A New Formulation of the Ricardian System-Carlo Casarosa

8/3/2019 A New Formulation of the Ricardian System-Carlo Casarosa

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A New Formulation of the Ricardian SystemAuthor(s): Carlo CasarosaSource: Oxford Economic Papers, New Series, Vol. 30, No. 1 (Mar., 1978), pp. 38-63Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2662848 .

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A NEW FORMULATION OF THE RICARDIANSYSTEM'

By CARLO CASAROSAIntroductionIN thelast twentyyears therehas been a renewed nterestn the Ricardiantheory.As a resulta generalconsensushas emergedon a 'version' oftheRicardian systemwhich accepts and gives full coherenceto a long inter-pretative traditionof Ricardo's thought. The most rigorous tatementofthis version' is, in our view, the mathematicalformulation resentedbyPasinettiin thebrilliant1960article[4] and in the followingwe shall dis-cuss the 'accepted' interpretation fRicardo's theoryreferringmainlytoPasinetti's formulation.According o us, Pasinetti's model differsn two mportant espectsfromRicardo's system. In this paper we point out the differences etweenPasinetti's model and Ricardo's analysis and we presentan alternative'version'oftheRicardiantheory,which s closerto Ricardo's thought hanPasinetti's formulation.1. Distributionand economic growthThe firstdifference etween Ricardo's thought and Pasinetti's modelhas to dowiththetheory f distributionnd, moreprecisely,with theleveltowards which thewage rate and the rate ofprofit onverge n a growingeconomy.Pasinetti remarks that in Ricardo's Principles there are two types ofequilibrium:marketequilibriumand natural equilibrium. Marketequili-brium,however, s a completely ransitory ituation, incein theeconomythereare mechanismswhich ensure ts quick convergence owardsnaturalequilibrium,whichtherefore epresents he 'attractionpoint' of marketequilibrium. As forthe distributivevariables, the mechanism of adjust-ment of marketequilibriumto natural equilibrium s representedby therelationshipbetweenthewage rate and the rate ofgrowth fpopulation.More precisely, fthe marketwage rate is higher lower)than the naturalwage rate, population increases (decreases); consequently,with a givenwages fund,the marketwage rate falls (increases)and converges owardsthenatural wagerate,and with t also therate ofprofit nd total rentcon1-verge to their natural levels. Accordingto Pasinetti, Ricardo is so con-vinced ofthe operationofthismechanismthat he carriesout his analysisas ifthe convergenceof marketequilibriumto natural equilibriumwere

1 I should like to thank Professor L. L. Pasinetti fordiscussion and comments on anearlier draftof thispaper. Needless to say I hold fullresponsibility orany mistake.


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CARLO CASAROSA 39almost immediate; thereforeRicardo disregardsmarket equilibrium andexamines he behaviourof he distributive ariables nterms f he dynamicequilibriumvalues.As for he dynamicsofthe wage rateand ofthe rate ofprofitnthe courseof economic growth,Pasinetti remarksthat the natural equilibriumisgenerallyunstable. In fact, whenthe rate of profit s positive, capital isaccumulated and employment xpands ineverysector. With the increaseinthe numberof workers mployed n agriculture hemarginalproductoflabour falls, nd, sincetheprofit erunitof abour s given bythedifferencebetween hemarginalproductof abour and the wage rate,and thenaturalequilibriumwage rate is constant, hereduction f the marginalproduct oflabour brings bout a fall n the natural equilibrium ate ofprofit. The rateofprofit oes onfalling s longas themarginalproductof abourcontinuesto falland, when the latter s equal to the naturalwage rate, the rate ofprofit ecomes zeroand theeconomic ystemfalls nto the stationary tate,which s the only stable equilibriumposition.As far s we are concerned,while we share Pasinetti's viewthatRicardodoes not consider hemarketequilibrium o be very mportant,we do notagreethat the natural equilibriumrepresents,n a growing conomy,the'attraction point' of the market equilibriumvalues of the distributivevariables.First of all, Ricardo states explicitly that in a growing economy themarketwage rate remainsconstantly bove the naturalwage rate:

Notwithstandinghe tendencyofwages to conform o theirnatural rate, theirmarket ate may, in an improving ociety,for an indefinite eriod, be constantlyabove it; forno soonermay the impulse,whichan increasedcapital givesto a newdemand or abourbe obeyed, han another ncreaseofcapital may producethesameeffect;nd thus, f the increase ofcapital be gradualand constant, he demandforlabourmay give a continued timulus o an increaseofpeople. [5, pp. 94>5]1Secondly,Ricardo maintains that, because of diminishingreturns nagriculture,heprocessof economicgrowthbrings bout a continuousfalloftherealwage rate and of the rate ofprofit, ntilthe economic systemfalls ntothestationary tate and thewagerate settlesdownat thenaturallevel:

In thenatural dvance ofsociety, hewagesof abourwillhave a tendency o fall,as far s theyare regulatedby supplyand demand; . . . If,for nstance,wageswereregulated ya yearly ncrease fcapital, t the rate of2percent., heywouldfallwhenit accumulated nly at the rate of I-: per cent. Theywould fallstill ower whenitincreased nly t the rate of1,or i-per cent., nd would continueto do so until thecapitalbecamestationary,whenwages also would become stationary, nd be onlysufficiento keep up thenumbersofthe actual population. I say that,under thesecircumstances, ageswouldfall, f heywere egulated nlybythesupply nd demandof labourers; .. . [5, p. 101]

1 See also [6], n. 145.


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40 A NEW FORMULATION OF THE RICARDIAN SYSTEMNow, since n the 'accepted version' ofthe Ricardian theorydiminishingreturns n agriculture ause a fall of the natural equilibriumrate of profit,but have no influence n the natural equilibriumreal wage rate, it is evi-dent that this 'version' is unable to account for Ricardo's results on thedynamics of the wage rate duringthe process of economic growth.'As a matter ffact, ccording o Blaug [1, pp. 25-6] there s no contradic-tion between heproposition hat during he processofgrowth he real wagerate tends to fall and the 'accepted version' of Ricardo's theory, ince, nBlaug's view, Ricardo is speaking of a fall of the 'corn' wage rate and notof the wage rate expressed in terms of the whole basket ofwage-goods.Therefore, s during heprocessofgrowth he relativepriceofcorn ncreases

because ofdiminishing eturnsnagriculture,we may wellhave at the sametime the fall of the cornwage rate and the constancy, t thenatural level,of the wage rate expressed n terms ofthewhole basket ofwage-goods.We agreewith Blaug that the fall of the cornwage rate is quite com-patible with the 'accepted version' of the Ricardian theory, but westrongly enythat Ricardo is speakingofa fall of the corn wage rate. Onthe contrary, rommanypassages ofthePrinciples2 t is clearthatRicardois speakingof a fallofthewage rateexpressed nterms f the whole basketofwage-goods. Ricardo, in fact,writes:As population ncreases hese necessarieswillbe constantly isingnprice,becausemore abour will be necessary oproducethem.... Instead, therefore,fthemoneywagesof abourfalling, heywouldrise;buttheywould not risesufficientlyo enablethe labourer o purchaseas manycomforts nd necessaries s he did before he riseinthepriceofthose commodities.... he would therefore eceivean addition nhismoneywages, thoughwiththat additionhewouldbe unable to furnish imselfwiththesame quantityof cornand other ommodities,whichhe had before onsumed nhis family. 5, pp. 101-2]And, a little further:. . . The fateofthe abourerwill be lesshappy;he willreceivemoremoneywages, t istrue,but his cornwageswill be reduced;and notonlyhis commandofcorn,buthisgeneral onditionwill be deteriorated . . the condition f the abourerwillgenerallydecline ... [5, pp. 102-3]

If our reading of Ricardo is correct,we are then confrontedwith thefollowingdilemma: eitherRicardo considered natural equilibriumas the'attractionpoint' of market quilibrium, s Pasinettimaintains,butinthiscase Ricardo's statementthat in the processof economicgrowth herealwage rate tends to fall does not make any sense; or this latterstatement

1 Further inconsistenciesbetween Ricardo's results and the 'accepted version' of histheory concernthe effects f taxation. In fact,while Ricardo repeatedly 5, pp. 116,221-2,225, 226, 233] states that everytax reduces thewage rate by slowing down therate of capitalaccumulation,within the 'accepted version' taxes have no effect n the (natural) equili-briumwage rate.2 See also [5, pp. 104, 112, 125, 225] and [6; n. 61, p. 124 and n. 63, pp. 126-7].


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CARLO CASAROSA 41is correct,but then we cannot say that Ricardo examines the behaviourofthe economic variables in termsofnatural equilibrium.In our opinion the latter s the case and in thenext threeparagraphs weshall presentourinterpretationf theRicardian theory nd we shall showthat,within ts framework, he Ricadian results,which are at odds withthe 'accepted version' of Ricardo's theory, re easily explained.2. Our 'version' of the Ricardian theory of distribution andeconomic growth

From someofthe passages quoted above and frommany otherswe canfind n the chapter on wages and elsewhere n the Principles and in otherwritings,t appears that in Ricardo's theorythe level towards which themarketwage rate (and, hence,the marketequilibrium) s attractedduringthe processof economicgrowth s determined ythe interplaybetween theaccumulationofcapital and thegrowth fpopulation,that is bythe inter-play betweendemand and supply of abour. And infact Ricardo's generalidea is that:... wagesare subject to a riseorfall from wocauses:1st The supply and demand of abourers.2dlyThe priceof the commodities n which the wages of labour are expended.[5, p. 97]

Moreprecisely, ccordingto our readingof Ricardo, at every level of'facilityfproduction' n agriculture he market quilibriumdoes not tendto natural equilibrium,but to what we have elsewhere [2] called the'dynamic quilibrium',that is a positionwherethewage rate and the rateof profit re such that the rate of increaseofpopulation and the rate ofcapital accumulation are equal. The convergenceof marketequilibriumto dynamic quilibrium s warrantedby the simultaneousworkingofthepopulation nd ofthe accumulationmechanisms. If at any moment themarketwage rate is above (and the rate of profitbelow) the dynamicequilibrium alue, the rate ofgrowth fpopulationis higher han therateof ncrease fcapital accumulation. Therefore,he marketwage rate fallsandthe rate ofprofit ncreases,and thisprocessgoes on till the values ofthetwodistributive ariables are such as to make the rate ofgrowthofpopulation qual to the rate of growthof capital. A similar mechanismof djustments at work fthe marketwage rate is below (and the marketrateofprofit bove) the dynamic equilibriumvalue.In conclusion,n ouropinionRicardo is convincedthat,at every evel of'facility fproduction' in agriculture, he marketequilibrium convergesto the dynamic equilibrium position. Consequently he examines thebehaviour of the distributive variables on the basis of the changing


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42 A NEW FORMULATION OF THE RICARDIAN SYSTEMcharacteristics f thedynamicequilibriumposition s the marginalproductof labour in agriculture r some otherparameterof the systemchanges.To be sure,Ricardo does not describe xplicitly he mechanism f adjust-ment of market equilibrium to dynamic equilibrium nor the positiontowardswhichmarketequilibriumtendsto converge.As for hefirst e says only that,whenthe rate of capital accumulationis higherthan the rate of increase ofpopulation,market wages increase,whilewhen the rate of growthof population is higherthan the rate ofcapital accumulationthe marketwage rate falls:... at suchperiods ccumulation s often o rapid,that labourers annot be suppliedwith the same rapidity s capital.... In thatcasewages ... wouldhave a tendencyto rise,because the demand for abourwould increasestill fasterthan the supply.[5, p. 98]... thewagesof abour willhave a tendency ofall .. for hesupplyof abourerswillcontinue o increase t the samerate,whilst hedemandfor hemwill ncreaseat aslowerrate. [5, p. 101]

On the 'attractionpoint' of market equilibriumRicardo is even lessexplicitand we can only pointat passages whichcan be understoodonly fread in termsofdynamic equilibrium.1However, fweput intoa coherentwholetherelationshipswhich, ccord-ing to Ricardo, link the market equilibriumvalues of the distributivevariables to the rate of growthof population and to the rate of capitalaccumulation, it is possible to show that the simultaneousworkingofthe population and of the accumulation mechanismsdrives the marketequilibrium owardsthedynamic quilibriumposition exactlythrough he

1 Particularly ignificant,n thisrespect, s thefirst art ofchapterXVI of thePrinciples(taxes on wages), whereRicardo is shownto be in full agreement with two passages fromSmith and Malthus in which it is clearlymaintained that, duringthe process of economicgrowth, he wage rate tendsto be such as to guaranteea rate of ncrease ofpopulation equalto the rate of capital accumulation. The passages, reportedfully n the Principles, are:'The demand for abour, according as it happens to be either ncreasing, tationaryordeclining,or to require an increasing, tationaryor declining population, regulates thesubsistence of the labourer, and determines n what degree it shall be either liberal,moderateor scanty' [5, p. 215],and: 'The priceof abour . . . expresses,clearly,the wants ofsociety respectingpopulation;that is, whatever may be the number of children to a marriagenecessary to maintainexactly the present population, the price of labour will be just sufficiento support thisnumber, or be above it, or below it, according to the state of the real funds, for the

maintenance oflabour, whether tationary,progressiveor retrograde' 5, pp. 218-19].Such is Ricardo's agreementwiththe opinions expressedby Smithand Malthus on thisparticular ssue, thathe uses Malthus's words to refuteBuchanan's argumentson the effectsproducedonthe wage rateby a tax on wages. Moreover, he conclusion ofRicardo's discus-sion is perfectlyn line withthe opinionsheld by Smith and Malthus:'Suppose thecircumstancesof thecountry o be such,that the lowest abourers are notonlycalled upon to continue theirrace,but to increase t; theirwageswould be regulatedaccordingly.Can they multiply n the degree required, f a tax takes from them a partof theirwages, and reduce themto bare necessaries?' [5, p. 220]. (The italics are ours.)


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CARLO CASAROSA 43mechanismofadjustmentsketchedat the beginning f the presentpara-graph. To provethis n thenext twoparagraphswepresent ndexaminetheworking f a verysimplemodel ofthe economicsystem,whosecharacteris-tics arestrictly ased onRicardo's ideas onthe determinationfthemarketequilibriumofthe distributive ariables and on therelationshipsbetweenthesevariables,the accumulationof capital, and thegrowth fpopulation.3. Market equilibrium

Let us first xamine Ricardo's views on the determination f marketequilibrium. As for the marketwage rate Ricardo writes:Themarketprice of abouris the pricewhich s reallypaid for t,from he natural

operationof the proportion f the supplyto the demand; labour is dear whenit isscarce,and cheap when it is plentiful. [5, p. 94]Hence, ifcapital increases:. themarket ate of wageswill rise,for n proportion o the increaseof capitalwillbe the increase in the demand forlabour; . . . [5, p. 95]In other words,accordingto Ricardo, the marketwage rate is deter-mined, n the short period,by the ratio betweenthe wages fund and thenumberof workers. Moreover,Ricardo usually assumes that the wages

fund s a constantproportion ftotal capital.Ricardo does not explain how the wage rate convergesto the marketequilibriumevel and refers enerically o the competition mongworkersand employers. However, his position can be easily rationalized, byassuming hat the workers ccept any wageraterather han be unemployedand theemployers pendthe wages fund entirely,whateverthewage ratetheyhave to pay.If weassume, for implicity, hatcapital consistsonlyof thewages fund,theRicardian theoryof the marketwage rate may be expressedthroughthefollowingquations: W = wN (I)

K-W (II)whereK isthe amountofcapital,N thenumber fworkers,w thewagerate,Wthetotalwage bill, and K and N are, in the shortperiod,constant.As for hemarketrate ofprofit, icardo's central dea is thatthevalueofthemarginalproduct of labour in agriculture s divided between theworkers nd the employers,that is, between wages and profits. If weassume,for implicity,hat the economicsystemproducesone commodityonly, et us say corn,therelationshipbetween the real wage rate and therateofprofitmay be expressed by:

= f'(N)-w (III)w


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44 A NEW FORMULATION OF THE RICARDIAN SYSTEMwherer is the rate of profit nd f'(N) the derivativeof the productionfunction orcorn:

X = f(N) withf'(N) > 0 and f"(N) < 0 (IV)whereX is the quantityof corn produced.Now,since nthe short unthe number fworkerss given,from quationIV we obtain the marginal product of labour which, togetherwith themarketwage rate, determined y theamount of capital and thenumberofworkers equations I-II), determines he marketrate of profit equationIII). We can therefore onclude that, n the Ricardian theory, hemarketequilibriumofthe distributivevariables is determinedby the amount ofcapital, the numberofworkers, nd the marginalproductivity f labourin agriculture.4. Market equilibriumand dynamicequilibrium

From theconclusionwehave just reachedon the forces'whichdeterminemarketequilibrium, t is evident that the dynamicsofthemarketequili-brium values ofthe distributive ariables duringthe process ofeconomicgrowthdepends on the dynamics of capital, the number of workers, ndthe marginalproductivity f labour in agriculture.As for hemarginalproductivity f abour,Ricardo accepts theso-calledprincipleof diminishing eturns accordingto which, as employment nagriculture ncreases,the marginalproductivityof labour tends to fall.However,thisprincipledoes not implya 'continuous' fall of themarginalproductof labour and in factRicardo implicitly ssumes large 'intervals'withconstant eturns. We assumethatthe intervals'withconstant eturnsare largeenoughand consider he dynamicsof marketequilibriumwithinthese intervals'. In thenextparagraphwe shall see whathappenswhenthesystemmoves fromone 'interval' to another,that is when the marginalproductivity f abour changes.Given the assumption of constant returns,the dynamics of marketequilibriumwithinany 'interval' is determinedby the accumulation ofcapital and the ncreaseofpopulation. Let us therefore onsider he laws'which,accordingto Ricardo, rule theprocessofcapital accumulationandthe growthofpopulation.Ricardo's theoryofcapital accumulation is very simple:

Thefarmer ndthemanufactureran no more ivewithout rofit,hanthe abourerwithoutwages. Theirmotive for ccumulationwilldiminishwitheverydiminutionof profit, . . [5, p. 122]We can thereforeay that,accordingto Ricardo, the rate of capital ac-cumulation s an increasing unction ftherateofprofit.This relationship


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CARLO CASAROSA 45may be expressed by:

1 dKK dt yr with O< y


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46 A NEW FORMULATION OF THE RICARDIAN SYSTEMOB0O0 may therefore e called the dynamic equilibriumvalues of thewage rate and ofthe rate of profit nd OA the dynamic equilibriumrateof growthof the economy.

L

A ----------0 S CCB IVK

FIG. 1.Let us nowsee whathappenstothemarketvariables fthe initial'valuesof capital and populationdetermine marketequilibriumwhich does notcoincide with the dynamic equilibrium. Assume, forinstance,that the

'initial' marketequilibrium mplies a marketwage rate higherthan 00(say 00') and, consequently,a rate of profit ower than the dynamicequilibriumrateofprofit. n thissituation, s emerges learlyfrom ig. 1,the rate ofcapital accumulation s lowerthan the rate ofgrowth fpopula-.tion. Therefore,s timegoesby,themarket quilibriumwageratedecreasesand themarketrateofprofitncreases. Consequently, he rate ofgrowthof population decreases and the rate of capital accumulation increases.However, as long as the former s higherthan the latter,the process ofchangeof market quilibriumgoesonin thesame direction nd it comestoa halt onlywhen the rate ofgrowthof capital and population are equal,that is onlywhenthe marketequilibriumvariables are at theirdynamicequilibrium evels.A similarmechanism s set inmotion f nitially he marketwage rate isbelow (and themarketrate ofprofit bove) itsdynamic equilibriumvalue.And also in this case the processofadjustment goes on until the market


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CARLO CASAROSA 47equilibriumof the distributivevariables coincides with their dynamicequilibrium.We can therefore oncludethat,at everyevelofthemarginalproductivityof labour in agriculture,he simultaneous working of the mechanisms ofpopulation growthand of capital accumulation, described by Ricardo,brings bout the convergence fmarketequilibrium owardsthe dynamicequilibriumposition,where he marketvalues ofthedistributive ariablesare such as to determine rate of increase of capital equal to the rate ofincreaseofpopulation. Consequently t would be sensibleto examine, asaccording o our nterpretation icardo does,thebehaviour oftheeconomicsystem ntermsof tsdynamic equilibriumposition, incethe latterrepre-sentsthe 'attractionpoint' of marketequilibrium.Formally, he dynamicequilibriumposition of the system s determinedby equations I-VI and by the dynamic equilibrium condition:

I dK I dN (VII)K d&t_N d-it-(The system now contains seven equations having eight unknowns.Thereforewe can solve it as a functionofN. that is as a functionof the

numberof workers mployedin agriculture.An explicitsolution forthedistributive ariables and therate ofgrowth s obtainedin theAppendix,wherewe also pointout some interesting roperties f thedynamic equili-briumof our system.However, ven within he framework f the simplemodelwe have beenworking ith, t is possibleto show that thedynamic equilibriumvalues ofthedistributive ariables and of the rate ofgrowth ependon themarginalproductivityf abour in agriculture. In fact,from quation V' it is clearthatthehigher he marginal productivity f labour,the higher, t everylevelof hewage rate,therateofprofitnd the rate ofcapital accumulation.In terms f ourdiagramthis means that the higherthemarginal produc-tivity f abour,thefurther o theright s the KK curve (and vice versa).Hence we mayconcludethat thedynamic equilibriumvalues of thewagerate, he rate ofprofit,nd the rate ofgrowth re increasingfunctions fthemarginal roductivity f labour in agriculture.5. Dynamic equilibriumand economic growth

In the last two paragraphswe have shown that our interpretation fRicardo'stheory s strictlybased on Ricardo's ideas about the interplaybetween he distributive ariables, population growth, nd capital accumu-lation. However,we have not yet 'proved' that Ricardo really worked nterms fdynamic equilibrium. To do this it is necessaryto compare the


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48 A NEW FORMULATION OF THE RICARDIAN SYSTEMresults hat we get fwe examine thebehaviour ofthe distributive ariablesduring the process of economic growth n termsof dynamicequilibrium,with the results obtained by Ricardo in his writings.Both in the 'accepted version' and in our 'version' of Ricardo's theorytheeconomic system s able to growonly f the marginalproduct of abouris 'initially' higher han the naturalwage rate. In termsof our diagram,a marginalproductivity f abourhigher han the natural wage rate meansthat the curve ofthe accumulationof capital cuts the horizontal xis ontheright f S. Let us assume that the initial'level ofthe marginalproductivityof abour s OT (see Fig. 2) and thatthe corresponding apital accumulation

KK'

S EK

FIG. 2.

curve is KK. Therefore we have a dynamic equilibrium position (E)characterizedby a positive rate of growth of capital and population),a wage ratehigher han the naturalwage rate, and a positive rate of profit.As growthproceeds, sooner or later the marginalproductivity f abourin agriculture eginsto fall. Consequently he capital accumulation curveshifts o the leftand the dynamicequilibriumvalues of the wage rate, therate ofprofit,nd the rateofgrowthdecrease. And, as long as the marginalproductivityof labour goes on falling, the capital accumulation curvecontinues o shift o the eft nd all the dynamic quilibrium alues continueto decrease.This process goes on up to the point when the marginal productivity flabour s equal to the natural wagerate and the capital accumulation curveassumes a position such as KK1: at this point the dynamic equilibrium


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CARLO CASAROSA 49positionofthe system s the stationary tate(1 K I dN )\-K t N dt=?'the dynamicequilibriumwage rate is the naturalwage rate, and the rateofprofits zero. Therefore, t this level of the marginalproductivity flabour,dynamic equilibriumand natural equilibriumcoincide.In conclusion, fwe examinethebehaviourof the distributive ariablesduring he process of economicgrowth n termsof dynamicequilibrium,we getthe following esults:

1. As longas themarginalproductivity f abour in agriculture s abovethe natural wage rate, the economic systemgrows at positiverates, thewage rate is higher han the naturalwage rate, and the rate of profit spositive.

2. As themarginalproductivity f abourfalls,we have a slowingdownoftherate ofgrowth nd a simultaneousdecrease ofboth thewagerate andtherate of profit.3. When the marginalproductivityof labour is equal to the naturalwagerate the economicsystem tops growing nd falls nto the stationarystate. Thewagerateis nowat its natural eveland therate ofprofits zero.It is easyto see that theseare the very same resultsreached byRicardoin hisanalysis. In particular,as faras the behaviourofthe wage rate isconcerned, ur interpretation f Ricardo's theoryin terms of dynamicequilibriumxplainsquitenaturallyboththe Ricardianpropositionswhich,as we have pointedout in the first aragraph,cannotbe explainedby the'acceptedversion':1. ... themarket wage]ratemay, nan improvingociety, or n indefiniteeriod,be constantlybove [thenaturalrate]. [5, pp. 94-5]2. In the naturaladvance ofsociety, hewagesof abourwill have a tendency ofall, ... [5, p. 101]Notice furthermorehat our analysis of the 'path' followedby thedynamic quilibriumof the systemduring heprocessofeconomicgrowthseems lmosta paraphrase of Ricardo's words:In differenttages of society,the accumulation of capital, or of the means of

employingabour, smoreor essrapid, nd must n all casesdependon theproductivepowers of labour . . . [5, p. 98]In thenatural dvanceofsociety, hewagesof abour willhave a tendency o fall,as far s they reregulatedby supplyand demand;for .. thedemand for hemwillincreaset a slowerrate. If,for nstance,wageswereregulatedby a yearly ncreaseof apital, t the rateof 2 per cent.,theywould fall when t accumulatedonlyat therateof11percent. Theywouldfallstill owerwhen t increasedonlyat therate of1or percent.... andwouldcontinue o do so untilthecapitalbecame stationary,O.E.P.301 E


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50 A NEW FORMULATION OF THE RICARDIAN SYSTEMwhen wages also would become stationary, nd be only sufficiento keep up thenumbersofthe actual population. [5, p. 101]We can therefore onclude that, since the traditional nterpretation fthe Ricardian theoryofdistribution nd economicgrowth s not capableof explaining ll Ricardo's results,whileour version' s, the latter version'shouldbe accepted as a better nterpretation fRicardo's thought.Beforewe examine further he characteristics f the 'dynamic equili-brium' of our model, it is necessary to clarifythe nature of dynamicequilibrium tself nd of its relationshipwiththe market equilibrium.In fact it is important o realize that in our model the dynamicequili-briumis not a steady growth equilibrium,but a moving equilibrium,asdefinedby Frisch [3, pp. 103-4]. In particular,the dynamicequilibriumpath is nota path on which the economicsystemcan remain all the time,since even ifthe economicsystemgets,at a certainmoment,on this path,and the rate ofgrowthofpopulation happens to be equal to the rate ofgrowth f capital, decreasingreturnsn food productioncause a change inthe dynamic equilibriumposition and break the equality between the(market)rate ofgrowthofpopulationand the (market)rate ofgrowthofcapital.

However,s we haveshown,marketquilibriumscontinuouslyattracted'towards hedynamicequilibrium osition,and it is thereforeegitimatetostudythe behaviour ofthesystem n termsofdynamicequilibrium, ven ifthe system tselfmightnever e on the dynamicequilibriumpath.6. Changes in the habits of the workers and in the propensity oaccumulate ofthecapitalists

It is interestingo consider he mpacton thedynamic quilibriumvaluesofthewage rate,the rate ofprofit,nd the rate ofgrowth, fchanges nthe'habits'of he workers nd inthepropensityo accumulateof hecapitalists.A changeofthe firstkind implies, n our model,either a change in thenaturalwagerate ora change n the reactivity'ofthegrowth fpopulationto the divergencebetweenthe marketrealwage rate and thenaturalwagerate, expressed by P3.If thenaturalwage rate changes,the LL curve shifts: o theright f thenatural wage rateincreases, o the left f t decreases. Therefore,s we cansee fromFig. 3, if the natural wage rises,at everylevel of the marginalproductivity f labour in agriculture he dynamicequilibrium wage raterises,while the rate ofprofit nd the rate ofgrowthdecrease. If, on thecontrary, he naturalwage rate decreases,the dynamic equilibriumwagerate decreases,while the profit ate and the rate ofgrowthrise.A change in the 'reactivity' of population growthto the divergencebetweenthemarketwagerateand thenaturalwageratecauses a 'rotation'


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CARLO CASAROSA 51ofthe LL curve: clockwise f the 'reactivity'decreases, anticlockwise ntheoppositecase. As we can see from ig. 4, inthe first ase, at every evelof the marginalproductivity f labour in agriculture, he dynamicequili-briumwage rate increases,whilethe rate ofprofit nd therate ofgrowthdecrease. In the second, the dynamic equilibriumwage rate decreases,while the rate ofprofit nd the rate ofgrowth ncrease.

L"1 L L

EI

, . // // IV/S"S 5' K IL" L L' FIG. 3.More rigorousproofs of these results are given in section 5 of theAppendix.Let us now consider he impacton the dynamicequilibriumvalues ofachange n thepropensity o accumulate of the capitalists.If thecapitalists' propensity o accumulate increases,at every evel ofthe rate of profit he rate of capital accumulation is higher. Hence, atevery evel ofthemarginalproductivity f abour in agriculture,we havea shift o therightof the capital accumulationcurvewhichbringsabout(seeFig. 5) an increase ofthedynamicequilibriumvalues of thewage rateand ofthe rate ofgrowth. The rate of profit, n the contrary, ecreases

since t every evel of themarginal productoflabourthewage rate is nowhigher see equation V').If the capitalists' propensity to accumulate decreases, the capitalaccumulation urve shifts o the leftand, therefore,t every evel of themarginal roductivityof labour the dynamic equilibriumvalues of thewage rate and of the rate of growthdecrease, while the rate of profitincreases.


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52 A NEW FORMULATION OF THE RICARDIAN SYSTEM

\K /L"w

LI

L LKFIG. 4.

K" K K'

0- SC V 5.L K" -K K'FIG. 5.


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CARLO CASAROSA 53Again, a more rigorousproofof these results s given in section 6 oftheAppendix.It is interesting o noticethat the results eached above and, in particular,those concerning he mpact on the wage rate, are in perfect greementwithRicardo's views. In factRicardo states clearlythat if the workers heckthe rateofgrowth fpopulation or the capitalists ncrease he rateofcapitalaccumulation, the wage rate increases:It is a truth hat admits not a doubt, that the comforts nd well-being f thepoorcannot be permanently ecuredwithout ome regardon theirpart, or some effort nthepart ofthe legislature, o regulatethe increase of theirnumbers, nd to renderless frequent mong them early and improvidentmarriages. [5, pp. 106-7]Witha populationpressing gainst the meansofsubsistence, he onlyremedies re

either reduction fpeople,or a morerapid accumulation fcapital. In rich ountries... the atterremedy sneither erypracticablenor verydesirable,because itseffectwould be, if pushed very far, o render ll classes equally poor. But in poor coun-tries,wherethere re abundant means of production n store,from ertileand notyet brought nto cultivation, t is the only safe and efficaciousmeans of removingthe evil, particularly s its effectwould be to elevate all classes of the people.[5, pp. 99-100]Thereforewe have a further ndirect proof of the correctnessof ourinterpretation f the Ricardian theoryof distribution.

7. Wage-goods, luxury-goods,and capital accumulationThe second difference etween Pasinetti's model and Ricardo's thoughthas to do with he role of uxury-goods. According o Pasinetti 4, pp. 90-1]theproduction onditions f wage-goods nfluence he determination fallthevariables of the system, he only exception being the real wage rate,while the production conditions of luxury-goods nfluence he variablesconcerning he luxury-goods,but not the other variables of the system,

such as therate of capital accumulation, the rate of profit, nd so on. Inour opinion wage-goods and luxury-goodshave really a differentole intheRicardian system,but it is not true that luxury-goods o not influencetherateof capital accumulation, since Ricardo states explicitlyon manyoccasions hatsavingand capital accumulationdepend on the ncomeofthecapitalists xpressed n termsof luxury-goods:As ... the power of saving from evenue to add to capital, must depend on theefficiencyfthenet revenue, o satisfy he wants of the capitalist, t couldnot failto

follow rom he reduction nthe price of commodities . . that withthesamewantshe wouldhave increasedmeans of saving,-increased facility f transferringevenueinto capital. [5, p. 390]Ricardo s pointofview on this matter merges learlywhen he considersthepossible effects f the introduction f machines and of foreign rade.In the first ase he says that, if the introduction f machines reduces thevalue ofwage-goods,the rate of profit ncreases and saving and the rate


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54 A NEW FORMULATION OF THE RICARDIAN SYSTEMof accumulation of capital increase with it; while, if the introductionofmachines reduces the value of luxury-goods, he rate of profitdoes notchange,but saving and the rate of capital accumulation ncrease, becauseofthehigher urchasing owerof he capitalists. SimilarlyRicardo declaresthat, if one or more wage-goods are imported at a price lower than theinternal price, the rate of profit and the rate of capital accumulationincrease; while, f one or more uxury-goods re imported at a price owerthan the internalprice, the rate of profit oes not change,but saving andthe rate of capital accumulation increase, since the purchasing power ofthe capitalists has increased:

There are two ways n which apital may be accumulated: t maybe saved either nconsequenceof ncreased evenue, r of diminished onsumption. .. If, by the ntro-duction of machinery, he generalityof the commoditieson which revenue wasexpended fell20per cent. n value, should be enabled to save as effectuallys ifmyrevenue has been raised 20 per cent.; but in one case the rate ofprofits s stationary,in the other t is raised 20 percent.-If, by introduction f cheap foreign oods,can save 20per cent. frommy expenditure, he effectwill be precisely he same asif machinery ad lowered he expense of theirproduction, ut profitswould not beraised.... If... bytheextension fforeign rade,orby mprovementsn machinery,the food and necessariesof the labourercan be brought o the marketat a reducedprice,profitswillrise .. but if the commodities btained at a cheaper rate, by theextension fforeign ommerce, rby improvementfmachinery, e exclusively hecommodities onsumedby the rich,no alterationwill take place in therate ofprofits.[5, pp. 131-2]8. Wage-goods, luxury-goods,and dynamic equilibrium

Let us now study the effects f changesin the productionconditionsofwage-goodsand luxury-goods n the dynamic equilibriumpositionof theeconomicsystem.As for thewage-goods it is quite easy to show the effects roduced bychanges n theirproductionconditionson the dynamic equilibriumvaluesof the wage-rate, he rate ofprofit, nd the rate ofgrowth. In fact, f atevery level of employment n agriculture the marginal productivityoflabour increases,the capital accumulation curve shiftsto the right. Asa consequence,at every evel ofemploymentn agriculturewe have higherequilibrium alues of hewage rate,therateofprofit,nd the rate ofgrowth.Ofcourse, n the case of a reduction nthemarginalproductivity f abourwe would have a reduction n the dynamic equilibriumvalues of all theabove variables.The problem s morecomplex fthe change in theproduction onditionsconcerns uxury-goods. In fact, n orderto analyse in a rigorousway theeffects fthis kind ofchangeon thedynamicequilibriumvalues,we shouldintroduce another commodity (a luxury-good) into our model and weshould also 'incorporate' n the model the Ricardian view that, at everylevel ofthe rate ofprofit, he propensity o accumulate of the capitalists


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CARLO CASAROSA 55is a decreasingfunction f the prices ofluxury-goods.All this s formallydone in the Appendix wherewe present completetwo-commoditymodelof the Ricardian system.However, it is possible to have a good idea of the effects f changes inthe productionconditionsof luxury-goods ven within the framework fthesimplemodel we have beenworkingwith so far. If there s an increasein the productivity f abour in the production f one ormore uxury-goodsthe prices of these goods fall, since the quantity of labour necessarytoproduceone unit of these goods has decreased. Therefore he capitalists'propensity o accumulate increases, so that at every evelofthemarginalproductivity f labour in agriculturewe have a shiftto the rightof thecapital accumulationcurve. As a consequence, we have an increase n thedynamic equilibriumvalues of the wage rate and of the rate ofgrowth.The rate ofprofit,nthecontrary, alls, inceat every evel ofthe marginalproductivity f labour the wage rate is now higher (see equation V').Naturally, f the productivityof labour in the luxury-goods producingsectordecreases ora tax is put uponthis kind ofcommodity) hedynamicequilibriumvalues of the wage rate and of the rate of growth decrease,whiletherate ofprofitncreases.

We can therefore oncludethat both wage-goodsand luxury-goodsn-fluence he dynamic equilibriumvalues ofthedistributive ariables and oftherateofgrowth.Nevertheless hetwo typesofcommodity ave differentroles n the economic system, as emerges fromthe fact that changes intheirproductionconditionsbringabout not onlyvariations of the samesign,but quantitativelydifferent,fthe real wage rate and oftherate ofgrowth see section4 oftheAppendix),but also variationsofopposite signoftherate ofprofit.The basic reason forthesedifferencess that,whilethe conditionsof production of wage-goods influence the relationshipbetween hewage rate and the rate ofprofit equation V'), the conditionsofproduction fluxury-goodsnfluence herelationshipbetween the rateofprofitnd the rate of capital accumulation,but not the relationshipbetween he wage rate and the rate of profit. In fact,in the case of anincrease f themarginalproductivity f abour in theproductionofwage-goods, t every evel ofthereal wage rate therecorresponds higherrateofprofit nd, hence,a higher ate ofgrowth. Consequently, n increase nthemarginal roductivity f abour in thewage-goodssectorbrings boutan increaseboth in thewage rate and in therate ofprofit.On the otherhand, f we have an increase in the productivity flabour in the luxury-goods sector, he rate of capital accumulationwhichcorresponds o eachlevel of therate ofprofit ncreases,but therelationshipbetween the wagerate and the rate of profit emainsunchanged. Therefore he real wagerate ncreases nd therate ofprofit alls.


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56 A NEW FORMULATION OF THE RICARDIAN SYSTEMIn Ricardo's writingswe may find some evidence that Ricardo wasaware of thefactthat both the wage-goods and the luxury-goods nfluencethe dynamicequilibrium values of the distributivevariables. In fact, inthe chapters on taxation Ricardo maintainsconsistently hat every kindoftax tends to reduce the rate of capital accumulation and hence to reducethewage rate. In particularRicardo denies that the taxes on 'rawproduce'are particularlyunfavourableto the working lass:

... no other nconvenience ould be suffered y thisclass, than that which heywouldsuffer rom ny othermode of taxation, namely, heriskthat the tax might nfringeon the fundsdestined forthe maintenanceof abour,and might herefore heck orabate the demandfor t....Nowthe sumrequired y thetax mustbe raised, ndthe question imply s,whetherthe same amountshall be taken from he individualsbydiminishing heirprofits, rby raising the prices of the commodities n whichtheir profitswill be expended.[5, pp. 166-7]

Ricardo makes his position perfectly lear in a letter to McCulloch, inwhich he explains that, from he point of view ofthe workers, here s nodifference etweentaxes on wage-goods and on luxury-goods:... I contend hat thepoorsuffer romhe dissipation fcapital, notonaccountof hepeculiar axes which re imposedupon them,butonaccount ofthe disturbancewhichit gives to the usual demandfor abour. It mattersnot, say,whether hetaxes belaid on wine, silksand velvets, the luxuriesof the rich,or on the cornand clothingconsumedbythelaboring lass,thespecific vil is inbothcases,notthetax, but theannihilation fcapital to whichthe tax givesrise .. raisethethirtymillionswithintheyear bytaxes on luxuries, r on the necessaries f thepoor,and theeffectwill bethe same-the poor will suffer ecause 30 millions of capital is withdrawnfromactiveemployment.'

We can therefore oncludethatthe resultswe have reached nthepresentparagraph are fullyconsistentwith Ricardo's own views.Conclusions

The mainresultswehave reached n ouranalysisofthe Ricardian theoryof distribution nd economicgrowthmay be summarized:1. In the courseofeconomicgrowth hemarketreal wage rate and therate ofprofit o not converge owards the natural equilibriumvalues, butrather owardsthedynamicequilibriumvalues; thatis,towardsthe values

thatguaranteethe equalitybetweenthe rate ofcapital accumulation andthe rate ofincreaseofpopulation.2. Profit s not a residual,since the rate ofprofit s determined imul-taneouslywiththewage rate.1 Letter n. 361 in P. Sraffa ed.), Work8 nd Corre8pondence f David Ricardo, vol. viii,p. 177.


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CARLO CASAROSA 573. In a growing conomy he dynamic quilibriumwage rate s constantlyabove the naturalwage rate and coincides withit onlyin the stationarystate.4. In the course of economic growth he equilibriumvalues of the realwage rate, he rate ofprofit,nd therateofgrowth end to falltogetherwiththe marginal product of labour.5. The conditions of production of wage-goods and of luxury-goodsinfluence he equilibrium values of the real wage rate, the rate of profit,and the rate of growth. However, the effects f similarchanges in theconditions ofproductionof the two types of goods are different nd, inparticular,of opposite sign as faras the rate of profit s concerned.6. An increase a reduction) of the natural wage rate or a reduction anincrease) of the 'reactivity' of population to the difference etween themarketand the natural wage rate, cause an increase (a reduction)of thedynamic equilibriumwage rate and a reduction an increase) ofthe rateofprofit nd of the rate ofgrowth.7. An increase (a reduction) of the 'propensityto accumulate' of thecapitalistscauses an increase (a reduction) of the dynamic equilibriumvalues of the wage rate and of the rate of growthand a reduction (anincrease)ofthe rate of profit.

FacoltM i Economita, isa

APPENDIXIN thisAppendixwe first ormulate mathematical two-commoditymodel which'incorporates'ur nterpretationf he Ricardiantheory fdistribution nd economicgrowthndthe notionthat therateofcapital accumulation s a decreasing unctionof heprices f he uxury-goods.Thenweinvestigate omeproperties f hedynamicequilibriumolutionforthe distributive ariables and the rate ofgrowth nd theconvergence f market equilibriumtowards the dynamic equilibrium position.Finally,we derive n a rigorouswaythe resultsverbally tated nsections5, 6,and 8.Al. 'Dynamic equilibrium'n a two-commodityystemLet us first ntroduce ur mathematicalmodelof theRicardian system.

THE MODELX= f(N1) where: X1 = physical quantityof cornproducedin one year;N1 = numberof workers mployed nthecornproduction; (1)


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58 A NEW FORMULATION OF THE RICARDIAN SYSTEMwiththefollowing roperties:f(O) > 0 (la)f'(0) > Ws where: w- = naturalwage rate in termsofcorn;(lb)f (N1) < 0 (ic)f'(o0) < W8 (Id)X2= N2 with a > 0 X2 physicalquantity fgold produced none year;N2 numberofworkers mployed n theproductionofgold;( physical quantityof gold producedby one worker n one year; (2)NM+N2= N N total numberofworkers; (3)W wN VW totalwage-bill, ntermsofcorn;w realwage rate, nterms fcorn; (4)K W K physicalstock of capital (corn); (5)R = f(N1)-N1f (N1) 1R yearlyrent, n realterms corn); (6)P1 = X1-R-N1w P1 yearly profits, n terms of corn,ob-tained in the corn-producingector;(7)p1 X1-P1R = N1 Pi priceofcorn; (8)P2X2 N2 P2 = price ofgold; (9)P2 2 P2X 2-N2pl w P2 = profits,n termsofgold,obtainedinthegold sector; (10)P = p1P1+P2 P2 P = totalprofits,nterms fthestandardofvalue; (11)8 p1w s monetarywage rate; (12)r pK r rate ofprofit; (13)PIKP =CP+PiI C(p constumptionf luxury-good y thecapitalists, xpressed n money;I = investments,n termsof corn; (14)

I- dK (15)P2X2 Cp~pPlR (16)K d yr with0 < y < 1 y shareofprofitswhich s invested;

(17)Y O(P2, a) with a a parameter,which expresses thea2Y < 0 and LY 0 propensity to accumulate of theOP2 Oacapitalists; (18)1 dN )1w-8 wt1 t = with = a constant; (19)0


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CARLO CASAROSA 59In the building fthe model we tookPasinetti's modelas a basis, since trepresentsin many respectsa rigorous translation'ofRicardo's thought. Pasinetti's model,we recall, s based on four implifyingssumptions:I. The systemproduces one wage-good only: corn.It. The productionprocess of corn takes one year.III. Capital consistsentirely f the wage-bill.IV. There s onlyone uxury-good,old,whoseproduction howsconstant eturns.The production rocess ofgold takes one year. Prices are expressed ntermsofgold and the monetary nit s the quantityofgold producedbyoneworkerin one year.Otherassumptionsrelevantto our model are the following:V. Workers pend their ncomeon corn.VI. Land-owners pend their ncome on gold.VIL. Capitalists spend part oftheir ncome on the luxury-good nd employtherest n the accumulationof capital.The first hirteen quations of our model are substantially quations (1)-( 11) and(13)-(14) of Pasinetti's model. The only difference oncernsproperty 1c) of theproduction unction f corn. In factwe thinkthat the assumptionofcontinuouslydecreasing eturns s too rigid and does not representRicardo's pointof view, sinceRicardo allows arge ntervals f constantreturns. Thereforewe softened asinetti'sassumption y assuming "(Nj) < 0.The last seven equations incorporate he ideas we put forth n paragraphs 1-4and 7 of our paper. More precisely:Equation (14) is the capitalists' budget constraint.Equation (15) is definitional.Equation (16) is the equilibrium ondition n the marketfor heluxury-good.Equations (17) and (18) show that the rate of capital accumulation s an increasingfunction f the profit ate and of the propensity o accumulate, and a decreasingfunction f the price of the luxury-good.Equation (19) shows the relationship etween the real wage rate and the rate ofincrease f population.Equation (20) is the dynamic quilibrium ondition f the system.The systemhas twenty quations and twenty-one nknowns:X1, X2, N1,N2, N,

W,w,KC R, P1,P2, P. P1, P2, s, r,Gp, , K dK Ia dN. Thereforewe can solve itas a function f N1. In particularwe can find he dynamicequilibrium alues ofthewage rate, he rate of profit, nd the rate of expansion.By substitutingn (13) from 3)-(12) we get:

r f'(N1)-w (21)wand, substituting rom 17), (19), and (21) in (20):f (Nj) -=w wew (22)

Hence: * - (y-P)wS+V{(y_-P)2w? + 4/3yfN1)ws} (23)and from 21): r* f (N1) w* (24)wherew*andr*are thedynamicequilibrium alues ofthe realwage rate and oftherateofprofit.The equilibrium ate ofgrowth fpopulationand capital is, then:

g* =yr* - (W-WS) (25)where *representsheequilibrium ateofexpansion.


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60 A NEW FORMULATION OF THE RICARDIAN SYSTEMAs for hemonetarywage rate, from6) and (8) weget:

a 1 (26)Pi f'(N1)and, substitutingn (12): s*

W*(27)Notice, ncidentally,hatbysolving 27) forw*and substitutingn (24) weget:

r* ---1 (28)whichrepresentshe (inverse)relationship etweenthemonetarywage rate and therateofprofit sed byRicardo in theanalysisoftheproblemof ncomedistribution.The 'dynamic equilibrium'solutionfortherealwage rate,the rate ofprofit,ndthe rateofexpansionallowsus to pointout some nteresting roperties f the Ricar-dian system.

The first ne concernsprofitsnd consists n thefact that the realwage rate andtherate ofprofit re determined imultaneously rom he same groupofequations,so thatprofits o nothave thecharacter f a residuum s inthe accepted version'ofthe Ricardiantheory nd in Pasinetti'smodel.The secondconcerns he equilibrium alue of the realwage rate. From (23)-(25)it is clear thattheequilibrium alues oftherealwage rate,ofthe rate ofprofit,nd oftherateofgrowth ependon themarginalproductivity f abourin agriculture. nparticular,we can distinguish hree cases:i:. iff'(N1) >ws w* >ws, r* >0, g* >0.II. iff'(N1) = ws w* = ws, r* = 0, g* = 0.III. iff'(N1) < ws w* < ws, r* < 0, g* < 0.In the first ase we have a growing,n the seconda stationary, nd in the thirda regressive conomy. Consequentlywe can say that, in a growing conomy,the'dynamic equilibrium'real wage rate is constantlyabove the natural wage andcoincideswith t onlywhenthe system s inthestationary tate.Notice, furthermore,hat the economicsystemdescribedby equations (1)-(20)is able to grow, incef'(0) > ws,but soonerorlater twillstop growing, ince

fP(co) < wvs.A2. Marketequilibriumnd dynamic quilibriumThe marketequilibriumofthe distributive ariables is determined y equations(21) and: w = K/N (29)obtainedfrom quations (4) and (5), whereK, N, and N1are given.To examine whethermarket quilibrium onverges o dynamicequilibriumnthe'intervals' where the productionfunction hows constant returnswe differentiate(29) and (21) withrespectto time:dw (1 dK I dN\ (30)dt =WK dat N dt (0

dr _ -'(Nl) 1 dK I dN (31)dt w dt N dt(From (30) and (31) we can see that, fthedistributive ariables (wand r) assumetheirdynamicequilibriumvalues,we have:1 dK_ 1 dN 0K tdt 0dtand, consequently:dw/dt 0 and dr/dt 0.


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CARLO CASAROSA 61If,on the contrary,he marketwagerate slower andthe rate ofprofit igher) hanthedynamicequilibrium evel, we have:I dN I d1fdN dT K dt

and, therefore; w/dt 0 and dr/dt 0. The marketwage rategoes on increasingand the rate ofprofit oes on fallinguntil we have:I dN * I dKN dt 9 k dtthat s,until hemarketwagerate and the market ateofprofitssume theirdynamicequilibrium alues.A similaradjustmentmechanism s at workif initially he marketwage rate isabove and the rate of profit s below the dynamicequilibrium evel.This is not a formalproof of the convergenceof marketequilibriumtowardsdynamicequilibrium,but is enough to state that, at everylevel of the marginalproductivity f abour nagriculture, ynamic quilibrium epresentshe attractionpoint' ofmarket quilibrium. Therefore,tmakessensetostudy, s Ricardodoes,theevolution fthedistributive ariables during he processofeconomicgrowthntermsof the dynamicequilibriumvalues.

A3. Dynamicequilibrium nd economic rowthSinceeconomicgrowth equiresan increase n the amount of cornproducedandthereforentheamountof abouremployedntheproduction fcorn,we canexaminethetendencies fthedynamic equilibriumvalues on thebasis ofthe derivativesofthesevariables withrespectto the amountof abouremployed n theproductionofcorn. The results re the following:

caw*= __________ ______ ____ < 0 asf "(Nl) < 0 (32)caN1 2J{(y 3)2wl+ 4yf '(Ni)ws}caN= f f(NN)(aw*D1)w}2W(1) > 0 asf"(Nl) < 0 (33)as* w*f"(Ni)-f'(N)(c@w*/1Ni) ? 0 asf"(N1) S 0 (34)c~N1 (w*)2 asf"N


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62 A NEW FORMULATION OF THE RICARDIAN SYSTEMA~s* _ f'(Ni){c@w*/c@f'(Ni)}-w* 0 (37a)af (N1) 1)}2ar* _ w*-f'(Ni){aw*/1f'(N1)} > ? (38a)

_____ c~r* __f ~aw= y ~~~> (39a)c3fN1) = 'f(N1) wS f '(N1)and:1 cw* 1f (y-_l)W2+2Pf'(Nl)ws v) > 0 (36b)Aos 2\ WmV{(-)2W2+4Pyf'(N)}Pa f'(Nl) 43a > ? (37b)

ai* __ -f(N)(aw*1/cac) < 0 (38b)

8a Y 43a+r*a- t > ?. (39b)Theseresults howthatchanges ntheconditions fproduction fboththewage-good and the luxury-goodnfluence he dynamicequilibriumvalues ofthe system.However,thetwotypesofchangescause variationsof the same sign,but quantita-tivelydifferent,fthe realwage and the rate ofgrowth, nd variationsofoppositesignof themonetarywage rate and of the rate ofprofit.A5. Changes n thehabits' oftheworkersA change in the habits' oftheworkersmplies, n ourmodel,either changeofthe natural wage rateor a change ofthe 'reactivity'of population's dynamicstothe evelof hewagerateexpressed yP. In thetwohypotheseswehave,respectively:

caw* -(Y P) (y-P)2WS+2/3yf'(N1)- 23 2flJ{(y-fl)2w2+4PyfN)w}> 0 (40a)csS* 1 cbw*a r*a (N1)(aW*1awS) < 0 (42a)

a Y 8-w pw (8W*(S)28- < 0 (43a)C9W8 c~W8 w)and:--aCh* I (y P)W + 2yf l)s]~~~~~~~ NW +hi -222t L 8~~ (_)W+4pyf (N,)WS}

+ (y-,B~w8-/{(y-P)2W2+ 4yf (N)wS) < 0 (40b)As* 1 3w* 0f'-(N- -ap (41b)Ar* -f'(N1)(E3)aw*/@) 0 (42b)apl (w*ycg* A cw* w* cr*GAB -s- =+--I _ y > 0 (43b)

l Notice hatfrom2) and (9) wehave:P2 1/c.1

Therefore, ince vpy< 0, aY > 0.


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CARLO CASAROSA 63In otherwords, fthe naturalwage increases decreases)orpopulation reacts' less(more) tronglyo thedivergence etween he market nd thenaturalwage rate,theequilibrium real and monetary)wage rate increases. decreases),whilethe rate ofprofit nd the rate ofgrowthfall.

A6. Changes n thepropensityoaccumulate fthe apitalistsIf thereis a change in the capitalists' propensity o accumulate, the dynamicequilibrium alues change as follows:aw* (, P)W Y 2+ 2pf(Nj)s S y~w* 1 WS[ (yl)IW?2fl'N1w 1- > 0 (44)- 2/L f+g)2w~?4/yf'(N1)ws}J aAs* 1 ew*-a f'(N1) Aa >0 (45)

er*= -f '(Nj)(w*1aa) < 0 (46)eg* __ f __ eY+ er* >0ea w8 ea r aa a ?Therefore,fthecapitalistswantto investa larger smaller) hare ofprofits t thegiven evelof thepriceofthe luxurygood,we have an increase a reduction)of theequilibriumalues of thewage rate and of therateofgrowth,whiletherate ofprofitfalls increases).

REFERENCES1. BLAUG, M. (1958), Ricardian Economics: a Historical Study. Yale UniversityPress.2. CASAROSA,. (1974), 'La teoria ricardiana della distribuzione dello sviluppoeconomico',Rivistadi PoliticaEconomica,959-1015.3. FRISCH,R. (1935-6), On the notionofequilibrium nd disequilibrium', eview fEconomic tudies,100-6.4. PASINETTI, L. L. (1960), 'A mathematicalformulationf theRicardian system',Review fEconomicStudies,78-98.5. RICARDO, ., On the rinciplesofPoliticalEconomy nd Taxation. In The WorksandCorrespondencef David Ricardo (P. Sraffa, d.), 1951,vol. i. CambridgeUniversityress.6. NotesonMalthus'sPrinciplesofPoliticalEconomy. In The Works,p. cit.,Vol. ii.

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A New Formulation of the Ricardian System-Carlo Casarosa