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Duopoly Model: Profit Maximization Strategies

Develop a simple repeated-game model of duopoly to analyze profits in cooperative and competitive cases, focusing on trigger strategies. Explore the Bertrand game scenario and compare profits under different pricing strategies. Further, consider maximizing joint profits and implement trigger strategies for long-term cooperation.

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Duopoly Model: Profit Maximization Strategies

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  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.

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