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Elluminate - 5
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sup u x  L  u  , u   0,    u0
u x  L  u   u x  L  u 
x  x
u x  L  u 
u x  L  u 
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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.
   x    as   
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The convexity property [P.8] of the input sets L(u) implies
that the production function Φ(x) is qausi-concave
  1    x   y   min   x  ,   y  
  y   u y  L  u 
  x   u x  L  u 
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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:
o [A.1] Φ(0) = 0
o [A.2] Φ(x) is finite for all x  D.
o [A.3] Φ(x’) ≥ Φ(x) if x’ ≥ x.
o [A.4] For any x > 0, or x ≥ 0 such that Φ(λx) > 0 for some scalar
λ > 0, Φ(λx) → +∞ as λ→ + ∞.
o [A.5] Φ(x) is upper semi-continuous on D.
o [A.6] Φ(x) is quasi-concave on D.
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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.
  x
u3
u2
u1
u0
0
1 2
3
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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?
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Prodfunct.gau is the Gauss code for the regressions in
Lecture 5.
o This Gauss code estimates a quadratic production function,
o A Transcedental production function, and
o A Cobb-Douglas production function.
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A major point of the code (probably not well programmed)
is: What to do about the zeros?
It uses a canned code (ols(.)).
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