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American Economic Association
Some International Evidence on OutputInflation TradeoffsAuthor(s): Robert E. Lucas, Jr.Source: The American Economic Review, Vol. 63, No. 3 (Jun., 1973), pp. 326334Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1914364
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S o m e International Evidence o nOutputInflationradeofs
By ROBERT E. LUCAS, JR.*This paper reports the results of anempirical study of real outputinflationtradeoffs, based on annual timeseries fromeighteen countries over the years 195167.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 concerned with the questions (i) does thenatural rate theory lead to expressions ofthe outputinflation relationship whichperform satisfactorily in an econometricsense for all, or most, of the countries inthe sample, (ii) what testable restrictionsdoes the theory impose on this relationship, 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 output 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 shortrun 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 coefficients 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 {"longrun"'in the usual distributed 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 suggests 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 aggregative model will be constructed usingthe elements sketched above. Resultsbased on this model are reported in Section II, followed by a discussion and conclusions.
1. An Economic ModelThe general structure of the model developed in this section may be describedvery simply. First, the aggregate pricequantity observations are viewed as intersection points of an aggregatedemand andan aggregate supply schedule. The formeris drawn up under the assumption of acleared money market and represents theoutputprice level relationship implicit in
* Graduate School of Industrial Administration, CarnegieMellon 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: OUTPUTINFLATIONTRADEOFF 327the standardISLM diagram. It is viewedas being shifted by the usual set ofdemandshift variables: monetary andfiscal policies and variation in export demands. The supply schedule is drawn under 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 additionalspecialassumption: 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 accounting 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 subsection 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 general 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 situation 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 suppliers.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, follows the trend line:(2) y.t a+? tThe cyclical component varies with perceived, relative prices and with its ownlagged value:(3) yet(z) y[Pt(z)  E(Pt It(Z))]+ NyC_t1(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 priceoutput relationship from the ISLM 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 verified 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 criticism, we found that this persistence in employmentcannot be fully explained by price expectations behavior. Both these effectsan expectations and a persistence effectwill 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,ti,yc,t2 . While this information doesnot permit exact inferenceof the log of thecurrent generalprice level, Pt, it does determinea "prior"distribution on Pt, common to tradersin all markets. We assumethat this distribution is known to be normal, with mean Pt (dependingin a knownway on the above history) and a constantvariance a2.
Second,we supposethat the actual pricedeviates from the (geometric) economywide average by an amount which is distributed 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 normally distributed,independent of Pt, withmean zero and variance r2. Then the observedprice in z, Pt(z) (in logs) is the sumof independent,normalvariates(4) Pt(z) = Pt + zThe informationIt(z) relevant for estimation 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 distribution 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,t1(z)
Averaging over markets (integrating withrespect to the distribution of z) gives theaggregate supply function:(7) = yYnt G7y(Pt Pt)
+ xLyt1  yn,t1IThe slope of the aggregate supply function (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 certain 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 development 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 demand side.As suggestedearlier,this will be done bypostulating a demandfunction forgoods ofthe form:(8) yt + Pt = xtwhere xt is an exogenous shift variableequal 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 elasticity 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: OUTPUTINFLATIONTRADEOFF 329The relevant history of the economythen consists (at most) of ynt (which fixescalendar time), the demand shifts xt,
xt1,... , and past actual real outputsYt1,Yt2. * 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(Xt1 + 5) + 72Xtl(10) + 7r3Xt2 + ... + ?lytl
+ 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
+  Xt1  (1  X)Y.t1 + OY+X0+ AxtI8 1 +O + 1 + 8 AX
+ Xyt1 + (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 + 7rAXti
Let us review these solutionsfor internalconsistency. Evidently, Pt is normallydistributed about Pt. The conditional variance 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 expectations. In order to avoid the introductionofan additional, spurious "expectations parameter,"one cannot solve for this intersection on a periodbyperiod basis; 4ccordingly, 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 immediate 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 timeseries structure (moments and laggedmoments) of the series Yet nd APt and is thus, in principle, 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 period. We note in particular that this lagpattern may wellproduceperiodsof simultaneous inflation and below average realoutput. Though these periods arise because 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 natural rate of output: the average ate of demand expansion, 5, appears in (11) with acoefficientequal in magnitude to the coefficient 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 depends on "fooling" suppliers (in the senseof subsectionA), one expects that 7rwill belarger the smaller the variance of thedemand shifts. We next develop this implication 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 obtained 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 deviation of output from trend, E(y,,), is invariant 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 observable 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 estimated 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 variations 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 theoutputinflation "tradeoff'' vary acrosscountriesin the way predicted by the naturalrate 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 estimated r values should decline as the sample 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 continuous series are available. The sample could thus bebroadened considerably by use of sources from individual 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 xtyt; yt is the residual from thetrend line yt= a+bt, fit by least squaresfrom the sample

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VOL. 63 NO. 3 LUCAS: OUTPUTINFLATIONTRADEOFF 331TABLE1DESCRIPTIVESTATISTICS,95267
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 output 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 expansive policies of the remaining sixteencountries. But if the sample provides onlytwo "points," they are indeed widelyseparated: the estimated variance of demand in the high inflation countries is onthe order of 10 times that in the stableprice countries.The first three columns of Table 2 summarize 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 outputdeterminingvariables have been omitted from the model. The R s for the inflation 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 coefficients 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 coefficient 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 goodnessoffitstatis
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 coefficients are imposed on (12) can be assessed formally byan approximate Chisquare test. By this measure, theloss is significant at the .05 level for Paraguay only. AsTable 2 shows, however, this test is somewhat deceptive: 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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332 THE AMERICAN ECONOMIC REVIEW JUNE 1973TABLE 2SUMMARY STATISTICS BY COUNTRY, 195367
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 timeseries models.In contrast to these somewhat mixed results, the behavior of the estimated 7rvalues across countries is in striking conformity 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 thisorderofmagnitude effect more sharply,let us examine the complete results for twocountries: the United States and Argen
tina. For the United States, the fitted versions of (11) and (12) are:yct =  .049 + (.910),Axt+ (.887)yc,tl
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,t1
In a stable price country like the UnitedStates, then, policies which increase nomi

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VOL. 63 NO. 3 LUCAS: OUTPUTINFLATION TRADEOFF 333nal income tend to have a large initialeffect on real output, together with a small,positive initial effect on the rate of inflation. Thus the apparent shortterm tradeoff is favorable, as long as it remains unused. In contrast, in a volatile pricecountry like Argentina, nominal incomechanges are associated with equal, contemporaneous 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 reported above is extremely simple, yet I amafraid it may have become obscured by therather special model in which it is embodied. 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 empirically refer to the effects of aggregate demand policies which tend to move inflation rates and output (relative to trend) inthe same direction, or alternatively, unemployment and inflation in oppositedirections. The conventional Phillips curveaccount of this observed comovement 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 misinterpret 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 crossnational comparison of these propositions would seemto be a direct examination of the association of average inflation rates and averageoutput, relative to "normal" or "full employment." Unfortunately, there seems tobe no satisfactory way to measure normaloutput. The deviationfromfittedtrendmethod I have used defines normal outputto be average output. The use of unemployment series suffers from the same difficulty, 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. Fortunately, the existence of a stable tradeoffalso implies a relationship between variances of inflation and output rates, asillustrated in Figure 1. With a stabletradeoff, policies which lead to wide variation 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
000
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 outputinflation association isentirely contemporaneous. In fact, however, it involves lagged effects which makea direct comparison of variances, as justsuggested, difficult in short timeseries.Accordingly, it has been necessary to impose a specific, simple structure on thedata. As we have seen, this structure accounts for output and inflation rate movements 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, 117.D. F. Gordon and A. Hynes, "Onthe Theoryof Price Dynamics," in E. S. Phelps et al.,Microeconomicsof Inflation and Employment Theory, New York 1969.R. E. Lucas, Jr., "Expectationsand the Neu
trality of Money," J. Econ. Theor., Apr.1972, 4, 10324., "EconometricTesting of the NaturalRate Hypothesis," Conferenceon the Econometrics of Price Determination, Washington 1972, 5059.and L. A. Rapping, "Real Wages,Employmentand the Price Level," J. Polit.Econ., Sept./Oct. 1969, 77, 72154._ and , "Unemployment in theGreat Depression: Is There a Full Explanation?",J. Polit. Econ.,Jan./Feb. 1972, 80,18691.J. F. Muth, "Rational Expectations and theTheory of Price Movements," Econometrica, July 1961, 29, 31535.E. S. Phelps, introductory chapter in E. S.Phelps et al., Microeconomicsof Inflationand Employment Theory, New York 1969.F. Raines, "MacroeconomicDemand and Supply: an Integrative Approach," Washington Univ. working paper, Apr. 1971.A. Rees, "OnEquilibrium in LaborMarkets,"J. Polit. Econ., Mar./Apr. 1970, 78, 30610.P. Taubman and M. Wilkinson, "User Cost,Capital Utilization and InvestmentTheory," Int. Econ. Rev., June 1970, 11,20915.United Nations, Department of Economicand Social Affairs, United Nations Statistical Office, Yearbook of National AccountsStatistics, 66 and 68, New York 1958.