Question

The southern Mail produces local newspapers. The company can rent its equipment and hire workers at competitive rates. Equipment needed for this can be rented at R4 per hour and labour at R3 per worker hour. The production function using technology can be expressed as Q= 2L⁰.⁵K⁰.⁵ . a) Determine the firm's optimal ratio of labour to capital b) Determine the cost minimizing level of capital and labour in the long run if the firm wants to produce 160 units. Calculate the cost c) Graphically illustrate this using isoquant and isocost lines

Answer 1

a) To optimize output, cost must be minimized

Therefore, the cost minimizing combination of labor and cost is where

$MRTS = \frac{MP_L}{MP_K} = \frac{w}{r}$

The marginal product of labor $= \frac{dq}{dl} = 2K$

The marginal capital of capital $= \frac{dq}{dK}= 2L$

$\therefore$ the marginal rate of technical substitution $= \frac{2K}{2l} = \frac{K}{L}$

To determine the optimal labor-capital ratio, the $MRTS$ is set toe be equal to the ratio of the wage rate of the rental rate of capital

$\frac{K}{L} = \frac{4}{3} or K = \frac{4}{3}L$

b) The cost minimizing level of capital and labor in the long run if the firm wants to produce 160 units

$Q = 2L^{0.5} .K^{0.5}$

$160 = 2L^{0.5}.(\frac{4}{3})L^{0.5}$

$160 = \frac{8}{3} L^{1}$

$\therefore L = 160/ \frac{8}{3}$

$L = 60.$

$Q = 2L^{0.5} .K^{0.5}$

$160 = 2(\frac{3}{4})K^{0.5} . K^{0.5}$

$160 = \frac{6}{4}K^{1}$

$K = 160 * \frac{4}{6}$

$K = 106.66$

c)