optimal monetary policy and downward nominal wage rigidity in frictional labor markets

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Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 37 (2013) 345–364

0165-18

http://d

n Tel.:

E-m1 D

Nomina

wage ch

prepond

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

Optimal monetary policy and downward nominal wage rigidityin frictional labor markets

Salem Abo-Zaid n

Texas Tech University, Department of Economics, Lubbock, TX 79409, USA

a r t i c l e i n f o

Article history:

Received 31 October 2010

Received in revised form

1 March 2012

Accepted 30 August 2012Available online 20 September 2012

JEL classification:

E31

E32

E52

E58

Keywords:

Downward nominal wage rigidity

Optimal monetary policy

Long-run inflation rate

Labor market frictions

Intertemporal wedge smoothing

89/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.jedc.2012.09.003

þ1 806 742 2466; fax:þ1 806 742 1137.

ail address: [email protected]

NWR means not only that wage increases a

l wages tend to increase in good times but t

anges. Note that the fact that wage increase

erance wage increases may reflect long-term

a b s t r a c t

This paper studies the optimal long-run inflation rate in a labor search and matching

framework in the presence of downward nominal wage rigidity. Optimal monetary

policy features positive inflation in the long run; the optimal annual long-run inflation

rate for the U.S. economy is slightly below 1 percent with a money demand motive and

around 2 percent otherwise. Positive inflation facilitates real wage adjustments and

hence it eases job creation and prevents excessive increase in unemployment following

adverse productivity shocks. The findings of the paper can also be related to standard

Ramsey theory of ‘‘wedge smoothing’’; with positive inflation under sticky prices, the

size and the volatility of the intertemporal wedge are significantly reduced.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

This paper studies optimal monetary policy in the presence of downward nominal wage rigidity (DNWR) within a laborsearch and matching model.1 When nominal wages are downwardly rigid, the optimal long-run inflation rate is strictlypositive. The optimal annual inflation rate is slightly below 2 percent with no money demand and slightly below 1 percentwhen a motive for money demand is introduced. A strictly positive long-run inflation rate is driven by precautionaryconsiderations in the expectations of adverse shocks. Positive inflation allows for downward real wage adjustments (thus‘‘greasing the wheels’’ of the labor market) that ease job creation and limit the increase in unemployment followingadverse shocks. The optimal annual long-run inflation rate suggested by this paper is higher than in a model withneoclassical labor markets, suggesting that the nature of the labor market in which DNWR is introduced does matter foroptimal monetary policy prescriptions.

ll rights reserved.

re more likely than wage cuts, but also that the distribution of wage changes is not symmetric.

hey do not tend to fall proportionally in bad times, thus generating an asymmetric distribution of

s are more common than wage cuts by itself is insufficient evidence for the presence of DNWR; a

productivity growth or steady state (positive) inflation.

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www.elsevier.com/locate/jedc

dx.doi.org/10.1016/j.jedc.2012.09.003

dx.doi.org/10.1016/j.jedc.2012.09.003

dx.doi.org/10.1016/j.jedc.2012.09.003

mailto:[email protected]

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dx.doi.org/10.1016/j.jedc.2012.09.003

S. Abo-Zaid / Journal of Economic Dynamics & Control 37 (2013) 345–364346

The results of the paper are related to standard Ramsey theory of smoothing distortions (or ‘‘wedges’’) over time.A virtually constant distortion across periods is the main insight of Barro (1979), in a partial equilibrium framework, andChari et al. (1991) in a quantitative general equilibrium model, among others. Recently, Arseneau and Chugh (2010) havedeveloped intertemporal and static notions of efficiency in general equilibrium models with labor search and matchingfrictions. They show that the intertemporal wedge should indeed be smoothed, but, contrarily to the cornerstone result oftax smoothing in the Ramsey literature, that occurs through volatility in the labor-income tax rate. In this paper, optimalmonetary policy, which calls for a positive inflation rate due to DNWR, reduces the size and the volatility of theintertemporal wedge when prices are sticky. This suggests that with both intertemporal and nominal distortions in place,the monetary authority cannot completely and simultaneously smooth both distortions. When prices are fully flexible,however, monetary policy keeps the intertemporal wedge virtually constant over the business cycle.

The results under zero-inflation policy at all dates and states are significantly different. In this case, the volatilityof the intertemporal wedge is substantially higher; the wedge absorbs the shock. Similar results are obtained forlabor market variables; the combination of DNWR and fully stable prices limit the decline in real wages considerably,thus generating unemployment increases and job creation declines far beyond their responses under a positiveinflation rate.

The optimality of a positive inflation rate with DNWR and money demand is particularly interesting. In this case, thereare three competing forces on inflation: DNWR calls for positive inflation, money demand calls for negative inflation andprice stickiness pushes towards zero inflation. This paper suggests that, with plausible calibration of the modelparameters, DNWR is the most dominant force of the three in the determination of the optimal long-run inflation rate.Money demand, however, remains very significant as it leads to a considerably lower optimal inflation rate compared tothe optimal inflation rate with no money demand motive.

The paper is motivated by the evidence on DNWR that has been suggested by several empirical studies. One of the mostnotable recent evidence on DNWR is the comprehensive work of the International Wage Flexibility Project (IWFP),reviewed in Dickens et al. (2007a,2007b). Their findings indicate asymmetry in the distribution of nominal wage changesin 16 OECD countries, with the U.S. being among the countries with very high degrees of DNWR. Gottschalk (2005) showsthat, after correcting for measurement errors that typically appear in wages reported in surveys, only about 5 percent ofworkers experienced wage cuts during a course of a year while working for the same employer. Card and Hyslop (1997)show a spike at the zero in the distribution of nominal wage changes, indicating DNWR. Altonji and Devereux (2000) findthat only 2.5 percent of hourly workers had wage reductions.

The idea that positive inflation is needed to ‘‘grease the wheels’’ of the labor market dates back at least to Tobin (1972).Following negative shocks that call for real wage drops, Tobin (1972) suggests setting a positive inflation rate, on one hand,and stabilizing nominal wages, on the other, to facilitate real wage adjustment in the presence of DNWR. Tobin’s idea hasgained more attention in recent years for two reasons. First, inflation rates have become very low in the last two decades.Clearly, DNWR is more relevant in low inflation environments and during recessions. Second, central banks around theworld target positive inflation rates, either explicitly or implicitly. DNWR creates a precautionary motive for positiveinflation: since the timing of (negative) shocks is not fully predicted, the monetary authority keeps the inflation ratepositive on average in order to ‘‘ensure’’ against those shocks once they materialize.

I allow for price rigidity, DNWR, money demand and labor market frictions, with the latter being consistent withpositive unemployment in equilibrium. To model DNWR, I follow Kim and Ruge-Murcia (2009,2011) by using the Linex

function as the nominal wage adjustment cost function. This function delivers higher costs in case of wage cutscompared to wage increases of the same magnitude. To see the significance of this setup, consider the response of aneconomy to an adverse productivity shock. If inflation is high, then DNWR is less likely to prevent real wage drops, andhence inflation mitigates the potential increase in unemployment. If inflation is low, however, DNWR will likelytranslate into Downward Real Wage Rigidity (DRWR). In this case, if the monetary authority seeks to keeps prices stable,downward rigidity in real wages implies higher unemployment than in the absence of DNWR. If the monetary authorityinstead chooses to stabilize employment, it inflates in order to achieve the desired cut in real wages. That is, the inflationrate needed ‘‘to grease the wheels’’ of the economy is higher than it would be if nominal wages were not downwardlyrigid. In short, the presence of labor market frictions may magnify the need for grease inflation if policy makers aretrying to keep unemployment low, or it may create excessive unemployment when attempting to keep prices close tofull stability.

The current study contributes to the recent literature that studies the optimal inflation rate in the presence of DNWR.Assuming frictionless labor markets, Kim and Ruge-Murcia (2009,2011) show that the optimal annual grease inflation inthe U.S. is positive (around 0.4 percent), regardless of whether a motive for money demand is introduced or not. Fagan andMessina (2009) introduce asymmetric menu costs for wage setting and show that the optimal inflation rate for the U.S.ranges between 2 percent and 5 percent. The optimal inflation rate in their model depends on the dataset used to measurethe degree of DNWR. The optimal long run inflation rate found in this paper is thus more in line with the results of Faganand Messina (2009). My paper differs from the work of Kim and Ruge-Murcia (2009,2011) mainly in the nature of the labormarkets. In the current paper, unemployment-inflation tradeoff is a concern for monetary policy makers, which is resolvedby setting a strictly positive inflation rate.

The fact that the inflation rate in this study is significantly higher than in Kim and Ruge-Murcia (2009,2011), despite theuse of a similar proxy for DNWR, suggests that the structure of the labor market in which DNWR is studied matters for

S. Abo-Zaid / Journal of Economic Dynamics & Control 37 (2013) 345–364 347

optimal policy recommendations. Since the discussion is over the long-run inflation rate, these differences areeconomically significant.

The remainder of this paper proceeds as follows. Section 2 outlines the search and matching model economy withDNWR. Section 3 discusses the search-based efficient allocations and the intertemporal wedge. Section 4 describes thecalibration methodology and the parameterization of the model. Section 5 presents the main results about the optimalinflation rate and the role of inflation in smoothing the intertemporal wedge. Impulse responses are presented in Section 6.Section 7 presents some sensitivity analyses and Section 8 concludes.

2. The model

The economy is populated by households, monopolistically-competitive firms that produce differentiated products anda monetary authority. Hiring labor by firms is subject to search and matching frictions. Following literature, the modelembeds the search and matching framework of Pissarides (2000), which has become the main framework within whichoptimal monetary policy is studied in the presence of labor market frictions. Each firm faces an asymmetric adjustmentcost function for nominal wages that implies higher costs of cutting nominal wages relative to increasing nominal wagesby the same magnitude. Changing prices by each firm is subject to a direct resource cost.

2.1. Households

The economy is populated by a representative household which consists of family members of measure one. At eachdate t a household member can be in either of two states: employed or unemployed and searching for a job. Employedindividuals are of measure nt and the unemployed individuals are of measure ut, where ut¼1�nt, as conventional in theliterature.

Following the assumption of consumption insurance in Merz (1995) and Andolfatto (1996), all family members in thishousehold have the same consumption. The disutility of work is the same for all employed individuals and the value ofnon-work is the same for all unemployed individuals. The members of the representative household also derive utilityfrom holding real money balances (‘‘Money in the Utility’’). Given these assumptions, the household’s problem is tomaximize:

E0

X1t ¼ 0

bt½uðctÞ�ntvðhtÞþgðmtÞ�, ð1Þ

where bo1is the standard subjective discount factor, E0 is the expectation operator, ct is the composite consumptionindex, ht denotes hours per worker, mt denotes real money balances, u(ct) is the period utility function of consumption,v(ht) is the period disutility from supplying labor and g(mt) is the period utility function of money holdings. These

functions satisfy the usual properties: @uðUÞ@c 40, @2uðUÞ

@c2 o0, @vðUÞ@h 40, @2vðUÞ

@h2 40, @gðUÞ@m 40 and @2gðUÞ

@m2 o0.

As standard in New Keynesian models, consumption (ct) is a Dixit–Stiglitz aggregator of differentiated products (cj,t)produced by monopolistically-competitive firms:

ct ¼

Z 1

0cððe�1Þ=eÞ

j,t dj

!ðe=ðe�1ÞÞ

, ð2Þ

where e41 is the elasticity of substitution between two varieties of final goods. In line with standard Dixit–Stiglitz basedNK models, the optimal allocation of expenditures on each variety is given by

cj,t ¼Pj,t

Pt

� ��ect , ð3Þ

where Pt ¼ ðR 1

0 P1�ej,t djÞð1=ð1�eÞÞ is the Dixit–Stiglitz price index that results from cost minimization.

Maximization is subject to the sequence of budget constraints of the form:

ctþBt

Ptþ

Mt

Pt¼

nthtWt

Ptþð1�ntÞbþ

Rt�1Bt�1

Ptþ

Mt�1

Ptþ

Tt

PtþYt

Pt, ð4Þ

where b stands for unemployment benefits, Wt is the nominal wage, Bt denotes nominal bonds, Mt denotes nominal moneyholdings, Rt is the nominal gross interest rate on bonds, Pt is the aggregate price level, Tt are net nominal transfers and Yt

stands for nominal profits from the ownership of firms.Household’s choices of ct, Bt and Mt yield the following optimality conditions:

uc,t ¼ bRtEtuc,tþ1

ptþ1

� �, ð5Þ

and

gm,t ¼ uc,tRt�1

Rt

� �, ð6Þ


where pt ¼ Pt=Pt�1 is the gross price inflation rate, uc,t is the marginal utility of consumption and gm,t is the marginal utilityfrom holding real money balances. Eq. (5) is the standard Euler equation in consumption. Eq. (6) is a money demandfunction that positively relates the quantity of real money demanded with current consumption and negatively with theopportunity cost of holding money. When Rt¼1 (the ‘‘Friedman Rule’’), the economy is satiated with real money balancesas the marginal utility of money holdings is zero.

2.2. Firms

There is a continuum of measure one of monopolistically-competitive firms. Each firm j hires labor as the only inputand produces a differentiated product yj,t using the following technology:

yj,t ¼ ztnj,tf ðhj,tÞ, ð7Þ

with zt denoting aggregate productivity (which is common to all firms), nj,t is employment at time t at firm j and hj,t

denotes hours per worker at the firm. The pricing of a firm is subject to a quadratic adjustment cost as in Rotemberg(1982), expressed in units of the final good.

Hiring workers by each firm is subject to search and matching costs. Each period, firms post vacancies and they meetunemployed workers searching for jobs. Nominal wages and hours per worker are determined in a Nash bargainingprocess between workers and firms to be outlined later. As discussed by Krause and Lubik (2007), the assumptions of aquadratic price adjustment cost function and symmetry among firms allow for integrating price decisions and employ-ment decisions at the same firm.2

Each firm faces an adjustment cost function for changing nominal wages. The cost of adjusting the nominal wage of oneworker by firm j is given by the following Linex function:

FWj,t ¼

fc2

exp �cðWj,t

Wj,t�1�1Þ

� �þcð

Wj,t

Wj,t�1�1Þ�1

� �ð8Þ

This adjustment cost is asymmetric- for any positive value of c, the cost of cutting nominal wages by a given amount ishigher than the cost of increasing nominal wages by the same magnitude. Also, as c approaches zero, this functionapproaches the quadratic adjustment cost function and hence it enables comparison with the case of symmetricadjustment cost function. In the other extreme, as c approaches infinity, this adjustment cost function becomes L-shaped,implying that nominal wages cannot fall.

In this model, the nominal wage rate is determined through bargaining between firms and workers. Hence, it is unclearwho should pay the costs of adjusting wages. Following Arseneau and Chugh (2008), I assume that firms entail these costsin my benchmark version of the paper. However, I also consider the case in which workers pay all the adjustment costs toshow that the main results of the paper are highly robust to the choice of who bear the costs. Notice also that theeconomy-wide resource constraint is the same under both specifications.

Posting a vacancy v entails a cost of g for the firm. Matches between unemployed individuals and vacant jobs aregoverned by a constant return-to-scale-matching function of the form:

mðvt ,utÞ ¼ smuzt v1�z

t , ð9Þ

where sm is a scaling parameter that measures the efficiency of the matching process and z measures the elasticity ofmatches with respect to unemployment.

Labor market tightness is defined as

yt ¼vt

utð10Þ

The probability of the firm to fill a job is given by qðytÞ ¼mðvt ,utÞ=vt , and the job finding rate is given bypðytÞ ¼mðvt ,utÞ=ut . As usual, the job filling rate is decreasing in labor market tightness and the job finding rate isincreasing in labor market tightness. Intuitively, the higher the ratio of vacancies to unemployment, the lower theprobability for a specific vacancy to be filled, but the higher the probability of an unemployed individual to find a job.

Finally, employment at each firm evolves according the following law of motion:

nj,tþ1 ¼ ð1�rÞðnj,tþmðvj,t ,utÞÞ, ð11Þ

with r denoting the separation rate from a match. This formulation assumes that a match formed at time t starts toproduce at time tþ1 given that it survived exogenous separations.

2 I assume, as in the canonical labor search model, that job stayers and new hires are paid the exact same wage. As discussed in Pissarides (2009),

wages of newly matched workers are more volatile than those of existing workers, given rise to distinguishing between the two groups of workers.

I abstract from this distinction since it does not change the main idea here, namely that (aggregate) nominal wages are rigid. This is particularly true

given that most workers at a certain point in time are job stayers. The implications of this distinction for optimal inflation policy are left to a future work.


Each firm j chooses its price, vacancies and employment for next period to maximize the expected present discountedstream of profits:

E0

X1t ¼ 0

btuc,t

uc,0

Pj,t

Ptyj,t�nj,twj,thj,t�gvj,t�

fc2

exp �cðWj,t

Wj,t�1�1Þ

� �þcð

Wj,t

Wj,t�1�1Þ�1

� �nj,t�

j2

Pj,t

Pj,t�1�1

� �2

yt

( ), ð12Þ

subject to the sequence of laws of motion of employment (with m(vj,t,ut) being substituted using the definition of the jobfilling rate):

nj,tþ1 ¼ ð1�rÞðnj,tþvj,tqðytÞÞ, ð13Þ

and the downward-sloping demand function for its product:

ztnj,tf ðhj,tÞ ¼Pj,t

Pt

� ��eyt ð14Þ

Since households own the firms, future profits of the firms are discounted by the stochastic discount factor ofhouseholds b uc,tþ 1

uc,t. The last term in (12) is the adjustment cost of prices, expressed in units of the final good. The

adjustment cost of prices is assumed to be quadratic rather than asymmetric in the benchmark model, but I consider thepossibility of asymmetric adjustment cost function in the robustness analysis section.

I assume symmetry across workers (they supply the same number of hours) and firms (they choose the same amount ofemployment and vacancies), and hence I suppress the index j in what follows. Then, combining the first-order conditionswith respect to ntþ1 and vt yields:

gqðytÞ

¼ bð1�rÞEtuc,tþ1

uc,t

� �mctþ1ztþ1f ðhtþ1Þ�wtþ1htþ1�

fc2ðexp �cðpw

tþ1�1Þ� �

þcðpwtþ1�1Þ�1Þþ

gqðytþ1Þ

" #( ), ð15Þ

where wtð ¼Wt=PtÞ is the real wage and mct is the Lagrange multiplier on the output constraint (14). This multipliermeasures the contribution of one additional unit of output to the revenue of the firm, and, in equilibrium, it equals the realmarginal cost of the firm.

Eq. (15) is the Job Creation (JC) condition (or Free-Entry condition), and it states that, in equilibrium, the firm equatesthe vacancy-creation cost to the expected present discounted value of profits from the match. As the term in bracketsmakes clear, the flow profit to a firm from a match equals output net of wage payments and the costs of adjusting wages.

In a symmetric equilibrium, in which all firms set the same price, Rotemberg pricing gives the standard forward-looking price Phillips curve:

1�jðpt�1ÞptþbjEtuc,tþ1

uc,t

� �ðptþ1�1Þptþ1

ytþ1

yt

� �¼ eð1�mctÞ ð16Þ

This equation suggests that the current price inflation rate is a function of the expected price inflation rate and thecurrent real marginal cost. When prices are fully flexible (j¼0) or fully stabilized (pt¼1 for all t), this equation collapsesto the familiar condition, mct ¼ e�1=e, which is the inverse of the steady state gross price markup.

The role of DNWR in driving inflation is better seen by substituting, using the time (t�1) version of (15), for the realmarginal cost mct in (16). The adjustment cost of nominal wages increases the marginal cost of the firm and leads, in turn,to an increase in inflation. DNWR plays here the role of an endogenous cost-push shock. Interestingly, this shock isasymmetric, which implies implications not only for the volatility of inflation but also for its long-run mean.

2.3. Nash bargaining

As is typical in the literature, wage payments and hours per employed individuals are determined by a Nash bargainingbetween firms and individuals. Firms and workers then split the surplus of a match according to their bargaining powers.Because of the monetary nature of my model and nominal wage rigidity, I follow Gertler et al. (2008), Arseneau and Chugh(2008) and Thomas (2008), among others, by assuming that bargaining is over nominal wages Wt rather than real wageswt. One caveat for this assumption is the possible existence of an indexation scheme. I, therefore, allow for indexation topast inflation in the robustness analysis of the paper.

As shown in the technical appendix, bargaining over nominal wages gives the following condition that characterizes thereal wage setting:3

ot

1�otmctztf ðhtÞ�wtht�FW

t þg

qðytÞ

� �¼wtht�

vðhtÞ

uc,t�bþEt

otþ1

1�otþ1

gqðytÞ

�gyt

� �� , ð17Þ

where Z denotes the share of workers in the match surplus (which is also their deterministic steady state bargaining

power), ot ¼Z

Zþð1�ZÞððDFt Þ=ðD

Wt ÞÞ

is the effective bargaining power of workers, DFt is the marginal change in the value of a filled

3 The formal derivations presented in this paper are available in a technical appendix in the author’s website at /http://sites.google.com/site/

salemabozaidshomepage/homeS, or by request.

http://sites.google.com/site/salemabozaidshomepage/home

http://sites.google.com/site/salemabozaidshomepage/home


job and DWt the marginal change in the value of being employed as the nominal wage varies. The wage adjustment cost

thus drives a wedge between the effective bargaining power and the ex-ante bargaining power of workers.Similarly, bargaining over hours per employed individual gives the following condition:

Gt

1�Gtmctztf ðhtÞ�wtht�FW

t þg

qðytÞ

� �¼wtht�

vðhtÞ

uc,t�bþEt

Gtþ1

1�Gtþ1

gqðytÞ

�gyt

� �� , ð18Þ

where Gt ¼Z

Zþð1�ZÞdF

t

dWt

is the effective bargaining power of workers in hours determination, dFt and dW

t are, respectively, the

marginal changes in the values of a filled job and being employed as hours per employed individual vary.Conditions (17) and (18) suggest that the current real wage is affected by the outside option of workers (unemployment

benefits), the disutility of work, the cost of adjusting nominal wages and the continuation value of being employed. Asstandard in this class of models, the real wage is increasing in the value of the outside option and disutility of work asworkers need higher real wages to compensate them for the disutility of work and the forgone outside option. Finally,when nominal wages are fully flexible (or fully stabilized), we have ot¼Z.4

2.4. The private-sector equilibrium

The private-sector’s equilibrium conditions are the Euler Eq. (5), the money demand function (6), the definition of labormarket tightness (10), the law of motion for employment (13), the job creation condition (15), the price Philips curve (16),the real wage setting Eq. (17), the hours determination Eq. (18), the resource constraint of the economy:

ntzthat�ct�gvt�

fc2ðexp �cðpw

t �1Þ� �

þcðpwt �1Þ�1Þnt�

j2ðpt�1Þ2ntzth

at ¼ 0, ð19Þ

the constraint on unemployment:

ut ¼ 1�nt , ð20Þ

and the identity describing real wage growth:

wt

wt�1¼pw

t

ptð21Þ

Condition (21) is typically introduced in models with sticky price and sticky nominal wages. This identity does not holdtrivially under sticky nominal wages and sticky prices, and hence it should be added to the equilibrium conditions of theprivate sector in order to tie the path of real wages to the paths of nominal wages and prices.5

Definition 1. Given the exogenous processes {Rt,zt}, the private-sector equilibrium is a sequence of allocationsfct , ht , mt , nt , ut , vt , yt , mct , wt , pt , pw

t g that satisfy the equilibrium conditions (5)–(6), (10), (13) and (15)–(21).

2.5. The optimal monetary policy problem

The monetary authority in this economy maximizes welfare subject to the resource constraint and the first-orderconditions of households and firms.

Definition 2. Given the exogenous process of technology zt, the monetary authority chooses fct , ht , mt , nt , ut , vt , yt ,mct , Rt , wt , pt , pw

t g to maximize (1) subject to (5)–(6), (10), (13) and (15)–(21).

3. Search efficiency and the intertemporal wedge

In the basic Ramsey theory of public finance, the planner aims at smoothing distortions (or ‘‘wedges’’) over the businesscycle. This is the main insight of the partial-equilibrium ‘‘tax-smoothing’’ result of Barro (1979). Chari et al. (1991) showthat this result is carried over to a general equilibrium framework. Judd (1985) and Chamley (1986) established that theoptimal capital income tax in the steady state is zero and that there are no intertemporal distortions. Albanesi andArmenter (2007) generalize this idea and show that, in the deterministic steady state of a general class of optimal policyproblems, it is optimal to achieve a zero intertemporal distortion. More recently, Arseneau and Chugh (2010) have shown,within a labor search and matching model, that the Ramsey planner aims indeed at smoothing intertemporal wedges, butthat is not mapped into labor tax smoothing. In the current paper, the only ‘‘tax’’ (or policy instrument) available to theRamsey planner is the inflation rate. Hence, I also examine whether the notion of ‘‘wedge smoothing’’ applies to thecurrent setup and, if it does, how that is mapped into smoothing price inflation.

4 The use of the term ‘‘effective bargaining power’’ in a model with wage stickiness follows Gertler and Trigari (2009) who assume staggered multi-

period wage contracting in a labor search and matching model.5 This constraint also appears in the study of Erceg et al. (2000) and Chugh (2006), among others.


The derivations give to the following definitions:

IMRS¼uc,t

buc,tþ1, ð22Þ

and,

IMRT ¼ ð1�rÞ ztþ1 f ðhtþ1Þ�ðvðhtþ1Þ=uc,tþ1Þþðgð1�mu,tþ1Þ=mv,tþ1Þ

ðg=mv,tÞ

� �ð23Þ

These definitions are borrowed from the recent work of Arseneau and Chugh (2010). IMRS is the intertemporal marginalrate of substitution between consumption choices across periods (put differently, the ratio of marginal utilities ofconsumption at time t and time tþ1). IMRT stands for the intertemporal marginal rate of transformation and it measuresthe increase in consumption at time tþ1 as a result of a forgone one unit of consumption at time t. In the Appendix, I showthat efficiency requires IMRT¼ IMRS.

The efficiency condition can also be written as

1¼ ð1�rÞEtbuc,tþ1

uc,t

� �ztþ1f ðhtþ1Þ�ðvðhtþ1Þ=uc,tþ1Þþðgð1�mu,tþ1Þ=mv,tþ1Þ

ðg=mv,tÞ

� �� ð24Þ

In the decentralized economy, the equivalent condition to (24) is given by

1¼ ð1�rÞEtbuc,tþ1

uc,t

� �mctþ1ztþ1f ðhtþ1Þ�ðvðhtþ1Þ=uc,tþ1Þ�FW

tþ1�bþðgð1�zÞ=mv,tþ1Þð1þEtþ1ððotþ2ðz�mu,tþ1Þ=zð1�otþ2ÞÞÞÞ

ðg=mv,tÞð1�z=1�otþ1Þ

" #( )ð25Þ

Comparing the square brackets in condition (24) with the square brackets in condition (25) implicitly defines the‘‘intertemporal wedge’’. By comparing (24) and (25), sufficient conditions for efficiency are: nominal wages are fully

flexible or fully stabilized (i.e. FWtþ1¼0, and hence otþ1¼Z), the Hosios condition holds (z¼Z), the unemployment benefits

are zero (b¼0) and no monopolistic power in the final-good sector (which implies mctþ1¼1).The fact that the adjustment cost function of nominal wages appears in the term defining the intertemporal wedge has

a special significance. When nominal wages are not fully flexible (or not fully stabilized), they have direct and indirect

effects on the intertemporal wedge, with the indirect effect being through the deviation of the effective bargaining powerof workers (ot) from its ex-ante value (Z). DNWR only magnifies the two effects in periods of downturn. Hence,‘‘smoothing’’ nominal wages are important for reducing and smoothing the wedge over time. Setting a positive inflationrate reduces the part of the intertemporal wedge induced by DNWR and it thus reduces and smoothes the intertemporalwedge. This can be easily seen by substituting for the real marginal cost (mct) using the price Phillips curve.

4. Calibration

In the first subsection, I describe the data used to estimate the model, the estimation methodology and the estimationresults. I then turn to discuss the calibration methodology applied in this study.

4.1. Functional forms

I assume the following period utility function:

uðct ,ht ,nt ,mtÞ ¼c1�s

t

1�s�ntwh1þW

t

1þWþmm1�t

t

1�t , ð26Þ

where s is the curvature parameter of the period utility function of consumption, t is the curvature parameter of theperiod utility function of real money, m and w are scaling parameters and W is the inverse of the intertemporal elasticity ofsubstitution of labor.

Output per worker has diminishing returns in hours per worker, as follows:

f ðhtÞ ¼ hat , ð27Þ

with a being the labor share of output.Total factor productivity is governed by the following AR(1) process:

logðztÞ ¼ ð1�rzÞlogðzÞþrzlogðzt�1Þþut , ð28Þ

where z¼1 is the steady state of technology and the innovation term ut is normally distributed with zero mean and astandard deviation of su.

4.2. Parameterization

This subsection describes the parameterization of the model. To estimate the model parameters, I assume thatmonetary policy is implemented using an interest rate rule. Alternatively, I could assume that policy makers in the U.S.have been always optimizing (maximizing a welfare criterion subject to the equilibrium conditions of the private sector


and the resource constraint), but this could be a strong assumption. The approach I follow here is more common in thisliterature (e.g. Kim and Ruge-Murcia, 2009,2011) and it can better explain the data (see Adolfson et al., 2011 for morediscussion about this issue).

I assume an interest rate rule of the form:

logRt

R

� �¼ rrlog

Rt�1

R

� �þrplog

pt

p

�þrulog

ut

u

�, ð29Þ

with undated variables denoting the deterministic steady state values of the corresponding variables, rr is the smoothingcoefficient of the interest rate, rp is the coefficient of inflation and ru is the coefficient of unemployment. The choice ofunemployment follows Faia (2008), who shows that the best interest rate rule in a model with labor market frictionsshould respond to unemployment alongside inflation (and that responding to output is welfare reducing).

I first divide the parameters of the model into two groups. The values of the first group of parameters, O1¼(a,b,e,r,t),are set based on US data. Setting those parameters to widely accepted values sharpens the estimation of otherparameters in what follows. I set a¼2/3 based on National Income and Product Accounts (NIPA) data. Assuming a time

unit of one quarter, the subjective discount factor b is set to 0.99, implying a yearly interest rate of roughly 4 percent. Theparameter t is set to 6.21, implying a interest-elasticity of money demand of about �0.16, which is the elasticityof money demand with respect to the nominal interest rate (measured by the Three-Month Treasury Bill) over the period1964:Q2-2007:Q4.6 The separation rate r is assumed to be 0.1, in line with the literature. This value is based on Hall(1995) and Davis et al. (1996). Finally, e is set to 11, implying a net steady state markup of 10 percent, in line with theliterature.

The second set of parameters, O2¼(j, f, c, s, b, W, m, z, g, ru, su, rr, rp, ru), is estimated using the Simulated Methods ofMoments (SMM) described in the appendix in the end of this paper. The essence of this method is to estimate O2 in orderto match observed U.S. data moments. Using the FRED database of the Federal Reserve Bank of St. Louis, I obtain aquarterly US data sample that covers the period 1964:Q2-2007:Q4.7 I extract data for the Gross Domestic Product, hoursper employed individual, consumption, the Consumer Price Index (CPI), civilian employment, the average hourly earningsof production and nonsupervisory employees (total private) and the M1 nominal money aggregate. I also use the findingsof Hagedorn and Manovskii (2008) for the standard deviations of vacancies, unemployment and labor market tightness.In estimating the parameters of the model, I match the standard deviations, autocorrelations and the covariances of thedata series.

The probabilities to fill a vacancy and to find a job are 0.7 and 0.6, respectively, in line with the literature. Given thosevalues, the scaling parameter sm is pinned down using either the expression for the job finding rate or the expression forthe job filling rate. The parameter w is calibrated so that the deterministic steady state value of h is 0.3.

As is standard in the literature, the benchmark calibration of the model assumes that the Hosios (1990) condition holds,and hence the Nash bargaining power of workers equals the contribution of an unemployed individual to the match(i.e. Z¼z). As shown in Hosios (1990), this condition guarantees the efficiency of the matching process. This assumptionalso allows for another degree of freedom in estimating the parameters of the model. In what follows, however, I alsopresent the results when this condition does not hold.

4.3. Estimation results

Table 1 presents the parameter estimates. I consider four different cases in turn- the optimal inflation rate under DNWRwith and without money demand and the optimal inflation rate in the absence of DNWR with and without moneydemand. The results with no DNWR help to assess the implications of DNWR for the optimal long-run inflation raterelative to a model with symmetric adjustment cost.

Almost all parameter values are statistically significant at the 95% confidence level. The statistical significance ofc suggests that nominal wages are downwardly rigid. Few parameters (e.g. s, W, z, ru and su) are in line with previousestimates and they are relatively stable across the different cases considered. The estimates of W suggest a relatively lowintertemporal labor elasticity (about 0.3–0.4), consistent with microeconomic evidence and with a model that assumes anexplicit extensive margin of the labor market.

The adjustment cost parameters of prices j and nominal wages f are the most sensitive to changes in thespecification of the model. The adjustment cost parameter of nominal wages f is higher in the absence of DNWR thanotherwise. This might be the case since, in the absence of DNWR, f is the only parameter that account for nominal wagerigidity in the model. The Price stickiness parameter is also higher in the presence of DNWR relative to the case withsymmetric adjustment cost. Since DNWR calls for further deviations from zero inflation, higher price rigidity is needed tomitigate this effect. Overall, the size of c suggests that DNWR is very important in accounting for the stickiness ofnominal wages.

6 This value is in line with Ireland (2001). The use of the Three-Month Treasury Bill follows Schmitt-Grohe and Uribe (2004a).7 My sample ends in the fourth quarter of 2007 in order to avoid any potential bias in the estimation of the interest-elasticity due to the sharp decline

in the interest rate and the extremely sharp increase in M1 in the last recession.

Table 1Parameter estimates, simulated method of moments.

Parameter No money demand With money demand

Symmetric adjustment cost Asymmetric adjustment cost Symmetric adjustment cost Asymmetric adjustment cost

s 1.5502n 1.8925n 1.6687n 1.8540n

(0.3225) (0.4207) (0.3466) (0.3723)

W 2.6030n 2.8830n 3.1088n 2.5932n

(0.5351) (0.6724) (0.6553) (0.4798)

b 0.4185nn 0.3497n 0.3508n 0.3842n

(0.2206) (0.1630) (0.1628) (0.1770)

m – – 0.0012n 0.0011n

(0.0002) (0.0001)

z 0.4427n 0.4093n 0.4469n 0.4377n

(0.0789) (0.0608) (0.0701) (0.0669)

g 0.3152nn 0.4024n 0.3461nn 0.3512n

(0.1749) (0.0926) (0.1807) (0.0889)

j 63.9145n 90.4717n 93.7010n 117.2947nn

(16.5754) (38.6846) (41.7055) (63.7387)

f 163.1643n 79.7170n 114.7843n 40.2530n

(72.6551) (33.9136) (45.3075) (12.5719)

c – 2480.1372n – 2383.3587n

(637.8995) (525.1455)

rr 0.8614n 0.8031n 0.9463n 0.8453n

(0.1862) (0.1347) (0.1434) (0.1647)

rp 1.2746n 1.6125n 1.4031n 1.4546n

(0.4752) (0.7541) (0.5782) (0.6841)

ru �0.1326n�0.1847n

�0.1147nn�0.2147n

(0.0623) (0.0872) (0.0641) (0.0931)

ru 0.9634n 0.9521n 0.9603n 0.9564n

(0.0191) (0.0156) (0.0168) (0.0179)

su 0.0076n 0.0067n 0.0079n 0.0077n

(0.0011) (0.0009) (0.0013) (0.0014)

Standard errors are in parentheses.n Denotes significance at 5% level.nn Denotes significance at 10% level.


4.4. Computational solution

Since Linearization eliminates the asymmetries present in the model, I apply second-order approximations for theequilibrium conditions of the monetary authority using the procedure developed by Schmitt-Grohe and Uribe (2004b).A second-order approximation also allows the unconditional mean of a variable in the ‘‘stochastic steady state’’ of themodel to be potentially different from its deterministic steady state value. In particular, it allows the unconditional meanof a variable to be affected by the size of the underlying shock.

5. The optimal long-run inflation rate

5.1. The optimal inflation rate in the deterministic steady state

In the absence of shocks and no money demand, the steady state inflation rate is exactly zero as the monetary authorityminimizes the resource cost of adjusting prices. When money demand is introduced, the optimal steady state inflation rateis negative, but near zero. This is the standard result in a model with money demand and sticky prices.

5.2. The optimal long-run inflation rate

The optimal long-run inflation rate in the ‘‘stochastic steady state’’ is discussed in this subsection. Before presentingresults with rigid wages, a note on the case with fully flexible nominal wages is in order (not shown). When nominal wagesare costless to adjust (f¼0), and prices are rigid, optimal monetary policy fully stabilizes prices at all dates and statesfollowing productivity shocks. Therefore, real wage adjustment in this case occurs entirely through instantaneousadjustment of the nominal wage.

When nominal wages are rigid, but the adjustment cost function is symmetric (f40, c¼0), the optimal long-runinflation rate is zero with no money demand motive (Panel I, Table 2) and nearly zero with money demand (Panel II,Table 2). The symmetry of the wage adjustment cost eliminates the precautionary motive for positive price inflation. If

Table 2Optimal annual long-run price inflation rate.

No money demand With money demand

Symmetric adjustment cost Asymmetric adjustment cost Symmetric adjustment cost Asymmetric adjustment cost

p 0.0010 1.8154 �0.0834 0.8622

(�0.0606, 0.0731) (1.5402, 2.2390) (�0.1492, �0.0356) (0.5478, 1.2483)

Std. Dev. of y 1.5936 1.6788 1.6632 1.7141

Std. Dev.of y 6.3046 14.8455 6.0157 9.4674

The 95% confidence intervals are in parentheses. The inflation rate is in annualized percentage terms.


money demand is introduced, the optimal inflation rate is negative, but it is closer to zero than to the Friedman Rule. Thisresult is in line with Schmitt-Grohe and Uribe (2004a) and Chugh (2006), among others.

When nominal wages are downwardly rigid (c40), the optimal long-run inflation rate is strictly positive. The optimalannual long-run inflation rate is slightly below 2 percent with no money demand and slightly below 1 percent otherwise.Both results suggest that DNWR is a very important factor for optimal inflation policy in a labor search and matchingmodel. With no money demand, DNWR is far more dominant than price rigidity as it pushes the optimal inflation rateaway from zero inflation. DNWR clearly dominates the Friedman effect and it thus gives a ground for positive inflationpolicy even in a model with a money demand motive. Money demand, however, is evidently very important as it leads to asignificant fall in the optimal long-run inflation rate relative to the case with no money demand.

The results reported in Table 2 are very important- they suggest that the optimal inflation rate is significantly higherthan in an otherwise model with neoclassical labor markets (as for example in the work of Kim and Ruge-Murcia,2009,2011). This result extends the recent literature that study optimal monetary policy in the presence of labor marketfrictions (e.g. Thomas, 2008; Faia, 2009). Following negative productivity shocks, the monetary authority faces anunemployment-inflation tradeoff, which is resolved by varying the inflation rate in order to avoid excessive increase inunemployment. The unemployment-inflation tradeoff is even stronger in a model with DNWR and, given the largevolatility of unemployment, it requires stronger response of inflation relative to a model with no explicit unemploymentmargin. This also applies to the long-run grease inflation rate. Therefore, the nature of the labor market in which DNWR isstudied does matter for optimal monetary policy recommendations.

Table 2 also reports the standard deviations of output and labor market tightness (which represents here the labormarket aggregates). The model does well in accounting for the volatility of output in all cases. The bottom line of Table 2presents an interesting result: the model with DNWR significantly improves the ability of the standard labor search andmatching model in accounting for labor market tightness. Relative to the model with a quadratic adjustment cost, thestandard deviation of y is more than doubled when no money demand motive is introduced, and it increases by more than50 percent with money demand (the difference between both cases is likely due to the difference in the size of c betweenthe two specifications as shown in Table 1). This finding is very noticeable given the size of the output’s standard deviationand the well-known failure of the canonical labor search and matching model in accounting for the true volatilities of labormarket variables since the seminal paper of Shimer (2005).

5.3. Discussion—the optimal inflation rate and intertemporal wedge smoothing

The strictly positive long-run inflation rate is due to precautionary behavior by the monetary authority. Since the timing(and magnitude) of adverse shocks cannot be fully predicted, positive inflation over time allows for real wage adjustmentswhen nominal wages are downwardly rigid, and hence it eases job creation and limits the increase in unemployment.

This result can also be rationalized by considering basic Ramsey theory of ‘‘wedge smoothing’’. DNWR contributes tothe wedge between the IMRS and the IMRT, thus positioning the economy away from the efficient state. The Ramseyplanner thus aims for closing this source of distortion, at least partially. This happens through stabilizing nominal wages tothe maximum possible extent. Holding a positive inflation rate over time helps in achieving real wage adjustments withless nominal wage adjustments and hence a smaller distortion.

To economize in presentation, I illustrate this idea in the general model with money demand, but the qualitative resultsholds also in the model with no money. Fig. 1 shows the standard deviation of the intertemporal wedge for various optimalinflation rates obtained under different values of the price rigidity parameter (j). The volatility of the intertemporal wedgeis falling with the optimal annual long-run inflation rate. Furthermore, Fig. 2 suggests that a higher inflation rate isassociated with a smaller size of the intertemporal wedge. Inflation, thus, reduces the size and the volatility of theintertemporal wedge. This happens through the direct and the indirect effects of the adjustment cost of nominal wages onthe wedge. The direct effect is through the explicit cost in the expression of the wedge and the indirect effect is by bringingthe effective bargaining power of workers closer to its efficient steady state value.

In the standard Ramsey theory, the intertemporal wedge should be completely smoothed over time. This is not the resulthere since the monetary authority does not have enough instruments to completely and simultaneously stabilize all distortions,including the intertemporal distortion, over the business cycle. Deviations from zero inflation are costly, and hence the

1.65 1.7 1.75 1.8 1.85 1.90.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Annual Inflation Rate

SD

of I

nter

tem

pora

l Wed

ge

Fig.1. Standard deviation (in percents) of the intertemporal wedge and the optimal annual inflation rate with various degrees of price rigidity.

1.65 1.7 1.75 1.8 1.85 1.90.026

0.027

0.028

0.029

0.03

0.031

0.032

0.033

0.034

Annual Inflation Rate

Siz

e of

Inte

rtem

pora

l Wed

ge

Fig.2. Size of the intertemporal wedge and the optimal annual inflation rate with various degree of price rigidity.


monetary authority trades-offs between stabilizing inflation and stabilizing the intertemporal wedge through stabilization ofnominal wages. Due to the convexity of the adjustment cost functions of nominal wages and prices, it is not optimal to fullystabilize one variable while allowing for the other to vary. Hence, the monetary authority chooses to spread the distortionsacross all margins. Spreading the distortions across margins is a well-known result in the literature (Dupor, 2002).

It is interesting to contrast these findings with the results under a policy that commits to zero-inflation at all dates andstates. Figs. 3 and 4 show, respectively, the volatility and the size of the intertemporal wedge for various degrees of DNWRunder optimal policy with rigid prices, optimal policy with flexible prices and under a zero-inflation rate policy. Both thesize and the volatility of the wedge are considerably smaller under optimal monetary policy than under zero- inflationpolicy, particularly for the relevant values of c (see Table 1).

The intertemporal wedge is almost entirely smoothed when prices are fully flexible, regardless of the size of c.8 In this case,the monetary authority has more room for policy actions to stabilize the intertemporal wedge. Put differently, the monetaryauthority faces less tradeoffs in conducting policy, which allows for complete smoothing of the intertemporal wedge.

8 For inflation to be determinate, prices in the current setup cannot be fully flexible. Hence, ‘‘fully flexible’’ prices in this experiment refers to the case

in which j is set to 10�5.

0 500 1000 1500 2000 2500 30000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ψ

SD

of I

nter

tem

pora

l Wed

ge

Optimal Policy, Rigid PricesZero-Inflation PolicyOptimal Policy, Flexible Prices

Fig.3. Standard deviation (in percents) of the intertemporal wedge as a function of the degree of DNWR.

0 500 1000 1500 2000 2500 30000

0.05

0.1

0.15

0.2

0.25

ψ

Siz

e of

Inte

rtem

pora

l Wed

ge

Optimal Policy, Rigid PricesZero-Inflation PolicyOptimal Policy, Flexible Prices

Fig.4. Size of the intertemporal wedge as a function of the degree of DNWR.


6. Impulse responses

This section describes, under optimal policy, the behavior of the economy following negative and positive productivityshocks of the same magnitude (the size of the shock is one standard deviation of the shock to TFP). Figs. 5 and 6 show thebehavior of the main variables of interest with no money demand motive and with a motive for money demand,respectively.

All variables display asymmetry in their responses to negative and positive shocks of the same magnitude. The responseto a negative shock is considerably stronger than the response to a positive shock. This is particularly the case of priceinflation; following a negative productivity shock, inflation responds strongly, allowing for the adjustment in the realwage. The positive response of price inflation to negative productivity shocks is stronger with no money demand. Whenmoney demand is introduced, the motive for holding money, which in isolation calls for negative inflation, limits thepositive response of inflation to shocks. In general, the response of inflation to negative shocks in the presence of DNWR isa short-run illustration of Tobin’s idea.

Wage inflation displays a relatively muted response to a negative productivity shock; with no money demand, wageinflation increases slightly relative to its steady state value, thus forcing all the reduction in nominal wages to occurthrough price inflation. With money demand, the response of price inflation is relatively muted, leading to a slight fall in

2 4 6 8 10 12 14 16 18 20 22 24-0.4

-0.2

0

0.2

Price Inflation

2 4 6 8 10 12 14 16 18 20 22 24

-2

0

2

Unemployment

2 4 6 8 10 12 14 16 18 20 22 24-0.2

0

0.2

Wage Inflation

Positive Negative

2 4 6 8 10 12 14 16 18 20 22 24-1

0

1

Real Wage

2 4 6 8 10 12 14 16 18 20 22 24-4

-2

0

2

4

Labor Market Tightness

2 4 6 8 10 12 14 16 18 20 22 24

-2

0

2

Vacancies

2 4 6 8 10 12 14 16 18 20 22 24-0.5

0

0.5

Hours

2 4 6 8 10 12 14 16 18 20 22 24

-1

0

1

Marginal Cost

2 4 6 8 10 12 14 16 18 20 22 24

-1

0

1

Consumption

2 4 6 8 10 12 14 16 18 20 22 24

-1

0

1

Output

Fig. 5. Response to positive and negative productivity shocks with asymmetric wage adjustment cost function (percentage deviations from SS levels)

without money demand.


wage inflation. The overall effect of the reduction in wage inflation and the increase in price inflation on the real wage issmaller than in the case with no money demand.

Unemployment displays asymmetry in the response to productivity shocks and it is more persistent followingnegative shocks. As standard in this class of models, the response of unemployment is hump-shaped, peaking after abouta year. The asymmetry in the response of unemployment is a reflection of the asymmetric behavior of the real wage.Therefore, a muted response of price inflation to a negative productivity shock generates a stronger response ofunemployment to the shock. Vacancies also respond asymmetrically to positive and negative TFP shocks because ofsimilar reasoning. Other variables display asymmetry in their responses to negative and positive shocks, includingoutput and consumption.9

9 In an earlier version of this paper, I show that with a quadratic adjustment cost of nominal wages, the economy displays symmetric response to

positive and negative productivity shocks of the same magnitude. I do not present the results of this experiment here to economize in space.

2 4 6 8 10 12 14 16 18 20 22 24

-0.2

0

0.2

Price Inflation

2 4 6 8 10 12 14 16 18 20 22 24

-2

0

2

Unemployment

2 4 6 8 10 12 14 16 18 20 22 24-0.1

0

0.1

Wage Inflation

Positive Negative

2 4 6 8 10 12 14 16 18 20 22 24-1

0

1

Real Wage

2 4 6 8 10 12 14 16 18 20 22 24-4

-2

0

2

4

Labor Market Tightness

2 4 6 8 10 12 14 16 18 20 22 24

-2

0

2

Vacancies

2 4 6 8 10 12 14 16 18 20 22 24-1

0

1

Hours

2 4 6 8 10 12 14 16 18 20 22 24

-1

0

1

Marginal Cost

2 4 6 8 10 12 14 16 18 20 22 24-2

0

2

Consumption

2 4 6 8 10 12 14 16 18 20 22 24-2

0

2

Output

Fig. 6. Response to positive and negative productivity shocks with asymmetric wage adjustment cost function (percentage deviations from SS levels)

with money demand.


7. Sensitivity analyses

7.1. Asymmetric price adjustment cost

I assume now the following asymmetric price adjustment cost for firm j:

FPj,t ¼

jk2

exp �kPj,t

Pj,t�1�1

� �� þk

Pj,t

Pj,t�1�1

� ��1

� �, ð30Þ

with j and k being the price stickiness parameter and the asymmetry parameter, respectively. The forward-looking pricePhillips curve reads:

1�jð1�exp �kðpt�1Þ½ �ÞptþbjEtuc,tþ1

uc,t

� �ð1�exp �kðptþ1�1Þ

� �Þptþ1

ytþ1

yt

� �¼ eð1�mctÞ ð31Þ

Table 3Parameter estimates and the optimal inflation rate.


s 1.9366n 1.8087n

(0.4807) (0.5036)

W 2.9159n 2.6805n

(0.6017) (0.6348)

b 0.3587n 0.3628nn

(0.1730) (0.1957)

m – 0.0012n

(0.0003)

z 0.4283n 0.4325n

(0.0679) (0.0843)

g 0.3761n 0.3729n

(0.0821) (0.1052)

j 83.5224n 97.3010n

(36.1648) (40.5993)

k 45.8037 61.2006

(39.0452) (43.3781)

f 81.1772n 45.3843n

(19.5042) (18.4668)

c 2452.1787n 2396.0531n

(641.1206) (712.3783)

rr 0.8268n 0.8562n

(0.1679) (0.1964)

rp 1.5621n 1.4873nn

(0.7638) (0.7726)

ru �0.1692nn�0.2243n

(0.0944) (0.1085)

ru 0.9530n 0.9557n

(0.0211) (0.0236)

su 0.0068n 0.0075n

(0.0012) (0.0015)

p 1.7241 0.7835

(1.4689, 2.1277) (0.4923, 1.1496)

Std. Dev. of y 1.6430 1.6924

Std. Dev.of y 13.9602 9.0251

Standard errors are in parentheses.n Indicates significance at 5% level.nn Indicates significance at 10% level.


When k approaches zero, the adjustment cost function restores the quadratic adjustment cost function of Rotemberg(1982) and Eq. (31) restores the familiar Phillips curve with the quadratic adjustment cost of prices (Eq. 16).10

The SMM results and the optimal inflation rate results assuming this adjustment cost are presented in Table 3.Noticeably, k is relatively small and it is statistically insignificant in both models. This observation supports the use of aquadratic adjustment cost for prices in the benchmark calibration of the model. It is also in line with some empiricalevidence that suggest symmetric (or approximately symmetric) price behavior (e.g. Chen et al., 2008, who show thatasymmetric behavior of prices ‘‘in the small’’, which are price changes up to 10 cents).

The estimates of other parameters are very similar to those in Table 1. Intuitively, since k is statistically insignificant,using a quadratic adjustment cost in the estimation of the parameters of Table 1 does not impose significant restrictions onthe estimation of other parameters. The optimal inflation rates, with and without money demand, are very similar to thecorresponding optimal inflation rates presented in Table 2.

7.2. Workers pay the adjustment cost of nominal wages

This subsection examines the other extreme case, in which workers pay all the adjustment cost of nominal wages.The equilibrium conditions in this case are presented in the appendix. When workers pay the adjustment costs and giventhat adjusting prices is costly, the incentives of firms to inflate are reduced (albeit not entirely as firms still wish to see realwage cuts following negative TFP shocks). Reductions in nominal wages are costly for workers because of two reasons.

10 This can be verified by applying l’Hopital’s rule whenever k-0.


First, given any price level, they imply lower real wages. Second, cutting nominal wages has a direct resource cost forworkers (hence, the household). Workers will thus be more reluctant to cutting nominal wages when bargaining overnominal wages takes place. Therefore, the firm may need higher inflation to get the desired fall in real wages. From optimalpolicy point of view, the planner aims at balancing those effects- inflating to allow for real wage adjustment (thus‘‘greasing the wheels’’ of the labor market) and not inflating in order to honor the incentives of firms for lower inflation.Ex ante, it is unclear which factor is more dominant.

If firms pay the adjustment cost of nominal wages, cutting nominal wages is costly for the firm, thus calling forinflating. However, the firm’s adjustment of prices is costly, which limits the use of inflation in reducing the real wage.The firm thus balances between those effects by both setting higher prices in the aftermath of shocks to productivity andby allowing for some nominal wage cuts. From the point of view of workers, cutting nominal wages or increasing prices bya given magnitude has a similar effect- a reduction in real wages. Workers are thus less reluctant, relative to the case inwhich they pay the adjustment cost of nominal wages, to accept nominal wage cuts. Following negative shocks, theplanner balances between allowing for lower nominal wages that hurt workers and firms (due to the asymmetricadjustment cost of nominal wages, with this effect outweighing the effect of lower nominal wages on the asset values offirms) and keeping nominal wages high and thus inflating (which also hurt workers and firms).

The above considerations make it difficult to judge, ex ante, which case is associated with higher inflation.The intertemporal wedge comes here into play. The most interesting observation is that the intertemporal wedge when workerspay the adjustment cost of nominal wages has the exact form as in the benchmark case in which firms bear the adjustment costs.More specifically, the adjustment cost function of nominal wages enters the wedge with the same sign in both cases. In fact, it ispossible to show that this form of the intertemporal wedge does not change if we consider the intermediate case in which firmsand workers share the adjustment costs of nominal wages. The only possible difference between the benchmark case and the casestudied here lies in the value of the effective bargaining power of workers ot. Therefore, if the aim of the Ramsey planner is tosmooth the intertemporal wedge, then we should not expect considerable differences between the two cases.



s 1.6927n 1.5130n

(0.4761) (0.4406)

W 2.4830n 2.5114n

(0.5473) (0.4967)

b 0.3221nn 0.3642n

(0.1731) (0.1684)

m – 0.0011n

(0.0002)

z 0.4196n 0.4323n

(0.0827) (0.0940)

g 0.3031n 0.3382n

(0.0784) (0.0984)

j 113.4717n 128.6022n

(53.4628) (61.7387)

f 87.7170n 46.2530n

(38.3214) (16.0362)

c 2561.1372n 2408.2451n

(613.6753) (507.3714)

rr 0.8231n 0.8761n

(0.1652) (0.2247)

rp 1.5680n 1.4893n

(0.6013) (0.6843)

ru �0.1937nn�0.1965nn

(0.1126) (0.1047)

ru 0.9573n 0.9613n

(0.0183) (0.0162)

su 0.0079n 0.0076n

(0.0019) (0.0015)

p 1.6501 0.8165

(1.3549, 2.0937) (0.5244, 1.2271)

Std. Dev. of y 1.6361 1.5682

Std. Dev.of y 18.7201 13.8251



The SMM results and the corresponding optimal inflation rate in the presence of DNWR when workers bear theadjustment costs of nominal wages are presented in Table 4. The optimal inflation rate in this case is only slightly lowerthan the optimal inflation rate in the benchmark case. Therefore, the assumption of the benchmark model does not haveconsiderable effects on the main results of the paper. This fact can be attributed to the various reasons discussed above(namely, similar expressions for the intertemporal wedge, the same resource constraint and the need for inflation underboth assumptions about who pay the adjustment cost of nominal wages).

7.3. Wage indexation

Following Gertler et al. (2008) and Christiano et al. (2005), this subsection introduces indexation of nominal wagesto past inflation. Indexation of nominal wages to past inflation reduces the effectiveness of inflation at the current period.The cost of adjusting the nominal wage of one worker by firm j is given by

FWj,t ¼

fc2

exp �cWj,t

Wj,t�1plt�1

�1

!" #þc

Wj,t

Wj,t�1plt�1

�1

!�1

!, ð32Þ

with l being the indexation rate. The main results of this experiment are summarized in Table 5.The estimated wage indexation rate is slightly below 50 percent in both models. This estimate is lower than the

estimate of about 60 percent in Smets and Wouters (2007), but it is line with Hofmann et al. (2010) who show that thewage indexation rate has been time varying over the last 5 decades, falling from about 60 percent to roughly 20 percent inthe mid 1990s. Ascari et al. (2011) also report a time varying wage indexation rate, suggesting that wage indexationdeclined to roughly zero during the ‘‘Great Moderation’’ years.



s 1.7236n 1.6103n

(0.5382) (0.4672)

W 2.6945n 2.7811n

(0.6794) (0.5392)

b 0.3281n 0.3460n

(0.1473) (0.1209)

m – 0.0010n

(0.0002)

z 0.4453n 0.4195n

(0.0964) (0.0816)

g 0.3130n 0.3476n

(0.1135) (0.1058)

j 176.4638n 136.0182n

(72.9006) (51.8290)

f 107.1472n 91.3592n

(48.5628) (32.5719)

c 2274.4618n 2103.8244n

(453.7706) (376.7720)

l 0.4457nn 0.4722n

(0.2452) (0.2231)

rr 0.8629n 0.8805n

(0.3347) (0.4162)

rp 1.4390n 1.4860n

(0.5541) (0.6513)

ru �0.1972n�0.1875n

(0.0814) (0.0724)

ru 0.9509n 0.9645n

(0.0204) (0.0141)

su 0.0064n 0.0073n

(0.0010) (0.0016)

p 1.0840 0.5274

(0.7692, 1.4893) (0.3168, 0.8405)

Std. Dev. of y 1.5631 1.5847

Std. Dev.of y 10.0997 8.1360



The optimal inflation rate in this version is considerably lower than in the model that abstracts from wage indexation,but it remains significantly positive and meaningful. Inflation remains a greasing tool mainly because the wage indexationrate is less than unity, leaving room for positive inflation to influence the behavior of real wages. Those results wellillustrate the role of inflation when nominal wages are downwardly rigid and labor markets are frictional. To avoidexcessive unemployment and in order to encourage job creation, the planner sets a quantitatively meaningful positiveinflation rate even with the existence of wage indexation.

7.4. Deviations from the Hosios condition

Faia (2009) shows that deviations from the Hosios condition have implications for optimal inflation volatility in a laborsearch and matching model similar to the one outlined in this paper, but with no wage stickiness. In this subsection, Istudy the case in which this condition may not hold and hence the deterministic steady state of the model is essentiallyinefficient.

The SMM results, reported in Table 6, suggest again that nominal wages are downwardly rigid. Generally speaking, theparameter estimates are similar to those reported in the second and forth columns of Table 1, suggesting that thismodification does not have a big impact on the main results of the paper. The optimal inflation rate in each case is onlyslightly higher than the corresponding optimal inflation rate when the Hosios condition holds. The relatively small changein the optimal long-run inflation has two potential explanations. First, the original model already captures the idea ofdeviating from the Hosios condition dynamically (i.e., ex post): as discussed in Section 2.3, nominal wage rigidity drives awedge between the contribution of workers to the match and the effective (i.e. ex post) bargaining power of workers.Ex-ante deviations from the Hosios condition do not influence the results significantly. Second, the results here cover the



s 1.6837n 1.7940n

(0.4390) (0.4432)

W 2.7193n 2.9325n

(0.6574) (0.6329)

b 0.3156nn 0.3342n

(0.1658) (0.1541)

m – 0.0012

(0.0002)

z 0.4057n 0.4277n

(0.0537) (0.0521)

g 0.3568n 0.3929nn

(0.1690) (0.2188)

Z 0.4429n 0.4678n

(0.0704) (0.0898)

j 85.8730n 115.2947n

(31.5449) (52.63841)

f 76.5079nn 43.2531n

(41.0640) (18.1363)

c 2441.5873n 2365.3587n

(776.5629) (582.7537)

rr 0.8614n 0.9463n

(0.1862) (0.1234)

rp 1.5406n 1.5146nn

(0.7241) (0.7841)

ru �0.1735n�0.1913nn

(0.0746) (0.1035)

ru 0.9575n 0.9512n

(0.0143) (0.0206)

su 0.0071n 0.0081n

(0.0011) (0.0011)

p 2.0386 0.9465

(1.7438, 2.5210) (0.6391, 1.3718)

Std. Dev. of y 1.6732 1.6374

Std. Dev.of y 14.1191 8.9541



case in which both Z and z are determined using the SMM method. The estimated differences between the two parametersare relatively small, which may also limit the implications of deviations from the Hosios condition for the optimal long-runinflation rate.

8. Conclusions

This paper studies the optimal long-run inflation rate within a labor search and matching framework in the presence ofdownward nominal wage rigidity. The optimal long-run inflation rate is positive- slightly below 1 percent annually withmoney demand and around 2 percent annually with no motive for money demand. With downwardly rigid nominalwages, positive inflation allows for real wage adjustments following adverse shocks, thus promoting job creation andpreventing an excessive increase in unemployment.

The results of this paper are related to Ramsey theory of smoothing wedges over time. In this study, the concern is overthe ‘‘intertemporal wedge’’, which is defined here as the deviation of the intertemporal marginal rate of substitution fromthe intertemporal marginal rate of transformation. Importantly, the asymmetric adjustment cost of nominal wages is partof this wedge. By setting a positive inflation rate, the Ramsey planner acts towards smoothing and reducing theintertemporal wedge and, as a result, taking the economy closer to the efficient allocation. Indeed, the results suggest thatthe size and the volatility of the wedge are both falling in the inflation rate as the degree of price rigidity is varied.The intertemporal wedge is virtually constant over time when prices are fully flexible. Those findings are in sharp contrastto the results under zero-inflation policy at all dates and states. Under a zero-inflation rate policy, the volatility and thesize of the intertemporal wedge are significantly higher.

The main results of the paper suggest that DNWR is a very important factor in accounting for the optimal rate ofinflation and that it is a major reason for the fact that central banks around the world target positive inflation rates.The combination of DNWR and an explicit unemployment margin only magnifies the role of inflation in greasing thewheels of the labor market following adverse shocks.

The current paper can be further extended. One natural extension is to evaluate the performance of different Taylor-typerules compared to the optimal policy. Another extension is to allow for endogenous participation in the labor force. Finally,future work may evaluate the role of DNWR in determining the optimal long-run inflation rate in a model that incorporatesother factors that call for positive inflation, such as the zero lower bound on the nominal interest rate and debt deflation.

Acknowledgments

I thank the editor Paul Klein and two anonymous referees for very invaluable comments and suggestions. I am also verygrateful to Sanjay Chugh for his invaluable guidance, John Shea, Boragan Aruoba and Pablo D’Erasmo for very usefulcomments and suggestions. Any errors are my own.

Appendix. Simulated method of moments

Let O be a sx1 vector of parameters of interest, whose values are unknown and xt be a rx1 vector of empiricalobservations obtained from the data. Let also xn(O) be the synthetic counterpart of xt, and it is computed from simulatingthe model using the parameters O. The length of the data series is assumed to be T, and the length of simulated series is N,with NZT. Then, the SMM estimator OSMM satisfies:

OSMM ¼ arg min DðOÞ0WDðOÞ,

with D(O) being the distances between the empirical values of the moments and their estimated values in the simulations.Formally:

DðOÞ ¼1

T

XT

t ¼ 1

xt�1

N

XN

n ¼ 1

xnðOÞ,

W is a positive semi definite weighting matrix given by the inverse of:

V ¼ limT-1

Var1ffiffiffiTp

XT

t ¼ 1

xt

!

A necessary, but not sufficient, condition for identification is that sZr. When the model is over- identified (i.e. s isstrictly greater than r), an over-identification test is run using Hansen’s (1982) chi-square test:

T 1þT

N

� �DðOSMMÞ

0WDðOSMMÞ �

-w2s�r ,

with D(OSMM)0WD(OSMM) being the minimized value of the objective function at the optimum. The results of this testindicate that, at the 95 significance level, the over-identification hypothesis is not rejected in any of the specificationsconsidered in the paper.


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