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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 (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 (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 (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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