Question

Suppose that the production function for widgets is Q = g(L,K) = 1L + 2K, and that the cost of one unit of capital is more than double the cost of one unit of labour. If a producer wishes to minimize his or her cost of producing 200 widgets, what combination of labour and capital should the producer use in production? Explain.

Answer 1

It is given that cost of one unit of capital is more than double the cost of one unit of labour. This implies that r > 2w or w/r < 1/2

Marginal rate of technical substitution is MPL/MPK or 1/2.

At the optimal input choice the marginal rate of technical substitution should be equal to the wage rental ratio. However in our case marginal rate of technical substitution is 1/2 but wage rental ratio is less than 1/2. This implies

MPL / MPK > w/r

Due to this reason, labour is more productive then capital and the firm should use only labour. this is because the labour and capital are perfect substitutes of each other.

Then we have Q = L or 200 = L

Therefore labour should be 200 units and capital should be 0 unit at the optimal input mix.