The production function for a firm is f (x, y) = 64x^3/4y^1/4, where x and y are the number of units of labor and capital utilized. Suppose that labor costs $96 per unit and capital costs $162 per unit and that the firm decides to produce 3456 units of goods.

(a) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of x, y that minimize 96x + 162y, subject to the constraint 3456 - 64x^3/4y^1/4 = 0.

(b) Find the value of l at the optimal level of production.

(c) Show that, at the optimal level of production, we have

Transcribed Image Text:

[marginal productivity of labor] [marginal productivity of capital] [unit price of labor] [unit price of capital]