More on Slutsky's Equation as Pareto's Solution

In an article published in this journal, Peter Dooley (1983) has pursued an intriguing footnote in Eugen Slutsky's famous 1915 paper on the comparative statics of consumer demand. Slutsky ([1915] 1952, 39 n. 3) credited Vilfredo Pareto (1892, [1909] 1971) with being the first to arrive at Slutsky's equation (43), which is equation (55) in Pareto 1893, equation (75) in Pareto [1909] 1971, and equation (3') in Dooley 1983. ¹ That equation, which shows the impact of a change in p_i, the price of good i, on x_i, the quantity of good i demanded, is:

where u´ is the marginal utility of income, R is the Hessian determinant of the utility function, u = u(x₁, x₂, . . . , x_n), M is the determinant of the Hessian bordered by the prices of the n goods, and M_0i and R_ii are cofactors of M and R, respectively² (see Dooley 1983 for details). The notation here follows Slutsky ([1915] 1952) and Dooley more closely than it does Pareto (1893, [1909] 1971, 1911).

Dooley discusses Pareto's derivation of equation (1) and notes that Slutsky followed this derivation closely. He also points out that Pareto did not separate his expressions for ∂x_i/∂p_i or ∂x_i/∂p_j into separate income and substitution effects. In addition, while Pareto (1893, [1909] 1971, 1911) did examine general, nonadditive utility functions, in order to arrive at meaningful comparative statics propositions he had to assume that the utility function is additive (so that the marginal utility of each good is independent of the quantity demanded of each of the other goods), and that the marginal utility of each good falls as more is consumed (Gossen's law). Under these restrictive assumptions, Pareto (1892, [1909] 1971, 1911) did show that demand curves slope downward.³

Thus, Slutsky's contribution to the comparative statics of consumer demand consisted of two parts. First, as Dooley notes, he divided the expression in (1) into separate income and substitution effects, and showed that the substitution effect is necessarily negative.⁴ Second, he provided the first successful analysis of general nonadditive utility functions. That is, he proceeded without assuming diminishing marginal utility and permitted the marginal utility of any good to depend generally on the quantity of any other consumed.⁵

Dooley has usefully reminded Anglophone economists of Pareto's pathbreaking comparative statics analysis of consumer demand. Here, I extend Dooley's analysis by suggesting why Pareto did not derive the income and substitution effects of a change in price while Slutsky did. The explanation lies in the fact that Pareto never derived an expression for the impact of a change in income on quantity demanded; without such an expression, he could not have derived Slutsky's equation from his solution. I examine some possible explanations as to why Pareto did not solve for the effect of a change in income on quantity demanded.

Pareto and Slutsky on Demand Functions

Pareto (1892, 1893, [1909] 1971, 1911) posed the consumer's problem as choosing n goods, x₁, x₂, . . . , x_n, to maximize the utility function:

subject to the budget constraint:

where x_i0 represents the household's initial endowment of x_i, and p_i (for i ≠ 1) represents the relative price of good i in terms of good 1. Good 1 is thus the numéraire. In this version of the budget constraint, the consumer is initially endowed with nonnegative quantities of each good. He sells in the market a portion (perhaps zero) of his endowments of some or all of these goods to finance his purchases of those goods that he wants to consume in quantities exceeding his endowment.

Given Pareto's statement and analysis of the consumer's problem, the question naturally arises of why Pareto should have proceeded so far with his mathematical analysis in 1892 and 1893 and repeated this analysis in the appendix to the 1909 French translation of the Manual and again in his 1911 article, "Economie Mathématique," published in the Encyclop...