Classical Models of Macro Econ

8/22/2019 Classical Models of Macro Econ

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A Keynesian Macroeconomic Model withNew-Classical Econometric Properties*JAMES PEERYCOVERUniversityof AlabamaTuscaloosa,Alabama

I. IntroductionThe developmentof New-Classical macroeconomic models with policy implications radicallydifferent romKeynesianmodels calls foreconometric estswhich candeterminewith whichtypeof model the availabledata s mostconsistent.At least twotypesof tests havebeenproposed.Eachone is based on thepresumptionhatNew-Classicalmodelsimplycertainrestrictions n estimatedcoefficients in reduced-form quations hatarenot impliedby Keynesianmodels. The purposeofthis paperis to demonstrate hatthere s a plausibleKeynesianmodel which is consistentwith twotestable restrictions mpliedby New-Classicalmodels.' The two restrictions re:(1) the argumentthat in New-Classical models the nominalquantityof moneyand othernominal variablesdo notGranger-causeoutputand other real variables,while this is not the case in Keynesianmodels;2and (2) the argumentthat New-Classical models imply cross-equation onstraintswhich do nothold in Keynesianmodels.3The results of this paper are important or two reasons. Firstly,proponents[10, 401; 17,403] of the above-mentioned ests believe thatthe rejectionof the "New-Classical"restrictionsis not evidence againstthe New-Classical structuren favor of the Keynesianstructure.Thatis,theremay be New-Classical models in which the restrictionsdo not apply.But at the same timeMcCallum[110] ndSargent[17]both believe thatacceptanceof these restrictions s clear evidencein favor of the New-Classical structurebecause such results, "wouldbe very difficult o explainaccordingto Keynesianmacroeconomicmodels" [17, 403].

*The authoris gratefulto the Universityof AlabamaResearchGrantsCommittee or a grant-in-aidreceived insupportof this research. The author hanksDavid Schutte,MatthewJ. Cushingand Donald Hooks for helpfulcommentson an earlier draft. The author s responsible or all remaining rrorsand omissions.1. In additionto the two restrictionsdiscussed below one may be temptedto bring up the sort of restrictionem-phasized by Barro [1; 2] in which he estimatesthe expected money supplyand tests the hypothesesthatthe expectedcomponentof the money supply does not affect output, while the unexpectedpartdoes. However,by examining howBarro'sresultsdiffer from those of Mishkin[12; 13], andMcCallum's 7] analysisandresults, t is obviousthat he resultsof tests of this restrictionare extremelysensitive to the manner n whichexpected money is defined. (Tobe moreto thepoint, and more in line with McCallum'sanalysis, the results are sensitive to the variables ncludedin the informationset.) The two tests emphasizedbelow do not suffer from this problem.Furthermore,uch a restriction s in principleconsistent with the restriction hatnominalvariablesdo not Granger-causeeal variables.2. See Sargent[15; 16; 17] and McCallum[9] for attempts o implement histest andarguments or its use.3. See McCallum[10] for arguments or the use of this test.

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832 JamesPeery CoverSecondly,proponents[6] of the increasinglypopular"real" heoriesof businesscycles haveused the failure of money to Granger-cause utputas evidence in favorof such real theories andas evidence against alternate heories. Below it is demonstrated hat it is actually quite easy to

derive these testable restrictions rom a Keynesianmacroeconomicmodel.The Keynesian model presentedbelow produces results different from other Keynesianmodels because it presumes that monetarypolicy is implemented n a manner which causesexpected aggregatedemand to equal expected aggregatesupply.In a disequilibriumKeynesianmodel aggregatedemandcan differ from aggregatesupplybecausepricesdo not adjustrapidlyenoughto equatethe two continuously.4Keynesianspresume hat n the realworldthe Walrasianauctioneerdoes not call out arraysof priceswhile time stands still. Rathereconomic actorsmustconsciously decide whether to bid prices upwardsor downwards.This takes time! Even if ex-pectationsare rational,there is no guarantee hat marketswill continuouslyclear. However, insuch a world, if policy-makershave a working knowledgeof how pricesare adjusting owardsthe equilibriumprice level, then they may be able to implementa monetarypolicy that causesthe expected value of aggregatedemand to equal the expected value of aggregate supply. Ifdisequilibriumsbetween aggregatedemand and aggregatesupplyare responsiblefor importantfluctuations n output, then in order for a policy to be optimalit must cause expected aggregatedemandto equalexpected aggregatesupply.If it is assumed that the policy implemented n a Keynesianmodel is one which causesexpected aggregate demand to equal expected aggregatesupply,then any differencesbetweenrealizedaggregatedemandand realizedaggregate upplyare random.Eventhoughpolicy is help-ing to keep the economy close to equilibrium,and is thereforeaffectingoutput,econometricallyit appearsthatpolicy is havingno effecton output; hat nominalvariablesdo not Granger-causerealvariables;and thatthe cross-equation onstraints ssociatedwith New-Classicalmodels hold.Considerationof this type of policy within a Keynesianmodel hasat leastone other nterest-ing implication-the findingthat the Lucasaggregate-supply quation s not necessary n ordertofind econometrically hatunexpectedchangesin the moneysupplyaffectoutput,while expectedchangesdo not.The reader who believes that considerationof such a policy is not empiricallyrelevant isurgedto withholdjudgmentuntilreading he final section.Section II presents a standardNew-Classicalmodel and demonstrateshow it implies theabove two testable restrictions. Section III introduces a form of price stickiness and explainswhy it is generallybelieved thatKeynesianmodels are inconsistentwith the above two testablerestrictions. Section IV changesthe natureof monetarypolicy in the Keynesianmodel from anarbitrary eedback rule to the policy thatcauses expected aggregatedemand to equal expectedaggregate supply. It is then demonstrated hat in a Keynesianmodel in which policy makersimplement this sort of optimal policy, the above two testable restrictionsalso hold. Section Vpoints out that the Lucasaggregate-supply quation s unnecessaryortheseresults,while sectionVI offers some supportingempiricalevidence and a conclusion.

4. For this view of Keynesianeconomics see Patinkin 14, 313-348], Clower[4], and BarroandGrossman 3].Note thatthe argument or price-stickiness mployedby Patinkinand used here does not dependuponthe existenceof long-termcontracts,rather t restson thepresumptionhat he Walrasian uctioneer,.e., thefree, perfectlycompetitivemarket,does not adjustwhile time stands still. Whetherone acceptsthis argument r not does not affect the result of thispaper,which is thatproposedempiricaltests are not capableof distinguishingNew-Classicalmodelsfrom a model withprice-levelstickiness.


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A KEYNESIAN MODELWITH NEW-CLASSICALPROPERTIES 833II. A New-Classical ModelThe New-Classicalmodel employedhere is a variantof thoseemployedby Sargent[15], Sargentand Wallace [18], andMcCallum[11]. It consists of the following equations:

Yt = bo + birt - blEt-l(Pt+l -Pt) + Yt, bl < 0; (1)mt - pt = co + cirt + C2Yt+ "t, C1< O, C2 > 0; (2)Yt = ao + alyt-1 + a2(Pt - Et-lpt) + ut, al, a2 > 0; (3)

mt = xo - xlmt-1 - x2mt-2 - x3Yt-1 I t, Xi > 0. (4)Yt m,, andPt respectivelyare the logrithmsof output,the nominalquantityof money, and theprice level, while r, is the nominalrate of interest.y,, e,, rt, andut are seriallyand mutuallyuncorrelateddisturbances.Et- denotes the mathematical xpectationof the variableon whichit operates, conditionalon informationavailable at the end of periodt - 1. Equation 1) is anIS curve;equation(2) is an LM curve;equation(3) is a Lucas-typeaggregatesupplyequation;and equation (4) is a money-supplyfeedbackrule. Althoughone can quarrelwith the abovespecification,reasonablemodificationsdo not affect the thesis of this paper(which is to showthata particular,reasonableKeynesianmodel impliesthe same testableconstraintsas the abovemodel).The solution for outputin the abovemodel is:

y, = ao + aly,-1 + [azcy, + a2b!(e, - i,) + bIu,]B, (5)whereB = bl + cla2 + czbla2 < 0. If equation 4) is solved forE, and the resultis inserted nto(5), the result is

y,=ao - (az2bxo B) + [a, + (a2bix3 B)]y,-l+(az2bIB)[m, -+x1m,-1 + x2zm,-2 + [a2zcy - az2br1, + blu,]l B. (6)

Equation(5) clearly implies that the above model is consistent with the propositionthatonly unexpectedchanges in the money supplyaffect the level of output.This follows because thedisturbance,c,, is the only term from the money-supply eedbackrule thatappears n equation(5). Equation 5) also implies thatoutput,y,, is notGranger-caused y the moneysupplyand theprice level. This follows because E,, y,,r9,. andu, areorthogonal o past values of the moneysupplyand the price level. This is the first testable mplicationof the New-Classicalmodel.

Equations 6) and(4) together mplythat he estimated oefficientsonpastvalues of moneyina regressionequationforoutputareproportionalo thosein a regression quation ormoney.Herethe proportion s (a2b /B). This simple-proportionality,ross-equationconstraint s the secondtestableimplicationof the abovemodel.

III. A Keynesian Model Inconsistent with the Two RestrictionsOne assumption mplicitin the above modelis that heprice eveladjusts o thataggregatedemandalways equals aggregatesupply.Many Keynesianstraditionally ave assumed thatprices adjust


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834 JamesPeery Coverprice level

aggregate supply

t-1AD'

t= p*+ (1- )t1 ADt- 1

ADoutput

Yt Y

Figure 1

sluggishly. FollowingMcCallum[8], one way in which the above model canbe modifiedso thatit is in principleconsistentwith this Keynesianpresumptions to assumethatPt*is theprice levelat which aggregatedemandequals aggregatesupply,but the price level which actuallyprevailsduringperiod t is definedby5

Pt = Ap* + (1 - A)pt-1, O< A< 1. (7)If aggregatedemandis less than aggregatesupply(Pt > Pt), then it is assumedoutputequalsaggregatedemand. If aggregatedemand s greater han or equalto aggregatesupply(Pt < Pt),then it is assumedthatoutputequals aggregatesupply.

Figure 1 illustrates he workingsof this price-adjustmentquation n a model with a fixed,vertical,aggregate-supply urve. Supposethatduringperiodt - 1 the economyis in equilibriumat price levelPt-1 andoutputy*. If the long-run,aggregate-supplyurveis fixed over time aty *,andthere is a decrease in aggregatedemand romADt 1 to ADt duringperiodt, outputdeclinesto yt because the price level does not decreaseall the way to pt*.If the aggregatedemandcurveremainsatADt and the aggregatesupplycurve remainsaty* during utureperiods,then the pricelevel graduallydeclines towardpt*and outputgradually ncreases towardy*. (If the aggregatedemandcurveshifts to AD' duringperiodt, thenoutputwould remainaty*, while the pricelevelgradually ncreases to p'.)As the above discussionimplies, and has been rigorouslydemonstrated y McCallum[8], ifoutput always equals aggregatesupply,thenthe testablerestrictions f New-Classicalmodels willcontinue to obtain. But it should be obvious that suchrestrictionswill not hold in generalunderthe Keynesianassumption hatoutput equalsaggregatedemand.However,as is demonstratednthe next section, undera reasonablyoptimalpolicy the restrictions ontinue o hold even if outputequals aggregatedemand.

5. The particular ype of price stickiness that exists is not important or the results of this paper,so long as it isa type that allows policy to affect aggregatedemand. The resultsbelow differfromMcCallum's 8] becausehe assumesoutputalways equals aggregatesupply,while here it is assumed thatif aggregatedemand s less thanaggregate supply,thenoutputequalsaggregatedemand.McCallum'sassumptions impossible n aneconomy n whicha large partof outputconsists of services. Furthermore, he assumption hatoutputequals aggregatedemand wheneveraggregatedemandisless than aggregatesupply is consistentwith Patinkin[14], Barroand Grossman 3], and manytextbook treatmentsofKeynesianmacroeconomics.


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A KEYNESIANMODELWITH NEW-CLASSICALPROPERTIES 835price level

s E ysY -1 t -1it

Ep=PE pEt-ltt t-- I

I t-iI outputt -1 EYt a01 t -

Figure 2IV. Rational Policy in a Keynesian ModelIn New-Classicalmodels demand-sidedisturbances ause fluctuationsn outputonly because ofthe effects on aggregate supply of unexpectedchangesin the price level. DisequilibriumKey-nesians in principle need not disagreewith the existence of such an effect. Be that as it may,disequilibriumKeynesiansemphasizethat demand-sidedisturbancesmay affectoutputbecauseof price-level stickiness. In particular, f there is a decreasein aggregatedemand, price-levelstickinesspreventsthe price level fromdecliningto the level at which aggregatedemandequalsaggregate supply.As a result,outputdeclinesto the level of aggregatedemand.6If price-level stickinessis a majorcauseof outputfluctuations, hen a rationalpolicy makerwould implementpolicy in a mannersuch thatthere is no need for the price level to change.(The exact policy dependson the type of price stickiness thatexists. Forexample, if the rate ofinflationis sticky, then the proper policy is one which does not requirethe rate of inflation tochange.)Considerthe model representedby equations 1)-(3) and(7). In Figure2 the curvelabeledyt-1 representsaggregatedemandduringperiodt - 1, while the curve labeledYf_1representsaggregate supply during period t - 1. It is assumedfor purposesof discussionthat aggregatedemandequals aggregate supplyduringperiodt - 1, although n generalthis is not necessarilythe case in a Keynesianmodel.

Althoughpolicy-makers annotprevent herefrombeinga disequilibrium etweenaggregatedemand and aggregate supply during period t, they can implementa policy which causes theexpected value of aggregatedemandto equal the expected value of aggregatesupply. But aslong as (7) holds, this can only be the case if the expectedvalueof the equilibriumprice level,pt*, is Pt-1. Hence, in Figure2 the optimal policy, or the policy which minimizesthe expecteddisequilibrium etweenaggregatedemandandaggregate upply s one which causestheexpected,aggregate-demand urvefor periodt, Et-lyd, to intersect he expected,aggregate-supplyurve,Et-lYt, at Pt-1. It is concluded that in the model consistingof equations(1)-(3) and (7), anoptimalpolicy must be one which causes Et- iP = Pt 1.If economic actorsare certainthat sucha policy is goingto be maintained, henthe expectedrate 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 Figure1.


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836 James Peery CoverPt = Apt* + (1 - A)pt,-1; (7)td= bo + birt + Yt; (8)

mt - Pt = co +Clrt + C2yt+ 71t; (9)

yt= ao + alyt-1 + a2(pt - Et-lPt) + ut; (10)

Et- IPt = Et- Ip; = Pt-1. (11)In equation(9) Ytequalsyd iifoutputequalsaggregatedemand;while it equalsyt if outputequals aggregate supply.There are severalways thatpolicy-makers an insurethat (11) holds.Since for present purposesit does not matterhow this is done, it is assumedthat the monetary

authority ries to set the moneysupplyat the level which causes(11) to hold.Applyingthe expectationsoperator o (9) andrearrangingields

Et-imt = Et-lp, + co + clEt,-rt + C2Et-lYt. (12)Applyingthe expectationsoperator o (8) andsolvingforEt- Irt yields

Et-lrt = -(bo/bl) + (1/bl)Et-ly,. (13)Substituting 13) into (12) yields

Et-lmt = Et-lpt + [(cobl - bocl)/b,] + [(C2bl + cl)/bl]Et-lyt. (14)Recall that so long as Et -Pt = Pt-1, then Et -Iy = Et-lyt = ao + alyt-1. Making these sub-stitutionsyields

Et-imt = [c2ao + co + cl(ao - bo)/bl] + pt-1 + (al/bi)(c2bl + cl)yt-1. (15)Equation(15) implies that the following money-supply ule will cause (11) to hold and causeexpected aggregatedemandto equalexpected aggregate upply:

mt = [c2ao + co + cl(ao - bo)/bl] + pt-1 + (al/bi)(c2bl + cl)yt-1 + Et. (16)

If the model consistingof equations(7)-(10) and (16) is solved, the solutions for the pricelevel, aggregatedemand andaggregatesupplyare as follows:Pt =Pt- + (A/B)[city + bi(et - t) - (c1 + C2bi)ut]; (17)Yd =ao + alyt-1 + (blA/B)ut +

{[az(cl + C2bl) + bil(1 - A)][bl(et - rt,) + clyt]/[B(cl + Czbl)]}; (18)YF=ao + alyt-1 + {Aa2Cly, + Aa2bl(et - rlt)+ [az(1 - A)(Cl + C2bi)

+ bi]ut } B; (19)where B = bl + a2cl + a2c2bl < 0.Notice from equations(17)-(19) thatthe conditionalexpectationof the price level equalsPt-1, while the conditionalexpectationsof both aggregatedemand and aggregatesupply equalao + alyt -1. Finally, note that if A = 1, then y/ = yts. These results imply that, under the policy


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A KEYNESIAN MODEL WITH NEW-CLASSICAL PROPERTIES 837considered here, output is not Granger-causedby any nominalvariables,even though outputequals aggregatedemand.Even thoughpolicy clearlyaffectsoutputwheneveroutput equals ag-gregatedemand,econometrically t appearsas if policy does not affectoutput.These resultsalsoimply thatonly unexpectedchangesin the money supplyaffectoutput n this Keynesianmodel.Do the simple-proportionality,ross-equation onstraintsmplied by New-Classicalmodelsapply in this particularKeynesianmodel? Noting thate, in equation(18) may be replaced bym, - Et,- mt, it is clearthat(18) impliesthat if one determinesE,-lm, by a linear,least-squaresprojectionof m, on its past values, then the coefficientson the pastvalue of mt in the regressionequation,

nYt =

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Classical Models of Macro Econ