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Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation The Stochastic Nature of Production: Lecture VII
Charles B. Moss1
1 University of Florida September 15, 2011 Charles B. Moss The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation 1 Stochastic Production Functions
Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation 2 Consistent Estimation 3 Panel Data Speciﬁcation Charles B. Moss The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Stochastic Production Functions
Just, Richard E. and Rulan D. Pope. 1978. Stochastic
Speciﬁcaiton of Production Functions and Economic
Implications Journal of Econometrics 7(1), 6786.
Our development of the random characteristics of the
production function was largely one of convenience.
We started with a production function that we wanted to
estimate
αα
f ( x1 , x2 ) = α 0 x1 1 x2 2
2
2
g (x1 , x2 ) = a0 + a1 x1 + a2 x2 + A11 x1 + 2A12 x1 x2 + A22 x2
(1) Charles B. Moss The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation In order to estimate each function, we multiplied or added a
random term to each speciﬁcation αα
f (x1 , x2 ) = α0 x1 1 x2 2 e u ⇒ ln (f (x1 , x2 )) = α0 + α1 ln (x1 ) + α2 ln (x
˜
2
2
g (x1 , x2 ) = a0 + a1 x1 + a2 x2 + A11 x1 + 2A12 x1 x2 + A22 x2 + ν
(2) Just and Pope discuss three diﬀerent speciﬁcations of the
stochastic production functions
y = F1 ( X ) = f ( X ) e E [ ] = 0
y = F2 ( X ) = f ( X ) E [ ] = 1
y = F3 ( X ) = f ( X ) + E [ ] = 0 Charles B. Moss (3) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Each of these speciﬁcations has “problematic” implications.
For example, the CobbDouglas speciﬁcation implies that all
inputs increase the risk of production:
2
αα
αα
V [f (x1 , x2 )] = E (α0 x1 1 x2 2 e ) − [E (α0 x1 1 x2 2 e )]
∂ V [f (x1 , x2 )]
⇒
>0
∂ x1 (4) Note that this expectation is complicated by the fact the
expectation of the exponential. Speciﬁcally, under lognormal
distributions
1 E [ e ] = e µ+ 2 σ Charles B. Moss 2 (5) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Just and Pope Propositions
Just and Pope propose 8 propositions that “seem reasonable
and, perhaps, necessary to reﬂect stochastic, technical
inputoutput relationships.”
Postulate 1: Positive production expectations E [y ] > 0.
Postulate 2: Positive marginal product expectations
∂ E [y ]
>0
∂ Xi (6) Postulate 3: Diminishing marginal product expectations
∂ 2 E [y ]
<0
∂ Xi2 Charles B. Moss (7) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Just and Pope Propositions  Continued
Continued
Postulate 4: A change in variance for random components in
production should not necessarily imply a change in expected
output when all production factors are held constant
∂ E [y ]
=0
∂ V [] (8) Postulate 5: Increasing, decreasing, or constant marginal risk
should all be possibilities
∂ V [y ] ≥
<0
∂ Xi Charles B. Moss (9) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Just and Pope Propositions  Continued
Continued
Postulate 6: A change in risk should not necessarily lead to a
change in factor use for a riskneutral (proﬁtmaximizing)
producer
∂ Xi∗
=0
∂ V [] (10) Postulate 7: The change in the variance of marginal product
with respect to a factor change should not be constrained in
sign a prior without regard to the nature of the input
∂y
∂V
∂ Xi ≥
<0
(11)
∂ Xj
Charles B. Moss The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Just and Pope Propositiosn  Continued Continued
Postulate 8: Constant stochastic returns to scale should be
possible
F (θ X ) = θ F (X ) Charles B. Moss (12) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation Individual Functions
The CobbDouglas, transcendental, and translog production
functions are consistent with postulates 1, 2, 3, and 8.
However, in the case of postulate 5
E [y ] = f (X ) E [e ] V [y ] = f 2 (X ) V [e ]
∂ E [y ]
∂ V [y ]
= fi E [ e ]
= 2ﬀi V [e ]
∂ Xi
∂ Xi (13) The marginal eﬀect of input use on risk must always be
positive. Thus, no inputs can be riskreducing.
For postulate 4, under normality
σ
∂ E [y ]
1
=
f (X ) e 2 > 0
(14)
∂ V []
2σ
Thus, it is obvious that our standard speciﬁcation of stochastic
production functions is inadequate. Charles B. Moss The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Just and Pope Propositions
Individual Functions
An Alternative Speciﬁcation An Alternative Speciﬁcation
Basic formulation for any function
y = F4 ( X ) = f ( X ) + h ( X )
E [] = 0, V [] = σ 2 (15) Increasing the abstraction slightly
yt =f ( t , α) + h (Zt , β ) t
Z
2
E [t ] , E t = 1, E [t s ] = 0 t = s (16) In the case of the CobbDouglas f (Zt , α) ≡ (ln (Zt )) α = zt α
h (Zt , β ) ≡ (ln (Zt )) β = zt β
Zt = Z ( X t )
Charles B. Moss (17) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Consistent Estimation
Rewriting the error term
u t = h ( Zt , β ) t (18) So the production function can be rewritten as
yt = f ( Z t , α ) + u t E ( u t ) = 0 (19) Where the disturbances are heteroscedastic.
Under appropriate assumptions, a nonlinear leastsquares
estimate of this expression yields consistent estimates of α.
Thus, these estimates can be used to derive consistent
estimates of ut
u t = yt − f ( Z t , α )
ˆ
ˆ
Charles B. Moss (20) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Heteroscedasticity Consistent estimates of β are obtained in the second stage by
regressions on u . Following the method suggested by Hildreth
and Houck
u t = h 2 ( Zt , β )
ˆ2 Charles B. Moss (21) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation Panel Data Speciﬁcation Going back to the simultaneity speciﬁcation
+ αβ
Y = Ax1 x2 e u0 (22) This expression becomes
+
ln (y ) − α ln (x1 ) − β ln (x2 ) = ln (A) + u1 + 1
+
ln (y ) − ln (x1 ) = ln (P ) − α − ln (W1 ) + u2 + 2
+
ln (y ) − ln (x2 ) = ln (P ) − β − ln (W2 ) + u3 + 3 Charles B. Moss (23) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation In order to discuss this speciﬁcation, we will begin with a brief
survey of estimation using panel data.
As a starting point of this model, we consider a panel
regression
yit = α + β xit + it i = 1, 2, · · · N
t = 1, 2, · · · T (24) This speciﬁcation is implicitly pooled, the value of the
coeﬃcients are the same for each individual at every point in
time.
As a starting point, we consider generalizing this
representation to include diﬀerences in constant of the
regression that are unique to each ﬁrm
yit = αi + β xit + it
Charles B. Moss (25) The Stochastic Nature of Production: Lecture VII Outline
Stochastic Production Functions
Consistent Estimation
Panel Data Speciﬁcation This speciﬁcation can be expanded further to allow for
diﬀerences in the slope coeﬃcients across ﬁrms
yit = αi + βi xit + it
(26)
Based on these alternative models, we 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.
Next, consider a random speciﬁcation for the individual
constants
yit = (¯ i + νt ) + β xit + it
α (27) Hsiao, Cheng. 1986. Analysis of Panel Data New York:
Cambridge University Press.
Charles B. Moss The Stochastic Nature of Production: Lecture VII ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.
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