Download 2.4 Multiproduct Monopoly

2.4 Multiproduct Monopoly
Matilde Machado
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2.4 Multiproduct Monopoly
„
„
„
„
„
„
The firm is a monopoly in all markets where it operates
i=1,….n goods sold by the monopolist
p=(p1,….pn) prices charged for each good (uniform)
q=(q1,….qn) quantities sold of each good
qi=Di(p) = demand of good i – Note that what is
important here is that demand for good i may
depend on the full price vector not only of pi
C(q1,…qn)= Cost function, depends on the quantities
produced of all goods. Note, quantities here
may not be added because the monopolist is
producing different goods.
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Examples:
Example 1: Launching Prices – e.g.: “imagénio” by
Telefónica, CNN plus (initial prices very cheap),
ING 1st deposit; cable TV (some extra channels
at very low prices).
Example 2: Learning-by-doing –
Example 3: New Product lines – Kmart, gas stations
at certain supermarkets.
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
A special case (theory)
¾
Suppose demands are independent i.e. they
only depend on their own price pi: qi=Di(pi).
¾
Separability in the Cost function :
C(q1,….qn)=C1(q1)+…Cn(qn)
In this case the monopolist’s maximization problem
may be written as n separate problems since the
n markets are independent.
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
A special case(cont.)
n
n
Max
Π = ∑ D ( p ) p − ∑ C ( D ( p ))
{
}
p1 ,... pn
i =1
i
i
i
i =1
i
i
i
∂Π
FOC:
= 0 for i = 1,..., n
∂pi
⇔ Di ( pi ) + Di′( pi ) pi =Ci′( Di ( pi )) Di′( pi )
⇔
pi − Ci′( Di ( pi )) 1
=
εi
pi
Lerner Index
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That is, the optimal pricing
strategy is to have a higher
margin in those markets in which
demand is less elastic. This is
the same result obtained in thirddegree price discrimination,
except that here the goods are
different while in third-degree
price discrimination we were
dealing with the same good
2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
More General case – w.l.o.g. assume n=2
Max
Π = D (p , p )p + D (p , p )p
{
}
1
1
2
1
2
1
2
2
− C ( D1 ( p1 , p2 ), D2 ( p1 , p2 ))
p1 , p2
FOC:
∂D ( p)
∂D ( p)
∂Π
∂C (•) ∂D1 ∂C (•) ∂D2
= 0 ⇔ D1 ( p ) + 1
+
p1 + 2
p2 =
∂p1
∂p1
∂p1
∂D1 ∂p1
∂D2 ∂p1
∂D ( p)
∂D ( p)
∂Π
∂C (•) ∂D1 ∂C (•) ∂D2
= 0 ⇔ D2 ( p ) + 2
+
p2 + 1
p1 =
∂p2
∂p2
∂p2
∂D1 ∂p2
∂D2 ∂p2
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Assume additive costs
C (q1 , q2 ) = C1 (q1 ) + C2 (q2 )
Hence, the first FOC simplifies to:
D1 ( p ) +
∂D1 ( p )
∂D ( p )
∂D
∂D
p1 + 2
p2 = C1′(•) 1 + C2′ (•) 2
∂p1
∂p1
∂p1
∂p1
⇔ D1 ( p ) +
D ∂D ( p )
D p
∂D1 ( p )
p1 × 1 + 2
p2 × 2 1 =
D1
D2 p1
∂p1
∂p1
= C1′(•)
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∂D1 D1 p1
∂D D p
×
+ C2′ (•) 2 × 2 1
∂p1 D1 p1
∂p1 D2 p1
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2.4 Multiproduct Monopoly
D1 ( p ) +
∂D1 ( p ) p1
∂D ( p) p1
p
D1 + 2
D2 2 =
∂p1 D1
∂p1 D2
p1
−ε11
−ε12
= C1′(•)
∂D1 p1 D1
∂D p D
+ C2′ (•) 2 1 2
∂p1 D1 p1
∂p1 D2 p1
−ε11
−ε12
The first FOC simplifies further to:
D1 ( p) − ε11 D1 − ε12 D2
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p2
D
D
= −C1′(•)ε11 1 − C2′ (•)ε12 2
p1
p1
p1
2.4. Multiproduct Monopoly
A
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2.4 Multiproduct Monopoly
D1 ( p ) − ε11 D1 − ε12 D2
p2
D
D
= −C1′(•)ε11 1 − C2′ (•)ε12 2
p1
p1
p1
A
Multiply both sides by p1/D1:
p1 − p1ε11 − p2
D2
D
ε12 = −C1′(•)ε11 − C2′ (•)ε12 2
D1
D1
⇔ − ( p1 − C1′(•) ) ε11 = − p1 + p2
⇔ ( p1 − C1′(•) ) = p1
⇔
1
ε11
D2
D
ε12 − C2′ (•) 2 ε12
D1
D1
− ( p2 − C2′ (•) )
D2
1
ε12
ε11
D1
p1 − C1′(•) 1 ( p2 − C2′ (•) ) ε12 D2
=
−
ε11
p1
p1ε11 D1
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2.4 Multiproduct Monopoly
Case 1: Independent goods ε12=0,
p1 − C1′(•) 1
=
p1
ε11
Case 2: Substitutes:
∂D2
∂D p
> 0 ⇒ ε12 < 0 because ε12 = − 2 1 < 0
∂p1
∂p1 D2
N
+
p1 − C1′(•) 1 ( p2 − C2′ (•) ) ε12 D2
1
=
−
>
p1
p1ε11 D1
ε11
ε11
−
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2.4. Multiproduct Monopoly
The monopolist’s
margin is higher
than with
independent goods
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2.4 Multiproduct Monopoly
Case 2 (cont.): intuition:
↑p1 ⇒↑D2 gives incentives to the monopolist to
↑p2
When maximizing the joint profit, the monopolist
internalizes the effects that the sale of one
good has on the demand of the others. In the
case of 2 substitute goods this implies that
the monopolist should increase the prices of
both goods relative to a situation where he
treated the two goods separately.
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Case 3: Complements
↑p1 ⇒↓D2 (because ↓D1 ) then we may guess
that the price of good 1 is lower than in the
case in which the monopolist would treat the
two goods independently. ⇒ ∂D2 < 0 ⇒ ε > 0
∂p1
12
p1 − C1′(•) 1 ( p2 − C2′ (•) ) ε12 D2
1
=
−
<
p1
p1ε11 D1
ε11
ε11
+
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2.4 Multiproduct Monopoly
Case 3 (cont.): Complements
↑p1 ⇒↓D2 (and so does ↓D1 ) therefore this gives
incentives to the monopolist to ↓p2
Note: If there is strong complementarity between the two
goods the monopolist sells, it may be optimal for the
monopolist to sell one of the goods, say good 1,
below its marginal cost in order to increase the
demand for good 2.
Example: Price of the mobile phone with and without
contract with the company
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Example 1: Launching prices, inter-temporal
production and imperfect information:
„
The Monopoly produces a single good
„
The good is sold in 2 consecutive periods
„
The first period’s demand is D1(p1) and costs
C1(q1)
For example,
„
Period 2: q2=D2(p2,p1) and C2(q2)
because when
there are more
„
↓p1
↑D1
consumers in
∂D2
period 1, there is
< 0 (complements)
↑D2 then
more information
∂p1
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2.4. Multiproduct Monopoly
about the product
in period 2.
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2.4 Multiproduct Monopoly
Example1: (cont.):
∂D1
Note: ∂p = 0
2
The profit of the monopolist is:
Max { p D ( p ) − C ( D ( p )) + δ p D ( p , p ) − δ C ( D ( p , p ))}
1
1
1
1
1
1
2
2
1
2
2
2
1
2
p1 , p2
∂D1
= 0 the problem in the 2nd period is standard:
∂p2
p − C2′ (.) 1
∂Π
FOC:
=0 ⇔ 2
= ⇔ monopoly price in period 2
p2
ε2
∂p2
p − C1′(.) 1
∂D2
(lower launching prices)
since
<
< 0 (complements) ⇒ ε12 > 0 ⇒ 1
p1
ε1
∂p1
given that by definition
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Example 1: (cont.):
Conclusion: The monopolist sacrifices some short-term
profits for higher long-term profits. Ex: launching
prices of CNN+, cable TV.
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2.4 Multiproduct Monopoly
Example 2: Learning by Doing – it is similar to a Multi-
„
„
„
„
product Monopolist with independent demands but
interdependent costs, i.e. costs decrease with quantity:
Monopolist produces a single good in two consecutive
periods
Demand in period t is qt=Dt(pt) (independent across
periods)
C1(q1) 1st period cost function
C2(q1,q2) second period cost function
∂C 2
∂C 2
< 0;
>0
∂ q1
∂q 2
The higher the amount produced in the
1st period, the lower are the costs in the
second period
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
C2
↑q1
q2
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2.4. Multiproduct Monopoly
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2.4 Multiproduct Monopoly
Example 2: (cont.): Monopolist maximizes:
Max { p D ( p ) − C ( D ( p )) + δ p D ( p ) − δ C ( D ( p ), D ( p ))}
1
1
1
1
1
1
2
2
2
2
1
1
2
2
p1 , p2
Again because period 2 does not have an effect in period 1's, the problem is standard:
∂C2
∂Π
FOC:
= 0 ⇔ δ D2 ( p2 ) + δ p2 D2′ ( p2 ) = δ
D2′ ( p2 ) ⇔ MR2 = MC2
∂p2
∂D2
∂Π
∂C
∂C2
= 0 ⇔ D1 ( p1 ) + p1 D1′( p1 ) = 1 D1′( p1 ) + δ
D1′( p1 )
∂p1
∂D1
∂D N1 −
−
∂p ()
∂C2
⇔ p1 + 1 q1 = MC1 + δ
⇒ MR1 < MC1
∂q1
∂D
N1
−
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2.4. Multiproduct Monopoly
q*1 is larger
than the
static
optimal
quantity.
Short-run
profits are
sacrificed.19
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