Consumer Theory and Axiomatics: A Note on an Early Contribution by Luigi Amoroso

In An Analysis of the System of Equations Defining the Consumer’s Behaviour (1928),¹ the Italian economist Luigi Amoroso proves an important proposition that can be summarized as follows: Let a consumer’s utility function be defined on an open consumption set, and let it be characterized by continuity up to second derivatives, concavity, and emptiness of the intersection of its indifference curves with the consumption set’s boundary. Then utility maximization under an appropriate budget constraint will always yield a unique solution.

While Joseph Schumpeter mentions this theorem with approval in his History of Economic Analysis (1954, 1007),² all other scholars have ignored it. Instead, a later exhaustive treatment of demand theory by the Swedish economist Herman Wold (1943–44) has become the standard reference in the field. In his paper Wold proves the existence of a solution to the consumer problem for the case in which the consumption set is the positive orthant. Given strictly positive prices, existence follows then from the compactness of the budget set and from the continuity of the utility function. Amoroso’s formulation is motivated by his interest, inherited from Léon Walras and Vilfredo Pareto, in the differentiability of the demand functions.³ Thus he considers the more difficult case in which the consumption set is open. As a consequence, his proof rests on a rather subtle argument that depends crucially on the assumption that the indifference curves do not intersect the boundary of the consumption set (Mas-Colell 1985, 69). In summary, Amoroso imposes more restrictions than Wold, but also obtains a stronger result (Wold’s solution needs not be in the interior of the consumption set). Curiously, in view of the return of interest in differentiable methods, Amoroso’s work acquires a more modern flavor.

Amoroso’s article is perhaps even more remarkable for the methodology used. It provides, in fact, one of the earliest applications of the axiomatic approach in economics.⁴ Following Gerard Debreu (quoted in Hildenbrand 1983, 4), this approach can be characterized as follows:

First, the primitive concepts of the economic analysis are selected and then, each one of these primitive concepts is represented by a mathematical object. Second, assumptions on the mathematical representation of the primitive concepts are made explicit and are fully specified. Mathematical analysis then establishes the consequences of these assumptions in the form of theorems.

This is exactly what Amoroso does. He associates mathematical objects to the notions of consumption set, utility function, initial endowment, and prices. Then he lists as assumptions all their relevant mathematical properties, all the while making a clear distinction between these objects and their economic interpretation. Finally, he states a number of propositions and proves them with an effective and intense use of calculus.

More than half a century after its publication in Italian, An Analysis of the System of Equations Defining the Consumer’s Behaviour has not yet been discovered by the profession. The availability of a translation into English has not changed matters. For example, neither the original text nor the translation is mentioned in Jan Horst Keppler’s (1994) recent survey of Amoroso’s contributions to economics. The purpose of this note then is not to point out omissions or to establish priorities, but rather to make historians of thought aware of an early example of the proper use of mathematics in economics.