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

2.4 Multiproduct Monopoly Matilde Machado Slides available from: http://www.eco.uc3m.es/OI-I-MEI/ 1 2.4 Multiproduct Monopoly The firm is a monopoly in all markets where it operates � i=1,….n goods sold by the monopolist � p=(p 1 ,….p n ) prices charged for each good (uniform) � q=(q 1 ,….q n ) quantities sold of each good � q i =D i (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 p i 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. Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 2 1

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 3 2.4 Multiproduct Monopoly A special case (theory) Suppose demands are independent i.e. they � only depend on their own price p i : q i =D i (p i ). Separability in the Cost function : � C(q 1 ,….q n )=C 1 (q 1 )+…C n (q n ) In this case the monopolist’s maximization problem may be written as n separate problems since the n markets are independent. Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 4 2

2.4 Multiproduct Monopoly A special case(cont.) n n ∑ ∑ Π = − Max D p p ( ) C D p ( ( )) i i i i i i { } = = p 1 ,... p i 1 i 1 n That is, the optimal pricing ∂Π = strategy is to have a higher = FOC: 0 for 1,..., i n margin in those markets in which ∂ p demand is less elastic. This is i ′ ′ ′ ⇔ + = the same result obtained in third- D p ( ) D p p ( ) C D p ( ( )) D p ( ) i i i i i i i i i i degree price discrimination, ′ − p C D p ( ( )) 1 ⇔ = except that here the goods are i i i i ε different while in third-degree p i i price discrimination we were Lerner Index dealing with the same good Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 5 2.4 Multiproduct Monopoly More General case – w.l.o.g. assume n=2 Π = + − Max ( , ) ( , ) ( ( , ), ( , )) D p p p D p p p C D p p D p p 1 1 2 1 2 1 2 2 1 1 2 2 1 2 { } p p , 1 2 ∂Π ∂ ∂ ∂ • ∂ ∂ • ∂ D p ( ) D ( ) p C ( ) D C ( ) D = ⇔ + + = + 1 2 1 2 FOC: 0 D p ( ) p p ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 1 2 p p p D p D p 1 1 1 1 1 2 1 ∂Π ∂ ∂ ∂ • ∂ ∂ • ∂ D ( ) p D p ( ) C ( ) D C ( ) D = ⇔ + + = + 2 1 1 2 0 D ( ) p p p ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 1 p p p D p D p 2 2 2 1 2 2 2 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 6 3

2.4 Multiproduct Monopoly Assume additive costs = + C q q ( , ) C q ( ) C q ( ) 1 2 1 1 2 2 Hence, the first FOC simplifies to: ∂ ∂ ∂ ∂ D p ( ) D ( ) p D D + + = ′ • + ′ • ( ) 1 2 ( ) 1 ( ) 2 D p p p C C ∂ ∂ ∂ ∂ 1 1 2 1 2 p p p p 1 1 1 1 ∂ ∂ D p ( ) D D ( ) p D p ⇔ + × + × = 1 1 2 2 1 D p ( ) p p 1 ∂ 1 ∂ 2 p D p D p 1 1 1 2 1 ∂ ∂ D D p D D p = ′ • × + ′ • × 1 1 1 2 2 1 C ( ) C ( ) ∂ ∂ 1 2 p D p p D p 1 1 1 1 2 1 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 7 2.4 Multiproduct Monopoly ∂ ∂ D p ( ) p D ( ) p p p + + = 1 1 2 1 2 D p ( ) D D 1 ∂ 1 ∂ 2 p D p D p 1 1 1 2 1 ∂ ∂ −ε 11 −ε 12 D p D D p D ′ ′ = • + • 1 1 1 2 1 2 C ( ) C ( ) ∂ ∂ 1 2 p D p p D p 1 1 1 1 2 1 −ε 11 −ε 12 The first FOC simplifies further to: p D D A − ε − ε = − ′ • ε − ′ • ε 2 1 2 D p ( ) D D C ( ) C ( ) 1 11 1 12 2 1 11 2 12 p p p 1 1 1 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 8 4

2.4 Multiproduct Monopoly A p D D − ε − ε = − ′ • ε − ′ • ε 2 1 2 D p ( ) D D C ( ) C ( ) 1 11 1 12 2 1 11 2 12 p p p 1 1 1 Multiply both sides by p 1 /D 1 : D D − ε − ε = − ′ • ε − ′ • ε 2 2 p p p C ( ) C ( ) 1 1 11 2 12 1 11 2 12 D D 1 1 ( ) D D ′ ′ ⇔ − − • ε = − + ε − • ε 2 2 p C ( ) p p C ( ) 1 1 11 1 2 12 2 12 D D 1 1 1 D 1 ( ) ( ) ′ ′ ⇔ − • = − − • ε 2 ( ) ( ) p C p p C ε ε 1 1 1 2 2 12 D 11 1 11 ( ) ′ ′ − • ε − • p C ( ) D p C ( ) 1 ⇔ = − 1 1 2 2 12 2 ε ε p p D 1 11 1 11 1 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 9 2.4 Multiproduct Monopoly Case 1: Independent goods ε 12 =0, ′ − • = p C ( ) 1 1 1 ε p 1 11 Case 2: Substitutes: ∂ ∂ D D p > ⇒ ε < ε = − < 2 2 1 0 0 because 0 ∂ 12 12 ∂ p p D � 1 1 2 + ( ) ′ ′ − • ε − • = The monopolist’s ( ) ( ) 1 p C D 1 p C − > 2 2 12 2 1 1 margin is higher ε ε ε p p D than with ��� � ���� � 1 11 1 11 1 11 independent goods − Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 10 5

2.4 Multiproduct Monopoly Case 2 (cont.): intuition: ↑ p 1 ⇒ ↑ D 2 gives incentives to the monopolist to ↑ p 2 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. Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 11 2.4 Multiproduct Monopoly Case 3: Complements ↑ p 1 ⇒ ↓ D 2 (because ↓ D 1 ) 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. ⇒ ∂ D < ⇒ ε > 2 0 0 ∂ 12 p 1 ( ) ′ ′ − • ε − • = p C ( ) D p C ( ) 1 1 − < 2 2 12 2 1 1 ε ε ε p p D ��� � ���� � 1 11 1 11 1 11 + Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 12 6

2.4 Multiproduct Monopoly Case 3 (cont.): Complements ↑ p 1 ⇒ ↓ D 2 (and so does ↓ D 1 ) therefore this gives incentives to the monopolist to ↓ p 2 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 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 13 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 D 1 (p 1 ) and costs � C 1 (q 1 ) For example, Period 2: q 2 =D 2 (p 2 ,p 1 ) and C 2 (q 2 ) � because when there are more ↓ p1 ↑ D 1 � consumers in ∂ D period 1, there is ↑ D 2 then (complements) < 2 0 more information ∂ p 1 about the product in period 2. Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 14 7

2.4 Multiproduct Monopoly Example1: (cont.): ∂ D = Note: 1 0 ∂ p 2 The profit of the monopolist is: { } − + δ − δ Max p D p ( ) C D p ( ( )) p D ( p p , ) C D ( ( p p , )) 1 1 1 1 1 1 2 2 1 2 2 2 1 2 p , p 1 2 ∂ D = 1 given that by definition 0 the problem in the 2nd period is standard: ∂ p 2 ∂Π = − ′ p C (.) 1 ⇔ = ⇔ 2 2 FOC: 0 monopoly price in period 2 ∂ ε p p 2 2 2 ∂ − ′ D p C (.) 1 < ⇒ ε > ⇒ < 2 1 1 since 0 (complements) 0 (lower launching prices) ∂ ε 12 p p 1 1 1 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 15 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. Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 16 8

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 q t =D t (p t ) (independent across � periods) C1(q1) 1st period cost function � C2(q1,q2) second period cost function � ∂ ∂ The higher the amount produced in the C C < > 2 0; 2 0 1st period, the lower are the costs in the ∂ ∂ q q 1 2 second period Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 17 2.4 Multiproduct Monopoly C2 ↑ q1 q2 Industrial Organization- Matilde Machado 2.4. Multiproduct Monopoly 18 9