# Extensive Form - London School of Economics Frank Cowell: Microeconomics March 2007 Exercise 10.15 MICROECONOMICS Principles and Analysis Frank Cowell Ex 10.15(1): Question Frank Cowell: Microeconomics purpose: Develop simple model repeated-game model of duopoly method: Find profits in cooperative and competitive cases. Build these into a trigger strategy. Ex 10.15(1): Bertrand game Frank Cowell: Microeconomics Suppose firm 2 sets price p2 > c

Firm 1 then has three options: 1= 0, if p1 > p2 1= [p2 c][k p2 ], if p1 = p2 1= [p2 c ][k p2 ], if p1 = p2 For small profits in case 3 exceed those in the other two it can set a price p1 > p2 it can match the price p1 = p2 it can undercut, p1 = p2 > c

The profits for firm 1 in the three cases are: implies that there exists an > 0 such p2 > c firm 1 undercuts firm 2 by a small and captures whole market If firms play a one-shot simultaneous move game firms share the market set p1 = p2 = c Ex 10.15(2): Question Frank Cowell: Microeconomics method: Consider joint output of the firms q = q1 + q2 Maximise sum of profits with respect to q Ex 10.15(2): Joint profit max Frank Cowell: Microeconomics

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 q = [k c] M Use inverse demand function to find price and the (joint) profit are, respectively p = [k + c] M Use pM and qM to find price (joint) profit: = [k c]2 M

Ex 10.15(3): Question Frank Cowell: Microeconomics 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 Ex 10.15(3): trigger strategy Frank Cowell: Microeconomics The trigger strategy is Example:

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 suppose firm 2 deviates at t = 3 by setting p = pM this triggers firm 1 response p = c then the best response by firm 2 is also p = c Time profile of prices is: firm 1: 1 pM 2 pM 3 pM 4 c

5 c ... firm 2: pM pM p c c t Ex 10.15(3): payoffs Frank Cowell: Microeconomics

If is small and firm 2 defects in one period then: If the firm had always cooperated it would have got M M [ + 2 + 3 +...] Simplifying this becomes 2 = M Present discounted value of the net gain from defecting is

for that one period firm 2 would get the whole market so, for one period, 2 = M thereafter 2 = 0 M [1 2] / [1 ] So the net gain is non-positive if and only if 1 Ex 10.15(1): Points to remember Frank Cowell: Microeconomics Set out clearly time pattern of profits Take care in discounting net gains back to a base period.