1. Multiproduct Monopoly, Commodity Bundling, and Correlation of Values Authors: R. Preston McAfee 1, John McMillan 2, Michael D. Whinston 3 1 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 2

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] 3

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 4

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 5

6. Model (1/4) A multiproduct monopolist sells two products, goods 1 and 2 – Produced at constant marginal costs c 1 and c 2 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 values 4 for each of the two goods, (v 1, v 2 ) ≧ 0 6 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(v 1, v 2 ) This distribution possesses no atoms (?) and represent its density function by f(v 1, v 2 ) Let g i (v i |v j ) and h i (v i ) denote the conditional and marginal densities derived from f( ．, ． ) G i (v i |v j ) and H i (v i ) denote the conditional and marginal distribution functions 7

8. Model (3/4) We assume that for each good i there exists a positive measure of consumers who have v i > c i (in order to avoid trivial outcomes) Resale by consumers is assumed to be impossible The monopolist can choose one of three pricing policies 1.Price each commodity separately 2.Offer the goods for sale only as a bundle, with a single bundle price (refer as pure bundling) 3.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) 8

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 P B and single-good prices P 1 = P B – c 2 and P 2 = P B – c 1 always yield profits at least as high as pure bundling with price P B, 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 9

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 P B which is no larger than the sum of individual goods prices, P 1 + P 2 – 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 10

11. Results (2/20) We employ one additional assumption on the distribution of preferences: – g i (P i |s) is continuous in P i at P i * for all s, where P i * 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 11

12. Results (3/20) Proposition 1. Let (P 1 *, P 2 *) be the optimal nonbundling prices. Mixed bundling dominates unbundled sales if Proof. First, introduce a bundle whose price is P B = P 1 * + P 2 * Profits are unchanged since the bundle is irrelevant due to its pricing 12

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 13

14. Results (5/20) 14 Decide not to demand the bundle. P B – P 1 * means the price of good 2 in the bundle

15. Results (6/20) 15 The actions of and profits from consumers with v 1 > P 1 * are unaffected by this change in P 2

16. Results (7/20) The actions of and profits from consumers with v 1 > P 1 * are unaffected by this change in P 2 We need only focus on the change in profitability from sales to those with v 1 < P 1 * Profits from this group as a function of ε are given by 16

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. 17

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) 18

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

20. Results (11/20) If P 2 * 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, (P 1 * - c 1 )[1 – H 2 (P 2 *)] > 0 Under our continuity assumption: h 1 (P 1 *) > 0 Thus, condition (3) holds, and a local gain from bundling is possible. Q.E.D. 20

21. Results (12/20) One can see three effects of locally raising P 2. 1.There is a direct price effect from raising revenues received from consumers in the set {(v 1, v 2 )|v 1 ≦ P 1 *, v 2 ≧ P 2 *} 2.Sales of good 2 fall by the measure of area (abcd) 3.Sales of good 1 increase by the measure of area (defg) due to consumers switching from purchasing only good 2 to purchasing the bundle 21

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 P 1 * With independence, however, this local gain must be zero at the optimal price P 2 * because the monopolist desires the same good 2 (unbundles) price regardless of a consumer’s level of v 1 Thus, the net effect of moving to mixed bundling when reservation values are independently distributed is positive 22

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 23

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 P i *(v j ) be the monopolist’s optimal price for good i conditional on knowing that a consumer’s valuation for good j is v j – If P 2 *(v 1 ) is decreasing in v 1, the first term must be positive 24

25. Results (16/20) It might be thought that this condition – that those with low values of v j 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 ε i (P i |v j ) = P i {g i (P i |v j ) / [1 – G i (P i |v j )]} to be the demand elasticity of good i conditional on valuation v j for good j 25

26. Results (17/20) (*) > 0 (respectively, < 0) implies a strictly negative (respectively, positive) correlation between v 1 and v 2 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 26 (*) (**) (?)

27. Results (18/20) What if purchases can be monitored? – That is, no need to satisfy P B ≦ P 1 + P 2 Proposition 2. Let (P 1 *, P 2 *) be the optimal nonbundling prices. Suppose that the monopolist can monitor sales. Then bundling dominates unbundled sales if 27

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 P B = P 1 * + P 2 * Lower P 2 slightly to P 2 – ε, where ε > 0 The resulting distribution of sales is 28

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 29

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: (v 1A, v 1B, v 2A, v 2B ) The firms engage in simultaneous prices choices – Independent pricing – Pure bundling – Mixed bundling 30

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 (P 1B, P 2B ) Then we can define each consumer’s “pseudo- reservation value” for firm A’s two products as 31

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 equilibrium 5 if 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 32 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) 33