Curva Philips

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JOHN M. ROBERTS

New KeynesianEconomics and the Phillips Curve

STICKYPRICESre an importantpart of monetarymodels of

business cycles. In recent years, a consensus has formed around he microfounda-

tions of sticky price models, and this consensus is an important artof New Keynes-

ian economics (Ball, Mankiw, and Romer 1988). In this paper, I show that several

of the New Keynesian models, including the models of staggered contractsdevel-oped by Taylor (1979, 1980) and Calvo (1983) and the quadraticprice adjustment

cost model of Rotemberg(1982), have a common formulation hat is similar to the

expectations-augmented hillips curve of Friedmanand Phelps.

I also present new estimates of this common model. Because prices are sticky in

the New Keynesian models, price setting must take into account futureprices, and

an important ssue in estimation s how tQdeal with expectationsaboutfutureprices.

Previous estimates of these models, such as those by Taylor(1980, 1989) and Ro-

temberg (1982), have been based on full-information echniques, in which expecta-

tions are solved under the assumptionof rationalexpectations. As is well known,full-information stimationhas the advantageof econometricefficiency if the model

is correctly specified, but the disadvantage that if any part of the model is

misspecified even a partof secondary nterest the estimateswill be inconsistent.

I explored two limited-informationapproaches. One was the technique, intro-

duced by McCallum(1976), of using the actualfuturevalue of a variableas a proxy

and then restricting the informationused in estimation such as any instrumental

variables to what was available at the time the expectationsof the variable were

The author is grateful to Dave Lebow, Maria Otoo, Dan Sichel, Dave Wilcox, and an anonymousreferee for helpful comments. The views expressed in this paperare not necessarilythose of the FederalReserve Board or its staff.

JOHNM. ROBERTSs an economist with the Federal Reserve Board.

Journal of Money, Credit, and Banking, Vol. 27, No. 4 (November 1995, Part 1)Copyright 1995 by The Ohio State University Press





JOHNM. ROBERTS : 977

Et {(Pt-Pt*) + C[(Pt-Pt- 1)-0(Pt+ t-Pt)]} = ° (2)

Assume that the discount rate, 0, is equal to one; for high frequencydata, this is

approximately orrect.Then we have

APt = Et5Pt+l-( l /C)(Pt-Pt*)

One way to specify p* is

Pt* = Pt + Yt + et,

where y is the percentdeviationof aggregateoutputfrom its trend, p is the pricechargedby otherfirms,e is an i.i.d. randomerror erm, ande is a positiveparame-ter.This equationassumesthat the firmhas anupward-sloping upplycurve, so thatit wants to raise its price if demand s high (forexample,when aggregate ncome ishigh).

Using this specification orp*, and assumingthatall firms areidentical,so thatp= p, equation(2) becomes

APt = EtAPt+X+ (I c) Yt-etl c . (3)

As discussedin section ld, equation 3) is a type of expectations-augmentedhillipscurve. Future nflationappearsbecauseprices aresticky,which meansthat the be-haviorof prices in the next period is importantn determiningcurrentprices. Notethat the coefficienton y is positive, because C and c arepositive.

Ib. Calvo's StaggeredContractsModet

In Calvo's (1983) model, each firmkeeps itspricefixed until it receivesa random

signal that it can change its price. Price changes are therefore"staggered,"and

when setting prices, the firmtakes into account the prices otherfirms will chargeuntil it has a chance to changeprices again. And because the prices of other firmswere set in the past, the firm takes into account past prices in setting currentones.

Theprobability hat a firmwill receive a signal in anyparticularime period s .2

Let's assume that there are enough firms so that fl also represents he fractionoffirms that can change prices in any period. Firms are assumedto have upward-sloping supplycurves. The firmthereforechooses its priceZt S° that

Zt = g Et E (1 -)J(Pt+y + Yt+y + st+j)

j=0,x

2. Calvo derived his model in continuoustime. To drawthe comparisonwith the other models moreclearly, I use Rotemberg's 1987) discrete-timecharacterization f Calvo's model.



978 : MONEY,CREDI1s,AND BANKING

where, as in the previous example, ,l3 s a positive constant.The aggregateprice

level will be

pt=5T E (l-fl)jzt_j.j=O,x

Assuming st is observablein period t, these two equationsimply the following

equationsof motion:

Z = ST(p + Yt + st) + (1-w) Etzt+

and

p = flZt-(1-)Pt-

Fromthese, we have

Et5Zt+l = wEtzt+l-(Pt + Yt + st)

and

APt=Zt+Pt-l (5)

Together, hese equationsimply

Et/\zt+1 = Et/\Pt+l-w Yt 6t (6)

Differencingequation(4) and leadingone periodgives

Et AZt+ 1 = ( 1 1fl) [Et APt+ 1 + (s-1 ) Apt]

Substituting nto (S) and rearranging ives

/\Pt = Et/\Pt+l + (fl2,8)/(1-AT) Yt + AT2/(1-T) Et * (7)

Again, we havean expectations-augmented hillipscurve. Again, the coefficienton

output is positive, as desired,because ,8 > O and fl is betweenzero andone.

Ic. Taylor's taggered-Contractsodel

In Taylor's(1979) model, wages are assumedto be set for two periods, so the

averagewage to the firmis

wt = (xt + xt_1 /2 ,



1OHNM.ROBERTS 979

wherew is the average,observablewage andx is the contractwage, chosen in peri-

od t, for the periodst andt + 1. The variablesw and x referto logarithms.

Workersare assumedto be concernedaboutreal wages and unemploymentoverthe life of theircontract.The laborsupplycurveis therefore

Xl-(Pt + EtPl+l)12 = Co-(RUl + ElRUt+l)12 + {l,

wherep is the log of price, RU is the unemploymentrate, e is a white-noiseerror

termcapturingvarious unobserveddeterminantsof wages, and c0 and ,3 are con-

stants. fi > 0, so higher unemployment s associatedwith lowerwages.

The firm sets prices as a constantmarkupover wages. Normalizing he markup o

zero,

pt = wt .

In this model, wages are the only sourceof nominalrigidity.This featuremakesthe

model an interestingcontrast o the firsttwo, which are basedon sticky prices.

The precedingthreeequationscan be combinedto form:

APt-EtAPt+l + Co-(RUt-l + RUt + Et_lRUt + EtRUt+l)

+ 2(et + et-1) + tIt,

wheren is an expectationalerror Et_ Pt-pt). Once again, we haveessentiallyan

expectations-augmented hillips curve. Again, the coefficienton the "realactivity"

variablehas the expectedsign:Whena moving averageof unemployment ises, in-

flation falls. Here, the residualerror erm has a moving-average omponent.

I d. TheNew KeynesianPhillips Curve

A common formulation hat capturesthe key elementof equations (4), (7), and

(8) is

APl-ElApl+, = Co+ WYl+ el s (9)

wheree is a residualerror erm,whichmay be seriallycorrelated,andw is a positive

constant. The relationshipof equation(9) to Rotemberg'sand Calvo's models is

clear. The following argumentmakesthe connectionbetweenequation(9) andTay-

lor's model: Because the unemploymentrate is stronglyseriallycorrelated,currentunemployment s an adequateproxy for the current, agged, and futureunemploy-

ment. Okun'slaw can be appliedto move fromunemploymentandoutput.Finally,

Taylor's model implies that the residual in equation (9) has a moving-average

component.

Equation(9) is similar to the expectations-augmentedhillipscurve of Friedman



980 : MONEY,CREDIT,AND BANKING

andPhelps. The contributionof the New Keynesianmodels in interpreting he Phil-lips curve is to emphasize the role of explicit nominalrigidities in interpretinghemodel, since in the models of sections la, lb, and lc, outcomes in labor andprod-

uct marketswould be classical in the absenceof rigidities.Equation(9) is also similarto the Lucassupplycurve(Lucas 1973). The two dif-

fer in thatequation(9) includesexpectationsof nextperiod'sinflation,whereas Lu-cas's supplycurve incorporates xpectationsof current nflation.The reasonfutureinflationmatters n the New Keynesianmodel is that pricesaresticky.

Becausechangesin real oil pricesarean important ndreadily dentifiable ourceof e variation, I made a slight modification o equation (9):

APt-Et5Pt+l = Co + z Yt + C1 Arpoilt + C2 Arpoilt_l + st,

whererpoil is the log of the realprice of crudeoil and cl andc2 areconstants.I alsoestimatedequationsusing the unemployment ate in place of thegap betweenactualand trendGNP. The unemploymentrate is an alternative ndicatorof overall eco-nomic activity,and if we view labormarketsas centralto aggregatenominalrigidi-ty, as they are in Taylor'smodel, thenunemploymentmay be a superior ndicator.Thus, I also exploredequationsof the form

APt-Et5Pt+l = Co-7'RUt + C1 Arpoilt + C2 Arpoilt_l + st .

2. PROXIESFOREXPECTATIONSAND OTHERDATA ISSUES

I used two types of proxies for expectedinflation.One approachwas to use sur-veys of price expectations. Two surveys of one-year-aheadnflationexpectationshave been conductedfor manyyears:The MichiganSurveyResearchCenter,in itsperiodicsurveys of consumerattitudes,asks individualswhatthey expect inflationto be over the coming year.3 And a survey of businesseconomists, called the Liv-ingstonsurveyafter its originator, s conductedby theFederalReserveBank of Philadelphia. Surveys have their limitations, however, the main one being thatrespondentshave little incentive to providethoughtfulanswers. Tothe extentthis isa problem,the surveysmay be poorproxies foractualexpectationsandthere will bea degree of noise introduced n the estimation.

In the second approach, ntroducedby McCallum(1976), the actualfuturevalueof inflation s used as a proxy for the expectation:

/\Pt-/\Pt+l = cO+ wyt+ st + (Et/\Pt+l-/\Pt+l)

This approachhas been popular in macroeconometricworkbecause explicit mea-suresof expectationsare notneeded. However,it introduces conometriccomplica-

3. Before1966,theMichiganurveyaskedonlyabout hefuturedirection f inflation;umericalvalues or theexpectations ere nferred yNobleandFields 1982).



JOHNM. ROBERTS : 981

tions, since thereis an additionalsourceof error,Vt = (EtApt+l-APt+l), that the

equationsthat use the surveysas expectationsproxies do not have, if the surveysare

good proxies for actualexpectations.As a consequence, theestimatesbased on Mc-

Callum'stechniquemay be less precisethan those that use the surveysas measures

of expectations.

Turning o otherdata issues, I used the twelve-monthpercentchangein the con-

sumer price index measuredfrom December to December as my measure of APt.

This measureof inflationmatches the inquiries n the Michigan andLivingston sur-

veys. I use the producerprice indexfor crudepetroleumdeflatedby the GNPdefla-

tor as my measureof real crude oil prices (rpoil).

I measuredy as the percentdeviationof real GNP froma deterministic rend. The

trendis estimatedby regressing the log of GNP on time and time squared.While

much research has focused on the potential importanceof a unit root in GNP,

Rudebusch 1993) and ChristianoandEichenbaum 1990) have shownthat, in prac-

tice, it is difficult to distinguishbetweena deterministic rend anda stochastic one.

And Perron(1989) has shown that a linear trendwith one break n the slope of the

trendwill renderGNP stationary. useda quadratic rendbecause it is a simple way

to introducean additionaldegree of freedom n the trendwithouthaving to choose a

break point.

To allow for the possibilitythat the e shocks could be correlatedwith output, I

estimatedusing instrumental ariables.Changes n the realprice of crudeoil servedas instruments or themselves. I used two variablesas instruments or output:the

currentand lagged level of real governmentpurchasesof goods and services, de-

trendedwith the same trendused for output, anda dummyvariablethat is equalto

one when a Democratis president.These arevalid instruments,since government

spendingis largely determinedby exogenousevents, such as the needto fighta war

or to expand schools. And althoughtherz is some indicationthateconomic condi-

tions can affect the outcomeof presidential lections, thiseffect is on the likelihood

of a change in party and not on the political party of the winner.The instruments

explainedabout 25 percentof the variation n detrendedoutput.Since the surveys were for one-year-aheadnflation expectations,I used annual

data. The estimationperiodwas from 1949 to 1990.

3. RESULTS

The results are shown in Table 1. The main parameter f interest,, has the ex-

pected sign, regardless of which proxy was used for inflationexpectationsand

whetheroutput or the unemploymentrate was used as the measureof economic ac-tivity. It is statisticallysignificant using both the Livingstonand Michigan survey

proxies, with eithereconomic activityvariable.

The estimate of w based on McCallum'sproxy is about the same size as the

survey-based stimates,but it is not significantbecause theestimatedstandard rror

is larger,both on the coefficientestimate and on the equationas a whole. Oneexpla-





JOHNM.ROBERTS 983

the substantialdiSerence in average nflation n the two partsof the sample(beforeandafter 1973). These resultssuggest thatthe changes in inflationdid not lead to achange in w and indicate thatthe estimates of w can be considered"structural," t

least for the rangeof inflationrates over the past fortyyears in the United States.

4. INTERPRETINGTHEMAGNITUDEOF y

Based on the resultsof Section 3, the valueof w was in therangeof 0.2 to 0.4 forthe deviation of actualoutputfromtrendandfrom -0.2 to -0.4 for the unemploy-mentrate.What do theseestimatesimplyin termsof theunderlyingmicroeconomicstructuresdescribed n section 1? Calvo'smodelprovides the richestinterpretation.

In Calvo's model, w is (Tr2T)/(l - Tr).So, w is positively related to both w thefirm's supplyelasticity ands the frequencyof pricechange. It is not possibletodisentanglethese parameters,butsome examplesillustrate he trade-oWsmpliedby.the estimates. If we assumethat Trs l/2, SOthat l/2 of firmschangetheirprice eachyear, a w estimate of l/4 (from an equationusingdetrendedoutput) mpliesan elas-ticity of the firm's supply curve of l/2. Thus, a 1 percentagepoint increasein ex-pecteddemand would lead to a l/2 percent ncrease n the firm'spricerelativeto theprices set by competitors.If a largerfractionof firmschangedpriceseach year-say, 90 percent then a w of l/4 would imply a supply elasticity of l/32, a much

flattersupply-curve lope.5Microeconomicevidence suggests thatthe latterinterpretationmay be closer to

the correctone: Roberts, Stockton, and Struckmeyer 1994), looking at industrydata,and Kashyap(1990) andCarlton 1986), looking at firm-leveldata,found thatprices are changed with relative ease; Roberts, Stockton, and Struckmeyeralsofound quite flat marginalcost. This interpretations also consistent with the NewKeynesian model of Ball andRomer(1990), in which flatsupplyschedulesplay asimportanta role in creatingaggregatenominalrigidity as does a reluctanceof firmsto changetheirprices.

LITERATURECITED

Ball, Laurence, and David Romer."RealRigiditiesandthe Non-Neutralityof Money."Re-view of EconomicStudies 57 (April 1990), 183-204.

Ball, Laurence, N. GregoryMankiw,and David Romer. "TheNew KeynesianEconomicsand the Output-InflationTrade-off."BrookingsPapers on EconomicActivity (1988:1),1-65.

Calvo, Guillermo A. "StaggeredContracts n a Utility-MaximizingFramework." ournalofA8onetaryEconomics 12 (September1983), 383-98.

5. In Taylor'smodel, a y (for unemployment)of 1/3 would be consistentwith a slope of the laborsupply schedulethatcorresponds o an elasticity of about 1/12. Thus, underthedegree of rigiditychosenfor theTaylor nterpretation, hereis also a very flataggregatesupplycurve. In Rotemberg'smodel, y isIc and therefore nvolves the inverse of the supplyelasticity of the typical good (^) andthe ratio of thecosts of changing price to the cost of being away from the optimumprice (c). Unfortunately, does nothavea natural nterpretation.



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