University of Minnesota – Twin Cities ECON4631-001
Assignment 4
Deadline: December 13th 11:30 pm
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Q1. Price discrimination – 1st degree (35 points)
Consider the market of some good with a unique firm in it. In this market
there are only two consumers, one of low valuation and one with high valuation.
The firm is able to perfectly distinguish between them. Each consumer’s gross
consumer surplus is given by the following expression:
U(θi, qi) = θi
for i = 1, 2.
The monopolist’s cost function is given by C(qi) = 3q
i + 100, for i = 1, 2.
a) (3 points) Set up the firm’s maximization problem, detailing the objective
function and the participants constraints.
b) (3 points) Briefly explain why, at a solution, both constraints must hold
with equality.
c) (6 points) Assume a = 5, w = 2, θ1 = 14 and θ2 = 24. Solve for the firm’s
optimal packages: (T ∗1 , q
d) (3 points) Find the firm’s profit under this solution.
e) (3 point) What is the market’s total surplus? (Hint: you do not need to
do any calculation to answer this question.)
University of Minnesota – Twin Cities ECON4631-001
f) (17 points) Solve the monopolist’s problem but suppose now that the
monopolist uses two-part tariffs: T1 = A1 + p1q1 and T2 = A2 + p2q2.
Follow these steps:
(a) (6 points) Set up the firm’s problem. Be very careful when defining
the objective function (profits should be a function of prices and
parameters).
(b) (5 points) Take FOCs with respect to prices to find optimal prices,
and then replace these values in the demand functions to find equi-
librium quantities.
(c) (3 points) Calculate the fixed part of each tariff.
(d) (3 points) Finally, conclude this exercise by showing that the solution
to this problem (under two-part tariffs) is exactly the same as the
solution obtained before when the firm was using packages.
Q2. Price discrimination – 2nd degree (35 points)
Consider the market of some good with a unique firm in it. In this market
there are only two types of consumers, one of low valuation and one with high
valuation. A fraction λ of the consumers correspond to the low valuation type
and the remaining 1 − λ are high valuation type. The firm cannot distinguish
between types. Each consumer’s utility function (or gross consumer surplus) is
given by the following expression:
U(θ, q) = θq −
The firm faces a unitary cost c for producing the good.
Assume θ1 = 75, θ2 = 80, λ =
and c = 25.
a) (10 points) Set up the firm’s problem and solve for the optimal packages
(Hint: When setting up the firm’s problem, you do not need to write down
all four constraints and show which ones hold and which ones do not. You
can just write down the ”relaxed version” as is done on the lecture notes).
University of Minnesota – Twin Cities ECON4631-001
b) (4 points) With these results, compute the firm’s profits and consumer
surplus for each type (verify that it’s zero for the low type). Finally,
compute total surplus.
c) (15 points) Suppose now that the monopolist is restricted to using a two-
part tariff system, namely T1 = A1+ p1q1 and T2 = A2+ p2q2. Set up the
firm’s problem and find the optimal tariff schemes (A∗1, p
(Hint: For this kind of preferences, indirect utility function (net consumer
surplus) is given by V (θi, pi) =
and demand function is qi =
θi − pi. for i = 1, 2).
d) (4 points) Compute firm’s profits, consumer surplus for each type and
total surplus under this pricing system.
e) (2 points) Provide a brief conceptual explanation as to why the firm is
making less profits under two-part tariffs than under packages.
Q3. Price discrimination: 3rd degree (15 points)
Suppose that there are N1 consumers of type θ1 and N2 consumers of type θ2.
Preferences are U(θi, qi) = θi
for i = 1, 2. The parameters are
given as θ1 = 12, θ2 = 15, and a = 4.
The monopolist can distinguish between types but it cannot distinguish between
consumers that belong to the same type. Therefore, the monopolist has to care
for two separate groups of consumers. Let’s assume that the monopolist has a
cost function C(Q) = cQ where Q = q1 + q2.
a) (3 points) Derive the demand function for each group.
b) (6 points) Once you have each group’s demand function, set up the firm’s
maximization problem and find the optimal choices of output for each
group, q∗1 and q
c) (3 points) Assume c = 6, N1 = 1, and N2 = 2 what are the optimal prices?
What are the firm’s profits?
d) (3 points) Calculate (net) consumer surplus for groups one and two. Con-
clude this exercise by computing aggregate welfare (total surplus)
University of Minnesota – Twin Cities ECON4631-001
Q4. Entry Deterrence: Stackelberg Revisited (15
Consider a market with only one firm operating in it, which we identify as
the ”leader” firm. There is however, a second firm evaluating whether or not to
enter to market. We shall identify this firm as the ”entrant” one. Cost functions
are given by: CL(qL) = 2qL and CE = 2qE + F where F is the fixed entry cost
that the entrant must pay in order to start producing.
Market demand is given by P (Q) = 20 − Q where Q = qL + qE (of course, if
the entrant is not in the market, then qE = 0).
As in the Stackelberg environment, competition takes place in two stages: in the
first stage, the leader chooses its quantity and in the second stage, the entrant
observes the leader’s quantity and decides to enter or not. If it does, the entrant
chooses at the same time how much to produce.
a) (7 points) Find an expression for the entrant’s reaction function.
b) (5 points) Assuming F = 16, what will the leader choose to do: accom-
modate to entry or deter it?
c) (3 points) What if F = 72? What can you say about entry in this market
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