Problem 1 Deriving a Demand Function from a Distribution Function

Consider a market with a "unit mass" of consumers. ("Unit" means that the number of consumers is normalized to 1. "Mass" means that each consumer is insignificantly small.) Suppose each consumer purchases either 0 or 1 unit of a product. Let v denote a consumer's value of the product. Each consumer has a different value v, with the consumer values of the population being uniformly distributed on [0, 1]. That is the population distribution functions for the values is F(v) = v. A consumer purchases a product if and only if her value is at least as great as the product's price p; that is, if v p.

a. What is the market demand function?

b. Graph the market demand function.

c. Suppose that the market has only one consumer, who purchases 0 or 1 unit of a product. The consumer knows her value v, which is drawn from a uniform distribution on [0, 1]. The firms in a market do not know the consumer's value. They believe the probability distribution function of v is F(v) = v. The consumer purchases the product if and only if v p. What is her (probabilistic) demand function? That is, what is her probability of purchasing the product as a function of p? Graph her demand function.

d. Suppose there are 10 consumers in the market, with each like the one in part c.

Each consumer independently draws a value v according to F(v). What is the

market (probabilistic) demand function?

e. Compare your answers to parts a and c