Throughout part A of the text, we used the technology we called more “realistic” in panel (b) of Graph 11.1.
A: Suppose now that the producer choice set was instead strictly convex everywhere.
(a) Illustrate what such a technology would look like in terms of a production frontier.
(b) Derive the output supply curve with price on the vertical and output on the horizontal axis (in graphs analogous to that in Graph 11.9) for this technology.
(c) Derive the labor demand curve for such a technology.
(d) Now suppose the technology were instead such that the marginal product of labor is always increasing. What does this imply for the shape of the producer choice set?
(e) How much should the firm produce if it is maximizing its profits in such a case?
B: Suppose that the production function a firm faces is x = f (ℓ) = 100ℓα.
(a) For what values of α is the producer choice set strictly convex? For what values is it non convex?
(b) Suppose α = 0.5. Derive the firm’s output supply and labor demand function.
(c) How much labor will the firm hire and how much will it produce if p = 10 and w = 20?
(d) How does labor demand and output supply respond to changes in w and p?
(e) Suppose that α = 1.5. How do your answers change?