Integrability of Pseudomonotone Differentiable Maps and the Revealed Preference Problem

The problem considered is as follows: given

and

F:C\rightarrow {{\mathbb R}}^n F : C → R n

differentiable, find

f:C\rightarrow {{\mathbb R}} f : C → R

differentiable such that

\|\nabla f(x)\|^{-1} \nabla f(x) = \|F(x) \|^{-1}F(x) ∥ ∇ f ( x ) ∥ − 1 ∇ f ( x ) = ∥ F ( x ) ∥ − 1 F ( x )

for all

x\in C x ∈ C

. Conditions for

f f

to be pseudoconvex or convex are given. The results are applied to the differentiable case of the revealed preference problem.