Simultaneity and Other “Simple” Problems Lecture VI Simultaneity and Estimation of the Production Function The above discussion (and estimates) makes the experimental plot design assumption regarding the data. Specifically, I essentially assumed that the data are being generated from some sort of experimental design so that the errors are truly random. If the data are actually the result of farm level decisions, the data are endogenous. Hock, Irving. “Simultaneous Equation Bias in the Context of the Cobb-Douglas Production Function.” Econometrica 26(4)(Oct 1958): 566-78. The basic firm-level model is that we have an empirical model under the assumption of: A Cobb-Douglas production function, and Competition. Q X 0 K0 X q 1 aq q X0 X 0 P0 P0 aq Pq Xq X q X 0 P0 aq X q Pq Y0 aq 1 Yq Klein demonstrates that the best linear unbiased estimate of aq is Yqi aˆq i 1 Y0 i I 1 I In this approach the “average” firm is defined to be the optimal firm. As an alternative X0 P0 a0 Rq Pq Xq where Rq is “some constant and the investigator wishes to test whether it is equal to one.” “The firm sets the value of the marginal product equal to the price augmented by the effect of any restrictions that exist.” “In this formulation, Rq can of course vary among firms; but, for a sample of firms, the investigator would be interested in testing whether the average Rq is equal to one.” Two models of simultaneity: Model 1: Production disturbance not transmitted to the “independent” variables. “If the disturbance in the production equation affects only the output and is not transmitted to the other variables in the system, then there is no simultaneous equation bias. Single equation estimates are consistent.” For example if inputs are fixed or are predetermined . Model 2: Production disturbance transmitted to the “independent variables.” “Simultaneous equation bias arises when disturbances in the production relations affect the observed values of all variables, and, as a result, single equation estimates are not consistent.” Empirical setup Q X 0 K0 X U q 1 aq q aq P0 Xq X 0Vq Rq Pq q 1, 2, Q Q x0 k0 aq xq u q 1 xq kq x0 vq x0 ln X 0 xk ln X q aq P0 kq ln Rq Pq q 1, 2, Q Indirect Least Square Solution Kmenta, J. “Some Properties of Alternative Estimates of the Cobb-Douglas Production Function.” Econometrica 32(1/2)(Jan-Apr 1964): 183–8. Simplifying the general system of equations x0i k0 a1 x1i a2 x2i v0i x1i k1 x0i v1i x2i k2 x0i v2i Transforming the estimation problem to x0i b0 b1 x1i x0i b2 x2i x0i ei yields estimates of b1 and b2 that are consistent. Note by the definitions x1i k1 x0i v1i x1i x0i k1 v1i x2i k2 x0i v2i x2i x0i k2 v2i Working through the mechanics x0i b0 b1 x1i b1 x0i b2 x2i b2 x0i ei 1 b1 b2 x0i b0 b1 x1i b2 x2i ei b0 b1 b2 1 x0i x1i x2i ei 1 b1 b2 1 b1 b2 1 b1 b2 1 b1 b2 bˆr ar 1 bˆ bˆ 1 2 r 1, 2 Zeros in the Cobb-Douglas Functional Form Moss, Charles B. “Estimation of the CobbDouglas with Zero Input Levels: Bootstrapping and Substitution.” Applied Economics Letters 7(10)(Oct 2000): 677–9. Zeros raise several difficulties in estimating the Cobb-Douglas production function. On the theoretical side, the presence of a zero-level input is that it violates weak necessity of inputs. On the empirical side, how do you take the natural logarithm of zero. Two assumptions: The existence of zeros is the result of measurement error. Agronomically, production of a crop is impossible without some level of each fertilizer. Thus, production occurs based on some true level of each nutrient available to the plant x xi i * i In fact we can think of the soil as a sponge that contains a variety of nutrients that we can augment by applying fertilizer. The actual level of fertilizer used by the crop could then be a function of what we add, the weather (i.e., if adequate moisture is not present the crop does not use the full potential), etc. Second, a production function that does not admit zero input could represent a misspecification. In this paper, I consider two techniques for adjusting the zero observations. First, I redraw from the sample averaging the result until the pseudo sample contains no zeros. Second, I substitute a small non-zero number for those observations that contain zeros (i.e., 0.1, 0.01, 0.001). The goodness of fit for each procedure is then compared using a Strobel measure of information si I si ln i 1 si N Where si is the theoretically appropriate budget share for each input and si is the budget share estimated using each empirical approximation of zero. Linear Response Plateau Y N