8. Oligopolistic Markets, Classical Duopolistic Models (Bertrand, Cournot, Stackelberg)

Question 1

Types of oligopolistic models

Answer 1

Models can be:
.
1. Collusive (e.g. cartel)
.

2. Competitive: no collusion (Bertrand, Cournot, Stackelberg).
   Simultaneous models (Bertrand, Cournot).
   Sequential models (Stackelberg).
   .
   Key variables:
   Prices adopted by each company (Bertrand)
   Quantities offered by each company (Cournot, Stackelberg)

Question 2

Oligopolistic models: Bertrand model

Answer 2

Bertrand 1822-1900

1. Sequential decisions:
   Decision on quantities: Quantity leadership (Stackelberg)
   Decisions on prices: Price leadership
   .
2. Simultaneous decisions:
   Decision on quantities: Quantity choice (Cournot)
   Decisions on prices: Price choice (Bertrand)
   .
3. Collusion
   Decision on quantities: Quantity joint decision
   Decisions on prices: Price joint decision

Question 3

Bertrand model

Answer 3

The Bertrand (1883) model analyzes firms’ behavior under conditions of oligopoly, adopting price as the focal strategic variable
In its simplest form, it is based on the following assumptions:
Only 2 companies: duopolistic competition
No potential entrants (closed markets)
Homogenous good
Perfect rationality
Perfect information
Same cost function (same technology) with MC=AC=c
Only 1 strategic variable: price
Price is decided simultaneously
.
Essentially, two identical, perfectly rational and perfectly informed firms i and j compete by simultaneously choosing price.
 Consumers, who are also perfectly rational and perfectly informed, demand the good from the company with the lowest price.

Question 4

Continued: What are the options for firm i? If i sets a price:

Answer 4

What are the options for firm i? If i sets a price:
.
- Lower than j, it captures the entire market demand
- Equal to j, it shares the market demand with j
- Greater than j, it has a null market demand (consumers demand the good from j)
.
The assumed cost function is: TC=c.q
.
This way:
Fixed costs are zero:
Average Cost (AC) and Marginal Cost (MC) coincide:
MC= row TC/row q=c ; AC= TC/q=c
Thus profit will be:
pi=(p-AC)q OR
pi=(p-c)q
.
i and j choose their price in order to maximize profits
.
Under our cost assumptions, profit functions are:
.(see presentation)
The game is simultaneous and competitive (each company tries to maximize its own profit)

Question 5

Bertrand model - equilibrium

Answer 5

Nash equilibrium:
Couple of strategies where none of the players find it convenient to change strategy given the other’s strategy
No one can unilaterally change its position and improve its situation
 Each company’s price maximizes profits given the other’s choice
.
It is possible to demonstrate that in the Nash equilibrium each firm chooses a price equal to c:
pi=pj=c
In this case, none of the two companies has an incentive to change its choice, given the other’s choice.
Price higher than c: loss of the entire demand
Price lower than c: the firms makes losses instead of profits
.
Given the market conditions, firms are identical and the game is symmetric; the reasoning developed for one player is perfectly applicable to the other

Question 6

Cont

Answer 6

As long as either of the two firms sets a price higher than c, there is no equilibrium. The firm setting p > c always has an incentive to lower the price to the point it is infinitesimally lower than the price charged by the other firm, in order to capture the whole demand.
.
 Each firm has always the incentive to revise its price decision, unless the price for both firms is equal to c=MC=AC

Question 7

Bertrand model - critique

Answer 7

In reality, most industries with only two competitors seem to make extra profits.
Why?
.

Question 8

Extra: Bertrand model (Pg 39)

Answer 8

Assumptions:

Answer 9

Assuming a linear cost function TC=c*y
The two firms choose a price in order to maximize profits:
The game is not cooperative: each firm maximizes its own profit.
The Bertrand’smodel solution is a Nash equilibrium: no one can unilaterally change its position and improve its situation􏰀. Each company’s price maximizes the profit, given the other’s choice.
It is possible to demonstrate that in the Nash equilibrium each firm chooses a price equal to its marginal costs. Pi=Pj=c
Firms cannot set different prices between each other and greater than c. Price cannot be lower than c, otherwise it would make losses instead of profits. Given the market conditions, firms are identical and the game is symmetric. The reasoning developed for one player is perfectly applicable to the other.

Question 10

We can analyze three cases:

Answer 10

We can analyze three cases:
1. Pi> Pj>c
In this case, firms set different prices and higher than the marginal cost. Firm i has null demand, while j captures the entire market demand and makes extra-profits. This solution is not a Nash equilibrium, i finds convenient to reduce the price in order to make extra- profits. This situation goes on until the equilibrium.
.
2. Pi> Pj = c
In this case, firms set a different price and equal to the marginal cost. For both the firms the profit is null and they have an incentive to deviate from their choices. The firm i has null demand that is an incentive to reduce the price. The firm j has a null extra-profit, thus it has an incentive to raise the price until it is only a bit lower than􏰀the competitor’s price in order to capture the entire market.
.
3.Pi = Pj > c
In this case, they both gain extra-profits but we assume that they are competing and not colluding. Therefore, they will set a lower price in order to get the entire market instead of sharing.
.
Each company has always the incentive to revise its price decision unless the price for both companies is equal to the marginal cost. The equilibrium is the following:
This is a Nash equilibrium: none of the two companies has an incentive to change its choice, given the other’s choice. If the price is higher than the marginal cost, the company loses the entire demand.
If the price is lower than the marginal cost, the firm makes losses instead of profits

Question 11

This solution depends on the assumptions of the model:

Answer 11

• If products are homogeneous, the demand depends only on price. With differentiation and information asymmetries, price can be higher without decreases in demand.
  

Question 12

Cournot intro- model assumptions

Answer 12

Developed in 1838 by a French philosopher, mathematician and economist Antoine Augustin Cournot
.
Cournot duopoly assumptions:
Only 2 firms, duopoly
No potential entrants (closed markets)
Homogenous good
Perfect rationality
Perfect information
Only 1 strategic variable: quantity (q)
Production levels are simultaneously decided
The price is determined by the market at a level where the demand equals the joint production of the two firms

Question 13

The strategic variable is quantity:

Answer 13

The strategic variable is quantity:
Firms choose how much they want to produce, and the price is given by the aggregated market demand (under the hypothesis of standard goods, the DD has a negative slope):

Question 14

Cournot model - equilibrium

Answer 14

The two firms strategically interact by influencing the (unique) market price through the quantity they set
.
Equilibrium: given the competitor’s choice, firms choose the best strategy to maximize their profits
.
Assume that:
Firms can choose the quantity they prefer in the interval [0,+infinity]
.
Both the profit functions can be differentiated in quantity

Question 15

Goal: derive the equilibrium (2 steps):

Answer 15

1. Determine the set of optimal choices of each firm given the rival’s behavior –> determine reaction functions
   .
2. Intersect the two reaction functions in order to find the combination of mutually compatible decisions (i.e., the Nash-Cournot equilibrium of the game)

Question 16

Cournot model – equilibriumGraphical derivation of the equilibrium (intersection)

Answer 16

See graph

Question 17

Cournot model – equilibrium - algebraic derivation

Answer 17

Given the following inverse demand function:
P(Q)=a-bQ
.
Where Q is the total industry output, equal to:
Q=q1+q2
.
And the following (assumed) cost functions:
TC1(q1)=c1q1
.
TC2(q2)=c2q2
.
Firm 1’s reaction function:
q1^* = a-c1/2b - 1/2*q2

Question 18

Cournot model – equilibrium – algebraic derivation

Answer 18

By simmetry, Firm 2’s reaction function can be found immediately:
.
q2(q1)=a-c2/2b - 1/2q1
.
The equilibrium is given by the couple of values (q1^N, q2^N)
In order to identify the equilibrium, Firm 1 decides its output on the basis of its conjecture regarding Firm 2’s behavior; for example, if Firm 1 expects that Firm 2 will produce the quantity q2^e , then Firm 1 will have to produce q1(q2^e)
.
(q1^N, q2^N) - equilibrium values
(q1^e, q2^e) - expected values

Question 19

Cournot model – equilibrium – algebraic derivation

Answer 19

Thus, the Nash equilibrium is the solution to the following system:
.
q1^N=a-2c1+c2/3b
.
q2^N=a-2c2+c1/3b

Question 20

Oligopolistic models and game theory

Answer 20

Quantity leadership (Stackelberg)
Decision on quantities
and
Sequential decisions

Question 21

Sequential models: 2-step competition

Answer 21

Sequential competition: the competition is not simultaneous anymore. Now, it articulates in two steps.
The second step depends on the first one
.
Extra:
The competitive environment changes according to the decisions taken in the first step. The second step inherits these conditions about price and/or quantity. The constraints and asymmetries in the first step can turn into positive extra-profits in the second step that means price is higher than the marginal cost. The results can be different if compared with the simultaneous competition.

Question 22

2-step competition: 2-step model solution: backward induction

Answer 22

2-step model solution: backward induction
.
The second step is contingent on the decisions taken in the first step: the follower maximizes profits given the leader’s choice.
The leader optimizes the first step by maximizing profits given the follower’s reaction in the second step (which is known a prior).
.
As usual, it is assumed that:
Both the leader and the follower know everything (perfect information)
Both the leader and the follower are perfectly rational
.
.
Extra:
The two-steps model solution is a backward induction: firstly, we solve the second step given the decision taken in the first step. Therefore, we solve the first step in order to maximize the payoff.
In the sequential model, one of the two firm precedes the other in choosing the strategy: it is called first-mover advantage. The first mover is the leader and the other is the follower, reacting to the leade􏰆􏰇s de􏰁isio􏰀.
The backward induction is based on the assumptions that the leader:

Question 23

Stackelberg model: profit maximization formula

Answer 23

The leader considers the (future) follower’s choice when it chooses its own output level. The leader take its decision considering the future choice of the follower􏰆: it considers the follower’s
profit maximization issue (game theory).
.
Its profit maximization depends on q2=f2 (q1) which is the follower’s reaction function.
.
max (q1) pi1 = p[q1 + f2(q1)]*q1-c(q1)
.
Extra:
By deriving the leader’s profit on the quantity, we can find its reaction function.
Putting together the two reaction functions, we can find the equilibrium. The total output will be the sum of the two firms’ outputs and it will be greater than the output under monopoly conditions.

Question 24

Other strategic variables

Answer 24

In the real world, firms’ strategic behaviors depend on several variables:
Price (Bertrand model)
Quantity (Cournot and Stackelberg models)
R&D investments; Product features; Commercialization modes
…
All these decisions imply strategic interaction and interdependence

Question 25

Exercise

Answer 25

The market for virtual reality devices has just emerged, and Zorn and Thorn are the only two companies competing in it, according to a classical Cournot duopoly. The market is characterized by the following inverse demand function: P = 200 – 3Q, where Q denotes the total quantity produced in the market. Each firm faces total costs equal to 20 multiplied by the quantity it decides to produce (TC = 20qi for both firms). Determine equilibrium quantity and price.
.
Suppose that Thorn has an advantage that allows it to enter the market before Zorn does. In this case, the competition would take place according to the Stackelberg paradigm with Thorn as the leader, everything else being equal to point c. However, Zorn has the possibility to employ a team of top-notch big data analysts that would allow it to overcome Thorn’s advantage. In this case, the competition would take place according to the Stackelberg paradigm with Zorn as the leader, everything else being equal to point c. Assume perfect rationality and perfect information on everything. No extraneous factors exist besides those strictly mentioned in the exercise. What is the maximum price that Zorn is willing to pay for the big data analysis?

Question 26

Solution (first point)

Answer 26

Denote Zorn with 1 and Thorn with 2.
.
P = 200 – 3Q
TC = 20qi
.
Π1 = q1 (200 – 3q1 – 3q2 ) – 20q1.
.
Π1 maximization → 200 – 6q1 – 3q2 – 20 = 0 → q1 = 30 – 0.5q2 (Zorn’s reaction function).
.
By symmetry, q2 = 30 – 0.5q1 (Thorn’s reaction function).
.
Equilibrium requires that q1 = 30 – 0.5(30 – 0.5q1) → q1 = q2 = 20.
.
Thus, Q = 20 + 20 = 40 and P = 200 – 120 = 80.

Question 27

Solution (second point)

Answer 27

Denote the leader with 1 and the follower with 2.
.
Reaction functions are equal and already known from the previous point.
.
Π1 = q1 [200 – 3q1 – 3 (30 – 0.5q1)] – 20q1. Π1 maximization → 200 – 6q1 – 90 + 3q1 – 20 = 0 → q1 = 30 → q2 = 15.
.
Thus, Q = 30 + 15 = 45 and P = 200 – 3*45 = 65.
Hence, the leader’s profits will be equal to 30 (65 – 20) = 1350.
.
Conversely, the follower’s profits will be equal to 15 (65 – 20) = 675.
.
The value that the big data analysis has for Zorn is tantamount to the difference between its profits as a leader and its profits as a follower = 1350 – 675 = 675. Hence, as long as the price for the analysis is lower than 675, Zorn will be willing to pay for it.

Question 28

References

Answer 28

POK: WEEK 5 - Monopoly and Oligopoly > Oligopoly > Types of Oligopolies.
Cabral (I edition) 4.1, 4.2, 4.3, 7, 7.1, 7.3, 7.4
Cabral (II edition) 8.1, 8.2, 8.3
Besanko et al. (Microeconomics) 13

Question 29

Extra: Stackelberg’s model