1. MICROECONOMICS Principles and Analysis Frank Cowell
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: t
firm 1: pM pM pM c c …
firm 2: pM pM p 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.