Imagine that consumer valuations vi are distributed uniformly between 0 and 100. Each consumer will buy at most one unit of the good depending on his or her willingness to pay. However, that willingness to pay depends on the fraction f of population that buys the good. In particular, consumer i will buy one unit of the good only if (0.4 + 6f 2)vi ≥ p. Otherwise, consumer i buys zero.
a. Assume that the price is p = $50. Show that the marginal consumer has basic valuation vM = 50/(0.4 + 6f 2).
b. Show that at this price, two non-zero market equilibria are possible: one with f = 0.1905 and one with f = 0.906. Which, if either of these, is stable?
