The conditions of Debreu and Mas-Colell

According to Mas-Colell, Whinston, and Green (1995: 49) “it is possible to give a condition purely in terms of preferences”, implying the existence of a utility representation of the same:

Intuitively, what is required is that indifference sets be smooth surfaces that fit together nicely so that the rates at which commodities substitute for each other depend differentially of the consumption levels.

(Ibid.)

The problem is that it is not at all clear which empirically meaningful (even if idealized) property must the preference relation of a consumer have so that its indifference sets “fit together nicely”. What is worse, C² differentiability is restrictive because some demand functions that are derivable do not come from a C² utility function, as the same authors have noticed (Mas-Colell, Whinston, and Green 1995: 95, n. 33). Furthermore, as Debreu (1983a: 201) has pointed out, it is enough for the utility function to be C¹ in order to guarantee that the corresponding Walrasian demand function be also C¹. Is it possible to find a con­dition that can be considered sufficiently natural and general for that purpose? My claim is that it is possible, and I intend to substantiate this claim by means of the intuitive discussion motivating Definition 7.7.1.

The differentiability condition has been interpreted by Mas-Colell in terms of the concept of a differentiable manifold, giving rise to the following important result.

I would like to conclude the present section with a reflection on the meaning of these conditions. In the first place, the definition of Gaussian curvature proposed

Preference and utility 93 by Debreu (taken from Hicks 1965, sec. 2.2) presupposes de facto that the indif­ference surfaces are already differentiable manifolds (actually, Debreu assumes that they are of class C²), and so the condition only translates the problem to a deeper level. For the question is, precisely, What is the property that must be attributed to the consumer in order to guarantee that the indifference surfaces are differentiable manifolds? In Debreu’s definition, the question whether the Gaussian curvature of the manifold is different from zero or not arises once the first problem has been solved. It seems clear that Mas-Colell’s condition is just a modified generalization of Debreu’s and so analogous considerations apply to it.

7.3