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, D 0 ( 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: ε = − D 0 ( p ) D ( p ) 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 ) + 2 P 0 ( q ) − C 00 ( q ) < 0 Result 1: MR ( q m ) = MC ( q m ) ⇒ p m > MC 2
• Program of the monopoly (in price) max { pD ( p ) − C ( D ( p )) } p FOC : pD 0 ( p ) + D ( p ) − C 0 ( D ( p )) D 0 ( p ) = 0 ⇒ p m SOC : pD 00 ( p ) + 2 D 0 ( p ) − C 00 ( D ( p ))( D 0 ( p )) 2 − C 0 ( D ( p )) D 00 ( p ) < • + assumptions on D 00 ( p ) and C 00 ( q ) (concavity or quasi concavity). $ ≡ p m − C 0 ( . ) = 1 Result 2: p m 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: < 1 because p m − C 0 ( . ) < p m p m − C 0 ( . ) p m so, 1 ε < 1 ⇒ ε > 1 Result 4: Monopoly price is a non decreasing function of marginal cost. 3
Proof: • 2 alternative cost functions C 1 ( . ) and C 2 ( . ) . • C 0 2 ( q ) > C 0 1 ( q ) for any q . • p m 1 , q m 1 if C 1 ( . ) , and p m 2 , q m 2 if C 2 ( . ) . • If cost is C 1 ( . ) (resp. C 2 ( . ) ), p m 1 (resp. p m 2 ) is charged by the monopolist p m 1 q m 1 − C 1 ( q m 1 ) ≥ p m 2 q m 2 − C 1 ( q m 2 ) p m 2 q m 2 − C 2 ( q m 2 ) ≥ p m 1 q m 1 − C 2 ( q m 1 ) • Sum these 2 inequalities C 2 ( q m 1 ) − C 2 ( q m 2 ) ≥ C 1 ( q m 1 ) − C 1 ( q m 2 ) C 2 ( q m 1 ) − C 1 ( q m 1 ) − [ C 2 ( q m 2 ) − C 1 ( q m 2 )] ≥ 0 R q m 2 [ C 0 2 ( x ) − C 0 1 1 ( x )] dx ≥ 0 q m • Because C 0 2 ( x ) > C 0 1 ( x ) then q m 1 ≥ q m 2 and p m 1 ≤ p m 2 . • 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 FOC : pD 0 ( p + t ) + D ( p + t ) − C 0 ( D ( q + t )) D 0 ( p + t ) = ⇔ [ D ( p + t ) − tD 0 ( p + t )] + D 0 ( p + t ) [ p + t − C 0 ( D ( q + t ))] = 0 • From the second term: p + t = C 0 ( . ) . 5
• First term: D ( p + t ) = tD 0 ( p + t ) ⇒ t = D ( p c ) D 0 ( p c ) • 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 con- cerning cost and demand. – Demand can be found with statistic studies. – Difficult to get information on costs. • Incentive theory. 6
1.2 Multi-Product Monopolist • Multi-product firm has a monopoly power over all goods. • q i = D i ( p ) , demand for good i = 1 , ...., n . • Prices, p = ( p 1 , p 2 , ...., p n ) . • Quantities q = ( q 1 , q 2 , ..., q n ) . • Cost C ( q 1 , q 2 , ..., q n ) . • Single-product firm ( n pricing problems) ⇔ multi- product firm with – independent demands: q i = D i ( p i ) – separable costs: C ( q 1 , q 2 , ..., q n ) = P n i =1 C i ( q i ) . • For each good i $ i = p m i − C 0 = 1 i p m ε i i Result 6: The markup is higher on goods with a lower elasticity of demand (Ramsey pricing). 7
• General multi-product monopolist program is ( n ) X max p i D i ( p ) − C ( D 1 ( p ) , D 2 ( p ) , ..., D n ( p )) p 1 ,p 2 ,....,p n i =1 X X n ∂D i ( p ) ∂D j ( p ) ∂C ( . ) ∂D k ( p ) FOC i : p i + D i ( p ) + p j − ∂p i ∂p i ∂q k ∂p k j 6 = i k =1 ∀ i , ∀ k 6 = i • SOC must be satisfied. 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 • q i = D i ( p ) • C ( q 1 , q 2 , ..., q n ) = P n i =1 C i ( q i ) • Program of the multi-product monopolist is 8
( n ) X X n max p i D i ( p ) − C i ( D i ( p )) p 1 ,p 2 ,....,p n i =1 i =1 from (1) D i ( p ) + P ∂D j ( p ) ∂p i ( p j − ∂C j ( . ) p i − ∂C i ( . ) ∂q j ) j 6 = i ∂q i ⇒ = − ∂D i ( p ) p i p i ∂p i • Own elasticity of demand p i ∂D i ( p ) ε ii = − D i ( p ) ∂p i • Cross elasticity of demand for good j p i ∂D j ( p ) ε ij = − D j ( p ) ∂p i • FOC becomes ³ ´ p j − ∂C j ( . ) p i − ∂C i ( . ) D j ( p ) ε ij X = 1 ∂q j ∂q i − p i ε ii ε ii p i D i ( p ) j 6 = i • Sign of the second term? (depends on the sign of ε ij ) 9
1 – If ( − ), $ i > ε ii , and thus higher price than in the case of a single-product monopolist. – if ( + ), $ i < 1 ε ii , and thus lower price. • If goods are substitutes , ∂D j ( p ) > 0 for j 6 = i so ∂p i ε ij < 0 . – thus $ i > 1 ε ii ⇒ p i > p m . • If goods are complements , ∂D j ( p ) < 0 for j 6 = i so ∂p i ε ij > 0 – thus $ i < 1 ε ii ⇒ p i < p m . • Example : Intertemporal pricing. • Single-product monopolist • 2 periods: t = 1 , t = 2 . • At t = 1 , – demand function q 1 = D 1 ( p 1 ) – cost function C ( q 1 ) • At t = 2 , – demand function q 2 = D 2 ( p 2 , p 1 ) – cost function C ( q 2 ) • Goodwill effect: ∂D 2 ( . ) < 0 ∂p 1 10
• Monopolist’s profit p 1 D 1 ( p 1 ) − C ( q 1 ) + δ [ p 2 D 2 ( p 2 , p 1 ) − C ( D 2 ( p 2 , p 1 ))] where δ is the discount factor. • ⇔ multi-product firm with interdependent demands. ∂p 1 + D 1 ( . ) − ∂C ( . ) ∂p 1 + δ ( p 2 − ∂C ( . ) ∂D 1 ∂D 1 ∂q 2 ) ∂D 2 FOC 1 : p 1 ∂p 1 = 0 ∂q 1 ∂p 2 + D 2 ( p 2 , p 1 ) − ∂C ( . ) ∂D 2 ∂D 2 FOC 2 : p 2 ∂p 2 = 0 ∂q 2 • In the second period, monopoly price as $ 2 = 1 ⇒ p m 2 ε 22 • In the first period, the monopolist sets a lower price as $ 1 < 1 ⇒ p 1 < p m 1 ε 11 • 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. 11
1.2.2 Independent demands and dependent costs • The demand functions are independent, q i = D i ( p i ) . • C ( q 1 , q 2 , ..., q n ) • The program is ( n ) X max p i D i ( p i ) − C ( q 1 , q 2 , ..., q n ) p 1 ,p 2 ,....,p n 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 q 1 = D 1 ( p 1 ) – cost function C 1 ( q 1 ) • At t = 2 , – demand function q 2 = D 2 ( p 2 ) – cost function C 2 ( q 2 , q 1 ) with ∂C 2 ( . ) < 0 ∂q 1 12
• Monopolist’s profit p 1 D 1 ( p 1 ) − C 1 ( D 1 ( p 1 ))+ δ [ p 2 D 2 ( p 2 ) − C 2 ( D 2 ( p 2 ) , D 1 ( p 1 ))] where δ is the discount factor. ∂p 1 + D 1 ( p 1 ) − ∂C 1 ( . ) ∂p 1 − δ ∂C 2 ( . ) ∂D 1 ∂D 1 ∂D 1 FOC 1 : p 1 ∂p 1 = 0 ∂q 1 ∂q 1 ∂p 2 + D 2 ( p 2 ) − ∂C 2 ( . ) ∂D 2 ∂D 2 FOC 2 : p 2 ∂p 2 = 0 ∂q 2 • In the second period, monopoly price as $ 2 = 1 ε 22 • In the first period, the monopolist sets a lower price as $ 1 < 1 ε 11 • 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. 13
• 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. 14
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 (compar- ison 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. 15
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