Below, we will investigate the profit maximizing choice in the two steps that first involve a strict focus on the cost side.
A: Consider again (as in the previous exercise) a production process that gives rise to a strictly convex producer choice set.
(a) Derive the cost curve from a picture of the production frontier.
(b) Derive the marginal and average cost curves from the cost curve.
(c) Illustrate the supply curve on your graph. How does it change if the wage rate increases?
(d) Now suppose the production process gives rise to increasing marginal product of labor throughout. Derive the cost curve and from it the marginal and average cost curves.
(e) Can you use these curves to derive a supply curve?
(f) The typical production process is one that has increasing marginal product initially but eventually turns to one where marginal product is diminishing. Can you see how the two cases considered in this exercise combine to form the typical case?
B: Consider again (as in the previous problem) the production function x = f (ℓ) = 100ℓα.
(a) Derive the firm’s cost function.
(b) Derive the marginal and average cost functions and determine how their relationship to one another differs depending on α.
(c) What is the supply function for this firm when α = 0.5? What is the firm’s labor demand curve?
(d) How do your answers change when α = 1.5?