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Simulating Mergers in a Vertical Supply Chain with Bargaining

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We model a two-level supply chain where Nash bargaining occurs upstream, and firms compete in a logit setting downstream, either via an auction or Bertrand price setting. The parameters can be calibrated with a discrete set of data on prices, margins, and market shares, facilitating use by antitrust practitioners. We perform numerical simulations to identify cases where modeling the full vertical structure is important and where consumer harm is likely. We also examine the proposed Anthem/Cigna merger and show how the model weighs the various arguments made by the government and the defendants.
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Simulating Mergers in a Vertical
Supply Chain with Bargaining
Gloria Sheu
Federal Reserve Board
Charles Taragin
Federal Trade Commission
July 2021
Abstract
We model a two-level supply chain where Nash bargaining occurs upstream and firms
compete in a logit setting downstream, either via Bertrand price setting or an auc-
tion. The parameters can be calibrated with a discrete set of data on prices, margins,
and market shares, facilitating use by antitrust practitioners. We perform numerical
simulations to identify cases where modeling the full vertical structure is important
and where harm is likely. We also examine the thwarted Anthem/Cigna merger and
show how the model weighs the various arguments made by the government and the
defendants.
Keywords: bargaining models; merger simulation; vertical markets
JEL classification: L13; L40; L41; L42
The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other
members of the Board research staff, by the Federal Reserve Board of Governors, by the Federal Trade
Commission, or by its Commissioners. This article has benefited from conversations with Randy Chugh,
Evan Gee, Nicholas Hill, Richard Langford, Nathan Miller, Joseph Podwol, and Nathan Wilson, along with
comments from numerous seminar participants at the Federal Communications Commission, the Federal
Trade Commission, the U.S. Department of Justice, the International Industrial Organization Conference
(April 21, 2018; Indianapolis, IN), and the DC Industrial Organization Day Conference (May 25, 2018;
Washington, DC).
Board of Governors of the Federal Reserve System, gloria.sheu@frb.gov.
Federal Trade Commission, ctaragin@ftc.gov
1 Introduction
The importance of vertical relationships was the central theme in two landmark merger
cases: Anthem/Cigna (2017) and AT&T/Time Warner (2018). In Anthem/Cigna, the
government argued that the reduction in horizontal downstream competition would raise
prices for health insurance, whereas the defendants argued that the combined firms would
obtain better upstream rates from healthcare providers and would hence lower insurance
prices.1In AT&T/Time Warner, the government argued that the vertical merger would cause
Time Warner to raise input prices for AT&T’s rivals, whereas the defendants argued that the
combination would lower consumer prices through the elimination of double marginalization
(EDM) and other synergies.2However, none of the economic experts in these cases used a
full vertical model that weighed effects upstream and downstream. As a result, the judges
deciding these cases were hampered in their ability to draw strong conclusions on the extent
of harm or benefit that would result.
This article develops a unified framework that nests the models used in these merger
cases.3,4 We provide two options for downstream competition: the Bertrand logit (Werden
and Froeb (1994)) and the logit second score auction (Miller (2014), Miller (2017)).5Whereas
the Bertrand setup is often used to study consumer retail markets in which customers are
price takers, the auction model is better suited for business-to-business transactions, where
the buyer collects quotes from suppliers.6We embed each of these in an upstream Nash-
in-Nash bargaining model. We have three main goals: (1) to provide a tractable merger
simulation tool for practitioners, (2) to highlight situations where accounting for vertical
aspects of the market are or are not likely to be important, and (3) to identify potential
indicators for the presence of consumer harm in vertical mergers.
Once it has been calibrated, the model can simulate the effects of both horizontal and ver-
tical mergers, as we show in a series of numerical experiments. Downstream horizontal merg-
ers feature the typical “upward pricing pressure” (UPP) effect in the downstream market,
due to the lessening of competition between substitute products. There is a countervailing
1These arguments are discussed in the Anthem/Cigna District-level opinion at pages 3 and 5.
2These contentions are discussed in the AT&T/Time Warner District-level opinion at pages 1-3.
3The government’s expert in the Anthem/Cigna trial relied on a second score auction to model the
downstream market. See the demonstrative exhibit used by David Dranove while testifying on behalf of the
plaintiffs, at slide 48, and the District-level opinion at pages 66-67 and 70-71.
4The government’s expert in the AT&T/Time Warner trial used Nash bargaining to model the upstream
market, and a separate Bertrand logit framework to model the downstream market. See Section 12 and
Appendix G of the expert report filed by Carl Shapiro on behalf of the plaintiffs.
5Miller (2014) builds upon work in Tschantz, Crooke, and Froeb (2000).
6Such purchasing behaviors are sometimes referred to as “request for proposal” (RFP) sales.
1
“recapture leverage effect” in the upstream market. In the event of a contract breakdown,
the merged downstream firms may recapture some lost sales through their merging partners,
which improves these firms’ negotiating positions with suppliers.7
For vertical mergers, downstream markets are again subject to the UPP effect, although
here the impact occurs through the lessening of competition between the downstream merged
firm and competitors that also distribute products sourced from its upstream merged partner.
This UPP is balanced against the additional effect of EDM, as now the downstream merged
firm can obtain some inputs at marginal cost.8In the upstream market, there are recapture
leverage effects that go in opposite directions. On the one hand, the downstream merged firm
may pay lower prices to unaffiliated suppliers, because a contract breakdown may increase
sales of the upstream merged partner. On the other hand, the upstream merged firm may
charge higher prices to unaffiliated downstream firms, as now a breakdown in negotiations
may increase sales for the downstream merged firm. The potential for higher input prices for
competitors is a phenomenon often called “raising rivals’ costs” (RRC). These many forces
are balanced by the model.
After exploring the magnitudes of these effects numerically, we then apply our framework
to the Anthem/Cigna merger, as a concrete, real world example.9We model Anthem and
Cigna as two of several insurance providers competing via second score auctions to administer
employers’ health plans. The insurers simultaneously bargain with hospitals upstream to
purchase medical services.
We have several key findings. First, we show that the model is easy to implement using
data typically seen in merger investigations, and we then demonstrate how the model behaves
in a variety of competitive environments. Using simulated data on thousands of different
markets, we identify instances where mergers can create net benefits for consumers. These
situations occur for a subset of vertical mergers and of downstream horizontal mergers with
second score auction competition.
Second, we find that the potential for downstream merging firms to bargain for lower input
prices only ever results in net consumer benefits under second score auction competition and
not under Bertrand. Thus, practitioners should be more concerned about modeling the
full vertical supply chain in cases where downstream competition occurs via second score
7This recapture leverage effect also appears in upstream horizontal mergers, but on the side of the
upstream suppliers, which leads them to negotiate for higher input prices.
8There is some argument over whether EDM should be credited legally as an efficiency. As Kwoka and
Slade (2020) point out, EDM is a pecuniary economy rather than an actual decrease in production costs.
Given that relatively few vertical mergers have been litigated, the legal standard is unclear.
9It is also possible to apply the model to the AT&T/Time Warner merger case. The in-progress research
of Gee, Sheu, and Taylor (2020) studies that merger.
2
auction instead of Bertrand price setting. With second score auctions, consumers receive
net benefits in about a third of our downstream horizontal merger simulations, primarily
when there are few downstream firms or when downstream firms have low bargaining power
relative to upstream suppliers.10
In turn, ignoring upstream interactions in the auction context can overstate consumer
harm from downstream horizontal mergers, as we demonstrate with our Anthem/Cigna
application. We find that the standard second score auction model used by the plaintiffs
predicts consumer harm that is three times greater than in our full model. We also find that
the input cost efficiencies proposed by the defendants are an order of magnitude larger than
those produced by the vertical model, meaning that we predict net harm to consumers of
$129 million per year, whereas the defendants argued there would be net benefits.
Third, for vertical mergers, we show that the relative bargaining power of upstream firms
compared to downstream firms has the ability to predict whether consumers are likely to be
harmed from a merger, whereas the number of firms and the Herfindahl-Hirschman Index
(HHI) do not. Identifying meaningful indicators of harm is an important policy issue, partic-
ularly given that the U.S. Department of Justice (DOJ) and the Federal Trade Commission
(FTC) recently released Vertical Merger Guidelines, but did not include any thresholds for
screening.11 We find that when upstream and downstream firms have equal power, the im-
pact of vertical mergers on consumer welfare is roughly neutral. When downstream firms
have more power, consumers are more likely to be harmed, and when upstream firms have
more power, consumers are more likely to benefit. When a downstream firm has higher rel-
ative bargaining power, its pre-merger input prices tend to be low, which limits the benefits
from EDM. Although it would seem that relative bargaining power is an abstract concept
that is difficult to measure, in our model it is equal to the ratio of upstream to downstream
gains from trade. We find that this ratio, in turn, is strongly correlated with relative up-
stream versus downstream profit margins, which is information that the antitrust agencies
typically collect as part of merger investigations.
The analysis we present has a few caveats. As mentioned above, we have assumed
a particular structural form for demand, relying on the logit, and for bargaining, relying
on the Nash-in-Nash setup. These structural assumptions are important for maintaining
the tractability of our framework, but we also discuss some alternatives. Furthermore, we
10We define bargaining power as the proportion of the gains from trade that accrue to a firm on one side
of a negotiation.
11The guidelines were released in June 2020. Slade (2020) argues that antitrust authorities need some
initial screening tools for vertical mergers, and proposes market share and concentration indices. She also
acknowledges that these measures have drawbacks.
3
also assume that upstream bargaining is over linear input prices, which is important for
generating impacts on final consumers. If optimal two-part tariffs were used instead, changes
in bargaining upstream would not shift downstream prices.12 The model also ignores the
possibility of exclusive contracts among a subset of suppliers or distributors (although that
could be incorporated as an extension) and of coordination or explicit collusion.
Our work is related to a large literature on merger simulation, including the aforemen-
tioned Miller (2014) and Werden and Froeb (1994). For a review of this topic, see Whinston
(2007) and Werden and Froeb (2008). We also draw upon the literature on vertical mergers.
See Riordan and Salop (1995) and Riordan (2008) for general summaries.
Bargaining models have already proven useful in analyzing a number of vertical sit-
uations, including retailer/wholesaler relationships (Draganska, Klapper, and Villas-Boas
(2010)), hospital/insurer contracting (Ho and Lee (2017) and Ho and Lee (2019)), and video
programmer/distributor negotiations (Crawford and Yurukoglu (2012) and Crawford, Lee,
Whinston, and Yurukoglu (2018)). These articles study several interesting aspects of bar-
gaining and competition in certain specific contexts. Our work complements this research
by taking a more general view of these types of models and studying how they behave in a
variety of scenarios.
Bargaining models have also appeared in the antitrust literature. Rogerson (2014) uses
Nash bargaining to analyze the 2011 merger between Comcast and NBC Universal. The
model presented by Rogerson is similar to that which we use for our upstream market, but
lacks an explicit formulation of the downstream level. Gaudin (2018) uses a Nash bargaining
setup to see whether reduced retail competition allows downstream firms to negotiate better
prices with upstream suppliers and hence increase consumer welfare. Consistent with our
results, he finds that whether such countervailing effects emerge depends on the extent of
the pass-through rate of input costs to prices. However, unlike Gaudin, our emphasis is
on how these effects emerge through changes in the firms’ disagreement payoffs should a
retailer/wholesaler negotiation break down.13 Dranove, Rothman, and Toniatti (2019) posit
a Nash bargaining model in order to study the Anthem/Cigna merger, but rely on measures
based on UPP instead of a formal structural simulation. Rogerson (2020) also proposes UPP
measures in a similar model, focusing on vertical mergers.
12In this case there would be no EDM from a vertical merger, but also no harm to consumers, assuming the
two-part tariffs remained in effect and that there were no changes to other incentives, such as for investment.
The merger could instead impact the size of lump-sum transfers between firms.
13Gaudin (2018) assumes that the downstream retailer’s disagreement payoff is zero. In related work, Spul-
ber (2017) examines mergers between upstream suppliers of complements that engage in Nash bargaining.
In contrast, we study competition between suppliers selling substitutes.
4
This article proceeds as follows. In Section 2 we describe the model, focusing on the
downstream Bertrand logit case so as to fix ideas. Section 3 shows how this model can be
calibrated and used to simulate horizontal and vertical mergers. We extend the model in
Section 4 to cover downstream auction competition. In Section 5 we provide results from a
series of numerical experiments and then present the Anthem/Cigna analysis in Section 6.
Section 7 concludes.
2 Theoretical Framework
We begin by presenting the baseline version of our model, featuring downstream Bertrand
logit competition and upstream Nash bargaining. Additional derivations using nested logit
demand appear in the Appendix.
Downstream Model
Let there be a set of consumers indexed by iwho can choose to buy a single product sold
by a single retailer. Retailers, indexed by r, source their merchandise from wholesalers
indexed by w. Each wholesaler offers only one product (meaning the product and wholesaler
indices are synonymous), but a retailer can purchase from multiple wholesalers.14 The set
of all retailers is denoted by R={1,...,|R|}, and the set of all wholesalers is denoted
by W={1,...,|W|}. The set Wis divided into |R|potentially overlapping subsets, each
labeled Wr, to indicate which wholesalers’ products are carried by which retailers. In turn,
the set of retailers Ris divided into |W|potentially overlapping subsets, each labeled Rw,
which indicate the retailers that carry the product sold by each wholesaler.
We assume that consumers choose which product to buy according to the multinomial
logit discrete choice model. The indirect utility function for consumer ipurchasing from
retailer rthe product owned by wholesaler whas the form uirw =δrw αprw +irw . The
parameter αmeasures consumer sensitivity to the retail price, denoted by prw. The δrw is a
demand shifter that captures average consumer tastes for the non-price aspects of product
wwhen purchased at retailer r. The final term, irw , is an independent and identically
distributed Type I extreme value error with a scale parameter of 1. We normalize the utility
of the outside good to be ui00 =i00. Integrating over the error term gives the market share
14Although we restrict our attention to single-product upstream firms for expositional simplicity, the model
can be extended to include multiproduct wholesalers. That case is discussed in more detail by Draganska,
Klapper, and Villas-Boas (2010).
5
among all available product-retailer combinations,
srw =exp(δrw αprw)
1 + PtRPxWtexp(δtx αptx),(1)
for product wsold by retailer r.
We assume that retailers simultaneously choose prices in Bertrand competition in order
to maximize profits. The retailer’s profit function takes the form πr=PwWr[prw pW
rw
cR
rw ]srw M, where pW
rw is the unit fee charged by wholesaler wto retailer r,cR
rw captures any
additional marginal costs borne by the retailer, and Mis the market size. The resulting first
order condition for the price prw takes the typical form,
X
xWr
[prx pW
rx cR
rx]srx
∂prw
+srw = 0.(2)
The series of first order conditions for each of the downstream prices together form a system
of equations that relates retail margins to market shares.
Upstream Model
We characterize the profits of wholesaler was πw=PrRw[pW
rw cW
rw ]srw M, where cW
rw is
the marginal cost borne by the wholesaler. The level of the price pW
rw is determined via a
bilateral negotiation between wholesaler wand retailer r.
Throughout this article, we assume that inputs are priced per unit. This linear pricing
structure generates meaningful feedback effects in the model, by allowing the bargaining
outcome to directly impact downstream marginal costs, and hence the price and quantity of
final units sold. Admittedly, this assumption may not be appropriate for some applications.
However, linear contracts are observed in a number of industries. Indeed, such pricing
behavior underpins one oft-cited justification for vertical mergers, EDM.
We assume that bargaining over the wholesale price pW
rw is characterized by the following
maximization problem:
max
pW
rw
(πrdr(Wr\ {w}))λ(πwdw(Rw\ {r}))1λ,(3)
where dr(Wr\ {w}) is the disagreement payoff for the retailer and dw(Rw\ {r}) is the
disagreement payoff for the wholesaler. The λ(which ranges from 0 to 1) measures the
6
relative bargaining power of retailers.15 In words, the wholesale price is chosen to maximize
the Nash product of two terms. The first term is the difference between the profits of the
retailer when it offers wholesaler w’s product versus when it does not. The second term is the
difference between the profits of the wholesaler when it sells to this retailer versus when it
does not. The entire expression within each parentheses pair is the retailer’s or wholesaler’s
gains from trade (GFT), respectively.
We assume that the disagreement payoff for the retailer is dr(Wr\{w}) = PxWr\{w}[prx
pW
rx cR
rx]sr x(Wr\ {w})M. The market share srx(Wr\ {w}) is computed in the case where
retailer rdoes not offer wholesaler w’s product.16 The disagreement payoff of the whole-
saler when it does not offer its product to retailer ris dw(Rw\ {r}) = PtRw\{r}[pW
tw
cW
tw]stw (Wr\ {w})M. Both the retailer’s and the wholesaler’s disagreement payoffs exhibit
forms of “recapture.” That is, when the two firms fail to come to an agreement, the retailer
can recoup some of its lost sales if customers substitute to other products instead of w, but
do not change which retail outlet they visit. Meanwhile, the wholesaler can regain some of
its lost sales if customers stay with the same product but switch to other retailers. In so far
as one or the other firm has a better payoff, that increases its relative bargaining leverage.
Throughout this article, we refer to “bargaining leverage” as relating to the strength of a
firm’s bargaining position based on its disagreement payoff. In contrast, we use the term
“bargaining power” to refer to a firm’s bargaining ability as captured in λ.
The bargaining setup involves a separate negotiation for each wholesaler-retailer pair.
However, the payoffs from the outcome of one negotiation are related to those from all
other negotiations due to competition in the downstream market. In order to simplify the
multilateral complexities this situation raises, we make two assumptions,
1. Simultaneous negotiations: when bargaining over a single input price, the wholesaler
and retailer act as if all other input price negotiations are taking place simultaneously.
Thus, all other wholesale prices are treated as fixed.
2. Simultaneous downstream pricing: when bargaining over a single input price, the
wholesaler and retailer act as if downstream prices are being set simultaneously. There-
fore, all retail prices are treated as fixed.
The benefit of both of these assumptions is that they produce a tractable solution to the
15The model can be extended to accommodate values of λthat differ across retailer-wholesaler pairs. See
Sheu and Taragin (2017), an earlier working paper version of this article, for additional details.
16That is, srx (Wr\ {w}) is calculated as in expression (1), but removing the term exp(δrw αprw ) from
the denominator.
7
series of first order conditions for the problem in expression (3). We discuss each of these
assumptions in turn.
The simultaneous negotiations assumption was developed by Horn and Wolinsky (1988)
in order to study situations with multiple firms engaged in bilateral contracting, where the
outcome of one negotiation affects the payoffs from other contracts. The resulting equilib-
rium is often called “Nash-in-Nash.” When firms in one bilateral negotiation treat all other
contracts as fixed, this means that the terms of these other agreements are viewed as un-
changed even if one negotiation breaks down. This assumption is admittedly restrictive, as
it implies that a firm that is party to multiple contracts treats each separately.17 Neverthe-
less, such simplification is important in our setting, where we are calibrating our model with
limited data. This assumption has also proven important in maintaining tractability even
in environments where more data are available, as seen in Crawford and Yurukoglu (2012),
Grennan (2013), and Gowrisankaran, Nevo, and Town (2015), among others.
The simultaneous downstream pricing assumption is common in the vertical bargaining
literature, appearing in, for example, Draganska, Klapper, and Villas-Boas (2010), Ho and
Lee (2017), and Crawford, Lee, Whinston, and Yurukoglu (2018). If the firms engaged in bi-
lateral bargaining assume that downstream prices are being set at the same time as upstream
prices, then these firms will view downstream prices as fixed. Although this assumption is
strong, it has some appeal in settings where upstream firms lack an obvious first-mover
advantage in pricing.18
An alternative assumption would be to model upstream fee negotiations as taking place
before downstream prices are chosen. In such a sequential framework, wholesalers could
strategically set their prices at a different level than is optimal under the simultaneous
solution in order to affect retail supply. As discussed in Rogerson (2020), estimating a
sequential model can be difficult.19 Given that downstream firms often cannot immediately
adjust their prices in many real world markets, the downstream simultaneity assumption
seems appropriate in many cases.20
17Collard-Wexler, Gowrisankaran, and Lee (2019) provide a microfoundation for this setup via an alter-
nating offers game that does not require such a stark separation between negotiations involving the same
firm. However, bargaining in their model is over lump sum transfers, not linear prices.
18Note that, although this assumption limits the way in which upstream and downstream prices interact,
retail prices still affect wholesale fees in equilibrium. When bargaining upstream, firms still take into account
how downstream prices will be set via the first order condition in equation (2).
19Existing examples in the antitrust literature limit themselves to one upstream and two downstream firms,
which greatly simplifies the problem. For instance, see Das Varma and De Stefano (2020) and Domnenko
and Sibley (2020).
20Furthermore, as discussed by Draganska, Klapper, and Villas-Boas (2010), relaxing this assumption can
create a tension with the assumption that all upstream negotiations are happening simultaneously and can
8
Define ∆stx(Wr\ {w})stx (Wr\ {w})stx, which is the difference in the share of good
xsold by retailer twhen good wis not offered by retailer rversus when good wis offered
by retailer r. Then under our assumptions, the bargaining first order condition simplifies to
wholesaler GFT
z }| {
[pW
rw cW
rw ]srw X
tRw\{r}
[pW
tw cW
tw]∆stw (Wr\ {w}) =
1λ
λ
[prw pW
rw cR
rw ]srw X
xWr\{w}
[prx pW
rx cR
rx]∆sr x(Wr\ {w})
| {z }
retailer GFT
.
(4)
This expression characterizes a system of first order conditions for upstream prices that
relates wholesale and retail margins to market shares. Together with the analogous conditions
for the downstream problem appearing in equation (2), this system can be solved for the
equilibrium outcome.21
Equation (4) implies that in equilibrium, the relative bargaining power of each whole-
saler/retailer pair, (1 λ), must equal their relative GFT. For example, if a wholesaler
and retailer have equal bargaining power, then in equilibrium this equates the wholesaler’s
and retailer’s GFT. This equilibrium relationship between bargaining power and bargaining
leverage is useful in identifying the bargaining power parameters, particularly when one has
access to information on margins.22
3 Merger Simulation
We now demonstrate how mergers, both horizontal and vertical, can be analyzed within this
framework. In what follows, we ignore the presence of efficiencies that cause marginal costs,
cR
rw and cW
rw , to decrease. However, incorporating such efficiencies can be done by adjusting
those costs inside the first order conditions we derive.
therefore be treated separately.
21Note that, so long as there are positive GFT between each retailer and wholesaler, all possible contracts
will be made. Ho and Lee (2019) provide an equilibrium extension that allows for partial networks of
upstream and downstream firms, which nests the equilibrium concept we use here.
22Crawford, Lee, Whinston, and Yurukoglu (2018) use the same relationship to estimate their bargaining
power parameters. Although they lack detailed margin data, their application, television programming, is
one in which wholesale marginal costs are plausibly zero.
9
Identification
We begin by explaining how one can calibrate the parameters of the downstream model using
data on margins, prices, and market shares. Assume that the researcher observes market
shares, {srw ;rR,wW}, retail prices, {prw ;rR,wW}, and one retail margin,
mR
rw =prw pW
rw cR
rw .23 Then the objects to be recovered in the downstream model are
the price coefficient, α, the demand shifters, {δrw ;rR,wW}, the remaining margins,
and their associated marginal costs.
Calibration proceeds following the methods used in a typical logit merger simulation, as
seen in Werden and Froeb (1994). The market share equation (1) has cross-price derivatives
given by ∂srx/∂prw =αsrxsrw and own-price derivatives given by srw /∂prw =αsrw (1
srw ). Thus, if shares and one margin are observed, the downstream first order conditions
provide a system of equations where the only unknowns are the parameter αand the other
margins. Solving these equations yields the coefficient αand the remaining unobserved
margins. Once margins have been computed, the underlying marginal costs (inclusive of
wholesale prices) are given by pW
rw +cR
rw =prw mR
rw . Then the demand shifters can be
recovered using the typical Berry (1994) relationship, ln(srw)ln(s00 ) = δrw αprw, as
retail prices are observed.
The above calibration routine assumes that one knows the share of the outside good,
s00. This share should be measured with care, as it impacts substitution patterns across
products and estimates of consumer welfare, which in turn can affect the results from merger
simulations. Whenever possible, researchers and practitioners should seek guidance from
data or other knowledge about the industry in order to choose an appropriate outside good
share. Werden and Froeb (1994) explain how the share can be recovered from a market price
elasticity, and Sheu and Taragin (2012) show how it can be identified from data on a second
retail margin.
Turning to the upstream model, assume that the researcher additionally observes whole-
sale prices, {pW
rw ;rR,wW}, for each retailer-wholesaler pair and margins, {mW
rw ;r
R}, for one wholesaler w.24 If all of the downstream parameters have been recovered, the
remaining unknown objects are the bargaining parameter, λ, and the other marginal costs,
{cW
rv ;rR,vW\ {w}}.
23If additional data are available, the model is overidentified. In that case, the parameters can be chosen to
give the closest possible match to the observed data, in a manner similar to method of moments estimation.
24Wholesale margins are defined as mW
rw =pW
rw cW
rw . If one observes margins for more wholesale firms,
these can be used to identify additional bargaining parameters that vary by wholesaler-retailer combination.
10
The form of the logit share equation implies that
stx(Wr\ {w}) = srw stx
1srw .(5)
For those familiar with the terminology of diversion ratios, the term in parentheses is the
diversion according to share from the excluded product wsold by retailer rto product x
sold by retailer t. Here, the logit diversion ratio dictates how consumers that buy product w
from retailer rwould substitute if it was withdrawn from the market.25 Given this equation,
expression (4) for wholesaler wis a function of observed market shares, margins, and the
unknown bargaining parameter. Solving this first order condition allows for the recovery of
λ, and the remaining first order conditions can then be used to identify the other wholesale
marginal costs.
Downstream Horizontal Mergers
Once the parameters of the model have been recovered, counterfactual merger simulations
can be performed. We focus here on two interesting cases that have countervailing effects:
downstream horizontal mergers and vertical mergers. The derivations for upstream horizon-
tal mergers appear in the Appendix.
We begin with a horizontal merger between two retailers, which we label firms rand s.
When setting downstream prices, the merged retailers now take into account the effect they
have on each other’s profits, as can be seen in the first order condition,
X
xWr
[prx pW
rx cR
rx]srx
∂prw
+srw +X
vWs
[psv pW
sv cR
sv]∂ssv
∂prw
| {z }
UPP effect
= 0,(6)
which is computed for a product sold by firm r.26 Compared to equation (2), the expression
above has an additional term that captures the effect that raising the price of one of retailer
r’s products has on the profits of retailer s. As the price prw increases, sales shift to retailer
s, which is reflected in the partial derivative ssv/∂prw. These increased sales earn the
margin given by psv pW
sv cR
sv. Greater sales recapture and/or higher margins increase
the incentive to raise price after the merger. This effect is referred to as “upward pricing
25For the logit, the diversion ratio is constant in prices. For other demand functional forms, the diversion
for a large price change, as is implicit when a product is withdrawn from the market, may differ from that
for marginal changes. Conlon and Mortimer (2020) discuss this issue.
26The condition for firm scan be derived analogously.
11
pressure” (UPP).27 The UPP effect is typical of most horizontal merger simulation models.
We now turn to the negotiations over upstream prices. With the merger, the retailer
disagreement payoff when firm rfails to reach an agreement with wholesaler wtakes into
account the profits of retailer s.28 The bargaining first order condition becomes
[pW
rw cW
rw ]srw X
tRw\{r}
[pW
tw cW
tw]∆stw (Wr\ {w}) =
1λ
λ
[prw pW
rw cR
rw ]srw X
xWr\{w}
[prx pW
rx cR
rx]∆sr x(Wr\ {w})
X
vWs
[psv pW
sv cR
sv]∆ssv(Wr\ {w})
| {z }
recapture leverage effect
,
(7)
which reflects the change in the disagreement payoff.29 Comparing equation (7) to equation
(4), we see that the difference is the last additional term. When retailer rloses access to
product w, some sales shift to retailer s, as measured by ∆ssv(Wr\ {w}). These sales raise
profit according to the margin given by psv pW
sv cR
sv. As a result of this recapture, the
merged retailers’ bargaining leverage increases, which then tends to lower input prices.30
Therefore, a downstream horizontal merger can have two possibly offsetting effects. The
UPP effect tends to increase final consumer prices, whereas increased bargaining leverage
can lower marginal costs and thus decrease final consumer prices.
Once we have recovered the predicted post-merger prices from a merger simulation, we
can quantify the resulting effect on consumers. We define the consumer surplus change due
to the difference in pre-merger prices (denoted by the subscript “pre”) and post-merger prices
(denoted by the subscript “post”) as follows:
CS =1
α(ln(s00,pre)ln(s00,post )) .(8)
This expression is the dollar value of the difference in expected utility before the merger
27Farrell and Shapiro (2010) have a broader discussion of UPP.
28The negotiation by retailer sworks similarly.
29We assume that when retailer rfails to reach an agreement with wholesaler w, retailer s’s contract
with wholesaler wremains in place. If instead wholesaler wwithholds its product from both of the merged
retailers, we would remove good wfrom the set Wsand sfrom the set Rwin the disagreement payoffs.
30We have assumed that λis unaffected by the merger, but changes in bargaining power could be accom-
modated as an extension.
12
versus after, after substituting in for the share of the outside good.
Vertical Mergers
Turning to vertical mergers, assume that retailer rand wholesaler wcombine into one firm.
When deciding what downstream price to set for product w, the merged firm now has a first
order condition given by
X
xWr\{w}
[prx pW
rx cR
rx]srx
∂prw
+srw +
EDM effect
z }| {
[prw cW
rw cR
rw ]∂sr w
∂prw
+X
tRw\{r}
[pW
tw cW
tw]stw
∂prw
| {z }
UPP effect
= 0.
(9)
This expression has two differences relative to the first order condition in equation (2).
First, the “elimination of double marginalization” (EDM) effect appears in the second to
last term. We assume that the wholesale price of good wto retailer ris a transfer price
between the merging parties, so its effective marginal cost becomes the sum of the upstream
and downstream costs, cR
rw +cW
rw . This force lowers the resulting retail price prw. Second, the
merged firm now takes into account the effect that lowering prw has on the wholesale profits
made by selling to other retailers besides firm r, as seen in the last term. This “wholesale
UPP” effect is similar to the standard UPP discussed in the previous section, except it
operates through diversion to the partner wholesaler rather than to a merged retailer. This
effect tends to raise the retail price prw . The net effect balances these two opposite forces.31
Turning to the upstream market, when wholesaler wis bargaining with a firm besides r
over what wholesale price to set, its disagreement payoff now has additional terms due to
31The first order condition for products besides wsold by the merged firm can be derived analogously.
13
the profits of its affiliated retailer. The wholesale price first order condition becomes
[pW
sw cW
sw]ssw X
tRw\{r,s}
[pW
tw cW
tw]∆stw (Ws\ {w})
RRC effect
z }| {
[prw cW
rw cR
rw ]∆srw (Ws\ {w})X
xWr\{w}
[prx pW
rx cR
rx]∆sr x(Ws\ {w}) =
1λ
λ
[psw pW
sw cR
sw]ssw X
vWs\{w}
[psv pW
sv cR
sv]∆ssv(Ws\ {w})
.
(10)
Compared to the pre-merger first order condition in equation (4), here when the merged
firm fails to agree with retailer s, it can recapture some of these lost profits through retailer
r. These extra profits will tend to be larger if the products sold by retailer rare closer
substitutes to the product wwhen sold by retailer s. This “raising rivals’ cost” (RRC) effect
will tend to increase the wholesale price firm wcharges to firm s, which in turn may increase
downstream prices.32
When the merged firm is bargaining with a wholesaler besides wover what input price
to pay as a retailer, the bargaining first order condition becomes
[pW
rv cW
rv ]srv X
sRv\{r}
[pW
sv cW
sv ]∆ssv(Wr\ {v}) =
1λ
λ
[prv pW
rv cR
rv ]srv X
xWr\{w,v}
[prx pW
rx cR
rx]∆sr x(Wr\ {v})
[prw cW
rw cR
rw ]∆srw (Wr\ {v})
| {z }
EDM effect
X
tRw\{r}
[pW
tw cW
tw]∆stw (Wr\ {v})
| {z }
recapture leverage effect
.
(11)
The second to last term in the above expression reflects the increased profits retailer rearns
on product wdue to decreased marginal cost from the EDM effect. The last term reflects the
recapture that stems from the profits of wholesaler w. In so far as sales shift to wholesaler
w’s clients when retailer rloses access to firm v’s product, the merged firm has a better
disagreement payoff than without the merger. Both of these effects increase the merged
32Rogerson (2020) calls this RRC term the “Bargaining Leverage Over Rivals” (BLR) effect, in order to
differentiate it from other raising rivals’ cost mechanisms seen in vertical models that feature take-it-or-leave-
it pricing instead of bargaining.
14
firm’s bargaining leverage, and can cause the fees it pays other wholesalers to fall. This in
turn may lower downstream prices to customers.
4 Downstream Auctions
Now that we have discussed the methodology behind simulating mergers in an environment
with upstream bargaining, we extend the framework to incorporate downstream auctions.
We follow Miller (2014) and Miller (2017) in using a second score auction setting. The
upstream model remains the same as in the previous sections.
We assume that consumer ihas an indirect utility function for product wsupplied by
retailer rof the same form seen in the Bertrand model, uirw =δr w αprw +irw .33 Each
consumer selects a single product to purchase by soliciting product-specific bids, brw , from
each retailer. The buyer chooses the option with the highest indirect utility, substituting
bids for prices. The probability that product wfrom retailer ris the best bid among all
product-retailer pairs is therefore
srw =exp(δrw αbrw)
1 + PtRPxWtexp(δtx αbtx),(12)
which is analogous to the market share function in equation (1). The buyer’s expected payoff
from the maximum of all offers is
1
αln 1 + X
tRX
xWt
exp(δtx αbtx)!.(13)
The expression for retailer profit remains the same as in the Bertrand framework.
In a second score auction, the equilibrium price equates the utility from the winning
bidder to what would have been achieved from the second best bid if the buyer were to
receive all the surplus. We assume that each retailer knows the value of irw for a prospective
customer of any of its products. The retailer does not observe this value for products sold
by other retailers. As shown in Miller (2014), the dominant strategy for any retailer in this
auction is to supply only the product wWrto consumer ithat gives the maximum possible
33Miller (2014) shows that this model can also be written in terms of suppliers having heterogeneous costs
to serve a particular customer, instead of buyers having heterogeneous demand for a particular product.
15
utility net of marginal cost. That is, a retailer will not outbid itself. Then price is such that
prw =1
αδrw +irw max
sR\{r},vWs{δsv +isv αbsv }(14)
in the case where product wfrom retailer rwins the auction. Furthermore, as also shown by
Miller (2014), the retailer will set its bid equal to its marginal cost, brw =pW
rw +cR
rw , because
of the second score pricing mechanism.
This auction structure gives the following expression for the conditional expected margin
when product wsold by retailer rwins,
EmR
rw rw wins=1
αPxWrsrx
ln 1X
xWr
srx!.(15)
This expression relates margins to market shares, analogous to equation (2).
The auction model does not introduce any additional parameters relative to the Bertrand
model. Therefore, it can be calibrated from equations (12) and (15) using the same data on
shares, prices, and margins.34
Once the model parameters have been recovered, the effects of potential mergers can be
simulated and the impact on consumer welfare measured in a similar fashion as with the
Bertrand model.35 Starting with a downstream horizontal merger, if two retailers rand s
combine, they will cease to bid against each other. That is, the merging companies will only
offer each customer the product out of both of their portfolios that has the largest utility
compared to marginal cost. Assume, without loss of generality, that this best product is
sourced from wholesaler wand sold by retailer r. Then the merged firms’ expected margin
conditional on winning the auction is
EmR
rw rw wins=1
αPt∈{r,s}PxWtstx
ln
1X
t∈{r,s}X
xWt
stx
.(16)
The merger will tend to raise prices for those customers for whom both retailers rand sare
highly valued. This is similar to the UPP effect seen in the Bertrand model, in that we see
greater impacts when the merging retailers have higher diversion between them.
Turning to vertical mergers, suppose retailer rand wholesaler wmerge. In terms of
34In fact, because in equilibrium bids are equal to the retailer’s marginal costs, the auction model does
not require retail prices to calibrate model parameters, only retail margins and costs.
35The Appendix contains the full expression used to calculate consumer surplus.
16
deciding on a downstream bid, the combined firm must balance two forces: the lower retail
marginal cost encourages it to decrease its bid (the EDM effect), but doing so decreases the
win probability for other retailers who purchase its wholesale product (the UPP effect). If
the possible profits from selling retailer r’s best product are higher than those that can be
earned from the wholesale market, then the merged firms will lower their bid to marginal
cost. If instead the profits from the wholesale market are greater, the merged firms will raise
their bid, effectively removing themselves from the retail choice set for this auction.
5 Numerical Simulations
In constructing our simulated data set, our aim is to generate variation in different metrics
of interest in order to gauge the impact they have on the welfare effects of mergers. We
focus on two variables: (1) the number of firms in the downstream or upstream market and
(2) the relative bargaining power between wholesalers and retailers. Marginal costs, cR
rw and
cW
rw , are assumed to remain unchanged post-merger.
Data Generating Process
This section provides an overview of our methodology, with additional details appearing in
the Appendix. We simulate markets by randomly sampling shares from a Dirichlet distri-
bution for 2, 4, 6, or 8 retailers or wholesalers, respectively.36 The price coefficient αis
calibrated by assuming that in the pre-merger world, there is a vertically integrated outside
option available to all customers. The other goods are differenced relative to this option,
which maintains the outside good normalization. The market size is set to 1.
We specify values for the bargaining parameter ranging from from 0.1 (wholesalers have
the advantage) to 0.9 (retailers have the advantage). To better understand the relative
bargaining strength of these parameter values, we report our results in terms of (1 λ),
which range from 9 (wholesaler power is nine times greater than retailer power) to 1/9
(retailer power is nine times greater than wholesaler power). The bargaining parameter is
identical for all of the retailers in each simulation, unless noted otherwise.
For each combination of number of retailers, number of wholesalers, and bargaining pa-
rameter, we draw 2,500 different sets of market primitives. This results in 2.16 million merger
simulations. We then eliminate mergers where the merger is unprofitable to the merging
36We parametrize the Dirichlet distribution so it is equivalent to a uniform distribution.
17
firms, as well as markets that do not pass the Hypothetical Monopolist Test, yielding 1.78
million markets.37 All 1.78 million markets treat as primitives the number of retailers, the
number of wholesalers, the bargaining parameter, and the wholesaler and retailer marginal
costs.
When simulating a horizontal merger, we assign the products produced by the two largest
firms in the market to a single entity post-merger. Similarly, when simulating a vertical
merger, we assign the products produced by the largest wholesaler and the largest retailer
to a single entity post-merger. This assignment is purposefully skewed towards mergers that
are more likely to have competitive effects and to come under agency review.
Table 1 provides summary statistics across our various simulations.38 The median average
wholesale pre-merger price is almost $5, and the median average retail pre-merger price
is $12. Because the market size is set to 1, these average prices are equal to total pre-
merger expenditures. Pre-merger HHIs range between 2,029 at the 25th percentile to 5,143
at the 75th, with a median of 2,828. HHIs for horizontal downstream mergers increase by
1,658 points at the median, resulting in a median post-merger HHI equal to 4,270. HHIs
for upstream mergers increase by 1,179 points at the median, resulting in a median post-
merger HHI equal to 4,243. HHIs for vertical mergers increase by 953 points at the median,
resulting in a median post-merger HHI equal to 5,174.39 Many of these markets fall into the
span designated by the DOJ/FTC Horizontal Merger Guidelines as “Highly Concentrated
Markets,” with HHIs over 2,500.40
Results Overview
Our overall results are depicted in Figure 1, which is divided into four panels, each showing
how the distribution of surplus changes for a particular set of agents (consumers, retailers,
or wholesalers), as well as the net effect on the market as a whole. Surplus is presented as a
percentage change relative to total pre-merger expenditure in the downstream market.
Each panel contains three pairs of box and whisker plots, with each pair corresponding
37The Hypothetical Monopolist Test requires that were a monopolist to jointly own all products in a
candidate market, that firm would raise the price of at least one of the merging producers’ products by at
least a “small but significant non-transitory increase in price” (SSNIP), which we take to be 5%.
38The antitrust R package contains the computer code needed to calibrate and simulate the effects of
mergers in a range of competitive scenarios, including the ones described here.
39We compute the post-merger HHI for vertical mergers by calculating the merged firms’ market share as
the sum of all the shares of downstream products that either incorporate the upstream partner’s input or
are sold by the downstream partner.
40HHI thresholds are discussed in the 2010 Horizontal Merger Guidelines, Section 5.3.
18
to a different type of merger. The blue box and whisker plots (on the left in each pair)
depict outcomes assuming that retailers are competing in a Bertrand model, and the orange
box and whisker plots (on the right in each pair) show outcomes assuming that retailers
compete in a second score auction. The whiskers display the 5th and 95th percentiles of the
outcome distribution, the boxes denote the 25th and 75th percentiles, and the solid horizontal
line marks the median. Note that negative outcome values imply agent harm, and positive
values imply agent benefits.
We focus first on the results for consumers in the left-most panel of Figure 1. The median
change is negative, indicating harm, across all three types of mergers for both the Bertrand
and auction models. However, the distributions and magnitudes differ. In Bertrand mar-
kets, there is only a partial rank-ordering of consumer harm across different types of mergers:
consumer harm from downstream and upstream mergers each first-order stochastically dom-
inate consumer harm from vertical mergers, but do not stochastically dominate one other.
Median harm from downstream mergers is 4.5% of pre-merger total expenditures, 1.3 times
the magnitude of that from upstream mergers, and 3.1 times the magnitude of that from
vertical mergers. By contrast, under second score, consumer harm in upstream mergers
stochastically dominates consumer harm in downstream mergers, which in turn dominates
harm in vertical mergers. The median harm from upstream mergers is 3.5% of pre-merger
total expenditures, 4.1 times that of downstream mergers, and 15 times that of vertical
mergers.
Three types of mergers are almost always harmful: upstream under Bertrand and sec-
ond score competition, and downstream under Bertrand. That upstream horizontal mergers
are net harmful is to be expected, as these do not have countervailing effects according to
the model. However, the fact that downstream mergers under Bertrand are almost never
beneficial and frequently quite harmful is somewhat surprising, given the potential for coun-
tervailing bargaining leverage to lower wholesale prices. Harm is less prevalent for verti-
cal mergers and downstream mergers under second score competition. Vertical mergers in
Bertrand markets benefit consumers in about 40% of all simulations and in second score mar-
kets in 48% of all simulations, and downstream horizontal mergers in second score markets
benefit consumers in 38% of all simulations. The median consumer benefit among positive
outcomes from vertical mergers equals 4.2% of pre-merger total expenditures (comparable
to the median consumer harm from downstream mergers under Bertrand), and the median
consumer benefit from downstream mergers in auction markets equals 1.7% of pre-merger
revenues (less than two-fifths of the median consumer harm from downstream mergers under
Bertrand).
19
Turning to retailers in the second panel of Figure 1, we find that downstream mergers
and vertical mergers are beneficial, whereas upstream mergers are harmful. Although it is
unsurprising that retailers gain from downstream mergers and lose from upstream mergers,
it is interesting that the harm experienced in the latter instances is quite small in relative
terms: median retailer harm from upstream mergers is about one-quarter of the median
consumer harm from an upstream merger. This, along with the narrow inter-quartile range
of retailer harm (less than 1% of pre-merger revenues) indicates that retailers are passing on
the bulk of the wholesale price increase to consumers.41
As for wholesaler surplus, which appears in the third panel, the effects seen there are
the mirror image of those for retailers, with net harm in almost all downstream mergers
and the majority of vertical mergers, and net benefits in all upstream mergers. Although
one would expect wholesalers to be harmed by downstream mergers, the amount of harm in
Bertrand markets is about one-third of that in second score markets. This difference partially
explains the difference in the magnitude of consumer harm between downstream mergers with
Bertrand versus auctions: namely, retailers’ greater ability to extract wholesaler surplus in
second score auction markets translates into more savings passed thru to consumers, which
offsets the harm from less retail competition.
Whereas wholesalers benefit from vertical mergers in about 17% of simulations, retailers
benefit from vertical mergers in about 99% of simulations. Moreover, the median benefit to
retailers is 7.5% of pre-merger expenditures, comparable to the median loss to wholesalers of
-9%. Together, these observations indicate that vertical mergers are largely a rent transfer
from wholesalers to retailers and consumers, with retailers keeping most of the gain.
In terms of total welfare, with the exception of second score downstream mergers, our
simulated mergers are typically net harmful, with second score vertical mergers yielding
the highest median total harm (-2.5% of pre-merger expenditures), followed by Bertrand
downstream mergers (-2.2%), Bertrand vertical mergers (-1.5%), and then upstream mergers.
Downstream mergers under auctions are slightly beneficial at the median, with a gain of 0.5%
of pre-merger expenditures. About 16% of Bertrand vertical mergers and 7% of second score
vertical mergers are net beneficial.
41This phenomena may also be seen by noting that the distribution of wholesaler gain from upstream
mergers is similar in magnitude to the distribution of consumer losses from upstream mergers, though less
skewed.
20
Downstream Horizontal Mergers
We now turn to a closer examination of the welfare effects of the two types of mergers
with the potential for countervailing harmful and beneficial effects; we discuss downstream
horizontal mergers here and vertical mergers in the next subsection. In the following figures,
we focus on the impact of two variables: changes in the number of firms and in relative
bargaining power. We provide analogous figures for upstream mergers, which do not involve
countervailing effects, in the Appendix.
We begin with Figure 2, which examines how the results change in terms of the number
of retailers. The four panels mirror those in Figure 1, with one each for consumers, retailers,
wholesalers, and the market as a whole. The horizontal axis varies the number of pre-merger
retailers that are present.
A key difference between the Bertrand and second score auction models is in how retail
market shares are determined. Although the shares have the same mathematical form, in
the Bertrand model, prices enter into shares, whereas in the auction, it is marginal costs that
enter. This distinction means that under second score, the impact of a horizontal merger
(ignoring changes in marginal costs and, for the moment, any effects stemming from upstream
bargaining over inputs) is observed only in realized prices, whereas market shares remain as
they were pre-merger. The merging retailers internalize the reduction in retail competition by
withholding a product only when they are the winner and runner-up for a buyer, which leaves
the shares of other firms unchanged. Meanwhile, with Bertrand competition, a horizontal
merger causes the merged firms to raise prices and simultaneously lose share. Their shrinking
shares mitigate the improved bargaining leverage that the merger creates in the upstream
negotiation, because it improves the likelihood of recapture for wholesalers selling to non-
merging firms. Therefore, we should expect that retailers competing via an auction would
experience a larger increase in bargaining leverage due to a downstream horizontal merger.
Furthermore, the structure of competition under a second score auction compared to
that under Bertrand typically leads to higher pass-through of any improvements in marginal
cost achieved through increased bargaining leverage. With the auction, the merging firms
lower their bids by the full amount of the marginal cost decrease, which then has a direct
impact on their combined market share. For Bertrand under logit demand (and indeed, for
a number of common demand systems), the merging firms only pass through a portion of
the cost decrease, which then attenuates their resulting increase in share.
The distinctions between the Bertrand and second score models are particularly apparent
in the effects on wholesaler surplus, seen in the third panel of Figure 2. In all cases wholesalers
21
are harmed due to the shift in bargaining leverage, but the median harm to wholesalers in
second score auctions is 2.4%, more than three times the median harm to wholesalers under
Bertrand. Moreover, the difference in median wholesaler harm between second score and
Bertrand markets increases as the number of retailers decreases, with mergers to monopoly
in second score markets reducing median wholesaler surplus by 4.9%, about 10.4 times the
corresponding median harm to wholesalers under Bertrand. As the number of retailers falls,
the merging retailers in second score markets are able to better extract more surplus from
wholesalers than merging retailers under Bertrand.
Moving to the impact on retailer surplus, in the second panel we see that retailers benefit
in all instances. These gains tend to be somewhat larger in absolute magnitude than the
losses experienced by wholesalers, which is due to the additional surplus retailers can extract
from consumers with the decrease in downstream competition. The effects tend toward zero
as more retailers are added.
In terms of the net impact on consumers, the first panel shows that Bertrand mergers are
always harmful to consumers in the 5th to 95th percentile range of the simulations. Moreover,
consumer harm decreases exponentially as additional retailers are added, with median con-
sumer harm decreasing by 3.75 times as the number of retailers increases from two to four.
By contrast, consumers benefit in some instances under a second score auction, particularly
when there are few retailers. With two retailers, about 38% of the simulated mergers with
auctions are harmful to consumers, whereas with four retailers about 79% are harmful. This
is due in part to the larger cost pass-through for second score auctions, combined with the
better ability of the merged firms to negotiate better input prices, particularly when there
are few retailers.
Turning to total surplus in the last panel, we see that Bertrand downstream mergers are
always net harmful, whereas second score mergers are nearly always net beneficial between
the 5th to 95th percentiles of our simulations. For both downstream frameworks, the net
effect diminishes with additional retailers, though less quickly for Bertrand markets than
second score markets: with four firms the median change in total surplus under Bertrand is
-3.4%, whereas under second score it is about 0.5%.
We shift to an examination of how bargaining power affects the results in Figure 3.
The horizontal axis tracks the ratio of wholesaler to retailer bargaining power, (1 λ),
meaning that retailers have an increasing advantage as we move to the right. The vertical
dotted line denotes the point where retailers and wholesalers have equal power. The form
of the bargaining first order condition in equation (4) implies that the ratio of bargaining
powers equals the ratio of wholesaler to retailer GFT, which in turn are functions of margins.
22
When wholesalers have higher margins than retailers, the calibration routine will typically
find that wholesalers have higher relative bargaining power.
The effect that bargaining power has on the ability of retailers to extract wholesaler
surplus is evident in the third panel of Figure 3. As we also saw in the previous figure, the
downstream merger under Bertrand competition does not greatly increase retailers’ ability
to bargain for better terms. With second score competition, retailers extract more surplus,
especially when their bargaining power is low. In instances where retailer bargaining power
is high, the retailers would have extracted much of the available surplus even before the
merger, so the change from the merger is smaller. A mirror image of this pattern can be
seen in retailer surplus, shown in the second panel of the figure. We see that, particularly
for second score auctions, mergers where retailers have more bargaining power are unlikely
to reduce wholesale prices by much, but allow retailers to capitalize on lost downstream
competition, on net increasing retailer surplus.
We next turn to the impact on consumers. As we saw in the previous figure, consumers
lose in the Bertrand world. With the second score auction model, consumers are more likely
to benefit when wholesaler bargaining power is relatively higher. We find that consumers
gain in the majority of second score simulations when the bargaining power ratio is 3/2
or greater, and in at least a quarter of all simulations when the bargaining power ratio is
2/3 or greater. Once the bargaining power ratio falls to 3/7, consumers lose in virtually
all simulations. Thus, the likelihood that harm will occur under second score depends on
whether one believes retailers have equal or better bargaining power than wholesalers. Once
the bargaining power ratio falls below 1, our simulations indicate that harm is more likely
than not. Because in our calibrations the ratio of pre-merger wholesaler to retailer dollar
margins correlates with bargaining power, more retailer power implies higher pre-merger
retailer margins in relative terms.
As for total welfare, Bertrand downstream mergers are always harmful to market par-
ticipants, whereas second score mergers are typically net beneficial to market participants
even when the bargaining power ratio is 4 or greater. Increasing retailer bargaining power
increases the total harm under Bertrand, and decreases the total benefit under second score.
Taken together, Figures 2 and 3 imply that downstream horizontal mergers are more likely
to benefit consumers when the number of retailers is low or the retailer has relatively less
bargaining power, but only under second score auctions. Bertrand mergers always generate
consumer harm within the 5th to 95th percentile range of our simulations, with the most harm
occurring when the number of retailers is low or the retailer has relatively more bargaining
power. Thus, harm indicia like firm counts and margin ratios can be informative, but only
23
when it is possible to determine how retailers interact strategically downstream.
Vertical Mergers
Figure 4 examines how variation in the number of retailers affects the outcomes of vertical
mergers. The greatest changes in welfare appear in the second and third panels. In the third
panel, wholesalers are harmed in the large majority of simulations. This negative effect is
due to the drop in wholesale prices because of EDM and the increased recapture leverage on
the part of the merged retailer. In the second panel, we see that retailers benefit in nearly
all simulations. This positive effect means that the losses for some retailers due to RRC are
dominated by the gains to the merged retailer from RRC and EDM. The impact on both
retailers and wholesalers tends towards zero as the number of retailers increases, although
the results are quite noisy. Unlike what we saw with downstream horizontal mergers, here
there are not stark differences between the effects under Bertrand versus second score. In
a number of instances the results for second score are larger, but the distributions for both
models are quite wide and substantially overlap.
Moving on to the effects felt by consumers, the first panel of the figure does not show a
clear pattern as the number of retailers varies. Regardless of the number of retailers or the
type of downstream competition model, the distribution of effects straddles the zero axis.
Thus, the number of retailers is a poor predictor of whether consumers lose or gain from
a vertical merger. Similarly, varying the number of retailers has almost no impact on the
distribution of total surplus. In most cases we find modest total welfare losses, typically less
than 5% of pre-merger expenditures.
Figure 5 examines the results in terms of variation in the number of wholesalers. It is
largely similar to the previous figure for number of retailers. The two noticeable differences
are that increasing the number of wholesalers more clearly decreases the median harm to
consumers and shrinks the inter-quartile range compared to increasing the number of re-
tailers, particularly when the number of wholesalers increases from 2 to 4. However, the
number of wholesalers is not an overall good indicator of whether consumers gain or lose
from a merger, as the distribution of surplus still straddles the zero axis close to the median
in most other cases.
Our finding that the number of wholesalers or retailers has limited predictive power for
the welfare impacts of vertical mergers is particularly interesting given the long tradition
of using the number of firms, market shares, and market concentration as guides for the
24
likely effects of horizontal mergers.42 Our results here suggest that similar indicators are
not informative for vertical mergers. As an additional check, we also generated versions
of Figures 4 and 5 by varying the number of effective equally sized firms, given by 10,000
divided by the HHI.43 Those additional figures are highly similar to the ones presented here.
This analysis indicates that rules of thumb for the likelihood or extent of harm from vertical
mergers based on the number of firms or market concentration may not be economically
meaningful.44,45
We now examine the effect that changing bargaining power has on the results of vertical
mergers. Figure 6 graphs surplus relative to the ratio of wholesaler to retailer bargaining
power, (1 λ). Compared to the previous two figures, these results are less noisy and
exhibit stronger patterns. As relative retailer bargaining power increases, median retailer
surplus falls (second panel) whereas median wholesaler surplus rises (third panel). These
changes occur because higher retailer bargaining power lessens the benefit to EDM, as re-
tailers are able to impose low wholesale prices even before the merger.
In terms of consumer welfare, the first panel of this figure indicates that when wholesalers
and retailers have relatively equal bargaining power (at the dotted line), vertical mergers
tend to be close to neutral, with second score mergers exhibiting slightly greater benefits than
Bertrand mergers. Increasing retailer relative bargaining power from parity raises consumer
harm, whereas increasing wholesaler relative bargaining power from parity raises consumer
benefit. The simulations indicate that vertical mergers in Bertrand markets are typically
harmful to consumers when relative bargaining power ratio is less than 2/3, and are always
harmful in the 5th to 95th percentile range for both Bertrand and second score markets when
relative bargaining power is less than 3/7. Therefore, we find that the ratio of wholesale to
retail bargaining power is a good predictor of harm for vertical mergers, especially compared
to the number of firms or concentration.
We also created additional versions of Figure 6 that fix the relative bargaining power of
all non-merging retailer-wholesaler pairs, and only allow the ratio for the merged retailer and
wholesaler to vary, and vice versa.46 In our model, each vertical merger balances EDM, RRC,
42For example, see Section 5 of the 2010 DOJ/FTC Horizontal Merger Guidelines. Slade (2020) advo-
cates extending these measures to vertical transactions as part of updating the DOJ/FTC Vertical Merger
Guidelines.
43These figures are available upon request from the authors.
44The Vertical Merger Guidelines do not contain screens based on concentration. The January 2020 draft
says that cases where the merging firms account for less than 20% of both the upstream and downstream
markets are unlikely to be challenged. This statement was removed from the final version.
45We checked whether a 20% threshold was a useful indicator in our simulations and found that condi-
tioning on it had little effect on the distribution of results.
46These figures are available upon request from the authors.
25
and other effects. Although making clean comparisons is therefore difficult, our purpose
here was to focus on cases where EDM was likely to be less and contrast them with other
situations. Holding all else equal, as the bargaining power of the merging retailer relative
to the merging wholesaler rises, the merging retailer’s pre-merger input prices from this
wholesaler are lower, and the scope for EDM is less. We found that changing the ratio of
bargaining power for the merging firms generated variation in consumer harm similar to
that seen in Figure 6, whereas the bargaining power involving non-merging firms generated
a much weaker pattern. This suggests that the presence of EDM between the merging firms
is an important determinant for whether there is consumer harm. These instances that have
a lower probability of substantial EDM appear to correlate with increased consumer harm.
In our model, relative bargaining power equals the ratio of pre-merger GFT, which in
our simulations is highly correlated with the ratio of pre-merger wholesaler dollar margins to
retailer dollar margins. Specifically, the correlation between (1λ)and the share-weighted
average of the ratio of wholesale margins to retailer margins is 0.91. A plot in terms of the
ratio of margins looks highly similar to Figure 6.47 Thus, information on margins can greatly
help practitioners attempting to measure the likely impact of a potential merger. Higher
relative wholesale margins are indicators that wholesalers have been able to extract more
surplus versus retailers, which speaks directly to the distribution of bargaining power.
Variations and Robustness
Here we compare our results to those from two alternative frameworks: (1) using nested
logit demand instead of logit and (2) modeling downstream horizontal mergers ignoring any
upstream bargaining effects. See the Appendix for results pertaining to simulating upstream
mergers when ignoring downstream retail competition.
Logit demand makes fairly strong assumptions regarding consumer substitution: namely,
that when consumers substitute to another product, they are more likely to choose a prod-
uct with a large market share, regardless of how similar it is to their current choice. This
phenomenon is often called “substitution according to share” or the Independence of Irrel-
evant Alternatives (IIA) property. To determine how sensitive our results are to this issue,
we also performed simulations using nested logit demand. The nested logit introduces a
nesting parameter, σ, which allows preferences for products in the same pre-defined nest to
be correlated and therefore closer substitutes. When σ= 0, the model is identical to the flat
logit. We provide additional mathematical details in the Appendix.
47Loudermilk and Taragin (2020) include such plots.
26
We simulate markets in the same manner done in our logit results, but set the nesting
parameter equal to either 0.3 or 0.6. At σ= 0.3 we create a moderate departure from
substitution according to share, whereas at 0.6 the departure is substantial.48 In order to
focus on instances that are more likely to result in impacts on competition, we place the
merging firms in their own nest. That is, for downstream horizontal mergers we assume
that all products sold by the merging retailers are in the same nest, for upstream horizontal
mergers we assume that all products sold by the merging wholesalers are in the same nest,
and for vertical mergers we assume that all products sold by either the merging retailer or the
merging wholesaler are in the same nest. All told, this allows us to examine the importance
of increasing the nesting parameter across 4.8 million simulated markets.
Figure 7 summarizes our findings, in a similar format to that seen in the previous figures.
The three panels on the left provide results for consumer welfare, and the three panels on the
right provide results for total welfare. The horizontal axis graphs σ, which increases from 0
to 0.6 from left to right.
The figure shows that for all types of mergers, as the merging firms become closer sub-
stitutes, the effects on welfare are magnified. In the case of consumers, harm increases by
a substantial amount. Raising the nesting parameter from 0 to 0.6 increases median harm
across both upstream and downstream mergers from 2.9% to 19.4%, and causes the median
outcome in vertical mergers to increase from slightly harmful (.75%) to substantially harmful
(14.6%). Moreover, the distributions tend to become more skewed as the nesting parameter
rises from 0 to 0.6. Overall, these results indicate that as the products sold by the merging
firms become closer substitutes, consumers harm increases. This is true even for vertical
mergers.
The right side of Figure 7 depicts total welfare for both horizontal and vertical mergers.
Here, we see that increasing the nesting parameter has similar effects on total surplus as
on consumer surplus, in that as σincreases, the effects move farther from zero. Therefore,
insofar as one believes that the products of the merging firms are closer substitutes than
indicated by market shares alone, our model would tend to under-predict the merger’s effect
on welfare. This feature could be helpful to antitrust practitioners in cases where our model
already shows a substantial negative impact on consumers, as then one could argue that
accounting for additional substitution would only show greater harm.
Turning to the impact of ignoring upstream bargaining when considering downstream
48In a three-product market with symmetric shares, diversion equals 0.5 when σ= 0, 0.62 when σ= 0.3
(a 24% increase), and 0.77 when σ= 0.6 (a 53% increase). In a six-product market, diversion equals 0.2
when σ= 0, 0.36 when σ= 0.3 (an 81% increase), and 0.58 when σ= 0.6 (a 190% increase).
27
mergers, we modify our existing simulations by re-solving for equilibrium post-merger out-
comes while holding fixed upstream prices. Figure 8 summarizes the differences in consumer
and total surplus changes between the full model and the partial model, expressed as a per-
centage of the full model. The first pair of box and whisker plots show consumer welfare
changes separately under Bertrand and second score auction competition, and the second
pair shows total welfare. A negative value in this figure means that the partial model predicts
a larger magnitude compared to the full model.
Focusing first on the results for the Bertrand model, here we find that there is a tendency
for the partial model to overstate the harm to consumers, but that the extent of the discrep-
ancy is modest. The effect on consumer surplus differs by -13.5% at the 25th percentile to
-0.3% at the 75th percentile, with a median of -3.6%. The impact for total welfare is close
to zero, differing by 0.02% at the 25th percentile and 4% at the 75th percentile. This result
is driven by our finding that merged retailers under Bertrand do not negotiate dramatically
better input prices, which in turn means that ignoring upstream bargaining has relatively
little effect. In cases where competition is plausibly Bertrand, this conclusion may provide
comfort to those who are not able to implement our full model.
The results are more dramatic for the second score model. Recall from Figure 1 that,
unlike in the Bertrand world, with second score auctions, we find that merged retailers can
extract meaningful wholesale discounts. Because this effect is not present in the partial
model, it overstates harm. The partial model always shows a net negative impact on con-
sumer and total welfare, whereas the full model sometimes shows a net gain. This greatly
increases the spread of the discrepancy. Change in consumer surplus differs by anywhere
from -65% at the 25th percentile and 159% at the 75th percentile, and total surplus differs
by anywhere from 0.4% at the 25th percentile and 21.1% at the 75th percentile.
Therefore, it seems that in cases where downstream competition is according to the second
score auction, accounting for the presence of upstream bargaining is important. In the next
section we discuss a specific example, the litigation over the proposed Anthem/Cigna merger,
where a second score auction model was used without incorporating upstream bargaining.
6 Example: The Anthem/Cigna Merger
In the summer of 2016, the DOJ, eleven states, and the District of Columbia sued to prevent
Anthem, the largest health insurer in the Blue Cross and Blue Shield Association, from
acquiring its rival Cigna for $54.2 billion. As part of their case, the plaintiffs argued that
28
Anthem and Cigna compete for the right to administer the health insurance plans of national
employers and that the merger would allow Anthem to raise fees on these administrative
services only (ASO) members and harm consumers. For their part, the defendants responded
that the merger would reduce Anthem and Cigna’s hospital payments by approximately $2.4
billion per year and that these payment reductions would be passed on to ASO members,
creating net benefits to consumers that the defendants claimed as efficiencies.
After a trial stretching from November 2016 to January 2017, the District Court issued
an opinion in February 2017 enjoining the merger. As part of her opinion, Judge Amy
Berman Jackson stated that the claimed medical cost savings where not merger-specific or
verifiable.49 The decision was subsequently affirmed upon appeal.
Here, we assess the plausibility of the plaintiffs’ and defendants’ arguments using three
models: (1) a second score auction model without efficiencies, which corresponds to what
the plaintiffs presented at trial; (2) a second score auction model where hospital prices are
assumed to decrease by $2.4 billion per year following the merger, which corresponds to
what the defendants presented at trial; and (3) a vertical supply chain model where insurers
bargain with a single large hospital over the price of hospital services and simultaneously
compete in a second score auction model to be an employer’s administrator, which com-
bines the plaintiffs’ and defendants’ claims into one unified framework. The third model
assumes that insurers bargain with only a single provider, which is admittedly a simplifi-
cation. Nonetheless, we feel that this model is informative because many of the anecdotal
instances of markets where insurers complain about high medical costs occur in highly con-
centrated provider markets.
Table 2 summarizes the inputs used to calibrate our demand parameters. All of these
inputs are constructed from publicly available sources, and therefore may differ somewhat
from the inputs used by the plaintiffs’ and defendants’ economists at trial, who had access to
proprietary information. Details on data sources are in the Appendix. Using these hospital
prices and margins, along with equation (4) implies a bargaining parameter for Anthem
equal to 0.83, or a relative bargaining power of about 0.2. We assume that this parameter
is the same for all insurers.
We pause for a moment to compare this example to the results from our previous simula-
tion exercises. There are four major insurers who offer an ASO product. Figure 2 indicates
that a little more than 75% of second score markets with four firms experience consumer
harm from downstream horizontal mergers, although the median harm is only 2.5% of pre-
merger total expenditures. Figure 3 indicates that simulated second score markets with a
49This is discussed in the District-level Memorandum Opinion at pages 5-6.
29
relative bargaining parameter of about 0.2 always yield harm in the 5th to 95th percentiles,
with a median harm of about 5% of pre-merger total expenditures. Meanwhile, Figure 8
suggests that ignoring the supply chain typically overstates consumer harm by about 13% at
the median, though this prediction is imprecise. Therefore, we expect that the most likely
outcome for the full model is to find net consumer harm from the merger, but of a smaller
magnitude than predicted by the plaintiffs.
Table 3 displays the firm-level effects of the Anthem/Cigna merger for each model. In the
first column, “None” refers to the standard second score auction model with no upstream
bargaining and no cost efficiencies, as was used by the plaintiffs; “Medical” takes the same
model and assumes that provider costs decrease by the amount put forward by the defen-
dants; and “Vertical” uses our full model. Here we see that the baseline second score model
predicts that Anthem’s price rises by about $20 and Cigna’s price rises by about $56 per
member annually. Absent efficiencies, the merger does not affect the equilibrium output or
prices of other insurers, because the merging firms only change their bidding strategies when
they are both first and second in an auction. By contrast, the Medical model predicts that
Cigna’s price falls by nearly $328 per member annually, and Cigna’s market share increases
by more than 47 percentage points, transforming Cigna into the dominant insurer. This
change occurs because of the large assumed annual cost decrease for Cigna, over $505 per
member. Interestingly, despite an assumed annual cost reduction of almost $85 per member,
Anthem’s price rises by about $56 dollars, almost three times the increase predicted by the
standard model. Concurrently, Anthem’s share declines by roughly 16 percentage points.
Turning to the full Vertical model, we see that both Anthem and Cigna negotiate better
hospital rates due to improved bargaining leverage, lowering input prices by approximately
$11 and $17 per member per year, respectively. These decreases in wholesale prices are much
smaller than those assumed in the Medical model. The hospital also lowers rates to other
insurers, which increases their attractiveness and thereby partially counteracts Anthem and
Cigna’s leverage. As a result, the Vertical model predicts only modest changes in output.
The medical cost savings are ultimately insufficient to outweigh the loss in competition from
the merger, resulting in a more than $10 price increase for Anthem members annually and
a more than $40 price increase for Cigna members.
Figure 9 summarizes the aggregate outcomes from each of the three models. The standard
baseline auction model (light grey) predicts that the merger would transfer $378 million
dollars of surplus from consumers to Anthem and Cigna through higher prices. Adding cost
savings via the Medical model (dark grey) causes the merger to benefit consumers by $1.48
billion, and also to increase insurer profits by $3.67 billion. By contrast, the Vertical model
30
(black) indicates that even with medical cost efficiencies from improved bargaining leverage,
the merger yields $129 million in consumer harm, reduces hospital profits by $230 million,
and increases insurer profits by $424 million. Therefore, we find that the baseline model
used by the plaintiffs overstated harm to consumers and that the Medical model used by
the defendants overstated benefits to consumers. On balance, we predict that consumers
would be harmed. Interestingly, the plaintiffs presented some evidence at trial suggesting
that Anthem and Cigna’s products may be closer substitutes than implied by their market
shares.50 If this is the case, then the nested logit analysis discussed in the previous section
would imply that the $129 million of consumer harm is a lower bound.
The plaintiffs also claimed that the acquisition would increase Anthem’s bargaining lever-
age with hospitals and physician groups, allowing Anthem to reduce payments to those en-
tities post-merger. The defendants viewed any reductions in medical costs as an efficiency,
whereas the plaintiffs viewed them as a harm. The plaintiffs supported this claim through
qualitative evidence rather than a structural simulation. However, the simulation of our full
model provides some quantitative support for this contention, in that hospital profits fall by
$230 million. In turn, the plaintiffs argued that decreased hospital revenues would lead to
harms due to decreased access and quality of medical care.51
7 Conclusion
We present a tractable merger simulation model that incorporates bargaining within a ver-
tical supply chain. We explore the properties of this model using a series of numerical sim-
ulations, and in the process identify relative bargaining power as a useful indicator of harm
in vertical mergers, and also highlight instances where accounting for the vertical structure
of a market is important. In particular, we show that ignoring the presence of upstream
bargaining overstates the extent of consumer harm from horizontal downstream mergers in
the second score auction setting. When we examine the proposed Anthem/Cigna merger,
we find that the plaintiffs’ model that does not account for upstream bargaining estimates
levels of consumer harm that are three times larger than from the full model. We also find
that the defendants assumed that their input costs would fall by far more than we observe
in our model, meaning that we predict net harm to consumers in contrast to the net benefits
the defendants claimed.
50This evidence can be seen in the demonstrative exhibit used by David Dranove while testifying on behalf
of the plaintiffs, at slide 47.
51This theory is discussed in the Anthem/Cigna Complaint at paragraph 64.
31
A fruitful area for future research would be to compare the predictions of our merger
simulation model to those from other methods, such as those based on a vertical gross up-
ward pricing pressure index (vGUPPI). These measures are developed for bargaining models
in Rogerson (2020) and for posted-price vertical mergers in Moresi and Salop (2013). Early
research in Das Varma and De Stefano (2020) and Domnenko and Sibley (2020) has identi-
fied situations where the Moresi-Salop vGUPPI departs from the results seen in structural
simulations.
32
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Table 1: Summary Statistics for Simulated Data
Merger Type Markets Variable 50% Min 25% 75% Max
All 1,782,467 Number of Wholesalers 4 2 2 6 8
Number of Retailers 4 2 2 6 8
Bargaining Power 0.6 0.1 0.4 0.8 0.9
Nesting Parameter 0 0 0 0 0
Average Upstream Price ($) 4.7 0.3 2 11 214
Average Downstream Price ($) 12 5.7 8.4 20 244
Market Elasticity -0.48 -16 -0.98 -0.31 -0.18
Upstream 660,182 Pre-Merger HHI 2,650 1,258 1,866 5,057 10,000
Post-Merger HHI 4,243 1,637 2,686 10,000 10,000
Delta HHI 1,179 0 822 2,048 5,000
Downstream 500,066 Pre-Merger HHI 2,592 1,259 1,722 5,149 10,000
Post-Merger HHI 4,270 1,643 2,568 10,000 10,000
Delta HHI 1,658 0 853 4,296 5,000
Vertical 622,219 Pre-Merger HHI 3,515 1,419 2,582 5,574 9,969
Post-Merger HHI 5,174 1,994 3,755 7,204 10,000
Delta HHI 953 21 675 1,517 4,409
Notes: This table summarizes the sample of simulated data used to perform the merger simulations in Section 5. A description of
how the data are constructed appears in Section 5 and in the Appendix. Average prices are reported pre-merger.
Table 2: Anthem/Cigna Merger Simulation Inputs
Insurer Prices ($) Subscribers (%) Margins ($) Efficiencies ($)
Anthem 4,356 39 239.58 -84.90
Cigna 11 -505.05
Aetna 15
United 30
Other 5
Notes: This table contains the data used for the Anthem/Cigna merger simulations. A de-
scription of the data sources appears in the Appendix. Efficiences are those claimed by the
defendants. Prices, margins, and efficiencies are denominated in dollars per member-year. The
market elasticity is assumed to equal 0.09.
36
Table 3: Anthem/Cigna Merger Simulation Results
Model Level Firm Price Change Output Change
($) (% Points)
None Downstream Anthem 20.03
Cigna 56.27
Medical Upstream Anthem -84.90
Cigna -505.05
Downstream Anthem 56.08 -16.09
Cigna -327.83 47.25
United -21.95 -18.70
Aetna -9.39 -9.35
Other -2.85 -3.12
Vertical Upstream Anthem -11.11 0.18
Cigna -17.38 0.42
United -7.92 -0.36
Aetna -7.96 -0.18
Other -7.98 -0.06
Downstream Anthem 10.51 0.18
Cigna 40.51 0.42
United -8.29 -0.36
Aetna -8.10 -0.18
Other -8.02 -0.06
Notes: This table reports the results for the Anthem/Cigna merger simulations. There
are three possible models: a downstream-only model without any efficiencies (“None”), a
downstream-only model with medical efficiencies estimated outside of the model (“Medical”),
and the full model of the vertical supply chain (“Vertical”). The model without efficiencies
results in changes to only the downstream prices of the merging firms and causes no changes in
output, so we only report results for those two prices. The Medical model assumes a decrease
in the upstream prices for Anthem and Cigna, and does not model what happens to the in-
teractions between hospitals and other insurers in the upstream market. Thus, we only report
upstream prices for the Medical model for Anthem and Cigna. Price and output changes are
reported in levels (dollars and percentage points, respectively).
37
Figure 1 The figure displays box and whisker plots summarizing the extent to which mergers affect consumer, retailer,
wholesaler, and total surplus. Each blue box (on the left in each pair) depicts the effects assuming that retailers are playing a
Bertrand pricing game, and each orange box (on the right in each pair) depicts the effects assuming that retailers are playing
a second score auction game. Whiskers depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and
75th percentiles, and the solid horizontal line depicts the median.
38
Figure 2 The figure displays box and whisker plots summarizing the extent to which mergers among two retailers affect
consumer, retailer, wholesaler, and total surplus as the number of retailers present in a market change. Each blue box (on the
left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and each orange box (on the
right in each pair) depicts the effects assuming that retailers are playing a second score auction game. Whiskers depict the 5th
and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal line depicts the
median.
39
Figure 3 The figure displays box and whisker plots summarizing the extent to which mergers among two retailers affect
consumer, retailer, wholesaler, and total surplus as the bargaining power of wholesalers relative to retailers changes. Each blue
box (on the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and each orange
box (on the right in each pair) depicts the effects assuming that retailers are playing a second score auction game. Whiskers
depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal
line depicts the median.
40
Figure 4 The figure displays box and whisker plots summarizing the extent to which vertical mergers between a wholesaler and
retailer affect consumer, retailer, wholesaler, and total surplus as the number of retailers present in a market change. Each blue
box (on the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and each orange
box (on the right in each pair) depicts the effects assuming that retailers are playing a second score auction game. Whiskers
depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal
line depicts the median.
41
Figure 5 The figure displays box and whisker plots summarizing the extent to which vertical mergers between a wholesaler
and retailer affect consumer, retailer, wholesaler, and total surplus as the number of wholesalers present in a market change.
Each blue box (on the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and
each orange box (on the right in each pair) depicts the effects assuming that retailers are playing a second score auction game.
Whiskers depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid
horizontal line depicts the median.
42
Figure 6 The figure displays box and whisker plots summarizing the extent to which vertical mergers between a wholesaler
and retailer affect consumer, retailer, wholesaler, and total surplus as the bargaining power of wholesalers relative to retailers
changes. Each blue box (on the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game,
and each orange box (on the right in each pair) depicts the effects assuming that retailers are playing a second score auction
game. Whiskers depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the
solid horizontal line depicts the median.
43
Figure 7 The figure displays box and whisker plots summarizing the extent to which chang-
ing the nesting parameter in the nested logit demand system affects consumer and total
harm for horizontal and vertical mergers. Each blue box (on the left in each pair) depicts
the effects assuming that retailers are playing a Bertrand pricing game, and each orange box
(on the right in each pair) depicts the effects assuming that retailers are playing a second
score auction game. Whiskers depict the 5th and 95th percentiles of a particular outcome,
boxes depict the 25th and 75th percentiles, and the solid horizontal line depicts the median.
44
Figure 8 The figure summarizes the extent to which failing to model the entire vertical
supply chain can impact changes in consumer and total surplus from downstream horizontal
mergers. The figure contains box and whisker plots of the gaps between the full model less
the partial model, expressed a percentage of the full model. Each blue box (on the left in
each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game,
and each orange box (on the right in each pair) depicts the effects assuming that retailers
are playing a second score auction game. Whiskers depict the 5th and 95th percentiles of a
particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal line
depicts the median.
45
Figure 9 The figure displays equilibrium market-level outcomes for the Anthem/Cigna
merger simulations. There are three different models: a downstream-only model without
any efficiencies (“None”), a downstream-only model with medical efficiencies estimated out-
side of the model (“Medical”), and the full model of the vertical supply chain (“Vertical”).
Consumer harm is measured in terms of compensating variation. “Down Producer Benefit”
refers to the change in downstream firms’ profits, and “Up Producer Benefit” refers to the
change in upstream firms’ profits. We only report results for upstream profits for the Vertical
model because the other two models do not incorporate an explicit upstream market and
are therefore silent on how upstream firms are impacted by the merger.
46
A Additional Mathematical Details
Upstream Horizontal Mergers
Assume that two wholesalers, firms wand v, merge. Then the first order condition for
wholesaler wbargaining with retailer rbecomes
[pW
rw cW
rw ]srw X
tRw\{r}
[pW
tw cW
tw]∆stw (Wr\ {w})
recapture leverage effect
z }| {
X
sRv
[pW
sv cW
sv ]∆ssv(Wr\ {w}) =
1λ
λ
[prw pW
rw cR
rw ]srw X
xWr\{w}
[prx pW
rx cR
rx]∆sr x(Wr\ {w})
.
(A.1)
An upstream horizontal merger increases a wholesaler’s bargaining leverage insofar as it is
able to recapture lost sales via its merging partner.52
Consumer Welfare for Second Score Auctions
Once we have simulated predicted post-merger prices, we can calculate the change in con-
sumer surplus using the following expression:
CS =1
α"ln(s00,pre)ln(s00,post ) + X
rRX
wWr
srw,pre EmR
rw,pre rw wins pre-merger
X
rRX
wWr
srw,post EmR
rw,post rw wins post-merger#.
(A.2)
The above equation comes from taking the expected maximum gross payoff to consumers in
equation (13), subtracting off the expected margin earned by firms, and comparing the result
pre-merger versus post-merger. This expression differs from that for Bertrand competition
(equation (8)) because, in the second score auction model, market shares are determined by
bids, not prices. The additional money that firms extract from consumers beyond the bids
must be accounted for.
52The expression for negotiations by wholesaler vis similar. The model could be extended to the case
where wholesalers wand vboth withhold their products from retailer r.
47
Nested Logit
Our nested logit formulation follows that seen in, for example, Berry (1994). We add a
parameter, σ, which measures how correlated preferences for products are within nests versus
between them. The form of the upstream negotiation remains as described in Section 2.
Downstream Bertrand Let the set of retailers be divided into a series of mutually exclusive
groups indexed by g. The set of groups is denoted by G, and the set of retailers within each
group is denoted by Rg.53 Define the following:
Dg=X
rRgX
wWR
exp((δrw αprw )/(1 σ)) gG,
where 0 σ < 1. If σ= 0, the model collapses to the logit. Then the share of product w
sold by retailer rwithin group gis
srw|g=exp((δrw αprw)/(1 σ))
Dg
,(A.3)
and the share for group goverall is
sg=D1σ
g
PfGD1σ
f
.(A.4)
Multiplying through gives the unconditional market share,
srw =exp((δrw αprw)/(1 σ))
Dσ
gPfGD1σ
f
,(A.5)
which replaces the logit share function in equation (1).
Downstream Auction The probability that a product wfrom retailer rhas the best bid
among all retailer-wholesaler pairs is given by equation (A.5), but substituting bids for retail
prices. The conditional expected margin when product wsold by retailer rwins is given by
EmR
rw rw wins=1
αPxWrsrx
ln
1sg
1 1X
xWr
srx!1σ
,(A.6)
which replaces the expression in equation (15).
53For expositional purposes, we have written the model here assuming that all products sold by a given
retailer fall in the same nest. However, this assumption can be relaxed.
48
B Additional Simulation Details and Results
Data Generation Process for the Numerical Exercises
We sample market shares for the Nretail products from a symmetric Dirichlet distribution
with a concentration parameter vector whose elements equal 1, which generates market
shares whose mean is 1/N and whose variance is (N1)/(N2(N+ 1)), and is equivalent to
a uniform distribution over the open standard N1 simplex. We assume that in the pre-
merger equilibrium, the outside option is sold by a vertically integrated firm that only offers
a single product to consumers. Pre-merger, this firm’s price is set according to the pricing
optimization condition for a single-product firm under Bertrand or under second score, as
appropriate. This firm’s price does not change as a result of the merger. Before the merger,
the outside option is assumed to have a 15% market share, earn a $5 retail margin per unit
sold, and be produced at zero marginal cost.54 The resulting simulated markets have fairly
inelastic demands, with an interquartile range for the market elasticity of -0.98 to -0.31 and
a median of -0.48.55 These elasticities vary as firm shares change. Our choice of outside good
is meant to help ensure that the resulting markets pass the Hypothetical Monopolist Test,
and are therefore well-defined antitrust markets. When we run the test assuming a 5% price
increase, about 4% of our simulated markets fail. These instances are excluded.
Once we have shares, the price coefficients, and the bargaining parameters in hand,
we can then use the upstream and downstream expressions that relate margins to shares
(equations (4) and (2) or (15), for the Bertrand and auction models, respectively) to recover
the wholesale and retail margins in levels. We then map margins to wholesale and retail
prices by setting wholesaler marginal costs equal to 25% of wholesaler pre-merger margins
and retailer marginal costs equal to 10% of pre-merger wholesale prices. Shares, pre-merger
prices (or, in the case of the second score auction model, marginal costs), and the price
coefficient are then used to impute the product-specific shifters δr w.
Once we calibrate the model parameters, we then simulate all three types of mergers as
described in the main text. We find that about 30% of our simulated horizontal mergers and
about 9% of our vertical mergers are unprofitable. These instances are excluded.
54We also ran simulations where we treated the margin as a uniformly distributed random variable over
the [$2,$9] interval and obtained similar results.
55For logit demand, the market elasticity is given by αps00, where pis the share-weighted average of
prices for the inside goods.
49
Data Sources for the Anthem/Cigna Merger Simulation
Our customer market size comes from the District Court opinion, which states that in 2016,
the 14 states where Anthem operated contained 27 million ASO Customers.56 Next, we ob-
tain firm-level ASO market shares from demonstrative slides used by the plaintiffs’ economic
expert.57 In addition, we use the large employer elasticity estimate of 0.09 in Abraham,
Feldman, and Graven (2016) to recover the outside good share.58 Finally, for the insurer
margin, we use the estimate in Dranove, Rothman, and Toniatti (2019), that Anthem earns
$239.58 per customer annually.59
Unfortunately, little information is publicly available on hospital prices and margins, so
for the purposes of this analysis we again follow Dranove, Rothman, and Toniatti (2019) and
assume that each of Anthem’s members can expect to pay $1,684 annually, yielding $556 per
member in profits. Dranove, Rothman, and Toniatti (2019) at page 677 arrive at this rule
by assuming that hospitals earn a 33% margin from Anthem, typically incur $2,000 in daily
average costs per patient, and that the average patient has 0.564 inpatient days per year
(.047 ×12). Together these figures imply that the typical Anthem enrollee costs a hospital
$1,128 annually.
Figures for Upstream Mergers
The first two figures display welfare results for upstream mergers. In contrast to downstream
horizontal and vertical mergers, an upstream horizontal merger does not exhibit countervail-
ing effects that could produce benefits for consumers. The increase in bargaining leverage
for wholesalers leads to higher input prices, which tends to increase prices downstream in
response. Thus, we expect harm to result from these mergers, so long as there are no cost
efficiencies. The third figure assesses the difference between our model and a partial one
where downstream firms are assumed to fully pass on any input cost changes.
56This appears in the District-level Memorandum Opinion at page 59.
57We reference the demonstrative exhibit used by David Dranove while testifying on behalf of the plaintiffs,
at slide 34.
58This source is cited on slide 16 in the demonstrative exhibit used by David Dranove during his testimony.
59Like Dranove, Rothman, and Toniatti (2019), we proxy for Anthem’s large-group premium by observing
that Anthem charges small-group customers $4,356 per year, earning Anthem a 5.5% margin. Comparable
estimates are arrived at in Froeb, Mares, Tschantz, and Taragin (2018), footnote 5.
50
Figure B.1 The figure displays box and whisker plots summarizing the extent to which mergers among two wholesalers affect
consumer, retailer, wholesaler, and total surplus as the number of wholesalers present in a market change. Each blue box (on
the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and each orange box (on
the right in each pair) depicts the effects assuming that retailers are playing a second score auction game. Whiskers depict the
5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal line depicts
the median.
51
Figure B.2 The figure displays box and whisker plots summarizing the extent to which mergers among two wholesalers affect
consumer, retailer, wholesaler, and total surplus as the bargaining power of wholesalers relative to retailers changes. Each blue
box (on the left in each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game, and each orange
box (on the right in each pair) depicts the effects assuming that retailers are playing a second score auction game. Whiskers
depict the 5th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal
line depicts the median.
52
Figure B.3 The figure summarizes the extent to which failing to model the entire vertical
supply chain can impact changes in consumer and total surplus from upstream horizontal
mergers. The figure contains box and whisker plots of the gaps between the full model less
the partial model, expressed a percentage of the full model. Each blue box (on the left in
each pair) depicts the effects assuming that retailers are playing a Bertrand pricing game,
and each orange box (on the right in each pair) depicts the effects assuming that retailers
are playing a second score auction game. Whiskers depict the 5th and 95th percentiles of a
particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal line
depicts the median.
53
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