Posted: December 2nd, 2022

Econ 100A: Microeconomics

Econ 100A: Microeconomics
“Midterm 3” homework exercise
Jim Campbell
University of California, Berkeley
Due by Dec. 2nd, 2022
11.59pm
ˆ Please complete the assignment on your own paper and scan and upload to
Gradescope. As part of your submission, you must indicate where on your
scan to find each answer.
ˆ You must handwrite (not type) your answers, unless you have a specific DSP
accommodation that allows typing.
ˆ You may discuss the homework with other students in groups, but each person
must upload their own work, in their own words, in their own handwriting.
ˆ You may not collaborate at large scale (i.e. on Ed Discussion, Discord etc.)
and you may not get ‘professional’ help (i.e. from Jim, the SLC etc.) with
the specific questions on the exam. I am asking you to use this opportunity
to demonstrate your own knowledge and understanding of the material.
1
Econ 1, Fall 2022 | “Midterm 3” Jim Campbell | UC Berkeley
1. Consider a two player, simultaneous move game. The strategies available to each player and
payoffs are shown in the following matrix form:
Player 1
Player 2
L C R
T 6, 4 5, 2 8, 5
M 4, 4 9, 3 5, 9
B 7, 5 4, 4 5, 2
Game 5
a) Find and write down the Nash equilibria in pure strategies, if any exist.
b) Say that instead of being a simultaneous move game this game had been sequential. First
player 1 will choose a strategy and then player 2 will observe the choice made by player
1 and choose a strategy. The full structure of the game, including all payoffs, is common
knowledge to both players. Sketch a game tree to represent the extensive form of this
game. Find and write down the subgame perfect Nash equilibria.
2. In this question we will combine a very simplified model of an externality that has long-lasting
impacts with the model of discounted utility that we learned in our game theory topic. Consider
a hypothetical environmental externality: produced as the byproduct of industrial activity, each
unit of emission of a pollutant that is emitted once, today, causes $10 of external harm to society
each and every year, starting immediately, forever. Let’s say that policymakers apply a discount
factor of δ ∈ (0, 1) to future gains and losses, in an analog of the discounted utility model except
for cash payoffs rather than utilities.
What would the socially efficient Pigouvian tax on the emission of this pollutant be if we applied
a discount factor of (i) δ = 0.5, (ii) δ = 0.9, and (iii) δ = 0.99? Give an intuitive explanation
of how to interpret the parameter δ in this context, and explain precisely but in simple terms
what the goal and effect of the Pigouvian tax would be.
3. Consider a model of Cournot duopoly. Two firms produce an identical product. The inverse
demand function for the product is given by p = 60 − y, where y = y1 + y2 is the sum of the
quantity produced by the two firms. Production costs are zero for both firms.
a) Find the quantity produced by each firm in the Nash equilibrium of this game. Find the
price of the product and the profit of each firm.
b) Now say that marketing guru Jim has a proposal for firm 1. He would be able to convince
people that the two firms’ products were in fact slightly different from each other, so that
the amount produced by each firm would not impact the price of the other’s product
as much as it did before. The inverse demand function for each product would become
p1 = 60 − y1 −
1
2
y2 and p2 = 60 − y2 −
1
2
y1. What is the most that firm 1 would be willing
to pay Jim to do this? Show your work and explain in a few sentences what you did to
find your answer.
2
Econ 1, Fall 2022 | “Midterm 3” Jim Campbell | UC Berkeley
4. A profit-maximizing monopolist produces a good and sells it to three consumers, A, B and C.
It has zero costs. The consumers have inverse demand curves given by
pA = 10 − qA, (1)
pB = 12 − 2qB, (2)
pC = 4 −
1
2
qC. (3)
a) If consumer B had been the only consumer and the monopolist had to set a single, uniform
price for its product, what price would it choose and how many units would it sell? What
would be the socially efficient price and quantity?
The monopolist has decided to price its good using a two part tariff. It knows that it has three
consumers, and it knows exactly what the demand curve is for each one.
b) Say that the monopolist could tell the consumers apart and set a personalized two part
tariff for each of them. What two part tariff would it set for each consumer?
c) Say that the monopolist cannot tell the three consumers apart and has to set just a single
two part tariff that all three consumers can then decide whether or not to pay. What two
part tariff would it set? Explain your answer.
3