Friday, September 30, 2022


Richard Wolff Interviews George DeMartino (About 15:15)

Sunday, September 25, 2022

Sowell, Kolakowski, Baumol, Schumpeter: Böhm Bawerk Was Mistaken

Empirically, prices are fairly close to proportional to labor values. But that is neither here nor there as far as the correctness of Marx's theory of value. To see that, you have to get to the last footnote in chapter 5 of Capital. (Most "refutations" of Marx are based on ignorance of the first few pages of chapter 1.)

From the foregoing investigation, the reader will see that this statement only means that the formation of capital must be possible even though the price and value of a commodity be the same; for its formation cannot be attributed to any deviation of the one from the other. If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately', because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe.

-- Karl Marx. 1887. Capital, first english edition.

Only volume 1 was published in Marx's lifetime. But the question of the validity of Marx's theory, say, the value theory of labor, cannot be discussed without looking at volume 3, unpublished in Marx's lifetime.

Wicksteed had an early discussion of the validity of Marx's theory from a marginalist standpoint. Eugen von Böhm Bawerk's Karl Marx and the Close of his System is problably better known. His book is definitely important from a historical perspective. But its content is not all that insightful. Some anti-Marxists agree with me.

"The classic 'refutation' of Capital was made by a leading figure in the new economics, Eugen von Bohn-Bawerk. His refutation repeatedly misunderstood what it was refuting, and unknowingly repeated criticisms that Marx had made of Ricardo in manuscripts still unpublished at that time. Bohm-Bawerk also made the claim, often echoed since then, that in his discussion of value Marx had attempted 'a stringent syllogistic conclusion allowing of no exception,' that Marx attempted 'a logical proof, a dialectical deduction.' As already noted, Marx considered the idea of proving a concept to be ridiculous. Moreover, Engels had asserted, long before Bohm-Bawerk, that one only proves one's ignorance of dialectics by thinking of it as a means by which things can be proved. This was typical of a tragi-comedy of errors that has plagued the interpretation of Marx ever since.

Contrary to some interpretations, Marx did not change his mind about value and price between volumes of Capital. He explicitly worked out the analysis to be followed in Volume III in a letter to Engels written several years before publication of Volume I."

-- Thomas Sowell. 1985. Marxism: Philosophy and Economics, Routledge.

Sowell refers to Marx's letter of 2 August 1862. I have all sorts of disagreements with Sowell, which I forget. In some, I am probably critical of Marx than Sowell. One should probably ignore the final chapter of this book.

My next example is not from an economist:

"Marx of course knew that prices are determined in practice by various factors, including labour productivity, supply and demand, and the average rate of profit. If he disregarded these in the first volume of Capital, it was for methodological reasons and not because he thought value and price were the same thing; thus he cannot be reproached with inconsistency as between Volume I and Volume III, which deals inter alia with the average rate of profit."

-- Leszek Kolakowski. 1978. Main Currents of Marxism: The Founders (Book 1), Chapter XIII, Section 6.

I turn back to an economist:

"Writers on 'the transformation problem' since L. Bortkiewicz have focussed on an issue that is largely periperal; and others like E. Bohm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless one reads into them an interpretation different from that which Marx repeatedly emphasized."

-- William J. Baumol. 1974. The transformation of values: what Marx 'really' Meant (An Interpretation). Journal of Economic Literature 12 (1): 51-62.

Even one of his students was not too impressed by Böhm Bawerk on Marx:

"As it was, most critics felt no hestitation in convicting him of having by the third volume flatly contradicted the doctrine of the first. On the face of it that verdict is not justified. If we place ourselves on Marx's standpoint, as it is our duty in a question of this kind, it is not absurd to look upon surplus value as a 'mass' produced by the social process of production considered as a unit and to make the rest a matter of distribution of that mass

Joseph A. Shumpeter, Chapter III. Marx the Economist, Capitalism, Socialism, and Democracy. Third Edition, New York: Harper and Brothers, 1942, 1947, 1950.

Valid, non-outdated criticisms of Marx exist. Böhm Bawerk does not have one.

Saturday, September 24, 2022

Saul Kripke (1940 - 2022)

I want to write an inadequate appreciation of Saul Kripke, a great analytical philosopher. The Guardian has an obituary.

I start from an example of a pratical use of his work. Chin and Older (2011) use modal logic to specify and reason about the security properties of (computer) systems. One wants to be able to assert who (or what) has access to certain data and who can grant access. Certain states of a system should never arrive.

A Kripke structure, as presented in Chin and Older (2011), is a three-tuple consisting of:

  • A set W, known as the set of all possible worlds.
  • An interpretation I, that maps each proposition to a subset of W. Informally, I(p) is the set of worlds in which the proposition p is true.
  • A function J that maps the name of each principal to a set of ordered pairs of possible worlds. Consider J(A), the set of ordered pairs (W1, W2) mapped to by this function. This is intended to be such that if the current world is W1, A believes the world might be W2.

For the above definition to be completely formal, one needs to specify the language for propositions p. In propositional logic, operators such as "and", "or", and "not" combine atomic propositions. Predicate calculus introduces such notions as "for all" and "there exists". Model theory is used to characterize semantics, to specify models in which a set of sentences is true. I guess a Kripke structure is a kind of structure, as structures are defined in model theory. Modal logic uses the apparatus of model theory to characterize propositions as "necessarily true", "possibly true", and so on. The textbook I have available for model theory, which I cannot get very far into, does not go into this.

Misleading talk, some of that by Kripke, about possible worlds raises the question of "transworld identification". Consider a proposition p(a) about an individual a. How do we identify a across possible worlds? If one uses the name Nixon to identify the winner of the 1968 U.S. presidential election, are we not requiring one to call Humphrey "Nixon" in some possible worlds? Amusingly, Kripke also raises the question of why we might say that 9 is necessarily greater than 7, but be unwilling to say that the number of planets is neccessarily greater than 7. In the current world, the number of planets is now eight. Pluto is now, by stipulation, not a planet. Kripke argues, in Naming and Necessity that Frege and Russell had mistaken theories about reference.

I will probably not be able to wrap my head around the idea of necessary contingent truths in a couple of weeks. Consider the length of a certain rod in Paris back when that was the definition of a meter. It was a necessary truth that a meter is that length. In some possible worlds, the room it is stored in might possibly be hotter. A meter would still be a meter in all possible worlds, according to Kripke, but the length of the standard meter might possibly be different.

I find Kripke (1982) to be easier to understand, maybe. I know that Kripke says that he is presenting an argument from Wittgenstein, not his own. If I try to orally summarize it, I tend to bring in Goodman's "grue" and "bleen", Putnam's "twater" on twin earth, or Wittgenstein's beetle in a box. The topic seems to be how words mean, not epistemology or ontology. How do you know that when talking to people they do not mean "grue" when they say "green"? Even more troublesome, how do you know you did not mean "grue" every time you said "green" in the past? By postulation in the argument, there does not seem to be any empirical fact one can point to. Meaning is not in an individual's mind.

  • Shiu-Kai Chin and Susan Older. 2011. Access Control, Security, and Trust: A Logical Approach. CRC Press.
  • Nelson Goodman. 1983. Fact, Fiction, and Forecast, 4th ed. Harvard University Press.
  • Wilfrid Hodges. 1997. A Shorter Model Theory. Cambridge University Press.
  • Saul A. Kripke. 1980. Naming and Necessity, 2nd ed. Harvard University Press.
  • Saul A. Kripke. 1982. Wittgenstein on Rules and Private Language: An Elementary Exposition. Harvard University Press.
  • Hilary Putnam. 1983. Reason, Truth, and History. Cambridge University Press.

Tuesday, September 20, 2022

Karl Marx To Abraham Lincoln On His Re-Election

Apparently, Lincoln responded to this congratulations from the International Workingmen's Association:


We congratulate the American people upon your re-election by a large majority.

If resistance to the Slave Power was the reserved watchword of your first election, the triumphant warcry of your re-election is, Death to Slavery.

From the commencement of the Titanic-American strife the working men of Europe felt instinctively that the star-spangled banner carried the destiny of their class. The contest for the territories which opened the dire epopee, was it not to decide whether the virgin soil of immense tracts should be wedded to the labour of the emigrant, or prostituted by the tramp of the slave-driver?

When an oligarchy of 300,000 slave-holders dared to inscribe, for the first time in the annals of the world, "slavery" on the banner of Armed Revolt; when on the very spots where hardly a century ago the idea of one great Democratic Republic had first sprung up, whence the first Declaration of the Rights of Man was issued, and the first impulse given to the European revolution of the 18th century; when on those very spots counter-revolution, with systematic thoroughness, gloried in rescinding "the [...] ideas entertained [...] at the time of the formation of the old Constitution", and maintained "slavery to be a beneficent institution", indeed the only solution of the great problem of "the relation of labour to capital", and cynically proclaimed property in man "the cornerstone of the new edifice", then the working classes of Europe understood at once, even before the fanatic partisanship of the upper classes for the Confederate gentry had given its dismal warning, that the slave-holders' rebellion was to sound the tocsin for a general holy crusade of property against labour, and that for the men of labour, with their hopes for the future, even their past conquests were at stake in that tremendous conflict on the other side of the Atlantic. Everywhere they bore therefore patiently the hardships imposed upon them by the cotton crisis, opposed enthusiastically the pro-slavery intervention, importunities of their betters - and, from most parts of Europe, contributed their quota of blood to the good cause.

While the working men, the true political power of the North, allowed slavery to defile their own republic; while before the Negro, mastered and sold without his concurrence, they boasted it the highest prerogative of the white-skinned labourer to sell himself and choose his own master; they were unable to attain the true freedom of labour or to support their European brethren in their struggle for emancipation, but this barrier to progress has been swept off by the red sea of civil war.

The working men of Europe feel sure that, as the American War of Independence initiated a new era of ascendancy for the middle class, so the American Anti-Slavery War will do for the working classes. They consider it an earnest of the epoch to come that it fell to the lot of Abraham Lincoln, the single-minded son of the working class, to lead his country through the matchless struggle for the rescue of an enchained race and the reconstruction of a social world.

Signed on behalf of the International Working Men's Association

The Central Council

I have noted before parallels between Marx and an American creed.

Friday, September 16, 2022

Three Quotations: Rousseau, Adam Smith, Engels

Here is Jean Jacques Rousseau:

"...whether those who command are necessarily better than those who obey, and if strength of body or of mind, wisdom or virtue are always found in particular individuals, in proportion to their power or wealth: a question fit perhaps to be discussed by slaves in the hearing of their masters, but highly unbecoming to reasonable and free men in search of the truth." -- Jean Jacques Rousseau, Discourse on Inequality (1755).

Early developers of political economy were not slavish. Here is Adam Smith explaining that returns to capital and land are the result of value added by labor not paid out in wages.

"In the early and rude state of society... the whole produce of labour belongs to the labourer; and the quantity of labour commonly employed in acquiring or producing any commodity, is the only circumstance which can regulate the quantity of labour which it ought commonly to purchase, command, or exchange for.

As soon as stock has accumulated in the hands of particular persons, some of them will naturally employ it in setting to work industrious people, whom they will supply with materials and subsistence, in order to make a profit by the sale of their work, or by what their labour adds to the value of the materials. In exchanging the complete manufacture either for money, for labour, or for other goods, over and above what may be sufficient to pay the price of the materials, and the wages of the workmen, something must be given for the profits of the undertaker of the work, who hazards his stock in this adventure. The value which the workmen add to the materials, therefore, resolves itself in this case into two parts, of which the one pays their wages, the other the profits of their employer upon the whole stock of materials and wages which he advanced. He could have no interest to employ them, unless he expected from the sale of their work something more than what was sufficient to replace his stock to him; and he could have no interest to employ a great stock rather than a small one, unless his profits were to bear some proportion to the extent of his stock...

...As soon as the land of any country has all become private property, the landlords, like all other men, love to reap where they never sowed, and demand a rent even for its natural produce. The wood of the forest, the grass of the field, and all the natural fruits of the earth, which, when land was in common, cost the labourer only the trouble of gathering them, come, even to him, to have an additional price fixed upon them. He must then pay for the licence to gather them, and must give up to the landlord a portion of what his labour either collects or produces. This portion, or, what comes to the same thing, the price of this portion, constitutes the rent of land, and in the price of the greater part of commodities, makes a third component part." -- Adam Smith, The Wealth of Nations, Book I, Chapter VI (1776).

Many read the above as an account of how labor is exploited under capitalism. I find something similar in an essay a young Engels wrote before his life-long partnership with Marx:

"We have seen that capital and labour are initially identical; we see further from the explanations of the economist himself that, in the process of production, capital, the result of labour, is immediately transformed again into the substratum, into the material of labour; and that therefore the momentarily postulated separation of capital from labour is immediately superseded hy the unity of both. And yet the economist separates capital from labour, and yet clings to the division without giving any other recognition to their unity than by his definition of capital as "stored-up labour". The split between capital and labour resulting from private property is nothing but the inner dichotomy of labour corresponding to this divided condition and arising out of it. And after this separation is accomplished, capital is divided once more into the original capital and profit-the increment of capital, which it receives in the process of production; although in practice profit is immediately lumped together with capital and set into motion with it. Indeed, even profit is in its turn split into interest and profit proper. In the case of interest, the absurdity of these splits is carried to the extreme. The immorality of lending at interest, of receiving without working, merely for making a loan, though already implied in private property, is only too obvious, and has long ago been recognised for what it is by unprejudiced popular consciousness, which in such matters is usually right. All these subtle splits and divisions stem from the original separation of capital from labour and from the culmination of this separation- the division of mankind into capitalists and workers-a division which daily becomes ever more acute, and which, as we shall see, is bound to deepen. This separation, however, like the separation already considered of land from capital and labour, is in the final analysis an impossible separation. What share land, capital and labour each have in any particular product cannot be determined. The three magnitudes are incommensurable. The land produces the raw material, but not without capital and labour. Capital presupposes land and labour. And labour presupposes at least land, and usually also capital. The functions of these three elements are completely different, and are not to be measured by a fourth common standard. Therefore, when it comes to dividing the proceeds among the three elements under existing conditions, there is no inherent standard; it is an entirely alien and with regard to them fortuitous standard that decides— competition, the cunning right of the stronger. Rent implies competition; profit on capital is solely determined by competition; and the position with regard to wages we shall see presently." -- Friedrich Engels, Outlines of a Critique of Political Economy (1844).

I think one can discard the conjectural history and moral overtones of the above and freely play in the mathematics of Leontief matrices. But many academic economists nowadays are imposing mind-forged manacles onto another generation.

Saturday, September 10, 2022

Marx On The Transformation Problem In 1847

This is the start of Section 5, "Strikes and combinations of workers", in the second chapter of The Poverty of Philosophy:

"'Every upward movement in wages can have no other effect than a rise in the price of corn, wine, etc., that is, the effect of a dearth. For what are wages? They are the cost price of corn, etc.; they are the integrant price of everything. We may go even further: wages are the proportion of the elements composing wealth and consumed reproductively every day by the mass of the workers. Now, to double wages ... is to attribute to each one of the producers a greater share than his product, which is contradictory and if the rise extends only to a small number of industries, it brings a general disturbance in exchange; in a word, a dearth.... It is impossible, I declare, for strikes followed by an increase in wages not to culminate in a general rise in prices: this is as certain as that two and two make four.' (Proudhon, Vol. I, pp. 110 and 111)

We deny all these assertions, except that two and two make four.

In the first place, there is no general rise in prices. If the price of everything doubles at the same time as wages, there is no change in price, the only change is in terms.

Then again, a general rise in wages can never produce a more or less general rise in the price of goods Actually, if every industry employed the same number of workers in relation to fixed capital or to the instruments used, a general rise in wages would produce a general fall in profits and the current price of goods would undergo no alteration.

But as the relation of manual labour to fixed capital is not the same in different industries, all the industries which employ a relatively greater mass of capital and fewer workers, will be forced sooner or later to lower the price of their goods. In the opposite case, in which the price of their goods is not lowered, their profit will rise above the common rate of profits. Machines are not wage-earners. Therefore, the general rise in wages will affect less those industries, which, compared with the others, employ more machines than workers. But as competition always tends to level the rate of profits, those profits which rise above the average rate cannot but be transitory. Thus, apart from a few fluctuations, a general rise in wages will lead, not as M. Proudhon says, to a general increase in prices, but to a partial fall - that is a fall in the current price of the goods that are made chiefly with the help of machines.

The rise and fall of profits and wages expresses merely the proportion in which capitalists and workers share in the product of a day's work, without influencing in most instances the price of the product. But that 'strikes followed by an increase in wages culminate in a general rise in prices, in a dearth even' - those are notions which can blossom only in the brain of a poet who has not been understood." -- Karl Marx

You can see why after this, Marx and Proudhon were no longer drinking buddies. Anyways, Marx here considers a rise in wages. Echoing Ricardo, Marx argues that some prices drop and others rise. So even though the labor embodied in commodities does not alter, relative prices of production vary because of a variation of wages. I take this to be a statement of the so-called transformation problem.

An early statement of Marx's solution can be found in his 2 August 1862 letter to Engels.

I see in the above an idea Marx takes over from Ricardo. Think of the yearly net output of a capitalist economy as produced by the labor employed during that year. The 'real wage', in Ricardo's terminology, is the proportion of that labor that goes to produce the commodities purchased and thereby consumed by the workers. Suppose productivity increases, and total employment does not change. The workers can thereby obtain a greater quantity of 'necessaries and luxuries', while the real wage declines, depending on how the results of this increased productivity are divided among the classes making up society. The current usage among mainstream economists of the term 'real wage' is an obstacle to reading Ricardo, if one is not careful.

By the way, in a comment on one of my posts on the Temporal Single System Interpretation, Hedlund recommends Robert A. Bryer's "Marx's accounting solution to the 'transformation problem'". This is a chapter in his book Accounting for Value in Marx's Capital: The Invisible Hand

Wednesday, September 07, 2022

Standards Organization As Partly A Non-Capitalist Logic

This post draws attention to the existence of a vast web of existing organizations that, perhaps, operate partly outside and partly inside a capitalist logic. These standards organizations define technologies vital to keeping our society running. I think I would find it interesting for some scholar interested in council communism or syndicalism to look into these. The examples I list are perhaps idiosyncratic, out-of-date, and reflect my personal history with computing.

Here is a list of some of the organizations I have in mind:

  • American National Standards Institute (ANSI)
  • International Electrotechnical Commission (IEC)
  • Institute of Electrical and Electronics Engineers (IEEE)
  • Internet Engineering Task Force (IETF)
  • International Standards Organization (ISO)
  • International Telecommunications Union (ITU)
  • National Institute for Standards and Technology (NIST)

The IEEE is a professional organization that provides an umbrella for a wide range of specialized societies. They publish academic journals, organize conferences, define certain certification programs, and have a professional code of ethics. IEEE-STD-754 is an example of a standard I am often conscious of. It defines how floating point numbers are stored in your computer, where floating point numbers are a finitary representation, in some sense, of real numbers. This standard was developed and is maintained by a committee of academics, representatives of companies, etc. under the auspices of the IEEE. The original standard and each update is circulated for comment and for voting by the wider membership.

Sometime a standard is developed and maintained by such a committee under the auspices of a government agency. For example, consider the Interplanetary Overlay Network (ION), a delay-tolerant network (DTN), implementing an interplanetary internet. As I understand it, ION was developed by the Interplanetary Networking Special Interest Group, a committee set up by the National Aeronautics and Space Administration (NASA) and chaired by Vincent Cerf.

Sometimes a standard is replaced by another one. The Perry memo, issued under Bill Clinton, directed the Department of Defense to prefer commercial standards to their own military standards, where feasible. For example, MIL-STD-2167A was replaced by ISO/IEC 12207 Information Technology – Software life cycle processes. This standard describes documentation to be produced during software development lifecycles, such as design documentation, user manuals, and test plans.

Sometimes a single corporation develops a standard. For example, Java was developed by Sun, now by Oracle. As those who develop enterprise systems know, a vast set of tools are available for Java, including integrated development environments (IDEs), application servers, deployment tools such as Ant and Gradle, and even other languages that compile to execute on the Java Virtual Machine (JVM).

Corporations have an interest in having their internal developments recognized by a standards body. I am not sure that this is a good example, but C was developed by Bell Labs before becoming ANSI C.

Sometimes, a standard is developed as a contest. (Dean Baker might find this of interest.) The Ada programming language (ANSI/MIL-STD-1815A) had four qualifiers for the first round: CII Honeywell Bull (green), Intermetrics (red), Softech (blue), and SRI International (yellow). Eventually, the U.S. Department of Defense decided "green is Ada". More recently, NIST developed the Advanced Encryption Standard (AES) with a contest.

The technical feasibility of a standard is sometimes in question, and a reference implementation is often developed alongside the standard. For some reason, I associate the term "reference model" with the Open Systems Interconnection (OSI) layered model for the Internet. A standards body might impose some control on who can claim to have implemented the standard. Ada was accompanied with, for example, test suites for verifying implementations.

Even in a society of hierarchical organizations, a need exists for people to cooperate across such organizations.

Thursday, September 01, 2022

On Sraffian Subsytems And Labor Values

1.0 Introduction

I recently stumbled across McKiernan (2017), an Austrian response to Sraffa's book. This is a weird working paper. He works through Sraffa, with apparently no knowledge of all the textbooks explaining the book. McKiernan correctly notes that Sraffa provides little context about his points. And the mathematics is not always explicit. Naturally, McKiernan makes mistakes. If he ever revisits this, I think he would want to break it up into several papers. (Fabra (1991) is the strangest published response to Sraffians of which I know.)

I only want to focus on one point, Sraffian subsytems. But before doing that, let me at least point out something insightful in McKiernan's paper. He points out one of what I later call a fluke case:

However, Sraffa makes no note of cases in which a scarcity of land is offset by using a more expensive method of production, and that employment meets present desire for the product exactly when applied to all available land, so that only one method is used. In this case, two variables, pcorn and ρ, correspond to one equation. (p. 22)

I think I could also construct a fluke case with intensive rent, which is closer to his point.

2.0 A Criticism Of Subsystems

McKieran mistakenly asserts:

But consider the pricing of a commodity that were not produced in surplus, as could happen in these models with a purely infrastructural commodity. A sub-system for this commodity would have a net production of 0, but these models have all presumed a need for active renewal, so there would be an expenditure of labor. If returns to scale were co-incident, so that the sub-system might be embodied, that sub-system standing alone would produce a wage of 0, but wages are presumed to be shared across sub-systems (else Sraffa’s argument falls apart). Hence, it appears that prices of commodities of this infrastructural sort must be whatever one makes of division by 0. (p. 8)

(If McKieran ever revisits this, I wish he would give a more explicit and formal definition of co-incident returns. I think I get his point, but I do not think I have ever seen this notion in the economics literature.)

Anyways, if the net output of a commodity is zero, the decomposition of the given quantity flows into Sraffian subsytems, with nothing left over, will result in no labor being directed towards producing a net output of that commodity. So one rather gets a quotient of zero divided by zero, not a division of a positive quantity of zero. Nevertheless, one can still find a Sraffian subsystem for producing that commodity.

3.0 A Decomposition of One Set of Quantity Flows

I take an example from my FAQ on the Labor Theory of Value. Consider an economy with the observed quantity flows shown in Table 1. In this little model economy, wheat, iron, and labor are used to produce a net output of 500 quarters of wheat and 8 tons iron. From the postulated observations, one cannot tell whether iron is somehow consumed or whether this economy is undergoing steady growth. Given, say, the real wage, one can calculate prices of production. But many questions are not adressed here.

Table 1: Given Quantity Flows
74 qr. wheat & 37 t. iron & 592 worker.->592 qr. wheat
18 qr. wheat & 3 t. iron & 48 workers->48 t. iron

I find Leontief coefficients of production useful. Table 2 results from dividing the first of Table 1 by 592 qr. wheat and the second row by 48 t. iron.

Table 2: Leontief Coefficients
1/8 qr. & 1/16 t. & 1 workers->1 qr. wheat
3/8 qr. & 1/16 t. & 1 workers->1 t. iron

Suppose these coefficients of production are used to calculate inputs when gross outputs of are approximately 588.2 quarters wheat and 39.2 tons iron. Table 3 results. The net output of the wheat subsytem is 500 quarters wheat, produced from an input of 32000/51 workers. That is 64/51 ≈ 1.25 workers are embodied in each quarter of wheat.

Table 3: Wheat Subsystem
1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers10000/17 qr. wheat
250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers2000/51 t. iron

Table 4 shows the iron subsystem. In this subsystem, 640/51 ≈ 12.5 workers produce a net output of 8 tons iron. In other words, 80/51 ≈ 1.57 workers are embodied in each ton iron.

Table 4: Iron Subsystem
8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers64/17 qr. wheat
56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers448/51 t. iron

The quantity flows shown in Tables 3 and 4 add up to the quantity flows in Table 1. In these subsystems are thought of as operating side-by-side, total quantity flows are as in the observed economy. No assumptions on returns to scale are needed for this decomposition into subsystems.

4.0 A Decomposition of Another Set of Quantity Flows

I now want to consider another set of quantity flows that might be observed. Suppose these quantity flows are as in Table 5. The net output of this economy consists of 500 quarters wheat. For ease of calculation, I have used the same coefficients of production as in Table 2. They very well could be different since gross outputs vary from those in Table 1. If coefficients of production vary, so do returns to scale. Anyways, in this example, the net output of iron is zero. But even so, one can find here a subsystem for producing iron.

Table 5: Another Set of Quantity Flows
1250/17 ≈ 73.5 qr. & 625/17 ≈ 36.8 t. & 10000/17 ≈ 588.2 workers10000/17 qr. wheat
250/17 ≈ 14.7 qr. & 125/51 ≈ 2.5 t. & 2000/51 ≈ 39.2 workers2000/51 t. iron

Consider the quantity flows shown in Table 6. The net output is 500 quarters wheat and -8 tons iron. This is not a Sraffian subsystem. It is unbalanced. But these quantity flows are constructed from the same coefficients of production as manifested in Table 5.

Table 6: Unbalanced Quantity Flows
1242/17 ≈ 73.1 qr. & 621/17 ≈ 36.5 t. & 9936/17 ≈ 584.5 workers9936/17 qr. wheat
194/17 ≈ 11.4 qr. & 97/51 ≈ 1.9 t. & 1552/51 ≈ 30.4 workers1552/51 t. iron

Now consider what quantity flows result from subtracting those in Table 6 from those in Table 5. Table 7 results. This is the same subsystem for producing iron as shown in Table 4. Even though the net output of iron in the observed quantity flows in Table 5 is zero, one can still find in them an iron subsystem.

Table 7: The Iron Subsystem Again
8/17 ≈ 0.5 qr. & 4/17 ≈ 0.2 t. & 64/17 ≈ 3.8 workers64/17 qr. wheat
56/17 ≈ 3.3 qr. & 28/51 ≈ 0.5 t. & 448/51 ≈ 8.8 workers448/51 t. iron

5.0 Results

No assumptions on returns to scale are made in analytically decomposing the observed quantity flows into subsystems.

Given observed physical quantity flows, one can ask how much more labor would be employed if net output of the economy was increased by a small quantity of a specified commodity. In other words, each commodity has an employment multiplier, to use the jargon of Leontief analysis. If non-constant returns to scale do not prevail, the error in this calculation will become more pronounced as the specified quantity increases. An insight behind the differential calculus is that, for continuous functions, a small enough variation has an approximately linear effect.