Cournot (1838) oligopoly
1
The basic model
The smallest possible step away from one supplier in a market is two suppliers
in a market.
Cournot analyzed two producers of a standardized product – mineral
water drawn from a common underground source.
In this model
• each firm knows the market demand curve
• the two firms have identical costs
• each firm knows that the other firm knows as much about the market
as it does.
• Each firm picks its own output to maximize its own profit, knowing that
the other firm acts in the same way and with the same information.
What outputs will the firms produce, and at what price will the product
sell?
Here we confront the notion of an equilibrium.
For those of you who saw the movie. . . Cournot’s equilibrium concept was
an anticipation of that of John Nash, and is often referred to in economics
as Cournot-Nash equilibrium.
Leaving motivation aside, what we require of an equilibrium pair of outputs is that each firm’s equilibrium output maximize its profit, given the
equilibrium output of the other firm.
For such an output pair, each firm is making as large a profit as it possibly
can, given what the other firm does.
In view of Cournot’s assumption that each firm seeks to maximize its own
profit, neither firm would wish to alter its own part of such an output pair.
We begin the task of finding equilibrium outputs by characterizing the
output that will maximize a firm’s profit for an arbitrary output level of the
other firm.
The schedule of all such output pairs is called the firm’s best response
function. We then look for mutually consistent output levels on the best
response functions of the two firms.
1
1.1
Best response functions
Use a specific example to illustrate the Cournot model. The equation of the
market inverse demand curve is
p = 100 − Q = 100 − (q1 + q2 ),
(1)
where q1 is the output of firm 1 and q2 is the output of firm 2.
Firms have identical cost functions, with constant average and marginal
cost, 10 per unit of output:
C(q1 ) = 10q1
C(q2 ) = 10q2 .
(2)
For simplicity, assume that there are no fixed costs.
If firm 2 produces an arbitrary output level q2 , the relation between firm
1’s output level q1 and the market-clearing price p is
p = (100 − q2 ) − q1 .
(3)
(3) is the equation of firm 1’s residual demand function, so-called because
it gives the relation between the quantity supplied by firm 1 and price in the
part of market left for firm 1 after firm 2 has disposed of its output.
In this left-over part of the market, firm 1 is a monopolist, or at least,
it acts as a monopolist, since firm 2’s output is assumed to be fixed at the
arbitrary level q2 . The output that maximizes a monopolist’s profit is that
which makes its marginal revenue equal to its marginal cost.
For a linear demand curve, the marginal revenue curve has the same
price axis intercept as the demand curve and a slope that is twice as great
in absolute value as the slope of the demand curve. The equation of firm 1’s
residual marginal revenue function is therefore
MR1 = (100 − q2 ) − 2q1 .
(4)
Firm 1’s profit-maximizing output makes its marginal revenue equal to
its marginal cost,
MR1 = (100 − q2 ) − 2q1 = 10 = MC.
This is shown in Figure 1, which is drawn for q2 = 30.
The equation
MR1 = (100 − q2 ) − 2q1 = 10 = MC.
2
(5)
p
Market demand
curve
100 .......
.....
..
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.
.....
...
.....
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.....
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.....
...
..... ........
..... ......
.....
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.....
.
1’s residual
demand curve
.....
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70 .......
..
.....
.......
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.
..... ....
... .....
..... ....
... ......
..
... .....
... ........
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... .....
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..... .......
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..... ... 1’s residual marginal
........
...
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revenue curve
...
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40
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Marginal .....
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cost .... .....
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curve .... ... .....
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10
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Q
{z
}
|
30 35
70
q2 = 30 100
Figure 1: Firm 1’s output decision, q2 = 30.
3
can be rewritten as
1
1
q1 = (90 − q2 ) = 45 − q2 .
(6)
2
2
Equation (6) is the equation of firm 1’s best response function for this example: it gives the profit-maximizing output of firm 1 for any output level of
firm 2.
To understand the shape of the best response curve, note that if firm 2
produces nothing, firm 1 is a monopolist, and maximizes profit by producing
the monopoly output. To make firm 1 choose to produce nothing, firm 2
would need to produce the output level that makes price equal to marginal
cost. Any sales by firm 1 would then push price below marginal cost and
mean losses for both firms; firm 1 would maximize profit by selling nothing.
Firm 1’s best response curve thus includes the points (qm , 0) on the horizontal
axis in Figure 2 and the point (0, Qc ) on the vertical axis in Figure 2. When
the inverse demand curve is linear and marginal cost is constant, the best
response curve is a straight line (this is clear from equation (6)), and it can
be drawn by connecting the two points (qm , 0) and (0, Qc ).
Going through the same procedure for firm 2, we obtain the equation of
firm 2’s best response function for this example,
1
q2 = 45 − q1 .
2
(7)
Firm 2’s best response curve is also graphed in Figure 2.
Cournot equilibrium
When firms are producing their equilibrium outputs, each firm is maximizing its profit, given the equilibrium output of the other firm.
In terms of the best response curve diagram Figure 2, the equilibrium
outputs are found at the intersection of the best response curves.
For this combination of outputs – and only at this combination of outputs – each firm is maximizing its own profit, given the output produced
by the other firm.
Analytically, the values of the Cournot equilibrium outputs are found by
solving the equations of the best response functions, here
1
q1 = 45 − q2
2
4
q2
90 ....
...
...
...
1’s best
response curve
...
....
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..
...
....
...
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... ......
... ......
...
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...
45 ..........
...
.........
2’s best
response curve
.........
...
......... .....
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......... ...
..
........... E
...
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............
..
30
... ........... .........
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... .........
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...
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.........
...
.........
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.........
.........
...
.........
...
.........
...
........
.
30
45
90
q1
Figure 2: Best response curves, Cournot duopoly.
5
1
q2 = 45 − q1
2
Since this example is symmetric, in the sense that the two firms have
identical cost functions and identical beliefs each about the other, the firms
will produce identical output levels in equilibrium.
Call this common equilibrium output qCour and set q1 = q2 = qCour in (6)
This allows us to find the Cournot equilibrium output per firm:
2qCour + qCour = 3qCour = 90,
(8)
qCour = 30.
1.2
Other characteristics of Cournot equilibrium
Other aspects of Cournot equilibrium that are of interest, in addition to the
outputs of the individual firms, are
•
total output
•
economic profit
•
price
•
consumers’ surplus
•
the degree of market power
•
deadweight welfare loss
These can all be determined in a straightforward way from the equilibrium
outputs.
Adding the outputs of the two firms gives total output:
QCour = 2qCour = 60.
(9)
Cournot equilibrium output is greater than monopoly output (45), but
less than long-run competitive equilibrium output (90).
From the equation of the inverse demand curve, the Cournot equilibrium
price is
pCour = 100 − 60 = 40 = 10 + 30.
(10)
This is greater than marginal cost (10), but less than the monopoly price
(55).
In Cournot equilibrium, the Lerner index of market power is
pCour − c
3
40 − 10
= .
=
pCour
40
4
6
(11)
Profit per firm is
π Cour = (pCour − 10)qCour = (30)(30) = 900.
(12)
Since there are two firms, total economic profit is twice πCour :
2π Cour = 1800.
(13)
Consumers’ surplus (CS ) is the area of the triangle the sides of which are
formed by the demand curve, the horizontal line pCour = 40, and the price
axis (see Figure 3). This area is
1
1
CS = (100 − 40)(60) = (60)2 = 1800.
2
2
(14)
Deadweight welfare loss (DWL) is the area of the triangle with sides
formed by the demand curve, the marginal and average cost line, and the
vertical line QCour = 60. This area of this triangle is
1
1
DW L = (40 − 10)(90 − 60) = (30)2 = 450.
2
2
2
(15)
Many firms
The general Cournot model – n firms rather than 2 – cannot be illustrated
graphically. Particularly in the symmetric firm case, however, the generalization from 2 to n firms is straightforward.
If there are n identical Cournot firms in the industry, write
Q−1 = q2 + ... + qn
(16)
for the combined output of all firms except firm 1.
Then we can write the equation of firm 1’s residual demand curve as
p = (100 − Q−1 ) − q1 .
(17)
This looks very much like the equation of firm 1’s residual demand curve
for the duopoly case, (3); the aggregate output of all other firms has been
substituted for the output of firm 2.
7
p
Demand
curve
100 .......
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Consumer
.....
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surplus
.....
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40
.....
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.....
.....
ΠCour
DWL ........
.....
.....
.....
10 ..........................................................................................................................................................................................................................................
.....
.....
....
Q
60
90 100
Figure 3: Market equilibrium, Cournot duopoly.
8
Proceeding in the same way as the duopoly case, we can find the equation
of firm 1’s best response curve by setting its marginal revenue along the
residual demand curve (17) equal to its marginal cost:
100 − Q−1 − 2q1 = 10,
(18)
2q1 + Q−1 = 90.
(19)
or
In the symmetric firm case, all firms will produce the same output in
equilibrium. (19) becomes
2qCour + (n − 1)qCour = 90,
so that
qCour =
90
.
n+1
(20)
(21)
If n = 2, (21) reduces to (8).
‘From the equilibrium output of a single firm, we can work out all the
other characteristics of n-firm equilibrium. Here we note two of these, total
output and price, which are
µ
¶
1
n
QCour =
90 = 1 −
90
(22)
n+1
n+1
and
pCour = 10 +
90
n+1
(23)
respectively.
If n = 1, these are the monopoly output and price. As n increases,
Cournot equilibrium output increases toward the long-run competitive equilibrium output level, and Cournot equilibrium price approaches marginal
cost. The symmetric Cournot model predicts that market performance will
improve as the number of firms increases, and in fact that market performance will approach that of long-run competitive equilibrium as the number
of firms approaches infinity.
9
3
Equilibrium market structure: Cournot oligopoly
We consider the case of a Cournot oligopoly with a linear demand curve:
p = a − bQ,
(24)
Suppose every firm operates with same cost function
c(q) = F + cq.
(25)
Fixed cost is F , marginal cost and average variable cost are c per unit.
We will have more to say about the determinants of fixed and variable cost.
The residual marginal revenue curve facing a single firm (say, firm 1) is
MR1 = a − b (Q−1 + 2q1 )
(26)
The firm maximizes profit by producing where marginal revenue equals
marginal cost:
a − b (Q−1 + 2q1 ) = c.
(27)
If we were to solve this for q1 , we would have the equation of firm 1’s best
response function.
Since all firms have the same knowledge about the market demand curve,
the same knowledge about the behavior of other firms (that they too maximize profit), and the same cost function, in equilibrium all firms will product
the same output.
Call this output qCour . Substitute in (27) and rearrange terms:
a − b [(n − 1) qCour + 2qCour ] = c
1 a−c
.
n+1 b
In equilibrium, the profit of any one firm is
qCour =
π Cour = (pCour − c) qCour − F.
From the equation of (say) firm 1’s residual marginal revenue curve
MR1 = a − b (Q−1 + 2q1 )
= a − b (Q−1 + q1 ) − bq1
10
(28)
(29)
= p − bq1 .
(30)
Then when marginal revenue equals marginal cost
p − bq1 = c
so that
p − c = bq1 ,
and in equilibrium
pCour − c = bqCour .
(31)
Substituting back in the expression for equilibrium profit, (29), and then
substituting from (28) to express qCour in terms of the demand curve parameters aand b and marginal cost c, we obtain
πCour = (pCour − c) qCour − F
2
−F
= bqCour
µ
¶2
1 a−c
=b
− F.
n+1 b
(32)
Suppose there is free and easy entry and exit.
There are (at least) to assumptions buried in this supposition.
The first is that no investments are sunk. We will see shortly what
happens if this assumption is abandoned.
The other assumption is that incumbents (firms in the market) cannot
or do not find it profitable to engage in strategic entry-deterring behavior.
Later in the course we will talk about the economics of and EU competition
policy toward such behavior.
If there is free and easy entry and exit, the number of firms will adjust
until equilibrium profit is zero.
This long-run Cournot equilibrium number of firms is
µ
¶2
1 a−c
b
−F =0
n+1 b
µ
¶2
1 a−c
F
=
n+1 b
b
r
1 a−c
F
=
n+1 b
b
11
a−c
n + 1 = qb
F
b
a−c
nCour = qb
F
b
− 1.
(33)
a−c
nCour = qb
F
b
−1
is one way to measure market size: it
To interpret this, notice that a−c
b
is long-run competitive equilibrium output, the quantity demanded if price
equals marginal cost. F/b is fixed cost normalized by the slope of the inverse
demand curve. What (33) says is that the equilibrium number of firms in an
imperfectly competitive Cournot market is larger, the larger is the market
and the smaller is fixed cost.
The expression given by (33) will not be an integer except by coincidence
(or by design, as for example on an exam question).1
The rest of the analysis: having worked out the equilibrium number of
firms, one can work out
³q ´
F
• long-run Cournot equilibrium output per firm
and long-run
b
q ´
³
equilibrium total output a−c
− Fb
b
³
√ ´
• long-run Cournot equilibrium price c + bF
• long-run Cournot equilibrium consumers’ surplus
These are important for a full analysis, but would take us away from the
immediate topic, which is the impact of market integration on equilibrium
market structure.
1
Hint.
12
4
Market integration and the equilibrium number of firms
Suppose there are two markets (countries), 1 and 2, with demand described
by the equations
pi = a − bqi ,
for i = 1, 2.
Then we know that the long-run equilibrium number of firms in each
market is
a−c
nCour = qb
F
b
− 1.
What happens if the two markets integrate and form a single market?
One might turn here to philosophical questions about what exactly it
is that market integration means. Since economists are not particularly
gifted in matters of philosophy, we take a prosaic approach and suppose
that “market integration” means “firms must charge the same price in both
formerly distinct markets.”
Then the quantity demanded at any price is
a−p
a−p a−p
+
=2
.
(34)
b
b
b
The equation of the inverse demand curve for the integrated market is
Q=
1
bQ = a − p
2
1
p = a − bQ.
2
Going through all the previous steps to find the long-run equilibrium
number of firms,2 after the markets are integrated,
nInt
Cour
nInt
Cour
a−c
b/2
=q
−1
√
2 a−c
+ 1 = √ qb = 2 (nCour + 1) .
2 F
b
2
F
b/2
You should actually do this.
13
If nCour is large, this is awfully close to "The equilibrium number of firms
in the integrated market is 1.4 times the equilibrium number of firms in each
of the pre-integration markets.”
Unfortunately for simplicity, it is the cases when nCour is small that are
of interest as far as the impact of integration are concerned: if nCour is large,
then equilibrium market performance is close to that of perfect competition
even before integration.
nCour
1
2
3
4
5
Int
n
2nCour − nInt
Cour
√Cour
2
(1
+
1)
−
1
=
1.
828
4
2
−
1
=
1
√
4−3=1
√2 (2 + 1) − 1 = 3. 242 6
6−4=2
√2 (3 + 1) − 1 = 4. 656 9
8−6=2
√2 (4 + 1) − 1 = 6. 071 1
2 (5 + 1) − 1 = 7. 485 3
10 − 7 = 3
Table 1: Pre-and post-integration Cournot equilibrium number of firms and
equilibrium number of firms that go out of business after integration.
Down the cost curve with gun and camera
Now we examine the kind of technology that might produce a cost function of the form (25)
c(q) = F + cq.
We make the assumption that a firm’s investment in capital goods is not
sunk: a firm can buy or sell a unit of physical capital for the same price.
After working out the implications of this assumption, we talk about the
differences that arise if a unit of physical capital, once purchased, cannot be
sold.
4.1
No sunk costs
Let the technology be described as follows:
• if a firm produces at all, it must have 1 manager, who is paid a wage
wm per time period.
• for every unit of output that the firm produces, it must hire aL workers
and use the services of aK units of capital.
14
This is called a Leontief or fixed-coefficient production function; for a
firm that has a manager, the relation between inputs (labor and capital) and
output is
µ
¶
K L
q = min
.
(35)
,
aK aL
Call the wage rate of labor w and suppose that unit of capital can be
bought or sold at a price pk .
Suppose also that capital wears out over time, and that the physical rate
of depreciation is δ per time period.
1
Think of δ as being small: if a machine lasts 10 years, δ = 10
or 10 per
1
cent. If a building lasts 50 years, δ = 50
or 2 per cent.3
It is natural to think of a firm as purchasing capital goods and using them
over time.
But if investment in capital goods are not sunk, then in an opportunity
cost sense, a firm that inherits a stock of capital goods from the past and
uses the capital for production is just the same as a firm that purchases a
new unit of capital and uses it for production.
A firm that purchases one unit of capital gives up pk at the start of the
period, is able to use the unit of capital during the period, and has 1 − δ
units of capital at the end of the period. At the end of the period, it can
either sell the capital for (1 − δ) pk or carry it forward into the future.
In an opportunity cost sense, a firm that inherits one unit of from the
past is in precisely the same situation. If it keeps the unit of capital and
uses it for production, then it does not sell the unit of capital at the start of
the period at a price pk , which it could do if it wanted to, since capital can
(by assumption) be bought or sold at the same price.
With this technology, what is the present discounted value of the income
stream of a firm that produces q units of output every period?4
Call this value Vq .
To produce q units of output, the firm needs the services of aK q units of
capital. It may well be that the firm inherits (1 − δ) aK q units of capital
from the past, so needs to make an out-of-pocket expenditure of δpk aK q to
3
Values of δ near one would mean that capital goods last just a little bit more than one
period, so that a firm must very nearly replace its entire capital stock every period. If a
firm is very nearly replacing its entire capital stock every period, it does not make much
difference whether or not the firm could sell off capital, should it wish to do so.
4
The assumption that output is the same in every period is made for simplicity only;
it is not essential for the result.
15
buy replacement capital to make up for depreciation, but if it refrains from
selling the capital inherited from the past, the opportunity cost of having the
services of the aK q units of capital on call is the full purchase price, −pk aK q.
At the end of the period, the firm sells its output, pays workers and the
manager, and has left over (1 − δ) aK q units of capital, which it can sell at
price pk or hold for use in the future; the opportunity cost of either choice is
the same.
If the firm produces q in every future period, then its value at the end
of the first period is the same as its value at the beginning of the period;
the contribution of income earned in the second to beginning-of-period value
must be must be discounted by 1 + r:
Vq = −pk aK q +
Collect terms:
Vq
pq − waL q + (1 − δ) pk aK q − wm
+
.
1+r
1+r
µ
1−
1
1+r
¶
(36)
Vq =
pq − waL q + (1 − δ) pk aK q − (1 + r) pk aK q − wm
1+r
r
pq − waL q − (r + δ) pk aK q − wm
Vq =
1+r
1+r
¤
£
pq − waL + (r + δ) pk aK q − wm
Vq =
(37)
r
Here (r + δ) pk is the rental cost of using the services of a unit of capital for
one period.
If we write
F = wm
(38)
and
c = waL + (r + δ) pk aK ,
(39)
then the value of the firm, the present-discounted value of its income stream,
is
pq − cq − F
.
(40)
Vq =
r
When investment in capital goods is not sunk, the fixed-coefficient technology (35), with the services of a manager required, yields a cost function
with fixed cost and constant marginal cost.
16
For completeness, we should make a remark here about the difference between accounting and economic concepts of cost and profit. In the expression
for the rental cost of capital services
(r + δ) pk ,
the first term,
rpk ,
is the normal rate of return on investment. From an economic point of
view, this in an opportunity cost of investing in the capital goods that are
needed to produce. Provided the accountant allows for depreciation that
that correct rate (δ), the accountant, in contrast, would record
rpk aK
as profit.
In practice, the difference between accounting and economic profit is more
complex: accounting depreciation is not the same as economic depreciation,
there are often special provisions of the tax code (for example, an investment
tax credit) that enter into the picture. All of these factors can be modelled
without changing the general nature of the results of the analysis that is
undertaken here.
4.2
Implications for market integration
If we consider the integration of identical markets with inverse demand curve
equations
pi = a − bqi ,
then the transition from pre-integration equilibria with
a−c
nCour = qb
F
b
− 1.
firms in each market to the pre-integration equilibrium with a total of
nInt
Cour
a−c
b/2
=q
F
b/2
− 1 < 2nCour
17
firms should be a smooth one. Some firms must go out of business in the passage to the new equilibrium, but they can sell off their undepreciated capital
for the same price as a new unit of capital, recover the funds invested in the
industry, and purchase some other kind of income-earning asset (possibly a
safe asset like a government bond, possibly starting a business in some other
market).
What if the investment in capital assets is
• partially sunk (resale is possible, but at something less than the purchase price)
• completely sunk (resale simply not possible)
The firms that survive in the integrated market will produce more, in
equilibrium, than they did in their home market:
r
r
F
F
2 >
.
b
b
To produce more, a firm will need to buy more capital. But some firms will
go out of business, meaning it is risky for them to make investments that
might, in essence, soon be thrown away.
Either
(a) firms delay investing until the capital stock of some firms depreciates
to such an extent that they can no longer cover their fixed cost, at which
point they shut down, or
(b) (some) firms invest, building up their capital stocks in an attempt to
ensure that they will be among the survivors.
Possibility (b) raises the spectre of industry overinvestment in capital
that will end up being thrown away.
In the Belgian coal industry, it seems to be (a) that took place.
The human capital of workers may also be sunk – incapable of application outside the original industrial sector.
Substantial retraining is required for a worker whose only skills regard
coal mining to find employment outside the coal industry.
This model: price before integration:
√
p = c + bF
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Price after integration:
p=c+
r
1
bF .
2
Price falls, so total output and therefore (given the assumptions we have
made about the production function) employment goes up.
If closures are concentrated more in one country than in another, there
may nonetheless be an employment problem in that country.
In the context of the simple model considered here, that is all one can
say.
In a more general model, if one admits the possibility that firms have
different costs, then firms with higher average cost will go out of business
after integration. Disruptions to the labor market will then be more severe
in higher (pre—integration) cost countries.
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