A. Profit maximization

Suppose a firm's production function is given by f(K, L) = K^1/4 L^1/3

Let rand w denote the input prices for K and L, and p denote the output price.

I. Choosing inputs to maximize profits (One person handling all operations for the firm):

(a) Set up the profit maximization problem with the two inputs as the choice variables. Find the optimal choices of K and L.

(b) Suppose w = 4 and r = 3. What is the profit maximizing level of output as a function of p? What is this relationship called?

(c) What is the maximum profit? [Note: This is also called the profit function. Just as the cost function is minimized cost, profit function is maximized profit. That's why I used the term "expression for profit" or just "profit" when talking about the difference between total revenue and total cost. "Profit function" is a more specific term. The terms 'difference between total revenue and total cost' and 'profit function' have the same relationship as 'utility' and 'indirect utility'.]

II. First choosing inputs and getting the lowest cost to produce each level of output (factory supervisor's problem - who reports this to the top brass of the firm), then choosing output to maximize profits (which is the top brass's job!)

(d) Find this firm's cost function as a function of w, r and y, where y denotes output level.

(e) How does the average cost change vary with y? What does this imply about the shape of the cost function?

(f) Now, use the cost function from part (d) to set up the profit maximization problem with y as the choice variable. What is the firm's supply function in terms of w, r and p?

(g) Suppose w = 4 and r = 3. What is the profit maximizing level of output? What is the maximum profit? [Verify that these answers match the answers to part (b) and part (c).]

(h) Compare the first order conditions from (a) and (d). Do you see the similarity between p in the former and in the latter? What is the interpretation of ? What does this mean for optimality condition for profit maximization?