The Return of the Wage Phillips Curve 

Jordi GalÌ

CREI, Universitat Pompeu Fabra, and Barcelona GSE

February 2010 (Örst draft: May 2009)

Abstract

The standard New Keynesian model with staggered wage setting is

shown to imply a simple dynamic relation between wage ináation and

unemployment. Under some assumptions, that relation takes a form

similar to that found in empirical applicationsñstarting with the original Phillips (1958) curveñand may thus be viewed as providing some

theoretical foundations to the latter. The structural wage equation

derived here is shown to account reasonably well for the comovement

of wage ináation and the unemployment rate in the U.S. economy,

even under the strong assumption of a constant natural rate of unemployment.

Keywords: staggered nominal wage setting, New Keynesian model,

unemployment áuctuations.

JEL ClassiÖcation No.: E31, E32

Correspondence: CREI, Ramon Trias Fargas 25; 08005 Barcelona (Spain). E-mail:

jgali@crei.cat. I have beneÖted from comments during presentations at the CREI

Macro Lunch, the Reserve Bank of Australia, Reserve Bank of New Zealand, U. Rovira i

Virgili, NBER Summer Institute, Kiel EES Workshop, New York Fed, Columbia, NYU and

Oxford University. Tomaz Cajner and Lien Laureys provided excellent research assistance.

I am grateful to the European Research Council, the Ministerio de Ciencia e InnovaciÛn,

the Barcelona GSE Research Network and the Government of Catalonia for Önancial

support.
1 Introduction

The past decade has witnessed the emergence of a new popular framework

for monetary policy analysis, the so called New Keynesian (NK) model. The

new framework combines some of the ingredients of Real Business Cycle

theory (e.g. dynamic optimization, general equilibrium) with others that

have a distinctive Keynesian áavor (e.g. monopolistic competition, nominal

rigidities).

Many important properties of the NK model hinge on the speciÖcation

of its wage-setting block. While basic versions of that model, intended for

classroom exposition, assume fully áexible wages and perfect competition in

labor markets, the larger, more realistic versions (including those developed

in-house at di§erent central banks and policy institutions) typically assume

staggered nominal wage setting which, following the lead of Erceg, Henderson and Levin (2000), are modeled in a way symmetric to price setting.1

The degree of nominal wage rigidities and other features of wage setting play

an important role in determining the response of the economy to monetary

and other shocks. Furthermore, the coexistence of price and wage rigidities has important implications for the optimal design of monetary policy.2

Yet, and despite the central role of the wage-setting block in the NK model,

the amount of work aimed at assessing its empirical relevance has been sur1See, e.g., Smets and Wouters (2003,2008) and Christiano, Eichenbaum and Evans

(2005). For a descriptions of versions of those models developed at policy institutions,

see Christo§el, Coenen, and Warne (2008), Edge, Kiley and Laforte (2007), and Erceg,

Guerrieri and Gust (2006), among others.

2See Erceg, Henderson and Levin (2000), Woodford (2003, chapter 6) or GalÌ (2008,

chapter 6) for a detailed discussion of the policy implications of the coexistence of nominal

wage and price stickiness.

1
prisingly scant.3 This is in stark contrast with the recent but already large

empirical literature on price ináation dynamics and Örmsípricing patterns,

which has been motivated to a large extent by the desire to evaluate the

price-setting block of the NK model.4

The present paper seeks to Öll part of that gap, by providing evidence on

the NK modelís ability to account for the observed patterns of wage ináation

in the U.S. economy. In order to do so, I reformulate the standard version

of the NK wage equation in terms of a (suitably deÖned) unemployment

rate. The main advantage of that reformulation is the observability of the

associated driving force (the unemployment rate), which contrasts with the

inherent unobservability of the wage markup or the output gap, which are the

driving forces in standard formulations of the NK wage ináation equation.

The staggered wage setting model ‡ la Calvo (1983) embedded in standard versions of the New Keynesian framework is shown to imply a simple

dynamic relation between wage ináation and the unemployment rate, which

I refer to as the New Keynesian Wage Phillips Curve. Under certain assumptions, that relation takes the same form as the original equation of Phillips

(1958). Furthermore, in the presence of wage indexation to past ináation

ñan assumption often made in extensions of the basic modelñ the resulting

wage dynamics are consistent with a speciÖcation often used in applied work.

The analysis developed here can thus be seen as providing some theoretical

3A recent exception is Sbordone (2006).

4See, e.g. GalÌ and Gertler (1999), GalÌ, Gertler and LÛpez-Salido (2001), Sbordone

(2002) and Eichenbaum and Fisher (2007) for examples of papers using aggregate data.

Micro evidence on price-setting patterns and its implications for aggregate models can be

found in Bils and Klenow (2004), Nakamura and Steinsson (2008), and Mackowiak and

Smets (2008), among others.

2
foundations for those speciÖcations, as well as a structural interpretation to

its coe¢ cients.

In the second part of the paper I turn to the empirical evidence, and show

how the New Keynesian Wage Phillips Curve accounts reasonably well for

the behavior of wage ináation in the U.S. economy, even under the strong

assumption of a constant natural rate of unemployment. In particular, the

model can account for the strong negative correlation between wage ináation

and the unemployment rate observed since the mid-1980s. On the other

hand, the lack of a signiÖcant correlation between the same variables for the

postwar period as a whole can be explained as a consequence of the large

áuctuations in price ináation in and around the 1970s, in combination with

wage indexation to past CPI ináation.

The remainder of the paper is organized as follows. Section 2 describes

the basic model of staggered nominal wage setting. Section 3 introduces

the measure of unemployment latent in that model, and reformulates the

wage ináation equation in terms of that variable. Section 4 provides an

empirical assessment of the modelís implied relation between wage ináation

and unemployment using postwar U.S. data. Section 5 concludes.

2 Staggered Wage Setting and Wage Ináation Dynamics

This section introduces a variant of the staggered wage setting model originally developed in Erceg, Henderson and Levin (2000; henceforth, EHL).

That model (and extensions thereof) constitutes one of the key building

blocks of the monetary DSGE frameworks that have become part of the

3
toolkit for policy analysis in both academic and policy circles. The variant

presented here assumes that labor is indivisible, with all variations in hired

labor input taking place at the extensive margin (i.e. in the form of variations

in employment). The assumption of indivisible labor leads to a deÖnition of

unemployment consistent with its empirical counterpart.

The model assumes a (large) representative household with a continuum

of members represented by the unit square and indexed by a pair (i; j) 2

[0; 1] [0; 1]. The Örst dimension, indexed by i 2 [0; 1], represents the type

of labor service in which a given household member is specialized. The

second dimension, indexed by j 2 [0; 1], determines his disutility from work.

The latter is given by tj

'

if he is employed and zero otherwise, where

'  0 determines the elasticity of the marginal disutility of work, and t > 0

is an exogenous preference shifter. Furthermore, utility is logarithmic in

consumption and there is full risk sharing among household members, as in

Merz (1995).

The household period utility corresponds to the integral of its membersí

utilities, and is thus given by

U(Ct; fNt(i)g; t)  log Ct  t

Z 1

0

Z Nt(i)

0

j

'

djdi

= log Ct  t

Z 1

0

Nt(i)

1+'

1 + '

di

where Ct denotes household consumption, and Nt(i) 2 [0; 1] is the fraction of

members specialized in type i labor who are employed in period t. Below I

discuss the robustness of the main Öndings to a generalization of the previous

utility function that is consistent with (empirically more plausible) smaller

wealth e§ects on labor supply.

4
The relevant decision unit is the household. The latter seeks to maximize

E0

X1

t=0



t U(Ct

; fNt(i)g; t)

subject to a sequence of budget constraints

PtCt + QtBt  Bt1 +

Z 1

0

Wt(i)Nt(i) di + t (1)

where Ptis the price of the consumption bundle, Wt(i) is the nominal wage

for labor of type i, Bt represents purchases of a nominally riskless one-period

bond (at a price Qt), and tis a lump-sum component of income (which

may include, among other items, dividends from ownership of Örms). The

above sequence of period budget constraints is supplemented with a solvency

condition which prevents the household from engaging in Ponzi schemes.

As in EHL, and following the formalism of Calvo (1983), workers supplying a labor service of a given type (or a union representing them) get to reset

their (nominal) wage with probability 1w each period. That probability is

independent of the time elapsed since they last reset their wage, in addition

to being independent across labor types. Thus, a fraction of workers w keep

their wage unchanged in any given period, making that parameter a natural

index of nominal wage rigidities. Once the wage has been set, the quantity of

workers employed is determined unilaterally by Örms, with households willingly meeting that demand (to the extent that the wage remains above the

disutility of work for the marginal worker), by sending its specialized workers

with the lowest work disutility.

When reoptimizing their wage in period t, workers choose a wage W

t

in

order to maximize household utility (as opposed to their individual utility),

subject to a sequence of isoelastic demand schedules for their labor type,

5
and the usual sequence of household áow budget constraints.5 The Örst

order condition associated with that problem can be written as:

X1

k=0

(w)

kEt



Nt+kjt

Ct+k



W

t

Pt+k

 MwMRSt+kjt

 = 0

where Nt+kjt denotes the quantity demanded in period t + k of a labor type

whose wage is being reset in period t, MRSt+kjt  t+kCt+kN

'

t+kjt

is the

relevant marginal rate of substitution between consumption and employment

in period t + k, and Mw 

w

w1

is the desired (or áexible wage) markup,

with w denoting the (constant) wage elasticity of demand for the services of

each labor type.

Log-linearizing the above optimality condition around a perfect foresight

zero ináation steady state, and using lower case letters to denote the logs of

the corresponding variable, we obtain the approximate wage setting rule

w



t = 

w + (1  w)

X1

k=0

(w)

kEt



mrst+kjt + pt+k

	

(2)

where 

w  logMw. Note that in the absence of nominal wage rigidities

(w = 0) we have w



t = wt = 

w + mrst + pt

, implying a constant markup



w of the wage wt over the price-adjusted marginal rate of substitution,

mrst + pt. When nominal wage rigidities are present, new wages are set

instead as a constant markup 

w over a weighted average of current and

expected future price-adjusted marginal rates of substitution.

Letting mrst  ct+' nt+t denote the economyís average (log) marginal

5Details of the derivation of the optimal wage setting condition can be found in EHL

(2000).

6
rate of substitution, where t  log t

, we can write

mrst+kjt = mrst+k + '(nt+kjt  nt+k) (3)

= mrst+k  w'(w



t  wt+k)

Furthermore, log-linearizing the expression for aggregate wage index around

a zero ináation steady state we obtain

wt = wwt1 + (1  w)w



t

(4)

As in EHL (2000), we can combine equations (2) through (4) and derive

the baseline wage ináation equation



w

t = Etf

w

t+1g  w(

w

t  

w

) (5)

where 

w

t  wt  wt1 is wage ináation, 

w

t  wt  pt  mrst denotes the

(average) wage markup, and w 

(1w)(1w)

w(1+w') > 0. In words, wage ináation

depends positively on expected one period ahead wage ináation and negatively on the deviation of the average wage markup from its desired value.6

Equivalently, and solving (5) forward, we have



w

t = w

X1

k=0



kEtf(

w

t+k  

w

)g (6)

i.e. wage ináation is proportional to the discounted sum of expected deviations of current and future average wage markups from their desired levels.

Intuitively, if average wage markups are below (above) their desired level,

workers that have a chance to reset their wage will tend to adjust it upward

(downward), thus generating positive (negative) wage ináation.

6Note that the previous equation is the wage analog to the price ináation equation

resulting from a model with staggered price setting ‡ la Calvo. See GalÌ and Gertler

(1999) and Sbordone (2002) for a derivation and empirical assessment.

7
Estimated versions of the model above found in the literature generally

allow for automatic indexation to price ináation of the wages that are not

reoptimized in any given period. Here I assume the following indexation rule:

wt+kjt = wt+k1jt + 


p

t+k1 + (1  
)

p + g (7)

for k = 1; 2; 3; :::where wt+kjtis the period t + k (log) wage for workers who

last re-optimized their wage in period t (with wtjt  w



t

), 

p

t

is the price

ináation variable to which wages are indexed, 

p denotes steady state price

ináation, and g is the rate of growth of productivity (and real wages) in

the steady state. In that case the following wage ináation equation can be

derived:



w

t  


p

t1 =  + Etf

w

t+1  


p

t g  w(

w

t  

w

) (8)

where   (1  )((1  
)

p + g).

While in existing applications it is often assumed 

p

t  

p

t  pt  pt1

(e.g. Smets and Wouters (2003, 2007)), it is important to note that the model

allows for ináation measures other than the one-period lagged ináation as an

indexing variable. In particular, some of the estimates below use the moving

average 

p

t  (1=4)(

p

t + 

p

t1 + 

p

t2 + 

p

t3

) as a "smoother" alternative

indexing variable.

8
3 Wage Ináation and Unemployment: A New

Keynesian Wage Phillips Curve

Next I introduce unemployment explicitly in the model above.7 Consider

household member (i; j), specialized in type i labor and with disutility of

work tj

'

. Using household welfare as a criterion, and taking as given current

labor market conditions (as summarized by the prevailing wage for his labor

type), he will Önd it optimal to participate in the labor market in period t if

and only if

Wt(i)

Pt

 tCt j

'

i.e. whenever the real wage prevailing in his trade is above his disutility from

working, expressed in terms of consumption using the householdís marginal

valuation of the latter.

Thus, the marginal supplier of type i labor (employed or unemployed),

which I denote by Lt(i), is implicitly given by

Wt(i)

Pt

= tCtLt(i)

'

Taking logs and integrating over i we obtain

wt  pt = ct + ' lt + t(9)

where lt 

R 1

0

lt(i)di can be interpreted as the modelís implied aggregate

participation rate, and wt 

R 1

0 wt(i)di is the average wage, both expressed

in logs.

7The general approach builds on GalÌ (1996). Other recent applications to the New

Keynesian model can be found in Blanchard and GalÌ (2007) and, more closely related

(although developed independently), Casares (2009).

9
I deÖne the unemployment rate ut as

ut  lt  nt (10)

which, for rates of unemployment of the magnitude observed in the postwar

U.S. economy, is a close (and algebraically convenient) approximation to the

more conventional measure (Lt  Nt)=Lt.

Combining (9) and (10) with the expression for the average wage markup



w

t  (wt  pt)  (ct + 'nt + t

) used above yields the following simple linear

relation between the wage markup and the unemployment rate



w

t = ' ut (11)

Let us deÖne the natural rate of unemployment, u

n

t

, as the rate of unemployment that would prevail in the absence of nominal wage rigidities. It

follows from the assumption of a constant desired wage markup that u

n

t

is

constant and given by

u

n =



w

'

(12)

Finally, combining (5), (11), and (12) we obtain the following New Keynesian Wage Phillips curve (NKWPC, for short):



w

t = Etf

w

t+1g  w' (ut  u

n

) (13)

Note that the simple linear relation between the wage markup and unemployment derived in this section holds irrespective of the details of the wage

setting process. In particular, it also holds in the presence of wage indexation

as described in equation (7). In that case we can derive the implied wage

Phillips curve by combining equations (8) and (11) to obtain:



w

t =  + 


p

t1 + Etf

w

t+1  


p

t g  w' (ut  u

n

) (14)

10
which I refer to henceforth as the augmented NKWPC.

3.1 Relation to the original wage Phillips curve

In his seminal paper, Phillips (1958) uncovered the existence of a strong inverse empirical relation between wage ináation and the unemployment rate

in the U.K. over the period 1861-1957. His analysis was subsequently replicated using U.S. data by Samuelson and Solow (1960), who showed that a

similar empirical relation had been prevalent in the U.S., with the exception

of the New Deal period and the early years of the Örst World War. Much of

subsequent empirical work turned its focus instead to the relation between

price ináation and unemployment, usually in the context of a discussion of

NAIRU and its changes (e.g. Gordon (1997) and Staiger, Stock and Watson

(1997)).8

Note that, like the original Phillips (1958) curve, the NKWPC establishes

a relationship between wage ináation and the unemployment rate. But two

key di§erences with respect to Phillipsíoriginal curve (and some of its subsequent amendments) are worth emphasizing.

Firstly, (13) is a microfounded structural relation between wage ináation and unemployment, with coe¢ cients that are functions of parameters

that have a structural interpretation, and which are independent of the policy regime.9

In particular, the steepness of the slope of the implied wage

8Gordon (1997) claims that the shift in focus towards price ináation has been "deliberate." In his words, "...[t]he earlier Öxation on wages was a mistake. The relation of prices

to wages has changed over time...The Fedís goal is to control ináation, not wage growth,

and models with separate wage growth and price markup equations do not perform as well

as the [price ináation] equation...in which wages are only implicit...."

9Needless to say this is only true to the extent that one is willing to take the assumptions

of the Calvo formalism literally, including the exogeneity of parameter w.

11
ináation-unemployment curve (given expected wage ináation) is decreasing

in the degree of wage rigidity w (which is inversely related to w). In the

limit, as w approaches zero (the case of full wage áexibility), the curve becomes vertical. Also, the slope of the (

w; u) relation is decreasing in the size

of the Frisch labor supply elasticity (which corresponds to the inverse of ').

That structural nature of (13) stands in contrast with the purely empirical

basis of Phillips (1958) original curve, whose only theoretical underpinning

was the plausibility of the principle that "when demand for labour is high

and there are very few unemployed we should expect employers to bid wage

rates up quite rapidly...".10

Secondly, note that (13) implies that wage ináation is a forward looking

variable, which is inversely related to current unemployment but also to its

expected future path. This feature, which reáects the forward looking nature

of wage setting, is immediately seen by solving (13) forward to obtain



w

t = w'

X1

k=0



k Etf(ut+k  u

n

)g (15)

which contrasts with the static, contemporaneous nature of the original

Phillips curve, in which expectations play no role.

Next I brieáy discuss two extensions of the previous framework. The Örst

one allows for changes over time in desired markups, whereas the second

introduces a speciÖcation of preferences that allows for limited short run

wealth e§ects on labor supply.

10Phillips (1958) also emphasized the likely existence of nonlinearities due to workersí

reluctance "to o§er their services at less than the prevailing rates when the demand for

labour is low and unemployment is high, so that wage rates fall only very slowly." In the

analysis of the present paper, as in standard versions of the New Keynesian model, the

possible existence of such asymmetries is ignored.

12
3.2 An Extension with Time-Varying Desired Wage

Markups

Estimated, medium-scale versions of the New Keynesian model often allow

for a time-varying, exogenous desired wage markup, f

w

t g (see, e.g. Smets

and Wouters (2003, 2007)). In that case, the wage ináation equation (shown

in its version without indexation, for simplicity) is given by



w

t = Etf

w

t+1g  w(

w

t  

w

t

) (16)

while the corresponding NKWPC now takes the form



w

t = Etf

w

t+1g  w'(ut  u

n

t

)

where u

n

t 



w

t

'

denotes the (now time-varying) natural rate of unemployment. Variations in the latter variable, resulting from changes in desired

wage markups, may thus potentially shift the relation between wage ináation and the unemployment rate.11

Note that we can also write



w

t = Etf

w

t+1g  w'ut + vt (17)

vt  w

w

t The previous speciÖcation can be compared against one often

used in the literature which relies on (16) combined with the deÖnition of

11Equivalently, we can write



w

t =  Etf

w

t+1g  w' ut + vt

vt  w

w

t

. In contrast with the representation of the wage equation found in Smets

and Wouters (2003, 2007), the error term in the wage ináation formulation proposed here

captures exclusively "wage markup shocks," and not preference shocks (even though the

latter have been allowed for in the model above). This feature should in principle allow

one to overcome the basic identiÖcation problem raised by Chari, Kehoe and McGrattan

(2008) in their critique of current New Keynesian models.

13
the average wage markup, and which takes the form



w

t = Etf

w

t+1g  w(wt  pt  ct  'nt) + v

0

t

where now the error term is given by v

0

t  w

w

t + wt

, i.e. it is ináuenced

by both wage markup shocks and preference shocks. That property contrasts

with (17), whose error term captures exclusively wage markup shocks, but

not preference shocks (even though the latter have been allowed for in the

model). This feature should in principle allow one to overcome the identiÖcation problem raised by Chari, Kehoe and McGrattan (2008) in their critique

of current New Keynesian models. That potential advantage of the present

formulation is discussed in detail in GalÌ, Smets and Wouters (2010), who

re-estimate the Smets and Wouters (2007) model using an unemploymentbased wage equation, and re-assess some of the Öndings therein in light of

the new estimates.

3.3 Robustness to a SpeciÖcation of Preferences with

Limited Short-Run Wealth E§ects

The assumptions on preferences made above, while analytically convenient,

have implications on labor supply that are rather implausible from an empirical viewpoint. In particular, the strong wealth e§ects implied by the

logarithmic speciÖcation, while seemingly needed in order to remain consistent with balanced growth, are likely to be counterfactual. This becomes

clear by looking at the labor participation equation (9), which implies that

the wage-consumption ratio (wtptct) should be positively correlated to labor participation, at least conditional on shocks other than preference shocks

being the source of áuctuations. In postwar U.S. data, and possibly due to

14
the wage rigidities of the kind emphasized in the present paper, wt pt ct

is

clearly countercyclical, while participation is procyclical (albeit moderately

so). Thus, and unless one is willing to attribute a dominant weight to preference shocks as a source of cyclical áuctuations in wages, consumption and

participation, equation (9) provides an unsatisfactory account of áuctuations

in the labor force.12

Here I consider a simple extension of the preferences assumed above that

can in principle overcome that problem, by allowing for arbitrarily small

short-run wealth e§ects while remaining consistent with a balanced growth

path. Thus, individual utility from consumption is now assumed to be given

by tlog Ct(i; j), where tis a preference shifter taken as exogenously given

by each household, but determined by the ratio of aggregate consumption Ct

to a measure of its trend level. More precisely, I assume t = Ct=Zt where

Zt = Z

#

t1C

1#

t

, for all t. Aggregation of individual utilities thus yields the

following period utility for the household

U(Ct; fNt(i)g; t)  tlog Ct  t

Z 1

0

Nt(i)

1+'

1 + '

di

The derivation of the wage ináation equation (5) (or (8), in the presence of indexation) carries over to this case, with the relevant marginal

rate of substitution in the optimal wage setting problem now being given

by MRSt+kjt  (t+k=t+k)Ct+kN

'

t+kjt

. Note, however, that in a symmetric

equilibrium Ct = Ct for all t, which allows us to write the equilibrium (log)

marginal rate of substitution as mrst = zt + 'nt + t, where zt  log Zt

12The previous observation is closely related to other implausible predictions of macro

models that rely on similar preferences, including the negative impact on activity of higher

anticipated productivity, as emphasized by the recent literature on "news shocks."

15
evolves over time according to zt = #zt1 + (1  #)ct

. The equations for the

average wage markup and participation are now respectively given by



w

t  (wt  pt)  (zt + 'nt + t

)

wt  pt = zt + ' lt + t

which can be combined to yield the same simple proportional relation between the wage markup and the unemployment rate as above, i.e. 

w

t = 'ut

.

Note that the previous speciÖcation is still consistent with a balanced growth

path since, in the long run, zt will grow at the same rate as consumption.

In the short run, however, the impact of changes in consumption on the

marginal rate of substitution may be rendered arbitrarily small by increasing

parameter #, thus yielding a more plausible labor supply model. Yet, and

most importantly for the purposes of the present paper, the speciÖcation of

the NKWPC in (13) (or (14)) remains una§ected.13

3.4 A Reduced Form Representation for the NKWPC

Next I derive a simple reduced form representation of the NKWPC, thus

setting the stage for the empirical analysis below. Consider the case of an

exogenous, stationary AR(2) process for the unemployment rate. As discussed below, this process turns out to provide a good approximation to the

13An identical robustness result can be shown to obtain under a period utility function

of the sort assumed by Jaimovich and Rebelo (2009) with a similar motivation, namely

U(Ct; fNt(i)g; Zt) = 1

1  



Ct  t Zt

Z 1

0

Nt(i)

1+'

1 + '

di1

where Zt = Z

#

t1C

1#

t

. The non-separability of the previous speciÖcation for household

utility, however, prevents one from interpreting it as the aggregation of the utilities of

individual household members.

16
behavior of U.S. unemployment in the postwar period. Formally,

ubt = 1ubt1 + 2ubt2 + "t (18)

where ubt  ut  u

n and f"tg is white noise. Combining (18) with (14) yields

the wage ináation equation



w

t =  + 


p

t1 + 0ubt + 1ubt1 (19)

where

 0  

w'

1  (1 + 2)

 1  

w' 2

1  (1 + 2)

or, equivalently,



w

t = 

0 + 


p

t1  ubt  1ubt (20)

where   ( 0 + 1).

Estimates of the unemployment process (18) using postwar U.S. data

unambiguously point to the following properties: 1 > 1 , 1 < 2 < 0

and 0 < 1 + 1 < 1, the latter being a requirement for stationarity If

those inequalities hold then we have 0 < 0, 1 > 0, and  > 0. Thus,

under the previous assumptions and conditional on the relevant lagged price

ináation measure used for indexation purposes, wage ináation should respond

negatively to both the level and the Örst di§erence of the unemployment

rate, with the size of that response being a well deÖned function of structural

parameters, including those characterizing the process for the unemployment

rate.

Interestingly, a speciÖcation like (20) has often been proposed and used

in empirical applications, albeit without any strong theoretical justiÖcation

17
(e.g. Blanchard and Katz (1999)), as well as in mainstream undergraduate

textbooks (though the latter typically omit lagged unemployment). In fact,

in his seminal paper Phillips (1958) himself argued that it was plausible that

wage ináation would depend negatively on both the level and the change of

the unemployment rate, since both captured important dimensions of the

degree of tightness or excess demand in labor markets, and tried to uncover

their joint ináuence on the unemployment rate.

The following section revisits and updates estimates of equations (19) and

(20) and reinterprets them through the lens of the New Keynesian model

developed above.

4 Empirical Evidence

The present section provides an empirical assessment of the NKWPC developed above. More speciÖcally, I want to evaluate to what extent a version of

the NKWPC with a constant natural rate can account for the joint behavior

of unemployment and wage ináation in the U.S. economy.

First, I use simple statistics and graphical tools to seek evidence of a prima

facie negative relationship between wage ináation and unemployment of the

sort predicted by the theory. Secondly, I compare the observed behavior

of wage ináation with that predicted by an estimated version of the model

above, conditional on the unemployment rate.14

The empirical analysis relies on quarterly postwar U.S. data drawn from

the Haver database. I use the civilian unemployment rate as my measure of

14The observability of the unemployment rate, may be viewed as an advantage of the

present framework relative to Sbordone (2006), who focuses instead on a parameterized

version of (5).

18
unemployment. All measures of price ináation are constructed using the consumer price index. There are two main alternative sources for average wage

data that one may use to construct measures of wage ináation: the earnings

data for production and nonsupervisory workers from the Establishment Survey (starting in 1964Q1), and the compensation data from the "Productivity

and Costs" publication of the Bureau of Labor Statistics (available from

1948Q1 onwards). Aside from their time span, the main di§erence between

the two is that "compensation" is a more encompassing measure of the cost

to the employer, and as such it includes employer contributions to employeebeneÖt plans or irregular bonuses, whereas "earnings" is restricted to wage

income proper (including premium pay for overtime). Unfortunately, and

as noted by several authors, the discrepancy between their implied ináation

measures is far from negligible.15 This is made clear by Figure 1, which plots

wage ináation based on both measures. Note that in this and subsequent

Ögures ñthough not in the formal econometric work belowñwage ináation is

measured as the centered four-quarter di§erence of the log nominal wage expressed in percent terms (i.e., 100*(wt+2  wt2)), in order to smooth the

high volatility associated with quarter-to-quarter log-di§erences.

Three features stand out in Figure 1. First, the two measures display

very similar medium-run patterns, both picking up during the Great Ináation episode, dropping during the Volcker disináation, and áuctuating around

a low and stable mean during the Great Moderation years. This is reáected

in their high contemporaneous correlation (0:82). Secondly, their mean values in the common sample period are noticeably di§erent: 4:44 percent for

15See, e.g. Abraham, Spletzer and Stewart (1999).

19
the earnings-based measure, 5:36 percent for the compensation-based one.

Finally, and perhaps more strikingly, the compensation-based measure appears far more volatile than the earnings-based measure, especially over the

past two decades. A possible explanation for the di§erence in volatility lies in

the inclusion of the payo§s from the exercise of employee stock options in the

compensation measure. As argued by Mehran and Tracy (2001), the impact

of changes in that (increasingly important) component of compensation on

wage ináation is substantial, even if it only accounts for a small fraction of

overall compensation.

Interestingly, and those di§erences notwithstanding, both sources are

used in empirical macro applications, often with little or no discussion regarding the choice. While the compensation-based measure has an advantage in

its longer span, its sizable high frequency variations render it somewhat suspicious, and suggest the presence of substantial measurement error. Thus, in

the empirical analysis below I will use the earnings-based wage ináation as a

baseline measure, but will check and discuss the robustness of the main Öndings to the use of the alternative, compensation-based, wage ináation measure. Perhaps surprisingly, given the seemingly large di§erences observed in

Figure 1, none of the key qualitative Öndings in the empirical analysis below

hinge on the choice of wage ináation measure.

4.1 A Quick Glance at the Data

The New Keynesian Wage Phillips curve derived in section 3 implies an

inverse relation between wage ináation and the unemployment rate, albeit

not a simple or contemporaneous one, especially in the presence of indexation.

20
As a Örst pass in the empirical assessment of the model it seems natural to

check whether the raw data hint at any such an inverse relation.

Figure 2 displays two scatterplots of wage ináation and the unemployment rate for the U.S. economy, using both measures of wage ináation. The

scatterplots reveal the absence of a stable negative relation between the two

variables. Similar graphs, though typically focusing on price ináation, have

often been used to demonstrate "the empirical failure of the Phillips curve."

That visual impression is conÖrmed by the correlation between the two series,

which is as low as 0:02 and 0:01, respectively.

Figure 3 displays the evolution of unemployment and wage ináation over

time, using the earnings-based measure of the latter. While no stable relation

seems evident at a Örst glance, a more careful examination points to a strong

inverse relation starting sometime around the mid-1980s and prevailing up to

the end of the sample. That inverse relation is conÖrmed by the scatterplots

of Figure 4, which are now restricted to the post-1984 period, i.e. the so-called

Great Moderation era. For the restricted sample period the correlations are

0:76 and 0:27, respectively.

Figure 5 adds a temporal dimension to the Phillips curve scatterplots initially shown in Figure 2, now restricted to the earnings-based wage ináation

It suggests that the paths of U.S. wage ináation and unemployment have

completed a full circle, returning in recent years to the same downward locus

that characterized the 1960s. The evidence thus points to the presence of

a stable negative relation between wage ináation and unemployment during

periods of relatively low and stable price ináation. That relation is broken

during transitions from low to high ináation (early 70s), or from high to low

21
ináation (the early 80s), leading to an overall lack of correlation, as suggested

by Figure 2.

Thus, it seems clear from the previous quick glance at the data that

any model that implies a simple inverse relation between wage ináation and

unemployment will be at odds with the behavior of those two variables during

the long 1970-1985 episode. Yet, one cannot rule out that extensions of such

a model which allow for indexation to price ináation may be consistent with

the evidence. I explore that hypothesis in the next subsection, using the

augmented version of the New Keynesian Wage Phillips curve.

Why has the re-emergence of a stable negative relation between wage ináation and unemployment over the past two decades gone unnoticed among

academic economists? A possible explanation lies in the focus on price ináation and away from wage ináation in much of the empirical research of recent

years, combined with a lack of a signiÖcant empirical relation between price

ináation and unemployment. The correlation between those two series over

the post-1984 period is low and insigniÖcant (0:13), and its negative value

is due exclusively to the most recent observations: if I end the sample period

in 2007Q4 the correlation becomes even smaller and with the wrong sign

(0:08). Of course, the theory developed above has nothing to say, by itself,

about the relation between price ináation and the unemployment rate, since

that relation is likely to be ináuenced by factors other than wage setting,

including features of price setting and the evolution of labor productivity,

among others.16

Next I turn to a more formal empirical assessment of the New Keynesian

16See, e.g. Blanchard and GalÌ (2009) and Thomas (2009) for an analysis of the relation

between price ináation and unemployment in a model with labor market frictions.

22
Wage Phillips curve.

4.2 Estimates of the Reduced Form New Keynesian

Wage Phillips Curve

In the present subsection I report estimates of the reduced form wage equation (19). The focus on (19) is motivated by the good Öt of an AR(2) process

for the unemployment rate. Using data starting in 1948Q1 I estimate the

following model

ut = 0:22

(0:08)

 + 1:66

(0:08)

 ut1  0:70

(0:08)

 ut2 + "t

with any further lags of the unemployment rate not being signiÖcant. Thus,

and as discussed above, under that simple univariate process, the wage ináation equation should have a negative coe¢ cient on the current unemployment

rate and a positive one on its lag, in addition to a positive coe¢ cient on lagged

price ináation in the presence of indexation. Furthermore, the model (19)

implies the cross-equation restriction 1 = 02, which can be easily tested

conditional on a value for , which I set to 0:99.

Table 1 reports OLS estimates of several speciÖcations of the wage ináation equation, each of which can be seen as a restricted version of (19).

The wage ináation data is based on the earnings series, and the sample period starts in 1964Q1. The Örst three columns report estimates including

only the current value of the unemployment rate, augmented in the case of

columns (2) and (3) with lagged quarter-to-quarter price ináation (column

(2)) and year-on-year price ináation (column (3)), with the latter expressed

as a quarterly rate to facilitate comparison of coe¢ cients. Note that when

price ináation is not controlled for, the coe¢ cient on unemployment is very

23
close to zero and statistically insigniÖcant. When lagged ináation is added as

a regressor its coe¢ cient is highly signiÖcant, while the coe¢ cient on unemployment increases in absolute value and becomes signiÖcant (though only at

the 10 percent level when quarter-to-quarter ináation is used as a regressor).

Columns (5) and (6) include the lagged unemployment rate, and are thus

consistent with the speciÖcation implied by the model. In both cases the coe¢ cients on the unemployment rates have the sign predicted by the theory,

though they are only signiÖcant when year-on-year price ináation is used as

the indexing variable. Also, as shown in the row labeled pvalue, the crossequation restriction speciÖed above cannot be rejected when both equations

are estimated jointly. The Önal two rows report the implied estimates of

the Calvo wage rigidity parameter w, conditional on calibrated values for '

and w, since the three parameters are not separately identiÖed. Note that

' is the inverse Frisch labor supply elasticity, a controversial parameter. I

consider two alternative calibrations, ' = 1 and ' = 5, which span the range

of values often assumed in the literature. Given ', I use (12) to set w to a

value consistent with a natural rate of unemployment of 5 percent, which is

roughly the average unemployment rate over the sample period considered.17

The point estimates range from 0:62 to 0:89, suggesting substantial wage

rigidities, with those associated with the ' = 5 calibration implying average

durations that may be viewed as implausibly long (though the non-negligible

size of the standard errors allow for more plausible underlying degrees of

wage rigidity).

A more detailed analysis of the Öt of the estimated wage equations re17Note that w = (1  expf'ung)

1

. This implies setting w = 20:5 when ' = 1, and

w = 4:52 when ' = 5.

24
ported in columns (5) and (6), suggests a poor Öt during the recent recession.

The reason is simple: the rapid increase in the unemployment rate and the

very low levels of price ináation (which became deáation for some quarters),

lead the Ötted wage equation to predict substantial nominal wage deáation.

While actual wage ináation was brought down by the recession, it has always

remained positive. The presence of downward nominal wage rigidities, which

are ignored in the standard wage setting model developed above, could in

principle account for that poor Öt. Motivated by that observation, columns

(7) and (8) in Table 1 report estimates of the wage equation using data up to

2007Q4, thus avoiding any distortion resulting from the use of recent data.

Note that for both speciÖcations, the coe¢ cients on current and lagged unemployment increase substantially and now become highly signiÖcant even

when quarter-to-quarter price ináation is used as a regressor. Figure 6 displays actual and Ötted wage ináation, using the estimates shown in column

(8). While the estimated model misses much of the high frequency variations,

it appears to capture well most movements at medium-term frequencies, with

the exception of the spikes in 1971-72 and 1976-77. The correlation between

the two series is 0:83.

Table 2 reports estimates of wage ináation equations using the measure

based on compensation, and with the sample period starting in 1948Q1. Most

of the Öndings in Table 1 appear to be robust to the use of the alternative

wage ináation measure, which is indeed surprising given the large discrepancies between the two ináation measures. Note however that the exclusion of

post-2007 data does not have much of an impact now, possibly because of

its reduced weight in the longer sample period. In particular, the coe¢ cients

25
on unemployment and its lagged value in column (5) are now signiÖcant, in

addition of having the right pattern of signs and relative magnitude.

4.3 A Measure of Fundamental Wage Ináation in the

Spirit of Campbell-Shiller

The previous subsection reported estimates of a reduced form wage ináation equation implied by the NKWPC, under the assumption that the unemployment rate follows an exogenous AR(2) process. While the previous

assumption provides a fairly good approximation to the dynamics of unemployment in the postwar period and leads to a reduced form speciÖcation

which makes contact with that used in Phillips (1958) and in subsequent

applied work, one may legitimately wonder whether the favorable empirical assessment of the NKWPC hinges on that assumption. Relaxing that

assumption has an additional justiÖcation: simple Granger-causality tests

reject the null of no-Granger causality from wage and price ináation to unemployment. Thus, in particular, the four lags of (earnings-based) wage

ináation and price ináation are signiÖcant at the one percent level in a regression of the unemployment rate on its own four lags and the lags of the

two ináation measures over the 1964Q1-2009Q3 sample period. An analogous

test using the compensation-based measure of wage ináation and extended

over the sample period 1948Q1-2009Q3 only rejects the null of no-Granger

causality at the 7 percent signiÖcance level.

Motivated by the previous observation, and in the spirit of Campbell

and Shillerís (1987) proposed assessment of present value relations, I start

by deÖning the following measure of "fundamental" or "model-based" wage

26
ináation:

e

w

t

()  


p

t1  w'

X1

k=0



k Efut+kj ztg

where vector  [
; w; ; w; '] collects the exogenous parameters of the

model and where zt = [ut; w

t 


p

t1

; :::; utq; w

tq


p

t1q

] for some Önite q.

Under the null hypothesis that model (19) is correct, it is easy to check that

e

w

t

() = 

w

t

, for all t. In other words, given the structure of the conditioning

variable zt the use of a limited information set is not restrictive under the

null.

Next I estimate e

w

t

() and plot it against actual wage ináation, to evaluate the extent to which the simple model developed here can explain observed áuctuations in that variable. I assume that the joint dynamics of

unemployment and wage ináation are well captured by the Örst order vector

autoregressive model

zt = A zt1 + "t

where Ef"tj zt1g = 0 for all t. Thus, and letting ei denote the i

th unit

vector in R

2q

, we have Efut+kj ztg = e

0

1Akzt

, implying

e

w

t

() = 
 

p

t1  w' e

0

1

(I  A)

1

zt (21)

I exploit the previous result to construct a time series for fundamental

ináation e

w

t

() using a minimum distance estimator. Since not all structural

parameters in are separately identiÖed, I calibrate three of them (; w; ')

and estimate the remaining two (
; w), which deÖne the degree of rigidities

and indexation. As above, I set  = 0:99 and report results for both ' = 1

27
and ' = 5, with w set to imply a natural unemployment rate of 5 percent in

each case. I estimate the two remaining parameters, w and 
, by minimizing

PT

t=0(

w

t  e

w

t

())2subject to (21), over all possible values (w; 
) 2 [0; 1] 

[0; 1], and given the calibrated values for (; w; ') and the OLS estimate

for matrix A (with q = 4). As in the empirical analysis above I use lagged

quarterly and annual ináation as an indexing variable, and both earningsbased and compensation-based measures of wage ináation.

Table 3 reports the main results for the exercise.18 The estimates of 
,

the degree of indexation, are always highly signiÖcant and lie between 0:52

and 0:83, depending on the speciÖcation, values which are slightly higher

than those obtained in the previous subsection. The point estimates for the

Calvo parameter w are also highly signiÖcant in all cases. Under the ' = 1

assumption they lie between 0:52 and 0:65, implying an average duration

between two and three quarters. When ' = 5 is assumed, the estimates are

substantially higher (between 0:75 and 0:82), but still within the range of

plausibility, given the evidence uncovered by micro studies.19 Interestingly,

my estimates for 
 and w are very close to those obtained by Smets and

Wouters (2007) using a very di§erent approach (and one that does not use

information on the unemployment rate, among other di§erences): 0:58 and

0:7, respectively.

The "multivariate" model analyzed here implies some restrictions that

can be subject to formal testing. In particular, note that if the model holds

18Standard errors are obtained by drawing from the empirical distribution of A, and

re-estimating w and 
 for each draw.

19See, e.g., Taylor (1999).

28
exactly, we must have

e

0

2 + w' e

0

1 = e

0

2A

Unfortunately the previous set of restrictions is rejected at very low signiÖcance levels for our sample and baseline calibration. This may not be

surprising, given the simplicity of the model. But, following Campbell and

Shiller (1987), I seek a more informal evaluation of the model by comparing

actual and fundamental wage ináation. The last row of Table 3 displays

the correlation between the four-quarter centered moving averages of both

variables: the correlations are positive and high (above 0:75) in all cases,

suggesting a good Öt of the model. This is also illustrated in Figure 7, which

displays actual (earnings-based) wage ináation and fundamental wage ináation, where the latter is based on the estimates using annual price ináation

as an indexing variable and ' = 1. While the Öt is far from perfect, it is

clear that the model-based series captures pretty well the bulk of the low and

medium frequency áuctuations in actual wage ináation (with the exception of

some episodes, including the 2008-09 recession). The fact that such a good Öt

is obtained using a model for wage ináation that assumes a constant natural

rate of unemployment makes that Önding perhaps even more surprising.

Given the large áuctuations in price ináation over the sample period

considered and the well known positive correlation between price and wage

ináation, one may wonder to what extent the high correlation between actual and fundamental ináation is largely a consequence of indexation to past

price ináation. In order to address that question I construct a measure of the

"cyclical" component of fundamental ináation, by subtracting from the latter

the ináation indexation component, 


p

t1

. The cyclical component is thus

29
driven exclusively by current and anticipated future unemployment rates.

Figure 8 displays the cyclical component of fundamental ináation thus constructed, together with actual wage ináation. It is clear from that evidence

that while the cyclical component did not have a dominant role in accounting

for the áuctuations in wage ináation in the Great Ináation era, one can still

detect several episodes in which it shapes the shorter-term áuctuations in observed wage ináation, including the 1968-69 hump and the 1974-75 trough.

It is not until the advent of the Great Moderation period and the associated

stability in price ináation that the cyclical component of fundamental wage

ináation emerges as a central factor behind áuctuations in wage ináation, as

Figure 8 makes clear.

5 Concluding comments

In his seminal 1958 paper, A.W. Phillips uncovered a tight inverse relation

between unemployment and wage ináation in the U.K.. That relation was

largely abandoned on both theoretical and empirical grounds. From a theoretical viewpoint, it was not clear why the rate of change of the nominal

wage (as opposed to the level of the real wage) should be related to unemployment. From an empirical viewpoint, economistsíattention shifted to the

relation between price ináation and unemployment, but hopes of establishing

a stable relationship between those variables faded with the stagáation of the

1970s.

The present paper has made two main contributions. First, it provides

some theoretical foundations to a Phillips-like relation between wage ináation

and unemployment. It does so not by developing a new model but, instead, by

30
showing that such a relation underlies a standard New Keynesian framework

with staggered wage setting, even though versions of the latter found in

the literature do not explicitly incorporate or even discuss unemployment.

Secondly, the implied wage equation is shown to account reasonably well for

the comovement of wage ináation and the unemployment rate in the U.S.

economy, even under the strong assumption of a constant natural rate of

unemployment. In particular, that equation can explain the strong negative

comovement between wage ináation and unemployment observed during the

past two decades of price stability.

It is far from the objective of the present paper to claim that the staggered wage setting model of Erceg, Henderson and Levin (2000) provides

an accurate description of U.S. labor markets. It is clear that some of its

underlying assumptions,ñmost noticeably, the unilateral setting of the wage

by a monopoly unionñare at odds with arrangements prevailing in most sectors. Yet, as a matter of fact, the EHL structure underlies most of the

medium-scale DSGE models that have been developed in recent years, by

both academics and institutions. Identifying and testing further predictions

coming out of those models would seem a worthy undertaking and a source of

guidance in any e§ort to improve the frameworks available for policy analysis.

This is, if nothing else, the spirit of the present paper.

31
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35
Table 1. Estimated Wage Ináation Equations: Earnings-based

(1) (2) (3) (5) (6) (7) (8)

ut 0:001

(0:019)

0:030

(0:017)

 0:079

(0:015)

 0:177

(0:114)

0:377

(0:083)

 0:334

(0:095)

 0:552

(0:076)



ut1 0:153

(0:112)

0:304

(0:078)

 0:294

(0:095)

 0:453

(0:073)



t1 0:415

(0:043)

 0:427

(0:052)

 0:503

(0:036)





(4)

t1

0:565

(0:038)

 0:611

(0:041)

 0:687

(0:038)



p  value 0:71 0:52 0:67 0:06

w (' = 1) 0:788

(0:083)

 0:629

(0:059)

 0:755

(0:061)

 0:607

(0:063)



w (' = 5) 0:896

(0:044)

 0:805

(0:036)

 0:879

(0:033)

 0:792

(0:039)


Table 2. Estimated Wage Ináation Equations: Compensation-based

(1) (2) (3) (5) (6) (7) (8)

ut 0:015

(0:028)

0:046

(0:024)

 0:087

(0:028)

 0:227

(0:099)

 0:387

(0:125)

 0:241

(0:091)

 0:397

(0:101)



ut1 0:189

(0:096)

 0:310

(0:116)

 0:189

(0:090)

 0:301

(0:098)



t1 0:500

(0:046)

 0:505

(0:046)

 0:522

(0:046)





(4)

t1

0:578

(0:058)

 0:610

(0:060)

 0:642

(0:052)



p  value 0:57 0:79 0:54 0:66

w (' = 1) 0:709

(0:086)

 0:637

(0:059)

 0:675

(0:069)

 0:569

(0:057)



w (' = 5) 0:853

(0:049)

 0:780

(0:043)

 0:833

(0:040)

 0:767

(0:037)


Table 3. Estimated Parameters using Minimum Distance Estimator

Earnings Compensation

t1 

(4)

t1 t1 

(4)

t1

' = 1 ' = 5 ' = 1 ' = 5 ' = 1 ' = 5 ' = 1 ' = 5


 0:52

(0:013)

0:52

(0:017)

0:83

(0:130)

0:81

(0:110)

0:58

(0:020)

0:59

(0:030)

0:80

(0:170)

0:79

(0:107)

w 0:65

(0:013)

0:82

(0:008)

0:52

(0:080)

0:74

(0:020)

0:64

(0:035)

0:81

(0:020)

0:54

(0:060)

0:75

(0:002)

(e

w

t

; w

t

) 0:82 0:82 0:91 0:91 0:77 0:78 0:77 0:76
Figure 1. Two Measures of Wage Inflation
Figure 2. Wage Inflation Inflation and Unemployment Unemployment
Figure 3. Wage Inflation Inflation and Unemployment Unemployment over 
Figure 4. Wage Inflation Inflation and Unemployment Unemployment during the Grea
Figure 5. The Wage Phillips Curve over Time

Early 70s Late 70s

Late 60s

Early 80s

GreatModeration
Figure 6. Actual vs. Fitted Wage Inflation 1964Q1‐20
Figure 7. Actual vs. Fundamental Wage Inflation
Figure 8. Wage Inflation Inflation and its Cyclical Cyclical Compo

Compo