Optimal Bundle Pricing under Correlated Valuations�

Bo Chen and Debing Niy

February 17, 2017

Abstract

We study optimal pricing issues for a monopolist selling two indivisible goods to a continuumof consumers with correlated private valuations over the goods, where the (positive or negative)correlation is modeled using copulas in the Fréchet family. We derive explicit optimal pricingschemes and comparative statics results for various environments in our setting. The optimalpricing schemes can take several forms, including pure bundling, partial mixed bundling, andmixed bundling, depending jointly on the degrees of asymmetry and correlation of the consumersvaluations. The explicit optimal pricing schemes also enable us to investigate whether and howthe monopolists prot can be further improved via random assignments.Keywords: Bundling, Correlated Valuations, Monopoly Pricing, Price Discrimination, Ran-

dom Mechanism.JEL Classication: D11, D42, D82, L12.

1 Introduction

Product bundle pricing typically refers to a strategy where a rm sells multiple products togetheras a combo package (and sometimes with component prices for individual products as well). Thisstrategy is a prevalent business practice in various industries, such as telecommunications services,digital information products, auto sales, and insurance products. While it is an important and heav-ily studied topic in the literature, our understanding of product bundle pricing has been somewhatimpeded by that even determining optimal bundling prices in simple settings can often be analyti-cally intractable. This paper provides a tractable multiproduct model of correlated valuations andderives closed-form solutions on optimal bundle pricing schemes in this framework.1

We consider a monopolist selling two indivisible products to a continuum of consumers withunit demand and correlated private valuations of the products. We model the buyersvaluationsvia bivariate copulas in the Fréchet family with uniform marginal distributions on asymmetric in-terval supports, which can be regarded as limitingprobability distributions given the marginal

�We thank two referees and the editor of this journal for very constructive comments. We are also grateful toYongmin Chen, Neil Gandal, Matthew Jackson, Stephen Morris, Max Stinchcombe, Caroline Thomas, Eyal Winter,Tom Wiseman, Haiqing Xu, and especially Jidong Zhou for helpful discussions. Remaining errors are ours.

yChen (Corresponding Author): Department of Economics, Southern Methodist University, USA(bochen@smu.edu); Ni: School of Management and Economics, University of Electronic Science & Technology ofChina, China (nidb@uestc.edu.cn).

1 In line with the bundling literature, we use correlated valuations and stochastically dependent valuations in-terchangeably (dependence here is a non-linear correlation based on copulas). The precise meaning of correlatedvaluations is presented in Section 2.

1

distributions and the degree of correlation.2 This enables us to x marginal distributions and con-sider joint valuation distributions with varying degrees of correlation. In particular, the consumersvaluations of the products can be independent, positively correlated, or negatively correlated.

The monopolist chooses a deterministic price-posting mechanism, or a pricing scheme, whichcan take the form of separate sales (only component prices being o¤ered), pure bundling (postinga single bundle price), partial mixed bundling (posting a bundle price and one component price),and mixed bundling where a bundle price and both component prices are o¤ered with the bundleprice not exceeding the sum of the component prices, a natural scheme when the monopolist cannotmonitor purchasing behavior of the buyers.3 Instead of treating pure bundling and partial mixedbundling as special cases of mixed bundling, we distinguish them explicitly in our study. Our mainpurpose is to analytically characterize the optimal pricing schemes and to analyze the e¢ cacy andcomparative statics of these pricing schemes in our setting.

We rst explicitly derive the optimal pricing schemes under correlated valuations, which canbe pure bundling, partial mixed bundling, or mixed bundling. The specic form of the optimalpricing schemes depends jointly on the magnitude of asymmetry in the supports and the degreeof correlation of the valuation distribution. Under both positive and negative correlation, mixedbundling is optimal when the support asymmetry and the degree of correlation are both small.Partial mixed bundling however becomes the optimal pricing scheme if the support asymmetry islarge (positive correlation) or if either the support asymmetry or the degree of correlation is large(negative correlation). Valuation correlation plays a salient role in driving partial mixed bundling tobe optimal with negative correlation with a higher degree of negative correlation, the monopoliststops selling the (dominant) product with possibly larger valuations separately, forcing consumersinterested mainly in the dominant good to purchase the entire bundle so as to achieve a better pricediscrimination outcome.4 Finally, pure bundling arises as an optimal pricing scheme only in thecase of symmetric distributions when the consumers valuations are su¢ ciently negatively correlated.The rationale is that pure bundling, treating both goods indiscriminately, is less e¤ective when thegoods are ex ante di¤erent to the consumers.

On comparative statics, we nd that the negative (resp., positive) correlation case is a more(resp., less) favorable setting for the monopolist, in that the optimal expected prot strictly increasesin the degree of negative correlation and strictly decreases in the degree of positive correlation. Suchcomparative statics results are widely accepted, but have not been analytically derived in the liter-ature. The driving force of this result is that with negatively correlated valuations, the monopolistcan always use both the bundle price and component prices e¤ectively to sort consumers into dif-ferent groups, while the component prices play rather passive roles when consumersvaluations arepositively correlated. Our analysis here provides a novel perspective on how valuation correlationa¤ects the e¢ cacy of price-posting mechanisms, particularly so for the positive correlation case,which is much less studied in the literature compared to negative correlation.5

2This copula has appeared in Chen and Riordan (2013). A copula is a joint probability distribution with givenmarginals (see Nelson (2006) for an introduction). Section 2 provides detailed justications for our copula choice.

3This is the standard no monitoring salesassumption in the literature, starting from Adams and Yellen (1976).4An (arguable) example of partial mixed bundling is the pricing scheme for base products and accessories. Con-

sumers with an old base product (i.e., razors, guitar or laptops) typically can only buy a new accessory (i.e., blades,guitar strings or batteries) or a new base product with a new accessory, but cannot buy a new base product withoutan accessory. We consider such consumers (with an old base product) because in our setting both products arevaluable for consumers and can be consumed separately and an accessory is only valuable to a consumer with a baseproduct (though the complementarity between a base product and an accessory is absent in our setting).

5We have also derived optimal pure bundling pricing schemes where the monopolist sells the products at a singlebundle price and there is certain similarity between the optimal pricing scheme and the optimal pure bundling schemein terms of the comparative statics in correlation. Detailed analysis on this is available upon request.

2

Finally, we go beyond deterministic price-posting mechanisms and investigate whether a randommechanism enables the monopolist to achieve strictly higher prots, which is a natural and directapplication of our characterization of the optimal pricing schemes. It is known in the literature thatcharacterizing the optimal (revenue-maximizing) selling mechanisms for a multi-product monopolistis signicantly more di¢ cult than that for a one-product monopolist, and the problem is in generalconsidered as intractable. Nevertheless, several previous studies present various examples to showthat random mechanisms can dominate price-posting mechanisms in revenue when a seller sellsmultiple indivisible goods (see, e.g., Thanassoulis (2004), Manelli and Vincent (2006, 2007), andHart and Reny (2015)).

Di¤erent from the previous literature, we investigate marginal deviations from our optimal(deterministic) pricing schemes toward a nearbyrandom mechanism. In the random mechanism,a consumer who only purchases an individual good under the optimal pricing scheme is insteado¤ered a lottery which awards the consumer with the individual good at the previous optimalcomponent price with probability close to one and the bundle at the previous optimal bundleprice with the remaining probability. The random mechanism intends to further exploit consumerswho purchase an individual good under the deterministic pricing scheme by selling more productsto such consumers and is in a sense close to the optimal deterministic pricing scheme. We ndthat the random mechanism strictly improves the monopolists prot whenever the valuations arenegatively correlated and the optimal pricing scheme features partial mixed bundling. While ourinvestigation here is restricted by the marginal-analysis approach, our analysis directly connectsthe random mechanism with an underlying optimal pricing scheme and our result reveals when anoptimal pricing scheme is not revenue-maximizing and how the deterministic pricing scheme canbe improved using lotteries, an approach that is new in the literature.

1.1 Related Literature

Various rationales and benets of product bundling have been identied in the business literature,including cost reduction, economies of scales, and complementarities among the underlying goods.From an economics perspective, bundling has been identied as a powerful price discriminationdevice, starting from the classic studies of Stigler (1963) and Adams and Yellen (1976), as well asa useful tool to protect and leverage market power (e.g., Stigler (1963), Whinston (1990)).6

On price discrimination, Schmalensee (1984) employs a bivariate normal distribution settingto show that bundling can dominate separate sales when consumer valuations are negatively, in-dependently, or positively correlated. Given the analytical di¢ culties, Schmalensees results arehowever mainly derived numerically. McAfee, McMillan and Whinston (1989) formally analyze atwo-product setting with general valuation distributions and establish su¢ cient conditions for prof-itability of mixed bundling (over separate sales) using a marginal analysis. Recently, using copulasto model stochastic dependence of valuations, Chen and Riordan (2013) demonstrate protabilityof bundling for settings much more general than the previous literature. In particular, the au-thors establish protability of mixed bundling when product valuations are negatively dependent,independent, or have su¢ ciently limited positive dependence.

This paper similarly focuses on the price discriminating aspect of bundling. While the previousbundling literature has addressed specic issues in general settings, we address multiple issues ina somewhat specic setting, so as to provide a complete picture on the optimal pricing schemeinstead of only evaluating whether/when separate sales is suboptimal. First, this paper is closelyrelated to Chen and Riordan (2013). Our copula choice is motivated by and a special case of Chen

6Our literature review focuses on bundling decisions by a monopolist. There is also an important literature onpricing and bundling decisions among competing rms. See Zhou (2017) and references therein.

3

and Riordan (2013) and we follow their methodology closely in our analysis. Our main objectiveis however di¤erent from that in Chen and Riordan (2013) in that we derive explicitly in oursetting the globally optimal pricing schemes and a corresponding comparative statics analysis incorrelation. Given such an objective, our study is also related to Eckalbar (2010) who developsanalytical results and insights for a mixed bundling problem in a two-good setting with independentand uniform marginal valuations.7 In comparison, our setting is more general and encompasses theentire ranges of valuation correlation and support asymmetry, which enables us to also derive somerelevant comparative statics results and to analyze the e¢ cacy of various pricing schemes undervarying degrees of valuation correlation. Indeed, our main motivation is to shed some light onthe optimal bundling and pricing issues when consumer valuations are correlated across products,which is largely under-explored in the current literature.

Our paper is also related to the multiproduct mechanism design literature (e.g., McAfee andMcMillan (1988), Armstrong (1996), Rochet and Choné (1998), Thanassoulis (2004), Manelli andVincent (2006, 2007), and Hart and Reny (2015)). The existing literature however lacks a generalcharacterization so far even for the case of two goods. Our result on randomization is mainlyon whether and how randomization can help improve revenue. The phenomenon where randomassignments can improve revenue has appeared in Thanassoulis (2004), Manelli and Vincent (2006,2007), and Hart and Reny (2015).8 Compared to these studies, our result is obtained in a specicsetting via a marginal analysis approach. The novelty of our approach however lies in that westart with a completely characterized deterministic price-posting mechanism and we ask when thissimple mechanism can be improved using randomization, thus establishing a novel link betweenprice-posting mechanisms and revenue-maximizing mechanisms.

2 Framework

We consider a classic framework where a prot-maximizing monopolist sells two indivisible productsindexed by i 2 f1; 2g to a continuum of buyers with measure one. The monopolist produces theproducts at a (normalized) zero marginal cost without capacity constraint.9 A representativeconsumer purchases at most one unit of each good and has private valuations v = (v1; v2) withvi being her valuation of good i. To focus on the price discrimination aspect of product bundling(rather than the complementarity aspect), we assume that a consumer with valuations v derivesutility v � I from allocation I = (I1; I2) where Ii 2 f0; 1g denotes whether the consumer consumesgood i (Ii = 1) or not (Ii = 0). A consumers willingness to pay on a good is hence independentof receiving the other good or not. We maintain the standard no monitoring sales assumptionthroughout, i.e., the monopolist cannot monitor consumers purchases and thus cannot preventthem from buying the two goods separately.

We employ two well-known copulas in the Fréchet family to model dependence of the consumersvaluations of the goods. Denote the marginal cumulative distribution function of vi as Hi (vi) over

7Eckalbar (2010) also considers the limiting cases of perfect (positive and negative) correlations, as well as ananalysis on consumer surplus.

8Thanassoulis (2004) shows that with two perfectly substitutable goods, randomization in the assignment ofgoods typically dominates deterministic assignments. Manelli and Vincent (2006) identify conditions where price-posting mechanisms are revenue maximizing, while Manelli and Vincent (2007) characterize revenue maximizingmultidimensional mechanisms for some valuation distributions. Recently, Hart and Reny (2015) among other thingsdemonstrate (via examples) revenue-nonmonotonicity and necessity of randomization in selling multiple goods.

9Our results continue to hold for the case of positive yet small marginal costs. In addition, notice that normalizingthe marginal costs to be zero is without loss of generality if the prices derived in the sequel are regarded as markups,i.e., prices net of marginal costs.

4

support [0; ai] with ai > 0, and the joint distribution function of (v1; v2) as F (v1; v2).10 Withoutloss of generality, we assume that a1 � a2 and call good 1 the dominant good in the sequel.11 Tomodel stochastic dependence of v1 and v2, we specify:

FN (v1; v2) = CN (H1 (v1) ;H2 (v2)) and FP (v1; v2) = CP (H1 (v1) ;H2 (v2)), where

CN (H1 (v1) ;H2 (v2)) = �maxfH1 (v1) +H2 (v2)� 1; 0g+ (1� �)H1 (v1)H2 (v2)CP (H1 (v1) ;H2 (v2)) = �minfH1 (v1) ;H2 (v2)g+ (1� �)H1 (v1)H2 (v2)

(1)

Here, superscripts denote types of correlation (Positive or Negative) and parameter � 2 [0; 1](resp., � 2 [0; 1]) captures the degree of negative dependence (resp., positive dependence). CopulasCN (H1 (v1) ;H2 (v2)) and CP (H1 (v1) ;H2 (v2)) are joint probability distributions of random vari-ables H1 (v1) and H2 (v2), each being uniformly distributed over [0; 1], while the joint probabilitydistribution functions FN (v1; v2) and FP (v1; v2) are dened on the rectangle [0; a1] � [0; a2]. Forour purpose of deriving explicit optimal pricing schemes, we shall focus on the benchmark case ofuniform (marginal) distributions throughout the paper.

Assumption (Uniform Margins). Valuation vi is uniformly distributed on [0; ai] for i 2 f1; 2g.Figure 1 provides a visual description of CN (H1 (v1) ;H2 (v2)) and CP (H1 (v1) ;H2 (v2)). To il-

lustrate, CN (H1 (v1) ;H2 (v2)) can be thought of resulting from drawing (H1(v1);H2(v2)) uniformlyfrom the negative 45-degree line in the unit square with probability � and drawing (H1 (v1) ;H2 (v2))uniformly from the rest of the unit square with probability (1� �). Accordingly, the distributionFN (v1; v2) can be seen as drawing (v1; v2) uniformly from the rectangle [0; a1]� [0; a2] with prob-ability (1� �), and uniformly from the negative diagonal line connecting (a1; 0) and (0; a2) withprobability �. Such a copula admits an interpretation that consumer valuations are driven jointlyby idiosyncratic shocks as well as a common shock.12

-@@

@@

@@

@@

@@

@@

H1(v1)

6H2(v2)

�

1

1

0-�

�����������

H1(v1)

6H2(v2)

�

1

1

0

Figure 1. illustrating CN (H1(v1);H2(v2)) (left) and CP (H1(v1);H2(v2)) (right).

There is a measure � (resp., �) consumers (uniformly) on the negative (resp., positive) diagonal line.

The motivation for our copula choice in (1) is threefold: First, given that there is insu¢ cientunderstanding of optimal pricing schemes under correlated valuations, a Fréchet copula is simple10With slight abuse of notation, we use v1, v2 to denote both random variables and their realized values.11Such a formulation of product asymmetry (via support asymmetry) is more general than it appears. In particular,

this formulation is equivalent to an alternative formulation where a representative consumer with valuations v =(v1; v2) derives utility kv1 + v2from buying the bundle with k � 1. The interpretation is that either the consumerlikes good 1 more and has an actual willingness-to-pay of kv1 from buying a unit of good 1, or for the same physicaltime period, the consumer needs good 1 more often (k times to be exact) than good 2 (the price of good 1 is thenthe payment for k units of good 1). We thank a referee for raising this issue to us.12We note that similar ideas of modelling valuation correlation have also appeared in Nalebu¤ (2004, Sec. III.C)

on pure bundling as an entry barrier and in Armstrong and Vickers (2010, Sec. 3.5) on competitive bundling issues.

5

and hence a good choice in modeling correlation valuations in a two-good framework. Second,by construction, the only di¤erence between joint distributions in our copulas is the degree ofinterdependence (parameters � and �), making it a natural environment for comparative staticsanalysis on such interdependence. Third, notice that our copulas are convex linear combinationsof the product copula and the Fréchet-Hoe¤ding upper and lower bounds for joint distributionswith the margins H1 (v1) and H2 (v2).13 The copulas are hence extreme joint distributions giventhe marginal distributions and correlation parameters, enabling us to interpret the derived resultsas the best/worstoutcomes for the monopolist.14

Our main goal is to characterize the optimal pricing schemes. Here a pricing scheme is adeterministic price-posting mechanism where the monopolist o¤ers take-it-or-leave-it prices (x; y; p),i.e., the component prices of goods 1 (x) and 2 (y) and the bundle price (p), so as to induce thebuyers to self-select into di¤erent market segments. To be specic, we focus on prices (x; y; p)satisfying x 2 [0; a1], y 2 [0; a2], and p 2 [max fx; yg ; x+ y], where the restriction on p is aconsequence of the no monitoring salesassumption.15

Given the representation (1) and the prices (x; y; p), we follow Chen and Riordan (2013) toderive the demands for good 1, good 2 and the bundle respectively as (l 2 fN;Pg)16

Q1(x; p) = H2(p� x)� C l(H1(x);H2(p� x));Q2(y; p) = H1(p� y)� C l(H1(p� y);H2(y));Q12(x; y; p) =

R H1(x)H1(p�y)[1� C

l1(z;H2(p�H�11 (z)))]dz + 1�H1(x)�Q1(x; p):

(2)

-1H1(x) H1(v1)

61

H2(y)

H2(v2)

@@@@H2(p� x)

H1(p� y)0Q1(x; p)

Q2(y; p)

Q12(x; y; p)

Figure 2. An illustration of The Demands in (2).

Figure 2 illustrates the demands heuristically for prices (x; y; p). Roughly each demand correspondsto the measure of buyers in the corresponding market segment For example, C l(H1(x);H2(p�x))denotes the measure with H1 (v1) < H1 (x) and H2 (v2) < H2 (p� x) in the unit square. Notice13 In our setting, the product copula is H1 (v1)H2 (v2), and the Fréchet-Hoe¤ding upper and lower bounds are

respectively minfH1 (v1) ; H2 (v2)g and maxfH1 (v1) +H2 (v2)� 1; 0g (Chapter 2.5 of Nelson (2006)).14A further comment on our copula model is perhaps important: Strictly speaking, given that we are considering

a specic class of distributions, it is by no means necessary to set up our model using copulas. However, our copulamodeling approach permits a compact formulation and it also enables us to further illustrate such distributionsthrough the lenses of the Fréchet-Hoe¤ding bounds as mentioned above.15 If the consumers purchasing behavior can be perfectly monitored, the monopolist can o¤er a pricing scheme

featuring bundling with a premium (instead of bundling with a discount), i.e., p > x + y. Absent monitored sales,such a pricing scheme is however problematic. The restriction p � max fx; yg on the other hand is an incentiveconstraint for the bundle price to have some e¤ect.16Here the derivative Cl1(z1; z2) = @C

l(z1; z2)=@z1 is exactly the conditional probability Cl (z2jz1) given z1 =H1 (v1) being uniformly distributed over [0; 1].

6

that we have implicitly accounted for the consumersoptimal purchasing decisions in deriving thedemand functions: Given prices (x; y; p), a consumer with valuations (v1; v2) purchases the bundleif v1 + v2 � p � max f0; v1 � x; v2 � yg, good 1 alone if v1 � x � max f0; v2 � y; v1 + v2 � pg, andgood 2 only if v2 � y � max f0; v1 � x; v1 + v2 � pg.

The monopolists objective is to choose prices x; y and p to maximize his expected prot�(x; y; p) given the associated demands in (2):

(P)maxx;y;p�(x; y; p) = xQ1(x; p) + yQ2(y; p) + pQ12(x; y; p)

s.t. x 2 [0; a1] ; y 2 [0; a2] ; p 2 [max fx; yg ; x+ y] :

The complexity in solving the problem (P) lies in multiple constraints and multiple parameters inthe maximization and that the copulas are not di¤erentiable, leading to too many cases to consider.Our approach to (P) is to search instead for various combinations of prices (x; y; p) that can ariseas optimal pricing schemes and then backtrack the parameter constellation for each pricing scheme.We explicitly identify four di¤erent pricing schemes: a separate sale strategy where the monopolistonly posts a price for each product, corresponding to prices (x; y; p) with x 2 (0; a1) ; y 2 (0; a2) andp � x+y; a pure bundling strategy with prices (x; y; p) satisfying x � minfa1; pg; y � minfa2; pg;and p 2 (0; a1 + a2), resulting in either purchasing the bundle or nothing for the consumers; amixed bundling strategy if x 2 (0; a1) ; y 2 (0; a2) and p 2 (max fx; yg ; x+ y), which featuresinteriorcomponent prices and a discount for purchasing the bundle; and nally, partial mixedbundling with prices (x; y; p) satisfying either x 2 (0; a1) ; y � minfa2; pg and p 2 (x; x+ a2)orx � minfa1; pg; y 2 (0; a2) and p 2 (y; y + a1), e¤ectively resulting in that the consumers eitherpurchase an individual product or both products, but not the other product individually.

3 Optimal Pricing Schemes

In this section, we explicitly characterize the optimal pricing schemes, analyze how the pricingschemes vary in the degree of valuation correlation, and discuss some welfare implications of thepricing schemes.

3.1 Negatively Correlated Valuations

It has long been recognized that (mixed) bundling is particularly attractive when the consumersvaluations are negatively correlated (e.g., Stigler (1963), Adams and Yellen (1976), and Chen andRiordan (2013)). The rough intuition is that absent perfect correlation, the e¤ects of loweringvaluation heterogeneity via the bundle price and capturing consumers with high demands for singleproducts via the component prices are jointly very powerful under negatively correlated valuations.We o¤er a complete characterization of the optimal pricing rule. In particular, we show that themonopolist unambiguously fares better with more negatively correlated valuations, and the exactform of the optimal pricing scheme is determined by both the support asymmetry and the degreeof correlation.

To build up useful intuition, we start with the simple case of symmetric supports a1 = a2 = a.Proposition 1 characterizes the monopolists optimal pricing scheme in this case:17

17All proofs are relegated to an Appendix. We adopt the usual tie-breaking rule throughout that consumers buythe good(s) if they are indi¤erent between buying and not buying. In addition, we note here that in the extremecases of perfectly correlation, pure bundling with bundle price a is equivalent to separate sales with component pricesa2and a

2, both being optimal when � = 1 and � = 1.

7

Proposition 1 The monopolists optimal pricing scheme (x�N ; y�N ; p

�N ) with degree-� negative cor-

relation and symmetric support (a1 = a2 = a) features

(i) Mixed Bundling for � 2�0; 13

�with x�N = y

�N =

2a3(1��) , and p

�N =

(4�p2+6�)a

3(1��) .

(ii) Pure Bundling for � 2�13 ; 1�with x�N = y

�N = p

�N = a.

Under symmetric supports, the monopolist sells both products as a pure bundle if (and onlyif as we shall see) the magnitude of correlation � is su¢ ciently large and the monopolist optsfor mixed bundling otherwise. The rough intuition is as follows: With independent valuations,mixed bundling with interior prices is optimal (Eckalbar (2010)). As � increases, a larger mass ofconsumersvaluations clusterson the negative 45-degree line, giving rise to a higher demand forboth the individual goods and the bundle. This induces the monopolist to increase both the bundleprice and component prices, with component prices increasing faster to push more consumers to buythe bundle. This process continues until the components prices (simultaneously) hit the choke price(a), leading to pure bundling. Hence, although mixed bundling is a more exible pricing strategywhich e¤ectively lowers valuation heterogeneity via a bundle price (i.e., capturing consumers withhigh v1+v2) and serves consumers with extreme values for individual goods via component prices(i.e., attracting consumers with high v1or v2only), a su¢ cient degree of negative correlationprompts the monopolist to abandon component prices entirely so as to increase prot by forcingconsumers to purchase the bundle.

While it has been known that pure bundling is optimal with perfect negative correlation (Eck-albar (2010)) and that the symmetric case is the most favorable to pure bundling (Schmalensee(1984)), Proposition 1 illustrates that pure bundling can be (uniquely) optimal for symmetric andsu¢ ciently negatively correlated valuations.18

Now consider the general case with asymmetric supports, i.e., a1 � a2. Intuitively, asymmetryin valuations creates at least two complications for the seller. First, with negative correlation, abundle price loses some of its grip in lowering valuation heterogeneity under asymmetric supportsin that a bundle price can no longer capture all consumers with valuations clustered on the negativediagonal line as e¤ectively and cleanly as before.19 We shall see that an immediate consequenceis that pure bundling ceases to be optimal whenever a1 > a2, indicating that pure bundling beingoptimal is indeed a rare event. Second, if a1 is su¢ ciently larger than a2, the component price forgood 1 should be much higher than that for good 2 but this creates a tension for the bundle price,which should be at least as large as the component price for good 1 but cannot be too large sothat enough buyers will buy the bundle. We shall see that this tension prompts the monopolist toopt for partial mixed bundling. With such intuition in place, we now present the optimal pricingscheme for the general case (a1 � a2):20

Proposition 2 (Optimal Pricing Scheme under Negative Correlation) The optimal pric-ing scheme (x�N ; y

�N ; p

�N ) with degree-� negative correlation has the following structure:

Case (i). If � < 2a2�a13a1 , the optimal pricing scheme is mixed bundling given by

x�N =2a1

3(1� �) ; y�N =

2a23(1� �) ; p

�N =

2(a1 + a2)�pa1a2(2 + 6�)

3(1� �) :

18Admittedly (as also pointed out by a referee), our result that pure bundling can be uniquely optimal is verymuch a consequence of our specic setting, where an �-measure of consumers have valuations clustering exactly onthe negative diagonal line.19Graphically, the bundle price is represented by a negative 45-degree line in [0; a1]� [0; a2], which is steeper than

the negative diagonal line of [0; a1]� [0; a2] with measure-� consumers when a1 > a2.20 In presenting the optimal pricing scheme, we sometimes choose to present the solutions (x�N ; y

�N ; p

�N ) implicitly,

as the explicit solutions are inconveniently long. We show in the proof that the solutions exist and are uniquelypinned down except in the extreme cases of � = 1 and � = 1.

8

If � � 2a2�a13a1 , then the optimal pricing scheme is pure bundling if a1 = a2, and is partialmixed bundling as long as a1 > a2 where the monopolist stops selling good 1 by itself. Specically,there is a unique threshold �(a1) 2 [0; 1) with �0(a1) > 0 such that

Case (ii). If � � �(a1), the optimal partial mixed bundling scheme is uniquely determined byx�N = p

�N ;

y�Na2+

p�N�y�Na1

= 1;

3a2(1� �)(1 + a1�a22a1 )(y�Na2)2 � [3(1� �)a2 + 2a1](

y�Na2) + a1 + a2 = 0:

Case (iii). If � < �(a1), the optimal partial mixed bundling scheme is uniquely determined by

x�N = p�N ;

p�N = �3(1��)(a1�a2)4a2[a1�(1��)a2] (y

�N )

2 + (1��)(a1�a2)a1�(1��)a2 y�N +

a12 ;

�(1� 2y�Na2) + (1� �)p

�N�y�Na1

(2� 3y�Na2) = 0:

Proposition 2 indicates that the monopolists optimal pricing scheme, which can take variousforms, is determined jointly by the correlation parameter (�) and the magnitudes of the supportsa1 and a2. Figure 3 graphically illustrates the optimal pricing schemes in Proposition 2.21

-a1x�N v1

6

a2

y�N

v2

Case (i)ccccccccccccc

@@@p�N � x�N

p�N � y�NGood 1

Good2

Bundle

-a1x�N v1

6

a2

y�N

v2

Case (ii)QQQQQQQQQQQQQQQ

@@@@@@

p�N � y�N

Good 2Bundle

-a1x�N v1

6

a2

y�N

v2

Case (iii)aaaaaaaaaaaaaaaaaaaaaaaaa

@@@@@@

p�N � y�N

Good 2

Bundle

Figure 3. An illustration of the optimal pricing schemes in Proposition 2.

Consumers with valuations in Area Good 1buy good 1 only (and similarly for AreaGood 2

and Area Bundle). The negative diagonal line has measure � consumers.

21Both case (ii) and case (iii) feature partial mixed bundling. The di¤erence is that in case (ii) the thresholdpoint (p�N � y�N ; y�N ) lies exactly on the negative diagonal line and all consumers with valuations on the negativediagonal line buy the bundle, while in case (iii), the point (p�N � y�N ; y�N ) lies above the negative diagonal line andsome consumers with valuations on the negative diagonal line are excluded from consumption. We can howevercategorize the cases alternatively into mixed bundling (� < 2a2�a1

3a1), pure bundling (� � 2a2�a1

3a1and a1 = a2), and

partial mixed bundling (� � 2a2�a13a1

and a1 > a2).

9

Proposition 1 corresponds to case (i) of Proposition 2 when a1 = a2. As noted earlier, purebundling is never optimal when a1 > a2, not surprising since pure bundling, designed to treatboth goods indiscriminately, is a crude pricing strategy, particularly so when the goods are ex antedi¤erent. In addition, Proposition 2 is consistent with the result in Eckalbar (2010) on the optimalpricing scheme under independence, as shown in Corollary 1 (where part (a) is related to case (i)and part (b) is related to case (iii) of Proposition 2):

Corollary 1 (Optimal Pricing Scheme under Independence) The monopolists optimal pric-ing scheme with independent valuations (� = 0) features

(a) Mixed bundling for a1 2 [a2; 2a2] with x�N =2a13 ; y

�N =

2a23 ; and p

�N =

2(a1+a2�p2a1a2)

3 :(b) Partial mixed bundling for a1 > 2a2 with x�N = p

�N =

a12 +

a23 and y

�N =

2a23 :

An implication of Proposition 2 is that mixed bundling is optimal when both support asymmetryand correlation are small while partial mixed bundling is optimal otherwise. Consider rst the e¤ectof asymmetric supports. Other things being equal, a larger a1 implies higher demands for both good1 and the bundle, resulting in higher component and bundle prices (x and p). To induce consumersto buy the bundle, x increases at a higher rate than p as a1 increases. This however creates atension with the constraint x � p, indicating that for some level of a1, the constraint x � pisbinding, i.e., good 1 is no longer sold separately. Next consider the e¤ect of correlation. A higherdegree of correlation typically makes it more likely for the monopolist to adopt a partial mixedbundling. Other things equal, a higher � implies a larger measure of consumers with valuations onthe negative diagonal line, and hence attracting such consumers becomes more important. Roughly,with a higher �, forcing consumers with high v1 to purchase the bundle is more protable sincesuch consumers are less likely to walk away and buy nothing. This prompts the monopolist to stopselling good 1 separately.

-a1x�N~x

�N v1

6

a2

y�N

v2

aaaaaaaaaaaaaaaaaaaaaaaaa

@@@@@@

" A

B C

E D

Good 2

Bundle

Figure 4. A marginal analysis of Proposition 2 with marginal variation ~x�N= p�N�".

We use a marginal analysis as in McAfee et al. (1989) to illustrate the rationale of partial mixedbundling from a di¤erent angle. Consider a partial mixed bundling scheme with prices x�N and p

�N

(cases (ii) and (iii) in Figure 3). If we decrease x�N to ~x�N = p

�N � " (keeping p�N and y�N the same

as before), two opposing e¤ects arise: There is extra revenue from some (new) buyers who nowbuy good 1 (i.e., consumers in ABC in Figure 4). At the same time, there is a net loss from some(old) buyers switching from buying the bundle to buying only good 1 (i.e., consumers in ACa1D inFigure 4).22 With su¢ ciently large a1 or �, the revenue increase from new consumers is dominated22To be specic, the extra revenue from the new consumers in ABC can be calculated explicitly as

"2 (1� �) (p�N � ") =2a1a2;while the revenue loss from switching consumers in ACa1D (including those on Ea1) is"2 (1� �) (2a1 � 2x�N + ") =2a1a2+ "2�=a2:The revenue loss and revenue gain are exactly equal when � = 2a2�a13a1 ,resulting the exact threshold between mixed bundling and partial mixed bundling in Proposition 2.

10

by the revenue loss from switching consumers, rendering mixed bundling to be suboptimal.We now briey describe the role of the two thresholds in Proposition 2. The e¤ective regions

where the three optimal pricing prevail are illustrated in the (a1; �)-parameter space in Figure5. The threshold � = 2a2�a13a1 separating case (i) and case (ii) of Proposition 2 is driven bythe constraint x � pin the maximization problem (P). This threshold determines whether theoptimal pricing scheme is mixed bundling or partial mixed bundling. The threshold � = � (a1)di¤erentiating case (ii) and case (iii) of Proposition 2 however is of less economic signicance andresults from a technical aspect of the Fréchet copula. In our setting, the probability distributionFN (v1; v2) of consumer valuations is not di¤erentiable along the negative diagonal line, which hasimplications on the location of the optimal solution point (p�N � y�N ; y�N ) in the rectangle, resultingin the threshold � = � (a1).

-113 �

6

a2

2a2

a1

(i)

(ii)

(iii)

Figure 5. The parameter space for the three cases of Proposition 2.

Proposition 2 enables us to obtain the following comparative statics for the general case a1 � a2:

Proposition 3 (Comparative Statics under Negative Correlation) In problem (P) with neg-ative correlation, the monopolists optimal expected prot strictly increases in �.

Hence, ceteris paribus the monopolist is strictly better o¤ if the consumer valuations are morenegative correlated. This is a well-expected result and the underlying rationale comes from the factthat other things xed, the demands for both the individual good(s) and the bundle increase as theconsumer valuations are more negatively correlated: While the monopolist faces an intricate taskof adjusting the prices in response to a higher degree of correlation, the component prices and thebundle price can all be e¤ectively adjusted as � changes (see Propositions 2).23 As a result, boththe component price(s) and the bundle price play an active role toward a better price discriminationoutcome, making negative correlation a favorable setting for our multi-product monopolist. As weshall see, this is in drastic contrast to the positive correlation case where component prices play arather passive role when consumersvaluations become more positively correlated.

3.2 Positively Correlated Valuations

Compared to the negative correlation case, the bundling literature o¤ers less insight on how positivecorrelation a¤ects the protability of bundling. Recently, Chen and Riordan (2013) establish in

23This can also be conrmed by a direct comparative statics analysis of the optimal prices with respect to �. Ourworking paper contains such comparative statics results (for the optimal prices) for both the negative correlation caseand the positive correlation case. These results are available upon request.

11

a general environment that bundling is strictly more protable than separate sales if the degreeof positive dependence of consumer valuations is not too large. As in Section 3.1, our objectivehere is to provide a complete characterization of the optimal pricing rule and a set of comparativestatics analysis for the positive correlation case. Our main result is that short of perfect correlation,the optimal pricing rule always takes the form of mixed or partial mixed bundling. In addition,everything else being equal the monopolists expected prot is always strictly decreasing in thedegree of positive correlation of consumersvaluations.

We again begin our analysis with the simpler case of symmetric supports a1 = a2 = a.

Proposition 4 The monopolists optimal pricing scheme (x�P ; y�P ; p

�P ) with degree-� positive cor-

relation and symmetric support features

(i) Mixed Bundling for � 2 [0; 1) : x�P = y�P = 2a3 ; p�P =

a(4�3��p2�2�+�2)

3(1��) with lim�!1 p�P = a.

(ii) Pure Bundling for � = 1 : x�P = y�P = p

�P = a.

Proposition 4 shows that the optimal pricing scheme under symmetric supports is mixedbundling for all � 2 [0; 1) and hence mixed bundling strictly dominates separate sales (and purebundling) for the entire range of positive dependence short of perfect correlation. An importantdi¤erence between Proposition 4 and Proposition 1 is that the component prices x�P and y

�P under

positive correlation, unlike x�N and y�N , are completely independent of the correlation parameter

�. While the forms of the component prices are specic to our setting, the main driving force ofsuch di¤erence comes from how di¤erently consumer valuations are distributed in the positive andnegative correlation cases. Such a phenomenon also arises in the general asymmetric support caseand we shall come back to this point later.

-aa

2v1

6

a

a2

v2

"

"

������������

@@@@@@@@@@@

A

B

C

d

Figure 6. A marginal analysis of bundling vs. separate sales for Proposition 4.

We now use a marginal analysis to illustrate Proposition 4. Starting from the optimal pricesfor separate sales, identically

�a2 ;a2

�for all � 2 [0; 1], consider a small discount of " for purchasing

the bundle. There are three marginal e¤ects from this discount: revenue gains from consumersswitching from buying one product to both products in areas Aand B, revenue losses fromconsumers who already purchase the bundle in area C, and revenue gain from consumers switchingfrom buying nothing to both products in area d. The net (rst-order) e¤ect on revenue can befound to be strictly positive for all � 2 [0; 1).24 Notice that the above analysis also conveys the

24Specically, the revenue gains from areas Aand Bis"( a2�")(1��)

a, the revenue loss from area Cis (1��)"

4+

�"2, and the revenue gain from area dcan be calculated as "

2(1��)(a�")2a2

+ "�(a�")2a

. Altogether, the net (rst-order)e¤ect on revenue is 1��

4.

12

intuition that for more general valuation distributions, separate sales may dominate bundled salesfor highly positively correlated valuations since revenue losses from o¤ering a bundle discount canbe signicant, as illustrated in Chen and Riordan (2013).

We now characterize the optimal pricing scheme for the general case with a1 � a2:

Proposition 5 (Optimal Pricing Scheme under Positive Correlation) The monopolists op-timal pricing scheme (x�P ; y

�P ; p

�P ) with degree-� positive correlation has the following structure:

Case (i) The optimal pricing strategy is mixed bundling when a1 2ha2;

a2�(�)

i:

x�P =2a13; y�P =

2a23;

p�P =2(1� �)(a1+a2)

2+2�a1a2�q2a1a2

�(1� �)

�a21 + a

22

�+ 2a1a2

��2 � � + 1

��3(1� �)(a1 + a2)

with lim�!1 p�P =a1+a22 .

Case (ii) The optimal pricing strategy is partial mixed bundling when a1 2�a2�(�) ;+1

�:

x�P = p�P =

(a1+a2)[2a2(1� �) + 3a1]6[a1 + (1� �)a2]

, y�P =2a23; and lim�!1 p

�P =

a12:

The threshold � (�) =p4�2�4�+9�(1+2�)

4(1��) 2�13 ;12

�is strictly decreasing and strictly convex.

Proposition 5 indicates that the monopolists optimal pricing scheme is either mixed bundling(case (i)) or partial mixed bundling (case (ii)). Figure 7 illustrates case (ii) and the (a1; �)-parameter space of Proposition 5.

-a1x�P v1

6

a2

y�P

v2

Case (ii)

"""""""""""""""""""

@@@@@@@@

p�P � y�P

Good 2

Bundle

-1 �

6

a2

2a2

3a23a2

a1

Case(i)

Case(ii)

Figure 7. An illustration of Proposition 5 and the corresponding parameter space.

The transition of the optimal pricing scheme from mixed bundling to partial mixed bundlingis again shaped jointly by the magnitudes of correlation and support asymmetry. First, xing �, alarge support asymmetry prompts the monopolist to stop selling good 1 separately. The intuitionis similar to that in the negative correlation case: As a1 increases, ceteris paribus, the monopolistcharges a higher price for good 1 to extract more surplus from consumers with large v1. Thisultimately conicts with the constraint x � p, leading to partial mixed bundling. The conditiona1 � a2=�(�)pins down exactly when this constraint is binding. Notice however that compared

13

to negative correlation, positive correlation is less likely to induce partial mixed bundling. Inparticular, with a small asymmetry between a1 and a2, mixed bundling remains to be optimal for all� 2 [0; 1), while partial mixed bundling is optimal for negative correlation whenever the correlation� is large enough (Figure 5 vs. the right panel of Figure 7). The rationale of the di¤erence isthat under positive correlation, consumer valuations cluster uniformly along the positive diagonalline connecting (0; 0) and (a1; a2). Fixing a small support asymmetry, a larger � implies a higherdemand for the bundle and lower demands for the individual goods, leading to strictly higher bundleprice p and (weakly) lower component prices x and y. Such price changes e¤ectively slacken theconstraint x � p, rendering mixed bundling to be optimal for all � 2 [0; 1).

Proposition 5 leads to the following comparative statics of the monopolists optimal prot:

Proposition 6 (Comparative Statics under Positive Correlation) In problem (P) with pos-itive correlation, the monopolists optimal expected prot strictly decreases in �.

Proposition 6 agrees with a common belief in the literature that positive valuation correlationis a less favorable setting for a multi-product monopolist. Proposition 6 is in contrast to the case ofnegative correlation. In particular, component prices are much less e¤ective here compared to thenegative correlation case: As consumer valuations cluster more along the positive diagonal line, thecomponent prices are independent of � and hence play a completely passive role (see Proposition 5).With negative correlation, however, a larger � implies more consumers with diametrically di¤erentvaluations for the two goods, and component prices are always e¤ective tools to induce the buyers toself-select into di¤erent market segments. Although the comparative statics result in Proposition 6is specic for our setting, we believe such results should hold more generally: In a general setting, ahigher degree of positive correlation implies more congregated consumer valuations around or nearthe positive diagonal line, resulting in fewer consumers with high valuations only for an individualgood. Therefore, component prices would similarly become less important or e¤ective in generatingprots. Our analysis in Proposition 6 hence provides a novel perspective on the e¢ cacy of pricingschemes when consumer valuations are positively correlated.

3.3 Welfare

One important issue missing in our analysis so far is a discussion of consumer surplus and totalwelfare under the optimal pricing schemes. The problem here is that it is technically di¢ cult toexplicitly calculate such welfare measures in our setting given that the form of the optimal pricingscheme changes as the parameters (the valuation correlation and asymmetric supports) change.To make progress, we conduct a set of Monte Carlo simulations to investigate how the (expected)consumer surplus and the (expected) total welfare vary with correlation and product asymmetry.25

The following gures (Figure 9 and Figure 10) present our simulated consumer surplus and totalwelfare as functions of � and � respectively. Each set of curves consists of 5 cases with di¤erent

25The Monte Carlo simulations are implemented using R-3.2.2: Take Figure 9 as an example. For each vector(�; a1; a2) with discretized � 2 (0; 1), we rst determine the optimal pricing scheme (x�N ; y�N ; p�N ) according toProposition 2. We then randomly generate 104 valuations (v1; v2) according to the copula in (1). The averageconsumer surplus and average total welfare are then calculated based on (x�N ; y

�N ; p

�N ). Finally, we use a simple

smoothing algorithm moving average with 20 points (to prevent overtting) to smooth the curves.

14

lengths of a1 where a1 2 f1; 1:5; 2; 2:5; 3g and a2 is xed to be 1.26

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

α

CS

a1=1

a1=1.5

a1=2

a1=2.5

a1=3

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

1.4

1.6

1.8

α

TW

a1=1

a1=1.5

a1=2

a1=2.5

a1=3

Consumer Surplus vs. � Total Welfare vs. �Figure 9. Simulated Welfare for Negative Correlation with a2 = 1.

In the negative correlation case in Figure 9, for the most part, the consumer surplus decreaseswhile total welfare increases in �. These phenomena are mainly due to that the consumersvalua-tions are more predictable as � increases, enabling the monopolist to more nely price discriminatethe consumers. Importantly, observe that such better price discrimination outcomes (as � increases)are achieved by adjusting the prices (x�N ; y

�N ; p

�N ) more e¤ectively, rather than by excluding more

consumers from consumption, since the total welfare typically increases in �. Notice that an im-portant caveat here is that the valuation distribution also changes as � increases and hence themonopolist can still serve relatively more consumers with higher prices. The same thing howeveris not true for positive correlation.

0.0 0.2 0.4 0.6 0.8 1.0

0.25

0.30

0.35

0.40

0.45

0.50

β

CS

a1=1

a1=1.5

a1=2

a1=2.5

a1=3

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

1.4

1.6

β

TW

a1=1

a1=1.5

a1=2

a1=2.5

a1=3

Consumer Surplus vs. � Total Welfare vs. �Figure 10. Simulated Welfare for Positive Correlation with a2 = 1.

26To provide a clean presentation, we have avoided labeling the cases in Propositions 2 and 5 in the gures (butthe thresholds on � and � separating the cases in the simulation are indeed consistent with Propositions 2 and 5).

15

For the positive correlation case (Figure 10), the total welfare strictly decreases as � increases,as a result of higher bundle prices (from larger �s) excluding more consumers from consumption.Unlike negative correlation, an increase in the bundle price here always excludes some consumers,given the � measure of consumer valuations on the positive diagonal line. For the consumer surplus,while it is di¢ cult to exactly disentangle its overall trend in � (other than its variations beingsmall), a comparison between the left panels of Figure 9 and Figure 10 reveals that more predictableconsumer valuations do not result in a better price discrimination outcome for the monopolist. Thisis consistent with our previous ndings in that component prices are less e¤ective in the positivecorrelation case compared with the negative correlation case.

Finally, in both Figure 9 and Figure 10, both the consumer surplus and total welfare strictlyincrease in the degree of product asymmetry between the goods. This result is a direct consequenceof that the marginal distribution of v1 improves in the sense of rst order stochastic dominancewhen a1 increases (with a2 being xed).

4 Selling More to the Consumers via Randomization

The deterministic optimal pricing scheme discussed in Section 3 is a simple and commonly usedmechanism. However, it is known in the literature that deterministic price-posting mechanisms maynot be revenue-maximizing for a multiproduct monopolist selling to a single buyer, while for a single-product monopolist, posting an optimal price is revenue maximizing among all feasible mechanisms(e.g., Myerson (1981) and Samuelson (1984)). Indeed, several previous studies discussed in Section1.1 show in various settings that for a multi-product monopolist, randomization in the assignmentof goods can be more protable than deterministic price-posting mechanisms.

We now present an interesting and direct application of our previous characterization of thepricing schemes. Our objective is not to characterize the revenue-maximizing mechanism in oursetting. Rather, we ask the question of whether it is possible to improve protability on thecharacterized optimal pricing schemes using a random assignment rule. Our starting point is thecommon slogan selling more to your customers to increase your companys protability.To thisend, we consider a random mechanism where the monopolist forces its previous single-good buyers(in the price-posting mechanism) to purchase the bundle by o¤ering a lottery. Similar to a partialmixed bundling which forces buyers with high valuations on good 1 to buy the bundle, such arandom mechanism entails a trade-o¤ between additional benets and additional costs. We employa marginal analysis to show that whenever the valuation distribution features negative correlation(� > 0) and the monopolists optimal pricing scheme is partial mixed bundling, the monopolist canstrictly improve its expected prot via such randomization.

We rst describe the random assignment which builds directly on the optimal pricing schemecharacterized in Section 3, denoted here as (x�; y�; p�).27 Now instead of posting deterministicprices (x�; y�; p�), the monopolist o¤ers a random mechanism, denoted as a price-allocation bundle(x�; p�; q), where a consumer can choose one of three options: (A) purchase good 1 only at pricex� (this option is not available when (x�; y�; p�) takes the form of partial mixed bundling), (B)purchase the bundle of both goods at price p�, and (C) enter a lottery where with probability(1� q), the consumer purchases good 2 only at price y�, and with probability q, the consumerpurchases the bundle at price p�.28

27Notice that our framework in Section 2 is equivalent to one where the monopolist sells the two indivisible goodsto a single buyer who has additive and correlated valuations over the goods. In addition, as our purpose is todemonstrate revenue improvement using some types of random mechanisms, we omit the description of a full-edgedmechanism design problem for the monopolist.28We can consider a more general random mechanism by replacing option (A) with a similar lottery as option (C).

16

We can equivalently state option (C) as purchase a lottery at price (1� q) y� + qp� to get thebundle with probability q and good 2 only with probability (1� q). Indeed, the random mecha-nism is more appropriately understood as deterministic pricing with random allocation rather thanrandom pricing.29 Notice that as standard in mechanism design, we require certain commitmentfrom the monopolist to carry out the transaction as stated. The lottery option, intermediate be-tween buying good 2 and buying the bundle, can be alternatively interpreted as a result of limitedquantity of good 2, as a slightly damaged or restrictive version of the bundle, or as a slightlyenhanced version of good 2.

-a1x� = p� v1

6a2

y�

v2

aaaaaaaaaaaaaaaaaaaaaaaaaaaaa

@@@@@@@

````````

`````̀y� + q(p� � y�)

Q�(y�; p�)

p� � y�

Random Good

Bundle

Figure 8. improving the optimal pricing scheme via randomizationConsumers with valuations in Area Random Goodbuy good 2 (resp., bundle) with probability(1� q) (resp., q), while consumers with valuations in Area Q�(y�; p�)stop buying anything.

Since the original optimal pricing scheme (x�; y�; p�) is obtained by setting q = 0 in the randommechanism (x�; p�; q), the random mechanism is close to the pricing scheme when q is small.Nevertheless, the random mechanism induces di¤erent consumption behavior from that of thepricing scheme: While the decisions of consumers who purchase the bundle (or good 1 if available)are the same as before, the random mechanism excludes some previous good-2 buyers with lowvaluations on good 2, and at the same time forces the remaining previous good-2 buyers to purchasethe bundle with positive probability. More precisely, for previous good-2 buyers with valuations(v1; v2) satisfying q(v1 + v2 � p�) + (1 � q)(v2 � y�) < 0, or v2 < y� + q(p� � y� � v1), theystop buying anything under the random mechanism. For the remaining previous good-2 buyers,they now purchase good 2 with probability (1� q) and the bundle with probability q. Accordingly,compared with the pricing scheme, the total demand reduction (Q�(y�; p�)) and the total demandincrease (Q+(y�; p�)) in the random mechanism are (see Figure 8):

Q�(y�; p�) =R H1(p��y�)0

�C l1(z;H2(y

� + q(p� � y� �H�11 (z))))� C l1(z;H2(y�))�dz;

Q+(y�; p�) = q [Q2(y�; p�)�Q�(y�; p�)] :(3)

The demand changes in (3) can be derived similarly as the demands in (2). Notice that while(3) o¤ers a compact representation in copulas, the explicit demand changes can be calculateddirectly as measures of buyers in the relevant market segments (the complication here only arisesin evaluating the measures of consumers whose valuations locate exactly on the negative diagonalline in Q�(y�; p�)). Finally, the net change in expected prot for the monopolist from the changes

This however does not have any e¤ect on our results in Proposition 7.29A case in point for the lottery option here is perhaps web shopping on Priceline where consumers can get killer

travel deals by taking some risks of not knowing exactly which good they will get in the end.

17

in demand can be expressed as:

��(q) = (p� � y�)Q+(y�; p�)� y�Q�(y�; p�): (4)

Our objective is to investigate whether the monopolist can improve its expected prot on thepricing scheme (x�; y�; p�) by choosing a su¢ ciently small but positive probability q. The nextproposition shows that the optimal partial mixed bundling pricing scheme identied in Proposition2 (cases (ii) and (iii)) may not be revenue maximizing. Moreover, the random mechanism improvesthe monopolists prot only in the negative correlation case.30

Proposition 7 (Prot Improvement via Randomization) Consider the random mechanism(x�; p�; q) based on the optimal pricing scheme (x�; y�; p�) in Section 3. For su¢ ciently small q > 0,the random mechanism (x�; p�; q) generates strictly higher expected prot than the deterministicpricing scheme (x�; y�; p�) if and only if consumers valuations are negative correlated and thepricing scheme features partial mixed bundling (i.e., a1 > a2, � > 0 and � � 2a2�a13a1 ).

The intuition of Proposition 7 is as follows: By Proposition 2, the monopolist adopts partialmixed bundling in the presence of a large support asymmetry or a high degree of negative correla-tion, leaving only good 2 to be sold separately. A reasonable conjecture is that forcing some of theprevious good-2 buyers to purchase the bundle improves prot so long as p� is su¢ ciently greaterthan y�, since the extra prot from an additional bundle buyer (induced by the random assign-ment) is p� � y�, while the lost revenue from excluding a previous good-2 buyer is y�, Introducingthe random component can improve prot only if the extra revenue exceeds the lost revenue.31

However, the mere requirement of p� su¢ ciently exceeding y� is not su¢ cient: Proposition 7 showsthat for the randomization to improve prot, the magnitude of Q+(y�; p�) also has to be large com-pared to Q�(y�; p�), which only happens when the consumersvaluations are negatively correlated(� > 0). As a result, prot improvement via randomization is possible only when the optimalpricing scheme is partial mixed bundling when either support asymmetry is large (so that p� isrelatively larger than y�) or correlation is su¢ ciently high (large �): From a technical point ofview, when the optimal pricing scheme is constrained to be partial mixed bundling, the monopolistis restricted from choosing component prices exibly to some degree. This is especially true in thenegative correlation case: While the optimal pricing scheme can also be partially mixed in the casesof independence and positive correlation, the optimal price of good 2 is xed and not a¤ected bywhether the optimal pricing scheme is constrained to be partially mixed or not.32

While we have not obtained the exact optimal selling mechanism, Proposition 7 demonstratesthat random mechanisms can dominate deterministic mechanisms in revenue under negative corre-lation. In particular, Proposition 7 provides directions through which the monopolist can furtherincrease its revenue on top of the deterministic pricing scheme. This is achieved along the lines ofselling more products to the consumers and in some sense, our random mechanism also resemblesthe practice of o¤ering both a standard version and a damaged version of a good by a seller.

30Proposition 7 does not contradict (nor is it implied by) the result in Manelli and Vincent (2006) where a buyersvaluations over multiple indivisible goods are independently distributed. Moreover, it is consistent with the sug-gestion in Manelli and Vincent (2006) that negative covariance of valuations poses problems(Sec. 8. Conclusion).31 It can indeed be veried that in Proposition 2 p�N � y�N < y�N in case (i) and p�N � y�N > y�N in cases (ii) and (iii).32To be specic, the optimal component price of good 2 always has y� = 2

3a2 regardless of whether the optimal

pricing scheme is mixed or partially mixed in Corollary 1 and Proposition 5.

18

5 Conclusion

Product bundling is a prevalent business strategy that is important for both academic researchand industry. We have proposed a useful multiproduct framework where consumersvaluations areex ante asymmetric and statistically correlated across the products. We have analytically derivedthe optimal pricing schemes, together with a set of comparative statics results in the framework.We have also demonstrated how random mechanisms constructed along the lines of selling moreproducts to the existing consumers can improve the monopolists prot over that from the optimaldeterministic price-posting mechanism. We have found that valuation correlation introduces severalissues for a multiproduct monopolist. In particular, valuation correlation can crucially a¤ect theform of the optimal pricing scheme and can raise the possibility that a random selling mechanismcan outperform a deterministic price-posting mechanism.

Our multiproduct setting is undoubtedly specic, as it should be given our objective and themathematical challenge of such problems in general. This however enables us to obtain explicitanalytical solutions and allows us to delineate the e¢ cacy of various pricing schemes under vary-ing degrees of correlation and asymmetry more precisely. Overall, our results can provide usefulintuition on optimal bundle pricing schemes for general multiproduct environments with correlatedvaluations.

Appendix33

Proof of Proposition 2. Given that problem (P) involves multiple constraints and parametersand the copula CN (H1(x);H2(p�x)) (resp., CN (H1(p�y);H2(y))) is not di¤erentiable on H1(x)+H2(p � x) = 1 (resp., H1(p � y) +H2(y) = 1), we search for combinations of (x; y; p) that can beoptimal, and then derive the associated parameter constellations for optimality. We consider fourspecic cases: N1. H1(x) +H2(p� x) > 1 and H1(p� y) +H2(y) > 1, N2. H1(x) +H2(p� x) � 1and H1(p � y) + H2(y) � 1, N3. H1(x) + H2(p � x) � 1 and H1(p � y) + H2(y) > 1 andN4. H1(x) +H2(p � x) > 1 and H1(p � y) +H2(y) � 1. Importantly, dividing the problem intothese cases also enables us to express CN (H1 (x) ;H2 (p� x)) and CN (H1 (p� y) ;H2 (y)) explicitlywithout the (inconvenient) maxoperator in (1). Roughly, Case N1 corresponds to the situationwhere p is relatively large compared to x and y, i.e., both the key points (x; p� x) and (p� y; y)lie above the negative diagonal line. Similarly, Case N2 roughly corresponds to the scenario wherep is relatively small. And p is relatively close to x and is much larger than y in Case N3. Finally,since a1 � a2 and p � x+ y, Case N4 is impossible and hence is omitted.

In our proof (and also in the proof for Proposition 5), we will repeatedly use the followingimportant argument: In all the maximization problems, the rst-order derivatives are quadraticand convex in the corresponding choice variables. Our analysis can hence be focused on the cor-responding rst-order conditions. Accordingly, we typically select the smaller root of a rst-ordercondition since the other root corresponds to a (local) minimizer.

Case N1: H1(x) +H2(p� x) > 1 and H1(p� y) +H2(y) > 1.

We will show that no price scheme (x; y; p) in Case N1 can be optimal.Given that CN (H1(x);H2(p� x)) = �(H1(x) +H2(p� x)� 1) + (1��)H1(x)H2(p� x) in this

33Since Proposition 1 and Proposition 4 are respectively special cases of Proposition 2 and Proposition 5, we onlypresent the proofs for Proposition 2 and Proposition 5 here.

19

case, the demands in (2) can be calculated as:

Q1(x; p) = (1� xa1 )[�+ (1� �)p�xa2]

Q2(y; p) = (1� ya2 )[�+ (1� �)p�ya1]

Q12(x; y; p) = (1� �)[x+y�pa1 �(x+y�p)(p�x+y)

2a1a2+ (1� xa1 )(1�

p�xa2)]

The corresponding rst-order conditions are (for example, @�(x;y;p)@x = x@Q1@x +Q1 (x; p) + p

@Q12@x )

@�(x;y;p)@x = �a2(a1 � 2x) + (1� �) (p� x) (2a1 � 3x) = 0 (A1)

@�(x;y;p)@y = �a1(a2 � 2y) + (1� �) (p� y) (2a2 � 3y) = 0 (A2)

@�(x;y;p)@p = 4(p� x)(a1 � x) + 2 (p� y) (2a2 � y) + (x+ y � p)(3p� x+ y) = 2a1a2 (A3)

Suppose there are optimal prices (x; y; p). First, the two conditions dening Case N1 immedi-ately imply that p > max fx; yg. Let xN (p; �) and yN (p; �) be the implicit solutions given p and� from (A1) (A2). Now (A1) and (A2) imply that xN (p; �) 2

�a12 ;

2a13

�and yN (p; �) 2

�a22 ;

2a23

�for all p and � 2 [0; 1]. Now consider the two conditions dening Case N1:

H (xN (p; �)) +H (p� xN (p; �)) > 1 =) p > a2 +a1 � a2a1

xN (p; �)

H (p� yN (p; �)) +H (yN (p; �)) > 1 =) p < a1 �a1 � a2a2

yN (p; �)

Hence, an optimal price p exists if

(a1 � a2) > (a1 � a2)�yN (p; �)

a2+xN (p; �)

a1

�(A4)

However, given that xN (p; �) � a12 and yN (p; �) �a22 , (A4) can never hold, a contradiction.

We hence conclude that no price scheme (x; y; p) in Case N1 can be optimal.

Case N2: H1(x) +H2(p� x) � 1 and H1(p� y) +H2(y) � 1.We will show that Case N2 characterizes cases (i) and (ii) of Proposition 2.We rst calculate the demands in this case as follows:

Q1(x; p) =p�xa2

h1� (1� �) xa1

iQ2(y; p) =

p�ya2

h1� (1� �) ya2

iQ12(x; y; p) =

x+y�pa1

� (1� �) (x+y�p)(p�x+y)2a1a2 �p�xa2

h1� (1� �) xa1

i� xa1 + 1

The corresponding rst-order conditions are

@�(x;y;p)@x =

p�xa2

h2� 3(1� �) xa1

i= 0 (A5)

@�(x;y;p)@y =

p�ya1

h2� 3(1� �) ya2

i= 0 (A6)

@�(x;y;p)@p =

2(p�x)a2

h(1��)xa1

� 1i+ p�ya1

h(1��)ya2

� 2i� (1��)(x+y�p)(3p�x+y)2a1a2 + 1 = 0 (A7)

With (A5)-(A7), any optimal bundle price should satisfy p�N >2a2

3(1��) : If not, we have x = y = p

at the optimum by (A5) and (A6). (A7) then implies that @�@p > 0, or the monopolist should increasep, a contradiction.

20

Consider rst the case of � < 2a2�a13a1 . Simultaneously solving (A5)-(A7), we have x�N , y

�N and p

�N

given in case (i) of Proposition 2, all corresponding to the smaller roots of (A5)-(A7) respectively.Further, one can immediately verify that the conditions x�N < p

�N , y

�N < p

�N , p

�N < x

�N + y

�N ,

H1(x�N ) +H2(p

�N � x�N ) < 1 and H1(p�N � y�N ) +H2(y�N ) < 1 are all satised here.

Next consider the case of � � 2a2�a13a1 . Notice that now x�N � p�N if x�N and p�N are given case (i)

of Proposition 2. Hence, the constraint x � p must be binding when � � 2a2�a13a1 . Moreover, noticethat we must also haveH1(p�y)+H2(y) = 1 at the optimum.34 Suppose thatH1(p�y)+H2(y) < 1.Then y = 2a23(1��) due to p

�N >

2a23(1��) and (A6). With such y and H1(p�y)+H2(y) =

p�ya1+ ya2 < 1,

we have p < (1�3�)a13(1��) +2a2

3(1��) . Substituting y =2a2

3(1��) and x = p into (A7) yields

@�@p =

13a1

h3a1(1��)+2a2

1�� � 6pi> 13a1(1��) [(1 + 3�)a1 � 2a2] � 0

where the last inequality is due to � � 2a2�a13a1 . This contradicts that such p is optimal.Given H1(p� y) +H2(y) = 1 and x = p, the Kuhn-Tucker theorem implies that35

@�@y � �(

@H1@y +

@H2@y ) =

p�ya1[2� 3(1� �) ya2 ]� �(

1a2� 1a1 ) = 0;

@�@p � �

@H1@p = �

2(p�y)a1

[1� (1� �) ya2 ]� (1� �)4py�y22a1a2

+ 1� �a1 = 0;

where � > 0 is the Lagrange multiplier for the constraint H1(p � y) + H2(y) � 1. For a1 > a2,we can derive the third equation in case (ii) of Proposition 2 by eliminating � and p from theKuhn-Tucker conditions using H1(p� y) +H2(y) = 1, i.e.,

3a2(1� �)(1 + a1�a22a1 )(y�Na2)2 � [3(1� �)a2 + 2a1](

y�Na2) + a1 + a2 = 0: (A8)

Dene the LHS of (A8) as � (y; �), which is quadratic and strictly convex in y for � < 1. Anapplication of the Intermediate Value theorem to � (y; �) shows the existence and uniqueness (bystrict convexity) of y�N 2 (0; a2) given � (0; �) > 0 and � (a2; �) < 0 when � �

2a2�a13a1

. Moreover,applying a1 = a2 = a to case (ii) of Proposition 2 immediately leads to part (ii) of Proposition 1.36

Case N3: H1(x) +H2(p� x) � 1 and H1(p� y) +H2(y) > 1.

We will show that Case N3 characterizes case (iii) of Proposition 2.Note rst that the demand Q1(x; p) (resp., Q2(y; p)) here is the same as that in Case N2 (resp.,

Case N1). For the bundle demand, we have

Q12(x; y; p) = 1� (1��)(p�y)a1 �(1��)(x+y�p)(p+y�x)

2a1a2� �(p�a2)a1�a2 �

p�xa2

h1� (1� �) xa1

i:

The rst-order conditions are ((A5) in Case N2 and (A2) in Case N1 are reproduced below)

@�(x;y;p)@x =

p�xa2[2� 3(1� �) xa1 ] = 0 (A5)

@�(x;y;p)@y = �(1�

2ya2) + (1� �)p�ya1 (2�

3ya2) = 0 (A2)

@�(x;y;p)@p = (1� �)

p�ya1( ya2�2)+

2(p�x)a2

[ (1��)xa1 �1]��(2p�a2)a1�a2 �

(1��)(x+y�p)(3p�x+y)2a1a2

+1 = 0: (A9)

34This also means that the solution point (p�N � y�N ; y�N ) lies on the negative diagonal line.35 It can be veried that the Hessian matrix of the corresponding Lagragian is negative denite if y � 2a2

3(1��) , which

holds here since if y > 2a23(1��) , we have (1) no (p; y) can satisfy H1(p� y) +H2(y) = 1 and (2)

@�@y< 0.

36The (unique) threshold � (a1) will be determined in Case N3 below.

21

We rst show that there is partial mixed bundling (x = p) at the optimum. Suppose not, orp > x. The condition H1(p � y) +H2(y) > 1 implies that y is also interior, i.e., y < p. We hencehave x = 2a13(1��) by (A5) and y 2

�a22 ;

2a23

�by (A2). Using the conditions in Case N3, we have

H1(x) +H2(p� x) � 1 : p � a2 + a1�a2a1 x = a2 +a1�a23(1��)

H1(p� y) +H2(y) > 1 : p > a1 � a1�a2a2 y �a1+a22

) a2 + a1�a23(1��) >a1+a22 or � >

13 :

However, this implies that x = 2a13(1��) > a1, which is never optimal. A contradiction. Notice that

given we have partial mixed bundling in this case, the condition � � 2a2�a13a1 is also satised here.We next show that given x = p, there are unique solutions y�N and p

�N , written here respectively

as y (�) and p (�), by simultaneously solving (A2) and (A9).37 This is immediate for the simplecases of � = 0 and � = 1, where y(0) = 2a23 , p(0) =

a12 +

a23 , y(1) =

a22 and p(1) =

a12 . For the case

of � 2 (0; 1), we derive two bundle prices p1 (y; �) and p2 (y; �) from (A9) and (A2) respectively as

p1 (y; �) = � 3(1��)(a1�a2)4a2[a1�(1��)a2]y2 + (1��)(a1�a2)a1�(1��)a2 y +

a12 ;

p2 (y; �) = y +�1��

a1(2y�a2)2a2�3y :

Dene a function �(y; �) = p1 (y; �)�p2 (y; �). Applying the Intermediate Value theorem to �(y; �)on y 2

�a22 ;

2a23

�and noting that �(y; �) strictly decreases in y for y > a22 , we obtain the existence

and uniqueness of solution y (�) solving the equations that dene � (y; �). The existence anduniqueness of y (�) also establishes the existence and uniqueness of solution p (�).

Finally, we are left to show that there is a unique threshold that separates Case N2 and CaseN3, i.e., there is a unique �(a1) 2 [0; 1) with �0(a1) > 0 such that H1(p(�)� y(�)) +H2(y(�)) > 1for all � < �(a1) and H1(p(�)�y(�))+H2(y(�)) � 1 for all � � �(a1). And �(a1) exists wheneverwe have 2a2 � a1. First, suppose we are in Case N3 and dene a function

w(�; a1) = H1(p(�)� y(�)) +H2(y(�))� 1 =p(�)� y(�)

a1+y(�)

a2� 1:

First, it can be veried that p0(�) = @p�N

@� =@x�N@� < 0 and y

0(�) =@y�N@� < 0, which imply that w is

strictly decreasing in �. Further, using our previous results on y(0), p(0), y (1) and p (1), we have

w(0; a1) =1

6� a23a1

� ( 0 if @w@a1 > 0. First, direct calculation implies that

@p1(y;�)@a1

= (1� 3y4a2 )�a2y(a1�a2)[a1�(1��)a2]2 +

12 > 0, and

@p2(y;�)@a1

= �(2y�a2)(1��)(2a2�3y) � 0;

37One can again verify that the associated Hessian matrix is negative denite if y � 2a23, which is implied by (A2).

22

which hold by y 2 [a22 ;2a23 ]. We hence have

@�@a1

> 0. Together with @�(y;�)@y < 0, we have@y(�)@a1

> 0.Further, using the rst-order condition (A2) directly, we have

@hp(�)�y(�)

a1

i@a1

= �a2(1��)[2a2�3y(�)]2

@y(�)@a1

> 0:

By the denition of w(�; a1), we have @w@a1 > 0, proving �0(a1) > 0.38

Finally, notice that our above discussion also implies that if either a1 < 2a2 (so that w(0; a1) < 0and H1(p(�)� y(�))+H2(y(�)) < 1 for all � � 0) or � � �(a1) holds, it is then impossible for themonopolists optimal pricing scheme to be in Case N3 or case (iii). This implies that if � < �(a1)(and hence a1 > 2a2 which automatically implies that � � 2a2�a13a1 ), the optimal pricing schemewill be in Case N3 representing case (iii) of Proposition 2, while if � � �(a1) and � � 2a2�a13a1 , theoptimal pricing scheme will be in Case N2, featuring case (ii) of Proposition 2.

Proof of Proposition 3. We apply the Envelope Theorem to show @�(x�N ;y

�N ;p

�N )

@� > 0.Case (i): � < 2a2�a13a1 .

@�(x�N ; y�N ; p

�N )

@�=

(x�NH1(x

�N )H2(p

�N � x�N ) + y�NH1(p�N � y�N )H2(y�N )

+p�N

hR H1(x�N )H1(p�N�y�N )

H2(p�N �H

�11 (z))dz �H1(p�N � y�N )H2(y�N )

i )

=(x�N )

2(p�N � x�N )a1a2

+(y�N )

2(p�N � y�N )a1a2

+p�N [(x

�N )

2 + (y�N )2 � (p�N )2]

2a1a2> 0;

where the last inequality is due to (x�N )2 + (y�N )

2 > (p�N )2, which can be veried directly.

Case (ii): � � �(a1) and � � 2a2�a13a1 .

@�(x�N ; y�N ; p

�N )

@�=

(y�N (1�H2(y�N ))(1�H1(p�N � y�N ))+p�N

hR H1(x�N )H1(p�N�y�N )

H2(p�N �H

�11 (z))dz +H1(p

�N � y�N )�H1

�a1(p�N�a2)a1�a2

�i )

= y�NH2(y�N )(1�H2(y�N )) + p�N

Z H1(x�N )H1(p�N�y�N )

H2(p�N �H�11 (z))dz > 0;

where the second equality follows from H2(y�N ) +H1(p�N � y�N ) = 1.

Case (iii): � < �(a1).

@�(x�N ; y�N ; p

�N )

@�=

(y�N (1�H2(y�N ))(1�H1(p�N � y�N ))+p�N

hR H1(x�N )H1(p�N�y�N )

H2(p�N �H

�11 (z))dz +H1(p

�N � y�N )�H1

�a1(p�N�a2)a1�a2

�i )

= p�N

�y�Np�N(1� y

�N

a2(1� p

�N � y�Na1

) +(y�N )

2

2a1a2+p�N � y�Na1

� p�N � a1a1 � a2

�(by x�N = p

�N )

� p�N�(y�N ; p�N ):

Next, notice that

@�(y�N ; p�N )

@y�N=

�3(y�N )2 � 2(a1 � a2)y�N + 3p�Ny�N + a2(a1 � 2p�N )a1a2p�N

<3y�N (p

�N � y�N )� a1a2a1a2p�N

�3y�N (

a12 �

a26 )� a1a2

a1a2p�N� �a23a1p�N

< 0;

38The result that �0 (a1) > 0 gives the shape of the curve that separates case (ii) and case (iii) in Figure 5, thoughnotice that the above mentioned curve in Figure 5 is the inverse of � (a1).

23

where the rst inequality is due to H2(y�N ) + H1(p�N � y�N ) =

y�Na2+

p�N�y�Na1

> 1, while the secondand third follow from p�N �

a12 +

a23 and

2a23 � y

�N �

a22 as � < �(a1). Similarly, we have

@�(y�N ; p�N )

@p�N=(y�N � a2)[y�N (a1 � a2)(a2 + y�N ) + a2(p�N )2]

a1a2p�N (a1 � a2)� y

�N

a1(a1 � a2)< 0:

Finally, one can check that

�(y�N j�=0; p�N j�=0) = ��2a23;a12+a23

�=3a1a2 + 2a2(a1 � a2)

18a1(a1 � a2)> 0;

implying that �(y�N ; p�N ) > 0 for all � < �(a1).

39

Proof of Proposition 5. Since copulas CP (H1(x);H2(p � x)) and CP (H1(p � y);H2(y)) arenot di¤erentiable when H1(x) = H2(p � x) and H1(p � y) = H2(y) respectively, we similarlydiscuss four cases: P1. H2(y) � H1(p � y) and H1(x) < H2(p � x), P2. H2(y) � H1(p � y) andH1(x) � H2(p � x), P3. H2(y) < H1(p � y) and H1(x) � H2(p � x), and P4. H2(y) < H1(p � y)and H1(x) < H2(p � x). Case P4 here is impossible and can be dropped as p � x + y. Noticefurther that Case P1 and Case P3 can be treated symmetrically, and we hence only discuss CaseP1 and Case P2 below. As in the proof of Proposition 2, dividing the problem into these casesenables us to write out CP (H1 (x) ;H2 (p� x)) and CP (H1 (p� y) ;H2 (y)) explicitly without the(inconvenient) minoperator in (1).

Case P1: H2(y) � H1(p� y) and H1(x) < H2(p� x).

The demand functions can be explicitly expressed as

Q1(x; p) =p�xa2

h1� (1� �) xa1

i� � xa1 ;

Q2(y; p) = (1� �)p�ya1�1� ya2

�;

Q12(x; y; p) = (1� �)hx+y�pa1

� (x+y�p)(p�x+y)2a1a2i+�1� p�xa2

� h1� (1� �) xa1

i:

The rst-order conditions with regard to x and y can be derived as

@�(x;y;p)@x =

p�xa2

h2� 3(1� �) xa1

i� 2� xa1 = 0 (A10)

@�(x;y;p)@y = (1� �)

p�ya1(2� 3ya2 ) = 0 (A11)

We will show that no optimal price scheme exists in Case P1. Suppose not and (x; y; p) areoptimal prices here. First, notice that

H1(x) < H2(p� x) and H2(y) � H1(p� y) =)(a1 + a2)

a1x < p � (a1 + a2)

a2y:

In addition, (A11) implies that y = 2a23 at the optimum. Consider an alternative rst-order condi-tion @�

0

@x = (1��)p�xa2[2� 3xa1 ] = 0: It can be veried that

@�@x �

@�0

@x = �2�

a1a2(xa1 � pa1 + xa2) > 0

given a1p > (a1 + a2)x. Since both @�@x and@�0

@x are convex in x, (A10) and@�@x >

@�0

@x jointly implythat x > 2a13 at the optimum. However, the result that x >

2a13 and y =

2a23 obviously contradicts

the above implication from the two conditions dening Case P1.

39Strictly speaking, this (�(y�N ; p�N ) > 0) also requires

@y�N@�

< 0 and @p�N

@�< 0. The latter can be veried directly

here. This has also been obtained in our working paper.

24

Case P2: H2(y) � H1(p� y) and H1(x) � H2(p� x).

The demand functions are as follows:

Q1(x; p) = (1� �)p�xa2 (1�xa1);

Q2(y; p) = (1� �)p�ya1 (1�ya2)

Q12(x; y; p) = (1� �)[x+y�pa1 �(x+y�p)(p�x+y)

2a1a2] + �( xa1 �

pa1+a2

) + (1� xa1 )[1� (1� �)p�xa2]

The corresponding rst-order conditions can then be derived as

@�(x;y;p)@x = (1� �)

p�xa2(2� 3xa1 ) = 0 (A12)

@�(x;y;p)@y = (1� �)

p�ya1(2� 3ya2 ) = 0 (A13)

@�(x;y;p)@p = 1� (1� �)

�2(1� x

a1)(p�x)

a2+

(p�y)(2� ya2)

a1+ (x+y�p)(3p�x+y)2a1a2

�� � 2pa1+a2 = 0 (A14)

We rst show that no p � 2a23 can be optimal. For any p �2a23 , (A12) and (A13) imply

respectively that x = p and y = p at the optimum. Plugging these prices into (A14), we have

@�@p = 1�

2�pa1+a2

� 3(1��)p2

2a1a2� 1� 4�a23(a1+a2) �

2(1��)a23a1

� 13 ;

where the two inequalities follow from p � 2a23 and a1 � a2 respectively. Hence, the monopolistsprot strictly increases in p for all p � 2a23 .

Dene a threshold �(�) =p4�2�4�+9�(1+2�)

4(1��) . It can be veried that �(0) =12 , lim�!1 �(�) =

13

and � 0(�) < 0. We now solve for the optimal prices (x�P ; y�P ; p

�P ). First consider mixed bundling

where p�P �2a13 . Such p

�P implies that x

�P =

2a13 and y

�P =

2a23 , both being the smaller solutions to

(A12) and (A13) respectively. Next, plug x�P =2a13 and y

�P =

2a23 into (A14) to obtain

@�

@p=3(1� �)2a1a2

p2 � 2((1� �)(a1 + a2)a1a2

+�

a1 + a2)p+

2(1� �)(a21 + a22)3a1a2

+ 1 = 0: (A15)

One can verify that p�P given in case (i) of Proposition 5 is the smaller root of the quadraticequation (A15). In addition, we have that in this case, a1 � a2�(�) implies p

�P �

2a13 . We next verify

that the prices (x�P ; y�P ; p

�P ) are consistent with H2(y

�P ) � H1(p�P �y�P ) and H1(x�P ) � H2(p�P �x�P ),

both identically reducing to p�P �2(a1+a1)

3 given x�P =

2a13 and y

�P =

2a23 . By (A15),

@�@p is convex

and quadratic in p. It can be veried that in (A15), @�@p = �13 when p =

2(a1+a2)3 , which immediately

implies that p�P <2(a1+a1)

3 . Finally, the constraint p�P <

2(a1+a2)3 = x

�P + y

�P also holds here.

Now consider partial mixed bundling, which arises when a1 > a2�(�) (and hence p�P <

2a13 ). Such

p�P implies that x�P = p

�P and y

�P =

2a23 by (A12) and (A13). Plug x

�P and y

�P into (A14) to obtain

@�

@p= 1�

�2(1� �)a1

+2�

a1 + a2

�p+

2a2(1� �)3a1

= 0: (A16)

One can verify that p�P =(a1+a2)[2a2(1��)+3a1]

6[a1+(1��)a2] in case (ii) of Proposition 5 solves (A16). Direct

calculations also imply that p�P >2a23 is equivalent to 3a

21+a1a2 > 2a

22+2�a2(a1�a2), which holds

here as a1 � a2 and � � 1. Finally, x�P = p�P and p�P >2a23 imply that H1(x

�P ) � H2(p�P � x�P ),

H2(y�P ) � H1(p�P � y�P ) and p�P < x�P + y�P are all satised.

25

Proof of Proposition 6. We again apply the Envelope theorem which implies

@�(x�P ;y�P ;p

�P )

@� =(p�P�x�P )2

a2(1� x

�Pa1)� y

�P (p

�P�y�P )a1

(1� y�Pa2) + p�P

h(x�P+y

�P�p�P )(p�P�x�P+y�P )

2a1a2� p

�P

a1+a2+

p�P�y�Pa1

i(A17)

When �(�) � a2a1 , plugging x�P =

2a13 and y

�P =

2a23 into (A17) to obtain

@�@�

���(2a13;2a23;p�P )

� ~� (p�P ) =�(p�P�

2a13)2

3a2� 2a2(p

�P�

2a23)

9a1

�+ p�P

�(2(a1+a2)

3�p�P )(p�P�

2(a1�a2)3

)

2a1a2� p

�P

a1+a2+

p�P�2a23

a1

�where ~� (p�P ) is dened to simplify notation and p

�P 2

h2a13 ;

2(a1+a2)3

�is given in case (i) of Proposi-

tion 5. We need to show that ~� (p�P ) < 0. Since p�P is inconveniently long, we prove this indirectly.

First, one can verify that ~��2(a1+a2)

3

�= 0. Hence to show ~� (p�P ) < 0, it su¢ ces to show that

function ~� (p) is increasing onh2a13 ;

2(a1+a2)3

�. Di¤erentiate ~� (p) to obtain

~�0 (p) = � 32a1a2

p2 + 2

�a1 + a2a1a2

� 1a1 + a2

�p� 2(a

21 + a

22)

3a1a2:

And ~�0 (p) is quadratic and strictly concave in p. Since ~�0�p = 2(a1+a2)3

�= 0 and ~�0

�p = 2a13

�=

2a2(a1�a2)3a1(a1+a2)

� 0, we immediately obtain that ~�0 (p) � 0 or ~� (p) is strictly increasing onh2a13 ;

2(a1+a2)3

�except possibly on p = 2a13 . Therefore,

~� (p�P ) < 0.When �(�) > a2a1 , substituting x

�P = p

�P and y

�P =

2a23 into (A17), we have

@�

@�

����(p�P ;

2a23;p�P )

� �� (p�P ) =4a2227a1

� 2a23a1