Growing Sub-systems, Vertically Hyper-integrated Sectors and the Labour Theory of Value

The concept of sub-system was originally presented by P. Sraffa in a brief non-mathematical appendix to his famous 1960 book. His purpose was that of providing and alternative, if less intuitive, line of approach to evaluate the quantities of labour that directly and indirectly enter the production of each commodity. To this effect, he used the more intuitive method of ‘reduction to dated quantities of labour’. This method works for the case of single products, but it does not work for joint production. Sraffa’s sub-system approach works in all cases. This concept has turned out to be of much wider relevance than Sraffa himself could have imagined. The Author of this paper was able to develop, from the concept of sub-system, two symmetrical concept of vertically integrated labour coefficient and vertically integrated unit of productive capacity. The present article takes a further step in the direction of a complete generalization of the concept of sub-system and vertically integrated sector. He shows that the analytical device of partitioning an economic system into sub-systems (and correspondingly of constructing vertically integrated sectors) can indeed be taken into dynamic analysis, with remarkable implications. See also "Growing Subsystems and Vertically Hyper-integrated Sectors: a Note of clarification", in Cambridge Journal of Economics, 1989, vol.13, pp.479-480.