2.4 Multiproduct Monopoly Matilde Machado Slides available from: - - PDF document

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2.4 Multiproduct Monopoly

Matilde Machado Slides available from:

http://www.eco.uc3m.es/OI-I-MEI/

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 2

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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Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 3

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.

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 4

2.4 Multiproduct Monopoly

A special case (theory)

• Suppose demands are independent i.e. they
• nly depend on their own price pi: qi=Di(pi).
• 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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Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 5

2.4 Multiproduct Monopoly

A special case(cont.)

{ }

1,...

1 1

( ) ( ( )) FOC: 0 for 1,..., ( ) ( ) ( ( )) ( ) ( ( )) 1

n

n n i i i i i i p p i i i i i i i i i i i i i i i i i i i

D p p C D p i n p D p D p p C D p D p p C D p p

Max

ε

= =

= − ∂Π = = ∂ ′ ′ ′ ⇔ + = ′ − ⇔ =

∑ ∑ Π

Lerner Index 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 third- degree price discrimination, except that here the goods are different while in third-degree price discrimination we were dealing with the same good

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 6

2.4 Multiproduct Monopoly

More General case – w.l.o.g. assume n=2

{ }

1 2

1 1 2 1 2 1 2 2 1 1 2 2 1 2 , 1 2 1 2 1 1 2 1 1 1 1 1 2 1 2 1 1 2 2 1 2 2 2 1 2 2

( , ) ( , ) ( ( , ), ( , )) ( ) ( ) ( ) ( ) FOC: ( ) ( ) ( ) ( ) ( ) ( )

p p

D p p p D p p p C D p p D p p D p D p D D C C D p p p p p p D p D p D p D p D C C D p p p p p p D p D

Max

= + − ∂ ∂ ∂ ∂ ∂Π ∂

• ∂
• =

⇔ + + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂Π ∂

• ∂
• =

⇔ + + = + ∂ ∂ ∂ ∂ ∂ ∂

Π

2 2

D p ∂

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Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 7

2.4 Multiproduct Monopoly

Assume additive costs Hence, the first FOC simplifies to:

1 2 1 1 2 2

( , ) ( ) ( ) C q q C q C q = +

1 2 1 2 1 1 2 1 2 1 1 1 1 1 1 2 2 1 1 1 2 1 1 1 2 1 1 1 1 2 2 1 1 2 1 1 1 1 2 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D p D p D D D p p p C C p p p p D p D D p D p D p p p p D p D p D D p D D p C C p D p p D p ∂ ∂ ∂ ∂ ′ ′ + + =

• +
• ∂

∂ ∂ ∂ ∂ ∂ ⇔ + × + × = ∂ ∂ ∂ ∂ ′ ′ =

• ×

+

• ×

∂ ∂

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 8

2.4 Multiproduct Monopoly

The first FOC simplifies further to:

1 1 2 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 2 1 2 1 1 1 1 2 1

( ) ( ) ( ) ( ) ( ) D p p D p p p D p D D p D p D p D p D D p D C C p D p p D p ∂ ∂ + + = ∂ ∂ ∂ ∂ ′ ′ =

• +
• ∂

∂

−ε11 −ε12 −ε11 −ε12

2 1 2 1 11 1 12 2 1 11 2 12 1 1 1

( ) ( ) ( ) p D D D p D D C C p p p ε ε ε ε ′ ′ − − = −

• −
• A

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Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 9

2.4 Multiproduct Monopoly

Multiply both sides by p1/D1:

2 1 2 1 11 1 12 2 1 11 2 12 1 1 1

( ) ( ) ( ) p D D D p D D C C p p p ε ε ε ε ′ ′ − − = −

• −
• A

( ) ( ) ( ) ( )

2 2 1 1 11 2 12 1 11 2 12 1 1 2 2 1 1 11 1 2 12 2 12 1 1 2 1 1 1 2 2 12 11 1 11 2 2 12 2 1 1 1 11 1 11 1

( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) 1 D D p p p C C D D D D p C p p C D D D p C p p C D p C D p C p p D ε ε ε ε ε ε ε ε ε ε ε ε ε ′ ′ − − = −

• −
• ′

′ ⇔ − −

• = −

+ −

• ′

′ ⇔ −

• =

− −

• ′

−

• ′

−

• ⇔

= −

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2.4 Multiproduct Monopoly

Case 1: Independent goods ε12=0, Case 2: Substitutes:

1 1 1 11

( ) 1 p C p ε ′ −

• =
• 2

2 1 12 12 1 1 2

0 because D D p p p D ε ε

+

∂ ∂ > ⇒ < = − < ∂ ∂

( )

2 2 12 2 1 1 1 11 1 11 1 11

( ) ( ) 1 1 p C D p C p p D ε ε ε ε

−

′ −

• ′

−

• =

− >

• 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

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. ⇒

( )

2 2 12 2 1 1 1 11 1 11 1 11

( ) ( ) 1 1 p C D p C p p D ε ε ε ε

+

′ −

• ′

−

• =

− <

• 2

12 1

D p ε ∂ < ⇒ > ∂

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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

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)

• Period 2: q2=D2(p2,p1) and C2(q2)
• ↓p1

↑D1 ↑D2 then (complements)

2 1

D p ∂ < ∂

For example, because when there are more consumers in period 1, there is more information about the product in period 2.

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2.4 Multiproduct Monopoly

Example1: (cont.):

Note: The profit of the monopolist is:

1 2

D p ∂ = ∂

{ }

1 2

1 1 1 1 1 1 2 2 1 2 2 2 1 2 , 1 2 2 2 2 2 2 2 1

( ) ( ( )) ( , ) ( ( , )) given that by definition 0 the problem in the 2nd period is standard: (.) 1 FOC: monopoly price in period 2 since

p p

p D p C D p p D p p C D p p D p p C p p D p

Max

δ δ ε − + − ∂ = ∂ ′ − ∂Π = ⇔ = ⇔ ∂ ∂ < ∂

1 1 12 1 1

(.) 1 (complements) (lower launching prices) p C p ε ε ′ − ⇒ > ⇒ <

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 16

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

2 2 1 2

0; C C q q ∂ ∂ < > ∂ ∂

The higher the amount produced in the 1st period, the lower are the costs in the second period

Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 18

2.4 Multiproduct Monopoly

↑q1 q2 C2

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2.4 Multiproduct Monopoly

Example 2: (cont.): Monopolist maximizes:

{ }

1 2

1 1 1 1 1 1 2 2 2 2 1 1 2 2 , 2 2 2 2 2 2 2 2 2 2 2 2

( ) ( ( )) ( ) ( ( ), ( )) Again because period 2 does not have an effect in period 1's, the problem is standard: FOC: ( ) ( ) ( )

p p

p D p C D p p D p C D p D p C D p p D p D p MR MC p D

Max

δ δ δ δ δ − + − ∂ ∂Π ′ ′ = ⇔ + = ⇔ = ∂ ∂

• 1

2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1

( ) ( ) ( ) ( ) () C C D p p D p D p D p p D D p C p q MC MR MC q D δ δ

− − −

∂ ∂ ∂Π ′ ′ ′ = ⇔ + = + ∂ ∂ ∂ ∂ ∂ ⇔ + = + ⇒ < ∂ ∂

• q*1 is larger

than the static

• ptimal

quantity. Short-run profits are sacrificed.