a linear model of cyclical growth

8/2/2019 A Linear Model of Cyclical Growth

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A Linear Model of Cyclical GrowthAuthor(s): Hyman P. MinskyReviewed work(s):Source: The Review of Economics and Statistics, Vol. 41, No. 2, Part 1 (May, 1959), pp. 133-145Published by: The MIT PressStable URL: http://www.jstor.org/stable/1927795 .

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A LINEAR MODEL OF CYCLICAL GROWTHHyman P. Minsky*

PROFESSOR SAMUELSON's ath-break-ing article on the "InteractionBetween theMultiplier Analysis and the Principle of Accel-eration" appearedin this REVIEW almost twen-ty years ago. A large literature has developedin which the basic ideas of that article havebeen appliedto both businesscycle and econom-ic growth problems. In a considerableportionof that literature, Samuelson'swarningthat "therepresentation s strictly a marginal analysis tobe appliedto the study of small oscillations"hasbeen overlooked.'Samuelson's warning can be interpreted asmeaning that the time series generatedby anyparticularsolution of the model will determineactual income for only a short time. Given themathematical model, the relevant particularsolution can change due either to (i) the accel-eratoror the multipliercoefficientschanging (asSamuelson suggests), or (2) the imposition ofnew initial conditions. Goodwin2has examinedvarious models in which the acceleratorcoeffi-cient is a variable. These non-linear modelsaremathematicallycomplex, and the specific limitcycles that Goodwinderives obviously are dueto the special assumptionshe makes about howthe path of income affects the accelerator co-efficient. Hicks3 has investigatedhow an other-wise explosiveacceleratormodelwill be affectedby floorsand ceilings. In this paper such floorsand ceilings will be interpretedas imposingnewinitial conditions, and therefore this paper canbe considered a reinterpretation of Hicks'ssetup.4

We will work with a slightly modifiedversionof Samuelson'smodel, and assume thatCt= ao + alYt-1)It -/ (Yt-1 - Yt-2) (2)so thatYt (al + P) Yt-1 Yt-2 + ao (3)where Ct, It, and Yt are the tth day's consump-tion, investment, and income; ao is the "zeroincome" consumption; a, is the marginal pro-pensity to consume; and /8 is the acceleratorcoefficient.5The solution to equation (3) is

t= Al1t+ A2+k0 (4)where the roots,tj and U2depend upona, and 8,the coefficientsA1 and A2 are determinedby theinitial conditions,and ko [ = ao/(i - a,) ] is de-termined by the "zero income" consumption.In the solution equation (4), ko has a naturalinterpretationas the income at which consump-tion equals income."The a, and /8 coefficients will be assumed tobe constants with values such that an explosivetime series will be generated, i.e., in equation(4) /k > 12 > I.7 As a1 and / are constants, so

* The major portion of this article was completed whilethe author was a Visiting Associate Professor at the Univer-sity of California, Berkeley, and he wishes to thank thosegraduate students who patiently sat through the presentationof this material, as well as Professors Irma Adelman andRoger Miller.IP. A. Samuelson, "Interactions Between the MultiplierAnalysis and The Principle of Acceleration," this REvIEw,xxi (May I939), 78. Reprinted in Readings in BusinessCycles(Philadelphia, I944), 269.

2R. M. Goodwin, "The Non-Linear Accelerator and thePersistence of Business Cycles," Econometrica, xix (Janu-ary I951), I-I7.'J. R. Hicks, A Contribution to the Theory of the TradeCycle (Oxford, I949).'References to the influence of initial conditions upon thetime series generated by an accelerator process are scarce.

One such is "But he [Hicks] (correctly) argues that disin-vestment is limited to the size of the depreciation allowances.This implies that after a downswing we do not follow thecourse of the cycle which produced the previous upswing.Instead we start a new cycle on the basis of new initialconditions." J. S. Duesenberry, "Hicks on the Trade Cycle,"Quarterly ournal of Economics,Lxiv (February I950), 465.

6 Samuelson's equation (i) is Ct = a, Yt-l and his equa-tion (2) is It = f3(Ct - Ct1). The modification of equation(2) does not change the essential character of the results andit is mathematically convenient. The modification of equa-tion (i) will enable us to introduce the effects of changes inko (either through a Duesenberry ratchet effect or a Tobin-Pigou asset effect) into our analysis. For the mathematicaldetails of the derivation of equation (4) see either W. J.Baumol, Economic Dynamics (New York, i95i) or R. G. D.Allen, Mathematical Economics (London and New York,I956).'In equation (3), if Yt = Yi- = Yt-2, then Yt = ko.7Thea, and j2 in the solution equation Y = A At+ A2 t

ko are the roots of the characteristic equation f(x) =x2- (al + p)x + p to the second-order difference equationyt = (a, +P)yt- - pyt -2. The roots are determined by set-ting f(x) = o. The following is known about f(x): (i)f(o)=P, (2) f(i) = i-a,>o, (3) f(al +,8) =j, (4) the(asi+8) ai?P3minimum value of f (x) is f ,(5) for o < x


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I34 THE REVIEW OF ECONOMICS AND STATISTICSare ul and t2. Rather than assume that theaccelerator and multiplier coefficients change,the marginal nature of the formulationwill betaken into account by intermittently imposingnew initial conditions. The initial conditionswill be interpretedas reflectingeffective supplyconstraints.The particularformulationto be used has theaggregate supply of income increase at an exog-enously given rate and the floor to income de-pend upon the capital consumption rate. Thenature of the time path of income generated bythe model depends upon whether the rate ofgrowth of ceiling income is greater than, equalto, or less than the minor (smaller) root of thesolution equation. By also taking into accounta "ratchet" in the consumption function, theapparatus to be developed can generate either(a) steady growth, (b) cycles, (c) booms, or(d) long depressions.Formally, it will be shown that the simplelinear accelerator-multipliermodel can be usedas a flexible frameworkfor the analysis of botheconomic growth and business cycles and thatthere is no analytical need to separate the twoproblems: that is, the model generatesboth thetrend and the cycle.8

EconomicAssumptionsIn order to use a linearaccelerator-multipliermodel as a flexible frameworkfor the analysisof growth and cycles, the flexibility must bebuilt into the economicassumptions. Of course

the fundamental assumption is that the accel-erator-multiplierapparatuscan be used to rep-resent the intertemporalgenerationof aggregatedemand. The validity of this assumption de-pends upon whether this apparatusyields satis-factory interpretationsor predictionsof events.No further argument for the validity of theaccelerator hypothesis is offered in this paper.The other economic assumptions made are:

i. The consumption function can be repre-sented by a straight line, Ct= ao+ a, Yt-1. Thevalue of a, is constant and the value of ao de-pends upon factorswhich, in turn, can be repre-sented as depending upon past income. By as-suming that ao changes intermittently and in-creasesmorereadily than it decreases,a ratchetis built into the consumptionfunction.2. At each date there exists an exogenouslydetermined ceiling to income which dependsupon population growth and technologicalchange, and which does not depend upon theexisting capital stock.3. During each time period, there exists amaximumpossible amount of net disinvestment,which depends upon the capital stock and atechnologically determined depreciation rate.The consumption function is a determinant ofaggregatedemand,whereas the ceiling and floorto incomeare determinantsof aggregate supply.The assumption that ao, the zero income con-sumption level, depends upon past incomes isbased upon two, not necessarily mutually exclu-sive, hypotheses. One is the Duesenberry chang-ing preference system hypothesis,9which in ourinterpretationhas aodepend uponprevious peakincome. The other is the Tobin-Pigou type ofasset (wealth) consumption relation,'0which inour interpretationhas ao depend upon the realvalue of assets. As far as the mechanics areconcerned either interpretation would suffice.However,the Duesenberryratchet is insensitiveto changes in the relative prices of assets andincome, whereas a Tobin-Pigou ratchet reactsto such changes. Thus, a Tobin-Pigou ratchetcan transmit financial market developments tothe real sector. As both preferencesystems and

creasing.For two distinctreal roots to exist it is necessarythat f( ) i and both

2> i and g2> i. If it is on the positively sloped arm,ai+Pthen - < I and both o < < I ando < I,2 < I. It is2impossibleto have Al> i and A2 < i; (ai+P)'- 4P> 0and p> i aretheconditions or g >U2 > I.8A. Smithies, "Economic Fluctuations and Growth,"Econometrica, xv (January 957), I-52, has a similar per-spectiveand usesmany of the same ngredients.S. S. Alexander,"The Acceleratoras a GeneratorofSteady Growth,"QuarterlyJournal of EconoMics, LxM(May I949), I74, demonstrated hat such an acceleratorapparatus ouldnot be used to generateboth steadygrowthand the businesscycle without additionalstabilizing actors.This paper negatesAlexander's esults.

9J. S. Duesenberry, Income, Saving and the Theory ofConsumer Behavior (Cambridge, I949)'.10 . Tobin, "Relative Income, Absolute Income and Sav-ing," in Money, Trade and Econo-mic Growth (New York,I951), I35-56; A. C. Pigou, Employment and Equilibrium,2nd ed. (London,I949), I3I-34.


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A LINEAR MODEL OF CYCLICAL GROWTHasset holdings do change over time, an eclecticinterpretationof the ratchet in the consumptionfunction is both useful and defensible.As the accelerator-multiplier apparatus ishere interpreted, t is simplerto consider chang-es in the relativeprices of incomeand wealth asexogenous factors which influence the genera-tion of aggregatedemandby affectingao.There-fore, in the formal model, aowill be assumedtodepend upon the previous peak income. In in-terpreting events, any developmentwhich raises(lowers) the ratio of consumingunits' wealth toincome will tend to increase (decrease) ao. Inmore concrete terms, a large increase in theratio of monetary assets to income (such astook place during World War II) will tend toraise ao,whereasa large fall in the marketpriceof assets relative to income (such as took placein the fall of I929) will tend to lower ao. It willbe shown that by feeding financial and moneymarket developments into the formal modelthrough the ratchet in the consumption func-tion, booms and depressions of varying ampli-tudeandlength can be generated.The form adopted for the consumptionfunc-tion implies that there is a positive equilibriumlevel of incomeat which consumptionequals in-come. If there is a trend in peak incomeand ifao is a linear function of peak income, theequilibrium ncome will exhibit the same trend.By appropriateassumptionswith regardto de-preciation, the same trend can be introducedinto the floor to income. With these trend as-sumptions a model is constructed which gen-erates a constant relative amplitude cycle. Itwill be shown that if these trendsare not postu-lated, the modelgeneratescycles with increasingrelativeamplitude.At each date, the ceiling real income is theincomewhich that date's labor force could pro-duce, given the date's technology, if the opti-mumcapital stock existed. As the actual capitalstock is less than or equal to the optimumcapi-tal stock, the ceiling income is not necessarilythe existing capacity income. Ceiling incomewill change due to changes in the labor forceand technology. It is assumed that a techno-logically progressivesociety is being consideredand that the labor force is growing. As a result,the ceiling real income, Yt(c), is always growingindependently of what the actual income and

capital stock may be. That is, y(c) = rY(c)where X > i and is also greater than the rate atwhich the labor force is growing. In the formalanalysis, the assumptionwill be made that X isa constant."The validity of the assumptionabout the be-havior of the income ceiling is questionable.This paper,at a formal level, can be interpretedas an examination of the joint implications ofthe accelerator-multiplierand the ceiling hy-potheses. However, as will be shown, the rateof growth of ceiling income can vary withinconsiderablelimits without affecting the quali-tative characteristicsof the model. Therefore,if desired, this rigid assumptioncan be relaxed.The maximumpossible disinvestmentduring

any period is determinedby the technical factthat at least a portionof the capital stock, fixedcapital, is long lived. Hence, duringany periodonly a small part of such capital can be con-sumed. We will assume that the ratio of capitalconsumptionto peak income capital stock is aconstant. Thus, given the capital stock beingconsumed, the maximum disinvestment whichcan take place per period is determined. To de-terminethe floor to income, this maximumpos-sible disinvestmenthas to be joined to behaviorassumptions which determine the behavior ofequilibrium ncome, ko, when income is falling.This maximum possible disinvestmentis a sup-ply condition, since it determines the greatestpossibleexcess of consumptionoverincome.The disinvestmentassumption s quite heroic.The ratio of possible disinvestmentper periodto the capital stock certainly depends upon thecompositionof thecapitalstock. As an example,the higher the ratio of working to fixed capital,the higher can be the ratio of a period's disin-vestment to the capital stock. It can be expect-ed that the ratio of working capital to fixedcapitalwill be higherat the beginningthan dur-

1 R. F. Harrod, Towards a Dynamic Economics (London,I948), defines a natural rate of growth which is the rate ofgrowth of the labor force. Our rate of growth of ceilingincome can be interpreted as a "converted" natural rate, toallow for technological change. However, Harrod's result,that there is a knife-edge equilibrium rate of growth, whenthe "natural" rate is equal to the "warranted" (full-employ-ment savings equal investment) rate of growth, is not borneout by our analysis.R. M. Solow, "A Contribution to the Theory of EconomicGrowth," Quarterly Journal of Economics, LXX (FebruaryI956), 65-94, also negates the Harrod conclusion of a knife-edge equilibrium rate of growth of income.


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136 THE REVIEW OF ECONOMICS AND STATISTICSing the later part of a downturn. Thus, there issome expectation that the maximum possibledisinvestmentper period decreasesas a depres-sion wears on.In addition,the size of the capital stock beingconsumedduringany period is open to debate.Is the maximum possible capital consumptionduringa period based upon the largest amountof capital which existed duringany previouspe-riod, or is it based upon the capital stock inexistenceat the beginningof the period? If thefirst assumptionis made, the maximumpossibledisinvestment will not decrease when net dis-investment is taking place, whereas the secondassumption implies that net disinvestmentperperiod will decrease as a depressioncontinues.Once again, since the mechanicsof the model isindependent of which assumption is made, wewill assume that the disinvestment ceiling isdeterminedby the maximumpriorexistentcapi-tal stock. The qualitative impact of variationin the amount of maximumdisinvestment willbe considered in the verbal discussion.

GeneralCharacteristicsf the ModelIn the models to be considered,the accelera-tor-multiplierprocess in conjunction with the

initial conditionsdeterminesaggregatedemand.Actual, or realized,incomeis determinedby theinteraction of aggregate demand and supply.We will assume that aggregatesupply is a lim-itational factor determinedeither by the auton-omously growingceiling or the floor to income.Aggregate demand determines actual incomeunless the income so determined s inconsistentwith the aggregate supply conditions. When-ever this occursactual income is determinedbyaggregatesupply, and simultaneouslya new re-lation will be determinedwhich generates sub-sequentaggregatedemand.Given that the acceleratorand the consump-tion coefficientsare constants, the two roots, Puland /2, of the solution equation are also con-stants; and we assume thatpl.and 2 arerealandgreater than one. With ko (the consumption-equals-income income) known, two observedincomes are needed as initial conditions to de-termineA1 and A2 in the solution equation. Asolution equationwith A1andA2determinedbyspecific initial conditions will be called a par-

ticular solution equation. As long as aggregatedemand generated by a particular solutionequation falls in the range determined by theceiling and floor to income, actual income willbe generated by this particular solution equa-tion. However, it is assumedthat /ju,the domi-nant root of the solution equation, is very largecomparedwith possible rates of growthof ceil-ing (decline of floor) income. Hence, incomeasgeneratedby a particularsolution equationwill,in time, exceed (fall below) the ceiling (floor)income; whenever this happens, supply condi-tions will determineactual income. Such actualincomes determined by supply conditions willbe the initial conditions in determining newvalues of A1 and A2 and this new particularsolution will generate successive aggregate de-mands.It will be shownthat the qualitativebehaviorof the model depends upon the relative valuesof the minor (smaller) root of the solutionequation and the rate of growth of ceiling (de-cline of floor) income. If the rate of growthofceiling (decline of floor) income is equal to orgreaterthan the minor root of the solutionequa-tion, then aggregate demand as determinedbythe particular solution equations will continu-ously press against the ceiling (floor) so thatsteady growth (decline) of actual income willtake place. If the rate of growthof ceiling (de-cline of floor) incomeis smaller than the minorroot, then actual income will bounce off theceiling (floor), that is, a turning point will begenerated. A succession of such bounces be-tween the ceiling and the floorgeneratesa cyclein actual income.When the ceiling becomes the effective de-terminantof income,a portionof real aggregatedemand is not realized. This cut-back can beeffected by (i) investment falling proportion-ately more than consumption, (2) consumptionfalling proportionately more than investment,or (3) investment and consumption falling tothe same extent. If investment is cut back pro-portionately more than consumption,a capitalgoods deficiencytends to be created. This couldincrease,B. If consumption s cut back propor-tionately more than investment, a deficiencyinfinal demand occurs, which could decrease /3.Such endogenously determined changes in j3make the model non-linear, of the type which


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A LINEAR MODEL OF CYCLICAL GROWTH 137Goodwin has studied. We will not examinethecase of changing B n detail. However, it can beshown that d,l/d,8 > o and d,d/3,8 < o, so thatif hitting the ceiling increases ,B, the minorrootof the solution equation decreases; this makesthe conditions for steady growtheasier. On theother hand, if ,B decreases, the condition forsteady growth becomes more difficult to sat-isfy.12In the third case, where investment and con-sumption fall to the same extent, the reasonsgiven above for ,Bchangingdo not apply. How-ever, the manner in which the adjustment be-tween the demand and ceiling incomes is effect-ed can be assumed to depend upon financialarrangements. If all investment and consump-tion demand is financed, money income willequal demand income even though real incomeis equal to the ceiling income. We can assumethat consumption and investment are reducedproportionately. The difference between de-mand income and ceiling income will be adjust-ed by a price rise. By assuming that the suc-ceeding period's real demand for consumptionand investment dependsupon the changein realincome, such price level changes can be ignoredin our analysis.'3Throughout the body of this paper it is as-sumedthat the roots of the characteristicequa-tion are real and greater than one. If the rootsare conjugate complexnumberswith a modulusgreater than one, the solution path is inherentlyoscillatory; and if the modulus is greater than

the rate of growthof ceiling income,in time theceiling will become the effective determinantofincome. In the Appendix t is shownthat in thiscase incomewill alwaysbounceoff the ceiling,sothat the qualitative characteristics of the timeseries are not affected by the existence of theceiling. The rich results we obtain by assumingthe roots to be real and greaterthan one are notavailablein the explosiveoscillatorycase.The income floor is determinedby the con-sumption-equals-income ncome and the maxi-mum possible capital consumptionper period.The maximumcapital consumption per periodis determinedby the capital necessary to pro-duce the previous peak income. With an in-variant consumption-equals-incomencome, therelative amplitude of the cycle generatedby in-come bouncingbetweenthe ceiling and the floorwill increase: the cycle is really explosive.How-ever, by adopting the Duesenberry hypothesisthat the consumption-equals-income incomeshifts upward whenever income turns downfrom a peak, the income floor is raised and aconstantrelativeamplitudecycle results.The accelerator-multiplierprocess generatesincome as a deviation from the consumption-equals-incomeincome, and the rate of growthof income as generated by any solution equa-tion refers to income so measured. The naturalinterpretation of the rate of growth of ceilingincomerefers to income measured from its nat-ural origin,i.e., fromzero. It will be shownthata given rate of growthof ceiling incomemust bemultiplied by the ratio of income measuredfrom the natural origin to income measuredasa deviation from equilibrium ncomebefore be-ing comparedwith the smaller root of the solu-tion equation to determinewhether incomewillcontinuously press against or bounce off theceiling. If the rate of growth of ceiling incomeso multiplied is equal to or greater than thesmaller root of the solution equation, steadygrowth of income measured from the naturalorigin will result. If the constant term in theconsumption function only increases intermit-tently, as the Duesenberryhypothesis suggests,then the ratio of income measured from thenatural origin to income measured as a devia-tion from the consumption-equals-income n-comedecreasesduringaperiodof steadygrowth.At some date, given that the rate of growth of

ai?+ +(ai+?P)2 - 4p8We have that /Ll = 2ai+8 - \/(aL+,8)2 - 4J8andt2 = 2

d1A:L (a,?+1-2)so that - = ?df8 \(ai+?1)2-413dIA2 1 1(a,?+1-2)and - -ad,B V\/(ai+?1)2- 4A8As 1> t2> I, > I and as \(aL+?)2 - 41>0,2d,ul ~~al?+-V/(cz+i? )2- 4I> o. Also U2 = >1,dj6 2(a? +P- 2) > \/(ai?3,8)2-4,l so that

a,+,8-2 dp2> i. Therefore, - < 0._\/(a,+,B) 2 _ 4.) d,8

'8 For an examination of monetary factors and such ac-celerator models see Hyman P. Minsky, "Monetary Systemsand Accelerator Models," American Economic Review, XLVII(December I957), 859-83.


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I38 THE REVIEW OF ECONOMICS AND STATISTICSceiling income is smaller than the smaller rootof the solution equation, the state of steadygrowthwill come to an end and income will turndown. Hence, it is possible for actual incometopress against the ceiling for a numberof periodsandthen turn down. This generatesa cycle witha long boom and the greater the ratio of incomemeasuredfrom natural origin to income meas-ured as a deviation from the consumption-equals-income ncome at the date the ceiling toincome becomes the effective determinant ofactual incomethe longerwill the economybe insuch a state of steady growth.The above result follows from the introduc-tion of the Duesenberryratchetin the consump-tion function. If the Tobin-Pigou effect is in-terpretedas affecting the ratio of the consump-tion-equals-income income to ceiling income,then financial market developmentswill resultin the generationof cycles with booms of vary-ing length. If the Tobin-Pigou effect is con-tinuously operative and the ratio of consump-tion-equals-income ncome to ceiling incomere-mains constant,steady growthcan result.A symmetricargumentappliesto the behaviorof the system with respect to the income floor,although the postulate that the income floor issteadily falling is not economicallynatural. Ifthe income floor is a constant during a depres-sion the only possible result of the processbeinganalyzed is that income bounces off the floor.Note that the prosperityphase generatedby thismodel can be longer than the depressionphase.Since the ratio of the consumption-equals-income income to ceiling income can be consid-ered a variable, the model can generate cyclesof varying amplitudes and periods. Also, byutilizing the Tobin-Pigou effect this apparatusmakes it possible to integraterealand monetaryaspects of cycles and growth.

The FormalModelObviously the results stated in the previoussection must be derived. It may surprise somethat newresultscan be achievedby lookingonceagain at a formulation that has been as wellexplored as the accelerator-multipliermodel.However, the typical formulations ignored theconstant term in the linear consumption func-tion and used the initial conditions only once,to start the solution equation off from the equi-

librium position. The possibility of booms ofdifferent lengths and amplitudes depends uponthe interpretation of the constant term in theconsumption function. The possibility thateither steady growth or a turningpoint will re-sult when the ceiling (floor) income constrainsan otherwise explosive acceleratorprocess fol-lows from imposing initial conditions at theceiling (floor) income rather than at the con-sumption-equals-income ncome.We must distinguish between actual, aggre-gate demand(as generatedby a particularsolu-tion equation), ceiling,and floorincomes.Thesedifferent incomes will be identified by super-scripts, thus:

y(a) is thetth date'sactual ncome,Y(d) is thetth date'saggregate emandncome,y(c) is the tth date'sceilingincome,Y(f) is the tth date's floor income.

We also take account of the intermittently im-posed initial conditions in our notation by writ-ing the date of the initial conditions in paren-thesis after y(d), A1, and A2 and subtractingthe date of the first initial condition from t inthe solution equation,e.g.,y(d) (n,n+I) )t-n

A,(2n8Y+I)ytttn + ko (n+ i). 5Equation 5 states that the particular solutionwas derived by using the nth and (n + i)stdates' actual incomes as the initial conditions.By writing Lt4 and iUt the initialconditionsdates are the zero and first powers of the roots.The writing of the equilibrium income asko(n+ i) denotes that the (n+ i)st date's in-come was the peak income which determinedaDuesenberry ratchet in the consumptionfunc-tion.In the next three sections, where the valuesof the A1 and A2 coefficientsand the effects ofthe income floor and ceiling are derived, theconsumption-equals-incomencomewill be con-stant. In these sections, it will be mathemati-cally convenient to measure income as a devia-tion from the consumption-equals-income n-come. A lower-case y will be used when incomeis so measured.Determination of A, and A2. To determineA1and A2it is necessary to know the actual in-comes of two dates, Y(a) and y(a). We have


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A LINEAR MODEL OF CYCLICAL GROWTHy(a) = y(a) - = A1(o, I) + A2(o, i)y(a) = y(a) - = A (o, i)ju + A2(o, i),2u

Assumingko is known and that y(a) = , (a) wehaveAi(o,I) = - I 2 y (a) (6)

1-2

A2(o, I) = y (a). (7)Assuming ya) > o. since /ul > /,2, if y > 12> thenAj(o, i) < o, A2(o, I) > Owhereasif 1> ja > /2then Al (, I) > o, A2(0, I) > d.)In any particular solution equation, y(d)= Aj(o, i)/jt + A2(o, i),Pt the incomeof anyperiod is an average of /uland I2 with weightsAj(o, i)/4t-1 and A2(o, I)U42. As t increases,A1(o, i)p- i/A2(o, i),t-1 increasesso that theweight of /jl in determiningaggregate demandincreases. In time, /ul dominates the behaviorof the time series. Nevertheless, if A2 o, I) I iSvery much greater than IA1(o, i)l for small tthe behaviorof the series will be dominatedbyA2(o, i)pt2 so that, even with A1(o, I) < o,Yt> Yt-i is possible.Assumey1 > yo > o and /ul > ju > J,2. ThenAj(o, i) and A2(o, i) are both positive, and allincomes generated by this particular solutionequationwill be positive and the rate of growthof income will increase, approaching pi as alimit. Onthe otherhand, if-2 > f then A1 o, I)is negative. For small t incomewill be positiveand can even be increasing,though at a dimin-ishing rate. However, as t increases,A1(o, I)ttakes over, so that incomewill fall and in timebecomes negative [y(d)(o, I) < ko]. Unlessconstrained,the rate of increaseof negative in-comewill increase,approachingul as a limit.Therefore, if y, > yo > o but Yl < I12Y0OA1(o, i), the coefficientof the dominant root,will be negative. This particularsolution equa-

tion, with roots (uq and I12) large enough togenerate an explosive time series, will in facthave one upper turningpoint. As a resultof theinitial displacement not being large enough, theexplosion will take place in the opposite direc-tion from its "start."Symmetrically,the above holds for y,, < Yn-i p2Yn-l1 (,L2 > A) then A1(n- i,n) will be positive andA2(n- i, n) will be nega-tive, so that one lower turning point will begenerated.The Ceiling to Income. The task of this sec-tion is to examine the relation between particu-lar solution equations that yield a monotonicallyincreasing income and the income ceiling. Thelarger root of the solution equation is assumedto be greater than the rate of growth of ceilingincome, so that in time the ceiling becomes ef-fective. The problemis whether, once the ceil-ing becomeseffective,incomewill bounce off theceiling, yielding an upper turning point, or willcontinue to press against the ceiling resultingina state of steady growth. Two cases must beexamined: (i) when the smaller root of thesolution equation is greater than the rate ofgrowth of ceiling income, and (2) when thesmaller root of the solution equation is equal toor smaller than the rate of growth of ceilingincome. It will be shown that the first case re-sults in income bouncing off the ceiling and thesecond case results in income continuingto pressagainst the ceiling.

i. Smaller Root GreaterThan Ceiling Rateof Growth. To make the exposition simpler, weassume that X is a constant rate of growth ofthe ceiling income measured as a deviationfromko, i.e., yc)n = Xny(c). We also have that y(d)A /ji + A2P; and we first assume thatJ > J > X > I.Since it does not matter where we break intothe time series, we begin with two positive, ar-

bitrarily dated incomes,y2a) and y1a). Bothincomesare less than the ceiling of their respec-tive dates and also p, > y(a)/y(a) > 2. Theseinitial conditions determine positive Ao(a, i)and A2(o, i). The particularsolution equationthus started will generate a monotonically in-creasingseries with an increasing (approaching-0) rate of increaseof income. In time, say atthe nth date, the income generatedby this par-ticular solution equation will become greater

14Note that in equations (6) and (7) only the numeratorsdiffer. If Au /21 < Ii,ul i then IA1(o,I)< IA2(0,I)I.It is also obvious that if 92 = A, then Ai(o, i) = o and ifji = #X A2(o, I) = o. Also if ,u > ,1 > ,U2 then Ai(o, i) > o,A2(0, ) < o with IA1(o,) I> IA2(0,)I. This leadsto theuninteresting case of growth at essentially g. aIf yo = o and y1 = a > o, thenA1(o, ) = - > 0and A2(o, I) = < O. This special case is the one con-

/2- [1lsidered by all those who, once and for all, impose initialconditions as a deviation from equilibrium.


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I40 THE REVIEW OF ECONOMICS AND STATISTICSthan the ceiling income, i.e., y(d)- =Al(o, I)p+A2 o, I2 > 0fy(C)As long as the particular solution equationyields an income less than the ceiling income (orgreater than the floor income), the income sodetermined will be the actual income. When-ever a particular solution equation tends togenerate an income inconsistent with the con-straint, actual income is equal to the constraint(ceiling or floor) income. And wheneveractualincome is determinedby a constraint, it will beinterpreted as imposing new initial conditions.Inasmuch as two initial incomes are needed todetermine A1 and A2 such an actual-equals-constraint incomeand one other income is need-ed. This other income will be either the incomejust prior to or the income immediatelyfollow-ing the first income at which the constraint be-comes effective.Hence, if y(d) > Xnyfc)hen the actual incomewill be y(a) = Xny(c). If y(a)/y(a) 1=i(n, n- )> tL2> X then the particular solution equationusing y(a) and ya)1 as the initial conditionswillgenerate an income yd)1 greater than the(n+i)st date's ceiling income.'5 Therefore,theactual income of (n + i) will be determinedbythe ceilingincome, and newinitial conditionsareimposed. As y(a) and y(a) are both ceiling in-comes y(a) /y(a) = X,so that anegativeA ,(n, n+ i)coefficient will be determined. This new par-ticular solution equation, y(d) (n, n + i) = Al(n,

n+i)/,l- 1+A2(n, n+ I)4tl-n, will generate fu-ture incomes smaller than the ceiling incomesof the respective dates. Therefore, the ceilingwill no longer be an effective constraint and,until such time as the floor becomes effective,actual incomes will be generatedby this solutionequation.If /2 > y(a)/y(a)I > X,then the solution equa-tion usingy(a)1 and y(a) as initial conditionswilldetermine Al(n - i, n) < o. If the absolutevalue of Al (n - i, n) is very small, the positivecomponent,A2(n - i, n) can dominate, so thaty(d) (n - i , n) > Xn+ly(c) . If this is true, thenonce again y(a) and y(a1 will be used to deter-mineA,(n, n+ I) andA2(n, n + i ) andthis casebecomes the same as the one above. On theotherhand,if IA1(n i, n) Iis not so small,then(d) -(n-i, n) -Xn+ly(c) , the ceiling is not ef-fective and this solution equationwill determinethe actual incomesof succeedingperiods.Regardless of whether the ceiling income isan effective constraintfor one or two periods,aslong as /12 > X> I, the "initial conditions"willdeterminea negativecoefficient or the dominantroot. Income will bounce off the ceiling and,unless constrainedby a floor, the solution equa-tion will in time generate an income that ap-proaches - oo, at a rate determined by /,u.

2. Smaller Root Equal to or Smaller ThanCeiling Rate of Growth. Assume that /jl > X> I.t2> i and that the actual incomes of the nthand (n + i ) st dates are ceiling incomes. A par-ticular solution equation determined by usingthese dates' incomes as initial conditions willhave positive A1 and A2. This particular solu-tion equation will yield only positive and in-creasing incomes and y(d)2 generated by thisequationwill be greaterthany(c)2.As a result of y(a) being determined by theceiling constraint rather than by the solutionequation, new initial conditions are effective.The new coefficients are A,(n + i, n + 2)

- [X h12 ]Xyn=XA,(n, n + i) and A2(n+ i,n+2) = ]Xy=X A2 n, n+ i ). Therefore,the new solution equation is y,(d) (n + i, n + 2)= A[A1(n, n+ i)ul-(fl+l) +A2(n, n+ i)u-(n+1)]This will be repeated as long as the ceiling in-come is the effective determinantof the actual

'5 The ratio of the two initialperiods ncome uAl(n-i, n)Al(n-i, n) + A2(n-i, n)A2(n-i, n)Ai(n-i, n) + A2(n-i, n)is a weightedaverageof j,uand ju2and the ratio of theAi(n-I, n)weightsof gi and U2 is . Fromthe solutionA2(n-I, n)

equation y( A (n- i, n)t2u + Aa(n-i, n) A2andy(d)/y(d) A1(n - i, n)pul

+ jl2n+I n A, (n- i, n),ul + A 2(n- I,n),U2+A2(n-i,n"),U2 2A, (n-i, nl)F'i + A2 (n- i, n)/,L2so that the ratioof the weightsof u,iand 12 iS

Al(n- n)# As /Li>Fu2 the weight of j1 in deter-A2(n - i n)/A2miningthe ratio y(d)/y(d) is greater han its weight indetermining the ratio y(d / y. Hence, y(d)1> AZy(d)As y(d) is a ceiling income, then y(d) > X (d) so that

(a) = (c) Yd)-n+1 n+1 n


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A LINEAR MODEL OF CYCLICAL GROWTHincome,so that y(d) (n+k, n+k+ I) = XhAl (n,n+ I )4+A2 (n, n+ I ) t4-] wheret is eitherhor h + i.By continuously imposing new initial condi-tions which are determined by a sufficientlylarge constant rate of growthof ceiling income,steady growth is generated. Note that thissteady growth state is not a "knife edge," forthe rate of growth of ceiling income can varybetween /uland y2 without causing either "run-away" expansionor a downturn.The Floor to Income. As was arguedearlier,a simple disinvestment assumption is that themaximumdisinvestmentper periodis some con-stant percentage,p, of the capitalstock requiredto produce the previous peak income. It hasbeen shown that a peak income is in the neigh-borhoodof a ceiling income so that the capitalstock to be depreciated is 3 [ko+ Xny(c) and themaximumpossible capital consumptionper pe-riod is pf3[ko+n y(0]. The floor income fol-lowing a particular peak income is given by

p/3ko n) + Xny(c))Y (f) (n) = ko(n) - Inal

orpf3(ko(n) + Xny(c))y(f) (n) - I - al

n+h a-lAssume that the income determined by theruling particular solution equation for the(n+h)th date is lower than the floor income,so that actual income equals the floor income.A new particularsolution equationwith Y4f)hasone of the initial conditionsis determined. Thearguments n the previoussectionexaminingtheratio between Ya)h and y(a+ could be repeat-ed; however, it is sufficient to argue from theassumptionthat both y(ah and Y(a)+ are equalto the floorincome.These two floor incomes determine a newparticularsolution equationwith

Al(n+k,n+ kI) =-p/3(k0 + xny(c))

b1 - 2 I-aland

A2(n?+h,n+h+I) =_pp (k o+ AnyO )c)

J 2 _ I_ - al-1 I ~< ./12 /11 Ial

As the coefficientof the dominantroot is posi-tive, Yn+h+2 will be greater than the floor in-come, and a monotonic expansionwill be set off.With a constant floor to income, it is impos-sible for income to glide along the floor. Forincome to glide along the floor it is necessarythat the rate of decrease of the negative floorincome be greater than or equal to /2. Thiswould require either an increase in p, the per-centage of capital that can be consumed perperiod,or a declinein kothe equilibrium ncome.To the extent that one wishes to interpretbusi-ness cycle events using this framework,a deepdepression - or a stagnation - could only bethe result of p or ko changing. These assump-tions would result in establishing formal sym-metrybetweenthe floorand the ceiling.Generation of an Explosive Cycle. In thissection it will be shown that in the case whereincome bounces between the ceiling and thefloor the assumption that the consumption-equals-incomeincome does not change impliesthat in time the absolute amplitudeof the cyclewill be greaterthan income. It follows that theassumption of an unchanging equilibrium in-come has to be abandoned,and in the next sec-tion the implicationsof various assumptionsasto the behavior of equilibriumincome will bederived.The amplitude of a cycle is defined as thedifferencebetween a peak income and the suc-ceeding trough income and is approximatelyequal to the differencebetween a ceiling and afloorincome. The floorincomeof a cycle whichbegins by either bouncing off or falling away

p3y (C)from a ceiling income y(c) is ko - and

p18Y,c)the amplitude of the cycle is Y(c) - (ko- - )=(I+ Pa )y) _ ko. If m periods elapsebefore the next peak income, the amplitude ofthe succeeding cycle will be (i + a)tmy(c)-ko where r is the constant rate of growth ofceiling incomemeasuredfrom its naturalorigin.In time p,8tmY(c)/I - a will be greater than ko,so that eventually the amplitude of the cyclewill be greaterthan peak income.The relative amplitude of the cycle, defined


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I42 THE REVIEW OF ECONOMICS AND STATISTICSas the ratio of the amplitude to the beginningpeak income, is (I + P )- Ini-al ,rmY (c)naddition to implying negative income at thetrough of the cycle, the assumption that kois aconstant implies that the amplitudeof the cyclewill be an increasing percentage of an exponen-tially growing ceiling income; the cycle gener-ated will be truly explosive.The Ratchet in the Consumption Function.The assumption hat ko, theconsumption-equals-income income, does not change leads to aneconomically unacceptable result. Hence, it isnecessary to abandon that assumption, and inthis section we shall derive the implicationsoftheassumptionthata ratchetin theconsumptionfunction exists. It will be shown that, dependingupon the assumptions made as to its determi-nants, a ratchet in the consumption functionleads to a number of interestingresults, namelycycles with (i) constant relative amplitude, (2)varying relative amplitude, and (3) booms (ordepressions) of differinglength. The first tworesults follow almost immediatelyfrom the dif-feringspecificationof how kochanges, the thirdresult requires an investigation of the relationbetweenthe rate of growthof income measuredfrom zero and the rate of growth of incomemeasuredas a deviation from the consumption-equals-income income.The specification of the Duesenberryratchetin the consumptionfunction to be used is that

ao(n) = yY(c) where y1(a) = y(c) , y(a)1 < Y(c)and Y (a) .Y(c) hk> I>i.e., that whenever a downturn (or a "fallingoff" fromthe ceiling) occurs,aoincreases.Fromthis specificationit follows that ao is a constantwithin a cycle, including any periods in whichincome glides along the ceiling, but it increaseswhenever ncomebouncesoff or falls away fromthe ceiling.As a result, koalso changes intermittently,infact ko(n) - aO(n) - y(c). It followsi -a, I - a,that the amplitude of each cycle is

PI+ P )y(c) - y(c) and the relativeial n i-al n 2

amplitudeis I+ P - which is a con-I-al I-alstant. The Duesenberryratchetin the consump-

tion function liquidates the embarrassing ex-plosiveness of the cycle generated by bouncingbetweenthe income ceilingand floor.If it is desired to consider ao (or y) as de-pendingupon financialor monetary phenomenain additionto previouspeak income,the relativeamplitude of the cycle can be a variable. If adownturn in income occurs when assets are alarge percentageof income, kowill be high andthe relative amplitudeof the cycle that followswill be small. On the other hand, if a downturnin income is associatedwith a financial crisis, sothat the ratio of the value of assets to incomefalls, then ko will also fall, leading to a cyclewith a high relative amplitude. By allowingfinancialdevelopments o affectthe consumptionfunction, the basic accelerator-multiplierproc-ess can generatecycles of varying amplitude.Note that if p, the depreciationratio, is in-terpretedas a behavior rather than a technicalcoefficient, t too would be affectedby the samemonetary and financial factors. That is, if adownturn n income is associated with financialease, the carrying costs of excess capacity andthe need by firms for additional liquidity areboth low, and p would tend to be small. On theother hand,if a financialcrisis occurs, the carry-ing costs of excess capacity are high and firmswould need additional liquidity, and p wouldtend to be high. Hence, financial ease would beassociated with low amplitude fluctuations andfinancial stringency with large amplitude fluc-tuations. This effect of financial developmentson the depreciationratio would tend to reinforcethe effect of financialdevelopmentson the con-sumptionfunction.In addition to differingin amplitude, cyclesdifferin duration. The ratchet in the consump-tion function and the Tobin-Pigou asset effectscan also affect the durationof the boom and, he'depression.In order to examine the determinantsof thedurationof a cycle, it is necessary to note thatthe solutionequationalways determines he rateof growth of income as a deviation from theequilibrium ncome,ko. The naturalinterpreta-tion of the rate of growthof ceiling income is asthe rate of growthof income measured from its'naturalorigin.'In previoussectionswe assumedthat there was a constant rate of growth of in-come measuredas a deviation from the equilib-


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A LINEAR MODEL OF CYCLICAL GROWTH I43riumincome. We will now investigate the rela-tion between T, the constant rate of growth ofincome measured from the natural origin, andX, the variable rate of growth of income meas-ured as a deviation from ko.The numerical growth of ceiling income be-tween any two dates is the same measuredfromeither origin, i.e.,

y(c) -y(C) = y(c) - y(c).y(c) - y(c)t+ tTherefore, X- I y(c)

andas Y(c) - y(C) - (T- I)y(C)y(c)twehave I+ ( - Iy(C)

If kodoes not change, Xis a variable, greaterthan r. As the limit of yVc)/y(c) is i, the lowerlimit of X s r.On the otherhand, if koincreasesat the samerate as y(c), which could be the result of inter-preting the Tobin-Pigou effect as reflecting theincrease in the net-worth of households thataccompanies nvestmentactivity, then T = X for

(c) - (c) - y(c)- -y() ytC) = YC y(c) -ko(t+i) + ko(t)so that (X - i)y(c) = ( - I)[Y(c) - ko(t)]= (T- I)y(c) 17

In determiningwhether income will bounceoff or press against the ceiling once it becomesthe effective determinant of income, it is x,ratherthan x, that is relevant. Once the incomeceiling becomes effectivethe necessaryand suf-ficient condition for AI o is forX(n,n+ i ) _ L2.Hence, if the income ceiling is the effective de-terminantof actual incomebeginningat the nthdate, steady growth will continue as long as

y(c)n+h-1X(n+h-I, n+h) =I + (7- I) [ y(c)n+h-1y(c) + g

I + (7- I)[ y(c)+ g-2The borderbetween a positive and a negative A1 is

7-I) Y(c) (2 I ) ylc)2 - T 2- TSince = (Thl -I )Y() wehave Th-i=

y(C)I+(-) (c_v) yfc)

h 1I =lg2( - y('r( ) onh = I +1logkO(n) [2 ] log T

where k is the number of periods after the ceil-ing first becomeseffective that income will pressagainst rather than fall away from (or bounceoff) the ceiling. Obviously, with a fixed rate ofgrowth of ceiling income the higher the ratio ofthe equilibriumto the actual income at the datethe ceiling first becomeseffective, the longer theboom; and the smaller the minor root of the so-lution equation, the longer the boom.'8

18If income grows at three per cent per period (T = I .03)and if the equilibrium income is 2/3 actual income, so thaty(c)/y(c) - 3, than X= I.O9. That is, a three per cent rateof growth of income in natural units is equivalent to a nineper cent rate of growth of income. measured as a deviationfrom equilibrium income.

"7Note that ko = ao/ (i - a,), so that if ko grows at thesame rate. as ceiling income so does ao. Since c") = ao(t-i)+a Y() at a date when actual income equals ceiling in-come then

a (t - i) + a Y(c6)C(c)/Y(c) =t t-l y(c)

t-1[a (o) + a Y(C)]Tt-

Tt-l y(c)0

a (o) + a Y(c)o 1 oy(c)

0which is a constant. Those authors who write Ct = ao Ytrather than Ct = ao+ al Yt-l can be interpreted as assumingthat the equilibrium income increases at the same rate as theceiling income, which can be identified as following fromthe specification that equilibrium income is affected by theresult of investment activity in increasing the net worth ofhouseholds.

'The followingtablegives the proximatevalue of h, thenumberof periodsfor which the modelwill exhibit steadygrowthafter reaching he ceiling ncome, for variousvaluesof k (n)/IY(', the ratio of equilibrium ncome to actualincome at the date the ceiling first becomes the effectivedeterminantof incomeand J2, the smallerroot of the solu-tion equationassuming hat T is I.03.

k(n)) / y(c) 2I.0 I .05 33.8 I.05 24.6 I .05 I 5I.0 I.07 20.8 I.07 I2.6 I .07 3I.0 I.I0 I3.8 I.I0 6.6 I.I0 I


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I44 THE REVIEW OF ECONOMICS AND STATISTICSAs the length of the period of sustained growthdepends upon the ratio of the equilibrium toactual income at the date the ceiling first be-comes effective, any phenomenonwhich tends toincrease this ratio will tend to stretch out this

period. The interpretation of the Tobin-Pigoueffect adopted in this paper would result infinancial market and monetary developmentsaffecting the equilibrium ncome. In particular,a period following a rise in the ratio of themoney value of households, assets to incomewould be characterized by long intervals ofsustained growth during which income glidesalong (or presses against) the ceiling, whereasa period following a fall in this ratio would becharacterizedby cycles in whichincome bouncesoff the ceiling.The monetary and financial developmentswhich tend to stretch out the boom also tend todecrease the amplitude of the fluctuations andthe developmentswhich tend to shorten boomsare associated with deep depressions. A deepdepression after a long boom can result if thefalling away from the ceiling incomewere asso-ciatedwith a financial crisis.The floor to income becomes effective wher-ever income generated by a solution equationtends to be lower than the floor. Measuringfrom the natural origin the floor to income ispf3Y(C3n7


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A LINEAR MODEL OF CYCLICAL GROWTHAPPENDIX

In the text the case where/%I t2> i is examined.In this appendixthe cyclically explosivecase, wherethe rootsof the solutionequationare conjugatecom-plex numbers with a modulus greater than i, isexamined.The solutionequationto Yt= (ai+f3)yt -I-Yt-2can be written as yt = Mt (A1cos tO+ A2 sin tO) inthe cyclical case, whereM, the modulus,is equal to/31and Bis defined by the conditionscos 0 = 2f81and sin 9 = --,84 (ai+8) . For the cyclicallyexplosive case the modulus must be greater than i,so that ,8 > i. Onceagain A1 and A2aredeterminedby the initial conditions.If ,B1 X the ceiling to income does not affect thetime series generated; the interesting case occurswhen 81 > X. To examine this case we will assumethat y(a) and y(a) are both ceiling incomes. Theproblem is whether y(d) > (c) so that y(a) = y(c)and a new particularsolution equation based upony(a) and y(a) as initial conditions is established. Itwill be shown that with a cyclically explosive solu-tion y(d) < y(c), so that incomealways bounces offthe ceiling.Using the initial conditions we have that y(c)

0/2(A1 cos o0+A2 sin o?) so that A1=y(c).Also, y(c) =Xy (c) =/3 (A1 cos O+A2 sin 0) so

2 (ai +f8),that2A= y(c). Therefore, the par-V4,P- (ai+13)2ticularsolutionequationis

y(d) = pt/2 (cos t 0 + 2X - (al+8) sin t O)y(C).\/4f8- (al +8) 2The problem is to determine the relation betweeny(d) and y(c).

y(d)=f(cs2+2X - (al+18)sn2yc.,8) (COS20@ ( sin 20)y (c).'\/48- (al+/8)2

We know that if cos 0 = a1+P2 -\/ V4X/ al___

and sin 0 - V4-(ai+1)22 -\/,then cos 20 = (al+/3)2-228

and sin 26 = (al + ) \/4/3- (al +/)22/8Therefore, y(d) = [X(a1+18) -#]y(c). If the ceilingis to be effective, it is necessary that y(d) > xy(c)= X2y(c). Therefore, y(d) = [X(a1+ ) ]yC> X2y(c) is a necessary condition for the ceiling to

be effective. This yields X2 _ (al+ P)x + f ? 0or -2 (al+/3)X + (/3+k) = o (k > o) andal++/ 4L /(al+1)2 - 4(G+k) However, x2

*ai+1 h V/(a +,8)2-4/is real, and since

al----- is as-2

sumed complex, (al+,8)2 - 4(,8+k) < o, so thatX(al+/8) - /3 > A2 contradicts our assumption.Therefore, [X(af1+#) - ]y(c) < X2y(c) and y(d)< y(c)-

a linear model of cyclical growth

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