Suppose you and your friend Bob are in the business of producing baseball cards.
A: Both of you face the same production technology which has the property that the marginal product of labor initially increases for the first workers you hire but eventually decreases. You both sell your cards in a competitive market where the price of cards is p, and you hire in a competitive labor market were the wage is w.
(a) Illustrate your profit maximizing production plan assuming that p and w are such that you and Bob can make a positive profit.
(b) Now suppose you find a costless way to improve the technology of your firm in a way that unambiguously expands your producer choice set. As a result, you end up producing more than Bob (who has not found this technology). Illustrate how the new technology might have changed your production frontier.
(c) Can you necessarily tell whether you will hire more or less labor with the new technology?
(d) Can you say for sure that adopting the new technology will result in more profit?
(e) Finally, suppose p falls. Illustrate how it might now be the case that Bob stops producing but you continue to stay in the business.
B: You and Bob initially face the production technology x = 3Aℓ2 − 0.1ℓ3, and you can sell your output for p and hire workers at a wage w.
(a) Derive the marginal product of labor and describe its properties.
(b) Calculate the optimal number of baseball cards as a function of A assuming output price is given by p and the wage is w = 20. (Use the quadratic formula.)
(c) How much will you each produce if A = 1 and p = 1, and how much profit do each of you earn?
(d) Now suppose you find a better technology—one that changes your production function from one where A = 1 to one where A = 1.1. How do your answers change?
(e) Now suppose that competition in the industry intensifies and the price of baseball cards falls to p = 0.88. How will you and Bob change your production decisions?