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+00. l is left to the reader. 3. ° Suppose pO and p' are positive and that ~ A ~ 1. Then C(u,pO) pOxO, where XO E L(u) and C(u,p') p'x' where x' E L(u). Moreover, C( u, Apo + (1 - A)p') + (1 - A)p')X*(A), where X*(A) E L( u) ApoXo + (1 - A)p'X' (ApO ~ AC(U,pO) + (1- A)C(U,p'). This proves that the cost function is concave in input prices.

In the F(xO). The figure shows that F(xO) = uO/Do(xO,uO) or equivalently, Thus in the scalar case the output distance function is merely the quotient between observed output UO and maximal output F(xO). ,N ~ 1. d u = o. 5). b models the situation where u ~ +00. Moreover, since 0,0 < Do(x,u). Thus in this case 0 < Do(x,u);'£ 1. Case 'I- P( x) but by taking a radial contraction of u, (Ju E P( x). In this case the infimum is also achieved but since u 'I- P( x), 1 < Do( x, u) < +00. since 0 E P(x),Vx E ~~, but there is no (J > 0 such that (Ju E P(x), Do(x,u) = case, since infimum is used to define the output distance function, Do(x,O) In the third case, +00.

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