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Multiproduct Monopoly, Commodity Bundling, and Correlation of Values

Multiproduct Monopoly, Commodity Bundling, and Correlation of Values. Authors: R. Preston McAfee 1 , John McMillan 2 , Michael D. Whinston 3. Source: The Quarterly Journal of Economics, Vol. 104, No. 2 (May, 1989), pp. 371-383.

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Multiproduct Monopoly, Commodity Bundling, and Correlation of Values

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  1. Multiproduct Monopoly, Commodity Bundling, and Correlation of Values Authors: R. Preston McAfee1, John McMillan2, Michael D. Whinston3 Source: The Quarterly Journal of Economics, Vol. 104, No. 2 (May, 1989), pp. 371-383 1 R. Preston McAfeem, VP and Research Fellow, Yahoo! Research, 2007-Present 2 John McMillan (January 22, 1951 – March 13, 2007) was the Jonathan B. Lovelace professor of economics in Stanford University's Graduate School of Business 3 Michael D. Whinston, Robert E. and Emily H. King Professor of Business Institutions, Northwestern University

  2. Outline • Introduction • Model • Results • Sufficient conditions to bundling dominates unbundled sales • Correlation of reservation values • Oligopoly • Comments

  3. Introduction (1/3) • Through what selling strategy can a multiproduct monopolist maximize his profits when his knowledge about individual consumers’ preference is limited? • Quantity-dependent pricing, studied in the context of a single-good monopoly [Oi, 1971][Maskin and Riley, 1984] • Package two or more products in bundles rather than selling them separately [Stigler, 1968][Adams and Yellen, 1976]

  4. Introduction (2/3) • We investigate the conditions under which bundling is an optimal strategy in the Adams and Yellen model • Our analysis distinguishes between cases where the monopolist can and cannot monitor the purchases of consumers • For each case we provide and interpret sufficient conditions for bundling to dominate unbundled sales

  5. Introduction (3/3) • Bundling is always an optimal strategy whenever reservation values for the various goods are independently distributed in the population of consumers • When purchases can be monitored, bundling dominates unbundled sales for virtually all joint distributions of reservation values

  6. Model (1/4) • A multiproduct monopolist sells two products, goods 1 and 2 • Produced at constant marginal costs c1 and c2 • Each consumer • desires at most one unit of each goods, • demands each independently of his consumption of the other, • and is characterized by his reservation values4 for each of the two goods, (v1, v2) ≧ 0 4 Reservation price: the biggest price a buyer is going to pay for a good or service, or the smallest price at which a seller is going to sell a good or service

  7. Model (2/4) • Reservation values are jointly distributed in the population according to the distribution function F(v1, v2) • This distribution possesses no atoms (?) and represent its density function by f(v1, v2) • Let gi(vi|vj) and hi(vi) denote the conditional and marginal densities derived from f(．, ．) • Gi(vi|vj) and Hi(vi) denote the conditional and marginal distribution functions

  8. Model (3/4) • We assume that for each good i there exists a positive measure of consumers who have vi > ci (in order to avoid trivial outcomes) • Resale by consumers is assumed to be impossible • The monopolist can choose one of three pricing policies • Price each commodity separately • Offer the goods for sale only as a bundle, with a single bundle price (refer as pure bundling) • Offer to sell either separately or bundled, with a price for the bundle that is different from the sum of the single good prices (refer as mixed bundling)

  9. Model (4/4) • We can rule out pure bundling as a (uniquely) optimal strategy, • Because mixed bundling is always (weakly) better: • Mixed bundling with a bundle price PB and single-good prices P1 = PB – c2 and P2 = PB – c1 always yield profits at least as high as pure bundling with price PB, and typically does better [Adams and Yellen, p. 483] • Thus, we now turn to a comparison of the profits obtainable from mixed bundling and unbundled sales

  10. Results (1/20) • It is important to distinguish between cases where the monopolist can and cannot monitor purchases • In the latter case he is effectively constrained to offer a bundle price PB which is no larger than the sum of individual goods prices, P1 + P2 • On the other hand, if the monopolist can monitor purchases, then he faces no such constraint since he can always prevent consumers from purchasing both good 1 and good 2 separately (how?) • This problem disappears when bundle discounts used

  11. Results (2/20) • We employ one additional assumption on the distribution of preferences: • gi(Pi|s) is continuous in Pi at Pi* for all s, where Pi* is the optimal nonbundling price for good I • We begin by establishing a general sufficient condition (valid regardless of whether purchases can be monitored) for bundling to dominate unbundled sales

  12. Results (3/20) • Proposition 1. Let (P1*, P2*) be the optimal nonbundling prices. Mixed bundling dominates unbundled sales if • Proof. First, introduce a bundle whose price is PB = P1* + P2* • Profits are unchanged since the bundle is irrelevant due to its pricing

  13. Results (4/20) • Consider a small increase in the price of good 2 to , where ε > 0 • The ability to monitor purchases is irrelevant since • The resulting pattern of purchases is give by

  14. Results (5/20) Decide not to demand the bundle. PB – P1* means the price of good 2 in the bundle

  15. Results (6/20) The actions of and profits from consumers with v1 > P1* are unaffected by this change in P2

  16. Results (7/20) • The actions of and profits from consumers with v1 > P1* are unaffected by this change in P2 • We need only focus on the change in profitability from sales to those with v1 < P1* • Profits from this group as a function of ε are given by

  17. Results (8/20) • Differentiating this expression with respect to ε and taking the limit as ε→ 0 yields the expression in condition (1) implying that ξ’(0) > 0 so that a strict improvement in profits would be possible through bundling. Q.E.D.

  18. Results (9/20) • In principle, one could also consider marginally raising the price of good 1 or marginally lowering the price of the bundle from the initial position • Either of these changes gives rise to a marginal change in profits that is identical to that in the expression in condition (1)

  19. Results (10/20) • One case of special interest is that of independently distributed reservation values • Corollary 1. If v1 and v2 are independently distributed, then bundling dominates unbundled sales. • Proof. For the case of independently distributed reservation values, condition (1) reduces to (note that hi(P) = gi(P|s) for all (P, s) and i = 1, 2):

  20. Results (11/20) • If P2* is the optimal unbundled price for good 2, then the first term in (2) is equal to zero (?), so that (1) reduces to • By the assumptions of no atoms (?) and existence of a positive measure of valuations above cost, (P1* - c1)[1 – H2(P2*)] > 0 • Under our continuity assumption: h1(P1*) > 0 • Thus, condition (3) holds, and a local gain from bundling is possible. Q.E.D.

  21. Results (12/20) • One can see three effects of locally raising P2. • There is a direct price effect from raising revenues received from consumers in the set {(v1, v2)|v1≦ P1*, v2 ≧ P2*} • Sales of good 2 fall by the measure of area (abcd) • Sales of good 1 increase by the measure of area (defg) due to consumers switching from purchasing only good 2 to purchasing the bundle

  22. Results (13/20) • The sum of the first two of these effects is exactly the local gain if the monopolist was able to slightly raise the price of good 2 only to consumers with valuations less than P1* • With independence, however, this local gain must be zero at the optimal price P2* because the monopolist desires the same good 2 (unbundles) price regardless of a consumer’s level of v1 • Thus, the net effect of moving to mixed bundling when reservation values are independently distributed is positive

  23. Results (14/20) • Proposition 1 implies that bundling is generally optimal in a much broader range of cases than just independence • The second term on the left side of inequality (1) corresponds to the rectangular area (efgd): This is the extra profit from consumers who buy both goods with bundling when without bundling they would buy only one good • This effect will always tend to generate gains from bundling

  24. Results (15/20) • The first term in (1), though, may be either positive or negative (in the independence case it is zero) • We can provide an intuitive sufficient condition for this term to be nonnegative: • Let Pi*(vj) be the monopolist’s optimal price for good i conditional on knowing that a consumer’s valuation for good j is vj • If P2*(v1) is decreasing in v1, the first term must be positive

  25. Results (16/20) • It might be thought that this condition – that those with low values of vj should be charged a higher price for good i – is directly tied to the presence of a negative correlation of reservation values • Not completely accurate • Defining εi(Pi|vj) = Pi{gi(Pi|vj) / [1 – Gi(Pi|vj)]} to be the demand elasticity of good i conditional on valuation vj for good j

  26. Results (17/20) (?) • (*) > 0 (respectively, < 0) implies a strictly negative (respectively, positive) correlation between v1 and v2 • The presence of (**) indicates that the sign of the first term in (1) cannot be tied solely to the level of correlation between reservation values (?) (**) (*)

  27. Results (18/20) • What if purchases can be monitored? • That is, no need to satisfy PB≦ P1 + P2 • Proposition 2. Let (P1*, P2*) be the optimal nonbundlingprices. Suppose that the monopolist can monitor sales. Then bundling dominates unbundled sales if

  28. Results (19/20) • Proof. Suppose the expression in (5) is positive, then Proposition 1 applies • Suppose the expression is negative • Introduce a bundle with price PB = P1* + P2* • Lower P2 slightly to P2 – ε, where ε > 0 • The resulting distribution of sales is

  29. Results (20/20) • Similar to the proof of Proposition 1, • Analysis of the limit of the derivative of profits with respect to ε as ε→ 0 then indicates that a gain in profits is available. Q.E.D. • From (5), independent goods pricing will virtually never be an optimal sales strategy here when purchases can be monitored

  30. Oligopoly (1/3) • Apply the results above to the case of multiproduct oligopoly • Consider a duopoly comprised of firm A and firm B, • Each of which produces a version of products 1 and 2 • Consumer valuations: (v1A, v1B, v2A, v2B) • The firms engage in simultaneous prices choices • Independent pricing • Pure bundling • Mixed bundling

  31. Oligopoly (2/3) • Given the pricing choice of its rival, each firm acts as a monopolist relative to the demand structure induced by its rival’s prices • Suppose that firm B is pricing its products independently at prices (P1B, P2B) • Then we can define each consumer’s “pseudo-reservation value” for firm A’s two products as

  32. Oligopoly (3/3) • If we let be the distribution of these induced reservation values, • it is not difficult (?) to show that independent pricing can only be a Nash equilibrium5if condition (1) (or (5) if monitoring of purchases is possible) fails to hold for the pseudo-reservation values induced at the independent pricing Nash equilibrium prices 5 A and B are in Nash equilibrium if A is making the best decision it can, taking into account B’s decision, and B is making the best decision it can , taking into account A’s decision

  33. Comments • In this paper’s model, each consumer desires at most one unit of each good • It is straightforward to suit to the Cloud Computing Services when fixed-fee services are provided • The unit of each service demanded by any consumer is up to one • Thus the possible strategies might be • Usage-based (light user) • Fixed-fee (heavy user) • Service-bundling (heavy user with multiple tastes)

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