(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.)