By Rolf Färe

Ebook by means of Färe, Rolf, Grosskopf, Shawna, Knox Lovell, C.A.

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Thus assume there exists A 2: such that Ax E: L( u). Since < F i( u, x);S I} = L( u). inputs are weakly disposable, A> 1, thus {x: Conversely, assume L(u) = {x: 0< F;(u, x);S 1}, u ~ 0. , 0< Flu, Ax);S 1. \:£ 1 => 0 < F j ( u, Ax):£ 1. Thus Ax E: L( u) proving that inputs are weakly disposable. 6: If x E: Isoq L(u), u ? 0, then clearly F;(u, x) = 1. \ < 1 such that Ax E: L(u) thus Fj(u, x) < 1. Hence if x t Isoq L(u), u? 0, then Fj(u, x) ~ I, or equivalently, if F;( u, x) = I then x E: Isoq L( u).

If the disposed inputs need not be paid a slightly larger cost reduction arises. Third, input i)verall efficiency (IOE), again in the sense of Farrell, is represented by IOE = OS/ OP. Of course, at S, u is not obtainable, since S L(u). But for the same input cost, W can be purchased, and W E: L(u). , W E: CM(u,p)), IOE also measures the cost saving resulting from moving from point P to point W. Unlike the two previous measures of cost saving, however, this overall measure of cost saving is input price dependent since CM(u, p) is input price dependent.

U)~ u E: P(x/O"'). Also, u E: lIO' P(x). Thus, lIO' P(x) = P(x/O"'). Now take All"': = lIO, then P(Ax) = All",. P(x). Similar arguments can be used to show the converse. • This theorem states that if the input or output correspondence is homogeneous of a positive degree, then its inverse correspondence is homogeneous of the reciprocal degree. For example, if ll' = 1, i. , of degree one. The linear technology introduced next will play an important role in this 43 THE STRUCTURE OF PRODUCTION TECHNOLOGY monograph.