Rational Expectations and the Optimal Money Supply Rule_Sargent_1975

7/27/2019 Rational Expectations and the Optimal Money Supply Rule_Sargent_1975

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"Rational" Expectations, the Optimal Monetary Instrument, and the Optimal Money SupplyRuleAuthor(s): Thomas J. Sargent and Neil WallaceSource: The Journal of Political Economy, Vol. 83, No. 2 (Apr., 1975), pp. 241-254Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1830921 .Accessed: 17/04/2011 19:26

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"Rational" Expectations,the OptimalMonetary nstrument, nd theOptimal Money Supply Rule

ThomasJ.Sargentand NeilWallaceUniversityfMinnesota

Alternativemonetary olicies re analyzed n an ad hoc macroeconomicmodel in which the public's expectations bout prices are rational.The ad hoc model s one in whichthere s long-run eutrality,ince tincorporates heaggregate upply cheduleproposedbyLucas. Follow-ing Poole, thepaperstudieswhether egging he nterest ate orpeggingthe money supplyperiod by period minimizes n ad hoc quadraticloss function.t turns ut thattheprobability istributionfoutput-dispersion s well as mean-is independent f the particulardeter-ministicmoney upplyrule in effect, nd that under an interest aterule the price evel s indeterminate.

This paper analyzes the effects falternative ways ofconducting monetarypolicywithinthe confinesof an ad hoc macroeconomic model. By ad hocwe mean that the model is notderivedfrom consistent etofassumptionsabout individuals' and firms' objective functions and the informationavailable to them. Despite thisdeplorable featureof the model, it closelyresembles themacroeconomic models currentlyn use, which is our excusefor tudying t. Following Poole (1970), we compare two alternative strat-egies available to the monetary authority. One is to peg the interestrateperiod by period, lettingthe supply ofmoney be whatever it must be tosatisfy he demand for t. The other is to set the money supply period byperiod, accepting whatever interest rate equilibrates the system. Westudy the effects f such policies fortwo versions of the model: an auto-regressiveversion in which the public's expectations are assumed formedvia fixed autoregressiveschemes on the variables being forecast, and arational-expectations version in which the public's expectations are

Workon thispaper was supported ytheFederal ReserveBank ofMinneapolis,whichdoes not necessarily ndorse the conclusions. Robert Barro, Milton Friedman,JohnKareken, and Robert E. Lucas made useful omments n an earlierversion f the paper.[Journal f PoliticalEconomy, 975, vol. 83, no. 2](C 1975 by The Universityof Chicago. All rights reserved.24I


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242 JOURNAL OF POLITICAL ECONOMYassumedequal toobjective mathematical) xpectationshatdependon,amongother hings,what s known bout the rulesgoverningmonetaryand fiscal olicy see Muth 1961).

The twoversionshave radicallydifferentolicy mplications.n therational-expectationsersion, a) the probability istributionfoutputis independent fthe deterministic oney upplyrule n effect,b) ifthelossfunctionncludes uadraticterms n theprice evel,then heoptimaldeterministiconey upply ule s thatwhich quatestheexpected alueofnextperiod'spricelevel to the targetvalue, and (c) a unique equi-libriumprice level does not exist f the monetary uthority egs theinterest ate periodby period,regardless fhow itsvalue variesfromperiodto period.None oftheseresults merges rom heautoregressiveversion. t, instead,exhibits ll theusual exploitable radeoffsetweenoutput nd inflation,mplies hatminimization fthe above lossfunctionis a well-defined ontrivial ynamicproblemgivingrise to a uniqueoptimaldeterministiceedbackrule eitherforthemoney upplyor fortheinterest ate,and has a unique period-by-periodquilibriumftheinterest ate s pegged.Thus, in theautoregressiveersion f themodel,whichin principle s merelya variant of Poole's model, whether ninterest atefeedback ule or a money upplyfeedback ule is superiordepends, ust as Poole asserted, n mostof theparameters fthemodelincluding hecovariancematrix fthedisturbances.In the rational-expectationsersionof the model, one deterministicmoney upplyrule s as good as anyother,nsofars concerns heprob-ability istributionfrealoutput. n thisweaksense, n X percent rowthrule forthemoneysupply s optimal n thismodel, from he point ofview ofminimizinghevarianceofrealoutput.Thus,switchingromheassumptionfautoregressivexpectationsothatofrational xpectationshas drastic olicy mplications.n particular,making hatchangetrans-forms hemodel in whichfollowing riedman'sX percentgrowth ulewould ngeneralbe foolishntooneinwhich uch a rule can be defendedas beingthe besttheauthority an do.1. The Ad Hoc ModelWe assume a structureescribed ythefollowingquations:'aggregateupply chedule:

Yt = arkt-1+ a2(p - p*) ? u1t, ai > O, = 1, 2; (1)aggregate emandscheduleor "IS" curve:Yt = b1kt-1 b2[rt (t+lP*-1- tI-p)] (2)+ b3Zt + U2 , b1 > O, b2 < ?;

1 The structureloselyresembles hemodelused by Sargent 1973).


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"RATIONAL EXPECTATIONS 243portfolioalance or "LM" schedule:

Mt= Pt + clYt+ C2rt U3 c1 > 0, c2 < 0; (3)determinationfproductive apacity:kt= d1kt-1 d2[rt (t+lP*-I - tp*-)] + d3Zt+ u4t, d2 < 0; (4)evolution ftheexogenous ariables:

qZt= E PZt + a,j=1 (5)q

Uit E Pijuist-j XiHerey, Pt,and mt re thenatural ogarithmsfoutput, heprice evel,and themoney upply, espectively;t s the nominal ateof nteresttself(not its logarithm);whileZt is the vectorofexogenousvariables. Thevariable t 1P*-1s the public's psychological xpectation f the log oftheprice evel to prevailat t + i, theexpectation eingheld as oftheend ofperiod - j. The variablekt 1 sa measure fproductiveapacity,such as the logarithm f the stockof capital or labor or some linearcombination fthe ogarithms f those tocks t the end ofperiodt - 1.Equation (1) is an aggregate upply chedulerelating utputdirectlytoproductive apacityand thegap between hecurrent rice evel andthe public's prior expectation f the currentprice level. Unexpectedincreasesn theprice evel thusboostaggregate upply, he reasonbeingthat uppliers f abor and goodsmistakenlynterpreturprisencreasesin theaggregate rice evel as increasesn the relative rices fthe aborand goods they are supplying. his happens because suppliers eceiveinformationbout thepricesoftheir wngoodsfaster hantheyreceiveinformationbout theaggregate rice evel. This is the kindofaggregatesupply chedule thatLucas (1973) has used to explainthe inverse or-relationbetween bserved nflationnd unemploymentepictedby thePhillips urve.Equation (2) is an aggregatedemand or "IS" scheduleshowing hedependence f ggregate emandonthereal rateof nterestndcapacity,a measureofwealth.The real rate of nterestquals the nominalratertminustherate of nflation etween and t + 1 expectedbythepublicas oftheend ofperiodt - 1, namely,t+,P~t - tP*-IIThe ratert sassumedto be theyieldto maturityn a one-periodbond. Aggregatedemand lsodepends na vector f xogenous ariablesZtwhichncludesgovernmentxpendituresnd taxrates.Equation (3) summarizeshe condition orportfolio alance. Ownersofbonds nd equities assumedtobe viewed s perfectubstitutesor neanother) resatisfied ith hedivision ftheir ortfoliosetweenmoney,


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244 JOURNAL OF POLITICAL ECONOMYon the one hand, and bonds and equities, on the otherhand, whenequation (3) is satisfied. quation (3) posits that the demand forrealbalancesdependsdirectly n real incomeand inverselyn thenominalrate of nterest.Equation (4) determinesroductive apacityfor he nextperiod,whileequation 5) describesutoregressiverocesses or he xogenous ariables.The 4's, whichare sometimes alled the "innovations" n theZ and uprocesses,re serially ncorrelatedandomvariables.To completethe model, we need equationsdescribing+lpt*-1 ndtP'-. Addingthose quations o (1)-(5) thenresultsn a systemapableofdetermining he evolutionover time ofyt,Pt,rt, P and tPt-and kt.2. The Stabilization ProblemIn order o discuss olicywithin hecontext f n ad hocmodel,we mustadopt an ad hoc loss function. he mostfamiliar uch function s thequadratic oss function

L = E E 6` [(Ytpt)K(YtPt)/ + (yt)py)( 1,K2)' + 4K + 4whereK is diagonal with elementsKii > 0, i = 1, 2; K1 and K2 areparameters; nd 0 < 6 < 1. This functions separatelyquadratic iny andp and implies hat L = 0, its owerbound,at particular onstantvaluesofyandp, thetarget alues -K1,/2K1,1or and -K2/2K22for .This functions easy toworkwithbecause it is quadratic, dditiveovertime, nd stationaryn that the functionfyandp whoseexpectationsto be minimized oes notdependon t.To minimize L, the monetary uthority ompares two strategies.The firsts topeg rt ia a deterministicinearfeedback ule

rt= GOt*, (6)whereO*representshesetofcurrent ndpastvaluesof all oftheendog-enous and exogenousvariables n thesystems ofthe end ofperiodt,and G is a vectorof parameters onformable o O*- The monetaryauthorityhooses heparametersn G to minimize . It mustthencom-pare theminimumossassociatedwith n interestaterulehavingthoseG's with he ossassociatedwith hebestmoney upplyfeedback uleoftheform mt= HOl (7)Whichever ule delivers he ower oss s the one that houldbe used.


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"RATIONAL EXPECTATIONS 2453. The Autoregressive Expectations VersionHere we assume that the psychological expectations p* I and t+lPt*-are governed by the distributed-lagor "adaptive" schemes

qt+ lPt A= - i (8)i=O

qt+21t = E 2i~t-i, (9)i=o

where the v 's and 2i5s are fixednumbers. Given that the money supplyis used as the monetary instrument,the system formed by equations(1)-(5), (8), and (9) can be reduced to a difference quation ofthe formq, q,Yl= Aiylt-i + Bimt-i + Olt) (10)i~~l ~i=O

where Y't = (y, Pt, rt,kt-1, Zt) and 4,t is a vector of serially un-correlated random variables, the components of which are functions ofthe It's in equations (5). The Ai's are vectors conformable with Y,, andthe B 's are scalars; both the A 's and B 's depend on the parameters ofequations (1)-(5), (8), and (9). To findthe best money-supplyfeedbackrule, the monetaryauthoritychooses the parameters H of the rule (7) tominimizethe loss L subject to (10). Where loss is quadratic and themodelis linearwithknowncoefficients,ules ofthe linear formof (7) are knownto be optimal.To findthe optimal interestrate rule, the systemformedby equations(1)-(5), (8), and (9) is writtenas

q, q,Y2= E CiY2ti + E Dirt-i + 02tn (11)i=l i=1

where Y2t = (yt,pt, mt,kt1, Zt). The optimal interestrate rule is theone with the G's of (6) chosen so as to minimize loss L subject to (11).3To show that (1)-(5), (8), and (9) yield versions of (10) and (1 1) thatgive rise to well-defined, nontrivial dynamic problems, it is enough toexamine the one-period reduced formsforytand Pt.With the money supply as the monetary instrument,we solve (1)-(3)fory,r, and p and get as a reduced formforptPt = Jo0(tt-l) + J(tQ+lPt*-1) + J2mt+ X, (12)

2 See Chow (1970).3 Chow (1970) describes owoptimalrulesofthe form6) or (7) are foundfor systemlike (10) or (11).


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246 JOURNAL OF POLITICAL ECONOMYwhereXt is a linearfunctioninvolving he parameters f [l]-[3]) ofkt-1,Zt, nd theuit's nd where

JO= [a2(1 + b2cI') + b2]/[a2(1+ b2cI') + b2cI'] < 1,J, = (1 - Jo)/(0 - C2'),J2 = -c_ 1Jl.

SubstitutionfPtfrom quation (12) into equation (1) gives the one-periodreducedform or t.TakingEt 1 ofPt ndytfrom hesereducedforms ndeliminatingmt ives he etofpairs Et-1yt,Et-1pt)attainablebychoiceofmt.The set s a line whoseslope s neithernfinityorzero.Itsposition, bviously, ependson laggedvaluesofp,via thep* variables,and on laggedvaluesofother ndogenous ariables, hedistributionsfwhichdependon laggedvalues of m. n otherwords, hechoice for hedeterministicartofmthas effectsn future eriods,which s whatwemeanwhenwe saythat 10) givesrise to a nontrivialynamicproblem.With the nterest ate as themonetarynstrument,quation (2) is theone-period educedform oryt hile hatfor t sobtainedbysubstitutingthe solution or t intoequation (1) and solving orPt.The solutionforPtisa2Pt= (a2 + b2)tp*_- b2(t?1p*_) + b2rt (13)+ (b, - a,)kt- + b3Zt - u"t + U2t.

Again, fwe takeEt 1 ofequation 2) and equation 13) and eliminate t,we find he setofpairs Et -1Yt,Etp- t) attainableby choiceofrt.Thatsetagaindependson laggedvaluesofp, which hows hat 11) also givesrisetoa nontrivial ynamicproblem.The monetaryuthoritys supposedto solve each of the twodynamicproblems,minimizingoss first nder n mrule and thenunder n rrule.Which policyis superiordepends on whichdeliversthe smaller oss,which n turndepends on all of theparameters f themodel, ncludingthe covariance matrix of the disturbances.Which rule is superior sthereforen empiricalmatter,n outcomewhich scompletelyonsistentwithPoole'sanalysis.4. The Rational-Expectations Version under a MoneySupply RuleHereweimpose herequirementhat hepublic'sexpectationse rationalbyrequiringhat = E (14)whereEt jpt+ is themathematical xpectation fPt+i calculatedusingthemodel (i.e., theprobability istributionfPt+ and all information


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"RATIONAL EXPECTATIONS 247assumedto be available as of the end of periodt - j. The availableinformations assumed to consist f data on current nd past values ofall endogenous nd exogenous ariablesobserved s of the end of periodt -j, that s, t*-j.To begin,we again solvethe system 1)-(3) fory, r,and p givenm.Withexpectations iven by (14), what is now a pseudo-reduced-form.equationfor is

Pt = JoEt-lpt + JEt-lpt+l + J2mt + Xt. (15)ComputingtIp tfrom 15) and subtractingheresult rom 15) we get

Pt - Et-lt = J2(mt Et1mt) + Xt - EtXt (16)= Xt - Et-1Xt,

wherethe ast equalityfollows rom he assumption hata deterministicrule of theform 7) is beingfollowed. utsinceXt - Et_1Xts a linearcombinationfthe nnovationsn theexogenous rocesses,tfollowshatPt - Et-pt is an exogenousprocess,unaffectedy therule chosenfordetermininghe money upply.Using 14) and (16), we can write quation (1) asYt = alk, l + a2(Xt - Et1Xt) + uat. (17)

Ifwe substitute he right-handide fort in equation (2), we can obtainthe real interest ate as a function f kt-1 and exogenousprocesses.Substitutinghatfunctionntoequation 4), wegeta differencequationin k drivenby exogenousprocesses.This provesthat k is an exogenousprocess,whichby (17) implies haty is an exogenous rocess, hat s, hasa distributionndependentfthedeterministicule for hemoney upply.So we have provedassertion a) above: thedistributionfoutputdoesnotdependon theparameters f the feedback ule for hemoney upply.To proveassertion b), we write he tth erm ftheloss function as

Lt = EO[Et-1(K2pt + K22Pt + Klyt + Kj1yt)],where the insertionof Et-1 is valid for t > 0. Using E (x2) =E[(x - Ex)2] + (Ex)2, we have

Lt = EO[Kot + K2Et-jpt + K22(Et-pt) 2],where = E - Et-pt)2 + Klyt + K11 2]and where,giventhe exogeneity fytand pt - Et -Pt proved above,Ko is an exogenousprocess.Moreover, t is possible, s we shall showbelow,tofind rule formthat mplies hoosing v-1ptto minimize

Kot + K2Etjpt + K22(Et-lPt)


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248 JOURNAL OF POLITICAL ECONOMYAnd, becauseKot s unaffected y settings or he money upply t anyother ime, rule whichminimizes t also minimizes .To showthat there xists ucha rule form,we must olve the model.Again, we takeEt_pt in (15) and write heresult s(1 - Jo)Et-lpt = JEt-lpt+1L + J2Et-1mt Et-,Xt, (18)Since thisholdsfor ll t, tfollowshat

(1 - Jo)Et-lpt+ = J1Et-lpt+j+l + J2Et-imt+j (19)+ Et-Xlt+j.Byrepeated ubstitutionrom 19) into 18), we obtain

n(1 - Jo)Et-lpt = E [J1/(l - Jo)]j(Et- Xt+j + J2Et-.mt+j)j=O (20)

+ [J1/(l - +'Et-lPt+n+l,where 0 < J1/(l - Jo) = 1/(1 - cI) < 1.We assume thatthe imit s n-+ o ofthe secondterm n the right-handside of (20) is zero,which s a terminal ondition hathas the effectfruling ut"speculative ubbles." Then, from20),

00(1 -Jo)Et- .pt = E [J,/(l - Jo)]jEt_.,(Xt+j+ J2mt+j). (21)j=oSince thisholdsfor ll t,we may replace t by t + 1 and computeEt 'oftheresult oget

00(1 - Jo)Et-lpt+l = E [J,/(lJO)]j__0 (22)x Et-,(Xt+j+, + J2mt+j+l).For an arbitrarymoney upplyrule of theform 7), substituting21)and (22) into 15) gives hesolution or t; substituting21) and (22) into(2) givesthe solution orrt.This assumes thatthe rule s not such as toimply ooexplosive process orXt j + J2m+j.4'A workable"reduced form"forptcan be obtainedby substituting20) into (15)and thenbyusing 5) and (7) toreplaceEt-Im,+ and E-IXt + j with he inearfunctionsofpast variablesthattheyequal. These linearfunctionsre easilycalculated from hefeedbackrule for mtand the autoregressions overning omponentsof X,. While theresultingreducedform"for tformallyesembles hecorrespondingquationin the sys-tem with "adaptive" expectations, here s a crucial difference.Now changes in theparameters fthe feedbackrule form,producechanges n theparameters f thereducedform orPt.This feature f the system s what rendersPoole's results napplicable. Foran explicit llustration fthedependence of thereduced-formarameters n theform fthepolicyrule,see Sargentand Wallace (1973, pp. 332-33).


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RATIONAL EXPECTATIONS 249To find heoptimalmoney upplyrule,multiply 22) by J1/(l - JO)and subtract he result rom21) toget

(1 - J0)E~t-lp - += Et -Xt + J2mt. (23)The value ofEt-1ptthatminimizes tfor ll t s

Et -pt = -K2/2K22, (24)so thatEt- lpt+ = -K2/2K22. (25)

The optimalrule forthe moneysupply s obtainedby substituting(24) and (25) into 23). The resulting xpressionormts a feedback uleoftheform7).Thus, in the rational-expectationsersionof our model, the choiceamongdeterministiculesforthe moneysupply s a trivialproblem.One argument f thelossfunction,, is unaffectedy therule,so theproblem ecomes he implest indofonetarget-onenstrumentroblem.Moreover, definite ule emergesonlybecause we have assumed inspecifying that here s a target aluefor heprice evel. f, nstead, osswere made dependent nlyon the varianceoftheprice evel,then onedeterministicule wouldbe as good as anyother.The reasonthat thedistributionf real output s independent f thesystematic oney upplyrule can be summarized s follows.n orderforthe monetary uthority o induce fluctuationsn real output, t mustinduceunexpectedmovementsn theprice evelbyvirtue fthe ggregatesupply urve 1). Butbyvirtue ftheassumptionhat xpectationsbouttheprice evel are rational, heunexpectedpartofprice movements sindependent fthesystematicartofthemoney upply, s longas theauthoritynd thepublicshare thesameinformation.here s no system-aticrulethattheauthorityan follow hatpermitst to affecthe unex-pectedpart of the price level. Of course, the authority ould add anunpredictable andomterm o thesystematicartofthemoney upply,so that 7) would be amendedtobecome

m=HO*. (7')mtHt*_ + A~t,whereAt is a randomvariableobeyingE/t IO* I = 0. Then the dis-tributionfunexpected ricemovements nd ofrealoutputwilldependon the distributionf t. But clearly, here s no way theauthorityanbase a countercyclicalolicyon thisparticular onneutrality,ince thereis no way theauthorityan regularlyhooseaftin response o the stateofeconomic ffairsn order to offsettherdisturbancesn thesystem.


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250 JOURNAL OF POLITICAL ECONOMYThis follows ince t, is that part of the money supply obeyingEI I *I =0. Furthermore, n our model it is optimal to set A = 0 for all t.5. The Rational-Expectations Version under an InterestRate RuleAbove we showed that a certain terminal condition implied the existenceof a unique equilibrium price level for the rational-expectationsversionunder a money supply rule that is not too explosive. That analysis tookasa starting point the difference equation (18). With the interest ratedetermined by the feedback rule (6), a seemingly analogous differenceequation is obtained by imposing rationality, equation (14), in (13) andtakingEt -1 of the resulto = b2(Et-pt -Etlpt+1) + b2rt+ (b1 - a,)kt- (26)

+ b3Et-1(Zt - Ut + u2t))6Ifwe solve (26) by recursion, proceeding as we did in deriving (20) from(18), we find

nEt-ipt= - E Et-I{rt+j + [(b1 - ai)/b2]kt+,-i + (b3/b2) (27)j=O x (Zt+j - Ult+j + U2t+j)} + Et-lPt+n+l.To obtain a particular solution forEt-1pt from 27) requires imposing aterminal condition in the formoftaking as exogenous a value ofEt.lpt+jfor omej > 0. This is obviouslya verymuch stronger erminalconditionthan we had to impose on (20), a consequence of (26) being a non-convergent difference quation. Thus, when the interestrate is pegged,the model cannot determine a path of expected prices Et -lPt+ij = 0, 1,..., and by implication cannot determine the price level pt.Neither can it determinethe money supply.The economics behind the underdetermined expected price level ispretty obvious. Under the interest rate rule (6), the public correctlyexpectsthat themonetary authoritywill accommodate whateverquantityof money is demanded at the pegged interestrate. The public thereforeexpects that, ceteris aribus,ny increase in pt will be met by an equalincrease in mt.But that means that one Et-,pt is as good as any otherfrom the point of view of being rational. There is nothingto anchor theexpected price level. And this is not simply a matter of choosing the"wrong" level or rule for the interestrate. There is no interestrate rulethat is associated with a determinateprice level.At least since the time of Wicksell it has been known that, in thecontext of a static analysis of a full-employmentmodel with wages andprices that are flexible nstantaneously, t can happen that the price level


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"RATIONAL EXPECTATIONS 25Iis indeterminate if the monetary authority pegs the interest rate.5 Insuch a static analysis, the indeterminacy of the price level dependscritically on output and employment being exogenous with respect toshocks to aggregate demand or portfolio balance; that is, the Phillipscurve must be vertical. In our model, however, the Phillips curve is notvertical,but Wicksell's indeterminacy still arises.6. An Information Advantage for the Monetary AuthorityHere we shall examine some consequences of the monetary authorityhaving more informationthan the public. We shall first how that if themonetary authorityfollows the money supply rule that is optimal if thereis no information discrepancy, then the loss attained is the same asattained when there is no information discrepancy. Then we considerwhether that rule is optimal given an informationdiscrepancy.We shall write Et_1 for the expectation conditional on what themonetary authority knows at the end of period t - 1 and EJ, -1 forthe expectation conditional on what the public knows at the end ofperiod t - 1, where 0 is a subset ofwhat the monetary authorityknows.Then in place of (14) we impose

t+ Pt-j = E0,t-jpt+ i, (28)so that in place of (15) we have

Pt = JoE6 tlpt + J1Eo0t-pt+l + J2mt+ Xt. (29)Then, takingE0 t- 1 ofpt and subtracting fromPt,we have

Pt - E0,t-pt = J2(Mt - EOtmt) + (Xt - Eot-.Xt). (30)The rule that we found to be optimal without an information dis-crepancy is

J2mt = (K2/2K22)(1 -Jo - J1) -Et - 1Xt (31)From this it follows that

J2 Mt -Ee, tmt) = -Et_ Xt + Es, t-Xt_ (32)Substituting nto (30), we have

Pt- E0stpt = Xt - Et-1Xt (33)5See Olivera (1970). In bothour model and thestandard taticmodel, theaggregatedemand schedulemust exclude any components freal wealth that varywith thepricelevel fWicksell'sndeterminacys to arise.For example, ftheanticipatedrate ofcapitalgains on real (outside) money balances is included in the aggregatedemand schedule,theprice level is determinatewitha pegged interest ate. However, such a systemhaspeculiar stability haracteristics,ince stabilityhinges on the signof the expected rateof nflation.


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252 JOURNAL OF POLITICAL ECONOMYUpon substitutingfrom (33) into equation (1) of the structure,we getequation (17). And ifwe substituteforyt n equation (2) the right-handside of (17), we obtain [r, - Eo,1(p,+1 - Pt)] as a functionof k,_1and the exogenous processes, the same functionthat we previouslyfoundforrt - Et-1(pt+l - Pt). Then, substituting hisfunction nto equation(5) of the structure,we get the same first-order ifference quation in kas we had without an informationdiscrepancy.This proves thatunder therulegiven by (31), thedistribution f k does not depend on theinformationdiscrepancy. It followsthen from quation (17) that the same is true fory.To find hedistribution fpt,we proceed to solve thedifference quation(1 - J0)E6 tp1pt J1EOst1Pt+l + J2E6 t-lmt + Eo t-,Xt) (34)which is obtained by taking E0 t-1 of (29). Then, proceeding as we didfor (18), we obtain expressionsexactly like (21) and (22) except that inplace ofEt -1 on the leftand rightwe have E0,t-1.But from 31)

Xtj + J2mt+j = -(K2-Jo - J1) + Xt+j -E Xt+so for > 0

Eot-,1(Xt+j + J2mt+j) -(K2/2K22)(1 - - J1)= Et-1(Xt+j + J2mt+j)-Thus, use of the rule given by (31) implies Eopt-lpt+j = Et_1tt+j)j = 0, 1, which by (29) implies that under the rule given by (31) thedistributionofp does not depend on 0, that is, does not depend on theinformationdiscrepancy. It follows that the loss attained under the rulegiven by (31) does not depend on the informationdiscrepancy.

This shows that the monetary authority can do as well given aninformation iscrepancyas it can do ifthere s none. But can it do better?Can it, as it were, take advantage of the presence of an informationdiscrepancy?We are not sure. But within our structure, he answer seemsto be that it can take advantage of a discrepancy,although necessarily na limitedand rathersubtle way.To indicate why, let us focus first n how the distributionofy dependson the rule form. Under presentassumptions,equation (1) of the structureisYt = a1kl-t + a2(pt - EOt1pt) + Ult (35)

It followsthat as of the end of t - 1,E0 t 1ytis unaffectedby the choiceofmt, inceE0,t-jyt = ajE0,t_1kt_1+ Eot-lult' (36)

To find the variance ofYt, we subtract (36) from (35) and obtainYt = alkt-, + a2kt + 1 V where t xt - E0,t1xt. The variance of


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"RATIONAL EXPECTATIONS 253yt aroundE0,tiyt is,therefore,

Es, -(_t2) = Eo,t.to[(ajkt-. + filt)t]+ Ea.- (fit)+ other erms,wherext_ Et_1xt 0,t_1xt nd where the omittedtermsare un-affected y the setting orthe deterministicartofmt.Thus, settingmtaccording o (31) (i.e., setting it 0) minimizesEo, -1(_i) only f thefirsterm n the right-handideof (37) cannotbe made negative.Thattermcan be made negativeby a rule differentrom 31) if alk/,_.Silt# 0, that is, if the monetary authorityknowsmore about either thekt_1or the u1process than does the public. Of course, to take advantage

of this information discrepancy, the monetary authority must knowprecisely how the public's informationdiffers rom ts own.Similar conclusions hold for the distributionof kt. The expectationE, t-lkt is unaffectedby the settingformt,but, in general, the varianceE0, 1 k2) depends on it and is notminimized by use of the rule given by(31).6 And since the settingformtaffectsthe distributionof (yt+j, Pt+j)for > 0 by way of its effect n the distributionofkt,this means that,given an informationdiscrepancy, our structure gives rise to a non-trivialdynamic problem.But this should not be taken to mean that we are back in the settingproduced by the assumption that expectations are formed on the basisof fixedautoregressive chemes. The informationdiscrepancyassumptiondoes not produce any simple tradeoffbetween the means of output andthe price level. The fact that E0 t-iyt and E, tlkt are unaffected bymt s very imiting f0 contains, say, as little as (1, pt 1,yt 1). Second, toexploit the informationdiscrepancy, the monetary authoritymust knowwhat it is. To assume that it does seems farfetched. ndeed, we suspectthat estimating the discrepancy is a verysubtle and perhaps intractableeconometric problem.For these reasons, we think some comfortcan be taken from the firstresult established in this section. Use of the rule given by (31) is optimalif the public is as well informed as the monetary authority. The lossattained under that rule does not depend on how well informed he publicis, and implementation of the rule does not required knowledge of howwell informed the public is.

This does not, of course, deny that there is a gain from earning moreabout the exogenous processes. Settingsfor the money supply under therule given by (31) depend on what the monetary authority knows.Operating under that rule, loss is smaller themore themonetary authorityknows about the exogenous processes.6 The reader may verify his by findingk, as a function fk,-1 and pt Eot-lptusing 35) and equations (2) and (5) of the structure.


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254 JOURNAL OF POLITICAL ECONOMY7. Concluding RemarksGiven that our conclusions re derivedfrom n ad hoc model, shouldthey e taken eriously?n one sense, hey houldnotbe. Because oftheirad hoc nature,neither he structureet out n section nor the loss func-tion of section 2 should be accepted as providing suitable contextwithinwhichto studymacroeconomic olicy.Nevertheless,ome aspectsof our model cannot be dismissed o easily. First,the hypothesis hatexpectations re rational must be taken seriously,f onlybecause itsalternatives,or xample,variousfixed-weightutoregressive odels, resubjectto so many objections. econd, the aggregate upplyhypothesisis one that has some macroeconomic oundations,7 nd it has proveddifficultodisposeof empirically.8t is precisely hese woaspectsofourmodel-rational expectations n conjunctionwith Lucas's aggregatesupplyhypothesis-that ccountformostof our results.We believe thattheresults oncerning ystematic ountercyclicalmacroeconomic olicyare fairly obust o alterations f other eatures f the model, uch as theaggregate demand schedule and the portfolio alance condition. nparticular, the dramatically different mplicationsassociated withassuming ational xpectations,n the one hand, or fixed utoregressiveexpectations,n the otherhand,willsurvive uch alterations.ReferencesChow, Gregory C. "Optimal Stochastic Control of Linear Economic Systems."J. Money,Credit nd Banking2 (August 1970): 291-302.Lucas, Robert E., Jr. "Some International Evidence on Output-InflationTradeoffs." A.E.R. 63 (June 1973): 326-34.Muth, John F. "Rational Expectations and the Theory of Price Movements."Econometrica9 (July 1961): 315-35.Olivera, Julio H. "On Passive Money." J.P.E. 78, no. 4, suppl. (July/August1970): 805-14.Poole, William. "Optimal Choice of Monetary Policy Instruments in a SimpleStochastic Macro Model." Q.J.E. 84 (May 1970): 197-216.Sargent, Thomas. "Rational" Expectations,heReal Rate of nterest, nd the Natural"Rate of Unemployment.rookings Papers on Economic Activity, no. 2. Washing-ton: Brookings Inst., 1973.Sargent, Thomas, and Wallace, Neil. "Rational Expectations and the Dynamicsof Hyperinflation." Internat. con. Rev. 14 (June 1973): 328-50.

7 For example, see Lucas (1973).8 Tests of the aggregate upply hypothesisre reportedby Lucas (1973) and Sargent(1973).

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Rational Expectations and the Optimal Money Supply Rule_Sargent_1975