Abstract labour in a model of joint production

AutorTakao Fijimoto - Fumiko Ekuni
CargoFukuoka University, Fukuoka, Japan - Shikoku-Gakuin University, Zentsuji, Japan
Páginas243-261
Abstract Labour in a Model
of Joint Production*
Takao Fujimoto**
Fumiko Ekuni***
Abstract
This paper presents a new mathematical method for constructing abstract labour as
conceived by Karl Marx, in a general input-output model with joint production and
heterogeneous labour. A brief history of previous efforts to define and compose abstract
labour using input-output models is also described. A numerical example is given to
show how our method works in a concrete way.
Key words: abstract labour, eigenvalue, eigenvector, heterogeneous labour, Kakutani
fixed point theorem.
Resumen
Este artículo presenta un nuevo método matemático para construir el trabajo abstracto
concebido por Karl Marx, mediante un modelo de insumo-producto con el trabajo
heterogéneo y la producción conjunta. También se describe una historia breve sobre
los esfuerzos para definir y componer una abstracción matemática usando los modelos
de entrada-salida. Se dan una serie de ejemplos numéricos que nos muestran cómo
funciona el método de manera concreta.
Palabras clave: trabajo abstracto, valor propio, vector propio, trabajo heterogéneo,
teorema de punto fijo de Kakutani.
Artículo recibido el 12-04-10
Artículo aceptado el 10-11-10
* Acknowledgments: thanks are due to professor Dr. Alejandro Valle Baeza who had
introduced this journal to the authors, and to our friend professor Dr Ul rich Krause for his
encouragement. The authors are grateful to the referees who have provided useful comments
and suggestions to improve this article.
** Fukuoka University, Fukuoka, Japan [takao@econ.fukuoka-u.ac.jp].
*** Shikoku-Gakuin University, Zentsuji, Japan [fumiko@sg-u.ac.jp].
244
Política y Cultura, primavera 2011, núm. 35, pp. 243-261
Takao Fujimoto y Fumiko Ekuni
roductivity can be enhanced by introducing division
of labour. This fact has been well known, and we
believe that in reality division of labour has played a
significant role in the development of human societies. Yet, theoretical analysis
of heterogeneity of labour has been given little attention by economists. When
heterogeneity of labour is allowed for in a model of economy, some basic
problems emerge. The first problem is concerned with how to measure labour
values of various commodities and labour services. The second one is whether
we could devise abstract labour, using given technological data. The third may
be how to define rates of exploitation for individual types of labour services
and for abstract labour. Some economists regard the reduction of heterogeneous
labour to abstract labour as unnecessary, while others think of the reduction
as essential to theory of exploitation.
Surely, it may sometimes be necessary to know the rate of exploitation for
the working class as a whole, in addition to the rates for individual types of
labour. Then, it becomes indispensable to carry out the reduction of various
types of labour to a common unit or abstract labour. Besides, it may be interesting
to study how to realize the uniform rate of exploitation among heterogeneous
groups of labour if at all the reduction is performed.
This paper represents a way of finding out reduction ratios, in a general
model with joint production and heterogeneous labour, by use of linear
programming problems and Kakutani fixed point theorem. Our method
guarantees the uniform rate of exploitation among all the types of labour. In
section 2, we include a brief history of the researches on the topic. Section 3
explains our model and how it is more general than the existing ones. Section
4 describes our method for somewhat restrictive cases. Section 5 presents a
numerical example to show how our method works. Then section 6 describes
our approach to a still more general model where each household activity can
produce a plural number of labour services jointly and some labour services
are produced by normal production processes. Final section 7 includes some
concluding remarks. It should be noted at the outset that we deal only with
mathematical aspects of abstract labour, and do not touch upon historical
and philosophical sides of the topic. We hope that this article will stimulate
discussions among those philosophers and historians who are interested in
abstract labour.
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