# solution (thanks to Kelby)

```Homework #6. Game Theory (10pts)
1. Find all pure strategy Nash equilibria in the following game, or show that none exist
a. Draw arrows representing each player’s choices
b. Show equilibrium (or equilibria) A,A B,B
Player 2
A
B
3,
2
2,
1
A
Player 1
B
2, 1
4, 2
IF P1=A then P2=A If P1=B then P2=B
2. For which values of X (if such X exists) the following game does not have a pure strategy
equilibrium, or show that this game always has one (or more) Nash equilibria
x&gt;4 then B,A if x&lt;4 then NONE if x=4 then B,A
Player 2
A
4,10
B
1, 12
A
Player 1
X, 14
B
2, 2
3. Cost of two firms is given by C  Q   Q , market demand is given by Q  25  P . Firm 1 can
credibly commit to a quantity choice before firm 2 makes its choice.
a. Fill out the rest of the payoffs
b. Find all pure strategy subgame perfect Nash equilibria in the following game 12,6
QFollower=6
_72__, _36__
Firm 2
QCournot=8
_48__, _32__
QFollower=6
_80__, _60__
QCournot=8
_64__, _64__
Firm 1
QCournot=8
Firm 2
4. Starting with the setup of the previous problem now assume that there is a fixed cost of 30 in the
production function and the leader can attempt to prevent entry by limit pricing.
a. Fill out the payoff matrix
b. Find all pure strategy subgame perfect Nash equilibria in the following game
14,0 and 14,0 in subgame
_42__, _6__
Enter, QFollower=6
Firm 2
Not enter, Q=0
__114___, 0
Enter, Q=_5_
__40__,__-5___
Not enter, Q=0
__110____, 0
Firm 1
QLimit=14
Firm 2
Notice that when the follower does not enter and the leader produces either 12 or 14, the market price will
be determined by the inverse demand as 25-12=13 or 25-14=11. Leader’s profits will be determined
correspondingly as (13-1)12-30=114 and (11-1)14-30=110
5. Solve problem 3 at the end of chapter 6 (solved in the book)
6. Solve problem 4 at the end of chapter 6. This problem is optional but a good attempt is required.
This problem demonstrates how the nature of the competition affects how firms react to changes
in prices. You may use formulas from the appendix to find the answer.
There are two ways to answer this question. Let me show you how to do the increase in price for Cournot
competition. From the equation 6A.7. p 
on p is
a  nm
a
n


m so the effect of 10% increase in m
n 1
n 1 n 1
n
.1m  . In the case of duopoly n  2 and the price will increase by 2/3 of the increase in the
n 1
marginal cost. Using equations for quantity and price of Stackelberg from the appendix of chapter 6 you
will be able to find the changes for the remaining price and quantity.
Another way would be to set up a very simple and familiar Cournot competition model, change marginal
cost by 10% and see what happens to the price. Say demand is Q  25  P; P  25  Q and the marginal
cost is 1. The price in Cournot duopoly is 9. Now if the marginal cost increases to 1.1 (a 10% increase)
Q
25  1.1 QB

 11.95  B . Using the familiar trick QA  QB we
2
2
2
can solve for quantity QA  QB  11.95*2 / 3  7.96667 so the total quantity is twice of that of each
the reaction functions become: QA 


2
3
firm’s and the price from the inverse demand is therefore P  25  2* 11.95*   9.06667 . In other
words, a 0.1 increase in the marginal cost caused a 0.06667 increase in the price. Notice that the answer is
the same whether you use the is 2/3
7. Solve problem 7 at the end of chapter 6
Substitute numbers into the formula for elasticity to find the elasticity. Cartel would price at the
monopoly price which is always in the elastic part of the demand.
```