Question

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

Answer 1

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

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

$C=wL+rK$

$C=f(x)$

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

Subject to $C=wL+rK.$

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

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

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

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

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

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

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

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

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

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

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

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

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