Question

# The production function for aluminum is given by Q= L1/4E1/4K1/2where L is the amount of labor...

The production function for aluminum is given by Q= L1/4E1/4K1/2where L is the amount of labor hired, E is the amount of energy consumed and K is the amount of capital used. Suppose that the price of energy is 4, the wage rate is 1 and the price of capital is also 1.

a)Suppose in the short run the number of units of capital is fixed at 10. Set up the factory's cost minimization problem and solve for the optimal number of units of energy to use and labor to hire as a function of Q. Find the cost, average and marginal cost curves. Illustrate the total, average, and marginal cost curves.

b)Suppose that in the long run, the number of units of capital is also variable. Set up the cost minimization problem and solve for the number of units of energy, labor, and capital used. Find the total, average marginal cost curves. Illustrate the cost curves.

c)Do your cost curves from (a) and (b) agree at any point? What is true at the points of agreement?

Part (A)

Short Run Analysis

capital fixed at 10 units

Part (B)

Long Run Analysis

substituting in production function

Part (C)

long run average cost and long run marginal cost are equal implying production at minimum efficient scale. At this point short run average cost is also minimized