30 August 2012

Digression on the Theory of Value

This post began as a series of footnotes to my fourth post on reading Sraffa. It seemed obvious that I was going to need to explain a lot of points that simply don't fit into a footnote.

Steve Keen is interested in Piero Sraffa' Production of Commodities by Means of Commodities because he believes that it explodes theories of value and price. Having read and re-read a lot of related material, I am increasingly skeptical of Keen's attempts to enlist Sraffa. But in order to explain why, I need to explain some things about rival theories of value.


In the past, the concept of a "just price" and its close cousin, the "natural price," stimulated debates about the fairness of capitalism itself. One provocation of debate--and much pre-19th century regulation--was the sudden increase of prices for goods in times of scarcity. Another was the frequently inverse proportion between the wealth of some people and the amount of productive work they did. Most food, for instance, was produced by people who were themselves desperately poor and subject to continual toil.

One approach for jurists was to reason from the principle of a "just price" for goods, which supposed to be derived from what that particular culture regarded as fair (or decent, moral) business practices. Another approach was to reason from what goods ought to cost. Adam Smith, for instance, reasoned that goods had a "natural price":
When the price of any commodity is neither more nor less than what is sufficient to pay the rent of the land, the wages of the labour, and the profits of the stock employed in raising, preparing, and bringing it to market, according to their natural rates, the commodity is then sold for what may be called its natural price.
Wealth of Nations, I.vii-§4."Of the Real and Nominal Price of Commodities"
The natural price of any item could be divided into the share paid to land, labor, and capital. Of these, capital and rent could be divided into accumulated past labor required to make them available to the general economy. So, for example, money borrowed at interest to pay the rental fees for the table saw and electric router used could be instead represented as labor performed in the past to create those items. Likewise, the wood, varnish, and glue (or nails).

The value of those inputs was all determined by what what required to make them forthcoming in the desired amounts. If the value placed on labor was too low, then of course humans would avoid commercial occupations and stick to traditional subsistence occupations; or, that denied them (as was the case in 18th century Britain), they would likely starve and have few children, leading to a dearth of labor. If the interest on borrowed money (capital) was too low, then too little would be saved and banking would be strangled by insufficient loanable funds, etc.

Shortly after Smith published The Wealth of Nations, David Ricardo decisively demonstrated that the concept of the "natural price" was actually a fallacy, and like the Farallon Plate, it is now completely subducted under Ricardo's labor theory of value (LTV).


Deciding what the true price of things ought to be required some objective principle that united all things of value. One point of view was that there was a unique attribute of all goods that could be quantified for purposes of comparison, and this attribute was labor. While the production of most goods require other goods (like machines and raw materials) in addition to labor, those other goods--in turn--can be reduced to earlier labor inputs, and so on backward in time.

This was mentioned by Adam Smith several times:
The annual labour of every nation is the fund which originally supplies it with all the necessaries and conveniencies of life which it annually consumes, and which consist always either in the immediate produce of that labour, or in what is purchased with that produce from other nations.
Wealth of Nations, I.i-§1. "Introduction and Plan of the Work"
leading to his distinction between value in exchange and use-value.
The word VALUE, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called 'value in use ;' the other, 'value in exchange.' The things which have the greatest value in use have frequently little or no value in exchange; and on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce any thing; scarce any thing can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it.
Wealth of Nations, I.iv-§13. "Of the Origin and Use of Money"

The value of any commodity, therefore, to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities.
Ibid, I.v-§1 "Of the Real and Nominal Price of Commodities" (emphasis added)
Historically (according to Smith) this would imply that capital accumulation (and hence, interest on capital), rent on land, and ultimately, profit were diversions of (exchange) value away from the actual producer, labor.1

Smith was not happy about this, but he accepted it as having an important payoff for society. The accumulation of capital led to the division of labor, massively increasing its productive powers. This included developments people usually associate with machinery as such: things like metallurgy allowed workers to contribute to food production from activities as far afield as coal mining and steel production. Because of this, economists who came after Smith were content to regard this diversion of revenue away from labor as morally acceptable, since it greatly enhanced the absolute benefits available to laborers.

After 1870, the situation changed.William Stanley Jevons, Leon Walras, and Carl Menger each published books that had a major impact on the field of economics--all of which attacked the labor theory of value.2

Instead of accepting economic growth as a sort of loan from the worker (in exchange for which the worker was paid "interest" in the form of rising real wages), the marginalists argued that the value of objects arose from its use value. One of Menger's followers, Eugen Böhm-Bawerk (The Positive Theory of Capital, III.iii.) argues that value in use is augmented by value in exchange: claiming that the above-mentioned diamond-water paradox had stumped economists from Adam Smith onward, the Austrians made it a paradox without a satisfactory solution, but then came up with the casuistry that the use-value of a diamond was really big because its exchange value was.3

Therefore, the value of economic inputs was determined by their contribution to the overall utility of goods they were used to produce. According to marginalists, profit was actually an inherent cost of production, and reflected the income "advanced" to workers before the production process was complete (according to marginalists, labor was unique among the factor of production in so far as it was not entitled to payment prior to the completion of the production process, whereas all other inputs were.)4


A fundamental doctrine of economics is that markets are totally spontaneous relations that always occur between free actors.

Now, as a historical fact it has be made absolutely clear that markets are things that have to be made by conscious effort; stock markets, for example, consist of very specialized experts called "market makers." This is a really basic fact well known to anthropologists and actual historians of economics. Spontaneous markets usually are confined to exotic situations where commerce has long been established and is suddenly disrupted by a disaster.

But let us suppose that, once markets are set up for many different goods, they perform the unique miracle of price discovery, by establishing an equilibrium between supply and demand.

The problem with declaring that this is what value really is, was obvious even to the Austrian school of economics: even Carl Menger or Eugen Böhm-Bawerk, to say nothing of William Stanley Jevons or Alfred Marshall, were going to claim that that was what value actually was. Value may have been "discovered" (i.e., revealed) by the process of commerce equalizing effective supply and effective demand, but in order for it to be revealed, value had to exist.


This is a tricky topic and what I need to say has already been said in this passage, so I'm merely going to quote at length:
[There is a very important] distinction between use value, the subjective valuation an individual gives to a commodity, which is contingent and arbitrary, and exchange value, which is the long-run ratio of exchange between a commodity and another, including commodity money. [...] Marx point[ed] out that the greatest classical economist, David Ricardo, assumed that a commodity produced would have greater exchange value than the total value of the wages paid to the workforce making that commodity (the value of the wages itself also expressed in commodities). As Marx noted, Ricardo did not explain how this came to be, i.e. where the difference came from. Marx then argued that the difference, "surplus value," arose from labor and labor alone in the process of capitalist production: it resulted from the use value of labor power, which is the labor of people as sold as a commodity in a competitive labor market, being that it had the ability to produce surplus value.
Krul (2012) (emphasis added)
As I've already mentioned above, Smith's distinction between use and exchange value absolves him or anyone else from having to plumb the metaphysical questions of who deserves to possess the "utility" of the social product. This is not, after all, a preoccupation of Smith, nor is it of Ricardo, nor even of Karl Marx.

Reading the writings of neoclassical economists, it is clear that they are usually mindful that their readers are evaluating various radical views about the existing social order. A major preoccupation was "debunking Marx," an activity that almost all popular writers on economics feel compelled to do. As we have seen, the most dogged and vociferous were members of the Austrian school of neoclassical economics (Menger, Böhm-Bawerk, Ludwig von Mises, Friedrich von Hayek, and Murray Rothbard, and the mostly-sympathetic Joseph A Schumpeter.) While the Austrian school is subject to differences of opinion like all the others, one common feature was this tendency to attribute to Marx, et al. the view that intrinsic worth of commodities was the thing being alienated, and therefore Marx was just an obnoxious simpleton.5


When talking about the labor theory of value, it sounds very simple: value in exchange will represent the sum of all past labor inputs into the creation of this final commodity. One might acknowledge that "in exchange" assumes a few points: one is that "exchange" involves a mostly fixed bundle of commodities (nominal values of money being held in low esteem); and that while the "value in exchange" of, say, 1000 yards of muslin, is subject to fluctuation depending on things like access to your primary markets, prevailing tastes in textiles, etc., it is stable enough to make long-term plans.

Another assumption is that labor-power may be treated as a homogenous, interchangeable input. This is no different that ordinary Taylorist methods for planning a new production process. Again, since the Marxist LTV is describing capitalist production, it follows that we can prima facie assume management policies optimizing worker output and even uniformity. Willfully substituting inefficient methods of production, for purposes of argument, is in bad faith.

Marxism is especially significant because it actually represents a crucial endpoint in the system of economic analysis. Logically, it is a more logically robust than neoclassical economics because it is compatible with a huge range of realistic market failures and neoclassical theory is not; it acknowledges (where neoclassical economics emphatically denies) the concept of sectional interests in society winning control of "the market," which apparently Sraffan economics implies must happen.


Marxist theory uses the term "commodity" in a specialized sense.  While I strongly recommend Robert Vienneau's FAQ page on the LTV, the entry for "commodity" says very little.  When writers like Matthijs Krul say that "The 'residue' exists because capitalism has not always existed, that is to say, that there has been non-commodity production before commodity production," readers understandably will wonder how this could possibly be true.6

Krul includes an extended quote from Marx's Capital that explains what he meant by "commodity," but I did not really think this was helpful: "A commodity is, in the first place, an object outside us, a thing that by its properties satisfies human wants of some sort or another"; this is a necessary condition of being a commodity, but it cannot possibly be sufficient, because all human societies have produced that sort of thing (and what about services?).  I wanted a succinct explanation of what it was about "commodities" that distinguished them from the goods and services that pre-capitalist economies presumably produced.  
The most important thing to realize about the concept of human capital is the direct meaning of the term itself: human capital. Conceiving of skills and education as an aspect of labor in fact reinforces the understanding of labor itself as a commodity, which happens to reside in human beings individually. This has two important effects. First, the individual human’s labor has under neoclassical theory already been assumed to be a commodity, a mere instrument for the production of capital. Now, even the aspects of the carrier of labor, the human individual, which were hitherto considered to be given and outside the sphere of neoclassical capital theory are now subsumed within it. All skills, all education, in fact any number of intangibles relating to character, outlook and so forth can now be subsumed into the theory as means of production.
Krul (2010) (emphasis added)
I am pretty sure that Krul has nailed it right here.  The essence of capitalism is that it seeks to accumulate capital, which is to say, it uses a combination of technology and social control to build up continuously growing pool of stuff required for the production of more stuff.

...[L]abor takes a specific form under capitalist production, because it is now a commodity (labor-power) which in the process of generalized competition is made into homogeneous, abstract labor. It is not, therefore, just like any other commodity, but it is one which, as essential prerequisite for the process of capitalist reproduction, becomes the very yardstick of its own accumulation: value, which is nothing else than socially necessary labor time needed for the reproduction of any commodity.
Krul (2012)

The "stuff" could be inventories, buildings and fixtures, machinery, or "human capital"; probably anything one could name has been used as capital at some time or another (including the state, whether of the USSR or the government of Lake County, Florida).  Idealistically, one could equally imagine nearly all of those things not used as capital: an Indian tribe whose cement factory provides employment and public revenue, or a research facility whose findings are public domain, and so on. At that point they would cease to be commodities, since commerce in them would not be driven by the process of accumulating social power for some capitalist.

  1. Smith, Wealth of Nations, I.vi-§5. Smith goes a little further in his Lectures on Jurisprudence:
    Of 10,000 families which are supported by each other, 100 perhaps labour not at all and do nothing to the common support. The others have them to maintain beside themselves, and besides [those] who labour have a far less share of ease, convenience, and abundance than those who work not at all. The rich and opulent merchant who does nothing but give a few directions, lives in far greater state and luxury and ease and plenty of all the conveniences and delicacies of life than his clerks, who do all the business. They too, excepting their confinement, are in a state of ease and plenty far superior to that of the artizan by whose labour these commodities were furnished... Thus he who as it were supports the whole frame of society and furnishes the means of the convenience and ease of all the rest is himself possessed of a very small share and is buried in obscurity. He bears on his shoulders the whole of mankind, and unable to sustain the load is buried by the weight of it and thrust down into the lowest parts of the earth, from whence he supports the rest.
    Lectures, 29 March 1763, p.341
    Page number is taken from the Glasgow Edition of the Works and Correspondence of Adam Smith, Vol. 5, Liberty Fund (1982).

  2. William Stanley Jevons, The Theory of Political Economy, London: Macmillan and Co. (1871); Leon Walras (translated William Jaffe), Elements of Pure Economics, London: Allen and Unwin (1954/1870); and Carl Menger (translated by J. Dingwall and B. F. Hoselitz), Principles of Economics, New York University Press, (1981/1871).

  3. It seems to be an article of faith among Austrian economists that they alone can explain why a diamond (which has little practical benefit) is expensive, but water is cheap. For a long time, economists explained that diamonds were extremely scarce, arising from the immense amount of labor required for "production"; whereas water could be captured and exploited with little labor. If social relations made water consumption more urgent than it is, then the value of water would rise relative to that of diamonds.

    This was not at all a puzzle for economists until the utility theory came along. The distinction between use-value and exchange value was used for explaining the mechanism by which the "natural price" was aligned with the "market price" of the good, and was not relevant to the question of what the intrinsic value of water or diamonds was. Relevant passage in Eugen Böhm-Bawerk is The Positive Theory of Capital, III.iii-§2, where he accuses [classical] economists of confusing usefulness and use value.

  4. According to Eugen Böhm-Bawerk, when an entrepreneur paid the workers, he did so long before their labor resulted in business revenue; therefore, their wages constituted a loan. Oddly, this was somehow an anomaly for labor, but not for things he bought from other entrepreneurs like inventory, raw materials, fixtures, machinery, and so on. See Böhm-Bawerk, The Positive Theory of Capital, VI.iv-§3, It would appear to me this explains the politically conservative view that contractual obligations to workers are frivolities that do not warrant respect, whereas less-than-contractual obligations to rich parties (like bondholders and shareholders) are sacrosanct.

  5. From Karl Marx, Critique of the Gotha Programme, I.:
    Labor is not the source of all wealth. Nature is just as much the source of use values [...] as labor, which itself is only the manifestation of a force of nature, human labor power. the above phrase is to be found in all children's primers and is correct insofar as it is implied that labor is performed with the appurtenant subjects and instruments. [...] And insofar as man from the beginning behaves toward nature [...] as an owner, treats her as belonging to him, his labor becomes the source of use values, therefore also of wealth.
    This leaves a lot of room for interpretation (and I am not the best interpreter!) but clearly Marx's "LTV" applies only to a society predominantly under a capitalist mode of production.

  6. Krul (2012)

Sources & Additional Reading

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Matthijs Krul, "Steve Keen’s Critique of Marx’s Theory of Value: A Rejoinder," Notes & Commentaries blog (4 July 2012)

 Matthijs Krul, "A Critique of ‘Human Capital’ theory," Notes & Commentaries blog (11 October 2010)

Adam Smith, Wealth of Nations (5th Edition), London: Methuen & Co., Ltd.(1904/1776)

Robert Vienneau, "Frequently Asked Questions about The Labor Theory of Value" version 1.2.4 (Last modified April 2004)

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07 August 2012

Reading Sraffa (3)

(Part 1 | Part 2)

In referring to passages from Sraffa (1960), I will hereafter refer to "paragraph" numbers used by Sraffa (thus: §11). These are numbered 1-96 through the entire book.


At this point it probably is a good idea to tell readers that I really want to avoid making any conclusions about Piero Sraffa's Production of Commodities by Means of Commodities, at least for now. One reason is that if conclusions are needed, I'm not a good source: I'm not an expert, and in intellectual pursuits like this I'm more interested in collaborating with other people to find out what manner of wisdom Sraffa has to offer.

In Chapter III (starting with §13), Sraffa introduces the variation technique he will use to develop the model. Using the standard methods of matrix algebra, he examines the effects on the solution sets of varying selected coefficients.1 The most important of these is variation in the wage rate in response to the introduction of profits; another is allowing for the effect of time discounting on the value of labor contributions to value (§47).

In §12 (immediately preceding Chapter III) he had dropped the bombshell that there were too many unknowns for the available equations; the system of linear equations has no unique solution, and in fact, no market economy can exist with a unique "market" value of labor or capital. I was led to wonder why Sraffa didn't just select one commodity at random to serve as the unit of value, the way Léon Walras had done. Sraffa could also eliminated another variable by exploiting w's inverse linear relation with r. Both questions are addressed by Sraffa in Chapter III.

First, Sraffa needed a monetary unit that actually said something explicit about value. Supposing he picked commodity a, which happens to be exceptionally labor intensive. If there is some macroeconomic event that leads to a shift in w from 1 to 0.90, then this will have a disproportionate effect on the price of a (pa).

Second, he is very concerned about the scenario of surplus production in which products as outputs are not in proportion to products as inputs. While it is reasonable to suppose that any complete economic system (i.e., one that imports/exports nothing) with raw inputs ak will produce outputs (a) → (k), such that each output is greater than or equal to its supply, it is not reasonable to assume the inputs enter the economy in the same proportions that they are released. The a industry might produce a very large surplus (from which a lot of non-basic products are made), while b produces scarcely any at all.

Proportional Relationships of Commodity Inputs & Outputs

By the way, we're interested here in ratios; so, for example, as inputs we have a ratio of b as 1.4 × a, and so on, whereas outputs leave a ratio of b as 1.9 × a (etc.)

. . .
. . .
. . .

In terms of outputs, (a) ≥ a, (b) ≥ b, (b) ≥ b, . . . (k) ≥ k; but notice the ratios of (a):(b):(c): . . . :(k) are quite different from the ratios a:b:c: . . . :k (Sraffa 1960, §4).

An obvious corollary is that the ratios of a:(a) are not necessarily equal to b:(b), or to c:(c), and so on.

The reason (a) has to equal, if not exceed, a, is that if it did not, a would have to be imported from somewhere--which would introduce a feature of the model for which no provision has been made. We could have the economy be changing relative proportions (for example, if Sraffa were interested in modelling the effects of peak oil, and a = petroleum, the annual supply of which is presumed to be declining), but for now we're not interested in that.

There could be likewise be substitution (as the supply of a is less than output, somehow (b) → (k) are used to produce more of a for the following production cycle, each year. For example, (b) → (k) might collectively be used to synthesize petroleum products from coal to make up the deficit in (a). But in that case, we would treat the synthetic a as its constituents, ak.2

It is not a problem for the realism of the model that outputs of commodities are not in the same proportion as inputs. In real life, of course, such proportions would be constantly changing somewhat, in response to depletion of some resources and new discoveries of others. Production functions would also change, and inevitably require different inputs of commodities.

More significantly, though, is that all of the surplus of production over inputs is the national income. That "income" takes the form of a mix of commodities that are not required to produce next year's supply; all consumption goods are made from the national income, whether consumer durables, food, public buildings and fixtures, and so on. Plausibly the massively complex arrax of millions of different consumption goods we actually consume is derived, á la Walras, using constrained optimization, utility functions, and so forth. But it is limited by the supply of raw materials available for use.

The two problems are interrelated.

When the wage is reduced to less than 1.00 of the economic surplus (AKA national income), the exchange values of the commodities required to permit the production cycle to begin anew, with the requisite surplus, will shift. This is a built-in problem of the matrix

(Aapa + Bapb + . . . + Kapk)(1 + r) + Law = Apa
(Abpa + Bbpb + . . . + Kbpk)(1 + r) + Lbw = Apb
. . . . . .
(Akpa + Bkpb + . . . + Kkpk)(1 + r) + Lkw = Apk

where the capital letters are all given, and the prices/wages are the unknowns. Sraffa laments that it is impossible to to say if changes in the overall price level reflects global change, or merely a peculiarity of the measuring standard. Moreover, any such peculiarities would be multiplied through the multistage production process that actually exists, in which a major non-basic input of most consumer goods consists of semi-finished parts produced in the past--the "residue"of prior production periods:
The relevant peculiarities, as we have just seen, can only consist in the inequality in the proportions of labour to means of production in the successive 'layers' into which a commodity and the aggregate of its means of production can be analysed; for it is such an inequality that makes it necessary for the commodity to change in value relative to its means of production as the wage changes.
Sraffa (1960), §23
The only way to avoid this problem is if we already know what the average contribution of labor is towards the production of the total national product, and find some commodity for which labor's contribution is the same as the overall average. Commodities for which labor played a larger role would (in the event of a wage reduction3) experience a comparative reduction in the cost of production. If prices in the matrix remained the same, and the values of the total amount of each commodity remained fixed, then the producers of the labor-intensive firms would be running a surplus, while those in the commodity input-intensive industries would be running a deficit (Sraffa 1960, §16).4 So we need to find an object that not only straddles the average in its direct consumption of labor, but also in its indirect consumption of labor.

This is not likely to exist, so Sraffa recommends instead a bundle of all the commodities (standard commodity), in proportions determined the following way: imagine a modified version of the matrix economy above, in which each of the production functions for AK are multiplied by a different constant. These constants are chosen so that the new, modified matrix produces goods (a) → (k) in the exact same ratios to each other as they appear (ak) as inputs. This new standard commodity, once derived, can be further reduced to the lowest common denominator: it now is guaranteed to remain unchanged in relative price by any change in the labor wage rate.

Sraffa will hereafter use the standard commodity for virtually all econometric observations about his system. For example, when talking about growth of the economy, he is not concerned about reducing the economy to monetary values for the purpose of comparing heterogeneous bundles of goods. Now expansion is applied uniformly to the standard commodity.


The net product is important because it represents the output that is available for consumption or for future investment. Everything else is required in order to keep the system running the way it is. But Sraffa also links it to the idea of the standard commodity, because he is keen to avoid ex post facto exchange values. The reason is that he is not interested in saying the market is good because it does what it does; one of the things the market needs to do is be capable of finding exchange values of goods that meet certain goals. If the market persistently undervalues some commodities relative to others, then the producers of the undervalued commodities will be compelled to underproduce, leading to the system's inability to reproduce itself.

Another point is that eventually he is going to need a uniform concept of value, which will survive technical or macroeconomic changes.5 The point of the standard commodity was that, instead of attempting to find out the value-weighted share of labor in the economy at the same time as finding the exchange values of all the various commodities (which would be impossible), or assuring "losers" that, by definition, their loss of income was "fair," it was necessary to restrict economists to unimpeachably homogeneous data.6 Instead of begging the question about the value of various components of net national product/economic surplus, Sraffa wants to rely on measures that are irrefragable.

One especially challenging discussion (Sraffa 1960, §26) refers to something called the standard net product (SNP). We suppose that the entire labor force is engaged in the production of the standard product (defined above), i.e., the entire laborforce is engaged in an economy that puts out exactly the same ratio of commodities as are used as inputs in the production cycle. Given whatever the prevailing rate of surplus is (R), a certain percent of this imaginary economy will be surplus of output over input.

That is the standard net product.

This represents an imaginary version of the economy--a virtual economy inside of a virtual economy. The standard ratio is the ratio of commodities to each other (reduced to a lowest common denominator). Sraffa is interested in the relationship of the SNP to the whole standard gross product (SGP), or R. This is the rate of surplus, and both profits and extra-subsistence wages must come from it.

The point of this exercise is that the ratios of wages and profits to total product always consist of multiples of the standard ratio/standard commodity, and therefore such ratios used to represent the share of wages or profits in the SNP are, in fact, literal ratios of apples to apples, oranges to oranges, and kilos of potash to kilos of potash.

Sraffa proceeds to prove that mathematical inferences made about wages and profits in regards to the SNP actually do apply to the "actual economy"--by which we mean, the virtual economy wherein the standard system is a virtual economy (Sraffa 1960 §§31-34). The end-user products that people actually consume, being non-basic goods not used in the production of other goods, play no role in this calculation (Sraffa 1960 §35).

(Part 4)

  1. In matrix algebra, one uses a set of n equations to solve for n unknown values. Hence, the solution of a problem in matrix algebra is known as a "solution set." The coefficients are the known values in the linear equations.

  2. Notice the use of parentheses to indicate outputs versus inputs. This is introduced in Sraffa (1960) in Chapter VII (§51), where the linear equations that produce a single commodity per each are superseded with equations that produce two.

    Supposing in 2012 a deficit occurs in the production of a, such that a < (a). In order to produce the same complement of commodities in 2013, some amount (a') = [ a - (a) ] has to be produced by the surplus of (b) → (k), which are all outputs. In 2013, the output (a') is now input a', which is added to (a) to permit the 2013 input of a to match what was available in 2012, plus a likely surplus. Readers see that a' is nothing more than a fraction of the 2012 surplus of (b) → (k) over bk. If we treat it as such, then a ≤ (a).

  3. The reason why Sraffa is always referring to wage reductions is that he is analyzing what happens to the solution set if you start out assuming w = 1.0, and go downward from there to permit a rate of profit. This progression is necessary because it permits Sraffa's model to capture more and more aspects of a real economy.

    For people who are agitated by the suggestion that wages are inversely related to profits (thereby implying the need for class struggle)--remember that in Sraffa's model, economic growth requires profit, and greater profits correspond to higher growth rates--which, in turn, cause higher absolute wages.

  4. Here is probably a good time to mention that Sraffa's matrices employ very advanced mathematical methods, and cannot be solved using the methods of linear algebra. See K. Vela Velupillai, "Sraffa's Mathematical Economics--a Constructive Interpretation", Discussion Paper, Universita' degli Studi di Trento—Dipartmento di Economica (Feb 2007). However, Velupillai mainly asserts that the systems of equations are good ones and Sraffa has included sufficient mathematical proofs. Another, much more helpful essay is Peter Newman (1962), pp. 58-75, which walks readers through the mathematical analysis used by Sraffa.

    That said, I think Newman's criticism of the model is mostly missing the point.

  5. By "technical changes," I am including changes in the production functions, whether of the sort that Sraffa can be reasonably expected to have provided for, or the ones he will introduce later in the book. By "macroeconomic changes," I am talking about changes in any of the various conditions that can lead to profits rising relative to wages (including, for example, an economic boom accompanied by rapid economic growth).

  6. A recurring theme in conservative political discourse about economics is that the government "shouldn't be picking winners and losers," because that is the job of the market. A logical corollary of this, somewhat less often spelled out, is that the decline in worker share of net national income reflects the irrefutable judgment of the market. As always, we need to ask, "In what sense is the judgment of the market irrefutable?" For an example of someone claiming that secular losses in worker income are immune to appeal, look up the long-simmering debate over the "skill-biased technical change" (SBTC)hypothesis. For a examination of SBTC claims (and debunking of them), see David Card & John E. DiNardo, "Skill Biased Technological Change and Rising Wage Inequality: Some Problems and Puzzles" , National Bureau of Economic Research (2002).

Sources & Additional Reading

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

Peter Newman's "Review of Production of Commodities by Means of Commodities," Schweizerische Zeitschrift für Volkswirtschaft und Statistik, Vol. XCVIII, March 1962, pp. 58-75.

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06 August 2012

Reading Sraffa (2)

(Part 1)

In referring to passages from Sraffa (1960), I will hereafter refer to "paragraph" numbers used by Sraffa (thus: §11). These are numbered 1-96 through the entire book.

"The Matrix" (1999) is an action movie written by Lana and Andy Wachowski; it is based on the premise that the world that we observe is actually a simulacrum of reality, and that the real world is a dystopian nightmare in which most humans are suspended in pods and force-fed neural lies. The protagonist, a computer hacker known only as Neo (Keanu Reaves), is introduced to the harsh reality when he accepts the invitation to ingest a red pill by Morpheus (Laurence Fishburne).

The title of the movie refers to the computer-generated simulacrum itself. Possibly the authors chose to this name because of the buzzword then popular for massively parallel multiprocessor computing used in applications like CGI and virtual reality.


Sraffa's Production with a Surplus

Let La, Lb, Lc,... represent the labor used in the production of commodities a, b, c... and w be the wage per unit of labor; the sum of all the labors is 1, representing the total labor available for production.

The production equations takes the form

(Aapa + Bapb + . . . + Kapk)(1 + r) + Law = Apa
(Abpa + Bbpb + . . . + Kbpk)(1 + r) + Lbw = Apb
. . . . . .
(Akpa + Bkpb + . . . + Kkpk)(1 + r) + Lkw = Apk

where r is the share of the total surplus (R) that goes to the non-labor residue--i.e, the profit.

The term "total surplus" (which Sraffa calls "net product" or "national income") has a special meaning that will be explained later.

Taken from Sraffa (1960), §10-11
As mentioned in Part 1, both Léon Walras and Piero Sraffa (initially) tended to treat production as if it were a single act of exchange. This may seem odd, especially since the outputs are always a different mix of inputs. Steel and coal are exchanged by producers of either to produce... steel and coal. In the case of steel and coal, that's actually fairly realistic, but for the huge variety of other objects produced, it isn't: bookshelves or chairs have no role in the production of bookshelves or chairs; table saws have no role in the production of table saws (but they are essential for bookshelves).

Most production processes are largely uni-directional, leading from a standard class of goods (raw materials, electricity, PNG, clean water) to finished goods. For Sraffa (and Walras) capital goods are represented symbolically, as bundles of heavily processed raw materials. The processing of pig iron into any of a large number of available grades of steel, followed by drop-forging, milling and hogging, assembly, and installation are all subsumed into various retail markups. For them, a marketed product like a piano represents its raw inputs plus many middlemen.

The purpose of this is to explain the process of abstraction: are such matrices inherently unrealistic in representing the economy? Sraffa occasionally refers to non-basics (e.g., §6, §35, §57-61)--commodities that are enter the economy only in the production of themselves, or else as endpoints of production. Because of the irreversibility of "commodity refinements" that transform raw materials into finished goods, one could argue that the vast majority of end-user goods are non-basics. Electricity, flour, and lumber are generic examples of end-user goods that are not confined to end-use; (most) manufactured goods, groceries, and services definitely are.

Non-basics are the dark matter of matrix economies: Sraffa argued that they could be ignored in resolving questions his matrix had been contrived to solve.1 Since they could be ignored, it was possible that Sraffa's conceptual economy might consist mostly of non-basics whose price was determined by the commodities used to produce everything else. In order for this to be physically possible, of course, there had to be an "economic surplus" consisting of goods or services that were "distributed" in some way to the general population depending on their contribution to production.

All of this relates, ultimately, to the question of whether or not the neoclassical school could actually explain the price of factors. If, according to winners Joan Robinson and Nicholas Kaldor (and even the main loser, Paul Samuelson), it could not, there was a compelling case to be made that no such equilibrium existed. While ideologues might hail the infallibility of the market, the problem here was that "the market" was at risk of being an idea without a coherent explanation. Mainstream economics teaches that "markets" find the socially optimum prices and quantities of goods and inputs spontaneously; but if the most rigorous economic analysis proves that such a thing is inherently unfindable, then "the market" is actually a figleaf for something else.


Several writers have attacked the approach Sraffa used for lack of realism. One that can dispensed with easily is the system ignores such questions as firms, consumer preferences, etc. However, Sraffa doesn't need to include things that make it harder for the market to identify clearing prices for goods. Sraffa's system is also missing technical shocks, such as a sudden fluctuation in the supply of a.

One author objected that his schemata lacked any marginal variation in output (say, as a result of a change in tastes).2 This is the result of walking into a conversation without knowing anything about what came before or after. Sraffa's object was to make the system explicitly independent of marginal anything. The marginalists (Walras, W. Stanley Jevons, and Carl Menger) had introduced the matrix economy to demonstrate the concept of general equilibrium; Sraffa's idea was to demonstrate the impossibility of any marginal theory of value determining the correct value of prices. For Sraffa to actually go on to deal with utility functions would be like General Sherman's march through Copenhagen: not terribly helpful for winning the American Civil War.

The absence of individuals, firms, or anything else but unimproved heaps of commodities serves to give the general equilibrium matrix every possible chance to find the prices (including the residue). The whole point of free markets is that they possess vastly more real-time information than anyone could possess. Assuming they are perfect, they will find whatever institutional arrangement is socially ideal, and Sraffa does not presume to limit this with his meager faith.3

Another possible objection I expected to see was that Sraffa had arbitrarily restricted the number of equations to the actual number of commodities, or that the number of commodity inputs (ak) was necessarily equal to the number of commodity outputs. At the very least, I expected someone to point out that Sraffa could have arbitrarily set the price of a as 1, and made the prices of bk in units of a.4 The reason for the identity of inputs and outputs is that the inputs have to come from someplace (so that Sraffa's economy, simple though it may be, is a global one); and once they have entered the economy, they never leave, they merely change form.

Taking an example, we can suppose a future cloud architecture-based computer that manages to gather all of the data required to simulate the global economy à la Sraffa. It includes tens of thousands of inputs and outputs, and a similar number of industrial procedures. For descriptive purposes, I'll pretend that the "industrial procedures" are linear equations that represent a recipe for the production of any one of those tens of thousands of outputs from a combination of specific quantities of the exact same items--as inputs. But rather than represent the billions of products available for sale (including things like replacement parts or billable services) in a modern global economy, a typical "production genome" is exclusively concerned with explaining how each of the various key inputs are combined to result in each other.

So, for example, an inventory of key industrial ores, energy resources, and farm commodities might actually suffice. The "economy" would consist of estimates of how much was required to recover all the inputs for the following year of production, adjusted to represent indirect inputs by way of imports.5 While there would be additional outputs not mentioned, these are subsumed into the "surplus," and since they are contain physical quantities of goods not actually used in reproduction of the original inputs, represent net national income.

So while there have been some objections regarding the realism of Sraffa's stylized economy, they don't prevent it from symbolically representing a vast range of plausible economies.

(Part 3)


wage unit: in Production of Commodities, Sraffa's wage unit has a very special significance. The wage unit is introduced as a share of the surplus produced by the economy (which itself has to be understood in a very special sense). Very loosely speaking, the economy may be described as either (a) operating at a subsistence level, in which case the wage can be ignored, and we can pretend the goods required to produce other goods are partly used to pay workers in kind; or (b) operating at a surplus, in which case we're interested in wages as a share of the excess of output over inputs. Again, very loosely speaking, the wage is expressed as a share of the economic surplus produced, with a potential range of 0 → 1.00.

The "very loosely speaking" qualifier, with red typeface, refers to Sraffa's attempts to address important mathematical quandaries I'll explain in a later part of this series.
  1. See Walras (1984), §167 (i.e., pp.211-212). Walras wrote the first edition in 1877; Sraffa wrote his first draft in 1926. I'm guessing both contained essential elements of the texts available to me. So Sraffa was writing about half a century after Walras, and was in part arguing with him over the ability of pre-marginalist Classical economics to explain the prices of factors of production (as, for example, the rate of wages and profits). The concepts of "non-basics" is Sraffa's; Walras had used a totally different approach.

    Walras divided physical valuables into income and capital; he used examples of livestock, that could produce eggs and milk (i.e., a stream of wealth), or be slaughtered to produce meat (i.e., objects of immediate consumption); or trees, that could produce fruit, or else be cut down to be burned as fuel. Buildings and machines "by their very nature" are items of capital, not of income; but he goes on to claim that "every kind of social wealth, whether from its intrinsic nature or the use to which it is put, can serve either more than once, or only once," and this determines if it is capital or income. In this sense, Walras simply regards social wealth as commodities that, through abstraction, may be either end-user or "basic" (as Sraffa would put it).

  2. Robert P. Murphy, "Sraffa's Production of Fallacies by Means of Fallacies" Mises Daily (Ludwig von Mises Institute) (7 April 2004). Murphy erroneously treats Sraffa as a quasi-Marxist popular amongst all manner of leftists. This is exactly wrong: Sraffa is generally regarded as having debunked the labor theory of value, and therefore burying academic Marxism in economics. See Steve Keen, Debunking Economics, 2nd Edition, Zed Books (2011), Chapter 17: "Nothing to Lose but their Minds." Murphy also jumps to the erroneous conclusion that because Sraffa proved "the market" could never find a unique market-clearing value for all necessary inputs, because there were too many variables, it therefore followed that it made no difference what these values were (with respect to productivity, anyway). See next post, footnote 3.

  3. In other words, modeling all possible [free market] economies with the greatest possible simplicity reduces the risk of a failure of imagination on the part of the person doing the modeling. If that is how you actually model the economy, then negative conclusions (statements about what is impossible) are more convincing than if you impose arbitrary restrictions on what sort of values the matrix is allowed to return.

  4. The reason why is that Sraffa needs to set the price in units of the standard commodity. The standard commodity and its importance will be explained in a future post in this series, but lest that future post never come, it forms the subject of Chapter IV in Sraffa (1960), p.20.

  5. For a list of key industrial ores, see "Commodity Statistics and Information" from the website of the USGS. For statistics on coal, natural gas, petroleum, and uranium, see the US Department of Energy's Energy Information Administration Web site. For farm commodities (including livestock and timber), see the National Agricultural Statistics Service (NASS).

Sources & Additional Reading

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

Graham Joncas, "Piero Sraffa’s Non-Economics: An Introduction, part I," Linguistic Capital (21 May 2012): analysis of Sraffa's interaction with interlocutors Gramsci, Hayek, and Wittgenstein. Sequel pending as of this writing. Includes a brief discussion of Production of Commodities by Means of Commodities and a recapitulation of the Cambridge Capital Controversy.

Alvaro Cencini, Macroeconomic Foundations of Macroeconomics, Psychology Press (2005)

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03 August 2012

Reading Sraffa (1)

In his book, Debunking Economics (2nd Ed), Steve Keen mentions Pierro Sraffa in connection to the dilemma of capital formation.

Capital formation is treated in [neoclassical] economics as occurring between static phases of production; capital itself is treated as if the financial meaning of "capital" were identical to the industrial sense of "capital." Economists seldom include any consideration of the industrial steps between financial "capital" (i.e., income diverted from present consumption into production, in the hopes of future income) and industrial capital (i.e., tools, buildings, fixtures, or inventories).1 They treat the provision of "spare" income as a hydraulic process, translating directly from money to productive instruments in a single step.

In the 1950s, a major preoccupation of economists was modeling simple economies using a set of equations. The idea was to simulate the behavior of a real economy with a virtual one. Obviously, a plausiblely complex economy complete with gas chromatographs and botts dots is not one that can be described with a manageable set of equations; so the economy was simplified immensely. Instead of acknowledging the existence of different modalities of capital, it was treated as a single fluid whole. Sraffa bucked this trend; he insisted on models of the economy that recognized the existence of physical objects used as capital.

The main historical impact of Sraffa was to attack the idea that a market economy can find a unique value of anything. A basic principle of neoclassical economics (and its successor, DGE) is that, in a free market, the share of economic surplus going to capital or labor will reach a socially optimum value.


Walrasian General Equilibrium

The pool of available commodities is represented as a matrix of n goods and m people. People are indicated as 1, 2, 3... and goods are represented as letters a, b, c...

Qa = qa, 1 + qa, 2 + qa, 3 . . .
Qb = qb, 1 + qb, 2 + qb, 3 . . .
Qc = qc, 1 + qc, 2 + qc, 3 . . .
. . . .
where Qa represents the total quantity of a, and qa, 2 represents the amount of a held by 1.

"Redistribution" takes place to meet the preferences of the original owners (who presumably made them in the first place). In this initial presentation of the problem, the prices of the commodities pb, pc,... are fixed (the price for commodity a is set to 1, and all other prices are in units of a).

Person (1) buys x'1, y'1, z'1,... of commodities a, b, c... (this is Walras' notation, and it wasn't thought out very well!) to wind up with
q'a, 1 + x'1 = qa, 1 + x'1,
q'b, 1 + x'1 = qb, 1 + x'1,
q'c, 1 + x'1 = qc, 1 + x'1,

. . . .

And Walras proceeds to demonstrate that markets will clear. Later he introduces the production process and utility functions, in a comprehensive effort to outline the system of equations the market must "solve."
Chart taken from Walras (1984), p.182-184
This is a loaded concept, and it's unlikely I'll explain it in a way that satisfies anyone. But neoclassical economics presupposes there is--at any given moment--an equilibrium value for prices and quantities of output that maximizes public welfare. If one had infinite knowledge of the production processes and utility functions, one could determine the amounts and prices of all the things the economy produced, such that everyone had the most satisfying mix of goods they could afford. An additional benefit is that the prices of goods would stimulate the correct amount of production, and cover the costs of inputs.

Lest anyone get the wrong idea, another crucial point of economics is that of course no one could possibly have remotely enough information to do this, but the free exchange of goods and productive inputs is constantly moving us closer to this equilibrium. Changing objective realities, such as shifting supplies of inputs, or changing preferences, pull us further away from it.

"Covering the costs of inputs" means--in this precise context--enough of a return to the supplier of each input that the supplier is motivated to provide enough to that particular goods producer to produce the market's demand for that particular good. So, for example, if the input is hours of productive labor, then there is a socially optimum wage that is best determined by the market.

The reason why I don't just say "Let the market decide!" is that, if economics is claiming to be a science instead of a religion, it has to claim there is some concrete social optimum out there to be discovered by the market, it has to explain what this social optimum will accomplish, and it has to explain why the market is likely to find it.


Sraffa's System

Superficially differs from Walras (above) in representing production rather than redistribution (though commerce), but mathematically the concept is similar: in Sraffa, redistribution occurs because producers of wheat, iron, and pigs are exchanging quantities of those commodities in order to produce another batch of each.

240 qr wheat + 12 ton iron + 18 pigs = 450 qr wheat
90 qr wheat + 6 ton iron + 12 pigs = 21 ton iron
120 qr wheat + 3 ton iron + 30 pigs = 60 ton pigs

In this example, the output is equal to the inputs, and exchange must take place again afterwards so that the same amount of everything (as before) can be produced. Subsequently, Sraffa switches to scalable matrix algebra to allow for many different commodities.

Here he shows the arrangement of identities required to ensure continuous production. The term Ba prefers to the amount of commodity b used in the production of a. The amounts used in the production of each commodity are fixed by available methods and the amount of each commodity "required" by Sraffa's "economy."

Aapa + Bapb + . . . + Kapk = Apa
Abpa + Bbpb + . . . + Kbpk = Bpb
. . . . .
Akpa + Bkpb + . . . + Kkpk = Kpk

When solved, the values of p ensure that the producers of each commodity will be able to buy the inputs they need in order to reproduce the output of the previous cycle.

Subsequently, Sraffa develops his imaginary economy to include wages and economic surplus.

Taken from Sraffa (1960), p.4
When Sraffa published his book, it triggered a major controversy over the question of general equilibrium in economics. With respect to mainstream economics in the USA, neoclassical economics had never really been displaced.2 In effect, Sraffa was writing a treatise on general equilibrium in a world that was still Walrasian; he had written a rough draft in 1926, then dusted off the manuscript, and published it in 1960--still in a Walrasian world.

The book is puzzling to most readers. It's not obvious if Sraffa is trying to carry a torch for his beloved David Ricardo, or if he really is making a pointed gesture for the benefit of Walrasians. Rather than construct a model economy with indifference curves and household endowments of capital/labor, Sraffa uses merely production functions and abstract commodities. The capital contribution to production represents all inputs besides labor. The commodities must be end products of production as well as inputs of production. Over successive chapters, Sraffa builds up his analysis to include a richer and more detailed model of the economy. But to what purpose?

In subsequent posts, I will attempt to explain what this purpose could be.

(Part 2)


  1. Classical economists, such as David Ricardo, made a major distinction between "circulating capital" and "fixed capital." Inventories, whether of merchandise or raw materials, are "circulating capital"; firm revenue is directly related to inventory moving out the door. Machinery, buildings, and fixtures are "fixed capital"; firm revenue may increase if the firm buys more of it, but the object is to keep it in working condition as long as possible before it is used up.

    Neoclassicals like Lionel Robbins noted that this was actually an ambiguous distinction, since the difference between f- and c-capital was really the time period of production one was looking at. Inventory of finished goods one hopes to sell is clearly circulating, but what about drill bits used in the process of recovering petroleum? The exploration company would like to conserve them, but over the long run, the more bits used, the more petroleum recovered (and sold to refineries).

  2. I actually discovered this when I was compelled to read Paul Samuelson (Foundations of Economic Analysis, Harvard University Press (1947). Subsequently, I noticed the persistence of neoclassical assumptions and methods all through the 1960s and 1970s. For readers shocked by my claim, this will seem like a weak defense. However, my posts on Steve Keen and Hyman Minsky will treat this in greater detail.

    For an account of the Cambridge Capital Controversy "provoked" by Sraffa's book, please see Eckhard Hein & Engelbert Stockhammer (2011), pp.3-4.
economic surplus: excess of output over input; not to be confused with "surplus value," a concept from Marxian economics.

Sources & Additional Reading

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

"Debunking Economics, Part V: The Holy War Over Capital," Unlearning Economics (26 July 2012): for a summary of Steve Keen's account of the Cambridge Capital Controversy

Eckhard Hein & Engelbert Stockhammer, A Modern Guide to Keynesian Macroeconomics and Economic Policies, Edward Elgar Publishing (2011). Hein & Stockhammer's account of the Cambridge Capital Controversy includes a plausible explanation of why Joan Robinson's victory over Samuelson and Solow was ignored by the economics profession.

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