Answer to Question #254927 in Microeconomics for Nessy

Question

The Southern Mail produces local newspapers. The company can rent its equipment and hire workers at competitive rates. Equipment needed for this operation can be rented at R4 per hour, and labour can be hired at R3 per worker hour. The production function using available technology can be expressed as: Q = 2L0.5 K 0.5 [8] 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)

The cost minimizing combination of capital and labor is the one where:

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

The marginal product of labor:

$\frac{dQ}{dL}=L^{-0.5}K^{0.5}$

The marginal product of capital:

$\frac{dQ}{dK}=L^{0.5}K^{-0.5}$

Set MRTS= Input price ratio, to determine the optimal labor to Capital ratio:

$\frac{L^{-0.5}K^{0.5}}{L^{0.5}K^{-0.5}}=\frac{3}{4}$

$L^{-0.5}K^{0.5}=\frac{3}{4}(L^{0.5}K^{-0.5})$

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

(b)

Substituting the value of L in the production function,

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

$Q=160$

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

$K=69$

$L=\frac{4}{3}(L)=\frac{4}{3}(69)=92.$

$TC=wL+rK$

$=3(92)+4(69)$

$=276+276$

$=552.$