1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 10.7
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2007

2. Ex 10.7(1): Question
purpose: examine equilibrium concepts in a very simple duopoly
method: determine best-response behaviour in a model where each firm takes other outputs as given

3. Ex 10.7(1): iso-profit curve
By definition, profits of firm 2 are
P2 = pq2  [C0 + cq2]
where q2 is the output of firm 2
C0, c are parameters of the cost function
Price depends on total output in the industry
p = p(q1 + q2)
= b0  b[q1 + q2]
So profits of firm 2 as a function of (q1, q2) are
P2 = b0q2  b[q1 + q2]q2  [C0 + cq2]
The iso-profit contour is found by
setting P2 as a constant
plotting q1 as a function of q2

4. Ex 10.7(1): firm 2’s iso-profit contours
Output space for the two firms
Contour for a given value of P
Contour map q2
b0q2  b[q1 + q2]q2  [C0 + cq2] = const
As q1 falls for given q2 price rises and firm 2’s profits rise
profit q1

5. Ex 10.7(2): Question method: Use the result from part 1
Use Cournot assumption to get firm 2’s best response to firm 1’s output (2’s reaction function)
By symmetry find the reaction function for firm 1
Nash Equilibrium where both these functions are satisfied

6. Ex 10.7(2): reaction functions and CNE
Firm 2 profits for given value`q of firm 1’s output:
P2 = b0q2  b[`q1 + q2]q2  [C0 + cq2]
Max this with respect to q2
Differentiate to find FOC for a maximum:
b0  b[`q1 + 2q2]  c = 0
Solve for firm 2’s output:
q2 = ½[b0  c]/b  ½`q1
this is firm 2’s reaction function c2
By symmetry, firm 1’s reaction function c1 is
q1 = ½[b0  c]/b  ½`q2
Substitute back into c2 to find Cournot-Nash solution
q1 = q2 = qC = ⅓[b0  c]/b

7. Ex 10.7(2): firm 2’s reaction function
Output space as before
Isoprofit map for firm 2
For given q1 find q2 to max 2’s profits
Repeat for other given values of q1 q1 q2
Plot locus of these points
Cournot assumption:
Each firm takes other’s output as given
profit
Firm 2’s reaction function
c2(q1) gives firm 2’s best output response to a given output q1 of firm 1 • • •
c2(∙) • • •

8. Ex 10.7(2): Cournot-Nash
Firm 2’s contours and reaction function
Firm 1’s contours
Firm 1’s reaction function
CN equilibrium at intersection q2
c1(∙)
c1(q2) gives firm 1’s best output response to a given output q2 of firm 2
Using the Cournot assumption…
…each firm is making best response to other exactly at qC qC •
c2(∙) q1

9. Ex 10.7(3): Question method: Use reaction functions from part 2
Find optimal output if one firm is a monopolist
Joint profit max is any output pair that sums to this monopolist output

10. Ex 10.7(3): joint profits Total output is q = q1 + q2
The sum of the firms’ profits can be written as:
P1 + P2 = b0q1  b[q1 + q2]q1  [C0 + cq1] +
b0q2  b[q1 + q2]q2  [C0 + cq2]
= b0q  b[q]2  [2C0 + cq]
Maximise this with respect to q
differentiate to find FOC for a maximum:
b0  2bq  c = 0
Solve for joint-profit maximising output:
q = ½[b0  c]/b
However, breakdown into (q1 , q2) components is undefined

11. Ex 10.7(3): Joint-profit max
Reaction functions of the two firms
Cournot-Nash equilibrium
Firm 1’s profit-max output if a monopolist
Firm 2’s profit-max output if a monopolist q2
Output combinations that max joint profit
Symmetric joint profit maximisation
c1(∙)
q1 + q2 = qM
(0,qM) • qC •
qJ = ½ qM qJ •
c2(∙) • q1
(qM,0)

12. Ex 10.7(4): Question method:
Use firm 2’s reaction function from part 2 (the “follower”)
Use this to determine opportunity set for firm 1 (the “leader”)

13. Ex 10.7(4): reaction functions and CNE
Firm 2’s reaction function c2:
q2 = ½[b0  c]/b  ½q1
Firm 1 uses this reaction in its calculation of profit:
P1 = b0q1  b[q1 + c2(q1)]q1  [C0 + cq1]
= b0q1  b[q1 + [½[b0  c]/b  ½q1 ] ]q1  [C0 + cq1]
= ½[b0  c  bq1] q1  C0
Max this with respect to q1
Differentiate to find FOC for a maximum:
½[b0  c ]  bq1 = 0
So, using firm 2’s reaction function again, Stackelberg outputs are
qS1 = ½[b0  c]/b (leader)
qS2 = ¼[b0  c]/b (follower)

14. Ex 10.7(4): Stackelberg • • • profit Firm 2’s reaction function
Firm 1’s opportunity set
Firm 1’s profit-max using this set q2 qC • qS •
profit
c2(∙) • q1
(qM,0)

15. Ex 10.7(5): Question method: compute profit
plot in a diagram with (P1 , P2) on axes

16. Ex 10.7(5): Possible payoffs
Profit space for the two firms •
(0, PM)
Attainable profits for two firms
Symmetric joint profit maximisation
max profits all to firm 1 (but with two firms present)
Monopoly profits (only one firm present)
Cournot profits
Stackelberg profits
PJ = [b0  c]2 /[8b]  C0
(PJ,PJ) •
2PJ = [b0  c]2 /[4b]  2C0
(PC,PC)
PM = [b0  c]2 /[4b]  C0
PC = [b0  c]2 /[9b]  C0 •
(PS,PS)
PS1 = [b0  c]2 /[8b]  C0 C0 { •
PS2 = [b0  c]2 /[16b]  C0
(PM,0) P1

17. Ex 10.7: Points to remember
Cournot best response embodied in c functions
Cooperative solution found by treating firm as a monopolist
Leader-Follower solution found by putting follower’s reaction into leader’s maximisation problem