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AEB 6184 – Shephard and Estimation
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1. AEB 6184 – Shephard and Estimation Elluminate - 5

2. Finishing Shephard’s Production Function • Let xD be arbitrarily chosen. • The input vector x belongs to L(0) [P1]. • There exists a finite value >0 such that x L() [P4] and [P6]. • Hence is finite. • It follows that x L(u) for u  [0,+∞). • Consequently, Φ(x) exists, it is finite for bounded x and Φ(0)=0. • Weak monotonicity • From property [P2] it follows that Φ(x’) ≥ Φ(x) for x’ ≥ x

3. Graphical Monotonicity

4. Ray Properties • 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.

5. Quasi-Concavity • The convexity property [P.8] of the input sets L(u) implies that the production function Φ(x) is qausi-concave

6. Properties of Production Function • 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: • [A.1] Φ(0) = 0 • [A.2] Φ(x) is finite for all x  D. • [A.3] Φ(x’) ≥ Φ(x) if x’ ≥ x. • [A.4] For any x > 0, or x ≥ 0 such that Φ(λx) > 0 for some scalar λ > 0, Φ(λx)→ +∞ as λ→ + ∞. • [A.5] Φ(x) is upper semi-continuous on D. • [A.6] Φ(x) is quasi-concave on D.

7. Production Technology • 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.

8. Profile of the Production Function

9. Transform of the Production Function • 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?

10. Empirical Estimation – Gauss Code • Prodfunct.gau is the Gauss code for the regressions in Lecture 5. • This Gauss code estimates a quadratic production function, • A Transcedental production function, and • 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(.)).