Marx's Error and Its Correction
Later scholars argued that Marx's formulas for competitive prices were mistaken.
First, competitive equilibrium requires a uniform rate of return over constant capital valued at its price, not its Marxian value, contrary to what is done in Table 2 above. Secondly, competitive prices result from the sum of costs valued at the prices of things, not as amounts of embodied labour. Thus, both Marx's calculation of and the sums of his price formulas do not add up in all the normal cases, where—as in the above example—relative competitive prices differ from relative Marxian values. Marx noted this, but thought that it was not significant, stating in Chapter 9 of Volume 3 of Capital that "Our present analysis does not necessitate a closer examination of this point."
The simultaneous linear equations method of computing competitive (relative) prices in an equilibrium economy is today very well known. In the greatly simplified model of Tables 1 and 2, where by assumption the wages rate is given and equal to the price of beavers, the most convenient way is to express such prices in units of beavers, which means normalising . This immediately yields the (relative) price of arrows as
Substituting this into the relative-price condition for beavers
gives the solution for the rate of return as
Finally, the price condition for deer can hence be written as
This latter result, which gives the correct competitive price of deer in units of beavers for the simple model used here, is generally inconsistent with Marx's price formulae of Table 2.
Ernest Mandel, defending Marx, explains this discrepancy in term of the time frame of production rather than as a logical error, i.e. in this simplified model, capital goods are purchased at a labour value price but final products are sold under prices which reflect redistributed surplus value.
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