The Determination of Prices of Production

Home , Prices of production, Value product

Paolo Giussani

The Determinationof Pricesof Production

Sincethe turn of the century a multitude of criticisms have been directedagainst the so-calledtransformation of quantities of commodityvalue into quantities of price, on theassumption of a uniformprofit rate for all industries,as setforth in chapter9 of Marx'sCapital, volume 3. Commonto all thesecriticisms is the idea thatMarx's procedure lacks a transformationof the quantitiesof value of the means of production and of labor (the inputs). If thisdilemma is to be resolved- accordingto all thecritics - thetwo fundamentalequations of theMarxian transformation (totalprice = totalvalue; total profits = totalsurplus value) can obviouslyno longerbe simultaneouslyvalid. The resultof this is thatthe general profit rate can no longerbe calculatedfrom Marx'slaw ofvalue, and the value quantities themselves cease to playany role in thecalculation of prices of production. This resultstems directlyfrom the assumptionthat the pricesof the productsand thoseof the meansof production mustbe determinedsimultaneously, an assumptionthat has always been accepteduncritically by almosteveryone. This has been trueeven of Marxists,especially given the accep- tance by "left-wing"economists of Sraffa'stheory of prices and distributionof revenue.Yet thisresult is muchless obvi- ous than it may seem. Once the assumption(which is not foundin Marx) is provedto be irrationaland is thereforedis- carded,the criticisms directed against the Marxian determina-

Paolo Giussaniis editorof the journal Plusvalore. 67

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 68 INTERNATIONALJOURNAL OF POLITICALECONOMY tionof the prices of production and the rate of profit lose all their force.

1. The simultaneousdetermination of prices

Noneof thediscussions of theprices of production that have ap- pearedso far,and certainly not Sraffa's 1960 treatment,1 hastaken thetrouble to provide a theoreticaljustification for the procedure of simultaneouscalculation of inputand outputprices. Nevertheless, thereis a kindof reasoning prevailing among the so-called experts, accordingto which "prices must necessarily be simultaneousprices, sincethe prices of all commoditiesare basednot on thecosts of production(or on thelabor) sustained in thepast, at thetime when thecommodities were produced, but on thecurrent costs (or the quantityof labor)." The astute observer will easily perceive that this is an entirelycircular argument. Ifprices are determined by current costs,it is necessaryto establishwhat these current costs are, but sincethe current costs are theprices themselves (of thevarious inputs),the whole thing comes down to saying that (current) prices aredetermined by(current) prices. Anotheridea, also expoundedby JoanRobinson, is much moreelementary - almost trivial. Since thecommodities of the sametype have the same price as outputand as input,this argu- mentgoes, the same price for those commodities is placed on the leftand on theright side of theequations used for price calculation.Unfortunately, as we shallsee, this argument forgets that a givencommodity cannot operate as an inputand as an outputat thesame time. In reality,the procedure of simultaneous calculationof pricescannot be givenany theoretical foundation; it is simplyan axiom,which works only so longas itis notshown to produceconsequences that are absurdand/or in plaincontradic- tionwith the facts. To findout whether this is indeedso, letus imaginea systemof production composed of three industries (A, B, C), each of whichuses onlyproductive inputs coming from theother two, and in whichsector C producesthe good serving as the real wages consumedby all the workers.The material

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 69 structureof this system is representedby means of the following schema: (1) Ba + La -> A

Ab+ Lb - B Ac+ Bc + Lc -> C

C = Ca + Ch+ Cc - La + Lb+ LC= L. If,for convenience, we nowsuppose that the system is repro- ducedon an unalteredscale (simplereproduction), it should turn outthat sector A exchangesits total product A againsta portion ofthe product of B andC, specifically,against the quantity Ba + Caywhile the commodity B should be exchangedagainst Ab + C¿>, andthe commodity C-Cc withAc + Bc.We thereforeshould have thefollowing series of equations (the last being a simpleidentity, sincethe quantity Cc is consumedwithin the same industryin whichit is produced): (2) A = Ba + Ca

B = Ab+ Cb c-cc = ca + cb.

As can be seen,the commodities that enter into the exchanges amongthe various productive sectors are only the outputs, that is, whatis foundon the rightside of expressions(1), whilethe inputs(the left side) have been exchanged in the previous period. It is quiteobvious that no outputcan be exchangedagainst an input,since therecannot be use values thatfunction simulta- neouslyas inputsand as commodities.If a particularproductive use value(e.g., oil) is playingthe role of input, this means that it is in theform of productive capital and thereforecannot at that timebe foundin the (capital) commodity form. Equations(2) representdirect exchanges of commoditiesfor

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 70 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY commodities;if we transformthese barterexchanges into exchanges for commodity-money,the situationis worsened.The outputsBa + Ca, Ab+ Cb,Ca + Cb mustnow all be computedfirst and thenrealized in money,that is, say,di fourth commodity, before theycan be used as productivecapital. The assumptionof the simultaneouscalculation of input and outputprices in factimplies the eliminationof thisnecessity, something justified at timesby the pretextthat "purchases, sales, and production always occur simulta- neouslyfor all sectorsand individualcapitals, without interruptions makingpossible the passage fromone phase to the next." The objectionis clearlyirrelevant, since if we are movingfrom di particular groupof inputsto a particulargroup of outputs,as in the so-calledlinear models of production,the phases of sale/purchase and of productionfor these groups obviously cannot occur at the same time.The variousproduction processes must take place at the same time withrespect to each otherin the varioussectors, and afterthis the products can all be exchanged.2To do away withthis "minor"detail is basicallyequivalent to assumingthat circulation is nota constituentelement of the market economy, or (analogouslyto Ricardo'squantitative theory of money)that inputs enter the pro- ductionprocess without prices and leave it withprices, that is, the productsdo nothave the form of commodities.

2. The simultaneouslinear equations

The moderntheory of pricesand distributionis clearlyforced by its own premisesto assume the simultaneousdetermination of theprices of inputsand outputs.If one wishesto obtainthe prices of productionfrom the known data concerningthe production technology,together with knowledge of one of theso-called dis- tributivevariables (e.g., the wage rate),without the aid of other unwelcomequantities (among which the "value" of thecommodities is obviouslyparamount), it is necessaryto adopt the same temporalindex for the prices of theinputs and theoutputs, or else it will no longerbe possibleto determinethe rate of profit. To illustratethis elementaryfact, let us imaginea systemof

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 71 two industries,A and B, each of whichproduces a productive input,with the commodity B also beingused as thewage good. Witha timespread between the price of the inputs and the price ofthe outputs (a spreadindicated by the indices t and t + 1), the systemof production prices would assume the following form: (3) + = [Ba Bta]pbt(l+r) Apat + i

[^"«í+V»i](1+r)=l>p»í+r

The expressionsin equation(3) forma systemof two linear first-orderdifference equations, whose solution requires prelimi- naryknowledge of thequantity r (the rate of profit), which will enterinto the coefficient 1+r. Alternatively,ifwe treatr as an unknownfunction of t (i.e.,r/+i), a furtherequation for the determinationof T/+1is required,an equationthat cannot be con- structedin theframework of theSraffian theory, that is, without theaddition of other quantities.

3. The physicalsurplus

Economistsof theSraffian school rarely miss an opportunityto assertthat the category of surplusvalue is unnecessaryfor the determinationofthe rate and the volume of monetary profits, the simplesurplus in physicalterms being sufficient for this. Not withoutreason, they like to designate their own theoretical orien- tationas "one based on thesurplus approach'' It is therefore appropriateto investigatewhether this concept of materialsur- plus is indeeddefined sufficiently to play the role that its sup- portersassign to it. Evenchildren know that the strict conception of a surplusin purelyphysical terms is an absurdity,since the quantity of matter (and/orenergy) in theuniverse is constant,or at least can be changedby there is no knownforce. One musttherefore employ a differentconcept of surplus,a surplusthat is a quantityof use

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 72 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY valuesthat can be utilizedby humanbeings, in excessof those consumedeither by individualsor by production.But how can thisexcess be measured?The matterseems simple. On one side, writethe total of use valuesthat are productively consumed dur- ing a givenperiod, and on the otherside, writethe totaluse values producedduring the same period:the differencebe- tweenthe latter quantity and the former is thephysical surplus obtained.An attemptto carryout this procedure,however, wouldquickly show thatit is impossible:the subtrahendele- mentsare notfound in theminuend, namely, all thosenatural forcesand entities (the elements hidden in theearth's crust, for example)that are utilizableuse values,to all intentsand pur- poses,and in factare absolutelyindispensable, yet which cannot have a price because they are not commodities.This circumstanceis enoughto explainwhy the concept of a surplus of use valuesis in and of itselfchimerical. It is therefore necessaryto have recourseto a thirdand differentconcept of surplusor net product. Inevitably,there is no otheralternative than the surplus formedby thedifference between groups of goods thathave a priceand thatare thereforethe only goods thatend up in the calculationof thedifference between gross revenues pocketed by the capitalistproducers of commoditiesand production costs whichthese same individualsmust sustain. However, if boththe outputsproduced and the inputsused changetheir physicaland technicalcharacteristics over time,it will no longerbe possible to calculate a surplusin materialterms givenonly which goods have a priceand whichdo not,since termswill appearon theright side of theproduction equations thatare nothomogeneous with those on theleft side, whose sole functionis to be replacedin successiveproduction cycles. It followsthat the material surplus cannot be used as a basis fordetermination of theprices and therate of profit.This is a necessaryconsequence of thesimultaneous calculation of the inputand outputprices, because (as we have seen) thistype of calculationimplicitly presupposes that the use values appear

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 73 simultaneouslyin thetwo positionsof commoditycapital and productivecapital, which is an impossiblepremise.

4. The mostrecent "solution" proposed

It is knownthat none of thesolutions proposed thus far for the problemof the transformation ofvalues into prices is compatible withthe simultaneousvalidity of the two Marxianequalities (totalvalue = totalprice, total surplus value = totalprofit). Nor have any of themsucceeded in usingvalue quantitiesas the essentialbasis for the determination ofthe prices of production. However,a new solutionrecently proposed by Dumenil,Foley, and others3seems to haveaccomplished an apparentquadrature ofthe circle, obeying the two Marxian equalities by means of new definitionsofboth the value of labor power and the group of com- moditiesto which the total of prices and the total of values refer. Dumeniland Foley oppose the traditional view of the value of laborpower (the value of the consumer goods that form the real wage ofthe workers), preferring to define this value in termsof to thesum of money received by the workers (value of the nomi- nal wages);they base thispreference on theargument that the workersmay spend this sum of moneyas theysee fit.If the quantity,the type, and the quality of the commodities consumed bythe wage laborers vary, then, according to thefirst definition of thevalue of laborpower, the averagerate of surplusvalue shouldalso change,regardless of changesin productivity;accordingto thesecond definition of the value of labor power, the rateof surplus value should be constant. Now, it is quiteevident that, with this new definitionof the valueof labor power, the value of this particular commodity will notmerely change its characteristics, but will end up by simply disappearing.Instead of the magnitudeof value of a definite groupof goods (thereal wage consumedby theworkers), we havea sumof money, established without any link to the value of thereal wage or to anythingelse. Becausethe prices of productionof the commodities are not yet known at thetime when this

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 74 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY quantityof money that is to representthe wages is fixed,there is no guaranteethat, once the prices are set, this quantity will actu- ally enablethe workers to purchasethe group of commodities comprising,at thattime, the socially established level of con- sumergoods required by the wage laborers, and, therefore, make possiblethe reproduction ofthe economic system. This allegedly new definitionof wagesas simplya quantityof moneyis in no way new,since it is in factexactly equivalent to thenominal wagethat we findin Sraffa,which causes so manyproblems for histheory. The same problems cannot fail to appearin the formulationof Dumenil and Foley. On theother hand, if this sum of moneythat is equivalentto thewages is fixedin relationto theprice of the composite com- modityconsumed by theworkers, we maywonder where this pricecould come from, since the prices are supposedto be the finalresult of thetransformation and surelynot its startingas- sumption.In thissecond case, we thusfind ourselves in thesim- ple Sraffianformulation ofthe system of prices with real wages, to whichis added(quite arbitrarily) the statement that the total nominalwages receivedby the workersand the value of the commoditiespurchased by the latter are one and the same. Yet thenew definition of the wage is necessaryif the circle is to be squaredwith respect to the two famous equations. In fact,it is enoughto add thatthe totalproduct, corresponding to the equationbetween the sum of values and the sum of prices, cannot be thegross product, as was believeduntil now, but rather the net product,if we are to avoida possibledouble counting of com- moditiesthat appear as inputsin a different,later process of production.Ifthe sum of the prices of the commodities that form thenet product of the economic system is equatedto thesum of thevalues of these same commodities, we obtainonly one ofthe infinitepossible numeraires of the Sraffian system of production prices,and if (as followsfrom the new definitionof wages)the sumof the wages is necessarilyequal to thevalue of theaggre- gatelabor power, it is a trivialresult that the sum of profits,or thetotal profit, should be equal to thesum of surplusvalues, or

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 75 the totalsurplus value, since the net productis knownto be dividedbetween wages and profits. Notonly could all ofthis be obtaineddirectly and easily from theSraffian system by the calculation of the prices of production, usingsuitable ad hoc definitions,without having to presentthe factsas thougha mysterycloaked in secrecyfor centuries were finallybeing explained, but also thisoperation runs the risk of defininga numerairethat is unacceptable,since it is notneutral withrespect to relativeprices, as has beenbrilliantly shown by Stamatis(1990) in his demonstrationthat the numeraires com- prisedin a compositecommodity cannot guarantee invariance of relativeprices with respect to thechoice of thesame numeraire inso-called linear systems of production.

5. The determinationof the prices of production

If it is necessaryto introducea timelag betweenthe prices of productionof the inputs and the prices of production of the out- putsin orderto takethe circulation process into account - some- thingthat would be essentialeven in thecase of a commodity circulationtime equal to zero4- assuminginput-output relations identicalto thoseof equation(3), the systemdetermining the pricesof production would assume the following form:

(4) + = [Ba ']Pbt(l+rt + i) ¿Pot + i

As we have alreadynoted, this system of equationsof itself does nothave a solution;in orderto be usefulfor something, it mustbe furnishedwith supplementary equations that give a pre- cise meaningto thefactor 1 + n + 1,that is, thegeneral rate of profit.This is in factthe purpose of theMarxian theory of the

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 76 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY exchangevalues of commodities as determinedby the labor time necessaryfor their production. The Marxiantheory declares that the magnitude of thevalue ofevery commodity is determinedby the sum of the value of the meansof production consumed in itsproduction (fixed constant capitalconsumed plus circulating constant capital consumed) and thevalue addedby laborin thelabor process (which is divided intothe value of labor power and surplus value). In theterms of our formulation,this means that the value magnitudesof the commoditiesA andB (whichwe denoterespectively Xa and Xb) aredetermined by the following system: (5)

Now,substituting the production prices at timet forthe quantitiesof value at timet of the constantcapital employed and consumedin thesystem (5), we obtainthe system of whatmay be calleddirect prices (designated by thequantities iia andtc¿>). Thissystem is indispensableif we areto derivethe equation for thegeneral rate of profit: (6)

The directprices express the quantity of total(gross) value createdin eachproduction sector, constituted by the price of the constantcapital consumed and the new value added, but not yet redistributedamong the various sectors in accordancewith the

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 77 criterionof the uniform temporal rate of profit on thetotal antici- patedcapital. These prices, far from being a fictitiousquantity or a merestep in theprocedure for calculating the prices of production,constitute the magnitudes to whichthe market prices of the commoditieswould tend if the mechanism by which money capi- talgoods is continuallytransferred from one production sector to anotherin the direction of the highest rate of profit should cease.5 Of course,the direct prices of thecommodities would coincide withthe value magnitudes if theprices of thecommodities that appearas constantcapital in (6) wereequal to the values of these commodities. At thispoint, it is quiteeasy to derivethe equation for the generalrate of profit. Given the starting prices of the commodities thatform the aggregatereal wages in the two sectors (Biapbt+ Bibpbt)and thequantities of thenew valuescreated, we obtainthe equation for the uniform rate of profit: (7)

rt+ 1 + + + • Abpat Baph Blapht Blapbt

whichwe mayinsert into each of the two equations of the system (4) toobtain the prices of production pö/+1 and p/?,+1.6 It is obviousthat our system of equations(4)-(7) cannotyet havea completesolution, as it stilllacks the establishment of an initialcondition pao , pbo- It maybe thoughtthat the initial condi- tionis givenby the value magnitudes of the commodities at the timet = 0, orat the moment of the passage from simple commod- ityproduction to capitalistproduction, or it maybe thoughtas givenin some other way. Whatever it is, it is irrelevant,since it is easyto see thatthe behavior of the solution of a systemof type (4), supplementedby equation (7), is entirelyindependent of the establishmentofany initial condition. This means, quite simply, thatthe prices of production can onlybe theredistribution ofthe

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 78 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY quantitiesof value (whichwe have heredesignated la and k) newlycreated in the course of the continuous succession of labor processesover time. Whateverthe initial condition chosen for the system (4)-(7), thisis necessarilyequal to that which is adoptedby the system of values(5). Whetherone starts at the dawn of commodity production,or sets out from a hypotheticaltransition from simple com- modityproduction (directly regulated by thevalue magnitudes) to productionbased on capital,the point of departuremust be commonto thethree systems of values,direct prices, and prices of production- that is, systems(5), (6), and (4), respectively; hence,the sum of the prices in a givenperiod will always neces- sarilybe identicalto thatof thevalues in thesame period, as is rathereasy to verifyfrom the solutions of thethree systems. In each period,the quantity that is distributedamong the various (two,in ourcase) capitalsis givenby: (8) ^ +l»,-(B^+BOp»«f where( Bia+ Bib) pbtcomprises the quantity of value produced thatgoes to laborpower. Probably it is exactlythis last magni- tudethat will somewhatperplex those Marxists who strongly desireto respecteverything that Marx said. But expression (8) is theonly way to remain within Marx's own theory of the prices of productionand of the value of labor power. Thequantity designated by ( Bia+ Bib) pbtis theprice of productionof the total of the commodities consumed by the workers (aggregatereal wages) at time t, resulting (in turn) from a redistribution ofquantity of labor performed inthe preceding period on thebasis ofthe criterion of the equal rate of profit for all typesof capitalist production;as such,it is theprice of production of labor power at timer. It is thereforeappropriate to ask whatrole is playedby the magnitudeof valueof laborpower in thetransformation process: doesit not end up disappearing,sic etsimpliciterl This value mag-

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 79 nitudeplays the same role as all the magnitudesof value in the generalprocess of productionand circulationof capital, being continuouslyand unceasinglyredistributed among the various capitals.The value and theprice of labor powermust be consid- ered not as the value and price of some metaphysicalentity, but ratheras thevalue and priceof a particulargroup of use values.It is quite obviousthat the total price of productionof laborpower, in thegeneral case, is different,in each period,from the total value of laborpower, as Marxhimself notes quite carefully in his discussion ofthe transformation ofvalues into prices: As forthe variable capital, the average daily wage is certainlyal- waysequal to thevalue product of thenumber of hoursthat the workermust work in order to produce his necessary means of subsistence;but this number of hoursis itselfdistorted by thefact thatthe productionprices of the necessarymeans of subsistencediverge fromtheir values. [Marx 1977c, p. 248; emphasisadded]

It is thereforepossible that even thecost-price of commoditiespro- ducedby capitals of averagecomposition may differ from the sum of valuesof theelements which make up thiscomponent of theirprice of production.Suppose the average composition is 80c + 20v. Now, it is possiblethat in theactual capitals of thiscomposition 80c may be greateror smallerthan the value of c, i.e., the constantcapital, because this c may be made up of commoditieswhose price of productiondiffers from their value. In the same way, 20v might divergefrom its value if the consumptionof the wages includes commoditieswhose price of production diverges from their value; in whichcase thelabourer would work a longer,or shorter,time to buy themback (to replace them)and wouldthus perform more, or less, necessarylabour than would be requiredof theprice ofproduction of such necessaitiesof lifecoincided with their value. [Marx 1977c, p. 253; emphasisadded]

In practice,Marx's statementmeans that the difference between themagnitude of value and theprice of productionof thetotal real wage is quiteirrelevant with respect to thetransformation of values intoprices. Since whatthe worker accomplishes in thelabor process is notto transferthe value of themeans of subsistenceto thevalue

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions 80 INTERNATIONAL JOURNAL OF POLITICAL ECONOMY of theproduct, but rather to re-createex novoa value equal to that ofthe means of subsistence (plus, obviously, a surplus), the division of theworking day betweennecessary labor and surplusvalue is adjustedin eachperiod by respect to theprice of production of the realwage, so thatany quantitative difference between the value andthe price of production of labor power loses all significance, a circumstanceexpressed in formula(8), whichgives the total amountof the surplus value produced (and redistributable among thevarious capitals) in a givenperiod. It followsautomatically thatin each individualperiod (circuit of capital)the sum of the surplusvalues produced in thevarious sectors can neverdiffer fromthe sum of the profits received in the various sectors.

6. Characteristicsof the solutions of the systemof prices of production

Althoughthe system of equations (4)-(7) is formallypresented as a nonlinearsystem, it is easy to linearizeand - if we holdcon- stantthe technical coefficients and thequantities of laborcon- sumed- itssolutions (for the two prices pat , Pbt and for the rate of profitn) are all absolutelystable and convergeon a pointof equilibriumreached from a certain i,whatever the initial conditions.7 The interestingfact is thatthe equilibrium solution of our system(4)-(7), exceptfor its independence from the initial conditions,looks identical to the solution of the corresponding Sraffian systemof simultaneous determination ofthe prices of inputs and theprices of outputs.This circumstance may easily lead one to thinkthat the two types of systemare, in theend, quite equivalent,and thatit is nottherefore worth the trouble to searchso zealouslyfor something new, in distinctionto thealready well- knownSraffian theory, which is so simpleand convenient.But thiswould be a seriouserror. The two systemsare equivalent (and thisonly under the assumptionof fixedtechnical coeffi- cientsand quantityof labor)only after the fixed point of the discretesystem (4)-(7) has beenreached8; in theprevious periods,the numerical values of theprices of productionin thetwo

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 81 systemsdiffer. Without the assumption of theconstancy of the technicalcoefficients and the labor inputs, and allowing the tech- nologiesadopted to vary,the two systemsnot only cease to be thesame, and thus to provideidentical numerical solutions, but actuallyreveal themselvesto be qualitativelydifferent. Our discretesystem of theprices of productionand of therate of profit(4)-(7) providesa clearand reliableway of determining thequantities of interest,moving from one periodto another, whilethe simultaneoussystem of the Sraffiantype (intrinsi- cally staticin its mathematicalnature) must ultimately break downinto as manynew systems as thereare periods of techno- logical changeconsidered, without providing a way to move directlyfrom one periodto the next,or fromone circuitof productivecapital to another[P . . . P'], a movementmediated bythe circuit of commodity-capital [C. . . C] andby that of the money-capital[M . . . M']. Thisis clearfrom the following schema, which puts the process of capitalcirculation, as expoundedby Marxin thesecond volumeof Capital,in relationwith our discrete system of equationsof prices of production: m cl . . . p ... c2 M' c' - P' . . . ^5- - ... + ^T + + ... ^X , APt it (Apt wPt)(l rt) =Qpt + l whereM standsfor money-capital; C' and Ci indicatethe commodity-capitalserving as inputand the commodity-capital that is theoutput of production,respectively; P is theproduction process; and theapostrophe (') denotesthe quantitative increment withrespect to theprevious period. In thebottom row, A is an indexfor the productive inputs, and w is an indexfor the aggregatereal wage, while Q indicatesthe aggregate output. A numericalexample of the effects of a continuousvariation in thetechniques and in labor productivity will serve much better than a generalargument toillustrate this important fact. Keeping in mind thesimultaneous (Sraffian) form of system (4)-(7), which is:

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(9) (1+r) =A*>a (*a +Bla)Pb

We mayassign these initial numerical values to the various coefficients: = 5; =7; + = £ = 20 . A6=10; Ba 5^=10; 5£ «a «6 For thesegiven values of thetechnical coefficients, the rela- tive Sraffianprices and the relativeequilibrium prices of the system(4)-(7) areequal topjpb = 1.0686,and the corresponding rateof profit to r = 1.2139. Ifwe nowlet all thetechnical coeffi- cientsvary for three consecutive periods and compare,for each period,the prices and the rates of profit obtained by our discrete system(pat , pbt , n) withthose that result from the static Sraffian system.The symbolsA%p and A%r indicate, for each period, the percentagedeviations in the discrete prices and rate of profit with respectto the static prices and rate of profit. 1.

= S; = 4; = 7 ; =7; « = 16. Ab Ba B^ 5£ = 1-0089 = 1.0427 Pa1/Pbl Pa/pb ra = 1.4721 r = 1.6082

A%p = - 3.242%

A%r = - 8.463%

Ab= 6 ; Ba = 3.5 ; B, = 7 ; B, = 5 ; I = 12 . = L2670 = im7 Pa2/í>62 Pa /Pb

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r2 = 2.2216 r = 2.3341 A%p = + 8.597%

A%r= -4.820%

3. = 5 ; Ba = 3.5 ; = 5 ; = 4 ; I = 10 . Ab B£a B£b

1.1149 pa/pb= 1.0320 pa/p6= r3 =2.9903 r = 3.4326

A%p = -7.436% A%r = - 12.885% 7. Conclusions

Thisarticle has demonstrated how Marx's theory of the prices of production,better known as the"transformation of values into prices,"not only presents no logicaldeficiency, but also is dy- namicallysuperior to Sraffa'swidely circulated and accepted formulationwhen expressed in its correspondingmathematical terms,which cannot be thosecustomarily used by economists of thestatic-linear tradition. In thecontext of a discretedynamic system,which is essentialif we areto consider the part played by circulationalongside production, the quantitiesof labor per- formedin the various production sectors (and thus the production of value) are absolutelyessential to definethe general rate of profit.This shows that the prices of production are nothing more thanvalues redistributed over the various invested capitals. The factthat the equilibrium solutions of a dynamicaldiscrete system andthe solutions of a staticsystem are identical when the technical coefficientsare constant is ofvery little practical importance andonly shows that the Sraffian system of equations amounts to no morethan the static case of Marx's theoryof theprices of production.9

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The discretesystem defined here, however, is insufficientand may stilllead to majormisunderstandings if its intrinsiclimits are notgrasped. The systemlacks a descriptionof themechanism that continuouslyleads to the formationof a uniformrate of profit- whichis a mereassumption here - fromthe different sectoral rates. As a result,it is not possible to studythoroughly the effectsof technologicalprogress and incrementsin productivityachieved by changesin themeans of productionand in thecomposition of the capitalwithin the individual branches of thesocial production.Fu- turework will extend the model presented here to themechanism of formationof theuniform rate of profitand to thevariation of labor productivityin thevarious sectors through the introduction of new andmore advanced means of production.

Notes

1. See Sraffa(1960). Here we are abstractingcompletely from the logical criticismsdirected against the Sraffatheory by Marxisteconomists, although this criticismhas demolishedthe mythof thattheory's coherence. See, for example,Savran (1979, 1980) and Stamatis(1990). 2. See Dumenil (1983), Foley (1982), and Glick-Ehrbar(1987). 3. Marx himselfimplicitly recognizes the need for a time spread in the formulationof theprocess of formationof theprices of productionon thebasis of values when,in chapter9 of volume 3 of Capital, he mentionsa "quantitative errorin thepast'* with respect to thepossible variancebetween the value of theconstant and variablecapital used in productionand its price of production.See Marx (1977c), p. 252. 4. Recent statementsof this argumentare to be foundin Freeman (1984) and in Carchedi (1991). While Kliman and McGlove (1989), althoughadher- ing to a viewpointrather similar to thatset forthin the presentwork on the problemof thetransformation of values intoprices, do not explainwhy a time lag should be placed between the prices of the inputsand the prices of the outputs.These authorsinvoke the "dialectic" and the "dialectical method"of Marx, as opposed to thestandard method of thebourgeois economists, but this is like the storyof Don Quixote and Sancho Panza, who saw the same phe- nomena appear in very differentforms depending on the differentattitudes withwhich they approached the adventure. 5. The importanceof what we have called here directprices follows directlyfrom the theoryof rent. Independentlyof differencesin the natural fertilityof the various typesof land and the differentamounts of capital invested,the prices of productssubject to groundrent will tendtoward the direct prices of equations (6), since privateproperty hinders the freecirculation of

This content downloaded from 139.184.14.159 on Fri, 11 Dec 2015 13:23:49 UTC All use subject to JSTOR Terms and Conditions WINTER1991-92 85 moneycapitals among the sectorswith respect to land, which is the practical mechanismthat leads to the equalization of the various rates of profitor the redistributionof the social surplusvalue to thedifferent branches of capitalist production.The fact thatagricultural and miningproducts are sold at direct pricesof equations(6) generatesabsolute rent, which is equal to thedifference between the directprice of the commodityyielding a rent and the price of productionat whichthis would be sold were thereno barrierto thecirculation of capital among the productionsectors. Disregard and ignoranceof the fact thatthe magnitudesof value are determiningfactors of prices have led the neo-Ricardiansand Sraffiansto rejectabsolute rent, due to theimpossibility of formulatinga theoryfor it, and have led many economistsof the "Third- Worldist"tradition (Amin, Dos Santos) utterlyto confuseabsolute rentwith monopolyrent. 6. Some readersmay doubtthat the numerator of equation(7) constitutesa sum of homogeneousterms, but theywould be wrong.In the entiresystem (4)-(7), quantitiesprovided with index t are eitherunknown magnitudes (functions)or knownmagnitudes (functions). In systemsof thiskind, the unknown magnitudes(functions), by thenature of things,assume thequality (time, work, weight,mass, etc.) of theknown magnitudes (functions), which in ourcase are the quantitiesof labor time la and & expendedin thetwo sectors of production. 7. Upon requestthe authorwill provide the completemathematical treat- mentof thediscrete dynamical system presented here, and itsrelations with the staticsystem of theSraffian type. 8. The essentialdifference between our discretesystem and thestatic Sraff- ian systememerges with great clarity in thecase of productionwithout the use of labor (fullyautomated production), which corresponds to zero value of the quantitiesla and h. For la, the Sraffiansystem (9) yieldspositive solutions for theprices of production,as forany other value of la and 1¿>,while thedifference system(4)-(7) has no solutions.In the absence of humanlabor expended in production,the two systems(with constanttechnical coefficients) no longer convergeto the same equilibriumsolution. This circumstancecorresponds to thefundamental principle of theMarxian theory, according to whichthe calculationof values is indissolublylinked to theexistence of thecommodity economy, which is incompatiblewith a fullyautomated production. The discrete systemdeveloped here,however, should not be confusedwith a simple itera- tive calculation of the prices of productionfrom values. It is a ratherwell knownfact that the Sraffian prices of productioncan eventuallybe obtainedby using the same technicalcoefficients, starting with the quantitiesof value of the commodities(or otherpositive quantities),and repeatingthe calculation (see Shaikh, 1977). However, the iterativecalculation from values is only a numericalcalculation procedure in whichthe intermediate values of theprices (beforethe final equilibrium value) do not count,since theyare purelyimagi- naryquantities; in our system(also assumingfixed technicalcoefficients and amount of labor), every intermediate(not equilibrium)value counts as an effectiveprice, since the equations are builton the assumptionof the circulation of capital and the necessarysuccession of the threecircuits of capital

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(productive,commodity, and money)(see Marx,1977b, pp. 26-123), an assumptionthat, as can be seen,exists neither in theSraffian theory nor in the variousstatic Marxist derivations of theof production prices. In thiswork we use theterm circuit of capital(productive, commodity, or money)instead of theterm cycle of capitalused by Marxin thefirst chapters of volume2 of Capitalto indicatethe various metamorphoses ofcapital, in order to avoidany possibleconfusion with the economic cycle, that is, thecyclical oscillations of capitalistproduction. 9. It is perhapsfor this reason that, whatever anyone said (e.g., Joan Robin- son),Piero Sraffa declared: "The labor theory of value is absolutelycorrect."

References

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