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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 + Pt∈RPx∈Wtexp(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=Pw∈Wr[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

x∈Wr

[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=Pr∈Rw[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

(πr−dr(Wr\ {w}))λ(πw−dw(Rw\ {r}))1−λ,(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}) = Px∈Wr\{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}) = Pt∈Rw\{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

t∈Rw\{r}

[pW

tw −cW

tw]∆stw (Wr\ {w}) =

1−λ

λ

[prw −pW

rw −cR

rw ]srw −X

x∈Wr\{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)/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 ;∀r∈R,∀w∈W}, retail prices, {prw ;∀r∈R,∀w∈W}, 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 ;∀r∈R,∀w∈W}, 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 ;∀r∈R,∀w∈W}, 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 ;∀r∈R,∀v∈W\ {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

1−srw .(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

x∈Wr

[prx −pW

rx −cR

rx]∂srx

∂prw

+srw +X

v∈Ws

[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

t∈Rw\{r}

[pW

tw −cW

tw]∆stw (Wr\ {w}) =

1−λ

λ

[prw −pW

rw −cR

rw ]srw −X

x∈Wr\{w}

[prx −pW

rx −cR

rx]∆sr x(Wr\ {w})

−X

v∈Ws

[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

x∈Wr\{w}

[prx −pW

rx −cR

rx]∂srx

∂prw

+srw +

EDM eﬀect

z }| {

[prw −cW

rw −cR

rw ]∂sr w

∂prw

+X

t∈Rw\{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

t∈Rw\{r,s}

[pW

tw −cW

tw]∆stw (Ws\ {w})

−

RRC eﬀect

z }| {

[prw −cW

rw −cR

rw ]∆srw (Ws\ {w})−X

x∈Wr\{w}

[prx −pW

rx −cR

rx]∆sr x(Ws\ {w}) =

1−λ

λ

[psw −pW

sw −cR

sw]ssw −X

v∈Ws\{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

s∈Rv\{r}

[pW

sv −cW

sv ]∆ssv(Wr\ {v}) =

1−λ

λ

[prv −pW

rv −cR

rv ]srv −X

x∈Wr\{w,v}

[prx −pW

rx −cR

rx]∆sr x(Wr\ {v})

−[prw −cW

rw −cR

rw ]∆srw (Wr\ {v})

| {z }

EDM eﬀect

−X

t∈Rw\{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 + Pt∈RPx∈Wtexp(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

t∈RX

x∈Wt

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 w∈Wrto 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

s∈R\{r},v∈Ws{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

αPx∈Wrsrx

ln 1−X

x∈Wr

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}Px∈Wtstx

ln

1−X

t∈{r,s}X

x∈Wt

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)/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)/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)/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 (1−λ)/λ 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

References

Abraham, J. M., R. Feldman, and P. Graven (2016). Employers’ Changing Economic

Incentives to Oﬀer Health Insurance under the Aﬀordable Care Act. American Journal

of Health Economics 2 (3), 273–299.

Berry, S. T. (1994). Estimating Discrete-Choice Models of Product Diﬀerentiation. RAND

Journal of Economics 25 (2), 242–262.

Collard-Wexler, A., G. Gowrisankaran, and R. S. Lee (2019). “Nash-in-Nash” Bargaining:

A Microfoundation for Applied Work. Journal of Political Economy 127 (1), 163–195.

Conlon, C. T. and J. H. Mortimer (2020). Empirical Properties of Diversion Ratios. forth-

coming in RAND Journal of Economics.

Crawford, G. S., R. S. Lee, M. D. Whinston, and A. Yurukoglu (2018). The Welfare Eﬀects

of Vertical Integration in Multichannel Television Markets. Econometrica 86 (3), 891–

954.

Crawford, G. S. and A. Yurukoglu (2012). The Welfare Eﬀects of Bundling in Multichannel

Television Markets. American Economic Review 102 (2), 643–685.

Das Varma, G. and M. De Stefano (2020). Equilibrium Analysis of Vertical Mergers.

Antitrust Bul letin 65 (3), 445–458.

Domnenko, G. and D. S. Sibley (2020). Simulating Vertical Mergers and the Vertical

GUPPI Approach. Working Paper, available at SSRN: https://ssrn.com/abstract=

3606641.

Draganska, M., D. Klapper, and S. B. Villas-Boas (2010). A Larger Slice or a Larger

Pie? An Empirical Investigation of Bargaining Power in the Distribution Channel.

Marketing Science 29 (1), 57–74.

Dranove, D., D. Rothman, and D. Toniatti (2019). Up or Down? The Price Eﬀects of

Mergers of Intermediaries. Antitrust Law Journal 82 (2), 643–677.

Farrell, J. and C. Shapiro (2010). Antitrust Evaluation of Horizontal Mergers: An Eco-

nomic Alternative to Market Deﬁnition. B.E. Journal of Theoretical Economics 10 (1),

1–39.

Froeb, L. M., V. Mares, S. Tschantz, and C. Taragin (2018). The Simple Algebra of

Surplus in Private Values Open Auctions: A Nested Logit Merger Model. Economics

Bulletin 38 (4), 2304–2312.

Gaudin, G. (2018). Vertical Bargaining and Retail Competition: What Drives Counter-

vailing Power? Economic Journal 128, 2380–2413.

33

Gee, E., G. Sheu, and Y. Taylor (2020). The Eﬀects of Vertical Integration Across Up-

stream and Downstream Markets: Theory and Simulation in the US TV Market.

Working Paper.

Gowrisankaran, G., A. Nevo, and R. Town (2015). Mergers When Prices are Negotiated:

Evidence from the Hospital Industry. American Economic Review 105 (1), 172–203.

Grennan, M. (2013). Price Discrimination and Bargaining: Empirical Evidence from Med-

ical Devices. American Economic Review 103 (1), 145–177.

Ho, K. and R. S. Lee (2017). Insurer Competition in Health Care Markets. Economet-

rica 85 (2), 379–417.

Ho, K. and R. S. Lee (2019). Equilibrium Provider Networks: Bargaining and Exclusion

in Health Care Markets. American Economic Review 109 (2), 473–522.

Horn, H. and A. Wolinsky (1988). Bilateral Monopolies and Incentives for Merger. RAND

Journal of Economics 19 (3), 408–419.

Kwoka, J. and M. Slade (2020). Second Thoughts on Double Marginalization. An-

titrust 34 (2), 51–56.

Loudermilk, M. and C. Taragin (2020). On the Eﬃcacy of Tools for Vertical Merger

Analysis. Working Paper.

Miller, N. H. (2014). Modeling the Eﬀects of Mergers in Procurement. International Jour-

nal of Industrial Organization 37, 201–208.

Miller, N. H. (2017). Modeling the Eﬀects of Mergers in Procurement: Addendum. Online

Appendix, available at SSRN: https://ssrn.com/abstract=3513510.

Moresi, S. and S. C. Salop (2013). vGUPPI: Scoring Unilateral Pricing Incentives in Ver-

tical Mergers. Antitrust Law Journal 79 (1), 185–214.

Riordan, M. H. (2008). Competitive Eﬀects of Vertical Integration. In P. Buccirossi (Ed.),

Handbook of Antitrust Economics, pp. 145–182. Cambridge: MIT Press.

Riordan, M. H. and S. C. Salop (1995). Evaluating Vertical Mergers: A Post-Chicago

Approach. Antitrust Law Journal 63 (2), 513–568.

Rogerson, W. P. (2014). A Vertical Merger in the video Programming and Distribution

Industry: Comcast-NBCU. In J. John E. Kwoka and L. J. White (Eds.), The Antitrust

Revolution, 6th Edition, pp. 543–575. Cambridge: Oxford University Press.

Rogerson, W. P. (2020). Modelling and Predicting the Competitive Eﬀects of Vertical

Mergers: The Bargaining Leverage over Rivals Eﬀect. Canadian Journal of Eco-

nomics 53 (2), 407–436.

34

Sheu, G. and C. Taragin (2012). Calibrating the AIDS and Multinomial Logit Models with

Observed Product Margins. EAG Discussion Papers 12-7, Economic Analysis Group.

Sheu, G. and C. Taragin (2017). Simulating Mergers in a Vertical Supply Chain with

Bargaining. EAG Discussion Papers 17-3, Economic Analysis Group.

Slade, M. E. (2020). Vertical Mergers: A Survey of Ex Post Evidence and Ex Ante Eval-

uation Methods. forthcoming in Review of Industrial Organization.

Spulber, D. F. (2017). Complementary Monopolies and Bargaining. Journal of Law and

Economics 60 (1), 29–74.

Tschantz, S., P. Crooke, and L. Froeb (2000). Mergers in Sealed versus Oral Auctions.

International Journal of the Economics of Business 7 (2), 201–212.

Werden, G. J. and L. M. Froeb (1994). The Eﬀects of Mergers in Diﬀerentiated Prod-

ucts Industries: Logit Demand and Merger Policy. Journal of Law, Economics, and

Organization 10 (2), 407–426.

Werden, G. J. and L. M. Froeb (2008). Unilateral Competitive Eﬀects of Horizontal Merg-

ers. In P. Buccirossi (Ed.), Handbook of Antitrust Economics, pp. 43–104. Cambridge:

MIT Press.

Whinston, M. D. (2007). Antitrust Policy Toward Horizontal Mergers. In M. Armstrong

and R. Porter (Eds.), Handbook of Industrial Organization, Volume 3, pp. 2369–2440.

Amsterdam: North-Holland.

35

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

t∈Rw\{r}

[pW

tw −cW

tw]∆stw (Wr\ {w})−

recapture leverage eﬀect

z }| {

X

s∈Rv

[pW

sv −cW

sv ]∆ssv(Wr\ {w}) =

1−λ

λ

[prw −pW

rw −cR

rw ]srw −X

x∈Wr\{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

r∈RX

w∈Wr

srw,pre EmR

rw,pre rw wins pre-merger

−X

r∈RX

w∈Wr

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

r∈RgX

w∈WR

exp((δrw −αprw )/(1 −σ)) ∀g∈G,

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=D1−σ

g

Pf∈GD1−σ

f

.(A.4)

Multiplying through gives the unconditional market share,

srw =exp((δrw −αprw)/(1 −σ))

Dσ

gPf∈GD1−σ

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

αPx∈Wrsrx

ln

1−sg

1− 1−X

x∈Wr

srx!1−σ

,(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 (N−1)/(N2(N+ 1)), and is equivalent to

a uniform distribution over the open standard N−1 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