Nonlinearity in monetary policy: A reconsideration of the opportunistic approach to disinflation

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Structural Change and Economic Dynamics 20 (2009) 288–300

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Nonlinearity in monetary policy: A reconsideration of theopportunistic approach to disinflation�

Massimiliano Marzoa,∗, Ingvar Stridb, Paolo Zagagliac

a Department of Economics, Università di Bologna, Piazza Scaravilli 2; 40126 Bologna, BO, Italyb Department of Economic Statistics and Decision Support, Stockholm School of Economics, Sveavägen 65; SE-113 83 Stockholm, Swedenc Modelling Division, Sveriges Riksbank, Brunkebergstorg 11; SE-103 37 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 19 February 2008Received in revised form 7 July 2009Accepted 7 August 2009Available online 30 September 2009

JEL classification:

a b s t r a c t

The proponents of the ‘opportunistic’ approach to disinflation suggest that, when infla-tion is close to the target, the central bank should not counteract inflationary pressures.Orphanides and Wilcox (2002) formalize this idea through a simple policy rule that pre-scribes a nonlinear adjustment to a history-dependent target for inflation. This embodies aregime change in monetary policy, which reacts to inflation only when this is far from theinflation target. Here we study the opportunistic approach in a New-Keynesian model withsizeable nominal and real rigidites in the form of a positive money demand and adjust-

E31E61

Keywords:DisinflationMonetary policyDynamic models

ment costs for investment. We find that the welfare gains delivered by the opportunisticrule arise from the time-varying inflation target, when welfare is measured by a quadraticapproximation of household utility. The nonlinear zone of inaction on inflation improveswelfare outcomes only when a central bank loss function with the absolute value of theoutput gap is used, as proposed by Orphanides and Wilcox (2002).

Regime change

“There are genuine issues here about going from mod-erate inflation to price stability. The public’s utility

� Marzo is grateful to MIUR (research grant) and to Università diBologna for the support coming from ‘Progetto Pluriennale d’Ateneo’.Zagaglia acknowledges partial financial support from Stockholm Univer-sity and MIUR. We are very grateful to the managing editor (H. Hagemann)and three anonymous referees for very useful and constructive com-ments that helped improving the paper to a major extent. We also thankKai Leitemo, Ulf Söderström and the participants to the 2006 SCE Meet-ing in Cyprus for helpful suggestions. Robert Kollmann provided verythought-provoking comments on an earlier draft that were key to improv-ing the quality of the paper. This paper is a revision of previous draftscirculated under the titles ‘Optimal Opportunistic Monetary Policy in aNew-Keynesian Model’ and ‘Optimal Simple Nonlinear Rules for Mone-tary Policy in a New-Keynesian Model’. The views expressed herein aresolely the responsibility of the authors and should not be interpreted asreflecting the views of the Executive Board of Sveriges Riksbank.

∗ Corresponding author Tel.: +39 051 209 8019.E-mail addresses: [email protected] (M. Marzo),

[email protected] (I. Strid), [email protected] (P. Zagaglia).

0954-349X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.strueco.2009.08.001

© 2009 Elsevier B.V. All rights reserved.

functions – how it values, or ought to value, the extradecline in inflation versus the output lost in gettingthere – is where the discussion should be, and is the cruxof the opportunism versus deliberate policy choice.”(Kohn, 1996)

1. Introduction

Former Fed Governors Alan Blinder and LawrenceMeyer have argued in favour of an ‘opportunistic approach’to disinflation. In their views, the central bank should notcounteract inflationary pressures when inflation is mod-erately above target. Rather, monetary policy should waitfor exogenous shocks to bring inflation back to the target,

and focus on output and employment stabilization alongthe transition. After the disinflation has taken place, thecentral bank should prevent inflation from rising to its pre-vious levels (see e.g. Meyer, 1996). This lays the groundfor a two-tier monetary policy strategy, where the reaction

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M. Marzo et al. / Structural Change a

f the central bank changes depending the strength of thenflationary pressures.

Orphanides and Wilcox (2002) provide the theoreti-al foundations for the opportunistic approach. Departingrom the linear-quadratic framework, they assume that theoss function of the monetary authority includes the abso-ute deviation of output from its natural level, along with aistory-dependent intermediate target for inflation. Thesewo elements generate a simple policy rule that prescribesnonlinear adjustment to a time-varying intermediate tar-et for inflation. Policy rates are set depending on a ‘zone ofnaction’. Within this range of inflation values, the centralank focuses on output rather than inflation stabilization.

nstead, when inflation falls outside the zone of inaction,he opportunistic rule prescribes a positive response to theeviations from the target. This generates a switch betweenwo regimes. In one regime, the central bank pursues anctivist monetary policy, and actively counteracts the infla-ionary pressures. In the other, a wait-and-see approachrevails.

To the best of our knowledge, Aksoy et al. (2006) ishe only academic contribution on opportunistic policyurrently available. Their study considers the long-runmplications of the opportunistic approach when inflationnds up outside the zone of inaction. The authors use amall estimated model of the U.S. economy to compare thetochastic distributions of inflation and the output gap aris-ng from the opportunistic rule with those of the policy ruledvocated by Taylor (1993).

This paper focuses on the business cycle properties ofhe opportunistic approach to disinflation. We formulatecalibrated New-Keynesian model with money demand,

apital accumulation, nominal and real rigidities. We solvehe model through a local approximation method, namelyhe second-order Taylor expansion proposed by Schmitt-rohé and Uribe (2004). This allows to incorporate theon-certainty equivalence that characterizes the structuralhange in the reaction function of the central bank. More-ver, this solution method is well suited for computingptimal monetary policy rules that maximize the intertem-oral utility of the households. We also contribute to thevailable analytical results on second-order approxima-ions by suggesting a numerical strategy for computing theelfare costs of alternative macroeconomic scenarios.

The quantitative findings indicate that the opportunisticule yields a Pareto-improvement in welfare in compari-on with the standard reaction function proposed by Taylor1993). We also introduce a rule for ‘deliberate’ disinfla-ion. Differently from the opportunistic approach, undereliberate disinflation, the central bank reacts to all the

u(

cit, mit, �it

)= 1

1 − 1/�

{[ac�−1

it

eviations from a time-varying inflation target. By compar-ng the macroeconomic performance of the opportunisticnd the deliberate disinflation, we can gain insight into

omic Dynamics 20 (2009) 288–300 289

the macroeconomic effects of the regime change incor-porated in the opportunistic approach. We show that thegains delivered by the opportunistic rule due solely to thetime variation of the intermediate inflation target.

Following Aksoy et al. (2006) we then investigate theimplications of the opportunistic central bank loss func-tion proposed by Orphanides and Wilcox (2002). We showthat in this case, the opportunistic policy rule with a zoneof inaction vis-à-vis small inflation deviations from tar-get results in lower central bank losses than linear policyrule with a time-varying inflation targets. In this regard,our paper confirms the findings of Aksoy et al. (2006) ina macroeconomic model with explicit micro-foundations.The opportunistic loss function of Orphanides and Wilcox(2002) may be considered a short-cut to account for theindivisibility of labor. Future work of interest would includethis indivisibility directly in the microfounded model toaccount for its welfare implications.

This paper is organized as follows. Section 2 describesthe model economy. The calibration strategy is presentedin Section 3. Section 4 deals with the computational aspectsof this work, including both the approximation techniqueto the first-order conditions of model and the welfare eval-uation. The results are discussed in Section 5. Section 6proposes some concluding remarks. In Appendix A, weshow how to compute our measure of welfare costs. InAppendix B, we discuss the state-space representation ofthe model.

2. The model

The structure of the model economy is standard inthe New-Keynesian tradition (see Woodford, 2003). Weconsider a closed economy. We include money demandthrough the money-in-the-utility function approach stud-ied by Feenstra (1986), and quadratic investment-adjustment costs like Kim (2000). Nominal price rigidityarises from quadratic-adjustment costs of changing prices.Although exchange rate interactions can lead to interestinginsights (see Laidler, 2005; Trautwein, 2005), we considerthe model of a closed economy.

2.1. Households

The model economy is populated by a large number ofinfinitely-lived agents indexed on the real line, i ∈ [0, 1],each maximizing the following stream of utility

Uit =∞∑

t=0

ˇtEtu(cit, mit, �it) (1)

subjected to the following specification for the instanta-neous utility function

− a)(

Mit

Pt

)�−1/�]�/�−1(

1 − �it

)�

}(1−1/�)

. (2)

The utility function considers money in a weakly separa-ble form with respect to consumption Cit . Consumptionand real money balances Mit/Pt are valued through anaggregator with constant elasticity of substitution, asdescribed by Chari et al. (2002). The advantage of this

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2.3. Government

Lump-sum taxes are adjusted to pursue a balanced

290 M. Marzo et al. / Structural Change a

modelling approach relies on the cross substitution effectsarising from the weak separability between consump-tion and money. It is worth noting that the equivalencebetween money-in-the-utility, transaction costs and cash-in-advance models has been proved by Feenstra (1986).

The ith household budget constraint (in real terms) isgiven by

cit + Mit − Mit−1

Pt+ Bit

Pt+ invit[1 + ACinv

t (i)]

≤ qitkit + wit�it + Rt−1Bit−1

Pt− �ls

t +∫ 1

0

�i(j)�t(j)dj. (3)

Households’ income derives from: (i) labor income in theform of wit�it , with wit real wage per unit of labor effort �it;(ii) capital income qitkit , with qit rental rate on capital stockkit; (iii) proceedings from investment in government bondsRt−1Bit−1/Pt , where Rt is the gross nominal rate, and Bit

is the stock of government’s bonds held by ith household.Each agent participates in the profit of the firm producinggood j via a constant share �i (j). We assume that this shareis constant and out of the control of the single agent. House-holds also pay taxes in the form of lump-sum transfers �ls

tto the government.

Households allocate their wealth among money Mit ,(nominal) bonds Bit and (real) investment invit . In orderto reduce the high investment volatility typical of New-Keynesian models, we follow the suggestion of Kim(2000) and introduce an investment-adjustment cost in thequadratic form

ACinvt (i) := K

2

(invit

kit

)2

. (4)

The assumption of quadratic cost of price adjustment sim-plifies algebra and delivers coherent results. The evolutionof capital accumulation is governed by

kit+1 = (1 − ı)kit + invit . (5)

Given the existence of differentiated goods and differentlabor inputs, there exists an intra-temporal optimizationprogram on the final goods-sector. The are j varieties offinal goods produced that are aggregated according to theconstant-elasticity of substitution technology proposed byDixit and Stiglitz (1977)

cit =[∫

ω2

cit(j)

�−1/�dj

]�/�−1

,

where � > 1 is the elasticity of substitution between differ-ent varieties of goods produced by each jth firm and ci

t (j) isthe consumption of varieties j by ith household. The inversedemand function for jth variety expressed by ith householdis

cit (j)cit

=[

Pt (j)Pt

]−�

,

where Pt (j) is the price of variety j and Pt is the generalprice index defined as

Pt =[∫ 1

0

Pt(j)1−�dj

]1/1−�

. (6)

omic Dynamics 20 (2009) 288–300

Aggregate consumption is ct =∫

ω1citdi, after summing

over the i ∈ ω1 households. The aggregate demand for vari-ety j can be written as

ct(j) + gt(j) = yt(j), (7)

such that the individual demand curve takes the form

Pt(j) =[

yt(j)yt

]−1/�

Pt. (8)

2.2. Firms

We assume the existence of a large number of firmsindexed by j ∈ ω2, each producing a single variety. Everyfirm acts as a price taker with respect to the varieties sup-plied by other competitors. The production function is

yjt = zt(kjt)˛(�jt)

1−˛ − �t, (9)

where kjt and �jt are capital and labor inputs, respectively.The productivity shock zt follows the autoregressive pro-cess

log zt = (1 − z) log z + z log zt−1 + εzt , (10)

where zt is normally distributed with zero mean and con-stant variance �2

z . Intermediate-good firms face also anadditive shock �t on markups

log �t = (1 − �) log � + � log �t−1 + ε�t , (11)

and �t is distributed as N(

0, �2�

). The steady-state level

� is calibrated so that profits are zero in the long-run, and there is neither entry nor exit dynamics in themonopolistically-competitive sector. Price stickiness arisesfrom the presence of quadratic cost of price adjustment àla Rotemberg and Woodford (1992). The adjustment costfunction is specified as

ACPt (j) := P

2

(Pt (j)

Pt−1 (j)− �

)2

yt. (12)

The optimal choice of labor and capital to be hired isdescribed as the maximization of the future stream of profitevaluated with the stochastic discount factor t

max{Pt (j),kt (j),�t (j)}E0

[ ∞∑t=0

�t�t (j)

](13)

s.t. �t (j) = Pt (j) yjt − Wt�jt − Ptqtkjt − PtACPt (j) , (14)

given the demand for differentiated products in (8), and theproduction function (9).

budget in every period. We assume that government con-sumption follows an exogenous autoregressive process

log(gt) = (1 − g) log(g) + g log(gt−1) + εgt , (15)

with εgt is distributed as N(0, �2

g ).

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M. Marzo et al. / Structural Change a

.4. Simple rules for monetary policy

We compare the macroeconomic performance of anrray of simple rules for monetary policy that have becometandard in the literature. The celebrated specification pro-osed by Taylor (1993) is the starting point for every studyf monetary policy

og[

iti

]= ˛�

[�t

�

]+ ˛y log

[yt

y

]+ ˛R log

[it−1

i

]. (16)

rphanides and Wilcox (2002) formalize the idea underly-ng the ‘opportunistic approach to disinflation’ through aolicy rule that is both time-dependent and nonlinear. Theonlinearity takes the form of a regime shift. When infla-ion is close to the time-varying target of the central bank,wait-and-see approach prevails, and the policy rates arenchanged. Instead, if inflation is out of a ‘zone of inac-ion’, the central banks acts to counteract the inflationaryressures in the economy.

The central bank pursues an intermediate target of infla-ion �op

t , which is a weighted average of long-run � andnherited past inflation �h

t

˜ opt := (1 − �)� + ��h

t . (17)

he term �ht is the source of history dependence for mon-

tary policy. For the purpose of computational simplicity,e assume that inherited inflation consists of the inflation

ate in the previous quarter—i.e., �ht = �t−1.

The nonlinearity of the opportunistic approach arisesrom a range of inflation deviations from the target withinhich output stabilization is the primary objective of mon-

tary policy. This is the zone of inflation inaction, wherebyhe central bank does not counteract changes in inflation.he larger the deviation of inflation from �t outside theone of inaction, the stronger the focus on price stability.ince the zone of inaction is defined around �op

t , it is alsoime-varying. If exogenous shocks bring down the inflationate within the zone of inflation, both �op

t and the zone ofnaction fall. This means that the central bank will preventnflation from adjusting to the past inflation targets.

The opportunistic policy rule proposed by Orphanidesnd Wilcox (2002) takes the form

it − ı] = G(

�t − �opt

)+ �3 [yt − y] + �4 [it−1 − ı] . (18)

heG(·) is a function that defines the transition between theegimes of inaction and action around the inflation target

(�t − �t)

:=

⎧⎪⎨⎪⎩

�1(�t − �opt − �2) if (�t − �op

t ) > �2

0 if �2 ≥ (�t − �opt ) ≥ −�2

�1(�t − �opt + �2) if (�t − �op

t ) < −�2(19)

he wait-and-see approach takes place when G(·) = 0,amely when the log-deviation of inflation around the tar-

et is within a band of �2. Summing up, the presence of aone of inaction is the mechanism that generates a struc-ural change in the reaction of the central bank to inflation.n the numerical exercise, we follow Aksoy et al. (2006) andse the twice continuously-differentiable approximation

omic Dynamics 20 (2009) 288–300 291

of G(·)G(·) ≈ �1[0.05(�t − �op

t ) + 0.475(−�2 + �t − �opt

+((−�2 + �t − �opt )

2)0.51

) + 0.475(�2 + �t − �opt

−((�2 + �t − �opt )

2)0.51

)]. (20)

Since we are modelling the U.S. economy, we assign thesame parameter values to the approximatedG(·) that Aksoyet al. (2006) use. There is a slightly positive slope even wheninflation is within the zone of inaction.

Bomfim and Rudebusch (2000) formalize the distinctionbetween opportunistic and deliberate disinflation. In thelatter, the central bank pursues an explicit path towards thefinal inflation target. Since the policy rate is set as a functionof the inflation deviation from target, also the linear Taylorrule 16 embodies the idea of explicit disinflation. However,that type of rule does not capture the structural change thatis typical of the opportunistic disinflation.

A consistent comparison between the linear and theopportunistic rule requires taking into consideration thefact that they include different inflation targets. In the lin-ear Taylor rules, the inflation target is constant, whereasit is time-varying in the opportunistic rule. Hence, itis important to disentangle the interaction betweenthe time-varying inflation target on one side, and thenonlinear-adjustment path on the other – prescribed by thefunction G(·) – in the setup of optimal policy. Hence, we for-malize the idea of deliberate disinflation as a modificationof the standard Taylor rule

log[

iti

]= ˛�

[�t

�dt

]+ ˛y log

[yt

y

]+ ˛R log

[it−1

i

], (21)

where the inflation target takes the form

�dt := (1 − �)� + ��h

t . (22)

Summing up, the difference between opportunistic anddeliberate disinflation is that the former prescribes aregime change around a time-varying inflation target,whereas the latter responds only to the time-varying infla-tion target.

2.5. Equilibrium and aggregation

A symmetric equilibrium consists of stationarysequences of prices, quantities, and values of the exoge-nous shocks that aggregate over ω1 = [0, 1] and ω2 = [0, 1],that are bound in a neighborhood of the steady state, thatclear the markets for goods and assets, and such that theresource constraint of the economy holds:

yt = ct + invt[1 + ACinvt ] + gt + ACP

t (23)

3. Calibration

The parameters are calibrated on quarterly data for

the U.S. economy. We assume that households have anintertemporal discount rate of 0.996. They devote 1/4 oftheir time to labor activities at the steady state. The weighton the consumption objective is 0.993. We assume thatthe intertemporal elasticity of substitution is 1/3, like in

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292 M. Marzo et al. / Structural Change and Econ

Table 1Calibration of the model.

Description Parameter Value

Subjective discount factor ˇ 0.9966Weight on leisure objective � 0.001Share of consumption objective a 0.99Interest elasticity � 0.39Coefficient of relative risk aversion 1/� 3Share of labor effort � 1/4Investment–output ratio i/y 0.25Capital–output ratio k/y 10.4Money–output ratio m/y 0.44

Steady-state inflation � 1.042(1/4)

Adjustment cost of prices P 60Adjustment cost of capital K 433Capital depreciation rate ı 0.024Capital elasticity of intermediate output ˛ 0.33Elasticity of substitution of interm. goods � 10

Persistence of productivity shock z 0.98Steady state of productivity shock z 1Standard dev. of productivity shock �2

z 0.055Persistence of markup shock � 0.911Standard dev. of markup shock �2

�0.141

period) moments conditional on an initial state vector.

Persistence of government-spending shock G 0.97Standard dev. of government-spending shock �G 0.1Public spending-output ratio g/y 0.148

Schmitt-Grohé and Uribe (2007). The calibration of theother parameters in the utility function is consistent witha consumption-output ratio of 0.57, and a money–outputratio of 0.44 in the long-run (see Table 1). The nominal rateof interest is 5% a year, and the inflation rate is 4.2%. Bothfigures are consistent with the U.S. postwar experience.

We set the investment–output and capital–outputratios as 0.25 and 10.4, respectively. Capital depreciates10% a year. Both the parameter K in the adjustmentcost for capital, and the persistence of the markup shockare from Kim (2000). Capital income has a share of1/3 in total output. The elasticity of substitution amongintermediate goods generates a steady-state markup ofapproximately 10%. Stochastic productivity shocks are cal-ibrated according to Chari et al. (2002). Finally, we assumethat government-spending is 14.8% of GDP. The calibrationfor the public spending shock is from Schmitt-Grohé andUribe (2007).

4. Computational aspects

4.1. Welfare evaluation

Aggregate welfare is the expected lifetime utility con-ditional on the initial values of the state variables s0

W0 := E

[ ∞∑ˇtu(ct, �t)|s0

]. (24)

t=0

In order to obtain an accurate welfare evaluation, theconditional welfare function is approximated through asecond-order Taylor expansion the distorted steady state.Thus we compute the second-order approximation to

omic Dynamics 20 (2009) 288–300

the system of nonlinear expectational equations.1 Thisamounts to using a local approximation method thataccounts for the uncertainty of households. As a result, theimpact on households decisions of the structural changebuilt into the model is fully taken into consideration.

The welfare costs of alternative policies are measuredas the permanent change in consumption, relative to thedeterministic steady state, that yields the expected util-ity level of the distorted economy. Given steady states ofconsumption c� and hours worked � of the model �, thistranslates into the number ��

c such that

∞∑t=0

ˇtu([1 + ��c]c, �) = W�

0, (25)

where � refers to the monetary-policy rule. FollowingKollmann (2008), we decompose the conditional welfarecost ��

c into two components denoted as ��E and ��

V . Giventhe second-order approximation to the utility function

u([1 + ��c]c, �) ≈ u(c, �)

+(1 − ˇ)∞∑

t=0

ˇt(

E[

ct − ��t |s0

]− 1

2VAR

[ct |s0

]), (26)

we compute the change in mean consumption ��E that

the household faces while giving up the total fraction ofcertainty-equivalent consumption ��

c

u([1 + ��E]c, �) = u(c, �) + (1 − ˇ)

∞∑t=0

ˇt

×(

E[ct |s0

]− �E

[�t |s0

]). (27)

Since the solution method is non-certainty equivalent, wecan also calculate the change in conditional variance of con-sumption that is consistent with the total welfare cost ofpolicies

u([1 + ��V ]c, �) = u(c, �) − (1 − ˇ)

12

∞∑t=0

ˇtVAR[ct |s0

],(28)

where hats denote log-deviations from the deterministicsteady states. Kollmann (2008) points out that the follow-ing relation holds

(1 + ��c) = (1 + ��

E)(1 + ��V ). (29)

In the simulation exercise, we compute ��c and ��

Vthrough Eqs. (25) and (28), and ��

E through 29. Since thereis no closed-form solution for the infinite summations,we simulate the conditional moments for 50,000 peri-ods, and compute the discounted sums. In Appendix A,we derive analytical formulas for computing the (finite-

These formulas are developed on the second-order solutionproposed by Schmitt-Grohé and Uribe (2004).

1 For the solution, we use the algorithm and computer codes describedin Schmitt-Grohé and Uribe (2004).

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M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300 293

Table 2Optimal policy rules based on W0.

˛� ˛y ˛R W0 % �c % �E % �V

Standard rule 1.2 0.3 0.4 −91.41 3.33 3.71 0.366

�0 �1 �3 W0 % �c % �E % �V

4

swsc

l

w

oatdoi

5

5

mmotw�tind

pott(oothi

rTn

policy is countercyclical on impact.3

Opportunistic rule 0.0 5.0 1.0

˛� ˛y ˛R

Deliberate disinflation 1.2 1.0 0.2

.2. Local validity of approximate solutions

Second-order perturbations are defined only aroundmall neighborhoods of the approximation points. Hence,e impose an ad hoc bound that restricts the stochastic

teady state of the nominal interest rate to be arbitrarilylose to its deterministic counterpart

n (R) > ��Rt, (30)

ith a constant � , and �Rtas the unconditional variance

f R. This constraint rules out policies that are excessivelyggressive, and that induce non-stationarity. The reason ishat large deviations of the nominal rate of interest from theeterministic steady state are likely to prescribe violationsf the zero bound on nominal interest rates at some pointn time. In what follows, we set � = 2.

. Quantitative results

.1. Main findings

Table 2 reports the coefficients of the policy rules thataximize the measure of conditional welfareW0. The opti-izing coefficients are computed through a grid search

ver {˛�, ˛y, ˛R} for the linear rules, over {�1, �3, �4} forhe opportunistic rule.2 Like Aksoy et al. (2006), we fix theeight on inherited inflation � to 0.5 and the parameter

2 to 0.0025, so that the width of the zone of inaction iswo percentage points around the quarterly target for grossnflation. This implies that the opportunistic rule 18 doesot nest either the standard linear rule 16 or the rule 21 foreliberate disinflation.

Table 2 suggests that the optimal standard rule com-lies with the Taylor principle. The optimized coefficientn the inflation target is higher than one, and larger thanhe coefficient on the output objective. The optimal oppor-unistic rule is similar to that obtained by Aksoy et al.2006). It entails a large coefficient on the nonlinear partf the rule, strong interest-rate inertia, and no response toutput fluctuations. One of the interesting results is thathe opportunistic rule achieves a conditional welfare leveligher than the standard Taylor rule. In what follows, we

nvestigate why that is the case.Fig. 1 shows the regions of parameters that yield unique

ational expectations equilibria as a function of ˛� and ˛R.here is no substantial difference between the determi-acy regions of the standard Taylor rule and the rule for

2 The grid spans from 0 to 3 for each parameter, with a step of 0.1.

−89.26 3.78 2.1 1.645

W0 % �c % �E % �V

−86.99 4.85 2.65 2.143

deliberate inflation, as these require that ˛�/(1 − ˛R) > 1.Again, this is the Taylor principle (see Schmitt-Grohé andUribe, 2007). Fig. 1(b) indicates that the Taylor principleholds also for the rule for opportunistic disinflation. How-ever, the shrinking region of determinacy indicates that theoptimal long-term response of monetary policy to inflationshould be stronger than for a standard Taylor rule.

The impulse responses for the optimized standard ruledepict well-known results. A positive productivity shockraises real output and lowers inflation (see Fig. 2(a)).Consumption and investment rise with respect to thedeterministic steady state. There is a liquidity effect,namely the fact that a fall in the short-term interest ratetriggers an increase in money holdings. Our model supportsthe idea that hours worked fall after a positive productivityshock, which is in line with the recent evidence of Rabanaland Gali (2004). The fall of the rental rate of capital onimpact indicates that our formulation of the adjustmentcost structure does not generate excessive sluggishnessin investment. The resulting persistence of investment isenough to generate a liquidity effect. Fig. 2(c) shows that agovernment-spending shock leads to an increase in output.However, public spending crowds out private consump-tion, and crowds in private investment. This indicates thatour model subscribes to the interpretation of government-spending shocks as productivity shocks. A markup shockcauses inflation to rise and output to fall, and triggers thecontractionary response of monetary policy (see Fig. 2(c)).The deviation of output from the deterministic steady statecloses down consistently as inflation returns to the baselinevalue.

Under optimal opportunistic policy, the impulseresponses are characterized by a small initial reaction ofoutput to a positive productivity shock. Since both con-sumption and investment move little, the time share ofmarket activity falls, and so does the rental rate of capital.Money holdings exhibit a very small change from baseline(see Fig. 3(a)). Two main aspects stand out. First, the pol-icy rates move in opposite direction with respect to the theinflation rate. Second, the initial increase in the interest rateis lower than both the increase in output and the fall of theinflation rate. With a positive productivity shock, monetary

It is also instructive to compare the responses to aninflationary markup shock under the standard and theopportunistic rule. The key aspect is that disinflation under

3 We do not report the impulse responses to a government-spendingshock because they are very similar to those of the standard Taylor rule,and no new patter arises.

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294 M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300

ce. (a) S

Fig. 1. Regions of determinate equilibria in the (˛�, ˛R)- and (�1, �0)-spa

the opportunistic policy follows a pace faster than underthe standard rule (see Fig. 3(b)). On impact, an opportunis-tic central bank opens up only a very small output deviationfrom the long-run value, as the policy rate falls mildly. Theresponse of output features an inverse hump shape, wherethe largest gap from the deterministic steady state is real-ized after four quarters. Hence, the fast drop in inflation isinduced by the continuous contraction of output up to thefourth quarter.

Table 3 reports some selected statistics for the modeleconomy with optimized policy rules when all the stochas-tic shocks are present. Two main points arise from panel(a). The first is that opportunism tolerates a sizeable devi-ation from perfect inflation stabilization, with a variance

Table 3Descriptive statistics.

yt �t

(a) Standard deviation (%)Standard rule 2.23 0.26Opportunistic rule 10.58 9.65Deliberate disinflation 1.46 10.46

(b) Correlation with outputStandard rule 1 −0.99Opportunistic rule 1 0.93Deliberate disinflation 1 0.91

tandard Taylor rule, (b) opportunistic rule and (c) deliberate disinflation.

of inflation almost ten times as large as that of the stan-dard Taylor rule. Second, panel (b) shows that opportunisticpolicy re-establishes the procyclicality in inflation that isabsent under the standard rule.

The last row of Table 2 shows that the policy rulefor deliberate disinflation achieves a conditional welfarelevel higher than that of the rest of the linear rules. Thisis due mostly to the gains in terms of lower variabilityof consumption. Most important, deliberate disinflation

is a preferable strategy with respect to the opportunisticapproach. This indicates that the main advantage of theopportunistic approach with respect to the standard linearrules arises from the time-varying inflation target. Hence,the regime switch around the intermediate target induced

Rt kt ct �t

0.26 0.75 3.30 10.250.85 1.69 2.06 7.935.94 7.45 1.38 7.01

−0.99 0.72 0.89 −0.79−0.47 0.26 0.99 0.83

0.88 0.34 −0.21 0.66

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M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300 295

F tandards

bw

ratmmdfotaei

5

d

where the expectations are conditioned on the initial statesector at time 0. Since there is no closed-form solution for

ig. 2. Impulse responses (%) for the optimized standard rule. (a) One spending shock and (c) one standard-deviation markup shock.

y the nonlinear part of the opportunistc rule yields noelfare gains.

With a positive productivity shock, the qualitativeesponses of the model economy under optimal deliber-te disinflation are very similar to those emerging underhe standard rule (see Fig. 4(a)). The main difference is that

onetary policy under deliberate disinflation respondsore strongly to the exogenous shock than under the stan-

ard rule. As a result, output rises on impact, and thenalls below its long-run value. This captures a key aspectulined by Kohn (1996), namely that “(. . .)i n contrast tohe opportunistic strategy, the deliberate strategy would bet least mildly restrictive even when inflation is only mod-rate, maintaining a small output gap until price stabilitys reached”.

.2. Indivisibility of labor and the role of nonlinearities

The utility function of households is based on the stan-ard assumption of divisibility of labor services. This is

-deviation productivity shock, (b) one standard-deviation government-

also reflected in the microfounded welfare criterion that isused to evaluate monetary policy.4Orphanides and Wilcox(2002) argue that relaxing the assumption of labor indivis-ibility along the lines of Hansen (1985) generates a centralbank loss function that penalizes the quadratic deviationsof inflation from the history-dependent inflation target,and the absolute deviations of output from potential:

L =∞∑

t=0

(�VAR0[�t − �t] + (1 − �)VAR0[yt − y]

+(1 − �)E0[|yt − y|]) , (31)

the infinite summation, we apply the computational strat-egy outlined in the previous section and approximate the

4 We are very grateful to one referee for suggesting to address this issue.

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296 M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300

Fig. 3. Impulse responses (%) for the optimized rule with opportunistic disinflation. (a) One standard-deviation productivity shock and (b) one standard-deviation markup shock.

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M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300 297

Fig. 4. Impulse responses (%) for the optimized rule with deliberate disinflation. (a) One standard-deviation productivity shock and (b) one standard-deviation markup shock.

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298 M. Marzo et al. / Structural Change and Economic Dynamics 20 (2009) 288–300

Table 4Optimal policy rules based on the opportunistic loss L.

� ˛� ˛y ˛R W0 % �c % �E % �V

Standard rule 0.6 1.04 0.9 0.5 −93.041 2.870 3.90 1.001

� �0 �1 �3 W0 % �c % �E % �V

Opportunistic rule 0.4 2.1 8.3 1.1 −91.820 3.122 4.17 1.016

� ˛� ˛y ˛R W0 % �c % �E % �V

Deliberate disinflation 0.6 1.05 1.3 0.7 −92.670 2.901 4.206 1.27

Table 5Sensitivity analysis on P .

˛� ˛y ˛R W0 % �c % �E % �V

(a) Linear policy rules(a.1) P = 90

Standard rule 1.2 0.2 0.4 −91.409 3.33 9.11 −5.227Deliberate disinflation 3.0 0.2 0.0 −91.408 3.32 6.45 −2.940

(a.2) P = 40Standard rule 1.2 0.2 0.4 −91.4103 3.33 6.35 −2.839Deliberate disinflation 3.0 0.2 0.0 −91.4099 3.33 6.48 −2.958

�0 �1 �3 W0 % �c % �E % �V

.0

.6

(b) Opportunistic policy ruleP = 90 0.0 5.0 1P = 40 0.0 3.2 0

summations in two steps. First, the model is solved overeach point of a grid including the parameters of the simplepolicy rule. Then, the second-order solution of the modelis simulated for 50,000 periods, and the discounted sum ofthe per-period losses is calculated.

Orphanides and Wilcox (2002) show that, with theloss function (31), the opportunistic rule is optimal in abackward-looking model. It should be stressed that thesecond-order welfare function disregards the role of non-linearities from the kink in output accounted for by theloss function (31). This may cause the welfare evaluationexperiment to downplay the role of nonlinearities. In orderto investigate the quantitative role of these nonlinearities,

we re-evaluate both the linear and the nonlinear policy ruleby minimizing the loss function (31).

In the experiments, we maximize over the parame-ters of the policy rule and the inflation weight in the lossfunction. Table 4 reports the optimizing parameters and

Table 6Sensitivity analysis on �.

˛� ˛y ˛R

(a) Linear policy rules(a.1) � = 2

Standard rule 1.2 0.8 0.2Deliberate disinflation 3.0 0.0 0.8

(a.2) � = 15Standard rule 1.2 0.2 0.4Deliberate disinflation 3.0 0.2 0.0

�0 �1 �3

(b) Opportunistic policy rule� = 2 0.0 5.0 1.0� = 15 0.0 3.2 0.2

−88.4801 3.92 8.39 −4.124−92.4998 1.28 8.90 −6.997

the microfounded welfare level associated with each rule.Accounting explicitly for the nonlinearity due to labor indi-visibility causes optimized policy to achieve welfare levelslower than those obtained from maximizing directly thehousehold’s utility. This is explained by the fact that opti-mal policy puts a large weight on output stabilization at theexpense of inflation stabilization. However, in this case, theopportunistic rule delivers the highest welfare level. Thissuggests that the second-order approximation is capable ofaccounting for the nonlinearity of the zone of inaction in asatisfactory way.

5.3. Sensitivity exercise

In what follows, we propose two sensitivity checks ofthe results outlined earlier. Table 5 reports the optimizedpolicy parameters subject to a degree of price rigidity

W0 % �c % �E % �V

−58.740 3.14 3.64 −0.482−50.293 3.47 2.73 0.722

−91.4103 2.82 6.86 −3.781−91.4100 2.82 6.99 −3.898

W0 % �c % �E % �V

−45.0633 3.55 7.67 −3.826−90.2043 5.37 6.11 −0.697

Page 12

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hdeisptapot

TtttscO(gtof

6

otecafmdmh

pmcrotoiwfatfuha

atr

M. Marzo et al. / Structural Change a

igher (P = 90) or lower (P = 40) than baseline.5 As theegree of price rigidity rises, the welfare level of the lin-ar policy rule remains unchanged. The interesting points that, with P = 90, the opportunistic rule delivers a sub-tantial increase in the welfare level. The opportunistic ruleerforms even better than the rule for deliberate disinfla-ion. This suggests that the contribution of the nonlineardjustment to the intermediate inflation target starts tolay a significant role. However, with less price stickiness,pportunistic policy yields quite a large fall in welfare, andhe gains from opportunism disappear.

The results of a sensitivity exercise on � are reported inable 6. Two general considerations arise. First, changes inhe degree of competition relax the welfare benefits fromhe deliberate approach to disinflation. This implies thathe case for the opportunistic approach is magnified. Theecond point is that a higher degree of market power (� = 2)orresponds to a higher welfare level for all the policy rules.n the other hand, the higher the extent of competition

� = 15), the lower the maximized welfare levels. This sug-ests that giving more – or taking away – market powero the the firms in the intermediate-good sector offsets –r strengthens – the welfare impact of some of the otherrictions included in the model.

. Final remarks

In this paper, we study the macroeconomic propertiesf a class of regime shift in monetary policy. We focus onhe opportunistic approach to disinflation, whereby mon-tary policy can follow either a wait-and-see approach, oran fight actively the inflationary expectations. We set upNew-Keynesian model with non-trivial money-demand

rictions and investment-adjustment costs. We solve theodel through a second-order approximation around the

eterministic steady state. Alternative simple rules foronetary policy are also considered that maximize the

ousehold’s intertemporal utility stream.Our results indicate that, when substantial rigidities are

resent, the opportunistic rule delivers a welfare improve-ent with respect to the traditional approaches. We also

ompare the opportunistic approach with a deliberateule for disinflation. We show that the main advantagef the opportunistic rule arises from the adoption of aime-varying intermediate target for inflation. The zonef inaction vis-à-vis small inflation deviations from targetn the opportunistic policy rule is only found to improve

elfare when we use the opportunistic central bank lossunction proposed by Orphanides and Wilcox (2002). Suchloss function may be motivated by a concern of the cen-

ral bank to account for the indivisibility of labor thatorces a few workers to bear a large burden in terms of

nemployment. In future work we are planning to includeis indivisibility directly in the microfounded model toccount for its welfare implications.

5 We should stress that we have performed the sensitivity checks forlarge number of different parameter values. For brevity, we report only

he results for two parameter variations. These results are however largelyepresentative of the outcome of the overall sensitivity exercise.

omic Dynamics 20 (2009) 288–300 299

Appendix A. Computing conditional moments

Kim et al. (2003) suggest that using the expressions ofthe full second-order approximation for computing con-ditional moments recursively introduces spurious higherorder terms. This problem can be avoided by exploiting thelinear (first-order) part of the solution. Let ê(2)

t denote the

full second-order solution, and ê(1)t denote the linear part.

We can re-write the system of solutions[ê(2)

t

ê(1)t ⊗ ê(1)

t

]= M1

[x(2)

t

x(1)t ⊗ x(1)

t

]+ K1 (32)

[x(2)

t+1

x(1)t+1 ⊗ x(1)

t+1

]= M2

[x(2)

t

x(1)t ⊗ x(1)

t

]+ K2 + ut+1 (33)

Define

Xt =(

x(2)t

x(1)t ⊗ x(1)

t

)(34)

Yt =(

ê(2)t

ê(1)t ⊗ ê(1)

t

)(35)

Paustian (2003) shows that ut takes the form

ut =(

�N�t

�2 (N ⊗ N) (vec(I) − �t ⊗ �t)

)

Eqs. (33) and (32) can be re-written by repeated substitu-tion as

Xt+k = Mk2Xt +

k−1∑i=0

iM2

(K2 + ut+k−i) (36)

Yt+k = M1Xt+k + K1 = K1 + M1Mk2Xt

+k−1∑i=0

M1i

M2

(K2 + ut+k−i) (37)

The expectation conditional on an initial state vector takesthe form

E (Yt+k|Xt) = K1 + M1Mk2Xt +

k−1∑i=0

M1i

M2

K2 (38)

The following two results can be established.

Proposition 1. The variance of the endogenous variableconditional on an initial vector of states Xt is

Yt+k − E (Yt+k|Xt) =k−1∑

M1i

M2

�u

(M1

iM2

)′(39)

i=0

where �u := E(utu′t).

Proof 1. From Eq. (38), we can write

Yt+k − E (Yt+k|Xt) =k−1∑i=0

M1i

M2

ut+k−i (40)

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Rochester Conference Series 39, 195–214.

300 M. Marzo et al. / Structural Change a

Cov(Yt+k|Xt)

= E{

[Yt+k − E (Yt+k|Xt)] [Yt+k − E (Yt+k|Xt)]′|Xt

}(41)

Using Eqs. (37) and (38) in (41), we obtain the result. �

Proposition 2. The variance matrix �u is

Eutu′t =

(�2NN′ 0

0 2�4 (N ⊗ N) vec(I)vec(I)′(N ⊗ N)′

)(42)

Proof 2. The mean of ut is Eut = 0. Since �t∼N(0, I), wehave the following

E�3it = 0 (43)

E�it�jt�kt = 0 (44)

if any of the indices i, j, k are different. This gives

�2E(

N�t�′tN

′) = �2NN′ (45)

�3E{

(N�t) [(N ⊗ N) (vec(I) − �t ⊗ �t)]′} = (46)

= �3E{

(N�t)(vec(I)′ − �′

t ⊗ �′t

)(N′ ⊗ N′)} = 0 (47)

Finally

�4E{

[(N ⊗ N) (vec(I) − �t ⊗ �t)]

× [(N ⊗ N) (vec(I) − �t ⊗ �t)]′} = (48)

= �4 (N ⊗ N) E{

[(vec(I) − �t ⊗ �t)]

× [(vec(I) − �t ⊗ �t)]′} (N ⊗ N)′ (49)

where

E{

[(vec(I) − �t ⊗ �t)] [(vec(I) − �t ⊗ �t)]′} = (50)

= E{

vec(I)vec(I)′ + �t�′t ⊗ �t�′

t − (�t ⊗ �t) vec(I)′

− vec(I) (�t ⊗ �t)}

= (51)

= E(�t�′

t ⊗ �t�′t

)− vec(I)vec(I)′ = (52)

= 2vec(I)vec(I)′ (53)

�

Appendix B. State-space form

Suppose that the first-order conditions of a model econ-omy can be arranged as

EtH (et+1, et, xt+1, xt |�) = 0 (54)

where y is a vector of co-state variables. The state variablesare collected in x

xt :=[

x1,t

x2,t

](55)

omic Dynamics 20 (2009) 288–300

with vectors of endogenous state variables x1,t , andexogenous state variables x2,t

x2,t+1 = �1x2,t + �2��t+1 (56)

with matrices �1 and �2. The scalar � > 0 is known.The model is re-written into the state-space form by

defining the vectors

x1,t = [kt Rt−1 mt−1 yt−1 �t−1]′, (57)

x2,t = [zt �t gt]′, (58)

et =[yt Rt mct invt ct mt �t �t qt wt �t �t

]′. (59)

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