Cost Minimization

The production function for a product is given by q = K1/2L1/4 where K is capital, L is labor and q is output.

a) Find the marginal products of labor and capital.

b) Is the marginal product of labor increasing or decreasing with labor? Is the marginal product of capital increasing or decreasing with capital? (Show your work to get credit. Just answering increasing or decreasing will not earn you any points)

c) Using your answers to part a) find the marginal rate of technical substitution between labor and capital.

d) Denote the wage of labor by w and the rental of capital by r. What is the cost minimization condition for a firm? Show it diagrammatically.

e) Now suppose w = 20 and r = 80. What is the minimum cost of producing q = 100. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.)