Introduction

This chapter deals with the foundations of individual rational choice, specifically with the foundations of consumer theory. Neoclassical consumer theory requires that the behavior of the consumer be explained by means of a preference relation, and that all the required properties of the corresponding utility representation be derived from the properties of this relation.

Just as classical dynamics purports to explain the motion of bodies by means of the concept of force, rational choice theory proposes to explain the behavior of the agents by means of the concept of preference. This is done by taking, as a starting point, a regular preference structure defined by axioms that actually attri­bute empirically meaningful (even though idealized) properties to the preference relation. Among these properties, strict convexity, non-satiation, or continuity can be mentioned. Restrictions on preference relations translate into restrictions on the form of the utility functions. For instance, if the preference relation is strictly convex, the corresponding utility representation is strictly quasi-concave; if the relation is nonsatiated, the corresponding utility representation is monotonically increasing; if the relation is continuous, the corresponding utility representation is also continuous. Certain specializations of the theory require, additionally, that the utility function representing the preference relation be differentiable, in order to apply methods of nonlinear programming to the derivation of the demand functions.

Even though some of the aforementioned properties are deemed as “non-sub- stantial” and “technical” by economists of a positivist and instrumentalist philo­sophical persuasion, nonetheless the tendency has been to formulate them by means of natural and intuitive conditions that depict an idealized consumer described by set-theoretical structures into which the empirical data can be imbedded.¹ Forthe actual meaning of the axioms defining the structures is impor­tant: the more idealized they are, the less precise are the empirical consequences of the same, and it is impossible to check intuitively their degree of idealization if their economic meaning is unknown. I think that the reason why it is said (for instance by Barten and Bohm 1981: 385-6) that even though “Axioms 1-3 [reflexivity, transitivity and completeness] describe order properties of a preference relation that have intuitive meaning in the context of the theory of choice... [this] is much less so with the topological conditions which are usually assumed as well” is that the language of topology obscures such intuitive meaning altogether because it is not suitable to express the economic meaning of such properties.

It is not really difficult to formulate conditions like continuity or convexity in intuitive terms, but the differentiability condition has turned out to be more resil­ient to such treatment. Certainly, Gerard Debreu (1983a, 1983b) and Andreu Mas-Colell (1985) have provided conditions over a preference relation that imply the existence of continuously differentiable utility functions. The problem is that - in contradistinction to the properties I referred to previously - these conditions are admittedly not intuitive. The first aim of the present chapter is to propose a language in which all the usual properties attributed to the preference relation, including differentiability, can be formulated in a natural, intuitive way. Even if differentiability is deemed as a mere technical com­putational convenience, without any actual empirical meaning, the condition pre­sented here is mathematically simpler (once the language has been assimilated) than the ones presented by Debreu and Mas-Colell (which rely upon the heavy machinery of differential topology), and is formulated within the framework of a unified language and conceptual apparatus that clarifies its relationship with the concept of preference strength.

After discussing, in the second section, the conditions proposed by Gerard Debreu (1983a, 1983b) and Andreu Mas-Colell (1985), in the third I will moti­vate and state, in intuitive numerical terms, the required differentiability condi­tion. The fourth section will be devoted to introduce the algebraic theory of difference as a preparation to present, in the fifth section, the conceptual and lin­guistic apparatus required to provide a geometric theory of preference strength within which differentiability (actually all the usual) conditions can be formulated in an intuitive way. The sixth section contains a development of preference theory within the proposed conceptual apparatus, up to the proof of the existence of a C¹ utility function for the preference relation. The seventh section introduces the dif­ferentiability condition and the eight and final one discusses the relevance and importance of having a continuously differentiable utility function. The chapter ends with a reflection on the convenience of formulating a non-standard version of Holder’s theory in order to formulate the differentiability condition in an even more intuitive way.

7.2