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The Production Coefficient Matrix and the Stolper-Samuelson Condition

Ken-ichi Inada

Abstract

The purpose of this paper is to generalize to the n-commodity, n-factor case the Stolper-Samuelson condition which has established a relationship between commodity prices and factor reward rates for the two-commodity, two-factor case, and to study some necessary and/or sufficient conditions for the generalized Stolper-Samuelson conditions. Two types of generalization are studied. One is the case where the inverse of the production coefficient matrix is a Minkowski matrix. Another is the case where it is a Metzler matrix. Some results about the former have already been obtained by some economists [1, 4, 8]. But the latter case has been left unexplored so far. The main purpose of this paper is to emphasize the necessity of studying the latter case and to obtain some results corresponding to those obtained for the former case. Another purpose of this paper is to establish a univalence theorem. When all principal minors of the Jacobian matrix are positive, univalence holds. This is the theorem by Gale and Nikaido [3]. In this paper, we prove that when all principal minors of the Jacobian matrix are negative, univalence holds. This theorem cannot be obtained trivially from Gale-Nikaido's theorem, but the technique employed by them for their proof can be used for our theorem.