Lecture07-2011 - Outline Stochastic Production Functions...

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Unformatted text preview: Outline 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), 67-86. 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 Cobb-Douglas 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 log-normal 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 input-output 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 risk-neutral (profit-maximizing) 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 Cobb-Douglas, 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 risk-reducing. 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 Cobb-Douglas 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 least-squares 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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