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Again assume that there are k = 1 , . . , iiT observation of inputs and outputs {x^^ y^) and let the output set for observation k satisfying strong disposability and constant returns to scale be given by K P{x^') ={y'. 66) /c=l K Yl ^k^kn ^ Xj^f^, n = 1, . . t. Y. ^kykm ^ OVk'm^ m = 1 , . . , M, k=l K ^ ZkXkn ^ x^>^, n = 1 , . . , A^, k=i Here we have imposed strong disposability of inputs and outputs as well as constant returns to scale. These may be modified to suit the application under investigation.

Ym > 0, Qy^ > 0, m = 1 , . . , M , and let o u t p u t s be freely disposable. If Do{x^ y\ Qy) = 0, then y G IsoqP{x). T h e first of the two propositions tells us t h a t if an output vector belongs to t h e output isoquant, then the directional distance function takes the value of zero when evaluated in any strictly positive direction. T h e second proposition is a partial converse t o t h e first. T h e 'partialness' is due t o t h e fact t h a t outputs must b e strictly positive in t h e second proposition, b u t not in the first.

Environmental Performance 47 In other words if an output vector (y, u) is feasible and there are no bad o u t p u t s produced, then under null jointness only zero good output can be produced. Equivalently if some positive amount of the good output is produced then some bad output must also be produced. W h a t this means is t h a t if we were to draw a production possibility set with one good output y and a single bad output u^ if they are null joint, t h e n t h e only part of the good output axis which is feasible is the origin (null).

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Advanced Macroeconomics Solutions by Romer

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