Question

Show that the quantity of labor(X1) and capital(X2) that a firm demands decreases with a factor’s own factor price (w for labor and r for capital) and increases with the output price (P) when the production function is a Cobb-Douglas of the form q=AX1αX2β

Answer 1

Given the Cobb Douglas equation:

$Q=AX_1^\alpha X_2^\beta$

For a given amount of labor and capital, the ratio $\frac{Q}{K}$ is the average amount of production for one unit of capital.

The change in production when labor is fixed and capital changes from $K$ to $K+∆K$ is:

$∆Q=f(L,K+∆K)-f(L,K).$

Dividing this quantity by $∆K$ gives the change in the production per unit change in capital.

$\frac{∆Q}{K}=\frac{f(L,K+∆K)-f(L,K)}{∆K}$

Taking the limit with infinitesimal changes in capital:

$\frac{\delta Q}{\delta K}$ is the marginal product of capital and $\frac{\delta Q}{\delta L}$ is the marginal product of labor.

$\frac{\delta Q}{\delta K}=\beta AL^\alpha K^{\beta-1}=\frac{\beta Q}{K}$

$\frac{\delta Q}{\delta L}=\alpha AL^{\alpha -1} K^\beta= \frac{\alpha Q}{K}$