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This document explores a modified approach to the Gauss-Seidel method tailored for the Constant Elasticity of Substitution (CES) production function. It formulates the optimization problem in terms of minimizing a squared-error objective function based on the first-order conditions for various inputs. The analysis includes derivations using Maxima to compute first-order and second-order derivatives, aimed at verifying the conditions of the Gauss-Seidel formulation. The study provides valuable insights for economists and mathematicians working with production functions.
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AEB 6184 – Allen Partial for the CES Elluminate - 4
Fitting the CES Function – Modified Gauss-Siedel • Another formulation of the Gauss-Siedel is to formulate the system as a squared-error system. • For example consider the CES production function • The first-order condition for each input then becomes
Error Objective Function • The objective function for the minimization problem then becomes • Left to your own, prove that the first-order conditions of Q(.) yields the same conditions as the Gauss-Siedel form.
Sample Maxima Program f(x1,x2,x3):=(0.6870*x1^(-0.0526)+0.0886*x2^(-0.0526)+0.1838*x3^(-0.0526))^(-19); f1(x1,x2,x3):=diff(f(x1,x2,x3),x1); f2(x1,x2,x3):=diff(f(x1,x2,x3),x2); f3(x1,x2,x3):=diff(f(x1,x2,x3),x3); f_1=subst(5,x1,subst(5,x2,subst(5,x3,f1(x1,x2,x3)))); f_2=subst(5,x1,subst(5,x2,subst(5,x3,f2(x1,x2,x3)))); f_3=subst(5,x1,subst(5,x2,subst(5,x3,f3(x1,x2,x3)))); f11(x1,x2,x3):=diff(f(x1,x2,x3),x1,1,x1,1); f12(x1,x2,x3):=diff(f(x1,x2,x3),x1,1,x2,1); f13(x1,x2,x3):=diff(f(x1,x2,x3),x1,1,x3,1); f22(x1,x2,x3):=diff(f(x1,x2,x3),x2,1,x2,1); f23(x1,x2,x3):=diff(f(x1,x2,x3),x2,1,x3,1); f33(x1,x2,x3):=diff(f(x1,x2,x3),x3,1,x3,1); f_11=subst(5,x1,subst(5,x2,subst(5,x3,f11(x1,x2,x3)))); f_12=subst(5,x1,subst(5,x2,subst(5,x3,f12(x1,x2,x3)))); f_13=subst(5,x1,subst(5,x2,subst(5,x3,f13(x1,x2,x3)))); f_22=subst(5,x1,subst(5,x2,subst(5,x3,f22(x1,x2,x3)))); f_23=subst(5,x1,subst(5,x2,subst(5,x3,f23(x1,x2,x3)))); f_33=subst(5,x1,subst(5,x2,subst(5,x3,f33(x1,x2,x3))));
