Scarf's conjecture and its validity

Abstract

In this chapter we will prove a substantial generalization of Gale-Nikaido's theorem on univalent mappings in which we assume P-property only on the boundary of the rectangular region and this was conjectured by Scarf. We will give two different proofs one due to Garcia and Zangwill and the other due to Mas-Colell. Proof of Garcia and Zangwill uses the norm-coerciveness theorem whereas Mas-Colell uses results from degree theory. There is a subtle difference between these results and Gale-Nikaido's fundamental theorem. The difference lies in the fact that the proofs of Garcia-Zangwill and Mas-Colell demand F to be a C⁽¹⁾ function whereas Gale-Nikaido's result holds good if we assume F to be a differentiable function not necessarily a C⁽¹⁾ function. It is not clear whether Garcia-Zangwill or Mas-Colell's result holds good if we assume F to be a differentiable function. This seems to be an interesting open problem in this area. Another problem which remains still unanswered is the following: Suppose F is a C⁽¹⁾ map from a compact convex set μ ⊂ Rⁿ to Rⁿ. Suppose Jacobian of F is a P-matrix for every x ε μ. Does it imply F is one-one in μ?