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Staggered W ag e Se t ting in a M a c r o M o d el Few economists now question the validityof the Friedman-Phelps accelerationist hy-pothesis that the Phillips curve is vertical inthe long run -at least as a first-order appro x-imation. Indeed. the once controversial hv-pothesis is now embodied in basic textboikmacro models (see Rudiger Dornbusch andStanlev Fischer. and Robert J. Gordon. forexam pie). This new accelerationist consensus,

however, has done little to settle the ongoingdeba te over aggregate dem and policy, wherethe crucial issues appear to depend on theshort-run Phillips curve and its dynamicproperties. The accelerationist theory pro-vided an elegant and concise representation ofthe inflationary process for the long run.However. it has proved distressingly unspe-cific as a framework for the development ofshort-run dynamics.Two sources of this incomplete specifica-tion have stimulated extensive research inrecen t years . T h e first-about which littlewill be said here-is th at the accele ration isttheory was not specific about the process ofexpectation formation. According to the theo-ry, the expected inflation rate A* should beadded to the right-hand side of the Phil l ipsequation. Hence the expectation processdetermining a* matters greatly for short-rundynamics. For example, if expectations areformed rationally, then as Thomas Sargentand Neil Wallace have shown (using theRobert Lucas supply model), the Phil l ipscurve will be vertical in the short run as wellas the long run from the point of view ofaggregate demand policy. On the othe r hand,if expe ctation s ar e adaptive-either byassumption or by derivation from a learningmodel-then the shor t-run slope mig ht be

*Columbia University. This research is beingsuppor ted by the National Science Foundation . Thispaper was completed while I was a consultan t to theFede ral Reserve Bank of Phila delph ia, which does notnecessarily endorse the views expressed. I ~ i s ho t ha nkMartin Baily , Phil ip Cagan, Guil lermo Calvo, andEdmund Phelps for usefu l comments .

very flat. But, if this were the only source ofambiguity in the accelerationist model, then itis likely that the controversy over the short-run properties would have been settled quick-ly: the attractiveness of rational expecta-tions-again as a first-order appr oxim ation-has become increasingly evident in theoreticala n d e m ~ i r i c a l o rk .The second source of imprecision is moretroublesome and is unlikely to be resolvedquickly. It involves the micro-economicdetails of wage and price adjustment whichar e just as much a part of the famous macro"expectations" adjus tmen t, as the expectationformation mechanism itself. While anextremely literal reading of the acceleration-ist theories would interpret x* as a pureforecast of inflation i nd e~ en d en t of thedynam ics of wage an d price contracts, a morepractical reading would suggest that T *represents the persistence of inflation due tothe gradual adjustments of outstanding wageand price contracts to new economic informa-tion. Som e modelling of this phenom enon canbe found in Edm und Phelps (19 70 ), especiallyhis Appendix 1, and in Arthur Okun ' scontract-based inflation model with accelera-tionist implications. Empirical work on priceand wage equations by Phil ip Cagan andMichael Wachter has emphasized the dy-namic implications of both contracts andexpectations. Policy-oriented studies by Wil-l iam Fellner (1978) and George Perry havealso taken this view of the accelerationisttheory, though with widely differing policysuggestions.Th e impact of aggreg ate demand on infla-tion and employment is crucially dependenton whether the contract mechanism or theexpectation mechanism dominates t he persis-tence effects commonly represented by T * .Hence, a resolution of the current macro-economic controversy requires some explicitmodels to disentangle the two mechanismstheoretically, if not em pirically, and to det er-mine how contract length and adjustment

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speeds affect aggregate dem and.The purpose of this paper is to discuss onesuch model which focuses on contracts andstaggered wage setting with rational expecta-tions. The model is based on some of myrecent research (see my 1978. 1979 papers)but is generalized here to permit alternativemixes of expectation and contract effects inthe wage equations.I . Staggered Wage Setting

A property of wage and price contractswhich has not typically been emphasized inmicro-economic analyses, but which is impor-tan t from th e viewpoint of ~ nacr oeco nom ics sthat contract decisions are staggered: al lcontract decisions in the economy are notmade at the same point in t ime. While somemonths are more popular than others foradjusting wage contracts, these adjustmentdecisions ar e generally staggered throughoutthe year. This property of contract formationis the central feature in the model discussedbelow.T o m ake th ings s imple suppose tha t wagecont racts last one year and th at decision date sare evenly staggered: half the contracts areset in January and half in July. If we letsix-month (semiannual) intervals be theperiod of measurement, and x,be the log ofthe contract wage for periods t and t + I , se tat th e star t of period t, then a sim ple model ofcontract wage determination is given by

where )I, is a measure of excess demand inperiod t, t , is a random shock, and b, d, and yar e positive parame ters. Th e "hat" over avariable represents its conditional expectationbased on period t - 1 information. Equation(1) s ta tes the assumpt ion tha t the contrac twage set a t the start of each semiannualperiod depends on three factors: the contractwage set in the previous period, the contractwage expected to be set in the next period,and a weighted ave rage of excess dem andexpected during the next two periods. Since,by assumption, x,will prevail for two periods,

firms and/or unions contemplating a wageadjus tme nt in period t will be concerned withwage rates which will be in effect duringperiods t and t + 1. Hen ce both x,-,and ?,+,are included in the equation. Note thatcontracts set before period t - 1 and a f te rperiod t + 1 are not included in the equation.Such contracts do not overlap with thecurre nt contract and ar e therefore not part ofthe relative wage structure.The b and d coefficients in equation (1)represent the elast ici ty of the cu rren t contractwage with respect to the previous contractwage and the next contract wage, respective-ly. Let us assume that b + d = 1 so tha t thecurrent contract decision is homogeneous ofdegree 1 in these lag and lead contracts. Ifb = d = 112 then th e lag and lead distribu-tion is symmetric. This has been the para-metric assumption used in my previous workand reflects the plausible assumption thatcurrent negotiations weight other contractsaccording to the number of periods that theyoverlap with the current contract . In thissense, when b an d d a r e equal to 112, contrac tdecisions are unbiased. Wage setters lookforward to the same degree they look back-ward. However, 1 will allow for the possibilityof biased weights in this paper by permitting band d to differ from 112. Thi s permits aspectrum of contract determination hypo-theses between the extremes of pure back-ward looking (b = I ) , and pure forwardlooking (d = I) . As will be demonstratedbelow the size of b vs. d is important for thedynamic behavior of contracts, and for thesensitivity of wage behavior to excessdem and . This im portance of forward lookingvs. backward looking has been emphasized ina recent paper by Perry in analyzing anhypothesis set forth by Fellner (1976).In order to derive a dynam ic representationfor the behavior of the contract wage fromequation (I), it is necessary to solve for j,,A+,, an d it+,.This involves specifying anaggregate demand relat ionship and a policyrule . Assume tha t the excess demand variable)I, is the percentage output gap (that is, thedeviation of the log of real output fromtrend), and that the demand for money isgiven by m, = J) , + w, - v , where the

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variables m,, w,, and v, are the logs of theaggr egate wage level, the money supply an d ashock, all measured as deviations from trend.Note that this money demand equation issimply the quanti ty equation with the wagesubstituted for the price level. This approxi-mation saves one equation and can easily bemodified. If the policy rule for the moneysupply is the log-linear form m, = gw,, thenwe can derive the simple aggregate demandrelation

where @ = 1 - g . No te t ha t /3 is a policyparame ter indicat ing the degree of accommo-dation of agg regate dema nd to wage changes.The model is closed by noting that w, is anaggregate of the contract wages x, and x,- ,outstanding at t ime t . If we use the geometricaverage, then

By substituting equations (3) and (2) into(1) and taking expectations conditional ont - 1 information we have thatwhere c = ( 1 + . 5 y p ) / ( l - . 5 y @ ) .Assum-ing that x, is stable yields a solution for x, ofthe form

c - [c ' - 4 d ( l - d)]"*where a = 2dAn equation for the average wage w, canreadily be derived from (5) using (3) and isgiven by

Equations (6) and (2) can be used toaddress a number of the issues raised above.From the parameter a we can determ ine howthe wage dynamics depends on aggregatedem and policy (P ), on the sensitivity of wagechange to excess demand (y ), and on thedegree of forward looking (d).Note, however, that in this model wecannot identify the two parameters y and dfrom a time-series on w, and y, withoutfurther assumptions. Given such time-series

we could easily estimate /3 and a from equa-tions (2) and (6). However, from the defini-tion of a, these estimates would not determ ined an d y uniquely. Of course this identificationproblem could be surmounted by makingadditional assumptions or by looking for shiftsin policy. For example, additional identifyingconstraints arise when contracts last for morethan two periods. Nevertheless, this potentialidentification problem should be kept in mindwhen at tempting to est imate the degree offorward looking using aggregate time-seriesd a t a .11. Forward Looking Contracts and Aggregate

Wage DynamicsThe parameter a in equation (5) cha racte r-izes the degree of persistence in aggregatewage behavior. Clearly the persistence willdepend on how accommodative aggregatedemand policy is to wage contract adjust-ments which are "too inflationary." Thisdependence is captured in the model by therelationship between a and P. The higher is /3(i.e., the less accommoda tive is policy) thelower is a (i.e., the less persistent are wagefluctuations). Hence by choosing @ large

enough, policy can achieve high degrees ofstability in the path of aggregate wages.However, since higher values of @ result inlarger f luctuations in the output gap (seeequation (2)), this wage stability must betraded off against real output and employ-ment stability. This stability tradeoff definesthe inflation unemployment dilemma in thismodel .In order to distinguish between the impactof co ntrac t effects and expectations effects onthis t radeoff, the parameter d can be variedover its range between 0 and 1 . Recall thatthe lower is d the more backward looking iscontract determination an d the less importantare expectations. For certain values of @ andy, Figure 1 illustrates how the wage dynam icsdepend on d . As one would e xpect, smallervalues of d a re associated with large r values ofa. That is , more backward-looking wagedetermination increases the persistence or theinertia of aggrega te wages. Th e shape of thisnegative relationship shows that increasing

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0

VOL. 6 9 N O . 2 W A G E S A N D E M P L O Y M E N T I I 1

1 0 -M o re A c c o m m d o t v e- P 01 1c y ( P = 1 1- 0 8 -

v1 L e s s A c c o m m o d a t ~ v ew P o l ~ c y 0 . 0 4 11 0 6 -LL

Wg 0 4 -a -?-V?B 0 2 -

1 1 / I 1 1 1 10 2 0 4 0 6 0 8 1 0

DEGREE OF FOR WAR D LOOK I N G ( d l

forward looking (d ) from 40-50 percentwould reduce persistence substantially. In-creasing d from 10-20 percent would onlyreduce persistence slightly, however.It can also be shown that the wage-outputstability tradeoff depends on d. Becauseforward looking increases the demand effectson wages, higher values of d improve thetradeoff. This corresponds to the intuitivenotion that more forward-looking contractdetermination increases the impact of aggre-gate demand policy on wages. Hence, infla-tion-stabilizing fluctuations in aggregate de-mand can be smaller and need not last aslong.

111. Contract Length, Empirical Regularities,and Micro Foundations

The wage and output dynamics generatedby this model share a num ber of featur es withthe actual behavior of these series and thislends some support to the idea that contractformation a s well as expectations is an im por-tant part of wage and price dynamics. Twofeatures are worth mentioning here. (Forfurther details see the author, 1978.)First the serial correlation struc ture for theoutput gap (or unemployment) in this modelis hump shaped: the impact of shocks onoutput rises before diminishing toward zero.This hump shaped property is also present inthe actual process for output or unemploy-

ment in the United States and a number ofother countries. Hence, the model is capableof explaining not only the serial persistence ofunemployment but also the shape of thepersistence.Second, a striking aspect of the U . S . quar -terly dat a is tha t the humped shape reaches apeak at about one year; this corresponds tocontract lengths in the model of about thesam e length. Hence, relatively short con tracts(much shorter than the frequently cited three-year union contracts) are capable of display-ing empirically observed serial persistence.Although other models might explain thesecorrelations just as well, this type of modelwith relatively short contracts appears to beconsistent with the data.Whether the model is consistent with arigorous micro theory is more difficult todetermine. Unfortunately, the assumed con-tract formation behavior is not explicitlyderived from a utility maximization model(see Robert Barro). While significant gainshave been made in our understanding ofcontracts through the work by Costas Azaria-dis , Martin Baily (19 74), and D. F. Gordon,the micro foundations of the staggeredcontract model presented here are far fromcomplete. 1 think there ar e important infor-mation reasons for contracts to be staggered(without an auctioneer some staggering isnecessary for firms to obtain informationabout the relative wage structure), but thesear e yet to be laid out rigorously. For thisreason such m odels should be used cautiou slysince contract length is not a da tum , and theoptimal length of contracts may change withchanges of policy. By way of comparison theinformation-based theories of aggregate dy-namics which have been developed by Lucasalso have ~ r o b l e m swith micro foundations.For example, disparate information is im-posed on such models with little discussion ofhow stabilization policy might affect mobilityor communication between markets whichwould alter this information structure.The theoretical approach to micro founda-tions has proved difficult and is likely toremain so. But there is also an empiricalapproach which has received little attention.An early example of this approach is the

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study of compensation policy made by Rich-ard Lester in the late 1940's. His aim was toinvestigate (through a survey of firms) anumber of alternative wage setting proce-dures: whether firms use wage surveys indetermining their wage scales, whether firmstry to anticipate future wage developments,and whether tight labor markets influencewage policy. While far from conclusive thestudy is suggestive of what might be doneusing modern techniques. For example,although wage surveys are now used almostuniversally by firms in determining wagescales, there is very little information avail-ab le concernin g how firms use these surveys.Such information would appear to be invalu-able in modelling the macroeconomics ofwage behavior.

IV . Concluding RemarksTh e theme of this paper has been th at theinflation dynamics typically associated withthe "expectations-augmented" Phill ips curveare significantly influenced by th e interac tionof staggered contracts as well as by expecta-tions effects. While these ideas are implicit inmuch accelerationist research, the aim here

has been to make them explicit in order thatalternative hypotheses concerning the infla-tion process can be stated more clearly. Theoverlapping contract model described in thepaper is closely related to a number of othermodels. (See George Akerlof, Baily, 1976,Fischer, Phelps, 1978, forthcoming, StephenRoss and Wachter, and J . C. R. Rowley andD. A. Wilton, for exa mple .) Wh ile the microfoundations of such models need to be devel-o ~ e dmore rigorously, they seem capab le ofimproving our understanding of the dynamicsof the inflationary process within a reasonablewell-specified rational setting.R E F E R E N C E S

G. Akerlof. "Relative Wag es and the Ra te of~nfla t ion,"Quart . J. E&., Aug. 1969, 83,353-74.C. Azariadis, "Implicit Contracts and Unem-ployment Equilibria," J. Polit. Econ., Dec.1975, 83, 1 183-202.

M. N . Baily, "Contract Theory and the Moder-ation of Inflation by Recession andControls," Brookings Papers, Washington1976, 3, 585-622., "Wages and Employment underUncertain Demand," Rev. Econ. Stat ist . ,Jan. 1974, 4 1 , 37-50.R. J. Barro, "Long-Term Con tracting, StickyPrices, and Monetary Policy," J. Motzet.Econ., July 1977, 3, 305-16.Philip Cagan, Th e Hydra-Headed Monster:the Problem of Injut ion in the UnitedStates , Washington, 1974.Rudiger Dornbusch and Stanley Fisher, Macro-economics, New York 1978.William J. Fellner, "The Core of the Contro-

versy about Reducing Inflation: An Intro-ductory Analysis," in his ContemporaryEconomic Problems, Washington 1978., Towards a Reconstruction of Ma-croeconomics: Problems of Theory andPolicy, Washington 1976.S. Fischer, "Long-Term Contracts , RationalExpectations, and the Optimal MoneySupply Rule," J. Polit. Econ., Feb. 1977,85, 191-205.M . Friedman, "The Role of Monetary Policy,"Amer. Econ. Rev., Mar. 1968, 58, 1-17.D. F . Gordon, "A Neo-Classical Theory ofKeynesian Unemployment," in Karl Brun-ner, and Allan H . Meltzer, eds. , The P hil-lips Cztrves and Labor Markets, Amster-da m 1976 , 65-97.Robert J. Gordon, Macroeconomics, Boston1978.Richard A. Lester, Companj' Wage Policies,Princeton 1948.R. E. Lucas, "Some Internation al Evidence onOutput Inflation Tradeoffs," Amer. Econ.Rev. , June 1973, 63, 326-34.A. M . Okun, "Inflation: Its Mechanism andWelfare Costs," Brookings Papers, Wash-ington 1975, 2, 351-401.G. L. Perry, "Slowing the Wage-P rice Spiral:The Macroeconomic View," BrookingsPapers, Washington 1978, 2, 259-91.E. S. Phelps, "Disinflation without Recession:Adaptive Guideposts and Monetary Poli-cy," Weltwirtsch. Ar ch., forthcoming., "Introduction: Developments in Non-Walrasian Theory," in his Studies in

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Macroeconomic Theory: Em ploym ent and New Efficient Estimates," Amer. Econ.Inflation, New York 1978. Rev. , June 1973, 63, 380-89., "Money Wage Dynamics and Labor T. J. Sargent and N. Wallace, " Rational 'Mark et Equilibrium," in Edmund S . Expectations, the Optim al Mon etary In-Phelps, et al., Microeconomic Foundations s t rument and the Opt imal Money Supplyoj'Employment and Inflation Theory, New Rule," J. Polit. Econ., Apr. 1975, 83,York 1970 . 241-54., "Phillips Curves, Exp ectations of J. B. Taylor, "Estimation and Control of aInflation, and Op tim al Em ploym ent over Macroeconomic Model with Ra tion al Ex-Time," Economica, Aug. 1967, 34 , 254- pectations," Econometrica, 1979.81. , "Aggregate Dynamics and Stag-S. A. Ross and M. L. Wachter, "Wage Determi- gered Contracts," unpublished paper, Co-nation, Inflation, and the Industrial Struc- lumbia Univ. 1978.ture," Amer. Econ. Rev. , Sept. 1973, 6 3 , M. L. Wachter, "The Changing Cyclical675-94. Responsiveness of Wage Inflation," Brook-J. C. Rowley and D. A. Wilton, "Quarterly i n g ~ apers, Washington 1976, 1 , 115-

Models of Wag e Determination: So me 59.