Editing Transformation problem (section)

=== 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>).