Exercise 10.15

1. Exercise 10.15 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

2. Ex 10.15(1): Question • purpose: Develop simple model repeated-game model of duopoly • method: Find profits in cooperative and competitive cases. Build these into a trigger strategy.

3. Ex 10.15(1): Bertrand game • Suppose firm 2 sets price p2 > c • implies that there exists an e > 0 such p2 e > c • Firm 1 then has three options: • it can set a price p1 > p2 • it can match the price p1 = p2 • it can undercut, p1 = p2 e > c • The profits for firm 1 in the three cases are: • P1= 0, if p1 > p2 • P1= ½[p2 c][k  p2], if p1 = p2 • P1= [p2 c e][k  p2], if p1 = p2 e • For small e profits in case 3 exceed those in the other two • firm 1 undercuts firm 2 by a small ε and captures whole market • If firms play a one-shot simultaneous move game • firms share the market • set p1 = p2=c

4. Ex 10.15(2): Question method: • Consider joint output of the firms q = q1 + q2 • Maximise sum of profits with respect to q

5. Ex 10.15(2): Joint profit max • If firms maximise joint profits the problem becomes • choose k to max [k  q]q  cq • The FOC is • k 2q  c = 0 • FOC implies that profit-maximising output is • qM = ½[k c] • Use inverse demand function to find price and the (joint) profit are, respectively • pM = ½[k+ c] • Use pM and qM to find price (joint) profit: • PM = ¼[k c]2

6. Ex 10.15(3): Question method: • Set up standard trigger strategy • Compute discounted present value of deviating in one period and being punished for the rest • Compare this with discounted present value of continuous cooperation

7. Ex 10.15(3): trigger strategy • The trigger strategy is • at each stage if other firm has not deviated set p = pM • if the other firm does deviate then in all subsequent stages set p=c • Example: • suppose firm 2 deviates at t = 3 by setting p = pM –ε • this triggers firm 1 response p = c • then the best response by firm 2 is also p = c • Time profile of prices is: 1 2 3 4 5 ... t firm 1: pMpMpMc c … firm 2: pMpMp c c …

8. Ex 10.15(3): payoffs • If ε is small and firm 2 defects in one period then: • for that one period firm 2 would get the whole market • so, for one period, P2 = PM • thereafter P2 = 0 • If the firm had always cooperated it would have got • P2 = ½PM • Present discounted value of the net gain from defecting is • ½ PM  ½PM [d + d2 + d3 +...] • Simplifying this becomes • ½ PM [1  2d] / [1  d] • So the net gain is non-positive if and only if ½ ≤ d ≤ 1

9. Ex 10.15(1): Points to remember • Set out clearly time pattern of profits • Take care in discounting net gains back to a base period.