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X' ~ x and x' =1= x). Thus, (x,y) is input efficient if x belongs to the efficient subset of L(y) which is defined by Eff L(y) = {x : x E L(y), x' ~ x =} x' f/. L(y)}, y ~ OM. 1) Thus, if x E Eff L(y) then any reduction in one or more of the inputs will render y an infeasible output vector. Distance Functions 29 A slightly weaker notion is input-isoquant efficiency. A feasible production plan, (x,y), is input-isoquant efficient if AX ¢ L(y) for all A < 1. 20)) IsoqL(y) = {x: x E L(y),A < 1 =?

For all x,x' in R~, if ~ A ~ 1 then P(x) n P(x') ~ P(AX + (1 - A)X'). 8 is equivalent to convexity of L(y), let x" = Ax + (1- A)X'. Then L(y) is convex ~ x,x' E L(y) implies x" E L(y) ~ Y E P(x), y E P(x') implies y E P(x") ~ Y E P(x) n P(x') implies y E P(x") ~ P(x) n P(x') ~ P(x"). g P(x) is convex for all x in R~. 4) the efficiency score for any observation, k', k' = 1, ... xk' E L(yk')} K LZkYkm ~ Yk'm, m = 1, ... Xk'n, n = 1, ... ,N, k=l Zk ~ 0, k= 1, ... , K}. 8. Normally, if the same output can be produced with proportionally less input then, alternatively, proportionally more output could be produced with the same input.

Condition (iii) requires that a positive amount of each input is used by at least one activity. Condition (iv) says that every activity must use a positive amount of at least one input. Using the techniques of activity analysis, various technologies can be constructed from the K observed, feasible activities. For example, suppose we want to specify a technology T that exhibits constant returns to scale globally. Then (xk,yk) E T, by feasibility, and (Zkxk, zkyk) E T, for all Zk ~ 0, by constant returns to scale.

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