Cournot, Bertrand, and Stackelberg
Oligopolies

by Anthony G. Mitchell

Introduction
An oligopoly is a market structure in which there are only a few firms, each of which is large relative to the total industry. There is no specific number of firms that defines an oligopoly, but the amount is usually between two and ten. Due to the complexity of oligopoly, there is no single model that can account for all oligopolies. I will differentiate amongst the models of Cournot, Bertrand, and Stackelberg, and elaborate on what circumstances each model should be applied. I will also discuss how each model can be applied in the world we live in today.

Cournot
A Cournot oligopoly is an industry in which the following conditions are present (Baye, 320):

1) there are few firms serving many customers
2) firms produce either differentiated or homogenous products
3) each firm believes rivals will hold their output constant if it changes its output
4) there are barriers to entry

This model is applied when each firm in the oligopoly determines what its output level is going to be simultaneously, or in other words, each firm in the oligopoly expects its own output decisions to have no impact on its competitors’ output decisions. The Cournot model is applicable to situations in which the products are either identical or differentiated.

I like my fine liquors, so I will use the global oligopoly of liquor producers for my example. For simplicity, we will assume a duopoly in the global liquor market of Diageo and Pernod Ricard.

As Diageo alters its output level, it operates under the assumption that Pernod Ricard will hold its output level constant. In seeking its determination of optimal output, Diageo will set marginal revenue equal to marginal cost. However, Diageo’s marginal revenue is impacted by Pernod Ricard’s output level. If Pernod Ricard increases its output, then the market price will decrease and lower Diageo’s marginal revenue. Therefore, the profit maximizing level of output for Diageo is dependent on Pernod Ricard’s output level. The profit maximizing output of Diageo and Pernod Ricard is a relationship known as a best-response function or a reaction function.

Suppose that the duopoly of Diageo (di) and Pernod Ricard (pr) face the total industry demand of:

P = 100-Q

Where Q = Qdi + Qpr. For ease of illustration, we will assume marginal costs of $0:

MCdi = MCpr = 0

As we discussed earlier, each firm takes the other firm’s output as fixed. Therefore, we will have an anticipated demand curve for Firm e (e=either firm, Diageo or Pernod Ricardo) of:

Pe = (100 – Qotr) – Qe

Qotr is the anticipated output of the other firm. The marginal revenue for Firm e is:

MRe = (100-Qotr) – 2Qe Note: The MR for a linear demand curve is a line with the same intercept, but twice the negative slope.

Firm e’s profits are maximized by setting marginal revenue equal to marginal cost (which are 0 in this example). Rearranging the expression yields the reaction curve:

Qe = 50 – 0.5Qotr

This reaction curve indicates either firm’s optimal output given the output choice of the other. For clarification:

Qdi = 50 – 0.5Qpr and
Qpr = 50 – 0.5Qdi

Inserting Qpr into the first reaction function yields

Qdi = 50 – 0.5(50 – 0.5Qdi)
Qdi = 50 – 25 + 0.25Qdi
Qdi = 25 + 0.25Qdi
Qdi – 0.25Qdi = 25
0.75Qdi = 25
Qdi = 33.33 units

Inserting Qdi into the reaction curve of Qpr will yield the same result.

So, total industry output is:

Q = Qdi + Qpr = 33.33 + 33.33 = 66.66 units

Price in the market is determined by the inverse demand function for 66.66 units (total industry output) produced:

P = 100 – Q
P = 100 – 66.66
P = $33.34

Figure_1_(Shrunk).jpg

The equilibrium is the intersection of the two reaction curves at Point A in Figure 1. Each firm is profit maximizing given the other firm’s output choice at these output levels. Diageo nor Pernod Ricard has an incentive to alter their output level. As we figured above, at equilibrium each firm produces 33.33 units for a total output of 66.66 units and the price is $33.34. The output is less than it would be in a perfect competition scenario. In perfect competition, the total output would be 100 units and the price would be $0, where P = MC (Recall that we set MC to zero). In the Cournot Equilibrium, the firms attain economic profits:

TR = P*Q
TR = 33.34*33.33
TR = $1,111.22
Profit = TR – TC (Recall that our average costs are zero)
Profit = $1,111.22 – 0
Profit = $1,111.22 This is the profit of each firm.

This profit is less than the two firms could obtain if they colluded and produced 25 units each (the monopolistic output – Point C in Figure 1). If this were the case, the joint profits would be $2,500, as opposed to $2,222.44. Specifically, recall that our total industry demand curve is:

P = 100 – Q

So, for our monopolistic output of 50 units (25 per), we have:

P = 100 – (50)
P = $50
50 units * $50 = $2,500 profit

Figure 2 illustrates the price and quantity outputs for the liquor industry under collusive, Cournot, and competitive equilibriums. As you can see, the output is largest and price is lowest under the competitive equilibrium and the output is lowest while the price is highest for the collusive equilibrium.



Figure_2_(Shrunk).jpg

Bertrand

In the Bertrand model, each firm assumes that its rivals will hold their price constant. So long as price exceeds marginal cost (P>MC), the Bertrand oligopolist will undercut its rivals by offering a slightly lower price.

An industry that is a Bertrand oligopoly has the following features (Baye, 336):

1) There are few firms that cater to many consumers.
2) The firms produce identical products at a constant marginal cost.
3) Firms engage in price competition and react optimally to prices charged by competitors.
4) Consumers have perfect information and there are no transaction costs.
5) Barriers to entry exist.

Consumers benefit from a Bertrand oligopoly, as it ultimately results in the exact same outcome as a perfectly competitive market, which will lead to zero economic profits for the producers. Since the products are homogenous and the consumers have perfect information and no transaction costs, all consumers will conduct their transactions with the firm that sells at the lowest price.

A good example of a Bertrand oligopoly is the airline industry. For simplicity, we will assume that the airline industry consists of two providers, Delta and Southwest.

If Delta were to charge the consumers the monopoly price, Southwest would slightly lower the price that it charges consumers to capture the entire market and attain positive profits. Thus, Delta would be unable to sell any airline tickets. To retaliate, Delta would lower its price slightly below that of Southwest and regain the market in its entirety. This price war would occur to the point that both Delta and Southwest charge a price that is equal to their marginal cost. At this point, neither Delta nor Southwest would lower its price any further because they would suffer a loss where P<MC and neither would raise the price, as this would negate them from selling any airline tickets at all. As stated earlier, a Bertrand oligopoly with identical products results in each firm setting P=MC and attaining zero economic profits.

A good example of this was the U.S. airline industry in the early 1990s. Major airlines during this time engaged in price wars that lowered the airfare to consumers while lowering the profits to the airline industry. Once one airline lowered its price, it was outmatched in a matter of only a few days by a competitor. Competition grew so fierce that many airlines during this period reported losses.

Generally speaking, the products that two duopolists produce are not perfect substitutes but have subtle differences that the consumer values, even at a slightly higher price relative to the other firm in the duopoly. This causes each firm to possess a downward sloping demand curve. Price exceeds marginal cost in equilibrium. Output is more than it would be in a Cournot competition, but less than it would be in a perfect competition scenario.

Figure_3_(Shrunk)2.jpg

At the point of intersection of Delta and Southwest airlines’ reaction functions is the point of equilibrium. It is at this point that each firm is maximizing its profits given its belief that the price of the other firm is fixed. As one firm increases its price, the profit-maximizing price of the rival firm increases. This results in positively-sloped reaction functions.

Stackelberg

The Stackelberg oligopoly model is utilized when firms differ as to when output decisions are made. One firm will be the leader and is assumed to make output determination decisions before the other firms in the oligopoly. The remaining firms in the industry (referred to as the followers), base their output levels that maximize their profits based on the leader’s output level. Followers behave in a fashion commensurate to a Cournot oligopolist. The industry leader selects an output that maximizes profits given that each follower will react to this output decision in accordance with a Cournot reaction function.

Stackelberg oligopolies have the following features (Baye, 332):

1) There are few firms serving many consumers.
2) The firms produce either differentiated or homogenous products.
3) A single firm (the leader) chooses an output before all other firms choose their outputs.
4) All other firms (the followers) take as given the output of the leader and choose outputs that
maximize profits given the leader’s output.
5) Barriers to entry exist.

In a Stackelberg oligopoly, the leader moves first and produces before the followers. The followers maximize profit given the leader’s output via their reaction function. In Figure 4, r2 is the follower’s reaction function. Since the leader knows the output reaction of the followers, it will select an output level that will maximize its profits with the assumption that the followers will react to the leader’s actions. The leader will choose the point along r2 that will achieve the highest profits. Note that the profits increase as the isoprofit curves get closer to the monopoly output. The leader will produce at the Stackelberg equilibrium, which is the point of tangency of π1S and r2 in Figure 4. As the followers gain knowledge of the leader’s output level, they will choose to produce at Q2s. At this point, the leader’s profits are higher than they would be at the Cournot Equilibrium (π1c) and the followers’ profits are lower (π2c).

Figure_4_(Shrunk).jpg


An example of a Stackelberg oligopoly is the pseudo-generic pharmaceutical drug industry. In this industry, brand name pharmaceutical companies enter into the generic drug competition by marketing their brand name drug with a pseudo-generic label under a separate company. The Federal Trade Commission has since stopped this in the U.S. market, following an investigation in the mid-1990s. Brand name drugs enter the market by their producer and shortly thereafter, the brand name producer establishes the pseudo-generic equivalent before the generic drug manufacturers can enter the generic market. This allows the pseudo-generic drug to attain most of the market share from the onset and establish itself as the market leader, quickly establishing a Stackelberg oligopoly in many cases.

Figure 5 below is a brief comparison of all of the models discussed above. This is a good snapshot of how each model is compared between price, total output, and consumer surplus.

Figure 5

Comparison.jpg











Questions

1. Which of the following is true about a Cournot oligopoly?
a. Firms produce homogenous goods only
b. Each firm assumes its rivals’ prices will remain constant
c. Output levels amongst the firms are determined simultaneously
d. Both A and C
e. Both B and C

ANSWER = C: Firms in a Cournot oligopoly assume their rivals will hold their output levels constant.

2. Which of the following is NOT true in a Bertrand oligopoly?
a. Producers benefit with economic profits in the long run
b. Reactions functions are positively-sloped
c. Followers in the market determine their output level once the market leader determines its
output level
d. Both A and B
e. Both A and C

ANSWER = E: Consumers benefit as they would in a perfect competition while producers will eventually gain zero economic profit. Choice C is a characteristic of a Stackelberg oligopoly. Therefore, choices A and C are false.

3. Which of the following statements are true?
a. In a Bertrand oligopoly, output is more than it would be in a Cournot competition, but less
than it would be in a perfect competition scenario.
b. Isoprofit curves illustrate greater profits the closer they are to the monopolistic output level.
c. In a Stackelberg oligopoly, the leader’s profits are higher than they would be at the Cournot
Equilibrium and the followers’ profits are lower.
d. All of the above are true.

ANSWER = D: All of the statements are true.

4. In a Stackelberg oligopoly:
a. All firms earn zero economic profits
b. The market leader produces at the point of tangency of its highest-value isoprofit curve and
the follower’s reaction curve.
c. All firms make output decisions simultaneously.
d. Followers in the market behave as Cournot oligopolists.
e. Both B and D are true.

ANSWER = E: The market leader chooses to produce an output that is tangent to the follower’s reaction curve and its (the market leader’s) highest-value isoprofit curve. Also, the follower’s in the market behave as Cournot oligopolists.

5. In a Cournot duopoly:
a. Firm 1’s marginal revenue is independent of Firm 2’s output level
b. Firms will produce at th e intersection of their highest-valued isoprofit curves.
c. Firms will produce at the intersection of their reaction functions.
d. Both A and C
e. Both B and C

ANSWER = C: Firms will produce at the intersection of their reaction functions. Each firm is
profit maximizing given the other firm’s output choice at these output levels. Answer choice A. is misleading. Firm 1's marginal revenue is dependent on Firm 2's output level.


6. Suppose that only two firms are in the market of producing single malt scotch, Glenlivet (g) and Balvenie (b).

If the total industry demand is: P = 150 - Q
And their MC = 0

(a) What is the reaction curve of Glenlivet?
a. Qg = 50 – 0.25Qb
b. Qg = 60 – 0.5 Qb
c. Qg = 70 – 0.5Qb
d. Qg = 75 – 0.5Qb

(b) How many units would Glenlivet produce?
a. 75 units
b. 50 units
c. 55 units
d. 125 units

(c) What would be the price?
a. $25 per unit
b. $65 per unit
c. $125 per unit
d. $50 per unit

(d) What would be the profit of Glenlivet?
a. $1,250
b. $2,500
c. $5,000
d. $5,500

ANSWER to Part (a):
d. Qg = 75 – 0.5Qb

Note that e=either firm and otr=anticipated output of the other firm.

P = 150 - Q
P = 150 - (Qg + Qb)
Pe = (150 - Qotr) - Qe
MRe = MCe
(150 – Qotr) – 2Qe = 0
150 – Qotr = 2Qe
Qe = 75 – 0.5Qotr
So, the reaction curve for each firm is:
Qg = 75 – 0.5Qb and
Qb = 75 – 0.5Qg

ANSWER to Part (b):
b. 50 units

Qg = 75 – 0.5 (75 – 0.5Qg)
Qg = 75 – 37.5 + 0.25Qg
Qg = 37.5 + 0.25Qg
Qg – 0.25Qg = 37.5
0.75Qg = 37.5
Qg = 50
Note that inserting Qg into the reaction curve of Qb will yield the same result.

ANSWER to Part (c):
d. $50 per unit

Total industry output is:
Q = Qg + Qb
Q = 50 + 50
Q = 100 units

P = 150 – Q
P = 150 – (100)
P = $50

ANSWER to Part (d):
b. $2,500

TR = P*Q
TR = $50*50units
TR = $2,500
π = TR – TC
π = $2,500 – 0
π = $2,500



References

Baye, Michael R., Managerial Economics and Business Strategy – 6e. New York: McGraw Hill Co., 2007.

Brickley, Smith, Zimmerman. Managerial Economics and Organizational Architecture. Boston: McGraw Hill Co., 1997.

Deutsch, Claudia H. “Pernod Ricard and Fortune to Buy Allied Domecq.” New York Times 21 Apr. 2005.

New York Times Online. http://www.nytimes.com/2005/04/21/business/worldbusiness/21drink.html.

Harris, McGuigan, Moye. Managerial Economics - Applications, Strategy, and Tactics - 8e. Cincinnati, OH: South-Western, 1999.

Landsburgh, Steven E. Price Theory & Applications – 5e. Ohio: South-Western, 2002.


Stiglitz, Joseph E. Economics – 2e. New York: W.W. Norton & Company, Inc., 1997.