Suppose that the gross utility of a representative consumer is V = N 1/2 where N is the number of products. Their net utility is U = V (N) - P = N 1/2 - P where P is the equilibrium price when there are N firms. Suppose that P = 1/N , so that as the number of firms increases, the
equilibrium price falls. Marginal cost of production is zero, but each firm incurs a fixed cost equal to f . The marginal benefit of another variety (firm) is dV/dN = 1/(2√N). The number of consumers is normalized to one.
(a) What is the socially optimal (first-best) number of firms?
(b) Assuming that the market share of each firm when there are N firms is 1/N , what is the free-entry number of firms?
(c) Compare the socially optimal number of firms with the free-entry number. When is there excess variety? Insufficient variety? Why?