Editing Transformation problem (section)

=== 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>