Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics
March 2007
Exercise 10.7
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 10.7(1): Question
Frank Cowell: Microeconomics
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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
Ex 10.7(1): iso-profit curve
Frank Cowell: Microeconomics
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By definition, profits of firm 2 are
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Price depends on total output in the industry
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p = p(q1 + q2)
= b0  b[q1 + q2]
So profits of firm 2 as a function of (q1, q2) are
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P2 = pq2  [C0 + cq2]
where q2 is the output of firm 2
C0, c are parameters of the cost function
P2 = b0q2  b[q1 + q2]q2  [C0 + cq2]
The iso-profit contour is found by
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setting P2 as a constant
plotting q1 as a function of q2
Ex 10.7(1): firm 2’s iso-profit contours
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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
q1
Ex 10.7(2): Question
Frank Cowell: Microeconomics
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
Ex 10.7(2): reaction functions and CNE
Frank Cowell: Microeconomics
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Firm 2 profits for given value`q of firm 1’s output:
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Max this with respect to q2
Differentiate to find FOC for a maximum:
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q2 = ½[b0  c]/b  ½`q1
this is firm 2’s reaction function c2
By symmetry, firm 1’s reaction function c1 is
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b0  b[`q1 + 2q2]  c = 0
Solve for firm 2’s output:
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P2 = b0q2  b[`q1 + q2]q2  [C0 + cq2]
q1 = ½[b0  c]/b  ½`q2
Substitute back into c2 to find Cournot-Nash solution
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q1 = q2 = qC = ⅓[b0  c]/b
Ex 10.7(2): firm 2’s reaction function
Frank Cowell: Microeconomics
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
Plot locus of these points
q2
 Cournot assumption:
 Each firm takes other’s output as
given
• •
•
• • •
 Firm 2’s reaction function
 c2(q1) gives firm 2’s best output
c2(∙)
response to a given output q1 of firm 1
q1
Ex 10.7(2): Cournot-Nash
Frank Cowell: Microeconomics
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
qC
•
to other exactly at qC
c2(∙)
q1
Ex 10.7(3): Question
Frank Cowell: Microeconomics
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
Ex 10.7(3): joint profits
Frank Cowell: Microeconomics
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Total output is q = q1 + q2
The sum of the firms’ profits can be written as:
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Maximise this with respect to q
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differentiate to find FOC for a maximum:
b0  2bq  c = 0
Solve for joint-profit maximising output:
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P1 + P2 = b0q1  b[q1 + q2]q1  [C0 + cq1] +
b0q2  b[q1 + q2]q2  [C0 + cq2]
= b0q  b[q]2  [2C0 + cq]
q = ½[b0  c]/b
However, breakdown into (q1 , q2) components is undefined
Ex 10.7(3): Joint-profit max
Frank Cowell: Microeconomics
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(∙)
(0,qM)
 q1 + q2 = qM
•
qC
•
 qJ = ½ qM
• qJ
c2(∙)
•
(qM,0)
q1
Ex 10.7(4): Question
Frank Cowell: Microeconomics
method:
 Use firm 2’s reaction function from part 2 (the “follower”)
 Use this to determine opportunity set for firm 1 (the “leader”)
Ex 10.7(4): reaction functions and CNE
Frank Cowell: Microeconomics
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Firm 2’s reaction function c2:
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Firm 1 uses this reaction in its calculation of profit:
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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:
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q2 = ½[b0  c]/b  ½q1
½[b0  c ]  bq1 = 0
So, using firm 2’s reaction function again, Stackelberg
outputs are
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qS1 = ½[b0  c]/b (leader)
qS2 = ¼[b0  c]/b (follower)
Ex 10.7(4): Stackelberg
Frank Cowell: Microeconomics
Firm 2’s reaction function
Firm 1’s opportunity set
Firm 1’s profit-max using this set
q2
qC
•
qS
•
c2(∙)
•
(qM,0)
q1
Ex 10.7(5): Question
Frank Cowell: Microeconomics
method:
 compute profit
 plot in a diagram with (P1 , P2) on axes
Ex 10.7(5): Possible payoffs
Frank Cowell: Microeconomics
P2
Profit space for the two firms
• (0, P )
Attainable profits for two firms
Symmetric joint profit maximisation
M
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
 2PJ = [b0  c]2 /[4b]  2C0
(PJ,PJ)
(PC,PC)
•
 PM = [b0  c]2 /[4b]  C0
 PC = [b0  c]2 /[9b]  C0
(PS1,PS2)
 PS1 = [b0  c]2 /[8b]  C0
{
C0
0
° (P• ,0)
M
P1
 PS2 = [b0  c]2 /[16b]  C0
Ex 10.7: Points to remember
Frank Cowell: Microeconomics
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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