Marx s transformation problem and Pasinetti s vertically integrated subsystems

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1 Cambrdge Journal of Economcs 208, of 8 do:0.093/cje/bex068 Marx s transformaton problem and Pasnett s vertcally ntegrated subsystems Ian Wrght* Pasnett (988) constructs a complete generalzaton of Marx s transformaton problem, n the context of a non-unform growth model, by provng that producton prces are not proportonal to the physcal quanttes of labour suppled to vertcally hyper-ntegrated subsystems. Pasnett therefore restrcts the labour theory of value to a pre-nsttutonal stage of analyss. In ths paper I demonstrate that the transformaton problem dssolves once we consder the vertcally ntegrated subsystems nduced by the specfc nsttutonal setup of a captalst economy. I prove that producton prces, both n steady-state and non-unform-growth models, are proportonal to the physcal quanttes of labour suppled to vertcally super-ntegrated subsystems. In consequence the labour theory of value, sutably generalzed, also apples at the nsttutonal stage of analyss. Key words: Pasnett, Labour theory of value, Marx, ransformaton problem JEL classfcaton: C67, E, D46 Pero Sraffa (960, ch. 3) demonstrated that the natural prces of reproducble commodtes necessarly vary wth the dstrbuton of ncome due to the nequalty of the proportons n whch labour and means of producton are employed n the varous ndustres, whereas the real costs of producton of commodtes, measured n terms of labour, do not. In consequence, labour costs cannot fully explan the structure of natural prces. hs explanatory gap creates varous problems for the classcal labour theory of value, most notably Karl Marx s transformaton problem (e.g. see Seton, 957; Desa, 988; and Hunt and Glck, 990). Many authors nterpret Sraffa s analyss to mply that the labour theory of value s, at best, ncomplete, or worse, logcally ncoherent (e.g. Samuelson, 97; Lpp, 979; Steedman, 98). Lug Pasnett, a follower of Sraffa, offers a dfferent nterpretaton. He proposes a separaton thess (Pasnett, 2007, ch. 9) that orders the study of economc systems nto a pre-nsttutonal or natural stage of nvestgaton, concerned wth the Manuscrpt receved 2 July 203; fnal verson receved 8 February 207. Address for correspondence: Ian Wrght, Economcs, Faculty of Socal Scences, the Open Unversty, Walton Hall, Mlton Keynes, MK7 6AA, UK; emal: wrght@acm.org *Open Unversty. I dedcate ths paper to the memory of Angelo Reat. My thanks to Phllp Adkns, Nada Garbelln, Andrew rgg, Arel Lus Wrkerman and the anonymous revewers for feedback and suggestons. Pasnett calls t a theorem, but snce no proof s nvolved, I prefer to call t a thess.

2 Page 2 of 8 I. Wrght foundatonal bases of economc relatons that reveal the fundamental constrants that any economc system must satsfy, followed by an nsttutonal stage (Pasnett, 2007, p. 276), whch s carred out at the level of the actual economc nsttutons (Pasnett, 2007, p. 275), whch dentfes how the constrants manfest n specfc nsttutonal setups. Pasnett s atttude to the labour theory of value s shaped by ths separaton. Pasnett argues, n a seres of works (e.g. Pasnett, 98, 988, 993), that the labour theory of value, rather than beng ncomplete or ncoherent, s a powerful analytcal tool at the pre-nsttutonal stage of nvestgaton, and therefore has to be taken as provdng a logcal frame of reference wth an extraordnarly hgh number of remarkable, analytcal, and normatve, propertes (Pasnett, 988, p. 32). For example, Pasnett (988) analyses the pre-nsttutonal cost structure of a nonunformly growng economy. Pasnett constructs a complete generalsaton of the pure labour theory of value (Pasnett, 988, p. 30) by provng that, n condtons where supply equals demand, the economy s natural prces are proportonal to the physcal quanttes of labour suppled to vertcally hyper-ntegrated subsystems that nclude the producton of net nvestment goods. In consequence, the labour emboded n a commodty, sutably generalsed, equals the labour t commands n the market. However, at the nsttutonal stage of analyss, the pure labour theory of value breaks down. Marx s prces of producton (Marx, [894] 97, ch. 9) are the steady-state prces that correspond to an nsttutonal setup n whch captalsts reallocate ther captal to seek hgher returns untl a unform, general rate of proft prevals across all sectors of producton. Pasnett constructs a complete generalsaton of Marx s transformaton problem (Pasnett, 988, p. 3) by provng that, n general, producton prces are not proportonal to the labour suppled to the hyper-ntegrated subsystems. Pasnett (98, p. 53) concludes that a theory of value n terms of pure labour can never reflect the prce structure that emerges from the operaton of the market n a captalst economy. Adam Smth restrcted the labour theory to an early and rude state of socety that precedes the accumulaton of stock (Smth, [776] 994, p. 53). Pasnett also restrcts the explanatory reach of the labour theory, but for a dfferent reason. Accordng to Pasnett, the labour theory provdes a natural or deal standard from whch to analyse and crtque the nsttutonal setups of actual economc systems. Pasnett, therefore, retans an essental normatve role for the labour theory. he argument of ths paper s that the labour theory of value s not merely normatve but also a postve theory that apples to the prce structure of a captalst economy. I solve, or more precsely dssolve, Marx s transformaton problem, and Pasnett s generalsaton of t, by extendng Pasnett s vertcally ntegrated approach to encompass the nsttutonal condtons of producton. I construct vertcally super-ntegrated subsystems that addtonally nclude the materal reproducton of the captalst class as part of the subsystem. I then prove that producton prces are proportonal to the labour suppled to the vertcally super-ntegrated subsystems. In consequence, the labour emboded n a commodty equals the labour t commands, even n the crcumstances of captalst producton. ransformaton problems necessarly arse when we compare nsttuton-dependent prces wth natural, or nsttuton-ndependent, vertcally ntegrated subsystems. he problems therefore dssolve once we compare prces wth vertcally ntegrated subsystems nduced by the specfc nsttutonal setup of an economy. A sutably generalsed labour theory of value s therefore nether ncomplete or ncoherent, and need not be

3 Marx s transformaton problem Page 3 of 8 restrcted to a normatve role, but spans both the natural and nsttutonal stages of analyss. he structure of the paper s as follows: Sectons to 3 summarse Pasnett s model and hs argument for restrctng the labour theory of value to a normatve role. Secton 4 proves that Marx s producton prces are proportonal to the labour suppled to the vertcally super-ntegrated subsystems. Secton 5 concludes by dscussng the mplcatons for the post Sraffan reconstructon of classcal economcs.. Hyper-subsystems and ther natural prces Sraffa (960, p. 89) proposed to decompose an ntegrated economc system nto as many parts as there are commodtes n ts net product, n such a way that each part forms a smaller self-replacng system the net product of whch conssts of only one knd of commodty. hese parts we shall call subsystems. A subsystem s a vertcally ntegrated slce of the economy that produces a sngle commodty as fnal output and replaces the used-up means of producton. Pasnett (988) generalses Sraffa s approach to a growng economy where each sector produces nvestment goods that ncrease the scale of producton. 2 Pasnett defnes growng vertcally ntegrated hyper-subsystems that addtonally nclude the producton of nvestment goods. Pasnett constructs an n-sector economy, whch exhbts unbalanced growth, n terms of n hyper-subsystems. he total output of a hyper-subsystem s q () t = q () t A + ( g + r ) q () t A + n () t, () where q () t s a vector of n quanttes, A = [ a ], j s a constant n n nput-output matrx, and n () t s a zero vector except for the th component, whch s a scalar n that represents the fnal demand for commodty. he total output of a hyper-subsystem, q (), t therefore breaks down nto () replacement for used-up means of producton, q () t A, () addtonal nvestment n means of producton, ( g + r ) q () t A, to meet ncreased demand for commodty due to the growth rate, g, of the populaton and the percapta growth rate, r, of consumpton demand for commodty (whch may be postve or negatve), and () the fnal output, or net product, n (), t whch s the quantty of commodty consumed. he total labour suppled to the hyper-subsystem s ( ) - L () t = lq () t = li-a( + g + r ) n () t, (2) where l = [ l ] s a vector of n drect labour coeffcents. 3 A hyper-subsystem ncludes the labour and means of producton necessary for the producton of ts fnal output and the labour and net nvestment n means of producton necessary for ts expanson at the growth rate ( g + r ). Pasnett defnes the trajectory of fnal demand as 2 For a dscusson of the geness of the concept of vertcal hyper-ntegraton, see Garbelln (200, pp ). 3 Equatons () and (2) are dentcal to equatons (2.5), (2.6) and (2.7) n Pasnett (988) except, n ths paper, we make the smplfyng assumpton that B = I.

4 Page 4 of 8 I. Wrght n g r t n () t n ( ) ( + = 0e ), (3).e. d = n ( g + r ), whch drves the growth of the subsystem startng from ts ntal dt scale at t = 0. For notatonal convenence I now drop explct tme parameters. All subsequent algebrac statements therefore hold at an mplct tme t. (I consder the mplcatons of the trajectory of fnal demand n Appendx 6.4.) Pasnett obtans the ntegrated economc system by composng the n hyper-subsystems. Defne the total output of the ntegrated economc system as the sum of ts n hyper-subsystems, q = = qa + qa + n q A n [ q ] g r + = (4) n n n where q = q, n = n, and L = lq = L, wth standard restrctons on the egenvalues of A and the feasblty of the growth rates. 4 Note that Pasnett s model ncludes = = = a steady-state economy as a specal case (by settng g = 0 and r = 0 for all ). Pasnett defnes natural prces that correspond to the pre-nsttutonal stage of nvestgaton. He stpulates that each hyper-subsystem has ts own natural proft rate, π, whch s equal to the rate of growth of demand for the correspondng consumpton good (Pasnett, 988, p. 29); that s, we have n natural proft rates, π, π,, π 2 n, where π = g + r and, n consequence, n vectors of natural prces, p p p,,, 2 n, one for each hyper-subsystem, such that p = p A+ p Aπ + l, (5) w where w s the wage rate. Defne X ( ) as the th column and X ( ) as the th row of X. he prce of commodty j n hyper-subsystem therefore breaks down nto () the cost of replacng used-up ( j means of producton, p A ), () the cost of net nvestment n addtonal means of producton, p ( j A ) π, and () the wage bll, l w. In general, each commodty type has a dfferent natural prce n each hyper-subsystem. he natural proft rates make possble the expanson of the producton of each fnal good accordng to the evoluton of ts fnal demand, whch, as Bellno (2009) explans, provdes a socal justfcaton for proft n terms of the structural necessty for mark-up rates that fund the growth of each hyper-subsystem. Proft and wages, n ths natural system, perform dfferent economc functons: proft s purchasng power that njects commodtes back nto the crcular flow, for the expanson of the system, whereas wages are purchasng power that ejects commodtes from the crcular flow, for fnal consumpton (Garbelln, 200, pp ). 2. A complete generalsaton of the pure labour theory of value he concept of a vertcally ntegrated subsystem and the real cost of producton of a commodty, measured n terms of labour, are closely connected. For example, Sraffa 4 Equaton (4) s dentcal to equaton (2.) n Pasnett (989), except, agan, we set B = I.

5 Marx s transformaton problem Page 5 of 8 (960, p. 3) defnes the labour cost of commodty as the total labour suppled to the subsystem that produces a sngle unt of as fnal output. he standard equaton for classcal labour-values, v = va+ l, mmedately follows, snce each labour-value, v, s the sum of the drect labour, l, suppled to sector and the vertcally ntegrated, ndrect labour, v A ( ), suppled to other sectors of the economy that replace the used-up means of producton (e.g. see Sraffa, 960; Samuelson, 97; Pasnett, 977). he natural prces, p, of each hyper-subsystem, gven by equaton (5), vary wth the natural proft rate, π = g + r, whereas classcal labour-values do not. Classcal labourvalues, therefore, cannot fully explan the structure of natural prces n Pasnett s growng economy. In the economy defned by (4), wth the set of natural prces (5), wages are the only type of ncome. Captalst proft, n the sense of ncome receved n vrtue of frm ownershp rather than labour suppled, s absent at the pre-nsttutonal stage of nvestgaton. As Reat (2000, p. 497) notes, the mere exstence of wages could presuppose two socal classes. However, on ths pont also Pasnett s model s flexble, because nothng prevents us from consderng a self-managed economy n whch workers decde on the amount and allocaton of a surplus. Pasnett therefore demonstrates that captalst proft s not the essental cause of the dvergence of natural prces and classcal labour-values. Even n the absence of captalst proft, a transformaton problem arses: the natural prces cannot be reduced to labour-values. Captalst socal relatons are therefore merely a suffcent, not a necessary, condton for the dvergence of natural prces and labour-values. 5 Pasnett constructs a more general defnton of labour cost that corresponds to hs more general defnton of a subsystem. he vertcally hyper-ntegrated labour coeffcents generalse classcal labour-values to nclude the labour suppled to produce net nvestment goods (the hyper-ndrect labour) as a real cost of producton: Defnton. Pasnett s vertcally hyper-ntegrated labour coeffcents, v, are v = l+ va+ va( g + r ), (6) whch s the sum of drect, ndrect and hyper-ndrect labour. 6 A hyper-ntegrated labour coeffcent s therefore the total labour suppled to the hyper-subsystem. Note that, n condtons of zero growth, the hyper-ntegrated labour coeffcents reduce to classcal labour-values,.e. v = l+ va. Pasnett then demonstrates that the natural prces of each hyper-subsystem are proportonal to the vertcally hyper-ntegrated labour coeffcents: ( ) = p = p A+ p Aπ + lw = li A( + π ) w v w. 5 hs pont was made, somewhat dfferently, by Samuelson (97), who demonstrates that the natural prces of a post-captalst economy, whch lacks captalst proft ncome, necessarly devate from classcal labour-values. 6 Equaton (6) s dentcal to equaton (2.9) n Pasnett (988), except B = I.

6 Page 6 of 8 I. Wrght he natural prces, therefore, reduce to the total wage bll of each hyper-subsystem,.e. the wages of the drect, ndrect and hyper-ndrect labour suppled to produce unt commodtes. Pasnett (988, p. 30) notes that ths s a complete generalsaton of the pure labour theory of value that recreates Smth s early and rude state of socety n whch labour-emboded equals labour-commanded. Furthermore, the analytcal step that allows the achevement of ths result s of course a re-defnton of the concept of labour emboded, whch must be ntended as the quantty of labour requred drectly, ndrectly and hyper-ndrectly to obtan the correspondng commodty as a consumpton good (Pasnett, 988, pp. 3 32). o summarse: f we relate classcal labour-values to the natural prces of a hypersubsystem, we encounter a new transformaton problem : labour costs and nomnal costs are ncommensurate. hs s not the classcal transformaton problem, but a further example of how classcal labour-values cannot account for general prce structures. he fundamental reason s smple: the natural prces of a hyper-subsystem nclude the nomnal cost of net nvestment as a component of the prce of commodtes, whereas classcal labour-values exclude the real cost of net nvestment as a component of the labour-value of commodtes. In ths case, the dual systems of prces and labour-values adopt dfferent, and ncommensurate, cost accountng conventons. Pasnett s transformaton problem dssolves once we relate natural prces to the hyper-ntegrated labour coeffcents. Hyper-ntegrated labour coeffcents adopt the same accountng conventon as the prce system, and therefore nclude the real cost of net nvestment as a component of the labour-value of commodtes. Commensurablty s thereby restored. Pasnett understands that, n certan crcumstances, classcal labour-values undercount the total labour costs of producton. He therefore constructs a more general measure of labour cost approprate to the more general economc settng. 3. A complete generalsaton of Marx s transformaton problem Pasnett now swtches to an nsttutonal stage of nvestgaton where captalsts, as owners of frms, receve proft ncome. Captalsts reallocate ther captal between sectors seekng the hghest returns untl a general, unform proft rate prevals across all sectors of the economy,.e. p = pa+ paπ + lω. (7) Marx ([894] 97, ch. 9) called these the prces of producton. he producton prces (7) n contrast to natural prces (5) mpose a sngle prce structure on the entre ntegrated economy. Also, at ths nsttutonal stage, the meanng of proft alters. Proft, n the context of captalst property relatons, s not merely a structural varable, determned by technology and growth requrements,.e. p A( g + r ), but s now a dstrbutonal varable receved n proporton to the money-captal nvested n means of producton wthn each sector of producton,.e. paπ. Pasnett then demonstrates that a pure labour theory of value breaks down n the nsttutonal crcumstances of captalsm. Pasnett wrtes equaton (7) n the equvalent form, p = pa+ pa( g + r ) + pa( π g r ) + lw, (8)

7 Marx s transformaton problem Page 7 of 8 where g + r s the growth rate of demand for any consumpton good we care to choose (here we have chosen the th commodty). For convenence, defne the matrx M = A I A g r ( ( + + ) ). We can therefore wrte producton prce equaton (7) n n dfferent, but equvalent, forms, 7 ( ) = 7 p = v I M ( π g r ) w, 2,,, n. (9) Consder, for a moment, the specal or accdental case, n whch the general proft rate equals the growth rate of demand for commodty ; that s, π = g + r. Equaton (9) then collapses to p = v w and producton prces are proportonal to the vertcally hyper-ntegrated labour coeffcents of hyper-subsystem. But n general, π g + r for any, and therefore producton prces are not proportonal to any hyper-ntegrated labour coeffcent of any hyper-subsystem. In fact, producton prces vary ndependently of the vertcally hyper-ntegrated labour coeffcents because prces are a functon of a global dstrbutonal varable, π, whereas the hyper-ntegrated labour coeffcents are not. Producton prces therefore cannot be reduced to labour costs, whether measured n terms of classcal labourvalues or Pasnett s hyper-ntegrated coeffcents. Pasnett (988, p. 3) notes, therefore, that equaton (9) can also be regarded as provdng a complete generalsaton of Marx s transformaton problem to the case of a non-unformly growng economy. Pasnett concludes, n an earler work, that ths analyss amounts to a demonstraton that a theory of value n terms of pure labour can never reflect the prce structure that emerges from the operaton of the market n a captalst economy, smply because the market s an nsttutonal mechansm that makes proportonalty to physcal quanttes of labour mpossble to realse (Pasnett, 98, p. 53). Pasnett therefore restrcts the pure labour theory of value to the pre-nsttutonal stage of nvestgaton, where t has to be taken as provdng a logcal frame of reference a conceptual constructon whch defnes a seres, actually a famly of seres, of deal natural prces, whch possess an extraordnarly hgh number of remarkable, analytcal, and normatve, propertes (my emphass) (Pasnett, 988, p. 32). In the next secton, I further generalse Pasnett s vertcally ntegrated approach. I defne non-natural, or nsttutonal subsystems, the vertcally super-ntegrated subsystems, whch correspond to the reproducton condtons of the specfc nsttutonal setup of captalsm. I prove that producton prces are proportonal to a more general measure of labour cost, the vertcally super-ntegrated labour coeffcents. In consequence, the labour theory of value, sutably generalsed, equally apples to the operaton of the market n a captalst economy. 7 Equaton (9) s dentcal to equaton (4.5) n Pasnett (988), and we derve t from equaton (8): p = pa+ pa( g + r ) + pa( π g r ) + lw pi ( A( + g + r )) = lw+ pa( π g r ) p = = l( I A( + g + r )) w+ pa( I A( + g + r )) ( π g r ) v w+ pm ( π g r ).

8 Page 8 of 8 I. Wrght 4. A general soluton to the transformaton problem We frst consder the specal case of a steady-state economy before generalsng to Pasnett s growth model. 4.. A specal case: the steady-state economy Producton prces are a functon of the dstrbuton of nomnal ncome between profts and wages. In order to defne the vertcally super-ntegrated subsystems we requre the correspondng physcal data that specfes the dstrbuton of real ncome. Assume, therefore, that workers receve the real wage, w = [ w ], and captalsts receve the consumpton bundle, c = [ c ], such that the net product n = w+ c. In condtons of zero growth,.e. g = 0 and r = 0 for all, Pasnett s quantty equaton (4) reduces to q = qa + n, whch we expand as q = qa + w+ c. (0) We analyse the followng specal-case, steady-state economy: Defnton 2. A steady-state economy wth producton prces produces quanttes, q = qa + w+ c, at prces, p = pa ( + π) + l w, where workers and captalsts spend what they earn, pw = lq w and pc = paq π. In ths economy, the net product s produced, dstrbuted and consumed wthn the perod of producton. Over multple perods, the economy self-replaces wth a constant composton and scale. 8 he producton and dstrbuton of the net product are necessarly related. For example, the quantty of commodty consumed by worker households per unt of wage ncome s w / lq w. he ncome receved by worker households, per unt output n sector j, s l j w. Hence, consumpton coeffcent w = wl / lq denotes the quantty, j j of commodty dstrbuted to worker households per unt output of j. Defne W = w l = [ w ], lq j (), as a matrx of worker consumpton coeffcents. W compactly descrbes the physcal flow rate of consumpton goods to worker households per unt outputs. Producton prce equaton (7) mples that proft s proportonal to the moneycaptal ted up n crculatng captal,.e. paq π. he quantty of commodty consumed by captalst households per unt of proft ncome s therefore c / paq π. he proft ncome receved by captalst households, per unt output n sector j, s pa ( j ) ( j π. Hence, consumpton coeffcent c = c pa ) / paq denotes the quantty, j of commodty dstrbuted to captalst households per unt output of j. Defne C = c pa = [ c ], paq j (2), 8 Pasnett s economy, n condtons of zero growth, reduces to a closed Leontef system wth fnal demand equal to the consumpton of workers and captalsts; see Pasnett (977, pp. 60 6).

9 Marx s transformaton problem Page 9 of 8 as a matrx of captalst consumpton coeffcents. C compactly descrbes the flow rate of consumpton goods to captalst households per unt outputs. Arrvng at matrx C va the prce system reveals the necessary connecton between captalsts real and nomnal ncome. Nonetheless, C s a physcal consumpton matrx, whch s ndependent of the system of prces, and solely determned by the real propertes of the economy: Proposton. he captalst consumpton matrx, C, s a functon of the real propertes of the economy, specfcally the technque, A and, and the real dstrbuton of ncome, gven by the real wage, w, and captalst consumpton bundle, c. Proof. See Appendx 6.. For example, the prce magntudes n equaton (2) cancel out yeldng physcal coeffcents. Each coeffcent c, j denotes the quantty of commodty dstrbuted to captalst households per unt output of commodty j. he technque A and consumpton matrces, W and C, together specfy the physcal flow rates of goods between sectors of producton and households. Recall that a subsystem s a self-replacng system that replaces used-up means of producton and produces a fnal output. A Sraffan subsystem, for example, s the drect and ndrect producton that produces a sngle component of the net product as fnal output, where the net product conssts of consumpton goods. All consumpton goods, n a Sraffan subsystem, are fnal outputs or surplus, and therefore not replaced by the subsystem. A gven economc system, however, can be decomposed nto alternatve knds of subsystems. Defne a vertcally super-ntegrated subsystem as the drect, ndrect and super-ndrect producton that produces a sngle component of the real wage as fnal output, where super-ndrect refers to the producton of captalst consumpton goods. Note that super-ntegrated subsystems, n contrast to Pasnett s subsystems, now nclude the reproducton condtons of a specfc nsttutonal setup. In a superntegrated subsystem, we only consder the real wage of workers as the fnal output or surplus. In consequence, a super-ntegrated subsystem also replaces the real ncome of captalsts. More formally, the total output of the th vertcally super-ntegrated subsystem s qˆ = qa ˆ + qc ˆ + w, where w s a zero vector except for the th component that equals w, whch s the wage demand for commodty. A super-ntegrated subsystem addtonally vertcally ntegrates the producton of captalst consumpton goods. he captalst consumpton matrx, C, therefore appears as a real cost of producton. he total output of the steady-state economy s then the composton of the vertcally super-ntegrated subsystems,.e. q = q ˆ. Sraffa s and Pasnett s natural subsystems are defned by technologcal and accumulaton condtons alone. In contrast, the super-ntegrated subsystems are also defned by socal and nsttutonal condtons. A super-ntegrated subsystem captures the nsttutonal fact that producton, n a captalst system, materally reproduces a captalst class at a gven level of real ncome.

10 Page 0 of 8 I. Wrght A vertcally super-ntegrated labour coeffcent, denoted v, s the total labour suppled to the th super-ntegrated subsystem when t produces a unt component of the real wage as fnal output,.e. when w =. For clarty, we now calculate ths quantty step-by-step. Consder the producton of unt of commodty n super-ntegrated subsystem. How much labour does ths producton requre? It requres l unts of drect labour, la ( ) unts of ndrect labour and lc ( ) unts of super-ndrect labour, gvng a total of l( A ( ) C ( + ) ) unts of labour operatng n parallel to produce the output, replace usedup means of producton and replace captalst consumpton goods, respectvely. Defne A = A + C as the technque augmented by captalst consumpton. Matrx A compactly represents the commodtes used up durng the producton of each commodty type ncludng the commodtes consumed by captalsts. he sum of drect, ndrect and super-ndrect labour s then l + la ( ). However, the ndrect and super-ndrect producton tself uses up means of producton and consumpton goods, specfcally the bundle AA ( ), whch s contemporaneously replaced by the supply of addtonal labour, laa ( ). o count all the drect, ndrect and super-ndrect labour, we must contnue the sum; that s, ( ) ( ) 2 ( ) v = l + la + laa + la A + n = l + l A A n= 0 ( ). hs sum represents the total labour suppled to the th super-ntegrated subsystem when t produces unt as fnal output. he vector v of super-ntegrated coeffcents s therefore v = l+ l n A A l A n =. n= 0 n= 0 Assumng that captalst consumpton s feasble, gven the technology, then matrx A s productve, and we may replace the nfnte seres wth the Leontef nverse, n l A li A = ( ) ; n consequence: n= 0 Defnton 3. he vertcally super-ntegrated labour coeffcents, v, n a steady-state economy wth producton prces, are v = l+ va whch s the sum of drect, ndrect and super-ndrect labour costs. = l+ va + vc, (3) he defnton of the super-ntegrated coeffcents does not provde or rely upon any theory of ncome dstrbuton or proft. However, n order to calculate the super-ntegrated coeffcents, the dstrbuton of real ncome must be a gven datum, n the same manner that, n order to calculate producton prces, the dstrbuton of nomnal ncome must be a gven datum. Conjectural varaton of ether the real or nomnal dstrbuton of ncome then affects both the super-ntegrated coeffcents and producton prces.

11 Marx s transformaton problem Page of 8 he super-ntegrated labour coeffcents, although more complex than classcal labour-values, nonetheless drectly relate, n a straghtforward manner, to the labour suppled durng the producton perod. For example, Pasnett (980, p. 2) classfes the total labour suppled n two ways: as () the sum of drect labour suppled to each sector of producton, lq = lq, or () the sum of drect and ndrect labour suppled to each Sraffan subsystem, vn = vn. he classfcatons are quanttatvely equal, that s, lq = vn, because the Sraffan subsystems collectvely produce the net product as fnal output and exhaust the total suppled labour: Proposton 2. he total labour suppled equals the classcal labour-value of the net product, lq = vn. Proof. From (0), q = qa + n = ni ( A ). Hence lq = ( I A) n = vn. he super-ntegrated subsystems provde another partton of the same economy. he total labour suppled can also be classfed as () the sum of drect, ndrect and superndrect labour suppled to each super-ntegrated sector, vw = vw. Agan, ths classfcaton s quanttatvely equal to the total labour suppled, that s, lq = v w, because the super-ntegrated subsystems collectvely produce the real wage as fnal output and exhaust the total suppled labour: Proposton 3. he total labour suppled equals the super-ntegrated labour-value of the real wage, vw = lq. Proof. From (2), Cq = ( / paq ) c paq = c. Substtute nto (0) to yeld, q = qa + qc + w = qa + w = w( I A ). Hence, lq = ( I A ) w = vw. Now that we ve defned the super-ntegrated coeffcents, we can relate them to producton prces. heorem. he producton prces of a steady-state economy are proportonal to the superntegrated labour coeffcents, p = vw. Proof. Snce captalsts spend what they earn, paq π = pc. Substtute for π nto prce equaton (7): p = pc pa ( + pc l pa paq ) + = + paq pa + l = p( A + paq c w w pa) + lw = pa+ pc+ lw = pa + lw = l I A ( ) w = v w. Producton prces equal the total wage bll of each super-ntegrated subsystem,.e. the wages of the drect, ndrect and super-ndrect labour suppled to produce unt commodtes. hs more general defnton of labour costs replcates the result, establshed by Adam Smth for an early and rude state of socety, that labour emboded equals labour commanded. he physcal confguraton of the steady-state economy, specfcally the prevalng technque and dstrbuton of real ncome, determnes both the structure of the vertcally superntegrated labour coeffcents and the structure of producton prces. Recall that Marx s transformaton problem arses because producton prces vary wth the dstrbuton of ncome but classcal labour-values do not. he super-ntegrated coeffcents, n contrast, also vary wth the dstrbuton of ncome because they vertcally ntegrate over the producton of the real ncome of captalsts. In consequence, producton prces and labour costs, sutably measured, are necessarly dual to each other and two sdes of the same con.

12 Page 2 of 8 I. Wrght he conceptual advance, compared to pror classcal analyss, s to consder not only techncal but also socal condtons of reproducton. A captalst economy, for nstance, functons accordng to specfc dstrbutonal rules, such as wage ncome from labour, and proft ncome from captal. By further generalsng vertcal ntegraton to nclude the socal condtons of reproducton, we embed the dstrbutonal rules of an economy wthn super-ntegrated subsystems. We can then compute the real costs of producton caused by specfc nsttutonal setups and compare those real costs to the system of prces caused by the same nsttutonal setup. heorem reveals the essental dualty between real costs of producton, measured n labour tme, and the natural prces of a captalst economy. In consequence, the value theory of classcal poltcal economy, rather than beng restrcted to specal cases, has broad applcablty and mportant mplcatons for socal theory as a whole. Next, I generalse ths result to Pasnett s growth model. he generalsaton does not requre any new arguments or deas. However, the super-ntegrated coeffcents, n the case of non-unform growth, nclude both hyper and super-ndrect labour he general case: Pasnett s non-unform growth model In the more general crcumstances of non-unform growth, the fnal demand s varable. he net product s therefore a functon of tme,.e. n() t = w() t + c() t. Assume an ntal dstrbuton of real ncome, w( 0 ) and c( 0 ). he trajectory of fnal demand, ( g+ r ) ( ) from equaton (3), s then n( t) [ w ( ) t g+ r t = 0 e ] + [ c ( 0) e ]. he vectors w and c now mplctly refer to tme-varyng consumpton bundles that, followng Pasnett, drve the growth of the economy. 9 For notatonal convenence, defne the non-unform captal nvestment vector, n g = rqa = [ g ], where q = [ q j ], s the total output of hyper-subsystem as defned = by equaton (), and each g s the quantty of commodty produced as addtonal means of producton, n the economy as a whole, n order to meet the total nonunformly growng demand. 0 Let Γ= dag( g) dag( q) = [ λ j ] be a dagonal nonunform captal nvestment matrx, where each element on the dagonal, λ, = g / q,, s the quantty of produced as addtonal means of producton, per unt output, to meet the total non-unformly growng demand (and λ, j = 0 for j). Rewrte Pasnett s quantty equaton (4) n the equvalent form, q = qa ( + g) + qγ + w+ c. (4) he total proft ncome remans paq π, as n the smpler case of a steady-state economy. But now a fracton of proft s nvested n addtonal means of producton to satsfy ncreased demand: paq g s nvested to satsfy the ncrease n demand due to populaton growth, and pγq s nvested to satsfy the non-unform change n demand. he resdual proft that remans for captalsts to spend on personal consumpton s therefore Y = pa ( ( π g) Γ) q. 9 Of course, n the context of an actual captalst economy, rather than Pasnett s system, growth s not drven by exogenous real demand. My goal here s to explore the full mplcatons of Pasnett s mposton of a captalst prce structure on hs model rather than develop a realstc growth model of captalsm. 0 Vector g features n Pasnett s equaton (4) that defnes the total output of the ntegrated economy.

13 Marx s transformaton problem Page 3 of 8 he full specfcaton of Pasnett s non-unformly growng economy wth producton prces s therefore: Defnton 4. A non-unformly growng economy wth producton prces produces quanttes, q = qa ( + g) + qγ + w+ c, at prces, p = pa( + π) + lw, where workers and captalsts spend what they earn, pw = lq w and pc = pa ( ( π g) Γ ) q. he producton and dstrbuton of the net product are, once agan, necessarly related. he matrx of worker consumpton coeffcents, W, s unchanged from the steady-state case. However, the matrx of captalst consumpton coeffcents, C, dffers because captalsts nvest a fracton of ther proft ncome n addtonal means of producton. he quantty of commodty consumed by captalsts per unt of resdual proft ncome s now c / Y. he resdual proft receved, per unt output n sector j, s ( j) ( j) ( j) ( j) pa ( ( π g) Γ ). Hence, consumpton coeffcent c = c pa ( ( π g) Γ )/ Y, j denotes the quantty of commodty dstrbuted to captalsts per unt output of j. Defne C = c p( A( π g) Γ) = [ c ] pa ( ( ) ) q, π g Γ j (5) as the matrx of captalst consumpton coeffcents. (Note that, when g = 0 and r = 0 for all, ths defnton of C reduces to the defnton for the steady-state economy.) Pasnett s hyper-ntegrated subsystems nclude the drect, ndrect and hyper-ndrect producton that produces a sngle component of the net product as fnal output. A vertcally super-ntegrated subsystem, n the context of non-unform growth, s the drect, ndrect, hyper and super-ndrect producton that produces a sngle component of the real wage as fnal output. he total output of the th vertcally super-ntegrated subsystem s qˆ = qˆ A + qˆ A g + qˆ Γ + qˆ C + w, where w s defned as before and q = ˆq. he net nvestment n a hyper-ntegrated subsystem, from equaton (), s q A ( g + r ), whch s ndependent of the cross-demand effects of the non-unform growth of the other hyper-ntegrated subsystems (.e. the r j for all j ). In contrast, the net nvestment n a super-ntegrated subsystem, qˆ A g + qˆ Γ, ncludes cross-demand effects. he technque augmented by hyper and super-ndrect real costs of producton s then A = A + A g + Γ + C. Matrx A compactly represents the commodtes used up durng the producton of each commodty type, ncludng the producton of net nvestment goods and captalst consumpton goods. Defnton 5. he vertcally super-ntegrated labour coeffcents, v, n a non-unformly growng economy wth producton prces, are

14 Page 4 of 8 I. Wrght vˆ = l+ vˆa = l+ vˆa+ vˆ( Ag + Γ ) + vˆ C, (6) whch s the sum of drect, ndrect, hyper and super-ndrect labour costs. he hyper and super-ntegrated coeffcents are dentcal n crcumstances of zero nonunform growth and zero captalst consumpton. And both the hyper and super-ntegrated coeffcents reduce to the classcal defnton of labour-value n crcumstances of zero growth and zero captalst consumpton (.e. Smth s early and rude state ). As before, the super-ntegrated subsystems collectvely produce the real wage as fnal output and exhaust the total suppled labour; n consequence, lq = ˆv w. We now state the man result of the paper: heorem 2. he producton prces of a non-unformly growng economy are proportonal to the super-ntegrated labour coeffcents, p = v w. Proof. From (5), pc = ( / Y) pc pa ( ( π g) Γ ). Hence, pa ( ( π g) Γ) = ( Y / pc ) pc. Wrte prce equaton (7) n the equvalent form, p = pa+ p( Ag + Γ) + p( A( π g) Γ) + lw, and then substtute to yeld p = pa+ p( Ag + Γ) + ( Y / pc ) pc+ lw. Snce captalsts spend what they earn, pc = Y. Hence, p = pa+ p( Ag + Γ) + pc+ lw = pa + lw = l( I A ) w = v w. Producton prces, n Pasnett s non-unformly growng economy, equal the total wage bll of each super-ntegrated subsystem,.e. the wages of the drect, ndrect, hyper and super-ndrect labour suppled to reproduce unt commodtes. Once agan, a more general defnton of labour costs replcates Adam Smth s result that labour emboded equals labour commanded. 2 heorem 2, t should be emphassed, undermnes the logcal bass for any clam that a labour theory of value s ncoherent because producton prces and labour-values are quanttatvely ncommensurate n lnear producton models (e.g. Samuelson, 97; Lpp, 979; Steedman, 98) echncal and socal cost structures Marx s transformaton problem, and Pasnett s generalsaton, reduce to msmatches between producton prces and labour costs. Producton prces nclude nsttutonal or socal costs, specfcally a proft rate that ncludes the ncome of a captalst class. In contrast, classcal labour-values, and Pasnett s hyper-ntegrated labour coeffcents, are purely techncal costs of producton, and therefore gnore the real cost of producng captalst ncome. ransformaton problems necessarly arse when we contravene Pasnett s separaton thess and compare a nomnal cost structure that belongs to an nsttutonal stage of analyss wth a real cost structure that belongs to a natural, or prensttutonal, stage of analyss. A commensurate relatonshp cannot obtan between cost structures defned by ncommensurate accountng conventons. Pasnett recognses the need to extend the classcal theory n order to explan the structure of natural prce systems. He constructs more general measures of labour cost Note that defnton 5 reduces to defnton 3 n condtons of zero growth. 2 Note that heorem 2 reduces to heorem n condtons of zero growth.

15 Marx s transformaton problem Page 5 of 8 that take nto account addtonal features of the crcumstances of producton, such as the labour cost of net captal nvestment. Despte ths mportant conceptual advance, Pasnett nonetheless beleves, n vrtue of the transformaton problems, that a labour theory of value can never reflect the prce structure that emerges from the operaton of the market n a captalst economy (Pasnett, 98, p. 53). heorems and 2, by generalsng the vertcally ntegrated approach to encompass socal and nsttutonal condtons, demonstrate the contrary. Producton prces, n both steady-state and non-unformly growng economes, are proportonal to physcal quanttes of labour, the vertcally super-ntegrated labour coeffcents, whch nclude the addtonal labour suppled to produce the real ncome of captalsts. he superntegrated labour coeffcents capture the real cost structure that emerges from the operaton of the market n a captalst economy (Pasnett, 98, p. 53). he transformaton problems therefore dssolve once we observe Pasnett s separaton thess and compare the nomnal and real cost structures that manfest at the same, nsttutonal stage of analyss. 5. Concluson Pasnett s separaton thess, and hs generalsaton of the vertcally ntegrated approach, are powerful analytc devces. However, Pasnett s specfc proposal to restrct the labour theory of value to a normatve role, a knd of logcal frame of reference, s unnecessary. Pasnett s theoretcal nnovatons nstead pont n the opposte drecton and toward a full generalsaton of the classcal labour theory and ts renstatement as the foundatonal theory of value for economc analyss wthn the producton paradgm (Pasnett, 986). he more general labour theory spans both the natural and nsttutonal stages of analyss and therefore can address normatve ssues n the crtque of poltcal economy, and postve ssues n the analyss of specfc economc systems. he more general theory, sketched here n an ntal and prelmnary manner, admts both techncal and socal measures of labour cost and apples both knds of measures n the approprate contexts. For example, n ths more general framework, classcal labour-values apply to dstrbuton-ndependent questons about an economy, such as measurng the techncal productvty of labour (e.g. Flaschel, 200, part ) or the surplus-labour suppled by workers (e.g. Marx, [867] 954), whereas the superntegrated labour coeffcents apply to dstrbuton-dependent questons, such as the relatonshp between relatve prces and the actual labour tme suppled to produce commodtes,.e. ssues n the theory of value. he post-sraffan separaton of the classcal surplus approach to ncome dstrbuton from ts labour theory of value does not consttute a sophstcated rejecton of nave substance theores of value but ndcates a falure to resolve the classcal contradctons, such as Marx s transformaton problem. he separaton ultmately derves from the classcal error of comparng techncal wth socal cost structures (see also Wrght, 204, 205). he post-sraffan reconstructon of classcal economcs therefore dspenses wth an essental am of a theory of economc value, whch s to explan what the unt of account mght measure or refer to. heorems and 2, whch demonstrate that producton prces are proportonal to physcal quanttes of labour, start to put the peces back together agan.

16 Page 6 of 8 Bblography I. Wrght Bellno, E Employment and ncome dstrbuton from a Classcal-Keynesan pont of vew: some tools to ground a normatve analyss, n Brancacco, E. and Fontana, G. (eds), he Global Economc Crss: New Perspectves on the Crtque of Economc heory and Polcy, London, Routledge Desa, M he transformaton problem, Journal of Economc Surveys, vol. 2, no. 4, Flaschel, P opcs n Classcal Mcro- and Macroeconomcs: Elements of a Crtque of Neorcardan heory, New York, Sprnger Garbelln, N Essays on the theory of structural economc dynamcs growth, techncal progress, and effectve demand, PhD thess, Unverstà Cattolca Del Sacro Cuore Mlano Hunt, E. K. and Glck, M ransformaton problem, pp n Eatwell, J., Mlgate, M. and Newman, P. (eds), Marxan Economcs, New York, Macmllan Press Lmted Lpp, M Value and Naturalsm, London, New Left Books Marx, K. [867] 954. Captal, vol., Moscow, Progress Publshers Marx, K. [894] 97. Captal, vol. 3, Moscow, Progress Publshers Pasnett, L. L Lectures on the heory of Producton, New York, Columba Unversty Press Pasnett, L. L he noton of vertcal ntegraton n economc analyss, n Pasnett, L. L. (ed), Essays on the heory of Jont Producton, New York, Cambrdge Unversty Press Pasnett, L. L. 98. Structural Change and Economc Growth A heoretcal Essay on the Dynamcs of the Wealth of Natons, Cambrdge, Cambrdge Unversty Press Pasnett, L. L heory of value a source of alternatve paradgms n economc analyss, pp n Baranzn, M. and Scazzer, R. (eds), Foundatons of Economcs: Structures of Enqury and Economc heory, Oxford, Basl Blackwell Pasnett, L. L Growng subsystems, vertcally hyper-ntegrated sectors and the labour theory of value, Cambrdge Journal of Economcs, vol. 2, Pasnett, L. L Growng subsystems, vertcally hyper-ntegrated sectors: a note of clarfcaton, Cambrdge Journal of Economcs, vol. 3, Pasnett, L. L Structural Economc Dynamcs A heory of the Economc Consequences of Human Learnng, Cambrdge, Cambrdge Unversty Press Pasnett, L. L Keynes and the Cambrdge Keynesans: A Revoluton n Economcs to Be Accomplshed, Cambrdge, Cambrdge Unversty Press Reat, A he complementarty of the post Keynesan and Marxan paradgms: the case of labour value, Cahers economques de Bruxelles, vol. 68, Samuelson, P. A. 97. Understandng the Marxan noton of explotaton: a summary of the so-called transformaton problem between Marxan values and compettve prces, Journal of Economc Lterature, vol. 9, no. 2, Seton, F he transformaton problem, Revew of Economc Studes, vol. 24, Smth, A. [776] 994. he Wealth of Natons, New York, Modern Lbrary Sraffa, P Producton of Commodtes by Means of Commodtes, Cambrdge, Cambrdge Unversty Press Steedman, I. 98. Marx after Sraffa, London, Verso von Wezsäcker, C. C. and Samuelson, P. 97. A new labour theory of value for ratonal plannng through use of the bourgeos proft rate, Proceedngs of the Natonal Academy of Scences USA, vol. 68, no. 6, Wrght, I A category-mstake n the classcal labour theory of value, Erasmus Journal for Phlosophy and Economcs, vol. 7, no., Wrght, I he law of value: a contrbuton to the classcal approach to economc analyss, PhD thess, Open Unversty 6. Appendx Here, for clarty, I also nclude numercal examples of heorems and 2 for an n = 2 economy, followed by a bref dscusson of tme-varyng fnal demand.

17 6.. Proof of Proposton Marx s transformaton problem Page 7 of 8 Lemma. In a steady-state economy wth producton prces, the rate of proft, π, s the domnant egenvalue of matrx ( I A WA ). Proof. Let X = ( I A W) A. hen px = λ p s an egenvalue equaton. Solve the characterstc equaton, det( X λi) = 0, to obtan λ = λ *, where λ * s the Perron- Frobenus root (domnant egenvalue). Multple the egenvalue equaton by A to obtan λ * * pa = p pa pw, whch mples p = pa( + λ ) + pw. Snce W = ( / lq ) w l, * then p = pa( + λ ) + lw, whch are the producton prces of a steady-state economy. Hence, π = λ *. Proposton. he captalst consumpton matrx, C, s a functon of the real propertes of the economy, specfcally the technque, A and, and the real dstrbuton of ncome, gven by the real wage, w, and captalst consumpton bundle, c. Proof. Defne the captalst consumpton matrx, c l( I A( + π)) A C = li ( A( + π)) A( w+ c) ( I A) Note that the proft rate, π, by lemma, s a functon of A and worker consumpton matrx W. Matrx W, by equaton (), s a functon of l, w and quanttes, q. And q, by equaton (0), s a functon of A, w and c. Hence, C, as defned n equaton (7), s a functon of the real propertes of the economy only. We now show that defnton (7) s equvalent to defnton (2). Note that p = l( I A( + π )) (from equaton (7)) and q = ( w+ c) ( I A) (from equaton (7)). Substtute nto equaton (7) to yeld C = c pa / paq Numercal example of heorem he followng proposton, whch lnks the technque, captalst consumpton and the proft rate, wll be useful: Proposton 4. In a steady-state economy wth producton prces, r( CA ) = π. Proof. Snce captalsts spend what they earn, paq π = pc. Hence, / paq = π / pc. From equaton (2), CA = ( / paq ) c p. Substtute for / paq to yeld CA = ( / π pc ) c p. An nner product s the trace of ts outer product,.e. r( c p) = pc. Hence, r( CA ) = π. Consder a statonary economy wth () technque, A = and l = [ 2 ]. and () captalst consumpton matrx, C = he technque and captalst consumpton matrx determne the super-ntegrated labour coeffcents,.e. from defnton 3, v = l( I ( A+ C)) = [ ]..

18 Page 8 of 8 I. Wrght he technque and captalst consumpton matrx determne the proft rate,.e. from Proposton 4, π = r = he proft rate determnes producton.. prces,.e. from equaton (7), p = l( I A( + π)) w = [ ] w. Hence, p = v w, as per heorem Numercal example of heorem 2 he followng proposton, whch lnks the technque, non-unform growth, captalst consumpton and the proft rate, wll be useful: Proposton 5. In a non-unformly growng economy wth producton prces, ( ( ) ) = r CA( π g) Γ. Proof. Snce captalsts spend what they earn, hence / Y = / pc. From equaton (5), CA ( ( π g) Γ) = ( / Y) c p. Substtute for / Y to yeld CA ( ( π g) Γ) = ( / pc ) c p. An nner product s the trace of ts outer product,.e. r( c p) = pc. Hence, r( CAπ ( ( g) Γ) ) =. Consder a non-unformly growng economy at tme t wth () technque, A = and l = [ 2 ], () captalst consumpton matrx, C = 0 0, () g = and (v) Γ = hs s suffcent nformaton to compute the super-ntegrated labour-values. From defnton 5, v ˆ = li ( A( + ) C) g Γ = [ ]. From Proposton 5, r ( CA ( ( π g) Γ) ) =. Solve to yeld the proft rate, π = Producton prces, from equaton (7), are then p = l( I A( + π)) w = [ ] w. Hence, p = ˆvw, as per heorem he trajectory of fnal demand Fnal demand, n( t ), grows exponentally, such that ( g+ r ) ( ) n() t w() t c() t [ w ( ) t g+ r t = + = 0 e ] + [ c ( 0) e ] (see Secton 4.2). In general, gven non-unform growth, that s, r r for some and j j, then matrces Γ and C and the proft rate, π, are not constant but vary wth tme. In consequence, both producton prces, p, and the vertcally super-ntegrated labour coeffcents, ˆv, vary along the growth trajectory. Nonetheless, heorem 2 holds at every nstant of tme. In consequence, the proportonalty of producton prces and superntegrated labour-values s a tme-nvarant property of Pasnett s model.