Frank Cowell: Microeconomics Exercise MICROECONOMICS Principles and Analysis Frank Cowell March 2007

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

Frank Cowell: Microeconomics Ex 10.15(1): Bertrand game Suppose firm 2 sets price p 2 > c Suppose firm 2 sets price p 2 > c  implies that there exists an  > 0 such p 2   > c Firm 1 then has three options: Firm 1 then has three options:  it can set a price p 1 > p 2  it can match the price p 1 = p 2  it can undercut, p 1 = p 2   > c The profits for firm 1 in the three cases are: The profits for firm 1 in the three cases are:   1  = 0, if p 1 > p 2   1  = ½[p 2  c][k  p 2 ], if p 1 = p 2   1  = [p 2  c   ][k  p 2 ], if p 1 = p 2   For small  profits in case 3 exceed those in the other two For small  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 If firms play a one-shot simultaneous move game  firms share the market  set p 1 = p 2 = c

Frank Cowell: Microeconomics Ex 10.15(2): Question method: Consider joint output of the firms q = q 1 + q 2 Consider joint output of the firms q = q 1 + q 2 Maximise sum of profits with respect to q Maximise sum of profits with respect to q

Frank Cowell: Microeconomics Ex 10.15(2): Joint profit max If firms maximise joint profits the problem becomes If firms maximise joint profits the problem becomes  choose k to max [k  q]q  cq The FOC is The FOC is  k  2q  c = 0 FOC implies that profit-maximising output is FOC implies that profit-maximising output is  q M = ½[k  c] Use inverse demand function to find price and the (joint) profit are, respectively Use inverse demand function to find price and the (joint) profit are, respectively  p M = ½[k + c] Use p M and q M to find price (joint) profit: Use p M and q M to find price (joint) profit:   M = ¼[k  c] 2

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

Frank Cowell: Microeconomics Ex 10.15(3): trigger strategy The trigger strategy is The trigger strategy is  at each stage if other firm has not deviated set p = p M  if the other firm does deviate then in all subsequent stages set p=c Example: Example:  suppose firm 2 deviates at t = 3 by setting p = p M –ε  this triggers firm 1 response p = c  then the best response by firm 2 is also p = c Time profile of prices is: Time profile of prices is: t firm 1:p M p M p M cc… firm 2:p M p M pcc…

Frank Cowell: Microeconomics Ex 10.15(3): payoffs If ε is small and firm 2 defects in one period then: 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,  2 =  M  thereafter  2 = 0 If the firm had always cooperated it would have got If the firm had always cooperated it would have got   2 = ½  M Present discounted value of the net gain from defecting is Present discounted value of the net gain from defecting is  ½  M  ½  M [  +  2 +  ] Simplifying this becomes Simplifying this becomes  ½  M [1  2  ] / [1   ] So the net gain is non-positive if and only if ½ ≤  ≤ 1 So the net gain is non-positive if and only if ½ ≤  ≤ 1

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