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Product Variety and Competitive Discounts

Daniel F. SpulberUniversity of Southern California, Los

Angeles, California 90089

Journal of Economic Theory 48, 510-525 (1989) Cited: 51 times

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Why choose this paper

• Nonlinear pricing• From monopoly to competing environment

• Application: spectrum management with multiple Mobile Network Operators (competing firms)

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Outline

• Introduction• Monopolistic Competition Model• Competitive Price Schedules• Discussion– Variety and efficiency– Equilibrium with free entry: an example– Nonlinear pricing vs. linear pricing

• Conclusion• Comment

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Introduction (1/4)

• Nonlinear pricing: the relation between quantities and total price is linear.

• Nonlinear pricing is generally making more profit than linear pricing.

• Traditionally, price discrimination had been studied in a monopoly setting.

• And then it is studied in a competitive setting.

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Introduction (2/4)

• Competitive setting– Nonlinear pricing• General multiproduct setting [7]• Free entry issue [24][19]

– Two part tariffs• Nash equilibrium in a differentiated product duopoly

[2]• Bertrand-Nash equilibrium in a Hotelling framework [3]

[2] P. S. CALEM AND D. F. SPULBER, Multiproduct two part tariffs, Inr. J. Ind. Organ. 2 (1984), 105-l 15.[3] P. C. COYTE AND C. R. LINDSEY, “Spatial Monopoly and Spatial Monopolistic Competition with Two-Part Pricing,” University of Alberta, Economics Department, May 1986.[7] E. GAL-OR, “Nonlinear Pricing-Oligopoly Case,” Working Paper 425, University of Pittsburgh, Graduate School of Business, 1981.[19] J. C. PANZAR AND A. W. POSTLEWAITE, Sustainable outlay schedules, Northwestern University discussion paper, 1984[24] D. F. SPULBER, Competition and multiplant monopoly with spatial nonlinear pricing, Int. Econ. Rev. 25 (1984), 425439.

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Introduction (3/4)

• In this paper, a model of monopolistic competition with free entry of firms and nonlinear pricing is presented. – Monopolistic competition: a form of imperfect

competition where many competing producers sell products that are differentiated from one another. – from Wikipedia

– Free entry: free for firms to enter the market

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Introduction (4/4)

• In the following of this paper, we would see– Nonlinear price equilibrium– Variety and efficiency– Equilibrium with free entry– Nonlinear pricing vs. linear pricing

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Monopolistic Competition Model (1/8)

• This section includes– The description of consumers– The equilibrium

• A. The description of consumers– The set of available brands is represented by

locations lj, j = 1, …, m in a circular brand space of unit length

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Monopolistic Competition Model (2/8)

– Consumers choose to purchase a single brand j– Each consumer has a most preferred good with

characteristics l* – Distance between the two, |l* - lj| – Consumers’ most preferred goods are uniformly

distributed in the brand space with density D– The number of units purchased, qj – Consumer’s utility, U = U(qj, |l* - lj|) + y• y is a numeraire commodity

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Monopolistic Competition Model (3/8)

– Each firm j offers a nonlinear price schedule Pj

(． ), j = 1, …, m.– Consumer’s net benefits from purchasing brand j

• where r = |l* - lj|

– Let qj(r) = qj(r, Pj(． )) denote the solution to (1)– The consumer selects from the available brands to

maximize net benefits

|)(|maxarg)( *** jjj llSlj

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Monopolistic Competition Model (4/8)

– Consumer’s preferences are assumed to be

• Marginal willingness to pay, v, is twice continuously differential and decreasing in q and r

– Let demand for q be normal (?) in r, vrq < 0– Without loss of generality, v may be

parameterized so that v is concave in r– The proceeding assumptions guarantee that a

complete separating equilibrium exists.

q

dxrxvrqU0

),(),(

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Monopolistic Competition Model (5/8)

– Consumers self-select by revealing the characteristics of their most preferred brand

– By well-known arguments, we have the following necessary conditions

– Let brands j be numbered clockwise in ascending order, j = 1, …, m around the brand space

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Monopolistic Competition Model (6/8)

– The price schedule Pj and Pj+1 induce a partition of consumers with preferred brands in the interval [lj, lj+1]

– Fig 1 represents two possible results of Lemma 2• Local monopoly• Competition

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Monopolistic Competition Model (7/8)

• B. Equilibrium– Firm cost functions are given by C(Q) = F + kQ– Firm strategies consist of a brand location lj and a

price schedule Pj(． )• Brand location coincides with another firm leads to

noexistence of equilibrium

– The present analysis considers a two-stage game• Firms commit to brand location lj • Firms compete with price schedules Pj(． )

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Monopolistic Competition Model (8/8)

– The perfect equilibrium consists of• A market structure j = 1, …, m* • A set of strategies (lj*, Pj*)

– Such that the following apply:• In the second stage, given locations lj*, firm price

schedules Pj* are chosen to maximize profits at a Bertrand-Nash (?) equilibrium• In the first stage, all firms in the market must anticipate

nonnegative profits• There is free entry in the first stage and any additional

entrant (m* + 1) earns negative profits

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Competitive Price Schedules (1/9)

• This section includes– The second stage equilibrium for a given market

structure m* and given firm locations lj* • Local monopoly• Competition

– The first stage equilibrium strategies

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Competitive Price Schedules (2/9)

• A. Local monopoly– Each firm chooses Pi(． ) subject to the individual

rationality constraint Si(r) >= 0 for all r <= B, where B is the firm’s market boundary

– The monopoly has incentive to raise the total outlays Pi until Si(B) = 0

– The firm’s problem is to choose its price schedule P (． ) to maximize profits

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Competitive Price Schedules (3/9)

– Proposition 1 gives the equilibrium strategy

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Competitive Price Schedules (4/9)

• B. Competitive equilibrium– Firm i‘s market boundaries Bi

+ and Bi- will depend

on its location li* and on the equilibrium nonlinear price strategies of rivals, Pi+1* and Pi-1*

– The marginal consumers Bi+ and Bi

- are defined by

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Competitive Price Schedules (5/9)

– From the consumer’s problem

– From Lemma 1

– So we have

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Competitive Price Schedules (6/9)

– Applying integration by parts to (9) and using (7), the competitive equilibrium strategy Pi* is obtained by choosing qi as follows

• subject to qi(r) nonincreasing

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Competitive Price Schedules (7/9)

– Proposition 2 gives the equilibrium strategy

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Competitive Price Schedules (8/9)

• C. First stage equilibrium strategy

Only one firm

(m*2BM <= 1) => (m <= 1/2BM)

(m*2BM > 1)

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Competitive Price Schedules (9/9)

• The purchase of the marginal consumer is raised since q(r) is nonincreasing, q(B*) > q(BM)

• Thus we have an immediate consequence of Proposition 3.

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Discussion (1/8)

• This section discusses three topics– Variety and efficiency– Equilibrium with free entry: an example– Nonlinear pricing vs. linear pricing

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Discussion (2/8)

• A. Variety and efficiency– The effect of increased variety• Increasing variety allows consumers to purchase goods

whose characteristics closely resemble their most preferred good• Given a sufficient condition vq(q, r)/r is nondecreasing

in r, quantity discounts exist (Lemma 3)

• With greater variety, the total output is greater (Proposition4)

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Discussion (3/8)

– Efficiency• With sufficient variety, the monopolistically competitive

equilibrium with nonlinear pricing approximates perfect competition (Proposition 5)

• As m → ∞, P*(q) approaches• Besides, as B* → 0, all consumer purchases approach

q(r)• So consumers pay only marginal cost k (marginal cost

pricing

q

qdxxxvkq

)0())(,()0(

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Discussion (4/8)

• B. Equilibrium with free entry: an example– A frequently observed property of monopolistic

competition is that as fixed costs become small, the equilibrium approaches the competitive outcome.

– We verify that this result holds for a given example• U(q, |l* - lj|) = αq – (1/2) βq2 - cq|l* - lj|• => q(r) = (α – k – 2cr)/ β• ckBM 2/)(

~

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Discussion (5/8)

– Given m firms and a competitive equilibrium, per firm profits are given by

– The derivative of profits with respect to m is

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Discussion (6/8)

– For sufficiently large m, π’(m) < 0• => π (m) is decreasing

– For F sufficiently small, there exists m(F/D) such that • π (m(F/D)) – F >= 0• π (m(F/D) + 1) – F < 0• m(F/D) nonincreasing in F/D

– Thus m(F/D) → ∞ as F/D → 0

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Discussion (7/8)

• C. Nonlinear pricing vs. linear pricing• Nonlinear pricing yields greater profits than linear pricing

for a monopoly– But it is not apparent at a competitive equilibrium

• We show that in the quadratic utility case, non linear pricing increases profits even under competition

• Let marginal cost k = 0– Profit at the competitive equilibrium with market structure m*

– Profit at a linear pricing equilibrium [17]

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Discussion (8/8)

– It can be shown that for m sufficiently large (m >= 5)

– For small fixed costs (? marginal costs), nonlinear pricing yields greater profits than linear pricing at a free entry equilibrium

)()( mm

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Conclusion

• This paper models monopolistic competition• The two strategies in the second stage

equilibrium are presented and necessary and sufficient conditions are obtained.

• In the end, nonlinear pricing is shown to have greater profit than linear pricing and is shown to approach the perfect competition outcome

• This implies that nonlinear pricing is a good approach when it comes to competing firms.

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Comment

• Still did not mention capacity constraint

• 調整 product的 location對MNO來說不是那麼直接，因為賣的物品是相同的，如果要讓每個MNO進入的 location不一樣，可能需要提供更多的服務給MVNO，例如保證不會賣給過多MVNO而影響 QoS(這與 trading model有關 )，或是額外提供monitor之類的服務等等

• MNO提供 unused spectrum給MVNO是很容易的嗎？ (free entry)