7.7 The empirical meaning of differentiability

Derivates are, and cannot be, but ratios between homogeneous magnitudes. That is why it is necessary to represent satisfaction differences by means of intervals

7.7.2 Theorem

If ≥ is uniformly differentiable, then there exists a utility function representing ≥ which is continuously differentiable in the interior of Ω.

PROOF: We will show, first, that there is a uniformly differentiable function φ on any open interval. Let ^f x₁, x₂^l be any such interval, x₃, x₄ any menus in ^f x₁, x₂^l with x₃ ¼ x₄, x any menu in ^f x₃, x₄^l, and ε any positive number, as small as you wish.

Since this derivative is continuous for every l (l = 1,..., L), we may conclude that φ is continuously differentiable at x₀. As x₀ was arbitrarily chosen, we may con­clude that φ is continuously differentiable in the interior of Ω. ?

7.8 New foundations of preference theory

The previous argument shows that we can provide a new conceptual apparatus for preference theory by means of which all the usual properties of the preference relation, as well as the differentiability condition, can be expressed. Is it possible to find another conceptual apparatus that allows the expression of some differ­entiability condition? That is unlikely because differentiability requires the com­parison of satisfaction distances with quantity distances within the same space.

By Theorem 7.5.9, a difference structure D is an algebraic-difference structure iff D admits a geometric representation. Hence, the new conceptual apparatus for preference theory can be briefly summarized as follows. I use the term ‘geomet­ric-difference’ to avoid confusion.

7.8.1 Definition

D is a geometric-difference structure iff there exist Ω, R and σ such that

As pointed out in section 6 above, the preference relation ≥ can be defined in terms of σ and all the properties usually attributed to it can be expressed using the language of geometric intervals (see the examples there). The novelty - as I have just shown - is that the smoothness condition can also be so expressed. Thus, using the notion introduced in Definition 7.7.1 we can define the concept of a smooth preference structure.

7.8.2 Definition

A is a smooth preference structure iff there exist Ω and ≥ such that

By virtue of Theorem 7.7.2, there is a continuously differentiable utility func­tion φ : Ω → R representing ≥. The relevance and usefulness of having such a function lies in that it allows the application of non-linear programming tech­niques in order to find the optimal points.

It would be desirable, because the empirical condition would be even more intuitive, to formulate entirely the smoothness condition in non-standard lan­guage, by means of the notion of an infinitesimal segment, as intimated in the informal discussion preceding the formal introduction of the condition. That is entirely feasible because Euclid’s Archimedian axiom is logically independent of the rest,⁷ but that would require a complete reformulation of Holder’s theory, as well as of the theory of algebraic-difference measurement.

Notes

1 See, for instance, Katzner (1970), Barten and Bohm (1981), and Mas-Colell, Whinston, and Green (1995).

2 Because the dimension of Ω is L, which is the same as that of R^l. This implies that the relative interior of Ω with respect to the linear space R^l is nonempty and so, for

Preference and utility 113 sufficiently small ε >0, the open ball B_ε(x) centered at x is contained in Ω. For a dis­cussion of the notion of relative interior that relates it to economic theory, see Koop- mans (1951), especially p. 45.

3 Precisely in the sense of Definition 3 in klst (1971: 151).

4 See Theorem 2 in klst (1971: 151); see p. 158 for a proof.

5 Von gleicher Richtung in the original.

6 Cf. Holder 1996: 241, §8.

7 Cf. Hilbert (1950: 21-2); Fleuriot (2001, ch. 4).

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