Part I: Exercise of Monopoly Power
Chapter 1: Monopoly
Two assumptions:
A1. Quality of goods is known by consumers;
A2. No price discrimination.
• Best known monopoly distortion: p > MC ⇒ DWL
(section 1).
• Other distortions:
– Monopolist has no control over costs (section 2);
– Rent dissipation behavior (section 3).
1
1 Pricing Behavior
• Distortion associated to monopolist pricing.
1.1 A Single-Product Monopolist
• q = D(p) demand function, D0(p) < 0, p = P (q) the
inverse demand function.
• C(q) the cost of producing q units of good, C 0(q) > 0.
• Elasticity of Demand:
D0(p)
ε=−
p
D(p)
• Program of the monopoly (in quantity)
max {qP (q) − C(q)}
q
FOC: qP 0(q) + P (q) − C 0(q) = 0 ⇒ q m
SOC: qP 00(q) + 2P 0(q) − C 00(q) < 0
Result 1: MR(q m) = MC(q m) ⇒ pm > MC
2
• Program of the monopoly (in price)
max {pD(p) − C(D(p))}
p
FOC : pD0(p) + D(p) − C 0(D(p))D0(p) = 0 ⇒ pm
SOC :
pD00(p) + 2D0(p) − C0000(D(p))(D000(p))2 − C 0(D(p))D00(p) <
• + assumptions on D (p) and C (q) (concavity or quasi
concavity).
pm − C 0(.) 1
Result 2: $ ≡
=
pm
ε
where $ is the Lerner index or relative markup.
Example Demand function q = kp−ε where k > 0.
Result 3: Monopolist always operates where ε > 1.
• See graph
• Formally:
pm −C 0(.)
pm
so,
< 1 because pm − C 0(.) < pm
1
< 1⇒ε>1
ε
Result 4: Monopoly price is a non decreasing function
of marginal cost.
3
Proof:
• 2 alternative cost functions C1(.) and C2(.).
• C20 (q) > C10 (q) for any q.
m
m m
• pm
1 , q1 if C1(.), and p2 , q2 if C2(.).
m
• If cost is C1(.) (resp. C2(.)), pm
1 (resp. p2 ) is charged
by the monopolist
m
m
m m
m
pm
q
−
C
(q
)
≥
p
q
−
C
(q
1
1
1 1
1
2 2
2 )
m
m
m m
m
pm
q
−
C
(q
)
≥
p
q
−
C
(q
2
2
2 2
2
1 1
1 )
• Sum these 2 inequalities
C2(q1m) − C2(q2m) ≥ C1(q1m) − C1(q2m)
C2(q1m) − C1(q1m) − [C2(q2m) − C1(q2m)] ≥ 0
R q1m
q2m
[C20 (x) − C10 (x)] dx ≥ 0
m
• Because C20 (x) > C10 (x) then q1m ≥ q2m and pm
1 ≤ p2 .
• Appropriate measure of distortion is the loss of social
welfare: the dead-weight loss.
• see graph
4
• Benchmark case: perfect competition
Result 5: The welfare loss does not necessarily decrease
with the elasticity of demand, even though the relative
markup does.
Proof: exercise 1.1. page 67 Tirole.
• DWL determines the loss from a monopoly to an
idealistic situation.
• DWL is one distortion created by a monopoly power.
• What kind of public intervention?
• Example: commodity taxation. Policy to restore social
optimum.
• Government imposes a tax on the output, t.
• The program of the monopolist becomes:
max {pD(p + t) − C(D(p + t))}
p
F OC : pD0(p + t) + D(p + t) − C 0(D(q + t))D0(p + t) =
⇔
[D(p + t) − tD0(p + t)]
+D0(p + t) [p + t − C 0(D(q + t))] = 0
• From the second term: p + t = C 0(.).
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• First term:
D(pc )
0
D(p + t) = tD (p + t) ⇒ t = D0(pc)
• and thus t < 0 = subsidy!
• Why?
– monopoly price induces consumers to consume too
little.
– If subsidy, consumption will increase.
• But the government needs to have information concerning cost and demand.
– Demand can be found with statistic studies.
– Difficult to get information on costs.
• Incentive theory.
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1.2 Multi-Product Monopolist
• Multi-product firm has a monopoly power over all
goods.
• qi = Di(p), demand for good i = 1, ...., n.
• Prices, p = (p1, p2, ...., pn).
• Quantities q = (q1, q2, ..., qn).
• Cost C(q1, q2, ..., qn).
• Single-product firm (n pricing problems) ⇔ multiproduct firm with
– independent demands: qi = Di(pi)
Pn
– separable costs: C(q1, q2, ..., qn) = i=1 Ci(qi).
• For each good i
0
pm
1
−
C
i
i
=
$i =
pm
εi
i
Result 6: The markup is higher on goods with a lower
elasticity of demand (Ramsey pricing).
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• General
monopolist program is
( multi-product
)
n
X
max
piDi(p) − C(D1(p), D2(p), ..., Dn(p))
p1 ,p2 ,....,pn
i=1
FOCi : pi ∂D∂pi(p)
+ Di(p) +
i
X
pj
∂Dj (p)
∂pi
j6=i
∀i, ∀k 6= i
• SOC must be satisfied.
−
n
X
∂C(.) ∂D (p)
k
∂qk
k=1
2 polar cases:
1. dependent demands, separable costs;
2. independent demands, dependent costs.
1.2.1 Dependent demands and separable costs
• Example: set of divisions
• qi = Di(p)
P
• C(q1, q2, ..., qn) = ni=1 Ci(qi)
• Program of the multi-product monopolist is
8
∂pk
max
p1 ,p2 ,....,pn
⇒
( n
X
i=1
from (1)
pi − ∂C∂qi(.)
i
pi
piDi(p) −
=−
Di(p) +
n
X
)
Ci(Di(p))
i=1
P
∂Dj (p)
j6=i ∂pi (pj
pi ∂D∂pi(p)
i
−
∂Cj (.)
∂qj )
• Own elasticity of demand
pi ∂Di(p)
Di(p) ∂pi
• Cross elasticity of demand for good j
εii = −
pi ∂Dj (p)
εij = −
Dj (p) ∂pi
• FOC becomes
pi − ∂C∂qi(.)
i
pi
=
1
−
εii
³
´
∂Cj (.)
X pj − ∂qj Dj (p)εij
εiipiDi(p)
j6=i
• Sign of the second term? (depends on the sign of εij )
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– If (−), $i > ε1ii , and thus higher price than in the
case of a single-product monopolist.
– if (+), $i < ε1ii , and thus lower price.
∂D (p)
• If goods are substitutes, ∂pj i
εij < 0.
– thus $i > ε1ii ⇒ pi > pm.
> 0 for j 6= i so
∂Dj (p)
∂pi
• If goods are complements,
εij > 0
– thus $i < ε1ii ⇒ pi < pm.
< 0 for j 6= i so
• Example: Intertemporal pricing.
• Single-product monopolist
• 2 periods: t = 1, t = 2.
• At t = 1,
– demand function q1 = D1(p1)
– cost function C(q1)
• At t = 2,
– demand function q2 = D2(p2, p1)
– cost function C(q2)
• Goodwill effect:
∂D2 (.)
∂p1
<0
10
• Monopolist’s profit
p1D1(p1) − C(q1) + δ[p2D2(p2, p1) − C(D2(p2, p1))]
where δ is the discount factor.
• ⇔ multi-product firm with interdependent demands.
∂C(.) ∂D1
∂C(.) ∂D2
1
F OC1 : p1 ∂D
+
D
(.)
−
+
δ(p
−
1
2
∂p1
∂q1 ∂p1
∂q2 ) ∂p1 = 0
∂C(.) ∂D2
2
+
D
(p
,
p
)
−
F OC2 : p2 ∂D
2
2
1
∂p2
∂q2 ∂p2 = 0
• In the second period, monopoly price as
1
$2 =
⇒ pm
2
ε22
• In the first period, the monopolist sets a lower price as
1
$1 <
⇒ p1 < pm
1
ε11
• Thus, the monopolist reduces the price at date 1
(sacrifice some short run profit) to increase the demand
(and thus the profit) at date 2.
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1.2.2 Independent demands and dependent costs
• The demand functions are independent, qi = Di(pi).
• C(q1, q2, ..., qn)
• The program(is
)
n
X
max
piDi(pi) − C(q1, q2, ..., qn)
p1 ,p2 ,....,pn
i=1
• Example: learning-by-doing
• Cost reduction can be achieved over time simply
because of learning.
• Example: semi-conductor industry, computers industry
• Single-Product monopolist
• 2 periods: t = 1, t = 2.
• At t = 1,
– demand function q1 = D1(p1)
– cost function C1(q1)
• At t = 2,
– demand function q2 = D2(p2)
– cost function C2(q2, q1) with ∂C∂q2(.)
<0
1
12
• Monopolist’s profit
p1D1(p1)−C1(D1(p1))+δ[p2D2(p2)−C2(D2(p2), D1(p1))]
where δ is the discount factor.
∂C1 (.) ∂D1
∂C2 (.) ∂D1
1
F OC1 : p1 ∂D
+
D
(p
)
−
−
δ
1
1
∂p1
∂q1 ∂p1
∂q1 ∂p1 = 0
∂C2 (.) ∂D2
2
+
D
(p
)
−
F OC2 : p2 ∂D
2
2
∂p2
∂q2 ∂p2 = 0
• In the second period, monopoly price as
1
$2 =
ε22
• In the first period, the monopolist sets a lower price as
1
$1 <
ε11
• Thus, the monopolist reduces the price at date 1 (and
sells more in the 1st period) to reduce the cost (and
thus increase the profit) in the 2d period.
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• However, in a more general setting (exercise 1.7) where
the output grows over time with stationary demand and
the cost decreases with experience, there are 2 effects:
a. myopic behavior: as MC decreases, quantity must
increase.
b. non myopic behavior: higher quantity at the
beginning.
• ⇒First effect dominates the second effect: The
monopolist does not reduce the price in the first period.
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2 Cost Distortion
• Distortion on the supply side.
• For given quantities, a monopolist may produce at a
higher cost than would a competitive firm.
• Delegation problem
– shareholders and manager do not have the same
objective.
– Thus, problem of monitoring and controlling.
– ⇒ inefficiency.
• How this inefficiency is affected by market power?
• Shareholders can use yardstick competition (comparison with other firms)
• Example: Ford management can be compared to GM.
• But you need to have another firm to be able to
compare!
• These extra costs add to the DWL.
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3 Rent Seeking behavior
• Third kind of distortion: the wasteful expenses incurred
by a firm to get and to maintain a monopoly position.
• The rent of the monopolist (profit) may lead to
rent-seeking behavior.
– Firms will tend to spend money and effort to acquire
the monopoly position;
– Once installed they will tend to keep on spending
money and exerting effort to maintain it.
• Different kinds of expenses:
– Strategic expenses
∗ R&D cost of obtaining a patent (chapter 10),
∗ accumulation of capital,
∗ barriers to entry (chapter 8).
– Administrative expenses
∗ cost of lobbying,
∗ advertising campaigns,
∗ legal expenses against charges of antitrust
violation.
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• Axiom (Porter, 75) says that
1. rent dissipation (total expenses to obtain rent = amount
of the rent). This is the zero-profit free entry condition.
2. socially wasteful dissipation: it is not socially valuable.
Regulated monopoly is allocated on the basis of
lobbying influence.
• So, rent-seeking behavior certainly wastes some of the
monopoly profit.
• The monopoly profit may be part of the welfare loss,
but what fraction, it is not clear...
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4 Conclusion
Distortions created by monopoly power
1. high prices, DWL
2. inefficiency (because objectives of managers are
different to those of the owner of the firm)
3. dissipation of the monopoly profit.
But a monopoly can have some advantages:
1. under increasing return to scale, production by a single
firm is technologically more efficient (it is less costly
to have only one firm, natural monopoly). It prevents a
wasteful duplication of fixed costs.
2. Schumpeter said that monopoly may be a necessary
condition to a decent amount of R&D (patents).
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