Abstract
It has been demonstrated that Marx’s major conclusions are valid in a Leontief economy or a narrow plain economy. In such an economy, however, alternative processes, fixed capital and joint-production are not admitted to their full extent.
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Notes
This was produced from the equation (1′ ) in Morishima (4),p.183.
Pd.C. is defined for Dx “>”0m in this chapter. The same note applies to Pf.C. and S.C. introduced later.
As for the generalised inverse, refer to the Mathematical Addenda, pp.147-8.
A Leontief economy in which alternative processes are permitted is usually called the generalised Leontief economy. Murata(2) discussed Marx’s theory of value in the generalised Leontief economy case. Murata defined value as a minimiser of a certain type of norm, and the value equation is approximately solved by using Penrose’s inverse. Penrose’s inverse is unique, if it exists, and hence the value can then be determined uniquely. The value may rest on the form of norms, nevertheless.
The axiom of impossibility of land of Cockaigne means that the primary factors of production are essential for production, and hence it comprehends the indispensability of labour. Nevertheless, the two are tantamount to each other here, because labour alone is the primary factor of production in the present discussion.
As seen in Proposition 2, L ≥ 0n does not suffice for the indispensability of labour.
This is an extension of the “efficient point” in activity analysis. Refer to Koopmans, p.60.
Kurz (2), in an attempt to criticise Steedman’s counterexample, rewrites Steedman’s value equation as follows: 5w1+l1=6w1+w2, 10w2+l2=3w1+12w2, where l1, and l2 stand for productivity indices.
From this, Kurz concluded that there exist l1, and l2 for which w1, and w2 are positive.
In Kurz’s modified value equation, however, the same labour creates an unequal amount of value, and hence his assertion seems to be against the law of value.
A similar theorem was proved by Shiozawa(2).
Equation (22) will make no sense, if it is overdetermined. Hence, it may be plausible to presuppose rank H = min (m, n), even if m ≦ n.
If n > m, (22) appears to be underdetermined: prices of m-n types of good can be determined a priori. This, however, raises no difficulty in the present discussion.
If the iteration formulae (I-23) are immediately extended, one can write: (23′) wt+1B=(1+μt)wtM, 1+μt=wtBx/wtMx. Even if rank B = m and hence BB- = I, MB- is not necessarily nonnegative.
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© 1982 Springer-Verlag Berlin Heidelberg
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Fujimori, Y. (1982). Joint-Production and the Theory of Value. In: Modern Analysis of Value Theory. Lecture Notes in Economics and Mathematical Systems, vol 207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45543-8_4
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DOI: https://doi.org/10.1007/978-3-642-45543-8_4
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