Question

A firm manufactures at home, and faces input prices for labor and capital produces q units of output using L units of labor and K units of capital. Abroad, the wage and cost of capital are half as much as at home. If the firm manufactures abroad, will it change the amount of labor and capital it uses to produce q? What happens to its cost of producing quantity q? The manager would also like you to explain the Cobb Douglas Production Function and how she would be able to use it to determine that production is efficient.

Answer 1

Suppose the production function is Q = KL and |MRTS| = K/L. At home the wage rate is 2w and capital rent is 2r. Abroad it is w and r respectively.

At home it will compare |MRTS| and wage rental ratio. Optimum input mix has |MRTS| = 2w/2r or K/L = w/r. Abroad also, it will use the same principle and will find that K/L = w/r.

Optimal input mix does not depend upon the wage rate or the capital price but on their ratio as satisfied by the equimarginal principle. Accordingly the ratio of the marginal product of an input to its price should be same across all the inputs for efficiency considerations. If wages and capital are half abroad, it does not indicate that the cost of production will be half abroad at all combinations of capital and labor. The marginal product of labour divided by wage rate should be equal to the marginal product of capital divided by the rental price.

If the marginal product of labour divided by its wage rate and the marginal product of capital divided by its rental price is a same in both home and abroad, then the manufacturing unit will use the same input combination both abroad as well as in home. But the cost of production abroad will be halved.

Cost abroad is given by C = wL + rK so when q units are to be produced, home uses K/L = w/r or K = L*w/r. Then the labor and capital units are L*(w/r)*L = q or L* = (qr/w)^0.5 and K* = (qw/r)^0.5. The cost function at home is C = w*(qr/w)^0.5 + r*(qw/r)^0.5

C = 2(qrw)^0.5

Cost of production at home is C = 2w*(qr/w)^0.5 + 2r*(qw/r)^0.5 or C = 4(qrw)^0.5

Hence under similar conditions cost of production is halved abroad.