Question

The production function of a product is given by q=100KL if the price of capital is ksh 150 per day and the price of labour is ksh 50 per day,calculate the cost of producing 1000 units of the product where K-capital and L-labour (10mks

Answer 1

Total cost (C) = wL + rK

The cost minimizing combination of capital and labour:

$MRTS = \frac{MPl}{MPq} = \frac{w}{r}$

The marginal product of labour:

$\frac{dQ}{dL} = 100K$

The marginal product of capital:

$\frac{dQ}{dK} = 100L$

Therefore the marginal rate of technical substitution:

$\frac{100K}{100L} = \frac{K}{L}$

Determine the optimal capital-labour ratio

Set the marginal rate of technical substitution equal to the ratio of labour rate of capital:

$\frac{K}{L} = \frac{50}{150} = L=3K$

Substitute for L in the production function and solve where K yields an output of 1000 units:

$1000=(100) (K)(3K) = where \; K =1.82$

Since L equals 3K, this means:

$L=(3)(1.82) =5.46$

Therefore Total Cost(T.C) = wL + rK

$(50) (5.46) + (150) (1.82)=273 + 273=546$

Total Cost of producing 1000 units = Ksh.546