# Cross-Market Mergers with Common Customers:
When (and Why) Do They Increase Negotiated Prices?

Enrique Ide IESE Business School, Carrer de Arnús i de Garí 3-7, 08034
Barcelona, Spain (email: eide@iese.edu). I have benefited enormously from the
comments and suggestions of Nano Barahona, Daniel Barron, Juan-Pablo Montero,
Sebastián Otero, and Eduard Talamàs. I am also thankful for the support
provided by the MCIN/AEI /10.13039/501100011033/ FEDER, UE for the project
with Ref. PID2021-123113NB-I00. I declare that I have no relevant or material
financial interests that relate to the research described in this paper.

###### Abstract

I examine the implications of cross-market mergers of suppliers to
intermediaries that bundle products for consumers. These mergers are
controversial. Some argue that suppliers’ products will be substitutes for
intermediaries, despite not being substitutes for consumers. Others contend
that because bundling makes products complements for consumers, products must
be complements for intermediaries. I contribute to this debate by showing that
two products can be complements for consumers but substitutes for
intermediaries when the products serve a similar role in attracting consumers
to purchase the bundle. This result leads to new recommendations and helps
explain why cross-market hospital mergers raise prices.

Keywords: Cross-Market Mergers, Conglomerate Mergers, Bargaining, Hospital
Mergers.

JEL classifications: D43, L13, L40.

##  1 Introduction

One of the most controversial current topics in merger enforcement is the
competitive implications of cross-market hospital mergers in the United
States. These mergers involve hospitals in distinct geographical areas or
different diagnostic markets. The controversy stems from the fact that such
transactions have been subject to little antitrust enforcement despite their
prevalence and the existence of credible empirical evidence suggesting that
they have led to increases in the reimbursement rates paid by insurers without
corresponding increases in quality (Lewis and Pflum, 2015; Dafny, Ho and Lee,
2019; Brand and Rosenbaum, 2019).

This lack of enforcement, however, has not been due to a lack of scrutiny or
regulatory neglect. Instead, authorities have concluded that such transactions
do not raise antitrust concerns since hospitals operating in different markets
are not substitutes for patients.111See Farrell et al. (2011) for an overview
of the FTC’s current approach when dealing with hospital mergers. The apparent
contradiction between the empirical findings and antitrust practice has
sparked interest among economists and practitioners alike, who, in recent
years, have proposed alternative mechanisms to explain these increases in
negotiated prices.

One theory that has gained increasing attention recently posits that hospitals
can be substitutes for insurers—even if they are not substitutes for
patients—in the presence of “common customers” (Vistnes and Sarafidis, 2013;
Dafny, Ho and Lee, 2019). Common customers are purchasing entities—such as
employers and households—that value the inclusion of multiple hospitals in the
insurers’ network.222Employers and households usually look for insurance plans
covering different diagnostic markets. Additionally, large employers with
offices in different geographical areas look for plans with facilities in the
locations where their employees live and work.

Nonetheless, this theory faces the challenge that the presence of common
customers is equally consistent with hospitals being complements for the
insurer, in which case the merger will lead to a reduction in negotiated
prices (Brand and Rosenbaum, 2019). This is because hospitals in an insurer’s
network offering non-overlapping services are likely complements for employers
and households, making them complements for insurers as well. The previous
argument is epitomized by Easterbrook et al. (2018, pp. 507-508) in the
following example:

> [Suppose there are] only two hospitals in an area—C and D… These two
> hospitals have no overlap in services… [However,] Because employers are
> likely to purchase health plans to meet the heterogeneous health needs of
> all of their employees… a plan with only hospital C or D may offer very
> limited value compared to a plan with both hospitals…
>
> This would then generate complementarities across the hospitals to employers
> selecting an insurance plan to offer their employees… Complementarity
> between hospitals at the second stage of competition [i.e., when employers
> and insurers meet] in turn drives complementarity at the first stage [i.e.,
> when insurers negotiate with hospitals].

The lack of a unified theory explaining these two different—and
contradictory—results has increased calls for more research on the topic. For
instance, Brand and Rosenbaum (2019) state that “a critical area for future
research is to go beyond possibility results and recover how insurer profits
change with network composition… determining how insurer profits shift with
the inclusion and exclusion of different types of hospitals should be a
priority.” My aim with this paper is to begin answering these calls.

More precisely, I analyze the effects of a merger of suppliers (e.g., two
hospitals) that bargain with a single intermediary333As I show in the online
Appendix, the results continue to hold with competing intermediaries. (e.g.,
an insurance company) over the carriage terms of products that are neither
direct substitutes nor technological complements (e.g., because the hospitals
operate in different markets). The intermediary bundles the products for sale
to consumers interested in multiple products, making consumers “common
customers” (e.g., they are employers looking for comprehensive health
insurance plans for their employees). A consumer then purchases the bundle
whenever the bundle’s expected net value exceeds the consumer’s private
outside option.

Although the model is inspired by the debate on cross-market hospital mergers
described above, it is general enough to encompass other relevant markets in
which similar transactions have arisen. For instance, the model accommodates
the retail industry equally well. In this market, consumer goods
manufacturers, which sell their products through large retail chains offering
one-stop shopping to consumers, have been consolidating during the last 20
years.444Two concrete examples of cross-market mergers in this industry are
Pernod Ricard and Diageo’s joint bid of USD 8.15 billion cash for Seagram
Spirits in 2001 (E.U. Case No COMP/M.2268) and Gillette’s acquisition by
Procter & Gamble for USD 57 billion in 2005 (E.U. Case No COMP/M.3732).

My main results are as follows. Consistent with the example in Easterbrook et
al. (2018), the fact that consumers are common customers makes products in the
bundle complementary to consumers even though these products are not direct
complements or substitutes.555More precisely, consumers’ indirect utility
function is supermodular in the inclusion of products in the bundle, which, in
turn, makes products gross complements when considering the intermediary’s
demand. However, I also show that Easterbrook et al.’s (2018) argument is
incorrect in that these consumer-level complementarities do not always
transfer upstream to the intermediary’s profit function. Thus, bundling can
make two products complements for consumers—the common customers—but the
products can still be substitutes for the intermediary. In this case, the
merger of suppliers leads to higher negotiated prices.

The underlying reason behind this result is subtle yet intuitive. The starting
point is realizing the dual role that products play in the intermediary’s
bundle/lineup: products directly contribute with revenue/profits and
indirectly contribute by attracting consumers to purchase the bundle (i.e.,
creating “consumer traffic”).

The widespread belief that consumer-level complementarities must drive
complementarities upstream can be explained by the direct profit
complementarities arising between two products in the intermediary’s bundle:
Because products are complements for consumers, the inclusion of, for example,
product C makes it more likely that the consumer purchases the bundle. This
then increases the demand for, for example, product D, increasing the
intermediary’s direct profits stemming from this last product.

The issue, however, is that the story does not end there. This is due to the
fact that, since each product can help enhance consumers’ participation, both
C and D can be used to increase the demand for other products in the
intermediary’s lineup. Thus, C and D can potentially be substitutes in
generating traffic for these additional products.666Products C and D are
substitutes in traffic when losing both simultaneously induces a larger drop
in the bundle’s sales than the sum of the reductions in sales from losing each
product individually. If that is the case, and this substitutability in
traffic dominates the direct profit complementarities described above, C and D
will be substitutes for the intermediary in overall profits despite the two
products being complements for the common customers.

I then focus on the antitrust implications of these results. First, I show
that the theory provides a preliminary screening test that can immediately
assess whether two products will be complements for the intermediary in
overall profits.

In particular, if the intermediary’s bundle solely comprises the merging
suppliers’ products or the merging products are weak complements in generating
consumer traffic—in the sense that the sum of the reductions in the bundle’s
sale from losing each product individually weakly exceeds the reduction in
sales from losing them both jointly—then the products will necessarily be
complements for the intermediary in overall profits. The beauty of this test
is that the first condition can be observed directly from the intermediary’s
portfolio, while the second condition can be readily assessed by empirically
estimating the consumer loss ratios of the merging products.777The consumer
loss ratio is defined as the fraction of all consumers lost by the
intermediary after removing an item or sets of items. This measure is already
being used in other antitrust contexts—most notably in the evaluation of
vertical mergers (e.g., U.S. v. AT&T, et al., Civil Case No. 17-2511, RJL,
D.D.C. Jun. 12, 2018). Although a precise estimation of these ratios would
most likely require a structural model, antitrust authorities have found
alternative ways to estimate them in practice (see U.S. v. AT&T, et al.).

Then, I show that even if two products fail this preliminary test so that the
products can potentially, but not always, be substitutes for the intermediary,
authorities can determine their complementarity/substitutability by assessing
the role played by the products in the intermediary’s lineup. For instance,
when two products mainly contribute to the intermediary by generating traffic,
the products are likely to be substitutes in overall profits if they are
substitutes in consumer traffic. In contrast, a product used to generate
traffic and one used to extract profits will likely be complements in overall
profits, even if they are substitutes in traffic.888Of course, this raises the
question of what determines a product’s role in the intermediary’s lineup. As
I later discuss, a critical determinant of a product’s role is the curvature
of its demand relative to the other products in the lineup. In particular,
products with more concave and rectangular-shaped demands will be used mainly
as direct revenue/profit contributors, while products with more convex demands
will generally be used as traffic generators. Hence, authorities should pay
particular attention to mergers involving two or more products with very
convex demands.

Finally, I use my results to propose a new divestiture framework for these
types of transactions. Indeed, suppose the merging suppliers offer portfolios
of products instead of single products. Assume, furthermore, that these
portfolios are known to be substitutes in overall profits, so the original
transaction leads to an increase in negotiated prices. In that case, to offset
the negative effects of a merger, authorities should force the merging parties
to divest those products that contribute toward intermediaries’ overall
profits mainly via consumer traffic.

This new divestiture approach, however, also comes with a note of caution: the
divestiture of assets can backfire if authorities select the wrong products,
i.e., those contributing to the intermediary via direct profits. This is
because the divestiture then decreases the direct profit complementarities
between the merging portfolios, making it more likely that the postdivestiture
portfolios will be substitutable in traffic and, therefore, in overall
profits.

###  1.1 Related Literature

This paper makes two important contributions to the literature on cross-market
mergers of upstream suppliers. First, it shows that in the presence of common
customers, two products can be substitutes for intermediaries despite being
complements for customers. Second, my paper proposes new antitrust tools to
deal with these types of transactions. I discuss the literature related to
these contributions here.

My paper is connected to the work of Vistnes and Sarafidis (2013) and Dafny,
Ho and Lee (2019), who propose the “holes in the network” theory of harm in
the context of the healthcare industry to explain why cross-market hospital
mergers can increase negotiated prices in the absence of foreclosure.999To
date, most theories stressing how portfolio effects can raise prices have
relied on monopolization or foreclosure arguments (e.g., Whinston, 1990;
Greenlee, Reitman and Sibley, 2008; Chambolle and Molina, 2023; Ide and
Montero, 2023). Hence, it is not surprising that antitrust authorities have
been evaluating cross-market mergers through those lenses (e.g., European
Commission, 2008). My paper, as well as the nascent literature on cross-market
mergers more broadly, urges caution in relying exclusively on such a
foreclosure-centric approach. The development of such a theory is important
because in most cross-market hospital mergers in the United States, as well as
in other conglomerate mergers, authorities have concluded that monopolization
concerns were minimal.

Briefly, the “holes in the network theory” states that in the presence of
common customers, two hospitals without overlapping services—for example,
hospitals C and D—will be substitutes for an insurer in overall profits if
customers continue to buy the plan without either hospital C or D but not
without both (i.e., customers tolerate one but not two “holes” in the
insurance plan).

As I formally show in Section 2.2, this theory is an extreme version of my
argument. In particular, the theory describes a situation where C and D are
“perfect substitutes” in creating traffic for the remaining items in the
bundle. This explains why the insurer views C and D as substitutes in overall
profits in this case.

Besides generalizing their results, my paper contributes to Vistnes and
Sarafidis’s (2013) and Dafny, Ho and Lee’s (2019) theory in three important
ways. First, neither paper addresses the apparent contradiction—raised by
Easterbrook et al. (2018) and Brand and Rosenbaum (2019)—of products being
complements for customers but substitutes for intermediaries. By providing a
unifying framework, my paper shows that there is, in fact, no contradiction
and clarifies under what conditions complementarities at the customer level
transfer to the intermediary level.

Second, neither Vistnes and Sarafidis (2013) nor Dafny, Ho and Lee (2019) make
the connection between the complementarity/substitutability of products at the
intermediary level and the role those products play in the intermediary’s
lineup. This is the key insight behind the new divestiture approach I develop
here. Third and finally, my general framework leads to new antitrust tools to
deal with these transactions: a preliminary test to screen whether two
products are complements for intermediaries and the aforementioned divestiture
framework.

My paper is also related to the work by Peters (2014). He provides a different
theoretical mechanism for explaining how cross-market mergers of upstream
suppliers can increase negotiated prices in the presence of common customers.
Unlike my theory, Peters’s (2014) theory posits that mergers can raise prices
even when products are not substitutes for intermediaries. Specifically, he
argues that cross-market mergers can increase suppliers’ reservation payoffs
in negotiations with intermediaries when intermediaries compete.

Concerning this alternative theory, in the online Appendix, I provide an
extension that combines this mechanism and mine. There, I show that the two
effects operate under different circumstances101010For example, Peters’s
(2014) effect will be more pronounced the more intense competition among
intermediaries is and the higher the bargaining weight/power of the latter
relative to suppliers. In contrast, the mechanism I develop here does not
require downstream competition and will have a more pronounced effect when
suppliers have a higher bargaining weight relative to intermediaries. and
provide a preliminary screening test—similar to the one I propose to screen
for complementarities at the intermediary level—that can be used to assess
whether this additional effect is present in a given merger.

##  2 A Simple Model

In this section, I present my main results in a simplified model. In this
model, I work with reduced-form payoffs (instead of dealing with retail and
wholesale prices and marginal costs) and assume that the intermediary and the
suppliers bargain over lump-sum fees. The latter implies that upstream mergers
will only affect firms’ profits, not consumer surplus.

In Section 3, I fully microfound this simplified setting and consider
negotiations over linear contracts (as in, e.g., Crawford and Yurukoglu, 2012;
Ho and Lee, 2017; Crawford et al., 2018; Luco and Marshall, 2020). There I
show that the simple model’s lessons remain unchanged with the exception that
consumer welfare is impacted.

###  2.1 The Setting

Setup.— Suppose the intermediary is a large monopoly retail chain offering
one-stop shopping to a unit mass of infinitesimal consumers (below, I show
that this model can be directly reinterpreted as one depicting the healthcare
market). This intermediary can potentially carry three products,
$i\in\\{0,1,2\\}$, that are neither direct substitutes nor technological
complements (i.e., they are “unrelated” in consumption). Each product,
moreover, is solely produced by a single supplier (hence, I can refer to
products or suppliers indistinguishably).

Suppliers can only reach final consumers indirectly via the intermediary.
Consumers have a unit demand for each product. I assume—in a reduced-form
fashion—that purchasing a unit of product $i$ gives the consumer a surplus of
$v_{i}\geq 0$, while it generates profits of $\pi_{i}\geq 0$ and
$\varphi_{i}\geq 0$ to the intermediary and supplier $i$, respectively.
Because products are unrelated, when the consumer buys multiple products, her
surplus—and the firms’ profits—are additive over these products.

When visiting the intermediary, consumers also incur a shopping cost $\xi\geq
0$. The latter is heterogeneous among the population of consumers and is
characterized by the cumulative distribution function $G(\xi)$. I assume that
consumers observe the intermediary’s portfolio of products before deciding
whether to incur $\xi$. This shopping cost could be interpreted literally as
the opportunity cost of time spent in traffic and parking, selecting products,
and so forth, or—taking a partial equilibrium view—as the value for the
consumer of visiting its best alternative among the intermediary’s
competitors.

Finally, I assume that the intermediary and the suppliers bargain over the
term of carriage. More precisely, each supplier engages in a bilateral Nash
bargain with the intermediary over a lump-sum fee, anticipating that the other
suppliers will also reach an agreement with the intermediary.111111The Nash-
in-Nash bargaining protocol (Horn and Wolinsky, 1988; Collard-Wexler,
Gowrisankaran and Lee, 2019) has been playing an ever more prominent role in
the theoretical and empirical literature analyzing the effect of upstream
negotiations on market outcomes (e.g., Dobson and Waterson, 1997; Ho and Lee,
2017). It has also become an integral part of authorities’ toolkit when
evaluating upstream mergers of suppliers and vertical mergers (e.g., Farrell
et al., 2011; Shapiro, 2018; Rogerson, 2020).

Payoffs.— Let $\mathcal{N}\equiv\\{0,1,2\\}$ be the set of all
products/suppliers. The indirect utility function of a consumer with a
shopping cost of $\xi$ if the intermediary’s portfolio is $\mathcal{G}\in
2^{\mathcal{N}}$ is given by
$U(\mathcal{G};\xi)=\max\\{\sum_{i\in\mathcal{G}}v_{i}-\xi,0\\}$. The overall
demand for product $i$, in turn, is equal to the fraction of consumers
visiting the intermediary:

$D_{i}(\mathcal{G})=\mathbbm{1}_{i\in\mathcal{G}}\mathbb{P}\big{(}\textstyle\sum_{j\in\mathcal{G}}v_{j}\geq\xi\big{)}=\mathbbm{1}_{i\in\mathcal{G}}\tilde{G}(\mathcal{G})$
(1)

where for future reference, $\tilde{G}:2^{\\{\emptyset,0,1,2\\}}\to[0,1]$
defined as $\tilde{G}(\mathcal{G})\equiv
G(\textstyle\sum_{i\in\mathcal{G}}v_{i})$ is the consumer traffic function and
$\mathbbm{1}_{X}$ denotes the indicator function of the event $X$. Thus, the
profits before negotiated fees made by the supplier of product $i$ are equal
to
$\Phi_{i}(\mathcal{G})=\varphi_{i}D_{i}(\mathcal{G})=\varphi_{i}\mathbbm{1}_{i\in\mathcal{G}}\tilde{G}(\mathcal{G})$.
On the other hand, since the intermediary makes a per-consumer profit of
$\mathbbm{1}_{i\in\mathcal{G}}\pi_{i}$ on product $i$, its overall profits
before fees are equal to:

$\Psi(\mathcal{G})=\big{(}\textstyle\sum_{i\in\mathcal{G}}\pi_{i}\big{)}\tilde{G}(\mathcal{G})$
(2)

Remark 1: Consumer Level Complementarities. The setting described exhibits
complementarities at the consumer level. More precisely, the indirect utility
of a consumer with shopping cost $\xi$, $U(\mathcal{G};\xi)$, is supermodular
in the inclusion of products in the bundle (see Appendix A for the formal
proof of this result). This, in turn, makes products gross complements when
considering the intermediary’s demand, in the sense that $i$’s demand,
$D_{i}(\mathcal{G})$, is increasing in the cardinality of the intermediary’s
product portfolio $\mathcal{G}$. Intuitively, the complementarity between
products at the consumer level arises due to one-stop shopping economies.

Remark 2: Healthcare Interpretation. The model presented above can be directly
reinterpreted as one depicting the healthcare market. Under this
interpretation, $v_{i}$ is the total surplus that all the employees of a given
employer receive from having access to hospital $i$. An employer will then
offer the intermediary’s insurance plan to their employees if the total
surplus employees receive from having access to the plan,
$\sum_{i\in\mathcal{G}}v_{i}$, exceeds the administrative cost or outside
option $\xi\in\mathbb{R}_{+}$.

Note that under this interpretation, the “consumer” choosing whether to “visit
the intermediary” is different from the one “consuming” the product. The
former is the employer (the “common customer” in this setting), while the
latter is the employer’s employees. Thus, under this interpretation,
complementarities between hospitals arise at the employer level, not at the
employee/patient level, and are explained by employers looking to “purchase
health plans to meet the heterogeneous health needs of all of their employees”
(as stated in the example by Easterbrook et al., 2018, quoted in the
Introduction).

###  2.2 The Effects of an Upstream Merger

I now analyze how the merger of suppliers $1$ and $2$ affects the terms
negotiated with the intermediary. Before the merger, the negotiated lump-sum
fee paid by the intermediary to supplier $i$, denoted by $t_{i}^{\text{pre}}$,
solves the following Nash bargaining problem:

$t_{i}^{\text{pre}}=\operatorname*{arg\,max}_{t_{i}}\left[\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})-t_{i}\right]\times\left[\Phi_{i}(\mathcal{N})+t_{i}\right]$

where for simplicity, I have assumed that the intermediary and supplier $i$
both have bargaining weights equal to $1/2$ (as shown in Section 3, this
assumption does not affect the results). Hence, the combined premerger
payments that suppliers $1$ and $2$ can secure are equal to:

$\textstyle
T^{\text{pre}}=\sum_{i=1,2}t_{i}^{\text{pre}}=\sum_{i=1,2}\Big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})-\Phi_{i}(\mathcal{N})\Big{)}\Big{/}2$
(3)

Suppose now that suppliers $1$ and $2$ merge. To highlight my results most
transparently, I assume that there are no cost efficiencies or synergies
related to the merger. Consequently, the newly negotiated terms for each
supplier within the conglomerate $\mathcal{U}=\\{1,2\\}$ (where $\mathcal{U}$
stands for “union” or “merger”) must solve the following reformulated Nash
bargaining problem:121212It is easy to see that the negotiated fee secured by
the nonmerging supplier does not change due to the merger, i.e.,
$t_{0}^{\text{pre}}=t_{0}^{\text{post}}$.

$\textstyle\bm{t}^{\text{post}}=\operatorname*{arg\,max}_{(t_{1},t_{2})}\left[\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\mathcal{U})-\sum_{i=1,2}t_{i}\right]\times\left[\sum_{i=1,2}\big{(}\Phi_{i}(\mathcal{N})+t_{i}\big{)}\right]$

where I have assumed—as in Crawford and Yurukoglu (2012), Gowrisankaran, Nevo
and Town (2015), Ho and Lee (2017), and Dafny, Ho and Lee (2019)—that upon
disagreement with any given merging supplier, the intermediary loses the
products of the entire conglomerate. The first-order condition of this problem
yields:

$\textstyle
T^{\text{post}}=\sum_{i=1,2}t_{i}^{\text{post}}=\bigg{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\mathcal{U})-\sum_{i=1,2}\Phi_{i}(\mathcal{N})\bigg{)}\bigg{/}2$
(4)

Comparing (3) and (4), the total payment to the conglomerate will be greater
than the sum of premerger payments (i.e., $T^{\text{post}}>T^{\text{pre}}$) if
the merging products are substitutes for the intermediary in terms of overall
profits:

$\textstyle\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\mathcal{U})>\sum_{i=1,2}\Big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})\Big{)}$
(5)

That is, if the loss in profits for the intermediary when simultaneously
losing both products exceeds the sum of the loss from losing each product
separately.

As noted by Dafny, Ho and Lee (2019), this condition will typically be
satisfied if products are direct substitutes for consumers. In the current
setting, however, this is not the case. Nevertheless, the “holes in the
network” theory for harm argues that products can still be substitutes for the
intermediary, given that consumers are interested in multiple products (i.e.,
they are “common customers”). If true, a merger of suppliers $1$ and $2$
should lead to higher negotiated prices.

The problem with this theory—as pointed out by Brand and Rosenbaum (2019)—is
that products are, in fact, complements for consumers (see Remark 1). Because
these consumer-level complementarities must transfer upstream (Easterbrook et
al., 2018), the latter implies that products must be complements—not
substitutes—for the intermediary in terms of profits (i.e., the inequality (5)
is reversed). Thus, the merger of suppliers $1$ and $2$ should lead to
lower—not higher—negotiated prices.131313In this simple model—where the
intermediary and the suppliers negotiate lump-sum fees—suppliers never have
incentives to merge if their products are complements for the intermediary in
overall profits (absent exogenous technological efficiencies). Hence, we would
only expect “anticompetitive” mergers to emerge in equilibrium. As I show in
Section 3, this changes when suppliers negotiate over linear contracts because
a double-markup problem emerges. Hence, in such a setting, both pro- and
anticompetitive mergers arise on-path.

Which is it, then? Are products complements or substitutes for the
intermediary? And how can we square the existence of consumer-level
complementarities with the “holes in the network” theory of harm that argues
that two products can sometimes be substitutable for the intermediary in this
context?

To try settling this debate, evaluate (2) on (5). Rearranging terms, one
obtains that products $1$ and $2$ will be substitutes for the intermediary in
overall profits (i.e., (5) holds) if and only if:

$\pi_{0}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\mathcal{U})-\textstyle\sum_{i=1,2}\big{(}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{i\\})\big{)}\big{]}\\\
>\pi_{1}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{2\\})\big{]}+\pi_{2}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{1\\})\big{]}$
(6)

This is the key equation of this paper; it summarizes the heart of my
contribution.

The right-hand side of (6)—which is always positive—represents the direct
profit complementarities that arise between products $1$ and $2$ in the
intermediary’s profit function. This complementarity is explained because
carrying product $1$ increases the number of consumers visiting the
intermediary and, therefore, increases the intermediary’s direct profits
stemming from product $2$ (and vice versa). The stronger this complementarity
of direct profits is, the less likely it will be that (6) holds. This side of
equation (6) is the source of the widespread belief that consumer-level
complementarities must also drive complementarities upstream.141414As
discussed by Collard-Wexler, Gowrisankaran and Lee (2019), the Nash-in-Nash
solution concept might not be an appropriate surplus division rule when there
are strong complementarities among contracting parties. That said, my entire
point is to show that such complementarities may not be such—i.e., that
products might be substitutes for the intermediary despite being complements
for consumers. Thus, this issue is less of a problem here. Moreover, as shown
in the online Appendix, similar results are obtained using the Shapley value
as the surplus division rule. This protocol does not have the same problems as
the Nash-in-Nash solution when strong complementarities are present and can
also be microfounded in a noncooperative way, as shown by Stole and Zwiebel
(1996).

The left-hand side of (6), in turn, represents the complementarity or
substitutability between the merging products in generating traffic for
product $0$, the product that is not involved in the transaction. In
particular, products $1$ and $2$ will be substitutes in generating traffic for
product $0$ if the intermediary’s loss in terms of consumer traffic from
simultaneously losing both products exceeds the sum of the reductions from
losing each product individually:151515A sufficient condition for this
condition to hold is for the consumer traffic function to be submodular in the
products carried by the intermediary, i.e., each additional product increases
the bundle’s sales but at a decreasing rate. In contrast, if the consumer
traffic function is weakly supermodular, products will always be weak
complements for the intermediary in terms of consumer traffic.

$\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\mathcal{U})>\textstyle\sum_{i=1,2}\big{(}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{i\\})\big{)}$
(7)

When that is the case, the left-hand side of (6) will also be positive.
Intuitively, products $1$ and $2$ can be substitutes in generating traffic for
product $0$, because each product can individually be used to increase overall
consumer traffic. The stronger this substitutability in traffic is, or the
higher the direct profits stemming from product $0$ are, the more likely it
will be that (6) holds.

Thus, if the merging products are substitutes in traffic and this
substitutability is strong enough, the products can be complements for
consumers but substitutes for the intermediary in overall profits. When this
happens, the merger of suppliers $1$ and $2$ leads to higher negotiated
prices, just as the “holes in the network” theory of harm predicts.

######  Proposition 1.

The merger of suppliers $1$ and $2$ can lead to higher negotiated prices
despite complementarities at the consumer level. A necessary condition for
this to occur is for the merging products to be substitutes in generating
consumer traffic.

###### Proof.

Immediate from the discussion above. ∎

In fact, it is easy to see that the “holes in the network” theory is an
extreme version of my argument. Indeed, recall that according to this theory,
two products will be substitutes for the intermediary when all consumers are
willing to purchase the bundle when one of the products is missing but not
when both are. That is, if
$\tilde{G}(\mathcal{N})=\tilde{G}(\mathcal{N}\setminus\\{i\\})=1$ for $i=1,2$
and $\tilde{G}(\mathcal{N}\setminus\mathcal{U})=0$, in which case (6) is
always satisfied as it becomes:

$\overbrace{\pi_{0}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\mathcal{U})-\textstyle\sum_{i=1,2}\big{(}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{i\\})\big{)}\big{]}}^{=\pi_{0}}\\\
>\underbrace{\pi_{1}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{2\\})\big{]}+\pi_{2}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{1\\})\big{]}}_{=0}$

Intuitively, if none of the consumers stop purchasing the bundle when either C
or D is missing, but all stop purchasing it when both C and D are missing, the
two products are, in essence, “perfect substitutes” in creating traffic for
the remaining items in the bundle. Moreover, because adding either product to
the bundle does not affect the bundle’s sales when the other product is
present, C and D create no direct profit complementarities to each other.

Thus, the situation described by this theory minimizes direct profit
complementarities among merging products and maximizes their substitutability
in creating traffic for other items in the intermediary’s lineup. This
explains why the intermediary always sees the merging products as substitutes
in terms of profits in this case. Equation (6) shows, however, that such an
extreme situation is not necessary for the merger to increase negotiated
prices; it is only sufficient. Much milder forms of substitutability in
traffic can also lead to increases in negotiated prices.

###  2.3 Some Novel Antitrust Implications

I now turn my attention to the antitrust implications of these results. First,
equation (6) immediately provides two different sufficient conditions that
guarentee that a given merge will not increase negotiated prices.

######  Proposition 2.

If either:

  * (i)

The intermediary’s bundle solely comprises the merging suppliers’ products,
i.e., product $0$ does not exist.

  * (ii)

Or the merging products are weak complements in generating traffic, i.e.,

$\textstyle\sum_{i=1,2}\big{(}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{i\\})\big{)}\geq\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\mathcal{U})$
(8)

Then products $1$ and $2$ are necessarily weak complements for the
intermediary in overall profits, and the proposed merger does not increase
negotiated prices.

###### Proof.

For (i), note that if product $0$ does not exist, then (6) becomes:

$0>\pi_{1}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{2\\})\big{]}+\pi_{2}\big{[}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{1\\})\big{]}$

which is never satisfied as the right-hand side is always weakly positive
(note that the same occurs if $\pi_{0}=0$). For (ii), note that if (8) holds,
then (6) is necessarily violated, as its left-hand side is weakly negative and
the right-hand side is weakly positive. ∎

Intuitively, if the intermediary’s bundle solely comprises the merging
parties’ products, whether the products are substitutes or complements in
generating traffic for other products in the intermediary’s lineup is
irrelevant; only the direct profits complementarities between the two merging
products matter. On the other hand, if the merging products are weak
complements in generating traffic, they are complements in the two dimensions
that matter for the intermediary’s profit function: direct profits and
traffic. Hence, the products must necessarily be complements for the
intermediary regarding overall profits as well.161616In a recent paper, Baye,
von Schlippenbach and Wey (2018) consider a Hotelling-type model with two
suppliers, one monopoly intermediary, and final consumers that one-stop-shop.
They find that suppliers’ products are always complements to the intermediary
in terms of profits, which is why an upstream merger always leads to lower
negotiated prices. My results show, however, that their results are not true
in general. They arise in their model because (i) the intermediary does not
carry products other than those of the merging suppliers, and (ii) in their
Hotelling-type of setting, consumers are uniformly distributed (so the
consumer traffic function is weakly supermodular).

These two sufficient conditions can then be used as a first-pass test to
immediately assess whether two products will be complements for the
intermediary in terms of profits. The beauty of this test is that the first
condition can be observed directly from the intermediary’s portfolio, while
the second condition can be readily assessed by empirically estimating the
consumer loss ratios of the merging products.171717As noted in footnote 7, the
consumer loss ratio is defined as the fraction of all consumers lost by the
intermediary after the removal of an item or sets of items (see, e.g., U.S. v.
AT&T, et al., Civil Case No. 17-2511, RJL, D.D.C. Jun. 12, 2018).

However, suppose two products in the intermediary’s lineup fail the test;
thus, the products can potentially, but not necessarily, be substitutes for
the intermediary in terms of profits. Is there another way for authorities to
assess their complementarity or substitutability for the intermediary? The
answer is positive. The key is to look at the products’ role in the
intermediary’s lineup.

To do this, let me begin by providing two definitions: product $i$ is said to
be a “pure” profit contributor if $v_{i}=0$ and $\pi_{i}>0$, while it is said
to be a “pure” traffic generator if $\pi_{i}=0$ and $v_{i}>0$. Intuitively,
pure profit contributors only contribute to the intermediary’s overall profits
with direct profits, while pure traffic generators only contribute by creating
consumer traffic. With these labels in mind, I can then state the following
proposition:

######  Proposition 3.

Suppose that $\pi_{0}>0$ and that $\tilde{G}(\cdot)$ is submodular in the
products carried by the intermediary (so two products are always substitutes
in generating traffic for a third unrelated product).

  * •

If product $1$ is a pure profit contributor and product $2$ a pure traffic
generator (or vice versa), the products are complements for the intermediary
in terms of overall profits.

  * •

If products $1$ and $2$ are pure traffic generators, the products are
substitutes for the intermediary in terms of overall profits.

###### Proof.

Consider a pure profit contributor with a pure traffic generator, i.e.,
$(\pi_{1}>0,v_{1}=0)$ and $(\pi_{2}=0,v_{2}>0)$. Then,
$\tilde{G}(\mathcal{N})=\tilde{G}(\mathcal{N}\setminus\\{1\\})=G(v_{0}+v_{2})$
and
$\tilde{G}(\mathcal{N}\setminus\\{2\\})=\tilde{G}(\mathcal{N}\setminus\mathcal{U})=G(v_{0})$.
The left-hand side of (6) is then equal to zero, while its right-hand side is
equal to $\pi_{1}[G(v_{0}+v_{2})-G(v_{0})]>0$. Thus, condition (6) is
violated, so the products are complements for the intermediary in terms of
profits.

Consider, in turn, two pure traffic generators, i.e., $(\pi_{1}=0,v_{1}>0)$
and $(\pi_{2}=0,v_{2}>0)$. Then, the right-hand side of (6) is zero, while the
left-hand side of (6) is equal to:

$\pi_{0}\left[\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\mathcal{U})-\textstyle\sum_{i=1,2}\big{(}\tilde{G}(\mathcal{N})-\tilde{G}(\mathcal{N}\setminus\\{i\\})\big{)}\right]\\\
=\pi_{0}\left[G(v_{0}+v_{1})+G(v_{0}+v_{2})-G(v_{0}+v_{1}+v_{2})-G(v_{0})\right]$

This last term is positive if $\tilde{G}(\cdot)$ is submodular in the products
included in the bundle. Thus, condition (6) is satisfied, and the products are
substitutes for the intermediary in terms of profits.∎

Intuitively, two pure traffic generators do not generate direct profit
complementarities to each other, as their direct profit contributions are
nonexistent. Hence, only their substitutability in traffic matters when
determining whether they are complements or substitutes for the intermediary.
In contrast, a pure profit contributor does not “compete” in consumer traffic
with a traffic generator, as the profit contributor does not generate traffic.
Therefore, the two products are complements in overall profits to the
intermediary, as only direct profit complementarities are present.

In reality, no product will ever be a pure profit contributor or a pure
traffic generator; products typically contribute with direct profits and
consumer traffic. However, simple continuity arguments imply that the results
of Proposition 3 extend beyond the limit cases considered there.

For a concrete example, suppose that
$(\pi_{i},v_{i})\in[0,\bar{\pi}]\times[0,\bar{v}]$ for $i=0,1,2$. Rhodes,
Watanabe and Zhou (2021) show how a relatively general family of demand
functions can generate this space (and below, I discuss what determines a
product’s position in it). Building on the intuition of Proposition 3, Figure
2 depicts the mergers that would increase negotiated prices for a given set of
parameters.

Figure 1: Mergers that Increase Prices According to the $(\pi,v)$ Space

Notes. In the figure, $\bar{\pi}=\bar{v}=1$. The first panel has
$(\pi_{1},v_{1})=(\nicefrac{{1}}{{4}},\nicefrac{{3}}{{4}})$, while the second
panel has $(\pi_{1},v_{1})=(\nicefrac{{1}}{{3}},\nicefrac{{2}}{{3}})$.
Remaining Parameters:
$(\pi_{0},v_{0})=(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})$,
$G(\xi)=1-e^{-\lambda\xi}$ with $\lambda=\nicefrac{{3}}{{2}}$.

More precisely, keeping $(\pi_{0},v_{0})$ and $(\pi_{1},v_{1})$ fixed, and for
$G(\xi)=1-e^{-\lambda\xi}$ (so that $\tilde{G}(\cdot)$ is submodular in the
products carried by the intermediary), the colored regions in the panels
represent all the combinations $(\pi_{2},v_{2})$ so that
$T^{\text{post}}-T^{\text{pre}}>0$. Moreover, the darker the color, the more
pronounced the price increase (i.e., the higher
$T^{\text{post}}-T^{\text{pre}}$).181818The noncolored regions, in turn,
represent the locations for product $2$ such that a merger of suppliers $1$
and $2$ would lead to lower negotiated terms. As the figure shows, the more
problematic mergers arise when both $(\pi_{1},v_{1})$ and $(\pi_{2},v_{2})$
are located towards the northwest corner of the unit square. These are mergers
of products that contribute little to the intermediary’s overall profit via
direct profits and significantly via consumer traffic.

The only remaining question—the elephant in the room—is what determines the
role different products play in the intermediary’s lineup. As shown by Rhodes,
Watanabe and Zhou (2021), and Ide and Montero (2023), products’ locations in
the $(\pi,v)$ space are intimately connected to the shape of products’ demand.
In particular, products with “more concave” and “rectangular-shaped” demands
will have a higher $\pi_{i}/v_{i}$ ratio.191919The concavity of a product’s
demand is measured by the demand’s curvature, formally defined as the
elasticity of the slope of the inverse demand (see, e.g., Rhodes, Watanabe and
Zhou, 2021, p. 455).

Intuitively, with more concave and rectangular-shaped demands, the
intermediary can extract a larger share of consumer surplus without
introducing significant monopoly distortions. As a result, the intermediary
uses those products mainly as profit contributors, leading to a higher
$\pi_{i}/v_{i}$ ratio. Conversely, the more convex a product’s demand is, the
more expensive it is to extract consumer surplus in terms of monopoly
distortions. Thus, the intermediary tends to use those products primarily as
traffic generators. I will illustrate this intuition in more detail in Section
3, once I introduce the full-fledged model.

Based on this understanding, it can be conjectured that highly specialized
hospitals primarily serve as traffic generators in an insurer’s network. This
is due to the fact that, for such hospitals, the option value of having access
to their services holds significant importance for patients, resulting in
highly convex demands. In contrast, routine healthcare providers, which offer
a wide range of routine procedures, are likely to be predominantly utilized by
insurers to generate direct profits, as their demand curves are expected to be
more concave and rectangular-shaped.

Consequently, my findings suggest that authorities should pay particular
attention to transactions involving two or more specialized healthcare
providers or cases where a large hospital system acquires a highly specialized
provider.

###  2.4 A New Divestiture Framework

Proposition 3 and Figure 1 also have an additional implication. If we
interpret suppliers’ offerings as portfolios of products instead of single
products, they lead to a new divestiture framework for these transactions. In
particular, they imply that to offset the negative effects of a merger,
authorities should force the merging entities to divest those products located
in the northwest corner of the $(\pi,v)$ space.

To see this point more formally, suppose that $\tilde{G}(\cdot)$ is submodular
and that there is an additional product in the market, product $3$. The latter
product is also manufactured by a single supplier, supplier $3$. Furthermore,
assume that suppliers $1$ and $2$ have already merged to form conglomerate
$\mathcal{U}=\\{1,2\\}$ and are now considering acquiring supplier $3$.

As before, denote by $\mathcal{N}\equiv\\{0,1,2,3\\}$ the set of all products
in the market, and let $T^{\text{pre}}\equiv\sum_{i=1,2,3}t_{i}^{\text{pre}}$
and $T^{\text{post}}\equiv\sum_{i=1,2,3}t_{i}^{\text{post}}$ be the merging
parties’ combined pre- and postmerger profits.202020As in the three-product
case, the negotiated fee secured by the nonmerging supplier does not change
due to the merger (irrespective of the divestiture measures implemented). See
Appendix B for details. Define also $\Delta T(\text{no divestiture})$, $\Delta
T(\text{divest}\ 1)$, and $\Delta T(\text{divest}\ 2)$ as the change in the
merging parties’ profits due to the merger, $\Delta T\equiv
T^{\text{post}}-T^{\text{pre}}$, if:

  * (i)

No divestiture remedy is imposed.

  * (ii)

$\mathcal{U}$ must divest product $1$ as a condition to acquire $3$.

  * (iii)

$\mathcal{U}$ must divest product $2$ as a condition to acquire $3$.

As I show in Appendix B, the relationship between $\Delta T(\text{no
divestiture})$, $\Delta T(\text{divest}\ 1)$, and $\Delta T(\text{divest}\ 2)$
is:

$\displaystyle\textstyle\Delta T(\text{divest}\ 1)=\Delta T(\text{no
divestiture})$ $\displaystyle\qquad\ \ \
-\frac{1}{2}\underbrace{\left[\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2,3\\})-\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1\\})\big{)}-\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{2,3\\})\big{)}\right]}_{\text{degree
of substitutability in profits between product $1$ and portfolio $\\{2,3\\}$
}}$ $\displaystyle\textstyle\Delta T(\text{divest}\ 2)=\Delta T(\text{no
divestiture})$ $\displaystyle\qquad\ \ \
-\frac{1}{2}\underbrace{\left[\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2,3\\})-\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{2\\})\big{)}-\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,3\\})\big{)}\right]}_{\text{degree
of substitutability in profits between product $2$ and portfolio $\\{1,3\\}$
}}$

These expressions have two important implications. First, forcing
$\mathcal{U}$ to divest product $1$ is more effective than forcing it to
divest product $2$—in the sense that it leads to a lower increase in
negotiated payments—when the degree of substitutability in profits between
product $1$ and portfolio $\\{2,3\\}$ is higher than the substitutability in
profits of product $2$ and portfolio $\\{1,3\\}$. This will typically be the
case when product $1$ is located closer to the northwest corner of the
$(\pi,v)$ space than product $2$, as the following example illustrates:

> Example 1. Suppose that products $1$ and $3$ are pure traffic generators,
> $(\pi_{1}=0,v_{1}>0)$ and $(\pi_{3}=0,v_{3}>0)$, while product $2$ is a pure
> profit contributor, $(\pi_{2}>0,v_{2}=0)$. The degree of substitutability in
> profits between product $1$ and portfolio $\\{2,3\\}$ is:
>
>
> $\pi_{0}\big{[}G(v_{0}+v_{1})+G(v_{0}+v_{3})-G(v_{0}+v_{1}+v_{3})-G(v_{0})\big{]}-\pi_{2}\big{[}G(v_{0}+v_{1}+v_{3})-G(v_{0}+v_{3})\big{]}$
>
> while the degree of substitutability in profits between product $2$ and
> portfolio $\\{1,3\\}$ is:
>
> $\pi_{2}[G(v_{0})-G(v_{0}+v_{1}+v_{3})]$
>
> Thus, product $1$ and portfolio $\\{2,3\\}$ have a higher degree of
> substitutability than product $2$ and portfolio $\\{1,3\\}$ given that:
>
>
> $G(v_{0}+v_{1})+G(v_{0}+v_{3})-G(v_{0}+v_{1}+v_{3})-G(v_{0})>0\quad\text{and}\quad
> G(v_{0}+v_{3})>G(v_{0})$
>
> Consequently, to offset the increases in negotiated payments arising due to
> $\mathcal{U}$’s acquisition of product $3$, an authority should aim its
> divestiture efforts toward product $1$—the traffic generator—rather than
> toward product $2$—the profit contributor.

The second important implication is that if authorities force the merging
entities to divest the wrong products (i.e., those originally located at the
southeast corner of the $(\pi,v)$ space), the policy can backfire. This is
because the divestiture negatively impacts the direct profit complementarities
between portfolios, making it more likely that the postdivestiture portfolios
will be substitutable in traffic. The previous point can be seen in the above
example, where products $1$ and $3$ are pure traffic generators and product
$2$ is a pure profit contributor:

> Example 1 (Continued). The degree of substitutability in profits between
> product $2$ and portfolio $\\{1,3\\}$ in this case is equal to
> $\pi_{2}[G(v_{0})-G(v_{0}+v_{1}+v_{3})]$. The latter is strictly
> negative—that is, portfolios $2$ and $\\{1,3\\}$ are complements in profits
> for the intermediary—implying that $\textstyle\Delta T(\text{divest}\
> 2)>\Delta T(\text{no divestiture})$.

The divestiture approach outlined above is novel and differs significantly
from the ones used in practice. For instance, in the 2005 cross-
market/conglomerate merger involving Procter & Gamble and Gillette, the
European Commission forced P&G to divest its battery toothbrush business
“SpinBrush,” since there were significant horizontal overlaps at the consumer
level between this brand and Gillette’s “OralB” brand.

In other words, the remedy focused on eliminating any horizontal overlaps as
viewed from the perspective of final consumers: SpinBrush and OralB were
direct “head-to-head” rivals at the point of sale. In contrast, the
divestiture measures presented here aim to tackle the more subtle “horizontal
overlap” arising at the intermediary’s level when two portfolios of products
are “substitutes” in generating traffic for other items in the intermediary’s
lineup, even though the products belonging to these portfolios are not direct
rivals at the point of sale.

This finishes the presentation of the article’s main results. In the next
section, I microfound the model presented above and show that the results
remain unchanged.

##  3 Microfounding the Simple Model

I now microfound the result of the previous section by employing a model
inspired by Ho and Lee (2017) and Ide and Montero (2023).

###  3.1 Setup

Products, Suppliers, and Technology.— There is a set
$\mathcal{N}=\\{1,...,n\\}$ of products (where $n\geq 2$), each produced by a
single supplier (thus I can continue referring to suppliers or products
indistinguishably).

To manufacture product $i\in\mathcal{N}$, the corresponding supplier uses a
constant-returns-to-scale technology. To avoid introducing merger synergies on
the technological side, I assume that production technologies are separable
across the different products. Thus, the marginal cost of producing
$i\in\mathcal{N}$, denoted by $c_{i}\geq 0$, is independent of the ownership
structure of the upstream market.

The Intermediary.— There is a single monopoly intermediary that buys products
from suppliers and resells them to consumers. This intermediary sets retail
prices for the products it carries. Moreover, it has no operating costs beyond
purchasing the products from suppliers.

Conglomerates.— Each supplier originally belongs to one of $m$ conglomerates,
where $2\leq m\leq n$. The set of all conglomerates is denoted by
$\bm{\mathcal{C}}=\\{\mathcal{C}_{1},...,\mathcal{C}_{m}\\}$. Note that
conglomerate $j$ can also represent a single supplier if
$|\mathcal{C}_{j}|=1$.

The intermediary must carry all or none of the products of a given
conglomerate, although it can negotiate different terms with each supplier
within the conglomerate. I also assume that all the suppliers belonging to the
same conglomerate perfectly internalize each other’s profits.

Final Consumers.— There is a unit mass of final consumers. They wish to buy
$Q_{i}(p_{i})$ units of product $i\in\mathcal{N}$ when its retail price is
$p_{i}$. Hence, the surplus a consumer obtains from purchasing $i$ is given by
$v_{i}(p_{i})=\int_{p_{i}}^{\infty}Q_{i}(x)dx$. I assume that products are
neither complements nor substitutes in consumption; thus, when a consumer buys
multiple products, her surplus is additive over these products.

Consumers also incur a shopping cost $\xi\geq 0$ when visiting the
intermediary. This shopping cost is characterized by the cumulative
distribution function $G(\xi)$. Below, I impose some regularity conditions
over $Q_{i}(p_{i})$ and $G(\xi)$ to ensure that the intermediary’s pricing
problem is well-behaved.

Supplier-Intermediary Bargaining.— I assume that the intermediary and the
suppliers bargain over linear contracts.212121It is straightforward to prove
that the upcoming results continue to hold under two-part tariffs. The only
difference is that, in this last case, upstream mergers only affect the
negotiated lump-sum fees, not wholesale prices. As a result—and similar to the
simple model of Section 2—when lump-sum transfers are feasible, mergers only
impact firms’ profits, not consumer surplus. For this reason, I focus on
linear contracts in what follows. I denote by $w_{i}$ the per-unit wholesale
price for product $i$.

As in the simple model of Section 2, I use the Nash-in-Nash protocol as my
bargaining solution concept. In contrast to that model, however, I now allow
the possibility of different bargaining weights within each bilateral pair
(though bargaining weights are the same across all negotiations). I denote by
$\beta\in(0,1)$ and $1-\beta$ the bargaining weights for the intermediary and
any given supplier, respectively, and assume that these weights do not depend
on the ownership structure of the upstream market.

Timing.— The timing of the game is as follows:

  * $t=1(a)$:

Negotiation over linear contracts: the intermediary and the suppliers bargain
over wholesale prices.

  * $t=1(b)$:

Price Setting: simultaneously with the determination of negotiated wholesale
prices, the intermediary sets retail prices.

  * $t=2$:

Consumers observe the intermediary’s product portfolio and its retail prices
and decide whether to visit the intermediary. Payoffs and profits accrue.

Note that I am assuming that wholesale and retail prices are simultaneously
determined. That is, retail prices remain fixed when maximizing the Nash
product of each intermediary-supplier pair, and wholesale prices remain fixed
when the intermediary sets the retail prices for its products. I follow this
timing for three reasons.

First, this is the timing that maps one-to-one to the simple model presented
in Section 2.222222If wholesale prices are negotiated before retail prices are
set, the outcome of the upstream negotiations depends not only in the
complementary or substitutability of products in overall profits for the
intermediary but also on the (local) sensitivity of retail prices to wholesale
costs. Hence, under such an alternative timing, there is no longer a one-to-
one mapping to the simple model of Section 2. Second, as argued by Ho and Lee
(2017), in the context of the healthcare industry (my motivating example),
this timing might be more realistic than one in which wholesale prices are set
before retail prices. Third and finally, this is the timing adopted by the
most recent empirical studies of the effect of upstream negotiations on market
outcomes (Ho and Lee, 2017; Crawford et al., 2018; Ho and Lee, 2019), and is
the timing implicitly assumed by the FTC when simulating the effects of
horizontal mergers of healthcare providers.232323Indeed, as can be seen from
equation (1) of Farrell et al. (2011, p. 277), the Nash Bargaining condition
estimated by the FTC disregards the local sensitivity of retail prices to
wholesale costs.

That said, in the online Appendix I provide simulations showing that my main
insights do not change when wholesale prices are negotiated before the
intermediary sets retail prices.

Payoffs.— Suppose the intermediary reaches an agreement with a subset
$\bm{\mathcal{G}}\subseteq 2^{\bm{\mathcal{C}}}$ of conglomerates. Let
$\bm{w}$ be the vector of wholesale prices resulting from the negotiations,
and $\bm{p}$ be the vector of retail prices set by the intermediary.242424I
use the convention that $p_{i}=w_{i}=+\infty$ if the conglomerate owning
product $i$ does not belong to $\bm{\mathcal{G}}$. The indirect utility
function of a consumer with a shopping cost of $\xi$ is then
$U(\bm{p},\bm{\mathcal{G}};\xi)=\max\\{\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})-\xi,0\\}$.
Hence, the fraction of consumers visiting the intermediary is
$\mathbb{P}\big{(}\textstyle\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})\geq\xi\big{)}=G\big{(}\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})\big{)}$,
and the overall demand for product/supplier $i\in\mathcal{C}_{j}$ is:

$\textstyle
D_{i}(\bm{p},\bm{\mathcal{G}})=\mathbbm{1}_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}Q_{i}(p_{i})G\big{(}\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})\big{)}$

Denoting by $\bm{w}_{\mathcal{C}_{j}}$ the vector of wholesale prices for the
products of conglomerate $\mathcal{C}_{j}$, i.e.,
$\bm{w}_{\mathcal{C}_{j}}\equiv(w_{i})_{i\in\mathcal{C}_{j}}$, the latter
implies that the profits made by conglomerate $\mathcal{C}_{j}$ are:

$\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{G}})=\textstyle\mathbbm{1}_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}(w_{i}-c_{i})D_{i}(\bm{p},\bm{\mathcal{G}})$

Moreover, since the intermediary makes a per-consumer profit of
$\mathbbm{1}_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\pi_{i}(p_{i};w_{i})$ in
product $i\in\mathcal{C}_{j}$ (recall that
$\pi_{i}(p_{i};w_{i})\equiv(p_{i}-w_{i})Q_{i}(p_{i})$), its overall profits
are equal to:

$\Psi(\bm{p},\bm{w},\bm{\mathcal{G}})=\big{(}\textstyle\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}\pi_{i}(p_{i};w_{i})\big{)}G\big{(}\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})\big{)}$
(9)

Regularity Conditions on $Q_{i}(p_{i})$ and $G(\xi)$.— Regarding products’
demands and the distribution of shopping costs, I assume that $Q_{i}(p_{i})$
is downward sloping with either a choke price of $\bar{p}_{i}$ or with the
property that $Q_{i}(p_{i})\to 0$ as $p_{i}\to\infty$. I also assume that
$Q_{i}(p_{i})$ and $G(\xi)$ are such that
$\max_{\bm{p}}\Psi(\bm{p},\bm{w},\bm{\mathcal{G}})$ has a solution and that
the latter is unique for any given $(\bm{w},\bm{\mathcal{G}})$ such that the
intermediary has a strictly positive demand if it sets its retail prices equal
to its wholesale marginal costs.252525If that is not the case, then the retail
pricing problem is trivial—the intermediary always obtain zero profits—but the
problem has many solutions. In the Online Appendix, I provide an example of a
relatively weak set of conditions that yield the sought properties.

Remark 3: Consumer Level Complementarities. As in the simple model, this
setting also exhibits complementarities at the consumer level. Indeed, it is
not difficult to prove that products are gross complements (i.e., overall
demand for product $i$ decreases in the price of any other product carried by
the intermediary) and that the indirect utility function of a consumer with
shopping cost $\xi$ is supermodular in the inclusion of products to the
intermediary’s lineup for any given vector of retail prices $\bm{p}$.262626The
proof of this last result is almost identical to that for the simple model.
For this reason, it is omitted.

###  3.2 Substitutability of Portfolios and The Effects of a Merger

I begin by showing that—just as in the simple model in Section 2—the
competitive implications of a merger between $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ depend on whether the merging portfolios are substitutes or
complements for the intermediary in terms of profits. I take a somewhat
informal approach in this part, as these results are well-known in the
literature (Farrell et al., 2011; Nevo, 2014; Easterbrook et al., 2018; Dafny,
Ho and Lee, 2019). In the next subsection, I generalize the new set of results
that constitute the main contribution of this paper.

Consider first the premerger scenario. In this case, the wholesale price
$w_{i}\in\bm{w}$ negotiated between the intermediary and supplier
$i\in\mathcal{C}_{j}$ maximizes the following Nash program:

$\max_{w_{i}}\Big{[}\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\Big{]}^{\beta}\times\Big{[}\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})\Big{]}^{1-\beta}$
(10)

Because for a fixed $\bm{p}$,
$\textstyle-\partial\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})/\partial
w_{i}=\partial\Phi_{i}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})/\partial
w_{i}=G\big{(}\sum_{\mathcal{C}_{j}\in\bm{\mathcal{C}}}\sum_{k\in\mathcal{C}_{j}}v_{k}(p_{k})\big{)}$
for $i\in\mathcal{C}_{j}$, then every supplier in conglomerate
$\mathcal{C}_{j}$ ends up having the same first-order condition:

$\beta\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})]\
\ \ \forall i\in\mathcal{C}_{j}$

Thus, the premerger equilibrium—defined as pair
$(\bm{w}^{\text{pre}},\bm{p}^{\text{pre}})$—is given by the solution to the
following system of equations:

$\begin{split}&\textstyle\beta\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})],\
\text{for $j=1,...,m$}\\\
&\bm{p}=\operatorname*{arg\,max}_{\bm{\tilde{p}}}\Psi(\bm{\tilde{p}},\bm{w},\bm{\mathcal{C}})\end{split}$
(11)

Unfortunately, as noted by Ho and Lee (2017, p.7 of the Online Appendix), this
system does not fully determine equilibrium wholesale prices, only
conglomerates’ total profits. This is because there is only an equation per
conglomerate-intermediary pair, not per supplier-intermediary pair. This is a
problem because negotiated wholesale prices impact the retail prices set by
the intermediary.

To proceed, I assume—similar in spirit to Ho and Lee (2017)—that the ratio of
suppliers’ profits belonging to the same conglomerate is equal to the ratio of
their marginal contributions to the intermediary’s profits at the prevailing
prices. That is, I assume that:272727To understand this refinement, consider
“the beginning of times” in which every conglomerate is composed of a single
supplier. In that case, the equilibrium is fully determined, and suppliers’
profits satisfy (12). Suppose that two suppliers bargain to create the first
conglomerate with more than one supplier. The refinement is imposing—in a
reduced-form fashion—that the outcome of this negotiation between suppliers is
such that their profits will continue to be proportional to the ones they
would obtain if the union is not formed. Repeating this procedure inductively
until reaching $\bm{\mathcal{C}}$—always matching one conglomerate with a
single supplier at a time—yields (12).,282828Let me explain why this
refinement is close in spirit to Ho and Lee (2017). In their empirical
estimation of the effect of insurance competition on healthcare markets, they
solve this underdetermination problem by matching the ratio of negotiated
wholesale prices $w_{i}/w_{i^{\prime}}$ to what they observe in the data and
assuming that this ratio is invariant to changes in the environment when they
compute counterfactual simulations. Since I do not have data to match, my
approach is to regress to the “origin” of times, “match my variables there”
(specifically match the ratio of suppliers’ profits), and then inductively
“move forward in time” letting conglomerates form keeping the matched ratio
fixed.

$\frac{(w_{i}-c_{i})D_{i}(\bm{p},\bm{\mathcal{C}})}{(w_{i^{\prime}}-c_{i^{\prime}})D_{i^{\prime}}(\bm{p},\bm{\mathcal{C}})}=\frac{\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\\{i\\})}{\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\\{i^{\prime}\\})}\
\ \ \forall\,i,i^{\prime}\in\mathcal{C}_{j}$ (12)

Putting everything together, $(\bm{w}^{\text{pre}},\bm{p}^{\text{pre}})$ then
comes from the system comprising equations in (11) and (12).

Consider now the negotiated terms after $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ merge and denote by
$\mathcal{U}\equiv\mathcal{C}_{1}\cup\mathcal{C}_{2}$ the entity created due
to the transaction. The wholesale price negotiated between the intermediary
and supplier $i\notin\mathcal{U}$ continues to be given by the solution to the
Nash problem (10). In contrast, the negotiated terms between the intermediary
and a supplier $i\in\mathcal{C}_{j}$, where $\mathcal{C}_{j}\in\mathcal{U}$,
must now solve the following reformulated Nash bargaining problem:

$\max_{w_{i}}\textstyle\Big{[}\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{U})\Big{]}^{\beta}\times\Big{[}\sum_{j=1,2}\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})\Big{]}^{1-\beta}$

Following a similar logic as in the premerger case, the postmerger equilibrium
$(\bm{w}^{\text{post}},\bm{p}^{\text{post}})$ is then given by the solution to
this modified system of equations:

$\begin{split}&\textstyle\beta\sum_{j=1,2}\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{U})]\\\
&\textstyle\beta\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})],\
\text{for $j=3,...,m$}\\\
&\frac{(w_{i}-c_{i})D_{i}(\bm{p},\bm{\mathcal{C}})}{(w_{i^{\prime}}-c_{i^{\prime}})D_{i^{\prime}}(\bm{p},\bm{\mathcal{C}})}=\frac{\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\\{i\\})}{\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\\{i^{\prime}\\})}\
\text{for all $i,i^{\prime}\in\mathcal{C}_{j}$ or
$i,i^{\prime}\in\mathcal{U}$}\\\
&\bm{p}=\operatorname*{arg\,max}_{\bm{\tilde{p}}}\Psi(\bm{\tilde{p}},\bm{w},\bm{\mathcal{C}})\end{split}$
(13)

To understand the role of the complementarity/substitutability of the merging
portfolios on the effects of the merger, let me compare the system given by
(11) and (12) with the system given by (LABEL:eq:system_post). The premerger
equations in (11) and (12) imply that the wholesale prices negotiated between
the intermediary and the suppliers in
$\mathcal{C}_{1}\cup\mathcal{C}_{2}\equiv\mathcal{U}$ must satisfy the
following condition:

$\textstyle\beta\sum_{j=1,2}\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)\sum_{j=1,2}[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})]$

That is, negotiated wholesale prices must equalize $\mathcal{U}$’s gains from
trade with the sum of the intermediary’s gains from dealing with each
conglomerate in $\mathcal{U}$ individually (weighted by the corresponding
bargaining weights).

On the other hand, by formally creating $\mathcal{U}$ and bargaining jointly,
suppliers in $\mathcal{U}$ can now secure wholesale prices that satisfy:

$\textstyle\beta\sum_{j=1,2}\Phi_{j}(\bm{p},\bm{w}_{\mathcal{C}_{j}},\bm{\mathcal{C}})=(1-\beta)[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{U})]$

which states that the postmerger equilibrium wholesale prices equalize
$\mathcal{U}$’s gains from trade with the intermediary’s gains from dealing
with $\mathcal{U}$ as a whole (again, weighted by the corresponding bargaining
weights).

A comparison of these two conditions indicates that the formal creation of
$\mathcal{U}$ allows $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ to appropriate a
higher share of the intermediary’s gains from trade if their portfolios are
substitutes for the intermediary in terms of overall profits, i.e., if:

$\textstyle\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{U})>\sum_{j=1,2}[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})]$
(14)

Conversely, the merging conglomerates appropriate a lower share of the
intermediary’s gains from trade after the merger if the inequality in (14) is
reversed. That is, when the intermediary views the merging portfolios as
complements in overall profits.

With this argument in mind, define:

$\textstyle\Delta\Psi\equiv\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{U})-\sum_{j=1,2}[\Psi(\bm{p},\bm{w},\bm{\mathcal{C}})-\Psi(\bm{p},\bm{w},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})]$
(15)

as the degree of substitutability in profits of $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ for the intermediary at prices $(\bm{p},\bm{w})$. If
$\Delta\Psi^{\text{pre}}\equiv\Delta\Psi\rvert_{(\bm{w}^{\text{pre}},\bm{p}^{\text{pre}})}>0$,
then, starting from the premerger equilibrium
$(\bm{w}^{\text{pre}},\bm{p}^{\text{pre}})$, and keeping the wholesale prices
of other conglomerates and retail prices fixed, a merger between
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ allows suppliers in $\mathcal{U}$ to
demand higher wholesale prices for their products.

As the remaining prices are then allowed to adjust, this demand for higher
wholesale prices puts upward pressure on retail prices and downward pressure
on the wholesale prices negotiated by the nonmerging conglomerates (indirectly
putting downward pressure on retail prices). If the first (direct) effect is
stronger than the second (indirect) effect, the merger will decrease the
surplus that consumers obtain—before shopping costs—from visiting the
intermediary
$u(\bm{p})=\sum_{\mathcal{C}_{j}\in\bm{\mathcal{C}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})$;
thus, overall consumer surplus is negatively impacted.

Conversely, if $\Delta\Psi^{\text{pre}}<0$, then starting from the premerger
equilibrium, the merger forces suppliers in $\mathcal{U}$ to accept lower
wholesale prices for their products. As other prices are allowed to change,
the latter puts downward pressure on retail prices and upward pressure on the
wholesale prices charged by other conglomerates. After all the adjustments,
the postmerger equilibrium usually exhibits a higher consumer surplus.

Figure 2: Substitutability/Complementarity between Portfolios and The Effects
of a Merger

(a) Consumer Surplus of Visiting the Intermediary

(b) Combined Profits of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$

(c) Wholesale Price of Product $1$

(d) Wholesale Price of Product $2$

(e) Wholesale Price of Product $3$

(f) Wholesale Price of Product $4$

Notes. Parameters: $n=4$, $m=3$,
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$ with
$\mathcal{C}_{1}=\\{1,2\\}$, $\mathcal{C}_{2}=\\{3\\}$, and
$\mathcal{C}_{3}=\\{4\\}$. $G(\xi)=1-e^{-\lambda\xi}$ with $\lambda=5$.
Demands:
$Q_{i}(p_{i})=\alpha_{i}(1-p_{i}(1-\sigma_{i})/(2-\sigma_{i}))^{1/(1-\sigma_{i})}$,
with $\bm{\alpha}=\\{1,1,1,1\\}$, $\bm{\sigma}=\\{0,1,1,0\\}$. Marginal costs:
$\bm{c}=\\{\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}}\\}$.

Figure 2 illustrates these results. The figure considers a setting with four
products and three initial conglomerates
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$, where
$\mathcal{C}_{1}=\\{1,2\\}$, $\mathcal{C}_{2}=\\{3\\}$, and
$\mathcal{C}_{3}=\\{4\\}$. It then plots the value of
$\Delta\Psi^{\text{pre}}$ and the effects of a merger between
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ over the surplus consumers obtain from
visiting the intermediary (before shopping costs), $\mathcal{C}_{1}$’s and
$\mathcal{C}_{2}$’s combined profits, and the negotiated wholesale
prices.292929The MATLAB code used to create this figure can be found on my
website (www.enriqueide.com). The code is flexible enough for readers to play
with different parameter values.

As the figure shows, the merger leads to a higher consumer surplus, lower
negotiated prices for $\mathcal{C}_{1}$’s and $\mathcal{C}_{2}$’s products,
and a higher wholesale price for $\mathcal{C}_{3}$’s product when
$\mathcal{C}_{1}$’s and $\mathcal{C}_{2}$’s portfolios are complements to the
intermediary at the premerger prices (i.e., when $\Delta\Psi^{\text{pre}}<0$).
However, these inequalities are reversed when the portfolios are seen as
substitutes by the intermediary at the premerger prices (i.e., when
$\Delta\Psi^{\text{pre}}>0$).

Before moving on, it is important to highlight that a merger between
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ can be profitable in this setting
irrespective of whether $\Delta\Psi^{\text{pre}}\gtrless 0$. The latter is
shown in panel (b) of Figure 2. However, the underlying reason the merger is
profitable when $\Delta\Psi^{\text{pre}}<0$ is different than that when
$\Delta\Psi^{\text{pre}}>0$.

Indeed, when $\beta$ is low enough—so suppliers’ bargaining weight is high
relative to the intermediary—the double-markup problem due to linear contracts
is severe. As a result, the merger can be profitable even if it weakens
suppliers’ bargaining leverage (i.e., if $\Delta\Psi^{\text{pre}}<0$), as it
helps suppliers commit to decreasing wholesale prices, mitigating the double-
markup problem. Thus, even though $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ end
up receiving a smaller fraction of the intermediary’s trade gains postmerger,
the transaction increases the size of these gains.

In contrast, when $\beta$ is high enough, the double-markup problem is less
severe. As a result, the merger will be profitable for $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ only if $\Delta\Psi^{\text{pre}}>0$. This is because
division-of-surplus considerations now trump the size-of-trade-gains
considerations. The transaction thus worsens double-marginalization but allows
the merging conglomerates to appropriate a higher fraction of the
intermediary’s surplus.

###  3.3 Portfolios can be Substitutes or Complements for Intermediaries

Let $\tilde{G}(\bm{p},\bm{\mathcal{G}})\equiv
G\big{(}\sum_{\mathcal{C}_{j}\in\bm{\mathcal{G}}}\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})\big{)}$
be the consumer traffic function given the set of agreements
$\bm{\mathcal{G}}\in 2^{\bm{\mathcal{C}}}$ and the vector of retail prices
$\bm{p}$. The next proposition shows that $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ can be substitutes or complements for the intermediary in
profits, despite the presence of consumer-level complementarities. The
proposition also shows that the first preliminary test to assess the
complementarity of two portfolios for the intermediary—based on the
composition of the intermediary’s lineup or the weak complementarity of the
portfolios in generating traffic—remains valid here.

######  Proposition 4.

Premerger, the merging portfolios can be complements or substitutes for the
intermediary despite the existence of complementarities at the consumer level,
i.e., $\Delta\Psi^{\text{pre}}\gtrless 0$. However, if either:

  * (i)

The intermediary’s lineup is comprised solely of the merging portfolios, i.e.,
$\bm{\mathcal{C}}\setminus\mathcal{U}=\emptyset$.

  * (ii)

Or if premerger the merging portfolios are complements in generating consumer
traffic, i.e.,

$\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}\geq\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{U})$

Then $\Delta\Psi^{\text{pre}}\leq 0$.

###### Proof.

First, that $\Delta\Psi^{\text{pre}}\gtrless 0$ can be immediately seen from
Figure 2. The proof of points (i) and (ii), in turn, closely follows the
analysis of Section 2. Fix an arbitrary $(\bm{w},\bm{p})$ and for notational
simplicity, let
$\Pi_{j}(\bm{p},\bm{w})\equiv\textstyle\sum_{i\in\mathcal{C}_{j}}\pi_{i}(p_{i};w_{i})$.
Evaluating (9) in the definition of $\Delta\Psi$ given by (15) and rearranging
terms leads to:

$\textstyle\Delta\Psi=\left(\sum_{k\in(\bm{\mathcal{C}}\setminus\mathcal{U})}\Pi_{k}(\bm{p},\bm{w})\right)\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{U})-\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}\big{]}\\\
-\Pi_{1}(\bm{p},\bm{w})\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{2})\big{]}-\Pi_{2}(\bm{p},\bm{w})\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{1})\big{]}$
(16)

Since the second and the third terms on the right-hand side of (16) are weakly
negative, it is clear that $\Delta\Psi\leq 0$ if either
$\bm{\mathcal{C}}\setminus\mathcal{U}=\emptyset$ (since then
$\sum_{k\in(\bm{\mathcal{C}}\setminus\mathcal{U})}\Pi_{k}(\bm{p},\bm{w})=0$)
or if:

$\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}\geq\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{U})$

Since the above results hold for an arbitrary $(\bm{w},\bm{p})$, they hold
when $(\bm{w},\bm{p})=(\bm{w}^{\text{pre}},\bm{p}^{\text{pre}})$.∎

I now show that the additional screening test—based on the role played by the
merging portfolios in the intermediary’s lineup—also holds in this setting. I
also elaborate on the divestiture measures proposed in Section 2.4. As a first
step, let me begin by defining the concept of a pure profit contributor and a
pure traffic generator in the current setting:

######  Definition.

Product $i$ is said to be a pure (premerger) profit contributor if
$v_{i}(p_{i}^{\text{pre}})=0$ and
$\pi_{i}(p_{i}^{\text{pre}};w_{i}^{\text{pre}})>0$, while it is said to be a
pure (premerger) traffic generator if $v_{i}(p_{i}^{\text{pre}})>0$ and
$\pi_{i}(p_{i}^{\text{pre}};w_{i}^{\text{pre}})=0$.

With these definitions at hand, I can then extend Proposition 3 of Section 2:

######  Proposition 5.

Suppose that $\bm{\mathcal{C}}\setminus\mathcal{U}\neq\emptyset$, and that for
any given $\bm{p}$, $\tilde{G}(\bm{p},\bm{\mathcal{G}})$ is submodular in
$\bm{\mathcal{G}}$. Then:

  * •

If $\mathcal{C}_{1}$’s portfolio solely comprises pure (premerger) profit
contributors and $\mathcal{C}_{2}$’s portfolio solely comprises pure
(premerger) traffic generators (or vice versa), then
$\Delta\Psi^{\text{pre}}<0$.

  * •

If both $\mathcal{C}_{1}$’s and $\mathcal{C}_{2}$’s portfolios are comprised
solely of pure (premerger) traffic generators, then
$\Delta\Psi^{\text{pre}}>0$.

###### Proof.

This proof is almost identical to that of Proposition 3. For an arbitrary
$(\bm{w},\bm{p})$, let
$\Pi_{j}(\bm{p},\bm{w})\equiv\textstyle\sum_{i\in\mathcal{C}_{j}}\pi_{i}(p_{i};w_{i})$
and
$V_{j}(\bm{p},\bm{w})\equiv\textstyle\sum_{i\in\mathcal{C}_{j}}v_{i}(p_{i})$.
Note then that $\Delta\Psi$ can be written as:

$\Delta\Psi\equiv\textstyle\left(\sum_{k\in(\bm{\mathcal{C}}\setminus\mathcal{U})}\Pi_{k}(\bm{p},\bm{w})\right)\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{U})-\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}\big{]}\\\
-\Pi_{1}(\bm{p},\bm{w})\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{2})\big{]}-\Pi_{2}(\bm{p},\bm{w})\big{[}\tilde{G}(\bm{p},\bm{\mathcal{C}})-\tilde{G}(\bm{p},\bm{\mathcal{C}}\setminus\mathcal{C}_{1})\big{]}$

If $\mathcal{C}_{1}$’s portfolio solely comprises pure (premerger) profit
contributors and $\mathcal{C}_{2}$’s portfolio comprises solely pure
(premerger) traffic generators, then
$\Pi_{1}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})>0$,
$V_{1}(\bm{p}^{\text{pre}})=0$,
$\Pi_{2}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})=0$, and
$V_{2}(\bm{p}^{\text{pre}})>0$. Thus,
$\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})>\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{2})$,
$\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})=\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{1})$,
and:

$\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{U})-\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}=0$

Therefore,
$\Delta\Psi^{\text{pre}}=-\Pi_{1}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})\big{[}\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{2})\big{]}<0$.

On the other hand, if both $\mathcal{C}_{1}$’s and $\mathcal{C}_{2}$’s
portfolios solely comprise pure (premerger) traffic generators, then
$\Pi_{1}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})=\Pi_{2}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})=0$,
$V_{1}(\bm{p}^{\text{pre}})>0$, and $V_{2}(\bm{p}^{\text{pre}})>0$. Thus,

$\Delta\Psi^{\text{pre}}=\textstyle\left(\sum_{k\in(\bm{\mathcal{C}}\setminus\mathcal{U})}\Pi_{k}(\bm{p}^{\text{pre}},\bm{w}^{\text{pre}})\right)\\\
\times\big{[}\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{U})-\textstyle\sum_{j=1,2}\big{(}\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}})-\tilde{G}(\bm{p}^{\text{pre}},\bm{\mathcal{C}}\setminus\mathcal{C}_{j})\big{)}\big{]}$

Hence, $\Delta\Psi^{\text{pre}}\geq 0$ as
$\bm{\mathcal{C}}\setminus\mathcal{U}\neq\emptyset$, and
$\tilde{G}(\bm{p},\mathcal{G})$ is submodular in the products carried by the
intermediary for any given $\bm{p}$ (in particular, when
$\bm{p}=\bm{p}^{\text{pre}}$). ∎

The intuition for Proposition 5 is the same as that for Proposition 3: the
complementarity in profits is maximized, and the substitutability of traffic
is minimized by combining a pure profit contributor and a pure traffic
generator portfolio. In contrast, the complementarity in profits is minimized,
and traffic substitutability is maximized by combining two pure traffic
generators.

As I stated in Section 2, no product will ever be a pure profit contributor or
a pure traffic generator. However, just as in the case of Proposition 3, the
results of Proposition 5 extend beyond the limit cases considered in the
proposition. The latter is illustrated in Figure 3, which depicts a situation
with four products and three conglomerates premerger
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$.

Figure 3: The Consumer Welfare Implications of Two Different Mergers

(a) Products’ Positions as $\beta$ Changes

(b) Change in Consumer Welfare

Notes. Panel (a) depicts the premerger location of the four products in the
$(\pi,v)$ space as a function of $\beta$. The arrows indicate how the location
of a product moves as $\beta$ decreases, starting from $\beta=0.99$. Panel (b)
plots the consumer welfare implications of two different potential mergers:
$\mathcal{U}=\\{1,2\\}\cup\\{3\\}$ and $\mathcal{U}=\\{1,2\\}\cup\\{4\\}$.
Parameters: $n=4$, $m=3$,
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$.
$G(\xi)=1-e^{-\lambda\xi}$ with $\lambda=5$. Demands:
$Q_{i}(p_{i})=\alpha_{i}(1-p_{i}(1-\sigma_{i})/(2-\sigma_{i}))^{1/(1-\sigma_{i})}$,
with $\bm{\alpha}=\\{1,1,1,1\\}$ and
$\bm{\sigma}=\\{1,-1,\nicefrac{{3}}{{2}},-6\\}$. Marginal costs:
$\bm{c}=\\{\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}}\\}$.

More precisely, panel (a) of that figure depicts the premerger location of the
four products in the $(\pi,v)$ space as a function of $\beta$. These locations
now change depending on the value of $\beta$ since they are endogenous
equilibrium objects. The arrows indicate how the location of a product moves
as $\beta$ decreases, starting from $\beta=0.99$.303030Note that as $\beta\to
0$, all four locations converge to $(0,0)$. Intuitively, as $\beta\to 0$, the
double-markup problem becomes infinitely severe and the negotiated wholesale
prices converge to the choke prices of the different demands. Panel (b), in
turn, plots the consumer welfare implications of two different potential
mergers, $\mathcal{U}=\\{1,2\\}\cup\\{3\\}$ and
$\mathcal{U}=\\{1,2\\}\cup\\{4\\}$

As the figure shows, when conglomerate $\mathcal{C}_{1}=\\{1,2\\}$ acquires
product $3$—which was contributing to the intermediary’s profits mainly via
traffic in the premerger scenario—the merger leads to a reduction in consumer
welfare. In contrast, the transaction unambiguously increases consumer surplus
when the conglomerate acquires product $4$ instead—a product that contributed
to the intermediary mainly through direct profits before the merger.

As noted in Section 2, Propositions 4 and 5 also lead to novel divestiture
measures. Indeed, they suggest that to mitigate the potential adverse effect
of the merger, authorities should aim to force the merging entities to divest
those products located (premerger) in the northwest corner of the $(\pi,v)$
space. The idea is to minimize any potential substitutability of traffic
between the merging portfolios while keeping direct profit complementarities
intact.

This last point is illustrated in Figure 4(a), which builds upon Figure 3. The
figure depicts the consumer welfare implications of forcing conglomerate
$\mathcal{C}_{1}=\\{1,2\\}$ to divest product $1$ as a precondition to
acquiring product $3$, comparing it to the case in which no remedy is imposed.
As the figure shows, the divestiture remedy almost completely offsets the
negative impact of the original transaction.

Unfortunately, however, just as in the simple model of Section 2, forcing the
merging entities to divest the wrong products can backfire. Figure 4(b) shows
an example of this phenomenon, depicting the implications of forcing
conglomerate $\mathcal{C}_{1}=\\{1,2\\}$ to divest product $2$ instead of
product $1$ as a precondition to acquiring product $3$.

Figure 4:  The Consumer Welfare Implications of Two Different Divestiture
Measures

(a) Divesting Product $1$

(b) Divesting Product $2$

Notes. Parameters: $n=4$, $m=3$,
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$.
$G(\xi)=1-e^{-\lambda\xi}$ with $\lambda=5$. Demands:
$Q_{i}(p_{i})=\alpha_{i}(1-p_{i}(1-\sigma_{i})/(2-\sigma_{i}))^{1/(1-\sigma_{i})}$,
with $\bm{\alpha}=\\{1,1,1,1\\}$ and
$\bm{\sigma}=\\{1,-1,\nicefrac{{3}}{{2}},-6\\}$. Marginal costs:
$\bm{c}=\\{\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}}\\}$.

As I mentioned in Section 2.3, products with “more concave” and “rectangular-
shaped” demands will have a higher
$\pi_{i}^{\textrm{pre}}/v_{i}^{\textrm{pre}}$ ratio. This was because the
intermediary can extract a larger share of consumer surplus without
introducing significant monopoly distortions the more concave and rectangular-
shaped a product’s demand is, leading to a higher
$\pi_{i}^{\textrm{pre}}/v_{i}^{\textrm{pre}}$.

Figure 5: Curvature of Demand and Products’ Premerger $(\pi,v)$

(a) Premerger $(\pi,v)$ Distribution (when $\beta=0.6$)

(b) Product $1$ ($\sigma_{1}=1$)

(c) Product $2$ ($\sigma_{2}=-1$)

(d) Product $3$ ($\sigma_{3}=\nicefrac{{3}}{{2}}$)

(e) Product $4$ ($\sigma_{4}=-6$)

Notes. Parameters: $n=4$, $m=3$, $\beta=0.6$,
$\bm{\mathcal{C}}=\big{\\{}\\{1,2\\},\\{3\\},\\{4\\}\big{\\}}$.
$G(\xi)=1-e^{-\lambda\xi}$ with $\lambda=5$. Demands:
$Q_{i}(p_{i})=\alpha_{i}(1-p_{i}(1-\sigma_{i})/(2-\sigma_{i}))^{1/(1-\sigma_{i})}$,
with $\bm{\alpha}=\\{1,1,1,1\\}$,
$\bm{\sigma}=\\{1,-1,\nicefrac{{3}}{{2}},-6\\}$. Marginal costs:
$\bm{c}=\\{\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{10}}\\}$.
Equilibrium outcome: $\bm{w}^{\text{pre}}=\\{0.87,0.53,3.85,0.48\\}$ and
$\bm{p}^{\text{pre}}=\\{1.57,1.10,6.47,1.03\\}$.

This intuition is illustrated in Figure 5, which depicts the premerger
scenario underlying Figures 3 and 4 when $\beta=0.6$. In this particular
example, consumers’ demands belong to the family of constant-passthrough
demands first introduced by Bulow and Pfleiderer (1983):313131It is easy to
prove that this family of demands satisfies the regularity conditions stated
in Section 3.1.

$Q_{i}(p_{i})=\alpha_{i}\left[1-\left(\frac{1-\sigma_{i}}{2-\sigma_{i}}\right)p_{i}\right]^{\frac{1}{1-\sigma_{i}}}\
\ \text{where $\alpha_{i}\in(0,\infty)$ and $\sigma_{i}\in(-\infty,2)$}$

This is a relatively general family of demand functions that has as special
cases (i) the unit (or perfectly inelastic) demand ($\sigma_{i}\to-\infty$),
(ii) linear demand ($\sigma_{i}=0$), and (iii) constant markup demand
($\sigma_{i}\to 1$), among others. The parameter $\alpha_{i}$ captures the
scale of the demand, while the parameter $\sigma_{i}$ captures its curvature.
More precisely, $Q_{i}(p_{i})$ will be concave when $\sigma_{i}<0$, linear
when $\sigma_{i}=0$, and convex when $\sigma_{i}>0$.

As the figure shows, products with “more concave” and “rectangular-shaped”
demands—such as products $2$ and $4$—have a higher
$\pi_{i}^{\textrm{pre}}/v_{i}^{\textrm{pre}}$ ratio as they are used by the
intermediary to mainly generate direct profits. Conversely, products with
“more convex” demands—such as products $1$ and $3$—have a lower
$\pi_{i}^{\textrm{pre}}/v_{i}^{\textrm{pre}}$ ratio, as the intermediary uses
those products to generate traffic.

##  4 Final Remarks

The effects of cross-market mergers of upstream suppliers in the presence of
common customers have been very controversial. Some researchers have argued
that common customers make products substitutes for intermediaries; thus, such
mergers likely increase negotiated prices. Other researchers posit that
because products are complements to customers, they must also be complements
for intermediaries. Hence, these types of transactions should lead to lower,
not higher, negotiated prices.

In this paper, I contribute to this debate by showing that two products can be
complements for customers but substitutes for the intermediary in terms of
profits. This seemingly contradictory result arises when two products serve a
similar role in attracting consumers to purchase the bundle and are, thus,
substitutes in generating consumer traffic for other products in the
intermediary’s lineup.

This result leads to novel antitrust recommendations. First, it suggests a
first-pass test to assess whether the portfolios of two merging suppliers will
be complements in profits for the intermediary. Second, it implies that
authorities can assess the likelihood of harm in a particular transaction by
assessing the role played by the merging products in the intermediary’s
lineup. Finally, the aforementioned result leads to a new divestiture
framework for these types of transactions: to offset the potential negative
effects of a merger, authorities should focus their divestiture efforts on the
products that contribute to the intermediaries’ overall profits primarily via
consumer traffic.

Appendix

##  Appendix A Supermodularity of $U(\mathcal{G};\xi)$

Instead of directly showing that
$U(\mathcal{G};\xi)=\max\\{\sum_{i\in\mathcal{G}}v_{i}-\xi,0\\}$ is a
supermodular set function (which requires going over all subsets of
$2^{\\{\emptyset,0,1,2\\}}$), I will use the following theorem due to Topkis
(1978) and reproduced in Brynjolfsson and Milgrom (2013, Theorem 1b., p.19):

######  Theorem A.0.1.

Suppose that $f:S\to\mathbb{R}$, where $S$ is a product set
$S=\times_{i=1}^{n}S_{i}$, and each $S_{i}\subseteq\mathbb{R}$. Then $f$ is
supermodular if and only if for all $1\leq i<j\leq n$ and all $x\in S$, the
function $g(z_{i},z_{j})\equiv f(z_{i},z_{j},x_{-ij})$ is supermodular.

To use this theorem, note that there is a one-to-one mapping between
$U(\mathcal{G};\xi)$ and the function $\tilde{U}:\\{0,1\\}^{3}\to\mathbb{R}$
defined as
$\tilde{U}(x_{i},x_{j},x_{k};\xi)=\max\\{x_{i}v_{i}+x_{j}v_{j}+x_{k}v_{k}-\xi,0\\}$
(intuitively, $x_{i}\in\\{0,1\\}$ denotes whether the intermediary stocks
product $i$). Thus—and given that $\tilde{U}(x_{i},x_{j},x_{k};\xi)$ is
symmetric upon permutations of $(x_{i},x_{j},x_{k})$—the indirect utility
function will be supermodular in the set of products if and only if for all
$x_{k}\in\\{0,1\\}$ the following condition holds:

$\tilde{U}(1,1,x_{k};\xi)+\tilde{U}(0,0,x_{k};\xi)-\tilde{U}(1,0,x_{k};\xi)-\tilde{U}(0,1,x_{k};\xi)\geq
0$ (17)

To show the latter, denote by $\delta$ the left-hand side of (17) and suppose,
without loss of generality, that $v_{i}\geq v_{j}$. Then:

  * •

If $x_{k}v_{k}\geq\xi$, then $\delta=0.$

  * •

If $v_{j}+x_{k}v_{k}\geq\xi>x_{k}v_{k}$, then $\delta=\xi-x_{k}v_{k}>0$.

  * •

If $v_{i}+x_{k}v_{k}\geq\xi>v_{j}+x_{k}v_{k}$, then $\delta=v_{j}\geq 0$.

  * •

If $v_{i}+v_{j}+x_{k}v_{k}\geq\xi>v_{i}+x_{k}v_{k}$, then
$\delta=v_{i}+v_{j}+x_{k}v_{k}-\xi\geq 0$.

  * •

If $\xi>v_{i}+v_{j}+x_{k}v_{k}$, then $\delta=0$.

Thus, $\delta\geq 0$ always (with strict inequality in some circumstances);
therefore, the indirect utility function of a consumer with a shopping cost
$\xi$ is supermodular in the set of products. $\square$

##  Appendix B Divestiture Measures with Four Products

In this appendix, I derive the equilibrium negotiated payments under four
scenarios: premerger, postmerger with no divestiture, postmerger with the
divestiture of product $1$, and postmerger with the divestiture of product
$2$. Then, I obtain the expressions for $\Delta T(\text{no divestiture})$,
$\Delta T(\text{divest}\ 1)$, and $\Delta T(\text{divest}\ 2)$, as stated in
Section 2.4 of the main text.

Premerger Equilibrium.— Consider first the premerger equilibrium. Following a
logic analogous to that used in Section 2.2, the equilibrium negotiated
payments are
$t_{i}^{\text{pre}}=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})-\Phi_{i}\big{)}\big{/}2$
for $i=0,3$, and
$\textstyle\sum_{i=1,2}t_{i}^{\text{pre}}=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2\\})-\sum_{i=1,2}\Phi_{i}\big{)}\big{/}2$
for $i\in\mathcal{U}=\\{1,2\\}$. Hence
$T^{\text{pre}}=\big{(}2\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2\\})-\Psi(\mathcal{N}\setminus\\{3\\})-\sum_{i=1,2,3}\Phi_{i}\big{)}\big{/}2$.

Postmerger - No Divestiture.— Consider the postmerger equilibrium without any
divestiture measures. The equilibrium negotiated payments then satisfy
$t_{0}^{\text{post}}(\text{no
divestiture})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{0\\})-\Phi_{i}\big{)}\big{/}2$,
and $\textstyle\sum_{i=1,2,3}t_{i}^{\text{post}}(\text{no
divestiture})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2,3\\})-\sum_{i=1,2,3}\Phi_{i}\big{)}\big{/}2$.
Thus, $T^{\text{post}}(\text{no
divestiture})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2,3\\})-\sum_{i=1,2,3}\Phi_{i}\big{)}\big{/}2$.

Postmerger - Divestiture of Product $1$.— Suppose that $\mathcal{U}$ must
divest product $1$ to acquire product $3$. The equilibrium negotiated payments
are then $t_{i}^{\text{post}}(\text{divest
1})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})-\Phi_{i}\big{)}\big{/}2$
for $i=0,1$, and $\textstyle\sum_{i=2,3}t_{i}^{\text{post}}(\text{divest
1})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{2,3\\})-\sum_{i=2,3}\Phi_{i}\big{)}/2$.
Hence, $T^{\text{post}}(\text{divest
1})=\big{(}2\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{2,3\\})-\Psi(\mathcal{N}\setminus\\{2\\})-\sum_{i=1,2,3}\Phi_{i}\big{)}\big{/}2$.

Postmerger - Divestiture of Product $2$.— Finally, suppose that $\mathcal{U}$
must divest product $2$ to acquire product $3$. The equilibrium negotiated
payments are then $t_{i}^{\text{post}}(\text{divest
2})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{i\\})-\Phi_{i}\big{)}\big{/}2$
for $i=0,2$, and $\textstyle\sum_{i=1,3}t_{i}^{\text{post}}(\text{divest
2})=\big{(}\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,3\\})-\sum_{i=1,3}\Phi_{i}\big{)}/2$.
Consequently, $T^{\text{post}}(\text{divest
2})=\big{(}2\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,3\\})-\Psi(\mathcal{N}\setminus\\{2\\})-\sum_{i=1,2,3}\Phi_{i}\big{)}\big{/}2$.

Expressions for $\Delta T(\text{no divestiture})$, $\Delta T(\text{divest}\
1)$, and $\Delta T(\text{divest}\ 2)$.— Combining the above results, I then
have that:

$\displaystyle\textstyle\Delta T(\text{no
divestiture})=\big{(}\Psi(\mathcal{N}\setminus\\{1,2\\})+\Psi(\mathcal{N}\setminus\\{3\\})-\Psi(\mathcal{N})-\Psi(\mathcal{N}\setminus\\{1,2,3\\})\big{)}\big{/}2$
$\displaystyle\textstyle\Delta T(\text{divest}\
1)=\big{(}\Psi(\mathcal{N}\setminus\\{1,2\\})+\Psi(\mathcal{N}\setminus\\{3\\})-\Psi(\mathcal{N}\setminus\\{1\\})-\Psi(\mathcal{N}\setminus\\{2,3\\})\big{)}\big{/}2$
$\displaystyle\textstyle\Delta T(\text{divest}\
2)=\big{(}\Psi(\mathcal{N}\setminus\\{1,2\\})+\Psi(\mathcal{N}\setminus\\{3\\})-\Psi(\mathcal{N}\setminus\\{2\\})-\Psi(\mathcal{N}\setminus\\{1,3\\})\big{)}\big{/}2$

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