Question

 A firm has a Cobb-Douglas production function given as q=AX1αX2β

a. Solve for the factor demand functions for labour (X1) and Capital (X2)

b. If the firms’ competitive output price is p find the wage rate

c. What is the share of the firm’s revenue paid to labour and capital?

d. If α=0.6, β=0.2 and A=1 find the LR labour and capital demand curve equations

Answer 1

Qn. A

For conditional factor demand of capital we haveα

$K=K(r, w,Q)$

r=rate, w=wage, Q=Quantity

We maintain w and Q as constants, so we have them as Wo and Qo

Thus resulting to our Capital factor demand function as;

$K=K(r)$

Similarly, for conditional factor demand in labour we have;

$L=L(r, W,Q)$

r=rate, W=wage, Q=Quantity

Having W and Q as constants, represented in the function as Wo and Qo

Thus resulting Labour factor demand function is;

$L=L(r)$

Qn. B

Wage rate= MPL

$MPL(AlαKβ)$

Wage rate AαKβ

Qn. C

Firm revenue paid to labour is;

$MPL\times Q\\ MPL=AαKβ\\ MPL\times Q=Q\times (AαKβ)\\ =Q(AαKβ)$

Firm revenue paid to capital is;

$MPK\times Q\\ MPK(AlαKβ)\\ MPK=ALαβ\\ MPK\times Q=Q\times (ALαβ)\\ =Q(ALαβ)$

Qn. D

$q=ALαkβ$

Demand equation for labour =(AαKβ)

Replacing the values of α, β and A we have;

$Q=0.6K0.2\\ Q=1.2K$

Demand equation for capital =(ALαβ)

Replacing the values of α, β and A we have;

$Q=L\times 0.6\times 0.2\\ Q=1.2L$