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Economics of Antitrust
Handout on Oligopoly Models
This handout is intended to give you a basic feel for how oligopoly models work. It will
cover both Cournot and Bertrand oligopolies. If there is demand for it, I’ll provide
further handouts with either extensions of these two models or details on other oligopoly
models.
Counot (1838) Oligopoly
Consider N firms, with the profits of the ith firm being given by the following expression:
[1]
πi = Pqi – TC(qi)
Price is determined by market demand, which is a function of the total quantity produced
by all of the firms, Q = ∑qi. The market demand can be described by an inverse demand
equation.
[2]
P = P(Q)
Before picking specific function for TC and P, it’s worth considering the general
approach to solving this model. Plug [2] into [1], take the derivative with respect to qi,
and set the result equal to zero. This is known as the first order condition (FOC) for firm
i, and must be true for any optimal choice of qi (assuming an interior solution – if it is
optimal for the firm to produce no output, the foc may not be satisfied). You can solve
the first-order condition for qi, in which case you will have firm i’s reaction function,
which gives its optimal qi as a function of the q’s chosen by each of the other firms.
Since we have N of these reaction functions (one for each firm) and N variables (one q
for each firm), we have N equations in N unknowns, which we can solve for all N of the
q’s. Once we have the optimal Q’s, it is a simple matter to plug them into [2] to get the
market clearing price, and into [1] to get the profits for each firm.
There is one trick that is sometimes useful in solving this system of equations. If the N
firms are identical, and we are at an interior solution (i.e., all of the firms are producing a
positive amount), then each firm will produce the same quantity. Thus, we can substitute
qi = q for all i, and reduce the system of equations to N identical equations, with one
variable. Note: when using this trick it is very important not to make this substitution
until after taking the derivative. If you make the substitution before taking the derivative,
you are assuming that the firm assumes that all of its competitors will match its quantity
choice, i.e., that the firm is picking one quantity for all of the firms, and not just its own
quantity. If you do this, you will get q equal to 1/N of the monopoly Q, leading the
monopoly pricing (which is wrong).
This may seem confusing, so let’s move away from the general case, and consider a
specific example that we can actually solve. Let’s start with some simplifying
assumptions:
A1: All of the firms have the identical cost function TC(qi) = MC*qi.
A2: The inverse demand function is P = AQ1/ε, where b<0 (A > 0 and ε < 0 are
constants).
First consider A2. Note that if you calculate the price elasticity of demand, you get that it
is constant and equals ε, the constant with a cleverly chosen name. [You can check this
as an exercise.]
Profits for firm 1 can then be written as follows:
[3]
π1 = A(∑qi)1/ε*q1 – MC*q1
Taking the derivative with respect to q1 gives us firm 1’s fist order condition:
[4]
q1*(A/ε)*(∑qi)-1+1/ε + A(∑qi)1/ε – MC = 0
Rather than solving for firm 1’s reaction function, I’ll apply the trick, setting qi equal to q
for all i (including i = 1).
[5]
q*(A/ε)*(Nq)-1+1/ε + A(Nq)1/ε – MC = 0
(Nq)1/ε*[q*(A/ε)*(Nq)-1 + A – MC*(Nq)-1/ε] = 0
q*(A/ε)*(Nq)-1 + A – MC*(Nq)-1/ε = 0
A/(Nε) + A = MC*(Nq)-1/ε
 
1 
 A * 1  N  

q 
MC







*
1
N
[A digression on HHI]
Economists some measure how concentrated an industry is using something called the
Herfindahl-Hirschman Index (HHI). HHI is defined to be the sum of the squares of the
market shares of the firms in the industry.
Example: Suppose there are four firms in an industry, with market shares of 45%, 40%,
10% and 5%, respectively. Then the industry’s HHI would be 0.452 + 0.42 + 0.12 + 0.052
= 0.3750. Alternatively, if market shares are measured as whole numbers, the industry’s
HHI would be 452 + 402 + 102 + 52 = 3750.
Note: Both methods of measuring market shares are used. It should be fairly clear from
context which method is being used. If shares are measured as fractions, the HHI will a
number between zero (perfect competition) and one (monopoly). If shares are measured
as whole numbers, the HHI will be a number between zero (perfect competition) and
10,000 (monopoly).
Note that HHI = 1/N if the N firms all have equal size, and with HHI > 1/N if there are N
firms with unequal size.
[End of digression.]
In this case, we can substitute HHI for 1/N into equation [5] to get
[6]

 HHI
 A * 1  

q
MC










* HHI
Thus, we have found the equilibrium quantities as a function of marginal costs, the HHI
measure of concentration, and marginal costs. From here it is pretty easy to find the
equilibrium price and profits per firm.
[7]
P = AQ1/ε
P = A(Nq)1/ε




1 



A * 1 



1
N  

P  A* N * 
* 
MC
N











[8]
P
MC
1 

1 

N 

P
MC
 HHI 
1 

 

πi = P*q – MC*q
πi = (P– MC)*q
1/ 


  
HHI
 MC
  A1  
i  
 MC  *  
MC
  HHI 
 
 1   
 

 


  HHI







1

 * HHI
 

 1  HHI  


* MC 1
  HHI 
*  A1 

 
 

Now we can analyze the solution. Unless otherwise stated, this analysis assumes that
demand is not “too” inelastic, specifically that ε < -HHI. Equations [6] to [8] then tell us
the following things:
1. P > MC, but P < PM (the monopoly price). Q is greater than the monopoly
quantity, but less then the competitive level.
2. As HHI goes to zero (i.e., N goes to infinity), P goes to MC, qi goes to zero, Q
goes to P(MC), πi goes to zero, and ∑πi goes to zero. In other words, as the
number of firms increases, the market outcomes approach the perfectly
competitive outcomes.
3. As MC increases, P increases, qi decreases, and Q decreases.
4. Suppose demand becomes more elastic. Then P goes to MC, qi goes to zero, Q
goes to P(MC), πi goes to zero, and ∑πi goes to zero. Just as with increasing the
number of firms, making demand more elastic causes the equilibrium to approach
the competitive level.
5. If ε > -HHI, then 1+HHI/ε is negative, and both price and quantity will be
negative. This obvious absurdity means that the model is invalid for values of
HHI and ε that make this happen. If you allow for entry and exit, the Cournot
model can be extended so as to be valid in these cases (basically entry will make
the HHI will fall enough to make ε < -HHI).
Bertrand (1883) Oligopoly
Bertrand oligopoly is a little different. Instead of each firm simultaneously choosing
its quantity, with the price being set by the market so that supply equals demand,
firms simultaneously choose their prices, with the market demanding an appropriate
quantity from each firm.
Let Q = Q(P) be the market demand function. Let N be the number of firms, and let
Ni be the number of firms that choose the same price as firm. All firms have zero
fixed costs and constant marginal costs. The demand for firm i’s output is
qi = Q(Pi)/Ni if Pi <= Pj
qi = 0 otherwise
In other words, if firm i charges no more than any other firm in the market, it will
share the market equally with all of the firms that had the lowest price. If it charges
more, then it will sell nothing (thus earning zero profits).
Now consider equilibrium costs. Suppose that the equilibrium price was above
marginal costs, so profits are positive for each firm charging the minimum price.
Each firm charging such a price gets 1/Ni of the total profits. If a firm were to
undercut this price by a single penny, the total industry profits would change only by
a very small amount (since P changes very little, total Q changes very little, which
implies that total costs change very little). But the firm that undercut the price would
earn all of the industry profits, and not just a share of them. For sufficiently small
price costs, the effect of going from a market share of 1/Ni to 100% will outweigh
any small change in total industry profits, so the firm will want to undercut the price.
But this argument holds for any price above marginal cost. Thus, the equilibrium
price can be no greater than the marginal cost. But the equilibrium price cannot be
below marginal cost, since that would mean that all firms would earn a negative
profit. Thus, the only equilibrium is for all firms to charge Pi = MC.
Comparing Cournot and Bertrand
So, when modeling an oligopoly, which model should you use? One approach is to
look at the industry in question, and to ask whether firms pick a price, and then sell as
much as the market wants at that price, or do they pick a quantity, and then try to find
the price at which exactly that quantity is sold? The problem with that approach is
that most firms fiddle with both price and quantity. For example, they have only an
imperfect estimate of demand, and the try and pick a point on their demand curve.
When the market starts running, they will find themselves either accumulating or
running down their inventory, and so they will adjust both their price and their
quantity in response.
Another approach is to attempt to reconcile the models, usually by making Bertrand
look more like Cournot. There are several ways to do this. For instance, if firms are
capactity constrained so that no one firm can serve the entire market, then a firm will
receive a positive market share even if its price is greater than the lowest price in the
market. This leads to equilibrium prices that are higher than marginal costs. In fact,
Kreps and Scheinkman (1983) showed that if firms first choose their capacity
constraints, and then play the Bertrand game with those capacity constraints, they will
choose capacity constraints such that the outcome of the capacity-constrained
Bertrand game is exactly the same as the outcome of the Cournot game.
Other ways to modify the Bertrand game include adding consumer search costs to the
model (so that consumers do not always find the firm with the lowest price), or addinf
product differentiation. The key to all of these modifications is they imply that a firm
can price slightly above the price of the low-price firm, without loosing all of its
sales. This blunts price competition, leading to prices above marginal costs and
positive profits.
Since the Bertrand model is not robust to these kinds of changes (i.e., even small
search cots, capacity constraints, or product differentiation leads to big changes in the
predicted outcomes), and since the Cournot model corresponds to our intuition that
having only two firms is not enough to get the competitive outcome, the Cournot
model is sometimes preferred by economists, at least in markets with homogenous
goods. However, there are cases in which the Bertrand model seems to fit the
industry quite well.
Other Oligopoly Models
There several other models of oligopoly, so Cournot and Bertrand are not the end of
the story. There are also many extensions to the basic Bertrand and Cournot models,
such as adding product differentiation, capacity constraints, consumer search costs,
consumer switching costs, entry/exit, advertising, cost-reducing or quality-enhancing
investments, and/or incomplete or asymmetric information. In general, economists
try to use the simplest possible model that captures the relevant features of the world.
Which features of the word are relevant will of course depend on the question the
model is being used to answer.
References
Bertrand, J. 1883. “Theorie Mathematique de la Richesse Sociale,” Journal des
Savants, pp. 499-588. Translated by James W. Friedman in Andrew F. Daughety,
ed., Cournot Oligopoly, Cambridge, United Kingdom: Cambridge University Press,
1988, pp. 73-81.
This is the original source of the Bertrand model.
Carleton, D. and J. Perloff. 2000. Modern Industrial Organization (New York:
Addison Wesley Longman).
This intermediate textbook includes analysis of many models of oligopoly.
Cournot, A. 1838. Recherches sur les Principes Mathematiques de la Theorie des
Richesses. English edition (ed. N. Bacon): Researches into the Mathematical
Principles of the Theory of Wealth (New York: Macmillan, 1897).
This is the original source of the Cournot model.
Kreps, D. and J. Scheinkman. 1983. “Quantity Precommitment and Bertrand
Competition Yield Cournot Outcomes,” Bell Journal of Economics, v. 14, pp. 326337.
This article relates the two models. The title pretty much gives away the conclusion.
Tirole, J. 1988. The Theory of Industrial Organization (Cambridge, Mass.: MIT
Press).
This is a good graduate level textbook that includes analysis of many oligopoly
models. For you, it should be very challenging, but comprehensible.
Viscusi, W., J. Vernon, and J. Harrington. 2000. Economics of Regulation and
Antitrust, 3rd edition (Cambridge, Mass.: MIT Press).
This is an undergraduate textbook that includes a very nice summary of oligopoly
models.