exercise 10.15
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Exercise 10.15. MICROECONOMICS Principles and Analysis Frank Cowell. March 2007. 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. Ex 10.15(1): Bertrand game. - PowerPoint PPT PresentationTRANSCRIPT
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Exercise 10.15
MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis
Frank Cowell Frank Cowell
March 2007 March 2007
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(1): Question
purposepurpose: Develop simple model repeated-game model of duopoly: Develop simple model repeated-game model of duopoly methodmethod: Find profits in cooperative and competitive cases. Build these into a trigger : Find profits in cooperative and competitive cases. Build these into a trigger
strategy.strategy.
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(1): Bertrand game Suppose firm 2 sets price Suppose firm 2 sets price pp22 > > cc
implies that there exists an implies that there exists an > 0 such > 0 such pp22 > > cc
Firm 1 then has three options: Firm 1 then has three options: it can set a price it can set a price pp11 > > pp22
it can match the price it can match the price pp11 = = pp22
it can undercut, it can undercut, pp11 = = pp22 > > cc
The profits for firm 1 in the three cases are:The profits for firm 1 in the three cases are: 11= 0, if = 0, if pp11 > > pp22 11= = ½[½[pp22 cc][][k k pp22 ],], if if pp11 = = pp22
11= = [[pp22 cc ][][k k pp22 ],], if if pp11 = = pp22 For small For small profits in case 3 exceed those in the other two profits in case 3 exceed those in the other two
firm 1 undercuts firm 2 by a small ε and captures whole marketfirm 1 undercuts firm 2 by a small ε and captures whole market
If firms play a one-shot simultaneous move gameIf firms play a one-shot simultaneous move game firms share the marketfirms share the market set set pp11 = = pp22 = = cc
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(2): Question
methodmethod: : Consider joint output of the firms Consider joint output of the firms q q = = qq11 + + qq22
Maximise sum of profits with respect to Maximise sum of profits with respect to qq
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 10.15(2): Joint profit max
If firms maximise joint profits the problem becomes If firms maximise joint profits the problem becomes choose choose kk to max [ to max [k k q q]]q q cq cq
The FOC isThe FOC is kk 22q q c c = 0= 0
FOC implies that profit-maximising output isFOC implies that profit-maximising output is qqM M = = ½½[[kk c c] ]
Use inverse demand function to find price and the (joint) Use inverse demand function to find price and the (joint) profit are, respectivelyprofit are, respectively ppM M = = ½½[[kk + c+ c] ]
Use Use ppMM and and qqM M to find price (joint) profit: to find price (joint) profit: M M = = ¼¼[[kk c c]]22
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(3): Question
methodmethod: : Set up standard trigger strategySet up standard trigger strategy Compute discounted present value of deviating in one period and being Compute discounted present value of deviating in one period and being
punished for the restpunished for the rest Compare this with Compare this with discounted present value of continuous cooperation discounted present value of continuous cooperation
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(3): trigger strategy
The trigger strategy is The trigger strategy is at each stage if other firm has not deviated set at each stage if other firm has not deviated set p p = = ppMM if the other firm does deviate then in all subsequent stages set if the other firm does deviate then in all subsequent stages set p=cp=c
Example:Example: suppose firm 2 deviates at suppose firm 2 deviates at t t = 3 by setting = 3 by setting pp = = ppMM –ε –ε this triggers firm 1 response this triggers firm 1 response p = cp = c then the best response by firm 2 is also then the best response by firm 2 is also p = cp = c
Time profile of prices is:Time profile of prices is:
11 22 33 44 55 ......tt
firm 1:firm 1: ppMM ppMM ppMM cc cc ……
firm 2:firm 2: ppMM ppMM pp cc cc ……
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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 marketfor that one period firm 2 would get the whole market so, for one period, so, for one period, 22 = = M M
thereafter thereafter 22 = 0 = 0 If the firm had always cooperated it would have got If the firm had always cooperated it would have got
22 = ½ = ½MM
Present discounted value of the net gain from defecting isPresent discounted value of the net gain from defecting is ½ ½ M M ½ ½M M [[ + + 22 + + 33 +...] +...]
Simplifying this becomesSimplifying this becomes ½ ½ M M [1 [1 2 2] / [1 ] / [1 ]]
So the net gain is non-positive if and only if ½ ≤ So the net gain is non-positive if and only if ½ ≤ ≤ 1 ≤ 1
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 10.15(1): Points to remember
Set out clearly time pattern of profitsSet out clearly time pattern of profits Take care in discounting net gains back to a base Take care in discounting net gains back to a base
period.period.