Question

Given a cobb – Douglas production functionQ = L0.5 K0.4And the prices of capital and labour are ksh3 and ksh 4 respectively while the firm outlay is Kshs.108, calculate the optimal combination of factors inputs.

Answer 1

$Q = L0.5 K0.4$

Cost minimization combination of labor and capital

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

Marginal product of labor

$\frac{dQ}{dL} = 0.2K$

Marginal product of capital

$\frac{dQ}{dK} = 0.2L$

$\therefore$ the marginal rate of technical substitution

$\frac{0.2K}{0.2L} = \frac{K}{L}$

Optimal capital - labor ratio

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

Substitution

$Q = L0.5 K0.4$

$108 =K \frac{3}{4}0.5 K 0.4$

$108 = 0.15K^2$

$K^2 = 720$

$K = 26.83$

$108 = L0.5 (26.83) 0.4$

$108 = 5.37L$

$L = 20.13$

$TC = wL + rK$

$TC = (3)(20.13) + (4)(26.83)$

$TC = Ksh 168$