Download Practice Problems II Answers

Practice Problems II Answers
1. The payoff matrix looks like this:
Player 1
R
P
S
R
0, 0
1, -1
-1, 1
Player 2
P
-1, 1
0, 0
1, -1
S
1, -1
-1, 1
0, 0
I’ve used yellow and green highlighting (respectively) to show player 1’s and player 2’s
best responses to each other. Since there is no cell in which both numbers are
highlighted, there is no pure-strategy Nash Equilibria. In other words, we can’t find any
combination of strategies such that neither player would wish to change his strategy.
However, it turns out there is a mixed-strategy Nash Equilibrium of this game, which is
for both players to randomize over R, P, and S with probabilities 1/3, 1/3, and 1/3. We
did not learn in class how to calculate a mixed-strategy equilibrium, but if you’ve ever
played this game, you probably understand and play this equilibrium intuitively.
2. The payoff matrix looks like this:
Akeem
Reach today
Don’t reach
Rojelio
Reach today
Don’t reach
$50, $50
$100, $0
$0, $100
$75, $75
(If both reach today, they split the $100, and there’s nothing left tomorrow. If both don’t
reach, the $100 grows to $150, which they split between them. If one reaches and the
other doesn’t, the one who reaches gets the whole $100, the other gets nothing, and
there’s nothing left for tomorrow.)
As with the previous problem, highlighting shows the players’ best responses. {Reach
today, Reach today} is the Nash Equilibrium of this game, since both payoffs are
highlighted in that cell.
This game is a Prisoners’ Dilemma, because both players have dominant strategies, and
the resulting dominant strategy equilibrium yields payoffs that are worse for both players
than they would have gotten if both had acted differently.
3. The payoff matrix looks like this:
Hector
Hard bargain
Easy bargain
Menelaus
Hard bargain
Easy bargain
$0, $0
$500, $100
$100, $500
$300, $300
(If both drive a hard bargain, there’s no deal, so there’s zero change relative to the
baseline. If both drive an easy bargain, Hector pays $700 for a $1000 gain, and Menelaus
receives $700 for accepting a $400 loss, yielding net $300 each. If Hector drives an easy
bargain and Menelaus a hard bargain, Hector pays $900 for a $1000 gain, and Menelaus
receives $900 for a $400 loss, yielding net $100 and $500 respectively. Similar
reasoning for a payment of $600 yields the numbers in the upper right cell.)
The highlighted best responses show that there are two Nash Equilibria: {Easy, Hard}
and {Hard, Easy}. This game is like the game of Chicken, because there are multiple
equilibria, one preferred by one party and one preferred by the other, as well as a
“disaster” outcome in which both parties lose out on possible gains.
4. (a) Under Cournot competition, firm 1 perceives its demand curve as P = [140 – 2q2] –
2q1. So firm 1’s marginal revenue is the same thing with twice the slope: MR = [140 –
2q2] – 4q1. Setting this equal to the MC of 20, we get:
[140 – 2q2] – 4q1 = 20
4q1 = 120 – 2q2
qR1 = 30 – (1/2)q2
That’s firm 1’s reaction function. Since the firms are identical, firm 2’s reaction is
similar:
qR2 = 30 – (1/2)q1
The equilibrium occurs where the reaction functions cross each other. Solve the system
of equations by substituting one into the other:
q1 = 30 – (1/2)[30 – (1/2)q1]
q1 = 15 + (1/4)q1
(3/4)q1 = 15
q*1 = 20
Since the firm’s are identical, q*2 = 20 as well, so Q* = 40, and P* = 140 – 2(40) = 60.
(b) Under Bertrand competition, the firms undercut each other’s prices until their prices
both equal the marginal cost: P*1 = P*2 = 20. Plugging this into the demand curve, we
get 20 = 140 – 2Q, so 2Q = 120, so Q* = 60, which is split between the two firms.
(c) Under Stackelberg competition, we use backward induction to find the solution.
Once firm 1 has chosen its quantity, firm 2 will respond according to the reaction
function we found earlier: qR2 = 30 – (1/2)q1. Predicting this, firm 1 perceives the
following demand curve:
P = 140 – 2q2 – 2q1
P = 140 – 2[30 – (1/2)q1] - 2q1
P = 80 – q1
Find the MR by doubling the slope, and set it equal to the MC:
MR = 80 - 2q1 = 20
2q1 = 60
q*1 = 30
Plug this into firm 2’s reaction function to get firm 2’s quantity:
q*2 = 30 – (1/2)(30) = 15
So the market quantity is Q* = 30 + 15 = 45, and the market price is P* = 140 – 2(45) =
50.
5. The calculations are similar to those above, so I’ll just give the answers:
(a) Cournot: q*1 = q*2 = 60, Q* = 120, P* = 50
(b) Bertrand: P*1 = P*2 = 30, Q* = 180.
(c) Stackelberg: q*1 = 90, q*2 = 45, Q* = 135, P* = 45
6. (a) Go through the same procedure as in the previous problems to find firm 1’s
reaction function. But since the firms’ marginal costs differ, you need to calculate firm
2’s reaction function as well. They turn out to be:
qR1 = 90 – (1/2)q2
qR2 = 75 – (1/2)q1
Then solve the system of equations by plugging one into the other.
q1 = 90 – (1/2)[ 75 – (1/2)q1]
q1 = 52.5 + (1/4)q1]
(3/4)q1 = 52.5
q*1 = 70
Plugging this into firm 2’s reaction function, we get q*2 = 75 – (1/2)(70) = 40. So Q* =
70 + 40 = 110, and P* = 210 – 110 = 100.
(b) Under Bertrand competition, the firms undercut each until firm 2 (the higher-cost
firm) leaves the market. Firm 1 sets a price just under firm 2’s MC; P*1 = 59, and Q* =
151 (all of it sold by firm 1). If it is costly for firm 2 to re-enter the market, firm 1 could
possibly raise its price higher after firm 2 leaves.
7. The calculations are similar to those above, so I’ll just give the answers:
(a) Cournot: q*1 = 14, q*2 = 11, Q* = 25, P* = 45
(b) Bertrand: P*1 = 17, Q* = 36.5, and firm 2 leaves the market.