Question

Suppose a producer faces the following production function: K^0,5.K^0,5 given that L is the unit of labour and K is the unit of capital. Suppose K is 16, labour wage rate is R10 and rental rate of capital is R50, derive the TC and MC functions

Answer 1

Production function is given as:$: Q=K^{0.5}L^{0.5}$

The amount of capital is fixed and equal to 16. So, the production function will look like:

$Q=16^{0.5}L^{0.5}$

$Q=4L^{0.5}$

$(\frac{Q}{4})^{2}=L$

$L=\frac{Q^2}{16}$

Total cost (TC) function of the firm is equal to:

$TC=w×L+r×K$

where w is wage rate, L is amount of labor, r is rental rate and K is amount of capital.

$TC=10L+16×50$

$TC=10L+800$

Putting value of L in this equation:

$TC=\frac{10Q^2}{16}+800$

Marginal cost function is equal to:

$MC=\frac{\delta TC}{\delta Q}$

$MC=\frac{ 20Q}{16}$

$MC= \frac{5Q}{4}$