Answer to Question #273611 in Microeconomics for Mamolly

Question #273611

Show that the cost function for a firm with the constant returns Cobb-Douglas production function  y=Az1αz21−α is given by  C(p,y)=yp1αp11−αB , where B is a function of A and  α only. Derive the conditional input demands


1
Expert's answer
2021-12-01T21:57:03-0500

"Y=AZ_1^\\alpha Z_2^{1-\\alpha}"

"X=b_0L^{b_1}K^{b_2}"

"C=wL+rK"

"C=f(x)"

Maximize "X=b_0L^{b_1}K^{b_2}"

Subject to "C=wL+rK."

"\\frac{\\delta \\phi}{\\delta L}=b_1\\frac{X}{L}-\\lambda w=0"

"\\frac{\\delta \\phi}{\\delta K}=b_2\\frac{X}{K}-\\lambda r=0"

"\\frac{\\delta \\phi}{\\delta \\lambda}=(C-wL-rK)=0"

"b_1\\frac{X}{L}=\\lambda w"

"b_2\\frac{X}{K}=\\lambda r"

"\\frac{b_1}{b_2}.\\frac{K}{L}=\\frac{w}{r}"

"K=\\frac{w}{r}.\\frac{b_2}{b_1}L"

"X=b_0[(\\frac{w}{r})(\\frac {b_2}{b_1})]^{b_2}L^{(b_1+b_2)}"

"L=(\\frac{rb_1}{wb_2})^\\frac{b_2}{b_1+b_2}(\\frac{X}{b_0})^\\frac{1}{b_1+b_2}"

"K=\\frac{w}{r}.\\frac{b_2}{b_1}.L"

"K=(\\frac{wb_2}{rb_1})^\\frac{1}{b_1+b_2}(\\frac{X}{b_0})^\\frac{1}{b_1+b_2}"

"C=(\\frac{1}{b_0})^\\frac{1}{b_1+b_2}[w(\\frac{rb_1}{wb_2})^\\frac{b_2}{b_1+b_2}+r(\\frac {wb_2}{rb_1})^\\frac{b_1}{b_1+b_2}].X^\\frac{1}{b_1+b_2}"

"C(p,y)=yP_1^\\alpha P_1^{1-\\alpha}B"



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