Estimation of the Primal Production FunctionOrdinary Least SquaresSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Transcendental MPP-NitrogenTranscendental MPP PhosphorousTranscendental MPP of PotashNonparametric Production FunctionsSlide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Estimation of the Primal Production Function Lecture VOrdinary Least Squares The most straightforward concept in the estimation of production function is the application of ordinary least squares. 20 1 1 2 2 3 3 11 1 12 1 22 213 1 3 22 2 23 2 3 33 3y a a x a x a x A x A x xA x x A x A x x A x e= + + + + + ++ + + +–Note that we have already applied symmetry on the quadratic. From an estimation perspective, since x1x2=x2x1 any other approach would not work. –Using data from Indiana and Illinois, we apply ordinary least squares to this specification to estimate–Do these estimates make any sense? What is wrong? Turning to the Cobb-Douglas form ( ) ( ) ( ) ( ) ( )1 2 31 2 3ln ln ln ln lny Ax x xy A x x xa b ga b g= �= + + +( ) ( ) ( ) ( )0 1 1 2 2 3 3ln ln ln lny x x xa a a a e= + + + +–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?04.5858*** (0.0561)a1 0.0126 (0.0118) 2 0.0168** (0.0073) 3 0.0132** (0.0063)–Estimating the Transcendental Production Function:•The transcendental production function has many of the same problems as the Cobb-Douglas. Specifically, the production function can be written as: 3 3 31 1 1 2 2 21 2 3a b xa b x a b xy Ax e x e x e=( ) ( ) ( )( )0 1 1 1 1 2 2 2 23 3 3 3ln ln lnlny a a x b x a x b xa x b x e= + + + + ++ +–Again, what are the assumptions about zeros or the distribution of error terms.Transcendental MPP-NitrogenTranscendental MPP PhosphorousTranscendental MPP of PotashNonparametric 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.Two approaches: –Fourier Expansions–Nonparametric regressions ( ) ( ) ( )( )0 01 1011cos sin2A Jk j jjAf x x x x jk x jk xk ka a a a aaa a aab b b b gb= ==� �� � � �= + + B + + -� �� ��B =-� ��A nonparametric regression is basically a moving weighted average where the weights of the moving average change for various input levels.–In this case y(x) is the estimated function value conditioned on the level of inputs x .( ) ( ) ( )ˆ, , ,y x y z f y z x dz�- �= d�–The value y(z) is the observed output level at observed input level z.–f(y,z,x,) is a kernel function which weights the observations based on a distance from the point of approximation.In this application, I use a Gaussian kernel.( )( )2121, , , exp22z xf y z x-� �-d = d� �dp� �� �The multivariate form of the Gaussian kernel function is expressed asBecause of the discrete nature of the expansion, I transform the continuous distribution into a discrete Gaussian distribution{ } { }( )( ) ( )11 121 1, , , , exp22f y z x A A z x A z x-- -� ��d = d - - -� �p� �The estimated value of the production function at point can then be computed as{ } { }( ){ } { }{ }{ }1, , , ,, , , ,, , , ,iiNjjf y x x Aw y x x Af y x x A=� �d� �d =� �d� ��( ) { } { }( )1ˆ, , ,Ni iiy x w y x x A y== d�Figure 5. Production Function for Corn Varying both Nitrogen and Potash100200300Nitrogen50100150200250Potash100110120130140Corn100200300Nitrogen50100150200250Potash0 50 100 150 200 250 300 350050100150200250300350Figure 6. Contour Plot of Corn Production varying both Nitrogen and Potash10 20 30 40 50 60 70Nitrogen-0.4-0.3-0.2-0.10.10.2Marginal Physical Product of NitrogenFigure 4. Distribution of the Marginal Physical Product of