In 20th-century discussions of Karl Marx's economics, the transformation problem is the problem of finding a general rule by which to transform the "values" of commodities (based on their socially necessary labour content, according to his labour theory of value) into the "competitive prices" of the marketplace. This problem was first introduced by Marxist economist Conrad Schmidt and later dealt with by Marx in chapter 9 of the draft of volume 3 of Capital. The essential difficulty was this: given that Marx derived profit, in the form of surplus value, from direct labour inputs, and that the ratio of direct labour input to capital input varied widely between commodities, how could he reconcile this with a tendency toward an average rate of profit on all capital invested among industries, if such a tendency (as predicted by Marx and Ricardo) exists?
Marx's theory
Marx defines value as the number of hours of labor socially necessary to produce a commodity. This includes two elements: First, it includes the hours that a worker of normal skill and dedication would take to produce a commodity under average conditions and with the usual equipment (Marx terms this "living labor"). Second, it includes the labor embodied in raw materials, tools, and machinery used up or worn away during its production (which Marx terms "dead labor" as it is not done by live beings). In capitalism, workers spend a portion of their working day reproducing the value of their means of subsistence, represented as wages (necessary labor), and a portion of their day producing value above and beyond that, referred to as surplus value, which goes to the capitalist (surplus labor). Marx uses this to establish that the value of a commodity is composed of the value of the dead labor, or constant capital , and the value of the necessary labor, or variable capital that was originally paid for prior to the production process, plus the value of the surplus labor, or surplus value generated by living labor but not paid for by the capitalist ().
According to Marx, the source of capitalist profit is this surplus value created by the surplus labor of the workers, and the rate of profit can be defined as , since in this theory only new, living labor produces value, and only surplus labor is not paid for by the capitalist. When this formula is combined with the assumption that the rate of surplus value () tends to equalize across industries, it should be the case that enterprises with a low organic composition (), or a higher proportion of capital spent on living labor, would have a higher rate of profit than enterprises with a high organic composition, or a higher proportion of capital spent on raw materials and means of production. However, in models of classical perfect competition, higher rates of profit are not generally found in enterprises with a low organic composition, and low rates of profit are not generally found in enterprises with a high organic composition. Instead, there is a tendency toward equalization of the rate of profit in industries of different organic compositions. That is, in such models with no barriers to entry, capitalists are free to disinvest or invest in any industry, a tendency exists towards the formation of a general rate of profits, constant across all industries.
Marx outlined the transformation problem as a theoretical solution to this discrepancy. The tendency of the rate of profit toward equalization means that, in this theory, there is no simple translation from value to money—e.g., 1 hour of value equals 20 dollars—that is the same across every sector of the economy. While such a simple translation may hold approximately true in general, Marx postulated that there is an economy-wide, systematic deviation according to the organic compositions of the different industries, such that 1 hour of value equals 20 dollars times T, where T represents a transformation factor that varies according to the organic composition of the industry in consideration.
In this theory, T is approximately 1 in industries where the organic composition is close to average, less than 1 in industries where the organic composition is below average, and greater than 1 in industries where the organic composition is higher than average.
Because Marx was considering only socially necessary labor, this variation among industries has nothing to do with higher-paid, skilled labor versus lower-paid, unskilled labor. This transformation factor varies only with respect to the organic compositions of different industries.
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 and deer are hunted. In the language of modern linear production models, call the unit labour-input requirement for the production of each good , where may be or (i.e., is the number of hours of uniform labour normally required to catch a beaver, and 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 hours) for beavers. The ratio —i.e., the relative quantity of labour embodied in (unit) deer production with respect to beaver production—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 . 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 (where the stands for absolute competitive prices in some arbitrary unit of account, and are defined as ).
Capital costs Things become more complicated if production uses some scarce capital good as well. Suppose that hunting requires also some arrows , with input coefficients equal to , meaning that to catch, for instance, one beaver you need to use arrows, besides hours of labour. Now the unit total cost (or absolute competitive price) of beavers and deer becomes
where 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 , or the competitive price of the arrows, multiplied by the proportion 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 , where is the (uniform) net rate of return of the system.
Summing up, and assuming a uniform replacement rate , the absolute competitive prices of beavers and deer may be written as
Yet we still have to determine the arrows' competitive price . Assuming arrows are produced by labor only, with man-hours per arrow, we have:
Assuming further, for simplicity, that (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:
Here, is the quantity of labor directly embodied in beaver and deer unit production, while is the labor indirectly thus embodied, through previous arrow production. The sum of the two,
,
gives the total quantity of labor embodied.
It is now obvious that the relative competitive price of deer can no longer be generally expressed as the ratio between total amounts of labour embodied. With the ratio will correspond to only in two very special cases: if either ; or, if . In general the two ratios will not only differ: may change for any given , if the net rate of return or the wages vary.
As it will now be seen, this general lack of any functional relationship between and , 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.
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:
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—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. 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. 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. 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, Capital, Volume I, Chapter 1
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 . 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, .