Transformation Problem

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 Marx in chapter 9 of the draft of volume 3 of ''Capital'', where he also sketched a solution. 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"). 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).

Since, according to Marx, the source of capitalist profit is this surplus labor of the workers, and since in this theory only new, living labor produces value, it would appear logical that enterprises with a low organic composition (a higher proportion of capital spent on living labor) would have a higher rate of profit than would enterprises with a high organic composition (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 profit rates 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 $(B)$ and deer $(D)$ are hunted. In the language of modern linear production models, call the unit labour-input requirement for the production of each good $l_i$, where $i$ may be $B$ or $D$ (i.e., $l_B$ is the number of hours of uniform labour normally required to catch a beaver, and $l_D$ 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 $l_D$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 $w$. 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 $P$ stands for absolute competitive prices in some arbitrary unit of account, and are defined as $P_i = wl_i$).

Capital costs

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

:$P_i = wl_i + k_A a_i , (i = B, D)$

where $k_A$ 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 $P_A$, or the competitive price of the arrows, multiplied by the proportion $h \le 1$ 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 $r P_A$, where $r$ is the (uniform) ''net rate of return'' of the system.

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

:$P_i = wl_i + (h + r) P_A a_i$

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

:$P_A = wl_A$

Assuming further, for simplicity, that $h = 1$ (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:

:$P_i = wl_i + (1 + r) wl_A a_i$

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

:$E_i = l_i + l_A a_i$,

gives the total quantity of labor embodied.

It is now obvious that the relative competitive price of deer $$ can no longer be generally expressed as the ratio between total amounts of labour embodied. With $a_i > 0$ the ratio $$ will correspond to $$ only in two very special cases: if either $r = 0$; or, if $=$. In general the two ratios will not only differ: $$ 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
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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 $v$. 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, $v_i = l_W l_i$.

By contrast, the value of other inputs—in our example, the indirect (or "dead") past labour embodied in the used-up arrows—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, $c_i = l_A a_i$.

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:

:$p_i = c_i + v_i + s_i = l_A a_i + l_W l_i + s_i$

Where $p_i$ 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:

:$p_i = l_A a_i + l_i = E_i$

Solving for $s_i$ the above two relationships one has:

:$= = \sigma$

for all $i$.

This necessarily uniform ratio $= \sigma$ is called by Marx the rate of exploitation, and it allows to re-write Marx's value equations as:

:$p_i = c_i + v_i (1 + \sigma) = l_A a_i + l_W l_i (1 + \sigma)$

Classical tableaux

Like Ricardo, Marx believed that ''relative'' labour values— $$ in the above example—do not generally correspond to relative competitive prices— $$ in the same example. However, in volume 3 of ''Capital'' he argued that competitive prices are obtained from values through a transformation'' process, whereby capitalists ''redistribute'' among themselves the given ''aggregate'' surplus value of the system in such a way as to bring about a tendency toward an equal rate of profit, $r$, among sectors of the economy. This happens because of the capitalists' tendency to shift their capital toward sectors where it earns higher returns. As competition becomes fierce in a given sector, the rate of return falls, while the opposite will happen in a sector with a low rate of return. Marx describes this process in detail.

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 $Q_B$ and $Q_D$ 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 $l_W = E_B = l_A a_B + l_B < 1$. Table 1 shows how the total amount of surplus value of the system, shown in the last row, is determined.

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.

Tables 1 and 2 parallel the tables in which Marx elaborated his numerical example.

Marx's supposed error and its correction

Later scholars argued that Marx's formulas for competitive prices were mistaken.

First, competitive equilibrium requires a uniform rate of return over constant capital valued at its ''price'', not its Marxian value, contrary to what is done in table 2 above. Second, competitive prices result from the sum of costs valued at the ''prices'' of things, not as amounts of embodied labour. Thus, both Marx's calculation of $r$ and the sums of his price formulas do not add up in all the normal cases, where, as in the above example, relative competitive prices differ from relative Marxian values. Marx noted this but thought that it was not significant, stating in chapter 9 of volume 3 of ''Capital'' that "Our present analysis does not necessitate a closer examination of this point."

The simultaneous linear equations method of computing competitive (relative) prices in an equilibrium economy is today very well known. In the greatly simplified model of tables 1 and 2, where the wage rate is assumed as given and equal to the price of beavers, the most convenient way is to express such prices is in units of beavers, which means normalising $w = P_B = 1$. This yields the (relative) price of arrows as

:$P_A = l_A$ beavers.

Substituting this into the relative-price condition for beavers,

:$1 = l_B + (1 + r) l_A a_B$,

gives the solution for the rate of return as

:$r = - 1$

Finally, the price condition for deer can hence be written as

:$P_D = l_D + (1 + r) l_A a_D = l_D +$.

This latter result, which gives the correct competitive price of deer in units of beavers for the simple model used here, is generally inconsistent with Marx's price formulae of table 2.

Ernest Mandel, defending Marx, explains this discrepancy in term of the time frame of production rather than as a logical error; i.e., in this simplified model, capital goods are purchased at a labour value price, but final products are sold under prices that reflect redistributed surplus value.

After Marx

Engels

Friedrich Engels