Thursday, November 05, 2020

Infeasibility Of All Machines Truncated

Figure 1: Factor Wage Curves For Feasible Techniques

There are 12 coefficients that can be varied in my minimum multisector model in which production in all sectors can require both fixed and circulating capital. I do not think I am being very orderly in exploring this twelve-dimensional space.

This is a fluke case in which the maximum rate of profits is zero for both the Alpha and the Beta techniques. If only new machines are used as means of production in producing new machines and in producing corn, no surplus product is available to pay out as wages and profits. Likewise, if a machine is run for its full physical lifetime of two years in the machine sector, but truncated in the corn sector, no surplus product is, once again, available. But this is a feasible technology in that a surplus product is available when machines are operated for a full two years in the corn sector. (If a1, 1 and a1, 2 are slightly increased, the maximum rate of profits is negative for both the Alpha and Gamma techniques. Then this would be a non-fluke case.)

Managers of firms choose among feasible techniques by deciding whether or not to operate machines for two years, or to truncate their use, in the machine sector. As shown by the wage frontier above, their decision varies with the distribution of income. Baldone (1980) notes the possibility that for a viable technology with fixed capital, the truncation of the economic life of a machine may result in a non-viable technology. In an already long paper, he does not have a numerical example, however. By the way, Baldone has an appendix with a numerical example of the recurrence of truncation.

For completeness, Table 1 specifies the coefficients of production. I also define the techniques, which are unchanged from previous posts exploring this model. I would be impressed if somewhere in this twelve-dimensional space, almost all phenomena noted in the literature for models of fixed capital and new cases could be found. I have yet to locate cases of reswitching in this model. By definition, I will not be able to find a 'one-good' model with reswitching in this model (albeit what happens if the coefficients of production are the same in the two sectors, perhaps extended to have the machine last three years?).

Table 1: Coefficients of Production for The Technology
New Machines1010
Old Machines A0100
Old Machines B0001
New Machines25/200
Old Machines A1000
Old Machines B0010

Table 2: Techniques
AlphaI, III
BetaI, II, III
GammaI, III, IV
DeltaI, II, III, IV

  • Salvatore Baldone (1980) Fixed capital in Sraffa's theoretical scheme. Trans. in Pasinetti (1980).
  • Christian Bidard (2020) The wage-minimisation property. Working paper 2020-17.
  • Luigi L. Pasinetti, ed. (1980) Essays on the Theory of Joint Production. New York: Columbia University Press.
  • Bertram Schefold (1980) Fixed capital as a joint proudct and the analysis of accumulation with different forms of technical progress. Trans. in Pasinetti (1980).
  • Paolo Varri (1980) Prices, rate of profit and life of machines in Sraffa's fixed capital model. Trans. in Pasinetti (1980).


Anonymous said...

I would encourage you to investigate the case with equal proportions as an extension of Steedman's latest paper. Also following the Steedmanite trend of using capital/output ratios would be a good appendix. Best regards.

Anonymous said...

About using as a modern tool of analysis the Capital-Output per industry it is interesting to see the differences between a model of Simple Production with equal coefficients and a model of Fixed Capital with equal coefficients. Capital-Output in the first is thought to follow a linear relation but nothing has been said about in the second model.

Robert Vienneau said...

Presumably, the latest Steedman paper is "Fixed capital in the corn-tractor model". This is the Samuelson-Garegnani model with depreciation done correctly. As is Steedman's wont, he has lots of homework problems that could do with some graphs.

I tend to worry about real Wicksell effects than price Wicksell effects. When I make comments about variations in output per person-year, I am implicitly drawing a conclusion about real Wicksell effects. It is surprising how complicated simple fixed capital models are and how room for research still exists here.

Doing my thing with perturbing coefficients of production would be appropriate here. And it would fit into my current exploration of fixed capital. One attribute of these simple models that I have been exploring is that I end up solving at most quartic polynomial equations. I use numeric methods to find fluke switch points. With Steedman's model, I would need to use numeric methods to even find the maximum rate of profits for a technique, I think. An extensive spreadsheet would probably not be enough.

I think I will put this off for a while more.

Anonymous said...

Thank you for the explicative comment. I was searching yesterday for topics on Fixed Capital and I bumped into this from old Steedman that it seems to me like an old version of the 2020 paper in different clothing:

On the other hand I also found a paper from Samuelson that seems like a reaction to that critique from Steedman's attack on the Fixed Capital. Samuelson uses the Fixed Capital case to suggest that if we abandon radioactive exponential depreciation our holy Standard Commodity ceases to exist. I find it very interesting.

Just to conclude I talked about the equal proportions because I saw in this post the reference to a case where "albeit what happens if the coefficients of production are the same in the two sectors" as being a special case of equal proportions.