The Technique of Comparative-Static Analysis in Whewell's "Mathematical Exposition"

This is an inquiry into the development of mathematical economics. In particular, it examines a part of William Whewell's 1829 work, "Mathematical Exposition of Some Doctrines of Political Economy," focusing on the technical aspect of it.

Whewell ([1829] 1968, 1) was explicit about what he intended to accomplish: "I hope . . . to make it appear, that some parts of this science of political economy may be presented in a more systematic and connected form, and . . . more simply and clearly, by the use of mathematical language than without such help." That is, his intention was not to develop any economic theory but to illustrate a new method of economic investigation.

In 1830 and 1850 Whewell published more papers for the same purpose, but neither these nor the earlier paper attracted much attention from economists. An exception was William Stanley Jevons, who himself was an enthusiast for mathematical economics. His review of Whewell's work in this field, however, was far from being enthusiastic. Jevons (1871, 16) might have missed the point when he remarked that Whewell's papers were "a mere translation in symbols of Ricardo's propositions." But Jevons was no less critical of the technical aspect: "[Whewell] regards questions in economy as little more than sums in arithmetic."¹

This "contemptuous verdict," as Joseph Schumpeter (1954, 146 n) disapprovingly called it, was seldom disputed until the 1960s. But we now could cite James L. Cochrane (1970) and Giuliana Campanelli (1982), who praised Whewell ([1831] 1968) for the first mathematical formulation of fixed capital. John Creedy (1989) refers to Whewell's "second memoir" (Whewell [1850] 1968) as "a genuine pioneer attempt to produce a general equilibrium model of international trade." In addition, James Henderson (1996) has published a book on Whewell's mathematical economics that covers various aspects including the "first mathematical formulation of the elasticity concept."

Nevertheless, there remains an aspect yet to be appreciated. It is that Whewell, in his 1829 essay, conducted a comparative-static analysis using differential calculus. He set out eight equations that implicitly but completely defined the functional relation between eight endogenous variables and one exogenous variable. He then successfully solved the equation system to deduce the implicitly defined functions, and he exactly calculated their derivative to obtain a linear approximation of each function.

Surprisingly or not, this remarkable aspect of Whewell's mathematical economics has so far been disregarded. As a result, the "contemptuous verdict" of Jevons is still prevailing as to the kind of mathematics Whewell used. The present article is in part a challenge to such a view.

Whewell's Analysis as He Presented It

The doctrines that Whewell mathematically expounded concern the incidence of taxes on the produce of land. According to "the followers of Mr. Ricardo," as Whewell called them, the consumers would ultimately pay all taxes. Others maintained that most taxes would fall upon the landlords. It is the reasoning of these two parties that Whewell reconstructed in a mathematical form, attributing their difference to some peculiarity of the suppositions introduced in the course of reasoning.

Axioms and Equations

Whewell ([1829] 1968, 15) started with a verbal exposition of six "axioms" before he put them in a more mathematical form.

1. 1. Rent is = produce minus return to capital with profits.

2. 2. Land will be cultivated if produce = or > than profits.

3. 3. There is always a limiting soil.

4. 4. Increase of price α diminution of supply.

5. 5. Price = cost of production + profits.

6. 6. Taxes do not affect the rate of profits.

Whewell then introduced various notations including those for three rates of change (u, w, v). In consequence of taxes on the produce of land, the whole number of acres in cultivation will diminish from a to a-a_n. Accordingly the whole produce will diminish from ar to ar(1-u) and the capital from ac to ac(1-v), where r and c respectively denote the average produce per acre and the capital employed on one acre. This change in production will lead to an increase in price from p to p(1+w) , but no movement in the...