Industrial Organization in Context
Oligopoly I
Stephen Martin
January 2010
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Why so many models?
The modeling approach of modern industrial economics is to tailor
oligopoly models to speci…c industries.
We will consider a series of speci…c models of oligopoly markets:
Cournot models: …rms decide how much to produce, and price adjusts
so that consumers demand what is produced (automobiles?);
Bertrand models: …rms set prices and sell whatever is demanded at
those prices (most services);
Horizontal product di¤erentiation models — some consumers prefer
apples, some oranges;
Vertical product di¤erentiation models (Chapter 4) — all consumers
‡ying between Chicago and London agree that business class is better
than economy class;
Search models: consumers must search to learn the prices o¤ered by
various …rms.
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Cournot (1838): the basic oligopoly model
The smallest possible step away from one supplier is two suppliers.
Cournot analyzed a market supplied by two producers of a standardized
product — mineral water drawn from a common underground source.
In this model
each …rm knows the market demand curve
the two …rms have identical costs
each …rm knows that the other …rm knows as much about the market
as it does.
Each …rm picks its own output to maximize its own pro…t, knowing
that the other …rm acts in the same way and with the same
information.
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What outputs will the …rms produce, and at what price will
the product sell?
Here we confront the notion of an equilibrium.
Cournot’s equilibrium concept was an anticipation of that of John
Nash, and is often referred to in economics as Cournot-Nash
equilibrium.
What we require of an equilibrium pair of outputs is that each …rm’s
equilibrium output maximize its pro…t, given the equilibrium output of
the other …rm.
For such an output pair, each …rm is making as large a pro…t as it
possibly can, given what the other …rm does.
In view of Cournot’s assumption that each …rm seeks to maximize its
own pro…t, neither …rm would wish to alter its own part of such an
output pair.
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Best response functions
We begin the task of …nding equilibrium outputs by characterizing the
output that will maximize a …rm’s pro…t for an arbitrary output level
of the other …rm.
The schedule of all such output pairs is called the …rm’s best response
function. We then look for mutually consistent output levels on the
best response functions of the two …rms.
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Example
Demand and cost
Let the equation of the market inverse demand curve be
p = 100
Q = 100
(q1 + q2 ),
where q1 is the output of …rm 1 and q2 is the output of …rm 2.
Firms have identical cost functions, with constant average and marginal
cost, 10 per unit of output:
C (q1 ) = 10q1
C (q2 ) = 10q2 .
For simplicity, assume that there are no …xed costs.
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Residual demand
If …rm 2 produces an arbitrary output level q2 , the relation between …rm
1’s output level q1 and the market-clearing price p is
p = (100
q2 )
q1 .
This is the equation of …rm 1’s residual demand function: it gives the
relation between the quantity supplied by …rm 1 and price in the part of
market left for …rm 1 after …rm 2 has disposed of its output.
In this left-over part of the market, …rm 1 is a monopolist, or at least, it
acts as a monopolist, since …rm 2’s output is assumed to be …xed at the
arbitrary level q2 . The output that maximizes a monopolist’s pro…t is that
which makes its marginal revenue equal to its marginal cost.
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Residual marginal revenue
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 …rm 1’s
residual marginal revenue function is therefore
MR1 = (100
q2 )
2q1 .
Firm 1’s pro…t-maximizing output makes its marginal revenue equal to its
marginal cost,
MR1 = (100
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q2 )
2q1 = 10 = MC .
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Residual marginal revenue
Figure: Firm 1’s residual demand curve, q2 = 30.
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Best response function
The equation
MR1 = (100
q2 )
2q1 = 10 = MC .
can be rewritten as
q1 =
1
(90
2
q2 ) = 45
1
q2 .
2
This is the equation of …rm 1’s best response function for this example: it
gives the pro…t-maximizing output of …rm 1 for any output level of …rm 2.
Going through the same procedure for …rm 2, we obtain the equation of
…rm 2’s best response function,
q2 = 45
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1
q1 .
2
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Best response curve diagram
Figure: Best response curves, Cournot duopoly.
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Cournot equilibrium
When …rms are producing their equilibrium outputs, each …rm is
maximizing its pro…t, given the equilibrium output of the other …rm.
The equilibrium outputs are at the intersection of the best response
curves.
For this combination of outputs — and only at this combination of
outputs — each …rm is maximizing its own pro…t, given the output
produced by the other …rm.
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Cournot equilibrium
Outputs
Analytically, the values of the Cournot equilibrium outputs are found by
solving the equations of the best response functions, here
q1 = 45
1
q2
2
q2 = 45
1
q1 .
2
Since this example is symmetric, in the sense that the two …rms have
identical cost functions and identical beliefs each about the other, the
…rms will produce identical output levels in equilibrium.
Call this common equilibrium output qCour and set q1 = q2 = qCour in the
best response equation. This allows us to …nd the Cournot equilibrium
output per …rm:
2qCour + qCour = 3qCour = 90,
qCour = 30.
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Cournot equilibrium market diagram
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Bertrand, andduopoly,
Generalizations
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Figure: Market equilibrium,
identical unit costs.
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Characteristics of Cournot equilibrium
Adding the outputs of the two …rms gives total output:
QCour = 2qCour = 60.
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.
pCour is greater than marginal cost (10), but less than the monopoly price
(55).
In Cournot equilibrium, the Lerner index of market power is
40 10
3
pCour c
=
= .
pCour
40
4
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Characteristics of Cournot equilibrium
Pro…t per …rm is
π Cour = (pCour
10)qCour = (30)(30) = 900.
Since there are two …rms, total economic pro…t is twice π Cour :
2π Cour = 1800.
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. This area is
1
1
CS = (100 40)(60) = (60)2 = 1800.
2
2
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
DWL =
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1
(40
2
10)(90
60) =
1
(30)2 = 450.
2
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Cournot duopoly with di¤erent unit costs
Suppose that each …rm has constant marginal cost, but allow di¤erent
…rms to have di¤erent marginal costs, the equations of the best response
functions become
Firm 1: residual marginal revenue = marginal cost
(100
q2 )
q1 =
1
(100
2
2q1 = c1
c1
q2 )
or
2q1 + q2 = 100
c1
q1 + 2q2 = 100
c2 .
and in the same way
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Best response curve diagram, cost di¤erences
Figure: Best response curves, c1 = 10, c2 = 40.
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Cournot equilibrium outputs, cost di¤erences
Equilibrium outputs (found by solving the equations of the best response
functions) are
1
q1 = (100 2c1 + c2 ),
3
1
q2 = (100 + c1 2c2 ),
3
and it follows that
q1 q2 = c2 c1 .
If c1 is less than c2 , then q1 is greater than q2 . In a Cournot duopoly, the
…rm with lower unit cost has greater equilibrium output.
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Cournot equilibrium price and Lerner indices, cost
di¤erences
This translates into a greater degree of market power for the …rm with
lower unit cost. Using the equation of …rm 1’s best response function, it
follows that
p = 100 q1 q2 = c1 + q1 ,
in equilibrium, so
p
c1
p
=
q1
q1 Q
s1
=
=
,
p
Q p
εQp
where
s1 = qQ1 is …rm 1’s market share and
εQp is the absolute value of the price elasticity of demand.
If …rm 1 has lower unit cost than …rm 2, it will have greater equilibrium
output than …rm 2, and will exercise more equilibrium market power than
…rm 2.
Lower cost improves market performance, in the sense of leading to greater
equilibrium output and lower equilibrium price (as it does in monopoly).
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Total output, Cournot duopoly, cost di¤erences
Add the best response equations: total output is
Q = q1 + q2 =
1
[200
3
(c1 + c2 )] ,
and this rises as c1 falls.
If output rises as c1 falls, then equilibrium price falls as c1 falls.
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Industry-average Lerner index
If we multiply …rm 1’s Lerner index p pc1 = εsQp1 by s1 , multiply …rm 2’s
Lerner index by s2 , and add the results, we obtain
p
p
where
b
c
=
s12 + s22
H
=
,
εQp
εQp
b
c = s1 c1 + s2 c2
is industry weighted-average unit cost.
H = s12 + s22
is the Her…ndahl index of seller concentration.
We thus obtain the result that in Cournot oligopoly with cost di¤erences,
the noncooperative equilibrium industry-average price-cost margin is higher
where seller concentration as measured by the Her…ndahl index is higher,
all else equal.
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The Her…ndahl index
Industry
1
2
3
4
s1
s1
s1
s1
Market Shares
= 75%, s2 = 25%
= s2 = 50%
= s2 = ... = s10 = 10%
= 91%, s2 = ... = s10 = 1%
H
0.625
0.500
0.100
0.829
1/H
1.6
2
10
1.206
Table: Her…ndahl index examples
If there are n equally-sized …rms in an industry, then H = 1/n and
1/H = n.
For this reason, the inverse of the H-index is said to be a
numbers-equivalent measure of seller concentration.
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The Her…ndahl index
Industry
1
2
3
4
s1
s1
s1
s1
Market Shares
= 75%, s2 = 25%
= s2 = 50%
= s2 = ... = s10 = 10%
= 91%, s2 = ... = s10 = 1%
H
0.625
0.500
0.100
0.829
1/H
1.6
2
10
1.206
Table: Her…ndahl index examples
Row 4: H = 0.829, 1/H = 1.206.
For such an industry, when we say its supply side is “as concentrated as an
industry supplied by 1.2 equally-sized …rms, we mean as a matter of
description that its structure is closer to that of monopoly than duopoly,
even though there are 10 active …rms.
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Recall the discussion in Chapter 2 of e¢ ciency rents in a
competitive (farm) industry with cost di¤erences:
Figure: Di¤erential rent.
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Welfare, Cournot duopoly with cost di¤erences
Figure: Market power and welfare losses with e¢ ciency di¤erentials; shaded area
is an e¢ ciency rent.
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Welfare, Cournot duopoly with cost di¤erences
In Cournot duopoly equilibrium, price is greater than c2 .
The net loss of social welfare is the deadweight welfare loss from output
restriction, plus the extra social cost of producing …rm 2’s output at unit
cost c2 rather than unit cost c1 :
( c2
c1 )q2 .
The higher-cost …rm has positive output only because the lower-cost …rm
does not act as a price-taker.
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Welfare, Cournot duopoly with cost di¤erences
Economic pro…ts are
π 1 = (pCour
c1 )q1 and π 2 = (pCour
c2 )q2 .
A portion of …rm 1’s pro…t,
( c2
c1 )q1 ,
shown as the shaded area in the graph, is an e¢ ciency rent (an income
stream that need not be received in order for the services of a factor of
production to be provided) collected by …rm 1 on the output that it does
produce.
The …rms’pro…ts are income transfers from consumers to producers.
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Conjectural variations
Thinking about the way best response curves shift is also a way to
understand what happens if the Cournot behavioral assumption — that
each …rm maximizes its own pro…t, given the equilibrium output of the
other …rm — is changed.
This leads to a simple way of modeling dynamic interactions among …rms
in imperfectly competitive markets using the static Cournot model.
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Best response function with conjectures
Suppose that …rm 1 expects that for every 1 per cent change in its own
output q1 , there will be an α per cent change in q2 :
α=
dq2
∆q2 /q2
or
q1 = αq2 .
∆q1 /q1
dq1
Firm 1’s total revenue:
TR1 = pq1 = (100
q1
q2 ) q1 ,
Firm 1’s conjectured marginal revenue
MR1 = 100
q1
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q2 +
1
dq2
dq1
q1 = 100
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2q1
(1 + α) q2 .
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Best response function with conjectures
Setting conjectured marginal revenue equal to marginal cost gives the
equation of …rm 1’s best response function in the conjectural variations
Cournot model,
MR1 = 100
2q1
(1 + α) q2 = 10.
2q1 + (1 + α)q2 = 90,
or
q1 = 45
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1
(1 + α)q2 .
2
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Best response function with conjectures
Figure: Firm 1’s best response curve, alternative conjectures.
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α =c 2010
0 (Cournot
conjectures),
slope
of the
best response curve is01/20101/2.32 / 98
Best response function with conjectures
Figure: Firm 1’s best response curve, alternative conjectures.
α > 0, …rm 1 expects that if it reduces its own output, …rm 2 will reduce
output as well. For any level of output of …rm 2, …rm 1 produces less
than it would with Cournot conjectures. Firm 1’s best response curve
rotates around the q1 -axis intercept toward the origin.
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Best response function with conjectures
Figure: Firm 1’s best response curve, alternative conjectures.
α < 0, …rm 1 expects that if it reduces its own output, …rm 2 will expand
output. For any level of output of …rm 2, …rm 1 produces more than it
would with Cournot conjectures. Firm 1’s best response curve rotates
around the q1 -axis intercept away from the origin.
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Equilibrium outputs with conjectures
If …rms have identical conjectures, equilibrium outputs are the same for
both …rms.
Setting q1 = q2 = qα in the equation of …rm 1’s best response function
2q1 + (1 + α)q2 = 90,
gives an expression for equilibrium outputs with identical conjectures:
qα =
90
.
3+α
Equilibrium outputs are smaller (total output is closer to the monopoly
level) for matching conjectures (α > 0) and equilibrium outputs are larger
(total output is closer to the long-run competitive equilibrium level) for
contrarian conjectures (α < 0). Matching conjectures move the market
toward the kind of outcome associated with collusion; contrarian
conjectures move the market toward the kind of outcome associated with
perfect competition. Smaller values of α mean tougher rivalry between
the two …rms.
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“Collusive” conjectures
If α = 1, then from
qα =
90
90
=
= 22.5,
3+α
3+1
in equilibrium each duopolist produces half the monopoly output. For this
reason, α = 1 is sometimes referred to as the case of collusive conjectures
(although if …rms make output decisions independently they would not
normally be considered to have colluded in a legal sense).
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Bertrand conjectures
If α =
1, then from
qα =
90
90
=
= 45,
3+α
3 1
in equilibrium each …rm produces half the perfectly competitive output.
Total output is what it would be in long-run perfectly competitive
equilibrium, with only two …rms supplying the market.
As we shall see, this is the same result as in the Bertrand model of
price-setting duopoly with standardized products. For this reason,
α = 1 is sometimes referred to as the case of Bertrand conjectures.
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Lerner index with conjectures
Another way to write …rm 1’s conjectured marginal revenue is (Cowling,
1976; Cowling and Waterson, 1976):
MR1 = p + q1
dp dQ
dp
= p + q1
dQ dq1
dQ
=p 1
1 q1 + αq2
εQP
Q
1+
dq2
dq1
=p 1
= p+p
Q dp q1 + αq2
p dQ
Q
α + ( 1 α ) s1
.
εQP
Here (as before) s1 = q1 /Q is …rm 1’s market share and
s2 = q2 /Q = 1 s1 is …rm 2’s market share.
Then setting marginal revenue equal to marginal cost and rearranging
terms slightly gives a further generalization of the Lerner index of market
power,
p c1
α + ( 1 α ) s1
=
.
p
εQP
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Industry average Lerner index with conjectures
If all …rms have the same conjecture,
p
c1
p
=
α + ( 1 α ) s1
.
εQP
can be aggregated from the …rm to the industry level as before.
This leads to the industry weighted-average Lerner index with conjectures,
p
p
b
c
=
α + (1 α ) H
.
εQP
The Cournot equilibrium industry-average price-cost margin is larger where
seller concentration (as measured by the Her…ndahl index) is larger and
the closer the conjectural elasticity α is to one, all else equal.
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Many …rms
Residual demand equation
If there are n identical Cournot …rms in the industry, write
Q
1
= q2 + ... + qn
for the combined output of all …rms except …rm 1.
Then we can write the equation of …rm 1’s residual demand curve as
p = (100
Q
1)
q1 .
This looks very much like the equation of …rm 1’s residual demand curve
for the duopoly case,
p = (100 q2 ) q1 .
The aggregate output of all other …rms has been substituted for the
output of …rm 2.
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Best response equation
Proceeding in the same way as the duopoly case, we can …nd the equation
of …rm 1’s best response curve by setting its marginal revenue along the
residual demand curve equal to its marginal cost:
100
Q
1
2q1 = 10,
or
2q1 + Q
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1
= 90.
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Equilibrium output per …rm
In the symmetric …rm case, all …rms will produce the same output in
equilibrium.
2q1 + Q 1 = 90
becomes
2qCour + (n
1)qCour = 90,
so that output per …rm is
qCour =
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90
.
n+1
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Equilibrium price
From the equilibrium output of a single …rm, we can work out all the other
characteristics of n-…rm equilibrium. Here we note two of these, total
output and price, which are
QCour =
1
n+1
n
90 =
n+1
1
pCour = 10 +
90
n+1
and
90
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 …rms increases, and in fact
that market performance will approach that of long-run competitive
equilibrium as the number of …rms approaches in…nity.
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Bertrand duopoly
In 1883, the French mathematician Joseph Bertrand wrote a review of
Cournot’s 1838 book, criticizing Cournot in particular for assuming that
…rms picked outputs and that price adjusted so that consumers would
willingly demand the total quantity supplied. Since that time, Bertrand’s
name has been associated with models of imperfectly competitive markets
in which …rms pick prices and sell whatever amount of output is demanded
at those prices.
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Residual demand,standardized products
Firm 1’s residual demand curve looks quite di¤erent if …rms set prices rather than
quantities.
Figure: Firm 1’s price decision, p2 = 40.
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Residual demand, standardized products
If …rm 2 has set a price p = 40, …rm 1:
Firm 1 can sell up to 60 units of output at a price slightly below 40.
For outputs less than 60 units, …rm 1’s marginal revenue is slightly
less than 40.
To sell more than 60 units of output, …rm 1 must reduce price and
move down the market demand curve.
For outputs greater than 60 units, …rm 1’s marginal revenue curve is
the same as the market marginal revenue curve.
The horizontal break in …rm 1’s residual demand curve at a price just
less than 40, p = 40 ε for some small number ε, means that there is
a vertical break in …rm 1’s marginal revenue curve at an output level
slightly greater than Q = 60.
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Pro…t maximization, standardized products
For outputs less than 60, …rm 1’s marginal revenue (40 ε) is greater than
its marginal cost (10).
For outputs greater than 60, …rm 1’s marginal revenue is less than its
marginal cost.
To maximize its pro…t, …rm 1 will set a price just a little below 40 and
supply the entire quantity demanded at that price.
Firm 2 will sell nothing.
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Figure: Firm 1’s price decision, p2 = 40.
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Equilibrium, standardized products
If …rm 2 has set a price equal to 40, …rm 1 will maximize its pro…t by
setting a price just a little below 40.
This cannot be an equilibrium, however.
Firm 2 would have an incentive to undercut …rm 1’s price slightly.
And if …rm 2 did this, …rm 1 would once again have an incentive to
set a price slightly below …rm 2’s new, lower, price.
If either …rm sets a price above marginal cost, the other has an
incentive to set a lower price. Neither …rm would set a price below
marginal cost, because that would mean losing money.
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Equilibrium, standardized products
These arguments show that the Bertrand equilibrium price is
p = 10.
When each …rm sets a price equal to marginal cost, each …rm is
maximizing its own pro…t, given that the other …rm sets a price equal
to marginal cost.
The Bertrand model therefore predicts that when the product is
standardized, market performance is the same as in the long-run
equilibrium of a perfectly competitive market, provided there are at
least two …rms supplying the market.
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Di¤erentiated products
The assumption that the product is standardized is essential for the result
that the Bertrand equilibrium price equals marginal cost with at least two
price-setting suppliers.
To see this, we introduce product di¤erentiation to the model of
price-setting duopoly.
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Payo¤ functions
Suppose that the two …rms produce di¤erentiated brands of mineral water,
with the inverse demand curves used for the discussion of Cournot
oligopoly with product di¤erentiation:
p1 = 100
(q1 + θq2 )
p2 = 100
(θq1 + q2 ) .
If, for example, θ = 1/2, the equations of the inverse demand functions are
p1 = 100
1
q1 + q2
2
p2 = 100
1
q1 + q2 .
2
If we solve these two equations for outputs in terms of prices, we obtain
the equations of the demand functions:
2
2
(100 2p1 + p2 )
q2 = (100 + p1
3
3
Firm 1’s pro…t as a function of p1 and p2 is
q1 =
π 1 = (p1
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10)q1 =
2
(p1
3
10)(100
IOIC: Cournot, Bertrand, and Generalizations
2p2 ).
2p1 + p2 ).
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Price best response function
From the …rst-order condition to maximize
π 1 = (p1
10)q1 =
2
(p1
3
10)(100
2p1 + p2 ).
with respect to p1 , we obtain the equation of …rm 1’s price best response
function :
1
p1 = 30 + p2 .
4
When …rms set prices, best response curves slope upward: if …rm 2 sets a
higher price, …rm 1’s pro…t-maximizing price is higher.
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Price best response function diagram
Figure: Price best response curves, Bertrand duopoly with product di¤erentiation.
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Equilibrium prices
1
p1 = 30 + p2 .
4
Bertrand equilibrium prices can be found (for this example) by symmetry:
the two …rms are identical, and in equilibrium they will charge the same
B . Setting p = p = p B in
price, p1/2
1
2
1/2
1
p1 = 30 + p2
4
and rearranging terms gives
B
p1/2
= 40.
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Equilibrium prices: the general case
For general values of the product di¤erentiation parameter θ, the equation
of …rm 1’s best response curve is
p1 =
1
(110
2
100θ + θp2 ) .
If θ = 0, the two products are independent in demand, and
p1 = 12 (110 100θ + θp2 ) reduces to
p1 = 55,
which is the monopoly price for a …rm that faces no substitute products.
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Equilibrium prices: the general case
If θ = 1, the two products are perfect substitutes, and
p1 = 12 (110 100θ + θp2 ) reduces to
1
p1 = 5 + p2 .
2
This is the homogeneous-product Bertrand case, and in equilibrium (set
p1 = p2 = p1B ) price equals marginal cost:
p1B = 10.
With price-setting …rms, the greater the degree of product di¤erentiation
(the lower is θ), the greater the equilibrium price-cost margin. As long as
products are di¤erentiated to some extent, equilibrium price-cost margins
fall as the number of …rms rises, when …rms set price as when they set
quantities.
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Contestable markets
A perfectly contestable market is (Baumol et al., 1982, p. 5)
one that is accessible to potential entrants and has the following
two properties: First, the potential entrants can, without
restriction, serve the same market demands and use the same
productive techniques as those available to the incumbent …rms.
. . . Second, the potential entrants evaluate the pro…tability of
entry at the incumbent …rms’pre-entry prices.
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Contestable markets
There is free and easy entry and exit (Baumol, 1982, pp. 3–4):
A contestable market is one into which entry is absolutely free,
and exit is absolutely costless. . . . the entrant su¤ers no
disadvantage in terms of production technique or perceived
quality relative to the incumbent, and that potential entrants
…nd it appropriate to evaluate the pro…tability of entry in terms
of the incumbent …rms’pre-entry prices. . . .
If the assumptions of perfect contestability hold and there are at least two
active …rms, then equilibrium price equals marginal cost.
The assumptions and the results of the theory of perfectly competitive
markets largely overlap with those of the Bertrand model of price-setting
oligopoly with standardized products.
As with the Bertrand model, equilibrium price is the same, and equal to
marginal cost, if there are 2, 3, 4, . . . suppliers.
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Contestability & the passenger airline industry
Early in the development of the theory of contestable markets, the
passenger airline industry was put forward as an case of a real-world
markets that might be approximately contestable. Super…cially, this
is not implausible: after all, aircraft can be ‡own into and out of
markets, suggesting that the cost of entry and exit ought to be low.
Empirical evidence suggests that neither the passenger airline industry
nor any other real-world industry is approximately contestable. Fares
typically rise as the number of airlines serving a route falls,
While aircraft may ‡y in and out of markets, to pick up and discharge
passengers, they must have landing slots and gates. An investment
in slots and gates may be partially sunk, if it is possible to acquire
them at all. Nor does the assumption that incumbents would expect
entrants to act as price takers …t well as a description of an industry
where computerized fare systems permit fares to be changed at will.
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Cournot & Bertrand compared
For price-setting …rms with the demand equations
2
2
(100 2p1 + p2 )
q2 = (100 + p1 2p2 ),
3
3
if we assume …rms set prices we reach the Bertrand equilibrium prices and
quantities
B
B
p1/2
= 40
q1/2
= 40.
q1 =
For the corresponding inverse demand equations,
1
1
p2 = 100
q1 + q2
q1 + q2
2
2
if we assume …rms set quantities we reach the Cournot equilibrium prices
and quantities
C
C
p1/2
= 46
q1/2
= 36.
p1 = 100
This is one example of a general result: for otherwise identical markets,
prices are lower and quantities higher if …rms set prices than if …rms set
quantities.
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Comparative residual demand curves
Figure: Firm 1’s residual demand curves, p1 = 100
p2 = pBert , alternatively q2 = qCour .
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q1 + 21 q2 , c = 10,
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Comparative residual demand curves
The graph shows the alternative residual demand curves for …rm 1,
depending on whether …rm 2 sets quantity or price:
p1 = 100
q1 = 23 (100
C = 36
q1 + 12 q2 with q2 = q1/2
B = 40
2p1 + p2 ) with p2 = p1/2
If …rm 2 is a quantity setter, …rm 1’s demand curve is relatively steep, less
inelastic.
If …rm 2 is a price setter, …rm 1’s demand curve is relatively ‡at, more
elastic.
From the Lerner index of market power
p
c
p
=
1
εqp
,
(where εqp is the price elasticity of residual demand) a …rm facing a more
elastic demand maximizes pro…t by setting a price closer to marginal cost,
all else equal.
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Markets with Consumer Search
Incomplete consumer information may dramatically a¤ect the results of the model of
perfect competition
Keep all but one of the assumptions of the standard model of perfect
competition:
The product is standardized.
There are many …rms, with identical cost functions.
There are many consumers.
Before allowing for search cost, the utility of a consumer who
purchases one unit of the good from store i at price pi is
u (pi ) = ρ
pi .
Each consumer will buy one unit of the product if the price is less
than or equal to the reservation price ρ.
If the price is greater than ρ, the consumer instead buys an outside
good that is traded in a perfectly competitive market.
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Search and Search Cost
To learn the price charged by a store, a consumer must go to the
store.
It is costly to search stores after the …rst store that is visited.
Firms know what it costs a consumer to search a new store.
Consumers know the distribution of prices over stores, but they do
not know (in advance of search) the price set by any one store. A
consumer remembers the prices set at all stores he or she has visited.
Because there are many small …rms and many consumers, no one …rm
can a¤ect consumers’search patterns, and consumers cannot a¤ect
…rms’price choices.
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Search Equilibrium
Could a set of prices in which one or more …rms had posted a price
less than ρ be an equilibrium? — Could all …rms be maximizing pro…t,
given consumer behavior and given the prices set by other …rms?
With such prices, a low-price store could charge a price that was
higher by an amount just below search cost without losing any sales:
Since it would cost a consumer already at the store more to search at
other stores than the amount of the increase in price, consumers in the
store would remain and purchase.
But then the original set of prices is not an equilibrium: a low price
…rm could increase its pro…t by raising its price.
In equilibrium, all …rms (no matter how many …rms there are) set the
reservation price ρ, which in this model is the monopoly price.
Consumers know the equilibrium distribution of prices — that all
…rms charge the monopoly price — and in equilibrium each consumer
visits one store and purchases one unit of the good.
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The Diamond Paradox
This result, due to Diamond (1971), is known as the Diamond
Paradox. A seemingly minor change in the assumptions of the
perfectly competitive model — the presence of small but positive
search costs — upends its predictions.
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Extensions: product di¤erentiation
We have seen that with just a small amount of product
di¤erentiation, the Bertrand Paradox ceases to hold.
Anderson and Renault (1999) combine product di¤erentiation with
search by introducing consumer uncertainty about product
characteristics as well as price. They write the utility (before allowing
for search cost) of consumer l who purchases at store i as
uli (pi ) = ρεli
pi .
εli is a scale factor indicating the way the particular variety of the
product o¤ered by store i satis…es consumer l’s preferences.
Before search, consumer l knows the distribution of ε across stores,
but does not know the value of εli at any one store.
Special cases of the Anderson & Renault model yield results
corresponding to those of the Bertrand and the Diamond models. In
the most general version of their model, equilibrium prices rise as
search costs rise and fall as the number of stores rises, so that the
Diamond paradox does not hold.
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Search in Markets for Perishable Goods
The Marseille Fish Market has many of the characteristics of a standard search model
Fish vary in quality, but quality is readily observable.
Around 40 sellers and 400 buyers (per week)
Sellers are wholesale distributors who purchase …sh for resale to
Customers who are …shmongers, retail grocers, or restauranteurs
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Institutional Details Matter
Marseille Fish Market
Market open between 3 A. M. and 8 A. M.
Sellers ordered their supplies two days before o¤ering them for resale
Fish are perishable and (by regulation) cannot be carried over for sale
the next day
Buyers & sellers interact on a one-to-one basis (Kirman and Vriend,
2000, p. 34):
Standing face-to-face with a seller, the buyer tells the seller
which type of …sh and which quantity he is interested in. The
seller then informs him about the price. Prices are not posted.
A seller may quote prices to di¤erent buyers, or di¤erent prices for
identical lots at di¤erent times of a day.
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Marseille Fish Market
In the standard search model, buyers are anonymous
At the Marseille Fish Market, sellers will learn to recognize buyers
Recognition may carry with it pro…table information: for example, if it
is known that a particular buyer is purchasing to supply a (say) a
restaurant that is some distance away, then as the morning wears on,
that buyer’s price elasticity of demand will fall, as time comes to get
back on the road and tend to business.
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Marseille Fish Market
Sellers’quantities are limited
A buyer who searches too diligently
incurs substantial search cost
runs the risk that when he returns to the stand that o¤ered the best
price, that seller has sold out for the day.
A seller who sets price too close to the monopoly level runs the risk of
driving too many buyers to search and being left with a stock of
unsold …sh at the end of the day.
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Loyalty Is Mutually Bene…cial
Ensures buyers of a source of supply without substantial investment in
search
Ensures sellers against unsold …sh at the end of the day
Because loyalty is mutually bene…cial, it is pro…table for sellers to
reward loyal buyers with moderate prices
But if no buyers ever searched, sellers would have little incentive to
reward loyal buyers.
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Buyers: Loyal vs. Searcher
Figure: Distribution of buyers of cod according to the number of sellers they visit
on average during one month in 1990. (Buyers visiting the market more than
once per month and in the market for more than six months.) Source: Weisbuch
et al.(2000, Figure 5).
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Buyers: Loyal vs. Searcher
Result: buyers divide themselves into two groups, loyal and searchers.
The …gure shows that 48 per cent of cod buyers per month, on average in
1990, visited one seller (not all visited the same seller).
Remaining buyers visited about four sellers per month on average — there
are enough searchers so that sellers have to reward the patronage of loyal
buyers.
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Two-sided markets
Newspapers and popular magazines sell one product to readers, but
also sell advertising space to businesses that want to deliver a
message to a potential client.
Private television and radio networks — likewise; their product is
given away to one side of the market.
Real estate agencies bring together home buyers and home sellers.
The producer of a computer operating system provides a platform for
transactions between writers and users of software packages.
Malls pro…t by providing a forum within which stores and shoppers
interact.
Dating clubs
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Distinctive features of two-sided markets
Uninternalized externalities
a credit card is more attractive to stores, the more consumers carry
the card.
a credit card is more attractive to consumers, the more stores accept
the card as a means of payment.
But no one consumer takes into account, in deciding to carry a credit
card, that the decision to do so makes the credit card more attractive
to store owners — an external bene…t to store owners who accept the
card. Nor does a store owner take into account that a decision to
accept payment by means of a credit card confers a bene…t on all
consumers who carry the card.
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Distinctive features of two-sided markets
Often, network externalities
A product exhibits network externalities if it is more valuable to any
one consumer, the more consumers use it.
Direct network externalities: a telefax machine is more valuable to
any one user, the more other people own telefax machines.
Indirect network externalities: a compact disc player is more
valuable to users, the more …lms are available in a compatible format.
There is no physical network, there is a virtual network — the more
consumers own CD players of a particular format, the more pro…table
it is for owners of …lm archives to license reproductions of their …lms
in that format, the greater the variety of …lms a consumer can view,
and the more attractive it is to own such a CD player.
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Bringing two sides of the market together
Firms that supply two-sided markets internalize these externalities by
structuring relative prices to bring both sides into the market.
Because their pro…table operation depends on balancing the two sides
of the market, intuition based on the way imperfectly competitive
one-sided markets work is often misleading when applied to two-sided
markets.
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Stylized example
There is a monopoly supplier of a credit card that may be used to pay for
lunch in the immediate neighborhood of a university.
Fees:
p B = fee to lunchers each time the card is used to pay for a meal.
p S = fee to restauranteurs each time a credit card is used to pay for a
meal.
In principle, either of these fees might be zero or negative (think of a cash
rebate for use of a credit card).
c = marginal cost of processing a transaction.
Quasi-demand functions:
N B = D B p B = number of buyers will to pay by credit card
N S = D S p S = number of sellers willing to accept payment by credit
card
Maximum number of possible credit-card transactions, if every luncher
goes to every restaurant:
D B pB D S pS .
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Number of transactions
Suppose that
in search of variety, lunchers who pay by credit card patronize
restaurants that accept credit cards one after another, and
visit all such restaurants once before patronizing any restaurant a
second time.
Measure the time period so that all lunchers are able to visit all
restaurants once and only once.
Then
D B pB D S pS
gives the actual number of credit card transactions.
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Payo¤ and pro…t maximization
Leaving aside …xed cost, the credit card company’s pro…t is
pB + pS
c D B pB D S pS .
The …rst-order conditions to maximize pro…t with respect to prices p B and
p S are
dD B p B
B
S
p +p
c
+ D B pB
0
dp B
and
pB + pS
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c
dD S p S
+ D S pS
dp S
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0.
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Lerner indices
The …rst-order conditions can be rewritten as
and
pB
c
pB
pS
pS
c
pS
pB
=
1
εB
=
1
,
εS
where
εB = price-elasticity of buyers’quasi-demand and
εS = price-elasticity of sellers’quasi-demand.
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Lerner indices
pB
c pS
pS
c pB
1
1
=
and
=
B
S
εB
εS
p
p
are Lerner indexes of market power, adapted to allow for the two-sided
nature of the market.
An increase in p B means slightly fewer buyers are willing to carry a credit
card.
Just how many depends on the elasticity εB .
For every buyer that does not use a credit card, a transaction does not
occur and the …rm saves the marginal cost of a transaction, c.
But the …rm also gives up the fee it would have received from the seller,
pS .
In a two-sided market, the opportunity cost of giving up the marginal
transaction by raising p B is c p S , not (as one would expect in a
one-sided market) c.
Similarly, c p B is the opportunity cost of giving up the marginal
transaction by raising p S .
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Lerner index, total price
From the …rst-order conditions, we can obtain a Lerner index that
characterizes the optimal total price, p B + p S :
pB + pS c
1
.
=
B
S
εB + εS
p +p
Total price, p B + p S , exceeds marginal cost.
Either element of total price, p B or p S , may be less than marginal cost, if
a low price is needed to recruit on one side of the market.
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Price ratio
From
pB
c
pB
pS
=
pS
1
and
εB
c
pS
pB
=
1
εS
we also obtain a relation between the pro…t-maximizing fees,
pB
ε
= B.
εS
pS
In an imperfectly-competitive one-sided market, we expect price and the
price-cost margin to be inversely related to the price-elasticity of demand.
A monopolist of a two-sided market, having set total price according to
p B +p S c
= εB +1 εS , will charge a higher fee to the side of the market that
p B +p S
has the highest quasi-demand elasticity.
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Experimental Tests of the Cournot model: Holt (1985)
Cournot duopoly experiments — zero production cost, laboratory inverse demand
equation
1
(q1 + q2 ) .
2
Cournot equilibrium outputs are 8 units each, a total of 16.
Joint-pro…t-maximizing outputs are six units each, a total of 12.
Competitive (or equivalently, Bertrand) total output is 24 units.
First session: 12 pairs of students picked outputs for 13 periods.
Second session (to investigate the e¤ects of experience): 16 of these
students were rematched and picked outputs in a second session for 9
trading periods.
p = 12
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Final period outputs for the 12 …rst-session Cournot pairs
(white circles) and the 8 second-sessions Cournot pairs
(black circles).
Figure: Final-period industry output, Cournot duopoly experiments. Source: Holt
(1985, Figure 2).
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Results
Holt: “Regardless of whether the …rst-market or second-market data are
considered, the mean and median (or medians) of the …nal-period industry
outputs are between 14 and 16.”
In other experiments as well, Holt reports that the Cournot model does
well in predicting the outcome of single-period experimental sessions.
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Cournot and Bertrand Compared
Huck et al. (2000) report the results of experimental tests of the Cournot
and Bertrand models in four-supplier markets.
The demand-side of the markets was simulated by the experimental
program. In all experiments, the equation of the inverse demand curve of
(for example) supplier 1 was
p1 = 300
q1
2
(q2 + q3 + q4 ) .
3
Marginal cost is 2 per unit.
Cournot equilibrium total supply: 74.5.
Bertrand equilibrium total supply: 86.9.
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Cournot and Bertrand Compared
80
70
60
50
40
30
20
10
112-
102-107
Bertrand
Cournot
107-112
97-102
87-92
92-97
77-82
82-87
67-72
72-77
57-62
62-67
0-52
52-57
0
Figure: Frequencies of average quantities in Cournot and in Bertrand oligopoly
experiments; equilibrium quantities are 74.5 (Cournot) and 86.9 (Bertrand).
Source: Huck et al. (2000, Figure 1).
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Results
Cournot equilibrium total supply: 74.5.
Bertrand equilibrium total supply: 86.9, price 39.25.
Results:
average quantity in the last 5 (of 40) rounds of 74.20 for the Cournot
sessions
an average price in the last 5 rounds of 41.91 for the Bertrand
sessions.
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Contestable markets
Perfect contestability
Coursey et al. (1984a):
duopoly experiments where each supplier produces with increasing
returns to scale (that is, average cost falls as output increases)
no sunk entry or exit costs.
The experiments support the predictions of the theory of contestable
markets (1984a, p. 108):
“Four duopoly experiments had price and quantity outcome that
converged directly to the competitive predictions. . . . The other
two duopoly experiments never achieved the competitive
outcomes, although a visual inspection suggests they were
tending in that direction.”
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Contestable markets
Some sunk cost
Coursey et al. (1984b):
require an experimental seller to purchase a $2 entry permit, valid for
…ve periods, in order to be allowed to post a price.
Introducing an entry cost did not stop entry: if it was pro…table for
experimental subjects to come into the market, taking entry costs
into account, they did so.
Sunk costs did seem to lead to temporary periods with market
performance farther from the competitive outcome than would have
been predicted in the absence of sunk costs.
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Contestable markets
Some sunk cost: results observed in di¤erent sessions
market performance like that of a perfectly contestable market (that
is, one without entry cost);
price swings: low with two sellers in the market, high after one seller
exits, low again after reentry;
limit pricing: after exit by one of two suppliers, the remaining supplier
keeps price in the competitive range;
(intermittent) tacit collusion: two suppliers, with episodes of prices
held above the competitive level.
Coursey et al. (1984b, p. 83) read their results as indicating
market structure matters
market structure is not all that matters: conduct has an independent
impact on market performance.
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Search models
Davis and Holt (1996) report the results of posted-o¤er experiments to
test the predictions of consumer search models.
three buyers and three sellers
competitive equilibrium price 25 cents, the monopoly price is 55 cents
sellers …rst set their prices and the maximum amounts they are willing
to sell during that trading period.
Posted-o¤er sessions: buyers were informed of all prices and of
remaining quantities for sale, o¤ered the chance to buy in random
order.
Search treatments: buyer had to pay a 15-cent search cost in order to
learn the terms o¤ered at any particular store.
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Results illustrated
Figure: The search design and mean transaction prices for six sessions. Source:
Davis and Holt (1996, Figure 1).
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01/2010
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Results interpreted
Results are illustrated on the right of the …gure.
There is a clear tendency for prices to be closer to the competitive level in
the posted o¤er sessions.
Davis and Holt conclude (1996, p. 145)
The two primary lessons of this research are (1) that the absence
of public price information raises prices when shopping is costly,
but (2) that the monopoly prices implied by the “Diamond
paradox” are not generally observed.
c 2010 (Purdue University)
IOIC: Cournot, Bertrand, and Generalizations
01/2010
98 / 98