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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