an equilibrium model of the business cycle

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An Equilibrium Model of the Business CycleAuthor(s): Robert E. Lucas, Jr.Source: The Journal of Political Economy, Vol. 83, No. 6 (Dec., 1975), pp. 1113-1144Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1830853

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An EquilibriumModel of theBusiness Cycle

Robert . Lucas,Jr.Universityf Chicago

This paper develops theoreticalxampleof a business ycle, hat s, amodel economy n which real output undergoes erially correlatedmovementsbout trendwhich re not explainableby movementsn theavailability ffactors f production. he mechanism eneratinghesemovementsnvolves nsystematic onetary-fiscalhocks, he effectsfwhich are distributed hrough ime due to informationags and anaccelerator ffect.Associatedwith these output movements re pro-cyclicalmovements n prices,procyclicalmovementsn the share ofoutputdevoted to investment,nd, in a somewhat imited ense,pro-cyclicalmovementsnnominalratesof nterest.

1. Introduction

This paper develops an exploratorybusiness cycle theoryin which un-

systematic monetaryshocksand an accelerator effectnteract to generate

serially correlated, "cyclical" movementsin real output. Associated withthese outputmovementsare procyclicalmovements n prices,in the ratioof nvestment o output, and, in a ratherspecial sense, in nominal interestrates. In contrast to conventional macroeconomic models, the modelstudied below has threedistinguishing haracteristics:prices and quanti-ties at each point in time are determined in competitivequilibrium;the expectations of agents are rational,given the information availableto them; information s imperfect,ot only in the sense that the future s

unknown,but also in the sense that no agent is perfectlynformed as tothe current state of the economy.The attempt to discover a competitive equilibrium account of the

business cycle may appear merely eccentric or, at best, an aesthetically

I would like to thankRobert Barro, FischerBlack, Edward Prescott, nd ThomasSargentformany veryhelpful omments n an earlierdraft.[Journal fPoliticalEconomv, 975, vol. 83, no. 6](C 1975 by The University of Chicago. All rightsreserved.

1113



III4 JOURNAL OF POLITICAL ECONOMY

motivated theoretical exercise. On the contrary, t is in fact motivatedentirely by practical considerations. The problem of quantitativelyassessing hypothetical countercyclical policies (say, a monetary growth

rule or a fiscal stabilizer) involves imagining how agents will behave in asituation which has never been observed. To do this successfully, onemust have some understanding of the way agents' decisions have beenmade in the past and some method of determining how these decisionswould be altered by the hypothetical change in policy. Insofar as ourdescriptionsofpast behavior relyon arbitrarymechanical rules ofthumb,adjustment rules, illusions, and unspecified institutional barriers, thistask will be made difficult, r impossible. Who knows how "illusions"

will be affectedby an investmenttax credit?'In all of the models discussed in the paper, real output fluctuationsare triggeredby unanticipated monetary-fiscal hocks.The first heoreticaltask-indeed, the central theoretical problem of macroeconomics isto find an analytical context n which this can occur and which does notat the same time imply the existenceofpersistent, ecurrent, nexploitedprofitopportunities. Section 2 develops a neoclassical monetary growth

model, with the aim ofillustratingwhy thisproblem cannot be resolved

within the class of aggregative models which view trade as taking placeeach period in a single, centralized market. This abstract environment,while analyticallyconvenient,places toomuch information t thedisposalof tradersforcyclical behavior to be consistentwithrationality.

In sections 3-5, this model is modifiedby viewing production and tradeas occurringin a large number of marketswhich are imperfectlyinkedboth physically and informationally.This is the analytical device firstproposed by Phelps (1969) and since utilized by myself (1972, 1973),Lucas and Prescott (1974), and Barro (1975). As shown in Lucas (1972),thismodificationof the information tructure f an otherwiseneoclassicalsystem eads to a real response to a purelynominal disturbance.

In Lucas (1972), and also in Sargent (1973b) and Sargent and Wallace(1973), however, these real movements are of no longer duration thanthe duration of the shock: no forcesare present to account forthe per-sistenceor cumulationof theeffects fthe nitialdisturbance. n thepresentstudy,two such forces re introduced: informationags, such as to preventeven relevantpast variables frombecoming perfectly nown,and physical

capital, introducinga formof the familiar accelerator effect.In the model set out in sections 3-5, agents' behavior is described by a

pair of asset demand functionsrelating decisions to expected yields.That is, the link between tastes, technology,and demand behavior is notmade explicit.On the otherhand, the inferenceproblem solved by agentsto relate available informationto expected yields is developed in somedetail in section 6.

1 This arguments muchmorefully eveloped n Lucas (1973b).



MODEL OF BUSINESS CYCLE I I I 5

Sections7-9 develop certain onditionswhichan equilibriumolutionmust atisfy. he mainnoveltyies in theexplication fthetheoreticallinks between structural" nd "reduced-form" arametersmplied by

the rationality f agents' expectations ormation: he "reduced form"depends on the "structure" or the usual reasons; and the "reducedform"determineshestochastic ehavior fprices, nd thereforeffectstheform foptimal orecastules, nd thereforehe"structuralquations"(decisionrules).

Sections 10-12 describe the nature of the "cycle" produced by themodel under three ets ofassumptionsn theparameters f the model.Section10 exhibits he modelofsection2 as a special case. Section11

develops purelymonetaryycle nwhich apitalplaysno role.Section12describes "monetary ver investment" ycle.2 It is the latterversionwhichexhibits he qualitative haracteristicsited n the first aragraphofthis ntroduction.

The cycles of sections11 and 12 occur in a settingwhichabstractsfrom he existence f economy-wide ecuritiesmarkets. n view of theimportance laced in the model on the partialnature fthe nformationconveyedby the "local" prices at which agents trade,this abstraction

may well be crucial. The informationalole of economy-widenterestrates s briefly,f nconclusively,iscussednsection13.Sections14 and 15 discuss,briefly,ome issuesoftesting nd policy

implications. ection 16 concludes he paper.

2. A Neoclassical Monetary Growth Model

Though the main concernof this paper is with oscillations f output

and prices bout a trend ath, t will be useful obeginon morefamiliargroundwiththe discussion fa fairly tandard, ndisturbed eoclassicalgrowthmodel. This willpermit he fixing fnotation nd theearlydis-posal ofcertain ide ssues.

Consider, o be specific,n economyproducing singleoutputto bedividedamong private onsumption, t,real governmentonsumption,

G,, nd nextperiod's apital,Kt ,.The production unctionsf, and

Ct + Gt+ Kt?1 =f(Kt, Nt) + (1 - 6)Kt (2.1)

holdswhereNt s employment.he functions as theusualmonotonicityand curvature ropertiesndis homogeneousfdegree1; 5 sa deprecia-tionrate.

There is a constant opulation f denticalhouseholdswhich ownallthe factorsfproduction. abor is hiredbyfirmst thewage Wt;capitalis rented t Ut; and output s sold (to households nd government)t Pt.

2 The terminologys takenfromHaberler's useful axonomy fcycletheories Haberler1960).



I I I6 JOURNAL OF POLITICAL ECONOMY

All threemarketsre competitive.irmsmaximize urrent-periodrofit,so that n equilibrium

fK(Kt,Nt) = ' (2.2)

fN(Kt,Nt) -= '. (2.3)

Householdssupply abor inelastically n quantityN, which fact, nconjunctionwith (2.2) and (2.3), determines quilibrium utput,realwage,andrealrental rice, ach as a function fKt. n addition o owningthecapital stock,households lso hold a stock,Mt, ofmoneybalancesand selectan end-of-periodalance, Mt+,.' Their budgetconstraint,givenfactormarket quilibrium,s then

Pt(Ct + Kt+1) + Mt+, < Ptf (Kt,N) + Pt(l - 3)Kt + M. (2.4)

The objective fthe household s tomaximize subjectively iscountedsumofcurrent eriodutilities,wherethe atterdependon consumptionand current oldings f realbalances,Mt+,IPt.

The model s completed ythe pecificationffiscal ndmoney upply

behavior.Let all ofgovernmentonsumption e financed ya monetaryexpansion t a constant, ivenratey. Then Gt s given mplicitlyy

PtGt=Mt+I -Mt = ,yMt. (2.5)

The dynamicbehaviorof the systemwill be determined nce thedemandonthepartofhouseholds or hetwoforms f ssetaccumulationisspecified. he most atisfactoryayto do this, romomepoints fview,is to make explicitthe household'spreference unctional nd then to

derive ssetdemandsfrom ouseholds'nfiniteeriodmaximum roblem.An alternative oute,takenhere so as to set the stageforsubsequentsections,s topostulate hesedemandsdirectly.4 hus, et the demandforcapital,Kt+1,dependon theexpected ne-periodates freturn ncapitaland money,rktnd rmt,nd the initial stateofthe household,Kt andMtIPt.The demand formoneywill dependon the same fourvariables.Given assetdemands, onsumptions implicit rom2.4).

3Here and in the remainder f the paper therewill be onlyone nominal assetand itwill be supplied exclusively y thegovernment. will call thisasset "money" withoutqualifying his usage each time it arises,but it mightas well be called "bonds" or"governmentiabilities."This meansthatthe analysiswill notbe able to deal withques-tions nvolving he relative mportanceofmonetary nd fiscaldisturbances, hough tcould withoutmuch difficultye modified o do so.

4 This is makingvirtue of analyticalnecessity, ut there are definite ntuitive d-vantagesto a parametric, certainty quivalent"approach as used here: theforecastingand choiceproblems olvedby agentsare separated thoughwe knowthis eparation sartificial), nd the distinct ffects f each on decisionrules are clearlyseen (see n. 15below).



MODEL OF BUSINESS CYCLE 1117

Again withan eye toward ater developments, oth relationshipsreassumedtobe log linear.Forthe ogofa variable,use thecorrespondinglowercase etter, o thatktmeans thelog ofcapital,and so forth. hen

thesecondequality n (2.5) becomes since og (1 + u) p ,]

Mt+1- Mt = . (2.6)

The two demandfunctionsor ssets re postulated s

kt+1 = oc + clrkt ?C2rmt + a3kt, (2.7)

Mt+1 - Pt = Pb - flrkt + f2rmt + #3kt. (2.8)

The elasticities a,2

2 3 and fl, f2, f3 are assumedto be positive;al > L2; P2 > Ih; and a3 and /3 are less than unity. or completeness,the ogofbeginning-of-periodeal balances,mt Pt, shouldalso appearon theright idesof 2.7) and (2.8) (since they iguren thebudget on-straint2.4]). Here and in subsequent ections shall neglect his"realbalance effect"norder o focus n the ffectsfmonetaryhanges n thetwoyields,ktnd mt.5

The real one-periodrate of return n capital is nextperiod's realrentalprice, K(Kt+i, N), less the depreciation ate. Approximatinghis

bya linearfunctionf the og ofcapital gives

ret 60- b1kt+1,1 > 0. (2.9)

The rate ofreturn n money s thepercentage ateofdeflation:

rmt Pt-Pt+i (2.10)Since both of theserates of returndepend on the values of future

variables, othare "expectations"t the timethey ffect he decisions f

traders. n thepresent ontext f certainty,t is naturalto take theseexpectationsobe correctorrational).Thisassumptionloses he ystem

(2.6)-(2. 10).Substituting2.6), (2.9), and (2.10) into (2.7) and (2.8), one obtains

a pairoffirst-orderifferencequations n capitaland realbalances.Theusual practice s to obtainthegeneral olution o this ystem nd thenapply the boundaryconditions hatcapital equal its historically iveninitialvalue and thatreal balancesremainboundedand boundedaway

fromeroas t tends o nfinity slightlyifferentethodsadoptedhere,one which urns ut to bemore onvenient henuncertaintys ntroduced.Given the structureftheeconomy the parameters ci, pj, , and [),

thepair (kt,mt)describes ompletelyhe state f thesystemt thebegin-ningofperiodt. This leads naturally o definingsolution o be a set

5 One can easily dd a term a4(mM - pt)" to theright f 2.7), and similarlyo (2.8),and traceout theconsequences.This leads topossiblynterestingtability roblemswhichare poorlyunderstood by me) and whichI do not wishto confoundwiththe cyclicalcomplicationswhich are introducedater.



ii i8 JOURNAL OF POLITICAL ECONOMY

of functionselating quilibrium ecisions nd priceto thesetwostatevariables.Since thesystems linear, t is naturalto conjecture he ex-istence fsolution unctionsf the form

kt+1= it10 + 7rllkt n1 2mV (2.11)

Pt -= 20 + 7t21kt 7r22mt. (2.12)

Then solving eans finding umberst0 ... ir22 such that 2.6)-(2.12)hold identicallyn (ks,mt).

Substitutingrom2.6) and (2.9)-(2.12) into 2.7) and (2.8) yields herequired ndentitiesn (kt,mt);equatingthe coefficientsields ix equa-

tions n theunknownUijs:it10 = T + a160 - (017E10 + X27t217t10 + L27t2214 (2.13)

711 -tibl~rll - L2)T21(l - I11) + LX3, (2.14)

7i12 -alblT12 + L2it21it12) (2.15)

I- '220 =0 - f130 + #fl13 0 -2it21it10 - fl27r225 (2.16)

- 7t21 = f1t17 + f27t21(1 - 7t11) + p3' (2.17)

1 -i22 =#161712 - f27t217t12. (2.18)

Equations 2.14) and (2.17) involve nly7tl and 7r21,; their olutions

diagrammednAppendixA. As seen n figure l, there re twosolutionpairs.One pair,with t1 > 1and 7i21 > 0,hasno economic ignificanceand willbe discarded.6The other s thedesired olution; t satisfies

CX3 < 711 < 1,(2.19)

1 + al l

it21 < 0. (2.20)

Inspection f 2.15) and (2.18) shows hatone solutions

it12 = 0, (2.21)

i22 = 1. (2.22)

Since 7r21 < 0, there s no solution ther han this lassical one. Finally,

the constantst10 and 7i20 are readily alculatedfrom2.13) and (2.16).With these solutionvalues, equation (2.1 1) is a stable first-order

differencequationin capital stock.Capital tendsmonotonicallyo itsstationaryalue, rt1 /(l - 71 ). The behavior fpricessgivenby 2.12),given he pathsofcapitaland money.

6 This discardingof an unstable root is, of course, the step which is customarily"justified"by a transversalityondition n models n whichagents'maximumproblemsare made explicit see, e.g., Brock1973).



MODEL OF BUSINESS CYCLE I I I 9

Note first hesense n whichmoney s "neutral" n this ystem. rom(2.21) and (2.22), a once-and-for-allhange n the evel fmoney alancesleads to a proportionalhange n the price evel in the current nd all

future eriods.There are no real effects. n the otherhand, tis evidentfrom 2.13) thatchanges n therate f ncreasefmoney, u,willhaverealconsequences: the higheryus the larger7Ii is and hence the largercapital is all along its time path and at its stationary oint.As Tobin(1965) and othershave noted, his ffect works"through he real yieldon money,which s,from2.10) and (2.12),

rmt 7r2l(kt k- 1) -A1,

or, in the stationary tate, imply he negativeof the rate of monetaryexpansion.Note, second,the peripheral ole played by the "flow variables"-

output, private and government onsumption, nd employment-indetermininghedynamicbehavior f thesystem. he model s analyzedbyfirst educingt to theequationsdescribinghe motion fassetsandtheir rices, olving hese, nd then eturningothedeterminationfflowequilibrium.This characteristic,ongfamiliarn moreabstract heory,

willcarry ver ntosubsequent ections.As a result, willbe discussingbusinessycleswith carcely referenceo suchkeymagnitudess employ-ment,consumption, overnmentpending, nd real output.This maygive an unfamiliar one to much of what follows, ut the translationbackintothe standard ocabulary s,Tthink, straightforwardxercise.

In particular, he reader mayverify hat the introduction fa tastefor eisure nd, consequently, variable abor supply nto themodel ofthis ection s easy to carry ut,withno effect n the form f (2.7) and

(2.8). This modifications obviously ssentialfor business ycletheoryand willbe takenforgranted elow.Finally,and in sharp contrast o traditionalmacroeconomicmodels,

the olution ound bove remains alid underverywidevariationsn whatis assumedabout the behavior f money.To take one example, upposeMt+1- mt is a sequence of independent, ormal variates,each withmean Itand varianceu2. If (2.9) and (2.10) are reinterpreteds expectedratesof return, onditional n informationvailable up through ,then(2.11) and (2.12) remaina solution,with the samecoefficients

ijas

found bove. In view oftheemphasis ften uton thedistinction etweenanticipated nd unanticipatedmonetary hanges, this fact may seem

7This nonneutralityf nflation id not appear in Lucas (1972) or Sargent (1973b),since both papers excluded capital formation. his led Tobin (1973), and perhapsothers,to wonder how monetary istortionsresent n models with certaintynd perfect ore-sight an disappear whenuncertaintys introduced. he answer s, they o not. The pointof Lucas (1972) and Sargent 1973b) is not that the ntroduction funcertainty emoveslong-familiar eoclassicalnonneutralitiesut, rather, hat t does not in itself ntroducenew ones.



II20 JOURNAL OF POLITICAL ECONOMY

paradoxical. It results rom he fact thatin a competitivemarket hecurrent rice is part of traders' nformationets.Thus, a trader whoknows he coefficientsf (2.12) and thecurrent eal capitalk,knowsm,

prior o committingimself,egardlessfwhethert s announced r not,or anticipated r not.

3. A Cycle Model: Introduction

The above discussion f a monetary rowthmodel concludedwith theobservationhatmerelyntroducingnoise" intomonetaryolicywas notsufficiento induce the sortofresponsesn real and nominal variableswhich ccurduring heobserved usiness ycle.The problems that n aneconomy n which all tradingoccursin a single competitivemarket,there s "too much" informationn thehandsof traders orthem everto be "fooled" nto altering eal decisionvariables.

To getaway from hisanalyticaldifficulty,ut not so far away as toprecludea simple description f aggregatebehavior, shall adopt thedeviceproposed yPhelps 1969) and,sinceutilized n similar ontextsyLucas (1972,1973)andLucas and Prescott1974), of thinking ftrading

as occurringndistinctmarkets,r "islands." Such a systems describedin this ection nd analyzed n the remainder f thepaper.

At thebeginningf a period, raders re distributedn somewayovera continuumfmarkets. ach market as capital nplace,as determinedbythepreceding eriod's rading. here s a stock fmoneyn thehandsoftraders; n addition,governmenturchasesntroduce ewmoneyn away whichvariesstochasticallyrommarket o market nd periodtoperiod.Within ach market, roduction,xchange, nd assetaccumula-

tiontakeplace exactly s describedn theprecedingection,with he oledifferenceeingthatthe twoyields, k,nd rmt,re conditional xpecta-tions ather hanknownnumbers. nce tradings complete,gents electa new market t random,new monetary hocksare realized,and theprocess ontinues.

Capitalaccumulatedn a particularmarkets assumedto remain hereinto thenextperiod,though tsownersmoveon. The dollar return ocapital s thenreceived yshareholdersfter radings complete. he size

ofthis ne-periodfloat" s taken obe proportionalo the tock fmoney(though,nfact, his annotholdexactly) nd isneglectedn whatfollows.

8 The idea behindthis slandabstractionsnot,ofcourse, ogaininsightntomaritimeaffairs, r to commenton the aimlessness f life. It is intended implyto capturein atractableway the factthat economicactivity ffersgentsa successionof ambiguous,unanticipated pportunities hich cannotbe expectedto stayfixedwhile more nforma-tion is collected. t seemssafeand, formy purposes, ensibleto abstracthere from hefactthat n reality his ituation an be slightlymitigated y thepurchaseofadditionalinformation.



MODEL OF BUSINESS CYCLE I I21

The financing f nvestments entirelyinternal":there re no economy-widemarkets or apital funds.

All exchange in this economytakes place at competitivemarket

clearing rices.The behavior feach trader s rationaloth nthe conven-tional sense of optimal,givenobjectives nd expectations, nd in theMuthian sense (Muth 1961) that available informations optimallyutilizednformingxpectations.n order hat he atter ssumption avean operationalmeaning, he analysiswill be restricted o the situationinwhich he relevant istributionsavesettled ownto stationary aluesand can thusbe "known" by traders.

The central conomic ngredientsfthismodelwill, s in thepreceding

section,be theasset demand functions2.7) and (2.8), whichwill nowdifferrommarket o marketdue to variations n capital stock nd ininformation.he aim of the analysiswill,also as above, be to obtaintheanalogue to the solutions 2.11) and (2.12) forthemotionofthe statevariablesand theirrelativeprice.The major differencenducedby theintroductionf relative nd aggregate noise"willbe in the calculation,by agents,of the expectedyieldsrk, nd rmt,which will now be meanvaluesconditioned n limited nformationather hanperfectlyoreseen

realizations.It will be convenient o develop these elementsn the reverse ftheusual order. n thenext ection, he nformationtructureftheeconomyis described nd the solutionof themodel is stated formally.Next, nsection5, the asset demand functionsre restated nd thetwo expectedyields redefined. he inference roblem olved by agents s treated nsection , completing he statementfthe model.

4. Notation and a Formal Solution

To move toward n explicit escription ftheeconomy escribed bove,thinkof trade as occurring n a continuumof separated marketsz, 0 < z < 1,where is an index of ocation.The systems drivenbystochasticnjections fnewmoney intheformfgovernmentalpending)whichvaryover timeand over markets t a giventime.Let theaverage(overmarkets) ercentage ncreasenmoney ex, wherex, - N(u, a2).

Market receives n increment hichdeviatesfrom heaverage bythepercentagemountO(z), where

O(Z) = POt-l(Z) + et(z), 0 < p < 1 (4.1)

and e,(z) - N(O, a2). Take Et(z) and x,to be independent or ll s, t,zand st(z) and 8s(Z') to be independent, nlesss = t and z = z'. Then

9 See sec. 13 for discussion f the probable effectsf ntroducingn economy-widebond market.



I I22 JOURNAL OF POLITICAL ECONOMY

the stationary istributionf [x,,O(z)] for nyfixed ocationz will benormalwithmean (ji, 0) and covariancematrix

(U2 o0

where a2 = a2/(l _ p2). None of the shockss,(z), O(z), and x, areever observedby agents,buttheirdistributionsre takento be constantand known y agents.'0

As a consequence f these hocks, heonlyonesaffectingheeconomy,capital stock may be expectedto varyover time and acrossmarkets.Let k,(z) denote the log ofbeginning-of-periodapital in z at t and let

ktbe theaveragevalue ofkt(z) over all markets.Use ut(z) = kt(z) -ktto denotethe deviation rom verageof market 's capital.These threevariableswill follow stochastic rocessto be determined. enote thestationary istributionfut(z) as N(0, o2). One would expecttheper-sistent, elative hocksOt(z) to affect apital movements,o that uuo=

E[Ot(z)ut(z)]willbe nonzero.These distributionalacts re also assumedknown o agents, hough t(z) and k,cannotbe directly bserved.

Also as a consequenceof the disturbances,ndividuals n different

marketswillacquire differingmounts fmoney uring trading eriod.Thinkof a large numberof agents, ach selecting extperiod'smarketat random, o that hedistributionf gentsby theirmoneybalanceswillbe thesame n all markets.n thenormal, og-lineartructureobe usedbelow, heonly hanging eature fthisdistributionill be its ogarithmicmean,denoted as in sec. 2) by mt. his averagefollows he randomwalk

mt+,= mt+ Xt. (4.2)

Assume that mt s notdirectly bservedby agents.Equations (4.1) and(4.2) together ivea complete escriptionfthe flows fmoney hroughthevariousmarketsn thiseconomy nd all therelevantnformationnthe distributionfmoney mong agents.

According o theabove description, hen, heaggregate or average)stateoftheeconomys describedby thevalueskt,mt, nd xtofcapitalstock,money, nd nominalgovernmentpending.The situation f anindividual market z is described by its capital relativeto average,

ut(z) = kt(z) - kt,and the government pending t receives relativetoaverage,Ot(z).As agentsdiffuse hrough his system, heyobservenone of these

variablesdirectly. ach period,however, hey radegoodsformoney t amarket clearing price pt(z). The historyof pricesPt(z), pt_,(z'),

10 The assumption hat theseunobserved istributionsre "known"need not be takenas a literaldescriptionftheway agents hink ftheir nvironment.t is usta convenientwayofassuming hatagentsuse the data available to them n the bestpossible way.




Pt- 2(z"), . . ., observedby an individual s his sourceof nformationnthecurrenttate fthe conomy ndof hemarket in whichhe currentlyfindshimself; quivalently,his history s his source of informationn

futureprices. Since tradersfollow differentaths, each will havedifferentnformationn hand, so that in generalone would need todescribe he nformationaltate ftheeconomy ya distributionf gentsby nformationeld.To complicatematterstill urther,his nformationalstatewill nfluencerices ndwillthen tself e an objectof peculationagentswillform xpectationsbout the expectationsf others.Two fur-ther onventions ill helptosimplifyhis omplexpicture. irst, ssumethateach agentsummarizes heprice historyPI-l P- 2, ... ) observed

byhim n a pair (kt,MP,),isunbiased estimate f thecurrent aluesofthe aggregate tatevariables, kr,m,).12 Second,priorto trading achperiod,theseestimatesre "pooled" bytraders y simpleaveraging,othat a single pair (kt,mt)of numbersdescribesthe perceptions f allagents.Let these perceptions e normally istributedbout the actualaggregatetate,withthe covariancematrix

(2 aak amk

mk am

The state f a particularmarket , then, s fullydescribedby sevennumbers:kt,Mt)kt,mt, t,Ot(z), ut(z). Agentsdo not knowthis state,though fcourse theydo know their wn expectationskt, m).On thebasisof he atter,hey ave a well-formedpinion f herelevant ariablesthey cannotobserve:kt,mt, t,Ot(z), ut(z). Specifically,hey believe(correctly)hat thisrandomvector s normally istributed ith mean(kt,Mt,A, 0, 0) and covariancematrix

akin0 0 2 43k Ukm? O

0km 0am 0 0- O m2 0 0 s ssssssssssssssss

L0 0 a2 ]O0a00 0 2a~ a

2

This completeshedescriptionfboth he ctualstate f heeconomynd

theopinions gentshave as to this tate.As in section2, the aim of the analysiswill be to define nd studytheequilibriummotion fthis ystemrom tate o state.Alsoas in section2, one has the choiceofthinkingfequilibrium s a setof timeaths fassets and prices,or as a set offunctionshichspecify ricesand asset

I This neglects, s Sargenthas pointedout tome, the nformationonveyed o traderswhen theyreceivethe dividend"check" for heir nvestmentf twoperiodsearlier.

12See n. 10 above.




movements,iven hecurrenttate.Takingthe atter oute,etan equilib-rium ake theform'

kt+l(z)= t10 +

tlikt 7r12rt+

7x13[kt ut(z)]+ 14mt + 7r15[Xt ? Ot(Z)], (4.4)

Pt(Z) = 7r2O + 7E21kt + 7t22rnt + 7r23[kt + ut(z)]

+ 1t24mt + it25[Xt + Ot(Z)]. (4.5)

Subsequent ectionswill be devotedfirst o developing set ofconditionswhichthesecoefficientsij must atisfynd thento developing he im-plications ftheseconditions.

5. Asset Demand Functions

Current-periodlowequilibriums determinedn each market xactlyas in section2. I shall focus, hen,on theassetdemandfunctions2.7)and (2.8), repeatedherein a notationwhichemphasizes heirmarketspecificityut s otherwise nchanged:

k1+1(Z) = LO + ,lrk,(z) - ci2rmt(Z) + oc3k,(z), (5.1)

mI(z) - PI(z) = P - flrkI(Z) + fl2rmt(z) + fl3kt(z). (5.2)

The parameters j, PJire restricteds in section . In addition to (5.1)and (5.2), money upplyfollows4.1) and (4.2), and

mI(z) = mt + X, + at(Z) (5.3)

holds.The two assetyields,

k,nd rMt,re conceptuallys in section2 but

in thepresent ase ofuncertainty ill be takento be conditionalmeans.The return nmoney s, again,theexpecteddeflation ate.Sincetraderswill choosenextperiod'smarket t random, heexpectedrate relevantinz at t s

rmt(Z) = Pt(z) - ?,1(z), (5.4)

wherept(z) is the observed urrent rice and i+ ,(z) is the expectedvalue of next period'saveragerice level, conditionalon information

availablein z at t.The return n capital is, as before, he expectedreal rentalprice.Since capital accumulated n z remains here ntothe nextperiod,thenominalrentalwill be proportional o the localprice prevailingnext

13 With a modest application of intuition, ne can specify ome of the solutionparameters n advance. (E.g., since two markets with the same total mt+ xt + 0should look alike, one should have 7t14 = 7l - and 7r24 = 7r2_5.) here is no harm incarrying long extraparameters, owever, nd since intuitions iffer,ne may as welldevelopsuchfacts ormally. his is done in sec. 9.



MODEL OF BUSINESS CYCLE I125

period.On the otherhand,since dividendswillbe spentelsewhere, heappropriatedeflators an expectedaverage rice. In addition to thesepriceeffects,hedampening ffect fdiminishingeturns illalso,as in

section ,be present.n viewoftheperipheral ole ofdiminishingeturnsoverthecycle, shallneglect he atter ffect ereand write

rk,(Z) Pt+1(z) - A+ 1(Z)) (5.5)wherept+1 z) is the priceexpectedto prevail locally,next period,onthebasisofcurrent-periodnformation.

6. The Formation of Expectations

Since the rates of returnwhichfiguren the demand functions,5.1)and (5.2), arenotdirectlybservable,hey ortheexpectedpriceswhichcomprise hem)mustbe inferredy agentsfrom vailable information.This inferenceroblem s thesubjectofthis ection.

The informationets nd "priors"ofagents re described nsection .Agentsknow the coefficientsf the solution 4.4)-(4.5) and take thejointdistributionfthestatevector kt,mV,t, t(z), ut(z)] tobe normal,withmean (ks,mt3 p, 0, 0) and covariancematrix as givenby (4.3).

Then, priorto trading, heyobservetheequilibrium rice,whichas afunction f the unobserved tatevector carries dditional nformation.On thebasis of thisnew nformation,gents orm posterioristributionon the statevectorto be used in forecasting.enote the mean of thisposterioristributionk,, t3 t) at(z), Ut(z)].

From (4.5), thepricewhich,priorto trading,had been expectedtoprevailwas

fit= 7t20 + ir21kt + rc22rt + n23kt + 7r247n + 7125!L-

Alsofrom4.5), thepricewhich nfactprevailss

pt(z) = at + 7r2 3(kt-k) + it23Ut(Z)

+ 7i24(mt -_Mt) + n25(Xt- ) + 7t25Ot(Z). (6.1)Thus, [kt,mt, t,Ot(z), ut(z)] andpt(z) - A are,from hepointof viewofagents,ointlynormally istributed ariateswitha covariancematrixgivenbyE and (6.1). A straightforwardalculationyields heconditionalmeans 14

= + ap (7r2 3 k + 7t247mk)[Pt(Z) - A]) (6.2)

mt= Mt + "r- (it2i3mk + 7t24m) [Pt(z) - A], (6.3)

= It + U2it25 2[pt(Z) -Pt], (6.4)

= - 2(i23o + 2 )o)[pt(Z) -fit]) (6.5)ut(z) = ai 2(i23uO + t2 a0)[p (z) -fit]) (6.6)

14 See, e.g., Graybill 1961, theorem .10, p. 63).



I i26 JOURNAL OF POLITICAL ECONOMY

where

2 2 2 q2 2 2 2up = 23(Fk + 2)T232t24Omk + )T24Um + 1t256

+ Tr232 + 27r231n25UO + 25a2 (6.7)

is the varianceof actual priceabout its priormean.One notesthateach posterior conditional)mean is simply he prior

mean,corrected ya termwhich ncorporateshe new informationon-tained n the market rice, t(z) -fit. In each case the weight ttachedto thenewinformation,(z) - in (6.2)-(6.6) is the simpleregressioncoefficientf the shock n question np,(z) - pt Thus, for xample, n

(6.4),U2it2 52 isthe ovariance f

t(z)- It and

xt- j dividedby the

variance ofprice.The estimates 6.2)-(6.6) are now used by tradersboth to update

their stimateskt,Mt) oftheaggregate tateof theeconomy nd toformunbiasedexpectationskt(z) nd rmt(z) ftheyieldswhichare relevantto the asset demand decision. Using E,( ) to denote an expectationformednz in t,onehas,from4.4),

kt+,(z) = Ez((lr0 + i1 jkt + 7r12rnt + 7r13kt + r14mt + 7t15Xt)

= 7t10 + it11kt + 7r126t + r13kt + 7r14rnt + 7t15Xtn

wheret 1 z) istheposteriorstimatefkt1 based on market informa-tion.Now substituterom6.2), (6.3), (6.4), and (6.1) and averageovermarkets toobtain theaveragestimate fkt+1:

kt+I = rl0 + (7TI + 7il3)kt + (n12 + 14) M + 7r15/i

+ B1[2 3 kt - k1) + 7c24(mt - 't) + ir25(Xt -j)], (6.8)

whereB1 isa functionftheelements fE and the ij, givenfor eferenceinAppendixB. Similar alculations ive

=t+1 t + JU+ B2 [ 23(kt - t)+ 724(mt - 7k) + 7r25(X - j)], (6.9)

whereB2 is given nAppendixB.The expectedyieldsas definedby (5.4) and (5.5) are calculatedin

thesameway.Forexample,r.tZ -

= p (z) - ft+1W

= 2 0 + 721 t + 2 2 t + c2 3[kt + ut(z)]

+ 7r24mt + 7t25[Xt + OM(z)]

-Ez(7r20 + Tr21kt+l+ 722M t+1 + 7R23kt+1

+ n24mt+1 + ir25Xt+l),




using the solution orprice, 4.5). ObservingthatE-[k,+,] = Ej[k,+?]and E.[M',+1] = E.[m,+1] and using the solutionforcapital (4.4) andthe monetary ule 4.2), one finds

Tmt(Z) = 7r2ikt + i222"t + 7t23[kt + ut(z)] + 7r24mt + 7t25[Xt + Ot(Z)]

- (721 + 7223)Ez(2t10 + 2llkt + '12't + )r13kt

+ n14mt + ic15Xt)

- (722 + 7r24)Ez(mt + Xt)

- 925-

Now usingthe estimates 6.3)-(6.5) and (6.1) and collecting erms, nefinds

rmt(Z) = (7r21 + 223)[1 - (71l + 7ti3)]kt

- (r21 + 7223)(7C12 + 7114)7t

? (1 - Ai)[7r23(kt - kt) + 7123Ut(Z) + r24(mt - t)

+ 7t25(Xt - u) + m250t(Z)]

+ cl, (6.10)

whereA1 sgivennAppendixB and C1 s a constant hichwillbe ignoredin the sequel. An analogous calculationgivesan expression orthe ex-pectedyieldon capital:

rkt(z) = A2[r23(k - kt) + r23Ut(z) + 24(mt -mt)

+ 7r25(Xt - i) + it250t(Z)] (6.11)

whereA2 isgiven nAppendixB.This completes he statement f the model, though mathematical

definitionf its solution s still two sections way. The giveneconomicparametersre the coefficientsn the assetdemandfunctions,0, . . ., 0C3

and Z) ...,,B3, theparameter , and thetwo variances 2 and o'. Theeconomic ssumptionsmply setofconditions elatingheseparametersto thesolution arameters: he coefficientsij n (4.4) and (4.5) and theremaininglements fthecovariancematrix

.Implicationsn the lope

coefficientsillbe developedn thefollowingection; hose n thecovari-ance matrixn section .

7. Implications on Slope Coefficients

Inserting heexpressionsor xpectedyieldsgiven by (6.10) and (6.11)intothecapitaldemand function5.1) yieldsk, 1(z) as a linearfunctionof the current tatevariables n market . A secondexpression f this



I i28 JOURNAL OF POLITICAL ECONOMY

functional relationship s given by (4.4). Since these two relationships areequivalent, their right-hand sides must be identically equal in kt, ht)kt,

mt Xt,Ot(z), and ut(z). Equating coefficients ives fiveconditions:

it11 = - [21X2 - 2(0 -AjJK23

- 12(t2l + it23)(1 - 11 - 7113), (7.1)

it12 - - [aA2- a2(l -A17f24

+ L2(7i2l + iT23)(7r12 + i14), (7.2)

it13 = [cxA2 - L2( - A1)]7r23 + a3 (7.3)

t14- [oLA2 - a2(l - A1)]ir24, (7.4)itl5 [a1A2 - a2(l - Al)]it25. (7.5)

Equating money demand and supply (eliminating mt'(z) between

[5.2] and [5.3]) and insertingthe yields (6.10) and (6.11) give an ex-pression for the current price, p,(z). Since the expression (4.5) must beequivalent, one obtains five more conditions:

7t21 = [P1A2 - 2(1 -Al)]723

- 92(7t21 + 7t23)(1 - - 7t13)) (7.6)

it22 = - [#1A2 - 2(1 -A11T24

+/2(7121 + 7r23)(7r12 + ir14), (7.7)

=t23 = [#1A2 2(1 - A1)]7t23 - /3, (7.8)

t24= [fl1A2 - /2(1 - A1)]ir24 + 1, (7.9)

=25 [/31A2 /2(1 - A1)]7r25 + 1. (7.10)

The two additional conditions forthe constant terms t10 and it20 will beneglected.

So far, then,we have ten equations involvingthe ten unknown rij and(via A1and A2) thefive unknown elementsof : Ck am, cink, (TuG, nd Ur2.

8. Implications on Covariances

Rationality of expectations also implies that the covariance matrix Zused by agents in forecasting s at the same time the truestationarycovariance matrix. For the exogenously given moments c2 and a', thisholds bydirectassumption.For theother elements ofE, some calculationsare involved.

From (4.4) one observes that

u,+1(z) = k,+1(z)-k, = t1ut(Z) + r150tZ)* (8.1)




Then, using the fact that

Ot+1(Z) = P0t(Z) + et)

the stationary moments U2 and aid are given by

22 i5 + PW13 2

1 - 1- (8.2)

1 - Wi3

provided 171131 1.

Averaging both sides of the solution (4.4) with respect to z gives anexpression for k,+1. Subtracting this equation from (6.8), one obtains

kt+-kt+l = (7r13 -723B,)(k- k,) + (714 - 7124B,)(Mrn - Mt)

- (1 -725B2)(Xt-s) (8.4)

Mt+ -t+l = -i21B2(kt - kt) + (1 -7r24B2)(t - m)

- (1 - 72B2)(Xt - 4) (8.5)

Provided the deterministicpart of the pair (8.4)-(8.5) (that is, thesystemobtained by settingxt = u for all t) is stable, it is a familiarcalculation to obtain three linear equations in the moments o2k a 2, and

Tmk'For reference, write it as

Uk kkamk KI1 ]f (8.6)

where the 3 x 3 matrix K1 is written in Appendix B.

9. Mathematical Solution: Preliminaries

The mathematical problem is now sharpened to: for given al,..., ,3)

2.3) oa2, a2 and p, find 7ra, . .. .* *25, v.,.2

2) U2, and umk which satisfy (7.1)-(7.10), (8.2), (8.3), and (8.6) such

that the difference equations (8.1), (8.4), and (8.5) are stable. In this

section, the size of the system will be drastically reduced by solving for

some coefficients in terms of others.

First, adding (7.1) to (7.3) and (7.6) to (7.8) gives two equations in

the sums 7t1 + 713 and '921 + 723- These are essentially the same equa-

tions solved for 7r, and 721 in section 2; their solution is diagrammed in

Appendix A. As in section 2, there are two solution pairs, one of which is

of economic interest. Denote these solution values 71r = 7trj + 713 and



1130 JOURNAL OF POLITICAL ECONOMY

- it2 = i21 + it23. FromfigureAl (AppendixA) one sees thatthesesolutionsatisfy

a3 < 91 < (9. 1)

and

< 7t2 < p3. (9.2)+ P2

Next,add (7.2) and (7.4) and concludethat

it12 + it14 = 0- (9.3)

Similarly,dd (7.7) and (7.9) and concludethat

it22 + i24 = 1- (9.4)Neitherof these classicalneutrality-of-moneyesults hould come as asurprise.

Third,from7.9) and (7.10), concludethat

it25 = it24 (9.5)

and from7.4), (7.5), and (9.5) that

it15 = it14-

(9.6)Fourth, olving7.8) and (7.9) for t23 gives

72 3 = -f3'924- (9.7)Then,from7.3), (7.4), and (9.7), oneobtains

iT13 = a3 - f3714- (9.8)

Reviewing he facts tated n (9.1)-(9.8), one sees thatall slope coeffi-cients7ij have been expressedn terms f CI14 and 7t24 and the now

"known" numbers r, and 7t2. Let me rename t14 and 7t24, it3 and 7i4

respectively.n terms f theseparameters1,..., 7r4, (8.4) and (8.5)become

kt+l -ktl = (3 - 1333 + f394B,) (kt kt) (9.9)

+ (it3 - n4B1)(?At-m) - (i3 - 4B,)(xt -It),

Mt+1 -Mt+ = /34B2(kt - kt) (9.10)

+ (1-

7t4B2)( m- Mt)-

(1 -74B2)(Xt-

The motion faggregate apitaland theprice evel,kt ndPt, s from(4.4) and (4.5):

kt+l = ir1o + ir1kt+ (it1 - a3 + fl3it3)(kt - kt)

- 3( -tmt) + it3Xt, (9.11)

Pt = i20 + Mt -2kt + (fl T4 - it2)(kt kt)+ (1 -4)(mz -m ) + it4XV. (9.12)




The informationontainedn the 10 equations 7.1)-(7.10) may nowbe convenientlyestated s

713 = [c(1A2 o2(1 - A1)]7r4, (9.13)74 = [:1A2 - 2(l - A1)]714 + 1. (9.14)

The termsAt, A2, B1, and B2 may similarly be expressed in terms of

7e12, 71r3, and 714; these implifiedxpressionsregiven nAppendixB.The problemofsolvingfortheequilibrium arameter alues is now

reducedto: find 13, 714, U2, a 2, Umk) and a such that 9.13), (9.14),(8.2), (8.3), and (8.6) aresatisfied. he fact hat he covariance tructure

and theresponse oefficients13 and 714 are mutuallydependentmakesthistaskdifficult,nd results ave beenobtainedfor pecial cases only.These willbe discussedn detail n subsequent ections.

At thispoint,however, he generalnatureof the dynamic ystem sfairly lear.Equations(9.9) and (9.10) describe heconsequences ftheunsystematichocks, t,on the deviations etween heperceived nd theactual aggregate tateof theeconomy, t-kt and m' mt.This auto-nomous,two-equation ystem onverts one-timepulse of monetary

"misinformation"nto an extended,distributedag effect.Equation(9.11) describeshemotion f apital stock s the umof "deterministic"part,which s essentiallyhesame as thecapital pathfoundn section ,and autocorrelated eviations bout thispath,determined ytheshocksand their agged effectsrom9.9) and (9.10). The effects n pricearegiven n (9.12).

10. Case 1: Centralized Market Clearing

The case in whichthe relative emandvariance '2 is zerocorrespondsexactly o the situation iscussed rieflyt theendof section inwhichmonetaryhocks re theonly xogenous isturbance o which he conomyissubject. incewithnovariation n0 all marketsre dentical, neconomywitha' = 0 maybe viewed as one in whichall trade takesplace in asinglemarket.

The algebraappropriate o thiscase is given n AppendixC. Briefly,

one observesirst

hat thefunction 2 is zero whenC2 =

0, implyingfrom 6.11) that expectedreal yieldson capital do not change withmonetaryhocks. t follows hat k =

2= = 0 is thesolution o

(8.6). Then,from9.13) and (9.14), thecoefficients13 and 714are 0 and1,respectively.

Insertinghese alues nto 9.9) and (9. 10) gives he olution or quilib-riumpriceand capital accumulation. n thiscase, k,(z) = k, = k1forall markets nd all periods.Similarly,mt= ml. In short, here s nomisinformation.he effectfmonetary hanges ncapital snil 7 3 = 0);




there s a proportional ffect n nominal prices (7t4 = 1). Monetarychanges are accurately conveyed to agents via price movements, eventhough unanticipated, nd the response s simply an adjustment n

nominalunits.While thiscase is of no particularnterest ubstantively,t does serve

to "justify" he apparatusset up in preceding ections, owhich shallshortly eturn.The introduction f separate, nformationallyistinctmarketss not a step toward"realism" or (obviously) elegance" but,rather, n analyticaldeparturewhichappears essential in some form)to an explanation fthewayinwhichbusiness ycles an ariseand per-sist n a competitiveconomy.

11. Case 2: A Purely Monetary Cycle15

The case in whichcapital stockdoes not respondto monetary hocksmay, n contrasto thepreceding ase,be ofpractical mportance,incecyclical variations in capital appear, at least at the casual level, to beof questionablequantitative ignificance. rithmetically,his case canbe obtainedfrom hepresentmodelby setting heelasticities finvest-

mentwithrespect oexpectedyields qual to zero. Ifa1 = 2 = 0 then,from 9.13), it3 = 0 and (see Appendix C) CZk akm 0. For allmarketsnd all t,kt(z) = kt= kt = 710/(l - i1).

The functions l and A2 are foundequal to Y/it4 nd p(l -y),respectively,here

a2 + a 2

a2 + a2 + a2

Then, from 9.14),74=

+ fl27 di

= 1 + f2 - thp(l -)

Letting o rangefrom toinfinity,hepriceresponse4 rangesfrom hehighvalue ofunity o a low value of (1 + ft2 pfll). In economicterms, s the fraction f demand variationdue to aggregatenominaldisturbances endsto unity, quilibriumpricestend to move in pro-

portion odemand hifts. s this ccurs, heoutput esponse ends ozero.Theirratio theslopeofthePhillips urve)tendsto infinity.

15 This is essentially parametric ersion f the model n Lucas (1972), except that nthe presentversion,monetary hanges are perceivedwitha distributedratherthan afixedone-period) ag. Setting 2 = 0 givesan exact counterpart o the model ofLucas(1972). A comparisonof the two givesa good idea of thecostsand benefits fworkingwith parametricallypecified emand functions ather hanwith preferenceunctionsfagents see n. 4 above).



I134 JOURNAL OF POLITICAL ECONOMY

which s impliedby the solution ound bove. Given exogenousmoneymovements s assumed n (4.2), (11.3) describes he way agents'beliefsaboutthe stateof theeconomymove throughimerelative o the motion

of the actual state.To get a more concretedea of the kind of "cycle" implied by this

solution, t is useful osimulate he response o a singleonce-and-for-alldemandshock even though he occurrence f sucha patternhas beenassumedto have zeroprobability).magine an initial ituationn whichperceivedand actual statesare equal: Mo = Mo. There is an initialshock to demand: x0 - = S. Thereafter,money growssmoothly tits average expansion rate: x, - = 0, t > 1. From (11.3), agents

will initially nderestimatehe true stock fmoneybutwill "catch on"at an exponential ate through ime:

Mt-Mt = (1 - y)tS, > 1. (11.4)

From (9.12), specializedto this case, the initial shockwill inducea price ncrease bovewhat had beenexpectednperiod0 intheamount7t4S. Priceswill continue ostay boveexpectationsue tolaggedadjust-ments n Mit utbyan exponentiallyecreasingmount.To be exact,

Ait Pt Z4(1 7)S t > 0 .

This motionwill not be affected y changes n the averagemonetarygrowth ate, lthough, fcourse, hepathofactual priceswillbe.

The movementsn flow ariables-output, mployment,nd consump-tion-which parallel thesepricemovements an be inferred rom heincome-expendituredentity2.1), the inkbetweenmonetaryxpansionand government pending, 2.5), and assumptions bout households'

preferencesoraborsupplied ndgoodsconsumed. orthe atter, ssumepurelyfor implicityhatconsumptionoes notvaryoverthecycle, othat fluctuationsn governmenturchases re absorbedby employmentfluctuations.n the case underdiscussion, apital and investmentrealso constant,o that 2.1) maythenbe solvedfor mployments a func-tionofg, = log (G,). Expandingthisfunction ieldstheapproximation

nt= lo + l17t, (11.5)

wherent s the og ofemployment.he elasticityl is theaverageratioof G to outputdividedby theelasticityfproduction withrespect olaborinput labor'sshare).

From 2.5) and (4.2), realgovernmentpendings in turngivenby

9t = Xt-aPt + Mr. (11.6)

Withcapitalfixed, 11.6) and (9.12) together ield

9t = (1 - 7r4)(Xt - mt + mt) + constant. (11.7)




Then,combining11.4), (11.5), and (11.7), the ime athof hepercentagedeviations f employmentrom tsnormal evel,resulting rom shockS,is given by

nt -ne = q1(1 - it4)(1 - Y)tS, t = 0, 1, 2,.. (11.8)

Similarly, he expectedyield on moneywill move in proportion oMt - 1t. The exact relationships, from 6.10),

rmt (1 - Y)(I + fll') T4(l - Y)tS, t > 0, (11.9)t - (+ 132Y)

where rat is the variable part of rmt(z) veraged over markets.The

relationship f thispattern n rmto observed yclicalpatternsn interestrates,which s by no meansa simple ssue, s discussed elow (sec. 13).One notes that the effects f an initial hock, n the purelymonetary

model,willpersist utcan never umulate:he argest ffectmustcome inthefirst eriod.To accountfor he observed radual cyclicalupswing, tappears that one must ntroduce ystematic atterns n the shocks ormodify he nternal tructuref the model.

12. Case 3: A Monetary Overinvestment Cycle

The preceding ectionexhibits unique solutionto the system 8.2),(8.3), (8.6), (9.11), and (9.12) forthe case when xl = a2 = 0. In thissection, pproximate olutions re developed and theirproperties is-cussedfor he ituationwhere c and x2 are smallbut positive. he detailsof thisexpansion re discussedn Appendix D; themain results re asfollows.

Forthe accelerator oefficientr3,one finds

73= [1 ? A _-

[Pal -x2 + (xfl2 - a2fll)y], (12.1)

wherey sthe varianceratiodefinedn theprecedingection.A sufficientconditionfor a positive cceleratoreffect,t3 > 0, is Pal - X2 > 0-

Sinceo, > a2, thiswillobtain fp is near 1. (Ifpwerenear ero,meaningthat relativedemand shifts ere nearlytransitory,ne would not expectan accelerator ffect,ince new capital can only be installed with aone-period ag). As withthe other eal consequences f monetary hocks,the accelerator ffectn investments larger he smaller s the fractionofdemand variancedue to nominal hocks. n summary,o be inducedto varytheinvestmentate, agentsmust i) be responsiveo perceivedfuture elative eturnsoa > 0), (ii) be convinced hatcurrent elativedemands are a good indicatorof these future eturns p large), and(iii) be convinced hat current rice movements ontain nformationncurrent elative emands y small).




The effects f introducing positiveaccelerator r3 on the laggedperceptionsfa monetarymovement re easy to describe n words. Theincreased capacitydue to an initial positive hock retardshe upward

adjustment f theprice evel to the new moneyntroduced y the shock.The adjustment f expectationso the shockwill then take place moreslowly han theexponential ace describedn (11.3). The detailsofthesemovementsn perceptionsre given n (9.9) and (9.10); expressions orthe coefficientsn theseequations,valid forLc, nd a2 small, are givenin AppendixD.

The characteristicootsof this system re near a3 and 1 - y,bothin the unit nterval, o thatfollowing one-time hock,perceptionsn

both capital and moneywill return o "normal" in a nonoscillatingfashion. he "crosseffects" re probablybothpositive:underestimationof capacity k, - k, < 0) leads tounderestimationfaggregate emand(mt+1 mt+i < 0), and, similarly, t+1- kt+1 ncreasesas m-mI

increases.n response oa pulseshock , bothk, - kt nd Mt- mtmoveproportionallyo S. One (butnotboth) oftheseerrors an continue omove n the same direction that s,errors ancumulate)whiletheotherdecays.Eventually, othtend to zero.

Given themotionof the perception rrorskt-kt and m' - mt, asjust discussed, he motionof actual apital stockktfollowingn initialshock S is given by (9.11). The initialeffects 7t3S; subsequent ffectsin the amedirectionre contributed ythetermr3 mt MtA). Offsettingeffectsrise romt kt. inceO3 < 711 < 1,and since ll three orcingterms end to zero,ktmust ventually eturn o itsnormal evel.

The consequences oremploymentfthesemovementsn actual andperceived tatevariables an be obtained s in theprecedingection.Thepresence f capitalmakesthese alculations oth morecomplicated ndmore interesting. gain, take consumption o be constant,"solve"(2.1) for the log of employment,nd expand to obtain the analogueof 11.5):

nt= 10 + 11gt + qi2(kt+1 kt) + 13kt. (12.2)

As before, heelasticity1 is theratioof G to outputdividedby labor'sshare; 12 is the veragecapital-outputatiodividedby abor's hare;andt3 iscapital's haredividedby abor's hare.Real spending,t, s obtainedfromI1.6) and (9.12):

=t 72kt - (#37r4 - 712)(kt - kt) + (1 -4)(xt-mt + me). (12.3)

Combining 12.2) and (12.3) yields hetimepathofemployment.The direct multiplier" ffect n employmentfa shock qgt) works

much as in theprecedingection: here s an initial ffect ue toa move-ment n xtfollowed yadditionaleffects ue to informationalags.Theeffectnew to this sectionis the accelerator term172(kt+1 - kt),which can




be relatively arge even for small values of it3. Further, ince capitalreturns o normal, the term it2(k,+1 - k,) must eventuallymake anegativeontributiono employment,ossibly riving mployment elow

its normal evel,even n the absence of a downward hock.Movementsn expectedyieldson both money nd capital will,as in

the preceding section, be procyclical. 6 These facts may be verifiedfrom 6.10) and (6.11) , but the exact expressionsneed not be given here.

13. The Role of Interest Rates

The procyclical pattern of interestrate movements has perhaps attracted

more theoretical and empirical attention in recent years than any other"stylized fact" concerning business cycles. The procyclical movement ofthe two expected yields, as shown in (11.9) and, for the general case, in(6.10) and (6.11), raises the hope that this fact too is accounted for bythe model developed above. The question is worth examining, althoughit will turn out that a satisfactory nswer remains beyond the scope ofthispaper.

Formally, the model above considers internal equity financing only,

in contrast with the established convention that, in theories which con-sider one source offinancing only, that one source should be bonds. Thisdeparture is obviously necessitated by the presence of uncertainty: theclaim to an uncertain yield cannot e a single type of bond. One couldadd private bonds as an additional source of financing. If bond trans-actions were localized, as are goods transactions (that is, exchangesamong agents in a single market), this would be easy to do, and onecan conjecture that bond yields would move as the expected yields in(6. 10) and (6.1 1). The interesting ssue, however, is to examine the con-sequences of a single economy-widearket for some standardized kind ofbond whichwould clear in an integralsense but not foreach fixed marketz. This modificationwould involve a major change in the informationstructure of the economy, since the equilibrium interestrate (or bondprice) would depend only on aggregate tate variables, and hence itsvalue would conveyto agentssome aggregate information ncontaminatedby local disturbances.

To see the effects f this, return to the inferenceproblem solved by

agents in section 6 and suppose that agents also observe the value of aknown linear function of kt - kt,Mt - mt, and xt. The extreme conse-quence occurswhen capital movementsare unimportant,as in the purely

16 Friedman (1971, p. 327) observes hat cyclical variations n the average marginalproductivityf capital are of slight quantitative mportance, o that variation n theexpected verage yield on capital mustplaya minor yclicalrole. This fact, s Friedmansuggests lsewhere n the same article, s thus entirely onsistentwith an importantcyclical rolefor verageexpectedreal yields.



1 J38 JOURNAL OF POLITICAL ECONOMY

monetarymodel of section11. In this ase, the nterest ate will conveythe aggregate tate of the economy erfectlyo agents,eliminating hereal part of the cyclealtogether.17With an accelerator ffect resent,

it seems ikely hattheexistence fan economy-wide ond marketwoulddampen cyclicalmovements utnoteliminatehem r alter heir ualita-tivecharacter.Without urthernalysis,however, he questionremainsopen and, clearly, rucial.

The interest ate question llustrates n interestingnalyticaltensionwhich mustarise n any cycle theory ased on incomplete nformation.On the one hand, it is easyto postulate gents nd market nstitutionswhich gnore rfoolishly aste nformation:heresults a theorywhich

seriously nderstatesgents' abilitiesto vary theirdecision rules withchanges n theenvironmentsuch as, for xample, hetheory nderlyingthemajoreconometricorecasting odels). t is equally easytopostulate"efficient" ecuritiesmarketswhichrapidly transmit ll informationoall traders: he result s a staticgeneralequilibriummodel. To observethat one mustavoid both extremes o understand he businesscycledoes not take one veryfar n discoveringhecorrect centrist"model,butit seemsnonethelessn essential ointofdeparture.

14. Remarks on Testability

The modeldescribedn section12,and any othervariant n thisgeneralclass,ascribes alues to all aggregatemoments: hecomplete ovariancefunction f thevector fobservable ariables. ince there remanymoresuchsamplemoments han there re freeparametersn thesystem,t isclearthatthe model has empirical ontent.

In the absenceof an economic heory n the behaviorof theshocks,one would in practice begin by describing he shocks tochastically ysomead hoc method,possibly singonly pastvaluesofthe series tself,possibly elatingt to movementsn other tate variables.Then, basedon thesefindings,ne wouldneed to redo thetheory bove (especiallythe nferenceroblemn sec. 6), assuming hat thesame patternn thedisturbances alsoknown o traders.f, s seemsikely, fairly omplicatedpattern say,three r four arameters)s required o describe heshocks,thiswill ead tomore,otfewer,estable estrictionsn the olution aram-eters.18 n short, here ppearsto be little isk fthevacuitywhichmarsso much of distributedag econometrics.

In additionto aggregatepredictions,hetheory lso "predicts"that

17 Could not a given nterest atemovement ndicateambiguously ither highx, ora highm, - Ma,? s a transientffect, es,butnot n thestationary istribution;ee (8.5)withkt-kt = 0.

18 See Sargent'sapplicationof the principleof rationality o the Fisherian nterestrate-inflationatedistributedag (Sargent1973a).



MODEL OF BUSINESS CYCLE I 39

deviations rom verage n the demandfor ndividualproductswill beindependent rom roduct o product nd through ime.One may (as Idid [Lucas 1972]) take these predictions"metaphorically'9as onetakes

the predictionhatall individuals re indistinguishablend live forever),but it is instructiveo ask which ovariance tructuresor ndividualde-mand shockswill ead to aggregate ehavior like" thatdescribed boveandwhichwillnot.The answer eems o be that neneedseach individualmarket hock obe expressables a linear ombinationf largenumberof roughly ommensuratendependent hocks so that the law oflargenumberspplies s usedabove) plus a singlehock ommon o all markets.This assumption, amely, hatthere xists ome singlerandomvariable

identifiables aggregate emand, s testablendsurely eservesystematicexamination.

15. Remarks on Policy Implications

All aggregateoutputmovementsn the models studiedabove resultfrommovementsn a singlemonetary-fiscalhockto aggregate emand.Evidently,hekey to any stabilization olicy n sucha settingwouldin-

volve the elimination f any avoidable components f the varianceofthis hock.The present tudy, hen,provides rationalizationorruleswhich smooth monetarypolicy,exactlyas did the earlierstudiesofLucas (1972), Sargent nd Wallace (1973), and Barro 1975). Similarly,it rationalizes heanalogousfiscalruleof continuous udgetbalancingand rules o stabilize hequantity fprivatemoney, uchas larger eserverequirementsor anks.Though tcouldbe extended o do so,thepresentmodel shedsno lighton therelativemportance fmonetarynd fiscal

effects,inceall shocks, yassumption,nvolveboth monetarynd fiscalelements.

If, as seems likely n fact, some components f aggregatedemandvariance are unavoidable, hepresentmodeloffers he additionalpossi-bility f tabilizationyaffectingheresponseharacteristicsf heprivatesector. For example,a fiscal stabilizerwhichreducedthe parametersac, nd C2 (theelasticitiesf nvestment ithrespect operceived, retaxrate-of-returnhanges) would convertthe economyof section 12 to

themore table conomy f ection11.Thisfeasibilityf tablebut reactivestabilization olicieswillobtain, t would appear, n anymodel n whichthe effectsf shocks ersist hroughffectsn capitalaccumulation.

In my view,thedesirabilityf reactivepolicyrules s a more seriousissue than s their easibility. tax policywhichreducedtheresponsive-nessof nvestmento aggregate emand changeswould,as can be seen

19 In his persuasive ommenton Lucas (1973a), Vining (1974) utilizesa literal n-terpretationf this ssumption o obtain a suggestivempirical est.




from the analysis in the preceding sections, reduce the variance ofaggregate output and employment. At the same time,such a policy wouldnecessarily reduce the responsiveness of investmentto relative demand

shifts,retarding the movement of resources into the socially most de-sirable activities.Since thepreferences nd production possibilities n thismodel economy have not been made explicit, one cannot conclude withthe presumption that the balancing achieved by the private sector inthis model is efficient.On the other hand, it has not been necessary tointroduce any of the standard types of "market failure" in order to ac-count for the main featuresof the observed cycle.

16. Conclusion

This paper develops a theoretical example of a business cycle, that is,a model economy in which real output undergoes serially correlatedmovements about trend which are not explainable by movements in theavailability of factors of production. The mechanism generating thesemovements involves unsystematic monetary-fiscal shocks, the effectsofwhich are distributed through time due to informational ags and an

accelerator effect. Associated with these output movements are (i)procyclical movements n prices, (ii) procyclical movements in the shareof output devoted to investment, nd (iii), in a somewhat limited sense,procyclical movements n nominal rates of interest.

This behavior is obtained under assumptions about expectationsformationwhich seem suited to the studyof a recurrent vent: agents arewell aware that the economy goes through recurrent "cycles" whichdistortperceived ratesofreturn.On theother hand, the transitory atureof real investment opportunities forces them to balance the risk of in-correctly responding to spurious price signals against the risk of failingto respond to meaningful signals.

Appendix A

Equations 2.14) and (2.17) aretwo quationsntheunknownarametersr11and7r21*Solvinghemmeans ssentiallyindingheroots f quadratic, hichcan ofcourse edone nseveralways.A particularlyonvenientay sto addf62times2.14)to 2 times2.17), btaininghe ine

f2 + (92al-

fla2)(51 7rj-a3fl2+ a2f3 (.1~21 = 2+ f2 11 a3f2+~~ (A.l1)

Rewrite2.14) as0(3 - (1 + aj3j)Dj (A.2)

a2(l - 711)

The two olutionso A.1)and A.2) (that s,to 2.14] nd 2.17])are llustratedin figure l. The relevant oot conomicallys in the southeastuadrant;tsatisfieshe nequalities2.19)and 2.20).



MODEL OF BUSINESS CYCLE 114I

72r

1+a181 I (A,2)

a2

az _ i/(A~~~~~~(,1)

FIG. Al

To obtain 71i = 7z1 + 71r3 and - 72 = 7Z21 + 7123 (as in sec. 9), one pro-ceeds in the same way. Adding (7.1) and (7.3) yields

7Zl = 3 - a2(- 7Z2) 1 -7ti)* (A.3)

Adding (7.6) and (7.8) gives

- 72 = - fl3 - I2(- 712)(1 - 71l). (A.4)

These are equivalent to (2.14) and (2.17), with c51= 0. Thus, (71k, - 712) satisfy(A. 1) and (A.2), with c1 = 0, and theirsolution values appear in figureAl, with

61 = 0. The inequalities (9.1) and (9.2) are easily verified.

Appendix B

Some expressionswhich arise in the text are

B1 = ?p [713(7123d + 7r24Umk) + 71l4(7123UTmk + 7124 m) + 71157125U] (BEl)

B r- 2 2 2, B2B2 = aP (7123Umnk 7124am +7125U ), (B.2)

A1 = (7r21 + 7123) B1 + (7122 + 7124)B2, (B.3)

A2 = [713;23(7123u2 + 7125UUO) + (71237115 + P7r25)(7123UuO + 71252)].

(B.4)



I142 JOURNAL OF POLITICAL ECONOMY

Define he quantitiesM, D1, D4 by

M = -3 - 2fJ3Umk + rmn 32 + /3au - 2/]3crO+ O, (B.5)

D=

M-'(fl22- _3Umk),

(B.6)D2 = M-1(- f3amk + Ur2 + a 2), (B.7)

D = M-1(fl2a 2- flauo), (B.8)

D4 = M-1( f 3Jo.+ aJ) (B.9)

Then, afterthe elimination f parameters escribed n section9, (B.1)-(B.4)can be rewritten

B1 = -3

D1 +'3

(D1 + D2), (B.l0)/337r4 7

B2 = - D2) (B.ll)7T4

A1 = 7r2 -3 fl3 3D1 + 1 - 3 D2' (B.12)l337(4 794

A2 = (3 - f337r3)D3 (p - 13373)D4. (B.13)

To obtainthe matrixK1 used in (8.6), firstbbreviate he coefficientsf 8.4)and (8.5) by

C1l = r13 - 2r23B1 = a3 fl3n3+ 133n4B1,

C12 = 714 - 7r24B1=g15 - r25B = 73 - 7t4B1,

C21 = -7(23B2 = 9374B2,

C22 = 1 - )T24B2 = 1 - 7r25B2 = 1 -74B2.

Thenl

C21 2C11C2 C2

K1 = C1 C21 C11C22 + C12C21 C12C221. (B.14)Co21 2C21C22 C22

Provided he matrix Cij) is stable (as requiredbythe definitionf a solutionused here),theprocess 8.4)-(8.5) has a unique stationary ovariancematrix.Sincethis ovariancematrixs also a solution o (8.6) (and viceversa), tfollowsthat 8.6) has a unique solution, r thatK1 - I is a nonsingularmatrix.

Appendix C

For the case a2 = 0o Ca2 = a = a 2 = 0 from 8.2) and (8.3). Then, from(B.8)-(B.9), D3 = D4 = Oso that romB.13) A2 = 0. It alsofollowshatC1l =

(a3/163)C21 and C12 = (M3/fl3)C22, o thatbydirectnspectionf 8.4) and (8.5)one sees thata2 = aa2U and Ukm = aC3a,. Thesefactspermit hecalculationofM, D1, and D2 as functionsf thetwovariances m2 nd a2. Insertinghese ntothe third quationof 8.6), one obtains cubic n theunknown m.The root erooccurstwice; the thirdrootis negative.Thus, the unique solutionof (8.6) isa2 = km = C2 = 0.

With thesevariances,D1 = 0 and D2 = 1. Inserting hesevalues into the




expression orAl from B.12), one finds hat713 = 0 and 74 = 1 is theuniquesolution o (9.13) and (9.14).

For thecase a, = a2 = 73 = 0, use (8.6) to solvefor r2and Ukm as functionsofc2 + a2. Evidently,0, 0) is a solution,ince n this ase DI = 0. If D1 = 0,theresnoother olution.fD1 t 0,one finds hat 2 < 0, an impossibility.hesolution iven n thetext s thusunique.

Appendix D

Equations 8.2), (8.3), (8.6), (9.11), and (9.12) are seven quationsn theunknownreduced-formarameters713, 714, Uk amk, a, u,(, c(ug). In section11 twasfoundthat 0, 1r4,0,0 , ,2 0, 0), with14given n (11.1) and j2 by (11.2), is theuniquesolutionwhen al = a2 = 0. Let a2 = Baj, where 4e(0, 1) is a constant. Then the

implicit unctionheoremmpliesthatforal sufficientlymall,a differentiablesolution xists.This solutioncan be approximated or a, small by expanding(713,714, Uk a 2,a,,uo) about the point (0, 714,0, 0, C, 0 0).

Carrying ut thisexpansion, 12.1) is immediate rom 9.3) and (11.1). Theapproximate oefficientsj (see AppendixB) ofthedifferencequations(9.9)and (9.10) are readily, f tediously, btained n thesame way. The expressionsgivenbeloware obtainedbyexpandingn al and bydiscarding ermsnvolvingpowers fyof2 or higher. he full xpressionsrenotdifficultoobtain,butthereis little o be gainedbyrepeating hemhere:

C11 = - (l -Y)73 +3 e n- 713

C12 = (1 -7)713 -a a,

C21 = f3y(l - l3LX 2 3P )1 a3 1 - a3

C22 1-

Y + ,3B ya3 + 2fl 1Y- C 73

Fora, (and hence 3) small, he roots f Cij) are seen tobe neara3 and 1 -y(the two diagonal elements).C21 will be positive, s asserted n the text.C12,treated lso as positiven thetext, an in facthave either ign, ven for , small.Forysmall, twill be positive.

References

Barro, Robert J. "Rational Expectations and the Role of Monetary Policy."Working paper, Univ. Chicago, 1975.

Brock, William A. "Money and Growth: The Case of Long-Run PerfectFore-

sight." Working paper, Univ. Chicago, 1975.Friedman, Milton. "A Monetary Theory ofNominal Income." J.P.E. 79, no. 2

(1971): 323-37.Graybill, Franklin A. An Introductiono LinearStatisticalModels. Vol. 1. New York:

McGraw-Hill, 1961.Haberler, Gottfried. Prosperity nd Depression. Rev. ed. Cambridge, Mass.:

Harvard Univ. Press, 1960.



1I44 JOURNAL OF POLITICAL ECONOMY

Lucas, Robert E., Jr. "Expectations nd the Neutrality f Money." J. Econ.Theory (April 1972): 103-24.

. "Some International videnceon Output-Inflationradeoffs."A.E.R.63 (1973): 326-34. (a)

. EconometricolicyEvaluation:A Critique. arnegie-MellonUniversityWorking aper. Pittsburgh: arnegie-Mellon niv., 1973. (b)

Lucas, RobertE., Jr.,and Prescott, dward C. "Equilibrium Search and Un-employment." . Econ.Theory (February1974): 188-209.

Muth, JohnF. "Rational Expectations nd theTheory of PriceMovements."Econometrica9 (July1961): 315-35.

Phelps,Edmund S., et al. Microeconomicoundationsf Employmentnd InflationTheory.ew York: Norton,1969.

Sargent,Thomas J. "InterestRates and Prices n the Long Run." J. Money,

Credit,ndBanking (February1973): 385-449. (a). "Rational Expectations,heReal Rate of nterest, nd theNaturalRateofUnemployment." rookingsapers con.Activity(1973): 429-72. (b)

Sargent,Thomas J., and Wallace, Neil. "'Rational' Expectations,heOptimalMonetary Instrument,nd the Optimal Money Supply Rule." Workingpaper,Univ. Minnesota,1973.

Tobin,James."Moneyand EconomicGrowth."Econometrica3 (October1965):671-84.

DiscussionofSargent 1973b). BrookingsapersEcon.Activity (1973):477-78.

Vining,Daniel R., Jr."The Relationship etweenRelative and General Prices."Working aper,Pittsburgh: arnegie-Mellon niv., 1974.

an equilibrium model of the business cycle

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