Question

Explain the mathematical condition for the firm maximizing output in the long run for a given level of cost outlay?

Answer 1

Suppose, firm's production function follows production function, q = KaL1-a and rent for capital is r and wages are w. The cost incurred by firm is C.

Then,

Marginal product of labour, MPL = dq/dL = (1-a)(K/L)a

And marginal product of capital, MPK = dq/dK = a*(L/K)1-a

=> MRTS = MPL/MPK = [(1-a)/a]*K/L

At equilibrium,

MRTS = w/r

=> L = (1/a - 1)Kr / w

Substituting this in cost constraint, we get

w((1/a - 1)K r/w) + rK = C

=> K = C/((1/a - 1)Kr + r) and L = (1/a - 1)C/((1/a -1)w + w)

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