider for highervalues of the policy reaction coefficient, with the highest ones being experienced when ggy0 ¼ 0:2: The rank ordering ofwelfare gains suggested by Fig. 2 is expected and consistent with the extent of non-internalized market distortions alongthe transition path to the socially optimum long-run in each country. For example, examination of Table 1 reveals thatGermany has the highest wage rigidity. 38

If we now turn to results in the area of the distorted long-run in Fig. 3, the first thing to note is that the welfare gainsfrom counter-cyclical fiscal action are greater than their corresponding values in Fig. 2. The larger welfare gains in Fig. 3 canbe explained by the fact that more non-internalized distortions are at work relative to the socially optimal case in Fig. 2.

30 The same procedure has been used in various forms by Rotemberg and Woodford (1997) and more recently by Leith and Wren-Lewis (2006) who

use the output subsidy to undo the distortion resulting from monopolistic power in the long-run.31 Recall that the wage setting Eq. (8) is specified so that there are no wage distortions in the long-run.32 This approximation is also taken around the steady-state of the social planner.33 We are grateful to the Editor and an anonymous referee for suggesting this measure.34 This approximation is also taken around the steady-state of the DCE.35 Perhaps this occurs because its output share is small in all three countries. Also, changes in public investment have direct allocation effects that

mitigate any potential stabilizing benefits.36 This happens because reaction to public debt requires a stronger counter-cyclical action than reaction to the output gap. To understand this,

consider Eq. (12a). Say that yt is above its long-run value and the government has to reduce public spending as share of output to slow the economy.

Ceteris paribus, this can happen even if gt rises; we only need the ratio gt=yt to fall; and vice versa when yt is below its long-run value. Now say that bt=yt is

above its long-run target. Ceteris paribus (i.e. given yt), the required fall in gt=yt implies a fall in gt; and vice versa when bt=yt is below its long-run target. In

other words, the required counter-cyclical action in gt is stronger when we respond to the debt target than to the output gap. Given this, it is not

surprising that reaction to debt creates indeterminacy problems (see also Schmitt-Grohé and Uribe, 1997 who find that more counter-cyclical policies

have a greater likelihood of being indeterminate in equilibrium).37 Note that for the solution linearized around the social planner’s steady-state, the only possible values of ggb0 and g

gb1 which lead to saddle-path

stable solutions are zero. In contrast, for the solution around the distorted steady-state, saddle-path stability can be found for parameter configurations

containing non-zero debt reactions in 12.8%, 8% and 16% of the cases for France, Germany and the UK, respectively. Since we aim to focus on

‘‘implementable’’ rules (see Schmitt-Grohé and Uribe, 1997, p. 1704), we exclude further consideration of the debt target. By focusing only on policies

which robustly deliver unique equilibria, we avoid potential dilemmas for policy makers who might inadvertently parameterize a non-implementable

rule.38 Recall that the other non-internalized market distortion, habits, are calibrated to the same value for all countries, i.e. x ¼ 0:7.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

Net

wel

fare

ζ

Public consumption reaction coefficient (current)

FRA GER UK

Fig. 3. Net welfare from fiscal stabilization of 1% temporary productivity shock (distorted steady state).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.000

0.001

0.002

0.003

0.004

Net

wel

fare

ζ

Public consumption reaction coefficient (current)

Fig. 2. Net welfare from fiscal stabilization of 1% temporary productivity shock (social planner’s steady state).

J. Malley et al. / European Economic Review 53 (2009) 689–714 703

These differences in distortions are now present both in the steady-state and along the transition path. For example,consumption habits were internalized in the steady-state of Fig. 2, but are not in the steady-state of Fig. 3. In addition,public spending is not optimally determined, and the distortions associated with the income and consumption taxes arenot undone, in the steady-state of Fig. 3. Finally, tax rates are not at their socially optimal values along the transition path inFig. 3.39

Another interesting contrast between Figs. 2 and 3 is that the cross-country ordering of welfare gains from active policychanges. For example in Fig. 3, the UK gains the most instead of the least, whilst the German gains are still greater than theFrench. A possible explanation for the change in the UK position around the distorted long-run case is that (with ad hocpolicies and with a relatively small size of government sector) its ability for automatic stabilization is lower than in bothFrance and Germany. For instance, all of its spending shares and tax rates (see Table 1), are relatively lower than thecorresponding magnitudes in France and Germany. Then, in Fig. 3, since automatic stabilizers can do relatively less work inthe UK than in the other two countries considered, there is more scope for gains from active policy intervention for the UK.In other words, although the UK is the least distorted economy in our setup, it is in most need of counter-cyclical fiscalaction because of inadequate automatic stabilizers.40 Germany still benefits more than France since it is more distorted (atboth the market and policy level). In other words, despite its bigger government size and automatic stabilizers, Germany’sdistortions also necessitate a stronger active policy intervention than in France. It should be pointed out that thegovernment size plays its conventional automatic stabilizing role only in Fig. 3, where policies are exogenous. This differsfrom the case in Fig. 2, where the analysis was around a solution in which the government size was optimally chosen onefficiency grounds in a non-stochastic non-distorted long-run environment.

39 Note, however, that there are market distortions (wage persistence and consumption habits) along the transition path in both Figs. 1 and 2.40 Andrés et al. (2008) also find that output and consumption volatility fall with the size of the government sector, provided there is a variety of

frictions.

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Table 4Net welfare from fiscal stabilization of 1% temporary productivity shock.

Reaction coefficient Reaction coefficient lag

Distorted steady-state Optical steady-state

0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20

FRA

0.05 0.021 0.032 0.042 0.052 0.001 0.001 0.002 0.002

0.10 0.031 0.042 0.052 0.061 0.001 0.002 0.002 0.003

0.15 0.041 0.051 0.061 0.070 0.002 0.003 0.003 0.004

0.20 0.050 0.060 0.069 0.079 0.003 0.003 0.004 0.004

GER

0.05 0.068 0.101 0.134 0.166 0.002 0.002 0.003 0.004

0.10 0.098 0.131 0.163 0.194 0.003 0.003 0.004 0.005

0.15 0.129 0.161 0.192 0.221 0.004 0.004 0.005 0.006

0.20 0.158 0.189 0.219 0.248 0.005 0.006 0.007 0.008

UK

0.05 0.075 0.112 0.147 0.182 0.001 0.001 0.001 0.001

0.10 0.111 0.147 0.181 0.215 0.001 0.001 0.001 0.002

0.15 0.146 0.180 0.214 0.247 0.001 0.001 0.002 0.002

0.20 0.180 0.214 0.246 0.278 0.002 0.002 0.002 0.002

J. Malley et al. / European Economic Review 53 (2009) 689–714704

5.4.2. Welfare and size of policy reaction (current and lagged output gaps)

To further explore the welfare implications of active versus passive fiscal policy, we next use the public consumptionreaction function with both current and lagged output gaps (see (12a)). These results are reported in Table 4. The first thingto note is that welfare gains increase when we react to both gaps and raise the degree of reaction, at least in the range ofparameter values reported here. Note that the magnitude of peak welfare gains (in bold) are substantially higher for eachcountry across both welfare measures. Further, note that the within- and across-measure country welfare rankings are thesame as reported in Figs. 2 and 3. Also, welfare around the optimal steady-state are lower than their corresponding valuesaround the distorted one, again as in Figs. 2 and 3. Thus, it appears that reasonable sized welfare gains can be expectedwhen business cycle fluctuations, induced by TFP shocks, are smoothed by using a reaction function for publicconsumption that takes account of both current and past output gaps.

The magnitude of welfare gains from counter-cyclical action are quite small when evaluated around the sociallyoptimum long-run (e.g. they reach 0.008% of private consumption for Germany), but noticeably larger when evaluatedaround the distorted long-run (e.g. they reach 0.278% of private consumption for the UK). We report that the effects in thedistorted long-run case can rise to nearly 1% of consumption when a permanent 1% productivity shock is consideredinstead of the temporary one applied in Table 4.41 We are aware, of course, that there can be combinations of distortionsthat may produce bigger quantitative effects depending on the model considered. Heathcote (2005), for instance, studiesthe effects of changes in tax policy in an economy with heterogeneous agents and credit constraints and finds substantialeffects on aggregate consumption. However, it is worth noting that he focuses on consumption and partly on investment,while our study is based on welfare that is affected by more endogenous variables and hence contains more compositionaleffects. For instance, stabilizing one variable may be achieved at the cost of destabilizing another.

5.4.3. Welfare and size of market distortion

To shed further light into the quantitative relationship between potential welfare gains from active policy and marketdistortions, we next condition on fixed values of the reaction coefficients (i.e. 0.2 for both the current and lagged-onceoutput gap) and allow wage rigidity, Z, to vary in the vicinity of the base calibration reported in Table 1.42 Figs. 4 and 5show, respectively, the case around the socially optimal steady-state and the case around the distorted steady-state. Notethat, in these experiments, the three countries have exactly the same degree of non-internalized market distortions, so theydiffer only in policy distortions, government size and other technology-preferences characteristics. The main result is that

41 If such gains appear small, recall that when Lucas (1990) compares the US economy, which has a capital income tax rate around 0.36, to the Ramsey

case in which the capital income tax rate drops to zero, he finds a welfare gain of around 2%.42 To conserve space, we do not report the experiment for habits as well, since the welfare gains are quantitative negligible for higher values of x in

the optimal case and can actually fall slightly in the distorted case. To explain the latter finding recall that (i) in the long run, private consumption, and

hence utility, decrease in x (see the expression for small u in the Appendix under (C.2)); (ii) consumption habits and the effects of x are not internalized (asthey are in the area of the long run socially optimum); and (iii) since active policy tries to close the gap between current values and long-run values, and

long-run utility decreases with x, it is natural that lifetime utility also decreases with x:

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0.86 0.88 0.9 0.92 0.94 0.96 0.980.000

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Net

wel

fare

ζ

η

FRA GER UK

Fig. 5. Net welfare from fiscal stabilization of 1% temporary productivity shock (distorted steady state).

0.86 0.88 0.9 0.92 0.94 0.96 0.980.00

0.01

0.02

0.03

0.04

Net

wel

fare

ζ

η

Fig. 4. Net welfare from fiscal stabilization of 1% temporary productivity shock (social planner’s steady state).

J. Malley et al. / European Economic Review 53 (2009) 689–714 705

the higher is Z, the higher the gain from active policy in all three countries. This result is intuitive, i.e. the higher the degreeof a particular market distortion, the stronger the argument for counter-cyclical policy action.

Concerning the cross-country ordering of welfare gains from active policy, Fig. 4 is like Fig. 2, although the magnitude ofwelfare differences across countries is smaller than in Fig. 2. In Fig. 4, differences in welfare are driven by countrycharacteristics other than wage and consumption rigidities. In Fig. 5, welfare gains from active policy are highest forGermany again, while the UK gains are now greater than the French. Germany seems to be the most distorted economy,even when we assume that non-internalized market distortions are the same. The UK benefits more than France becauseactive fiscal action makes up for inadequateautomatic stabilizers, as discussed in Fig. 3.

6. Conclusions

In this paper, we studied two of the main responsibilities of the government, namely the stabilization of themacroeconomy, as well as the re-allocation of resources via the provision of public goods and services, in the EU-3.

To this end, we developed a DSGE model which (i) allowed fiscal policy instruments to react to two key fundamentals,i.e. cyclical output and public debt; (ii) justified feedback policy by assuming real wage rigidity, which is widely believed tobe one of the main scleroses in Europe, and persistence in consumption habits; (iii) deliberately employed a minimal setupwhere the only distorting effect from the part of policy was the non-availability of lump-sum policy instruments; (iv)studied how each fiscal policy instrument, as well as the choice between an output gap target and a public finance target,(de)stabilize the economy, where by stabilization we mean both the internal stability of the system and the standarddeviation of macroeconomic variables when the economy is subjected to supply shocks; and (v) calculated the generalequilibrium welfare difference when moving from passive to active policy for each policy instrument and each target.

The main policy messages arising from our study are that (i) policymakers should avoid using tax rates for counter-cyclical policy because it usually leads to indeterminacy; (ii) reaction to a public debt-to-output target can also lead to

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J. Malley et al. / European Economic Review 53 (2009) 689–714706

indeterminacy or produce negligible welfare benefits; (iii) using public investment for stabilization also yields negligiblewelfare benefits, probably because its output share is very small in the data; (iv) the most effective instrument-targetcombination is to use public consumption spending, as a share of output, to stabilize the output gap; (v) the higher thedegree of a wage rigidity, the stronger the argument for counter-cyclical fiscal action; and (vi) since active policy andautomatic stabilizers are substitutes, it is possible that relatively undistorted economies may be in much need of counter-cyclical fiscal action because of inadequate automatic stabilizers.

A natural extension is to introduce more distortions (at market and/or policy level) and reevaluate the desirability ofactive versus passive policy. In general, the design of tax-spending policies, and in particular the policy instruments andtargets (if any) for the government to use in an attempt to fine-tune the economy, remains at the heart of macroeconomics.

Acknowledgments

We would like to thank the editor, Juergen von Hagen, an associate editor and two anonymous referees for constructivesuggestions. We also thank Konstantinos Angelopoulos, Paul De Grauwe, Harris Dellas, Jayasri Dutta, Marco Ercolani, BurkhardHeer, Campbell Leith, Ioana Moldovan, Johann Scharler, Peter Sinclair, Mike Wickens and seminar participants at the Universitiesof Bern, Birmingham, Stirling and the CESifo MacroArea Conference (2006) for helpful comments. All errors are our own.

Appendix A. Steady-state of DCE

In the long-run, all variables in Sections 2.5 and 2.6 do not change and there are no shocks. For any variable xt , let xdenote its long-run value. This implies that a ¼ a0, gc ¼ gc0y, gi ¼ gi0y, gtr ¼ gtr0 y, ty ¼ t

y0, t

c ¼ tc0 and ts ¼ ts0. Straightforwardsubstitutions in Eqs. (10a)–(10j) imply that the long-run solution for ðh; y; c; i; k; kg ; rk;w; rb; gtr0 Þ is

43:

h ¼a2m1ð1� t

y0Þð1� t

s0Þ

a2m1ð1� ty0Þð1� ts0Þ þ m2ð1þ tc0Þ 1�

xz

� �c

y

(A.1)

y ¼ a0a1ebð1� ty0Þð1� ts0Þ

z� ebð1� dpÞ !a1

ðhÞa2gi0

zþ dg � 1

!1�a1�a224 351=a2 (A.2)

c ¼ 1� gc0 � gi0 �

a1ebð1� ty0Þð1� ts0Þðzþ dp � 1Þz� ebð1� dpÞ

!y (A.3)

i ¼a1ebð1� ty0Þð1� ts0Þðzþ dp � 1Þ

z� ebð1� dpÞ !

y (A.4)

k ¼a1ebð1� ty0Þð1� ts0Þ

z� ebð1� dpÞ !

y (A.5)

kg ¼gi0

zþ dg � 1

!y (A.6)

rk ¼a1ð1� ts0Þy

k(A.7)

w ¼a2ð1� ts0Þy

h(A.8)

rb ¼ z�ebebð1� ty0Þ (A.9)

gtr0 ¼ 0:6z 1�1eb

!� gc0 � g

i0 þ t

s0 þ t

y0ð1� t

s0Þ þ

c

ytc0 (A.10)

where eb � bzð1�m2Þð1�sÞ.

43 As discussed in the text, consistent with the SGP, we set b ¼ 0:6y in the long run. This implies that, one of the other fiscal policy instruments has to

become endogenous to satisfy the government budget constraint. Here we choose, to residually determine government transfers as a share of GDP, gtr0 .

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J. Malley et al. / European Economic Review 53 (2009) 689–714 707

Appendix B. First-order approximate DCE

We take a first-order Taylor series expansion of the stationary DCE in Sections 2.5 and 2.6 around steady-state to obtainthe following first-order representation:

byt ¼ bat þ a1bkt þ a2bht þ ð1� a1 � a2Þbkgt (B.1)byt ¼ cybct þ iybit þ gc0bgct þ gi0bgit (B.2)zbktþ1 ¼ ð1� dpÞbkt þ ikbit (B.3)Et bwtþ1 ¼ Zbwt þ ð1� ZÞ

1� xz

� � Et bctþ1 � xz bctþ1� �

þ Etð1� ZÞh

1� h

� �bhtþ1þð1� ZÞty0

1� ty0

!Etbtytþ1 þ ð1� ZÞtc01þ tc0

� �Etbtctþ1 (B.4)

m1ð1� sÞ � 1

1� xz

0BB@1CCA Etbctþ1 � xz bct� �

� m2ð1� sÞh1� h

� �Etbhtþ1

þ ð1� m1 � m2Þð1� sÞEtbgctþ1 � tc01þ tc0� �

Etbtctþ1¼ m1ð1� sÞ � 1

1� xz

0BB@1CCA bct � xz bct� �

� m2ð1� sÞh1� h

� �bht þ ð1� m1 � m2Þð1� sÞbgct � tc01þ tc0� �btct

�ebzð1� ty0Þr

k

" #Etbrktþ1 þ ebz ty0rk

" #Etbtytþ1 þ ap ik ðbit � bktÞ

� apebz

i

k1� dp þ i

k

� � !ðEtbitþ1 � bktþ1Þ (B.5)

zbkgtþ1 ¼ ð1� dgÞbkgt þ gi0 ykg bgit (B.6)gc0bgct þ gi0bgit þ gtr0 bgtrt þ zeb by bbt þ ð1� ty0Þrb bybrbt¼ z b

ybbtþ1 þ ty0 1þ rb by� ts0

� �btyt þ ½ts0 þ ty0ð1� ts0Þ�bytþ ts0ð1� t

y0Þbtst þ tc0 cybct þ tc0 cybtct (B.7)

Etbrktþ1 ¼ Etbytþ1 � bktþ1 (B.8)zeb� 1

!Etbrbtþ1 ¼ zeb� 1þ dp

!Etbrktþ1 � dpty01� ty0

!Etbtytþ1

� ap ik

zeb ðbit � bktÞ þ ap ik 1� dp þ ik� �

ðEtbitþ1 � bktþ1Þ (B.9)bwt ¼ byt � bht (B.10)bctþ1 ¼ bct (B.11)

where for any variable xt , ðxt � xÞ=x ’ lnðxt=xÞ � bxt is a first-order Taylor approximation and x is the long-run value of xt .Note that (B.11) defines an auxiliary variable used for lagged-once consumption in (B.5).

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Log-linearizing the stationary laws of motion for the fiscal policy instruments in Section 2.6 around the steady-stateyields44:

bgct ¼ ð1þ ggy0 � ggb0Þbyt þ ggb0bbt (B.12)bgit ¼ ð1þ giy0 � gib0Þbyt þ gib0bbt (B.13)bgtrt ¼ byt (B.14)btyt ¼ rtbtyt�1 (B.15)

Finally, the log-linearized process for technology in (11) is given by

bat ¼ rabat�1 þ et (B.16)Appendix C. Second-order approximate welfare (socially optimal steady-state)

In this Appendix, we construct a welfare measure by following the methodology introduced by Rotemberg andWoodford (1997). This measure will have the characteristics discussed in Section 5.2. We work as usually in the relatedliterature. Thus, (a) we first derive the second-order approximation of the within-period stationary utility function; (b) wederive the second-order approximation of the economy’s constraints; (c) we use the latter into the approximate utilityderived in (a); (d) we solve the social planner’s problem and then derive the long-run values of the output subsidy and theconsumption tax rate that can help to make the long-run DCE equivalent to the long-run solution of the social planner’sproblem; and (e) we use steps (a–d) to obtain a measure of utility that depends only on quadratic terms.

C.1. Second-order approximation of within-period utility

We start by re-expressing household h’s instantaneous utility function, (2), in stationary form. Using our notation inSection 2.5 and since Zt ¼ zt , we have for (2) omitting h-superscripts in a symmetric equilibrium:

Ut ¼ zð1�m2Þð1�sÞtðct �

xzct�1Þm1ð1�sÞð1� htÞm2ð1�sÞ

1� s ðgct Þð1�m1�m2Þð1�sÞ

264375 (C.1)

Hence, intertemporal welfare in (1) can be re-expressed as

E0X1t¼0ðebÞtut (C.2)

where 0oeb � bzð1�m2Þð1�sÞo1 can be thought as the effective discount rate and we defineut � ðct � ðx=zÞct�1Þm1ð1�sÞð1� htÞm2ð1�sÞðgct Þ

ð1�m1�m2Þð1�sÞ=ð1� sÞ.The second-order approximation of ut around its long-run value, u, is (variables without time subscripts denote long-

run values):

ut ’ uþ@ut@ctþ eb @utþ1

@ct

� �c

� �bct þ @ut@ht

h

� �bht þ @ut@gct

gc� �bgct

þ 12

@ut@ctþ eb @utþ1

@ct

� �c þ @

2ut@c2tþ eb @u2tþ1

@c2t

" #c2

!ðbctÞ2

þ 12

@ut@ht

hþ @2ut

@h2th2

!ðbhtÞ2 þ 1

2

@ut@gct

gc þ @2ut@gc2tðgcÞ2

!ðbgct Þ2

þ @2ut

@ct@htch

!bctbht þ @2ut@ct@gct

cgc

!bctbgct þ @2ut@ht@gct hgc !bhtbgct þ O½3� (C.3)

where for any variable xt , bxt � lnðxt=xÞ; ðxt � xÞ=x ’ bxt þ 12 ðbxtÞ2; at t, we take ct�1 as given, while we take into account howthe choice of ct affects next period’s utility (i.e. as also done by the social planner—see below), and we set the given ct�1 atits long-run value; all partial derivatives in (C.3) are specified by using the functional form in (2) and are then evaluatedat a long-run; O½3� contains all terms of order higher than two (following the literature, from here forward, these terms willbe omitted).

44 To save on space, Eqs. (B.12)–(B.13) are written with response to current targets only.

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C.2. Second-order approximation of economy’s constraints

A second-order approximation of the economy’s stationary constraints around their steady-state is45

byt ¼ bat þ a1bkt þ a2bht þ ð1� a1 � a2Þbkgt (C.4)c

ybct ’ byt � iybit � gcy bgct � giy bgit � 12 cy ðbctÞ2 þ iy ðbitÞ2 þ gcy ðbgct Þ2 þ giy ðbgitÞ2 � ðbytÞ2

� �(C.5)

i

kbit ’ zbktþ1 � ð1� dpÞbkt þþ12 zðbktþ1Þ2 � ð1� dpÞðbktÞ2 � ik ðbitÞ2

� �(C.6)

gi

kgbgit ’ zbkgtþ1 � ð1� dgÞbkgt þþ12 zðbkgtþ1Þ2 � ð1� dgÞðbkgt Þ2 � gikg ðbgitÞ2

� �(C.7)

where gc ¼ gc0y and gi ¼ gi0y.It is straightforward to show that Eqs. (C.4)–(C.7) imply that we have for bct in a second-order approximation:bct ’ y

cbat þ a2bht � gcy bgct � ky zbktþ1 � 1� dp þ a1ykh ibkt � 12 zð1� dpÞ

���k

iðbktþ1 � bktÞ2�� kgy zbkgtþ1 � 1� dg þ ð1� a1 � a2Þykg

� �bkgt � 12 z�

�ð1� dgÞ kg

giðbkgtþ1 � bkgt Þ2�� 12 cy ðbctÞ2 þ iy ðbitÞ2 þ gcy ðbgct Þ2 þ giy ðbgitÞ2 � ðbytÞ2

�� �(C.8)

C.3. Second-order approximation of utility (revisited)

Using (C.8) for bct into (C.3), the latter becomesut ’ uþ

@ut@ctþ eb @utþ1

@ct

� �c

� �y

cbat þ a2bht � gcy bgct � ky zbktþ1

��� 1� dp þ a1y

k

h ibkt � 12

zð1� dpÞ kiðbktþ1 � bktÞ2�� kgy zbkgtþ1n

� 1� dg þ ð1� a1 � a2Þykg

� �bkgt � 12 zð1� dgÞ kg

giðbkgtþ1 � bkgt Þ2�

�12

c

yðbctÞ2 þ i

yðbitÞ2 þ gc

yðbgct Þ2 þ giy ðbgitÞ2 � ðbytÞ2

� ��þ @ut

@hth

� �bhtþ @ut

@gctgc

� �bgct þ 12 @ut@ct þ eb @utþ1@ct� �

c þ @2ut@c2tþ eb @u2tþ1

@c2t

" #c2

!ðbctÞ2

þ 12

@ut@ht

hþ @2ut

@h2th2

!ðbhtÞ2 þ 1

2

@ut@gct

gc þ @2ut@gc2tðgcÞ2

!ðbgct Þ2

þ @2ut

@ct@htch

!bctbht þ @2ut@ct@gct

cgc !bctbgct þ @2ut@ht@gct hgc

!bhtbgct (C.9)which can be used into (C.2). But this expression also includes endogenous linear (deviation) terms, bht , bgct , bktþ1, bkt , bkgtþ1and bkgt :C.4. Social planner’s (SP) problem

We now solve the associated social planner’s (SP) problem. The SP maximizes the representative household’s welfare,(C.2), subject to the production function, (10a), the resource constraint, (10b), and the two capital evolution Eqs. (10c) and(10g). To do so, the planner chooses the paths of all allocations in the economy fc�t , h

�t , i�t , k�tþ1, y

�t , g

c�t , g

i�t , k

g�tþ1g

1t¼0. Note that

to distinguish between the DCEs and the SPs solution, a * superscript is employed for the latter. Also note that, in contrast tothe DCE in Section 2.5: (i) there are no market failures (here in the form of non-internalized wage rigidities andconsumption habits); (ii) there are no policy failures (here in the form of distorting taxes); (iii) public consumption and

45 Note that relevant constraints include the production function, (10a), the resource constraint, (10b), and the two capital evolution Eqs. (10c) and

(10g). Further note that (C.4) is an exact expression.

ARTICLE IN PRESS

J. Malley et al. / European Economic Review 53 (2009) 689–714710

public investment are also chosen optimally instead of being exogenous; and (iv) allocations are directly chosen so that theSP does not face any prices. It is straightforward to show that the solution to this problem gives a system of eight nonlinearstochastic difference equations in fc�t , h

�t , i�t , k�tþ1, y

�t , g

c�t , g

i�t , k

g�tþ1g

1t¼0 for a given path of technology, fatg1t¼0. In the long-run of

this system, where allstationary variables do not change and there are no shocks, we have the following solution:

h� ¼ a2m1

a2m1 þ m21�xz

� �c�

y�

1�ebxz

� �(C.10a)

y� ¼ a0a1eb

z� ebð1� dpÞ !a1

ðh�Þa2gi�0

zþ dg � 1

!1�a1�a224 351=a2 (C.10b)

c� ¼ 1� gc�0 � gi�0 �

a1ebðzþ dp � 1Þz� ebð1� dpÞ

!y� (C.10c)

i� ¼ a1ebðzþ dp � 1Þ

z� ebð1� dpÞ !

y� (C.10d)

k� ¼ a1eb

z� ebð1� dpÞ !

y� (C.10e)

kg� ¼gi�0

zþ dg � 1

!y� (C.10f)

rk� �