1. Prove proposition 8.1.
2. (a) Prove that it is inconsistent with the theory of perfect competition that a firm in a long-run equilibrium would ever produce a (positive) quantity at which its long-run average costs are declining. What are the implications of this for an industry in which all firms have average costs that are perpertually declining?
(b) Prove that in any long-run equilibrium, firms produce at efficient scale or larger, and if a firm is making positive profits, it must be producing at larger than efficient scale. (You may assume that average and marginal cost curves are all continuous.)
Endless variations on the example given in section 8.2 and 8.3 can be played. Problem 3 is quite difficult; in it you consider the "equilibrium dynamics" in the second example from the text. In problem 4, you see what happens if there is limited entry into a competitive industry; in problem 5 we consider a case in which firms possess a number of different technologies; and in problem 6 you work through "equilibrium dynamics" when a factor price changes.