Product Variety and Competitive Discounts

Cited: 51 times Journal of Economic Theory 48, 510-525 (1989) Product Variety and Competitive Discounts Daniel F. Spulber University of Southern California, Los Angeles, California 90089

Why choose this paper • Nonlinear pricing • From monopoly to competing environment • Application: spectrum management with multiple Mobile Network Operators (competing firms)

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

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.

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.

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

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

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

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

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

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.

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

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

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(．)

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

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

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

Competitive Price Schedules (3/9) • Proposition 1 gives the equilibrium strategy

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

Competitive Price Schedules (5/9) • From the consumer’s problem • From Lemma 1 • So we have

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

Competitive Price Schedules (7/9) • Proposition 2 gives the equilibrium strategy

Competitive Price Schedules (8/9) • C. First stage equilibrium strategy (m*2BM <= 1) => (m <= 1/2BM) Only one firm (m*2BM > 1)

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.

Discussion (1/8) • This section discusses three topics • Variety and efficiency • Equilibrium with free entry: an example • Nonlinear pricing vs. linear pricing

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)

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

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

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

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

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]

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

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.

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)

Product Variety and Competitive Discounts