Transcript PowerPoint
AEB 6184 – SHEPHARD
AND VON LIEBIG
ELLUMINATE - 3
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
D x x 0, x R n
• 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,+).
PRODUCTION INPUT SET
x2
x1, x2
L u
x1
TECHNOLOGICALLY EFFICIENT SET
•
PROPOSITION 3
x2
x x1 , x2
L u
L u
x1
EFFICIENT SETS
• From the definition of
the efficient subset E(u)
x2
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.
x
S x
x1
• The first point is to
define a closed ball.
x2
BR 0 x x R, x R n , R 0
• Given this definition
of the closed ball,
there exists some
distance measure R
where the ball is
tangent to the level
set.
x1
x0 min x x BR 0 L u
•
Dy x | x 0, x y
K u x | x E u , 0
• The intersection of L(u) Dy is a bounded, closed
subset of L(u).
K u
K u
y
z
y
x
x
L u
(a)
L u
(b)
• In the second case (b)
min zi z y, z K u Dy L u
i
• 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
x x 0, x L u , x L u for 0,1
DIFFERENT ISOQUANTS
u3
u2
u1
u3
u2
u1
u3
u2
u1
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
x max u x L u , u 0, , x D
• Giving to the production function Φ(x) the
traditional meaning as the largest output rate for x.
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
Y 0 1 N 2 P 3 N 2 4 P 2 5 NP
• Von Liebig Functional Form
Y max Y * , 1 2 N , 3 4 P
• Mitscherlich-Baule
Y 0 1 exp 1 2 n 1 exp 3 4 P