Lecture05-2011 - Estimation of the Primal Production...

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Unformatted text preview: Estimation of the Primal Production Function: Lecture V Charles B. Moss September 6, 2011 Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 1 / 19 Outline 1 Ordinary Least Squares Empirical Estimates of the Quadratic The Cobb-Douglas Production Function Empirical Estimates of the Cobb-Douglas Transcendental Production Function Empirical Results for the Transcendental 2 Nonparametric Production Functions Two Nonparametric Approaches Nonparametric Regression Kernel Production Frontier Contour Plot Marginal Physical Product of Nitrogen Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 2 / 19 Ordinary Least Squares Ordinary Least Squares The most straightforward concept in the estimation of production function is the application of ordinary least squares. 2 y = a0 + a1 x1 + a2 x2 + a3 x3 + A11 x1 + A12 x1 x2 + 2 2 A13 x1 x3 + A22 x2 + A23 x2 x3 + A33 x3 + ￿ (1) Note that we have already applied symmetry on the quadratic. From an estimation perspective since x1 x2 = x2 x1 any other approach would not work. Using data from Indiana and Illinois, we apply ordinary least squares to this specification. Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 3 / 19 Ordinary Least Squares Empirical Estimates of the Quadratic Empirical Estimates of the Quadratic Parameter α0 α1 α2 α3 A11 Estimate 100.2819 (6.8117) -0.0054 (0.0791) 0.2344 (0.1063) 0.0590 (0.0795) 0.000138 (0.00030) Parameter A12 A13 A22 A23 A33 Estimate -0.00089 (0.00030) 0.00073 (0.00047) 0.00019 (0.00038) -0.00087 (0.00049) -0.00007 (0.00031) Do these estimates make any sense? What is wrong? Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 4 / 19 Ordinary Least Squares The Cobb-Douglas Production Function The Cobb-Douglas Production Function Turning to the Cobb-Douglas form αβγ y = Ax1 x2 x3 ⇒ ln (y ) = ln (A) + α ln (x1 ) + β ln (x2 ) + γ ln (x3 ) ln (y ) = ln (A) + α ln (x1 ) + β ln (x2 ) + γ ln (x3 ) + ￿ (2) What are some of the problems with this specification? First, the one problem is that there may be zero input levels. What is the production theoretic problem with zero input levels? What is the econometric problem with zero input levels? Second, what is the assumption of the error term? Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 5 / 19 Ordinary Least Squares Empirical Estimates of the Cobb-Douglas Empirical Estimates of the Cobb-Douglas Parameter A α1 α2 α3 Charles B. Moss () Estimate 4.5858∗∗∗ (0.0561) 0.0126 (0.0118) 0.168∗∗ (0.0073) 0.0132∗∗ (0.0063) Estimation of the Primal Production Function: Lecture September 6, 2011 V 6 / 19 Ordinary Least Squares Transcendental Production Function Transcendental Production Function The transcendental production function has many of the same problems as the Cobb-Douglas. The production function can be written as: a a a y = Ax1 1 e b1 x1 x2 2 e b2 x2 x3 3 e b3 x3 ln (y ) = a0 + a1 ln (x1 ) + b1 x1 + a2 ln (x2 ) + b2 x2 + a3 ln (x3 ) + b3 x3 + ￿ (3) Again, what are the assumptions about zeros or the distribution of error terms. Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 7 / 19 Ordinary Least Squares Empirical Results for the Transcendental Empirical Results for the Transcendental Parameter a0 a1 b1 a2 Charles B. Moss () Estimate 4.5520 (0.05875) -0.00006 (0.01564) -.00039 (0.00038) 0.0166 (0.00887) Parameter b2 a3 b3 Estimate -0.00004 (0.00045) 0.0025 (0.0090) 0.00080 (0.00047) Estimation of the Primal Production Function: Lecture September 6, 2011 V 8 / 19 Ordinary Least Squares Empirical Results for the Transcendental Transcendental MPP - Nitrogen Marginal Physical Product 0.0022 0.0021 0.0021 0.0020 0.0020 0.0019 0.0019 0 50 100 150 200 250 300 Pounds of Nitrogen Charles B. Moss () Estimation of the Primal Production Function: Lecture September 6, 2011 V 9 / 19 Ordinary Least Squares Empirical Results for the Transcendental Marginal Physical Product Transcendental MPP - Phosphorous 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 0 50 100 150 200 Pounds of Phosphorous Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 10 / 19 Ordinary Least Squares Empirical Results for the Transcendental Marginal Physical Product of Potash Transcendental MPP - Potash 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 0 50 100 150 200 Pounds of Potash Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 11 / 19 Nonparametric Production Functions Nonparametric Production Functions It is clear from our discussions on production functions that the choice of production function may have significant implications for the economic results from the model. The Cobb-Douglas function has linear isoquants that has implications for the input demand functions. While the Cobb-Douglas function has no stage III, the quadratic production function is practically guaranteed a stage III. Thus, one approach is to generate nonparametric functional forms. These nonparametric functional forms are intended to impose allow for the maximum flexibility in the input-output map. The approach is different that the nonparametric production function suggested by Varian. Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 12 / 19 Nonparametric Production Functions Two Nonparametric Approaches Two Nonparametric Approaches Fourier Expansions 1 f (xk ) = β0 + β ￿ x + x ￿ Bx + 2 A J ￿ ￿￿ ￿￿￿ ￿ ￿ ￿￿ β 0α + βj α cos jkα x − γj α sin jkα x α=1 j =1 B=− Nonparametric regressions Charles B. Moss () A ￿ (4) ￿ β 0α k α k α α=1 Estimation of the Primal Production Function: Lecture V September 6, 2011 13 / 19 Nonparametric Production Functions Nonparametric Regression Nonparametric Regression A nonparametric regression is basically a moving weighted average where the weights of the moving average change for various input levels. ￿∞ y (x ) = ˆ y (z ) f (y , z , x , δ ) dz (5) −∞ In this case y (x ) is the estimated function value conditioned on the level of inputs x . The value y (z ) is the observed output level at observed input level z . f (y , z , x , δ ) is a kernel function which weights the observations based on a distance from the point of approximation. Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 14 / 19 Nonparametric Production Functions Kernel Kernel In this application, I use a Gaussian kernel. ￿ ￿ 1 −1 (z − x )2 f (y , z , x , δ ) = √ δ exp 2δ 2 2π (6) The multivariate form of the Gaussian kernel function is expressed as 1 f ( y , { z } { x } A , δ ) = √ δ − 1 | A | − 1/2 2π ￿ ￿ 1 ￿ −1 × exp − (z − x ) A (z − x ) 2 Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 (7) 15 / 19 Nonparametric Production Functions Kernel Because of the discrete nature of the expansion, I transform the continuous distribution into a discrete Gaussian distribution w ( y , { xi } { x } A , δ ) = f [ y , { xi } { x } A , δ ] N ￿ f [ y , { xj } { x } A , δ ] (8) j =1 The estimated value of the production function at point x can then be computed as y (x ) = ˆ N ￿ i =1 Charles B. Moss () w ( y , { xi } { x } , A , δ ) yi Estimation of the Primal Production Function: Lecture V September 6, 2011 (9) 16 / 19 Nonparametric Production Functions Production Frontier Production Frontier Potash Figure 5. Production 150 100 50 Function for Corn Varying both Nitrogen 250 200 and Potash 140 130 Corn 120 110 100 100 200 300 Nitrogen Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 17 / 19 Nonparametric Production Functions Contour Plot Contour Plot Figure 6. Contour Plot of Corn Production varying both Nitrogen and Potash 350 300 250 200 150 100 50 0 0 Charles B. Moss () 50 100 150 200 250 300 350 Estimation of the Primal Production Function: Lecture V September 6, 2011 18 / 19 Nonparametric Production Functions Marginal Physical Product of Nitrogen Marginal Physical Product of Nitrogen Marginal Physical Product of Nitrogen Figure 4. Distribution of the Marginal Physical Product of Nigrogen 0.2 0.1 Nitrogen 10 20 30 40 50 60 70  0.1  0.2  0.3  0.4 Charles B. Moss () Estimation of the Primal Production Function: Lecture V September 6, 2011 19 / 19 ...
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