Question

A firm has a Cobb-Douglas production function given as

q=AX1

αX2

β

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

(a)

$q=AX_{1}^\alpha X_{2}^\beta$

Price of good = p

Wage=w.

Price of capital =r.

The input demand for labor:

$L=f(w,r,p)$

The input demand for Capital:

$K=f(w,r,p)$

(b)

Wage rate $=\frac{w}{p}=MPL$

Change in the production when Capital is fixed and labor is changed from $X_{1}$ to $X_{1}+∆X_{1}$ is :

$∆Q=f(X_{1}+∆X_{1},X_{2})-f(X_{1},X_{2})$

Dividing this by $∆L$ gives the change in production per unit change in labor:

$\frac{∆Q}{∆X_{1}}=\frac{f(X_{1}+∆X_{1},X_{2})-f(X_{1},X_{2})}{∆X_{1}}$

Taking the limit with infinitesimal changes in labor, $\frac{\delta Q}{\delta X_{1}}$ is the marginal product of labor.

For the Cobb Douglas function:

$\frac{\delta Q}{\delta X_{1}}=\alpha AX_{1}^{\alpha -1} X_{2}^b=\frac {\alpha Q}{X_{2}}$

(c)

$\frac{\delta Y}{\delta X_{1}}=\frac {w}{p}$ ...equation 2.

$\frac{\delta Y}{\delta X_{2}}=\frac {r}{p}$ ... equation 3.

The above equations show that profit is maximized when marginal revenue product of labor is equal to the marginal cost of labor and marginal revenue product of capital is equal to the marginal cost of capital.

$\frac{\delta Y}{\delta X_{1}}=\delta A(\frac{X_{2}}{X_{1}})^\beta$ ... equation 4.

$\frac{\delta Y}{\delta X_{2}}=\beta A(\frac{X_{1}}{X_{2}})^\alpha$ ... equation 5.

Equating 2 and 3, and 3 and 5:

$\frac{w}{p}=\alpha A(\frac {X_{2}}{X_{1}})^\beta$

$\frac{r}{p}=\beta A(\frac {X_{1}}{X_{2}})^\alpha$

The income spent on a resource is the expenditure on the resource divided by the total income Y.

$\therefore$ Labor's share $=\frac {(\frac{w}{p}) X_{1}}{Y}$

Capital's share $= \frac{(\frac{r}{p}) X_{2}}{Y}$ .

(d)

$q=AX_{1}^\alpha X_{2}^\beta$

Demand equation for labour ;

$Q=A\alpha X_{2}^\beta$

$=1\times0.6\times\ X_{2}\times0.2$

$=1.2X_{2}$

Demand equation for capital ;

$Q=A\alpha X_{1}^\beta$

$=1\times0.6\times\ X_{1}\times0.2$

$=1.2X_{1}$