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AEB 6184 – ALLEN PARTIAL FOR THE CES Elluminate - 4

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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

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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.

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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))));

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Output – First Order

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Output – Second Order

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