Consider a monopolist with the constant marginal cost c > 0 who faces a continuum of consumers represented by the linear demand function q = a − bp, where a > c. Each consumer demands at most one unit of the output good. The interval [0, a/b] can be viewed as the interval where valuations of consumers are distributed. A consumer with valuation v ∈ [0, a/b] will buy the good iff v ≥ p. Therefore if p < a/b, then every consumer with valuation v ∈ [p, a/b] will buy the good, and we can say that the monopolist sells q = a − bp units of the good at the price p per unit. Consumers with valuations v ∈ [0, p) do not buy the good.
(a) Suppose the monopolist charges the same unit price independent of the consumers' valuations. What price is charged? What is the monopolist's profit? What is the lowest valuation of the consumer who buys the good? What is the consumers' surplus?
a) P = c, Profit = 0, Lowest valuation = 0, Consumers' surplus = 0
b) P = a/b, Profit = (a²)/(2b), Lowest valuation = a/b, Consumers' surplus = (a²)/(2b)
c) P = a, Profit = (a²)/(2b), Lowest valuation = a/b, Consumers' surplus = (a²)/(2b)
d) P = b, Profit = (a²)/(2b), Lowest valuation = b, Consumers' surplus = (a²)/(2b)