= = Consider two duopolists whose cost functions are Ci 10qi, i 1,2, where C₁ and qi are their costs and outputs, respectively. The two duopolists have identical products and face a market demand function: p=402q, where q = 91 +92. Compare the outputs and profits in the following three cases: (i) simulta- neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii) cooperative solution
= = Consider two duopolists whose cost functions are Ci 10qi, i 1,2, where C₁ and qi are their costs and outputs, respectively. The two duopolists have identical products and face a market demand function: p=402q, where q = 91 +92. Compare the outputs and profits in the following three cases: (i) simulta- neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii) cooperative solution
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Question:
=
Consider two duopolists whose cost functions are Ci 10qi, i
where C₁ and qi are their costs and outputs, respectively.
duopolists have identical products and face a market demand function:
p=40 - 2q, where q
=
Compare the outputs and profits in the following three cases: (i) simulta-
neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii)
cooperative solution.
Use below methods to answer above question:
3 Collusion in Quantity Competition
• Assume that the two duopolists collude in their choice of quantities.
● That is, they solve the joint maximization problem:
Thus, we have:
• Let the solution be given by: q*,92*.
. In other words, we have:
: 91 +92.
Thus, dividing the two equations, we get:
Thus, from equation (2) it follows that:
Similarly, for firm 2, we have:
Max 91,92 {π¹ (91, 92) + π²(91, 92)}
3.1 Example - Linear Case:
• Do this as an exercise.
Əπ¹ (q*, q2*)
Əq₁
C*
On¹ (qi*, q2*)
əq2
● Cost:
● Thus, profits are given by:
π}(9₁*, 92*)
π (9₁*, 92*)
But, note that the slopes of the iso-profit curves are given by:
• Firm 1's problem is:
● This yields:
. These are shown in the diagram.
. Given the symmetry, the solution is:
● or simply:
+
Ən¹ (91, 92)
əqi
• Case 1: Simultaneous game.
• The corresponding best reply functions are:
+
dq2
dq₁
SEL
dq2
dq₁
● That is, the collusive solution lies on the contract curve.
It is, therefore, Pareto efficient.
. But, is it sustainable?
• Assume that we have the collusive solution.
● Since we usually have,
dq2
dq₁
q²
*
9₁ =
=
• Thus, for firm 1, the first-order condition above implies that at the collusive solution, we have:
π}(qî*, q2*) = −π}(qî*, q2*) > 0
||
π2
• Quantity competition and linear demand - strategic substitutes.
• Demand:
=
• In other words, as far as both firms are concerned, they each increase their profits by "cheating" and
deviating from the collusive solution.
• Namely, since ¹(qî*, 92*) > 0 and ²(q*, 92*) > 0, both firms can do better by increasing their outputs.
² (¶¶*, 92*)
Əq1
² (q¶*, 92*)
Əq2
qi* = q³*
9²
π²(qi*, q2*) = −π²(q¶*, 92*) > 0
-
||
=
p = a - b(q² + q¹)
a -
< 0, i, j = 1,2, i ‡ j
(a - c)
q²:
26
=
NEN
26
π² = q¹ [a − b(q² + q¹)] − cqi
=
Ci = cqi
-π² (qi*, 92*)
-π²² (9₁*, 92*)
C*
dq2
dq₁
c)
T
=
Case 2: Sequential game.
• The solution to Firm 2's second stage problem is given by its best reply function:
(a - c) 1
26
2
"
q² +q¹=
=
(a
1
-
max{q¹ (a − b(q¹ + [a= c) — 2¹-0¹
cq²¹
26
1
=
= 0
NI
(a − c)
36
*
92
= 0
¹
=
4b
• The solutions for the simultaneous and sequential games are shown in the diagrams above.
• Using this example, we can also derive the contract curve as:
-
(a — c)
26
(a - c)
26
1,2,
The two
• i.e., along the line such that total output equals monopoly output ((a-c)).
26
. This is also shown in the diagrams above.
(3)
(4)
(5)
(6)
(7)
(8)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09238b20-1e79-43dc-b0a6-3d40e032e374%2Fb8988b2c-3fc2-4127-94ca-b6e3bc8419a8%2Fnj9m95a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question:
=
Consider two duopolists whose cost functions are Ci 10qi, i
where C₁ and qi are their costs and outputs, respectively.
duopolists have identical products and face a market demand function:
p=40 - 2q, where q
=
Compare the outputs and profits in the following three cases: (i) simulta-
neous move Nash equilibrium, (ii) sequential move (leader/follower), (iii)
cooperative solution.
Use below methods to answer above question:
3 Collusion in Quantity Competition
• Assume that the two duopolists collude in their choice of quantities.
● That is, they solve the joint maximization problem:
Thus, we have:
• Let the solution be given by: q*,92*.
. In other words, we have:
: 91 +92.
Thus, dividing the two equations, we get:
Thus, from equation (2) it follows that:
Similarly, for firm 2, we have:
Max 91,92 {π¹ (91, 92) + π²(91, 92)}
3.1 Example - Linear Case:
• Do this as an exercise.
Əπ¹ (q*, q2*)
Əq₁
C*
On¹ (qi*, q2*)
əq2
● Cost:
● Thus, profits are given by:
π}(9₁*, 92*)
π (9₁*, 92*)
But, note that the slopes of the iso-profit curves are given by:
• Firm 1's problem is:
● This yields:
. These are shown in the diagram.
. Given the symmetry, the solution is:
● or simply:
+
Ən¹ (91, 92)
əqi
• Case 1: Simultaneous game.
• The corresponding best reply functions are:
+
dq2
dq₁
SEL
dq2
dq₁
● That is, the collusive solution lies on the contract curve.
It is, therefore, Pareto efficient.
. But, is it sustainable?
• Assume that we have the collusive solution.
● Since we usually have,
dq2
dq₁
q²
*
9₁ =
=
• Thus, for firm 1, the first-order condition above implies that at the collusive solution, we have:
π}(qî*, q2*) = −π}(qî*, q2*) > 0
||
π2
• Quantity competition and linear demand - strategic substitutes.
• Demand:
=
• In other words, as far as both firms are concerned, they each increase their profits by "cheating" and
deviating from the collusive solution.
• Namely, since ¹(qî*, 92*) > 0 and ²(q*, 92*) > 0, both firms can do better by increasing their outputs.
² (¶¶*, 92*)
Əq1
² (q¶*, 92*)
Əq2
qi* = q³*
9²
π²(qi*, q2*) = −π²(q¶*, 92*) > 0
-
||
=
p = a - b(q² + q¹)
a -
< 0, i, j = 1,2, i ‡ j
(a - c)
q²:
26
=
NEN
26
π² = q¹ [a − b(q² + q¹)] − cqi
=
Ci = cqi
-π² (qi*, 92*)
-π²² (9₁*, 92*)
C*
dq2
dq₁
c)
T
=
Case 2: Sequential game.
• The solution to Firm 2's second stage problem is given by its best reply function:
(a - c) 1
26
2
"
q² +q¹=
=
(a
1
-
max{q¹ (a − b(q¹ + [a= c) — 2¹-0¹
cq²¹
26
1
=
= 0
NI
(a − c)
36
*
92
= 0
¹
=
4b
• The solutions for the simultaneous and sequential games are shown in the diagrams above.
• Using this example, we can also derive the contract curve as:
-
(a — c)
26
(a - c)
26
1,2,
The two
• i.e., along the line such that total output equals monopoly output ((a-c)).
26
. This is also shown in the diagrams above.
(3)
(4)
(5)
(6)
(7)
(8)
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