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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   