Editing Transformation problem (section)

== Classical tableaux ==
Like Ricardo, Marx believed that ''relative'' labour values&mdash; <math>{p_D \over p_B}</math> in the above example&mdash;do not generally correspond to relative competitive prices&mdash; <math>{P_D \over P_B}</math> in the same example. However, in volume 3 of ''Capital'' he argued that competitive prices are obtained from values through a '''transformation'' process, whereby capitalists ''redistribute'' among themselves the given ''aggregate'' surplus value of the system in such a way as to bring about a tendency toward an equal rate of profit, <math>r</math>, among sectors of the economy. This happens because of the capitalists' tendency to shift their capital toward sectors where it earns higher returns. As competition becomes fierce in a given sector, the rate of return falls, while the opposite will happen in a sector with a low rate of return. Marx describes this process in detail.<ref>[http://www.marxists.org/archive/marx/works/1894-c3/ch09.htm Capital III, Ch. 9]</ref>

=== Marx's reasoning ===
The following two tables adapt the deer-beaver-arrow example seen above (which, of course, is not found in Marx, and is only a useful simplification) to illustrate Marx's approach. In both cases it is assumed that the total quantities of beavers and deer captured are <math>Q_B</math> and <math>Q_D</math> respectively. It is also supposed that the subsistence real wage is one beaver per unit of labour, so that the amount of labour embodied in it is <math>l_W = E_B = l_A a_B + l_B < 1</math>. Table 1 shows how the total amount of surplus value of the system, shown in the last row, is determined.

{| border="2" cellpadding="4" cellspacing="0" style="vertical-align:center;text-align:center; border: 1px #aaa solid; border-collapse: collapse;"
|+'''''Table 1&mdash;Composition of Marxian values in the deer-beaver-arrow production model'''''
|-
 ! Sector
 ! Total Constant Capital <br/> <math>Q_i c_i</math>
 ! Total Variable Capital <br/> <math>Q_i v_i</math>
 ! Total Surplus Value <br/><math> \sigma Q_i v_i</math>
 ! Unit Value <br/><math>c_i + (1 + \sigma) v_i</math>
|-
 ! Beavers
 | <math>Q_B l_A a_B</math>
 | <math>Q_B(l_A a_B + l_B) l_B</math>
 | <math>\sigma Q_B (l_A a_B + l_B) l_B </math>
 | <math>l_A a_B + (1 + \sigma) (l_A a_B + l_B) l_B</math>
|-
 ! Deer
 | <math>Q_D l_A a_D</math>
 | <math> Q_D (l_A a_B + l_B) l_D </math>
 | <math> \sigma Q_D (l_A a_B + l_B) l_D </math>
 | <math> l_A a_D + (1 + \sigma) (l_A a_B + l_B) l_D </math>
|-
 ! Total
 |
 |
 | <math> \sigma (l_A a_B + l_B) (Q_B l_B + Q_D l_D) </math>
 |
|}

Table 2 illustrates how Marx thought this total would be redistributed between the two industries, as "profit" at a uniform return rate, ''r'', over constant capital. First, the condition that total "profit" must equal total surplus value—in the final row of table 2—is used to determine ''r''. The result is then multiplied by the value of the constant capital of each industry to get its "profit". Finally, each (absolute) competitive price in labour units is obtained, as the sum of constant capital, variable capital, and "profit" per unit of output, in the last column of table 2.

{| border="2" cellpadding="4" cellspacing="0" style="vertical-align:center;text-align:center; border: 1px #aaa solid; border-collapse: collapse;"
|+'''''Table 2&mdash;Marx's transformation formulas in the deer-beaver-arrow production model'''''
|-
 ! Sector
 ! Total Constant Capital <br/> <math>Q_i c_i</math>
 ! Total Variable Capital <br/> <math>Q_i v_i</math>
 ! Redistributed Total <br/>Surplus Value <br/> <math> rQ_i c_i</math>
 ! Resulting <br/> Competitive <br/>Price <br/> <math> v_i + (1 + r) c_i</math>
|-
 ! Beavers
 | <math> Q_B l_A a_B </math>
 | <math> Q_B (l_A a_B + l_B) l_B </math>
 | <math> rQ_B l_A a_B </math>
 | <math> (l_A a_B + l_B) l_B + (1 + r) l_A a_B </math>
|-
 ! Deer
 | <math> Q_D l_A a_D</math>
 | <math> Q_D (l_A a_B + l_B) l_D </math>
 | <math> rQ_D l_A a_D </math>
 | <math> (l_A a_B + l_B) l_D + (1 + r) l_A a_D </math>
|-
 ! Total
 |
 |
 |<math>r l_A(Q_B a_B + Q_D a_D) = \sigma (l_A a_B + l_B) (Q_B l_B + Q_D l_D)</math>
 |
|}

Tables 1 and 2 parallel the tables in which Marx elaborated his numerical example.<ref>[http://www.marxists.org/archive/marx/works/1894-c3/ch09.htm Capital, III Chapter 9]</ref>

=== 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>