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Using measures of competitive harm for optimal screening of

horizontal mergers?,??

Charles Taragina, Margaret Loudermilkb,∗

aFederal Trade Commission, Bureau of Economics, 600 Pennsylvania Ave NW, Washington DC 20580

bUS Department of Justice, Antitrust Division, Economic Analysis Group, 450 5th St. NW, Washington DC

20530

Abstract

Antitrust practitioners often use measures like the Herﬁndahl-Hirschman Index and Up-

ward Pricing Pressure in order to predict whether a horizontal merger is likely to be

anti-competitive. However, such measures suﬀer from two drawbacks. First, there is

little empirical work analyzing the accuracy of these measures in predicting competitive

eﬀects. Second, these measures either do not have associated decision rules for deter-

mining whether a horizontal merger is likely to be anti-competitive, or the relationship

between the decision rules and results from theoretical or empirical economic models is

unclear. To address these issues, this paper investigates the relationship between standard

merger enforcement screening measures and predicted competitive eﬀects in depth and

uses decision-theoretic models to develop optimal decision rules for merger enforcement

based on these measures. We ﬁnd that across a range of common strategic interactions all

measures correctly predict when a merger is likely to have little anti-competitive eﬀect. In

contrast, we ﬁnd that these measures do a poor job of predicting the magnitude of these ef-

fects when they are likely to be large. Finally, the optimal decision rules we derive depend

critically on the mix of competitive models that are believed to exist in the population of

potential mergers.

Keywords: Herﬁndahl-Hirschman Index, upward pricing pressure, merger simulation,

?The views expressed herein are entirely those of the authors and should not be purported to reﬂect those

of either the Federal Trade Commission or the US Department of Justice.

??We would like to thank Chris Adams, Yong Chao, Eric Lewis, Nathan Miller, Gloria Sheu, Phillippe

Sulger, Nathan Wilson, and participants at the 2019 International Industrial Organization Conference, DC

IO Conference, and US Department of Justice Antitrust Division seminar.

∗Corresponding author

Email addresses: ctaragin@ftc.gov (Charles Taragin), margaret.loudermilk@usdoj.gov

(Margaret Loudermilk)

Preprint submitted to Elsevier October 2, 2019

merger enforcement, unilateral eﬀects

1. Introduction

Antitrust practitioners employ a variety of measures to predict whether a horizontal

merger is likely to be anti-competitive. Some measures, like the number of signiﬁcant

competitors in the market and the combined market share of the merging ﬁrms, are based

on the documented positive correlation between these concentration measures and industry

prices across markets and over time. Other measures, like Upward Pricing Pressure (UPP)

and Compensating Marginal Cost Reduction (CMCR), are derived from non-cooperative

game theoretic models whose aim is to capture how a horizontal merger can change the in-

centives of the merging parties. One commonly used measure, the Herﬁndahl-Hirschman

Index (HHI), has not only been derived from theoretical models but has also been shown

to be positively correlated with industry prices. However, the accuracy of these measures

in predicting the the potential anti-competitive, unilateral eﬀects of horizontal mergers is

not well understood.

Accurately predicting whether a horizontal merger is likely to substantially lessen com-

petition is important. The harm to competition and consumers from not identifying poten-

tially anti-competitive mergers can be large. In addition, it is costly to investigate poten-

tially anti-competitive mergers and even more costly to stop mergers that are thought to

be anti-competitive. Nonetheless, there is little empirical work analyzing how accurately

these measures predict either the likelihood of signiﬁcant competitive eﬀects or the mag-

nitude of harm or beneﬁt from horizontal mergers. This paucity stems from two related

problems. First, for many markets, it is diﬃcult to collect the requisite information before

and after the merger in order to conduct an evaluation. Second, it is often intractable to

suﬃciently control for the myriad of factors that change in a market concurrent with the

merger.1

As a result, the majority of research on the performance of screening measures arises

from the academic literature on the structure-conduct-performance paradigm, which il-

lustrates a positive relationship between concentration measures and industry prices. In

particular, the positive correlation between HHI and and merging ﬁrm prices is well docu-

mented.2However, it is unclear to what extent an increase in the HHI causes market out-

1As an alternative, the FTC uses the case study method in its evaluation of merger remedies. For exam-

ple, “The FTC’s Merger Remedies 2006-2012: A Report of the Bureaus of Competition and Economics”

includes case studies of 50 merger settlements.

2Examples include Borenstein (1989), Manuszak and Moul (2009), Manuszak and Moul (2008), Pinkse

et al. (2002), and Schmalensee (1989).

2

comes like industry prices or welfare to change post-merger. In fact, Farrell and Shapiro

(1990) show that reliance solely on HHI can be misleading in merger analysis. Similarly,

Hosken et al. (2011) ﬁnd no consistent relationship between HHI and industry wholesale

or retail prices after two changes in ownership among San Francisco Bay Area gasoline

reﬁneries.

A separate strand of the literature addressing the eﬃcacy of screening measures fo-

cuses on merger simulation methods, arguably the primary modern tool for assessing the

predicted unilateral competitive eﬀects of horizontal mergers.3Werden and Froeb (1996)

investigates in a simulation study whether the combined share of the merging ﬁrms or post-

merger HHI are useful predictors of price and welfare eﬀects. In a diﬀerentiated product,

Nash-Bertrand setting with Logit demand, Werden and Froeb ﬁnd that both measures are at

best mediocre predictors of price and welfare eﬀects, and both measures exhibit variation

that increases with the change in market concentration. In a retrospective analysis, Haus-

man and Sidak (2007) ﬁnd no relationship between HHI and consumer prices for wireless

telephone service in the EU. Miller et al. (2017) demonstrate that UPP acts as an accurate

screen with log-concave demand speciﬁcations in Monte Carlo experiments but ﬁnds that

UPP understates price eﬀects under more convex demand assumptions. Similarly, Che-

ung (2016) used a cross-section of data on airline market routes to compare hypothetical

horizontal merger predictions between UPP and merger simulation. Cheung found that

the correlation between the UPP and merger simulation results increased as the size of

the merging ﬁrms or cost savings from the merger decreased. Dutra and Sabarwal (2019)

investigate the accuracy of UPP as a screening measure in the presence of merger-speciﬁc

cost eﬃciencies by Monte Carlo simulation, concluding that the standard UPP formulation

substantially over predicts price eﬀects.

This paper extends the eﬀorts of this research stream, further investigating the perfor-

mance of common measures employed by antitrust practitioners using simulation meth-

ods. We simulate markets from a set of commonly used demand and supply models and

use these markets to compute the price eﬀects from a merger.4It is then simple to compare

the simulated merger eﬀects to those predicted by the screening measures.5We ﬁnd that

3Surveys of simulation methods for merger analysis can be found in Werden and Froeb (2006) and

Budzinski and Ruhmer (2009). See also Nevo and Whinston (2010) for a discussion of the use of merger

simulation in antitrust analysis as well as Peters (2006), Weinberg (2011), and Bjornerstedt and Verboven

(2016) for comparison of results from merger simulation and retrospective estimates.

4The sensitivity of merger simulation estimates to assumptions about demand and supply speciﬁcations

is well-known. See Weinberg and Hosken (2013), Crooke et al. (2003), Bass et al. (2008), Bokhari and

Mariuzzo (2018) and Knittel and Metaxoglou (2011).

5It is important to note that in our simulation study because all the models assume that market participants

are playing a non-cooperative game our analysis can only be used to assess how well these measures predict

3

across a range of common demand and supply speciﬁcations all the measures correctly

predict when a merger is likely to have little anti-competitive eﬀect. In contrast, we ﬁnd

that these measures do a poor job of predicting the magnitude of these eﬀects when they

are likely to be large.

A second serious drawback with the aforementioned measures is that many lack a de-

cision rule for determining whether a horizontal merger is likely to be anti-competitive

ex-ante. For those that do have an associated rule, the underlying theoretical or empiri-

cal economic justiﬁcation is often unclear. To remedy this, we use two decision-theoretic

models to develop optimal decision rules for merger enforcement based on these mea-

sures. We ﬁnd that the optimal decision rules depend critically on the mix of competitive

models that are believed to exist in the population of potential mergers. In general, higher

thresholds would be warranted under the belief that the population of potential mergers is

largely comprised of ﬁrms competing according to a diﬀerentiated products auction model.

We also ﬁnd that the current thresholds would be most relevant in a population with the

majority of industries engaged in Cournot competition.

Two papers are most similar in spirit to ours, each with a somewhat diﬀerent focus.

Garmon (2017) also assesses the accuracy of merger screening methods. However, Gar-

mon’s study is speciﬁc to methods used in hospital mergers and employs a retrospective

analysis. Coate (2011) compares concentration measures to the outcomes of FTC merger

challenge decisions and uses these to infer a benchmark merger screen level for UPP of

less than 15%.

The paper proceeds as follows: Section 2 summarizes the set of models, concentra-

tion measures, and data generating processes employed in the simulation study. Section 3

reports the results of these numerical simulations. Section 4 considers two optimal screen-

ing rules for initiating horizontal merger investigations, using concentration as a proxy for

anti-competitive harm, and section 5 concludes.

2. Monte Carlo Experiments

2.1. Demand and Supply Speciﬁcations

Here, we describe the game-theoretic models that underpin our numerical experiments.

These include: Cournot with quadratic costs and log-linear as well as linear demand;

Bertrand with Almost Ideal Demand (AIDS), Constant Elasticity of Substitution (CES)

demand, and Logit demand; and a second score auction with Logit demand.6We selected

the Cournot, Bertrand and 2nd score auction models for two reasons. First, versions of

unilateral merger eﬀects. Potential coordinated merger eﬀects are not addressed.

6See Appendix A for details of the demand and supply model speciﬁcations.

4

these models have been employed by both the FTC and the DOJ in publicly litigated mat-

ters. For example, in 2011 the Division used a Bertrand pricing game with linear demand

to simulate the eﬀects of a merger between HR Block and TaxAct, two ﬁrms that spe-

cialize in tax preparation. In 2017, The Division also employed a Bertrand model, this

time to simulate the eﬀects of a horizontal merger between two health insurers, Aetna

and Humana. In the same year, the Division used a 2nd score auction model to simulate

the eﬀects of a merger between Anthem and Cigna, two other large heath insurers. Fi-

nally, in 2018, the FTC used a Cournot model to simulate the eﬀects of a merger between

Tronox and Cristal, two of the largest producers of titanium dioxide. Second, while other

papers have established that for the Bertrand model, the curvature of the demand curve

can signiﬁcantly aﬀect the magnitude of the post-merger price changes, to our knowledge,

little research has been done analyzing the eﬀect that the assumed form of the suppliers’

strategic game has on post-merger price changes.

We selected the linear and log-linear demand systems for study under Cournot com-

petition because in our experience, the Cournot model is often used to simulate merger

eﬀects in homogeneous good industries and the linear and log-linear demand systems cap-

ture this feature nicely, while also allowing us to explore two extreme cases with regard

to curvature. Linear demand has no curvature (second derivatives equal 0), whereas log-

linear demand exhibits substantial curvature, particularly at high prices. Logit demand is

used in the 2nd score auction model largely for convenience and tractability but is also one

of the few functional forms previously used in published work.

For the Nash-Bertrand game, we selected the AIDS, Logit and CES demand systems.

First, like log-linear demand in Cournot, AIDS can have large second derivatives, partic-

ularly at high prices. In contrast, the Logit model and to a lesser extent the CES model

generally have smaller second derivatives. Including these three speciﬁcations allows us

to assess the eﬀect of demand curvature for a given model of supply. Second, while the

choice of demand speciﬁcation is heavily inﬂuenced by the particulars of an industry, data

limitations often inﬂuence the set of models that can be implemented in practice since the

data requirements of these models diﬀer.7

Also of interest are the demand models that we did not include, most notably log-linear

and linear demand for Bertrand. Without additional restrictions these demand systems

require estimation of a large number of parameters, and it is often the case in practice that

there is insuﬃcient data to obtain reliable estimates. Further, unlike AIDS, Logit, and CES,

without further restrictions log-linear and linear demand are not necessarily consistent with

7Typically, both AIDS and CES demand are estimated using expenditure shares while Logit demand is

estimated using quantity shares, for example.

5

the axioms of consumer choice theory8. As a result, it is diﬃcult to calculate the welfare

measures that are important in assessing the competitive eﬀects of a merger.

2.2. Measures of Competitive Harm

A number of indicia are commonly used as proxies for the anticipated competitive

eﬀects of a horizontal merger, including industry ﬁrm counts, merging party size rank,

combined party market share, HHI, UPP, and CMCR. Many of these indicia consist of two

parts: an algorithm for calculating the index and a threshold decision rule for determining

whether a horizontal merger is likely to raise competitive concerns. Here we describe the

“thought experiment” underlying each of these indicia, as well as some of their properties,

advantages and drawbacks.

Before doing so, it is important to reiterate that we are only examining the ability of

these indicia to predict relatively short run unilateral eﬀects from horizontal mergers. It is

possible that some of these measures may also be useful predictors of other adverse eﬀects

from horizontal mergers, like coordinated eﬀects. However, testing these properties would

require implementation of more sophisticated models. Such extensions are beyond the

scope of this paper.

2.2.1. Firm Count

The intuition underlying the “ﬁrm count” indicia is that a merger reduces the number

of ﬁrms in a market from Nto N−1, which would tend to raise prices and harm consumers,

ceteris paribus. The magnitude of the price eﬀect is expected be small when Nis large

and large when when Nis small. Calculating this indicia is easy: 1) Identify ﬁrms in the

market; 2) Count the ﬁrms (N); and 3) Reduce the number of ﬁrms by 1 (N−1).

Interpreting the Firm Count measure is more diﬃcult. On one hand, it seems plausible

that a merger reducing the number of ﬁrms from 100 to 99 is unlikely to yield substantial

harm. On the other hand, it is plausible to believe that a merger reducing the number

of ﬁrms from 2 to 1 yields substantial harm. But what about a 10 to 9 merger or a 5

to 4 merger? In addition, this measure assumes each ﬁrm included in the market is as

competitively signiﬁcant as every other ﬁrm (i.e., Firms are in some sense “symmetric”.),

but what if some ﬁrms have more sales than others? What if manufacturing costs vary

across ﬁrms, consumers value the attributes of each product diﬀerently, or some ﬁrms

oﬀer a portfolio of products while others only sell a single product? It seems plausible that

most if not all of these factors distinguish one ﬁrm from another in ways that render this

measure noisy.

8von Haefen (2002) describes how Slutsky Symmetry is typically violated by these demand models and

derives the parameter restrictions needed to satisfy this assumption.

6

The ﬁrm count measure also provides no guidance as to which ﬁrms should be included

in the antitrust market of interest. Rather, it assumes that the contours of the market are

clearly identiﬁed, which is rarely true in diﬀerentiated product markets. This criticism of

Firm Count is true of nearly all the measures we consider and thus is not repeated in each

of the subsequent sections.

2.2.2. Merging Party Size Rank

The “Party Rank” measure, which identiﬁes the size rank of the merging parties (e.g.

the merging parties are the 1st and 3rd largest ﬁrms in the market), addresses one criticism

of the ﬁrm count metric by partially internalizing the potentially asymmetric nature of the

merging parties. A merger reducing the number of asymmetric ﬁrms in a market from N

to N−1 may raise prices by a substantial amount, depending upon the sizes of the merging

ﬁrms. The magnitude of the price eﬀect will be small, all other things equal, when either

both merging parties are small and therefore have low rank OR one party has a high rank

and the other a low rank. Thus, Party Rank attempts to capture the diﬀerence between the

the 1st and 2nd largest ﬁrms merging out of ten from the 9th and 10th largest merging. In

this example, the Firm Count for both mergers is 10 to 9 although we would expect larger

competitive eﬀects from a merger of the market’s largest ﬁrms.

2.2.3. Merging Party Combined Share

One important drawback of both the Firm Count and Party Rank measures is that they

do not capture information about the magnitude of the size diﬀerences between ﬁrms. The

“Party Share” measure, which equals the combined market share of the merging parties,

remedies this by explicitly using information on the size of the merging parties. The

intuition behind this measure is straightforward: all else equal, mergers where the parties

have a higher combined share control more of the market post-merger and therefore have

a greater incentive to raise prices.

Aside from its intuitive appeal, the importance of this indicia has been recognized

at times by both the U.S. Courts and antitrust enforcement agencies. In United states v.

Philadelphia National Bank, the U.S. Supreme Court ruled that mergers where the com-

bined party share is greater than 30% are presumptively anti-competitive.9Moreover, the

now superseded 1992 Horizontal Merger Guidelines (HMG) published by the U.S. FTC

and DOJ indicate that if the combined party share is greater than 35%, then the agencies

will view the merger as presumptively anti-competitive. While the most recently revised

2010 HMG no longer enumerates a combined share threshold, the HMG still recognizes

the import of merging party market shares, stating “The Agencies also may consider the

9United States v. Philadelphia Nat’l Bank, 374 U.S. 321 (1963)

7

combined market share of the merging ﬁrms as an indicator of the extent to which others

in the market may not be able readily to replace competition between the merging ﬁrms

that is lost through the merger.”10

Although intuitive, the Party Share measure has two drawbacks. First, it does not dis-

tinguish between diﬀerent market compositions that could aﬀect competitive interactions

such as between markets with a few large non-merging parties versus a market with only

many small non-merging ﬁrms. Second, this measure does not provide a way to quantify

the magnitude of harm associated with a particular concentration level. This criticism is

also shared by nearly all of the measures studied.

2.2.4. Herﬁndahl-Hirschman Index

The Herﬁndahl-Hirschman Index explicitly uses information on ﬁrm size and quanti-

ﬁes the diﬀerences in market composition, remedying some weaknesses of the previous

measures. Speciﬁcally, the HHI is calculated as the sum of the squared shares of ﬁrms in

the relevant market, yielding an indicia that is between 0 (perfect competition) and 10,000

(monopoly). It can be shown algebraically that the post-merger change in the HHI, ∆HHI,

is given by 2 times the product of the merging ﬁrms’ market shares.

The HHI has a number of useful properties. First, it may be derived from a standard

Cournot quantity-setting game, one of work-horse game-theoretic models in the IO litera-

ture. Second, the HMG articulate criteria describing under what conditions a merger may

or may not be cause for concern in terms of the HHI. As shown in Figure 1 if the post-

merger HHI >2,500 and ∆HHI >200 then, according to the HMG, the merger should

be considered presumptively anti-competitive. If the post-merger HHI is >2,500 and 100

<∆HHI ≤200 OR post-merger 2,500 ≥HHI ≥1,500 and ∆HHI >100 the merger may

potentially raise signiﬁcant competitive concerns. If the post-merger HHI <1,500 or ∆

HHI ≤100 then it is considered unlikely to have adverse competitive eﬀects. A ∆HHI =

200 could result, for example, from the merger of two ﬁrms each with a 10% market share.

Another feature of the HHI is that it can be readily converted into a measure of the

eﬀective number of symmetric ﬁrms in a given market, Ne f f =10,000/HHI. Among

other things, this useful result aids in interpreting the HMG HHI thresholds since a post-

merger HHI of 2,500 implies 4 equal-sized ﬁrms, while a 1,500 HHI implies roughly 7

equal-sized ﬁrms. Therefore, loosely speaking, the HMG states that mergers resulting in 4

or fewer equal-sized ﬁrms in a market are presumptively anti-competitive, while mergers

where 7 or more ﬁrms remain in the market are unlikely to be anti-competitive. Despite

these useful properties, the HHI does not quantify how much harm is likely to be generated

at the Guidelines’ thresholds.

10U.S. Department of Justice and Federal Trade Commission (2010), p.18

8

2.2.5. Upward Pricing Pressure

Upward Pricing Pressure (UPP) attempts to directly quantify the competitive eﬀects

of a horizontal merger and is derived as a ﬁrst order approximation of the merging parties

equilibrium pricing strategy under the Nash-Bertrand pricing game. Intuitively, the UPP

measure asks ‘When a ﬁrm raises its price on a product, how much of the sales it loses

will shift to its merging partner (diversion), and how much are those sales by the merging

partner worth (margin)?’ Recapturing sales diverted from the merging partner lessens the

cost of raising price, creating additional incentive to increase price, referred to as “upward

pricing pressure”.

UPP is often quantiﬁed by the Generalized Upward Pricing Pressure Index (GUPPI).

The GUPPI is obtained by expressing the value of a ﬁrm’s sales diverted to the merging

partner as a fraction of the ﬁrm’s price. Calculating the GUPPI requires the prices of

both ﬁrms, one of the ﬁrm’s price-cost margins, and a measure of diversion between the

merging ﬁrms under the assumption that ﬁrms are competing through price setting. For

example, if Firm 1 with pre-merger price p1is acquiring Firm 2 with pre-merger price p2,

price-cost margins, m2, and diversion from ﬁrm 1 to 2, d12, then the GUPPI for Firm 1 can

be calculated as

GUPPI1=(d12)∗(m2)∗ p2

p1!

While there is no direct link between a particular value of the GUPPI and the magni-

tude of price eﬀects or other quantiﬁcations of harm resulting from a horizontal merger, a

larger number indicates a larger incentive to raise price post-merger and has been shown to

be positively correlated with post-merger party prices11. UPP has the beneﬁt of potentially

allowing comparisons to a common benchmark, and some practitioners have suggested

a “safe-harbor” level of 5-10%.12 However, each merger can result in a multiplicity of

GUPPI ﬁgures. For example, two merging ﬁrms each selling 3 products in a single market

results in 6 UPP numbers. There is no theoretical or empirical guidance regarding which

of the resulting numbers to use in evaluating the competitive eﬀects or on how to weigh

diﬀerent results across products or markets.

11See Miller et al. (2017) for further details.

12One example is found in former FTC Commissioner Joshua Wright’s dissenting statement in the Fam-

ily Dollar/Dollar Tree merger. See “Statement of Commissioner Joshua D. Wright Dissenting in Part and

Concurring in Part In the Matter of Dollar Tree, Inc. and Family Dollar Stores, Inc.”, FTC File No.

141-0207, July 13, 2015, https://www.ftc.gov/system/files/documents/public_statements/

681781/150713dollartree-jdwstmt.pdf

9

The main diﬃculty in calculating the GUPPI is obtaining reasonably accurate estimates

of the merging party’s margins, prices and diversions. While margin and price estimates

can often be obtained from accounting data, care must be taken to ensure that the margins

and prices match as closely as possible to the products that are explicitly included in the

merger simulation and the theoretical margins in the model approximated by the GUPPI.

Accurate diversion estimates can also be diﬃcult to obtain. Common sources for diversion

include: party win-loss data, bidding opportunity data, or own- and cross-price elasticities

from published industry or academic studies. Care must be taken to ensure as much as

possible that the diversion estimates reﬂect losses to a particular product due to that prod-

uct experiencing a relative price increase. Observing that customers switched between the

parties’ products, without knowing the reason for switching is only a proxy for diversion,

and the diﬀerence can have substantial eﬀects on estimates of anticompetitive harm.

2.2.6. Compensating Marginal Cost Reduction

The compensating marginal cost reduction (CMCR) considers the dual problem to

those discussed previously. Instead of asking by how much prices will rise after a merger,

CMCR can be thought of as the proportional reduction in the merging ﬁrms’ marginal costs

needed to oﬀset a price increase. CMCR indicates that the merging parties’ will lower

their product prices post-merger if anticipated marginal cost reductions are greater than

what CMCR predicts and raise their product prices post-merger if anticipated marginal

cost reductions are less than what CMCR predicts for all the parties’ products. CMCR

makes no prediction regarding post-merger price eﬀects if some of the parties’ anticipated

cost reductions are above CMCR predictions while others are below CMCR predictions.

CMCR uses the same inputs as the GUPPI and as a result, all the issues with obtaining

accurate estimates for margins, prices, and diversions apply to CMCR as well. However,

CMCR is an exact prediction of the unilateral eﬀects of a horizontal merger - not a ﬁrst

approximation like UPP. A further beneﬁt of CMCR is that the formula can be derived un-

der various forms of strategic interaction among ﬁrms, including Bertrand (Werden (1996)

) and Cournot (Froeb and Werden (1998) ) competition. For a merger between two single-

product ﬁrms under Bertrand competition with price-cost margins (mi), prices (pi), and

diversion from ﬁrm i to j (dij ), CMCR for Firm 1 is given by

CMCR1=m1d12d21 +m2d12 (p2/p1)

(1 −m1)(1 −d12d21 ).

2.3. Data Generation

In order to assess the eﬃcacy of the measures in Section 2.2, we use the models de-

scribed in Section 2.1 to simulate the eﬀects of a large number of horizontal mergers

across diﬀerent market conditions. For all simulations, we assume that in the pre-merger

10

state there are N∈ {3,4,5,6,7}single-product ﬁrms “inside” the market of interest and

an outside good. We restrict our attention to markets with fewer than 7 ﬁrms post-merger

because, given the other assumptions underlying our simulations, mergers in markets with

more ﬁrms rarely yield more than a 5% industry-wide price increase and frequently yield

price increases of less than a 1%. We restrict our attention to markets with at lease 3 ﬁrms

pre-merger to exclude the case of merger to monopoly.

The ﬁrms and the outside good are assumed to interact strategically in the pre-merger

state, but only the inside goods change their prices in the post-merger state. The single-

product ﬁrm shares are drawn from a Dirichlet distribution with all Nconcentration pa-

rameters equal to 2.5. Setting the concentration parameters in this fashion yields ﬁrm

shares with mean 1/Nand variance equal to N−1

2.5N+1. In addition, we assume that in ev-

ery simulated market, the outside ﬁrm’s product costs $6 and that the outside ﬁrm earns

a margin of $2. Finally, the outside ﬁrm’s share is drawn from a Uniform Distribution

restricted to be between 0.1 and 0.7.13 We then use the pre-merger ﬁrst-order condition

for the outside good as well as the the speciﬁed demand system to calibrate the model’s

demand parameters.

For Cournot, these assumptions are also suﬃcient to identify plant-speciﬁc cost pa-

rameters. For, Bertrand and the second score auction, however, these assumptions are not

suﬃcient to separately identify ﬁrm marginal costs and prices. To remedy this, we assume

for the Bertrand and Auction models that marginal costs are equal to $2 for all ﬁrms14.

We simulate 50,000 markets for each demand-supply model speciﬁcation described in

Section 2.1. For each market, a horizontal merger is created by randomly assigning two

ﬁrms as the merging parties. Among these two ﬁrms, the one with the largest market share

is designated as the acquirer and the other as the target ﬁrm. In all our models, a horizontal

merger is represented as an ownership change, with the merging parties’ products (or

plants) placed under common control.

Next we test that each simulated market meets the criteria to be both a valid antitrust

market, using the Hypothetical Monopolist Test (HMT) at the 5% level, and to require

pre-merger notiﬁcation under the Hart-Scott-Rodino Antitrust Improvements Act of 1976,

referred to subsequently as an HSR ﬁling. In the context of our numerical simulations, the

13We chose this range in order to to determine how sensitive merger eﬀects are to the attractiveness of the

outside good. We limit the range of outside shares to [0.1, 0.7] because our parameter calibration strategies

rely heavily on the value of the outside share, and outside shares beyond this range tended to yield more

markets with implausible model parameters (i.e. negative marginal costs for Cournot) or implausibly large

price eﬀects.

14Doing so guarantees that all ﬁrm prices and marginal costs are positive and that ﬁrms’ pre-merger prices

are not too close to zero, thereby preventing price changes expressed as a percentage of pre-merger prices

from becoming explosively large.

11

HMT identiﬁes markets that do not contain all the alternatives consumers would plausibly

view as substitutes to products included in the market. Both the FTC and DOJ use this

test to evaluate whether a set of products constitutes a relevant antitrust market.15 This

test is important because all of the indicia discussed here are calculated using data from

ﬁrms included in the market, and if the market does not contain all plausible alternatives,

the indicia are more likely to incorrectly identify a merger as anti-competitive when it is

not. More than 6% of the simulated markets failed the HMT at the 5% level and were

excluded.16

As of 2018, an HSR ﬁling is required if one party has annual revenues of at least 168.8

million USD and the other has revenues of at least 16.9 million USD.17 To implement this

restriction we assign the acquiring ﬁrm the HSR threshold of 168.8 million USD. Doing

so implies both a total market size and the revenues of the target ﬁrm. If the revenues of

the target ﬁrm are greater than or equal to 16.9 million USD, we assume that the merger

meets the HSR ﬁling requirements. About one-quarter of the simulated markets failed

the notiﬁcation thresholds and were excluded. Eliminating markets that do not meet both

the HMT and HSR criteria results in approximately 222,000 simulated HSR reportable,

antitrust markets. Henceforth, this set is referred to as the ‘sample markets’. Restricting

our sample in this way allows us to focus on the performance of indicia in the set of

markets that would be relevant to antitrust enforcement authorities.

Table 1 summarizes market revenue, elasticity, and share of the outside goods for the

sample markets at the 5th, 25th, 50th, 75th, and 95th percentiles. Market revenues are re-

ported in millions of US dollars. The share of the outside good is reported as a percentage.

This table reveals that our numerical simulations represent a diverse set of markets with

annual market revenues typically between $300 million and $2.8 billion, the distribution

of outside shares appearing roughly uniformly distributed between 0.1 and 0.7, and market

elasticities ranging from very elastic to very inelastic.

Table 2 summarizes the competitive eﬀects implied from the simulated mergers deﬁned

above for the sample markets at the 5th, 25th, 50th, 75th, and 95th percentiles by demand-

supply model. Consumer harm and producer beneﬁt are reported in millions of US dollars.

The sensitivity of the merger simulation results to the assumed demand-supply model

is clearly illustrated with Cournot competition with log demand consistently producing

15U.S. Department of Justice and Federal Trade Commission (2010), p.8-13

16Over 90% of markets that failed the HMT were Cournot with linear demand.

17There are addition requirements for “size of transaction” that consider the value of voting se-

curities and assets of the merging parties. These considerations are outside the scope of the pa-

per. See the FTC’s Pre-merger Notiﬁcation Program website at https://www.ftc.gov/enforcement/

premerger-notification-program.

12

Table 1: Sample market features

Feature 5% 25% 50% 75% 95%

Market revenues 318 500 741 1,171 2,762

Outside share 13 24 38 52 66

Market elasticity -5.4 -2.1 -1.3 -0.76 -0.33

the largest competitive eﬀects estimates. Once again, this table shows that the simulated

markets exhibit a wide range of competitive eﬀects with predicted industry and merging

party price changes ranging from approximately zero to 40%.

3. Results

Given the set of sample markets derived from our numerical simulations, we construct

the measures of competitive harm described in Section 2.2. Note that all the measures are

constructed using data from products included in the market. For example the diversion

ratios used for both UPP and CMCR are based on shares from products included in the

market, excluding the share of the outside good. We make this assumption for two rea-

sons. First, it allows for the various measures considered here to be readily compared. For

example, shares used to construct HHI ( both in levels and in changes) are same as the

shares used to construct diversion ratios. This essentially allows us to examine the beneﬁt

that adding an incremental piece of information, such as margins, has in terms of predict-

ing post-merger prices. Second, proponents of UPP have advocated its use as a merger

screening tool in the early stages of an investigation. As investigators are unlikely to have

access to either reasonable estimates of diversion or the share of the outside good early on

in an investigation, using diversion according to inside share is likely to be the best proxy

available for actual diversion when a screening tool is the most useful.

Table 3 presents summary statistics for the indicia. The range of values for Firm Count,

Party Rank, Party Share, and HHI arise largely by construction from the data generating

process described above. The distribution of these values for the 5th, 25th, 50th, 75th, and

95th percentiles are reported along with those for ∆HHI, UPP, and CMCR.

There is signiﬁcant positive correlation between all pairs of measures, as shown by the

correlation matrix in Table 4, which is unsurprising due to the fact that all the measures

to some degree rely on market shares. The correlations provide some quantiﬁcation of

the potential for combinations of indicia to provide additional information in the merger

screening process. For example, if the merging parties’ combined share is known then

little additional information is gained by calculating the change in HHI resulting from the

merger (correlation is 0.96 between the measures), but also knowing the upward pricing

pressure could add valuable information if UPP does a good job of screening for unilateral

13

Table 2: Sample market outcomes, by model

Outcome Supply Demand 5% 25% 50% 75% 95%

Log 8.1 23 53 142 601Cournot

Linear 4.3 13 31 72 235

Aids 1.4 4.4 10 23 88

Logit 2.3 5.6 10 18 37

Bertrand

Ces 1.1 2.8 5.9 11 27

Consumer Harm ($)

Auction Logit 1.6 3.8 6.9 12 28

Log 0.6 1.8 4.3 10 41Cournot

Linear 0.29 0.98 2.3 5 13

Aids 0.13 0.56 1.6 4.4 22

Logit 0.2 0.69 1.6 3.4 9.2

Bertrand

Ces 0.19 0.6 1.2 2.8 7.7

Industry Price Change (%)

Auction Logit 0.13 0.47 1.1 2.4 7.5

Log 0.6 1.8 4.3 10 41Cournot

Linear 0.29 0.98 2.3 5 13

Aids 0.42 1.3 3 6.9 29

Logit 0.96 2.1 3.8 6.5 13

Bertrand

Ces 0.9 1.9 3.1 5.2 11

Merging Party Price Change (%)

Auction Logit 0.71 1.5 2.6 4.6 10

Log 5.9 15 33 86 353Cournot

Linear 3.3 8.8 21 47 148

Aids 0.48 1.7 5 14 50

Logit 0.74 2 4.8 10 23

Bertrand

Ces 1.2 2.7 5.6 13 39

Producer Beneﬁt ($)

Auction Logit 1.6 3.8 6.9 12 28

Table 3: Indicia summary

Index 5% 25% 50% 75% 95%

Firm Count 3 4 5 6 7

Party Share 18 30 41 56 79

Post-Merger HHI 2,002 2,523 3,306 4,693 6,711

HHI Change 120 355 703 1,339 2,762

UPP 1.2 3.4 6.3 11 24

CMCR 1.7 5.7 13 30 119

14

Table 4: Correlations among indicia (Spearman)

Party Share Post-merger HHI HHI Change UPP CMCR

Firm Count 0.69 0.90 0.65 0.56 0.57

Party Share 0.85 0.96 0.78 0.80

Post-merger HHI 0.81 0.67 0.69

HHI Change 0.82 0.81

UPP 0.99

competitive eﬀects (correlation is 0.78). Similarly, if the only measure known is Firm

Count then any indicia other than HHI could provide signiﬁcant additional information.

However, the question of interest is whether such concentration measures contain suﬃ-

cient information to accurately predict harm from unilateral merger eﬀects. This question

is addressed in Figures 2 - 9. Except for Figures 3 and 7, all ﬁgures are violin plots that

depict the distribution of simulated industry price changes as a function of the discretized

values of a particular indicia for the 6 diﬀerent model speciﬁcations. We chose violin plots

because they do an excellent job at displaying asymmetric distributions. For a particular

discretized value, the horizontal lines in each violin plot depict the 25th, 50th, and 75th

percentile industry price changes. To make these plots more readable, we also chose to

censor the industry price changes at the 5th and 95th percentile across all model speciﬁca-

tions. 18

3.1. Firm Count

Figure 2 depicts the distribution of industry price changes by Firm Count. First and

foremost, the plot reveals that across all models, markets with fewer ﬁrms have signiﬁ-

cantly larger average eﬀects on industry-wide prices than markets with more ﬁrms. Mar-

kets with fewer ﬁrms also have a larger range of plausible price eﬀects than markets with

a larger number of ﬁrms. Thus, while the distribution of price eﬀects is positively skewed

for all values (i.e., skewed toward larger industry price eﬀects), the distributions are most

skewed in markets with fewer ﬁrms. This result is intuitive since ﬁrms have the greatest

potential to exercise market power in concentrated markets.

Relatedly, holding the number of ﬁrms constant, skewness appears to decrease as one

moves from the left-most model (Cournot, log-linear demand) to the right-most model

(Auction, Logit demand). This is partially due to the fact that Firm Count contains no

information about substitution to the outside good (i.e. market elasticity). Apparently,

merger eﬀects under Cournot, where the product is homogeneous, are particularly sen-

18Interactive box and whisker plots using the same data may be found at https://daag.shinyapps.

io/ct_shiny/, under Mergers →numerical simulations.

15

sitive to the magnitude of market elasticity, even after we exclude markets that do not

pass the Hypothetical Monopolist test at the 5% level. By contrast, merger eﬀects under

diﬀerentiated product regimes are less sensitive to the magnitude of the market elasticity.19

Notice that at the 5 to 4 level (highlighted in orange), representing the implied current

threshold, all model speciﬁcations produce mean industry price changes of less than 5%,

and more than 95% of all the simulations have industry-wide price eﬀects less than 10%.

By contrast, the likelihood of a negligible industry price eﬀect (less than 1%) is very small

for a 3 to 2 merger. Additionally, the probability of a negligible eﬀect increases with the

number of pre-merger ﬁrms in the market as illustrated by the increasing mass below 1%

in each plot as Firm Count increases in each panel.

Taken together these observations suggest that a signiﬁcant amount of information

about potential price eﬀects is conveyed by the Firm Count measure. Both the likelihood of

a signiﬁcant price eﬀect and the predicted size of the eﬀect increase as the number of pre-

merger ﬁrms in the market decreases. However, the measure also becomes increasingly

noisy as the number of ﬁrms decreases. This is consistent with the observations in Section

2.2.1 of factors that could substantially vary competitive conditions between markets with

the same number of ﬁrms.

3.2. Party Rank

Figure 3 displays a heat map summarizing the relationship between each of the merg-

ing parties’ size rank, as measured by market share, and median industry price changes,

with darker blue shading corresponding to larger price eﬀects. These plots reveal that

that under the models considered the largest price eﬀects typically occur when ﬁrms are

closely ranked. For example, in the Cournot model with log-linear demand, the largest

median price eﬀects occur when the merging parties are ranked 2nd and 3rd, while some

of the smallest price eﬀects occur when one party is ranked 1st and the other is ranked 7th

in terms of size. Notice, however, that while markets where the merging parties are ranked

sequentially typically have the largest price eﬀects, the magnitude of the price eﬀects tend

to decrease as the rankings of both parties increase. For example, the median price eﬀects

are typically larger when the parties are ranked 2nd and 3rd than when they are ranked

4th and 5th. These results are intuitive for largely algebraic reasons since the greatest

gain in combined market share, and thus concentration, occurs when ﬁrms are similar in

size. Finally, notice that the rate at which these merger eﬀects decay appears to vary across

19To see this more formally, note the Cournot ﬁrst order condition in A.1 shows that the Cournot equi-

librium price is determined by weighted average equilibrium margins, the Herﬁndahl-Hirschman Index and

the equilibrium market elasticity. Hence, knowing the Herﬁndahl-Hirschman Index, which we have shown

is related to Firm Count, is insuﬃcient to identify equilibrium price levels.

16

models, with the left-most model (Cournot,log-linear) exhibiting the slowest rate of decay,

and the right-most model (Auction, Logit) suggesting the fastest rate of decay.

3.3. Party Share

Figure 4 displays the relationship between the pre-merger combined share of the merg-

ing parties and industry price changes. Highlighted in orange is the 30% threshold, which

has been historically identiﬁed as the threshold for establishing that a merger is presump-

tively anti-competitive. Similar to the “Firm Count” measure, we see that while median

price eﬀects from the simulated mergers increase as party combined shares increase, so

does the plausible range of price eﬀects. Critically, at the 30% threshold, notice that across

all the models, approximately 95% or more of the simulated markets experience less than

a 5% price eﬀect, and half of all markets experience less than a 2% price eﬀect. This

suggests that, if used alone, this threshold would lead to investigating a large number of

mergers with very small predicted price eﬀects as compared to merger simulation results.

3.4. HHI

Figure 5 presents the distribution of estimated industry price changes by the level of

the post-merger HHI for each model. Across all models, both the level and the variance of

the quantiles of the estimated industry price changes increase with the change post-merger

HHI. However, it is apparent that the choice of model matters with the greatest variance in

the eﬀects exhibited by the model with Cournot supply and log demand. In contrast, the

Logit demand model - the workhorse model for merger simulation - exhibits signiﬁcantly

less variation in price eﬀects in both the Bertrand and auction supply settings. Figure 5 also

illustrates that all models produce signiﬁcantly larger price eﬀects in a small fraction of

the simulated markets with price eﬀects at the 95th percentile generally more than double

the median value.

The results are nearly identical in Figure 6, which presents the distribution of estimated

industry price changes by the change in HHI (∆HHI). However, the change in HHI has an

additional beneﬁt, as the change in the HHI has a stronger negative correlation with a neg-

ligible price eﬀect. This makes the change in HHI more useful as a screening metric since

it is likely to lead to fewer investigations of mergers that would not prompt an enforcement

action.

However, the HMG categorizes mergers by the combination of the level and change in

industry concentration. Figure 7 investigates whether there is an informative relationship

between pre-merger HHI and post-merger change in HHI across models as illustrated by

a heatmap. While Figure 7 shows price eﬀects are more likely as both measures increase,

signiﬁcant eﬀects are only predicted on average at very high values. Thus, there is no

relationship between the two measures to clearly inform merger enforcement decisions.

17

3.5. UPP

Figure 8 displays the relationship between UPP and industry-wide price changes across

models. Before discussing these results, it is important to reiterate that UPP is at best

able to predict the amount by which the merging parties’ prices would increase, and is

therefore not explicitly designed to predict industry-wide price eﬀects. With this caveat

in mind, Figure 8 conﬁrms previous ﬁndings that UPP’s predictive power substantially

degrades as demand curvature increases. The predictions exhibiting the greatest varia-

tion in price eﬀects are given by the Cournot-log and Bertrand-Aids speciﬁcation, which

have substantial curvature. With the exception of Cournot-log and Bertrand-AIDS speci-

ﬁcations, UPP tends to over-predict merging party price eﬀects and industry price eﬀects

from a merger. For example, even in the Cournot model with linear demand, where UPP

performs generally well, industry price changes in the [8,10) interval are below 7% for

more than three-quarters of simulated markets, and for the Bertrand-Logit and Auction

models are almost never above 5%.

In general, we ﬁnd that UPP typically over-states industry-wide price eﬀects by any-

where from about 65% (25th percentile) to 537% (75th percentile), with UPP overstating

the industry-wide price eﬀects for the median market by by 255%. We also ﬁnd that UPP

typically over-states the merging party price eﬀects by anywhere from about 36% (25th

percentile) to 185% (75th percentile), with UPP overstating the party eﬀects for the median

market by 91%. These results were calculated using a diversion measure that excluded the

outside good, thereby overstating the degree to which the merging parties’ products are

substitutes. When we examine the relationship between UPP calculated with actual diver-

sion, we ﬁnd that that gap between industry price change and UPP is typically between

-220% (UPP overstates the eﬀect) and 3% (UPP understates the eﬀect), with UPP overstat-

ing the industry-wide price eﬀects for the median market by 145%. We also ﬁnd that that

gap between merging party price changes and UPP is typically between -24% and 22%,

with UPP overstating the party price eﬀects for the median market by about 8%.

3.6. Bertrand CMCR

Figure 9 displays the relationship between the CMCR, derived under Bertrand com-

petition, and industry-wide price changes across demand-supply models. As with the

other indicia, CMCR does have some predictive power. Like UPP, CMCR tends to over-

predict industry-wide price eﬀects and models with substantial demand curvature exhibit

the largest increases in price dispersion. For the Bertrand model with Logit demand, where

one might expect CMCR to perform well, industry price changes in the highest reported

bin, 50-60%, are never above 10%.

In general, we ﬁnd that CMCR is typically anywhere from 3.8 times (25th percentile)

to 15.4 times (75th percentile) larger than the industry-wide price eﬀects, with CMCR

18

about 9.1 times larger than the industry-wide price eﬀect for the median market. We also

ﬁnd that CMCR is typically anywhere from about 2.5 times (25th percentile) to 6.5 (75th

percentile) larger than the merging party price eﬀects, with CMCR about 4 times larger

than the party eﬀects for the median market. As with UPP, these results were calculated

using a diversion measure that excluded the outside good, thereby overstating the degree

to which the merging parties’ products are substitutes. When we examine the relationship

between CMCR calculated with actual diversion, we ﬁnd that CMCR is typically anywhere

from 1.5 times (25th percentile) to 6.7 times (75th percentile) larger than the industry-wide

price eﬀects, with CMCR about 3.9 times larger than the industry-wide price eﬀect for the

median market. We also ﬁnd that CMCR is typically anywhere from about 1.25 times

(25th percentile) to 2.9 (75th percentile) larger than the merging party price eﬀects, with

CMCR about 1.7 times larger than the party eﬀects for the median market.

4. Optimal screening rules for investigating horizontal mergers

In this section, we develop two “optimal” decision rules for determining whether to

investigate a horizontal merger based on the measures of competitive harm previously

analyzed. It is important to emphasize that all of the rules discussed hinge on two impor-

tant assumptions: 1) the data generating process underlying the population of horizontal

mergers considered, and 2) the behavioral model for weighing the costs and beneﬁts of a

particular decision rule. Initially, we will assume that the population of potential mergers

consists of the markets in our numerical experiments. In other words, we will assume that

all horizontal mergers can be described by the data generating process described in Section

2.3. Later, we will relax this assumption and consider how changing the composition of

models aﬀects the decision rules.

It is important to stress that the behavioral models we consider here are relatively sim-

ple heuristics designed to weigh the harm from not pursuing a potentially anti-competitive

merger against the beneﬁt from not investigating relatively benign mergers with both mea-

sures construed very narrowly. Neither behavioral model analyzed takes into account the

wealth of other considerations that antitrust enforcement agencies might also incorporate.

4.1. Type I/Type II error analysis

Suppose that the agencies are interested in investigating horizontal mergers that yield

substantial harm to consumers but are also interested in not investigating horizontal merg-

ers that are unlikely to yield substantial harm. Further, suppose that thresholds for deter-

mining what constitutes substantial harm are well deﬁned.

Formally, Let VHdenote the average beneﬁt an agency receives from investigating a

harmful merger and VSdenote the average cost an agency incurs from investigating an

19

innocuous merger. Further, suppose that a merger is considered harmful if it is expected to

increase average industry prices, ∆p, by at least ∆pHpercent, while a merger is considered

innocuous if it expected to increase ∆pby no more than ∆pS. Then, for a given index I

agencies choose threshold DIto maximize the ex ante net beneﬁt from investigating future

mergers:

max

DI

VHPr(∆p≥∆pH|DI)−VSPr(∆p<∆pS|DI).

For continuous decision rules, taking the derivative of the above w.r.t to DIand re-

arranging yields:

VH

VS

=

dPr(∆p<∆pS|DI)

dDI

dPr(∆p≥∆pH|DI)

dDI

,(1)

VH

VSrepresents how enforcement agencies value preventing anti-competitive mergers

relative to not investigating benign mergers, which we will refer to as the “enforcement

ratio”. Here we explore two cases. First, we consider the case when VH

VS=1. In this

case, agencies place equal weight on these two outcomes and consequently the thresholds

are only a function of the relative incidence of harmful and benign mergers. Second, we

consider the case when VH

VS=11, which is the value at which the Party Share and Post-

merger HHI decision thresholds predicted by this model closely match the 30% combined

share threshold articulated by the U.S. Supreme Court in United states v. Philadelphia

National Bank and the 2,500 post-merger HHI threshold articulated in the 2010 HMG.

This value of the enforcement ratio would suggest that agencies put much greater weight

on preventing potentially harmful mergers than on investigating benign ones.

In determining the optimal screening thresholds, we assume that any horizontal merger

that yields at least a 5% industry-wide price increase is anti-competitive, while any hori-

zontal merger yielding no more than a 1% industry-wide price increase is benign.20 Figure

10 displays the cumulative proportion of simulated markets for a given value of an index

that have 1) more than a 5% price eﬀect (solid dark blue) and 2) less than a 1% price eﬀect

(solid light blue). The green (dot-dashed) vertical line displays the most current threshold

for each index (if available). For example, the 30% “Party Share” rule occurs about where

the 5% and 1% thresholds cross near the maximum of the 5% rule, indicating that under

20We also experimented with a 10%/5% threshold and found that at those values no threshold existed for

most of the measures.

20

Table 5: Optimal thresholds,by model

Model Indicia Current Optimum Value Variance

Firm Count 5 6 1.9 17

Party Share 30 32 2.1 9.5

HHI 2,500 2,508 2 8.4

HHI Change 200 456 2.1 7.2

UPP 4.5 2.1 3.8

Type I/Type II

CMCR 8.4 2.1 1.7

Firm Count 5 5 92,043,644 42

Party Share 30 39 92,047,832 9.5

HHI 2,500 2,998 92,045,696 23

HHI Change 200 649 92,048,516 19

UPP 6.9 92,049,158 8.3

Agency

CMCR 14 92,049,315 6.8

our modelling assumptions, the 30% rule ensures that all harmful mergers are investigated

while about half of all benign mergers are also investigated. By contrast, the 200 point

“HHI Rule” also investigates all the harmful mergers, but also investigates almost 70% of

all benign mergers.

The pink (dashed) vertical line displays the optimal decision rule for each index under

the assumption that VH

VS=11 , so the agencies value preventing merger harm 11 times more

than not investigating benign matters. As this relative valuation was obtained by matching

the predicted thresholds to the observed thresholds for Party Share and Post-merger HHI,

it is unsurprising that those thresholds match so closely. Interestingly, however, the Firm

Count threshold is predicted to be 6, which is greater than the threshold of 5 that is inferred

from the HHI level with symmetric ﬁrms. Likewise, the HHI Change threshold is greater

than 400, more than twice the magnitude of the current 200 threshold. Also of interest are

the UPP and CMCR thresholds, which are about 5% and 9%, respectively, and in line with

proposed “safe-harbor” values in the literature. The red (dotted) vertical line displays the

optimal decision rule for each index under the assumption thatVH

VS=1 , indicating agen-

cies value preventing merger harm the same as investigating benign matters. Under this

assumption, the optimal thresholds are uniformly and substantially larger in magnitude.

The top half of Table 5 depicts the current rule (if available), the optimal rule, the

value of the objective function at the optimum, and the variance of industry-wide prices

in a neighborhood around the optimal decision rules when VH

VS=11. All the thresholds

that maximize the Type I/Type II decision rule are greater than the existing thresholds,

although the optimal values for Firm Count, Party Share and post-merger HHI are quite

similar under these assumptions. Further, note that the variance of industry-wide prices

changes around the optimum varies across the indices, with Firm Count having the largest

21

variance and CMCR the lowest.

4.2. Antitrust Agency Cost-Beneﬁt Analysis

While a useful starting point for exploring optimal decision rules, the Type I/Type II

error framework suﬀers from a two main drawbacks. First, this framework does not reﬂect

any of the resource constraints that agencies typically operate under. While agencies do

have substantial budgets, they are also tasked with evaluating a large and growing number

of mergers each year. For example, the DOJ Antitrust Division’s budget increased by

about 12% from $148 million in 2008 to $165 million in 2017, while the number of HSR

ﬁlings the Division received during that period increased by about 19% from 1,726 ﬁlings

in 2008 to 2,057 in 2017. These numbers suggest that the Division has fewer resources to

spend per HSR ﬁling.

Second, by focusing only on the percentage change in industry prices, the above frame-

work ignores the fact that some mergers can have larger adverse eﬀects because they oc-

cur in larger markets. For example, the above framework treats a 1% industry-wide price

increase in a $1 billion dollar market as being benign while a 5% industry-wide price in-

crease in a $100 million dollar market as being anti-competitive, even though the ﬁrst mar-

ket yields roughly $10 million of harm per year while the second market yields roughly $5

million. Here we develop screening thresholds that are more closely based on the amount

of harm from a merger as well as the costs associated with investigating a merger.

Suppose in each year t, ﬁrms ﬁle NtHart-Scott-Rodino notices to merge with an en-

forcement agency. Further, suppose that when the agency is determining whether to open

investigations, they weigh the expected beneﬁt from opening an investigation into a merger

against the costs associated with investigating the merger. We begin by identifying an op-

timal decision rule for determining whether to investigate a merger based on the change in

the HHI.

Formally, suppose that candidate decision rule DIbased on index I, the expected net

beneﬁt to the agency is

E[NBt|DI]=

∞

X

n=1

E[NBt|DI,N(t)=n]e−λt(λt)n

n!(2)

E[NBt|DI,N(t)=n]=E[

d(n)

X

j=1Hj−j

κe−ρSj|DI] (3)

where Hjis the harm from the jth merger, ρis the discount rate, Sjis is the arrival

time of notice n,j

κis the incremental cost associated with investigating a merger, and the

agency chooses to further investigate any merger with a change in index Igreater than

22

DI. Since the number of Hart-Scott-Rodino ﬁlings varies from year to year, we suppose

that N(t) follows a Poisson distribution with rate λ. Assuming that the size of the merging

parties is independent from when Hart-Scott-Rodino is ﬁled, it can be shown that equation

(2) reduces to

E[NBt|DI]=2κE[DIHn]−λPr(I>DI)(λtPr(I>DI)+2)

2κρ λ(1 −e−ρt) (4)

where E[DIHn] is the expected amount of harm from a merger when the agency

chooses to investigate it further under rule DI. Hence, an optimal decision threshold D∗

Iis

one which maximizes equation (4) for a given indicia I.

To calculate these thresholds, we ﬁrst assume that the actual distribution of horizontal

mergers corresponds to the data generating process underlying our Monte Carlo simula-

tions. Next, we calibrate κby assuming that the 2017 average variable costs for the DOJ

Antitrust Division are equal to the total number of HSRs received by the Division in 2017,

divided by the Division’s expenditures on HSRs. This is calculated by multiplying the per-

centage of total matters initiated in 2017 that were HSRs by the Division’s 2017 budget.

The value of κis then backed out from the deﬁnition of average variable costs.

Figure 11 summarizes the results. The black curve depicts agency expected net beneﬁt

for a given threshold using (2). The green vertical line displays the current threshold, if

available. The pink vertical line displays the threshold that maximizes (2). This ﬁgure

depicts some interesting features of the optimization problem. First, the objective func-

tion achieves its highest values under UPP and CMCR, and its lowest value under Firm

Count, suggesting that at least on average, UPP and CMCR outperform the other indicia.

Second, CMCR has the “ﬂattest” objective function, suggesting that it is the most robust

to relatively small deviations away from the optimum.21

The bottom half of Table 5 depicts the current rule (if available), the optimal rule, the

value of the objective function at the optimum, and the variance of industry-wide prices in

a neighborhood around the optimal decision rules when VH

VS=11. Note that CMCR is the

decision rule that maximizes the Agency decision rule. In addition, note that the variance

of industry-wide price changes varies across the indices, with Firm Count again having

the largest variance and CMCR the lowest.

4.3. Model composition and thresholds

One drawback of the decision rules depicted in Figures 10 and 11 is that they assume

that the underlying population of mergers follows a particular model distribution: 30%

21This is likely due to it’s explicit accounting for higher order eﬀects on prices.

23

Table 6: Thresholds with diﬀerent model weights, VH

VS=11

Cournot Bertrand Auction Firm Count Party Share HHI HHI Change UPP CMCR

0.9 0.05 0.05 7 23 1,856 215 2.1 3.4

0.7 0.1 0.2 7 24 1,986 272 2.4 3.8

0.5 0.3 0.2 7 27 2,144 328 3.2 5.4

0.5 0.1 0.4 7 27 2,154 328 2.6 4.2

0.3 0.5 0.2 7 31 2,403 413 4.2 7.6

0.3 0.3 0.4 6 32 2,446 453 4.2 7.5

0.3 0.1 0.6 6 32 2,499 452 4.2 6.7

0.27 0.55 0.18 6 32 2,508 456 4.5 8.4

0.1 0.7 0.2 5 39 2,914 633 5.7 12

0.1 0.5 0.4 5 39 2,996 648 6.5 15

0.1 0.3 0.6 5 40 3,053 667 8 18

0.1 0.1 0.8 5 42 3,505 676 8.4 21

Cournot, 50% Bertrand, and 20% Auction. Here, we explore how changing the model

composition can aﬀect these thresholds.

Table 6 reports that under the simple model with VH

VS=11, the decision rules for

each indicia vary substantially as the model mix changes. For example, the Firm Count

threshold varies from 5 to 7 pre-merger ﬁrms, depending on the mix of competitive models.

The Party Share threshold varies from 23% when Cournot competition characterizes 90%

of markets to 42% when 80% of markets follow the Auction model. Interestingly, the

HHI Change threshold only approaches the current level of 200 when Cournot comprises

90% of markets. Table 6 also illustrates the range of potential UPP and CMCR thresholds

derived under this decision rule, approximately 2-8% and 3-21%, respectively.

Similar patterns occur in Table 7, which reports comparable results for the Agency

model. For example, the Firm Count threshold varies from 4 to 7 ﬁrms. The Party Share

threshold varies from 27% to 50%. Interestingly, the post-merger HHI threshold only

approaches the current threshold of 2,500 when Cournot competition characterizes about

50-70% of markets, and the HHI Change threshold never comes close to the current level

of 200. The UPP and CMCR thresholds exhibit a wider range as well at 2-13% and 4-36%,

respectively.

Taken together, these tables yield two observations of interest. First, they suggest that

the current HHI thresholds are best suited to a population of mergers in which industries

engaged in Cournot competition dominate. To the extent that the composition of the pop-

ulation of interest shifts toward ﬁrms playing either a Bertrand or Auction game, then

this analysis suggests that the HHI thresholds discussed in the HMG should be revides

accordingly.

Finally, these tables indicate that the Agency model, which includes more informa-

24

Table 7: Thresholds with diﬀerent model weights

Cournot Bertrand Auction Firm Count Party Share HHI HHI Change UPP CMCR

0.9 0.05 0.05 7 27 2,101 315 2.5 4

0.7 0.1 0.2 7 30 2,334 385 3.1 5.1

0.5 0.3 0.2 6 33 2,634 467 4.4 8.3

0.5 0.1 0.4 6 33 2,646 465 4.4 9.3

0.3 0.5 0.2 5 38 3,094 606 6.7 14

0.3 0.3 0.4 5 37 2,932 605 7.4 18

0.3 0.1 0.6 5 39 2,973 630 8.5 22

0.27 0.55 0.18 5 39 2,998 649 6.9 14

0.1 0.7 0.2 4 46 3,776 920 9.3 22

0.1 0.5 0.4 4 47 3,834 969 10 27

0.1 0.3 0.6 4 47 3,854 986 11 31

0.1 0.1 0.8 4 50 3,854 975 13 36

tion regarding Agency caseload and investigative costs than the Simple model, also yields

thresholds that are typically 17% (25th percentile) to 36% higher (75th percentile) than

the simple model, with the median diﬀerence between the models equal to about 29%.22

5. Conclusion

This paper contributes to understanding the relationship between horizontal merger in-

vestigation screening measures and predicted industry price eﬀects from standard merger

simulation models. We ﬁnd that across a range of common demand and supply speci-

ﬁcations all the concentration measures and other indicia of competitive harm correctly

predict when a merger is likely to have little anti-competitive eﬀect (i.e small values of the

indicia are associated with small price eﬀects). However, while large indicia values are as-

sociated with anti-competitive eﬀects, in general these indicia do a poor job of accurately

predicting price eﬀects when they are likely to be large. It is not surprising that simple

screens do a poor job of predicting continuous outcomes since much of the variation is

driven by factors not captured by the measures. Further, when merger eﬀects are large the

outcome is farther away from the pre-merger equilibrium, making out of sample prediction

accuracy poor.

We also consider two optimal decision rules for merger enforcement based on com-

monly used measures of concentration and compare them to current screening thresholds

employed by US antitrust enforcement agencies. We ﬁnd that in our ﬁrst framework the

current thresholds suggest a signiﬁcantly higher value is placed on investigating potentially

22These statistics exclude the “Firm Count” measure.

25

anti-competitive mergers than on avoiding investigating mergers that are unlikely to raise

competitive concerns. In addition, we ﬁnd that across both frameworks optimal decision

rules depend critically on the mix of competitive models that are believed to exist in the

population of potential mergers. In general, higher thresholds would be warranted under

the belief that the population of potential mergers is largely comprised of ﬁrms competing

according to a diﬀerentiated products auction model. In addition, the current thresholds

would be most relevant in a population with the majority of industries engaged in Cournot

competition.

An important implication of our Monte Carlo experiments and the results of the decision-

theoretic models is that measures that include information on substitution to the out-

side good (UPP, CMCR) appear to perform better than those that do not. This reﬂects

the importance of appropriately deﬁned alternatives to the products at issue in a merger,

which is an under-appreciated driver of merger simulation results. While optimal decision

thresholds can be developed for any competitive screening measure, the resulting decision

thresholds perform better (both on average and in terms of dispersion) using indicia that

incorporate substitution to the outside good.

26

Figure 1: HMG HHI-based decision rule

27

Figure 2: summarizes the relationship between number of pre-merger ﬁrms and industry price changes. Highlighted in orange is the 5-4

decision rule implied by the HHI thresholds in the 2010 HMG. Industry price changes are censored at the 5th and 95th percentile. Horizontal

lines within each plot represent the 25th, 50th and 75th percentiles. Interactive box and whisker plots using the same data may be found at

https://daag.shinyapps.io/ct_shiny/, under Mergers →numerical simulations.

28

Figure 3: displays a heatmap summarizing the relationship between the share ranks of the merging parties and median industry price

changes.

29

Figure 4: summarizes the relationship between the cod share of the merging parties and simulated industry price changes. Industry

price changes have been censored at the 5th and 95th percentile. Horizontal lines in each plot depict the 25th, 50th and 75th percentiles.

The bin containing combined shares of 30% and 35%, historically identiﬁed as thresholds for a presumptively anti-competitive merger,

is colored orange.Interactive box and whisker plots using the same data may be found at https://daag.shinyapps.io/ct_shiny/,

under Mergers →numerical simulations.

30

Figure 5: displays violin plots summarizing the relationship between the HHI and simulated industry price changes. Industry price changes

have been censored at the 5th and 95th percentile, and the horizontal lines depicted in each plot represent the 25th, 50th and 75th percentiles.

The bin containing the post-merger HHI of 2500, a component of establishing that a merger is presumptively anti-competitive under the

HMG, is colored orange. Interactive box and whisker plots using the same data may be found at https://daag.shinyapps.io/ct_

shiny/, under Mergers →numerical simulations.

31

Figure 6: displays violin plots summarizing the relationship between changes in HHI and simulated industry price changes. Industry

price changes have been censored at the 5th and 95th percentile, and the horizontal lines depicted in each plot represent the 25th, 50th

and 75th percentiles. HHI change of 200, a component of establishing that a merger is presumptively anti-competitive under the HMG,

is colored orange. Interactive box and whisker plots using the same data may be found at https://daag.shinyapps.io/ct_shiny/,

under Mergers →numerical simulations.

32

Figure 7: displays a heatmap summarizing the relationship between pre-merger HHI, changes in HHI and median industry price changes.

33

Figure 8: displays violin plots summarizing the relationship between the Upward Pricing Pressure index (UPP) and simulated industry

price changes. Each panel summarizes the results from 50,000 Monte Carlo Simulations. Industry price changes have been censored at the

5th and 95th percentile, and the horizontal lines depicted in each plot represent the 25th, 50th and 75th percentiles. All simulations are run

assuming that the outside good has a price of $6 and $2 marginal costs. Interactive box and whisker plots using the same data may be

found at https://daag.shinyapps.io/ct_shiny/, under Mergers →numerical simulations.

34

Figure 9: displays violin plots summarizing the relationship between the compensating marginal cost reduction (CMCR) and simulated

industry price changes. Each panel summarizes the results from 50,000 Monte Carlo Simulations. Industry price changes have been censored

at the 5th and 95th percentile, and the horizontal lines depicted in each plot represent the 25th, 50th and 75th percentiles. All simulations

are run assuming that the outside good has a price of $6 and $2 marginal costs. Interactive box and whisker plots using the same data may

be found at https://daag.shinyapps.io/ct_shiny/, under Mergers →numerical simulations.

35

Figure 10: displays the cumulative percentage of markets exhibiting less than a 1% price increase (light blue) or greater than a 5% price

increase (dark blue) for a given level of each indicia. Green (dot-dashed) vertical lines depict the currently used threshold where available.

Pink (dashed) vertical lines depict the threshold that maximizes the gap between the light and dark blue lines, when the enforcement ratio

equals 11. The red (dotted) vertical lines depict the threshold that maximizes the gap between the light and dark blue lines, when the

enforcement ratio equals 1.

36

Figure 11: displays net beneﬁt (harm averted less enforcement costs) to a hypothetical antitrust agency facing linear enforcement costs as

function of each indicia. Green vertical lines depict the current threshold for each (where available). Pink vertical lines depict the threshold

that maximize the agency’s net beneﬁt.

37

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Appendix A Demand and Supply Speciﬁcations

A.1 Cournot

Suppose there are N∈ {3,4,5,6,7}ﬁrms playing a Cournot quantity setting game,

where single-plant ﬁrms with quadratic cost technology Cn(qn)=q2

n

2knsimultaneously pro-

duce qnunits of a homogeneous product once to maximize their proﬁts.

Mathematically, each ﬁrm nsolves

max

qn

Pqn−Cn(qn)

with demand equal to

P=

a−b−1Q,if linear

exp (a)Q1

,if log-linear,

where Q=P

j∈N

qjdenotes equilibrium output and =dQ

dP

P

Qdenotes the aggregate

elasticity.

Deﬁne mn=−P−dC

dqn

Pas plant n’s proportional margin and sn=qn

Qas plant n’s quantity

share. Pre-merger, the ﬁrst order condition (FOC) of each ﬁrm can be arranged to yield

the Lerner condition mn=sn

, which is an implicit function of all ﬁrms quantities. Un-

der quadratic costs, it can be shown that this system of Nequations has a unique Nash

Equilibrium in quantities and that this equilibrium has a closed-form solution.

Moreover, the Lerner condition may be rearranged to yield an equilibrium relationship

between industry average margins, HHI, and market elasticities. Speciﬁcally, taking the

share-weighted average of all Nﬁrms’ Lerner conditions yields

X

n∈N

snmn=−P

n∈N

s2

n

=HHI

To calibrate the model parameters, we ﬁrst assume that market size Mis known and

draw Nshares from a Dirichlet distribution. Second, we assume that in the pre-merger

state, the equilibrium price is $6 and there is an outside Firm 0 playing the Cournot game

earning an equilibrium margin of $2 with quantity share s0drawn from a uniform distri-

bution between 0.1 and 0.7. We use Firm 0’s Lerner condition to solve for the aggregate

41

elasticity, and then the assumed price and aggregate output to solve for the remaining de-

mand parameters. Finally, we again use the Lerner condition, quantities and price to solve

for the cost parameters kn23.

Next, suppose that Firm 1 acquires Firm 2. We simulate the eﬀect of the merger by

assuming that Firm 1 sets output on plants 1 and 2 jointly. Mathematically, Firm 1 solves

max

q1,q2

P(q1+q2)−C1(q1)−C2(q2)

Taking the FOC and rearranging yields

m1=m2=s1+s2

The functional form of the non-merging parties’ Lerner condition does not change as

a result of the merger. The outside product’s output is assumed not to adjust post-merger.

A.2 Bertrand

Suppose there are N∈ {3,4,5,6,7}ﬁrms playing a Bertrand price setting game, where

single-product ﬁrms with constant marginal cost technology Cn(qn)=cnqnsimultaneously

set their prices pnonce to maximize their proﬁts.

Mathematically, each ﬁrm n∈Nsolves

max

pn

(pn−cn)qn

Like Cournot, the pre-merger ﬁrst order condition (FOC) of each ﬁrm can be arranged

to yield the Lerner condition mn=1

nn , which is an implicit function of all ﬁrms prices. In

general, this system of equations has no closed form solution, but a Nash equilibrium in

prices may be solved for numerically.

Moreover, it has been shown that the magnitude of the price eﬀects in the Bertrand

model depends critically on the second order properties of the demand curve. These prop-

erties are captured by the own-price elasticities nn =−∂qn

∂pn

pn

qn. Speciﬁcally, we will consider

the properties of three popular demand systems: Logit, CES, and AIDS

A.2.1 Logit Demand

Logit demand is based on a discrete choice model that assumes that each consumer is

willing to purchase at most a single unit of one product from the Nproducts available in

23From time to time, this procedure yielded plant margins that were greater than 1. When this occurred,

we simply discarded the market and sampled another one.

42

the market. The assumptions underlying Logit demand imply that the probability that a

consumer purchases product n∈Nis given by

sn=exp(Vi)

1+P

k∈N

exp(Vk)

where snis product n’s quantity share and Vnis the (average) indirect utility that a

consumer receives from purchasing product n. We assume that Vntakes on the following

form

Vn=δn+αpn, α < 0

The Logit demand system yields the following own- and cross-price elasticities:

nn =α(1 −si)pi

n j =−αsjpj

It is helpful to note that because snis a probability, sn=sn|N(1 −s0). sn|Ndenotes the

probability that product nis selected, conditional on one of the Nproducts in the market

being selected. s0denotes the probability that the outside good is selected.

A.2.2 CES Demand

Like the Logit, CES demand is based on a discrete choice model. However, CES

diﬀers from the Logit model in that under CES consumers do not purchase a single unit

of a product but instead spend a ﬁxed proportion of their budget on one of the nproducts

available in the market.24

The assumptions underlying CES demand imply that the probability that a consumer

purchases product n∈Nis given by

24Formally, each consumer chooses the product n∈Nthat yields the maximum utility Un=ln(δnqn)+

αln(q0)+i, subject to the budget constraint y=pnqn+q0. Here, qnis the amount of product nconsumed by

a consumer, δnis a measure of product n’s quality, q0is the amount of the numeraire, yis consumer income,

and nare random variables independently and identically distributed according to the Type I Extreme Value

distribution.

43

where rnis product n’s expenditure share and Vnis the (average) indirect utility that a

consumer receives from purchasing product n. We assume that Vntakes on the following

form

Vn=δnp1−γ

n, γ > 1

The CES demand system yields the following own- and cross-price elasticities:

nn =−γ+(γ−1)rn

n j =(γ−1)rj

Functional form diﬀerences aside, one important diﬀerence between the CES and Logit

demand systems is that the Logit model’s choice probabilities are expressed in terms of

quantity shares, while the CES model’s choice probabilities are expressed in terms of

expenditure shares.

A.2.3 AIDS

AIDS without income eﬀects assumes that the expenditure share rnfor each product

n∈ {N,0}in the market is given by

rn=αn+X

n∈{N,0}

βn j log( pj) for all n∈ {N,0}, βnn <0,

where rn=pnqn

x. Total expenditure xis given by

log (x)=η+X

j∈N

αjlog (pj)+X

k∈NX

j∈N

βk j log ( pk) log (pj)

The AIDS model yields the following own- and cross-price elasticities:

nn =−1+βnn

rn

+rn, nn <0

n j =βn j

rn

+rj, n j ≥0

This version of the AIDS model assumes that βn j =βj j , satisﬁes homogeneity of degree

zero in prices, and that diversion occurs according to revenue share djn =−rn

1−rj=−βn j

βj j .

44