optimal monetary policy with the sticky information model of price adjustment: inflation or...

C© 2012 The Author. Bulletin of Economic Research C© 2012 Blackwell Publishing Ltd and the Boardof Trustees of the Bulletin of Economic Research. Published by Blackwell Publishing, 9600 GarsingtonRoad, Oxford OX4 2DQ, UK and 350 Main St., Malden, MA 02148, USA.

Bulletin of Economic Research 00:0, 2012, 0307-3378DOI: 10.1111/j.1467-8586.2012.00441.x

OPTIMAL MONETARY POLICY WITH THE STICKYINFORMATION MODEL OF PRICE ADJUSTMENT:

INFLATION OR PRICE-LEVEL TARGETING?

M. Murat Arslan

Yildirim Beyazit University, Ankara, Turkey

ABSTRACT

I investigate the optimal monetary policy in a New Keynesian macroeconomic framework withthe sticky information model of price adjustment. The model is solved for optimal policy, andwelfare implications of three alternative monetary policy regimes under this optimal policy arecompared when there is a cost-push shock to the economy. These monetary policy regimes are theunconstrained policy, price-level targeting and inflation targeting regimes. The results illustrate thatoptimal policy depends on the degree of price stickiness and the persistence of the shock. Inflationtargeting emerges as the optimal policy if prices are flexible enough or the shock is persistentenough. However, the unconstrained policy or price-level targeting might be preferable to inflationtargeting if prices are not very flexible and the shock is not very persistent. The results also showthat as prices become more flexible, the welfare loss usually gets bigger.

Keywords: inflation targeting, optimal policy, price-level targeting, sticky information

JEL classification numbers: E31, E37, E52

I. INTRODUCTION

Currently there is wide consensus on the central banks’ objectives of keeping inflation close tozero and output close to its natural level. Therefore it is generally accepted that the objective ofany monetary policy is to achieve a low expected value of a discounted loss function. However,there is considerable disagreement on the details of this general objective and on the specificform of the loss function, which depends on the model’s specification. This loss function needsto be specified to be able to evaluate alternative policy rules and to decide on which pricestabilization policy to pursue, such as price-level targeting or inflation targeting.

This paper investigates the optimal monetary policy in a New Keynesian macroeconomicframework with the sticky information model of price adjustment. This price setting model isproposed by Mankiw and Reis (2002) as an alternative to the sticky price model that leads to thewell-known New Keynesian Phillips curve (NKPC), which has been criticized for producingimplausible results regarding inflation and output dynamics. Criticism includes the fact that the

Correspondence: M. Murat Arslan, Yildirim Beyazit University, SBF, Department of Economics, CinnahCad. No:16, 06690 Cankaya-Ankara, Turkey. Tel.: +90-312-3241555; Fax: +90-312-3241505. Email:[email protected].

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2 Bulletin of Economic Research

sticky price model does not exhibit the inflation persistence and delayed and gradual effects ofshocks observed in data.1 It is also unable to account for correlation between inflation and theoutput gap. The sticky price model, contrary to the empirical evidence, implies that inflationshould lead the output gap over the cycle. This model also violates the natural rate hypothesisand implies that an increasing inflation rate will tend to keep output permanently low. Ball(1994) showed that credible disinflations cause booms rather than recession with this model.Because of such problems with the sticky price model, there have been some extensions andalternatives to this model.2 One alternative is the sticky information model. The main premiseof this model is that information about macroeconomic conditions spreads slowly throughoutthe population; although prices are set every period, information collecting and processing taketime. In this model, a fraction of firms get complete information about the economy in eachperiod randomly and independent of waiting time, and set their prices according to this newinformation, while the remaining firms set their prices according to the old information.

In this study, the sticky information model in a dynamic stochastic general equilibrium(DSGE) framework is solved for optimal policy that shows a relationship between the outputgap and inflation or the price level. To be able to find out some policy implications, this optimalpolicy must be evaluated under alternative policy regimes, thereby making it possible to saywhich policy regime constitutes the optimal policy. There could be many alternative policyregimes to be evaluated, but in this study only three general policy regimes are consideredbecause of their relevance in central banking and academia. Therefore, in this paper, welfareimplications of the three alternative monetary policy regimes are compared in the presenceof a cost-push shock to the economy.3 These monetary policy regimes are the unconstrainedpolicy, price-level targeting, and inflation targeting regimes. The unconstrained policy regimeimplies that when a central bank tries to minimize its welfare loss function, it does not have anycommitments or targets to manipulate private-sector expectations. However, there is a targetlevel for prices or inflation in price-level and inflation targeting regimes, and the expectationscan be manipulated if the commitment to the target is credible in these regimes. The results showthat optimal policy depends on the degree of price stickiness and the persistence of the shock.Inflation targeting becomes the optimal policy when prices are flexible enough or the shock ispersistent enough. However, unconstrained policy or price-level targeting might be preferred toinflation targeting if prices are not very flexible and the shock is not very persistent.

Targeting regimes have become the most common type of monetary policies in developedcountries in recent years, and inflation targeting has been adopted by many central banks forthe conduct of monetary policy.4 The rationale for targeting regimes might be to guaranteethat monetary policy avoids mistakes easily by identifying a clear nominal anchor for thepolicy. There is a voluminous literature on optimal policy and targeting regimes for the sticky

1 However, as Mankiw and Reis (2002) discussed, the key problem with the sticky price model is not thepersistence of the dynamic responses but the delayed and gradual responses to shocks.

2 See Woodford (2003).3 Such shocks are the supply shocks and are also called the markup shocks. They appear in price adjustment

equations, and represent everything other than the output gap that may affect the expected marginal costs.They cause the model to generate inflation variation independent of the demand shocks, as observed in data.

4 Despite the common use of the term ‘targeting’, there is no agreement on the meaning of it in theliterature. In one terminology, such as in Svensson (1997, 2002) or Clarida et al. (1999), targeting meansminimizing an objective function in which a target variable shows up. In the other, such as in McCallum andNelson (1999) or Reis (2009), targeting means that the target variable is used in a feedback policy rule, likeTaylor type rules. In this study, targeting term is used in the spirit of the first definition. There is also noagreement on how targeting regimes should be implemented, either by using ‘targeting rules’, as studied inSvensson (1997, 2002), or by using some mechanical instrument rules, such as Taylor type rules. In thosepapers, Svensson proposes that inflation targeting is better described as a commitment to a ‘targeting rule’rather than following a mechanical instrument rule.

C© 2012 The Author. Bulletin of Economic ResearchC© 2012 Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research

Optimal Monetary Policy with Sticky Information 3

price model in a New Keynesian framework that usually favours inflation targeting. Price-leveltargeting, which may be thought of as an extreme version of the inflation targeting, is alsodiscussed in the literature, but has not received much support.5 The main difference betweeninflation targeting and price-level targeting is that a base drift in the price level is allowed ininflation targeting.6

Since Mankiw and Reis (2002) proposed the sticky information model, there have been agrowing number of studies investigating the dynamic implications of this model and comparingit with the sticky price models. However, to the best of my knowledge, optimal policy and welfareimplications of the sticky information model have only been studied in Ball et al. (2005) andReis (2009). Ball et al. (2005) find optimal policy as the price-level targeting and the inflationtargeting is just a suboptimal policy in a simple model that is not a DSGE framework. In contrastto our study they use a simple quantity equation to represent the demand side of their model,and take the money supply as the monetary policy instrument. Another difference between thisstudy and their study is the approach taken to solve the model. In this study, unlike in Ballet al. (2005), an optimal policy rule is obtained independent from the policy regimes, andthen each specific regime is imposed on this optimal solution to see the welfare implications.Reis (2009) tries to obtain the optimal policy rule when there is pervasive information stickiness.He considers the Taylor rule and price-level targeting rule as alternative policies. He assumesan unchanged monetary policy rule of a given interest-rate rule. In his model the performancesof different policies are compared by a measure of the social welfare, which under the priorpolicy rule leads to the same unconditionally expected utility as in the alternative policy beingconsidered. Unlike his approach, in this study a utility based welfare is derived independentfrom the policy rules or regimes, and the optimal policy is taken to be the one that minimizesthis welfare loss function. Reis (2009) finds that the Taylor rule is usually better than theprice-level targeting rule, but if only firms are inattentive the price-level targeting dominates theTaylor rule.

Because there is still an active debate on different price adjustment models, and optimalmonetary policy has usually been studied in the sticky price models, the main contributionof this study is to provide additional information on optimal monetary policy by studying theoptimal policy in the sticky information model that has a different stickiness approach. Thisstudy addresses the optimal policy problem in a DSGE model, and unlike the papers by Ballet al. (2005) and Reis (2009), it first derives a welfare loss function for such a framework andthen finds out the optimal monetary policy rule that minimizes this specific loss function. Afterhaving this optimal monetary policy rule, specific policy regime constraints are imposed on itand then those regimes can be evaluated and compared. This different approach is important

5 Among others, Clarida et al. (1999, 2001, Clarida et al. 2002) identify optimal policy as inflationtargeting. For an open economy, they obtain optimal policy as domestic inflation targeting under discretionand domestic price-level targeting under commitment. Gali and Monacelli (2002) find domestic inflationtargeting is optimal for a small open economy. Svensson and Woodford (2005) show that inflation targetingis an effective means of maintaining low and stable inflation and inflation expectations, without causing anyinstability in the output gap. Svensson (2003) argues that in general targeting rules may be superior to theTaylor rule. Hovewer, Schmitt-Grohe and Uribe (2007) find the optimal policy as a real interest rate targetingrule; this is a Taylor type feedback rule with an aggressive response to inflation and no response to output.Juillard et al. (2006) find the optimal policy as an interest rate rule that lies close to the Taylor efficiencyfrontier. Levin et al. (2006) find a simple interest rate rule as optimal that focuses on wage inflation.

6 As mentioned in Clarida et al. (1999), if the price level overshoots its target, the central bank may haveto contract economic activity to return the price level to its target. So, there should be some inflation belowthe level implied by the price-level target to return the price level to its target. Under inflation targeting,bygones are bygones: overshooting of inflation in one year does not require an inflation level below thetarget in the following year.



since it enables us to compare the alternative policy regimes in a DSGE framework accordingto a welfare criterion which is completely independent from those policy regimes.

The rest of the paper is organized as follows. The next section summarizes the relatedliterature. The model is briefly outlined in Section III. In Section IV, utility approximation ofwelfare is described under the sticky information model. The model is solved, and optimal policyand welfare implications of the alternative policy regimes for a cost-push shock are investigatedin Section V. The results are given in Section VI, and Section VII concludes.

II. RELATED LITERATURE

The sticky information model of price adjustment is a relatively new model and proposed byMankiw and Reis (2002). Since then there have been many studies investigating the dynamicimplications of this model and comparing it with the sticky price models.7 However, optimalpolicy and welfare implications of the sticky information model have been studied only in a fewpapers, such as Ball et al. (2005), where only firms are inattentive, and Reis (2009), where allagents are inattentive. In contrast to our findings in this study, Ball et al. (2005) find price-leveltargeting to be the optimal policy while inflation targeting is suboptimal in a simple model thatis not a DSGE framework.8 They represent the demand side of their model by a simple quantityequation, and take the money supply as the monetary policy instrument. In our framework,unlike Ball et al. (2005), an expectational IS equation, which is obtained from the optimizationof households and relates the interest rate to output and inflation, is used to represent thedemand side of the economy. So the monetary policy instrument becomes the interest rate asin monetary practice.9 This study also differs from their study by using a different approachto solve the model and obtain the optimal policy. They do not solve their model explicitly toobtain the optimal money-supply rule, that is they do not characterize the optimal monetarypolicy rule independent from the policy regimes. They instead obtain reduced form for theprice level and interpret the expected price level as the policy instrument. They first impose thepolicy constraints on the price adjustment equation, and then find the welfare losses. However,in our study, since we have an explicit policy instrument given by the interest rate, a generalequilibrium model is solved first for an optimal solution, and then the regime constraints areimposed on this solution to obtain the welfare implications of these regimes. Ball et al. (2005)

7 Mankiw and Reis (2002) obtain the dynamic responses to a monetary policy shock and compare theirmodel with the sticky price model. Due to their results, the sticky information model can explain a long lagbetween monetary policy and inflation, while the sticky price model cannot. They also show that disinflationsare always contractionary. Arslan (2008) examines and compares dynamic responses of the sticky price andsticky information models to a cost-push shock in a New Keynesian DSGE framework. He finds out that thesticky information model produces more reasonable dynamics through lagged, gradual and hump-shapedresponses as observed in data. Reis (2009) investigates dynamic properties of a sticky information generalequilibrium model. To a monetary shock he obtains delayed and hump-shaped responses of inflation and theoutput gap, and inflation follows the output gap. His model also generates enough persistence of inflationand output. In contrast with the predictions of the NKPC model, he obtains the accelerationist Phillips curverelation. See also Korenok (2008), Keen (2007), Dupor et al. (2010), and Kiley (2007).

8 Their exact finding is that flexible price-level targeting (price level gradually returns to its target level)is optimal to cost-push shocks. This implies that allowing base drift in the price level is not optimal in theiranalysis.

9 In a DSGE model, the price adjustment equation is the crucial part for optimal policy and dynamics ofoutput and inflation, because it determines the specific relationship between the output gap and inflation. Inthis study, while the price adjustment is modelled by the sticky information, the demand side is representedby an IS equation, which allows the nominal interest rate to be used as the instrument of monetary policy.Interest rate may also be considered as a variable to be stabilized along with the output gap and inflation.Therefore, volatility of interest rate, so the specific form of the IS equation, may be important to determinethe optimal policy or compare alternative monetary policies.



also make their analysis just for one value of the price stickiness parameter, however our studyinvestigates the optimal policy for different values of this parameter, and indeed finds that theresults are sensitive to this parameter.

Reis (2009) examines the impact of the sticky information on monetary policy and tries tofind out which rules are optimal, whether the interest-rate rules (Taylor-type rules) or price-targeting rules in a model that has pervasive information stickiness, that is sticky informationexists in all markets. He first estimates the model with a priori parameter set and then withposterior parameters obtained from the US data, then characterizes optimal monetary policy.In his model the Central Bank can target the nominal interest rate and monetary policy followsa given interest-rate rule (a Taylor rule, in which the interest rate depends on inflation and theoutput gap, so it can be considered a kind of inflation targeting rule). Thus, his model assumesan unchanged monetary policy rule. However, the IS curve is derived from the optimization ofconsumers. In Reis (2009), the performances of alternative policies are compared by a measureof the social welfare that is different from the one used in this study. Reis (2009), measures thesocial welfare as the percentage increase in steady-state consumption under the prior policy rulethat would lead to the same unconditionally expected utility as in the alternative policy beingconsidered. However, in our study the utility based welfare is derived independent from thepolicy rules or regimes, and the optimal policy is taken to be the one that minimizes this welfareloss function. So in our study the optimal policy is independent from the policy regimes, andalternative policy regimes would be compared by calculating the welfare losses after imposingthe regimes constraint on the optimal policy. In Reis (2009) two policy rules are considered asalternative policies. The first one is the Taylor rule with no monetary shock. The second ruleis a price-level targeting rule. His results show that the Taylor rule is usually better, and bothrules require a strong response to output deviations. However, if only firms are inattentive theprice-level targeting dominates the Taylor rule,10 if only workers are inattentive the Taylor ruleis better, and if only consumers are inattentive any monetary policy rule is as good as another.

This paper is another study on optimal monetary policy under the sticky information modelof price setting, and thus provides additional information and a different approach for thisissue. This study, unlike the above studies, first derives a welfare loss function under the stickyinformation price setting, and then obtains the optimal monetary policy rule that minimizes thiswelfare loss function. When this optimal monetary policy rule is derived, the specific policyregime constraints are imposed on it to evaluate and compare these alternative policy regimesin a DSGE framework according to a welfare criterion which is completely independent fromthose policy regimes.

III. THE MODEL

III.1 Households

The model is a version of the standard New Keynesian dynamic general equilibrium model withprice rigidities, which has been used extensively for theoretical analysis of monetary policy.11

Households are assumed to be a monopolistically competitive supplier of their labour that results

10 Reis (2009) shows that when only firms are inattentive, serial correlation of output growth becomesnegative in contrast with data. Mankiw and Reis (2006) showed another problem with inattentive firms only:real wage growth is much more volatile than productivity, but post-1986 US data shows their volatilities arealmost the same.

11 See Rotemberg and Woodford (1997), McCallum and Nelson (1999), Gali and Monacelli (2002), andArslan (2008).



in a cost-push shock term in the model. The economy is closed and composed of a continuum ofidentical infinitely lived households indexed by i ∈ [0, 1] and a continuum of firms indexed byj ∈ [0, 1]. The households supply labour, which is an imperfect substitute of other labour inputs,purchase consumption goods, and hold financial assets. The firms hire labour and specializein the production of a single good that is an imperfect substitute of other goods. Since eachfirm and household has some monopoly power, the economy has monopolistically competitivemarkets similar to those studied in Dixit and Stiglitz (1977) or Blanchard and Kiyotaki (1987).

The households and firms behave optimally and maximize their utility and profits, respec-tively. There is also a financial market in the economy in which households can trade in a rangeof securities that is large enough to completely cover all states of nature; that is, complete marketis assumed and the households can insure themselves against idiosyncratic uncertainty.

The households derive utility from composite consumption goods and leisure, and the utilityof household i in period t is given by

Uit = C1−σi t

1 − σ− N 1+ϕ

i t

1 + ϕ(1)

where Cit is a Dixit–Stiglitz type CES aggregator of composite consumption of household i andis defined over production C j

it of firm j, and Nit is the household i’s composite labour supplyindex and is defined over the labour demand N j

it of firm j. These are defined as

Cit =(∫ 1

0

(C j

it

) ε−1ε dj

) εε−1

(2)

Nit =(∫ 1

0

(N j

it

) ηt −1ηt dj

) ηtηt −1

(3)

The parameter ε is the elasticity of substitution among the goods, and ηt is the elasticity oflabour demand. These two parameters are greater than one and the same across households.The parameter ϕ is the marginal disutility of labour and is positive.12 The parameter σ is therisk aversion factor and 1/σ represents the elasticity of intertemporal substitution in aggregateconsumption evaluated at steady state.13

Each household i seeks to maximize the lifetime utility

E0

∞∑t=0

β tUit (4)

subject to the intertemporal budget constraint, where β < 1 is the discount factor. Since thehouseholds are identical and the financial markets are complete, the households will have thesame wealth and there will be complete risk sharing in consumption. As in Woodford (2003),this implies the same consumption decisions, and the common level of consumption denoted byCt although the labour supply and output may change. Therefore, Cit = Ct , and the first orderconditions can be obtained as

Wt

Pt

= (1 + μw

t

)Cσ

t N ϕ

i t (5)

β It Et

{(Ct+1

Ct

)−σ (Pt

Pt+1

)}= 1 (6)

12 It can be interpreted as the inverse of the Frisch elasticity of labour supply.13 Intertemporal elasticity of substitution is defined as : 1/σ ≡ −Uc/UccC .



where μwt = 1

ηt −1is the optimal wage markup, and I −1

t is the price of a riskless one-period asset.The consumption of household i can be obtained as

cit = Et cit+1 − 1

σ(it − Etπt+1 + ln β) (7)

where small characters represent the logarithm of those variables, πt = pt − pt−1 is the inflationrate, and it is the nominal interest rate at period t. The demand functions for the consumptionand labour indices can also be obtained as

C jit =

(Pjt

Pt

)−ε

Cit (8)

N jit =

(W jt

Wt

)−ηt

Nit (9)

where Pjt is the price of the goods produced by firm j, Pt is the aggregate price index, W jt isthe nominal wage paid by firm j, and Wt is the aggregate wage index.

III.2 Firms

Each firm j produces its specialized product with a linear technology according to the productionfunction

Yjt = At N jt (10)

That is, the output is only the function of the labour input N jt and the aggregate productivitydisturbance At . The labour input to firm j is given by a CES aggregator of individual householdlabour N j

it as

N jt =(∫ 1

0

(N j

it

) ηt −1ηt di

) ηtηt −1

(11)

The firms hire labour, produce and sell their differentiated products in the monopolisticallycompetitive market, and try to minimize their cost of production. The cost minimization of firmj subject to the labour input N jt yields

N jit =

(Wit

Wt

)−ηt

N jt (12)

where Wit is the nominal wage earned by household i. Since wages are flexible, each householdwill charge the same wage and provide the same amount of labour. Therefore, Wit = W jt = Wt ,and this implies N j

it = Nit = N jt through Equations (9) and (12).In a monopolistically competitive model, it is assumed that each firm knows that its sale

depends on the price of its product. When all purchases are made for private consumption,then the aggregate demand Yt corresponds to the households, total consumption index. So, thedemand function can be written from (8) as

yjt = yt − ε(pjt − pt ) (13)

where yjt is the output produced by firm j, and yt is the aggregate output.The firms also decide what optimal price P∗

j t to charge to maximize their profit given theabove demand function. So, the firm j’s decision problem gives

P∗j t

Pt

= ε

ε − 1

Wt/Pt

At

= μ γt (14)



where μ = ε

ε−1is the constant price markup, and γt = Wt /Pt

Atis the firm’s real marginal cost. Since

the technology is constant returns to scale and the shocks are the same across the firms, the realmarginal cost γt is the same across all firms. This is the standard result in a monopolisticallycompetitive market when all firms are able to adjust their price in every period; that is, eachfirm sets its optimal price P∗

j t equal to a markup over its nominal marginal cost Ptγt . By usingequation (5), the above expression can be rewritten as

P∗j t

Pt

= ε

ε − 1

(1 + μw

t

) Cσt N ϕ

i t

At

= μ(1 + μw

t

) Cσt N ϕ

i t

At

(15)

The market clearing conditions require that consumption should be equal to output, so Cit =Ct = Yt . Then, by using the production function and the demand function, the above equationcan be rewritten as

p∗j t = pt + σ + ϕ

1 + ε(σ + ϕ)yt − 1 + ϕ

1 + ε(σ + ϕ)at + log μ + log

(1 + μw

t

)1 + ε(σ + ϕ)

(16)

When all firms can set their prices freely each period, that is when prices are flexible, all firmsset the same price p∗

j t = pt . The natural level of output yNt is defined as the level where prices

are flexible and the wage markup is fixed at its steady state value μw. This setup means thatthere are no wage markup shocks, and variations in the natural level of output do not reflectthe variations in wage markup. Therefore, under flexible price equilibrium, the natural level ofoutput can be obtained from (16) as

yNt = 1 + ϕ

σ + ϕat − log μ + log(1 + μw )

σ + ϕ(17)

If this is used in Equation (16), one can obtain

p∗j t = pt + α

(yt − yN

t

) + ut (18)

where α and the cost-push shock term ut are defined as

α = σ + ϕ

1 + ε(σ + ϕ), ut = log

[(1 + μw

t

)/(1 + μw )

]1 + ε(σ + ϕ)

.

Therefore in the framework of this study, the cost-push shock term can be interpreted to representthe bargaining power of households in the labour market.

When Equation (7) is aggregated over all households, it can be rewritten in terms of aggregateoutput index as

yt = Et yt+1 − 1

σ(it − Etπt+1 + ln β) (19)

With the assumption of an exogenous AR(1) technology shock process at = ρaat−1 + ξt , thisequation can be written in terms of the output gap xt = yt − yN

t as

xt = Et xt+1 − 1

σ(it − Etπt+1) + νt (20)

This is the IS equation used in the New Keynesian framework, and the disturbance term is givenas νt = − log β

σ− (1−ρa )(1+ϕ)

σ+ϕat.

III.3 Price setting

Mankiw and Reis (2002) proposed the sticky information model as an alternative to the stickyprice model. They argue that the dynamics of their model are similar to those of backward-looking expectations models, and the expectation structure is close to Fisher’s contracting



model. In the sticky information model prices are set every period, but information collectingand processing, that is optimal price computing, occur slowly over time. A randomly selectedfraction 1 − θ of firms receive complete information about the state of the economy in eachperiod, and adjust their prices according to this new information, while the remaining fraction θ

of firms adjust their prices according to old information. The parameter θ measures the degreeof price stickiness; a large one shows that few firms get new information, so fewer firms adjusttheir prices, and the expected time between price adjustments will be longer. The firms that donot adjust their prices will adjust their output according to demand function of the market.14

When a firm j sets its price in period t, it will set it to its optimal expected price according tothe last information it has at period t − k as

pkjt = Et−k p∗

j t (21)

Since new information arrives at the rate of 1 − θ , the share of the firms that last adjusted theirplan k periods ago will be (1 − θ )θ k . Therefore, the aggregate price index that is the average ofall prices in the economy can be written as15

pt = (1 − θ )∞∑

k=0

θ k Et−k p∗j t (22)

By using p∗j t from Equation (18), the sticky information price adjustment equation can be

obtained in terms of prices as

pt = (1 − θ )∞∑

k=0

θ k Et−k(pt + αxt + ut ) (23)

and in terms of inflation as

πt = φα xt + φ ut + (1 − θ )∞∑

k=0

θ k Et−1−k(πt + α�xt + �ut ) (24)

where φ = (1 − θ )/θ . Thus, in the sticky information model expectations are the past expec-tations of current economic conditions. This model also satisfies the natural rate hypothesis,because without any surprises the model implies that pt = Et− j pt , and this implies xt = 0.

IV. WELFARE AND UTILITY APPROXIMATION

IV.1 Measuring the welfare

Welfare is defined as the utility of the representative agent within the model. Such a utilityoptimization based welfare criterion is useful when comparing the consequences of alternativepolicy rules. Woodford (2001) shows that this approach can justify the traditional assumption of

14 In this study, only firms have sticky information as in Ball et al. (2005). Of course all agents andmarkets could have sticky information as in Reis (2009), and in such a case the results might be different.One reason of assuming that only firms are subject to sticky information here is to be in parallel and thesame framework with Ball et al. (2005), and to be able to make a comparison with their results. Anotherreason is that assuming firms have sticky information implies a sticky information price adjustment andPhillips curve, that is the most important part in any new Keynesian model for evaluating the monetarypolicy regimes. However, the case for which information stickiness is pervasive and all markets have stickyinformation might be the subject of another study, as Reis did in his 2009 paper.

15 The aggregate price index Pt can be defined in the model as Pt = (∫ 1

0 P1−εj t d j)

11−ε . A first order

approximation of this linearizes and gives the aggregate price index as the averages of all prices aspt = ∫ 1

0 p jt d j .



price stability, which assumes a quadratic loss function in some form. Also, a precise formulationof the appropriate loss function can be derived in this approach depending upon the model’sassumptions, especially upon the specification of price adjustment. Woodford (2001) shows thatquadratic approximation to the utility function of households in a monopolistically competitiveframework can be expressed as

Ut = 1

2[λ(xt − x∗)2 + vari (pi − pt )] (25)

where x∗ is the efficient level of the output gap,16 pi is the price of differentiated goods, and pt

is the general price level. This expression is valid for any specification of price stickiness.17

In a New Keynesian framework with the standard sticky price (NKPC) model, this utilityfunction reduces to the familiar form of

Ut = 1

2

[λ(xt − x∗)2 + π 2

t

](26)

So, when x∗ is assumed to be exogenous, then the objective function of monetary policy can beexpressed as

max − 1

2Et

{ ∞∑i=0

β i(λx2

t+i + π 2t+i

)}(27)

where λ represents the relative weight on the output gap. Since this objective function takespotential output as the target, it also implicitly takes zero as the target inflation. Much of theliterature has assumed a priori this functional form as the objective of monetary policy. In such arepresentation the problem is what the relative weights of the output and inflation losses shouldbe. There is no specific answer to this, but it is usually accepted that the primary objectiveof monetary policy should be controlling inflation. Therefore, the weight of the inflation losswould be much greater than that of the output gap.

IV.2 Welfare for the sticky information model

The utility-based welfare criterion is the level of expected utility

E0

∞∑t=0

β tUt (28)

approximated around the equilibrium point, where there are no real disturbances. Ut is the periodutility function for any household and is given by Equation (1). The period utility function Ut

can be approximated as

Ut = −e(1−σ )ytσ + ϕ

2

[x2

t + ε−1 + ϕ

σ + ϕvar j (yjt − yt )

]+ t .i .p. (29)

where xt is the output gap, yt is the equilibrium point around which the approximation takesplace, and the t.i.p term represents the terms independent of policy. The details of this welfareapproximation procedure are given in the Appendix.18

The demand function in Equation (13) yields

var j (yjt − yt ) = ε2var j (pjt − pt ) .

16 The level of the output gap when there is no distortion such as due to taxes or market power.17 See Woodford (1999, 2001) and Rotemberg and Woodford (1999).18 See also Woodford (2001, 2005) and Ball et al. (2005).



If this expression is used in Equation (29), then the utility function becomes


2

[x2

t + ε(1 + εϕ)

σ + ϕvar j (pjt − pt )

]+ t .i .p (30)

This is the quadratic approximation to the period utility function and is valid for any model withmonopolistic competition and price stickiness. It shows that policy should aim to stabilize theoutput gap and reduce the price variability. However, the relationship between the price variabil-ity and the stabilization of general price level depends on the price adjustment mechanism. Ballet al. (2005) show that the cross-sectional price variability for the sticky information model ofprice adjustment can be expressed as

var j (pjt − pt ) =∞∑

i=1

ηi (pt − Et−i pt )2 (31)

where ηi = θ i (1−θ)(1−θ i )(1−θ i+1)

. It demonstrates that the variance of relative prices depends upon thequadratic deviations of the price level from the levels expected at all past dates. Only unexpectedcomponents of the price level relative to past expectations affect the equilibrium price variability.This relationship can be rewritten in terms of inflation rate as

var j (pjt − pt ) =∞∑

i=1

ηi

(i−1∑l=0

(πt−l − Et−iπt−l)

)2

(32)

Thus, the equilibrium price variability depends on the sum of squared deviations of inflationrates from the levels expected at all past periods, and only unexpected components of theinflation rates at all previous periods matter for the equilibrium price variability.

Therefore, the quadratic approximation to the utility of the representative agent, by ignoringt.i.p., is given in terms of prices as

Ut = −e(1−σ )ytε(1 + εϕ)

2

[λx2

t +∞∑

i=1

ηi (pt − Et−i pt )2

](33)

and in terms of inflation as

Ut = −e(1−σ )ytε(1 + εϕ)

2

⎡⎣λx2

t +∞∑

i=1

ηi

(i−1∑l=0


)2⎤⎦ (34)

where λ = σ+ϕ

ε(1+εϕ)is the relative weight of the output gap.19

V. OPTIMAL POLICY

The sticky information Phillips curve proposed by Mankiw and Reis (2002) is given in terms ofprices and inflation in (23) and (24), respectively. It is clear from Section III that the modelling

19 This parameter or weight is much smaller than the weight on price deviations. So this implies thatwelfare cost of the deviations from the potential output is much smaller than the cost due to the deviations ofprice levels from the past expected levels. In such a representation, it is usually accepted and observed that theprimary objective of monetary policy is to control inflation. Therefore, the fact that the weight of the lossesdue to prices or inflation is much greater than that of the output gap would conform to today’s monetarypolicy on both theoretical and empirical grounds. Schmitt-Grohe and Uribe (2007) to find a Taylor-type ruleas an optimal policy, in which there is an aggressive response to inflation term but no response to output.Also in the New Keynesian price adjustment models (Phillips curves), the coefficients of the output gap areusually estimated as very small when compared with the coefficients of the price or inflation term.



framework includes the possibility of other shocks, such as technology or productivity shockand demand shock; however, in this study only a cost-push shock is assumed to happen in theeconomy, and so the optimal policy for this shock is investigated.20 Therefore, when there is nodemand or technology shock, the IS curve of the model given in (20) can be rewritten in termsof prices as

xt = Et xt+1 − 1

σ(it − Et pt+1 + pt ) (35)

Since the terms at the outside of the square brackets in the period utility functions (33) and(34) do not affect the optimization problem, they can be dropped. So, the intertemporal lossfunction can be written in terms of prices as

−1

2E0

∞∑t=0

β t

(λx2

t +∞∑

i=1


)(36)

According to this loss function monetary policy should target the potential output and the pricelevel target depends on the squared deviations of the price level from all the levels expectedin the past. Because the price level in a given period is determined based on all informationup to that period, any surprise that causes a deviation of price from the expected level leads toa welfare loss. Since the size of the weights decreases as we go into the past, the effect of asurprise decreases as this happens further in the past.

The optimal monetary policy will be choosing xt , pt , and it , which maximizes the objectivefunction (36) subject to the sticky information Phillips curve and the IS equation. The first stepof the optimization maximizes (36) subject to (23) and gets a relationship between xt and pt ,then the optimum it implied by the IS equation can be obtained.

The first step of this optimization problem yields the optimality condition, which relates xt

and pt as

xt = −�

∞∑i=1

ηi (pt − Et−i pt ) = −�N pt + �

∞∑i=1

ηi Et−i pt (37)

where � = α(1−θ)θλ

and N = ∑∞i=1 ηi.

This optimality condition is valid under both discretionary and commitment monetarypolicies.21 In this study, monetary policy can be thought of as discretionary in the sense thata central bank optimizes its objective function each period. However, this central bank canmake some commitments to manipulate the private sector expectations. This approach also isin accord with today’s modern central banking practice.

This study investigates the optimal policies for a cost-push shock under three policy regimes.First, the unconstrained optimal monetary policy is investigated. A central bank optimizeswithout any commitments to manipulate private-sector expectations under this policy, so thoseexpectations are free to change. In the other two policy regimes, the central bank makes

20 The reasons to investigate the optimal policy only under the cost-push shock are twofold. First, it isconsidered the most important shock in the New Keynesian literature to evaluate the optimal policy. Ballet al. (2005) and Reis (2009) are also focused more on this shock. Second, adding the other shocks andmaking the optimal policy analysis for each of these shocks would make the study too voluminous and large.So, the analysis with other shocks is performed in another study. See Arslan (2009) for an optimal policywith sticky information model for a demand or technology shock.

21 Discretionary policy implies that a central bank optimizes (36) period by period without any com-mitment, therefore it is not able to manipulate the private sector expectations. However, commitmentto a policy implies the global optimization of (36) under a specific commitment, which affects private-sector expectations. In both cases, expectations are predetermined, and are taken as given; therefore, bothof the period by period and global optimizations of (36) yield the same optimality condition given byEquation (37).



commitments to affect expectations, and targets some level of prices or inflation, so theseregimes might be interpreted as the ones under commitment, although optimization takes placeevery period.22 It is also assumed that the cost-push shock term ut follows an AR(1) process andis given by ut = ρut−1 + εt , where εt is the white noise, and ρ is the correlation coefficient ofthe shock. This cost-push shock term can also be expressed in an infinite moving average formas ut = ∑∞

i=0 ρ iεt−i .

V.1 Unconstrained optimal policy

The monetary policy does not have any announced target or credible commitment to manipulateprivate-sector expectations in this case. Therefore, the central bank maximizes the objectivefunction (36) without any constraint on private-sector expectations that are free to change. Theoptimality condition for this regime is represented by Equation (37). The sticky informationPhillips curve in (23) can be rewritten as

pt = (1 − θ )(pt + αxt + ut ) + (1 − θ )∞∑

k=1

θ k Et−k(pt + αxt + ut ) (38)

If the optimality condition (37) is substituted into (38) to get rid of xt , one can obtain

pt (1 − (1 − θ ) + α(1 − θ )�N ) = (1 − θ )ut + α(1 − θ )�∞∑

i=1

ηi Et−i pt

+ (1 − θ )∞∑

k=1

θ k Et−k

[pt + α

(−�N pt + �

∞∑i=1

ηi Et−i pt

)+ ut

].

(39)

If pt is represented by pt = ∑∞i=0 γiεt−i , Equation (39) can be solved by the method of

undetermined coefficients. To find pt , the coefficients γi are obtained in terms of the correlationcoefficient ρ of the cost-push shock. These serial correlation coefficients γk of the prices forany k are obtained as23

γk = ϒk

� − �k

(40)

where ϒk = ρk∑k

i=0 θ i, � = (θ + α(1 − θ )�N )/(1 − θ ), and

�k = α�

k∑i=1

ηi + (1 − α�N )k∑

i=1

θ i + α�

(k∑

i=1

ηi

) (k∑

i=1

θ i

).

22 This might be thought, in the terminology of Svensson, as a ‘targeting rule’ but implementation isdiscretionary. However, there is no specific commitment-based rule in this study although the central bankmakes commitments and has some objectives to form expectations. In the new Keynesian literature, acommitment rule usually implies a predetermined, mechanical rule like a Taylor-type rule. In this study herepolicy regimes, which have implications on expectations through some type of policy rule, are compared,but the specific forms of these rules are not important in the framework of this study. The policy regimescompared here also have the same objective function and optimality conditions as given above, independentfrom any specific form of policy rule. Welfare losses are calculated and compared after the expectations areformed according to these policy regimes.

23 In expression (40), both the numerator and denominator approach to zero when k approaches infinity.The convergence of the terms ϒk and � − �k depends on the values of ρ and θ , respectively. Therefore,an optimal solution for the price path depends on the convergence speed of these terms. If ρ is largerthan a critical value for any given θ , then the numerator term ϒk converges to zero more slowly than thedenominator term � − �k ; therefore, the solution to γk diverges, and a bounded solution cannot be obtained.A bounded solution requires that ρ is smaller than this critical value, which implies the faster convergenceof ϒk to zero than � − �k .



When a solution to the price path is found, the output gap can be obtained from the optimalitycondition in (37). By expressing the output gap as an infinite MA process, xt = ∑∞

i=0 βiεt−i ,and substituting it into (37), one can obtain

∞∑i=0

βiεt−i = −�N∞∑

i=0

γiεt−i + �

∞∑j=1

∞∑i= j

η jγiεt−i (41)

The coefficients βk for any k can be obtained from this expression again by the method ofundetermined coefficients as

βk = −�Nγk + �γk

k∑i=1

ηi (42)

The optimal nominal interest rate it can be found from the IS equation by using the solutionsto the output gap and prices as

it = σ (Et xt+1 − xt ) + Et pt+1 − pt

If it is expressed as it = ∑∞i=0 φiεt−i , then the serial correlation coefficients φk for any k can be

found as

φk = σ (βk+1 − βk) + γk+1 − γk (43)

Having the output gap and prices, the welfare loss of the model under the unconstrained optimalpolicy can be calculated from Equation (36) as

Welfare Loss = −E0

∞∑t=0

β t

(λx2

t +∞∑

i=1


)

= −∞∑

t=0

β t

⎡⎣λx2

t +∞∑

i=1

ηi

(i−1∑k=0

γkεt−k

)2⎤⎦

(44)

V.2 Optimal policy with price-level targeting

In this subsection, the optimal policy for a cost-push shock is investigated under the price-leveltargeting regime. In this regime, monetary authority commits to keep the price level constantand takes actions to return the price level to its steady state value after a shock happens. Sinceit is assumed that the central bank has a credible commitment to a targeting regime, it canmanipulate the private-sector expectations. Therefore, the expectations about the price levelunder the price-level targeting regime satisfy24

Et−i pt = 0 (45)

Under the price-level targeting, the optimality condition (37) reduces to

xt = −�

∞∑i=1

ηi pt = −�N pt (46)

So the price adjustment Equation (23) becomes

pt = (1 − θ )

θ + (1 − θ )α�N

[ut +

∞∑k=1

θ k

∞∑i=k

ρ iεt−i

](47)

24 It is assumed here that the target price level is normalized to be 1.



If pt = ∑∞i=0 γiεt−i , then the coefficients γk for any k can be found in terms of the correla-

tion coefficient ρ of the cost-push shock by the method of undetermined coefficients fromEquation (47), and given by

γk = (1 − θ )

θ + (1 − θ )α�Nρk

k∑i=0

θ i (48)

The output gap can be obtained from Equation (46) as

xt = −�N (1 − θ )

θ + (1 − θ )α�N

[ut +

∞∑k=1

θ k

∞∑i=k

ρ iεt−i

](49)

If it is assumed that xt = ∑∞i=0 βiεt−i , then the coefficients βk for any k can be found in terms

of ρ as

βk = −�N (1 − θ )

θ + (1 − θ )α�Nρk

k∑i=0

θ k (50)

The nominal interest rate it can be obtained from the IS equation. If it = ∑∞i=0 φiεt−i , then the

coefficients φk for any k are given by

φk = σ (βk+1 − βk) + γk+1 − γk (51)

The welfare loss can be calculated after obtaining the output gap and prices. Therefore, the lossfunction given in Equation (36) takes the following form under the price-level targeting

Welfare Loss = −∞∑

t=0

β t

(λx2

t +∞∑

i=1

ηi p2t

)= −

∞∑t=0

β t(λx2

t + N p2t

)(52)

V.3 Optimal policy with inflation targeting

In an inflation targeting regime, the monetary authority commits to maintain a stable inflationlevel around zero. Any base drift in price level is allowed. Because of the central bank’s crediblecommitment to the inflation target, the private-sector expectations about inflation are preset,and do not change when a temporary shock hits the economy. Therefore, expectations about theinflation under the inflation targeting regime satisfy

Et−iπt = 0 (53)

When the variable of interest is inflation, the intertemporal loss function given in (36) can berewritten in terms of inflation as

−1

2E0

∞∑t=0

β t

⎡⎣λx2

t +∞∑

i=1

ηi

(i−1∑l=0


)2⎤⎦ (54)

The optimality condition in (37) can be written in terms of inflation by using Equation (32) as

xt = −�

∞∑i=1

ηi

i−1∑k=0

(πt−k − Et−iπt−k)

Under the inflation targeting, this expression reduces to

xt = −�

∞∑i=1

ηi

i−1∑k=0

πt−k (55)



and the Phillips curve in equation (24) can be rewritten as

πt = φα

(−�

∞∑i=1

ηi

i−1∑k=0

πt−k

)+ φ ut + (1 − θ )

∞∑k=0

θ k Et−1−k

(�ut − α�

∞∑k=1

ηk (πt − πt−k

)

(56)

If πt = ∑∞i=0 γiεt−i , then the above equation can be solved to obtain the inflation path by finding

the coefficients γk for any k in terms of the model’s parameters and the correlation coefficientρ of the cost-push shock. These coefficients are obtained from the expression

γk(1 + αφ�N ) = − αφ�Nk−1∑i=0

γi + αφ�

k−1∑i=0

γi

k−i∑j=1

η j + (1 − θ )α�Nk−1∑i=0

γiθk−1−i

− (1 − θ )α�θ

k−2∑i=0

γiθk−2−i

k−1−i∑j=1

η j + φρk + (ρ − 1)ρk−1

k−1∑i=0

θ i

(57)

Once inflation is obtained, the output gap can be found from Equation (55). If xt = ∑∞i=0 βiεt−i ,

then the serial correlation coefficients βk for any k are given by

βk = −�

k∑i=0

γk

(N −

k−i∑j=1

η j

)(58)

The nominal interest rate it can be obtained from the IS equation, and if it = ∑∞i=0 φiεt−i , then

the coefficients φk for any k are given by

φk = σ (βk+1 − βk) + γk+1 (59)

By having inflation and the output gap, the welfare loss under the inflation targeting can becalculated from (54) as

Welfare Loss = −∞∑

t=0

β t

⎡⎣λx2

t +∞∑

i=1

ηi

(i−1∑k=0

πt−k

)2⎤⎦ . (60)

VI. SIMULATION AND RESULTS

The parameters of the model need to be calibrated to make simulations and get results for optimalpolicies. These parameters are calibrated as follows: the elasticity of substitution among goods,ε = 6; risk aversion factor, σ = 1, which is also the inverse of the intertemporal elasticityof substitution of consumption; marginal disutility of labour, ϕ = 1.25; and discount factorβ = 0.99.25 The calibration of the parameters σ and ϕ follows Chari et al. (2000).

The dynamic paths of the output gap, the price level, inflation and the nominal interestrate are given for ρ = 0.8 and ρ = 0.4 when θ = 0.75 in Figure 1. These two values for thepersistence of the shock are chosen to represent low and high persistence, and they seemed

25 This value of ε implies a price markup of 20 percent, which is consistent with accepted values for thisparameter. There is no consensus on the value of parameter σ in literature and there are different estimationsfor this parameter; see Arslan (2008) for a review. However, the simulations with different parameter valueshave been performed to check the robustness of the results; although the results and pictures have beenaffected from those changes, the main conclusion have not. The parameter α given in Equation (18) is takenas 0.1 in Ball et al. (2005); with my calibration this parameter becomes around 0.15.



0 10 20 30 40 50 60 70 80 90-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 = 0.8

0 2 4 6 8 10 12-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6 = 0.4

0 10 20 30 40 50-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4 = 0.8

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

= 0.4

0 10 20 30 40 50-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 = 0.8

0 5 10 15 20 25 30-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 = 0.4

PriceOutput gap Inflation Nom. Interest rate

(a) Unconstrained Policy

(b) Price-level Targeting

(c) Inflation Targeting

Fig. 1. Optimal policy impulse responses to a unit cost-push shock when θ = 0.75.

enough to depict the main picture in impulse responses to the shock. The chosen value of theprice stickiness parameter θ implies that prices on average are fixed for four quarters; this is inthe accepted range and in accord with theoretical and empirical literature.26 The figure shows

26 Gali and Gertler (1999) found this duration as five to six quarters, while Sbordone (2002)found it as nine to 14 months. Some survey evidence shows this duration is somewhat less than



0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5

0W

elfa

re L

oss

= 0.85

0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5

0

Wel

fare

Los

s

= 0.75

0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5

0 = 0.4

Wel

fare

Los

s

Unconstrained Inflation Targ. Price-level Targ.

0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5

0 = 0.2

Wel

fare

Los

s

Fig. 2. Sensitivity of welfare loss to persistence of the cost-push shock.

that all variables initially deviate from their steady state values when there is a shock to theeconomy. The price level converges back to its target level under the unconstrained policy andprice-level targeting regimes, however there is a base drift in the price level while the inflationreturns to its target level under the inflation targeting. The output gap decreases while theprice level and inflation increase at the beginning, and then they all return to their steady-statevalues. Inflation overshoots in all policy regimes; but the output gap overshoots only in inflationtargeting. It can also be seen that fluctuations in the nominal interest rate increase when theshock becomes less persistent. This figure also illustrates that the persistence of the variablesdepends on the persistence of the shock term. However, these impulse responses do not say muchabout which policy is better, so the welfare implications of these alternative policy regimes needto be investigated.

The welfare implications of the policy regimes are illustrated in Figures 2 and 3. The panelsin Figure 2 show the sensitivity of welfare losses to the persistence of the shock for differentvalues of θ under alternative policy regimes.27 Since there are no bounded solutions for somevalues of θ and ρ, the loss functions diverge for those values as shown in the figure. The figure

the above figures and around three to four quarters; see Blinder et al. (1998) and Bils and Klenow(2004).

27 For the cases shown in Figure 2, the most plausible value of θ would be θ = 0.75, because this valueimplies a price rigidity of four quarters, which is in the accepted range; see footnote 24.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-6

-5

-4

-3

-2

-1

0 = 0.1

Wel

fare

Los

s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-6

-5

-4

-3

-2

-1

0 = 0.5

Wel

fare

Los

s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-6

-5

-4

-3

-2

-1

0 = 0.8

Wel

fare

Los

s

Unconstrained Inflation Targ. Price-level Targ.

Fig. 3. Sensitivity of welfare loss to price stickiness.

shows that losses are similar and move together in the unconstrained policy and the price-leveltargeting regimes, especially when prices are stickier (larger θ ). The loss in the inflation targetingregime behaves similarly to others when θ values are low; however it is steady and somewhatincreasing function of ρ until it becomes very close to one for higher θ values.

These graphs illustrate that the best policy depends on the degree of price stickiness θ and thepersistence of the cost-push shock ρ. The inflation targeting emerges as the best policy whenprices are more flexible, that is when θ is small. However, when prices become stickier, that is forlarge values of θ , the best policy depends on the persistence of the shock, and the unconstrainedpolicy and the price-level targeting dominate the inflation targeting when the shock is notvery persistent and ρ is less than some threshold value.28 Therefore, it can be concluded thatthe inflation targeting is the best policy and dominates the other regimes by having a smallerwelfare loss when prices are more flexible or the shock is persistent. However, the price-leveltargeting dominates the inflation targeting when prices become stickier and the shock is notvery persistent.29 Therefore, a policy regime could not emerge as the best one in all situations,

28 As can be seen from the two top panels of Figure 2, this threshold value of ρ increases with an increasein stickiness.

29 When the shock is very persistent the inflation targeting dominates in all cases. However, for the lesspersistent shock the intuition may imply the reverse of those results. That is, one may expect the price-level



since one policy regime could dominate the other depending on the model’s parameters (i.e.,price stickiness and/or shock persistence). These findings can also be obtained to some extentfrom Figure 3, which shows the welfare losses when the price stickiness parameter changes fora given value of the persistence of the shock.30

Both figures, especially Figure 3, show that as prices become more flexible (θ gets smaller),the welfare loss usually gets bigger under all three policy regimes examined here. This is becausewhen prices become more flexible they can easily fluctuate and deviate from the target levelswith a shock, so this causes large welfare losses. As can be seen from Equation (30) or (36),welfare loss occurs when output deviates from the potential level and price variability increases.When prices become more flexible; they can change easily and frequently; this would cause autility loss for households, and thus a welfare loss for the whole economy due to increase inprice variability.

VII. CONCLUSIONS AND POLICY IMPLICATIONS

In this study, optimal monetary policy for a closed economy within a New Keynesianmacroeconomic model with the sticky information model of price adjustment is investigated.Specifically, the welfare implications of three alternative monetary policy regimes are examinedand compared for a unit cost-push shock. The first one is a discretionary policy regime in a sensethat the central bank optimizes each period without any commitment or target to manipulate theprivate-sector expectations, and it is called the unconstrained policy regime. The second policyregime is the price-level targeting in which the central bank still optimizes each period, but thistime it commits to a targeting rule that puts a specific target level for prices, so it is able tomanipulate the private-sector expectations. The third one is the inflation targeting in which thecentral bank has a specific target level for the inflation rate rather than prices, so a base drift inthe price level is allowed.

The results of this study show that optimal policy depends on the degree of price stickinessand the persistence of the shock. The inflation targeting emerges as the best or the optimal policywhen prices are flexible enough or the shock is persistent enough. However, the unconstrainedpolicy or the price-level targeting might be preferable to the inflation targeting if prices are notvery flexible and the shock is not very persistent. The results also show that as prices becomemore flexible, the welfare loss usually gets bigger under all three policy regimes examined here.

This study finds strong support for the inflation targeting policy regime in the stickyinformation model only for a specific set of parameters. Therefore, when the sticky informationmodel is used as a different price setting mechanism instead of the standard sticky price model,

targeting to be optimal for flexible prices, and the inflation targeting to be optimal for sticky prices. However,the difference between inflation and price-level targeting is that a base drift in the price level is allowed ininflation targeting. It might be thought that when prices are flexible, allowing a drift in the price level dueto a shock, that is following an inflation targeting regime, will not be too costly. On the other hand, whenprices are sticky, following a price-level targeting policy would be less costly than following the inflationtargeting policy. This is because when prices are sticky a base drift would be more costly, so by following aprice-level targeting regime such drifts may be avoided and the cost would be smaller. But, it must be keptin mind that those results are due to the combination of many factors, such as credibility, past expectations,and information stickiness.

30 In Figure 2 sensitivity of welfare losses is drawn for a few stickiness parameters while the persistence ofthe shock changes between 0 and 1. The similar exercise is performed in Figure 3 by keeping the persistenceparameter fixed and changing the stickiness parameter between 0 and 1. However, the former one wouldbe preferred and be more relevant to the real world because the shock persistence is likely to exhibit morevariability than the price stickiness parameter.



the results support the general practice of inflation targeting regime in today’s modern centralbanking only under special circumstances.

REFERENCES

Arslan, M. M. (2008). ‘Dynamics of the sticky information and sticky price models in a NewKeynesian DSGE framework’, Economic Modelling, 25, pp. 1276–94.

Arslan, M. M. (2009). ‘Optimal monetary policy with the sticky information model of priceadjustment: inflation or price-level targeting?’, Working Paper 12-2009, Baskent University.

Ball, L. (1994). ‘Credible disinflation with staggered price setting’, American Economic Review,84, pp. 282–9.

Ball, L., Mankiw, N. G., Reis, R. (2005). ‘Monetary policy for inattentive economies’, Journalof Monetary Economics, 52, pp. 703–25.

Bils, M., Klenow, P. (2004). ‘Some evidence on the importance of sticky prices’, Journal ofPolitical Economy, 112, pp. 947–85.

Blanchard, O. J., Kiyotaki, N. (1987). ‘Monopolistic competition and the effects of aggregatedemand’, American Economic Review, 77, pp. 647–66.

Blinder, A. S., Canetti, E. R., Lebow, D., Rudd, J. B. (1998). Asking About Prices: A NewApproach to Understand Price Stickiness, New York: Russell Sage Foundation.

Chari, V. V., Kehoe, P. J. and McGrattan, E. R. (2000). ‘Sticky price models of the busi-ness cycle: can the contract multiplier solve the persistence problem?’, Econometrica, 68,pp. 1151–79.

Clarida, R., Gali, J., Gertler, M. (1999). ‘The science of monetary policy: a New Keynesianperspective’, Journal of Economic Literature, 37, pp. 1661–707.

Clarida, R., Gali, J., Gertler, M. (2001). ‘Optimal monetary policy in open versus closedeconomies: an integrated approach’, American Economic Review, 91, pp. 253–57.

Clarida, R., Gali, J. and Gertler, M. (2002). ‘A simple framework for international monetarypolicy analysis’, NBER Working Paper No. 8870.

Dixit, A. K., Stiglitz, J. E. (1977). ‘Monopolistic competition and optimum product diversity’,American Economic Review, 67, pp. 297–308.

Dupor, B., Kitamura, T., Tsuruga, T. (2010). ‘Integrating sticky prices and sticky information’,Review of Economics and Statistics, 92, pp. 657–69.

Gali, J., Gertler, M. (1999). ‘Inflation dynamics: a structural econometric analysis’, Journal ofMonetary Economics, 44, pp. 195–222.

Gali, J. and Monacelli, T. (2002). ‘Monetary policy and exchange rate volatility in a small openeconomy’, NBER Working Paper No. 8905, Universitat Pompeu Fabra.

Juillard, M., Karam, P., Laxton, D. and Pesenti, P. (2006). ‘Welfare-based monetary policy rulesin an estimated DSGE model of the U.S. economy’, European Central Bank Working PaperNo. 613.

Keen, B. D. (2007). ‘Sticky price and sticky information price setting models: what is thedifference?’, Economic Inquiry, 45, pp. 770–86.

Kiley, M. (2007). ‘A quantitative comparison of sticky price and sticky information models ofprice setting’, Journal of Money, Credit, and Banking, 39, pp. 102–25.

Korenok, O. (2008). ‘Empirical comparison of sticky price and sticky information models’,Journal of Macroeconomics, 30, pp. 906–27.

Levin, A. T., Onatski, A., Williams, J. C. and Williams, N. (2006). ‘Monetary policy underuncertainty in micro-founded macroeconometric models’, in Gertler, M., Rogoff, K. (eds),NBER Macroeconomics Annual 2005, Cambridge: MIT Press, pp. 229–87.

Mankiw, N. G., Reis, R. (2002). ‘Sticky information versus sticky prices: a proposal to replacethe New Keynesian Phillips curve’, Quarterly Journal of Economics, 117, pp. 1295–328.

Mankiw, N. G., Reis, R. (2006). ‘Pervasive stickiness’, American Economic Review, 96,pp. 164–9.



McCallum, B. T., Nelson, E. (1999). ‘An optimizing IS-LM specification for monetary policyand business cycle analysis’, Journal of Money, Credit and Banking, 31, pp. 296–316.

Reis, R. (2009). ‘Optimal monetary policy rules in an estimated sticky-information model’,American Economic Journal: Macroeconomics, 1, pp. 1–28.

Rotemberg, J., Woodford, M. (1997). ‘An optimization based econometric framework for theevaluation of monetary policy’, NBER Macroeconomics Annual, 12, pp. 297–346.

Rotemberg, J. and Woodford, M. (1999). ‘Interest rate rules in an estimated sticky price mode’,NBER Working Paper No. 6618.

Sbordone, A. (2002). ‘Prices and unit labour costs: a new test of price stickiness’, Journal ofMonetary Economics, 49, pp. 265–92.

Schmitt-Grohe, S. and Uribe, M. (2007). ‘Optimal inflation stabilization in a medium-scalemacroeconomic model’, in Schmidt-Hebbel, K. and Mishkin, F. (eds), Monetary PolicyUnder Inflation Targeting, Santiago, Chile: Central Bank of Chile, pp. 125–86.

Svensson, L. E. (1997). ‘Inflation forecast targeting: implementing and monitoring inflationtargets’, European Economic Review, 41, pp. 1111–46.

Svensson, L. E. (2002). ‘Inflation targeting: should it be modeled as an instrument rule or atargeting rule?’, NBER Working Paper No. 8925.

Svensson, L. E. (2003). ‘What is wrong with Taylor rules? Using judgment in monetary policythrough targeting rules’, Journal of Economic Literature, 41, pp. 426–77.

Svensson, L. E. and Woodford, M. (2005). ‘Implementing optimal policy through inflationforecast targeting’, in Bernanke, B. S. and Woodford, M. (eds), The Inflation-TargetingDebate, Chicago: University of Chicago, pp. 19–83.

Woodford, M. (1999). ‘Optimal monetary policy inertia’, NBER Working Paper No. 7261.Woodford, M. (2001). ‘Inflation stabilization and welfare’, NBER Working Paper No. 8071.Woodford, M. (2005). Interest and Prices, Princeton, NJ: Princeton University Press.

APPENDIX

Quadratic approximation to welfare

Welfare is defined as the total utility of households in the model and given by

E0

∞∑t=0

β tUt (61)

where Ut is the period utility function for any household given in Equation (1). This can beapproximated around the equilibrium point where there are no real disturbances. Under theequilibrium with no real disturbances, there should not be any shock resulting from the marketpowers of firms and households. Such a situation can be obtained if it is assumed μ = 1 andμw = 0, then the equilibrium point from Equation (17) will be

yt = 1 + ϕ

σ + ϕat (62)

Under the equilibrium condition, and given the aggregate production function, the period utilityfunction can be expressed as

Ut = e(1−σ )yt

1 − σ−

∫ 1

0

e(1+ϕ)(y jt −at )

1 + ϕdj (63)



The quadratic Taylor-series approximation of the first term on the right around the equilibriumvalue yt is given by

e(1−σ )yt

1 − σ= e(1−σ )yt

1 − σ+ e(1−σ )yt (yt − yt ) + 1

2(1 − σ )e(1−σ )yt (yt − yt )

2

= e(1−σ )yt

1 − σ+ e(1−σ )yt

[(yt − yt ) + 1

2(1 − σ )(yt − yt )

2

] (64)

The second term can be approximated as∫ 1

0

e1+ϕ)(y jt −at )

1 + ϕd j =

∫ 1

0

[e1+ϕ)(yt −at )

1 + ϕ+ e(1+ϕ)(yt −at )(yjt − yt ) − e(1+ϕ)(yt −at )(at − at )

+ 1

2e(1+ϕ)(yt −at )(1 + ϕ)(yjt − yt )

2 + 1

2e(1+ϕ)(yt −at )(at − at )

2

− e(1+ϕ)(yt −at )(1 + ϕ)(yjt − yt )(at − at )]

d j

(65)

If yt = yt − yt , at = at − at , y j t = yjt − y j t , and t.i.p. is the terms independent of policy, theperiod utility function in (63) can be written as

Ut = e(1−σ )yt

(yt + 1

2(1 − σ )y2

t

)− e(1+ϕ)(yt −at )

∫ 1

0

[y j t + 1 + ϕ

2y2

j t − (1 + ϕ)y j t at

]dj + t .i .p.

Ut = e(1−σ )yt

(yt + 1

2(1 − σ )y2

t

)− e(1+ϕ)(yt −at )

[E j y jt + 1 + ϕ

2(var j y j t + (E j y jt )2)

− (1 + ϕ)at E j y j t

]+ t .i .p.

(66)

where E j y jt = ∫y j t d j and var j y j t are the mean value and the variance of y j t across all goods

jat date t, respectively. In the model, Dixit-Stiglitz index of aggregate demand can be written interms of output as

Yt =(∫ 1

0

(Yjt )ε−1ε dj

) εε−1

(67)

This expression can be rewritten in a log-linear form, and be approximated as a second orderTaylor series expansion around yt as

eε−1ε yt = ∫ 1

0e

ε−1ε y jt dj

=∫ 1

0

eε−1ε yt dj +

∫ 1

0

[ε − 1

εe

ε−1ε yt (yjt − yt ) + 1

2

(ε − 1

ε

)2

eε−1ε yt (yjt − yt )

2

]dj

= eε−1ε yt + ε − 1

εe

ε−1ε yt

[∫ 1

0

(y j t + 1

2

ε − 1

εy2

j t

)dj

]

= eε−1ε yt

[1 + ε − 1

ε

(E j y jt + 1

2

ε − 1

εE j y

2j t

)].

(68)

Since var j y j t = E j y2j t − (E j y jt )2, expression (68) can be written as

ε − 1

εyt = log

[1 + ε − 1

ε

(E j y jt + 1

2

ε − 1

ε(var j y j t + (E j y jt )

2)

)](69)



When yt is close enough to yt , as expected, the above expression can be approximated as

yt = E j y jt + 1

2

ε − 1

εvar j y j t (70)

If this expression is used in the period utility function (66) to eliminate the E j y jt terms, andsome small terms are ignored, then one can obtain

Ut = e(1−σ )yt

(yt + 1

2(1 − σ )y2

t

)− e(1+ϕ)(yt −at )

×[

yt + 1 + ϕ

2y2

t + ε−1 + ϕ

2var j y j t − (1 + ϕ)yt at

]+ t .i .p.

(71)

This equation can be rewritten by using Equation (62) as


2

[y2

t − 2(1 + ϕ)

σ + ϕat yt + ε−1 + ϕ

σ + ϕvar j y j t

]+ t .i .p (72)

It can be written from the natural rate expression and Equation (62) that

yNt = yN

t − yt = 1 + ϕ

σ + ϕat (73)

Then by plugging this expression into (72), one can get


2

[y2

t − 2yNt yt + ε−1 + ϕ

σ + ϕvar j y j t

]+ t .i .p (74)

If −(yNt )2 and yt (it is constant with respect to j) terms are added into the t.i.p. and into the

variance of the above equation, respectively, (74) can be rewritten as


2

[(yt − yN

t

)2 + ε−1 + ϕ

σ + ϕvar j (y j t − yt )

]+ t .i .p (75)

If the hat terms are expended by using their definition, the above expression can be written as


2

[x2

t + ε−1 + ϕ

σ + ϕvar j (yjt − yt )

]+ t .i .p (76)

This is the Equation (29) given in the text.


optimal monetary policy with the sticky information model of price adjustment: inflation or...

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