By Prof. Rolf Färe (auth.)

This graduate textual content develops construction concept from a collection of moderate axioms. the speculation is gifted either in a primal and twin in addition to in an oblique (constrained) framework. the fundamental version ends up in a collection of potency measures which might be comfortably hired in empirical paintings. a primary draft of the textual content was once used to coach scholars at Vanderbilt college. The textual content contains a number of workout problems.

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Thus assume that u > 0 and that 36 > 0 such that 6u£P(x). Now, DO(x,6U) inf{~ > 0: inf{(~6)/6 (6u/~)£P(x)) > 0: (6u/~)£P(x)) 6D o (x,u). 3). S): This proof is left to the reader. 5): Recall that a function is lower semi-continuous if and only if its lower level sets are closed for all values in its range. (a), the lower level set of Do is closed. 5 holds. ~ 1. ~ Thus let ~ > 0, then 1). Thus (x,u)£GR <=> Do(x,u) Thus, since v~ > 0, Do(x,u/~) ~ ~ 1. 5 holds. 6. The 33 eUEP(AeX) implies DO(Aex,eu) from the equality p(x) = ~ 1.

4) Definition: An input vector xO~R~ is called a Threshold if (x: P(x) ~ (OIl, is a cone with vertex at XC. Let XO be a threshold and let x ~ 0, and A ) 0. 6) POx) where r: R++ ~ rO)p(x), R++ is a function consistent with the axioms on P. 1) can be generalized by making r into a more general function, for example by letting r depend on Factor Intensity (xiii xII) or the level of the input vector (x). 6). 8) is a multiplicative Cauchy functional equation, with the solution (Acz~l, 1966, pp. 6) are equivalent.

F(Ax) First, suppose (1) holds, then max{u: UEPOX) I max{u: UEAg(x/llxll )P(x)} Ag(x/llxli )max{v: ve:P(x)} Ag ( x III xii ) F ( x ) . 21) FOx) Suppose next that the input correspondence is ray-homothetic, and that a scalar output is produced. If ue:Dom Land u > 0, 55 L(u) 1. 22) max{A > 0: (x/A)eH(u)L(1») (H(u»- 1D i(1,x). 23) F(x) max (u: (H (u) ) -1 Di (1 ,x) ~ 1) max(u: H-1 (Di (1 ,x) ~ u) H-1 (Di (1 ,x)). The input distance function is homogeneous of degree +1 in x, and H-1 is a monotone transformation.