Prices of Production - Facts and Logic - Lack of A Formal Proof

Lack of A Formal Proof

Marx's critics interpreting his models often argue he keeps assuming what he needs to explain, because rather than really "transforming values into prices" by some quantitative mapping procedure, such that prices are truly deduced from labour-values, he either (1) equates values and prices, or else (2) he combines both values and prices in one equation.

Thus, for example, either Marx infers a rate of profit from a given capital composition and a given quantity of surplus-value, or else he assumes a rate of profit in order to find the amount of surplus-value applying to a given quantity of capital invested. That might be fine if the aim is to just investigate what profit an enterprise or sector would receive on average, having produced a certain output value with a certain capital composition. But this manoeuvre of itself cannot contain any formal proof of a necessary quantitative relationship between values and prices, nor a formal proof that capitals of the same size but different compositions (and consequently different expenditures of labour-time) must obtain the same rate of profit. It remains only a theory.

Marx insists both that output prices obtained will necessarily deviate from values produced, but also that the sum of prices would equal to the sum of values in the pure case, yet, critics claim, he fails to show quantitatively how a distribution process could then occur such that price magnitudes map onto value magnitudes, and such that a uniform profit rate returns equal profits to capitals of equal sizes (a mapping relation is used here in the mathematical sense of a bijective morphism, involving one-to-one correspondence between value quantities and price quantities via mathematical equations). In that case, there is again no formal proof of any necessary relationship between values and prices, and Marx's manuscript really seems an endless, pointless theoretical detour leading nowhere. In modelling, simple logical paradoxes appear of the type that:

• in a static model, it is impossible to uphold the postulate of a uniform rate of profit and the postulate of total values=total prices at the same time;
• to find production-prices, a uniform rate of profit must be assumed, while at the same time to find a uniform rate of profit, production-prices must already be assumed;
• a price level must be assumed, rather than be deduced from labour-values.

All these conceptual and logical issues mentioned become crucial when attempts are made to model value and price aggregates mathematically to study capitalist competition. Different kinds of theoretical assumptions or interpretations will obviously lead to very different results.