Problem
We consider a market in which consumers can buy a network good. The utility of a consumer can be written as U = θ+vn-p, where p is the price, θ a random variable for each consumer (with θ following a uniform distribution over [0,1]), v a positive parameter and n the number of consumers buying the good.
i. If ne is the anticipated number of users at the time of purchase, what is the characteristic (the value of θ) of the consumer indifferent between buying or not the good?
ii. Assuming rational expectations, what is then the demand function when v < 1?
iii. Taking as given the market price and still assuming that v < 1, what is the equilibrium (that the equilibrium number of consumers)?
iv. Suppose now that v > 1. What is the shape of the inverse demand function (hint: consider the cases n = 0 and n = 1 closely)? For a given price, what are the possible equilibria?