Production Sets by Murray C. Kemp

By Murray C. Kemp

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I ie/* «. = onl y if (<**> ßue τί9 i e i* e e ^*. , n where /* is the set of i for which sf > 0 and E* is the set of e such that resource e is fully employed. This definition is more complicated than Definition 2 to allow for the possibility that in equilibrium some producers may not be in business, some outputs may not be entirely consumed, and some resources may not be fully employed. Each of the bits of the definition is now explained. J* identifies those producers who are actually using inputs to produce output.

R where 01 is the «-dimensional null vector. Following the proof of Kemp, Khang, and Uekawa, it can be shown that DkC\\\ . . , DkC\gk) are linearly independent. , gr}· Suppose that all factors are internationally immobile. , r, lke { 1 , . . , gk) with «-dimensional null vectors. Clearly, any vector Σ ϊ = ι àkDkC%lk), where ôk is a scalar taking the value 1 or 0, is linearly dependent on the Σ ϊ = ι 9k vectors DkC\lk). , Ck(gk)). Remark 3. If the vectors dk satisfying conditions (i) and (ii) of Corollary 1 are such that dk φ 0 for k = 1, .

TAWADA (z, z1), there corresponds a vector dl satisfying (a)-(c) of Theorem 1 ; moreover, z — z1 = Dd\ Hence the set {Dd G R":d satisfies (a) and (c) of Theorem 1} contains r linearly independent vectors. ■ Several specializations of Theorem 2(i) are of interest, (a) Let k = n = m = 2. If A is singular (that is, if the same factor intensities prevail in both industries), then S contains at most one linearly independent vector and the production substitution surface is locally either linear or strictly concave; thus the familiar result concerning nonjoint production is weakened.

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