Arizona State University Economics Competing Varieties Questions


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Economics Letters 151 (2017) 28–30
Contents lists available at ScienceDirect
Economics Letters
journal homepage:
Complementarity without superadditivity?
Steven Berry a , Philip Haile a, *, Mark Israel b , Michael Katz c
Yale University, United States
Compass Lexecon, United States
University of California, Berkeley, United States
Article history:
Received 26 October 2016
Accepted 18 November 2016
Available online 1 December 2016
Discrete choice
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
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: (P. Haile).
1 See Cournot (1838) and Economides and Salop (1992).
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 greengrocer’s apple sales. Thus, an
increase in the price of beer at the liquor store reduces demand
for the greengrocer’s apples. Symmetrically, an increase in the
greengrocer’s apple price reduces demand for the liquor store’s
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 good’s
price; thus, for example, an increase in the price of A corresponds
S. Berry et al. / Economics Letters 151 (2017) 28–30
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 i’s utility from
purchasing both goods is
a +b +?
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) +
Pr (a + b + ? ? 0|a) dFA (a),
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 + ?
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
5 Our result extends to the case of more than one pair of goods with superadditive
utilities by letting a1 + b1 + ? represent a consumer’s 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.
S. Berry et al. / Economics Letters 151 (2017) 28–30
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 )
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?
aj + max bk? ? a1 + b1 + ?.
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? ,
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, 105–123.
Gans, J.S., King, S.P., 2006. Paying for loyalty: Product bundling in oligopoly. J. Ind.
Econ. 54, 43–62.
Gentzkow, M., 2007. Valuing new goods in a model with complementarities: Online
newspapers. Amer. Econ. Rev. 97, 713–744.
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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