Price Discrimination. A monopolist produces a good with constant marginal cost normalized to zero. The monopolist knows that her customers consist of two groups. After normalization, the population of the high types is 1 ; the population of the low types 3 . The utility function of an individual of type i is Ui(q,y)=aq(1/2)biq2+y, where y is the outside good, their demand for the monopoly good qi=biap, and consumer surplus is CSi(p)=2bi(ap)2, where i=h,l,bl>bh, and q is consumption of the monopoly good. Let a=1,bl=2/3, and bh=1/3. (a) Verify the demand function for type i. [Hint: Marginal utility for type i is abiq.] (b) Show that consumer surplus for type i is equivalent to the increase in utility at the optimal choice from optimally consuming q relative to q=0 (c) What are the profit-maximizing uniform price, quantities, and profits? [Hint: Market demand at price p is the sum of type l and type h demand at that price.] (d) Suppose that the monopolist can charge type specific two-part tariffs. Find the menu of profit-maximizing two-part tariffs. Explain why this menu is not incentive compatible. (e) Find the profit-maximizing two-part tariff if the monopolist cannot distinguish types. Verify that the usage price is 1/6. [Hint: The rate of decrease in consumer surplus for type i when the price increases is biap.] Price Discrimination. A monopolist produces a good with constant marginal cost normalized to zero. The monopolist knows that her customers consist of two groups. After normalization, the population of the high types is 1 ; the population of the low types 3 . The utility function of an individual of type i is Ui(q,y)=aq(1/2)biq2+y, where y is the outside good, their demand for the monopoly good qi=biap, and consumer surplus is CSi(p)=2bi(ap)2, where i=h,l,bl>bh, and q is consumption of the monopoly good. Let a=1,bl=2/3, and bh=1/3. (a) Verify the demand function for type i. [Hint: Marginal utility for type i is abiq.] (b) Show that consumer surplus for type i is equivalent to the increase in utility at the optimal choice from optimally consuming q relative to q=0 (c) What are the profit-maximizing uniform price, quantities, and profits? [Hint: Market demand at price p is the sum of type l and type h demand at that price.] (d) Suppose that the monopolist can charge type specific two-part tariffs. Find the menu of profit-maximizing two-part tariffs. Explain why this menu is not incentive compatible. (e) Find the profit-maximizing two-part tariff if the monopolist cannot distinguish types. Verify that the usage price is 1/6. [Hint: The rate of decrease in consumer surplus for type i when the price increases is biap.]