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

4. Suppose a firm produces according to the production function Q = AL0.6K0.2, and faces wage rate ₵10, a rental cost of capital ₵5, and sells output at a price of ₵20.

a. Obtain and expression for the factor demand functions if A=1.

b. Compute the profit-maximizing factor demands for capital and labour if A=1.

5. In the short run, a competitive firm has a production function,

Q = f(L) = 2.6667L0.75. The output price is $4 per unit and the wage is $5 per hour. Find the short-

run labor demand curve of the firm.

Answer 1

(a)

Factor demand functions for labor and capital

conditioned factor demand of capital:

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

where r=rate, w=wage and Q=Quantity.

To get the factor demand function of capital, we keep $w$ and $Q$ constant.

This results to:

$X_2=X_2(r)$

The conditional factor demand of labor:

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

where r=rate, W= wage and Q=Quantity

when we have $W$ and $Q$ as constants, the resulting labor force demand function is:

$X_1=X_1(r)$ .

(b)

Wage rate = Marginal Productivity of Labor.

$MPL(Al\alpha X_2\beta)$

the wage rate will be: $A\alpha X_2\beta$ .

(c)

Firm revenue paid to labor:

$=MPL\times Q$

but $MPL=A^\alpha X_2^\beta$

$\therefore MPL\times Q=(A\alpha X_2\beta)=Q(A\alpha X_2\beta)$

Firm revenue paid to capital :

$=MPK\times Q$

$=AX_1\alpha \beta\times Q$

$=Q(AX_1\alpha \beta )$ .

(d)

$Q=AX_1\alpha X_2\beta$

The demand equation for labor $=(A\alpha X_2\beta)$

Substituting the values of $\alpha$ ,$\beta$ and $A$ :

$Q=0.6K0.2=1.2K$

The demand equation for capital$=(AX_1 \alpha \beta)$

substituting the values of $\alpha$ ,$\beta$ and $A$ :

$Q=X_1\times 0.6\times 0.2= 1.2X_1$