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Elluminate - 5 sup u x L u , u 0, u0 u x L u u x L u x x u x L u u x L u If x > 0, or if x ≥ 0 and Φ(λx) > 0 for some positive scalar λ, the value of Φ(λx) may be made to exceed any output rate u [0,+∞) by choosing a sufficiently large λ. x as The convexity property [P.8] of the input sets L(u) implies that the production function Φ(x) is qausi-concave 1 x y min x , y y u y L u x u x L u Proposition 3: The production function Φ(x) = Max {u | x L(u), u [0,+∞)}, x D, defined on the input sets L(u) of a technology with Properties [P.1], … [P.9] has the following properties: o [A.1] Φ(0) = 0 o [A.2] Φ(x) is finite for all x D. o [A.3] Φ(x’) ≥ Φ(x) if x’ ≥ x. o [A.4] For any x > 0, or x ≥ 0 such that Φ(λx) > 0 for some scalar λ > 0, Φ(λx) → +∞ as λ→ + ∞. o [A.5] Φ(x) is upper semi-continuous on D. o [A.6] Φ(x) is quasi-concave on D. Proposition 3.1: The level set LΦ(u) = {x | Φ(x) ≥ u} of a production function Φ(x) defined on a technology T: L(u), u [0,+∞) is identical to the input set L(u) for each u ≥ 0. From the definition of the production function Φ(x), it is clear that this function is unique for any technology T on which it is defined. Hence, the structure of the production function may be defined either in terms of a suitable production function Φ(x) or by a family of production input sets L(u), since one is uniquely determined from the other. x u3 u2 u1 u0 0 1 2 3 Definition: A function F(Φ(x)), where F(.) is a finite, nonnegative, upper semi-continuous and non-decreasing function of Φ(x) with F(0) = 0, is a transform of the production function Φ(x). What is this transform function? What about a profit or revenue function? Prodfunct.gau is the Gauss code for the regressions in Lecture 5. o This Gauss code estimates a quadratic production function, o A Transcedental production function, and o A Cobb-Douglas production function. A major point of the code (probably not well programmed) is: What to do about the zeros? It uses a canned code (ols(.)).