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Lecture 07-2005

# Lecture 07-2005 - The Stochastic Nature of Production...

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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 Cobb-Douglas 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 input-output relationships.” input-output 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 risk-neutral (profit-maximizing) 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 Cobb-Douglas, 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 risk-reducing. risk-reducing. 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 least-squares 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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