Reduction of Heterogeneous Labour

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 207)


Heterogeneous capital has already come up for a wide discussion in economic literature, but various types of heterogeneous labour having unequal wage rates have been rarely discussed, although they are also observable in an economy. The reason for this seems to be, at the outset, that one of the aims of economic analysis is to make clear the distributional relations between capital and labour. As for capital goods, there is no distributional difficulty, even if they are heterogeneous, because an equal rate of profit is applied to them.


Reduction Ratio Skilled Labour Capital Good Education Sector Input Matrix 
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  1. 1).
    Note that when “skilled labour j” is mentioned, j ≠ 1. Also remark that education and training necessary for skilled labour are regarded as being the same.Google Scholar
  2. 2).
    In Okishio(3), the counterpart of (2) is written as γ = wE + γT + σ, (in the present notation). That is, the value of capital goods in the education sectors enters into the value-creating force of skilled labour. This, however, is incorrect as pointed out here.Google Scholar
  3. 3).
    As shown here, if the exclusive training period is considered, self-efforts become greater than unity. This is because selfefforts exerted in the education period are accumulated in worker j and reveal themselves when worker j starts working.Google Scholar
  4. 4).
    The following figure illustrates the accumulation of self-efforts to skilled labour.Google Scholar
  5. It should be taken note of that, as mentioned before, γ plays the role of a peculiar operator. Write, for instance x̃ − Ax̃ = ỹ and premultiply (w,v,Y), and γyIII = o, because no value is produced in the third sphere.Google Scholar
  6. 5).
    (L,T) in A does not give the labour matrix of the hyper-closed system, because T here is not related to the production of value.Google Scholar
  7. 6).
    Let m = n = s =1, and all the variables and coefficients of (LP.I) and (LP.II) are scalars. Put B = 1, and one has Λo = L/ (I-A)Na and γo = 1/Na.Google Scholar
  8. 7).
    In a closed Leontief economy, one has a stronger result as max μ. >; o.Google Scholar
  9. 8).
    It will be convenient to make a note on the difference between the assumptions: Lxa ≥ 0s in §2 and Na> 0s in §3. Since productive labour alone is conducive to the production of value, a strict inequality with respect to actual employment seems to create no problem. A similar distinction between productive and nonproductive labour, as made in the prpof of Proposition 10 can be applied to that of Proposition 17, if B, A, L and F are specified as in this subsection.Google Scholar
  10. 9).
    Krause’s original theorem is proved for “r ≧ o”. This proposition is a simplified version. For details, see Krause (1).Google Scholar
  11. 10).
    Hollander’s original theorem is the conclusion (ii) here.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  1. 1.Josai UniversitySakado,SaitamaJapan

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