Preference Dependence and Product Bundling Yongmin Cheny and Michael H. Riordanz August 20, 2010 Preliminary and Incomplete; Please do not quote, cite, or circulate without permission.

Abstract. Using copulas to model the stochastic dependence of values, this paper establishes new general results on the pro…tability of product bundling. A multiproduct monopolist achieves higher pro…t from mixed bundling than from separate selling if consumer values for two products are negatively dependent, independent, or have limited positive dependence. With more than two goods, step bundling, a partial form of mixed bundling, achieves the higher pro…t and possesses other interesting properties. Preference dependence is also important for the price and welfare e¤ects of bundling, and situations where bundling decreases or increases consumer welfare are discussed.

Keywords:

y University z Columbia

of Colorado; [email protected] University; [email protected]

We thank participants of the 2010 Shanghai Microeconomics Workshop (SHUEF), the 2010 Summer Conference on Industrial Organization (UBC, Vancouver), and of seminars at Columbia University, University of Rochester, and Yonsei University for helpful comments.

1

1. INTRODUCTION For a multiproduct monopolist, when is bundling more pro…table than separate selling? This question has long intrigued economists. Stigler (1963) showed with a simple example of two goods that bundling can be pro…table even without demand complementarity or economies of scope. Adams and Yellen (1976) expanded on this view, showing mostly with examples that mixed bundling can be a pro…table way to segment markets. Schmalensse (1984) studied pro…tability and prices under pure and mixed bundle when consumer values for two goods have a bivariate normal distribution, and found su¢ cient conditions on the marginal distributions for pure bundling to dominate separate sales for any correlation coe¢ cient. McAfee, McMillan, and Whinston (1989; henceforth MMW) moved the literature forward by considering a general model where consumer values for two products follow arbitrary joint distributions and proving that mixed bundling dominates separate selling when values are distributed independently.1 Much attention in the economic analysis of bundling has been directed towards the issue of how the correlation of values between products matters for the pro…tability of bundling. However, analytical results under general distributions of product values have been lacking in the literature, other than the …ndings in MMW. In this paper, we revisit this issue using a new approach that represents the distribution of consumer values for multiple products by copulas and marginal distributions. A copula is a function that couples marginal distributions of random variables to form a joint distribution, making it straightforward to vary dependence while holding marginal distributions constant. We are then able to develop a strong new analytical result under general distributions of consumer preferences. Speci…cally, advancing the analysis of MMW, we show that mixed bundling dominates separate selling if values for two products are negatively dependent, independent, or have limited positive dependence.2 When the multiproduct monopolist sells an arbitrary number (n mixed bundling would involve up to

2n

2) of goods, X1 ; :::;Xn ;

1 distinctive prices, which may be impractical for

not too small n. We propose a partial form of mixed bundling, called step bundling, that 1

MMW in fact provides a more general but hard-to-interpret su¢ cient condition for mixed bundling to be more pro…table than separate selling, which is satis…ed under independent values. While the condition implies that bundling is optimal in a broader range of cases than just independence, it is not clear what these cases are. 2 Note that mixed bundling would have the same pro…t as separate selling if consumer values for the two products are perfectly positively dependent. When the copula density is bound from above so that consumer preferences are strictly away from this extreme case, we also show that bundling is pro…table under any dependence relationship, if in addtion market coverages for both goods are su¢ ciently high under separate selling.

2

extends the pro…tability result of monopoly bundling to any number of goods. A k-step bundle o¤ers for sale the following: n individual goods with possibly n prices and a single price rl for any l-good bundle among Jl goods, l = 2; :::; k, and Jl

J

fX1 ,...,Xn g ;

with k 2 [2; n] : We further say that k-step bundling is monotonic if Jl contains exactly l goods, l = 2; :::; k: When n = 2; step bundling is the same as mixed bundling. More generally, step bundling has at most 2n substantially fewer than

2n

1 distinct prices for n goods, which for large n is

1. We show that if there are at least two goods the value of

which are negatively dependent, independent, or have limited positive dependence, then 2step bundling (and by extension step bundling) will have higher pro…t than separate selling. Furthermore, we establish a general su¢ cient condition for pro…t from k-step bundling to be strictly increasing in k for k < n; and this condition is satis…ed for any monotonic kstep bundling if consumer values for k + 1 goods are independently distributed. It is also interesting to note, as we shall discuss later, that step bundling is widely used in practice. Our …ndings on step bundling nicely complement the recent contribution on bundlesize-pricing by Chu, Leslie, and Sorensen (forthcoming, henceforth CLS). Step bundling is similar to bundle-size-pricing, with the di¤erences being that step bundling sets separate prices for all n individual goods and Jl may contain fewer than n goods, whereas under bundle-size-pricing only a single price is set for all individual goods and Jl = J for all l. Thus step bundling nests bundle-size-pricing. CLS provides the interesting insight, through plausible numerical examples, that pro…t under bundle-size-pricing is very close to pro…t under mixed bundling. Unfortunately, no explicit analytical result about the pro…tability of bundle-size-pricing relative to separate selling is established in CLS, and their numerical analysis indicates that pro…t under bundle-size-pricing is sometimes lower than that under separate selling even when consumer values for the goods are negatively correlated. Since it nests bundle-size-pricing, step bundling must also achieve pro…t that is close to pro…t from mixed bundling in plausible settings, without the need to set a burdensome number of prices. But, as we establish in this paper, there are explicit and general su¢ cient conditions under which step bundling dominates separate selling, and step bundling has the additional desirable property that it always has higher pro…t than separate selling when consumer values for at least two goods are negatively dependent. Our approach also leads to new insights on how bundling a¤ects prices and welfare. The bundle is sold under a discount relative to the individually-priced goods under mixed bundling. Intuitively, one suspects that the discount on the bundle might be based on in‡ated prices on goods being o¤ered individually, but the literature has not addressed this

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possibility in a setting with general distributions of consumer values.3 Under dependence properties similar to those for the pro…tability of bundling and with certain quali…cations, we show that a multiproduct …rm selling two goods will charge higher standalone prices for each good under mixed bundling than under separate selling. Therefore, whereas bundling often leads to higher pro…t, it also frequently reduces the surplus of at least some consumers. For a class of preference distributions described by the Fairlie-Gumbel-Morgenstern (FGM) family of copulas and uniform marginal distributions, we further identify situations where bundling reduces or increases aggregate consumer welfare and social welfare, and discuss how prices, pro…ts, and welfare under bundling vary with preference dependence. The rest of the paper is organized as follows. In Section 2, we set up a basic model of multiproduct monopoly with two goods, where the joint distribution of consumer values for the goods are described by a copula and the marginal distributions. Section 3 establishes our main result on the pro…tability of bundling. Section 4 shows that this result generalizes to a multiproduct monopoly selling any number of goods, by introducing the concept of step bundling and establishing its properties. Section 5 studies the price and welfare e¤ects of bundling as well as how outcomes under bundling vary with preference dependence. Section 6 shows that the pro…tability result of monopoly bundling extends to markets where a multiproduct …rm competes against a single-product …rm. Section 7 concludes. 2. THE BASIC MODEL There are two goods, X and Y. The size of consumer population is normalized to 1. Each consumer demands at most one unit of each good, and her consumption of one good does not a¤ect her decision to purchase the other good. A consumer’s value for X is u and for Y is v; which has marginal distributions F (u) and G (v) on respective supports [u; u] and [v; v] ; with corresponding density functions f (u) > 0 and g (v) > 0 on (u; u) and (v; v). The constant marginal costs for X and Y are mx and my ; respectively. The value of two goods together is u + v; with marginal cost mx + my . Following Adams and Yellen (1976) and MMW, this framework rules out product complementarity or economies of scale as explanations for bundling. As in MMW, to avoid trivial situations, we assume that there are positive measures of u and v for which u > mx and v > my . The main departure of our modeling approach from the bundling literature is to use 3 An exception is Choi (2008), who …nds that prices for the indidually-priced goods are higher under mixed bunding by a merged …rm than under separate selling by pre-merger …rms; however, the products in his model are complemenary goods, and the price comparison is not made to the same …rm with or without bundling, but to di¤erent …rms where the price change also contains the merger e¤ect.

4

copulas to describe the stochastic relationship between values of di¤erent products. A copula, C (x; y) ; is a bivariate uniform distribution that“couples” marginal distributions to form joint distributions. Standard uniform margins for x and y imply C(x; 1) = x and C(1; y) = y.

A copula additionally satis…es C(x; 0) = 0 = C(0; y). By Sklar’s Theorem

(Nelsen, 2006), it is without loss of generality to represent the joint distribution of consumer values for the two products by a copula and the marginal distributions (Nelsen, 2006).4 Let x = F (u) ; y = G (v) ; and the copula associated with the joint distribution of (u; v) be C (x; y) : Then the joint distribution of (u; v) is C(F (u); G(v)); with u (x) v (y)

G

1 (y)

for (x; y) 2

F

1 (x)

and

2

I2

[0; 1] . We assume that C (x; y) is a twice-di¤erentiable

function with density C12 (x; y)

@ 2 C(x; y)=@x@y > 0 on the interior of I 2

[0; 1]2 .

The copula describes the statistical dependence of consumer values for the two products. In particular, C1 (x; y)

@C(x; y)=@x is the conditional distribution of y given x. Value

v is stochastically increasing (SI) in u if C11 (x; y)

@ 2 C(x; y)=@x2

0 for all (x; y) 2 I 2 ;

@ 2 C(x;y) @x2

conversely, v is stochastically decreasing (SD) in u if C11 (x; y) (x; y) 2

I 2.

0 for all

Value u is stochastically increasing (decreasing) in v by analogous de…nitions.

Values u and v are stochastically independent if C (x; y) = xy for all (x; y) 2 I 2 : For the

rest of the paper, we shall omit the word "stochastically", denote Cii (x; y) by Cii if a

relationship holds for all (x; y), and say that consumer values for X and Y are positively dependent when, for at least one i = 1; 2; Cii

0; conversely, u and v are negatively

0:5

dependent when, for at least one i = 1; 2; Cii

As a benchmark, if the two goods are sold separately at prices p and q; consumers will purchase X if u (x)

p; or if x

F (p) ; and will purchase Y if v (y)

The optimal prices for X and Y under separate sales,

ps

and

qs;

q; or if y

G (q) :

are assumed to satisfy

…rst-order conditions 1

F (ps )

(ps

mx ) f (ps ) = 0;

(1)

1

G (q s )

(q s

my ) g (q s ) = 0:

(2)

As long as u and v are not too much above the marginal costs, ps and q s will indeed be interior values satisfying (1) and (2). Since F ( ) ; G ( ) ; and costs are all given, ps and q s 4

Sklar’s Theorm holds also for more than two values, where any joint distribution can be represented by marginal distributions and a multivariate copula. We will use this later in generalizing our results to any n 2 products and to our analysis of bundling under competition. 5 Among the various dependence concepts, SI (or SD) implies positive (or negative) quadrant dependence: C (x; y) xy (or C (x; y) xy) for all (x; y) 2 I 2 ; which further implies that the Pearson’s correlation coe¢ cient is non-negative (or non-positive). On the other hand, SI is in turn implied by positive likelihood ratio dependence (Nelson, 2006).

5

are given for our analysis that follows. By bundling, we include both mixed and pure bundling.6 Suppose that under mixed bundling X and Y are o¤ered individually at prices p and q, respectively, and the XY bundle is o¤ered at price r

p + q.

Without loss of generality, we formulate bundling

as mixed bundling, which allows pure bundling as a special case with p and q being high enough. The value of the outside option is normalized to zero. Consumers are willing to purchase the bundle if u(x) + v(y) and v(y)

r

r

p, and Y alone if v(y)

max f0; u(x) q

p; v(y)

0 and u(x)

r

qg, X alone if u(x)

p

0

q. Consequently, demands

for each good and the bundle are, respectively: QX (p; r)

G(r

QY (q; r)

F (r q) C(F (r q); G(q)); R F (p) C1 (x; G(r u(x))] dx + [1 F (r q) [1

QXY (p; q; r)

p)

C(F (p); G(r

p));

(3) (4) F (p)]

QX (p; r):

(5)

my ) QB (p; q; r) :

(6)

Pro…t function under mixed bundling is (p; q; r) = (p

mx ) Qx (p; r) + (q

my ) Qy (q; r) + (r

mx

The monopolist chooses (p; q; r) to maximize pro…ts subject to r

p + q: Bundling has

higher pro…t than separate selling if r < p + q at the solution; pure bundling is optimal if p and q are such that separate sales for X and Y are zero, which is equivalent to not o¤ering the goods separately, otherwise mixed bundling is optimal. 3. PROFITABILITY OF BUNDLING In this section, we develop a general condition on when a multiproduct monopolist achieves higher pro…t from mixed bundling than from separate selling. This new result substantially extends the su¢ cient condition in MMW that mixed bundling dominates separate selling under independent values. As in MMW, the condition is developed by considering the possibility of a local improvement through bundling, starting from separate selling. De…ne

Notice that

2 [1 F (ps )] [1 [1 G (q s )]2 + [1

> 0, is independent of C (x; y) ; and

6

G (q s )] : F (ps )]2

(7)

= 1 if F ( ) = G ( ).

In the tradition of the economics literature on bundling, pure bundling refers to the sale of two goods only through a bundle, whereas mixed bundling refers to the selling of two goods individually as well as a bundle.

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Theorem 1 Mixed bundling has higher pro…ts than separate selling if consumer values for two products are negatively dependent, independent, or positively dependence but min fC11 ; C22 g > .

Proof. Consider (") =

(ps ; q s ; ps + q s

") for "

0;

where recall ps and q s are the optimal prices when the two goods are sold separately. Mixed 0

bundling is more pro…table than separate selling if 0

(ps

(0) =

mx )

(ps + q s

(0) > 0: We have

@Qy (q s ; ps + q s ) @Qx (ps ; ps + q s ) (q s my ) @b @b @Qb (ps + q s ; ps ; q s ) mx my ) Qb (ps + q s ; ps ; q s ) : @b

From (3), (4), and (5), simple di¤erentiation and substitution yield: (ps

mx ) g (q s ) [1

+ (ps + q s [1 = (ps

mx

F (ps )]

my ) f[1

[G (ps )

mx ) f (ps ) [1

[1

C2 (F (ps ) ; G (q s ))]

(q s

C1 (F (ps ) ; G (q s ))] f (ps ) + [1

F (ps )] [1

(0) = [1

= [1

G (q s )] [1

C2 (F (ps ) ; G (q s ))]

C (F (ps ) ; G (q s ))] :

C1 (F (ps ) ; G (q s ))] + [1

F (ps )] + [G (q s )

[1

C1 (F (ps ) ; G (q s ))]

C2 (F (ps ) ; G (q s ))] g (q s )g

my ) g (ps ) [1

Using the …rst-order conditions for ps and q s ; (1) and (2), 0

(0) =

C (F (ps ) ; G (q s ))]

C1 (F (ps ) ; G (q s ))] + (q s

F (ps )] + [G (q s )

my ) f (ps ) [1

0

0

(0) becomes

G (q s )] [1

C2 (F (ps ) ; G (q s ))]

C (F (ps ) ; G (q s ))]

C2 (F (ps ) ; G (q s ))] +

Z

1

(8)

[C1 (x; G (q s ))

C1 (F (ps ) ; G (q s ))] dx

F (ps )

= [1

s

G (q )] [1

s

s

C2 (F (p ) ; G (q ))] +

Z

1

F (ps )

Thus, since C2 (F (ps ) ; G (q s )) = Pr ( X negatively dependent (i.e., C11

Z

x

C11 (z; G (q s )) dzdx:

F (ps )j G (q s )) < 1,

0

0) or independent (i.e. C11 = 0):

7

(9)

F (ps )

(0) > 0 if u and v are

Next, suppose that u and v are positively dependent (i.e., C11 . Re-writing (9) as s

1

s

G (q )

F (p ) +

Z

1

C2 (F (ps ) ; y) dy + C (F (ps ) ; G (q s ))

G(q s )

s

[1

s

s

G (q )] C2 (F (p ) ; G (q )) +

Z

1

F (ps )

G (q s )

= 1 +

Z

1

F (ps )

Z

x

C11 (z; G (q s )) dzdx

F (ps )

F (ps ) + C (F (ps ) ; G (q s )) +

Z

1

G(q s )

Z

x

0) but min fC11 ; C22 g >

Z

y

C22 (F (ps ) ; z) dzdy

G(q s )

C11 (z; G (q s )) dzdx:

F (ps )

But since positive dependence implies G (q s )

1

F (ps ) + C (F (ps ) ; G (q s ))

= Pr (X > F (ps ) ; Y > G (q s )) we have, with min fC11 ; C22 g > 0

(0)

s

[1 +

F (p )] [1 Z

1

F (ps )

> [1

"

Z

(10)

s

G (q )] +

Z

1

Z

y

C22 (F (ps ) ; z) dzdy

G(q s )

C11 (z; G (q s )) dzdx

F (ps )

F (ps )] [1 [1

G (q s )] ;

:

G(q s )

x

F (ps )] [1

[1

G (q s )]

G (q s )]2 [1 + 2

F (ps )]2 2

#

2 [1 F (ps )] [1 [1 G (q s )]2 + [1

G (q s )] = 0: F (ps )]2

If a family of symmetric copulas, C (x; y; ) ; indexed by parameter satis…es the monotonic dependence ranking property

(MDR),7

2

;

R;

then the weak inequality

in (10) becomes strict and Theorem 1 can be stated more concisely as that bundling has higher pro…t than separate sales if C11

(i.e., if values for X and Y are not too positively

dependent): Notice that dependence is entirely a property of the copula, and the limited positive dependence allowed in the su¢ cient condition, ; depends only on the marginal distributions 7

If a copula is symmetric, then C (x; y) = C (y; x) ; which implies C11 (x; y) = C22 (y; x) : If copula family C (x; y; ) satis…es MDR, then it is symmetric and C11 (x; y; ) < 0 for 0 < x; y < 1: Thus, under MDR, a higher leads to less negative dependence or more positive dependence (Chen and Riordan, 2010).

8

and marginal costs that jointly determine the prices under separate selling. Previous studies have found by various examples that bundling is more pro…table than separate selling when values for products are negatively dependence, but it has remained an open question whether this is always the case. Theorem 1 completely settles this issue. Theorem 1 also identi…es a fairly wide range of positive dependence on which bundling is more pro…table. For example, for the FGM family of copulas discussed later, which satis…es MDR, C11 =

2 y (1

y) 2 [ 1=2; 1=2] >

1; thus bundling is always more pro…table

than separate selling for the FGM family of copulas if F ( ) = G ( ) (so that

= 1):

To see why in general some restriction on the degree of positive dependence between products might be needed for the pro…tability of bundling, we note that if values of X and Y were perfectly positively dependent, bundling would have the same pro…t as separate selling. If the distribution of consumer values is bound away from this extreme situation, then the restriction on the range of positive dependence in Theorem 1 can be relaxed. Speci…cally, suppose that the copula density is bound from above by some positive number M . This, together with the condition that market participation under separate selling is relatively high (i.e., F (ps ) and G (q s ) are relatively low, which would be the case if mx and my are relatively low), ensures that bundling will be more pro…table than separate selling under any dependence relations. Formally: Proposition 1 If (i) for all (x; y) 2 I 2 ; C12 (x; y) max fF (ps ) ; G (q s )g

1 2M ;

then bundling is more pro…table than separate selling. M for some …xed M > 0: If G (q s )

Proof. Suppose that C12 (x; y) 0

(0)

F (ps )] [1

[1 [1

C1 (F (ps ) ; F (ps ))] + [1

F (ps )] + [G (q s ) F (ps )] [1

= [1

= [1

F (ps )] 1 s

+ [F (p ) [1

Z

F (ps )

0 s

F (ps )] [1

F (ps ) ; from 8:

C2 (F (ps ) ; F (ps ))]

C (F (ps ) ; G (q s ))]

C1 (F (ps ) ; F (ps )) "

M for some …xed M > 0 and (ii)

C2 (F (ps ) ; F (ps ))] + [F (ps )

C12 (F (ps ) ; y) dy

Z

F (ps )

#

C12 (x; F (ps )) dx

0

s

C (F (p ) ; F (p ))] F (ps )] [1

> [1

2F (ps ) M ] + [F (ps )

F (ps )] [1

2F (ps ) M ]

9

C (F (ps ) ; F (ps ))]

0 if F (ps )

C (F (ps ) ; F (ps ))]

1 : 2M

The proof for the case where G (q s ) 0

G (q s )] [1

(0) = [1 [1

F (ps ) is similar, since from 8 we can rewrite

C2 (F (ps ) ; G (q s ))] + [1

G (q s )] + [F (ps )

F (ps )] [1

C1 (F (ps ) ; G (q s ))]

C (F (ps ) ; G (q s ))] :

For example, suppose that the copula belongs to the Frank copula family (Nelson, 2006), which satis…es MDR, and, for

2 ( 1; 1) = f0g

C (x; y; ) =

1

x

e

ln 1 +

1 (e

1)

where C (x; y; ) is positively (negatively) dependent if pendence as ( !

y

e

1

!

;

> 0 ( < 0); C (x; y; ) ! inde-

! 0; and C (x; y; ) ! perfectly positively (negatively) dependent as

1). Numerical analysis indicates that for

Therefore, If F ( ) = G ( ) and

1; C11

0:5: For

> 1; C12

!1 2 :

1; bundling is more pro…table than separate selling

according to Theorem 1. For any given has the higher pro…t from Proposition 1.

> 1; if max fF (ps ) ; G (q s )g

1 2M

=

1 4

; bundling

However, some copula families do not have a copula density that is bound from above. For example, for the Clayton copula family, where C (x; y; ) = max

h

x

+y

1

i

1

; 0 ; for

2 [ 1; 1)= f0g ;

C12 (x; y) ! 1 if x or y ! 0:

(Check numerically: For the Clayton copula family, suppose that F ( ) and G ( ) are

the standard uniform distribution, F (t) = G (t) = t: When

is high enough, will mixed

bundling have the same pro…t as separate selling?) 4. GENERALIZING TO ANY NUMBER OF GOODS: STEP BUNDLING Our pro…tability condition of bundling by a monopolist with two goods can be extended to situations where the monopolist sells any number of products. To show this, we …rst 2 products: X1 ; ..., Xn :8 Let the consumer

generalize our model in Section 2 to any n 8

In reality, a …rm sometimes sells multiple groups of products, and goods within each product group could be substitutes such that a consumer may purchase only one of them. For ease of exposition, we do not explicitly model this situation, but we can accommodate this possibility by allowing the interpretation of Xi ; if appropriate, as any (symmetric) good from product group i; i = 1; :::; n; where goods within group i are substitutes:

10

value for Xi be ui ; the marginal distribution of ui be Fi (ui ) ; and the multivariate copula associated with the joint distribution of consumer values be C (x1 ; :::xx ) : We use Xi to denote the random variable corresponding to xi : By Sklar’s Theorem, it is without loss of generality to represent the joint distribution of consumer values for the n goods by Fi (ui ) and C ( ) ; so that the joint distribution is written as C (F1 (u1 ) ; :::; Fn (un )). Assume that the value of two goods Xi and Xj together is ui + uj ; with constant marginal cost mi + mj ; and the values and marginal costs are similarly obtained for l goods together, l

n. As before, to avoid trivial situations, we assume that there are positive measures of

ui for which ui > mi for all i. Again, this framework rules out product complementarity or economies of scale as explanations for bundling.9 Let x

(x1 ; :::xn ) and x

(x1 ; :::; xi

i

1 ; xi+1 ; :::xn ) ;

i = 1; :::; n: Since xi has the

standard uniform margin, Ci (x) is the joint distribution of x

1

conditional on Xi = xi ;

and the values of product Xj ; j 6= i; are positively dependent, independent, or negatively dependent in the values of Xi (in the stochastic sense) if Ci (x)

@C (x) =@xi = Pr ( X1

x1 ; :::Xi

1

xi

1 ; Xi+1

xi+1 ; :::Xn

xn j Xi = xi )

is decreasing, independent, or increasing in xi ; respectively: Thus the dependence concept under bivariate distribution extends naturally to the multivariate distribution; and, we say that values for X

i

when for all x 2

I n;

are positively dependent, independent, or negatively dependent on Xi Ci (x)

0; Ci (x) = 0 or Ci (x)

0; respectively. As we shall see

shortly, our results in this subsection actually only require a dependence property between two of the products, which is a weaker requirement than the dependence property between one product and all other products. Denote the optimal prices for Xi under separate selling by psi . Then, as when n = 2; psi is assumed to be interior. Hence pi satis…es 1

Fi (psi )

(psi

mi ) fi (psi ) = 0:

With n goods, mixed bundling involves up to 2n practical when n is large (for instance, if n = 10; 210

1 distinct prices, which may not be 1 = 1023): Thus, even if we can

show that mixed bundling is more pro…table than separate selling; the result may have little practical relevance when n is not a small number. We propose a partial form of 9 We abstract away from situations where there could be complementarity between goods (or for goods across product groups), so that our result on the pro…tability of bunding will not be driven (but can be reinforced) by the complementarity between goods. We shall return to this when discussing examples of bundling later.

11

mixed bundling, step bundling, that achieves higher pro…t than separate selling but without a burdensome number of prices. A k-step bundling o¤ers the following for sale: n goods individually for prices p1 ; :::; pn , and any l-good bundle (B2 ) from a set of goods Jl under a bundle price rl , for l = 2; :::; k, Jk

J

fX1 ,...,Xn g ; and k 2 [2; n] : Step bundling may

include part or all k-step bundling. Notice that step bundling also includes o¤ering a single bundle of 2 or more goods, together with n goods individually.10

If n = 2; then step bundling and mixed bundling are all the same. Since mixed bundling nests step bundling, if step bundling achieves higher pro…t than separate selling, so must mixed bundling. We next extend Theorem 1 in the following result, which uses the fact that the joint distribution of consumer values for any two goods, say X1 and X2 ; is represented by C~ (F1 (u1 ) ; F2 (u2 )) ; where C~ (x1 ; x2 ) C (x1 ; x2 ; 1; :::; 1) is a bivariate copula: Thus, for all (x1 ; x2 ) ; the values of X1 and X2 are negatively dependent when C11 (x1 ; x2 ; 1; :::; 1)

0;

independent if C11 (x1 ; x2 ; 1; :::; 1) = 0; and positively dependent if C11 (x1 ; x2 ; 1; :::; 1)

0:

As in (7), similarly de…ne = where

= 1 if F ( )

2 [1 F1 (ps1 )] [1 [1 F2 (ps2 )]2 + [1

F2 (ps2 )] > 0; F1 (ps1 )]2

F2 ( ) ; we have:

Proposition 2 If consumer values for two of the n goods, denoted as X1 and X2 , are negatively dependent, independent, or have positive dependence but minfC11 (x1 ; x2 ; 1; :::; 1) ; C22 (x1 ; x2 ; 1; :::; 1)g >

for all (x1 ; x2 ) 2 I 2 ; then 2-step bundling (and by extension step

bundling) will have higher pro…t than separate selling.

Proof. Consider 2-step bundling with B2 = fX1 ,X2 g ; prices pi for the i = 1; :::; n goods individually, and r2 for B2 . Since pro…ts from X3 ; :::Xn under the 2-step bundling is the

same as pro…ts from separate selling, with prices pi = psi ; i = 3; :::; n, the 2-step bundling will have higher pro…t than separate selling if pro…t from goods X1 and X2 ; with associated copula C~ (x1 ; x2 ) ; is higher under the usual mixed bundling for the two goods than under separate selling, which, from Theorem 1, is true if C~ (x1 ; x2 )

C (x1 ; x2 ; 1; :::; 1) possesses

one of the stated dependence properties. Moreover, since the speci…c 2-step bundling is a special case of step bundling, by extension step bundling must also be more pro…table than separate selling under the same su¢ cient condition. 10 For example, selling a bundle B3 = fX1 ,X2 ,X3 g together with n goods individually is a speical case of 3-step bundling, with J2 = fX1 ,X2 g, J3 = fX1 ,X2 ,X3 g ; and r2 p1 + p2 :

12

Therefore, when there are any number of goods, under fairly general conditions, a bundle of two or more goods, together with the individually priced goods, will lead to higher pro…t than under separate selling. Obviously, if the dependence properties hold for more than two goods, the …rm can also increase pro…t by o¤ering several 2-good bundles. We further consider the question of when pro…t from a k-step bundling will be strictly increasing at k; for k = 2; :::; n 1: First consider 2-step bundling, where there is a bundle ˜2 that contains, say, goods X1 and X2 : Denote the optimal prices and the optimal J2 under B ˜2 by z2 ; 2-step bundling by p ; p ; :::; pn ; r ; and J : Also denote the (induced) value of B 1

2

2

2

given pi ; r2 and J2 ; where, for instance, if J2 = fX1 ,X2 g ; z2 = u 1 + u 2

max f0; u1

p1 ; u 2

p2 g :

In general, z2 may also depend on J2 and pj for j = 3; :::; n: Denote the cumulative distri˜2 as a new good. Suppose that bution function of z2 by G2 (z2 ) : Then we can consider B there is another good, say X3 ; whose induced value is u ~3 with cdf F~3 (u3 ) :11 If z2 and u ~3 possess the dependence property described in Proposition 2, then 3-step bundling, where ˜2 ; J3 = fX1 ,X2 ,X3 g ; and B ˜3 contains B ˜2 and X3 ; must have higher J2 = fX1 ; X2 g with B

pro…t than 2-step bundling. To see this more clearly, we can start from the prices under ˜2 and X3 , r ; p ; and price r3 = r + p for the B3 bundle: Under "separate selling" for B 2 3 2 3 the assumed dependence properties, a slight increase of p3 ; which has the equivalent e¤ect ˜2 on pro…ts under mixed bundling as a slight decrease of r3 ; will increase the pro…t from B and X3 , from consumers who were previously purchasing both X3 as a separate good and B2 . Furthermore, since all other prices have not changed, this slight increase in p3 cannot reduce (but can potentially increase) the pro…ts from all other goods under the original 2-step bundling (including possibly other 2-good bundles). We can repeat the above argument at any k-step bundling for k < n, denoting the optimal prices and the optimal J2 ; :::; Jk at k-step bundling by p1 ; :::; pn ; r2 ; :::; rk ; and J2 ; :::; Jk 12 ˜k = fX1 ,...,Xk g is being o¤ered where, if necessary, the products are re-named so that B under the k-step bundling. Let z1

˜k by zk and u1 ; and de…ne the (induced) value of B

the cdf of zk be Gk (zk ) : Then, k+1-step bundling will have higher pro…t than the kstep bundling if there is some product; say Xk+1 ; whose induced value is u ~k+1 with cdf F~ (~ uk+1 ) ; such that zk and u ~k+1 are negatively dependent, independent, or have limited If X3 2 = J2 ; then u ~3 = u3 and F~ ( ) = F ( ) : So as not to introduce additional notations, we use p1 ; :::; pn to denote the optimal standalone prices under k-step bundling for all k, even though these prices could change with k: The optimal bundle prices, r2 ; r3 ; :::; may also change with k: 11 12

13

positive dependence: To summarize, we have: Proposition 3 Under k-step bundling for k = 2; :::; n

1, denote the optimal prices by

p1 ; :::; pn ; r2 ; :::; rk ; and denote the optimal sets of included goods at each step by J2 ; :::; Jk : De…ne

fp1 ; :::; pn ; r2 ; :::; rk ; J2 ; :::; Jk g : Pro…t is strictly increasing in k at the k-step ˜k ; and a good, say Xk+1 ; such that under k both B ˜k bundling if there exist a k-bundle, B k

and Xk+1 have positive sales and the joint distribution of their values, zk and u ~K+1 ; is k k ~ uk+1 ) ; where C (x; y) is the associated copula, with the property that C Gk (zk ) ; Fk+1 (~ k C11

k 0; or C11

k 0; or C11

k

k ; Ck 0 but min C11 22 >

k

; where

h i Gk (rk )] 1 F~k+1 pk+1 h i2 > 0: 2 Gk rk + 1 F~k+1 pk+1

2 [1 1

If consumer values for the n goods are independently distributed and if there is a good, say Xk+1 ; with Xk+1 2 = fJ2 ; :::; Jk g, then zk and uk+1 are independent for all k = 1; :::; n 1: The result below then follows immediately from Proposition 3, where we say that k-step bundling is monotonic if Jl contains exactly l goods for l

k:

Corollary 1 Assume that consumer values for k + 1 2 [3; n] goods are independently dis-

tributed. If there is at least one good, say Xk+1 ; such that Xk+1 2 = fJ2 ; :::; Jk g, then pro…t

from k-step bundling strictly increases in k for all k 2 [2; n

1]. In particular, pro…t from

monotonic k-step bundling strictly increases in k.

Step bundling is easy to implement and widely used in practice. For instance, a telecom company o¤ers phone, Internet, and TV services, among other products. For new customers, in addition to o¤ering each of these three services under individual prices, it also o¤ers a 2good bundle including Internet service and DirectTV, and a 3-good bundle that additionally includes home phone service. So this is an example of monotonic 3-step bundling. A satellite TV company o¤ers …ve individually-priced premium channels (with one price for each of 4 channels and a higher price for the remaining channel), and additionally, a bundle price rl for any l channels, l = 2; 3; 4; 5: So this is an example of 5-step bundling where Jl

J.

Restaurants frequently o¤er combination meals, in addition to individually priced items. A combination dinner may consist of two, three, or even more items from the menu, and there are often restrictions on the categories of items that can be chosen from.13 Moreover, 13 For example, under 3-step bundling, the 2-good bundle may allow one choice from seveal main courses and a salad, and the three-good bundle may add a soup (or a dissert) to the 2-good bundle. The restaurant

14

all kinds of businesses practice the simple step-bundling where a single bundle of 2 or more goods is sold alongside individually priced goods. In practice, step bundling can be implemented even more easily with small variations. For instance, stores sometimes practice the simple variant of step bundling where a consumer receives a discount (credit) when the purchase reaches a certain amount. For instance a $5 discount is o¤ered for every $50 purchase, a $15 discount for every $100 purchase, and a $30 discount for every $150 purchase; some times there are also restrictions on goods that can be included towards the purchased amount.14 Although the details of these examples may di¤er from our model, they illustrate that step-bundling not only has interesting theoretical properties, but is also frequently used in various markets. Our …ndings nicely complement Chu, Leslie, and Sorensen (forthcoming), who provide the interesting insight, through plausible numerical examples, that pro…t under bundle-sizepricing is very close to pro…t under mixed bundling. Unfortunately, no explicit analytical result about the pro…tability of bundle-size-pricing is established in CLS, and their numerical analysis indicates that bundle-size-pricing sometimes has lower pro…t than separate selling even when consumer values for the goods are negatively correlated. Step bundling di¤ers from bundle-size-pricing in that under the former there can be di¤erent prices for the n individual goods and there can also be restrictions on what goods are allowed to construct a bundle. Thus step bundling nests bundle-size-pricing, and the two are identical if the marginal distributions of consumer values for the n goods are the same and J2 = ::: = Jn

J. Hence, as in CLS, pro…t from step bundling would also be close to that from

mixed bundling, without a burdensome number of prices. However, as we have established, there are explicit and general analytical conditions under which step bundling dominates separate selling, and step bundling always has higher pro…t than separate selling when consumer values for at least two goods are negatively dependent. 5. PRICE AND WELFARE EFFECTS OF BUNDLING Our copula framework is also useful to derive new insights on the price and welfare e¤ects of bundling. Focusing on a monopolist producing two goods, we …rst show that, may not want the customer to choose any three goods for the 3-good bundle, possibly to prevent a dining family from choosing 3 main course items under the single bundle price. Notice that there might be complementarity between goods from di¤erent catogories, which could reinforce the pro…tability of bundling. 14 It can be useful to exclude certain goods from a bundle, and it might matter what goods can be included in each step of bundling. In practive, sometimes it may be di¢ cult to specify varying Jl ; possibly because it might confuse consumers. Thus, when the di¤erence to pro…ts is small, for convenience a …rm may simply allow consumers to choose from all goods for any bundle

15

under similar dependence properties ensuring the pro…tability of bundling and with additional restrictions on the marginal distributions, the standalone prices for each good are higher under mixed bundling than under separate selling. We then identify situations where bundling decreases or increases consumer and/or social welfare in the FGM-uniform Case, and illustrate how prices, pro…ts, and welfare vary with preference dependence. E¤ects of Bundling on Standalone Prices Denote the optimal prices under bundling by p ; q and r : Recall that under mixed bundling the standalone prices p and q are interior values such that QX > 0 and QY > 0.15 To compare (p ; q ) with (ps ; q s ) ; we consider the following conditions: h i h i 0 and d 1 g(v) 0: A1. d 1 f F(u)(u) G(v)

A2. r

p

my > 0 and r

q

mx > 0:

A1 is the familiar monotonic hazard rate condition, which ensures the unique existence of ps

and q s . A2, which says that the bundle earns a positive margin on both goods, is based

on variables that are observable in equilibrium. It is satis…ed, for instance, if u v

mx and

my , or consumer values for each product is relatively high compared to the marginal

costs16 The result below refers to the following de…nition, where we note that f ( ) > 0; g ( ) > 0; and C12 ( ) > 0 on interior values:

1

min

(

(r

p

my ) g (r

(r

q

mx ) f (r

p ) q

R1

s C12 (x; G (r RF 1(p ) ) F (ps ) C12 (F (r

p )) dx; p ) ; y) dy

)

> 0:

(11)

Proposition 4 Under A1 and A2, p > ps and q > q s if values for X and Y are negatively dependent, independent, or positively dependent but min fC11 ; C22 g

2 1:

Proof. See the appendix. Thus, although under mixed bundling the bundle is sold at a discount relative to individuallypriced goods, with certain quali…cations, the standalone prices for X and Y are higher under mixed bundling than under separate selling. The dependence properties that are su¢ cient for this result are similar to those ensuring the pro…tability of bundling. Since the optimal 15

The stand-alone prices for each good are trivially higher under pure bundling than under separate sales. As it will be clear from its proof, Proposition 4 also holds when only one product has positive standalone sales under mixed bundling, in which case the price for the good with zero sales is trivially higher than that under separate sales. 16 Since G (r p ) > 0; r p > v: Thus r p my r p v > 0 if v my : Similarly, r q mx > 0 if u mx : The condition u mx and v my ; however, is much too strong for what is needed. Notice that (r ; p ; q ) are observable under mixed bundling whereas ps and q s are not.

16

prices p and q depend on the copula, the limited positive dependence that is allowed in the su¢ cient condition is only given implicitly:17 As we shall see in the FGM-Uniform Case discussed next,

1

can be quite large, and r

p

my > 0 and r

q

mx > 0 always

hold. An immediate implication of Proposition 4 is that mixed bundling, while often increasing a monopolist’s pro…t relative to separate selling, will also often reduce the welfare of at least some consumers. It would be interesting to see how aggregate consumer welfare and social welfare under bundling compares with those under separate selling: We address this issue with the numerical analysis of the FGM-Uniform case, which also illustrates how pro…ts, prices, and welfare vary with preference dependence. FGM-Uniform Case The FGM copula family is given by C (u; v; ) = uv + uv (1 with

u) (1

v) ;

2 [ 1; 1] ; C (u; v; ) = C (v; u; ) ; and density C12 (u; v; ) =

@ 2 (uv + uv (1 u) (1 @u@v

v))

= 1 + (2v

1) (2u

1) :

The FGM family is useful for modelling a limited range of positive and negative dependence, including independence ( = 0). Kendall’s tau, which is a measure of dependence and ranges from ( ) =

2 9

1 (perfectly negative dependence) to +1 (perfectly positive dependence), is

for the FGM family; with

< 0 and

> 0 indicating negative and positive

dependence, respectively, and @C11 ( ) =@ < 0 (so it satis…es the MDR property). For ease of computation, we further assume mx = my = m and F ( ) = G ( ) ; with F (u) =

u

4 5

; 4 < u < 9:

Tables 1-4 in the Appendix details results with m = 0; m = 2; m = 4; and m = 6: The speci…c uniform distribution and values of m are chosen to consider situations where either all consumers or only some consumers value the products more than their marginal costs. 17

With (minor) additional restrictions, the lower bound on C11 and C22 can be given as a …xed number. For instance, if r p my , r q mx ; f ( ) ; g ( ) ; and C12 ( ) are all bound above zero, then 1 is bound ^1 above zero: The lower bound on C11 and C22 in Proposition 4 can then be written as min fC11 ; C22 g ^ for some …xed 1 > 0.

17

The following is a summary of our …ndings: First, on pro…ts, we observe: (i) As predicted by Proposition 1, with F ( ) = G ( ) ; bundling always increases pro…t in the FGM-uniform case; for which C11 2 [ 1=2; 1=2]. (ii)

Pure bundling is optimal if m is low relative to valuations (m is optimal when m is relatively high (m

2), whereas mixed bundling

4; in which case not every consumer’s product

valuation is strictly above m). An advantage of mixed bundling over pure bundling is that it limits ine¢ cient production for consumers who would purchase the bundle but have a low value for one of the goods. This explains why mixed bundling dominates pure bundling when m is high enough. (iii) As

increases, pro…t under bundling always decreases. With

greater preference dependence, consumer preferences become more similar in some sense, which diminishes the usefulness of bundling as a tool to reduce the e¤ective dispersion of values under pure bundling (Schmalensee, 1984) or to segment the consumer population for price discrimination under mixed bundling (Adams and Yellen, 1976).18 This might explain why pro…t decreases with

(under either pure or mixed bundling):

Second, on prices, we observe: (i) The standalone prices under mixed bundling are always higher than the corresponding prices under separate selling. Thus the positive dependence allowed in Proposition 4 need not be small: (ii) The price for the bundle can be higher than the sum of prices under separate selling (in which case bundling raises prices to all consumers), but it appears to occur only under pure bundling (when m = 0 or when m = 2 and

is relatively low). (iii) As

increases, the bundle price decreases under pure bundling

(which occurs when m is relatively low) but increases under mixed bundling. As mentioned earlier, pure bundling works by reducing the e¤ective dispersion of values. As

increases,

this mechanism is less useful, which might explain why price under pure bundling goes down in order to maintain the quantity of sales. On the other hand, under mixed bundling, with higher

consumers are more likely to have similar values for both products, which might

motivate the …rm to raise the bundle price to sell to consumers with high values for both products, and it still can capture the consumers who have high values only for one product through the individually-priced goods (and the consumers with low values for both products won’t purchase anyway). (iv) The standalone prices under mixed bundling decrease in when m = 4 but increase in

when m = 6: (v) As

increases, under mixed bundling the

di¤erence between the bundle price (r ) and the sum of prices of the separately-o¤ered goods (2p ) decreases. Intuitively, this di¤erence is a measure of the degree of price discrimination 18

The mechansims through which pure bundling or mixed bundling incerases a monopolist’s pro…t are di¤erent. Despite the common perception that monopoly bundling is pro…table as a price discrimination device, pure bundling extracts consumer surplus through a single bundle price by reducing the e¤ective dispersion of consumer values, instead of segmenting consumers for price discrimination.

18

under mixed bundling (where consumers purchase both goods receive a discount through the bundle), and there is less room for price discrimination with greater price dependence. Third, on consumer welfare, we observe: (i) Both pure bundling and mixed bundling can reduce (aggregate) consumer welfare. Pure bundling reduces consumer welfare when it raises prices to all consumers (r > 2ps ); which occurs when m = 0; or when m = 2 and is relatively low (

0:5) ; it also reduces consumer welfare when r < 2ps but r is

su¢ ciently high (when m = 2 but

= 0:5 or 1), in which case while some consumers have

higher surplus, the consumer loss dominates, and there are two sources of consumer loss: some consumers no longer purchase, and some purchasing the bundle prefer to purchase only one good at ps : Mixed bundling reduces consumer welfare when m = 4 and

= 1

or when m = 6: (ii) Both pure bundling and mixed bundling can also increase consumer welfare. Pure bundling increases consumer welfare when m = 2 and bundling increases consumer welfare when m = 4 and increases with

= 1; whereas mixed

0:5: (iii) Consumer surplus

under pure bundling, partly due to the fact that price deceases with

under pure bundling. On the other hand, consumer surplus varies non-monotonically with under mixed bundling, partly due to the fact that the standalone prices under mixed bundling is non-monotonic in . Finally, on social welfare, we observe: (i) Bundling can either reduce or increase social welfare. It reduces social welfare when m = 0 and increases social welfare when m = 0 and

=

is in the intermediate range, but

1 or 1, and when m = 2; m = 4 and m = 6,

for all : It is interesting that for a relatively larger range of parameter values bundling increases social welfare. (ii) Social welfare under bundling (and its di¤erence from that under separate selling) is non-monotonic in

when m = 0; but decreases in

for m = 2;

m = 4; and m = 6: Thus, social welfare under bundling is often lower with greater preference dependence, partly due to the fact that pro…t always decreases in : 6. PROFITABILITY OF BUNDLING UNDER COMPETITION The pro…tability of bundling under multiproduct monopoly extends to markets where a multiproduct …rm competes against a single-product …rm. We focus on the case where the multiproduct …rm, A; o¤ers two products X and YA , whereas a single-product …rm, B; o¤ers a di¤erent version of product Y, YB .19 Yi is v(yi ) with (x; yA ; yB ) 2

I 3.

A consumer’s value for X is u(x), and for

Therefore, the marginal distribution of consumer values

for X is F (u), and the symmetric distribution for each variety of product Y is G(v); with 19

From our analysis on step bundling in Section 4, the results also readily extend to settings where a multiproduct …rm producing any n > 2 products compete with a single-product rival.

19

corresponding density functions f (u) > 0 and g (v) > 0 on their respective supports. The copula C(x; yA ; yB ), with yA and yB exchangeable, describes the population of consumers. We assume that C(x; yA ; yB ) is twice-di¤erentiable and admits a joint density function C123 ( ) > 0 on interior values of I 3 : As for the case with n

2 products, we say that values

for X and Y are positively dependent, independent, or negatively dependent when for all (x; yA ; yB ) 2 I 3 ; C11 (x; yA ; yB )

0; C11 (x; yA ; yB ) = 0 or C11 (x; yA ; yB )

0; respectively.

Under mixed bundling, let p denote the standalone price of X, qi the standalone price of

Yi , and r

p + qA the price of Firm A’s bundle. Consumers will purchase the bundle if u(x) + v(yA )

r

max f0; v(yB )

qB g ;

u(x) + v(yA )

r

u(x)

u(x) + v(yA )

r

v(yA )

yA

G (r

u(x) + max f0; v(yB )

yA

G (r

p + max f0; v(yB )

x

F (r

qA ) :

p + max f0; v(yB )

qB g ;

qA ;

or, equivalently, qB g) ;

qB g) ;

Consumers will purchase X as a standalone product, rather than as part of the bundle, if x yA

F (p) ; G (r

p + max f0; v(yB )

qB g) ;

and YA as a standalone product if x < F (r yA

qA ) ;

G (qA + max f0; v(yB )

qB g) :

The standalone demands for X and YA are QX (p; qB ; r)

(12)

= C (1; G (r p) ; G (qB )) C (F (p) ; G (r p) ; G (qB )) R1 + G(qB ) [C3 (1; G (r p + v(y) qB ) ; y) C3 (F (p) ; G (r

20

p + v(y)

qB ) ; y)] dy;

QYA (qA ; qB ; r)

(13)

= C (F (r qA ) ; 1; G (qB )) C (F (r qA ) ; G (qA ) ; G (qB )) R1 + G(qB ) [C3 (F (r qA ) ; 1; y) C3 (F (r qA ) ; G (qA + v(y)

qB ) ; y)] dy;

and the demand for the bundle is QXYA (p; qA ; qB ; r) = 1 + +

F (p) R F (p)

(14)

Qx (p; qB ; r)

F (r qA ) [C1 (x; 1; G (qB ))

R F (p)

F (r qA )

C1 (x; G (r

R1

G(qB ) [C13 (x; 1; y)

u(x)) ; G (qB ))] dx

C13 (x; G (r

u(x) + v(y)

qB ) ; y)] dydx:

The demand for YB is analogous. The pro…t of Firm A is A (p; qA ; qB ; r)

= (p + (r

mx ) QX (qA ; qB ; r) + (qA mx

my ) QYA (qA ; qB ; r)

my ) QXYA (p; qA ; qB ; r) :

Separate Pricing As a benchmark, we …rst consider equilibrium under separate pricing, which is equivalent to r = p + qA : In this case, the demands for X and YA are respectively QsX (p) = 1 QsYA

(qA ; qB ) = 1

F (p) ; C (1; G (qA ) ; G (qB ))

Equilibrium separate prices (ps ; q s ) satisfy 1 =

(q s

my )

h

F (ps ) = (ps

R1

G(qB ) C3 (1; G (qA

+ v(y)

qB ) ; y) dy:

mx ) f (ps ) ;

1 C (1; G (q s ) ; G (q s ))] 2 [1 i R1 C2 (1; G (q s ) ; G (q s )) g(q s ) + G(qs ) C23 (1; y; y) g(v(y))dy :

(15)

We assume that an interior equilibrium satisfying these conditions exists. See Chen and Riordan (2010) for details on symmetric equilibrium in the Y market.

21

Pro…tability of Bundling We next establish the pro…tability of bundling that extends Theorem 1. The result below refers to the following de…nition:

2

(q s

my ) g (q s )

Z

1

F (ps )

"Z

G(q s )

#

C123 (x; G (q s ) ; z) dz dx > 0;

0

(16)

where ps and q s are the equilibrium prices for X and YA under separate pricing from (15):20 Proposition 5 In competing against a single-product rival, bundling is more pro…table than separate selling for the multiproduct …rm if values for X and Y are negatively dependent, independent, or positively dependent but C11

4 2.

Proof. See the appendix. Thus, similarly as in the multiproduct monopoly model, with competition bundling also dominates separate sales for the multiproduct …rm if values for X and Y are negatively dependent, independent, or have (su¢ ciently) limited positive dependence.21 Notice also that Proposition 5 holds for any dependence relationship between YA and YB : When competing with a single-product …rm, a multiproduct …rm may also pro…tably use bundling to foreclose competition. The foreclosure theory of bundling was …rst formalized in Whinston (1990). Our result here shows that even without the foreclosure motive, a multiproduct …rm can often pro…t from bundling its products in competing with a singleproduct rival.22 We have con…ned our analysis of bundling under competition to situations where a multiproduct …rm competes with a single-product rival. The issue of whether the multiproduct …rm pro…ts from bundling is relatively simple in this case, because only the multiproduct …rm can choose to bundle its products. In markets where there is competition between multiproduct …rms, the issue of bundling is more complex, since the pro…tability of bundling for one …rm may also depend on whether the other …rm bundles or not. The issue is also more 20

Since ps and q s are interior values, q s my > 0: This, together with g ( ) > 0 and C123 ( ) > 0, implies > 0. 2 21 Unlike the in Theorem 1, here 2 depends on C ( ) : With (minor) additional restrictions, it’s possible to …nd a lower bound on C11 that is a …xed number. For instance, if QsYA , g ( ) ; and C123 ( ) are all bound above zero, then 2 is bound above zero; and the lower bound on C11 in Proposition 5 can be stated as ^2 for some …xed ^2 > 0. C11 22 Other important contributions on the foreclosure e¤ects of bundling include, for example, Carlton and Waldman (2002), and Choi and Stefanadis (2001). See also Nalebu¤ (2004) for the large pro…t gains that a multiproduct …rm can obtain from using (pure) bundling to deter the entry of a potential single-product entrant.

22

complex because there are dependence relations both between values for di¤erent products and between values for products by di¤erent …rms. We leave it for future research to address the issue of equilibrium product bundling by competing multiproduct …rms, under general preference dependence relations.23 7. CONCLUSION This paper has established a new general su¢ cient condition for the pro…tability of bundling: a multiproduct …rm achieves higher pro…t from mixed bundling than from separate selling if consumer values for two products are negatively dependent, independent, or have limited positive dependence. When the …rm sells more than two products, step bundling will achieve the higher pro…t, and pro…t from monotonic k-step bundling strictly increases in all k < n if values for k + 1 products are independently distributed. Under similar dependence properties that ensure the pro…tability of bundling, a …rm selling two products will charge higher standalone prices for each good under mixed bundling than under separate selling. Numerical analysis of the FGM family of copulas indicates that bundling can either increase or reduce consumer and social welfare. Finally, the pro…tability result of monopoly bundling extends to markets where a multiproduct …rm competes against a single-product …rm. There are several important directions for future research. For instance, although we have identi…ed situations where monopoly bundling decreases or increases social welfare in numerical examples, it would be desirable to …nd general conditions with which the welfare e¤ects of monopoly bundling can be evaluated. It is also interesting to further study the incentives and e¤ects of bundling under competition. For instance, according to the existing literature, whereas bundling can be e¤ective entry barriers, it sometimes may also be entry-accommodating by creating (or increasing) product di¤erentiation. It would be desirable to develop a general understanding of when bundling deters entry and when it softens competition, in a general framework of preference dependence. 23

As MMW observed, under competition between multiproduct …rms, if consumer values for all goods are independently distributed, then a …rm will …nd it optimal to engage in mixed bundling if the other …rm does not, so that it cannot be an equilibrium for all …rms to choose separate selling. With competition between multiproduct …rms, …rms may also choose to o¤er (di¤erent) bundles in order to create product di¤erentiation (Carbajo, De Meza, and Seidman, 1990; and Chen, 1997).

23

APPENDIX Proof of Proposition 4.— We prove that p > ps under the stated conditions. The proof for q > q s is analogous: Under mixed bundling, p satis…es @ (p ; q ; r ) = Qx (p ; r ) + (p @p

mx )

@Qx (p ; r ) + (r @p

mx

my )

@Qb (p ; q ; r ) = 0: @p

From (3), (4), and (5), simple di¤erentiation and substitution yield: 0 = G (r

p )

+ (p

mx ) [ g (r

+ (r

mx

= G (r

C (F (p ) ; G (r

+ (r

p ) [1

my ) [g (r

p ) p

p )) C2 (F (p ) ; G (r

p ) [1

C2 (F (p ) ; G (r

C (F (p ) ; G (r my ) g (r

p ))

p ) [1

(p

C1 (F (p ) ; G (r

p )) f (p )]

p ))]]

mx ) f (p ) C1 (F (p ) ; G (r

C2 (F (p ) ; G (r

p ))

p ))] :

ps : Then, from A1, (p

Now, suppose that to the contrary, p and 0 =

p ))]

mx ) f (p )

1

F (p )

@ (p ;q ;r ) @p

=

G (r

p )

+ (r Z 1

p

C (F (p ) ; G (r my ) g (r

[C1 (x; G (r

p ) [1 p ))

p ))

[1

F (p )] C1 (F (p ) ; G (r

C2 (F (p ) ; G (r

C1 (F (p ) ; G (r

p ))

p ))]

p ))] dx

F (p )

+ (r

p

my ) g (r =

Z

1

F (p )

p ) [1

Z

C2 (F (p ) ; G (r

x

C11 (z; G (r

(r

p

p )) dzdx +

1;

(17)

F (p )

where 1

p ))]

my ) g (r

p )

Z

1

C12 (x; G (r

p )) dx > 0:

F (ps )

If u and v are negatively dependent or independent (i.e., for (x; y) 2 I 2 ; C11 (x; y)

C11 (x; y) = 0); then (17) becomes 0 > 0; a contradiction.

24

0 or

Next, suppose that u and v are positively dependent, but C11 Z

0

1

F (p )

Z

[1

2 2 . Then, from (17),

x

C11 (z; G (r

p )) dzdx +

1

F (p )

F (p )]2 2

1

1

+

2

>

1

2

+

1

2

= 0;

a contradiction. Q:E:D: Proof of Proposition 5.— Let (ps ; q s ) denote the prices of X and Y products in a separate pricing equilibrium. Then (")

A (p

s

+ ; q s ; q s ; ps + q s )

is Firm A’s pro…t from increasing the standalone price of X, while holding constant the standalone prices of the Y product and the price of the bundle at r = ps + q s . Bundling is 0

pro…table in equilibrium if 0

(0) =

@

(0) > 0. Noticing that @QYA (qA ; qB ; r) =@p = 0; we have

A (p

s ; q s ; q s ; ps

+ qs)

@p @QX (p ; q ; p + q ) @p @QXYA (p ; q ; q ; p + q ) my ) @p

= QX (ps ; q s ; ps + q s ) + (ps + (ps + q s

mx

mx )

with QX (ps ; q s ; ps + q s ) = C (1; G (q s ) ; G (q s )) R1 + G(qs ) [C3 (1; y; y) =

@QX (ps ; q s ; ps + q s ) @p

C (F (ps ) ; G (q s ) ; G (q s )) C3 (F (ps ) ; y; y)] dy

1 [1 2

F (ps )] +

1 [C (1; G (q s ) ; G (q s )) 2

=

[C2 (1; G (q s ) ; G (q s ))

C (F (ps ) ; G (q s ) ; G (q s ))] ;

C2 (F (ps ) ; G (q s ) ; G (q s ))] g (q s )

C1 (F (ps ) ; G (q s ) ; G (q s )) f (ps ) R1 C23 (F (ps ) ; y; y)] g (v (y)) dy G(q s ) [C23 (1; y; y) R1 s s G(q s ) C13 (F (p ) ; y; y) dyf (p ) ;

25

@QXYA (ps ; q s ; ps + q s ) @p

=

Therefore,

=

f (ps ) + [C2 (1; G (q s ) ; G (q s )) C2 (F (ps ) ; G (q s ) ; G (q s ))] g (q s ) R1 + G(qs ) [C23 (1; y; y) C23 (F (ps ) ; y; y)] g (v (y)) dy + C1 (F (p) ; 1; 1) f (p) : 0

(p

(q

s

s

@QX (ps ; q s ; ps + q s ) @p s + [C1 (F (p ) ; 1; G (q s )) C1 (F (ps ) ; G (q s ) ; G (q s ))] f (ps ) R1 + G(qs ) [C13 (F (ps ) ; 1; y) C13 (F (ps ) ; y; y)] dyf (ps ) f (ps )

mx ) my )

(

(

(0) = Qx ( ) + (ps

@Qx ( ) + @p

f (ps ) + [C1 (F (ps ) ; 1; G (q s )) C1 (F (ps ) ; G (q s ) ; G (q s ))] f (ps ) R1 x( ) + G(qs ) [C13 (F (p) ; 1; y2 ) C13 (F (p) ; y2 ; y2 )] f (p) dy2 @Q@p

R1

f (ps ) + [C2 (1; G (q s ) ; G (q s ))

we have (

0

)

+

C2 (F (ps ) ; G (q s ) ; G (q s ))] g (q s ) +

C23 (F (ps ) ; y; y)] g (v (y)) dy + C1 (F (p) ; 1; 1) f (p)

G(q s ) [C23 (1; y; y)

After cancelling out two terms of (ps 1 dC1 (1;y2 ;y2 ) ; 2 dy2

mx )

s x m) @Q @p ; noticing C1 (F (p ) ; 1; 1) = 1 and C13 (1; y2 ; y2 ) =

(0) = Qx ( ) +

C1 (F (ps ) ; G (q s ) ; G (q s ))] f (ps ) R1 (p mx ) 1 +f (ps ) C1 (F (p) ; 1; G (q s )) f (ps ) G(q s ) 2 dC1 (F (p) ; y2 ; y2 ) f (p) ( ) s ) ; G (q s )) s ) ; G (q s ) ; G (q s ))] g (q s ) [C (1; G (q C (F (p 2 2 R1 + (q s my ) + G(qs ) [C23 (1; y; y) C23 (F (ps ) ; y; y)] g (v (y)) dy s

= Qx ( )

f (ps ) + [C1 (F (ps ) ; 1; G (q s ))

+ (p + (q

s

s

mx ) my )

(

(

C1 (F (ps ) ; G (q s ) ; G (q s )) f (ps ) 1 2

[C1 (F (p) ; 1; 1)

C1 (F (p) ; G (q s ) ; G (q s ))] f (ps )

)

[C2 (1; G (q s ) ; G (q s )) C2 (F (ps ) ; G (q s ) ; G (q s ))] g (q s ) R1 + G(qs ) [C23 (1; y; y) C23 (F (ps ) ; y; y)] g (v (y)) dy

Substituting Qx ( ) ; and using (ps

)

m) f (ps ) = 1

26

F (ps ) ;

)

)

:

0

(0) =

Z 1 1 1 1 [1 F (ps )] + C1 (x; G (q s ) ; G (q s )) dx [1 F (ps )] 2 2 F (qs ) 2 1 C1 (F (ps ) ; G (q s ) ; G (q s )) [1 F (ps )] 2 ( ) s ) ; G (q s )) s ) ; G (q s ) ; G (q s ))] g (q s ) [C (1; G (q C (F (p 2 2 hR i R1 + (q s my ) 1 + G(qs ) F (ps ) C123 (x; y; y) dx g (v (y)) dy >

since

R1

1 2

Z

F (q s )

F (ps ) C123 (x; y; y) dx

2

= (q s

1

Z

x

C11 (z; G (q s ) ; G (q s )) dzdx +

> 0; where

my ) g (q s )

Z

1

F (ps )

since q s

2;

F (q s )

"Z

G(q s )

#

C123 (x; G (q s ) ; z) dz dx > 0

0

my > 0 due to ps and q s being interior values; and f ( ) > 0; g ( ) > 0; and

C123 ( ) > 0 for interior values. Thus,

0

(0) > 0 if values for X and Y are negatively dependent (C11

0) or independent

(C11 = 0): Now, suppose that values for X and Y are positively dependent but C11

4 2:

Then 0

Z Z x 1 1 4 2 dzdx + 2 F (qs ) F (qs ) 1 (1 F (q s ))2 4 2 + 2 > 4

(0) > =

2

2

+

2

= 0:

Q:E:D: Tables for the FGM-Uniform Case.— When m = 0; ps = 4:5;

s

= 8: 1; and W s = 4: 05: T =

bundling is optimal, and r >

2ps

+ W;

T =

+

W: Pure

= 9: Bundling makes all consumers worse o¤. Both

the bundle price and pro…t decrease as

increase, whereas consumer surplus increases as

increases. Table 1. m = 0

27

m=0

p

r

W

W

T

=

1

9

10: 564

9: 736 1

1: 636 1

2: 487 7

1: 562 3

0:073 8

=

:5

9

10: 412

9: 463 6

1: 363 6

2: 655 2

1: 394 8

0:031 2

=0

9

10: 210

9: 212 7

1: 112 7

2: 862 0

1: 188

0:075 3

= :5

9

9: 955 6

8: 992 5

0:892 5

3: 110 2

0:939 8

0:047 3

=1 9 9: 675 2 8: 811 4 0:711 4 3: 377 7 0:672 3 0:039 1 When m = 2; ps = 5:5; s = 4:9; and W s = 2:45: Pure bundling is optimal, and r > 2ps = 11 when decrease as

0:5 but r < 2ps when

0: Both the bundle price and pro…t

increase, whereas consumer surplus increases as

consumer surplus is lower under bundling when Table 2. m = 2 m=2 p r

increases. (Aggregate)

0:5 but higher when W

W

= 1:

T

=

1

9

11: 100

6: 146 8

1: 246 8

2: 007 9

0:442 1

0:804 7

=

:5

9

11: 042

5: 942 1

1: 042 1

2: 102 0

0:348

0:694 1

=0

9

10: 961

5: 740 4

0:840 4

2: 212 1

0:237 9

0:602 5

= :5

9

10: 846

5: 543 4

0:643 4

2: 345 8

0:104 2

0:539 2

=1 9 10: 680 When m = 4; ps = 6:5; and r < 2ps = 13: As

5: 354 7 0:454 7 2: 515 2 0:065 2 0:519 9 = 2:5; and W s = 1:25: Mixed bundling is optimal, p > ps ;

s

increases, the standalone price and pro…t decrease, the bundle

price increases, whereas consumer surplus …rst increases then decreases. Consumer surplus is higher under bundling when Table 3. m = 4 m=4 p

0:5 but lower when

r

= 1:

W

W

T

=

1

7: 499 9

12: 072

2: 917 4

0:417 4

1: 263 6

0:013 6

0:431

=

:5

7: 431 1

12: 170

2: 828 6

0:328 6

1: 278

0:028

0:356 6

=0

7: 333 3

12: 310

2: 746

0:246

1: 274 9

0:024 9

0:270 9

= :5

7: 211 1

12: 496

2: 674 1

0:174 1

1: 253 0

0:003

0:177 1

=1 7: 091 7 12: 700 2: 617 1 0:117 1 1: 220 9 0:029 1 0:088 s s s When m = 6; p = 7:5; = 0:9; and W = 0:45: Mixed bundling is optimal, p > ps ; and r < 2ps = 15: As

increases, the standalone price and bundle price increase, pro…t

decreases, and consumer surplus varies non-monotonically. Consumer surplus is always lower under bundling. Table 4. m = 6

28

m=6

p

r

W

W

T

=

1

7: 682

14: 226

0:956 09

0:056 09

0:444 33

0:005 67

0:050 42

=

:5

7: 684 6

14: 381

0:945 48

0:045 48

0:446 77

0:003 23

0:042 25

=0

7: 691 6

14: 52

0:938 61

0:038 61

0:444 78

0:005 22

0:033 39

= :5

7: 704 6

14: 636

0:937 51

0:037 51

0:421 57

0:028 43

0:009 08

=1

7: 724 4

14: 730

0:932 41

0:032 41

0:435 18

0:014 82

0:017 59

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[12] Schmalensee, Richard (1984), "Gaussian Demand and Commodity Bundling,"Journal of Business, 57, S211-S230. [13] Stigler, George (1963), "A Note on Block Booking," U.S. Supreme Court Review, 1963, 152-175. [14] Whinston, Michael D. (1990), “Tying, Foreclosure, and Exclusion,” American Economic Review, 80, pp. 837-59.

30