Editing Transformation problem (section)

=== Marx's supposed 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. Second, 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 <math>r</math> 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 the wage rate is assumed as given and equal to the price of beavers, the most convenient way is to express such prices is in units of beavers, which means normalising <math>w = P_B = 1</math>. This yields the (relative) price of arrows as

:<math>P_A = l_A</math> beavers.

Substituting this into the relative-price condition for beavers,

:<math> 1 = l_B + (1 + r) l_A a_B</math>,

gives the solution for the rate of return as

:<math>r = {(1 - l_B) \over (l_A a_B)} - 1</math>

Finally, the price condition for deer can hence be written as

:<math>P_D = l_D + (1 + r) l_A a_D = l_D + {a_D (1 - l_B) \over a_B} </math>.

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 that reflect redistributed surplus value.<ref>Ernest Mandel [http://www.marxists.org/archive/mandel/19xx/marx/ch04.htm Marx's Theory of Value]</ref>