Eco205 Chapter 12 – Imperfect Competition 18th January 2013
Oligopoly A market with few firms but more than one
Overview: Pricing Of Homogeneous Goods
It is difficult to predict exactly the possible outcomes for prices when there are few firms; prices depend
on how aggressively firms compete, which in turn depends on which strategic variables firms choose,
how much information firms have about rivals, and how often firms interact with each other in the
The Bertrand model — identical firms choose prices simultaneously in their one meeting in the market,
has Nash equilibrium. This figure assumes that marginal cost (and average cost) is constant for all output
levels. Even though there may be only two firms in the market, in this equilibrium they behave as if they
were perfectly competitive, setting price equal to marginal cost and earning zero profit.
Perfect Cartel Outcome
At the other extreme, firms as a group may act as a cartel, recognizing that they can affect price and
coordinate their decisions. Indeed, they may be able to act as a perfect cartel, achieving the highest
possible profits, namely, the profit a monopoly would earn in the market.
One way to maintain a cartel is to bind firms with explicit pricing rules. Such explicit pricing rules are
often prohibited by antitrust law. Firms do not need to resort to explicit pricing rules if they interact on
the market repeatedly. They can collude tacitly.
The Bertrand and cartel models determine the outer limits between which actual prices in an
imperfectly competitive market are set. This band of outcomes may be very wide, and such is the wealth
of models available that there may be a model for nearly every point within the band.
Cournot model - An oligopoly model in which firms simultaneously choose quantities
Nash Equilibrium in the Cournot Model
For a pair of quantities, qA and qB, to be a Nash equilibrium, qA must be a best response to qB and vice
versa. We have to start by computing the best-response function for firm A. Its best-response function
tells us the value of qA that maximizes A’s profit given for each possible choice qB by firm B. Nash equilibrium requires each firm to play its best response to the other. The only point where both
are playing best responses is the intersection between their best-response functions. No other point
would be stable because one firm or the other or both would have an incentive to deviate
Comparisons and Antitrust Considerations
The equilibrium price and industry profit in the Cournot model is above the perfectly competitive level
and below the monopoly level; industry output is below the perfectly competitive level and above the
monopoly level. The firms manage not to compete away all the profits as in perfect competition. But the
firms do not do as well as a monopoly would, either
The industry does not attain the monopoly profit in the Cournot model because firms do not take into
account the fact that an increase in their output lowers price and thus lowers the other firm’s revenue.
Firms ‘‘overproduce’’ in this sense. According to this model, firms would have an incentive to form a
cartel with explicit rules limiting output. Another way to increase profits would be for the firms to
merge, essentially turning a Cournot model with two firms into a monopoly model with one firm.
Consumers benefit from the higher output and lower prices in the Cournot model compared to
monopoly. Government authorities, through the antitrust laws, often prohibit conspiracies to form
cartels and mergers that would increase concentration in the industry. Assuming the government
authorities act in the interest of consumers, the Cournot model provides some justification for these
As the number of firms grows large, it can be shown that the Nash equilibrium approaches the
competitive case, with price approaching marginal cost
Bertrand model - An oligopoly model in which firms simultaneously choose prices
To state the model formally, suppose there are two firms in the market, A and B. They produce a
homogeneous product at a constant marginal cost (and constant average cost), c. Note that this is a
generalization of our assumption in the Cournot model that production was costless. Firms choose
prices PA and PB simultaneously in a single period of competition. Firms’ outputs are perfect substitutes,
so all sales go to the firm with the lowest price, and sales are split evenly if PA = PB.
We will generalize the demand curve beyond the particular linear one that we assumed in the Cournot
model to be any downward-sloping demand curve. We will look for the Nash equilibrium of the Bertrand
model. It turns out that the marginal analysis (MR = MC) we used to derive the best-response functions in the Cournot model will not work here since the profit functions are not smooth. Starting from equal
prices, if one firm lowers its price by the smallest amount, its sales and profit would essentially double
Nash Equilibrium in the Bertrand Model
The only Nash equilibrium in the Bertrand game is for both firms to charge marginal cost: PA = PB = c. In
saying that this is the only Nash equilibrium, we are really making two statements that both need to be
verified: (1) that this outcome is a Nash equilibrium, and (2) that there is no other Nash equilibrium.
To verify that this outcome is a Nash equilibrium, we need to show that both firms are playing a best
response to each other - that neither firm has an incentive to deviate to some other strategy.
In equilibrium, firms charge a price equal to marginal cost, which in turn is equal to average cost. But a
price equal to average cost means firms earn zero profit in equilibrium. It will make no sales and
therefore no profit, not strictly more than in equilibrium. If it deviates to a lower price, it will make sales
but will earn a negative margin on each unit sold since price will be below marginal cost. So the firm will
earn negative profit, less than in equilibrium. Because there is no possible profitable deviation for the
firm, we have succeeded in verifying that both firms’ charging marginal cost is a Nash equilibrium.
To verify that this outcome is the only Nash equilibrium, there are a number of cases to consider. It
cannot be a Nash equilibrium for both firms to price above marginal cost. If the prices were unequal, the
higher-pricing firm, which would get no demand and thus would earn no profit, would make positive
sales and profit by lowering its price to undercut the other.
If the above-marginal-cost prices were equal, either firm would have an incentive to deviate. By
undercutting the price ever so slightly, price would hardly fall but sales would essentially double because
the firm would no longer need to split sales with the other. Nash equilibrium cannot involve a price less
than marginal cost either because the low-price firm would earn negative profit or could gain by
deviating to a higher price. For example, it could deviate by raising price to marginal cost, which, since it
also equals average cost, would guarantee the firm zero profit
Price is set to marginal cost, and firms earn zero profit. This is the same as perfect competition that’s
why its called a paradox. It is paradoxical that competition would be so tough with as few as two firms in
The Bertrand Paradox could also be avoided by making other assumptions, including the assumption
that the marginal cost is higher for one firm than another, the assumption that products are slightly differentiated rather than being perfect substitutes, or the assumption that firms engage in repeated
interaction rather than one round of competition.
Capacity Choice and Cournot Equilibrium
Capacity constraint - A limit to the quantity a firm can produce given the firm’s capital and other
The firm can satisfy this increased demand because it has no capacity constraints, giving firms a big
incentive to undercut. If the undercutting firm could not serve all the demand at its lower price because
of capacity constraints, that would leave some residual demand for the higher-priced firm, and would
decrease the incentive to undercut
Firms cannot sell more in the second stage than the capacity built in the first stage. If the cost of building
capacity is sufficiently high, it turns out that the subgame-perfect equilibrium of this sequen- tial game
leads to the same outcome as the Nash equilibrium of the Cournot model.
The principal lesson of the two-stage capacity/price game is that, even with Bertrand price competition,
decisions made prior to this final (price-setting) stage of a game can have an important impact on
Comparing the Bertrand and Cournot Results
The Bertrand model predicts competitive outcomes in a duopoly situation, whereas the Cournot model
predicts prices above marginal cost and positive profits; that is, an outcome somewhere between
competition and monopoly. These results suggest that actual behavior in duopoly markets may exhibit a
wide variety of outcomes depending on he precise way in which competition occurs
the equilibrium outcomes from the two games resemble that from the Prisoners’ Dilemma. The Nash
equilibrium in all three games is not the best outcome for the players. Players could do better if they
could cooperate on an outcome with lower outputs in Cournot, higher prices in Bertrand, or being Silent
in the Prisoners’ Dilemma. But cooperation is not stable because players have an individual incentive to
deviate. In equilibrium of both the Cournot and Bertrand games, firms in a sense compete too hard for
their own good (to the benefit of consumers)
We assume that the market is composed of a few slightly differentiated products that can be usefully grouped together because they are more substitutable for each other than for goods outside the group
Bertrand Model with Differentiated Products
One way to model differentiated products is to specify demand curves that are functions of the
product’s own price and also of the price of the other good.
Example given on page 456
With differentiated products, the profit functions are smooth, so one can use marginal analysis to
compute the best-response functions, similar to the analysis of the Cournot model.
The best-response functions show the profit-maximizing price for a firm given a price charged by its
competitor. The best-response functions tend to be upward-sloping: an increase in, for example, B’s
price increases A’s demand, which would lead A to respond by raising its price. This contrasts with the
Cournot case, where the best- response functions were downward sloping
Product characteristics—including color, size, functionality, quality of materials, etc.—are strategic
choices for the firms just as are price and quantity.
Consider a two-stage game in which firms choose product characteristics in the first stage and price in
There are two offsetting effects at work in the first-stage product-characteristics game. One effect is
that firms prefer to locate near the greatest concentration of consumers because that is where demand
is greatest. For example, if consumers’ favorite colors are beige and metallic gray automakers will tend
to produce beige and metallic-gray cars. This effect leads firms to locate near each other, that is, to
produce very similar products. There is an offsetting strategic effect. Firms realize that if their products
are close substitutes, they will compete aggressively in the second-stage price game. Locating further
apart softens competition, leading to higher prices. An increase in product differentiation between the
two firms shifts their best-response functions out and leads to a Nash equilibrium with higher prices for
If firms’ products become too specialized, they risk the entry of another firm that might locate in the
product space between them.
Prices may differ across goods if products are d