Week 5 - simultaneous move games with continuous srategies

Question 1

What is the method to find the best response in a bertrand oligopoly with simultaneous moves.

Answer 1

1. set up the payoff functions
2. first the FOC wrt their own price
3. set FOC = 0 and solve for price
4. check second order condition to make sure it’s a maximum
5. repeat for firm 2

Question 2

True or false : When plotting best response curves, everything above the intersection isn’t rationalizable

Answer 2

False

Question 3

What is the belief hierarchy?

Answer 3

Very large prices in the restaurant game are rationalizable e.g. if p2=4000, this would only happen if p1 = 8000 which would only happen if p2 = 12,000 etc..

Question 4

How do you work out the NE in a bertrand oligopoly with simultaneous moves?

Answer 4

Where the BR functions intersect

Question 5

Why would all prices above the best response curves intersection never happen?

Answer 5

Even though they are rationalizable, they will not be stable because someone always has incentive to deviate so they will not be played in a NE

Question 6

What happens when you put a price cap in a bertrand oligopoly with simultaneous moves?

Answer 6

Everything above the NE is no longer rationalizable

Question 7

How would you find the collusive outcome in a bertrand oligopoly with simultaneous moves?

Answer 7

1. Set up a single payoff function which depends on both firms profits. big pi(p1,p2) = pi1(p1,p2) + pi2(p1,p2)
2. Find FOC wrt p1 and p2
3. solve for p1 and p2

Question 8

Why could collusion never be sustained in a NE.

Answer 8

It is not a stable outcome - players have an incentive to unilaterally deviate. Collusion is also not on the best response cures

Question 9

How might firms sustain collusion?

Answer 9

communication
price-matching guarantee - in some countries price matching is illegal
repeated game strategies

Question 10

How do you find the rollback equilibrium in a bertrand oligopoly with sequential moves?

Answer 10

1.Work out the simultaneous move best responses first i.e. p2 = …
2. Work out p1’s best response subbing in p2 from the simultaneous moves game
3. FOC, solve for p1
4. put p1 back into p2’s simultaneous move BR.

Question 11

In a bertrand oligopoly with perfect substitutes, what are the 3 outcomes (depending on how you price wrt the other price)

Answer 11

1. All the q of customers (if p1<p2)
2. no customers (if p1>p2)
3. exactly half the customers (if p1=p2)

Question 12

How do you work out the demand function of the whole market in a bertrand oligopoly with perfect substitutes?

Answer 12

Add all of the demand functions of all the firms together

Question 13

How do you work out the profit functions in a bertrand oligopoly with perfect substitutes, for the 3 different outcomes?

Answer 13

Multiply the q of customers you will serve by (pi - x), with x = mc

Question 14

What is the NE in a bertrand oligopoly with perfect substitutes?

Answer 14

prices = mc

Question 15

In a bertrand oligopoly with perfect substitutes, if 1 firm has lower mc than the other, what is the outcome?

Answer 15

the firm with lower costs will choose the highest price the other firm cannot match - 1 pence below the mc of the other firm.

Question 16

When working out the NE in Cournot competition, what do you have to do with the profit function?

Answer 16

Take out your quantity from the whole market quantity. i.e. express sum of all quantities as your quantity + the others.

Question 17

In Cournot, if the number of firms increase what happens to the price?

Answer 17

Decreases

Question 18

What is the significance of BRF in Cournot to be downward sloping?

Answer 18

If other firms cut production, you want to increase yours - these games are strategic substitutes

Question 19

What is the significance of BRF in Bertrand to be upward sloping?

Answer 19

If other firms cut prices, you want to cut yours - these games are strategic complements.