Editing Transformation problem (section)

== British classical labor theory of value ==
Marx's value theory was developed from the [[labor theory of value]] discussed by [[Adam Smith]] and used by many British [[classical economists]]. It became central to his economics.

=== Simplest case: labor costs only ===
Consider the simple example used by Adam Smith to introduce the subject. Assume a hunters’ economy with free land, no slavery, and no significant current production of tools, in which beavers <math>(B)</math> and deer <math>(D)</math> are hunted. In the language of modern [[linear production models]], call the unit labour-input requirement for the production of each good <math>l_i</math>, where <math>i</math> may be <math>B</math> or <math>D</math> (i.e., <math>l_B</math> is the number of hours of uniform labour normally required to catch a beaver, and <math>l_D</math> a deer; notice that we need to assume labour as uniform in order to be able, later on, to use a uniform wage rate).

In this case, Smith noticed, each hunter will be willing to exchange one deer (which costs him <math>l_D</math>hours) for <math>{l_D \over l_B}</math> beavers. The ratio <math>{l_D \over l_B}</math>&mdash;i.e., the relative quantity of labour embodied in (unit) deer production with respect to beaver production&mdash;gives thus the exchange ratio between deer and beavers, the "relative price" of deer in units of beavers. Moreover, since the only costs are here labor costs, this ratio is also the "relative unit cost" of deer for any given competitive uniform wage rate <math>w</math>. Hence the relative quantity of labor embodied in deer production coincides with the ''competitive relative price'' of deer in units of beavers, which can be written as <math>{P_D \over P_B}</math> (where the <math>P</math> stands for absolute competitive prices in some arbitrary unit of account, and are defined as <math>P_i = wl_i</math>).

=== Capital costs ===
Things become more complicated if production uses some scarce [[capital good]] as well. Suppose that hunting requires also some arrows <math>(A)</math>, with input coefficients equal to <math>a_i</math>, meaning that to catch, for instance, one beaver you need to use <math>a_B</math> arrows, besides <math>l_B</math> hours of labour. Now the unit total cost (or absolute competitive price) of beavers and deer becomes

:<math>P_i = wl_i + k_A a_i , (i = B, D) </math>

where <math>k_A</math> denotes the capital cost incurred in using each arrow.

This capital cost is made up of two parts. First, there is the replacement cost of substituting the arrow when it is lost in production. This is <math>P_A</math>, or the competitive price of the arrows, multiplied by the proportion <math>h \le 1</math> of arrows lost after each shot. Second, there is the net rental or return required by the arrows' owner (who may or may not be the same person as the hunter using it). This can be expressed as the product <math>r P_A</math>, where <math>r</math> is the (uniform) ''net rate of return'' of the system.

Summing up, and assuming a uniform replacement rate <math>h</math>, the absolute competitive prices of beavers and deer may be written as

:<math>P_i = wl_i + (h + r) P_A a_i</math>

Yet we still have to determine the arrows' competitive price <math>P_A</math>. Assuming arrows are produced by labor only, with <math>l_A</math> man-hours per arrow, we have:

:<math>P_A = wl_A</math>

Assuming further, for simplicity, that <math>h = 1</math> (i.e., all arrows are lost after just one shot, so that they are [[circulating capital]]), the absolute competitive prices of beavers and deer become:

:<math>P_i = wl_i + (1 + r) wl_A a_i</math>

Here, <math>l_i</math> is the quantity of labor directly embodied in beaver and deer unit production, while <math>l_A a_i</math> is the labor indirectly thus embodied, through previous arrow production. The sum of the two,

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

gives the total quantity of labor embodied.

It is now obvious that the relative competitive price of deer <math>{P_D \over P_B}</math> can no longer be generally expressed as the ratio between total amounts of labour embodied.  With <math>a_i > 0 </math> the ratio <math>{E_D \over E_B}</math> will correspond to <math>{P_D \over P_B}</math> only in two very special cases: if either <math>r = 0</math>; or, if <math>{l_B \over l_D} = {a_B \over a_D}</math>. In general the two ratios will not only differ: <math>{P_D \over P_B}</math> may change for any given <math>{E_D \over E_B}</math>, if the net rate of return or the wages vary.

As it will now be seen, this general lack of any functional relationship between <math>{E_D \over E_B}</math> and <math>{P_D \over P_B}</math>, of which Ricardo had been particularly well aware, is at the heart of Marx's transformation problem. For Marx, r is the quotient of surplus value to the value of capital advanced to non-labor inputs, and is typically positive in a competitive capitalist economy.