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ECON20120

Exercises on dynamic games

Question 1 Three competitors, firms 1, 2 and 3 face an inverse demand:

p = 10 − Q

Q = q1 + q2 + q3

where qi is the quantity produced by firm i = 1, 2, 3.

All firms face constant marginal costs c1 = 2 and c2 = c3 = 1 respectively.

Find the equilibrium prices and quantities in this market.

Question 2 (From past exam) Two competitors, firm 1 and firm 2, who produce differentiated products face demand functions of the form:

D1 (p1, p2) = 50 − 2p1 + p2

D2 (p1, p2) = 50 − 2p2 + p1

where pi is the price of good i = 1, 2.

Both firms face constant marginal costs c1 = 10 and c2 = 8 respectively.

Firm 1 has the opportunity to invest in new technology which reduces its costs to c1 = 0. Investment in new technology involves a one-off payment of f. The two firms are in a two stage game where, at the first stage firm 1 chooses whether to invest or not. If firm 1 does not invest in the first stage then in the second stage the two firms compete by simultaneously setting prices (Bertrand style). If firm 1 invests in the first stage, then in the second stage firm 2 sets its price first, followed by firm 1, which sets its own price. Find the range of values of f for which firm 1 will choose to invest.

Question 3 Suppose you bargain with someone over splitting £100. This amount is reduced by 6/1 every minute. The negotiation proceeds with one offer made each minute. There will be 6 rounds of offers after which the negotiation is over and nobody gets nothing. If you make the first pitch how much would you offer to your opponent?

Question 4 Mammy is out of the kitchen and it’s a great opportunity to get you hands on that cake inside the fridge. Alas, Daddy has the same idea and also rushes to the same end! Your Dad wishes to negotiate and offers you 3 out of the 20 pieces of the pie. You figure that offers back and forth will take 5 seconds each to respond. Meanwhile the cake melts at a rate of 50/1 every second and Mammy will be back in the next 20 seconds. Should you accept Daddy’s offer or not?

Question 5 Take the money and run: you and your mate see a sack of money that some pirates left on the dock while running away. The sack contains 100000 but it is torn apart and the wind blows away 10% every minute. Capt. Jack Sparrow also arrives there and proposes to split the money and run before the other pirates come back. You figure that there is time for four rounds of offers before the pirates come back and it’s ’goodbye companiero’.... You promise your mate half of what you will negotiate if she lets you ’make an offer he can’t refuse’. Answer: (a) Should you let Jack make an offer first or make yourself an offer he can’t refuse? (b) What would you offer to/accept from Capt. Jack Sparrow?