E L LU M I N AT E - 3

AEB 6184 – SHEPHARD AND VON LIEBIG

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,+).

0, nD x x x R

PRODUCTION INPUT SET

1 2,x x

L u

1x

2x

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 xL(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,+).

PROPOSITION 3

1 2,x x x

L u

2x

L u

1x

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.

S xx

2x

1x

• 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.

0 , , 0nRB x x R x R R

0 min 0Rx x x B L u

1x

2x

• 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)

| 0,

| , 0

yD x x x y

K u x x E u

• The intersection of L(u) Dy is a bounded, closed subset of L(u).

K u

y

x

L u

K uy

x

L u

z

(a) (b)

• 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

min ,i yi

z z y z K u D L u

0, , for 0,1x x x L u x L u

DIFFERENT ISOQUANTS

1u2u

3u

1u2u

3u

1u2u

3u

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.

max , 0, ,x u x L u u x D

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

2 20 1 2 3 4 5Y N P N P NP

*1 2 3 4max , ,Y Y N P

0 1 2 3 41 exp 1 expY n P