WORKING 

 PAPERS

Consequences of Model Choice in Predicting Horizontal

Merger Effects

Matthew Panhans 

Charles Taragin

WORKING PAPER NO. 348

July 2022

___________________________________________________________________________

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______________________________________________________________________________

BUREAU OF ECONOMICS

FEDERAL TRADE COMMISSION

WASHINGTON, DC 20580
Consequences of Model Choice in Predicting Horizontal

Merger Effects

Matthew Panhans and Charles Taragin

Federal Trade Commission∗

2022-07-11

Abstract

How practitioners model competition influences the predicted effects of a merger.

We consider three models that are commonly used to evaluate horizontal mergers:

Bertrand price setting, second score auction, and Nash bargaining. We first show how

these models relate to one another, and specifically that the Bertrand and second score

auction models can both be nested within a bargaining framework. Second, we show

through numerical simulations how the predicted merger effects vary with model choice.

Third, we show that two commonly used strategies for obtaining demand parameters

can yield markedly different outcomes across the three models. Finally, we show how

model and calibration strategy choices affect the magnitude of predicted harm in the

2012 Bazaarvoice/Power Reviews merger.

Keywords: bargaining models; merger simulation; horizontal markets

JEL classification: L41, L13, K21

∗The views expressed in this article are those of the authors and do not necessarily reflect those of the

Federal Trade Commission or any individual Commissioner. We thank Dave Schmidt, Nathan Petek, Matt

Leisten, Dan Hosken, and an anonymous reviewer for helpful comments. Corresponding email: mpanhans

@ftc.gov.

1
1 Introduction

Antitrust practitioners frequently analyze the effect of a proposed horizontal merger using

merger simulation.1In order to simulate a merger’s effects, practitioners must decide which

models of competition best characterize the market in question, and then decide how to

obtain demand and cost parameters required by the chosen models.

In business-to-business markets, where firms procure inputs from other firms, three models

of differentiated product competition are often used by antitrust practitioners to assess competitive interactions: Bertrand price setting, second score auction, and Nash bargaining.2

However, it is often not obvious which of the three models is best suited to a particular

business-to-business context, and often an argument can be made for the applicability of

each one.3 This article demonstrates how the choice of model can play an important role in

the predicted merger effects, even given the same observable inputs. The article also shows

how the strategy used to obtain the model inputs can influence the predicted merger effects.

The specific context we consider is a business-to-business market with a small number of

sellers and a large number of buyers, a common occurrence in many wholesale and procurement markets. Examples include: a retailer contracting for supply from a private label

product manufacturer; a restaurant contracting with a food ingredient supplier; a company

contracting for office supplies at corporate offices. In this context, we examine the horizontal

competitive effects of mergers between two sellers of differentiated goods or services. We

quantify the differences in predicted price effects resulting from a merger simulation of the

1Antitrust practitioners also employ indicia like the Herfindahl-Hirschman Index, Upward Pricing Pressure Index, and Compensating Marginal Cost Reduction to gauge merger effects. Because these indicia are

all derived from particular models of competition, the analysis presented in this paper applies to these indicia

as well.

2For example, a Bertrand price setting model was used by the plaintiffs in H&R Block (2011) and Aetna

(2017), second score auction models were used by the plaintiffs in Sysco (2015), Bazaarvoice (2014), and

Anthem (2017), while Nash bargaining models have been used by the defense in Anthem (2017).

3Bajari, McMillan, and Tadelis (2008) discuss why firms might choose auctions or negotiations for procurement. And see for example Sweeting et al. (2020) regarding the analysis of procurement of private label

breakfast cereal by retailers: “Several reasonable models that are commonly used by antitrust practitioners

may fit the procurement setting... Unfortunately... these models often generate meaningfully different results,

which they did in this matter’ ’ (p. 772).

2
same market characteristics (prices, shares, number of firms) when changing only the model

of competition.4 Assuming that customer demand is given by the logit distribution allows

us to propose a simple framework with parameters that can be calibrated with minimal data

requirements (Werden and Froeb (2002)). The model then allows for the quantification of

effects of a hypothetical merger as the change in pricing incentives when going from two

independent firms to two jointly owned firms.

We show theoretically that a Nash bargaining game can nest the Bertrand model of competition, and that the models are equivalent when buyer bargaining power is zero. We also show

that a second score bargaining model can nest the second score auction, and that again the

models are equivalent when buyer bargaining power is zero. Previous literature has noted

each of these relationships separately; we show how a general bargaining framework relates

to both a Bertrand model and a second score auction, and we show exactly how the payoffs

specified can yield a bargaining process that generalizes Bertrand price setting, and which

alternative payoff specifications will yield a generalization of the second score auction. We

then use numerical simulation to quantify the magnitude of the differences in consumer,

producer, and total welfare across the models after a merger of two sellers.

We also consider how predicted merger outcomes vary under two approaches for obtaining

demand parameters. The first approach, which we denote as the fixed demand parameters

approach, assumes that the demand parameter estimates are known. The demand parameters, prices, and market shares are used as inputs, and the firm marginal costs required

for an equilibrium to hold are then determined by the model of competition. For example, a practitioner might simulate the results of a merger using parameters from demand

estimation with appropriate exogenous variation.5 The second approach, which we call the

4The Gross Upward Pricing Pressure Index or “GUPPI’ ’ approximates the price effects from a horizontal

merger between firms playing a Bertrand Pricing Game (Farrell and Shapiro (2010)). Taragin and Loudermilk

(2019) show how the GUPPI-based on the Bertrand model can substantially over or under-predict merger

effects, depending upon the true model.

5This approach, for instance, is the primary implementation in the mergersim Stata package (Björnerstedt and Verboven (2014)). Valletti and Zenger (2021) refer to this approach as “merger simulation based

on demand estimation’ ’

3
calibrated demand parameters approach, uses as inputs the prices and market shares, and

also assumes that at least some firm costs are also observed. Then for a particular market,

and given a model of competition, this approach entails calibrating demand parameters that

are consistent with those market characteristics and the model of competition.6 We consider

both approaches to treating the demand parameters because they are both commonly used,

depending on the information available during a merger investigation.

We find that these two approaches can yield substantially different merger effects even assuming the same model of competition. Specifically, when fixing demand parameters, median

consumer harm is 39% lower in a Bargaining model than a Bertrand model, and 44% lower in

an auction relative to a Bertrand model. By contrast, when calibrating demand parameters,

median consumer harm is 15% greater in a Bargaining model than a Bertrand model, and

29% lower in an auction relative to a Bertrand model. Thus it is not only model choice that

matters, but also the decision about which data elements are taken as inputs.

The reason that model choice matters differently under each approach is because under the

fixed demand parameters approach, the same demand parameters are held fixed across all

considered models, causing calibrated pre-merger equilibrium margins and marginal costs

to vary across models. By contrast, under the calibrated demand parameters approach, the

demand parameters and inside good margins will vary. Thus, the consequences for model

choice differ across the two approaches to the demand parameters. In Section 3, we show

the theoretical relationships of model parameters across models for both the fixed demand

and calibrated demand approaches. We show that with fixed demand parameters, different

models lead to different predictions about upstream firm margins. With the calibrated

demand approach, different models leads to different price coefficients and thus different

demand elasticities.

To give a concrete illustration of the consequences of model choice, we examine the 2012

6This approach is the primary implementation in the Antitrust R package (Taragin and Sandfort (2021)).

4
merger between Bazaarvoice and Power Reviews. Treating the demand parameters as fixed

across all models, the Bertrand model yields the greatest consumer harm; simulating this

merger under other models yields consumer harm estimates that are anywhere between 26%

to 68% of the magnitude of the Bertrand model. Calibrating the demand parameters, the

Bargaining model yields slightly greater harm than the Bertrand model, but the second score

auction and second score bargaining model still yield lower harm than the Bertrand model.

Previous literature has noted, but not systematically explored, relationships among these

models. Werden and Froeb (1994) introduces the Bertrand price-setting game with Logit

demand for horizontal merger analysis. Grennan (2013) uses a bargaining model and notes

that the model generalizes the standard Bertrand-Nash price-setting model. Froeb and

Tschantz (2002) introduces the second price auction model for merger analysis, and Miller

(2014) shows how that model can be reframed as a second score auction when products are

differentiated. Miller (2014) also introduces a Nash bargaining model that can generalize

the second score auction, but does not specify how that bargaining model relates to an

analogous bargaining model that nests the Bertrand model, as we do in this paper. Miller

and Sheu (2020) describe Bertrand and second score models of competition and their use

in merger review. The present article proposes a framework that can be used to compare

outcomes across all three models, and specifies how the Bargaining model can be adjusted

to nest the Bertrand model or the second score auction. We also quantify the magnitude of

the consequence of these model choices for predicted merger effects.

2 The Bargaining Game

Denote the supplier/seller/wholesaler by w and the buyer/customer/retailer by r. Each

supplier offers one or more products that are indexed by j. Profits are denoted by Π. Πw

j

denotes the profits to supplier w only due to the sales of product j. And similarly Πr

j

is the

buyer’s utility from being supplied by product j.

5
We begin with a general form of the bargaining equation:

max

pj

[Πr

j

(pj) − Π̃r

j

]

λ

r

j

⋅ [Πw

j

(pj) − Π̃w

j

]

(1−λr

j

)

(1)

which yields a general form of first order condition:

(1 − λr

j

) ⋅ [Πr

j

(pj) − Π̃r

j

] ⋅ ∂Πw

j

∂pj

+ λr

j

⋅ (∂Πr

j

∂pj

) [Πw

j

(pj) − Π̃w

j

] = 0

Π̃r

j

denotes the utility to the buyer in the case of a breakdown in negotiation between the

buyer and the supplier. Likewise, Π̃w

j

is the payoff to the supplier in disagreement. λ

r

j

is

the bargaining power of the buyer, and in the most general formulation can vary by buyerproduct combinations; for most of this paper, however, we will restrict it to be the same

across both inside products and buyers (λ

r

j = λ, for all j, r ≠ 0).

2.1 Nash Bertrand Bargaining Model

As we shall see, a bargaining model with λ

r

j = 0 is equivalent to the Bertrand model of

competition. When λ

r

j = 1, the merger effect falls to zero.

2.1.1 Model Setup

Denote a seller n that owns a portfolio of products Wn that includes product j. cjis

the marginal cost of supplying product j. To nest the Bertrand model in this bargaining

framework, define:

• Πw

j = ∑j∈Wn

(pj − cj) ⋅ sj(p)

• Π̃w

j = ∑k∈Wn/j (pk − ck

) ̃sk

• Πr

j =

1

−α

log(1 + ∑jexp(δj + αpj))

• Π̃r

j =

1

−α

log(1 + ∑k≠j exp(δk + αpk))

6
sj =

exp δj+αpj

1+∑kδk+αpk

denotes the Logit probability that a buyer either chooses inside product j

for its supply agreement, or instead selects an outside option with p0, δ0 normalized to 0. In

this context, sjis interpreted as a probability of being chosen in any given negotiation, and

thus the profits should be interpreted as expected profits. This exact same model can be

used in other contexts where sjshould be interpreted as a share of quantity sold, and these

derivations will still hold with only a change in the interpretation of sj.

Further, let ̃sk =

sk

1−sj

denote the disagreement choice probabilities when product j is excluded from the choice set. The buyer payoffs are multiplied by 1

−α

to translate buyer utility

into dollar units, and α < 0 is the marginal utility of income. Note that this implies that:

• Πr

j − Π̃r

j =

ln(1−sj)

α

•

∂Πr

j

∂pj

= −sj(p)

•

∂Πw

j

∂pj

= (sj + (pj − cj) ⋅ (αsj(1 − sj)) − ∑k∈Wn/j(pk − ck) ⋅ αsjsk)

We then have all the necessary terms to specify the FOC from the bargaining model, which

recall from above is given by:

(1 − λr

j

) ⋅ [Πr

j

(pj) − Π̃r

j

] ⋅ ∂Πw

j

∂pj

+ λr

j

⋅ (∂Πr

j

∂pj

) [Πw

j

(pj) − Π̃w

j

] = 0

From this FOC, it can be seen that this bargaining model will nest the Bertrand solution.

When λ

r

j = 0, and the supplier has all of the bargaining power, the FOC simplifies to:

[Πr

j

(pj) − Π̃r

j

] ⋅ ∂Πw

j

∂pw

j

= 0

But in this setup, the buyer always will have some positive payoff, meaning that Πr

j −Π̃r

j > 0

will always hold. As a consequence, this FOC is equal to zero if and only if ∂Πw

j

∂pw

j

= 0, which

is exactly equal to solution from the Bertrand FOC. This must hold, even for multi-product

7
firms, because Πw

j

is equivalent to the Bertrand objective function, and Π̃w

j

is not a function

of pjand thus drops out of the partial derivative.

There are a few differences between the bargaining setup here and that of other papers such

as Draganska, Klapper, and Villas-Boas (2010), Gowrisankaran, Nevo, and Town (2015),

Ho and Lee (2017), Miller and Sheu (2020), and Sheu and Taragin (2021). While those

papers model negotiation between a set of upstream firms and a set of downstream firms,

the model in this paper can be thought of as the same model but with only a single downstream monopolist. This more limited setting allows for the model to directly have a simple

Bertrand price-setting game as the limiting model as the bargaining power shifts to the

wholesaler.7 Having a single retailer in the bargaining model shuts down the channel for

one retailer’s negotiation to affect another retailer, which is present in the other models.

Secondly, rather than explicitly model the final consumer demand for the product of the

downstream firm, we model the downstream firm (the retailer) with a reduced-form profit

function that reflects the integration of downstream consumer demand and the retailer production technology. One advantage of employing a reduced-form approach is that it has

fewer input requirements, since the downstream firm margins and final consumer demand

need not be separately estimated.

2.1.2 Logit Equations for Simulation

Plugging in the values defined above into the general formulation of the bargaining problem,

result in:

7We think this model appropriate for settings such as procurement. Although there is a monopolist

retailer in the market, there will be many procurement or bargaining events. Wholesalers in this model

make offers to maximize their expected profits, and once the preference shocks are realized, the retailer will

choose their most preferred option, and the price will be observed. Because retailers are identical (other

than the logit shock), the observed price will be the same every time a given wholesaler is chosen. Over

many such events, the shares are the frequencies with which each wholesaler is chosen.

8
max

pj

[

ln(1 − sj)

α

⋅ M]λ

r

j

⋅

⎡

⎢

⎣

(pj − cj) ⋅ sj⋅ M − ∑

k∈Wj/j

(pk − ck) ⋅

sksj

(1 − sj)

⋅ M⎤

⎥

⎦

(1−λr

j

)

(2)

The logit assumptions then allow for the derivation of the first order condition with respect

to price that we use in the merger simulations:

⎡

⎢

⎣

(pj − cj) − ∑

k∈Wj

(pk − ck) ⋅ sk

⎤

⎥

⎦

= [ λ

r

j

(1 − λr

j

)

⋅

sj

(1 − sj)

− ln(1 − sj)]−1⋅ [ln(1 − sj

)

α

] (3)

This first order condition holds for each product j, and these can be stacked and then

expressed using matrix notation. Define the margin for each product as mj ≡ pj − cj, and

define the function f(sj) as the right hand side of Equation 3. Letting Ω denote a matrix of

product ownership,

m − Ω(m ∘ s) = F(s)

This is the equation directly implemented for the simulations.

2.2 Second score bargaining Model

This section shows that a second score bargaining model can nest a second score auction

model, and specifies exactly what modifications must be made to the classic formulation

discussed above. Analogous to the relationship between the original bargaining model and

the Bertrand outcome, the second score bargaining framework when λ

r

j = 0 is equivalent to

the second score auction. The predicted effects of a merger reach zero when λ

r

j = 1.

9
2.2.1 Model Setup

Three modifications need to be made to the bargaining model specified above so that the

model can nest the second score auction solution:

1. We now assume that in disagreement, the second-best option gets the contract at cost.

As a consequence, supplier payoff in disagreement is zero, so Π̃w

j = 0.

2. Buyer payoffs are specified to reflect that the equilibrium price will be such that buyer

utility equals the surplus that could have been created from the second-best offer.

3. Shares are now a function of costs and not prices. So before, sj(p), and now sj(c).

Shares might be a function of costs and not equilibrium prices in cases when the contract

price will not directly affect downstream demand. Rather, the negotiated prices would serve

only to split a fixed amount of agreement surplus, and then the choice probabilities are a

function only of costs, with more efficient suppliers being more likely to win contracts.

In the second score bargaining problem, define:

• Πw

j = (pj − cj

) ⋅ sj(c)

• Π̃w

j = 0

• Πr

j =

1

−αE[max(δk + αpk

)|j wins] = 1

−α (δj + α ⋅ pj + γ − log(sj

))

• Π̃r

j =

1

−αE[maxk≠j(δk + αck

)|j wins]

We set Π̃w

j = 0 because we assume that in disagreement, the buyer gets the second best

option at cost. This means that the supplier will receive zero payoff in disagreement.

The buyer payoffs are specified so as to follow the payoffs in a second score auction.8In

that model, the seller’s dominant strategy is to bid their true costs, and the buyer chooses

the seller that will create the most total surplus. In this sense, this outcome of this model is

8For further details of the second score auction and why the wholesaler’s dominant strategy is to bid

their true costs, see Miller (2014).

10
efficient. The total surplus is then split between buyer and seller through the price, which is

set so that the buyer utility equals the surplus that could have been created by the secondbest seller. Thus, if the second-best option is a very similar value to the buyer as the best

option, the buyer will capture most of the surplus; if the second-best option is a distant

substitute to the best option, the seller will capture most of the surplus.

In the more general second score bargaining framework we propose here, a bargaining parameter splits the seller’s share of the surplus from the second score auction. When the seller has

all of the bargaining power (λ

r

j = 0), the surplus split and equilibrium price exactly match

the second score auction. When the buyer has all of the surplus (λ

r

j = 1), then the buyer is

able to extract all available surplus from the best option, and so there are no merger effects

because the same best option is still chosen and at the same price.

Because the price from an auction event is only observed for the winning seller, the retailer

payoffs are conditioned on a particular seller having won the auction. Πr

j

takes the form

given above as this is the expression for the buyer utility from product j, conditional on j

being the best option.

The buyer disagreement payoff Π̃r

j

reflect the value of the second best surplus, conditional

on j being the best option. However, that conditional second best surplus cannot be expressed directly in a closed-form equation. Therefore, we use an implication from the logit

distribution to express the conditional second best surplus in terms of objects which do

have a closed form solution. Specifically, it is a property of the logit distribution that

E[max(Vk

)|j wins] − E[maxk≠j(Vk)|j wins] = 1

sj

(E[max(Vk

)] − E[maxk≠j(Vk)]). Or in other

words, the difference between the conditional total surplus and the conditional second best

surplus is equal to the difference between the unconditional total surplus and the unconditional second best surplus, divided by the share of the option being conditioned on. This

means that the conditional second best surplus can be expressed in terms of other objects

for which closed form solutions are available under the Type 1 extreme value distribution.

11
In particular:

E[max

k≠j

(δk + αck)|j wins] =

1

−α ((δj + α ⋅ cj + γ + log(

1

sj

)) − ( 1

∑k∈Wjsk

Emax − 1

∑k∈Wjsk

Emax̃k∉Wj))

where Emax denotes the familiar log-sum term from logit preferences, Emax ≡

log(∑jexp(δj + αcj)).

Thus the equation we use for the buyer disagreement payoff Π̃r

j

has two pieces. The first

piece is the total surplus available conditional on choice j being the best option; the second

piece is the unconditional first best surplus (Emax) minus the unconditional second best

surplus (Emax̃k∉Wj). The value Emax is the logsum term based on the logit assumption,

and Emax̃k∉Wjis the logsum term excluding product j as well as any other products in the

same portfolio as j.

Defining payoffs this way implies that

• Πr

j − Π̃r

j =

1

−α (α(pw

j − cw

j

) + ( 1

∑k∈Wjsk

Emax − 1

∑k∈Wjsk

Emax̃k∉Wj))

The remaining terms required in the FOC are then derived:

•

∂Πr

j−Π̃r

j

∂pj

= −1

•

∂Πw

j

∂pj

= sj(c)

We then have all the necessary terms to express the general bargaining FOC, reproduced

here from above:

(1 − λr

j

) ⋅ [Πr

j

(pj) − Π̃r

j

] ⋅ ∂Πw

j

∂pj

+ λr

j

⋅ (∂Πr

j

∂pj

) [Πw

j

(pj) − Π̃w

j

] = 0

12
Now when λ

r

j = 0, the FOC simplifies to

Πr

j − Π̃r

j = 0

because it is always the case that

∂Πw

j

∂pj

> 0

This gives a solution exactly equal to the second score auction FOC when λ

r

j = 0.

2.2.2 Logit Equations for Simulation

Similar to before, we make the assumption of logit demand to obtain a simple form of the

first-order condition that we use in the simulations.

We can use the general form FOC to derive an expression of the FOC in terms of margins.

To begin, first define the constant term as:

EmaxDiff ≡

Emax − Emax̃k∉Wj

∑k∈Wj

sk

Then, substituting in the objects into the FOC and some rearrangement yields the FOC

in terms of margins and model parameters. This is the equation we use directly for the

simulations.

mj =

(1 − λr

j

) ⋅ EmaxDiff

−α

(4)

3 Model relationships

For a given approach to obtaining demand parameters, the choice of model will have implications for other model parameters, such as predicted margins, costs, or price sensitivities.

Specifically, in the fixed demand parameters approach, demand parameters are assumed to

13
be known. Those demand parameters, prices, and market shares are used as inputs, and the

firm marginal costs required for an equilibrium to hold are then determined by the model

of competition. In the calibrated demand parameters approach, at least one firm’s marginal

costs need to be observed. The inputs are prices, market shares, and marginal costs. Then

for a particular market, and given a model of competition, demand parameters are calibrated

to be consistent with the inputs and the model of competition.

In this section, we show relationships of model parameters across models for both the fixed

demand parameter and calibrated demand parameter strategy. The reason that the treatment of the demand parameters matters is because how these model parameters vary across

models is different under the fixed demand parameters versus the calibrated demand parameters approach. In the next section we use numerical simulations to explore how the

differences in these model parameters affect merger outcomes.

First, we explore the relationship in the models’ predicted margins and marginal costs when

the demand parameters are fixed across models.

Proposition 3.1 (Fixed demand parameters). Assume as known and fixed the demand

parameters α, δ and market prices and shares. Let mb

j

, mnb

j

, m2a

j

, m2b

j

denote the the equilibrium pre-merger margin for a single-product firm j under the Bertrand, Nash bargaining,

second score auction, and second score bargaining models with Logit demand, respectively.

Let c

b

j

, cnb

j

, c2a

j

, c2b

j

denote the corresponding constant marginal cost for the firm. For any

observed market share 1 > sj > 0, price pj > 0, price coefficient α < 0, and bargaining

power parameter λ ∈ (0, 1),

mb

j > mnb

j

,

mb

j > m2a

j > m2b

j

and

c

b

j < cnb

j

,

c

b

j < c2a

j < c2b

j

14
Furthermore, when λ

1−λ < log(1 − sj

)

1−sj

sj

+ 1, the full ordering across models is:

mb

j > mnbj > m2aj > m2bj

,

c

b

j < cnbj < c2aj < c2bj

Proof. See appendix.

Note that with the exception of the Nash bargaining game’s margins (mnb

j

) and marginal

costs (c

nb

j

), all the inequalities hold for multi-product firms as well. Also, λ

1−λ < log(1 −

sj)

1−sj

sj

+ 1 never holds when λ ≥ 0.5.

This proposition shows that given the same demand parameters, margins will be highest in a

Bertrand model, followed by Nash bargaining, second score auction, and finally second score

auction bargaining. As we will see in the next section, this ordering of margins by magnitude

matches the average ordering of the magnitude of consumer harm in numerical simulations

under the fixed demand parameter approach. This result is important to realize because even

fixing the demand parameters, different models of competition can yield different predictions

about firm margins; if one particular model is predicting margins that contradict available

information about the industry, that might suggest that a different model of competition

may be more appropriate.

The ordering of marginal costs follows from the ordering of margins because in this approach,

observed prices are assumed to equal pre-merger equilibrium prices implied by each model.

The marginal costs will only matter for merger analysis in this approach when marginal cost

efficiencies are being evaluated.

Next, we explore the relationship between price coefficients α implied by each model when

observed margins, prices, and shares are used to infer α.

Proposition 3.2 (Calibrated demand parameters ). Assume as known and fixed the market

prices and shares, and marginal cost of at least one firm in the market. Let α

b

, αnb, α2a, α2b

denote the price coefficient α < 0 under the Bertrand, Nash bargaining, second score auction

and second score bargaining models with Logit demand, respectively. Then for any observed

15
market share 1 > sj > 0, price pj > 0, margin mj > 0, and bargaining power parameter

λ ∈ (0, 1),

α

b < αnb

,

α

b < α2a < α2b

Furthermore, when λ is large enough relative to the share to satisfy λ

1−λ >

1

sj

+

1−sj

sj

log(1−sj),

then the full ordering across models is:

α

b < αnb < α2a < α2b

Proof. See appendix.

When α approaches zero, demand in that model becomes more inelastic. Therefore, the

ordering of the magnitude of α also illustrates an important driver of differences in merger

effects across models in the calibrated demand parameters approach. In models with an

outside option that makes for a decent substitute, demand elasticity is high, and harm from

a merger in that model will be small. When demand is inelastic, the outside option is a poor

alternative for the firms in the market, and harm from a merger has scope to be much larger.

This proposition shows that even for the same inputs, the calibration of α will be in part

driven by the model of competition.

4 Numerical Simulations

We use numerical simulation to explore how merger outcomes differ across the four aforementioned models: Bertrand, second score auction, Nash bargaining, and second score bargaining.

In particular, we are interested in using these simulations to answer three questions. First,

how do consumer, producer and total surplus vary both within and across models, holding

demand parameters constant across models? Second, how does surplus vary both within and

across models when the demand parameters are calibrated to be consistent with the model?

Third, how do market outcomes vary as the bargaining power parameter vary?

16
To answer these questions, we simulate about 450,000 markets with logit demand, and then

use the data from these generated markets to simulate horizontal merger effects under the

four different models using two different demand parameter calibration strategies: the fixed

demand parameters strategy, which uses the demand parameters from the Nash Bertrand

model, and the calibrated demand parameters strategy, which calibrates demand parameters

from the equilibrium margin condition of each model. For details of the numerical simulation

exercise, see Appendix A.

Each market has three data elements: (i) prices and shares, (ii) costs, and (iii) demand

parameters. When generating the raw data for the numerical simulations, we generate all

three elements for each market using a Bertrand model, so that these elements are consistent

with each other within that competitive environment. To compare merger effects across

models, only two elements can be taken as inputs, while the third element needs to be

recovered to be consistent with the model and the two input elements. In the fixed demand

parameters approach, for each market we take the (i) prices and shares and (iii) demand

parameters as inputs, and back out (ii) costs to be consistent with the given model of

competition. In the calibrated demand parameters approach, we take (i) prices and shares

and (ii) one firm’s cost as the data inputs, and calibrate the (iii) demand parameters to be

consistent with the given model of competition.9

Although we generated the data elements with a Bertrand model, this is not a test of model

misspecification, as conditional on prices and either demand parameters or a single margin

and shares, model-specific product marginal costs are just-identified and the Bertrand, second score, and Nash bargaining models are therefore observationally equivalent pre-merger.

Accordingly, in this exercise, we take each model as true when conducting the merger simulations with that model. The only consequence of using Bertrand to generate the data

is that both the fixed demand and calibrated demand approaches will give the exact same

9There is a third approach, which is to take (ii) costs and (iii) demand parameters as inputs, and recover

(iii) prices and shares for each model. We do not consider that approach in this paper.

17
merger effects for the Bertrand model, because the data element recovered in each approach

for Bertrand is equivalent in value to the input used in the other approach.

Figure 1 displays box and whisker plots summarizing the extent to which horizontal mergers

affect consumer, producer and total surplus across the Nash bargaining, auction, and second

score bargaining models. Outcomes are expressed relative to the Bertrand outcome, as

the percent change relative to Bertrand, or Effectmodel−EffectBertrand

EffectBertrand

. The Bargaining and

second score bargaining categories show effects for models with a bargaining parameter

ranging from 0 to 1 in 0.1 increments.

Within each panel, red (left) plots depict outcomes using the fixed demand parameters

strategy, where the same demand parameters are used across all models in a given simulation.

The blue (right) plots depict outcomes using calibrated demand parameters strategy, where

the demand parameters are model-specific. Whiskers depict the 5

th and 95th percentiles of

a particular outcome, boxes depict the 25th and 75th percentiles, and the solid horizontal

line depicts the median.

The red plots in the “Consumer Harm” panel show that with fixed demand parameters, the

Bertrand model typically generates more merger harm than either the Nash bargaining, second score auction, or second score bargaining models. In particular, relative to the Bertrand

model, median consumer harm from horizontal mergers is 39% less under Bargaining, 44%

less under second score auction, and 63% less under second score bargaining. The ordering of

consumer harm across these models matches the ordering of margins predicted by each margin shown in the previous section. By contrast, the blue plots show that with model-specific

calibrated parameters, the Bertrand model generates more consumer harm than the Nash

bargaining model in 58% of markets, more consumer harm than the second score auction

model 99% of markets, and more consumer harm than than the second score bargaining

model in 26% of markets. Moreover, with model-specific calibrated demand parameters, the

inter-quartile range is more than twice as large relative to models where the same set of

18
demand parameters are used. This may be caused by, as shown in the previous section, the

calibrated price coefficient varies by model, and that appears to lead to larger variations in

merger effects than in the fixed demand approach.

Turning to the “Producer Benefit” panel, the red plots suggest that for the same demand

parameters, there is no systematic relationship between the models. Relative to Bertrand,

median producer surplus is about 15% lower under Nash bargaining, but 5% higher under

second score auction and 37% lower under second score bargaining. However, it is difficult

to know what to infer from the median as 26% of the Nash bargaining mergers have greater

surplus than Bertrand, 46% of the second score auction mergers have less surplus than

Bertrand, and 22% of second score bargaining mergers have greater surplus than Bertrand.

By contrast, the blue plots indicate that with model-specific calibrated parameters, median

producer surplus is 70% higher under the Nash bargaining model, 30% higher under the

second score auction, and 22% higher under the second score bargaining model than the

Bertrand model.

Finally, the “Total” panel reveals that under both demand parameter assumptions, Nash

bargaining models typically generate less harm. For the fixed demand parameters, median

total harm was 67% less under Nash bargaining than Bertrand, while for model-specific

demand parameters, median harm was 44% less. The -100% difference in total surplus

between the Bertrand model and both the second score auction and second score auction

bargaining models is entirely due to the fact that under these two models consumer harm

exactly equals producer benefit, implying no change in total surplus.

4.1 Bargaining Parameter

Having examined how these models differ, we next investigate the important role that the

bargaining parameter plays in the outcomes of these simulated mergers. Figure 2 displays

box and whisker plots summarizing how the value of the bargaining power affects consumer,

19
producer and total surplus in a merger for the Nash bargaining (top panel) and second score

bargaining (bottom panel) models. This figure displays the same merger simulations shown

in Figure 1 for the bargaining models, but broken out by the bargaining parameter. The

bargaining parameter for all inside goods is the value on the x-axis, while the bargaining

power parameter for the outside good is held fixed and equal to 0.5. The left panel in Figure

2 reveal that relative to the Bertrand model, median consumer harm always decreases. In

particular, note that under the Nash bargaining model with fixed demand parameters, the

bargaining power parameter is a good predictor of relative consumer harm. For example,

under the Nash bargaining model, median consumer harm when the bargaining power parameter is 0.5 is 46% lower than the harm under the Bertrand model. The same cannot be

said either for the Nash bargaining model with model-calibrated parameters or for the second

score bargaining model. For example, when the bargaining power parameters is 0.5, median

consumer harm from the Nash bargaining model with model-calibrated demand parameters

is 25% greater than the Bertrand model. Likewise, under the second score bargaining model,

consumer harm is 72% less than the harm under the Bertrand model under the fixed demand

parameters, but about 29% less than the Bertrand model with model-calibrated parameters.

The middle panel of Figure 2 reveals an analogous pattern for producer surplus. There

are, however, two major differences. First, for sufficiently low bargaining weights, relative

producer benefit is frequently greater under both bargaining models than under Bertrand.

For example, under the same demand parameters, when λ = 0.1, 51% of the Nash bargaining

markets and 56% of the second score bargaining markets yield greater surplus than Bertrand.

However, increasing λ to 0.5 reverses this trend, with surplus under Bertrand greater than

Bargaining in 78% of markets and greater than second score bargaining in 95% of markets.

A similar pattern occurs with calibrated demand parameters, though at higher values of

λ. Second, relative to consumer harm, producer benefit exhibits substantial variation even

after conditioning on bargaining weights. For example, under the same demand parameters,

at λ = 0.1, the producer surplus inter-quartile range is 10% for Nash bargaining markets

20
and 53% for second score bargaining markets, compared to the 2% consumer surplus interquartile and 8% for second score bargaining. This variation in producer surplus is largely

driven by the differences in the different models’ implied marginal costs, which persist despite

the fact that all models employ the same demand parameters.

Like consumer surplus, the total surplus under Nash bargaining increases as λ increases.

However, unlike consumer surplus, the bargaining power is a poorer predictor of total harm

than consumer harm. For example, under the Nash bargaining model with the same demand parameters, median consumer harm when the bargaining power parameter is 0.5 is

73% smaller under the Nash bargaining model than the Bertrand model. Moreover, as the

bargaining parameter increases to 0.7, consumer harm becomes 90% smaller. Because consumer harm and producer benefit exactly offset each other under the second score bargaining

model, conditioning on the bargaining parameter provides no additional information.

5 Bazaarvoice Revisited

Here, we revisit the 2012 merger between Bazaarvoice and Power Reviews, two customer

rating and reviews platforms, in order to explore how predicted merger effects can vary based

on model and calibration strategy in an actual merger investigation (see Miller (2014)).

Table 1 summarizes the quantity shares, prices, margins and bargaining parameters for the

three platforms that we use for this exercise: Bazaarvoice, Power Reviews, and an aggregated

“Self-Supply/Other” alternative. We assume that λ equals 0.5 for all firms in the bargaining

models.10

10With only one observed margin, λ is not separately identified from the demand parameters.

21
Table 1: Bazaarvoice/Power Reviews Simulation Inputs

Firm Share (%) Price ($) Margin (%) λ

Bazaarvoice 40 154,000 34 0.5

Power Reviews 28 84,080 0.5

Self/Other 32 37,183 0.5

Source: prices, shares, and margins are from Miller (2014) backup.

Figure 3 displays the simulation results for all four models. The “Fixed Parameters” panel

(left) uses the same demand parameters from all the models, specifically the parameters calibrated from assuming Bertrand competition. Here we see results that are broadly consistent

with Figures 1 and 2. The Bertrand model yields about twice the consumer harm as the

auction and bargaining models and more that three times the harm of the second score bargaining model. By contrast, the “Calibrated Parameters” panel (right), which uses demand

parameters calibrated by each model, tells a different story. Consumer harm is now greatest

under the Nash bargaining model. Moreover, consumer harm under both bargaining models is more than twice as large using model-calibrated parameters as the same parameters.

Finally, because of the assumption that λ is the same for all firms, the model-calibrated

parameters yield identical results for the second score auction and second score bargaining

models.

6 Conclusion

We have described three models of competition commonly used in merger analysis, and

clarified the relationship among the models under commonly used strategies for obtaining

demand parameter estimates. The R package ‘Antitrust: Tools for Antitrust Practitioners’

contains code to calibrate and simulate all the models discussed in this paper using either

22
calibration strategy. Further research remains to be done on how to better incorporate model

uncertainty into the merger effects estimates, as well as which contexts best suit the different

models.

Our paper generates two key sets of results of particular importance to economists using

merger simulation tools for policy analysis. The first is that for purposes of evaluating horizontal merger effects, how a practitioner models the strategic interaction between firms matters. In particular, we show that merger effects not only vary substantially across Bertrand,

auction, and bargaining models, but also within bargaining models as well. For the two

bargaining models considered, the assumed model of competition has different implications

for how a merger can affect total surplus. The second score bargaining model has the agents

bargaining over a fixed pie of surplus, while the Bertrand-based Nash bargaining model has

agents bargaining over a total surplus amount that will change with the resulting equilibrium

negotiated prices. This is because the choice probabilities in the Bertrand-based bargaining

model are functions of prices, whereas the second score bargaining model are functions of

marginal costs. Thus depending on whether the surplus is fixed, or whether the demand

and therefore surplus will change with prices, could be informative about which model is

appropriate in a given context.

The second is that the decision of how to treat the demand parameters matters a great

deal for how predicted merger effects vary across models. In the fixed demand approach, the

demand parameters, prices, and market shares are used as inputs, and the firm marginal costs

required for an equilibrium to hold are then determined by the model of competition. With

that approach, we find that the Bertrand model predicts the greatest amount of harm from

a merger, while the Nash bargaining model predicts less consumer harm. In the calibrated

demand approach, the inputs are prices, market shares, and at least one firm cost, and the

demand parameters required for an equilibrium to hold are then determined by the model

of competition. In contrast to the fixed demand approach, the calibrated demand approach

23
results in the Nash bargaining model predicting greater harm than the Bertrand model.

Moreover, while the second score auction always predicts less harm from a merger than a

Bertrand model, under the fixed demand parameter approach the second score bargaining

model predicts even less harm that it does under the calibrated demand approach. Using

calibrated demand parameters, the second score auction and second score bargaining models

predict equivalent harm.

In sum, both the choice of model of competition and the treatment of demand parameters

will affect the conclusions of a merger analysis.

24
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27
Figure 1: The figure displays box and whisker plots summarizing the extent to which

horizontal mergers affect consumer, producer and total surplus across the Bargaining, Auction, and second score bargaining models. For the Bargaining and second score bargaining

models, the effects shown reflect models with bargaining parameters ranging from 0 to 1 in

0.1 increments. Outcomes are expressed relative to the Nash Bertrand model outcome, as

(Effectmodel − EffectBertrand)/(EffectBertrand). Red (left) plots depict outcomes using

demand parameters calibrated from the Bertrand Model, while the blue (right) plots depict

outcomes using demand parameters calibrated from the corresponding model. Whiskers depict the 5

th and 95th percentiles of a particular outcome, boxes depict the 25th and 75th

percentiles, and the solid horizontal line depicts the median.

28
Figure 2: The figure displays box and whisker plots summarizing how the value of the

bargaining power parameter affects consumer, producer and total surplus across the Bargaining, and second score bargaining models. The bargaining parameter of one firm is

fixed at λ1 = 0.5, while the bargaining parameter for the other firms is given by the xaxis. Outcomes are expressed relative to the Nash Bertrand outcome, as (Effectmodel −

EffectBertrand)/(EffectBertrand). Red (left) plots depict outcomes using demand parameters calibrated from the Bertrand Model, while the blue (right) plots depict outcomes using

demand parameters calibrated from the corresponding model.Whiskers depict the 5

th and

95th percentiles of a particular outcome, boxes depict the 25th and 75th percentiles, and the

solid horizontal line depicts the median.

29
Figure 3: Simulated outcomes across models. “Fixed parameters” panel (left) depicts

simulation results using fixed demand parameters across all models. “Calibrated parameters”

panel (right) depicts simulation results using demand parameters calibrated from each model.

30
A Numerical Simulation

Here we describe our two-part strategy for numerically simulating the market data underpinning Figures 1 and 2. First, we simulate pre-merger market prices and shares from the Nash

Bertrand pricing game with Logit demand. We then use the simulated pre-merger market

characteristics to simulate merger effects for each model under both the fixed and calibrated

demand parameter strategies.

A.1 Data Generating Process (DGP)

For each simulated market, first we draw the number of firms N = {3, ..., 7} and separately

sample N + 1 market shares for both the inside and outside goods. The outside share s0is

drawn from a uniform distribution bounded between .1 and .6, while conditional inside shares

are drawn from a symmetric Dirichlet distribution with concentration parameter equal to

2.5 and then multiplied by 1 − s0to calculate the unconditional shares.11

Second, we assume that in the pre-merger state, all firms, including the outside firm, are

playing a Nash Bertrand pricing game with Logit demand, and that pre-merger, the outside

firm sets a price p0of $100, and earns a per-unit margin m0 drawn from a uniform distribution

bounded between $10 and $90.12

Third, we simulate N equilibrium pre-merger prices. To accomplish this, we use the Bertrand

Lerner condition m0 = (α(1 − s0))−1 to recover the price coefficient α, and then use the

sampled unconditional shares, the Lerner condition, and α to calculate Bertrand margins for

the N inside goods. For each inside firm, we randomly draw marginal costs from a uniform

distribution bounded between 85% and 115% of the outside firm’s marginal costs. We then

recover pre-merger equilibrium prices by adding these marginal costs to the corresponding

11When the concentration parameter equal 1, the Dirichlet samples from a uniform distribution over

an open standard K-1 simplex. In general, a concentration parameter above 1 favors dense symmetric

distributions from the simplex.

12Post-merger, the outside firm is assumed to fix its price at pre-merger levels.

31
margin.

Finally, with the price coefficient α and prices and shares, mean values δjcan be recovered

from the Logit share equation.

A.2 Calibration Strategies

Having simulated market characteristics, we then recover missing inputs for each market

using two approaches to treating demand parameters. Under the fixed demand parameters

approach (red box and whisker plots), we treat the demand parameters from the DGP as the

true underlying parameters and conduct merger simulations changing the model of competition but holding these parameters fixed. Under the calibrated demand parameters approach

(blue box and whisker plots), we instead treat the market prices, shares, and one product

margin from the DGP as observed, use those inputs to calibrate demand parameters for each

model separately, and then conduct merger simulations holding market characteristics fixed

but with demand parameters varying across models.

The nature of the two different treatments of the demand parameters means that each

approach uses different inputs from the DGP. The fixed demand parameter approach takes

as the input the prices and shares from the DGP, and the corresponding α and δjfor that

market. Model-specific marginal costs are recovered from the first-order conditions of the

corresponding model, providing the final requirement for a merger simulation exercise. In

the calibrated demand approach, the market prices and shares from the DGP and the drawn

margin of the outside good m0

are taken as the inputs. The model-specific α is recovered

from the outside firm’s equilibrium margin equation.13 Given α, the product-level Logit mean

valuations δj may be recovered from the Logit share equation and the observed shares and

prices. Finally, model-specific inside good marginal costs are recovered from observed prices

and the equilibrium margin equation, providing the required input for merger simulations

13For the second score auction model, this is m0 =

log (1−s0)

αs0

. For the Bargaining models, we assume

either equation 3 or 4 holds for the outside firm in the pre-merger state, with λ0 = 0.5.

32
under this approach.

A.2.1 Identification when calibrating demand parameters

Suppose product-level prices, margins, and shares are observed for a single time period. Then

there are 2 ∗ |J| equations available for estimation (|J| first order conditions and |J| choice

probabilities). If the bargaining weights freely varied across products, the model would be

under-identified, as there would be 2 ∗ |J| + 1 parameters to estimate with only 2 ∗ |J|

equations. If addition assumptions are made to restrict λ, then the model can be identified.

For example, if it is assumed that the bargaining parameter does not vary at all across the j

products, and λ

r

j = λ ∀j, then the model is over-identified and can be calibrated with inputs

from only one time period. In this case, there would be only |J| + 2 parameters (α, λ, and

δjfor j ∈ J), and these parameters can be calibrated using a minimum distance estimator

that stacks the J first order conditions and J choice probabilities, subject to the constraints

that λ ∈ (0, 1), α < 0.

Calibration proceeds in a fashion similar in second score bargaining, but with the first order

condition given in Equation 4. One consequence of the form of that FOC is that in the case

when the bargaining power parameter is assumed to be constant across products, then the

parameter values and therefore merger effects will be identical to the effects from the second

score auction.

A.3 Merger Simulation

For all models we simulate a merger by randomly selecting two inside products and change

the ownership matrix accordingly. We then re-solve each model for post-merger equilibrium

prices.

For each N, we simulate 100,000 markets, yielding 500,000 markets. For the Nash Bargaining

and second score bargaining models, we explore how changing the bargaining parameter for

33
all N inside firms affects merger outcomes by splitting the 500,000 markets into ten groups

of 50,000 and then assuming that the inside firms in each group have a bargaining power

parameter value between 0 and 1 in 0.1 increments.

We exclude simulated markets where any firm has negative marginal costs, the merger is

unprofitable, or where a Hypothetical Monopolist over all inside products could not impose

a 5% price increase on one of the merging parties’ products. We also exclude unconcentrated

markets (pre-merger HHI < 1,500) and markets where the merger yields small changes in

concentration (HHI changes < 100) as these markets are “unlikely to have adverse competitive effects” (2010 Horizontal Merger Guidelines). Doing so eliminates approximately 10% of

the simulated markets, yielding about 452,000 markets for our analysis. Table 2 summarizes

the HHI, change in HHI, as well as market-wide price changes, firm-specific price changes,

and welfare changes (Bertrand only). Also reported is the distribution of market elasticity

ε = αs0p where p is the share weighted average of the prices.

Table 2: Merger simulation summary. Effects are for the Bertrand model only. Consumer,

producer, and total surplus are reported as a percentage of pre-merger revenues.

Variable 50% 5% 25% 75% 95%

HHI 2,604 1,698 2,078 3,417 4,510

HHI Change 687 151 350 1,328 2,804

Industry Price Change (%) 1.9 0.3 0.8 4.2 11.7

Merging Party Price Change (%) 4.5 0.9 2.4 8.1 16.6

Consumer Harm (%) 1.9 0.3 0.9 4.2 11.3

Producer Benefit (%) 1.0 0.1 0.4 2.2 6.2

Total (%) 0.8 0.1 0.3 1.9 5.8

ε -0.9 -3.7 -1.6 -0.6 -0.3

34
B Proofs

B.1 Proof for Proposition 3.1 (Fixed demand parameters).

Fix α, δ, p, and s. Let 0 ≤ λ ≤ 1.

First, we show that c

b

j < cnbj

. From equation 3, we have that for a single-product firm the

margin can be expressed as:

(pj − cj) − (pj − cj) ⋅ sj = (pj − cj)(1 − sj) =

1

α

⋅ log(1 − sj)

− log(1 − sj) + λ

1−λ

sj

1−sj

Equivalently,

mj ≡ (pj − cj) =

1

α

⋅ log(1 − sj)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

(5)

In Bertrand, λ = 0, and so the term λ

1−λ ⋅ sj

in the denominator drops out. In a bargaining

model with any λ > 0, this term in the denominator is positive. Thus, the right-hand side

of Equation 5 is larger in the Bertrand model than in any Nash Bargaining model. Thus,

mb

j > mnb

j

, ∀λ. Since prices are fixed, c

b

j < cnb

j

, ∀λ.

Second, we show that c

2a

j < c2b

j

. From equation 4, we have that the margin can be expressed

as:

mj ≡ (pj − cj) =

(1 − λr

j

) ⋅ EmaxDiff

−α

(6)

In the second score auction, the margin is given by this equation with λ = 0. In a second

score bargaining model with any λ > 0, the margin is given directly by this equation. Since

0 < (1 − λ) < 1, and EmaxDiff is the same under both models, it must be the case that

m2a

j > m2b

j

for all λ. Since prices are fixed, this implies that c

2a

j < c2b

j

.

Finally, we show when mnb > m2a. Recall the expression for single-product firm margins in

these two models:

mnb

j =

1

α

⋅ log(1 − sj)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

(7)

35
m2a

j =

1

α

⋅ EmaxDiff = 1

α

⋅ [

Emax − Emax̃k∉Wj

sj

] (8)

And note that for a single-product firm, [Emax − Emax̃k∉Wj] = − log(1 − sj) = log(1/(1 −

sj)). To see this, note that for single-product firms:

Emax = log (

J

∑

k=0

e

δk+αpk)

Emax̃k∉Wj= log (∑k≠je

δk+αpk)

− [Emax − Emax̃k∉Wj] = log (

∑k≠j e

δk+αpk

∑

J

k=0 e

δk+αpk

)

= log (

∑

J

k=0 e

δk+αpk − eδj+αpj

∑

J

k=0 e

δk+αpk

)

= log(1 − sj)

Therefore, Emax − Emax̃k∉Wj= − log(1 − sj) = log(1/(1 − sj)).

Now to compare margins under Nash Bargaining (nb) and the second score auction (2a),

36
mnb

j > m2aj

⟺

1

α

⋅ log(1 − sj)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

>

1

α

⋅ (−1) ⋅ [Emax − Emax̃k∉Wj

sj

]

⟺

log(1 − sj)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

< (−1) ⋅ [Emax − Emax̃k∉Wj

sj

]

⟺ log(1 − sj) ⋅ sj < (−1) ⋅ [Emax − Emax̃k∉Wj] ⋅ [−(1 − sj)log(1 − sj) + λ

1 − λ ⋅ sj]

⟺ log(1 − sj) ⋅ sj < (−1) ⋅ [− log(1 − sj)] ⋅ [−(1 − sj)log(1 − sj) + λ

1 − λ ⋅ sj]

⟺ sj > −(1 − sj)log(1 − sj) + λ

1 − λ ⋅ sj

⟺ sj + (1 − sj)log(1 − sj) > λ

1 − λ ⋅ sj

⟺ 1 +

(1 − sj)

sj

log(1 − sj) > λ

1 − λ

⟺

(1 − sj)

sj

log(1 − sj) > λ

1 − λ − 1

⟺ log(1 − sj) >

sj

(1 − sj)

[

λ

1 − λ − 1]

⟺ − log(1 − sj) <

sj

(1 − sj)

[1 − λ

1 − λ]

⟺

λ

1 − λ < log(1 − sj

)

1 − sj

sj

+ 1

B.2 Proof for Proposition 3.2 (Calibrated demand parameters).

Fix pj, cj, sj. We will show under what conditions α

b < αnb < α2a < α2b will hold.

First, we show that α

b < αnb for all λ ∈ (0, 1). Because prices and costs are fixed across

models, margins are also fixed across models. Thus we have that:

37
mnb

j = mbj

⟹

1

αnb ⋅ log(1 − sj

)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

=

1

αb ⋅ log(1 − sj

)

−(1 − sj)log(1 − sj)

⟹ αb⋅ (−(1 − sj)log(1 − sj)) = αnb ⋅ (−(1 − sj)log(1 − sj) + λ

1 − λsj

)

Because λ

1−λsj > 0 This implies that α

b < αnb

.

Second, we show that α

2a < α2b for all λ ∈ (0, 1).

m2a

j = m2bj

⟹ EmaxDiff2a

−α2a =

(1 − λ)EmaxDiff2b

−α2b

⟹ α2bEmaxDiff2a = α2a(1 − λ)EmaxDiff2b

⟹ α2b[− log(1 − sj)] = α2a(1 − λ)[− log(1 − sj)]

⟹ α2b = α2a(1 − λ) ⟹ α2b > α2a

Finally, we show the condition under which α

nb < α2a

.

mnb

j = m2aj

⟹

1

αnb ⋅ log(1 − sj

)

−(1 − sj)log(1 − sj) + λ

1−λ ⋅ sj

=

− log(1 − sj)

−α2a

⟹ α2a = αnb ⋅ [−(1 − sj)log(1 − sj) + λ

1 − λ ⋅ sj]

This means that

38
α

nb < α2a

⟺ [−(1 − sj)log(1 − sj) + λ

1 − λ ⋅ sj] > 1

⟺

λ

1 − λ >

1

sj

+

1 − sj

sj

log(1 − sj)

39