A: Suppose the production process for a firm is homothetic and has decreasing returns to scale.
(a) On a graph with labor ℓ on the horizontal and capital k on the vertical axis, draw an isoquant corresponding to output level . For some wage rate w and rental rate r, indicate the cost minimizing input bundle for producing .
(b) Indicate in your graph the slice of the production frontier along which all cost minimizing input bundles lie for this wage and rental rate.
(c) In two separate graphs, draw the (total) cost curve and the average cost curve withthe marginal cost curve.
(d) Suppose that, in addition to paying for labor and capital, the firm has to pay a recurring fixed cost (such as a license fee). What changes in your graphs?
(e) What is the firm’s exit price in the absence of fixed costs? What happens to that exit price when a fixed cost is added?
(f) Does the firm’s supply curve shift as we add a fixed cost?
(g) Suppose that the cost minimizing input bundle for producing x that you graphed in part (a) is also the profit maximizing production plan before a fixed cost is considered. Will it still be the profit maximizing production plan after we include the fixed cost in our analysis?
B: As in exercises 13.1 and 13.2, suppose the production process is again characterized by the production function x = f (ℓ,k) = Aℓαkβ with 0 < α,β ≤ 1 and α + β < 1.
(a) If you have not already done so in a previous exercise, derive the (long run) cost function for this firm.
(b) Now suppose that, in addition to the cost associated with inputs, the firm has to pay a recur- ring fixed cost of FC. Write down the cost minimization problem that includes this FC. Will the conditional input demand functions change as a result of the FC being included?
(c) Write down the new cost function and derive the marginal and average cost functions from it.
(d) What is the shape of the average cost curve? How does its lowest point change with changes in the FC?
(e) Does the addition of a FC term change the (long run) marginal cost curve? Does it change the long run supply curve?
(f) How would you write out the profit maximization problem for this firm including fixed costs? If you were to solve this problem, what role would the FC term play?
(g) Considering not just the math but also the underlying economics, does the addition of the FC have any implications for the input demand and output supply functions?