Lecture 14: Duopoly and Oligopoly
Consider the case of a firm selling 10 units of its product at $10 a unit, who now
decides to produce and sell an 11th unit. How does it calculate its extra revenue?
Extra Revenue from selling an incremental unit for a pricetaking firm
Revenue Gain
= $10
$10
$100
10
If a price-taking firm selling 10 units at $10 each decides to increase
output by a unit, its revenue will increase by $10, as indicated by the
shaded area above.
But if the firm is a price-taker, the firm has a downward sloping demand function. If
the firm must charge all customers the same price, it can only sell that unit by lowering the
price it charges for the first ten units. This price reduction will offset – and, under some
conditions more than offset the revenue gain on the 11th unit.
Extra Revenue from selling an incremental unit for a pricemaking firm
$10
B
$100
A
D
10
If a price-making firm selling 10 units at $10 each decides to
increase output by a unit, its revenue will increase by the gain in
revenue on the 11th unit less the loss in revenue on the first 10 units,
as given by area A less area B.
Still, there is a third case. We know that the demand function of any firm depends on
the prices of substitutes, particularly close substitutes. Suppose, for instance that our pricetaking firm, in an effort to increase sales, cuts is price. There may well be a reaction from its
close substitutes. Suppose for instance, that its effort to increase sales from 10 to 11 units
leads its competitor to cut its price.
How do you price when you must take the reactions of your competitors into account?
Extra Revenue from selling an incremental unit for a pricemaking firm, taking competitive reactions into account
$10
B
$100
A
D
D'
10
If a price-making firm selling 10 units at $10 each decides to
increase output by a unit, its revenue will increase by the gain in
revenue on the 11th unit less the loss in revenue on the first 10 units,
as given by area A less area B. However, that answer becomes
incomplete if competitors react and begin cutting their prices in
response. Then the demand function shits to D'.
Economists refer to this as the duopoly or oligopoly problem. A Duopoly is a firm
whose profits depend on the behavior of another firm, while an oligopoly is a firm whose
profits depend on the behavior of a few firms. It tends to simplify things to concentrate on
the case of Duopoly, because nothing, or at least very little, is gained by consider oligopoly,
except to make the mathematics slightly more complicated.
When do you have a Duopoly/Oligopoly?
A duopoly/oligopoly exists if a firm’s profits depend on the behavior of another firm or
firms. Another approach commonly used is to define oligopoly in terms of something called
the four firm concentration ratio, a number conveniently computed by the US Department of
Commerce. Simply, it measures the fraction of the market for a particular product held by
the top four firms. (There are also eight firm concentration ratios, etc., but the one that gets
the attention is the four firm concentration ratio).
However a four firm concentration ratio can misses the point.
Automobile Industry and also Joe’s Corner Grocery.
Consider the US
Thus, there is no firm that can ignore the lessons of the competitive market and there
are almost no firms that can ignore the lessons of the duopoly model, for it faces situations
where the behavior of other firms impacts its profits.
The Cooperative Solution
Consider a duopoly for widgets. Suppose the demand curve for widgets is given by q
= 100-2p, and that the marginal cost of producing widgets is $5. We know that the following
solutions apply for a monopolist and for a competitive firm
The Basic Data
Price
Monopolist
Competitive Industry
$27.50
$5.00
Quantity
45
90
If you don’t recall these results, go back to your notes.
The best the two firms can do together is to cooperate and set total production to 45
widgets (maybe at 22.5 each) and reap the profits. A monopolist can do no better. This
arrangement will maximize their combined profits.
Economists are dubious of these arrangements working out. While certainly firms do
attempt to cooperate, they cannot cooperate by sitting down and working out a formal
agreement. Such a contract is not enforceable and its mere existence would be a violation of
the Sherman Act. What must be developed is a strategy of cooperation that does not involve
explicit cooperation. Several ideals have been suggested.

First, firms adopt strategies that appear to be legal, but do not involve explicit
cooperation. For example, I might post a sign “I will meet my competitor’s price”. This
strategy is perfectly legal, and essentially signals a cooperative strategy.

Second, firms find more complicated ways of signaling. For example, announce will cut
(or raise) prices 10 days hence and then see what other firms do. In some cases this is
legal, while in others it clearly is not.

Third, firms find back door ways of cooperating. Trade associations founded for
perfectly legal cooperative efforts often provide a venue for cutting deals.
The Cournot Duopoly Model
Suppose firms cannot cooperate by these devices. Economists have developed a series
of models designed to explain Duopoly behavior. These give us some guides to how a firm
ought to behave. They also give us some guides to the deficiencies in the models.
We will consider two basic models: the

The Cournot Model and

The Bertrand Model (or Cournot in Prices Model)
The Basic Cournot Model
Each firm maximizes profits taking the output of the other firms as fixed. Suppose
the market demand for widgets is given by the demand function D. Suppose that firm "B" is
now producing qB widgets. Then firm A takes its demand function as
A firm's demand function under the Cournot Model
P
q
b
D
DA
Q
In a Cournot Model, each firm takes its demand function as the
market demand function less the quantity currently being produced
by its competitors. This the demand function for A, is simply the
market demand function shifted to the left by q b, the quantity
currently being produced by firm b
D-qB
Is this realistic? Surely firm A knows that firm B will respond to any changes it
makes. Surely no one would want to make decisions without taking into account how others
will react to those changes. Thus, we should modify the model to allow for these reactions.
Perhaps, but this is the Cournot model and it does not consider such reactions.
A Graphical Solution
This makes the problem of profit maximization quite simple. Suppose the firms have
straightforward linear cost functions and linear demand functions. The firm's demand
function is the industry demand curve shifted to the left by the other firm's output. Then
profit maximization proceeds in a straightforward manner.
Profit maximization under the Cournot Model
D
P
D’
MC
MR
q
a
The firm's demand function is simply the market demand function
shifted over by the amount of the other firm's production. It then
proceeds to set MR = MC.
A Mathematical Illustration
From the example above, Firm A maximizes profits given by
 = (50-qb/2)q - q2/2 - 5q.
This requires that A set output according to the equation
q = 45 - qb/2
Notice, we have solved for q under the assumption that B's output is q b. Thus, we know how
A's output will change as B's output changes. This equation is referred to as the reaction
function.
This equation also gives us insight into how B will react and what will be the final
solution. B will be going through the same calculations and come up with its own reaction
function
q = 45 – qa/2.
Assuming that this is exactly what the second firm does, we will probably see a series
of rounds of quantity adjusting.
There is another way we could arrive at the same solution. Consider our two reaction
functions:
Firm A: q = 45 - qb/2.
Firm B: q = 45 – qa/2.
We know that the solution requires that qa = qb and we can solve the reaction
functions for the solution qa = qb = 30.
The General Solution to the Problem
There is a more general solution to the problem. First, when we have a demand
curve of the form
q=a–bp
And in the case of two firms with constant marginal cost c, the two reaction
functions will be
Firm A: q = a/2 –2c/b – qb/2
and
Firm B: q = a/2 –2c/b – qa/2
We can always solve these equations.
A Graphical Solution to our Numerical Example
We have two reaction functions, and we can simply plot the two reaction functions
and see where they intersect.
Cournot Equilibrium as the Intersection of
Two Reaction Functions
A's Output
90
B's Reaction Function
45
A's Reaction Function
B's Output
45
90
Each firm has a reaction function giving its output as a function of the
others. There is only one intersection.
To read this graph, suppose B's output is 90. Then A will produce 0.
And, if A produces 90, B will produce 0. The Cournot Equilibrium
occurs when A and B are each producing 30.
The Mechanics of Plotting the Reaction Function
Put A’s output on the Y-axis and B’s on the X-axis. Two points will get us going:
A’s Reaction Function
If B Produces
Then A Produces
0
(Monopoly Output) 45
(Competitive Output) 90
0.0
And ditto for B's reaction function:
B’s Reaction Function
If A Produces
Then B Produces
0
(Monopoly Output) 45
(Competitive Output) 90
0.0
Combining these two functions we have a graphical way of getting output
equilibrium. Note the equilibrium is at qa = qb = 30.
This graphical technique works only for linear demand curves.
A Second Example
Suppose the industry demand curve is
Q = 200 – 5p
and that the product can be produced for a marginal cost of $10 each. We know that the
competitive solution will involve 150 units of the product selling for $10 each. In the case of
the monopoly solution, the inverse demand function is
P = 40 – 0.2Q
Thus a monopolist's profits would be
 = (40- 0.2 Q) Q – 10 Q = 30 Q – 0.2Q2
Profit maximization requires that
' = 30 – 0.4Q = 0,
or Q = 75. We can then plot the reaction function quite simply:
A’s Reaction Function
If B Produces
Then A Produces
0
(Monopoly Output) 75
(Competitive Output) 150
0.0
And ditto in the opposite reaction.
B’s Reaction Function
If A Produces
Then B Produces
0
(Monopoly Output) 75
(Competitive Output) 150
0.0
I leave it as an exercise to show that the reaction functions cross at Q=50. That is,
each firm produces 50 units of output. The easiest way to do this is to plot the two reaction
functions and then take it from there.
The General Case
Consider the more general case. Suppose that the industry demand curve is
q=a–bp
In the case of two firms with constant marginal cost c, the two reaction functions will be
Firm A: q = a/2 –2c/b – qb/2
and
Firm B: q = a/2 –2c/b – qa/2
Suppose our market demand function is a straight-line demand function. Marginal
cost is c. The demand function hits the marginal cost line at Q 1. Intuitively this is the
quantity bought and sold in a competitive industry, with price equal to marginal cost.
Reaction Functions for the Cournot Model:
First Steps
D
MR
MC = c
Q1/2
Q1
Qo
To plot the general reaction functions for the Cournot model, notice
that the monopoly firm will always produce at Q 1/2, half the output of
a competitive industry, when marginal costs are constant.
Suppose the industry is monopolistic. Output will be determined by the quantity
where MR = MC. Thus, we know for a linear demand function the intersection of MR and
MC will occur at Q1/2.
To plot the general reaction functions, suppose firm B is producing at the competitive
level, Q1. Then A's profit maximizing level of output is zero. On the other hand, if B is not
producing, then A's profit maximizing level of output is the monopoly level of output, Q 1/2. If
we connect these two points, we have A's reaction function. In turn, the same logic will lead
us to B's reaction function.
A’s Reaction Function
If B Produces
Then A Produces
0
(Monopoly Output) Q1/2
(Competitive Output) Q1
0.0
And ditto in the opposite reaction.
B’s Reaction Function
If A Produces
Then B Produces
0
(Monopoly Output) Q1/2
(Competitive Output) Q1
0.0
The point where the two reaction functions cross is Q 1/3. In short, if there are two
firms sharing a Cournot Duopoly (and where there is a linear demand function and where
there is constant marginal cost), then each firm will produce 1/3 of the output that will be
produced in a competitive industry.
There is a pattern:
 With a linear demand curve and constant marginal cost, the monopolist produces half as
much output as would be demanded if the product sold at marginal cost.
 With two Cournot duopolists, they produce 2/3 as much output as would be demanded if
the product sold at marginal cost.
Generalized Cournot Reaction Functions, Linear Demand
Functions and constant marginal cost
A
Q1
B's Reaction Function
Q1/2
Q1/3
A's Reaction Function
B
Q1/3 Q1/2
Q1
Suppose there are two firms producing a product with a linear
demand function, and constant marginal cost, where Q 1 is the
amount that would be produced under competition. These are the
Cournot reaction functions. Note that each firm produces 1/3 of
competitive output.
Extensions to Multiple Firms
You might ask if the 2/3 rule generalizes. It does

When there are n Cournot duopolists, they produce n/(n+1) as much output as would
be demanded if the product sold at marginal cost.

Thus if there are three Cournot duopolists, each will produce 1/4 as much as
output as would be demanded if the product sold at marginal cost. Total output
will be ¾ as much as would be demanded if the product sold at marginal cost.

If there are four, they will produce 4/5 as much output as would be demanded if
the product sold at marginal cost, etc.
I leave the proof as an exercise.
An Application to the Wudget/Widget Example
Assuming a marginal cost of production of $500 a unit, and given the following
demand curve, the manufacturer would price it at $1,300 a unit.
Q = 630,000 – 300p
If the marginal cost is $500 a unit, a monopolist maximizes profits by selling 240,000 units at
a price of $1300.
Suppose that next year marginal cost drops to $400 a unit. The monopolist will cut is
price to $1,250 and sales rise to 255,000 units. If the firm were so foolish as to price at
marginal cost, the price would be $400 and 510,000 units would be sold.
Our Cournot model provides us a means of predicting how prices will change over
time when additional firms join the market. To be specific, suppose in the third year,
another producer enters the market, and a Cournot duopoly arises. Then total production
will rise to 340,000, as the price drops to $967 a unit. Note that we take advantage of the
notion that output will be 2/3 that of the competitive solution.
Finally, suppose that, each year after than a new producer enters the market
Sales and Price Data, Wudget and New Competitors
Year
Price
0
1
2
3
4
5
$1300
$1250
$967
$825
$740
$683
Quantity
240,000
255,000
340,000
382,500
408,000
425,000
As you can see, the solution is getting closer and closer to the competitive solution, as
the price approaches $400, the competitive price.
How many bidders to get competition?
Our discussion of Cournot Equilibrium suggests that as the number of firms grows,
the price approaches the competitive price. Leonard Wiess studied how many bidders are
required to get the lowest bid. The idea is simple. If there is one bidder, they will charge the
monopoly price. As the number of bidders increases, the price should decline at a slower
rate. When the curve flattens out, we are presumably at the competitive price, and the
number of bidders is an estimate of the number of bidders ensuring competition. The data
suggest that 4-8 bidders are enough.
Price as a Function of the Number of Bidders
Pm
Pc
Number of Bidders
When there is one bidder, the monopoly price, P m, prevails. As the
number of bidders increases, the price drops. At some point, the
price stops dropping, indicating that we are getting a competitive
equilibrium.
Nash Equilibrium
This Cournot solution is also a Nash Equilibrium, after the mathematician John
Nash (another winner of the Nobel Prize in Economics). Nash Equilibrium involves the
following steps:



Each person is acting rationally
Each person believes his actions will not change the decision of others
No person has an incentive to change.
Bertrand Duopoly Model (Cournot in Prices)
Here, instead of calling out quantities, the two firms call out prices in sort of a
winner-take-all sort of world. It also involves Nash Equilibrium.
Consider
q = 100-2p
Suppose that each firm is to call out a price p a and pb, with the “winner”, the firm
calling out the lowest price, to be the one that gets the market. You know, as firm A that the
demand function you face is as follows:
Reactions in the
Bertrand Model
Your Price
Your Market
pa > pb
pa = pb
0
(0.5)(100-2pa)
That is, the two
firms
split the market
(100-2pa)
pa > pb
How then should you call out your price? Clearly there is only one Nash Equilibrium,
and that is at pa = pb = 5. Only at marginal cost pricing do the two competitors have no
incentive to change.
When do you have a Bertrand duopoly and when do you have a Cournot
duopoly?
Some people ask when to use the Bertrand model and when to use the Cournot
Model. The answer is that it depends. However as a rule, the following will do the trick:

If it is a winner takes all situations then the Bertrand duopoly model is
applicable.

Otherwise, Cournot.
Some Applications
As a customer of a duopoly
You are in the business of purchasing a commodity from one of two firms. You know
that there is a possibility for the firms to exercise duopoly powers. You would like to
structure your dealings with them so as to prevent them from doing so. In particular, you
want to get them to act as competitive firms and thus charge you marginal cost.
Pashigian presents the problem of getting bids on getting a building cleaned. You
have three strategies:

Take bids from each firm on doing a certain number of square feet. I.e., Acme
Cleaners offers to clean A square feet and Brilliant Cleaners then offers to clean
B square feet. You determine your demand price for having A + B square feet
cleaned and then give them the job at that price.

You offer the job to the one with the lowest bid. You tell both to submit a bid per
square foot and you will then give the job to low bidder.

You invite the two firms to submit a bid. The job will then be split between the
two firms but at the lower bid. Of course, the lower the bid, the more square feet
you will have done.
The first strategy results in Cournot Equilibrium; the second results in Bertrand
Equilibrium, while the third is just a bad strategy. Clearly, as the manager of the firm
putting the job out for bid, you want to go the second strategy. The two bidders would like
you to go the first strategy (or, if they are particularly greedy, the third strategy).
As a manager of such a firm
Assume you are one of a few firms in an industry. You want to make the duopoly
work. You want to cooperate with your counterpart at the other firm. One solution, of
course, is simply to operate like a cartel. That is, adopt the cooperative arrangement. Here,
you have two problems.
Ethical and Legal Problems
Cartels are illegal, and efforts to collude overtly with your counterpart clearly violate
the law.
The Cheating Problem
You will need ways of keeping cheating to a minimum. You need to adopt policies
that (a) give you evidence that the other party is cheating and (b) conduct your business in
such a way that the other party feels you are not.
How to do it

If possible, get the customer to rely on sealed bids, which are publicly opened.

Try to make repeat sales to the same customers.

Try to get the customer to break a job into many auctions.

Try to find ways to signal your fellow firm(s) that you will cooperate.

Get the government on your side.
Note: this is not a complete list.