# [OPE-L] Koopmans versus Kantorovich

From: Paul Cockshott (clyder@GN.APC.ORG)
Date: Sat May 19 2007 - 08:40:13 EDT

```I have just been re-reading Kantorovich's 1939 paper on mathematical
methods of organising and planning production, and was struck by the
way in which his formulation of linear programming is significantly different
from that in standard western presentations.

Kantorovich himself says:

" I want to emphasize again that the greater part of the problems of which I
shall speak, relating to the organization and planning of production, are con-
nected specifically with the Soviet system of economy and in the majority of
cases do not arise in the economy of a capitalist society. There the choice of
output is determined not by the plan but by the interests and profits of indi-
vidual capitalists. The owner of the enterprise chooses for production those
goods which at a given moment have the highest price, can most easily be sold,
and therefore give the largest profit. The raw material used is not that of
which
there are huge supplies in the country, but that which the entrepreneur can buy
most cheaply. The question of the maximum utilization of equipment is not
raised; in any case, the majority of enterprises work at half capacity."

Recent Russian commentaries on his work make out that this is nothing
but a token bit of marxist ideology to cloak a work that actually is
quite independent of or non marxist. At first sight this would be
reinforced by the fact that 10 years later linear programming or linear
optimisation techniques were developed in the west as well.
But the approach taken by Kantorovich seems significantly different.
Western texts emphasise that the objective function being maximised is
a linear combination of the outputs aX_1+ bX_2+ cX_3
this makes sense if you view a,b,c as prices at which the outputs
X_i will be sold.

Kantorovich on the other hand assumes that the outputs must in
fixed ratios so that x_1 = a m, x_2= b m, x_3 = c m
where m is the scale of production and a,b,c are multipliers

In the western form the combination of outputs is a matter
of indifference since they take commodity production for granted,
and all they are concerned with is maximising money income.
For Kantorovich the proportions of outputs are taken as given
as he takes planned production for granted. Kantorovich optimises
along a ray, but the western version attempts to reach a maximal
hyperplane.

I get the impression that there is some way of translating
Kantorvich's ray formulation into the Planar formulation of Koopmans
and Danzig, but I have not found a source explaining how to do it,
and it looks to me as if a simplex solution would not actually
meet Kantorovich's requirement.

Does anyone know how one utilizes the simplex method to solve
Kantorovich's problem?

Paul Cockshott

www.dcs.gla.ac.uk/~wpc
reality.gn.apc.org

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