Editing Transformation problem (section)

== Marx's labour theory of value ==
=== Surplus value and exploitation ===
Marx distinguishes between [[labour power]] as the potential to work, and labour, which is its actual use. He describes labour power as a commodity, and like all commodities, Marx assumes that on average it is exchanged at its value. Its value is determined by the value of the quantity of goods required for its reproduction.

Yet there is a difference between the value of labour power and the value produced by that labour power in its use. Unlike other commodities, in its use, labour power produces new value beyond that used up by its use. This difference is called [[surplus value]] and is for Marx the source of profit for the capitalists. The appropriation of surplus labor is what Marx denoted the exploitation of labour.

=== Labour as the "value-creating substance" ===
Marx defined the "value" of a [[commodity]] as the total amount of socially necessary labour embodied in its production. He developed this special brand of the labour theory of value in the first chapter of volume 1 of ''Capital'''. Due to the influence of Marx's particular definition of value on the transformation problem, he is quoted at length where he argues as follows:

<blockquote>Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever those proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things&mdash;in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other. Each of them, so far as it is exchange value, must therefore be reducible to this third.</blockquote>
<blockquote>This common 'something' cannot be either a geometrical, a chemical, or any other natural property of commodities. Such properties claim our attention only in so far as they affect the utility of those commodities, make them use values. But the exchange of commodities is evidently an act characterised by a total abstraction from use value.</blockquote>
<blockquote>If then we leave out of consideration the use value of commodities, they have only one common property left, that of being products of labour. […] Along with the useful qualities of the products themselves, we put out of sight both the useful character of the various kinds of labour embodied in them, and the concrete forms of that labour; there is nothing left but what is common to them all; all are reduced to one and the same sort of labour, human labour in the abstract.</blockquote>
<blockquote>A use value, or useful article, therefore, has value only because human labour in the abstract has been embodied or materialised in it. How, then, is the magnitude of this value to be measured? Plainly, by the quantity of the value-creating substance, the labour, contained in the article.
:—Karl Marx, [http://www.marxists.org/archive/marx/works/1867-c1/ch01.htm#35b/ ''Capital'', Volume I, Chapter 1]</blockquote>

=== Variable and constant capital ===
As labour produces in this sense more than its own value, the direct-labour input is called [[variable capital]] and denoted as <math>v</math>. The quantity of value that living labour transmits to the deer, in our previous example, varies according to the intensity of the exploitation. In the previous example, <math>v_i  = l_W l_i</math>.

By contrast, the value of other inputs&mdash;in our example, the indirect (or "dead") past labour embodied in the used-up arrows&mdash;is transmitted to the product as it stands, without additions. It is hence called [[constant capital]] and denoted as ''c''. The value transmitted by the arrow to the deer can never be greater than the value of the arrow itself. In our previous example, <math>c_i = l_A a_i</math>.

=== Value formulas ===
The total value of each produced good is the sum of the above three elements: constant capital, variable capital, and surplus value. In our previous example:

:<math>p_i = c_i + v_i + s_i = l_A a_i + l_W l_i + s_i </math>

Where <math>p_i</math> stands for the (unit) Marxian value of beavers and deer.

However, from Marx's definition of value as total labour embodied, it must also be true that:

:<math>p_i = l_A a_i + l_i = E_i</math>

Solving for <math>s_i</math> the above two relationships one has:

:<math>{s_i \over v_i} = {(1- l_W) \over l_W} = \sigma </math>

for all <math>i</math>.

This necessarily uniform ratio <math>{s_i \over v_i} = \sigma</math> is called by Marx the [[rate of exploitation]], and it allows to re-write Marx's value equations as:

:<math>p_i = c_i + v_i (1 + \sigma) = l_A a_i + l_W l_i (1 + \sigma)</math>