Download A Keynesian Macroeconomic Model with New

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].
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