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A Keynesian Macroeconomic Model with New-Classical Econometric Properties* JAMES PEERY COVER Universityof Alabama Tuscaloosa,Alabama I. Introduction The development of New-Classical macroeconomicmodels with policy implications radically differentfrom Keynesianmodels calls for econometrictests which can determinewith which type of model the availabledatais most consistent.At least two types of tests havebeen proposed.Each one is based on the presumptionthatNew-Classicalmodels implycertainrestrictionson estimated coefficients in reduced-formequationsthatare not impliedby Keynesianmodels. The purposeof this paper is to demonstratethatthereis a plausibleKeynesianmodel which is consistentwith two testable restrictionsimplied by New-Classicalmodels.' The two restrictionsare:(1) the argument that in New-Classical models the nominalquantityof money and othernominal variablesdo not Granger-causeoutput and other real variables, while this is not the case in Keynesianmodels;2 and (2) the argumentthat New-Classical models imply cross-equationconstraintswhich do not hold in Keynesian models.3 The results of this paper are importantfor two reasons. Firstly, proponents[10, 401; 17, 403] of the above-mentionedtests believe that the rejectionof the "New-Classical"restrictions is not evidence against the New-Classical structurein favorof the Keynesianstructure.That is, there may be New-Classical models in which the restrictionsdo not apply. But at the same time McCallum[110]and Sargent[17] both believe thatacceptanceof these restrictionsis clear evidence in favor of the New-Classical structurebecause such results, "would be very difficultto explain accordingto Keynesianmacroeconomicmodels" [17, 403]. *The author is grateful to the University of Alabama ResearchGrantsCommitteefor a grant-in-aidreceived in supportof this research. The authorthanksDavid Schutte, MatthewJ. Cushingand Donald Hooks for helpful comments on an earlier draft. The authoris responsiblefor all remainingerrorsand omissions. 1. In addition to the two restrictionsdiscussed below one may be temptedto bring up the sort of restrictionemphasized by Barro [1; 2] in which he estimates the expected money supply and tests the hypotheses that the expected component of the money supply does not affect output, while the unexpectedpart does. However, by examining how Barro'sresults differfrom those of Mishkin [12; 13], and McCallum's[7] analysisandresults,it is obvious thatthe results of tests of this restrictionare extremely sensitive to the mannerin which expected money is defined. (To be more to the point, and more in line with McCallum's analysis, the results are sensitive to the variablesincluded in the information set.) The two tests emphasized below do not suffer from this problem. Furthermore,such a restrictionis in principle consistent with the restrictionthat nominal variablesdo not Granger-causereal variables. 2. See Sargent [15; 16; 17] and McCallum[9] for attemptsto implementthis test and argumentsfor its use. 3. See McCallum [10] for argumentsfor the use of this test. 831 832 James Peery Cover Secondly, proponents[6] of the increasinglypopular"real"theoriesof businesscycles have used the failure of money to Granger-causeoutputas evidence in favorof such real theories and as evidence against alternatetheories. Below it is demonstratedthat it is actuallyquite easy to derive these testable restrictionsfrom a Keynesianmacroeconomicmodel. The Keynesian model presented below produces results different from other Keynesian models because it presumes that monetarypolicy is implementedin a manner which causes expected aggregate demand to equal expected aggregatesupply.In a disequilibriumKeynesian model aggregate demand can differ from aggregatesupply because prices do not adjustrapidly enough to equate the two continuously.4Keynesianspresumethatin the real world the Walrasian auctioneerdoes not call out arraysof prices while time standsstill. Rathereconomic actors must consciously decide whether to bid prices upwardsor downwards.This takes time! Even if expectations are rational, there is no guaranteethat marketswill continuouslyclear. However, in such a world, if policy-makershave a workingknowledge of how prices are adjustingtowards the equilibriumprice level, then they may be able to implementa monetarypolicy that causes the expected value of aggregate demand to equal the expected value of aggregate supply. If disequilibriumsbetween aggregate demand and aggregatesupply are responsible for important fluctuationsin output, then in order for a policy to be optimal it must cause expected aggregate demand to equal expected aggregatesupply. If it is assumed that the policy implementedin a Keynesian model is one which causes expected aggregate demand to equal expected aggregate supply, then any differencesbetween realized aggregatedemandand realizedaggregatesupplyarerandom.Eventhoughpolicy is helping to keep the economy close to equilibrium,and is thereforeaffectingoutput, econometrically it appearsthat policy is having no effect on output;that nominalvariablesdo not Granger-cause real variables;and thatthe cross-equationconstraintsassociatedwith New-Classicalmodels hold. Considerationof this type of policy within a Keynesianmodel has at least one otherinteresting implication-the findingthatthe Lucas aggregate-supplyequationis not necessaryin orderto find econometricallythat unexpectedchanges in the money supply affect output, while expected changes do not. The reader who believes that considerationof such a policy is not empiricallyrelevant is urged to withholdjudgmentuntil readingthe final section. Section II presents a standardNew-Classical model and demonstrateshow it implies the above two testable restrictions. Section III introduces a form of price stickiness and explains why it is generally believed that Keynesianmodels are inconsistentwith the above two testable restrictions. Section IV changes the natureof monetarypolicy in the Keynesianmodel from an arbitraryfeedback rule to the policy that causes expected aggregatedemandto equal expected aggregate supply. It is then demonstratedthat in a Keynesianmodel in which policy makers implement this sort of optimal policy, the above two testable restrictionsalso hold. Section V points out that the Lucas aggregate-supplyequationis unnecessaryfor these results, while section VI offers some supportingempiricalevidence and a conclusion. 4. For this view of Keynesianeconomics see Patinkin[14, 313-348], Clower[4], and Barroand Grossman[3]. Note that the argumentfor price-stickinessemployedby Patinkinand used here does not dependupon the existence of long-termcontracts,ratherit rests on the presumptionthatthe Walrasianauctioneer,i.e., the free, perfectlycompetitive market,does not adjustwhile time standsstill. Whetherone accepts this argumentor not does not affectthe result of this paper, which is that proposed empiricaltests are not capableof distinguishingNew-Classicalmodels from a model with price-level stickiness. A KEYNESIAN MODELWITH NEW-CLASSICALPROPERTIES 833 II. A New-Classical Model The New-Classical model employed here is a variantof those employedby Sargent[15], Sargent and Wallace [18], and McCallum [11]. It consists of the followingequations: Yt = bo + birt - blEt-l(Pt+l mt - pt = co + cirt + C2Yt + "t, -Pt) + Yt, < 0; bl < > C1 O, C2 0; Yt = ao + alyt-1 + a2(Pt - Et-lpt) + ut, mt = xo xlmt-1 - x2mt-2 - x3Yt-1 I t, (1) (2) al, a2 > 0; (3) Xi > 0. (4) Yt, m,, and Pt respectively are the logrithmsof output, the nominalquantityof money, and the price level, while r, is the nominal rate of interest. y,, e,, rt, and ut are serially and mutually uncorrelateddisturbances.Et- denotes the mathematicalexpectationof the variableon which it operates, conditional on informationavailableat the end of period t - 1. Equation(1) is an IS curve; equation (2) is an LM curve; equation(3) is a Lucas-typeaggregatesupply equation; and equation (4) is a money-supply feedback rule. Although one can quarrelwith the above specification, reasonable modificationsdo not affect the thesis of this paper (which is to show that a particular,reasonableKeynesianmodel implies the same testableconstraintsas the above model). The solution for output in the above model is: y, = ao + aly,-1 + [azcy, + a2b!(e, - i,) + bIu,]B, (5) where B = bl + cla2 + czbla2 < 0. If equation(4) is solved for E, and the result is insertedinto (5), the result is y,=ao - (az2bxo B) + [a, + (a2bix3 B)]y,-l+ + + + B. [a2zcy - az2br1, blu,]l x2zm,-2 (az2bIB)[m, -+ x1m,-1 (6) Equation (5) clearly implies that the above model is consistent with the proposition that only unexpected changes in the money supply affect the level of output.This follows because the disturbance,c,, is the only term from the money-supplyfeedbackrule that appearsin equation (5). Equation(5) also implies thatoutput,y,, is not Granger-causedby the money supplyand the price level. This follows because E,, y,,r9,. and u, are orthogonalto past values of the money supply and the price level. This is the firsttestableimplicationof the New-Classicalmodel. Equations(6) and (4) togetherimply thatthe estimatedcoefficientson past values of money in a regressionequationfor outputare proportionalto those in a regressionequationfor money. Here the proportionis (a2b /B). This simple-proportionality, cross-equationconstraintis the second testable implicationof the above model. III. A Keynesian Model Inconsistent with the Two Restrictions One assumptionimplicit in the above model is thatthe pricelevel adjustsso thataggregatedemand always equals aggregate supply. Many Keynesianstraditionallyhave assumedthat prices adjust 834 James Peery Cover price level aggregate supply t-1 AD' t= p* + (1- )t1 ADt- 1 AD output Yt Y Figure 1 sluggishly. Following McCallum [8], one way in which the abovemodel can be modified so that it is in principleconsistent with this Keynesianpresumptionis to assumethatPt* is the price level at which aggregate demand equals aggregatesupply,but the price level which actuallyprevails duringperiod t is defined by5 Pt = Ap* + (1 - A)pt-1, O < A < 1. (7) then it is assumed output equals If aggregate demand is less than aggregate supply (Pt > Pt), is demand. If demand than or equal to aggregatesupply (Pt < Pt), aggregate greater aggregate then it is assumedthat outputequals aggregatesupply. Figure 1 illustratesthe workingsof this price-adjustmentequationin a model with a fixed, vertical, aggregate-supplycurve. Suppose thatduringperiod t - 1 the economy is in equilibrium at price level Pt-1 and outputy*. If the long-run,aggregate-supplycurve is fixed over time at y *, and there is a decrease in aggregatedemandfromADt-1 to ADt duringperiod t, outputdeclines If the aggregatedemandcurve to yt because the price level does not decrease all the way to pt*. futureperiods, then the price remainsat ADt and the aggregatesupplycurveremainsat y* during level gradually declines toward and output graduallyincreasestowardy*. (If the aggregate pt*period t, then outputwould remainat y*, while the price level demandcurve shifts to AD ' during graduallyincreases to p'.) As the above discussion implies, and has been rigorouslydemonstratedby McCallum[8], if outputalways equals aggregatesupply,then the testablerestrictionsof New-Classicalmodels will continue to obtain. But it should be obvious that such restrictionswill not hold in general under the Keynesian assumptionthat outputequals aggregatedemand. However,as is demonstratedin the next section, undera reasonablyoptimalpolicy the restrictionscontinueto hold even if output equals aggregatedemand. 5. The particulartype of price stickiness that exists is not importantfor the results of this paper, so long as it is a type that allows policy to affect aggregatedemand. The resultsbelow differfrom McCallum's[8] because he assumes output always equals aggregate supply, while here it is assumedthat if aggregatedemandis less than aggregate supply, then outputequals aggregatedemand. McCallum'sassumptionis impossiblein an economyin which a large partof output consists of services. Furthermore,the assumptionthat output equals aggregatedemandwheneveraggregatedemand is less than aggregate supply is consistent with Patinkin[14], Barroand Grossman[3], and many textbook treatmentsof Keynesianmacroeconomics. A KEYNESIAN MODELWITH NEW-CLASSICALPROPERTIES 835 price level s Y -1 E ys t -1it -E Ep=P p t-ltt I t-- I t -1 I EYt =a01 E t-i output ot - Figure 2 IV. Rational Policy in a Keynesian Model In New-Classical models demand-sidedisturbancescause fluctuationsin outputonly because of the effects on aggregate supply of unexpectedchanges in the price level. DisequilibriumKeynesians in principle need not disagree with the existence of such an effect. Be that as it may, disequilibriumKeynesians emphasize that demand-sidedisturbancesmay affect output because of price-level stickiness. In particular,if there is a decrease in aggregate demand, price-level stickiness prevents the price level from decliningto the level at which aggregatedemandequals aggregate supply. As a result, outputdeclines to the level of aggregatedemand.6 If price-level stickiness is a majorcause of outputfluctuations,then a rationalpolicy maker would implement policy in a manner such that there is no need for the price level to change. (The exact policy depends on the type of price stickinessthat exists. For example, if the rate of inflation is sticky, then the proper policy is one which does not requirethe rate of inflation to change.) Consider the model representedby equations(1)-(3) and (7). In Figure 2 the curve labeled demand duringperiod t - 1, while the curve labeled Yf_1 represents yt-1 representsaggregate aggregate supply during period t - 1. It is assumed for purposes of discussion that aggregate demand equals aggregate supply duringperiod t - 1, althoughin general this is not necessarily the case in a Keynesian model. Althoughpolicy-makerscannotpreventtherefrombeing a disequilibriumbetween aggregate demand and aggregate supply during period t, they can implementa policy which causes the expected value of aggregate demand to equal the expected value of aggregate supply. But as long as (7) holds, this can only be the case if the expected value of the equilibriumprice level, is Hence, in Figure 2 the optimal policy, or the policy which minimizes the expected pt*, Pt-1. disequilibriumbetween aggregatedemandand aggregatesupplyis one which causesthe expected, aggregate-demandcurve for period t, Et-lyd, to intersectthe expected, aggregate-supplycurve, and (7), an Et-lYt, at Pt-1. It is concluded that in the model consisting of equations (1)-(3) optimal policy must be one which causes Et- iP = Pt-1. If economic actors are certainthatsuch a policy is going to be maintained,then the expected rate of inflation is zero, or Et -l (Pt +1 - Pt) = 0. Hence the model becomes 6. See Patinkin [14, 316-24] and the above discussionof Figure 1. 836 James Peery Cover Pt = Apt* + (1 - (7) A)pt,-1; td = bo + birt + Yt; (8) mt - Pt = co + Clrt + C2yt + 71t; = ao + alyt-1 + a2(pt + ut; Et-lPt) yt Et- IPt = Et- Ip; = Pt -1. (9) (10) (11) In equation (9) Yt equals yd iif outputequals aggregatedemand;while it equals if output yt equals aggregate supply. There are several ways that policy-makerscan insure that (11) holds. Since for present purposes it does not matterhow this is done, it is assumedthat the monetary authoritytries to set the money supply at the level which causes (11) to hold. Applying the expectationsoperatorto (9) and rearrangingyields Et-imt = Et-lp, + co + clEt,-rt + C2Et-lYt. (12) Applying the expectationsoperatorto (8) and solving for Et - Irt yields Et-lrt = -(bo/bl) + (13) (1/bl)Et-ly,. Substituting(13) into (12) yields Et-lmt = Et-lpt + [(cobl Recall that so long as Et -Pt + [(C2bl + cl)/bl]Et-lyt. bocl)/b,] = Pt-1, then Et -I y = stitutionsyields (14) = ao + alyt-1. Making these subEt-lyt Et-imt = [c2ao + co + cl(ao - bo)/bl] + pt-1 + (al/bi)(c2bl + cl)yt-1. (15) Equation (15) implies that the following money-supplyrule will cause (11) to hold and cause expected aggregatedemandto equal expected aggregatesupply: mt = [c2ao + co + cl(ao - bo)/bl] + pt-1 + (al/bi)(c2bl + cl)yt-1 + Et. (16) If the model consisting of equations(7)-(10) and (16) is solved, the solutions for the price level, aggregatedemand and aggregatesupplyare as follows: + + (A/B)[city bi(et + + Yd =ao alyt-1 (blA/B)ut + Pt =Pt- t) - (c1 + C2bi)ut]; {[az(cl + C2bl) + bil(1 - A)][bl(et - rt,) + clyt]/[B(cl YF=ao + alyt-1 (17) + Czbl)]}; (18) + {Aa2Cly, + Aa2bl(et - rlt)+ [az(1 - A)(Cl + C2bi) + bi } ]ut /B; (19) where B = bl + a2cl + a2c2bl < 0. Notice from equations (17)-(19) that the conditionalexpectationof the price level equals Pt-1, while the conditional expectationsof both aggregatedemandand aggregatesupply equal These results imply that, under the policy ao + alyt -1. Finally, note that if A = 1, then y/ = yts. A KEYNESIAN MODEL WITH NEW-CLASSICAL PROPERTIES 837 considered here, output is not Granger-causedby any nominal variables, even though output equals aggregate demand. Even though policy clearly affects outputwheneveroutputequals aggregate demand, econometricallyit appearsas if policy does not affect output.These results also imply that only unexpectedchanges in the money supply affectoutputin this Keynesianmodel. Do the simple-proportionality,cross-equationconstraintsimplied by New-Classicalmodels apply in this particularKeynesian model? Noting that e, in equation(18) may be replaced by m, - Et,- mt, it is clear that (18) implies that if one determinesE,-lm, by a linear, least-squares projectionof m, on its past values, then the coefficientson the past value of mt in the regression equation, n Yt = < i=0 + ?t, -imt-i are proportionateto those on past values of mt in a linear, least-squaresprojectionof mt on its past values, or proportionateto the coefficientsin n mt= I3imt-1 + Et. i=1 V. Wither the Lucas Aggregate-Supply Equation? One interesting implication of the above discussion is that it implies that the Lucas aggregatesupply equation-an aggregate-supplyequationthat implies expectationalerrorsare responsible for fluctuationsin aggregatesupply-is not necessaryfor unexpectedchangesin the money supply and the price level to have an econometriceffect on outputthat differs from expected changes. For note that in the Keynesianmodel discussed above in section IV if the equationfor aggregate supply is changed to YS = ao + aly,-1 + (20) ut, then the solution for aggregatedemandbecomes, ytd = ao + alyt-1 + Aut + (1 - A)[bl(et - t) + Clyt]/(cl + C2bi). (21) According to (21) unexpected changes in the money supply, e,, affect output if output equals aggregate demand. Therefore, empiricalevidence that unexpectedchanges in the money supply have an effect on output different from the effect of expected changes, or evidence that only unexpected changes in money have an effect on outputdo not provideunambiguoussupportfor the Lucas aggregate-supplyequation.That is, such evidence in generalcannotdistinguisha NewClassical model with a Lucas aggregate-supplyequationfrom a Keynesianmodel withouta Lucas aggregate-supplyequation. VI. Relevance of the Results The above demonstratesthat there is a Keynesianmodel with econometricpropertiessimilar to those of New-Classical models. In particular,in the Keynesianmodelpresentedhere nominalvari- 838 James Peery Cover ables do not Granger-causereal variablesand the simple-proportionality, cross-equationconstraint associated with New-Classical models holds. The importanceof this finding obviously depends upon the plausibilityof this Keynesianmodel-which is defendedin the followingparagraphs. As is pointed out in the above, the key featureof the Keynesianmodel thathas econometric propertiessimilar to New-Classical models is that policy-makersimplementmonetarypolicy in a mannerthat causes expected aggregatedemandto equal expected aggregatesupply. If policymakersreally do believe thatdisequilibriumsbetweenaggregatedemandand aggregatesupplyare an importantcause of fluctuationsin output,then it would seem thatrationalpolicy-makerswould implementpolicy in a mannerthat achievesthis result. This in itself makesthe model plausible. But is there any reason to believe that such a policy has been applied?In other words, are the results of this paper applicableto U.S. time-seriesdata? It is maintainedhere that they are because of the results of a formal statisticaltest implemented below. Considerthe generalpartial-adjustment specification n Pt = AoPt*+ i=1 AiPt-1i; (22) wherept* is the price level at which aggregatedemandequals aggregatesupply and (Ao + A + ... An) = 1. In orderfor expected aggregatedemandto equal expected aggregatesupply,then it must be thatEt,-IPt = Et-l p*. If this conditionis imposed on (22) then it becomes n Pt = (1/(1 -Ao)) i=1 AiPt-. (23) (23) implies that the price level follows a nonstationaryAR process with no drift. Although there may be a numberof reasonswhy the price level mightfollow a nonstationary stochastic process, the discussion does point to a reasonabletest of whether policy has been implemented in a manner that causes expected aggregatedemand to equal expected aggregate supply. To get at such a test define the rate of inflationto be 7Tt = Pt - Pt--. Now assume that price stickiness takes the form of a partialadjustmentof the rate of inflationwith an MA error term, or 7Tt = AT* + (1 - A)7rt_1 + +t -+ •t-1; (24) where 7ri = p t*- P -1; and rt is a white noise disturbance.If policy makersimplementpolicy so that Et,- Pt = Et,- Pt*, then t =7T-i + (8/(1- A)),-1. E,-17 (25) Equation(25) implies that the rate of inflationfollows an (1,0,1) process with unit root and no constant, clearly a testable hypothesis. A standardtime-series analysis of the quarterlyrate of inflationfor 1967.1 to 1986.2 suggests that it does indeed follow a (1,0,1) process. The fitted equationis rt, = .002 + .87wr-1 + s, .31sr_1; (.002) (.11) (.16) (26) where standarderrorsare in parentheses.In (26) not only is the constantnot significantlydifferent from zero, but also the coefficient on lagged inflationis not significantlydifferentfrom one. The A KEYNESIAN MODELWITH NEW-CLASSICALPROPERTIES 839 F-statistic of the joint null hypothesisis only 1.85, hence it is concludedthatone cannotreject the joint hypothesis that during 1967.1-1986.2 the price-level exhibitedstickiness of the form (24) and monetaryauthoritiesimplementedpolicy in a mannerthatcausedexpectedaggregatedemand to equal expected aggregate supply.7 7. This is the case whether one uses standardtables of the F- and t-distributions,or the empiricaltables found in Dickey and Fuller [5]. References 1. Barro, RobertJ., "UnanticipatedMoney GrowthandUnemploymentin the UnitedStates."AmericanEconomic Review, March 1977, 101-15. 2. , "UnanticipatedMoney, Output,andthe PriceLevel in the UnitedStates."Journalof PoliticalEconomy, August 1978, 549-80. 3. and Herschel I. Grossman, "A GeneralDisequilibriumModel of Income and Employment."American EconomicReview, March 1971, 82-93. A TheoreticalAppraisal,"in The Theoryof Interest 4. Clower, Robert W. "The KeynesianCounter-Revolution: Rates, edited by F Hahn and F Brechling. London:McMillan, 1965. 5. Dickey, David A. and Wayne A. Fuller, "LikelihoodRatio Statisticsfor AutoregressiveTime Series with Unit Root." Econometrica,July 1981, 1057-72. 6. Eichenbaum,Martinand KennethJ. Singleton. "Do EquilibriumBusiness-CycleModels ExplainPostwarU.S. Business Cycles?" in MacroeconomicsAnnual: 1986, edited by StanleyFischer.Cambridge,Mass.: MIT Press, 1986. 7. McCallum, Bennett T., "RationalExpectationsand the NaturalRate Hypothesis:Some ConsistentEstimates." Econometrica,January1976, 43-52. 8. , "Price-Level Adjustmentsand the RationalExpectationsApproachto MacroeconomicStabilization Policy." Journalof Money, Credit, and Banking,November 1978, 418-36. 9. , "Monetarism,RationalExpectations,OligopolisticPricing, and the MPS EconometricModel." Journal of Political Economy, February1979, 57-73. 10. , "On the ObservationalInequivalenceof Classical and KeynesianModels." Journalof Political Economy, April 1979, 395-402. 11. , "Price-Level Determinacywith an Interest-RatePolicy Rule and RationalExpectations."Journal of MonetaryEconomics, November 1981, 319-29. 12. Mishkin, Fredric S., "Does AnticipatedPolicy Matter?An EconometricInvestigation."Journal of Political Economy, February1982, 22-51. 13. , "Does AnticipatedAggregate DemandPolicy Matter?FurtherEconometricResults."AmericanEconomic Review, September 1982, 788-802. 14. Patinkin, Don. Money, Interest,and Prices, 2nd edition. New York:Harperand Row, 1965. 15. Sargent,ThomasJ., "RationalExpectations,the Real Rateof Interest,andthe NaturalRateof Unemployment." BrookingsPapers on EconomicActivity, 2, 1973, 429-72. 16. , "A Classical MacroeconometricModel for the United States." Journal of Political Economy, April 1976, 207-37. 17. , "Causality, Exogeneity, and Natural Rate Models: Reply to C. R. Nelson and B. T. McCallum." Journalof Political Economy, April 1979, 403-9. 18. and Neil Wallace, " 'Rational'Expectations,the OptimalMonetaryInstrument,and the OptimalMoney Supply Rule." Journalof PoliticalEconomy, April 1975, 241-54.