Unformatted text preview: The Stochastic Nature of
Production
Production
Lecture VII Stochastic Production
Functions
Functions Just, Richard E. and Rulan D. Pope.
Just,
“Stochastic Specification of Production
Functions and Economic Implications.”
Journal of Econometrics 7(1)(Feb 1978):
Journal
67–86
67–86 Our development of the random
Our
characteristics of the production function
was largely one of convenience.
was We started with a production function
We
that we wanted to estimate
that
αα
f ( x1 , x2 ) = α 0 x1 1 x2 2 g ( x1 , x2 ) = a0 + a1 x1 + a2 x2 + A x + 2 A12 x1 x2 + A x
2
11 1 2
22 2 In order to estimate each function, we
In
multiplied or added a random term to
eachα specification
each α u
f ( x1 , x2 ) = α 0 x1 x2 e ⇒ ln ( f ( x1 , x2 ) ) = α 0 + α1 ln ( x1 ) + α 2 ln ( x2 ) + u
2
g ( x1 , x2 ) = a0 + a1 x1 + a2 x2 + A11 x12 + 2 A12 x1 x2 + A22 x2 + v
1 2 Just and Pope discuss three different
Just
specifications of the stochastic
production function
production
y = F1 ( X ) = f ( X ) e ε y = F2 ( X ) = f ( X ) ε y = F3 ( X ) = f ( X ) + ε E(ε ) = 0
E( ε ) =1 E(ε ) = 0 Each of these specifications has
Each
“problematic” implications. For example,
the CobbDouglas specification implies
that all inputs increase the risk of
production:
production: ( α xα1 xα 2 eε ) 2 − E ( α xα1 xα 2 eε ) 2 ⇒ ∂V f ( x1 , x2 ) > 0
V f ( x1 , x2 ) = E
01
2 01 2 ∂x1 Note that this expectation is complicated
Note
by the fact the expectation of the
exponential. Specifically, under logexponential.
normal distributions E eε = e µ + 1 2σ 2 Just and Pope propose 8 propositions
Just
that “seem reasonable and, perhaps,
necessary to reflect stochastic, technical
inputoutput relationships.”
inputoutput Postulate 1: Positive production
Postulate
expectations E[y]>0 Postulate 2: Positive marginal product
Postulate
expectations ∂E ( y ) ∂X i >0 Postulate 3: Diminishing marginal product
Postulate
expectations
∂2 E ( y )
2 <0
∂X i Postulate 4: A change in variance for
Postulate
random components in production should
not necessarily imply a change in expected
output when all production factors are held
constant
∂E ( y )
=0
∂V ( ε ) Postulate 5: Increasing, decreasing, or
Postulate
constant marginal risk should all be
possibilities
>
∂V ( y )
=0
∂X i
< Postulate 6: A change in risk should not
Postulate
necessarily lead to a change in factor use for
a riskneutral (profitmaximizing) producer ∂X i* ∂V ( ε ) =0 Postulate 7: The change in the variance of
Postulate
marginal product with respect to a factor
change should not be constrained in sign a
prior without regard to the nature of the input
prior
>
∂V ( ∂y ∂X i )
=0
∂X j
< Postulate 8: Constant stochastic returns to
Postulate
scale should be possible F (θ X ) =θF ( X ) The CobbDouglas, transcendental, and
The
translog production functions are
consistent with postulates 1, 2, 3, and 8.
However, in the case of postulate 5
However,
E ( y ) = f ( X ) E ( eε ) ∂E ( y ) ∂X i = fi E ( e ε V ( y ) = f 2 ( X ) V ( eε ) ) ∂V ( y ) ∂X i = 2 f f iV ( eε ) The marginal effect of input use on risk must
The
always be positive. Thus, no inputs can be
riskreducing.
riskreducing. For postulate 4, under normality
∂E ( y ) σ
1
=
f ( X)e 2 >0
∂V ( ε ) 2σ Thus, it is obvious that our standard
Thus,
specification of stochastic production
functions is inadequate.
functions An alternative specification y = F4 ( X ) = f ( X ) + h ( X ) ε
E ( ε ) = 0,V ( ε ) = σ 2 Econometric Specification
Econometric
yt = f ( Z t ,α ) + h ( Z t , β ) ε t E ( ε t ) = 0, E ( ε t2 ) = 1, E ( ε t ε s ) = 0 t ≠ s
ln f ( Z t ,α ) ≡ ( ln ( Z t ) ) ′ α ≡ zt′α ln h ( Z t , β ) ≡ ( ln ( Z t ) ) ′ β = zt′ β Zt = Z ( X t ) Consistent estimation Rewriting the error term ut = h ( Z t , β ) ε t So the production function can be
So
rewrittenZas ) + u
rewritten ,α
y =f(
E( u ) = 0
t t t t Where the disturbances are
Where
heteroscedastic.
heteroscedastic. b. Under appropriate assumptions, a
Under
nonlinear leastsquares estimate of this
expression yields consistent estimates of
α. Thus, these estimates can be used to
Thus,
derive consistent estimates of ut ˆ
ˆ
ut = yt − f ( Z t ,α ) Consistent estimates of β are obtained
Consistent
in the second stage by regressions on u.
Following the method suggested by
Hildreth and Houck
Hildreth ˆ
u = h ( Zt , β )
2
t 2 Expanding the Specification to
Panel Data
Panel Going back to the simultaneity
Going
specification
specification
α
1 +
β u0
2 Y = Ax x e
This expression becomes
ln ( y ) − α ln ( x1 ) − β ln ( x2 ) = ln ( A ) + u1+ + ε1
+
ln ( y ) − ln ( x1 ) = ln ( P ) − α − ln ( W1 ) + u2 + ε 2
+
ln ( y ) − ln ( x2 ) = ln ( P ) − β − ln ( W2 ) + u3 + ε 3 In order to discuss this specification, we
In
will begin with a brief survey of
estimation using panel data.
estimation As a starting point of this model, we consider
As
a panel regression
panel yit = α + β xit + ε it i = 1, 2,L N
t = 1, 2,LT This specification is implicitly pooled, the
This
value of the coefficients are the same for
each individual at every point in time.
each As a starting point, we consider generalizing this
As
representation to include differences in constant
of the regression that are unique to each firm
of yit = α i + β xit + ε it This specification can be expanded further to
This
allow for differences in the slope coefficients
across firms
across yit = α i + βi xit + ε it Based on these alternative models, we
Based
conceptualize a set of nested tests. First we
test for overall pooling (i.e., the production
function have the same constant and slope
parameters for every firm). If pooling is
rejected for both sets of parameters, we
hypothesize that the constants differ for
each firm, while the slope coefficients are
the same
the Next, consider a random specification for the
Next,
individual constants
individual yit = ( α i + ν t ) + β xit + ε it Hsiao, Cheng Analysis of Panel Data
Hsiao,
Analysis
New York: Cambridge University Press,
1986.
1986. ...
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