AEB 6184 – Shephard and Von Liebig

1. AEB 6184 – Shephard and Von Liebig Elluminate - 3

2. Shephard’s Production FUnction • Let u  [0,+) denote the output rate. • Let x = (x1, x2,…xn) denote factors of production. • The domain of inputs can then be depicted as • Definition: A production input set L(u) of a technology is the set of all input vectors x yielding at least the output rate u, for u  [0,+).

3. Production Input Set

4. Technologically Efficient Set • Definition: E(u) = { x | x  L(u), y  x  y  L(u) }. • Definition: A production technology is a family of input sets T: L(u), u  [0,+] satisfying • P.1 L(0) = D, 0  L(u) for u> 0. • P.2 x  L(u) and x x imply xL(u). • P.3 If (a) x > 0, or (b) x  0 and (x)  L() for some > 0 and > 0, the ray intersects L(u) for all u [0,+). • P.4 u2  u1  0 implies L(u2) L(u1). • P.5 for u0. • P.6 is empty. • P.7 L(u) is closed for all u [0,+). • P.8 L(u) is convex for all u [0,+). • P.9 E(u) is bounded for all u [0,+).

5. Proposition 3

6. Efficient Sets • From the definition of the efficient subset E(u) of the production set L(u)is the boundary of the set. • Suppose xL(u), then a sphere S(x), centered on x composed entirely of point in x exists. • Thus, yL(u) where y  x, contradicting the efficient set.

7. The first point is to define a closed ball. • Given this definition of the closed ball, there exists some distance measure R where the ball is tangent to the level set.

8. Proposition 1. The efficient subset E(u) of a production input set L(u) is nonempty for all u [0,+). • Each production input set L(u) may be partitioned into the sum of the efficient subset E(u) and the set D = {x | x  0, x  Rn}. • Proposition 2. L(u) = E(u) + D = (u) + D. • We show that L(u)  (E(u) + D). • Let y  L(u) be arbitrary chosen. • The vector y belongs to a closed ball B||y||(0)

9. The intersection of L(u)  Dyis a bounded, closed subset of L(u). (a) (b)

10. In the second case (b) • Let x denote the minimum. • Then x  E(u) and y = x + y with y ≥ x, so y  (E(u) + D). • Definition: The production isoquant corresponds to an output rate u > 0 is a subset of the boundary of the input set L(u) defined by

11. Different Isoquants

12. Definition of Production Functions • The production function is a mathematical form defined on the production input sets of a technology, with properties following from those of the family of sets L(u), u  [0,+∞) which can be best understood this way instead of making assumptions ab initio on a mathematical function. • For any input vector x  D, consider a function Φ(x) defined on the sets L(u) by • Giving to the production function Φ(x) the traditional meaning as the largest output rate for x.

13. A Comparison of Alternative Crops Response Models • This paper compares a response function based on a quadratic functional form and specifications of the von Liebig including the Mitscherlich-Baule. • Quadratic Functional Form • Von Liebig Functional Form • Mitscherlich-Baule