Abstract Labour in a Model of Joint Production*

Transcription

1 Abstract Labour n a Model of Jont Producton* Takao Fujmoto** Fumko Ekun*** Abstract Ths paper presents a new mathematcal method for constructng abstract labour as conceved by Karl Marx, n a general nput-output model wth jont producton and heterogeneous labour. A bref hstory of prevous efforts to defne and compose abstract labour usng nput-output models s also descrbed. A numercal example s gven to show how our method works n a concrete way. Key words: abstract labour, egenvalue, egenvector, heterogeneous labour, Kakutan fxed pont theorem. Resumen Este artículo presenta un nuevo método matemátco para construr el trabajo abstracto concebdo por Karl Marx, medante un modelo de nsumo-producto con el trabajo heterogéneo y la produccón conjunta. Tambén se descrbe una hstora breve sobre los esfuerzos para defnr y componer una abstraccón matemátca usando los modelos de entrada-salda. Se dan una sere de ejemplos numércos que nos muestran cómo funcona el método de manera concreta. Palabras clave: trabajo abstracto, valor propo, vector propo, trabajo heterogéneo, teorema de punto fjo de Kakutan. Artículo recbdo el Artículo aceptado el * Acknowledgments: thanks are due to professor Dr. Alejandro Valle Baeza who had ntroduced ths journal to the authors, and to our frend professor Dr Ulrch Krause for hs encouragement. The authors are grateful to the referees who have provded useful comments and suggestons to mprove ths artcle. ** Fukuoka Unversty, Fukuoka, Japan [takao@econ.fukuoka-u.ac.jp]. *** Shkoku-Gakun Unversty, Zentsuj, Japan [fumko@sg-u.ac.jp].

2 244 Takao Fujmoto y Fumko Ekun 1. ntroduccón P roductvty can be enhanced by ntroducng dvson of labour. Ths fact has been well known, and we beleve that n realty dvson of labour has played a sgnfcant role n the development of human socetes. Yet, theoretcal analyss of heterogenety of labour has been gven lttle attenton by economsts. When heterogenety of labour s allowed for n a model of economy, some basc problems emerge. The frst problem s concerned wth how to measure labour values of varous commodtes and labour servces. The second one s whether we could devse abstract labour, usng gven technologcal data. The thrd may be how to defne rates of explotaton for ndvdual types of labour servces and for abstract labour. Some economsts regard the reducton of heterogeneous labour to abstract labour as unnecessary, whle others thnk of the reducton as essental to theory of explotaton. Surely, t may sometmes be necessary to know the rate of explotaton for the workng class as a whole, n addton to the rates for ndvdual types of labour. Then, t becomes ndspensable to carry out the reducton of varous types of labour to a common unt or abstract labour. Besdes, t may be nterestng to study how to realze the unform rate of explotaton among heterogeneous groups of labour f at all the reducton s performed. Ths paper represents a way of fndng out reducton ratos, n a general model wth jont producton and heterogeneous labour, by use of lnear programmng problems and Kakutan fxed pont theorem. Our method guarantees the unform rate of explotaton among all the types of labour. In secton 2, we nclude a bref hstory of the researches on the topc. Secton 3 explans our model and how t s more general than the exstng ones. Secton 4 descrbes our method for somewhat restrctve cases. Secton 5 presents a numercal example to show how our method works. Then secton 6 descrbes our approach to a stll more general model where each household actvty can produce a plural number of labour servces jontly and some labour servces are produced by normal producton processes. Fnal secton 7 ncludes some concludng remarks. It should be noted at the outset that we deal only wth mathematcal aspects of abstract labour, and do not touch upon hstorcal and phlosophcal sdes of the topc. We hope that ths artcle wll stmulate dscussons among those phlosophers and hstorans who are nterested n abstract labour.

3 Abstract Labour n a Model of Jont Producton bref hstory Whle Marx (1865) stressed the heterogenety of human labour when he consdered the values of varous labour servces and ther prces, Marx (1867) shfted the focus to abstract labour or human labour n the abstract n order to unte heterogeneous labour nto one workng class. 1 He wrote: Sklled labour counts only as smple labour ntensfed, or rather, as multpled smple labour, a gven quantty of sklled beng consdered equal to a greater quantty of smple labour [...] For smplcty s sake we shall henceforth account every knd of labour to be unsklled, smple labour. 2 Thus, Marx dd not explan how to derve reducton ratos. As early as n 1913, Potron (1913) used a model wth heterogeneous labour wthout jont producton. 3 Potron s paper attracted lttle attenton n hs days, whle Bowles-Gnts s (1977), 65 years after Potron, was the starter of a seres of more recent research efforts thereafter for a short perod. 4 Most of contrbutons dealt wth models wthout jont producton, and two notable exceptons are Steedman (1980) and Krause (1980). 5 Steedman (1980) nssts that there s no need to reduce heterogeneous labour to abstract labour, smple labour, or common unt. On the other hand, Krause (1980) proposes hs standard reducton of labour, whle representng many nterestng propertes concernng prces, reducton ratos, proft rates, and explotaton rates. Krause (1980), however, shows the exstence of the standard reducton only for specal cases. In Krause (1981), reducton ratos are gven as an egenvector of a certan matrx for a model wthout jont producton. 6 Indeed, the reducton method n Krause (1981) 1 K. Marx, Wages, Prce and Proft, speech n 1865, publshed n 1898: Secton 7; K. Marx, Captal I, the orgnal German edton n 1867 [ 2 K. Marx, Captal I, 1867, op. ct., Part 1, Chapter 1, Secton 2, p M. Potron, Quelques proprétés des substtutons lnéares à coeffcents 0 et leur applcaton aux problèmes de la producton et des salares, Annales Scentfques de l É.N.S., 3e sére, t. 30, 1913, pp See also K. Mor, Maurce Potron s lnear economc model: A de facto proof of fundamental Marxan theorem, Metroeconomca, vol. 59, 2008, pp S. Bowles and H. Gnts, The Marxan theory of value and heterogeneous labour: a crtque and reformulaton, Cambrdge Journal of Economcs, vol. 1, 1977, pp I. Steedman, Heterogeneous labour and «classcal» theory, Metroeconomca, vol. 32, 1980, pp U. Krause, Abstract labour n general jont systems, Metroeconomca, vol. 32, 1980, pp U. Krause, Heterogeneous labour and the fundamental Marxan theorem, Revew of Economc Studes, vol. 48, 1981, pp

4 246 Takao Fujmoto y Fumko Ekun s mathematcally the same as that n Potron (1913). 7 After these artcles, few theoretcal works have been publshed for three decades. In Fujmoto and Opocher (2010), a new defnton of labour values s gven for a general model of jont producton wth heterogeneous labour, mprovng on a method n Fujta (1985). 8 Based upon ths defnton, Fujmoto (2009) consders the concept of explotaton among workers themselves wthout reducng heterogenety to abstract labour. 9 It s noted that whle Steedman (1980) and Krause (1980, 1981) adopt a system of equatons, Fujmoto and Opocher (2010) use systems of nequaltes. Besdes, Fujmoto and Opocher s method s completely ndependent of actual outputs or actvty levels, and therefore can be employed also n dsequlbrum states. 3. a general model of jont producton wth heterogeneous labour Let us repeat what s gven n Fujmoto (2009) as a descrpton of our model. Ours s a sort of Morshma-von Neumann model (Morshma (1964)) slghtly generalzed. 10 Jont producton may be nvolved n each producton process. Thus, we can deal wth durable captal goods as well. Moreover, we allow for the exstence of heterogeneous labour. Varous types of labour servces are treated exactly lke normal commodtes, whch enables us to use the symbols B and A as the output and nput coeffcent matrces, both of whch have (m+k) rows and (n+h) columns. Each column of these matrces stands for a producton process or a household actvty wth the nputs gven n A and the outputs n B. There are altogether m knds of normal commodtes and k types of labour servces. On the other hand, there exst n normal producton processes and h household actvtes to produce labour servces. Household actvtes are descrbed as are observed n statstcal surveys, lke normal producton processes. Ths way of formulaton makes t possble to take nto consderaton durable consumpton 7 See K. Mor, Maurce Potron s lnear economc model..., op. ct., p It seems to the authors, however, that Potron (1913) had no noton of explotaton. 8 T. Fujmoto and A. Opocher, Commodty content n a general nput-output model, Metroeconomca, vol. 61, 2010, pp Y. Fujta, On the reducton problem n Marxan value theory, Fukuoka Unversty Revew of Economcs, vol. 30, 1985, pp T. Fujmoto, The concept of explotaton n a general lnear model wth heterogeneous labour, Investgacón Económca, vol. 68, UNAM, 2009, pp Some notatonal typos reman. 10 M. Morshma, Equlbrum, Stablty, and Growth: A Mult-sectoral Analyss, Oxford Unversty Press, See also M. Morshma, Marx s Economcs: A Dual Theory of Value and Growth, Cambrdge Unversty Press, 1973.

5 Abstract Labour n a Model of Jont Producton 247 commodtes n household actvtes: a durable consumpton commodty n a column of household actvty of A wll appear n the correspondng column of B as one perod, say one year, older commodty. For each type of labour, there can be more than one household actvty to reproduce that labour. One household actvty may produce a plural number of labour servces jontly. Workers may save a part of ther ncomes, and may have propertes. These complcatng elements from the real world do not dsturb our study whle we deal wth values and explotaton: ths should be true n any models ncludng Leontef models. At ths pont, t seems to be more readable f we express our symbols B and A for a specal case of Morshma-von Neumann model, usng more famlar notaton. 11 Employng the symbols n Krause (1980), we have B 0 B = and A = A C, (1) 0 l k L 0 where B s the m by n materal output coeffcent matrx, l k s the k by k dentty matrx, A the m by n materal nput coeffcent matrx, L the k by n labour nput coeffcent matrx, and C the m by k consumpton basket matrx. 12 In Krause (1980), there s only one household process to reproduce each type of labour so that the number of household processes reduces from h to k. Note that n our framework, household actvtes to produce labour servces are juxtaposed wth the normal producton processes n the matrces B and A. In our model, the zero elements of B and A for ths specal case can be postve, thus allowng for durable consumpton goods, jont producton of labour servces, and drect labour nputs to produce labour servces. Moreover, n our model, the number of columns of C need not be k, admttng of alternatve household actvtes to produce labour servces when k<h. Now t s possble to select any commodty or a type of labour as the standard of value because goods and labour are treated n a completely symmetrc way. Here, however, we choose a type of labour as the standard and let t be the -th element among the (m+k) rows of B and A (m < < m + k). We gve our defnton of labour values as follows. 11 The readers who are accustomed to Leontef models are referred to T. Fujmoto, The concept of explotaton n a general lnear model wth heterogeneous labour, op. ct., for an explanaton, easer to grasp, of our model. 12 U. Krause, Abstract labour n general jont systems, op. ct., used the symbol F n place of our C.

6 248 Takao Fujmoto y Fumko Ekun Defnton of values for our general model: Values n a general nput-output model are nonnegatve magntudes assgned to commodtes (ncludng normal commodtes and varous types of labour) such that the value of the standard labour be maxmzed under the condton that the total value of the output of each possble process should not exceed that of the nput. When calculatng the total value of the nput of a process, unty s assgned to the drect nput of the standard labour. 13 Our productveness assumpton here s Productveness Assumpton: There exsts an (B A) x e, x R n+h + such that: R n+h + where s the Eucldean space of dmenson (n+h), and e s the (m+k)- column vector whose -th entry s unty wth all the remanng elements beng zero. Ths assumpton tells us that t s possble to organze an economy-wde producton plan n whch more than one unt of -th labour servce s produced as net output. Havng defned values as above, we can now explan how to compute values n a systematc way. Let us frst defne the followng vectors: (λ, λ,..., λ, λ, λ,..., λ ) and (2) m+k (λ, λ,..., λ, 1, λ,..., λ ) m+k The vector s the vector of values wth -th element (labour) beng the standard of value, and the entry λ stands for the value of commodty j j wth -th labour as the standard of value. On the other hand, s wth ts 13 Among a plural number of solutons, we adopt those whch realze the maxmum number of equaltes n the constrants. And yet, a soluton may not be unque.

7 Abstract Labour n a Model of Jont Producton 249 -th entry replaced by unty so that t can embody the requrement that unty s assgned to the drect nput of the standard labour. Our defnton above s rewrtten lke ths: Fnd out 0 such that λ should be maxmzed subject to. B <. A. (3) The constrant n ths problem can be transformed frst through addng (1 - λ ). b () to both sdes, then multplyng both sdes by nonzero nonnegatve scalar v, yeldng v.. B < v.. A + v. (1- λ ). b (), where b () s the -th row of B. Then, we set v. (1 - λ ) = or λ = 1-1. v Ths normalzaton yelds as our constrant v.. B < v.. A + b (). Snce we have λ = 1-1/v from our normalzaton, maxmzng v s equvalent to maxmzng λ. Wrtng v. smply as a varable vector q, thus q v, we have the lnear programmng problem (LP): (LP) max q subject to q. B < q. A + b () and q R m+k. + Fnally, the values can be calculated as q* λ = - 1 q* and λ j = for j = 1,...n, j, (4) q * j q *

8 250 Takao Fujmoto y Fumko Ekun where q* s an optmal soluton to (LP). Dual to (LP) above s the problem (DP): (DP) mn b (). x subject to Bx A x + e and x R n+h. + Thanks to our Productveness Assumpton, the problem (DP) has an optmal soluton, and ts optmal value s not less than one by the very formulaton of the problem. By use of dualty n lnear programmng, t follows that q* = b (). x * 1, and we have shown n Fujmoto (2009) that n fact, 0 < λ < 1 from eq. (4). 14 And ths fact was used to dscuss the exstence of explotaton n Fujmoto (2009) n relaton to prces and wage rates. The rate of explotaton for labour type s gven (1 - λ )/ λ as a postve magntude when λ reducton to a common unt There may be at least two reasons for the necessty of reducton of heterogeneous labour to a common unt. Frst, we wsh to know the overall rate of explotaton for the whole workng class wthout usng the actual employment data. Second, we may have to construct a model wth one homogeneous labour, gven dsaggregated data based upon heterogeneous labour. 14 Proposton 2 n T. Fujmoto, The concept of explotaton n a general lnear model wth heterogeneous labour, op. ct., p When λ = 0, the rate of explotaton can be regarded as nfntely large f a worker of ths type s employed at a postve wage rate.

9 Abstract Labour n a Model of Jont Producton 251 Now we are ready to propose a method of reducton of heterogeneous labour to a common unt or abstract labour. To do so, we have to make more restrctve our model explaned n the prevous secton. That s, our matrces B and A are represented as: B D B = and A = A C. (5) F J L E Here A s the m by n materal nput coeffcent matrx, B the m by n materal output coeffcent matrx, C the m by h consumpton basket matrx, D the m by h matrx of durable commodtes remanng from C after one perod, E the k by h matrx of drect labour nputs n household actvtes, F the k by n labour servce output matrx from normal producton processes, L the k by n labour servce nput matrx n the normal producton processes, and J the k by h labour servce output matrx from household actvtes. We assume that F = 0, and that the matrx J has a specal form as follows: 1,1,...,1 0,0,...,0 0,0,...,0 0,0,...,0 0,0,...,0 1,1,...,1 0,0,...,0 0,0,...,0 J = ,0,...,0 0,0,...,0 0,0,...,0 1,1,...,1 Thus, there s no jont producton of labour servces among all the household actvtes. It should be noted agan we have made one more restrctve assumptons that labour servces are produced only n household actvtes,.e., F = 0. Our way of fndng reducton ratos proceeds lke ths. Frst we select a row k-vector α S k-1, where S k-1 s the smplex of dmenson (k - 1),.e., S k-1 { w w R k k +, w = 1 }. =1 Usng ths α as a vector of converson rates, we reduce our matrces B and A to

10 252 Takao Fujmoto y Fumko Ekun B D B r = and A r = A C. 0 αj αl αe In ths modfcaton, varous types of labour nputs n each process are fused nto one scalar, and the sze of two matrces B r and A r are now (m+1) by (n+h). Wth B r and A r gven as data, we calculate labour values [m+1], by solvng (LP) and usng eq.(4). 16 Denote smply by the m-vector formed by the frst m elements of [m+1]. Ths represents the values of normal commodtes n terms of fused labour. The nequalty constrant (3) now turns nto From ths, we get [m+1]. B r < [m+1]. A r. [m+1] λ [m+1]. α. J < (C - D) + λ [m+1]. α. E, (6) m+1 m+1 where at least one strct equalty holds because we maxmze λ [m+1] m+1 subject to ths constrant and F = 0. Now the last element of the vector [m+1] shows the value of fused labour wth the fused labour beng the standard, and we know 0 < λ [m+1] < 1 as has m+1 been found n the prevous secton. We are then able to calculate the value of each labour type n terms of fused labour. The row h-vector (C - D) + λ [m+1]. α. m+1 E means the vector of values n terms of fused labour actually consumed n household actvtes. Snce the matrx J s of the supposed specal form, the value of -type of labour s obtaned as the mnmum of the elements of (C - D) + λ [m+1]. α. m+1 E whch correspond to the columns contanng unty n the -th row of J. More precsely, when we defne the set S() S() { J j = 1 }, 16 For [m+1], see eq. (2) above. The ndex s here replaced by [m+1].

11 Abstract Labour n a Model of Jont Producton 253 the value of -type labour servce n terms of fused labour, λ, s λ mn { ( (C - D) + λ [m+1]. α. E) j j S ()} m+1 We wrte w (λ 1, λ 2,..., λ k ), and assume n addton that w Normalze w 0 so that the sum of ts elements s unty,.e., the normalzed vector f(α) S k Snce a soluton to (LP) may not be unque, we have a correspondence from S k - 1 nto tself. The mage set f(α) s compact and convex, and the correspondence s evdently upper sem-contnuous. Therefore, Kakutan fxed pont theorem guarantees the exstence of a vector α* S k - 1 such that α* f (α*). Ths mples that we can fnd converson rates whch are proportonal to values n terms of fused labour. That s, pα = w, where p s a postve scalar. From (6) and one strct equalty theren, we have p = λ [m+1] < 1. Then, the rate m+1 of explotaton of -type of labour n terms of fused labour s gven as: α λ λ = α pα = 1 p pα p 0, whch s shown to be common among all types. Wth ths property just stated, α*l may be called abstract labour, not fused labour n a naïve way. As the reader may have notced, we can rescale α* as we lke. Thus, when we set a new α** cα* so that cα* j = 1wth a postve scalar c, t may be nterpreted as choosng j-type of labour servce as a common unt. Any type can be a common unt so long as α* j 0. At ths pont, t s better to show that our reducton method ncludes that of Krause (1981) as a specal case. In hs model, B = I (the dentty matrx) wth m = n, D = 0, E = 0, and J = I (the dentty matrx) wth k = h, referrng to our model (5) and hs (1). Besdes, the nequalty (3) n the defnton of values becomes an equalty. 19 Hence, our fxed pont property becomes an egenvalue problem pα = αl(l A) -1 C. 17 It s suffcent to assume that there exsts at least one type of labour servce, smple or unsklled, whch s requred drectly or ndrectly to produce each type of labour servce. 18 Snce ths vector depends on the parameter α chosen at the start, t s wrtten as n the text. 19 Ths s related to the non-substtuton theorems. See T. Fujmoto et al., A complete characterzaton of economes wth the non-substtuton property, Economc Issues, vol. 8, 2003, pp [

12 254 Takao Fujmoto y Fumko Ekun The standard reductons explctly represented n Krause (1980) for models wth jont producton are also specal cases of ours. To have α strctly postve, Krause (1981) assumes that two matrces M A + CL and H L(l A) 1 C to be Sraffa matrces, whch s weaker than rreducblty or ndecomposablty. 20 It seems natural, however, to allow α to be a nonnegatve and nonzero vector, allowng for zero converson rates for a set of labour types. 5. a numercal example Let us use an example n Bowles and Gnts (1977) to show how our approach works. In ther example, there are four normal commodtes and three types of labour servces: m=n=4 and k=h=3. The data theren gven as Table 1 n Bowles and Gnts (1977) are A , L , and C 0 0 0, B = l 4, J = l 3, D = 0, E = 0, and F = 0, where the symbols are those n eq. (5). Bowles and Gnts 21 named the four commodtes Food, Steel, Housng and Mercedes, and three types of labour servces Supervsory, Prmary and Secondary. 20 In the case of jont producton, U. Krause, Abstract labour n general jont systems, op. ct., assumes the system (A, B, L) s connected. For ndecomposablty, see T. Fujmoto and F. Ekun, Indecomposablty and prmtvty of nonnegatve matrces, Polítca y Cultura, núm. 21, 2004, pp S. Bowles and H. Gnts, The Marxan theory of value and heterogeneous labour: a crtque and reformulaton, op. ct., p. 182.

13 Abstract Labour n a Model of Jont Producton 255 Ther labour values of commodtes (Table 2 n Bowles and Gnts) 22 are L(l A) 1 = ~ : Table Ther labour contents wthn each type of labour (Table 3 n Bowles and Gnts) 23 are H L(l A) 1 C = ~ (7) The Frobenus root of ths matrx s μ(h) = < 1. Therefore, as Potron (1913) and Krause (1981) proved, the equlbrum rate of proft s postve f the wage rate of each type of labour s exactly equal to the money value of ts consumpton basket. To show that Bowles and Gnts (1977), Potron (1913), and Krause (1981) all made a wrong nterpretaton about the matrces and H above, let us here compute the correct labour values n terms of Secondary labour based upon conventonal method a la von Neumann. Frst we create the augmented nput coeffcent matrx A + whch ncludes n each process the necessary materal nputs to employ Supervsory and Prmary labour,.e., A + A + C (3) 1 L (3) = = , Ibd., p Ibd., p. 184.

14 256 Takao Fujmoto y Fumko Ekun where C (3) and L (3) are the matrces C and L wth the 3rd column and the 3rd row removed respectvely. Then, the tradtonal way to calculate the labour values of normal commodtes s by [7] L3 (l - A+)-1 = ( ), (8) c where s the vector of values of normal commodtes n terms of c[7] Secondary labour, L 3 the 3rd row of the matrx L. It s clear that the correct values are greater than those obtaned by Bowles and Gnts (1977), the 3rd row of Table 2 above. Ths s smply because the requrements through the use of drect labour n each process are neglected. Now let us proceed to our own way. Our matrx B s the 7 by 7 dentty matrx I, whle A becomes A Our Productveness Assumpton n secton 3 s satsfed by x = ( ), where a prme stands for transposton of the vector. In the specal case here wthout jont producton, the lnear programmng problem (LP) s solved by the followng: q. I = q. A + e and q R+ 7. The soluton q to ths equaton, wth =5 ( Supervsory labour), =6 ( Prmary labour), and =7 ( Secondary labour) respectvely, are approxmately ( ), ( ), and ( ).

15 Abstract Labour n a Model of Jont Producton 257 Hence, by use of eq. (4), we compute the labour values n terms of labour type =5,6 and 7, respectvely, as: [5] ( ), =~ [6] ( ), and =~ [7] ( ), =~ We can here see at once that the values of normal commodtes n terms of Secondary labour,.e., the frst four elements of [7], concde wth those computed above n the conventonal way,.e., eq. (8). Ths s natural enough because of our defnton eq. (3) wth the strct equalty holdng and B beng the dentty matrx I. It can be seen that all the labour values of commodtes turn out larger than those n Table 2 by Bowles and Gnts, 24 agan because they neglected necessary labour va the nputs of other types of labour. 25 Incdentally, f we adopt the method n Fujmoto and Opocher (2010), to tell f a type of labour s more sklled than another, Supervsory labour s the most sklled and the Secondary labour the least. When each type of labour receves ther respectve no-savngs wage rate, we can calculate the rates of explotaton for ndvdual types as E 5 = = ~ E 6 = = ~ E 7 = = ~ , and Ibd., p Based upon the matrx S. Bowles and H. Gnts, The Marxan theory of value and heterogeneous labour: a crtque and reformulaton, op. ct., gave two defntons of explotaton rate for each labour type. Both are, however, ether nadequate or arbtrary as are crtczed by Catephores. In short, they had to make up for the underestmaton of values. See G. Catephores, On heterogeneous labour and the labour theory of value, Cambrdge Journal of Economcs, vol. 5, 1981, pp

16 258 Takao Fujmoto y Fumko Ekun We have agan strkngly dfferent results from those by Bowles and Gnts, 26 n whch the rate of explotaton for Supervsory labour s negatve,.e., -46%: that rate here s postve and 170%, the largest among three rates. Ths dscrepancy comes from ther arbtrary weghts, actually equal weghts, gven to labour types when convertng labour hours to some common unt. If we look at the values closely, the labour contents of housng (commodty 3) n terms of Supervsory labour s as low as 0.316, whle those n terms of Prmary labour and Secondary labour are and 1.70, respectvely: the latter are much greater. So, the value of Supervsory labour n terms of tself s small, though one unt of Supervsory labour consumes housng as large as 1.0 unt. The reader s remnded that n order to calculate the rates of explotaton for ndvdual types of labour, we need no data on how many workers of each type are employed, nor converson of varous types of labour to abstract labour (or a common unt). And yet, we can actually construct our abstract labour followng the method n Krause (1981) for the above specal case. All we need s to fnd out the Frobenus root μ(h) and the row egenvector assocated wth t for the matrx (7),.e., H L(l - A) -1 C. They are computed as =~ = ~ p = μ(h) < 1 and α ( ) S 2. Hence, the unform rate of explotaton s 1 - p E = = ~ = ~ p The common rate of explotaton s by far smaller than the ndvdual rates of explotaton obtaned above. Ths s because labour servces of other types are regarded as normal commodtes, horse/cow work, or even slaves tol when we calculate the value of a partcular type of labour n terms of that labour. In a sense, that type of labour s dctatoral n realzng the mnmum amount of labour of the type whle producng one unt of net labour servce of the same type,.e., n determnng ts value wth the standard beng ts labour type. Ths s a weak as well as nterestng pont n our defnton of values of ndvdual heterogeneous labour types. On the other hand, when we reduce varous types 26 S. Bowles and H. Gnts, The Marxan theory of value and heterogeneous labour: a crtque and reformulaton, op. ct., 184.

17 Abstract Labour n a Model of Jont Producton 259 of labour servces to abstract one, each type enters commodtes as well as labour servces on equal terms,.e., no slaves at all, thus ncreasng the amount of necessary labour n terms of abstract labour, whch mples a smaller rate of explotaton. Indeed, the noton of abstract labour should be useful to combne the whole spectrum of heterogeneous labour nto one workng class. 6. a stll more general model A remanng problem s how we can treat more general models n whch household actvtes can produce varous labour servces jontly, and household actvtes may requre labour servces as drect nputs. Ths we explan here n ths secton n a teleologcal manner. Let α be agan a sem-postve row k- vector n the smplex of dmenson (k 1),.e., α S k-1. Some more symbols are defned as follows: (λ 1, λ 2,..., λ m, λ m+1, λ m+2,..., λ m+2 ) α (λ 1, λ 2,..., λ m, α 1, α 2,..., α k ), (λ, λ,..., λ C 1 2 m ), and L (λ m+1, λ m+2,..., λ m+k ). In ths secton, superscrpts are left out from s. To obtan the reducton ratos n a general model, we frst consder m+k (LPG) max λ subject to. B < α. A for R m+k and =m+1 + L = p. α for some nonnegatve scalar p. Here α s a parameter vector to the problem (LPG). The meanng of the problem s that we should maxmze the sum of values of all the labour servces whle the total value of outputs s not larger than that of nputs n each process / household actvty wth the drect labour nputs beng gven the weghts represented by α. One more constrant tells us that values of labour servces

18 260 Takao Fujmoto y Fumko Ekun are proportonal to the elements of a gven parameter vector α. Substtutng ths requrement nto the frst constrant, the problem becomes B D (LPG*) max p subject to ( C,P ). < ( A,1 ). C for αf αj C αl αe C Rm and p 0. + Note that here E and F are now not necessarly the zero matrx, and J s of a general type. Exactly n the same way, ths problem (LPG*) has a feasble vector by vrtue of our Productveness Assumpton, and an optmal soluton p* (α) s less than one. Ths p* (α) can be postve under a certan condton. 27 Fnally, the desred reducton ratos α* are to be obtaned by solvng max p* (α) for α Sk -1. And the unform rate of explotaton s defned by 1 - p* (α*) e = p* (α*) Our method n ths secton may nclude the reducton ratos htherto proposed as specal cases. Let us take up the case where B and A are m by m square matrces, and the nonnegatve nverse (B - A) -1 exsts; D, E, and F are all zero matrces; h=k, and J= l k. In ths case, t follows from the two constrants of (LPG*) that C = αl (B - A) -1 and pα < C. C = αl (B - A) -1 C for α S k-1. From Perron-Frobenus theorem, we know that the maxmum p s realzed when t s the Perron-Frobenus root of L(B - A) -1 wth α beng ts assocate egenvector. 27 See footnote 18.

19 Abstract Labour n a Model of Jont Producton remarks An alternatve lnear programmng problem to (LP) above can be obtaned drectly from (3): (LP2) max. e subject to. (B A ) < a () and R m+k, where A s A wth ts -th row replaced by zero vector, and a () s the -th row of A. And the dual one s: (DP2) mn a (). x subject to (B A ). x e and x R n+h. Usng these, we are able to obtan the same propostons. Ths problem (LP2) gves a more drect nterpretaton of labour value, and can be an alternatve defnton. We have, however, adopted a tradtonal defnton of drect and ndrect requrement. In sum, we have shown a method to dscover the reducton ratos whch Marx descrbed n Captal I, for a general model wth jont producton n producton processes as well as household actvtes, and durable consumpton goods. A mathematcal key s the generalzed egenvalue problem. Supposng such reducton ratos, Marx was able to conceve the one workng class as a whole n spte of the heterogenety of labour power, and an equal rate of explotaton among all the types of labour servces s observed. It s now easy to tell whch labour type s more sklled than another. In a sense, the method explaned n secton 3 s a separatst method, whle the reducton to abstract labour presented n secton 4 s a unonst approach. Surely the latter s more favorable to Marx because all the workers are regarded as frends n a captalst producton system. The separatst method does not necessarly thnk of workers of dfferent types as enemes to each other, but certanly not as frends. One fnal remark s concerned wth the fact that labour values can be computed by usng abstract labour, whle prces are descrbed by usng heterogeneous labour servces. Our method s applcable to a theory of aggregaton of commodtes nto composte ones wthout dependng prces n a model wth jont producton. Ths s another theme we wll tackle wth n the future research M. Morshma and F. Seton, Aggregaton n Leontef matrces and the labour theory of value, Econometrca, vol. 29, 1961, pp