Duncan Foley - On Two Specifications of Asset Equilibrium in Macroeconomics Models

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On Two Specifications of Asset Equilibrium in Macroeconomic ModelsAuthor(s): Duncan K. FoleyReviewed work(s):Source: Journal of Political Economy, Vol. 83, No. 2 (Apr., 1975), pp. 303-324Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1830924 .

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On Two Specificationsof AssetEquilibriumnMacroeconomic Models

DuncanK.FoleyStanford niversity

The problem of specifying sset marketequilibrium onditionsn amacroeconomicmodel is studied. Two specificationsre proposed:"beginning-of-period"nd "end-of-period" ormulationsn periodmodels. In each case the limit of the model as the time period isshorteneds rigorouslyalculated. t is shown hatthe twoformulationsare internally onsistent, ut consistentwitheach other only underassumptionsfperfect oresight. brief eview f he iquidity reference

versus oanable funds iterature hows that theorists ave used thesemodelswithout xplicitly oting he differencesnd inconsistencies.

1. Introduction

The purpose of this paper is to state clearly and point out some connec-tions between three related conceptual problems in macroeconomicmodels: the choice of continuous versus period analysis; perfectforesight

versus imperfect expectations; and stock versus flow equilibrium in themarketsfor ssets. Although current research focuseson "disequilibrium"models, mostwritershave an "equilibrium" notion that lies behind theirdisequilibrium formulation. For this reason I will study equilibriummodels.

To exhibit clearly and definitelythe relations between these ideas, asimple model will be helpful. I hope it will be clear that the propositionsdemonstrated n thispaper apply to a wide class of macroeconomic models,

I would like to thank Karen Johnson,Martin Hellwig, students nd faculty n themonetary eminarsat StanfordUniversity nd the University f California at SantaBarbara, and my students n monetary heory t the Massachusetts nstitute fTech-nologyforhelpful omments nd criticisms t various tages n the writing f thispaper.Any errors emain my responsibility.his work was supportedby the National ScienceFoundationthrough rantGS-39937 and in part through rantGS-40104 at the nstituteforMathematicalStudies n theSocial Sciences,StanfordUniversity.[Journal fPoliticalEconomy, 975, vol. 83, no. 2]?O 1975 by The Universityof Chicago. All rightsreserved.

303



304 JOURNAL OF POLITICAL ECONOMY

both static and dynamic, nd that the questionsraised here mustberesolvednsomeway n all macroeconomicmodels.The particularmodeloutlinedhere is meant to be only an illustrative uide to make the

argumentmoreconcrete.The argument s directed o thefollowingesults: hatthere re twodifferentossiblespecifications f assetmarketequilibrium n macro-economicmodels, nbothcontinuousime nd periodversions; hat hesetwo specificationsre each internally onsistent ut inconsistent itheach otherexceptunder the assumption fperfect oresight;hat theexistingmacroeconomiciterature oesnotalwaysrecognize hedistinc-tionbetween hetwospecificationsnd occasionally onfuseshem; andthat ertain heoreticalropositionsretrue noneversion nd not n theother. One specifications "beginning-of-period"quilibrium,whichcorresponds n continuous ime to "stock" equilibrium; the other is"end-of-period"quilibrium,whichcorrespondsn continuous ime to"flow"equilibrium.

2. A Simple Model

Time in thismodel can flowcontinuouslyr in periods. f periodsare

used,the ength ftheperiodwill be calledAt.Any tock ariablewillbemeasured t thebeginningf theperiod;flowvariableswill be assumedconstanthrough period nd convertedostock ariablesbymultiplyingbyAt;and prices re assumed onstanthrough heperiod nd chosen oas tosatisfyertainmarket learing onditions.

There are twoproducedgoods n thismodel: consumptionoods,QC,whichwillbe thenumeraire,nd investmentoods,Q1, ssumedto beindistinguishablerom xistingapital.The priceof nvestmentoods n

terms fconsumptionoodswillbe Pk. The twogoodsare produced bytwofactors:abor,N,which s inelasticallyupplied nd fixed vertime;and capital services,which are proportionateo the stock of physicalcapital,K, the um ofpastoutput nthe nvestmentoodssector. here sno depreciation fcapital. The rental in consumptionoods unitsperunittime) earnedbyoneunitofcapitalserviceswillbe called r.

Under theassumptions fperfectmobility ffactors etween ectorsand, forthemoment, ullemployment f bothfactors,we can derivesupplyfunctions orflowsof consumption nd investment oods thatdependonthetotal apital ervices, , and therelative riceof nvestmentgoods, Pk: QC(K, Pk); Q1(K, Pk)- The rental to capital in consumptiongoods units will depend onlyon the relativeprice k (see Rybczynski1955; Uzawa 1961; and Foleyand Sidrauski1971,chap. 1). The rateofreturn o capital as an assetwillhave a component,/pk,representingcurrent ental arnings,whichwillalso dependonlyonpk.

There are twoassets n themodel: money representingiquid assets



ASSET EQUILIBRIUM 305

ingeneral),M, and the stock fphysical apital, K. The value ofmoneyin terms fconsumption oods,Pm)s the inverse f theprice level (ofconsumer oods n terms fmoney).

The expected uture athsof kandPm illbe denotedApm(t) ndJk(t).These paths take as unchanged n real timewhentheperiod changesand as equal to current rices t thecurrent eriod,which willcall time0. Thus, Pk(0) = Pk(Q), Pm(O) = Pm(O).

The stock fphysical apital s the um or ntegral fpast output ftheinvestmentector, ssumingno depreciation, nd money s issuedbythegovernments a "lump sum" transferohouseholds, .

The laws governing hegrowth fnominalmoney nd physical apitaland theconsumptionmarket quilibrium ondition an be written

M(At) - M(O) = D(O)At, (1)

K(At) -K(O) = QI[K(O),Pk(O)]At, (2)

Qc[K(O), pk(O)]At CD(. .. )At, (3)

whereCD is a functionwhich reports he flowdemand for onsumptiongoods by households. ts argumentsmayinclude both stocks nd flows,that s,wealth ncluding rexcluding human"wealth, nd variousdefi-

nitions f ncome.For present urposes, t does notmuchmatterwhattheform rarguments fCD actually re. These three quations re thebasisof twodifferent odels n what follows.

Essentially hismodel has two market-determinedrices:Pm' he in-verse of the price level, and pk, which in this versionrepresents he"interest ate," since the value ofPkdetermines he rental omponent fthe rate ofreturn o capital. The pair (Pm)k) s constrainedythecon-sumptionmarket learing quation (3). The quantity mmayenterthe

demandforconsumptionhroughwealth, incechanges nPm lterthevalue of outside iabilities say, high-poweredmoneyor thegovernmentdebt).

There is onlyone market learing elation o farto determine hepair(Pnl,,k). The additional ondition aturally oncerns he asset markets.

3. Two Specifications of Asset Market Equilibrium

Given theirexpectations bout the futurepath of pricesand rates ofreturn,gentswillchoose particular athfor heir onsumptionpendingandholdingsf ssets.Atanymomentlongthis athfor ny etof urrentprices,we can ask the question, What configuration f assetswouldagents ike to hold at thatmomentgiventhattheyare constrained ochoose ssets hathavethe ame valueat thegivenprices s the ssets heyactuallyhold?The answers o thisthought xperimentefine demandfunctionor ssets t each moment.The changesthatagentswould like




to make n their ssetholdings vertime an be expressedn terms fthesemomentary emandsfor ssets perhapswith the additionof constraintsorcosts ssociatedwith ertainpaths).

In general, hesemomentary emand functions ill depend on time,on total marketablewealth, nd on the rates of return xpectedon thevarious ssets ver the near future. Unless there s some "irreversibility"to assets, herewillbe no need for he agentsto consider he fardistantfuture.) he expected ates freturnmay be point xpectations eld withcertainty,or distributions. n any of these circumstances,we still have awell-defined answer to the original thought experiment.

For the purposes of this paper, I will assume that the momentarydemand functionsfor money and capital are stationary n time (that is,the agents, whenever they have the same wealth and face the same meanexpected rates of return, choose the Dame portfolios)and I will compressany effects funcertaintyon the demands into theshapes ofthefunctions,so thatonlymean expected rates of return ppear explicitlyas arguments.Since throughout this paper I deal only with equilibrium in a singleperiod, if themomentary demand functionswere not stationary,the basicpointsI am trying o establishwould stillhold, thoughthe situation wouldbe more complicated. Introducing additional momentsoftheexpectations

of prices and rates of return would also complicate matters withoutresolving any of the problems I want to point out.

I beginwithaggregate momentary stock) demand functions ormoneyand capital, which will depend on total nonhuman wealth, W, and ex-pected rates ofreturn to the two competing assets:

( 1 ) [ P (t + At) _(Pt )

Ad(t) Am~) LW(t), At (t

P(t) Pk + At) -k ( t 1Pk(t) Atfik(t) )

Kd(t) = (1 )! W~)fm(t + At) -Pim(t)

r(t) + Pk(t + At) -pk(t),

Pk(t) At~k(t)

Notice that the functionsL and J report the real value of money andcapital in terms of consumption good units that the agents want to hold,given their expected wealth and expected rates ofreturn.

The rate of return to holding money is the rate of expected change inthe value of money per unit time. The rate of return to holding capitalis the rental rate of returnplus the expected change in the value of capitalper unit time. Nonhuman wealth at time t has to be the value ofmoney




and capital n existence t time . For any values ofPmAk. tmustbe truethat

L + J = pm(t)Ad(t) + p)(t)Kd(t) = Pm(t)AI(t) + Pk(t)Ks(t) = W(t),

(A)

whereMs and KSare expectedholdings f money nd capital at time t.Agentsmust plan to hold theirwealth n the form feithermoneyorcapitalat each instant. hisbalancesheet elationsprior o and separatefrom ny Walras's law-type relationwe may derive concerning lowdemands. will call it the "balance sheetconstraint."

The question ow arises s to how equilibriumn themarkets ormoneyand capital ought to be characterized. believe that there are twodistinct haracterizationsn use by theorists hich have quite differenttheoreticalonsequences.Theorists re notalwaysexplicit bout whichversion hey re using,nor are they nvariably arefuln copingwith hedifficultiesach version resents. willconsider ach in turn.

A. End-of-Periodquilibrium

In a period model, it seems reasonable that agentswill look ahead in

makingtheirtradingplans and tryto anticipatetheirdesiredpositionas of the end ofthe period nformulatingheirmarket emands. n thisview,theoperative emand for apital (or money)willbe the amountofcapital or money) heagents hink,t time , that heywillwant to holdat thebeginningf thenextperiod, hat s,at timeAt.The appropriatesupply fcapitalwill be the sum of theexistingtockofcapitaland theadditionsto it cumulated over the period. This view leads to assetequilibriumonditions:

JV(At), -A Pm(At)K(O) + QI[K(O), Pk(O)]At = 1 W(At) A)Pk(At) LAtPm(At)

r(At) Pk(2At) Pk(At) KD(t),Pk A^t) AtPkAt )

(4)

M(O) + D(O)At A1 L FJ'v(At), 2At) A-

pm(At) LA t~mAt)

+(t 2A ) _ (t)](At) +Pk(2t ) - Pk = MD (At).Pk(At) AtPk(At) j

(5)

Theseequations eterminem(0)andPk(O) iven xpectations. oticethatthe ppropriateates freturnn money ndcapitalare those xpected ornextperiod,not forthis period,since the agent is trying o adjust itsportfolioobe correct t thebeginningf nextperiod.




The quantityW s wealth xpected t thebeginningfthenextperiod.Itclearlymust e related oplanned aving ver heperiod.The differencebetween urrentwealth nd wealth xpected t the beginning f the next

period has threecomponents: the value of saving at current-periodmarketprices;the capital gain or loss expectedon beginning-of-periodcapital and moneydue tochanges np,,, ndPk t theend of theperiod;and the capitalgain on themoney nd capital accumulatedduringtheperioddue to changes nPmndPk t theendoftheperiod.Since ncome,Y, s equal to thevalue of utput lusthevalue ofgovernmentransfers,ecan write

W(At) W(O) = [QC + Pk(O)QI + pm(O)D - C ]At

+ EPk(At) - Pk(O)]K(O)

+ Eok(At) - Pk(O)][K (At) - K(O)]

+ [PmAt) - pmO) Al (O) (6)

+ [Pm(At) _Pm(O)][MD(At) - M(O)]

= [QC - CD]At

+ {Pk(O)QIAt + [kk(At) -pk(O)]KD(At)}+ {Pm(O)DAt + [Lim(At) -Pm(O)]MD(At)}.

This expression leads directly to the demonstration that "Walras'slaw" holds forthe threemarketsin consumption,money, and capital inthe sense that the sum ofthe values of excess demand at any (Pm, k) pairin three marketswill be zero, since, using equation (A), and thefacts that

Pk(At)K (At)+

Pmm(At)MD(At) WJ(At)and

Pm(O)M(O) + Pk(O)K(O) = W(O),

(QC - CD)At = PkK (At) + pmMD(At) - W(O) - PkQIAt - pmDAt

= Pk[K (At) - K(O) - QuAt] (7)

+ pm[MD(At) - M(O) - DAt],

which is just the "Walras's law" statement. If households plan their

savings and consumption behavior using the same expected paths forprices, the sum of excess demands in the threemarketswill be equal.

In this end-of-period equilibrium model, one of the three marketclearing equations is redundant, and any one of (3), (4), or (5) could beeliminated, leaving two market equilibrium conditions to determine thetwopricesPm nd Pk. We could even, for nstance,eliminate the consump-




tion marketclearing relationand operate only with the marketsforcapital and money.

B. Beginning-of-PeriodquilibriumThere san alternativepecificationf ssetmarket quilibrium.We maysuppose hatprices nd rates freturn djust o as to makewealthholderscontent t thebeginning ftheperiodwithcurrentsset upplies.

This idea leads to the following ssetmarket quilibriumonditions:

K(?) (0 JF[k(O)K(O) + pm(Q)M(Q),Pm(At) pm(O)Pk0) LAtpm (O)

r(O) Pk(At) - Pk(0) 1Pk(O) Atpk(O) (4b)

M(?) p O)L Pk(O)K(O) + pm(O)M(O), m(At)- Pm(O)

Pm(0) L~~~~ () AtmO)

r(O) Pk(At) - Pk(0) (5b)PkO) Atpk(O) j

Either one of theseconditions s implied by the otherthrough hebalance sheetconstraint,inceL + J Apn(O)M(O)+ Pk(O)K(O).How-ever, Walras's aw" doesnot oldbetween hethree onditions3), (4b),and (5b). It is easy to find Pm)k) pairs that atisfy4b) and necessarily(5b) without atisfying3). The reasonfor his s that thebalance sheetconstraintorces he sum of theexcessdemands n theassetsmarkets nthisbeginning-of-periodenseto be zero,regardless f excessdemand in

themarket or onsumptionoods. Whereas n theend-of-periodquilib-riumcase we can eliminate ny one of thethreemarket learing ondi-tions, ncluding heconsumptionmarket,n thebeginning-of-periodasewe can eliminate nyassetmarket learing ondition ut we must etainthe consumptionmarket clearing condition, which always containsseparate nformation.

C. The"Scenarios"

What are theeconomicpictures hat ie behind these womathematicalformulations?n the end-of-periodquilibrium, emandsand suppliesare offereds ofthe end of theperiod.Agents an offer osell capital,forinstance,whichdoes not exist t the tradingmoment utwhich hey lantoproduce during he period.Contracts re made for abor and capitalservices ndconsumption uring heperiod nd assetdeliveries t the nd.



3IO JOURNAL OF POLITICAL ECONOMY

In the beginning-of-periodequilibrium, trading in and delivery ofassets are assumed to take place at the same time, that is, the beginningof the period. Within-period consumption is contracted for, but within-

period production of new capital is done on a kind ofspeculation, takingthe current-periodprice of capital as a guide.It is somewhat surprising from an economic (though not from a

mathematical) point of view that thisvery minor difference n concept-ualization of equilibrium trading should give rise to sharply differentmodels.

4. Continuous and Period Analysis

The arguments of this section are based on a methodological preceptconcerning macroeconomic period models: No substantiverediction rexplanationn a well-defined acroeconomiceriodmodel houlddepend n therealtimeength fthe eriod.

In reality, economic transactions do not take place continuously.Purchases of finalgoods, capital account transactions,wage payments, anddividend payments all take place at discrete times. When the rhythmofthese transactions s veryrapid, however, and when transactionsoverlap

for different conomic agents, it is tempting to model transactions astaking place continuously.We must remember, however, that this pro-cedure is only an approximation to the real situation.

Althoughindividual transactions re discrete,theyare notsynchronizedas among various agents. Therefore, period models, which assume thatall transactionsofa certain class occur in the same, synchronizedrhythm,are also an approximation to reality. "First, it must be admitted that theperiod analysis is highlyartificial, ince while people may take decisionsdiscontinuously, not all people take decisions at the same time" (Hahn1955, p. 64). The proper method of modeling transactions would pre-sumably be to allow time to flowcontinuouslyand to view transactions asdiscreteeventsofzero duration, like the "arrivals" in a Poisson process.

There might be some circumstances in which a "natural period" foreconomic activitycan be identified. n a largely agricultural,single-cropeconomy, almost all plans fora single growingseason mightbe made atthe same moment. But in modern industrialeconomies there seems to beno presumptionthat plans are all made at the same calendar time. Some

writershave argued that the period in the period model is economicallydetermined, say, as the planning period ofthe agents. This approach as ajustification for a macroeconomic-period model is fallacious because itdoes not speak to the question of synchronization.If all agents have thesame planning period but their initial planning dates are distributed

' See May (1970) for a lucid commenton Patinkin'smodel,which introducedthisprinciple s a basis for rigorous riticism fmacroeconomicmodels.



ASSET EQUILIBRIUM 3 I I

uniformly ver calendar time, a continuousmodel would be moreplausible.

A theorist sing periodmodelmust ither stablish naturalperiod

inwhichdecisions fmanydifferentgents resynchronizedraccepttheposition hat periodmodel s, ikethe continuousmodel, n approxima-tionto reality,nwhichcase theoutcomes fthemacroeconomicmodelshould not dependin any importantway on theperiodused.A periodmodel houldgivethe ame resultsallowing or econd-ordernaccuraciesdue to coarsenessf pproximation) hentheperiod shalvedordoubled.

Period models are attractive ecause theyare easy to conceptualize.This advantage s bought t a price, however,which s thedangerthathidden assumptionsmay enterthemodel under the cover of the fixedperiodlength. f the results f a periodmodel do not depend in anyimportant ay on theperiod,themodel can be formulateds a contin-uous model. The methodused to accomplish his s to retainthe engthofthe period s an explicit ariable n themathematicalormulationf aperiod model and to make sure that it is possibleto findmeaningfullimiting orms f theequationsas theperiod goes to zero. (This is theprocedureMay [1970] uses to criticizePatinkin'smodel.) In my view,this rocedure houldbe routinelyppliedas a test hat ny periodmodel

isconsistentnd wellformed hereno particular alendar ime sspecifiedas the naturalperiod.

The remainder fthis ectionwillbe devoted oapplying his est o thetwo versions f assetequilibrium ntroducedn section .

First, here s nodifficultyith quations 1 -(3) ingoing othe imitingcase where At becomes very small, since by inspection heyhave thelimiting orms

d = D(0), (1*)dtdK = QJEK(0), pk(0)] (2*)dt

QCEK(0), pk()] = CD( ...). (3*

A. End-of-Periodquilibriumnd"Flow" Equilibrium

The end-of-periodquilibrium onditions s writtennequations 4) and

(5) do notpass the testproposed bove, sincetheequilibrium ricesofmoney nd capitalwillchangedrasticallys theperiodbecomes horterand will nfactnotexist t all forvery mall At. To see this, ewrite 4)in thefollowingorm:

KD(At) - KD(O) + KD(0) - K(O) = QI[K(O),pk(O)]At.

We have added and subtractedKD (0), desiredholdingsof capitalat the beginning f the currentperiod at currentperiodprices.The



3I2 JOURNAL OF POLITICAL ECONOMY

quantityKD(O) mayor maynotbe equal to K(O), since current eriodprices, ccording o (4) and (5), are chosento equalize prospectivend-of-periodemandand supplies ndmaynotequalizebeginning-of-period

demands nd supplies.Ifwe now divide byAt,we get

K(A)- KD(O- FKD 0) -K 0[KD(At) AtD(0] J+ L At K(O)] QI[K(0),Pk(0)]I

Ifwe let At hrink,hefirst erm n the eft-handide has a finiteimit,dKD/dt,hichwill be derivablefrom he function . Butthe secondtermhas no finiteimit, o that the equilibrium ondition s revealedto beill-formed.

The problemcan be seen intuitively.he household, n planning tsend-of-periodtockofcapital,has twogoals in mind:first,o correcttscurrent ortfoliombalances nd, second,toallowfor tsplannedsavingover theperiod.The hiddenassumptions thatthehousehold djusts tsportfoliombalancecompletely ithin heperiod.2 Iftheperiodbecomesshort, he size of this djustment erm, ince t stays onstantn absoluteterms, verwhelmshe flow f nvestmentnd saving.

One solutionto thisproblem s to make the speed ofadjustment f

current ortfoliombalance ndependent f the modelperiod. ntroducea new parameter,k, measured n percentper unit time,the speed ofadjustment f existingportfoliombalance. Then the market learingconditionsn capitaland moneymarketswill be

KD(At) - KD(0) + kAt[KD(0) - K(0)] = QI[K(0),Pk(0)]At, (4a)

MD(At) - MD(0) + kAt[MD(0) - M(0)] = D(0)At. (5a)

These revised lowdemands the eft-handides of [4a] and [5a]) also

imply hat he um ofthevaluesof xcessdemands t current ricesnthethreemarketswillbe zero. To see this,note thatthatbalance sheet on-straintmplies hat

PkKD(0) + pmMD 0) = PkK(0) + pmM(0) = W(0),

which mplies hat

kAtpk[KD(0) - K(O)] + kAtpm[MD(0) - M(0)] = 0,

so that,by (7),

Pk[KD(At) - KD(0)] + kAtpk[K(0) - K(O)] - PkQ,[K(O), Pk]At+ pm[MD At) - MD(0)] + kAtPm[MD(O)- M(O)] - pmD(O)At

= PkKD(At) - pkKD(?) - PkQI[K(Q), Pk(0)]At (8)

+ PmMD (At) - pMD(0) - pmD(O)At= (QC - CD)At.

2 This point and the discussionmmediatelyollowing ere first rought o myattentionn a private ommunicationromollyReynolds llen.



ASSET EQUILIBRIUM 3I3

Introduction of the parameter k also allows us to derive a limitingcontinuous formofthe model. Dividing both sides of (4a) and (5a) by At,we get

KD(At) -K D(O) + k[KD(O) -K(O)] = QI[K(O),Pk(O)]1At

MD(At) - MD(O) + k[MD(O) - M(O)] = D(O).At

We can now take the limits of these equilibrium conditions as At goesto zero.3

3 We can approximate fm(2At) as

Pmi(2At) Pmi(At) + dA.

Pm(O)+ dfim At + (dfim + dt2m At Atdt 0 \ dt 0 dt 0 /

= Pm(O)+ 2 dfm At + d-2m At2,dt 0 dt 0

using

Pm(At) pm(O) + dfm Atdt 0

and

Pm Pm d Pmdt At dt tt

Therefore,

li J2 [pm(2At) - fm(At) fim(At) - Pm(O)]At -0 At AtimAt A pm?)

=lim J2 - [Pm(2At)pm(O) At -im(At) Atp,,(O)

- fm(At)fm(At) At + fm(At) Atpm(O)]

= lim "2 ~ p,.(O)[pmO) ? 2 dfiA + dtm (At 2]At-? (At)2pm(O)fim(At)~l P() dt OAt dt o()

[Pm) + dt Ao t] [pm(O) + dT| At])dt 0 ~~~dt

=I' "(2 [Pm(O)pm(O)2pm(O)d At

+ Pm0) d2m ((Att)2- Pm (O)pm(O) - 2pm(O) djt At (d-i )2 (At)21d 0 dt ~~~~~~0 0t J

At-0 (At)2pm(O)fi (At) [ dt2p

dt(O)(At)2- (+:

0(At)2]

=lirn J2 [d im p( (dkn ) 2]

0t

where 7Cm (dfim/dt)!pm.he other terms in the limit are derived in a similar fashion.




The continuous time version of the end-of-period or flow equilibriumasset market clearing conditions are

-kKD ( O)+ ?-d

J d-t J2+c d(r/p) + d J3kpl dtl dt Ldt +dt

+ k[KD(O) -K(O)] (4a*)

QI[Kf(O), Pk(?)]b

7r M(O)+ IdW eld%(r/pk) 1)r-ThmMD(?) p-{ dt L1 +

dmL ( + -k L3>

~pm dt dt L dt dt3

+ k[MD(O) - M(O)] (5a*)=D(O),

where 7rk = (dAi/dt)/pknd 7mm (dfin,/dt)/Pmre the expected rates ofchange in capital and money prices in terms ofconsumptiongoods.

It seemsnatural to call (4a*) and (5a*) "flow" equilibrium conditions,since they equate an instantaneous flow demand for an asset with aninstantaneous flow supply. It is clear that in continuous time the flowequilibrium condition corresponds to end-of-period equilibrium in a

period model.The first term represents the fact that expected price changes will

satisfy ome of the increased desire to hold the asset. The second termshows the flow demand for the asset that arises fromcurrent plannedsaving. The thirdand fourth ermsrepresentrevisionofportfoliobalancein responseto expected changes in ratesof return to the two assets. Noticethat theseterms nvolve second-orderxpectations,hat is, expectations aboutthe second time derivative of prices and about the rate of change (the

first imederivative) ofrates ofreturn. (Since there are no bonds in thismodel, the only term that reflects his is [d/dt][r/Pk].f therewere bondstherewould be terms nvolving di/dt, he expected change of the interestrate.)

The closed continuous flow equilibrium model consists of equations(3*), (4a*), and (5a*). Walras's law holds in these three markets takentogether, o that any one is redundant and we are leftwith two equilib-rium conditionsto determinethe two market variables Pm nd Pk.

In order to achieve a consistent formulation of this model, we havebeen forced, however, to introduce an additional arbitraryparameter, k,the speed at which wealth holders adjust their portfolios.Ideally thisspeed should be derived froman underlying theoryof the real costs toportfolio djustment and should be an economically determined variablein the system.4

4This point arose in a conversation with Martin Hellwig.




B. Beginning-of-periodquilibrium nd "Stock" Equilibrium

The continuous time version of beginning-of-period quilibrium is easyto derive. As At becomes small, the expected rates of return approach

their nstantaneousderivative values and nothingelse changes. The result-ing continuous time asset market equilibrium conditionsare

K(O) = J Pk(O)K(O) + Pm(O)M(O)5 E r(o) + 7Ek 3PkO) LPk (O) I(b*

(4b* )

M(O) = - L [Pk(O)K(O) + P,(O)M(O)5 ;(O) +? k 5

(5b*)where 7Cm (dfim/dt)/Pmnd 7Ck = (dlk/dt)/Pkre as before.

It seemsnatural tocall (4b*) and (5b*) "stock" equilibrium conditions,since they equate an instantaneous demand to hold a stock of an assetwith an instantaneousstocksupply. It is clear that in continuous time thestock equilibrium conditions correspond to beginning-of-period quilib-rium in a period model.

In the continuousmodel (3*), (4b*), and (5b*), eitherone ofthe asset

market clearing conditions can be eliminated, but the consumptionmarketclearing condition must be retained.

5. Stock and Flow Equilibrium and Perfect Foresight

Though we seem at thispoint to be faced with the necessity of choosingbetween the beginning-of-period or stock equilibrium model and theend-of-period or flow equilibrium model of assets markets, there is still

one glimmerof hope that we can avoid making the choice. It might bethat both hold simultaneously,that theyare not inconsistentbut in factare only two differentways ofwriting the same condition.

There is an additional assumption that makes the two equilibriumconditions consistent,the assumption that the paths of expected prices(Pm)Pk in the model I have been using as an example) coincide with the

paths of actual prices. I will call this assumption the "perfect-foresight"assumption. Perfectforesightmay hold over all time or only over certain

periods of time.5

That perfectforesightmakes the two equilibrium conditionsconsistentcan be seen intuitively by considering the period model (4a) and (5a),

5Notice that"perfect oresight"n this ense s a property nlyof the mean expectedprice. In morecomplexmodelswhere expectations re represented s probabilitydis-tributions,hisnotionof "perfect oresight"orresponds o theassumption hat themeanofthatdistributions correct. t does not correspond o the idea that as theperiod be-comes short hevariance of thatdistributionoes to zero.



3i6 JOURNAL OF POLITICAL ECONOMY

where theperiod s chosento makekAt= 1. If thebeginning-of-periodequilibrium onditionholds n boththefirst nd secondperiod, gents'portfoliost the end ofthefirst eriodwillbe exactly he onestheywould

have chosen f they had expectedpricespm(,At)nd pk(At).But on aperfect-foresightart of thepath,thesewillbe the valuesofPm(At) ndPk(At), so the end-of-periodequilibrium condition will also be fulfilled nthe first eriod.

Itmayat firsteem that ncontinuousime, erfect oresight ustholdbecausePm(?)= pm(O) nd Ak(?) = AM) So that as Atbecomessmalltheexpected evel ofpricescomes close to the actual level. In fact,theproblem s transferrednto the first erivative fpriceexpectationsncontinuous ime:although gentsalwaysknowwhatpricerules at anymoment, heymaybe mistakenbout the current ateofchange n theprice.To require gents ncontinuous ime o havecorrect otions ftherate of change of prices s mathematically quivalent to requiring hatthey avecorrectxpectationsf thenextperiod's evel ofprices nperiodanalysis.Anyobjection otheassumptionfperfectoresightn a periodmodel applies withequal force o the assumption f correctnotionsoffirsterivatives fprices n a continuousmodel.

The preciserelations etween tock nd flow quilibrium nd perfect

foresightan be summedup in thefollowingropositions:Proposition: The assumptionf tock quilibriumnd theassumption

ofperfect oresightogethermplythat the flowequilibrium onditionswillbe met.

Proof: will argue this in the continuous ime model. The stockequilibriumondition,whichholds at every nstant,s

K = JPkK + PmM,m) + Irk) (4b*)

Differentiatehiswithrespect o timetoget

dK = _ dh (I/p~k)J+ I {dJl ( Jr K + dpm + P dM

+ k(I p2 J dpk( +tk + k

Fd(r/Pk) dicrk1+ J2+J3 i + Il

Lt dtJtThis can be rewritten

QI(K, Pk) = - ( dk) ( /pk)J

+1 dW

J1 + J2 dt + J3 dr + d-tk]}Pk dt dt dt ~~~dt




which is exactly the same as (4a*) if expectations are correct (which

implies Thk = [dPk/dtl/Pknd 7tm= [dpm/dt]/pm),ince KD(0) - K(O) = 0by the hypothesisof stock equilibrium.

PropositionI: The assumption of stock equilibrium and the assumptionof flow equilibrium, if they hold for arbitrary asset demand functions,implyperfect oresightnPM,k. and r/pk.

Proof: Differentiating he stock equilibrium conditions as in the proofof Proposition I, letting KD(0) - K(O) = MD(O) - M(O) = 0 in theflowequilibrium conditionsand equating the two equilibrium conditions,yields

(dk dW dW\ ~ d(r/pl d(r/P)J

___ __ _

JN Fdrpk drpk)1J

=0dt 1 - 7Cm)L + (dt - dL1 + -dt0

( dt Pm ) dt dt L dt dt ]

From equation (6) we can show that

dJW dW - dA d\ tdfim dPm\---_ I--=

Adpk K + Mdt dt dt dt/ dt dt/

when KD(0) = K(O) and MD(0) = M(0). In the capital market thisyields,after a little manipulation,

(r dpkl\ /I TdpmI Li _~ Fd(r/Pk) d(r/Pk)]

- P)JLdtkdt Pm [dt dtj

This equation can hold for arbitraryJL1, LJ1, and J3 nly if

dPk 1 dp 1 and d(r/pk) _ d(r/Pk)

7k=dt pkT

dtmad

dt dt

so that expectations are in fact correct.Clearly therewill be some special cases where a particular set of asset

demand functions nd particular nonperfectforesight aths for expecta-tions are compatible with both stock and flow equilibrium conditionsholding simultaneously, but, in general, stock and flow equilibrium

together mply perfectforesight.It is not true that flow equilibrium plus perfectforesightwill imply

stock equilibrium in any finite time, because the general flow equilib-rium condition assumes only a gradual adjustment of an initial stockdisequilibrium.

However, the theoretical moral of these propositions is that we canavoid choosing between the stock equilibrium model and the flow




equilibrium model only ifwe are willing to assume perfectforesightas isoftendone in macroeconomiccapital theory models).

The decisive objection, in my opinion, to the assumption of perfect

foresightn a macroeconomic model is that macroeconomics is in largepart directed toward understanding the effectsof discretionary govern-ment policy changes on the economy and perfectforesight ules out allgovernmentpolicy changes except those that are fully anticipated andannounced far ahead of the time they are taken. It also seems odd toassume that the private economy predicts the course of asset pricescorrectlywhen the historyofindividual firms, he distributionofwealth,and asset prices themselves are written argely in terms of unanticipatedmovements of asset prices.

While it is logically satisfying o assume perfectforesight nd in thatway reconcile the two possible asset market equilibrium models, thepractical need to treat mperfect nticipations forcesus to choose in manypractical applications between stock and flow equilibrium models.

6. Stock versus Flow Equilibrium

Given thata choice must be made between thestockand flowequilibrium

conditionsin assetsmarkets,on what grounds can it be made?In the continuous time versionsof the two models, we can show that

in a formalsense the stockequilibrium conditions are the limit ofthe flowequilibrium conditions as the parameter k becomes infinite.Rewriting(4a*), for nstance,we get

KD(O) - K(O) (QI[K(O), Pk(o)1 + 7rkK (O)

! dW +d~m d(r/Pik). dT~if-{dJ1 + d7mJ2 + rL(A) k J3}-Pk dt' dt Ldt dtj if

Since the expression in large parentheses is bounded, as k --* oo thisequation will read KD(O) - K(O) = 0, which is the same as (4b*).

However, it is not clear that the same equivalence holdsmathematicallyin a period model. If we keep At constant,the modified"end-of-period"equilibrium conditions (4a) and (5a) are difficult o interpret fkAt > 1,since theyimply an overadjustmentof initial portfolio

mbalance withinthe period. (If kAt = 2, for nstance,wealth holders would be adding totheir demand for end-of-periodassets twice the amount of their initialportfolio mbalance.)

In lightof the precept discussed in section4, the fact that thislimitingequivalence of stock and flowequilibrium holds only in the continuoustime versionsof themodels should make us rathercautious in interpretingthe equivalence result.We may be confrontedhere with a difference n




"visions" of the economic process: one, stock equilibrium, that conceivesof thewhole ofexistingassets being at least potentiallyon the market atany moment; and another,flowequilibrium, that conceives ofonly a tiny

fractionof the existing ssets,ofthe same order ofmagnitude as additionsto stocks ofassets,being on the market at any moment.The equivalence does suggest, however, that there will be very little

differencebetween the predictionsof the models when speeds of adjust-ment of portfolio imbalance are high or, equivalently, when costs ofportfolioadjustment are low. Asset markets are in fact among the bestorganized of markets; information about prices of many (especiallyfinancial) assets is disseminated widely and rapidly, and the great bulkof the total wealth in industrialized capitalist economies is held in verylarge portfoliosfor which fixed transaction costs will be negligible inrelation to portfolioshifts.These observations suggest that the vision ofstock equilibrium may be a good approximation to the real situation.Empirical evidence of large transactioncostswould, of course, upset thisconclusion.

In passing, we might note some other differencesbetween the twomodels,which have more pedagogic than researchsignificance.The stockequilibrium conditions are uniformly asier to manipulate than the flow

conditions. Comparative statics in the flow model require assumptionsabout the second derivatives of asset demand functions,while the sameexercises with stock equilibrium require only assumptions about firstderivatives.The much used (by stock theorists)and much criticized (byflowtheorists)"IS-LM" apparatus can be rigorously ustified n the stockmodel, since asset market equilibrium is separate fromthe consumptiongoods marketbut no sense in the flowequilibrium model, whereWalras'slaw operates in the threemarketstogether.

My own feelingas I write this s that the flowequilibrium model com-plicates our view of reality without explaining many of the significantphenomena left unexplained by the stock equilibrium model. This isprobably a minorityview among workers n the field.

7. Confusions over the Distinction

A word may be in order about how easy it is forthese issues to becomeconfused n scholarlydebate. The choice between stockand flow

equilib-rium is oftensuppressed to the level of the "model in the back of one'smind" rather than broughtout explicitly s part of thespecification.Evenin the back of people's minds, the issue may be confused by implicitreference o theperfect-foresightase where the two models become one,especially since the perfect-foresightssumption is the natural inter-temporal generalization ofthe assumption in macroeconomictheorythatagents have perfect nformationabout prices in all markets. A theorist




consulting isor her"economic ntuition,"hat s,thebasic micromodel,may come up witha modelin which ncompatible ragmentsf a stockequilibriummodel and a flow quilibriummodel are combined.

The confusion roblem s compoundedby the fact that,because thestock excess demands are part of the flowdemands,most ordinarytheoreticaltatementssuch s a fall n the nterestate, ther hings qual,increases xcess demand formoney)will be true n both specifications.The disagreement ill howup only ndiscussionsfWalras's aworsuchcontroversiess: does excess upply n the (consumption) oods marketimply n offsettingxcessdemand formoney?does an increase n savingtend, holding ncomeconstant, nstantaneouslyo depressthe interestrate?In the flow quilibrium ense, he answer o both questions s yes;in the stock quilibrium ense, t is no.

To illustratehesepoints, willreviewbriefly smallsubsequence fthe literature n "loanable funds"and "liquiditypreference" heoriesoftherateof nterest. have chosenonlya fewofthemanyarticles ocommentn, and I have tried o choose omewhich hrow he ssues aisedin thispaperintosharprelief.

The controversyegan withan exchangebetweenOhlin and Keynesin theEconomicournaln 1937. There s no doubt that his xchange ep-

resents mutual misunderstandingt various evels.Keynes continuallyargued that the interest ate was determined y the demand to holdmoney,while Ohlin referredo the nterest ateas beingdeterminednthemarket orflows fcredit. ubsequentdiscussion ixed n themoneyversus reditdichotomyather han on the stockversus lowdichotomy.As we have seen above, in bothbeginning-of-periodnd end-of-periodequilibriumhere s a symmetryetween ll assetsmarkets.t ispointlesstoarguewhich ssetmarket ondition eterminesny particular riceor

rate freturn: hey eterminell prices nd rates freturn imultaneously.The question f whether he nterest ate s determinedin" themoneymarket rthe bondmarkets a redherringat leastfor quilibrium nal-ysis) n bothversions.

However, think t is clear that Ohlin had an "end-of-period" ndKeynes a "beginning-of-period"ision of asset market equilibrium.Ohlinwrites1937, pp. 224-25):

The willingnessfcertain ndividuals o ncreaseheir oldings fvariousclaimsand otherkindsof assetsminushe willingness fothers o reduce heircorresponding oldingsgivesthe supplycurvesforthe differentindsofnewcreditduringtheperiod.Naturally, he quantities ach individual s willingto supplydependon the nterest ate. . . Similarlyhe total upply fnewclaimsminus hereductionntheoutstanding olumeofoldonesgivesthe demand-also a functionfthe ratesof nterest-forthedifferentinds f creditduring he period.The pricesfixed



ASSET EQUILIBRIUM 32 I

on the market or hesedifferentlaims-and thereby he ratesofinterest-aregovernedby supplyand demand in the usualway. [Italics n original]

This passage seems to me to be a good description f what I call"end-of-period"quilibrium.Keynes expressed imselfmore crypticallyand never xplicitly eferredo thebalance sheetconstraints a way ofreconcilinghemoneyversus reditdebate. His language, however, itsverywell the model I have called stockequilibrium Keynes 1937,p. 250) 6 "The resulting heory i.e., Keynes'sown],whether ight rwrong,s exceedinglyimple-namelythatthe rate of nterest.. has tobe established t thelevelwhich, n theopinion of thosewho have the

opportunityfchoice-i.e., ofwealth-holders-equalizes heattractionsofholdingdlecash and ofholding he oan."In succeedingyearstwo schools eem tohave grownup without om-

pletelyrecognizing he differencesetween them. Tobin (1969), forexample, invariably nd instinctivelyormulateshe determinationfasset prices s an equilibrium etween tockdemands nd stock uppliesof assets.Patinkin, n theotherhand,devotedMoney,nterest,ndPricestoan elaboration ftheend-of-periodquilibriummodel. That there s a

certainlack of awareness that end-of-periodnd beginning-of-periodequilibriumre distinctnd inconsistentxceptundertheassumption fperfectoresights clear from assages ike thefollowingPatinkin1958,p. 302): "True thedemandformoney s a flow squite differentrom hedemand formoney s a stock; n fact, heydo not evenhave the samedimensions!Nevertheless, he excess emand formoneyas a flowisidenticalwiththe excess emandformoney s a stock."

Thisstatementbout excessdemands s true nlyundervery estrictiveconditions,s referenceoequation 5a) willmake clear.First, hedeficitisclearly ssumed o be zerobyPatinkin:hiseconomys na steady tate.Second,there s no savingand no anticipated hanges n prices o thatM D(At) = M D(0). Third,kAt s assumedtobe 1,so thattheonlytermlefts the stock xcessdemand. The viewthat ssetmarketsperateonlyto adjuststockdisequilibrium ppears to underlie hispassage.Patinkinlaterwrites 1958, p. 306): "Stockanalysis, s wellas flow nalysis, re-supposes periodof time: namelytheperiod betweenthemoment twhich he ndividual smakinghisplans, nd themomentfor hichhe is

making hem.Hence iftheperiodspresupposed y theanalyses re thesame,theexcess-demandunctionf tock nalysismustbe identicalwiththat fflow nalysis. hispropositionolds lsointhe imitingasewheretheperiod s instantaneous."

This passageseems to ignorethe problemof choosingwhether gents

6 Keynes triedto patch thingsup between himself nd Ohlin by introducing heplannedflow f nvestments a newargument thefinancemotive) n thestockdemandformoney.This invention eaves thestockversusflowproblemunresolved, f course.




plan for the beginning or the end of the period in trading assets. It pre-judges the issue by hiding in the words "if the periods presupposed by theanalyses are the same" the idea that the analyses also identify he same

part of the period as the moment for which the agent is making plans.Finally, Patinkin sums up his views (1958, p. 307): "It should also beclear fromthe above that the distinction that is sometimes attemptedbetween flow equilibrium and stock equilibrium is an inadvisable one.Once again, if the periods are the same, the existence of one type ofequilibrium implies the existence of the other." This way of talkingignoresthe distinction have been trying o establish above.

A final passage from Johnson may be in orier, since it seems torepresent a widely accepted "compromise" view (Johnson 1961, pp.

6-7) 7

The liquidity preference-loanable funds debate turns on thequestion of whether the rate of interest s better regarded asequilibrating the flow of funds onto and off the market forsecuritiesor as equilibrating the demand forand supply of thestock of cash. The answer,which is now so deeply embedded inmathematical argument that no one can be sure he has got itright, seems to be that the stock-flowdistinction s irrelevant,since either theorycan be expressed in stockor flowterms; andthat it makes no differencewhether one workswith money orsecurities,provided first, hat one is concerned only with thedetermination of the equilibrium rate of interest, nd, second,that one realizes that this is a general equilibrium problemwhich can be reduced only by artifice to a problem ofequilib-rium in one market. In more formal terms, f one assumes tobegin withthat themarketsforgoods and factors re in equilib-

rium, equality between the demand and supply for moneyimplies equality between the (stock and flow) demand forandsupply of loans and vice versa. The two theoriesbecome differ-ent, however, when applied to dynamic analysis of disequilib-rium situations, since liquidity preferencetheory implies thatthe rate of interest rises only in response to an excess ofthe demand for over the supply of money, whereas loanablefunds theory implies that it rises only in response to an excess

supply of over demand forsecurities,and when the goods andfactorsmarkets are out of equilibrium an excess demand formoney does not necessarily mply an excess supply ofsecurities.In a dynamic context,theloanable fundstheorydefinitelymakesmore economic sense; and the sustainedresistance ofKeynesians

7The lastsentence s quoted (approvingly, iven thecontext n which t appears) byLeijonhufvud1968, p. 65, n. 13).




to admittingt,evidentmostnotably n theprolongeddefensein theEnglish iteraturefthepropositionhat n increase n thepropensityo save lowers he nterest ateonlyby reducing he

level of ncome, s a credit o their ngenuity ather han theirscientific pirit.

It may be worthwhile o point out in detailwhichargumentsn thispassage seem to be correct,nd whichdo not, n light fthe analysis fthispaper.The stockflowdistinctions relevant, ven though n eitherstockor flow quilibrium ither hemoneyor securitiesmarket an beeliminated.The remaining xpressions illnot be identical n the twoversions, owever.) Even if markets orgoods and factors re not in

equilibrium, qualitybetween hestockdemandfor nd stock upplyofmoney mplies equality betweenthe stockdemand forand supplyofsecurities,nd vice versa.Only ifthegoods and factormarkets re inequilibriumdoes flowequilibrium n the moneymarket mplyflowequilibrium n the securitiesmarket.The two theories stockand flowequilibrium) redifferentven n static nalysis fequilibrium ituations,as long as perfect oresights notassumed.

In the ast part of thispassage,Johnson mplicitlyssumes he flow r

end-of-periodquilibriummodel, and his remarks re correct n thatcontext. f the economists e mentionsn the lastquoted sentencehadbeginning-of-periodquilibriumnmind,perhaps hey idnot deserve oharsh judgment n their cientificntegrity.

Perhaps the only constructive ay out of this confusion, hort ofdevelopingmore exact modelsofaggregate ransactions,s fortheoriststo become familiarwith both specifications,ring the choice betweenthemup to the conscious evel, and make explicitwhichvisionunderliestheir esult.

8. Conclusions

We can drawthefollowingmainconclusions rom hese rguments:1. There existtwo different ays of specifyingsset equilibrium on-

ditionsna macromodel, ach internallyonsistent oth nperiodmodelsand in continuous ime.

2. The two pecificationsre consistent ith ach other nlywhenthereis perfect oresightn thepartof gents oncerning ear future rices ndratesofreturn.

3. There s no easychoicebetween he woversions f sset quilibrium,becausethey rebothhighlybstractpproximationsoa realitynwhichtransactionsre discrete nd unsynchronized.

4. Certaintheoretical ropositions old in one model but not in theother. For instance, "Walras's law" holds in end-of-periodr flow




equilibrium between goods and asset markets. It does not apply inbeginning-of-period r stock equilibrium, although in stock equilibriumit is possible to eliminate any asset market. For another instance, it is

possible to have asset market equilibrium in the stock version withoutgoods marketequilibrium, so that an IS-LM-type division of the equilib-riumconditions s natural. No such division is possible in flow equilibriummodels.

These results lead me to think that the most fruitful ine of futureinquirymay be intomodels where the explicitstructure f the transactionsprocess is studied more precisely, rather than into "disequilibrium"variants ofeither of these models.

References

Foley, D., and Sidrauski, M. Monetary nd Fiscal Policy in a GrowingEconomy.New York: Macmillan, 1971.

Hahn, F. "The Rate of Interest and General Equilibrium Analysis." Econ. J. 65(March 1955): 52-66.

Johnson, H. G. "The General Theory after Twenty-five Years." A.E.R. 50(May 1961): 1 ff.

Keynes, J. M. "Alternative Theories of the Rate of Interest." Econ. J. 47 (June

1937): 241-52.Leijonhufvud, A. On KeynesianEconomics nd the Economicsof Keynes.London:Oxford Univ. Press, 1968.

May, J. "Period Analysis and Continuous Analysis in Patinkin's MacroeconomicModel." J. Econ. Theory , no. 1 (March 1970): 1-9.

Ohlin, B. "Some Notes on the Stockholm Theory of Savings and Investment. II."Econ. J. 47 (June 1937): 221-40.

Patinkin, D. "Liquidity Preference and Loanable Funds: Stock and FlowAnalysis." Economica, .s. 25 (November 1958): 300-318.

Rybczynski, T. M. "Factor Endowments and Relative Commodity Prices."

Economica 2 (November 1955): 336-41.Tobin, J. "A General Equilibrium Approach to Monetary Theory." J. Money,Credit ndBanking 1, no. 1 (February 1969): 15-29.

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Duncan Foley - On Two Specifications of Asset Equilibrium in Macroeconomics Models