Transformation problem
Encyclopedia
In 20th century discussions of Karl Marx
Karl Marx
Karl Heinrich Marx was a German philosopher, economist, sociologist, historian, journalist, and revolutionary socialist. His ideas played a significant role in the development of social science and the socialist political movement...

's economics
Marxian economics
Marxian economics refers to economic theories on the functioning of capitalism based on the works of Karl Marx. Adherents of Marxian economics, particularly in academia, distinguish it from Marxism as a political ideology and sociological theory, arguing that Marx's approach to understanding the...

 the transformation problem is the problem of finding a general rule to transform the "values" of commodities (based on 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 Capital
Das Kapital
Das Kapital, Kritik der politischen Ökonomie , by Karl Marx, is a critical analysis of capitalism as political economy, meant to reveal the economic laws of the capitalist mode of production, and how it was the precursor of the socialist mode of production.- Themes :In Capital: Critique of...

's
draft Volume III, where he also sketched a solution. The essential difficulty was this: given that Marx derived profit, in the form of surplus value
Surplus value
Surplus value is a concept used famously by Karl Marx in his critique of political economy. Although Marx did not himself invent the term, he developed the concept...

, 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 tendency to an average rate of profit on all capital invested?

Marx's theory

Marx defines value
Exchange value
In political economy and especially Marxian economics, exchange value refers to one of four major attributes of a commodity, i.e., an item or service produced for, and sold on the market...

 as the number of hours of labor contained in 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"). Workers work a portion of their day reproducing the value of their means of subsistence (represented as wages), and a portion of their day producing value above and beyond that, referred to as surplus value
Surplus value
Surplus value is a concept used famously by Karl Marx in his critique of political economy. Although Marx did not himself invent the term, he developed the concept...

.

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 profit, it would appear to be logical that enterprises with a low organic composition
Organic composition of capital
The organic composition of capital is a concept created by Karl Marx in his critique of political economy and used in Marxian economics as a theoretical alternative to neo-classical concepts of factors of production, production functions, capital productivity and capital-output ratios. Marx first...

 (a higher proportion of capital spent on new living labor) would have a higher rate of profit than enterprises with a high organic composition (a higher proportion of capital spent on raw materials and means of production). However, higher rates of profit are not generally found in low-organic composition enterprises or vice versa.[Evidence??] Instead, there is a tendency toward equalization of the rate of profit in industries of different organic compositions.

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 – which 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 in society, such that 1 hour of value equals 20 dollars times T, where T represents a transformation factor which 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, it is less than 1 in industries where the organic composition is lower than average, and greater than 1 in industries where the organic composition is higher than average.

Note: It should be recalled that because Marx is considering only socially necessary, simple labor
Socially necessary labour time
Socially necessary labour time in Marx's critique of political economy is what regulates the exchange value of commodities in trade and consequently guides producers in their attempt to economise on labour....

, this variation from industry to industry has nothing to do with higher-paid skilled, versus lower-paid unskilled labor. This transformation factor varies only with respect to the organic compositions of different industries.

British classical labour theory of value

Marx's value theory was developed from the labour theory of value discussed by Adam Smith
Adam Smith
Adam Smith was a Scottish social philosopher and a pioneer of political economy. One of the key figures of the Scottish Enlightenment, Smith is the author of The Theory of Moral Sentiments and An Inquiry into the Nature and Causes of the Wealth of Nations...

 and used by many British classical economists. It became central to his economics.

Simplest case: labour costs only

Consider the very 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, where 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 either a beaver or a deer; notice that we need to assume labour as uniform in order to be able, later on, to use a uniform wage rate).

Then—Smith noticed—each hunter will be willing to exchange one deer (which costs him hours) for beavers. The ratio —i.e. the relative amount of labour embodied in (unit) deer production with respect to beaver—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 labour costs, that ratio is also the "relative unit cost" of deer for any given competitive uniform wage rate . Hence the relative amount of labour 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 get less simple if production uses some scarce capital good
Capital good
A capital good, or simply capital in economics, is a manufactured means of production. Capital goods are acquired by a society by saving wealth which can be invested in the means of production....

 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 stands for the capital cost incurred in using each arrow.

Now, 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, times the proportion of arrows lost after each shot. Second, there is the net rental or return required by the arrows' owner (who might or might 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 labour only, with man-hours per arrow, we have:


Assuming further for simplicity (all arrows are lost after just one shot, so that they are circulating capital
Circulating capital
Circulating capital refers to physical capital and operating expenses, i.e., short-lived items that are used in production and used up in the process of creating other goods or services. This is roughly equal to Intermediate consumption. It includes raw materials, intermediate goods, inventories,...

), the absolute competitive prices of beavers and deer become:


Here is the amount of labour directly embodied in beaver and deer unit production, while is the labour indirectly thus embodied, through previous arrow production. The sum of the two,


gives the total amount of labour 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 from to —of which Ricardo had been particularly well aware—is at the heart of Marx's transformation problem. Although Marx was aware that at equilibrium with no monopoly r=0. For Marx r was a portion of surplus 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 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
Surplus value
Surplus value is a concept used famously by Karl Marx in his critique of political economy. Although Marx did not himself invent the term, he developed the concept...

 and is for Marx the source of profit for the capitalists. The appropriation of surplus labor is what Marx denoted as exploitation of labour.

Labour as the "value-creating substance"

Marx defined the "value" of commodities as the total amount of socially necessary labour embodied in their production. He developed this special brand of the labour theory of value in the first Chapter of Capitals Volume I. 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 amount of value which labour transmits to the deer in our previous example, varies according to the intensity of the exploitation. In our previous example one has .

By contrast, the value of other inputs—in our example the indirect (or “dead”) past labour embodied in used up arrows—is transmitted to the product as it stands, without additions. It is hence called constant capital
Constant capital
Constant capital , is a concept created by Karl Marx and used in Marxian political economy. It refers to one of the forms of capital invested in production, which contrasts with variable 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 one has .

Value formulas

The total value of each produced good is obtained as the sum of the above three elements: constant capital plus variable capital plus surplus value. In our previous example:


Where stands for the (unit) Marxian value of beavers and deer.

However, from the definition of Marxian value as total labour embodied it must also be true that:


Solving for the above two relationships one has:


for all .

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

Transformation of values into prices

Like Ricardo, Marx knew that relative labour values— in the above example—do not generally tally with relative competitive prices— in the same example. However, in the third volume of Capital he argued that competitive prices were obtained from his values through a transformation process, whereby capitalists redistributed among themselves the given aggregate surplus value of the system, in such a way as to bring about a tendency toward a more equal rate of profit between sectors of the economy. This happened because of the capitalists' tendency to shift their capital towards the sectors where it earned higher returns. As competition become fierce in a given sector, the rate of return comes down, while the opposite will happen in a sector with a low rate of return. Marx describes this process in detail in chapter 9 of Volume III.

Marx's reasoning

The two following tables adapt the deer-beaver-arrow example already 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 and 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 .
! Sector
! Total Constant Capital

! Total Variable Capital

! Total Surplus Value

! Unit Value
> ! Beavers
|
|
|
| > ! Deer
|
|
|
| > ! Total
|
|
|
|>
Table 1—Composition of Marxian values in the deer-beaver-arrow production model


Table 1 shows how the total amount of surplus value of the system—in the last row—is determined.
! Sector
! Total Constant Capital

! Total Variable Capital

! Redistributed Total
Surplus Value

! Resulting
Competitive
Price
> ! Beavers
|
|
|
| > ! Deer
|
|
|
| > ! Total
|
|
|
|>
Table 2—Marx's transformation formulas in the deer-beaver-arrow production model


Table 2 then illustrates how Marx thought that 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 last 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 in Chapter 9 of Capital's Volume III.

Marx's error and its correction

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

First, competitive equilibrium
Competitive equilibrium
Competitive market equilibrium is the traditional concept of economic equilibrium, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis...

 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. Secondly, 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 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 by assumption the wages rate is given and equal to the price of beavers, the most convenient way is to express such prices in units of beavers, which means normalising . This immediately yields the (relative) price of arrows as
beavers.

Substituting this into the relative-price condition for beavers


gives the solution for the rate of return as


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


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
Ernest Mandel
Ernest Ezra Mandel, also known by various pseudonyms such as Ernest Germain, Pierre Gousset, Henri Vallin, Walter , was a revolutionary Marxist theorist.-Life:...

, 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 which reflect redistributed surplus value. http://www.marxists.org/archive/mandel/19xx/marx/ch04.htm

Mathematical vs historical transformation

Frederick Engels – the editor of Capitals Volume III – hinted since 1894 at a possibly alternative way to look at the whole matter. His view was that the pure Marxian "law of value" of Volume I and the "transformed" prices of Volume III applied to different periods of economic history. In particular, the "law of value" would have prevailed in the pre-capitalist exchange economies – from Babylon to the fifteenth century of the common era – while the "transformed" prices would have materialized under capitalism: see Engels' quotation by Morishima and Catephores (1975) p. 310.

Engels' reasoning was later taken up by Meek (1956) and Nell (1973). These authors argued that – whatever one must say of his interpretation of capitalism – Marx's "value" theory keeps its usefulness as a tool to interpret pre-capitalist societies, because – they maintained – in pre-capitalist exchange economies there were no "prices of production" with a uniform rate of return (or "profit") on capital. It hence follows that Marx's transformation must have had a historical dimension, given by the actual transition to capitalist production (an no more Marxian "values") at the beginning of the modern era. Then this true "historical transformation" could and should take the place of the mathematical transformation postulated by Marx in Chapter 9 of Volume III.

As Althusser and Balibar (1970) and others have pointed out, this proposal runs counter to Marx's own ideas, as expressed in Marx (1859), where one reads:
"The point at issue is not the role that various economic relations have played in the succession of various social formations appearing in the course of history […], but their position within modern society." (En. trans. p.210, italics in the original.)


Moreover, the available quantitative research by economic historians has not generally endorsed Engels' view of antiquity and the Middle Ages as a "value epoch" in the Marxian sense: see for instance Hicks (1969) and Godelier (1973).

Other Marxist views

There are several schools of thought among those who see themselves as upholding or furthering Marx on the question of transformation from values to prices, or modifying his theory in ways to make it more consistent.

According to the Temporal single-system interpretation
Temporal single-system interpretation
The temporal single-system interpretation of Karl Marx's value theory emerged in the early 1980s in response to renewed allegations that his theory was "riven with internal inconsistencies," and that it must therefore be rejected or corrected. The inconsistency allegations had been a prominent...

 of "Capital" advanced by Alan Freeman, Andrew Kliman and others, Marx's writings on the subject can be interpreted in such a way as to remove any supposed inconsistencies (Choonara, 2007). Modern traditional Marxists argue that not only does the Labour Theory of Value hold up today, but that Marx's understanding of the Transformation Problem was in the main also correct .

Political-economic readings of Capital, such as Harry Cleaver
Harry Cleaver
Harry Cleaver is Associate Professor in the Department of Economics at the University of Texas at Austin, where Cleaver teaches Marxism and Marxist economics. He is best known as the author of Reading Capital Politically, an autonomist reading of Karl Marx's Capital. Dr...

's Reading Capital Politically re-define exploitation as a direct control of worked time, unrelated as such to distribution. These readings are usually associated with the autonomist strand of Marxism, which focuses on production as the key economic site within society. These readings of Capital are typically hostile to economics as such, and consider the transformation problem unimportant because they see all social arrangements in capitalism (in particular, profit and distribution) as politically determined contests between classes.

The probabilistic interpretation of Marx advanced by Emmanuel Farjoun and Moshe Machover in Laws of Chaos (see references). They "dissolve" the transformation problem by reconceptualising the relevant quantities as random variables. In particular, they consider profit rates to reach an equilibrium distribution. A heuristic analogy with the statistical mechanics of an ideal gas leads them to the hypothesis that this equilibrium distribution should be a gamma distribution.

Finally, there are Marxist scholars (e.g., Anwar Shaikh
Anwar Shaikh (economist)
Anwar M. Shaikh is an American economist, and currently Professor of Economics at the Graduate Faculty of The New School in New York City. His work in political economy has focused on the economic theory and empirical patterns of developed capitalism. He has written on international trade,...

, Fred Moseley, Alan Freeman, Makoto Itoh
Makoto Itoh
is a Japanese economist and is considered internationally to be one of the most important scholars of Marx's theory of value. He teaches at Kokugakuin University, Tokyo, and is professor emeritus of the University of Tokyo....

, Gerard Dumenil & Dominique Levy, Duncan Foley) who hold that there exists no incontestable logical procedure by which to derive price magnitudes from value magnitudes, but still think that it has no lethal consequences on his system as a whole. In a few very special cases, Marx's idea of labour as the "substance" of (exchangeable) value would not be openly at odds with the facts of market competitive equilibrium. These authors have argued that such cases—though not generally observed—throw light on the "hidden" or "pure" nature of capitalist society. Thus Marx's related notions of surplus value and unpaid labour can still be treated as basically true, although they hold that the practical details of their workings are more complicated than Marx thought.

In particular, some (e.g. Anwar Shaikh
Anwar Shaikh (economist)
Anwar M. Shaikh is an American economist, and currently Professor of Economics at the Graduate Faculty of The New School in New York City. His work in political economy has focused on the economic theory and empirical patterns of developed capitalism. He has written on international trade,...

) have suggested that—since aggregate surplus value will generally differ from aggregate "profit"—the former should be in fact treated as a mere pre-condition for the latter, rather than a full explanation of it. Using input-output data and empirical proxies for labour-values, Anwar Shaikh
Anwar Shaikh (economist)
Anwar M. Shaikh is an American economist, and currently Professor of Economics at the Graduate Faculty of The New School in New York City. His work in political economy has focused on the economic theory and empirical patterns of developed capitalism. He has written on international trade,...

 & Ochoa have provided some statistical evidence to show that, although there may be no incontestable logical deduction possible of specific price magnitudes from specific value magnitudes even within a complex model (in contrast to a probabilistic prediction), even a "93% Ricardian theory" of labour-value appears to be a better empirical predictor of price than its rivals.

Critics of the theory

Many mathematical economists assert that a set of functions in which Marx's equalities hold does not generally exist at the individual enterprise or aggregate level, so that Chapter 9's transformation problem has no general solution, outside two classes of very special cases. This was first pointed out by, among others, Böhm-Bawerk
Eugen von Böhm-Bawerk
Eugen Ritter von Böhm-Bawerk was an Austrian economist who made important contributions to the development of the Austrian School of economics.-Biography:...

 (1896) and Bortkiewicz
Ladislaus Bortkiewicz
Ladislaus Josephovich Bortkiewicz , August 7, 1868 – July 15, 1931) was a Russian economist and statistician of Polish descent, who lived most of his professional life in Germany, where he taught at Strassburg University and Berlin University...

 (1906). In the second half of the twentieth century, Leontief
Wassily Leontief
Wassily Wassilyovich Leontief , was a Russian-American economist notable for his research on how changes in one economic sector may have an effect on other sectors. Leontief won the Nobel Committee's Nobel Memorial Prize in Economic Sciences in 1973, and three of his doctoral students have also...

’s and Sraffa
Piero Sraffa
Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...

’s work on linear production models provided a framework within which to prove this result in a simple and general way.

Although he never actually mentioned the transformation problem, Sraffa’s (1960) Chapter VI on the "reduction" of prices to "dated" amounts of current and past embodied labour gave implicitly the first general proof, showing that the competitive price of the produced good can be expressed as
,

where is the time lag, is the lagged-labour input coefficient, is the wage and is the "profit" (or net return) rate. Since total embodied labour is defined as
,

it follows from Sraffa’s result that there is generally no function from to , as was made explicit and elaborated upon by later writers, notably Ian Steedman
Neo-Ricardianism
The neo-Ricardian school is an economic schoolthat derives from the close reading and interpretation of David Ricardo by Piero Sraffa, and from Sraffa's critique of Neoclassical economics as presented in his The Production of Commodities by Means of Commodities, and further developed by the...

 in Marx after Sraffa.

A standard reference – with an extensive survey of the entire literature pre-1971 and a comprehensive bibliography – is Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

's (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices" Journal of Economic Literature 9 2 399–431.

Since the 1970s several major schools of Marxian economics
Marxian economics
Marxian economics refers to economic theories on the functioning of capitalism based on the works of Karl Marx. Adherents of Marxian economics, particularly in academia, distinguish it from Marxism as a political ideology and sociological theory, arguing that Marx's approach to understanding the...

 have arisen in response to the transformation problem-related challenges of the neoclassical and Sraffian schools. Analytical Marxists held that the transformation problem disproved the Labour Theory of Value and based their Marxian social theory on a combination of the Fundamental Marxian theorem, game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, and other neoclassical and mathematical tools. Empirical Marxists, including Anwar Shaikh
Anwar Shaikh (economist)
Anwar M. Shaikh is an American economist, and currently Professor of Economics at the Graduate Faculty of The New School in New York City. His work in political economy has focused on the economic theory and empirical patterns of developed capitalism. He has written on international trade,...

, Moshe Machover
Moshé Machover
Moshé Machover is a mathematician, philosopher, and socialist activist, noted for his writings against Zionism. Born to a Jewish family in Tel Aviv, then part of the British Mandate of Palestine, Machover moved to Britain in 1968 where he became a naturalised citizen...

, and Paul Cockshott, maintain that since empirical data bears out the correspondence of prices and labour values, the transformation problem is irrelevant. Followers of the Temporal Single System Interpretation and the New Interpretation argue that critics have misunderstood Marx's definition of value and that, correctly defined, there is no difference between value and price.

The lack of any function to transform Marx's "values" to competitive prices has important implications for Marx's theory of labour exploitation
Exploitation
This article discusses the term exploitation in the meaning of using something in an unjust or cruel manner.- As unjust benefit :In political economy, economics, and sociology, exploitation involves a persistent social relationship in which certain persons are being mistreated or unfairly used for...

 and economic dynamics
Capital accumulation
The accumulation of capital refers to the gathering or amassing of objects of value; the increase in wealth through concentration; or the creation of wealth. Capital is money or a financial asset invested for the purpose of making more money...

, namely that there is no Tendency of the rate of profit to fall
Tendency of the rate of profit to fall
The tendency of the rate of profit to fall is a hypothesis in economics and political economy, most famously expounded by Karl Marx in chapter 13 of Das Kapital Vol. 3. It was generally accepted in the 19th century...

. This means that it is not pre-ordained that capitalists must exploit labour to compensate for a declining rate of profit. This implies that Marx's prophecy that worsening labour exploitation would result in an eventual revolution against the capitalist system and the establishment of communism
Communism
Communism is a social, political and economic ideology that aims at the establishment of a classless, moneyless, revolutionary and stateless socialist society structured upon common ownership of the means of production...

 is logically and mathematically false.

The mathematical proof that Marx's transformation problem has no general solution has been formally questioned http://www.mtholyoke.edu/~fmoseley/CRITIQUE.pdf by proponents of the Temporal Single System Interpretation, who argue that the determination of prices by simultaneous linear equations (which assumes that prices are the same at the start and end of the production period) is logically inconsistent with the determination of value by labour time. Other Marxian economists accept the proof, but reject its relevance for some key elements of Marxian political economy. Still others reject Marxian economics outright, and emphasise the politics of the assumed relations of production
Relations of production
Relations of production is a concept frequently used by Karl Marx and Friedrich Engels in their theory of historical materialism, and in Das Kapital...

 instead. To this extent, the transformation problem—or rather its implications—is still today a controversial question.

Non-Marxian critiques

Mainstream scholars such as Samuelson question the assumption that the basic nature of capitalist production and distribution can be gleaned from unrealistic special cases. For example, in special cases where it apply, Marx's reasoning can be easily turned upside down, through an inverse transformation process, Samuelson argue that Marx's inference:


"Profit is therefore the [bourgeois] disguise of surplus value which must be removed before the real nature of surplus value can be discovered." (Capital Volume III, Chapter 2)


could with equal cogency be “transformed” into:


"Surplus value is therefore the [Marxist] disguise of profit which must be removed before the real nature of profit can be discovered." [Samuelson (1971) p.417]


To further clarify this point, it may be noticed that the special cases in question are also precisely those where J.B. Clark
John Bates Clark
John Bates Clark was an American neoclassical economist. He was one of the pioneers of the marginalist revolution and opponent to the Institutionalist school of economics, and spent most of his career teaching at Columbia University.-Biography:Clark was born and raised in Providence, Rhode...

’s old model of aggregate marginal productivity holds strictly true, leading to equality between the equilibrium levels of the real wage rate and labour's aggregate marginal product, a hypothesis regarded as disproved by all sides during the Cambridge capital controversy
Cambridge capital controversy
The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists concerning the nature and role of capital goods and the critique of the dominant neoclassical vision of aggregate...

. One would thus have a "pure" state of capitalist society where Marx's exploitation theory
Exploitation theory
The exploitation theory is the theory, most associated with Marxists, that profit is the result of the exploitation of wage earners by their employers....

 and its main supposed confutation were both true.

This remarkable result, it is maintained, leads straight to the heart of the matter. Like Clark’s contention about the "fairness" of marginal-productivity wages, so Marx's basic argument—from the "substance" of value to the concept of exploitation—is claimed to be a set of non-analytical and non-empirical propositions. That is why, being non-falsifiable, both theories may be found to apply to the same formal and/or empirical object, though they are supposed to negate each other, as Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...

and many others had argued.

Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...

 not only dismissed the labour theory of value because of the transformation problem, but provided himself, in cooperation with economists like Carl Christian von Weizsäcker, solutions. Carl Christian von Weizsäcker 1962, and, together with Paul Samuelson, 1971 analysed the problem under the assumption that the economy grows at a constant rate following the Golden Rule of Accumulation
Golden Rule savings rate
In economics, the Golden Rule savings rate is the rate of savings which maximizes steady state level or growth of consumption , as for example in the Solow growth model...

. Weizsäcker concludes:


"The price of the commodity today is equal to the sum of the 'present' values of the different labour inputs,..." [Weizsäcker (2010 [1962]) p.262]


The "rate of interest" necessary to compute the "present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

s" of the labour inputs is the rate of growth of the economy equal to the rate of profit.

Marxian reply to non-Marxian critiques

The Marxian reply to this mainstream view is as follows:
  • It is important to understand that the attempt to discard the theoretical relevance of the necessary preconditions of Marx's value analysis in volume 1 of Capital through a reductio ad absurdum
    Reductio ad absurdum
    In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

     is superficial. By first identifying that the preconditions which are necessary for J.B. Clark
    John Bates Clark
    John Bates Clark was an American neoclassical economist. He was one of the pioneers of the marginalist revolution and opponent to the Institutionalist school of economics, and spent most of his career teaching at Columbia University.-Biography:Clark was born and raised in Providence, Rhode...

    ’s old model of aggregate marginal productivity to hold true are the same as those which are necessary for Marxian values to conform to relative prices we are then, on a leap of faith which is passed as logic, supposedly to conclude that the foundation of Marx's analysis as based in these preconditions is faulty because Clark's model had been proven wrong in the Cambridge capital controversy
    Cambridge capital controversy
    The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists concerning the nature and role of capital goods and the critique of the dominant neoclassical vision of aggregate...

    . The superficiality stems from the fact that those who support this reduction forget that the Cambridge capital controversy
    Cambridge capital controversy
    The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists concerning the nature and role of capital goods and the critique of the dominant neoclassical vision of aggregate...

     put the entire concept of marginal productivity into question by attacking not the special case assumptions of Clark but the notion that physical capital can be aggregated. Marx simply does not run into this problem because his analysis does not rely on an aggregation of physical quantities which receive a return based on their contribution as 'factors' of production. The fact that marginal productivity in its aggregate form is 'a hypothesis regarded as disproved by all sides during the Cambridge capital controversy
    Cambridge capital controversy
    The Cambridge capital controversy – sometimes simply called "the capital controversy" – refers to a theoretical and mathematical debate during the 1960s among economists concerning the nature and role of capital goods and the critique of the dominant neoclassical vision of aggregate...

    ' simply has nothing to do with the validity of the special cases of Marx, and thus we WOULD NOT "have a "pure" state of capitalist society where Marx's exploitation theory
    Exploitation theory
    The exploitation theory is the theory, most associated with Marxists, that profit is the result of the exploitation of wage earners by their employers....

     and its main supposed confutation (Clark) were both true", as is concluded from this view, because the 'correctness' or 'incorrectness' of Clark's aggregate marginal productivity scheme in this case flows not from special case assumptions but from the fact that he is aggregating physical units of capital - i.e., Clark would still not hold true even with the assumed special cases. Once again, the bourgeois theorists manage to impress us their erudition while completely sidestepping the substance of the debate.


To further clarify this point, it may be noticed that
  • it is firstly argued that it is never possible to provide any absolute scientific proof for the truth of any particular concept of economic value in economics, because the attribution of economic value itself always involves human and moral interpretations which go beyond facts and logic. By nature, the concept of economic value is not itself a scientifically provable concept but an assumption. Marx himself explicitly ridiculed the idea that he should be required to "prove his concept of value".

  • Secondly, however, the validity of any proposed theory of value depends on its explanatory, heuristic
    Heuristic
    Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

     and predictive power, i.e. whether it makes possible a coherent interpretation of the known facts that can at least to some extent predict observable trends. In this sense, Marx evidently felt that he had "proved" the validity of his concept of value by the integrated theory of capitalist development which it made possible (see also law of value
    Law of value
    -General:The law of value is a central concept in Karl Marx's critique of political economy, first expounded in his polemic The Poverty of Philosophy against Pierre-Joseph Proudhon, with reference to David Ricardo's economics...

    ). What mattered was the application of the concept.

  • Thirdly, once a certain concept of economic value is assumed, certain predictions or explanations can be made on the basis of it, and those explanations or predictions can at least in principle be falsified by reference to logic and observables. And that concept of value can be compared with rival concepts and the rival theories they make possible, in order to establish which has more explanatory or predictive capacity.

  • Fourthly, modern philosophy of science
    Philosophy of science
    The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...

     rejects Popper's falsification theory as an adequate portrayal of science. Scientific statements are not necessarily falsifiable statements but instead fallible statements (they could be wrong) which, in principle, can be tested against observables, even if contingently we do not yet know technically how to do this. Scientists do not aim mainly to falsify theories, but confirm them to provide usable knowledge.

  • Finally, as Piero Sraffa
    Piero Sraffa
    Piero Sraffa was an influential Italian economist whose book Production of Commodities by Means of Commodities is taken as founding the Neo-Ricardian school of Economics.- Early life :...

     showed clearly, the theory of the production and distribution of a surplus, however it might be devised, is logically independent of any particular theory of labour-exploitation. Labour-exploitation may occur and be conceptualised in various ways, regardless of which theory of value is held to be true. Consequently, if Marx's theory of labour-exploitation is false, this is a quite separate issue.

See also

  • labour theory of value
  • law of value
    Law of value
    -General:The law of value is a central concept in Karl Marx's critique of political economy, first expounded in his polemic The Poverty of Philosophy against Pierre-Joseph Proudhon, with reference to David Ricardo's economics...

  • surplus value
    Surplus value
    Surplus value is a concept used famously by Karl Marx in his critique of political economy. Although Marx did not himself invent the term, he developed the concept...

  • socially necessary labour time
    Socially necessary labour time
    Socially necessary labour time in Marx's critique of political economy is what regulates the exchange value of commodities in trade and consequently guides producers in their attempt to economise on labour....

  • prices of production
    Prices of production
    Prices of production refers to a concept in Karl Marx's critique of political economy. It is introduced in the third volume of Das Kapital, where Marx considers the operation of capitalist production as the unity of a production process and a circulation process involving commodities, money and...

  • capital accumulation
    Capital accumulation
    The accumulation of capital refers to the gathering or amassing of objects of value; the increase in wealth through concentration; or the creation of wealth. Capital is money or a financial asset invested for the purpose of making more money...

  • return of capital
    Return of capital
    Return of capital refers to payments back to "capital owners" that exceed the growth of a business. It should not be confused with return on capital which measures a 'rate of return'....

  • Temporal Single System Interpretation
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