# Econ 302 Quiz — solution

```Simon Fraser University
Fall 2012
Econ 302 Quiz — solution
Section D200
Thursday Oct. 4, 2012
Due: end of the class, 9:20 am.
Write your name, SFU ID number, and tutorial section number on the exam booklet.
This is a closed-book exam. You may use a non-graphing calculator.
Please leave the room quietly if you finish and want to leave early. I will deduct
points off your quiz if you cause too much distractions to your neighbors.
1
1. Multiple-choice / True-false (no penalty for wrong guesses)
(a) (2 points) A mixed strategy is a/an
(A) strategy profile, (B) action profiles, (C) probability distribution over actions, (D)
collection of numbers over action profiles, or (E) none of the above.
(b) (2 points) A Nash equilibrium
(A) always involves strictly dominated actions, (B) may not exist in a finite game, (C)
is a symmetric strategy profile, (D) makes a coarser prediction than ISD, or (E) none of
the above.
(c) (1 points) True or False: Quantity competition usually leads to a lower price than price
competition.
(d) (1 points) True or False: In a two-player game written out in the “matrix” form, Player
1 controls the columns, and Player 2 controls the rows.
For Problem 2 to 4 show all your work.
2. (8 points) The market (inverse) demand function for a good is P (q) = 18 − q. There
is a monopolist with a constant marginal cost of 2 for producing each unit of the good.
Assume that the government imposes a sales tax of 50%, and the sales tax does not impact
the monopolist’s cost. Calculate the monopolist’s price and quantity, and calculate the
corresponding consumer’s price.
The monopolist solves
2
max
q
1
(18 − q)q − 2q.
1 + 0.5
The first order condition is:
2
(18 − 2qm ) − 2 = 0.
3
Solving this gives the monopolist’s quantity qm = 15/2 = 7.5. The consumer’s price is then
1
P (qm ) = 18 − 7.5 = 10.5. The monopolist’s price is 1+0.5
P (qm ) = 7.
3. (8 points) For the following game, (a) find the strategies that survive iterated deletions
of strictly dominated strategies (ISD), and (b) find all Nash equilibria (pure and mixed).
W
X
Y
Z
A 1, 2 -1, 1 -1, 0
8, 2
B
0, 0
1, 1
-1, 0
7, 1/3
C 2, -1 -1, 0
1, 1
10, 1/3
D 5, 3/2 0, 1 -1/3, 4
8, 2
Part (a):
First round: delete A (strictly dominated by 1/2C + 1/2D).
Second round: delete W (strictly dominated by Z) and Z (strictly dominated by 1/2X +
1/2Y ).
Third round: delete D (strictly dominated by 0.6B + 0.4C).
ISD strategies: {B, C} and {X, Y }
Part (b):
There are two pure-strategy Nash equilibria: (B, X) and (C, Y ). There is one non-trivial
mixed-strategy Nash equilibrium: (1/2B + 1/2C, 1/2X + 1/2Y ).
4. (8 points) The market (inverse) demand function for a good is P (q) = 18 − q. There
are 7 firms, each with a constant marginal cost of 2 for producing each unit of the good.
The 7 firms compete by setting their quantities of production, and the price of the good is
determined by the market demand function given the total quantity. Calculate the symmetric
pure-strategy Nash equilibrium in this game and the corresponding market price.
3
Fix a symmetric strategy profile in which every firm produces q units.
The profit of any firm (say firm 1) when he produces q1 and the others each produce q is:
Π1 (q1 ) = q1 (P (6q + q1 ) − 2) = q1 (18 − 6q − q1 − 2).
Given q, firm 1 wants to maximize Π1 (q1 ). The first order condition with respect to q1 is
18 − 6q − 2q1 − 2 = 0,
which gives the best-response function:
q1 = 8 − 3q.
At a symmetric Nash equilibrium, we must have q1 = q. Therefore, q = 8 − 3q, which gives
q = 2.
In summary, each firm produces q = 2 is the symmetric pure-strategy Nash equilibrium.
The equilibrium price is 18 − 2 &times; 7 = 4.
4
5
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