Question

1. Suppose a good is produced according to the following production function:

Q = L^1/2K^1/2

            so that the marginal product of labor and capital are

                                                MP_L = (1/2)(K/L)^1/2

MP_K = (1/2)(L/K)^1/2

If w = $8 and r = $4, determine the necessary conditions for the input choices, K and E to be cost-minimizing. In other words, what is the cost-minimizing ratio of K to E for this firm? Your answer will be in the form of 2L: 5K. You do not need to solve for specific values of L&K.

Answer 1

Answer

In order to minimize cost a firm hires that amount of Inputs(L and K) such that MRTS = Slope of Isocost curve.

MRTS = Marginal rate of Technical Substitution

Total cost(TC) = wL + rK = 8L + 4K.

Let Labor be on horizontal axis and Capital be on Vertical axis.

Then MRTS = Slope pf Isoquant = -MPL/MPK = -(1/2)(K/L)^1/2 / [(1/2)(L/K)^1/2] = -K/L

Also TC = 8L + 4K and at isocost curve TC is constant. and as we assume Labor be on horizontal axis and Capital be on Vertical axis. Lets Calculate Slope of Isocost curve :

TC = 8L + 4K => 0 = 8 + 4dK/dL

=> dK/dL = -2 --------------Slope of isocost curve

Thus MRTS = Marginal rate of Substitution

=> -K/L = -2/1

=> K/L = 2/1

=> K : L = 2 : 1

Hence, the cost-minimizing ratio of K to L for this firm is 2 : 1