Lump of Labor Day Special: Advanced (Elementary) Concepts in Mathematics

"…if there be but a certain proportion of work to be done; and that the same be already done by the not-Beggars; then to employ the Beggars about it, will but transfer the want from one hand to another…" – John Graunt, 1662.

In a previous post, Graunt Work, I discussed the theological foundations of John Graunt's Natural and Political Observations made upon the Bills of Mortality and hinted at the unacknowledged survival of assumptions of divine providence in expectations of a harmonious social order. This post is concerned with the distinction between the "certain proportion of work" assumed by Graunt (or possibly inserted by Petty) in his discussion of employing beggars and the "fixed amount of work" assumption, alleged by economists to be a widespread fallacy.

Proportion and number are two distinct concepts in mathematics. The distinction is fundamental to mathematical reasoning. The title of Fra Luca Pacioli's classic Summa de Arithmetica, Geometrica, Proportioni et Proportionalita suggests the importance of the concept of proportion.

"One of Graunt's prevailing concerns," Philip Kreager has argued, "was to show that the body politic was integrated in the divine order in virtue of the numerical symmetry, intrinsic proportionality, and correspondences which could be found in its constituent parts." The word "proportion" appears 40 times in Graunt's text, where it consistently refers to the relation of one part to another or of a part to a whole -- not to simple quantities or amounts. "Quantity" occurs only once in the book and "amount" not at all. Quantities and amounts are indicated in Graunt's text by the word "number."

There has been some conjecture regarding the extent of William Petty's contribution to Graunt's book. In his 1939 introduction, Walter Willcox suggested passages may be ascribed to Petty:

"...wherever conjectures, whether numerical or not, are made but no evidence offered in support of them… Furthermore, numerical statements which apply to matters of popular or political interest, but are of no importance for science, are supposed to be his." 

Using Willcox's criteria, the passage about employing beggars is arguably Petty's. Petty shared with Graunt the idea of a unique and profound significance to proportionality. Petty was educated as a physician and had been an anatomy instructor at Oxford. In his preface to The Political Anatomy of Ireland, Petty wrote:

"Sir Francis Bacon, in his Advancement of Learning, hath made a judicious Parallel in many particulars, between the Body Natural, and Body Politick, and between the Arts of preserving both in Health and Strength: And it is as reasonable, that as Anatomy is the best foundation of one, so also of the other; and that to practice upon the Politick, without knowing the Symmetry, Fabrick, and Proportion of it, is as casual as the practice of Old-women and Empyricks."

So if proportion has such a conspicuous significance, how does it get confused with quantity? For example, after citing Graunt on the "proportion of work to be done," historian Peter Buck went on to state that "Graunt presupposes that there is only a fixed amount of work to be done…" apparently oblivious to the crucial distinction between a certain proportion and a fixed amount.

One might suppose that economics has inherited a special concern for proportion from Graunt and Petty, the pioneers of political arithmetic. Unemployment and interest rates; productivity and price indices; efficiency, equality and inequality; elasticity of demand and of substitution are all proportional. Even the intersection of supply and demand is a proportion (an equality). So where does the attributed assumption of a "fixed amount of work" fit in with all this proportionality?

"There is, say they, a certain quantity of labour to be performed. This used to be performed by hands, without machines, or with very little help from them. But if now machines perform a larger share than before, suppose one fourth part, so many hands as are necessary to work that fourth part, will be thrown out of work, or suffer in their wages. The principle itself is false. There is not a precise limited quantity of labour, beyond which there is no demand." – Dorning Rasbotham, 1780. 

"In accordance with this theory it is held that there is a certain fixed amount of work to be done, and that it is best in the interests of the workmen that each shall take care not to do too much work, in order that thus the Lump of Labour may be spread out thin over the whole body of work-people." – David F. Schloss, 1891. 

"Economists have historically rejected the concerns of the Luddites as an example of the 'lump of labor' fallacy, the supposition that an increase in labor productivity inevitably reduces employment because there is only a finite amount of work to do." – David H. Autor, 2014.

Characteristic of the hundreds of denunciations of the supposed lump-of-labor fallacy (Sandwichman has amassed over 500 examples) is a obstinate refusal to substantiate allegations of such a belief with documentary evidence. "We don't have to show you any stinking badges!" Oddly enough, the denunciations also omit reference to particular authorities on the fallacy claim, settling for allusions to a nebulous view held by unnamed "economists" -- the above Autor quote is typical. Moreover, various arguments offered as "proof" of the fallacy of the assumption of a fixed amount of work evaporate with the simple substitution of a "certain proportion" for the conventional "fixed amount."

Of course a "certain proportion of work" is also vague. It might refer to a proportion of work to population, to industrial output, to resource consumption or to virtually anything else. Nevertheless, the alternatives of an "infinite amount of work to do" or of demand for labor that automatically adjusts to changes in the supply of labor are even less tenable.

One might also suppose that the increasing reliance of economics on mathematical modeling would make economists sensitive to the not-so-subtle distinction between quantity and proportion. Such a supposition would be premature, however. Liping Ma's study of math teachers' understanding of math fundamentals in China and the United States, reviewed by Roger Howe, found that:

"…successful completion of college coursework is not evidence of thorough understanding of elementary mathematics. Most university mathematicians see much of advanced mathematics as a deepening and broadening, a refinement and clarification, an extension and fulfillment of elementary mathematics. However, it seems that it is possible to take and pass advanced courses without understanding how they illuminate more elementary material, particularly if one’s understanding of that material is superficial."

So much for rigor. In addition, it's not a sure thing that understanding of math guarantees an unbiased use of it. Kahan et al. found that "more numerate subjects would use their quantitative reasoning capacity selectively to conform their interpretation of the data to the result most consistent with their political outlooks."

In a reply to comments from four economists on his General Theory of Employment, Keynes doubted that "many modern economists really accept Say's Law that supply creates its own demand. But they have not been aware that they were tacitly assuming it." Similarly, although economists may not have expressly believed, "that every individual spends the whole of his income either on consumption or on buying, directly or indirectly, newly produced capital goods," nevertheless, they tacitly assumed it. "They have discarded these older ideas without becoming aware of the consequences."

Keynes was wrong. Economists didn't "discard" the older ideas, they renounced the form in which they were expressed and devised new bottles in which to store the coveted old wine.

The tacit assumptions that Keynes highlighted are assumptions of inherent proportionality (the divinely ordained harmonious social order). Say's Law can be restated as: demand increases in proportion to an increase in supply. The tacit assumption about spending is that an increase in income will result in proportional increases, respectively, in consumption and investment.

One of the inevitable consequences of these tacit assumptions is that the proportion of work to be done must also be a constant. In other words, the alleged lump of labor, which in its distorted "fixed amount of work" formula is refuted by Say's Law is, in its original form of a "certain proportion of work," a corollary to Say's Law!