Types of Duopoly Models

4TH SEM
MICRO EONOMICS

The uncertainty is respect of behaviour pattern of a firm under

oligopoly arising our of their unpredictable action and reaction
makes a systematic analysis of oligopoly difficult.
However, classical and modern economists have developed a
variety of models based on different behavior assumptions.
These models can broadly be classified into two categories (I)
classical duopoly models and modern oligopoly Duopoly
Models, when there are only two sellers a product, there, exists
duopoly.
Duopoly is a special case of oligopoly. Duopoly is a special case
in the sense that it is limiting case of oligopoly as there must be
at least two sellers to make the market oligopolistic in nature.
1. The Cournot’s Duopoly Model
2. The Chamberlin Duopoly Model
3. The Bertrand’s Duopoly Model
4. The Edgeworth Duopoly Model
1. Cournot’s Duopoly Model:
Augustin Cournot, a French economist, was the first to develop
a formal duopoly model in 1838.
To illustrate his model, Cournot assumed:
(a) Tow firms, each owing an artesian mineral water well;
 (b) Both operate their wells at zero marginal cost2;
(c) Both face a demand curve with constant negative slope;
(d) Each seller acts on the assumption that his competitor will
not react to his decision to change his and price. This is
Cournot’s behavioural assumption.
On the basis of this model, Cournot has concluded that each
seller ultimately supplies one-third of the market and charges the
same price. While one-third of the market remains unsupplied.
Diagram Representation:
Cournot’s duopoly model is presented in Fig. 1. To begin the
analysis, suppose that there are only two firms. A and B, and
that, initially. A is the only seller of mineral water in the market.
In order to maximize his profits (or revenue), he sells quantity
OQ where his MC = O MR, at price OP2 His total profit is
OP2PQ.

Now let B enters the market. The market open to him is QM

which is half of the total market. He can sell his product in the
remaining half of the market. He assumes that A will not change
his price and output as he is making the maximum profit i.e., A
will continue to sell OQ at price OP2 Thus, the market available
to B is QM and the demand curve is PM.
When to get maximize revenue, B sells ON at price OP1, His
total revenue is maximum at QRP’N. Note that B supplies only
QN = 1/4 = (l/2)/2 of the market.) With the entry of B, price
falls to OP1 Therefore, A’s expected profit falls to OP1 PQ
Faced with this situation, A attempts to adjust his price and
output to the changed conditions. He assumes that B will not
change his output QN and price OP1 as he is making maximum
profit.
Accordingly, A assumes that B will continue to supply 1/4 of
market and he has 3/4 (= 1 – 14) of the market available to him.
To maximise his profit. Supplies 1/2 of (3/4), i.e., 3/8 of the
market. Note that A’s market share has fallen from 1/2 to 3/8.
Now it is B’s turn to react. Considering Cournot’s assumption, B
assumes that A will continue to supply only 3/8 of the market
and market open to him equals 1 – 3/8 = 5/8.
In order to maximise his profit under the new conditions B
supplies 1/2 x 5/8 = 5/16 of the market. It is now for A to
reappraise the situation and adjust his price and output
accordingly.
This process of action and reaction continues in successive
periods. In the process, A continues to lose his market share and
B continues to gain. Finally situation is reached when their
market shares equal at 1/3 each.
Any further attempt to adjust output produces the same result.
The firms, therefore, reach their equilibrium position where each
one supplies one-third of the market.
The equilibrium of firms, according to Cournot’s model, has
been presented in table below:

Cournot’s equilibrium solution is stable. For given the action

and reaction, it is not possible for any of the two sellers to
increase their market share.
It can be shown as follows:
A’s share= 1/2(1 – 1/3) = 1/3.
Similarly B’s share = 1/2 (1 – 1/3) = 1/3.
Cournot’s model of duopoly can be extended to the general
oligopoly. For example, if there are three sellers, the industry,
and firms will be in equilibrium when each firm supplies 1/3 of
the market. Thus, the three sellers together supply 3/4 of the
market, 1/4 of the market remaining unsupplied. The formula for
determining the share of each seller in an oligopolistic market is:
Q -f- (n + 1), where Q = market size, and n = number of sellers.
Criticism of the Model:
Although ournot’s model yields a stable equilibrium, it has been
criticised on the following grounds:
(1) Curnot’s behavioural assumption [assumption (d) above] is
naive to the extent that it implies that firms continue to make
wrong calculations about the competitor’s behaviour. Each seller
continues to assume that his rival will not change his output
even though he reportedly observes that his revel firm does
change its output.
(2) The assumption of zero cost of production is totally
unrealistic. If this assumption is dropped, it does not alter his
position.

2. Chamberlin’s Duopoly Model- A Small Group Model:

Chamberlin’s model of duopoly recognizes interdependence if
firms in such a market. Chamberlin argues that in the real world
of oligopoly firms are not so native that they will not learn from
the past experience. However, he makes the same assumptions
as the exponents of old classical models have done. In other
words, his model is also based on the assumption of
homogeneous products, firms of equal size with identical costs,
no entry by new firms and full knowledge of demand.
Recognition of interdependence of firms in an oligopolistic
market given us a result quite different from that of Cournot.
Chambrilin argues that firms are aware of the fact that their
output or price decision will definitely invite reactions of other
firms. Therefore, he goes not visualize any price war in
oligopolistic markets. He also rules out the possibility of firms
adjusting their outputs over a period of time and thus reaching
the equilibrium at an output level lower than that would be
reached under monopoly.
According to Chamberlin, recognition of possible sharp
reactions to an oligopolistic firm’s price or output manipulations
would avert harmful competition amongst the firms in such a
market and would result in a stable industry equilibrium with the
monopoly price and monopoly output. He further stated that no
collusion is required for obtained this solution.
In case farms in an oligopolistic market are aware of their
mutual dependence, and willing to learn from their past
experience, then in order to maximize their individual and joint
profits they will charge the monopoly price.
Chamberlin’s model can be explained in the frame work of a
dupoly market. Chamberlin, like Cournot, assumes linear
demand for the product. For simplicity we assume that even in
this case the cost of producing the good is zero.
Chamberlin model has been illustrated in Figure 2. In this figure
DQ is the market demand curve. If firm A is first to enter the
market, it will produce output OQ1 because at this level of
output its marginal revenue is equal to marginal cost (MR = MC
= 0). The firm can charge price OP1 which is the monopoly
price.
This will maximise its profits. At price OP) elasticity of demand
is unity. Firm B entering market at this stage considers that its
demand curve is CQ and will thus produce Q1Q2 so as to
maximise its profit. It will charge price OP2.
It now realizes that it cannot sell QQ1 quantity at the monopoly
price and thus decides to reduce the output to QQ3, which is
one-half of the monopoly output QQ1. Firm B can continue to
produce quantity Q1Q2 which is same as Q3Q1.
The industry output thus is OQ1 and the price rises to the level
OP1. This is an ideal situation from the point of view of both
firms A and B. In this case, the joint output of the two firms is
monopoly output and they charge monopoly price. Thus,
considering the assumption of equal costs (costs = 0) the market
will be shared equally between firms A and B.
Appraisal of the Model:
Chamberlin’s model is certainly more realistic than earlier
models. It assumes that firms recognize interdependence and
then act in a manner that monopoly solution is reached. In the
real world of oligopoly there are certain difficulties in reaching
this solution. In the absence of collusion, firms must have a
good knowledge of market demand curve which is almost
impossible to obtain. In case this information is lacking, firms
will not know how to reach monopoly solution.
Further, Chamberlin ignores entry. In real practice, oligoplistic
markets are rarely closed. So if we recognize the fact of entry, it
would not be certain that the stable monopoly solution will ever
be reached. Differences in costs and market opportunities are
also hindrance for attaining a monopoly-type outcome by the
independent actions of firms in oligopolies.

3. Bertrand’s Duopoly Model:

Bertrand, a French Mathematician developed his own model of
duopoly in 1883. Bertrand’s model differs from Cournot’s
model in respect of its behavioural assumption. While under
Cournot’s model, each seller assumes his rival’s output to
remain constant, under Bertrand’s model each seller determines
his price on the assumption that his rival’s price, rather than his
output, remains constant.
Bertrand’s model focuses on price competition. His analytical
tools are reaction function of the duopolists. Reaction functions
are derived on the basis of iso-profit curves. An iso-profit curve,
for a give level of profit, is drawn on the basis of various
combinations of prices charged by the rival firms. He assumed
only two firms, A and B and their prices are measured along the
horizontal and vertical axes, respectively.
Their iso-profit curves are drawn on the basis of the prices of the
two firms. Iso-profit curves of the two firms are concave to their
respective prices axis, as shown in Fig. 3 and 4. Iso- profit
curves of firm A are convex to its price axis PA (Fig. 3) and
those of firm B are convex to PB (Fig. 4).
In Figure 4, we have curve A, which shows that A can earn a
given profit from the various combinations of its own and its
rival’s price. For example, price combinations at points, a, b and
c yield the same level of profit indicated by the iso-profit curve
A1. If firms B fixes its prices Pb1– firm A has two alternative
prices, Pa1 and Pa2, to make the same level of profits.
When B reduces its price, A may either raise its price or reduce
it. A will reduce its price when he is at point c and raise its price
when he is at point a. But there is a limit to which this price
adjustment is possible. This point is shown by point b. So there
is a unique price for A to maximize its profits. This unique price
lies at the lowest point of iso-profit curve.
The same analysis applies to all other iso-profit curves, A1 A2
and A3 we get A’s reaction curve. Note that A’s reaction curve
has a rightward slant. This is so because, iso-profit curve tends
to shift rightward when A gains market from his rival B.
Following the same process, B’s reaction curve may be drawn as
shown in Fig. 4.
The equilibrium of duopolists suggested by Bertrand’s model
may be obtained by putting together the reaction curves of the
firms A and B as shown in Fig. 5.
The reaction curves of A and B intersect at point E where their
expectations materialize, point E is therefore equilibrium point.
This equilibrium is stable. Fo, if any one of the firms disagrees
to this point, it will create a series of actions and reactions
between the firms which will lead them back to point E.
Criticism of the Model:
Bertrand’s model has been criticised on the same grounds as
Cournot’s model. Bert- rand’s implicit behavioural assumption
that firms never learn from their past experience seems to be
unrealistic. If cost is assumed to be zero, price will fluctuate
between zero and the upper limit of the price, instead of
stabilizing at a point.

4. Edgeworth’s Duopoly Model:

Edgeworth developed his model of duopoly in 1897.
Edgeworth’s model follows Bertrand’s assumption that each
seller assumes his rival’s price, instead of his output, to remain
constant. His model is illustrated in Fig. 6.
In this figure we have supposed that there are two sellers, A and
B, in the market who face identical demand curves. A has his
demand curve DDA and as DDA Let us also assume that seller
A has a maximum capacity of output OM and B has a maximum
output capacity of OM’. The ordinate ODA measures the price.
To explain Edgeworth’s model, let us assume, to begin with,
that A is the only seller in the market. Following the profit
maximising rule of a monopoly seller, he sells OQ and charges a
price, OP2. His monopoly profit under zero cost, equals OP2EQ
Now, let B enter the market. B assumes that A will not change
his price since he is making maximum profit. He sets his price
slightly below A’s price (OP2) and is able to sell his total
output. At this price, he captures a substantial part of A’s
market.
Seller A, on the other hand, that his sales have gone down. In
order to regain his market, A sets his price slightly below B’s
price. This leads to price-war between the sellers.
The price- war takes the form of price-cutting which continues
until price reaches OP1 At this price both A and B are able to
sell their entire output- A sells OQ and B sells OQ’ The price
OP1 could therefore be expected to be stable. But, according to
Edgeworth, price OP1 should not be stable.
Simple reason is that, once price OP is set in the market, the
sellers observe an interesting fact. This is, each seller realise that
his rival is selling his entire output and he will therefore not
change his price, and each seller thinks that he can raise his
price to OP2 and can make pure profit.
This realisation forms the basis of their action and reaction. For
examples, let seller A take the initiative and raise his price to
OP2. Assuming A to retain his price OP2.B finds that if he
raises his price at a level slightly below OP2 he can sell his
entire output at a higher price and make greater profit.
Therefore, B raises his price according to his plan.
Now it is A’s turn to know the situation and react. A finds that
his price is higher than B’s price and his total sale has fallen.
Therefore assuming B to retain his price, A reduces his price
slightly below B’s price.
Thus, the price-war between A and B begins once again. This
process continues indefinitely and price keeps moving up and
down between OP1 and OP2 Obviously, according to
Edgeworth’s model of duopoly, equilibrium is unstable and
indeterminate since price and output are never determined. In
the words form Edgeworth, “there will be an indeterminate
tract through which the index of value will oscillate, or,
rather will vibrate irregularly for an indefinite length of
time”.
In a net shell Edgeworth’s model, like Cournot’s is based on a
naive assumption, i.e. each seller continues to assume that his
rival will never change his price even though they are proved
repeatedly wrong. But according to Hotelling Edgeworth’s
model is definitely an improvement upon Cournot’s model in
that it assumes price, rather than output, to be the relevant
decision variable for the sellers.

REF:
1. Economic Discussion
2. J.Sarkhel
3. F.Gold
4. Banerjee & Majumdar