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Economics Letters 151 (2017) 2830

Contents lists available at ScienceDirect

Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

Complementarity without superadditivity?

Steven Berry a , Philip Haile a, *, Mark Israel b , Michael Katz c

a

b

c

Yale University, United States

Compass Lexecon, United States

University of California, Berkeley, United States

article

info

Article history:

Received 26 October 2016

Accepted 18 November 2016

Available online 1 December 2016

Keywords:

Complements

Substitutes

Discrete choice

Mergers

a b s t r a c t

The distinction between complements, substitutes, and independent goods is important in many contexts.

It is well known that when consumers conditional indirect utilities for two goods are superadditive,

the goods are gross complements. Generalizing insights in Gans and King (2006) and Gentzkow (2007),

we show that superadditivity between one pair of goods can also introduce complementarity between

competing pairs of goods. One implication is that lower prices can result from a merger between producers

of goods that themselves offer no superadditivity.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

Whether two good are complements is often of interest. For example, if two goods are complements, a merger of their producers

tends to reduce prices because the merged firm will internalize the

benefits that lowering the price of one good has on demand for the

other.1

A sufficient condition for two goods to be complements is that

they have superadditive conditional indirect utilities. Sources of

such superadditivity include direct consumption synergies, bundle

discounts, convex loyalty rewards, price club memberships, convex shipping discounts (e.g., Amazon Prime), and benefits of onestop shopping. We show that, whatever the source, superadditivity

between one pair of goods creates complementarity, not only

between that pair but also between pairs of competing goods. An

important implication is that a merger between producers of goods

that exhibit no superadditivity can lead to lower prices.

To suggest why, consider two categories of goods, apples and

beer, with no intrinsic complementarities in consumption: once

purchased, the value a consumer places on consuming a good from

each category is the sum of the values she places on consuming

each good alone. Apples are offered by a greengrocer; beer by a

liquor store. A supermarket offers both apples and beer. A consumer experiences travel cost t from each visit to one of these sellers. Consider a consumer who buys apples from the greengrocer

? The authors served as consultants to AT&T in its merger with DIRECTV. Early

versions of the results presented here were discussed in filings with the U.S.

Department of Justice and Federal Communications Commission on behalf of AT&T.

We thank Joshua Gans for helpful comments and Wayne Gao for research assistance.

Corresponding author.

E-mail address: philip.haile@yale.edu (P. Haile).

1 See Cournot (1838) and Economides and Salop (1992).

*

http://dx.doi.org/10.1016/j.econlet.2016.11.020

0165-1765/© 2016 Elsevier B.V. All rights reserved.

and beer from the liquor store but, at current prices, finds this only

slightly preferable to her second-best option of buying both from

the supermarket. If the liquor store raises its beer price, this consumer will switch to purchasing both goods from the supermarket,

because doing so saves t in travel cost. This is the only response to

the price change affecting the greengrocers apple sales. Thus, an

increase in the price of beer at the liquor store reduces demand

for the greengrocers apples. Symmetrically, an increase in the

greengrocers apple price reduces demand for the liquor stores

beer. Therefore, these two goods are gross complements.

Our result generalizes an observation in Gans and King (2006),

who studied pricing in a stylized linear city model with no

outside good, assuming that all consumers purchase one good from

each of two product categories. They note that a key force in their

analysis is the fact that bundle discounts offered for one pair of

goods create negative cross-price elasticities between the goods of

competing firms. We focus on this phenomenon itself and show

that it extends to a much more general random utility model,

with no restriction on the form of competition or the source of

superadditivity in consumers preferences.

2. Baseline: two goods

We begin with the two-good random utility model of Gentzkow

(2007), dropping his functional form and distributional assumptions. Consider a market with two goods, A and B, and a continuum of consumers with mass normalized to one. Each consumer

desires at most one unit of each good. A given consumer i obtains

conditional indirect utility (utility) ai when purchasing A alone,

and bi from purchasing B alone. Each utility accounts for the goods

price; thus, for example, an increase in the price of A corresponds

S. Berry et al. / Economics Letters 151 (2017) 2830

29

producing indifference between the AB bundle and the outside

good. For consumers whose utilities are bounded away from this

line segment, a small change in b has no effect on the decision

to purchase A. However, when the price of good B increases, a

mass of consumer utilities just above this locus of indifference will

move downward across the line segment, leading the associated

consumers to switch from AB to the outside good.

3. Competing varieties

Fig. 1. Two-good case.

to a reduction in ai for every consumer. Throughout we normalize

the utility of the outside good to zero. Consumer is utility from

purchasing both goods is

a +b +?

i

i

where ? ? 0 and, for simplicity, ? is constant across consumers.

When ? > 0, consumers have superadditive utilities.

To ease notation, we henceforth drop the superscript i from utilities. Let F denote the joint distribution of (a, b) across consumers.

For expositional convenience, we assume F has support R2 and is

absolutely continuous with respect to the Lebesgue measure.

Consumers in different regions of R2 will make different

choices. As in Gentzkow (2007), an examination of how price

changes alter the measure of consumers in each choice region

demonstrates that superadditivity leads to complementarity.

Proposition 1. Goods A and B are strict gross complements if ? > 0,

and independent goods if ? = 0.

Proof. A is purchased (alone or with B) whenever a > 0. When

a ? 0, A is purchased if and only if both ?? ? a ? 0 and

a + b + ? ? 0.2 Total demand for A is therefore

?

1 ? FA (0) +

0

??

Pr (a + b + ? ? 0|a) dFA (a),

(1)

where FA denotes the marginal distribution of a. An in increase in

the price of good B implies a reduction of b in terms of first-order

stochastic dominance; if ? > 0, this means that demand for good

A falls. If ? = 0, however, (1) does not depend on b. ?

Fig. 1 illustrates the product choice regions with ? > 0.3 The

thick diagonal line segment shows the set of consumer utilities

2 We ignore ties since these occur with probability zero and therefore have no

effect on demand.

3 This figure is also found as the ? > 0 panel of Fig. 1 in [4].

The equivalence between complementarity and superadditivity

of the conditional indirect utilities breaks down when consumers

can choose

varieties within each product category.

( from multiple

)

Let goods A1 , . . . , AJ denote different varieties in category A, with

(B1 , . . . , BK ) denoting varieties in category B. Each consumer again

values at most one unit from each category.4 For example, a consumer might wish to have both home television service (category

A) and home internet service (category B), but does not benefit

sufficiently from a second variety of either category of service to

offset its price.

Each consumer has (conditional indirect) utilities for the standalone goods given by aj and bk . As before, utilities are net of

prices and vary(randomly across consumers.

Let H denote the joint

)

distribution of a1 , . . . , aJ , b1 , . . . , bK . We assume H has support

RJ × RK and is absolutely continuous. As a notational convention,

we introduce a0 ? 0 and b0 ? 0.

The utility from purchasing A1 and B1 is superadditive, equal to

a1 + b 1 + ?

(2)

where ? is nonnegative and, for simplicity, constant across consumers.5 For the remaining combinations of goods, utilities are additive.6 Thus, by Proposition 1, for j ? = 1 and k ? = 1, Aj and Bk would

be independent goods in a market without competing varieties.

However, in the multi-variety setting, superadditive utilities for A1

and B1 induce complementarity between these Aj and Bk as well.7

Proposition 2. For all j ? = 1 and k ? = 1, Aj and Bk are strict gross

complements iff ? > 0.

Proof. Take j ? = 1 and k ? = 1 and consider the effect of a change

in the price of Bk on demand for Aj (a symmetric argument applies

to the effect of Aj s price on demand for Bk ). Aj is purchased by a

consumer if and only if both

4 Formally this is a restriction on the utilities of consumer choices involving more

than one unit of A or B. This places no restriction on the joint distribution H defined

below.

5 Our result extends to the case of more than one pair of goods with superadditive

utilities by letting a1 + b1 + ? represent a consumers maximum utility among all

superadditive pairs and, in the proof, considering j and k from the goods not part of

any superadditive pair.

6 The asymmetry in superadditivity modeled here arises naturally when ? reflects, e.g., one-stop shopping, bundle discounts, convex shipping discounts, or

loyalty rewards. Less extreme asymmetry may sometimes arise in the case of

intrinsic consumption synergies between different varieties of A and B, e.g., due to

varying aesthetic or technological compatibility.

7 To reconcile these observations, consider the case J = K = 2 and note that one

cannot construct the total demand for good A2 from the demand for A2 , B2 , A2 B2

and outside option all other goods. For example, the option A2 B1 would be then

be part of the outside option.

30

S. Berry et al. / Economics Letters 151 (2017) 2830

aj ? (a1 + b1 + ? ? bk , a1 + b1 + ? ? bk + ?)

will switch from purchasing Aj (along with Bk ) to purchasing A1

(along with B1 ). Our support assumption guarantees that there is a

positive measure of such consumers. No other consumers will alter

their choices with respect to Aj. ?

Fig. 2 illustrates in the case J = K = 2. The figure shows

choice regions in the

( space

) of (a2 , b2 ), with (a1 , b1 ) held fixed at

particular values, a?1 , b?1 . In general different values of (a1 , b1 )

8

create different

(

) graphs and product choices. For Fig. 2 we have

selected a?1 , b?1 such that the pair (A1 , B1 ) is preferred to the outside good and either A1 or B1 alone. Note that A2 is never purchased

when a2 < 0; but when a2 > a?1 + ?, A2 is purchased regardless

of b2 . Consumers in the lower left region choose (A1 , B1 ) ; those

in the upper right region choose (A2 , B2 ). The thick diagonal line

segment represents the consumers who are indifferent between

the two pairs. Substitution between these pairs creates the strict

complementary between A2 and B2 .

Fig. 2. Four-good case.

aj ? max aj?

j?

References

(3)

and

aj + max bk? ? a1 + b1 + ?.

k?

(4)

If ? = 0, (3) implies (4), so that the values of bk have no effect

on demand for Aj . Now suppose ? > 0. If the price of Bk rises by

? ? (0, ?), all consumers for whom aj = maxj? aj? , bk = maxk? bk? ,

and

Cournot, A., 1838. Researches into the Mathematical Principles of the Theory of

Wealth. Macmillan, New York. Translation (1897) by Nathaniel Bacon.

Economides, N., Salop, S.C., 1992. Competition and integration among complements, and network market structure. J. Ind. Organ. 40, 105123.

Gans, J.S., King, S.P., 2006. Paying for loyalty: Product bundling in oligopoly. J. Ind.

Econ. 54, 4362.

Gentzkow, M., 2007. Valuing new goods in a model with complementarities: Online

newspapers. Amer. Econ. Rev. 97, 713744.

8 Although Fig. 2 has a form similar to Fig. 2 in Gans and King (2006) , there are

important differences. Because Gans and King (2006) assume that every consumer

purchases either A1 or A2 and either B1 or B2 , a consumer chooses among only

four options rather than 9 (more generally, (J + S ) × (K + S ), where S denotes the

number of pairs exhibiting superadditivity). And in their two-dimensional linear

city model, a single two-dimensional diagram fully characterizes the map from

consumer types to the four choice probabilities.

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