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    American Economic Association

    Some International Evidence on Output-Inflation TradeoffsAuthor(s): Robert E. Lucas, Jr.Source: The American Economic Review, Vol. 63, No. 3 (Jun., 1973), pp. 326-334Published by: American Economic AssociationStable URL:

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  • 8/2/2019 Lucas Tradeoff 1973


    S o m e International Evidence o nOutput-Inflationradeofs

    By ROBERT E. LUCAS, JR.*This paper reports the results of anempirical study of real output-inflationtradeoffs, based on annual time-series fromeighteen countries over the years 1951-67.These data are examined from the pointof view of the hypothesis that averagereal output levels are invariant underchanges in the time pattern of the rate ofinflation, or that there exists a "naturalrate" of real output. That is, we are con-cerned with the questions (i) does thenatural rate theory lead to expressions ofthe output-inflation relationship whichperform satisfactorily in an econometricsense for all, or most, of the countries inthe sample, (ii) what testable restrictionsdoes the theory impose on this relation-ship, and (iii) are these restrictions con-

    sistent with recent experience?Since the term "'natural rate theory"refers to varied aggregation of models andverbal developments,' it may be helpfulto sketch the key elements of the particularversion used in this paper. The firstessential presumption is that nominal out-put is determined on the aggregate demandside of the economy, with the divisioninto real output and the price level largelydependent on the behavior of suppliers oflabor and goods. The second is that thepartial "rigidities" which dominate short-run supply behavior result from suppliers'lack of information on some of the pricesrelevant to their decisions. The third

    presumption is that inferences on theserelevant, unobserved prices are madeoptimally (or "rationally") in light of thestochastic character of the economy.As I have argued elsewhere (1972),theories developed along these lines willnot place testable restrictions on the co-efficients of estimated Phillips curves orother single equation expressions of thetradeoff.They will not, for example, implythat money wage changes are linked toprice level changes with a unit coefficient,or that {"long-run"'in the usual distrib-uted lag sense) Phillips curves must bevertical. They will (as we shall see below)link supply parameters to parametersgoverningthe stochastic nature of demandshifts. The fact that the implicationsof thenatural rate theory come in this form sug-gests an attempt to test it using a sample,such as the one employed in this study, inwhich a wide variety of aggregatedemandbehavioris exhibited.In the following section, a simple ag-gregative model will be constructed usingthe elements sketched above. Resultsbased on this model are reported in Sec-tion II, followed by a discussion and con-clusions.

    1. An Economic ModelThe general structure of the model de-veloped in this section may be describedvery simply. First, the aggregate price-quantity observations are viewed as inter-section points of an aggregatedemand andan aggregate supply schedule. The formeris drawn up under the assumption of acleared money market and represents theoutput-price level relationship implicit in

    * Graduate School of Industrial Administration, Car-negie-Mellon University.1 The most useful, general statements are those of .Milton Friedman (1968) and Edmund Phelps. Specificillustrative examples are provided by Donald Gordonand Allan Hynes and Lucas (April 1972).326

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    VOL. 63 NO. 3 LUCAS: OUTPUT-INFLATIONTRADEOFF 327the standardIS-LM diagram. It is viewedas being shifted by the usual set ofdemand-shift variables: monetary andfiscal policies and variation in export de-mands. The supply schedule is drawn un-der the assumption of a cleared labormarket; its slope therefore reflects laborand product market "rigidities."The structure of this model, which isessentially that suggested in Lucas andLeonard Rapping (1969), will be greatlysimplified by an additionalspecialassump-tion: that the aggregate demand curve isunit elastic.2 In this case, the level ofnominal output can be treated as an"exogenous"'variable with respect to thegoods market, and the entire burden of ac-counting for the breakdown of nominalincomeinto real output and price is placedon the aggregate supply side. In the nextsubsection, A, a supply model designed toserve this purpose is developed. In subsec-tion B, solutions to the full (demand andsupply) model are obtained.

    A. AggregateSupplyAll formulations of the natural ratetheory postulate rational agents, whosedecisions depend on relative prices only,placed in an economic setting in whichthey cannot distinguish relative from gen-eral price movements. Obviously, there isno limit to the number of models one canconstruct where agents are placed in thissituation of imperfect information; thetrick is to find tractable schemes with thisfeature. One such model is developedbelow.We imagine suppliers as located in alarge number of scattered, competitivemarkets. Demand for goods in each period

    is distributed unevenly over markets,leading to relative as well as generalpricemovements. As a consequence, the situa-tion as perceived by individual supplierswill be quite different from the aggregatesituation as seen by an outside observer.Accordingly,we shall attempt to keep thesetwo points of view separate, turning firstto the situation faced by individual sup-pliers.Quantity suppliedin each market will beviewed as the product of a normal (orsecular)componentcommon to all marketsand a cyclical component which variesfrom market to market. Letting z indexmarkets, and using ynt and yet to denotethe logs of these components, supply inmarket z is:(1) yt(Z) = Ynt + yet(Z)The secular component, reflecting capitalaccumulation and population change, fol-lows the trend line:(2) y.t a+? tThe cyclical component varies with per-ceived, relative prices and with its ownlagged value:(3) yet(z) y[Pt(z) - E(Pt It(Z))]+ NyC_t-1(z)wherePt(z) is the actual pricein z at t andE(PtI It(z)) is the mean current, generalprice level, conditioned on informationavailable in z at t, It(z).3 Since yet is adeviation from trend, f I< 1.

    2 An explicit derivation of the price-output relation-ship from the IS-LM framework is given by FredericRaines. Of course, this framework does not imply anelasticity of unity, though it is consistent with it. Sincethe unit elasticity hypothesis is primarily a matter ofconvenience in the present study, I shall comment belowon the probable consequences of relaxing it.

    3 A supply function for labor which varies with theratio of actual to expected prices is developed and veri-fied empirically by Lucas and Rapping (1969). Theeffect of lagged on actual employment is also shown.In our 1972 paper, in response to Albert Rees's criti-cism, we found that this persistence in employmentcannot be fully explained by price expectations behav-ior. Both these effects-an expectations and a persis-tence effect-will be transmitted by firms to the goodsmarket. In addition, they are probably augmented byspeculative behavior on the part of firms (as analyzedfor example, by Paul Taubman and Maurice Wilkinson).For a general equilibrium model in which suppliersbehave essentially as given by (3), see my 1972 papers.

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    328 THE AMERICAN ECONOMIC REVIEW JUNE 1973The information available to suppliersin z at t comes from two sources. First,traders enter period t with knowledge of

    the past course of demandshifts, of normalsupply ynt, and of past deviations ye,t-i,yc,t-2 . While this information doesnot permit exact inferenceof the log of thecurrent generalprice level, Pt, it does de-terminea "prior"distribution on Pt, com-mon to tradersin all markets. We assumethat this distribution is known to be nor-mal, with mean Pt (dependingin a knownway on the above history) and a constantvariance a2.

    Second,we supposethat the actual pricedeviates from the (geometric) economy-wide average by an amount which is dis-tributed independently of Pt. Specifically,let the percentagedeviation of the price inz from the average Pt be denoted by z (sothat markets are indexed by their pricedeviations from average) where z is nor-mally distributed,independent of Pt, withmean zero and variance r2. Then the ob-servedprice in z, Pt(z) (in logs) is the sumof independent,normalvariates(4) Pt(z) = Pt + zThe informationIt(z) relevant for estima-tion of the unobserved (by suppliers in zat t) Pt, consists then of the observedprice Pt(z) and the history summarizedin P.tTo utilize this information,suppliersuse(4) to calculate the distribution of Pt,conditionalon Pt(z) and Pt. This distribu-tion is (by straightforward calculation)normal with mean:

    E(Pt I(z)) = E(Pt Pt(z), iPt)(~ ~ ~~~ = (1-)Pt (z) + 07Pt

    where 0= i2/(u2+i2) , and variance Ooa2.Combining (1), (3), and (5) yields thesupply function for market z:(6) yt(Z) = Ynt+ O@Y[Pt(z) PTt]

    + XYc,t-1(z)

    Averaging over markets (integrating withrespect to the distribution of z) gives theaggregate supply function:(7) = yYnt G7y(Pt- Pt)

    + xLyt-1 - yn,t-1IThe slope of the aggregate supply func-tion (7) thus varies with the fraction 0 oftotal individual price variance, a2+r2,which is due to relative price variation,,Incases where r2 is relatively small, so thatindividual price changes are virtually cer-tain to reflect general price changes, thesupply curve is nearly vertical. At theother extreme when general prices arestable (a2 is relatively small) the slope ofthe supply curve approaches the limitingvalueof y.4

    B. Completionand Solution of the ModelA central assumption in the develop-ment above is that supply behavior isbased on the correctdistribution of theunobserved current price level, Pt. To

    proceed, then, it is necessary to determinewhat this correct distribution is, a stepwhich requires the completion of themodel by inclusion of an aggregate de-mand side.As suggestedearlier,this will be done bypostulating a demandfunction forgoods ofthe form:(8) yt + Pt = xtwhere xt is an exogenous shift variable-equal to the observable log of nominalGNP. Further, let Axt} be a sequence ofindependent, normal variates with mean 6

    and variane2 rande ar'lance x04This predicted relationship between a supply elas-ticity and the variance of a component of the price seriesis analogous to the link between the income elasticity ofconsumption demand and the variances of permanentand transitory income components which Friedman(1957) observes. As will be seen in Section II, it worksin empirical testing in much the same way as well.This particular characterization of the "shocks" to

    the economy is not central to the theory, but to discuss

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    VOL. 63 NO. 3 LUCAS: OUTPUT-INFLATIONTRADEOFF 329The relevant history of the economythen consists (at most) of ynt (which fixescalendar time), the demand shifts xt,

    xt1,... , and past actual real outputsYt-1,Yt-2. * Since the model is linear inlogs, it is reasonable to conjecture a pricesolution of the form: 6(9) Pt= 7rO+rlXt+7r2Xt_l+7r3Xt_2+ *

    +l1 Yt_1+?12Yty2+ +tOyntThen 7Ptwill be the expectation of Pt,based on all information except Xt (thecurrentdemand level) or:

    Pt = 750 + 7rl(Xt-1 + 5) + 72Xt-l(10) + 7r3Xt-2 + ... + ?lyt-l

    + n72yt.-2 + + tOyntTo solve for the unknown parameters7ri, Ij and 4o we first eliminate yt between(7) and (8), or equate quantity demandedand supplied. Then inserting the rightsides of (9) and (10) in place of Pt and Pt,one obtains an identity in Xt}, {yt , and

    ynt, which is then used to obtain theparametervalues. The resulting solutionsfor price and output are:'Pt= - + X- ,1+GY 1 +OY

    + - Xt-1 - (1 - X)Y.t1 + O-Y+X0+ -AxtI8 1- +O + 1 + 8 AX

    + Xyt-1 + (1 - X)yntIn terms of APt and yet, and letting7r= y/(1+G'y),the solutions are:(11) Yet = - irb + 7rAxt + Xye,t_1(12) APt = - d + (1 - r)AXt + 7rAXt-i

    Let us review these solutionsfor internalconsistency. Evidently, Pt is normallydis-tributed about Pt. The conditional vari-ance of Pt will have the constant (asassumed) variance 1/(1+0Gy)2o-,. Thusthose features of the behavior of priceswhich were assumed "known"by suppliersin subsection A are, in fact, true in thiseconomy.To review, equations (11) and (12) arethe equilibrium values of the inflation rateand real output (as a percentage deviationfrom trend). They give the intersectionpoints of an aggregate demand schedule,shifted by changes in xt, and an aggregatesupply schedule shifted by variables(lagged prices) which determine expecta-tions. In order to avoid the introductionofan additional, spurious "expectations pa-rameter,"one cannot solve for this inter-section on a period-by-period basis; 4c-cordingly, we have adopted a methodwhich yields equilibrium"paths" of pricesand output. Otherwise, the interpretationof (11) and (12) is entirely conventional.Not surprisingly, the solution values ofinflation and the cyclical component ofreal output are indicated by (11) and (12)to be distributed lags of current and pastchanges n nominaloutput. A changein thenominal expansion rate, Axt, has an im-mediate effect on real output, and laggedeffects which decay geometrically. The

    rational expectations formation at all, some explicitstochastic description is clearly required. Independenceis used here partly for simplicity, partly because it isempirically roughly accurate for most countries in thesample. The effect of autocorrelation in the shockswould, as can be easily traced out, be to add higher orderlag terms to the solutions found below.

    6 This solution method is adapted from Lucas (1972),which is in turn based on the ideas of John Muth.7 If a demand function of the form Yt=-Pt+xt hadbeen used, these solutions would assume the same form,with different expressions for the coefficients. If t $1,however, xt is an unobserved shock, unequal in generalto observed nominal income. In this case, the model stillpredicts the time-series structure (moments and laggedmoments) of the series Yet nd APt and is thus, in princi-ple, testable. I have found empirical experimentingalong these lines suggestive, but the series used are

    simply too short to yield results of any reliability.

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    330 THE AMERICANECONOMIC REVIEW JUNE 1973immediate effect on pricesis one minus thereal output effect, with the remainderofthe impact coming in the succeeding pe-riod. We note in particular that this lagpattern may wellproduceperiodsof simul-taneous inflation and below average realoutput. Though these periods arise be-cause of supply shifts, the shifts resultfromlaggedperception of demandchanges,and not from autonomous changes in thecost structure of suppliers.In addition to these features, the modeldoes indeed assert the existence of a nat-ural rate of output: the average ate of de-mand expansion, 5, appears in (11) with acoefficientequal in magnitude to the co-efficient of the current rate, and with theopposite sign. Thus changesin the averagerate of nominalincomegrowth will have noeffect on average real output. On the otherhand, unanticipated demand shifts dohave output effects, with magnitude givenby the parameter ir. Since this effect de-pends on "fooling" suppliers (in the senseof subsectionA), one expects that 7rwill belarger the smaller the variance of thedemand shifts. We next develop this im-plication explicitly.From the definition of ir in terms of 0and y, and the definition of 6 in terms ofq2 and r2 we have

    r27a2 + r2(1 + y)

    Combiningwith the expressionfor u2 ob-tained above, this gives(13) r =-*, (1 - )2a2 + r2(1 + y)For fixedr2and y, then, ir takes the valuey/(1 +y) at ax= 0 and tends monotonicallyto zero as o2 tends to infinity.The prediction that the average devia-tion of output from trend, E(y,,), is in-variant under demand policies is not, ofcourse,subject to test: the deviations from

    a fitted trend line must average to zero.Accordingly, we must base tests of thenatural rate hypothesis (in this context)on (13): a relationship between an ob-servable variance and a slope parameter.

    II. Test ResultsTesting the hypothesis advanced aboveinvolves two steps. First, within eachcountry (11) and (12) should performreasonably well. In particular, under thepresumption that demand fluctuations arethe major source of variation in APt and

    yct, the fits should be "good." The esti-mated values of 7rand Xshouldbe betweenzero and one. Finally, since (11) and (12)involve five slope parametersbut only twotheoretical ones, the estimated ir and Xvalues obtained from fitting (11) shouldwork reasonably well in explaining varia-tions in APt.The main object of this study, however,is not to "explain" output and price levelmovements within a given country, butrather to see whether the terms of theoutput-inflation "tradeoff'' vary acrosscountriesin the way predicted by the nat-uralrate theory. For this purpose, we shallutilize the theoretical relationship (13)and the estimated values of r and a'2.Under the assumption that r2 and y arerelatively stable across countries, the esti-mated r values should decline as the sam-ple variance of Axt increases.Descriptive statistics for the eighteencountries in the sample are given in Table1., As is evident, there is no association

    8 The raw data on real and nominal GNP are fromYearbookof National Accounts Statistics, where seriesfrom many countries are collected and put on a uniformbasis. The choice of countries is by no means random:the eighteen used are all the countries from which con-tinuous series are available. The sample could thus bebroadened considerably by use of sources from indi-vidual countries. To obtain the variables used in thetests, the logs of real and nominal output, Ytand xt, arelogs of the series in the source. The log of the price level,Pt, is the difference xt-yt; yt is the residual from thetrend line yt= a+bt, fit by least squaresfrom the sample

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    Mean Mean Variance. Variance VarianceCyt Apt Yet AxtArgentina .026 .220 .00096 .01998 .01555Austria .048 .038 .00104 .00113 .00124Belgium .034 .021 .00075 .00033 .00072Canada .043 .024 .00109 .00018 .00139Denmark .039 .041 .00082 .00038 .00084West Germany .056 .026 .00147 .00026 .00073Guatemala .046 .004 .00111 .00079 .00096Honduras .044 .012 .00042 .00084 .00109Ireland .025 .038 .00139 .00060 .00111Italy .053 .032 .00022 .00044 .00040Netherlands .047 .036 .00055 .00043 .00101Norway .038 .034 .00092 .00033 .00098Paraguay .054 .157 .00488 .03192 .03450Puerto Rico .058 .024 .00205 .00021 .00077Sweden .039 .036 .00030 .00043 .00041United Kingdom .028 .034 .00022 .00037 .00014United States .036 .019 .00105 .00007 .00064Venezuela .060 .016 .00175 .00068 .00127

    between average real growth rates andaverage rates of inflation: this fact seemsto be consistent with both the conventionaland natural rate views of the tradeoff.Since our interest is in comparingreal out-put andpricebehaviorunderdifferenttimepatternsof nominalincome,these statisticsare somewhat disappointing. Essentiallytwo types of nominal income behavior areobserved:the highlyvolatile and expansivepolicies of Argentina and Paraguay, andthe relatively smooth and moderately ex-pansive policies of the remaining sixteencountries. But if the sample provides onlytwo "points," they are indeed widelyseparated: the estimated variance of de-mand in the high inflation countries is onthe order of 10 times that in the stableprice countries.The first three columns of Table 2 sum-marize the performanceof equation (11)in accounting for movements in yet. Theestimated values for ir all lie between zeroand one; with the exceptions of Argentina

    and Puerto Rico, so do the estimated Xvalues. The R's indicate that for many, orperhaps most countries, important out-put-determiningvariables have been omit-ted from the model. The R s for the infla-tion rate equation, (12), are given incolumn (4) of Table 2. In general, thesetend to be lower than for equation (11),and not surprisingly the estimated co-efficients from (12) (which are not shown)tend to behave erratically. Column (5) ofTable 2 gives the fraction of the varianceof AP' explained by (12) when the co-efficient estimates from (11) are imposed.(A "-" indicates a negative value.)9With respect to its performanceas anintracountry model of income and pricedetermination,then, the system (11)-(12)passes the formal tests of significance. Onthe other hand, the goodness-of-fitstatis-

    period. The moments given in Table 1 are maximumlikelihood estimates based on these series. The estimatesreported in Table 2 are by ordinary least squares.

    9 The loss of explanatory power when these coeffi-cients are imposed on (12) can be assessed formally byan approximate Chi-square test. By this measure, theloss is significant at the .05 level for Paraguay only. AsTable 2 shows, however, this test is somewhat decep-tive: for several countries the least squares estimates of(12) are so poor that there is little explanatory power tolose, and the test is "passed" vacuously.

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    Country 7r R R R2Argentina .011 -.126 .018 .929 .914(.070) (.258)Austria .319 .703 .507 .518 -

    (.179) (.209)Belgium .502 .741 .875 .772 .661(.100) (.093)Canada .759 .736 .936 .418 -(.064) (.075)Denmark .571 .679 .812 .498 .282(.118) (.110)West Germany .820 .784 .881 .130 -(.136) ( .110)Guatemala .674 .695 .356 .016(.301) (.274)

    Honduras .287 .414 .274 .521 .358(.152) (.250)Ireland .430 .858 .847 .499 .192(.121) (.111)Italy .622 .042 .746 .934 .914(.134) (.183)Netherlands .531 .571 .711 .627 .580(.111) (.149)Norway .530 .841 .893 .633 .427(.088) (.096)Paraguay .022 .742 .568 .941 .751(.079) (.201)Puerto Rico .689 1.029 .939 .419(.121) (.072)

    Sweden .287 .584 .525 .648 .405(.166) (.186)United Kingdom .665 .178 .394 .266 .115(.290) (.209)United States .910 .887 .945 .571 .464(.086) (.070)Venezuela .514 .937 .755 .425(.183) (.148)

    tics are generally considerably poorer thanwe have come to expect from annual time-series models.In contrast to these somewhat mixed re-sults, the behavior of the estimated 7rvalues across countries is in striking con-formity with the natural rate hypothesis.For the sixteen stable price countries, *ranges from .287 to .910; for the twovolatile price countries, this estimate issmaller by a factor of 10! To illustrate thisorder-of-magnitude effect more sharply,let us examine the complete results for twocountries: the United States and Argen-

    tina. For the United States, the fitted ver-sions of (11) and (12) are:yct = - .049 + (.910),Axt+ (.887)yc,t-l

    APt = - .028 + (.119) Axt + (.758) Axt_- (.637) Aycet_1

    The comparable results for Argentina are:yct = - .006 + (.011)Axt - (.126)y,.,t_j

    APt = - .047 + (1.140)Axt - (.083) Axt-+ (.102)Ayc,t-1

    In a stable price country like the UnitedStates, then, policies which increase nomi-

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    VOL. 63 NO. 3 LUCAS: OUTPUT-INFLATION TRADEOFF 333nal income tend to have a large initialeffect on real output, together with a small,positive initial effect on the rate of infla-tion. Thus the apparent short-term trade-off is favorable, as long as it remains un-used. In contrast, in a volatile pricecountry like Argentina, nominal incomechanges are associated with equal, con-temporaneous price movements with nodiscernible effect on real output. Theseresults are, of course, inconsistent with theexistence of even moderately stable Phillipscurves. On the other hand, they followdirectly from the view that inflationstimulates real output if, and only if, itsucceeds in "fooling" suppliers of laborand goods into thinking relative prices aremoving in their favor.

    III. Concluding RemarksThe basic idea underlying the tests re-ported above is extremely simple, yet I amafraid it may have become obscured by therather special model in which it is em-bodied. In this section, I shall try to re-

    state this idea in a way which, though notquite accurate enough to form the basis foreconometric work, conveys its essentialfeature more directly.The propositions to be compared empiri-cally refer to the effects of aggregate de-mand policies which tend to move infla-tion rates and output (relative to trend) inthe same direction, or alternatively, un-employment and inflation in oppositedirections. The conventional Phillips curveaccount of this observed co-movement saysthat the terms of the tradeoff arise fromrelatively stable structural features of theeconomy, and are thus independent of thenature of the aggregate demand policypursued. The alternative explanation ofthe same observed tradeoff is that thepositive association of price changes andoutput arises because suppliers misinter-pret general price movements for relativeprice changes. It follows from this view,

    first, that changes in average inflationrates will not increase average output, andsecondly, that the higher the variance inaverage prices, the less "favorable" will bethe observed tradeoff.The most natural cross-national com-parison of these propositions would seemto be a direct examination of the associa-tion of average inflation rates and averageoutput, relative to "normal" or "full em-ployment." Unfortunately, there seems tobe no satisfactory way to measure normaloutput. The deviation-from-fitted-trendmethod I have used defines normal outputto be average output. The use of unem-ployment series suffers from the same diffi-culty, since one must somehow select the(obviously positive) rate to be denoted fullemployment.Thus although the issue revolves aroundthe relation between means of inflationand output rates, it cannot be resolved byexamination of sample averages. Fortu-nately, the existence of a stable tradeoffalso implies a relationship between vari-ances of inflation and output rates, asillustrated in Figure 1. With a stabletradeoff, policies which lead to wide varia-tion in prices must also induce comparablevariation in real output.' If these samplevariances do not tend to move together(and, as Table 1 shows, they do not) one

    APt'Var(yct) 'a2 Varf(APt) 0 0


    0 Stable Price Country0 Volatile Price Country

    YctFIGURE 1

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    334 THE AMERICAN ECONOMIC REVIEW JUNE 1973can only conclude that the tradeoff tendsto fade away the more frequently it isused, or abused.

    This simple argument leads to a formaltest if the output-inflation association isentirely contemporaneous. In fact, how-ever, it involves lagged effects which makea direct comparison of variances, as justsuggested, difficult in short time-series.Accordingly, it has been necessary to im-pose a specific, simple structure on thedata. As we have seen, this structure ac-counts for output and inflation rate move-ments only moderately well, but wellenough to capture the main phenomenonpredicted by the natural rate theory: thehigher the variance of demand, the moreunfavorable are the terms of the Phillipstradeoff.

    REFERENCESM. Friedman, A Theory of the ConsumptionFunction, Princeton1957., "The Role of Monetary Policy,"Amer. Econ. Rev., Mar. 1968, 58, 1-17.D. F. Gordon and A. Hynes, "Onthe Theoryof Price Dynamics," in E. S. Phelps et al.,Micro-economicsof Inflation and Employ-ment Theory, New York 1969.R. E. Lucas, Jr., "Expectationsand the Neu-

    trality of Money," J. Econ. Theor., Apr.1972, 4, 103-24., "EconometricTesting of the NaturalRate Hypothesis," Conferenceon the Econ-ometrics of Price Determination, Washing-ton 1972, 50-59.and L. A. Rapping, "Real Wages,Employmentand the Price Level," J. Polit.Econ., Sept./Oct. 1969, 77, 721-54._ and , "Unemployment in theGreat Depression: Is There a Full Expla-nation?",J. Polit. Econ.,Jan./Feb. 1972, 80,186-91.J. F. Muth, "Rational Expectations and theTheory of Price Movements," Economet-rica, July 1961, 29, 315-35.E. S. Phelps, introductory chapter in E. S.Phelps et al., Micro-economicsof Inflationand Employment Theory, New York 1969.F. Raines, "MacroeconomicDemand and Sup-ply: an Integrative Approach," Washing-ton Univ. working paper, Apr. 1971.A. Rees, "OnEquilibrium in LaborMarkets,"J. Polit. Econ., Mar./Apr. 1970, 78, 306-10.P. Taubman and M. Wilkinson, "User Cost,Capital Utilization and InvestmentTheory," Int. Econ. Rev., June 1970, 11,209-15.United Nations, Department of Economicand Social Affairs, United Nations Statisti-cal Office, Yearbook of National AccountsStatistics, 66 and 68, New York 1958.