2. OLIGOPOLY
• Oligopoly is a market structure characterized by a
small number of firms and a great deal of
interdependence, is perceived among them
• It is a market structure in which there are a few
large firms selling homogeneous product or a
market structure where a few large firms
dominate.
• The number of firm may be two or more, but not
more than twenty.
4. OLIGOPOLY
• In oligopoly, firms have the power to fix price and control
output, because of the large share of product they control.
• A change in one firm’s price or output influences the sales
and profits of other firms in the oligopoly market, therefore
firms in oligopoly cannot act independently
• Firms are always mindful of the reaction of other firms to
their actions or decisions, this creates uncertainty in
oligopoly
• Competition among firms in oligopolistic market is
inherently a setting of strategic interaction
Examples; Oil industries, banks, soft drinks, (coke and pepsi),
telecommunication companies, airline industry etc
5. OLIGOPOLY
• The two level of models used in oligopoly analysis
are Multi-period models and Static model
For now we will focus on static model
• The static model includes:
Cournot model
Stackelberg model
Bertrant model
Kinked model
6. COURNOT MODEL
• This was developed by a French economist called Augustine
Cournot in early 1800.
• The Cournot oligopoly model is the most popular model of
imperfect competition.
Assumptions of Cournot model
• In the Cournot model, there are two interdependence firms, who
simultaneously set quantities.
• The firms are producing homogeneous goods.
• The market price is set at a level such that demand equals the total
quantity produced by all firm
• The Cournot model assumes that each firm chooses quantity that
will maximize output, with the belief that the other firm will
continue to produce the same rate of output
7. Assumptions of Cournot model
• Each firm makes an output plan at the beginning of each period,
and it cannot revise the plan within the duration of the period.
• Price are determined by market force and each accepts the price
at which the total planned output can be sold.
• Each firm wants to maximize profit subject to the price and the
quantity produced by the rival firm
• Each firm assumes that, irrespective of his output plan in any
period, his rival will maintain his output at the same level as in
the previous period.
8. Cournot model
• To illustrates the Cournot model we assume at first that
there is only one firm (Monopolistic market) and then
we allow additional firms to enter
If we recall the demand function for a linear demand
curve is:
• Qd = a – b(P)
• Where Qd is quantity demanded,
• a the intercept on the Q axis. (The quantity demanded
when price is zero)
• b is a factor that show how quantity demanded will
change for every change in price.
• The negative sign is reflecting the law of demand, that
price is an inverse of quantity demanded
9. Cournot model
• In deriving the Cournot model, we take the demand function of the
linear demand curve to be
• X = a – a/b (Px)
X = x1 + x2
a = intercept
a/b = slope
Px = price
• Now suppose a second firm assumes that monopolist will produce (½)a
of the total output X, that is x1 = (½)a
• He automatically assume the remaining (½)a of the demand curve as his
share of the total output and behaves like a monopolist on that share
(½)a, that is x2 = (½)a
• Therefore the total output in the oligopolistic market well be
X = x1 + x2 = (½)a + (½)a
• These shares are known by the firms, and the shares are expected to be
unchanged in a production period
10. Cournot equilibrium
• In a Cournot equilibrium neither firm will find it
profitable to change its output once it discovers
the choice actually made by the other firm.
• Cournot equilibrium is simply the pair of
outputs at which the two reaction curves cross.
• At such a point, each firm is producing a profit-
maximizing level of output given the output
choice of the other firm.
11. Cournot equilibrium
• To find the Cournot equilibrium we assume a zero
marginal cost and linear demand function as
X =x1 + x2
Px = Price
We then derive firms reaction functions
• Firstly, we take the inverse of the demand function
• That is
13. Profit of Firm 1
• To solve for the profit of firm 1
• TR for firm 1 = Pxx1
Recall that
• Therefore firm 1 total revenue (TR) will be
• Since we assumed that MC is zero,
• Profit for firm 1 will be
14. Profit maximization for firm 1
• To solve for profit maximization for firm, we take the
first order derivative of the profit function with respect
to X1 and equating to 0
• That is
16. Reaction Function
• This is the reaction function for firm one
• It describes how much firm 1 will produce X1 as a function of
each given output by firm 2 (X2)
• To derive the reaction function for X2, we use the same
procedure
• Taking the first order derivative of firm 2’s profit in respect to
X2 and equating to zero, you solve for X2
17. Cournot equilibrium
• To solve for equilibrium of this process, we substitute
the reaction function of X1 into reaction function of X2
•
18. • Similarly, substituting X2 into X1 by following the same
procedure will yield
• Substituting X2 into X1 = ½(a – X2), we
X2
X1
1/3a
1/3a
C.E
CE (Cournot Nash
equilibrium), is the pair of
outputs at which the two
reaction curves cross. At the
equilibrium both firms produce
the same level of output and
none of them wants to
unilaterally deviate from this
output level
19. Worked example
i. Suppose an industry is faced with a demand function, q = 1000 – Pq.
Each firm has a constant marginal cost equal to MC = MK600. Find
the Cournot reaction function and equilibrium for two firms.
Solution
Demand function is q = 1000 – Pq, the inverse demand function is Pq =
1000 – q
• q = q1 + q2 Since we have two firms
• Therefore
• Pq = 1000 – (q1 +q2)
• Removing the bracket we have
• Pq = 1000 – q1 – q2
• Recall that TR = Price X Quantity = Pq X q
• Therefore TR = (1000 – q1 – q2)q
• To solve for quantity that maximises profit we equate MR = MC
20. Worked example
For firm 1
TR = (1000 – q1 - q2)q1
TR = 1000q1 - q1
2 - q1q2
• we equate TR to zero and differentiate in respect to q1
MR = δTR/δq1= 1000 – 2q1 – q
MC = MK600
• To solve for Profit maximization of firm 1
MR = MC
That is
1000 – 2q1 – q2 = 600
Solving for q1
q1 = (400 -q2)/2 = 200 – 1/2q2 (Firm 1’s reaction function)
Similarly to solve for firm 2 reaction function
• We equate MR = MC
21. Worked example
TR = 1000q2 - q1q2
2 - q2
2
MR =δTR2/δq2= 1000 – q1 - 2q2= 0
MC = 600
Equate MR = MC
1000 – q1 - 2q2 = 600
• Solve for q2 to get firm 2 reaction function, we get
2q2 = 1000 – 600 –q1
q2 = 200 – 1/2q1
• To solve for cournot equilibrium we substitute the reaction
function for firm 2 into that of firm 1
That is,
substitute q2 = 200 – 1/2q1 into q1 = 200 – ½( q2), we get
q1 = 200 – ½(200 -1/2q1)
q1 = 200 – 100 +1/4q1
q1 -1/4q1 = 100
3/4q1 = 100
q1 = 100*4/3 = 133.33
22. Worked example
• Similarly, if we substitute the reaction function for firm 1 into
that of firm 2, that is substitute q1 = 200 – 1/2q2 into q2 = 200
– ½( q1), we get
q2 = 200 – ½(200- 1/2q2)
q2 = 200 – 100 +1/4q2
q2 -1/4q2 = 100
3/4q2 = 100
q2 = 100*4/3 = 133.33
• This means that cournot equilibrium is attained at output q1 =
q2 = 133.33
We can solve for price
Pq = 1000 – (q1 +q2)
output q1 = q2 = 133.33
Pq = 1000 – (133.33 + 133.33)
Pq = 733.34
• Equilibrium price is MK733.34
23. Worked example
ii. In a Cournot duopoly firms 1 and 2 have a constant marginal costs c1
and c2 respectively. Inverse demand is linear, given by p = 1 – Q. show
that quantities in Cournot equilibrium are given by
• q1 = (1-2c + c2)/3 and q2 = (1 – 2c2 + c1)/3
Solution
• Recall that to solve for profit
• (π) = Pq - cq
• Now, p = 1 – Q and Q = q1 + q2,
• Therefore P = 1 – q1 –q2
• cost for firm 1 and firm 2 = c1 and c2 respectively
• Profit for firm one will be
• π1 = (1 – q1 – q2 )q1 – c1q1
• π1 = q1 – q12 – q1q2 – c1q1
• δπ1/δq1 = 1 – 2q1 –q2 –c1 = 0
• 1 – 2q1 –q2 –c1 = 0
• 2q1 = 1 –q2 –c1
25. Worked example
To solve for q2 we substitute q1 = (1 – 2c1 + c2)/ 3 into
q2 = (1 –q1 –c2)/2
q2 = (1 –q1 –c2)/2
2q2 = 1 – q1 – c2
2q2 = 1 – (1 – 2c1 + c2)/ 3 – c2
Multiply both sides by 3
6q2 = 3 – 1+ 2c1 – C2 – 3c2
6q2 = 2 + 2C1 – 4C2
Divide both sides by 6
q2 = (2 + 2C1 – 4C2)/6
q2 = 2(1 + C1 – 2C2)/6
q2 = (1 – 2C2 + C1)/3
26. STACKELBERG MODEL
• The Stackelberg assumes that one firm is a quantity
leader, while the other is the follower
• It is a dominant firm model or a leader – follower model
• Unlike the Cournot model where firm chooses output
simultaneously, firms chooses output level sequentially
in the Stackelberg model.
• The dominant firm which is seen as the leader chooses
its output level and the follower after observing the
leader’s output level choose their own output level that
will give the best response functions.
27. STACKELBERG MODEL
• The main difference between cournot and
stackelberg model is that unlike the cournot where
there is equal share of output, in stackelberg model
the leader may have a higher level of output.
• The leader produce a high level of output while the
follower’s best response is to produce a low level
of output
• The market leader may be a large firm or may have
a better information than other firms.
28. STACKELBERG MODEL
• The leader chooses its output q1 before the follower
chooses it output level q2.
• By the Stackelberg assumption the leader recognizes that
his follower acts on the cournot assumption.
• Given the committed production level q1 for firm 1, firm
2 will select q2 using the same reaction function as in
cournot, firm 2 finds q2 to maximize profit
• Recall that profit is
• TR - TC
• Inverse demand is given as
• P(q1 + q2)
• π = P(q1 + q2)q1 – Cq1
29. STACKELBERG MODEL
• For firm 2 to maximize profit we have
Profit 2 = P(q1 + q2)q2 – C2q2
• P(q1 + q2) is the inverse demand function that will
yield the reaction function Q2(q1) for firm 2.
• Firm 1 which is the leader accounts for firm 2 reaction,
since firm 1 chooses its output first and firm 2 reacts to
firm 1 output level.
• It is assumed that firm 1 knows what Firm 2 reaction
will be, which is Q2(q1), so for firm 1 to maximize
profit, it substitute firm 2 reaction function into its
profit expression
30. STACKELBERG MODEL
Recall that
Profit expression for q1 is
Profit 1 = P(q1 + q2)q1 – C2q1
• We substitute firm 2 reaction function [Q2(q1)] into firm
1’s (leader) profit expression
That is,
• Profit 1 = P(q1 + Q2q1)q1 – C2q1
• In stackelberg model, it is assumed that one duopolist
recognizes that the rival acts on the Cournot assumption.
• This sophisticated duopolist can then anticipate the
reaction curve of the rival and take into account in
seeking its own profit-maximazing strategy.
31. STACKELBERG’S EDGEWORTH CONTRACT
CURVE
• At any point on the Edgeworth contract curve, at least one firm or both
firms or both firms will be better off if they enter into some form of
agreement or contract than they would otherwise be in cournot
equilibrium.
• The model shows that a bargaining procedure and a collusive agreement
becomes advantageous to both duopolists.
• With such a collusive agreement the duopolists may reach a point on the
Edge-worth contract curve, thus attaining joint profit maximisation.
• The firms recognizes their interdependence as well as each other
reaction, by so doing each duopolist can reach a higher level of profit
32. STACKELBERG MODEL
• If only one firm is sophisticated, it will emerge as the leader, and a stable
equilibrium will emerge, since the naive firm will act as a follower.
• If both firms are sophisticated, then both will want to act as leaders,
because this action yields a greater profit to them, this will lead to
Stackelberg’s disequilibrium and the effect will either be a price war until
one of the firms surrenders and agrees to act as follower, or a collusion is
reached, with both firms abandoning their naive reaction functions and
moving to a point closer to (or on) the Edge-worth contract curve with
both of them attaining higher profits.
• If the final equilibrium lies on the Edge-worth contract curve the industry
profits (joint profits) are maximised
• This shows clearly that naive behaviour does not pay. By recognizing the
other’s reactions each duopolist can reach a higher level of profit for
himself
• The rivals should recognize their interdependence, If each ignores the
other, a price war will be inevitable, as a result of which both will be
worse off.
33. Possible outcomes in Stackelberg model
1. Firm 1 wants to be leader and firm 2 wants to be follower.
2. Firm 2 wants to be leader and firm 1 wants to be follower.
3. Both firms want to be followers
4. Both firms desire to be leaders
• In outcome 1 & 2 it is possible to reach a stable equilibrium
• Outcome 3 can lead to Cournot equilibrium
• Outcome 4 will lead to Stackelberg disequilibrium
• Equilibrium will be reached either by collusion, or after the
‘weaker’ firm is eliminated or succumbs to the leadership of the
other
34. Worked example
• Let p = 1000 - Q q = q1 +q2
• Therfore
• P = 1000 – (q1 + q2)
• P = 1000 – q1 – q2
• MC = MK600
• We assume that firm 1 considers how firm2 will react, that is he
already knows its reaction function. From the first example in
cournot model firm 2 reaction function was q2 = 200 – 1/2q1, so we
assume that firm 1 already knows this
• Therefore to calculate the TR for firm 1 we first solve for P using
firm 2 reaction function, that is;
• P = 1000 – q1 – q2
•
• We substitute q2 = 200 – 1/2q1 in the above equation, we have
• P = 1000 – q1 – (200 – 1/2q1)
35. Worked example
P = 800 – q1 + 1/2q1
P = 800 -1/2q1
TR = Pq
TR for firm 1 = (800 -1/2q1)q1
TR = 800q1 – 1/2q12
δTR1/δq1 = MR = 800 – q1
MC = MK600
To solve for profit maximization quantity of firm 1 we equate MR to MC
MR = MC
800 – q1 = 600
q1 = 200
Substitute q1 into firm 2 reaction function q2 = 200 – 1/2q1
q2 = 200 – ½(200)
q2 = 200 – 100
q2 = 100
So
q1 = 200 and q2 = 100
36. Worked example
Therefore to solve for P
P = 1000 – (q1 +q2)
q1 = 200 and q2 = 100
P = 1000 – 200 + 100)
P = 1000 – 300
P = MK700
Now we know P, q1 and q2 we can clearly illustrate the profit of each firm
Recall profit = TR – TC
For firm 1
Profit = Pq1 –cq1
P = 700 and C = 600, q1 = 200
Therefore,
= (MK700)(200) – MK600(200)
= MK140,000 – MK120,000
= MK20,000 (Firm 1’s profit)
Profit for firm 2 = Mk700) (100) – MK600 (100)
= MK70000 – MK60000
= MK10,000 (Firm 2’s profit)
38. Worked Example
Suppose an industry is faced with a demand function, q= 1000 – P. each
firm has a constant marginal cost equal to MC = MK600. using the
Cournot reaction function in example one, find the Stackelberg equilibrium
quantities and price for two firms.
Solution
P = 1000 – (q1 +q2) = 1000 –q1 – q2
Recall from the Cournot reaction function that
q1 = 200 – 1/2q2
q2 = 200 – 1/2q1
For firm 1, profit will be
TR – TC = Pq – Cq
Profit = [1000 – q1 – q2]q1 – 600q1
= 1000q1 – q12 - q1q2 - 600q1
Recall that q2 = 200 – 1/2q1, substitute q2 into q1 profit equation
We have,
Profit = 1000q1 – q12 - q1(200 – 1/2q1) - 600q1
π1 = 1000q1 –q1 – 200q1 – 1/2q12 – q1
39. Worked example
To solve for profit maximization we take the first order
condition of the profit equation
π1 =1000q1 –q1 – 200q1 – 1/2q12 – q1
Therefore,
δ π1 /δq1 = 1000 – 200 – q1 – 600 = 0
q1 = 200
In a similar way you solve for Firm 2 profit
π2 =( 1000 – q2 –q1)q2 – 600q2
π2= 1000q2 – q22 – q1q2 – 600q2
Reaction function of firm 1 = 200 – 1/2q2
Substitute firm 1 reaction function into the profit equation we
have
π2 = 1000q2 – q22 – (200 – 1/2q2)q2 – 600q2
π2 = 1000q2 – q22 – 200q2 – 1/2q22 – 600q2
40. Worked example
Taking the first order condition for firm 2’s profit
maximization we have
δ π2 /δq2 = 1000 -2q2 – 200 – q2 – 600 = 0
q2 = 200
To solve for price
Recall that
Price = 1000 – q
q1 = 200 and q2 = 200
P= 1000 – (q1 + q2)
P = 1000 – q1 – q2
P = 1000 – 200 – 200
P = 600
41. Worked example
In a stackelberg duopoly, an industry is faced with a market
demand function, p = 100 – (q1 + q2) firm cost are c1 = 10q1
and c2 = q22
• Calculate the market price and each firm’s profit assuming that firm 1
is the leader and firm 2 is the follower
• Calculate the market price and each firm’s profit assuming that firm 2
is the leader and firm 1 is the follower
• Calculate the market price and each firm’s profit if both firm assume
they are leaders
• Find the Cournot Nash equilibrium
Solution
i. Firm 1 is the leader and firm two is the follower
• We first look for reaction function of firm 2, so we can substitute it into
the profit maximization equation of firm 1
• To solve for firm’s 2 reaction we solve for total revenue
42. Worked example
Total revenue for firm 2 (TR2):
TR2 = p.q2
Price = 100 – q
= 1000 – (q1 + q2)
TR2 = (100 – q1 – q2)q2
1000q2 – q1q2 – q22
MR for firm 2 will be
MR = δTR2/δq2 = 100 – q1 – 2q2
TC for firm 2 was given as q22
MC = δTC2/δq2 = 2q2
For profit maximization
MR = MC
100 – q1 – 2q2 = 2q2
From this equation firms 2 reaction function will be
100 – q1 = 4q2
q2 = 25 – q1/4
43. Worked example
For firm 1, the total revenue is
TR1 = p.q1
= [100 – (q1 + q2)] q1
{100 – (q1 + q2)}q1
100q1 – q12 + q1q2
Substituting reaction function for firm 2 (q2 = 25 – q1/4) into TR1 yields
TR1 = 75q1 – ¾ q12
MR = 75 – 3/2q1
TC is given as 10q1
MC = 10
Equating marginal revenue for firm 1 to its marginal cost yields
MR1 = MC1
75 – 3/2q1 = 10
q1 = 130/3,
we solve for q2
q2 = 25 – q1/4
q2 = 25 – q1/4 = 25 – (130/3)/4
q2 = 85/6
44. Worked example
The price is
P = 100 – (q1 + q2)
P = 100 – (130/3 + 85/6)
= 255/6
The profit is
π1 = pq1 – c1
P = 255/6
q1 = 130/3
C1 = 10q1
For firm 1: Profit = (255/6 X 130/3) – (1300/3)
= 4225/3 = 1408.33
For firm 2:
P = 255/6
Q2 = 85/6
C2 = q22
Profit (π2) = 255/6 X 85/6 – 7225/36 = 14450/36 = 401.39
45. Worked example
ii. When firm 2 is the leader and firm 1 is the follower
We solve for reaction fuction of firm 1 and substitute into the
profit maximization equation of firm 2
Total Revenue for firm 1 is
P.q1
P = 100 – (q1 + q2) = 100 – q1 – q2
TR1 = p.q1 = 100q1 – q12 q1q2
Marginal Revenue for firm 1 is
MR1 = 100 – 2q1 – q2
Equating MR = MC for firm 1 yields
C1 = 10q1
MC = 10
Therefore MR = MC is
100 – 2q1 – q2 = 10
Solving for q1
q1 = 45 – 1/2q2
47. Worked example
Solving for q1 using Q1 = 45 – 1/2q2 and q2 = 55/3 yields
q1 = 45 – ½(55/3)
= 215/6
The price charged is
P = 100 – q1 – q2
P= 100– 215/6– 55/3
P= 275/6
The profit for firm 1 is
Recall
• Profit 1 = 275/6 X 215/6 – 2150/6 = 46,225/36 = 1284.03
• For firm 2
• Profit 2 = 275/6 X 55/3 – (110/6)2 = 18,150/36 = 504.17
48. Worked example
iii. If both firm assume to be leaders
Recall their outputs for firm 1 and firm 2 are;
q1 = 260/6 and q2 = 55/3 or 110/6
• The price will be
• P = 100 – q1 – q2
• = 100 – 260/6 – 110/6 = 230/6
• Now, the profit realized by each firm is
• Profit for firm 1
• 230/6 X 260/6 – 10(260/6) = 44200/36 = 1227.78
• For firm 2
• 230/6 X 110/6 – (110/6)2 = 13200/36 = 366.67
49. Worked example
iv. Cournot Nash equilibrium
For firm 1, revenue is
TR1 = 100q1 – q12 – q1q2
MR = 100 – 2q1 –q2
C = 10q1
MC = 10
MR = MC
100 – 2q1 –q2 = 10
100 – 10 –q2 = 2q1
q1 = (90 –q2)/2
q1 = 45 –q2/2 (Reaction function for firm 1)
For firm 2 revenue is
R2 = 100q2 –q1q2 – q22 and marginal revenue is
MR2 = 100 – q1 – 2q2
C = q22
MC = 2q2
MR = MC
100 – q1 – 2q2 = 2q2
100 – q1 = 2q2 + 2q2
4q2 + q1 = 100
50. Worked example
Substituting the reaction function reaction for firm 1 yields
4q2 + (45 –q2/2) = 100
4q2 + 45 – q2/2 = 100
4q2 – 1/2q2 = 55
7/2q2 = 55
q2 = 110/7
We then solve for q1
q1 = 45 –q2/2
q1 = 45 – (110/7)/2
q1 = 45 – 110/14
q1 = 520/14 = 260/7
The price will be
P = 100 – q1 – q2
q1 = 260/7 and q2 = 110/7
= 100 – 260/7 – 110/7
= 330/7
51. Worked example
The profit will be
P = 330/7
q1 = 260/7
C = 10q
For firm 1
330/7 X 260/7 – 10(260/7)
= 1379.59 = 1380
For firm 2
• C = q22
• P = 330/7
• q2 = 110/7
•
• Profit = 330/7 X 110/7 – (110/7)2
• = 493.88 = 494
