oligopolistic markets
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1 BUSINESS AND INDUSTRIAL ECONOMICS 20152016 Boris Mrkajic, PhD Analysis of oligopolistic markets – Exercises’ solutions Exercise 1 solution As firms compete in quantities simultaneously, we solve a Cournot model.3 As both cost functions are linear, fixed costs of both firms are null ( = 0). Both firms want to maximize their profits. Firm 1 solves the following profit function: max ! ! ! = ! + ! ∙ ! − ! ! max ! ! ! = (10 − 2 ! + ! ) ∙ ! − 5 ! max ! ! ! = 10 ! − 2 ! ! − 2 ! ∙ ! − 5 ! Firm 2 solves the following profit function: max ! ! ! = ! + ! ∙ ! − ! ! max ! ! ! = (10 − 2 ! + ! ) ∙ ! − 2 ! max ! ! ! = 10 ! − 2 ! ! − 2 ! ∙ ! − 2 ! First order-condition for profit maximisation are: ! ! = 0 & ! ! = 0. In particular, ! ! = 10 − 4 ! − 2 ! − 5 = 0 ! ! = 10 − 4 ! − 2 ! − 2 = 0 Implying that:
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oligopolistic market theory
Transcript of oligopolistic markets
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BUSINESS AND INDUSTRIAL ECONOMICS 2015-‐2016
Boris Mrkajic, PhD
Analysis of oligopolistic markets – Exercises’ solutions
Exercise 1 solution
As firms compete in quantities simultaneously, we solve a Cournot model.3
As both cost functions are linear, fixed costs of both firms are null (𝐹𝐶 = 0).
Both firms want to maximize their profits.
Firm 1 solves the following profit function:
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = (10− 2 𝑞! + 𝑞! ) ∙ 𝑞! − 5𝑞!
max!!
𝜋! = 10𝑞! − 2𝑞!! − 2𝑞! ∙ 𝑞! − 5𝑞!
Firm 2 solves the following profit function:
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = (10− 2 𝑞! + 𝑞! ) ∙ 𝑞! − 2𝑞!
max!!
𝜋! = 10𝑞! − 2𝑞!! − 2𝑞! ∙ 𝑞! − 2𝑞!
First order-condition for profit maximisation are:
𝜕𝜋!𝜕𝑞!
= 0 & 𝜕𝜋!𝜕𝑞!
= 0.
In particular,
𝜕𝜋!𝜕𝑞!
= 10− 4𝑞! − 2𝑞! − 5 = 0
𝜕𝜋!𝜕𝑞!
= 10− 4𝑞! − 2𝑞! − 2 = 0
Implying that:
2
𝑞! =5− 2𝑞!
4 =54−
𝑞!2
𝑞! =8− 2𝑞!
4 = 2−𝑞!2
Solving this system of linear equations we get:
𝑞! = 2−54−
𝑞!2
2
2𝑞! = 4−5− 2𝑞!
4
8𝑞! = 16− 5+ 2𝑞!
𝒒𝟐∗ =𝟏𝟏𝟔
And
𝒒𝟏∗ =𝟏𝟑
Making the Cournot-equailbrium: 𝑞!∗, 𝑞!∗ = (!!, !!!).
Total quantity provided by the market is: 𝑄 = !"!
, for the price of 𝑃 = !"!
.
Profits of the firms are: 𝜋! =!!, 𝜋! =
!"!!"
.
Exercise 2 solution
As the firms compete in quantities simultaneously, we solve a Cournot model. Both firms want to maximize their profits.
a) If Firm 1 decides not to invest, it pays nothing (𝐹𝐶! = 0), while it incurs cost of 1 for each unit produced, meaning that its average and marginal costs are the same and they equal: 𝐴𝐶! = 𝑀𝐶! = 1. At the same time, the total costs are: 𝑇𝐶! = 𝑞!. Firm 2’s total costs are: 𝐹𝐶! = 0, 𝐴𝐶! = 𝑀𝐶! = 1. At the same time, the total costs are: 𝑇𝐶! = 𝑞!. In this case, both firms solve the following profit function
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
First order conditions yield:
𝜕𝜋!𝜕𝑞!
= 3− 2𝑞! − 𝑞! − 1 = 0
3
𝜕𝜋!𝜕𝑞!
= 3− 2𝑞! − 𝑞! − 1 = 0
Solving this system of linear equations we get the Cournout-equalibrium: 𝑞!∗, 𝑞!∗ = (!!, !!).
Total quantity provided by the market is: 𝑄 = !!, for the price of 𝑃 = !
!.
Profits of the firms are: 𝜋! =!!, 𝜋! =
!!.
b) If Firm 1 decides to invest, it pays initial investment (𝐹𝐶! = 𝐼 > 0), while it incurs no costs of each unit produced, meaning that its average and marginal costs are the same and they equal: 𝐴𝐶! = 𝑀𝐶! = 0. At the same time, the total costs are: 𝑇𝐶! = 𝐼. Firm 2’s total costs are still: 𝐹𝐶! = 0, 𝐴𝐶! = 𝑀𝐶! = 1. At the same time, the total costs are: 𝑇𝐶! = 𝑞!. In this case, Firm 1 solves the following profit function
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = (3− 𝑞! + 𝑞! ) ∙ 𝑞! − 𝐼
While Firm 2 solves the same profit function as in the previous case
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = (3− 𝑞! + 𝑞! ) ∙ 𝑞! − 𝑞!
First order conditions yield:
𝜕𝜋!𝜕𝑞!
= 3− 2𝑞! − 𝑞! = 0
𝜕𝜋!𝜕𝑞!
= 3− 2𝑞! − 𝑞! − 1 = 0
Solving this system of linear equations we get the new Cournout-equalibrium: 𝑞!∗, 𝑞!∗ = (!!, !!).
Total quantity provided by the market is: 𝑄 = !!, for the price of 𝑃 = !
!.
Profits of the firms are: 𝜋! =!"!− 𝐼, 𝜋! =
!!.
c) Based on the profits of Firm 1 with and without invest made, we can state that Firm 1 will find appealing to invests iif the investment is lower than !"
! (𝜋!!"#$%& > 𝜋!!"#!$%&' only if 𝐼 < !"
!).
In the case of initial investmest, Firm 1 reduces its product costs (AC! = MC! = 0, after the investment), and hence also benefits from increaseing its production (𝑞!!"#$%& > 𝑞!!"#!$%&'). As the reaction function of Firm 2 in Cournot model is down-sloping (𝑞! = 1− !
!𝑞!, in this case), Firm 2 will have to decraese its
production. This is a consequence of the fact that quantities are strategic complements in Cournot model.
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Exercise 3 solution
As the game is sequential, we solve a Stackelberg model, and one firm makes the decision first. We assume Firm 1 is the first mover (leader), and Firm 2 is the second mover (follower). Both firms want to maximize their profits.
We first derive the reaction function of the follower with respect to the assumed choice of the leader:
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = (100− 𝑞! − 𝑞!) ∙ 𝑞! − 40𝑞!
First order condition yields:
𝜕𝜋!𝜕𝑞!
= 100− 𝑞! − 2𝑞! − 40 = 0
Firm 2 reaction function is:
𝑞!(𝑞!) = 30−12 𝑞!
The total market demand is now:
𝑃 𝑄 = 70−12 𝑞!
Then, we derive the reaction function of the leader with respect to the reaction of the follower:
max!!
𝜋! = 𝑃 𝑞! + 𝑞! ∙ 𝑞! − 𝑇𝐶! 𝑞!
max!!
𝜋! = 70−12 𝑞! ∙ 𝑞! − 40𝑞!
First order condition yields:
𝜕𝜋!𝜕𝑞!
= 70− 𝑞! − 40 = 0
𝒒𝟏∗ = 𝟑𝟎
𝒒𝟐∗ = 𝟏𝟓
The Stackelberg-equalibrium is: 𝑞!∗, 𝑞!∗ = (30,15).
Total quantity provided by the market is: 𝑄 = 45, for the price of 𝑃 = 55.
Profits of the firms are: 𝜋! = 450, 𝜋! = 225.
Exercise 4 solution
As the firms compete in prices simultaneously, we solve a Bertrand model. Both firms want to maximize their profits.
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a) The firms solve the following profit function
max!!
𝜋! = 𝑝! ∙ 𝑞! − 𝑇𝐶!
First order conditions yield:
𝜕𝜋!𝜕𝑝!
= 12− 12𝑝! + 5𝑝! = 0
𝜕𝜋!𝜕𝑝!
= 10+ 2𝛼 − 12𝑝! + 5𝑝! = 0
Firm 1 reaction function is:
𝑝! 𝑝! = 1+512𝑝!
Firm 2 reaction function is:
𝑝! 𝑝! =5+ 𝛼6 +
512𝑝!
b) Given 𝛼 = 1, rection function of Firm 2 is:
𝑝! 𝑝! = 1+512𝑝!
Solving this system of linear equations we get the Bertrand-equalibrium: 𝑝!∗,𝑝!∗ = (!"!, !"!).
Total quantity provided by each firm is: 𝑞! = 𝑞! =!!.
Profits of the firms are: 𝜋! = 𝜋! =!"!"
.
c) If the firms are symmetric (in the case of 𝛼 = 1, they indeed are), the prices are set equal to marginal costs. This result is known as the “Bertrand paradox”, when the two firms charge a price equal to marginal cost and hence incur zero extra-profits, while in other oligopolistic models (e.g. Cournot) the price is higher than the Marginal Cost and guarantees to earn positive extra profits.
