operational rule is backward looking and responds to ination and output deviations fromtheir long-run levels.

2013 Elsevier Inc. All rights reserved.

monetary policy in NK frameworks, instead, (e.g. Khan et al., 2003; Schmitt-Groh and Uribe, 2007a,b; Faia, 2008b, 2009),ich is, instead, ofh studies Rt no growt

0164-0704/$ - see front matter 2013 Elsevier Inc. All rights reserved.

q We are very grateful to an anonymous referee for very helpful comments and suggestions on an earlier version of this paper. The usual diapplies. Corresponding author. Tel.: +39 0672595731.E-mail addresses: barbara.annicchiarico@uniroma2.it (B. Annicchiarico), lorenza.rossi@eco.unipv.it (L. Rossi).1 Tel.: +39 0382986483.2 Blackburn and Pelloni (2004, 2005) and Annicchiarico et al. (2011a).3 See also Schmitt-Groh and Uribe, 2004a,b. An exception is given by Mattesini and Nistic (2010) who explore the optimal behavior of the monetary

authorities in an NK model with trend (exogenous) growth.

Journal of Macroeconomics 38 (2013) 274285

Contents lists available at ScienceDirect

Journal of Macroeconomics

journal homepage: www.elsevier .com/ locate/ jmacrohttp://dx.doi.org/10.1016/j.jmacro.2013.10.001usually abstract from growth, so disregarding the interaction between short-run dynamics and growth whinterest for the optimal monetary policy analysis.3 The paper most related to ours is Faia (2008b), whicmonetary policy in a basic NK model with capital accumulation and sticky prices la Rotemberg (1982), buamseyh.

sclaimerThe traditional NK literature features exogenous growth or no growth at all. Yet, business cycle uctuations affectgrowth-enhancing activities and modify the growth trend of the entire economy. However, very few papers analyze theinteraction between growth and uncertainty in the context of monetary models (e.g. Dotsey and Sarte, 2000; Varvarigos,2008). An even smaller subset introduce nominal rigidities, but in the form of one-period nominal wage contracts.2 An excep-tion are Annicchiarico et al. (2011b), who consider an NK model and study the interplay between nominal rigidities, nominaluncertainty and growth under different Taylor rules, but do not study optimal monetary policy. Those papers studying optimalJEL classication:E32E52O42

Keywords:Monetary policyEndogenous growthRamsey problemOptimal simple rules

1. IntroductionOptimal monetary policy in a New Keynesian modelwith endogenous growthq

Barbara Annicchiarico a,, Lorenza Rossi b,1aDepartment of Economics, Law and Institutions, University of Rome Tor Vergata, Via Columbia 2, 00133 Roma, ItalybDepartment of Economics and Management, University of Pavia, Via San Felice 5, 27100 Pavia, Italy

a r t i c l e i n f o

Article history:Received 29 March 2013Accepted 2 October 2013Available online 17 October 2013

a b s t r a c t

We study optimal monetary policy in a New Keynesian (NK) model with endogenousgrowth and knowledge spillovers external to each rm. We nd that, in contrast withthe standard NK model, the Ramsey dynamics implies deviation from full inationtargeting in response to technology and government spending shocks, while the optimal

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In this paper we ll this gap and consider an NK model with endogenous growth a` la Romer (1986) and nominal rigiditiesdue to staggered prices a` la Calvo (1983) to study optimal monetary policy. Since in this paper we want to deviate fromthe mainstream NK model only for the inclusion of an endogenous growth mechanism, we opt for the Calvo settingwhich among the various models of price rigidities is the most widely used in the derivation of New Keynesian PhillipsCurves and represents a key ingredient of the standard NK textbook model (see e.g. Gal, 2008; Walsh, 2010). Moreover,Ascari et al. (2011) provide evidence in favor of the statistical superiority of the Calvo setting with respect to the Rotembergone.4

In particular, we study the Ramsey optimal monetary policy and characterize the monetary policy rules that are optimalwithin a family of implementable and simple rules in a calibrated model of the business cycle under a positive steady-stateination rate. In this respect we depart from the standard NK literature which studies optimal monetary policy in economieswhere long-run ination is nil or there is some form of wide-spread indexation.5 From an empirical point of view, neither ofthese two assumptions is realistic for economies like the United States or the Euro Area. Thus, it is of interest to investigate thecharacteristics of optimal policy in their absence and their relationship with growth.

The economy we consider in this paper features three sources of inefciency which provide a rationale for the conduct ofmonetary policy. The rst two distortions are the ones which characterize the basic NK model, namely: (i) monopolisticcompetition, which generates an average markup, which lowers output with respect to the efcient economy; (ii) nominal

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 275rigidities due to staggered prices, which generate price dispersion. The third source of inefciency is the one thatdifferentiates the present model from the standard NK model, i.e. the presence of knowledge spillovers which are externalto each rm. In other words, a sort of serendipitous learning mechanism characterizes the production activity. In thiscontext, the decentralized equilibrium is Pareto suboptimal and the economy grows at a lower rate than under the allocationthat would maximize the representative households lifetime utility. The following main results characterize our modeleconomy.

First, even in the presence of the additional distortion due to knowledge spillovers, we nd that the Ramsey steady-stateination rate is zero.6 The reason is the following. In the long run, a higher ination rate, by increasing the average markup andby introducing price dispersion, would imply a lower return on capital and a reduced level of economic activity, thus loweringsavings and growth. The increase in consumption and in growth rate more than compensates the increase in hours worked andthus households welfare increases as trend ination decreases.

Second, despite the long-run value of ination is zero, the Ramsey dynamics requires deviation from full ination target-ing in response to technology and government spending shocks. However, the intensity of the reaction crucially depends onthe nature of the shock. Following a positive technology shock the central bank tolerates moderate deviations of the inationrate from its optimal steady state in order to push the short-run economy growth rate toward the efcient one. In this caseoptimality calls for an increase in the real interest rate so as to moderate consumption, foster capital accumulation and sogrowth.7 Also in response to a government spending shock, the optimal monetary requires an increase in the real rate, so as togenerate a fall in consumption and mitigate the expansionary effects of the demand shock.

Finally, the optimal operational monetary rule is backward-looking, features a strong positive reaction to output move-ments and a mild response to ination, contrary to the previous ndings in the literature.8 As will be clear in the paper, allthese results strongly depend on the role played by the endogenous growth mechanism and the implied inefciency due to thepresence of external knowledge spillovers.

Summing up, while the NK literature assumes that growth is an exogenous and independent process with respect to thebusiness cycle, the literature that studies the interplay between growth and business cycle concentrates on the relationshipbetween volatility and growth and disregards the implied optimal monetary policy prescriptions. Thus, to the best of ourknowledge we are the rst to study the monetary policy implication of this setup. 9

The paper proceeds as follows. Section 2 describes the model. Section 3 analyzes the Ramsey optimal policy. Section 4shows results from the search of an optimal operational interest rate rule. Section 5 concludes.

4 We are aware that the Calvo and Rotemberg price-setting mechanisms, despite the strong similarities to a rst order of approximation (provided that thereis no trend ination, as shown by Ascari and Rossi, 2012), may have very different welfare implications at higher order of approximation, even if almostnegligible, when the steady state is distorted (see e.g. Lombardo and Vestin, 2008). Hence, for robustness check and to make our ndings more comparable tothose of Faia (2008b), in a separate appendix, available on the authors webpages, we also consider an NK model with AK technology and Rotemberg pricing.

5 An exception are Schmitt-Groh and Uribe (2007a). The authors show that, by assuming zero steady-state ination or full price indexation, nominalrigidities have no real consequences for economic activity and thus welfare in the long run. Thus, the assumptions of zero long-run ination or indexationshould not be expected to be inconsequential for the form that optimal monetary policy takes.

6 It is well known in the literature that in a model with Calvo pricing the rst two distortions require a zero steady-state ination (see King and Wolman,1999).

7 We will see that this result stands in sharp contrast with that obtained by Faia (2008b) in the context of a NK model with capital accumulation. In this sensewe argue that the explicit introduction of an endogenous growth mechanism itself may be source of non-trivial implications for the optimal monetary policyanalysis. As already anticipated, for robustness check, we have also solved the Ramsey problem under Rotemberg pricing and found that the optimal steady-state ination is zero (as in Faia, 2008b), while the optimal response to productivity and public consumption shock is always counter-cyclical.

8 See for example Schmitt-Groh and Uribe (2004a, 2007a,b)) who, in different models, nd that the optimal interest-rate rule features a muted response tooutput.

9 In a similar framework Vaona (2012) explores the relationship between ination and growth.

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2. A s

level o

In

nal g

2.2. In

where

set itsLet

276 B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285The typical rm, able to reset its price at time t, will choose the price Pt so as to maximize the expected presentdiscounted value of prots given the demand schedule and the marginal cost MCt. At the optimum

PtPt hp

hp 1EtP1

i0nipQt;tiMCti

PtiPt

hpYti

EtP1

i0nipQt;ti

PtiPt

hp1Yti

; 4

where Qt,t+i is the stochastic discount factor used at time t by shareholders to value date t + i prots.

Dene the two articial variables xt EtP1

i0nipQ t;tiMCti

PtiPt

hpYti and zt Et

P1i0n

ipQt;ti

PtiPt

hp1Yti and denote

pt Pt

Pt. The optimal price Eq. (4) now reads as follows

pt hp

hp 1xtzt; 5

where xt can be written recursively as:

xt C1t YtMCt npbEtphpt1xt1; 6while zt can be written as:

zt C1t Yt npbEtphp1t1 zt1; 7taa1 a1a AtZat Pt Ptcapital RKt as given, yields the standard optimality conditions, Wt aMCNj;t Yj;tNj;t and RKt 1 aMCNj;t Yj;tKj;t which, in turn, imply

that real marginal cost, MCt MCNt =Pt , is common to all rms:

MC 1 1 RKt

!1aWt a

: 3optimal price Pj;t otherwise the price is unchanged.MCNj;t denote the nominal marginal cost, the cost minimization, taking the nominal wage rateWt and the rental cost ofacquired through learning-by-doing. In particular, we assume Zt = Kt, where Kt 0 Kj;tdj. Following convention, productivityZt is taken as given by each rm, so that learning takes the form of a pure externality. The term At is an aggregate productivityshock, which follows the following process

logAt 1 qA logA qA logAt1 eA;t; 2with 0 < qA < 1 and eA;t i:i:d:N0;r2A.

Prices are modeled la Calvo. In each period there is a xed probability 1 np that a rm in the intermediate sector canYj;t AtK1aj;t ZtNj;ta;a 2 0;1;A > 0; 1Zt represents an index of knowledge, taken as given by each rm, which is freely available to all rms and which isR 1The market is populated by a continuum of rms acting as monopolistic competitors. We assume that this continuum ofintermediate good-producing rms j 2 [0,1] employ labor Nt and capital Kt from households to produce Yt units of theintermediate good using the following technology:ood producers leads the aggregate price index Pt 0 P pj;t dj .

termediate good-sector and externalitiesintermediate sector, with the standard CES technology: Yt R 10 Y

hp1=hpj;t dj

hp=hp1, with hp > 1 being the elasticity of substi-

tution between differentiated goods. Taking prices as given, the typical nal good producer assembles intermediate goodquantities Yj,t to maximize prots, resulting in the usual demand schedule: Yj;t Pj;t=PthpYt . The zero-prot condition ofR 1 1h 1=1hpeach period, the nal good Yt is produced by perfectly competitive rms, using the intermediate inputs produced by theh i

2.1. Final good-sectorf total factor productivity and government spending, which is assumed to be fully nanced by lump-sum taxes.The economy is described by a standard New Keynesian model with nominal prices rigidities la Calvo (1983), includingan endogenous growth mechanism with serendipitous learning la Romer (1986). There are two sources of uncertainty: theticky price endogenous growth model

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where

Pt h

and th

where

Physic

where

2.4. M

In

t : Nt aggreg

where

whereas a c

2.5. St

cointegrated with Kt+1. We divide variables by the appropriate cointegrating factor and denote the corresponding stationaryvariab

Cap

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 277eRKt 1 a MCtyt; 18wt aMCt ytNt ; 19les with lowercase letters. Eqs. (5), (8), (16), (2) and (28) are already expressed in terms of stationary variables.ital and labor demands are now expressed asIn this economy a number of variables, such as output and consumption. will not be stationary along the balanced-growthpath. We therefore perform a change of variables, so as to obtain a set of equilibrium conditions that involve only stationaryvariables. We note that non stationary variables at time t are cointegrated with Kt, while the same variables at time t + 1areGt is public consumption, fully nanced by lump-sum taxation Tt, and, on balanced growth path, is assumed to evolveonstant fraction of output.

ationary competitive equilibriumDp;t 1 npphpt npphpt Dp;t1: 16

Finally, the following aggregate resource constraint must hold:

Yt Ct It Gt ; 17it is easy to see that Dp,t evolves according to a non-linear rst-order difference equation:Yt AtKtNat Dp;t1; 15arket clearing

equilibrium factor and good markets clear, hence the following conditions are satised for allR 10 Nj;tdj;Kt

R 10 Kj;tdj and YtDp;t

R 10 Yj;t where Dp;t

R 10

Pj;tPt

hpdj is a measure of price dispersion. Using (1)

ate production is found to be:1Rt EtQt;t1; 12

WtPt

lnCtN/t ; 13

C1t bEtC1t1 eRkt1 1 d ; 14where Qt;t1 b CtPtCt1Pt1 is the stochastic discount factor for nominal payoffs and eRkt1 Rkt1=Pt1.d is the depreciation rate of capital. The rst-order conditions for the consumers problem can be written as:Kt1 1 dKt It; 11the gross nominal return on capital, Tt denotes lump-sum taxation and Dt are dividends from ownership of rms.al capital accumulates according to:purchases of riskless one-period bonds, and paying one unit of the num raire in the following period t + 1, while Bt isthe quantity of bonds carried over from period t 1. Rt is the gross nominal return on riskless bonds purchased in periodt;RKt isCt is consumption and Nt labor hours at time t, Kt is physical capital, It denotes investments and Bt+1 represents1 npphp1t 1 np pt 1hp : 8

2.3. Households

The representative household maximizes the following lifetime utility subject to a sequence of ow budget constraints:

E0X1t0

bt logCt lnN1/t1 /

!;/;ln > 0 and b < 1; 9

PtCt R1t Bt1 Bt WtNt Dt RKt Kt PtIt Tt ; 10pt = Pt/Pt1. Finally, the aggregate price level Pt R 10 P

1hpj;t dj

1=1hpevolves according to

npP1hpt1 1 npP1hpt

i1=1hp, that is to say that the price level is just a weighted average of the last periods price level

e price set by rms adjusting in the current period. This equation can be rewritten as follows:

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where y = Y/K and w =W/PK. In terms of stationary variables the price related Eqs. (6) and (7) are

xt c1t ytMCt npbEtphpt1xt1; 20zt c1t yt npbEtphp1t1 zt1: 21

where i = I/K.

Th

Eqs. (5

2.5.1.

rigiditto eac

omy.

Thknowlvalue

10 Schis that plevel of

278 B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285Graham and Snower, 2004).11 Further notice that the marginal productivity of capital, 1 aMCtAtNat Dp;t1, in the absence of these two distortions, would turn out to be 1 aAtNat .mitt-Groh and Uribe (2007a) show that price dispersion is bounded below at one, so that it is always costly in terms of aggregate output. The intuitionrice dispersion causes rms, despite being symmetric, charge different prices and thus produce different levels of output. This, in turn, decreases theaggregate output by Jensen inequality because the elasticity of substitution among goods is larger than one (see Ascari, 2004, King and Wolman (1999),function has constant returns to scale in capital, due to knowledge spillovers, rms will accumulate capital as if they aree third additional source of inefciency differentiates the present model from a standard NK model, i.e. the presence ofedge spilloverswhich are external to each rm. In these circumstances capital accumulation is below its social optimumsince agents do not price the role of capital stock in increasing productivity. Hence, although, the aggregate productionaggregate production since the higher is price dispersion the more inputs are needed to produce a given level of output.10

It should be noted that in this AK setting, since growth is due to external learning, a higher (lower) level of economic activityleads to higher (lower) growth. That is why eliminating the rst two distortions has also positive effects on growth.11intermediate goods sector generates an average markup, which lowers output with respect to the competitive econ-Nominal rigidities due to staggered prices generates price dispersion which, in turn, results in an inefciency loss inThe rst two distortions are the ones which characterize the basic NK model and act as follows. Monopolistic competitionin theale for the conduct of monetary policy: (i) monopolistic competition in the intermediate goods sector; (ii) nominalies due to staggered prices introduced la Calvo (1983); (iii) the presence of knowledge spillovers which are externalh rm.The competitive economy considered so far is distorted. In particular, it features three sources of inefciency providing arationInefciencies of the competitive equilibriumt0exogenous stochastic processes fAt ; gtg1t0.t;pt ;Dp; t; pt ; xt; ztg1t0 that remain bounded in some neighborhood around the deterministic steady state and satisfy), (8), (16), (18)(26), (and) (27), given a sequence of nominal interest rate fRtg1 , initial value for Dp,t1 and a set ofDenition 1. A stationary competitive equilibrium is a sequence of allocations and prices fct; it ; gk; t 1;Nt ; yt ; eRKt ;MCt ;we competitive equilibrium of the economy under study can now be formally dened.

where 0 < qG < 1 and eGt i:i:d:N 0;r2G

.log gt 1 qG log g qG log gt1 eG;t; 28The production function (15) is simply

yt AtNat Dp;t1: 26Finally, the resource constraint of the economy (17) in stationary terms is

yt ct it gt; 27where gt = Gt/Kt and evolves asThe Euler Eq. (12) can be expressed as

c1t EtbRt ct1gk;t1 1 1

pt1; 22

where c = C/K and gk,t+1 = Kt+1/Kt. It should be noted that with growth the economy displays a lower marginal rate ofintertemporal substitution for consumption, which, in turn, implies a lower effective discount factor.

The labor supply (13) can be written as

wt lnctN/t : 23The capital Euler Eq. (14) becomes:

c1t Etb ct1gk;t1 1 eRKt1 1 d : 24

The capital accumulation Eq. (11) becomes

gk;t1 1 d it ; 25

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facing a production function with decreasing returns to scale and therefore investments will be sub-optimal and the econ-omy will grow at a lower level than the optimal one.

In this context the decentralized equilibrium is Pareto suboptimal and the economy grows at a lower rate than under theallocation that would maximize the representative households lifetime utility.

ing pain steaY = 0.1

Simgovernare se

3. Ram

Thagentsa distopossible to combine all constraints in a single implementability constraint, thus, as common in the literature, we follow a hybrid

hp 1 zt

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 279npphp1t 1 1 np

hphp 1

xtzt

1hp 0; 32

xt lnN/1t

a npbEtphpt1xt1 0; 33

npbEtphp1t1 zt1 zt c1t AtNat Dp;t1 0: 34

In addition it should be noted that since public consumption is fully nanced by lump-sum taxes, the government budgetconstraint does not appear in the implementability constraints for the Ramsey planner. Finally, by noticing that in the ab-sence of monetary frictions the nominal interest rate only enters into the consumption Euler Eq. (22), this last condition canbe omitted from the set of constraints. Without monetary frictions the solution to the Ramsey problem consists of a realequilibrium which is determined for given nominal interest rate (see e.g. Faia, 2009; Faia and Rossi, 2013). Putting it differ-ently, Rt is set so as to always satisfy (22), given the outcome of the Ramsey plan.16 In this context, the central bank choosesthe policy instrument, namely the ination rate, to implement the optimal allocation obtained as solution to the Ramseyproblem.

12 All the simulation results have been obtained by following the pure perturbation method devised by Schmitt-Groh and Uribe (2004c). In particular, theresults on the optimal simple rules have been obtained adapting the codes developed by Faia (2008a) and Faia and Rossi (2013).13 On this see Rotemberg and Woodford (1999) and the original microeconometric papers by MaCurdy (1981) and Altonji (1986).14 Sensitivity analysis has been done on alternative preference parameters and results are qualitatively unchanged.15 See Khan et al. (2003), Schmitt-Groh and Uribe (2007a), Faia (2009) for a discussion on welfare analysis with a distorted steady state.16 The Lagrange multiplier associated to this last constraint, in fact, would always be equal to zero.approach in which the competitive equilibrium conditions are summarized via a minimal set of equations. Starting from theoptimality conditions for households and rms and the resource constraint of the economy, namely (5), (8), (16), (18)(26),(and) (27), the number of constraints to the Ramsey planners optimal problem can be reduced by substitution so to have:

bEt1

ct11 alnN

/1t1 ct1a

1 d" #

gk;t1ct

0; 29

ct gk;t1 gt AtNat Dp;t 1 1 d 0; 30

Dp;t 1 nphp xt

hp npphpt Dp;t1 0; 31rameters to have C/Y = 0.65 in steady state and a quarterly growth rate of output of 0.5% along the BGP. This implies thatdy state the public consumption capital ratio g is equal to 0.021, while the output-capital ratio y is 0.1430, so that G/446.ilarly to Schmitt-Groh and Uribe (2007a) the persistence of the technology shock is qa = 0.8556, while that of thement spending shock is qg = 0.87. The standard deviations of productivity and of the government purchases processest equal to ra = 0.0064 and rg = 0.016, respectively.

sey monetary policy

e Ramsey optimal policy is determined by the central bank which maximizes the discounted sum of utilities of allgiven the constraints of the competitive economy. The Ramsey approach allows to study the optimal policy aroundrted steady state, as it is in our model.15 We assume that ex-ante commitment is feasible. In most NK models it is not2.6. Calibration

Starting from the stationary model it is then possible to compute the deterministic steady state of the transformed modelaround which the model is approximated and solved numerically.12 We set the benchmark parameters in line with the exist-ing literature. Time is measured in quarters.

The discount factor b is set to 0.99, so that the annual interest rate is equal to 4%. The initial steady state ination rate isset equal to 4% at annual level. In our calibration we opt to set the Frisch elasticity to the upper end of the microdata esti-mates ranging from 0.05 to 0.5, hence / is equal to 2.13 The parameter ln on labor disutility is calibrated to get a steady statevalue of labor hours equal to 1/3.14 The price elasticity hp is set equal to 6 and the probability that rms do not revise prices np isset equal to 0.75. Labor return to scale a is set equal to 2/3. Finally, capital depreciation rate dis 0.025. We calibrate the remain-

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Let {k1,t, k2,t, k3,t, k4,t, k5,t, k6,t} represent sequences of the Lagrange multipliers on the constraints (29)(34), respectively.Given an initial value for the price dispersion, Dp,t1, and a set of exogenous stochastic processes for productivity and publicconsumption, fAt ; gtg1t0, the allocations plans for the control variables dt fct ; gk;t1;Nt ;pt ;Dp;t ; xt; ztg1t0 and for the co-statevariables Kt fk1;t ; k2;t ; k3;t ; k4;t ; k5;t ; k6;tg1t0 represent an optimal allocation if they solve the following maximization problem:

MinfKtg1t0Maxfdtg1t0E0X1

bt log ct lnN1/t b log gk;t1

!; 35

dy-state ination rate associated with the Ramsey optimal policy turns out to be zero in an NK economy characterized byendogenous growth and knowledge spillovers. This happens because the central bank chooses the ination rate that reducesthe diinefccicalis sligstate i

ThgoodsWolmward-and oinduci

closer

17 The18

280 B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285The optimality of the zero steady-state ination turns out to be conrmed also under alternative parametrization. Results are available from the authorsupon request.19 The average markup, i.e. P

MCN(MCN denoting the nominal marginal cost), can be split in two terms: P

MCN PPi

PiMCN

. The rst term, i.e., PPi , is the price adjustmentgap, i.e. the ratio between the CPI index P and the newly adjusted price Pi , and is shown to decrease with steady state ination. The second term, i.e.,

PiMCN

, is themarginal markup and is shown to increase with steady state ination. King and Wolman (1996) demonstrate that the marginal markup effect generallydominates the price adjustment gap effect, while only for extremely low values of trend ination the opposite is true.to its efcient level.

Ramsey steady-state equilibrium was calculated numerically by using an OLS approach, as described by Schmitt-Groh and Uribe (2012).of economic activity. Second, the price dispersion increases rapidly with trend ination, so that the negative effect of trend ina-tion on output and growth through this channel is quite powerful (see Ascari, 2004; Yun, 2005).

Overall, both the average markup and price dispersion are therefore increasing with trend ination, generating a negativesteady-state relationship between ination and output and so between ination and growth. In such circumstances, theRamsey planner would nd it optimal to set the steady-state ination equal to zero. In addition, in the presence of knowl-edge spillovers which are external to each rm, a zero steady-state ination, by reducing the average markup and by elim-inating price dispersion implies a higher return on capital and thus higher savings, so pushing the steady-state growth ratestortion induced by the monopolistically competitive behavior of the intermediate-goods producers and eliminates theiency resulting from the relative price dispersion, so pushing the steady-state growth rate closer to the efcient one. Spe-ly, given the parametrization used in the previous Section, under the Ramsey plan, the balanced-growth path growth ratehtly higher than under the competitive equilibrium and equal to 0.53% at quarterly level, so conrming that zero steady-nation is benecial for growth.18

e intuition for this result is straightforward. First, as already pointed out, monopolistic competition in the intermediatesector generates an average markup which lowers the level of economic activity and growth. As shown by King andan (1996), with staggered prices the average markup tends to increase with trend ination, since price resetting for-looking rms will set higher prices relatively to their current marginal costs, exactly to offset the erosion of markupsf relative prices that trend ination creates.19 In other words, higher trend ination yields a larger average markup,ng a larger distortion related to the lack of competition in the intermediate good sector and so a lower steady-state levelt0 1 / 1 b

subject to constraints (29)(34), where the discounted sum of utilities of all agents has been expressed in stationary terms(see the Appendix for the full derivation). Since the above maximization problem exhibits forward-looking constraints isintrinsically non-recursive (see Kydland and Prescott, 1980). As common practice in the literature (i.e. Khan et al., 2003),we circumvent this problem by augmenting the policy problem with a full set of lagged multipliers, corresponding to theconstraints exhibiting future expectations of control variables.

The augmented Lagrangian for the optimal policy problem then reads as follows:

MinfKtg1t0Maxfdtg1t0E0X1t0

bt log ct lnN1/t1 /

b1 b log gk;t1

! k1;t1 1ct 1 a

lnN/1t cta

1 d !"(

k1;tgk;t1ct

k2;t ct gk;t1 gt AtNat Dp;t 1 1 d k3;t Dp;t

phpt 1 np 1

phpt

hphp 1

xtzt

hp npDp;t1

!

k4;t npphp1t 1 1 np hp

hp 1xtzt

1hp ! k5;t xt lnN

/1t

a

! k5;t1npbphpt xt k6;t1npphp1t zt

k6;t zt c1t AtNat Dp;t 1 #)

:

3.1. The Ramsey optimal steady state

In what follows we analyze the optimal monetary policy in the long run by looking at the Ramsey optimal steady-stateination rate. This amounts to computing the modied golden rulesteady-state ination, i.e. the steady-state ination rateobtained imposing steady-state conditions ex post on the rst-order conditions of the Ramsey plan.17 We nd that the stea-

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0.3

0.4

0.5

0.6

0.7Consumption

2

4

6

8

10x 10-3 Inflation

0.1

0.15

0.2

0.25Hours

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 2813.2. The Ramsey optimal dynamics

Lets now analyze the dynamic properties of the Ramsey plan in a calibrated version of the model. The dynamic responsesof the Ramsey plan are computed by taking second-order approximations of the set of rst-order conditions around thesteady state.20

Fig. 1 shows the Ramsey optimal impulse response functions to a one percent positive productivity shock for consump-tion, ination, employment, output, rate of growth and nominal interest rates. All results are reported as percentagedeviations from the steady state, except ination, growth and nominal interest rates, which are expressed as percentage-point deviations. As in the competitive economy, output and consumption increase. Ination rate decreases, but the nominalinterest rate is above its long-run level, implying a higher real interest rate and so a counter-cyclical policy. Our ndingsstand in contrast with those of Faia (2008b) who shows that by simply embedding capital into an NK model the optimalRamsey monetary policy turns out to be pro-cyclical in response to a technology shock.21

0 10 20 300

0.1

0.2

0 10 20 30-2

0

0 10 20 300

0.05

0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4Output

0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12Rate of Growth

0 10 20 300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04Nominal interest rate

Fig. 1. Dynamic responses to a 1% increase in productivity under Ramsey monetary policy.To understand the logic of the different behavior of the Ramsey planner in this context, a few remarks are needed. First, asalready emphasized, the serendipitous learning mechanism is the source of an additional distortion which characterizes thiseconomy. Firms, in fact, do not internalize the positive externalities deriving from the use of capital, hence the private mar-ginal productivity of capital is lower than that prevailing with rms fully internalizing the benecial effects of capital utili-zation.22 Thus, in order to reach the second-best allocation, the Ramsey planner creates the conditions to generate an increase inthe real interest rate. This, in turn, would imply a higher growth rate and would place consumption on a higher path of growth.Second, in a growing economy the effective discount factor is lower than in an economy with no growth, implying that agentsdiscount more the future.23 This means that, by inducing an increase in the real interest rate, the central bank would lead to ahigher growth rate, as well as to higher consumption. Finally, the stationarized welfare function (35) depends on both growthrate and consumption. Thus, a higher growth rate, would imply higher consumption and so higher welfare. Further, from thewelfare equation, it is clear that the weight assigned to growth is higher than that assigned to the stationarized level ofconsumption.

Having said this, it comes as no surprise that the Ramsey monetary policy turns out to be counter-cyclical instead ofpro-cyclical in response to a positive technology shock. In these circumstances, in fact, the real interest rate prevailing inthe market does not capture the positive externalities of capital utilization, hence consumption is too high during a periodof expansion triggered by a positive productivity shock. In addition, in the competitive economy the effective discount factorwill be reduced following the higher growth rate, therefore consumers will tend to consume too much instead of exploiting

20 See Schmitt-Groh and Uribe (2004b,c,d).21 For robustness check we have also solved the present model under the assumption of quadratic cost of adjusting prices la Rotemberg (1982) as in Faia(2008b). We nd that the optimal monetary policy still prescribes a counter-cyclical response to both shocks. These additional results are available in a separateappendix on the authors webpages.22 The private marginal productivity of capital is eRKt 1 aMCtyt , while with rms fully internalizing the positive externalities we would have eRKt MCtyt .23 From the Euler Eq. (22) we observe, that the effective discount factor is bgk;t1 , where b captures the relative weight placed on the future versus today and

1gk;t1

captures the fact that, thanks to economic growth, agents expect to enjoy a higher consumption in the future.

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-0.02-0.01

0Consumption

0

5x 10-4 Inflation

0.025

0.03Hours

282 B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285the temporary period of expansion to enhance current and future growth prospects by accumulating more capital andmoving to a higher consumption stream. Thus, the Ramsey planner will nd it optimal to offset these effects, by creatingthe conditions that induce households to build up the capital stock during the early phases of the adjustment process, soboosting growth and, ultimately, moving the economy to a higher balanced growth path.

Fig. 2 shows impulse response functions to a one percent positive government spending shock. The government spendingshock crowds out consumption and investments. The ination and the nominal interest rate responses are such that the realrate is always positive along the adjustment path. In this context, the optimal monetary policy calls for an increase in the realrate so as to moderate the temporary expansionary effects of aggregate demand on output. Intuitively, following an increasein public consumption (a pure waste in this economy), the welfare-maximizing planner will nd it preferable to suffer ashort-run decrease of the growth rate, than a sharp increase in the disutility deriving from non-leisure activity. In this caseour ndings are consistent with Faia (2008b), where following a positive government expenditure shock, the Ramsey plan-

0 10 20 30-0.09-0.08-0.07-0.06-0.05-0.04-0.03

0 10 20 30-20

-15

-10

-5

0 10 20 300

0.005

0.01

0.015

0.02

0 10 20 300

0.005

0.01

0.015

0.02Output

0 10 20 30-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0Rate of Growth

0 10 20 300

0.2

0.4

0.6

0.8

1x 10-3 Nominal interest rate

Fig. 2. Dynamic responses to a 1% increase in public spending under Ramsey monetary policy.ner will nd it optimal to increase the nominal interest rate so to reduce consumption and implying a fall in the price level.Summing up, the optimal Ramsey dynamics requires a deviation from price stability in an economy characterized by

endogenous growth and knowledge spillovers. The central bank tolerates a moderately negative ination rate in order topush the short-run economy growth rate toward the efcient one in response to a positive technology, while increasesthe real interest rate to mitigate the effects of a positive government spending shock on output and labor hours.

Overall, a model with knowledge spillovers, and thus endogenous growth, is characterized by having the capital growthrate into the welfare function. As a consequence, it turns out that a full ination targeting policy is far from being optimal.

4. Optimal operational interest rate rules

Despite the Ramsey policy delivers the optimal policy functions in response to shocks, in practice most of the centralbanks nowadays implement simple feedback interest rate rules. For this reason, we now study the optimal operational inter-est rate rules. Such a rule is obtained by searching, within the class of Taylor-type rules, for the parameters that maximizehouseholds conditional welfare subject to the competitive equilibrium conditions that characterize the model economy. Aswell explained by Schmitt-Groh and Uribe (2004a), Schmitt-Groh and Uribe, 2007a, Schmitt-Groh and Uribe, 2007b andFaia (2008a) among others, this class of rules must satisfy the following four criteria: (a) they must be simple, i.e. they mustinvolve only observable variables; (b) they have to guarantee the uniqueness of the rational expectation equilibrium; (c) theymust be optimal, i.e. they have to maximize the expected life-time utility of the representative agent; (d) they must respectthe zero bound on nominal interest rates.

Following Schmitt-Groh and Uribe (2004a), Schmitt-Groh and Uribe, 2007a, Schmitt-Groh and Uribe, 2007b we focuson the conditional expected discounted utility of the representative agent, that is24:

24 Specically, the conditional measure of welfare assumes an initial state of the economy and allows to take into account the transitional effects from thatinitial condition to the stochastic steady state implied by the policy rule adopted by the monetary authorities. As commonly assumed, the initial state of theeconomy coincides with the deterministic steady state.

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ferentthis pbe pre

plicityboth pcircumrate inthe faccal poductivreal in

Table 1Optimal monetary policy rules.

/p /y /r Conditional welfare

25 Seewage inmuted

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 283, let us consider each shock in turn. An increase in public consumption boosts output and ination, but tends to crowd outrivate consumption and investment, so diminishing current and future growth, and future consumption as well. In thesestances, the positive ination and output coefcients of the optimal rule work in the same direction and the real interestRamsey monetary policy, where we observe an increase in the real interest rate in response to positive demand and supplyshocks. The optimal rule must be designed so as to balance the effects of shocks uncertainty on the main macrovariables,accounting for the benets from faster growth. To understand the optimality of a counter-cyclical behavior, for the sake of sim-from zero. The coefcient relative to the response to past ination is, instead, higher than 1, but not too strong. Fromoint of view this result differs from what is generally found in the literature, where ination targeting policies tend tovail.25 Intuitively, a rule such as (39) entails a strong counter-cyclical component, which is in line with that implied by theThus, optimized interest-rate rule: (i) requires a vigorous response to output and a quite aggressive reaction to ination;(ii) is a backward-looking rule. These results can be explained as follows.

First, it is worth noticing the following. We nd that the coefcient attached to past output is positive and strongly dif-lnRtR

1:3125 ln pt1p

0:9375 ln yt1y

: 39where pis the deterministic balanced growth path value of pt, y is the deterministic BGP value of yt = Yt/Kt. R is the determin-istic BGP value of Rt and /r, /p, /y are policy parameters.

The central bank searches for the optimal rule by maximizing the welfare (37), subject to the constraints represented bythe competitive economy relations. Numerically, the search is conducted over the parameter space given by {/p, /y, /r}. Theparameter i in the Taylor rule is alternatively set to i = 0 for contemporary rules, i = 1 for backward-looking rules and nallyto i = 1 for forward-looking rules. We resort to constrained optimal rules, that is we restrict the grid-search to consider onlyempirically relevant values for the policy parameters /p and /y: we search for /p restricted to lie in the interval [1,3] and for/y in [0,2] with a step of size 0.0625 and for /r restricted in [0,0.9] with a step of size 0.1. As in Schmitt-Groh and Uribe(2007a) we approximate the non-negativity constraint on the nominal interest rate by requiring that a rule must inducea low volatility of the nominal interest rate around its target level, that is we impose the condition: 2rR < R, where rR denotesthe standard deviation of the nominal interest rate. Table 1 summarizes the results.

The optimal operational rule takes the form: t

R t1

Rtip

tiy

with i 0;1;1; 38V0 E0X1t0

bt logCt lnN1/t1 /

!; 36

however, in a model with endogenous growth the value function (36) needs to be stationarized. Thus, after some algebra wend the following expression:

tt log ct lnN1/t1 /

b1 b log gk;t1 bEttt1: 37

where tt Vt 11b lnKt.The analysis of the optimal rules and the welfare comparison with ad-hoc rules is done based on the Taylor-type class of

rules

R R /r p /p y /y" #1/r

Ramsey policy 217.6216Optimal rulesCurrent looking 1.4375 0.0625 0.9000 221.3196Backward looking 1.3125 0.9375 0 217.6375Forward looking 1.3125 0.0625 0.70000 220.0160creases so as to moderate the inationary pressure and re-establish the incentives to invest more. Putting it differently, ine of public consumption shocks it is possible to counteract by trying to increase savings by implementing a counter-cycli-licy and by stabilizing ination variability, so moderating the upward pressure on the markups. Following a positive pro-ity shock, as already argued, the existence of positive externalities in the production function calls for an increase in theterest rate to promote capital accumulation and enhance current and future growth prospects.

for example Schmitt-Groh and Uribe (2007b) who, in a medium scale model, nd that the optimal interest-rate rule responds to current price andation, while it is mostly mute in output, and implies only moderate inertia. In Schmitt-Groh and Uribe (2007a, 2004a) optimized policy rules featureresponse to output and no inertia.

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Second, the optimal rule is backward looking, implying that an inertial response of the nominal interest rate is required tomaximize welfare. According to rule (39), in fact, only past conditions are taken into account in choosing the current policysetting. In addition, the inertial behavior of the interest rate turns out to be necessary for every rule considered. Indeed, asshown in T, the optimal operational rule exhibits a strong level of inertia in interest rates and a muted response to outputwhen the feedback part of the rule is current-looking or forward-looking. Thus, the optimal operational interests rules re-quire either to target past ination and past output, or a high response to past interest rate when the policy is current-look-ing or forward-looking.26 Putting it differently, it is then desirable that agents be able to fully anticipate the central bank policy

rigidities due to staggered price la Calvo. An additional source of inefciency differentiates our model from the standard

284 B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285NK model, i.e. knowledge spillovers which are external to each rm. The decentralized equilibrium is Pareto suboptimaland the economy grows at a lower rate than under the allocation that would maximize the representative householdslifetime utility. We show that despite the optimal long-run value of ination is zero, the Ramsey dynamics requires devi-ation from full ination targeting in response to technology and government spending shocks. However, the intensity ofthe reaction crucially depends on the source of uctuations. Following a positive technology shock the central bank toler-ates moderate deviations of the ination rate below its optimal steady state coupled with a higher nominal rate in order tofoster savings and push up the short-run economy growth rate. In response to a positive government shock, optimalitycalls for an increase in the real interest rate, so as to moderate the effects of the expansionary policy. The optimaloperational monetary rule is found to be backward-looking, featuring a strong response to output deviations and a mildreaction to ination movements. In general, we nd that the inertial behavior of the interest rate turns out to be necessaryfor every rule considered. This history dependent attitude enables the Central Bank to steer expectations in a way thatfacilitates its stabilization target, because of the reduced uncertainty. These results differ from what is generally foundin the literature.

Overall, we nd that macroeconomic stabilization policy must explicitly consider the additional transmission channelintroduced by an endogenous growth mechanism. In this sense, our analysis provides a further step towards the understand-ing of the non-trivial interconnections between macroeconomic uctuations and growth. Therefore, we believe that the NKliterature cannot disregard the additional transmission channel introduced by endogenous growth.

The analysis of the present paper has been deliberately restricted to the analysis of monetary policy in the context of avery simple endogenous growth model. We argue that future research should be oriented to explore in more depth theseissues considering different and more realistic growth models as well as the implications for both monetary and scal policy.

26 As a matter of fact, the higher the coefcient /r the higher the weight assigned to past events, the higher the long-run response to shocks and the moreexpectations are stabilized.27 On this point see Woodford, 2003, chapter 7.28 On the effect that uncertainty on pricing decisions and on the consequently detrimental effect on long-run growth, see Annicchiarico and Pelloni(forthcoming) and also Annicchiarico et al. (2011b).conduct one period earlier. This history dependent attitude enables the monetary authority to manage expectations in a waythat furthers its stabilization target.27 As a matter of fact in a forward-looking model, in which agents expectations about futurepolicy are one of the determinants of current outcomes, the reduced uncertainty of the monetary policy conduct itself affectsthe decision of rms resetting their prices, which will tend to set lower markups so implying an augmented level of economicactivity and higher growth. Notably, in the face of higher uncertainty rms able to reset their price will tend to set a highermarkup than that prevailing in a deterministic context. The existence of nominal rigidities with imperfect competition is, in fact,conducive of a negative relationship between uncertainty and growth, since higher uncertainty tends to boost averagemarkup.28 But why should higher uncertainty lead to higher markups? Intuitively, the Calvos pricing mechanism implies thatproducers resetting their prices will choose a price that is a positive function of the weighted average of current and expectedfuture nominal marginal costs. Diminishing marginal productivity of labor implies that the nominal marginal cost is a convexfunction of labor inputs. By Jensens inequality this, in turn, suggests that a higher variability in labor inputs due to uncertainty,raises average nominal marginal costs and so increases the price set by rms, implying that a higher markup will prevail in theeconomy. Overall, a more inertial interest rate rule implies a lower degree of uncertainty in the economy and thus highergrowth.

Finally, as argued by many authors, both the current and the forward-looking rule are not truly operational. If on the onehand, the forward-looking rules require information on future ination expectation, which is not directly observable, on theother hand the current rule requires information on the current value of ination and output which are not in the informa-tion set of the central bank. Thus, according to these critics, a rule is truly operational only when is based on past informa-tion. Thus, we can claim that the backward-looking rule implied by our model with endogenous growth is not only theoptimal one, but it is indeed the truly operational rule.

5. Conclusions

We consider an NK model characterized by endogenous growth with serendipitous learning la Romer, and nominal

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Appendix A. Welfare measure

The welfare of the typical individual (36) can be written in recursive form as:

Vt logCt lnN1/t bEtVt1: 40

Schmitt-Groh, S., Uribe, M., 2004c. Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal ofEco

SchmittSchmittSchmitt

Vaona,VarvariWalsh,Woodfo

B. Annicchiarico, L. Rossi / Journal of Macroeconomics 38 (2013) 274285 285Yun, T., 2005. Optimal monetary policy with relative price distortions. American Economic Review 95, 89108.A., 2012. Ination and growth in the long run: a new Keynesian theory and further semiparametric evidence. Macroeconomic Dynamics 16, 94132.gos, D., 2008. Ination, variability, and the evolution of human capital in a model with transactions costs. Economics Letters 98, 320326.C.E., 2010. Monetary Theory and Policy. The MIT Press, Cambdridge, MA and London.rd, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton and Oxford.Monetary Policy under Ination Targeting, Central Bank of Chile, pp. 125186.Schmitt-Groh, S., Uribe, M., 2012. An OLS approach to computing Ramsey equilibria in medium-scale macroeconomic models. Economics Letters 115, 128

129.nomic Dynamics and Control 28, 755775.-Groh, S., Uribe, M., 2004d. Optimal scal and monetary policy under imperfect competition. Journal of Macroeconomics 26, 183209.-Groh, S., Uribe, M., 2007a. Optimal, simple, and implementable monetary and scal rules. Journal of Monetary Economics 54, 17021725.-Groh, S., Uribe, M., 2007b. Optimal ination stabilization in a medium-scale macroeconomic model. In: Miskin, F.S., Schmidt-Hebbel, K. (Eds.),1 /By adding and subtracting 11b logKt and

b1b logKt1 we get

Vt logCt lnN1/t1 / logKt

11 b logKt

b1 b logKt

b1 b logKt1

b1 b logKt1 bEtVt1; 41

where we have used the fact that 11b logKt logKt b1b logKt . Collecting terms and dening tt Vt 11b lnKt yield (37)which can be also expressed as:

tt EtX1j0

bj log ctj lnN1/tj1 /

b1 b log gk;t1j

!: 42

References

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doi:10.1093/oep/gpt013.Annicchiarico, B., Pelloni, A., Rossi, L., 2011b. Endogenous growth, monetary shocks and nominal rigidities. Economics Letters 113, 103107.Ascari, G., 2004. Staggered prices and trend ination: some nuisances. Review of Economic Dynamics 7, 642667.Ascari, G., Rossi, L., 2012. Trend ination and rms price-setting: Rotemberg versus Calvo. Economic Journal 122, 11151141.Ascari, G., Castelnuovo, E., Rossi, L., 2011. Calvo vs. Rotemberg in a trend ination world: an empirical investigation. Journal of Economic Dynamics and

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Optimal monetary policy in a New Keynesian model with endogenous growth1 Introduction2 A sticky price endogenous growth model2.1 Final good-sector2.2 Intermediate good-sector and externalities2.3 Households2.4 Market clearing2.5 Stationary competitive equilibrium2.5.1 Inefficiencies of the competitive equilibrium

2.6 Calibration

3 Ramsey monetary policy3.1 The Ramsey optimal steady state3.2 The Ramsey optimal dynamics

4 Optimal operational interest rate rules5 ConclusionsAppendix A Welfare measureReferences