ECO380H1 Lecture Notes - Lecture 8: Price Discrimination, Incentive Compatibility
ECO380: Tutorial Problem Set - For Wednesday, January 31st, 2018
Second-degree Price Discrimination
1. [Optimal 2nd Degree Price Discrimination] As promised in lecture, you will now show that under 2nd
degree price discrimination, the prot-maximizing menu of packages that our nightclub owner can oer
is {(11 drinks for $99.50), (1 drink for $9.50)}. As a reminder, the monopolist knows that there are
two types of people in the economy, those with high income and those with low income. Demand for
drinks in this nightclub for the high-income type is given by P= 15 −QH. For the low-income type,
demand is given by P= 10 −QL. The monopolist knows that there is one person of each type, but
cannot tell who is of which type, so each consumer gets to choose the package that he or she likes best.
(a) Show that the prot-maximizing menu of packages for this rm is to oer the two options {(11
drinks for $99.50), (1 drink for $9.50)}. To get you started, consider the following:
i. How much is a low-type willing to pay for an arbitrary number of drinks? How much prot
does the monopolist make o of selling that number of drinks to the low-type?
ii. Does the monopolist have any incentive to oer a package with a given number of drinks to
the low type at a price less than the low-type's maximum willingness to pay?
iii. For that low-type package, how much surplus would the high-type receive if he purchases that
package? What does this tell us about incentive compatibility?
iv. What is the most that the high-type would pay for a bundle with a given number of drinks,
given his incentive compatibility constraints? How much prot does the monopolist make o
of that transaction?
v. With that in mind, what quantitiy should the monopolist have in the packages aimed at
low-types? High-types? What price can he charge, and what is his overall prot?
(b) For comparison, how much more prot does the monopolist make if he can implement this optimal
2nd degree price discrimination scheme than if he could only charge a uniform price per drink
(and no entrance fee, as in our baseline linear monopoly pricing)?
(c) How much has it cost the monopolist to solve the identication problem? That is, how much less
prot does he make under 2nd degree price discrimination than under 1st degree price discrimi-
nation?
2. [2nd Degree - continued] Suppose now that instead of having one person of each type, our monopolist
from the previous example faces:
(a) two people of each type. What is the optimal menu to oer to consumers now that there are two
of each type? What does this tell us about the importance of the actual numbers of consumers?
(b) four people of the low-income type, and two people of the high-income type. Now what is the
optimal menu of bundles to oer to the consumers?
(c) two people of the low-income type, and four people of the high-income type. Now what is the
optimal menu of bundles to oer to the consumers?
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