The Marxian Theory of Profits
As explained in chapter 1, Marx believed he had discovered the answer to the mystery of where profits come from even when capitalists must pay for all inputs according to their labor values and sell their outputs according to their labor values. Marx argued that the answer lies in one special commodity capitalists buy, labor power, which has the unique ability to produce more value when used than the number of hours it takes to produce it, and therefore capitalists have to pay for it. Michio Morishima (1974) provided a formal representation of this explanation of the origin of profits in what became known as the “fundamental Marxian theorem” (FMT).
It is important to understand how the rate of exploitation is defined in Marxian theory. Marx measured labor time in days, but it can also be measured in hours, as is now commonly done by Marxists who have formalized the labor theory of value using linear algebra, and as we have done throughout this book. Using hours as our unit of labor time, Marxists define the “exchange value” of labor power as the number of hours it takes to produce a worker’s hourly wage bundle, i.e. the number of hours it takes to “produce” a worker ready and able to work for an hour. So if the real hourly wage bundle consists of b(1) units of good 1 and b(2) units of good 2, and if the labor value of good 1 is V(1) and the labor value of good 2 is V(2), then the number of hours it takes to produce a worker’s real hourly wage bundle, and the “exchange value” of labor power, is V(1)b(1) + V(2)b(2). In modern Marxist theory this is represented as Vb where V is a row vector of labor values and b is a column vector of commodities in workers’ real hourly wage bundle. And if all commodities sell according to their exchange values, as Marx assumed at this stage of his analysis, this is what employers must pay for an hour of labor power. Marx then observed that the “use value” of labor power to its new owner, the employer, is the number of hours it incorporates into, or “embodies,” in the goods produced in an hour, which will determine what the capitalist can sell them for. In sum, the use value of an hour of labor power is 1 hour, while the exchange value of an hour of labor power is the number of hours it takes to produce the worker’s wage bundle, Vb. The difference between the “use value” of labor power and the “exchange value” of labor power is what Marx called “surplus value,” which in modern Marxist notation is (1−Vb). Finally, Marx defined the “rate of exploitation” as the ratio of surplus value to exchange value of labor power, i.e. the fraction of each hour the worker is, in effect, working for her employer (her unpaid labor), divided by the fraction of each hour she is, in effect, working for herself (her paid labor.) Or, in modern Marxist notation the rate of exploitation is (1−Vb)/Vb. What Morishima proved in 1974 is that if and only if (1−Vb)/Vb > 0 will r > 0, i.e. that if and only if what Marx defined as his rate of exploitation is positive will the rate of profit in the economy be positive.1 Morishima’s FMT is commonly verbalized as: If and only if the rate of exploitation of labor is positive will the rate of profit be positive, i.e. profits come from exploiting labor.
There is no problem with Morishima’s proof of the FMT. However, there are four problems with the Marxian explanation of profits as deriving from the exploitation of one input to production, labor power. (1) As in the case of prices, labor values are not necessary to explain profits, i.e. they are redundant. (2) The choice of labor power as the input capitalists “exploit” is arbitrary since any other input can be used to tell the same story. (3) The implication that only one input is an “exploitable” source of profits is misleading because in fact capitalists mark-up on the cost of every input they buy. And finally, (4) belief that profits derive only from exploiting labor can mislead one to think that automation will depress profits, which, as we will see next chapter, it does not. After presenting the Sraffian theory of profits we discuss the first three problems with the Marxian theory in this chapter, leaving the last problem to chapter 3 where we discuss technical change.
The Sraffian Theory of Profits
The Sraffian theory of profits starts with what it means for the economy to be “productive” and its ability to produce a physical economic “surplus.” Consider our example of a two sector economy:
a(11) = .3 a(12) = .2
a(21) = .2 a(22) = .4
L(1) = 1.0 L(2) = .5
Notice what happens if we produce one unit of each good. We will use up 0.3 units of good 1 making 1 unit of good 1, and 0.2 units of good 1 making 1 unit of good 2. So in total (adding across the first row) we will use up 0.5 units of good 1, and therefore have 1−0.5 = 0.5 units of good 1 left over as “net output,” or surplus, when we produce 1 unit of good 1 and 1 unit of good 2 as “gross output.” We will also use up 0.2 units of good 2 making 1 unit of good 1, and 0.4 units of good 2 making 1 unit of good 2. So in total (adding across the second row) we will use up 0.6 units of good 2, and therefore have 1−0.6 = 0.4 units of good 2 left over as “net output,” or surplus, when we produce 1 unit of good 1 and 1 unit of good 2 as “gross output.” We describe this by saying that our technology is such that the economy is “productive” or capable of producing a physical “surplus.” Indeed, Sraffian theory is sometimes referred to as “the surplus approach.”
It is clear from our example that if the economy were not productive, if it were incapable of producing a surplus of any good after replacement of all inputs used up, there could be no profits and the rate of profit would necessarily be zero.2 However, in our case the economy is productive and capable of producing a physical surplus of goods. Does this mean the rate of profit will be positive? That depends on how much of the “surplus” goes to workers as wages, and therefore how much is left over for capitalists as profits.
In our example, to produce 1 unit of good 1 requires 1 hour of labor, and to produce 1 unit of good 2 requires 0.5 hours of labor. So when we apply 1.5 hours of labor to our economy (summing across the third row) to produce 1 unit of gross output of each good we get a surplus of 0.5 units of good 1 and 0.4 units of good 2. If the real wage per hour consists of 0.3333 units of good 1 and 0.266 units of good 2, the entire surplus of goods produced will go to the workers who produced it. This is because 0.333 units of good 1 per hour times 1.5 hours = 0.5 units of good 1, the entire surplus of good 1, and 0.266 units of good 2 per hour times 1.5 hours = 0.4 units of good 2, the entire surplus of good 2. If the real wage per hour is anything less than this, some of the physical surplus of goods 1 and 2 will go to capitalists and profits will be positive. This result is generalized and proved as theorem 11 in Hahnel 2017. To contrast it with the fundamental Marxian theorem, FMT, I have named it the fundamental Sraffian theorem, FST, which says: If and only if there is a physical surplus of goods after wages have been paid will profits be positive.4 In other words, if and only if those who produce goods are deprived of some of the surplus goods they produce can capitalists have positive profits.
There is another way to see this which also sheds light on the relationship between the wage rate and the rate of profit in the economy. In chapter 1 we calculated that when the wage is equal to 0.691 units of good 2 per hour, the rate of profit in the economy will be zero. This is because if we apply one hour of labor to the economy in such a way that we produce no surplus of good 1 – that is we produce only enough good 1 to replace all we use up – and we produce all of the economic surplus as a physical surplus of good 2, the surplus will be precisely 0.691 units of good 2.5 Therefore, if the wage per hour is 0.691 units of good 2, there is no surplus left over for profits, and the rate of profit will be zero. On the other hand, if w falls to 0.500 units of good 2 per hour, then 0.691−0.500 = 0.191 units of good 2 will be left over for profits, which is sufficient to yield a rate of profit of 12.6%. And if w falls to 0.400, then 0.691−0.400 = 0.291 units of good 2 will be left over for profits, which is sufficient to yield a profit rate of 20.8%.
Clearly the fact that the rate of profit rises when the wage rate falls in our example is no accident. The negative relation between the hourly real wage rate and the rate of profit is a well-known result in Sraffian economics, frequently represented graphically as the “wage-profit frontier.”6
The Sraffa model makes clear that it is the economy that is productive. Or at least it is the entire economy as a whole that is potentially productive, i.e. capable of producing a surplus of physical goods even after all produced inputs are replaced. In fact we might say the framework makes this not just clear, but “crystal clear” to use a phrase made famous by the movie A Few Good Men. In other words, the productivity of the economy is a characteristic of the known technologies for producing all goods and services. Of course the economy is only potentially productive, and produces nothing until real people actually go to work and turn this potential into an actual surplus of useful goods and services. So a more careful and accurate wording would be that while it is the economy as a whole that is potentially productive at any point in time, the physical surplus of goods is produced by those who work, and thereby realize the economy and their own potentials. Edward Bellamy explained it this way in 1897:
The main factor in the production of wealth among civilized men is the social organism, the machinery of associated labor and exchange by which hundreds of millions of individuals provide the demand for one another’s product and mutually complement one another’s labors…. The element in the total industrial product which is due to the social organism is represented by the difference between the value of what one man produces as a worker in connection with the social organization and what he could produce in a condition of isolation…. It is estimated, I believe, that the average daily product of a worker in America today is some fifty dollars. The product of the same man working in isolation would probably be highly estimated on the same basis of calculation if put at a quarter of a dollar…. Now tell me… to whom belongs the social organism, this vast machinery of human association, which enhances some two hundredfold the product of every one’s labor?
(Bellamy 1970: 88)
FIGURE 2.1 The wage-profit frontier
Moreover, the productivity of Bellamy’s “social organism” has continually increased. Based on research by Richard Sutch, William Nordhaus, Angus Maddison, William Baumol, Nathan Rosenberg, Moses Abramovitz, and others, Gar Alperovitz and Lew Daly summarize the consensus among economic historians: “A person working today the same number of hours as a similar person in 1800 – and working just as hard and no harder – can produce many, many times the economic output. Recent estimates suggest that national output per capita has increased more than twenty fold since 1800. Output per hour worked has increased an estimated fifteen fold since 1870 alone” (Alperovitz and Daly 2008: 3). But if individuals do not really improve, i.e. if individual intelligence and effort change little over time, where does all this increase in productivity come from?
Robert Solow opened economists’ eyes to how little mainstream growth models explain when he estimated that growth in the supply of capital goods and labor explained perhaps as little as 10%, and at most 20% of the growth in US output in the first half of the twentieth century, leaving a “residual” of as much as 80 to 90% – which Solow observed could only be explained by “technical change in the broadest sense.” Those who have tried ever since to pin down exactly what is responsible for such a large residual have discovered the extraordinary role knowledge plays in generating economic growth. When Paul Romer searched for an answer to the puzzle that a college educated engineer today is far, far more productive than one working 100 years ago, despite the fact that they each have the same human capital, he concluded that the reason was obvious: “He or she can take advantage of all the additional knowledge accumulated as design problems that were solved during the last 100 years” (Romer 1990: 83–84).
More broadly, the efficiency of storing and retrieving the scientific, technical, and cultural cumulative knowledge available to each of us has created what Douglas North called the “scaffolding” of economic growth. Moreover, the popular image of this creative process as a sequence of “great inventions” by “extraordinary heroes” appears to be largely a myth. Instead, the process of technological change is most often far more collective, collaborative, and cumulative – diverse contributions slowly build up over time until a breakthrough becomes all but inevitable.
While the Sraffian framework represents it simply as the existing technologies, or recipes available to produce each and every good, {A, L}, it is, of course, much more complicated. {A, L} represent not only the known recipes for making goods and services but also the knowledge and skills necessary to use them, the elaborate divisions of labor they require, and all of the institutions, both formal and informal, necessary for maintaining and coordinating this elaborate division of labor. Moreover, all of this was worked out by countless people, going back over countless years.
Economic historian Joel Mokyr refers to all this productive knowledge as a “gift from Athena,” explaining that “technological progress… has provided society… an increase in output that is not commensurate with the increase in effort… necessary to bring it about” (Mokyr 1992: 3). A character in Bellamy’s famous utopian novel Looking Backward explained it in simple terms to a time traveler from a capitalist past:
How happened it that your workers were able to produce more than so many savages would have done? Was it not wholly on account of the heritage of the past knowledge and the achievements of the race, the machinery of society, thousands of years in contriving, found by you ready-made to your hand? How did you come to be possessors of this knowledge and this machinery, which represent nine parts to one contributed by yourself to the value of your product? You inherited it, did you not?
(Bellamy 1960: 100)
In any case, whatever we call it, the important point is that what allows us to be as productive as we are is something that each generation inherits collectively from all who went before us – irrespective of whether or not some among us appropriate parts of our common inheritance and extract tribute from others before we are permitted to use it.
In case it is not already “crystal clear” that it is the entire economy that is productive, in case it is not “crystal clear” that this is our generation’s common inheritance from all generations before us, in case it is not “crystal clear” that any debts which may be owed by some today to others today for this common inheritance are trivial and inconsequential compared to the unpayable debt all today owe to all who preceded us – the Sraffa framework provides a final useful reminder: Although any particular technical change necessarily takes place in some particular industry, how much any technical change increases labor productivity can only be meaningfully measured as an increase in labor productivity in the economy as a whole. Which is why how much any technical change increases productivity can be measured by a single number, ρ(l), the percentage increase in the productivity of the economy as a whole which a particular technical change in a particular industry creates.7
In conclusion, Sraffians first explain what it means for an economy to be productive and capable of producing a physical surplus of goods. After which the explanation of the origins of profits and the relationship between the wage rate and profit rate are straightforward: (1) Profits result when some of the physical surplus of goods produced in a productive economy is taken away from those who produced it by their employers. (2) The lower the real wage, the more of the surplus is taken away from workers, and therefore the higher the rate of profit will be.
Comparing Theories of Profits
Marx believed labor values were necessary to explain where profits come from. Sraffians explain where profits come from without reference to labor values. Marxians explain profits as the result of exploiting a unique input to production, labor power. Sraffians explain profits as the result of denying workers in a productive economy all of the surplus goods they produce.
But how can the FMT and FST be reconciled, since they seem to provide different answers to the question of where profits come from? Because both theorems are true it must be the case that if and only if the economy is productive after wages are paid will the rate of exploitation as defined in Marxian theory be positive. And indeed this can be shown to be the case.8 In other words, just as values proved to be redundant when explaining capitalist prices, the Marxian rate of exploitation proves to be redundant when explaining capitalist profits.
Moreover, we could just as easily define a theory of value based on any basic good in the economy.9 In fact, one can prove that the rate of profit is positive if and only if every basic input is “exploited” when we choose to define values embodied in terms of that input. As John Roemer explained: “One cannot maintain, as is frequently done, that labor power is that one special commodity that mysteriously produces more value than is embodied in it, and hence its exploitation is the sole cause of profits. For, as an alternative to labor value, one could choose corn to denominate value, defining the embodied corn values of all commodities, and the following would be true: The economy is productive in the sense of being capable of producing a surplus if and only if corn is exploited” (Roemer 1981: 52). In which case the verbal statement of Morishima’s FMT theorem would be: Profits come from exploiting corn – hardly a ringing call to arms for workers of the world to unite! In short, positive profits have nothing to do with exploitation of one input to production rather than another. The notion that there is one special input, labor, whose exploitation is the source of capitalist profits is arbitrary since any basic input can be chosen to tell the same story.
Moreover, the fact that capitalists obtain profit based on every input to production, not just the labor they hire, is a fundamental characteristic of capitalism, not some sort of deception we must avoid being tricked by, as Marxists would have us believe. Capitalists achieve positive profits by markups on every input, not just labor. And it is anyone who does not understand this who is deceived, and fails to understand something important about how capitalism works!
To be clear, there is something qualitatively unique about labor compared to other inputs. When a capitalist pays for an hour of what Marx called “labor power,” what he gets is the right to try and extract an hour of “labor done,” but exactly how that will work out is unknown in advance. Whereas, when a capitalist buys non-labor inputs he can be much more sure what using those inputs will accomplish. The struggle between employers and employees over extracting labor from labor power is crucial to many aspects of how capitalism functions, and forms the basis for the very insightful radical “conflict theory” of the firm.10 However, while labor is qualitatively unique it is not quantitatively unique in the way the Marxian labor theory of value and Morishima’s fundamental Marxian theorem imply. An hour of labor done is no different than a ton of steel as far as being “exploitable” and a source of capitalist profits. In fact, one could argue that the traditional Marxist focus on a false quantitative distinction between labor done and other inputs to production has sometimes distracted attention from a true qualitative distinction between labor power and all other inputs, which does have profound implications.
Instead of searching for an input which is magically capable of “expanding value” during production, it turns out that reality is much simpler: It is the productivity of the economy after wages are paid that allows for positive profits. It is because workers are denied part of the surplus goods they produce that profits are positive. There is no need to elaborate a labor theory of value to make this point; no need to define a complicated technical ratio defined in terms of labor values; and no misleading identification of one input in particular that holds the key to the origins of profit, when in fact every input could be used to tell the same story, and markups on all inputs are where profits come from. Sraffian theory identifies the actual surplus of goods workers produce which capitalists manage to expropriate by markups on all inputs they purchase, and establishes a strong prima facie case that those who do no work, but nonetheless consume part of the physical surplus others produce, are parasites.
Marxian and Sraffian Theories of Wages
Sraffians make clear that they believe the level of the real wage is determined by a host of factors, including political factors, that lie beyond the scope of what a formal economic model or analysis can predict, but can be summarized as “bargaining power,” or “class struggle.” And in particular, Sraffians point out that the real wage need not be a subsistence wage, as it is implicitly assumed to be in formal Marxian theory. In fact, Sraffians emphasize that the real wage can climb a great deal above subsistence if the economy becomes sufficiently productive and workers gain sufficient bargaining power to keep their employers’ rate of profit in check so that increases in labor productivity translate into real wage increases instead of increases in the profit rate.
In truth, both Marx and his followers emphasize the importance of class struggle over the real wage. Nonetheless, in their formal theory the real wage is determined by an assumption having to do with their (provisional) theory of prices, namely that all commodities, including labor power, will exchange according to the number of hours it took to produce them. By doing so Marxian formal theory implicitly treats labor power as if it were produced under similar conditions to other commodities, and therefore, like other commodities will exchange for the amount of time it took to produce it – which in the case of labor power is the amount of time it takes to produce a subsistence wage bundle.
Malthus famously argued that any rise in the real wage above a subsistence level would lead to an increase in birth rates and/or fall in death rates sufficient to push the wage rate back down to subsistence. But Malthus’ “iron law of wages” has fallen into disrepute among modern demographers as real wages in the advanced capitalist economies have risen considerably over the past two hundred years. And to be fair, Marx dismissed the Malthusian “iron law of wages” as a “libel against the human race.” However, to make reality square with the implicit assumption in their formal theory that labor power is paid a subsistence wage, Marxists have to explain why the “value of labor power” is not simply a matter of biology and counting calories. They must explain how cultural and historical influences affect what the value of labor power will be in any society. Sraffians, on the other hand, take a more direct approach to the real wage: It’s all about bargaining power, stupid!
Labor Values, Profits, and the Economic Surplus
Last chapter we concluded that labor values were neither necessary nor useful for understanding price formation in capitalist economies. However, it has been argued that even if labor values are not needed to explain capitalist prices, they are needed to understand profits and/or production. We deferred discussion of those rationales for conducting analysis in terms of labor values to this chapter.
Some Marxists argue that labor values are necessary because the origin of profits can only be understood by studying labor values. They argue that without Marx’s key insight that the value of labor power is less than the labor time it imparts to goods during production it is impossible to understand where profits come from – except as the result of capitalists “buying cheap and selling dear,” which no doubt explains where some capitalists’ profits sometimes come from, but does not explain why normal, or average, profit rates for capitalists as a whole are positive. And Marxists cite the FMT as further proof that labor values are needed to explain profits.
However, as we have seen the FST provides a more straightforward explanation of where profits come from, and FST ↔ FMT because in formal terms saying that it takes less than an hour to produce an hourly subsistence wage bundle, Vb < 1, is the same as saying the economy is still productive after wages are paid, dom(A+bL) < 1. Moreover, we have also seen that if labor is “exploited” then every other basic input in production is also “exploited” if we define the value of commodities as the amount of that input needed directly and indirectly to produce it. In short, the key to understand profits is that (1) the economy must be productive, i.e. capable of producing a surplus of goods after all goods used up are replaced, and (2) workers must be denied part of the surplus of goods they produce. Moreover, this has nothing to do with labor values or any particular input to production.
Finally, other Marxists argue that capitalism is best understood by studying production first and exchange second, and that values are essential to understanding the former, while prices of production are only needed to understand the latter.11 It is very important to be clear here: There is a major difference between “classical” economic theory, which begins by focusing on production and the surplus of goods produced after all produced inputs are replaced, and neoclassical theory which does not identify and focus on a physical surplus from production prior to analyzing price and income determination. Both Marxian and Sraffian theories are in the “classical tradition” in the sense that they begin with production of a surplus. So beginning with production, thinking clearly about what it means for an economy to be productive, and recognizing there is a surplus of goods which emerges from production, which then gets distributed among different classes, is extremely important in classical economics, and both Marxian and Sraffian theory proceed in this way.
However, the belief that production can only or best be analyzed in terms of labor values does not follow. It does not follow that to understand why we end up with a surplus of goods as the result of production we must identify where some extra “value” came from. Marxian economics uses the labor theory of value to identify and analyze what happens to the surplus value created in production.12 Sraffian economics identifies and analyzes what happens to the physical surplus of goods created in production without resort to labor values. In other words, Sraffian theory simply explains where the extra goods came from. The fact that we can identify and quantitatively measure a physical surplus from production in a multi-good economy without recourse to values demonstrates that values are redundant for identifying the surplus which does, indeed, emerge from the production process.
Unlike neoclassical economists, both Marxians and Sraffians emphasize that aggregate profits are only possible if a surplus is generated in production, i.e. that aggregate profits cannot emerge from the exchange of commodities which merely distributes whatever surplus was already there. The difference is that Marxists claim the surplus that emerges from production after replacing all produced inputs used up and paying workers a subsistence wage, can only be demonstrated and quantitatively measured as a magnitude of “surplus value.” Sraffians, on the other hand, emphasize that a “physical surplus” of actual goods emerges from production, show how the magnitude of this physical surplus can be measured rigorously without resort to values, and explain that the price system distributes this physical surplus of goods among capitalists as profits.13
For those who recognized that a surplus is generated in production the dilemma was always how to quantify and measure it. In a single good world this is straightforward: Production requires both labor and some amount of a single good in order to produce a larger quantity of the same good. Simply subtract the amount of the good used as an input from the amount of the good produced as output, and you have a quantitative measure of the surplus in physical units of the only good in the economy, which can also be expressed as a fraction of the amount of the good produced. But in a multi-good world, where production requires both labor and specific quantities of different goods in order to produce larger quantities of many different goods, it was never obvious how to measure the size of the surplus. In the nineteenth century the labor theory of value provided a preliminary way to achieve in a multi-good environment what was simple to accomplish in a single good environment. In the twentieth century, first Sraffa, and then Sraffians using the Frobenius–Perron theorem, provided a better solution to what we might think of as the “surplus aggregation problem.” Sraffa demonstrated that if different goods were produced in specific proportions which he called the “standard commodity,” the amount of every good left over as “surplus” divided by the amount of that good produced would be the same for every good. In other words, for a particular vector of gross outputs we would get the same answer to the question “how large is the surplus” – is it 3%, 5%, or 9%? – no matter which good we used to calculate the answer. Following Sraffa others used the Frobenius–Perron theorem to generalize Sraffa’s insight. They noticed that Sraffa’s standard commodity was simply the right eigenvector for the input coefficient matrix for the economy including the consumption of workers, (A+bL), and the dominant eigenvalue for this matrix, dom(A+bL), represents the fraction of production required to replace all inputs used and pay workers in general. For example, suppose dom(A+bL) = 0.95. This means that after replacement and wage payments, 0.05 of production is left over, as surplus, to be distributed among capitalists.
Sraffian theory has now demonstrated that values are not necessary to identify and measure the magnitude of the surplus that emerges from production. In other words, we don’t need concepts that were initially designed to help understand why goods exchange in particular proportions in market economies – labor values – in order to solve the “surplus aggregation problem” in production. Moreover, as we will see next chapter, thinking in terms of surplus value instead of a physical surplus of goods can mislead one into drawing false conclusions about the impact of capital-using, labor-saving technological change on the rate of profit.
Notes
1 This is not as obvious as it may appear because as we know, inputs and outputs will not actually sell according to their labor values in capitalism. Instead, they will sell at “prices of production” that equalize profit rates across industries. Nonetheless, what Morishima proved was that if and only if (1−Vb)/Vb > 0 will the uniform rate of profit in the economy, r, be positive.
2 For a thorough discussion of alternative formal definitions of what it means for the economy to be productive and their equivalence, see appendix A in Hahnel 2017. Mainstream economists refer to this condition as the Hawkins-Simon condition. For proof that the economy must be productive in order for there to be any possibility for positive profits see theorem 10 in Hahnel 2017.
3 In the decimal .333 underlining the last 3 means that the number “3” which is underlined continues indefinitely. In other words .333 means .33333333 … etc. This convention will be used throughout.
4 Formally: If and only if the dominant eigenvalue of A* = [A+bL] < 1, i.e., if and only if A* is productive and therefore yields a physical surplus, will profits be positive. A is the matrix of produced input coefficients, a(ij), L is the row vector of direct labor input coefficients, l(j), b is a column vector of the quantities of different goods in the hourly wage bundle, and A* = [A+bL] is often referred to as the “socio-technology matrix” for the economy because it is determined by the economy’s technology, {A, L} and also by “sociological” factors which determine the real wage bundle, b.
5 To do this we must distribute the single hour of labor between the two industries precisely so as to generate no surplus in the first industry.
6 See theorem 13 in Hahnel 2017 for proof that the relationship between w and r is necessarily negative.
7 See Hahnel (2016a) for proof that increases in overall labor productivity due to any technological change in any industry can be calculated as ρ(l) = (1–β’) in a Sraffian framework, where β’ = dom(A’+b̃L’), the dominant eigenvalue for the economy with the new technology, {A’, L’}, and a real wage vector b̃ that reduces the rate of profit in the old economy to zero.
8 Formally, if and only if dom[A+bL] < 1 will Vb be less than 1, (1−Vb) > 0, and the rate of exploitation, (1−Vb)/Vb > 0. This result is proved as theorem 22 in Hahnel 2017.
9 One of Sraffa’s achievements was to explain that “basic goods” – goods which enter into the production either directly or indirectly of every good – play a different role in the economy than “non-basic” goods. In all the examples in this book all goods are basic. See chapter 1 in Hahnel 2017 for discussion of how the existence of non-basics can affect outcomes.
10 See chapter 8 in Hahnel and Albert 1990 for a rigorous presentation of the conflict theory of the firm and evaluation of many insights it provides.
11 Ira Gerstein (1976) presented this argument when presenting a rationale for why bother to transform from values to prices of production: “The transformation is actually between two theoretical levels of the construction of the economic region of the capitalist mode of production. The first of these levels is production in itself (Capital, Volume I), [where values prove useful] while the second is the complex unity of production and circulation (Capital, Volume III) [where prices of production are necessary].”
12 This is the core argument of the so-called “new interpretation” (NI) school of Marxists for why we should bother with labor values. NI Marxists do not argue that values are necessary to explain prices, but instead refer to the “law of conservation of value” in which value (socially necessary abstract labor time, or SNALT) is created in production and conserved in exchange. Whereas Sraffians simply analyze the surplus from production in physical terms, y = x − Ax = (I−A)x is the net output vector, i.e. the physical surplus of each good produced; NI theorists insist on conceptualizing the amount of SNALT embodied in these goods, Vy. Their particular twist is to normalize the price vector – that is convert the relative price vector, p, into an absolute price vector, by setting the monetary value of net output, mpy, equal to the total amount of hours embodied in the net output, Vy. This normalization choice means that 1/m = py/Vy, which NI Marxists call the “monetary equivalent of labor time” (MELT). See Dumenil 1980, 1984, Foley 1982, 1986, 2000, and Mohun 1994, 2004. As explained below, because Sraffian theory can identify, quantify, and measure changes in the size of the physical surplus even in a multi-good world, none of this is any longer necessary.
13 If there are no barriers to entry and exit among industries eventually prices will adjust until the surplus of goods that emerges from production yields equal rates of profit to capitalists in all industries. If there are barriers, and capitalists cannot move freely among industries, a different set of prices will distribute the surplus of goods from production so as to yield different rates of profits in different industries.