Okayama Economic Review
Published by the Economic Association of Okayama University

Online ISSN 2433-4146
Print ISSN 0386-3069

A Note on Qualitative Economics for Univalence of Nonlinear Mappings

Fujimoto, Takao
Ranade, Ravindra R. Department of Economics, University of Kagawa
This note is aimed at presenting an easy and simple proposition on the univalence of a given nonlinear differentiable mapping whose Jacobian matrix has sign-regularity. First the notion of sign-regularity of Jacobian matrix on a domain is defined. We classifY the sign patterns into four categories: plus, minus, zero, and the rest. The plus sign is given to the (i, j) entry of the Jacobian matrix when the i-th component function is always increasing with respect to the j-th coordinate variable, the negative sign when the function is always decreasing, and the sign of zero when the function does not include the j-th coordinate variable. Otherwise, the sign is set as an asterisk *. Our proof is simple and elementary by use ofthe mean value theorem. In the final section, we give a list of our future research topics, some of which are under way. Especially a generalization to discontinuous mappings should be interesting.
研究ノート (Note)