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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