When we discussed optimal behavior for consumers in Chapter 6, we illustrated that there may be two optimal solutions for consumers whenever there are non-convexities in either tastes or choice sets. We can now explore conditions under which multiple optimal production plans might appear in our producer model.
A: Consider only profit maximizing firms whose tastes (or isoprofits) are shaped by prices.
(a) Consider first the standard production frontier that has initially increasing marginal product of labor and eventually decreasing marginal product of labor. True or False: If there are two points at which isoprofits are tangent to the production frontier in this model, the lower output quantity cannot possibly be part of a truly optimal production plan.
(b) Could it be that neither of the tangencies represents a truly optimal production plan?
(c) Illustrate a case where there are two truly optimal solutions where one of these does not occur at a tangency.
(d) What would a production frontier have to look like in order for there to be two truly optimal production plans which both involve positive levels of output?
(e) True or False: If the producer choice set is convex, there can only be one optimal production plan.
(f) Where does the optimal production plan lie if the production frontier is such that the marginal product of labor is always increasing?
(g) Finally, suppose that the marginal product of labor is constant throughout. What production plans might be optimal in this case?
B: In the text, we used a cosine function to illustrate a production process that has initially increasing and then decreasing marginal product of labor. In many of the end-of-chapter exercises, we will instead use a function of the form x = f (ℓ) = βℓ2 −γℓ3 where β and γ are both greater than zero.
(a) Illustrate how the profit maximization problem results in two "solutions". (Use the quadratic formula to solve for these.)
(b) Which of your two "solutions" is unambiguously not the actual profit maximizing solution?
(c) What else would you have to check to be sure that the other "solution" is profit maximizing?
(d) Now consider instead a production process characterized by the equation x = Aℓα. Suppose α < 1. Determine the profit maximizing production plan.
(e) What if α > 1?
(f) What if α = 1?