ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
expand_more
expand_more
format_list_bulleted
Related questions
Question
advanced
![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)](https://content.bartleby.com/qna-images/question/09238b20-1e79-43dc-b0a6-3d40e032e374/b8988b2c-3fc2-4127-94ca-b6e3bc8419a8/nj9m95a_processed.jpeg)
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)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 5 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Similar questions
- Which of the following products/markets is most consistent with the perfect competition model? Apple iPhones a bustling farmers' market Kellogg's Frosted Flakes electric utilities automobiles 2 Adam Smith’s “invisible hand” refers to The mechanism that moves market price and quantity to equilibrium The natural tendency of markets to avoid monopolies and ensure competition The market’s incentive to lower price in order to increase quantity sold The lack of government role in the free market due to the market’s ability to self-regulate The tendency of firm’s to seek to merge in order to realize synergies and market dominance 3 A company facing inelastic demand for a product sees an increase in its costs after a worker strike forces a wage increase. What is likely to happen to the price and quantity sold of that product? Price will fall slightly while…arrow_forwardHow do a capitalist economic system influence safety and healtharrow_forwardSelect the following statement that is accurate regarding entrepreneurs. Entrepreneurs must always make decisions for the company they started. If an entrepreneur wants to reduce their risks, they can pass the risks on to a CEO. Entrepreneurs only need a willingness to take a risk to start a business. An entrepreneur can fire and replace the managers of their company.arrow_forward
- In at least two paragraphs, tell the class what you've learned in Business 101 and how this will make you a better professional. Business 101 includes; basic and concept of business such as revenue,profit. basic and concepts of economic and business evironment. Basic marketing Business Planning Thanks.arrow_forwardWhich organization would you choose for farming ?arrow_forwardMatch the market structure and its characteristics.arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
Principles of Economics (12th Edition)EconomicsISBN:9780134078779Author:Karl E. Case, Ray C. Fair, Sharon E. OsterPublisher:PEARSON
Engineering Economy (17th Edition)EconomicsISBN:9780134870069Author:William G. Sullivan, Elin M. Wicks, C. Patrick KoellingPublisher:PEARSON
Principles of Economics (MindTap Course List)EconomicsISBN:9781305585126Author:N. Gregory MankiwPublisher:Cengage Learning
Managerial Economics: A Problem Solving ApproachEconomicsISBN:9781337106665Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike ShorPublisher:Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-...EconomicsISBN:9781259290619Author:Michael Baye, Jeff PrincePublisher:McGraw-Hill Education
Principles of Economics (12th Edition)
Economics
ISBN:9780134078779
Author:Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:9780134870069
Author:William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:9781305585126
Author:N. Gregory Mankiw
Publisher:Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-...
Economics
ISBN:9781259290619
Author:Michael Baye, Jeff Prince
Publisher:McGraw-Hill Education