ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER

Upward Pricing Pressure as a

Predictor of Merger Price Effects

by

Nathan H. Miller, Marc Remer, Conor Ryan, and Gloria Sheu*

EAG 16-2 March 2016

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*Miller: Georgetown University, McDonough School of Business, 37th and O Streets NW, Washington DC 20057, nhm27@georgetown.edu. Remer: Swarthmore College, Department of Economics,

500 College Avenue, Swarthmore PA 19081, mremer1@swarthmore.edu. Ryan: University of Minnesota, Department of Economics, 4-101 Hanson Hall, 1925 Fourth Street South, Minneapolis

MN 55455, ryan0463@umn.edu. Sheu: U.S. Department of Justice, Antitrust Division, Economic

Analysis Group, 450 5th St. NW, Washington DC 20530, gloria.sheu@usdoj.gov.
Abstract

We use Monte Carlo experiments to evaluate whether “upward pricing pressure”

(UPP) accurately predicts the price effects of mergers, motivated by the observation

that UPP is a restricted form of the first order approximation derived in Jaffe and

Weyl (2013). Results indicate that UPP is quite accurate with standard log-concave

demand systems, but understates price effects if demand exhibits greater convexity.

Prediction error does not systematically exceed that of misspecified simulation models,

nor is it much greater than that of correctly-specified models simulated with imprecise

demand elasticities. The results also support that both UPP and the HHI change

provide accurate screens for anticompetitive mergers.
1 Introduction

In a number of recent antitrust enforcement actions, the U.S. Department of Justice (DOJ)

and the Federal Trade Commission (FTC) have alleged that mergers between producers

of competing differentiated products would adversely affect unilateral pricing incentives.1

This follows a decades-long trend that has both spurred on and been informed by academic

research on how mergers affect prices (e.g., Davidson and Deneckere (1985); Berry and Pakes

(1993); Hausman, Leonard and Zona (1994); Werden and Froeb (1994); Nevo (2000); Jaffe

and Weyl (2013); Carlton and Keating (2015)). Continuing this evolution, the DOJ and

the FTC updated their Horizontal Merger Guidelines in 2010, in part motivated by a desire

to better align the document with economic theory and antitrust practice as they relate to

markets with differentiated products (Shapiro (2010)).

One point of emphasis in the 2010 Horizontal Merger Guidelines is that mergers between competitors create opportunity costs, which in turn place upward pricing pressure (or

“UPP”) on the combining firms. This principle is easily derived from basic economic models,

and the magnitude of the opportunity costs often can be quantified with information from

only the merging parties. This combination of theoretical and practical simplicity make UPP

a useful diagnostic tool. Referring to UPP as the value of diverted sales, the Guidelines state

that “[t]he Agencies rely more on the value of diverted sales than on the level of the HHI for

diagnosing unilateral price effects in markets with differentiated products.”2 The FTC has

employed UPP calculations to support arguments in court (FTC v. Sysco Corporation, et

al.) and to justify enforcement decisions (Family Dollar/Dollar Tree).3

Although UPP has a clear relationship to firms’ pricing incentives, antitrust economists

have been wary about using it as a prediction of price effects. UPP does not incorporate the

manner in which the pass-through of costs to prices depends on the higher-order properties

of the underlying demand system. Nor does it account for the possibility that non-merging

competitors change their prices as the market shifts to a new equilibrium. Two of the

principal authors of the 2010 Horizontal Merger Guidelines, Joseph Farrell and Carl Shapiro,

emphasize in their academic work that “UPP does not predict post-merger prices, but only

predicts the sign of changes in price” (Farrell and Shapiro (2010)).4 Furthermore, Jaffe and

1E.g., U.S. v. H& R Block Inc., et al.; U.S. v. AT&T Inc., et al.; U.S. v. Bazaarvoice, Inc.; FTC v.

Sysco Corporation, et al.; U.S. v. AB Electrolux, et al.

2See Section 6.1.

3The FTC Press Release is available online: https://www.ftc.gov/system/files/documents/public_

statements/681781/150713dollartree-jdwstmt.pdf.

4See also Shapiro (2010), who writes that:

The value of diverted sales, taken alone, does not purport to quantify the magnitude of any

1
Weyl (2013) show that UPP must be scaled by an appropriate measure of pass-through to

provide a first order approximation to price effects. Obtaining estimates of the requisite passthrough information can be difficult even in advantageous empirical settings (e.g., Miller,

Osborne and Sheu (2015)).

In this paper, we revisit whether UPP can be applied to predict the magnitude of

price effects. We begin with a theoretical discussion in which we develop that, in some

standard settings, using an identity matrix to proxy for the relevant pass-through matrix

may introduce only limited misspecification error. By implication, UPP itself may provide

a reasonable approximation to the true price effects. We then explore this possibility using

a large-scale Monte Carlo experiment that simulates mergers in markets with differentiated

products. Results indicate that UPP is quite accurate with standard log-concave demand

systems, but understates price effects if demand exhibits greater convexity. The prediction

error that arises with UPP does not systematically exceed the prediction error that occurs

due to functional form misspecification in simulation models, nor is it much greater than the

prediction error that arises in correctly specified simulation models that rely on imprecise

econometric estimates of their structural parameters.

The Monte Carlo experiments follow the data generating process developed in Miller,

Remer, Ryan and Sheu (2016). We repeatedly draw randomized market shares and a single

price-cost margin, and use these data to calibrate the parameters of a logit demand system.

We then calibrate the less restrictive linear, almost ideal, and log-linear demand systems to

match the elasticities of the logit model, sometimes incorporating a degree of measurement

error in the elasticities. The analysis thus features two log-concave demand systems (linear

and logit demand) alongside two demand systems that exhibit greater convexity (e.g., almost ideal and log-linear demand). These four demand systems are commonly employed in

antitrust analyses (Werden, Froeb and Scheffman (2004); Werden and Froeb (2008)), and

also have been used in academic studies that examine the effect of demand curvature on the

precision of counterfactual simulations (e.g., Crooke, Froeb, Tschantz and Werden (1999);

Huang, Rojas and Bass (2008)).5

The analysis is subject to a number of caveats and limitations. The most serious pertain

post-merger price increase.... The value of diverted sales is a measure of the extra (opportunity)

cost the merged firm bears in selling units of Product 1. Higher costs give the merged firm an

incentive to raise the price of Product 1. But further analysis is needed to determine how that

cost increase translated into a price increase.

5Our research also is related to Cheung (2013), who compares the performance of predictions arising from

a structurally estimated merger simulation model with those from UPP in the context of the airline industry.

2
to external validity. First, we focus exclusively on pricing in differentiated products markets,

although the UPP calculation itself can be generalized to other settings (e.g., Jaffe and Weyl

(2013)). Second, we impose the Nash-Bertrand equilibrium concept throughout the data

generating process in order to focus the analysis on the “unilateral effects” of mergers. UPP

is unlikely to perform as well for mergers that create coordinated effects. Lastly, we make

a number specific assumptions about the demand systems and marginal cost functions that

are necessitated by the Monte Carlo approach. We do not seek to provide the most general

results available. Instead, we seek to establish certain relationships that advance the dialog

on UPP and motivate future research.

The remainder of the paper proceeds as follows. Section 2 details the theoretical

connection between UPP, first order approximation, and the price effects of mergers. Section

3 describes the Monte Carlo experimental design and provides summary statistics. Section

4 presents the results. There we plot the raw data, compare the prediction error that arises

from UPP with that from merger simulation, and also discuss the use of UPP and HHI-based

measures as early-stage screens. Section 5 concludes with a summary and a discussion of

the appropriate scope of application for UPP.

2 Theoretical Framework

2.1 Merger price effects and UPP

We examine the connection between different methods of merger price prediction within the

context of Nash-Bertrand price competition between multi-product firms. Assume that each

firm, i, produces a subset of products available to consumers, faces a twice-differentiable

demand function, and maximizes the following profit function:

πi = P

T

i Qi(P) − Ci(Qi(P))

where Piis a vector of firm i’s prices, Qiis a vector of firm i’s unit sales, P is a vector

containing the prices of every product, and Ciis the cost function. The superscript T

denotes the vector/matrix transpose. Profit-maximizing prices are characterized by firstorder conditions:

fi(P) ≡ − "

∂Qi(P)

∂Pi

T

#−1

Qi(P) − (Pi − MCi) = 0, (1)

3
where MCi = ∂Ci/∂Qi

is a vector of firm i’s marginal costs. Now consider a merger between

two firms j and k that, for simplicity, does not affect the cost functions. The post-merger

first-order conditions are given by

hi(P) ≡ fi(P) + gi(P) = 0 ∀i ∈ I (2)

where

gj (P) = −



∂Qj (P)

T

∂Pj

−1 

∂Qk(P)

T

∂Pj



| {z }

Matrix of Diversion from j to k

(Pk − MCk)

| {z }

Markup of k

(3)

and gk(P) is defined analogously, while gi(P) = 0 for all i 6= j, k. Prices that satisfy the postmerger first order conditions can be computed given sufficient information on the demand

system and marginal costs. Directly computing post-merger prices using this information

is referred to as merger simulation and has been a main focus of research spanning more

than two decades; numerous literature reviews summarize the topic (e.g., Werden and Froeb

(2008); Budzinski and Ruhmer (2010); Baker and Reitman (2013)).

The merger can be interpreted as creating an opportunity cost within the joined firm.

Aggressive pricing from one merging partner creates forgone profits that otherwise would be

earned by the other. The magnitude of these opportunity costs – given by the g(P) function

– depends multiplicatively on the customer diversion rates between the merging firms and

their markups. Reinforcing this interpretation is the fact that both marginal costs and gi(P)

are additively separable in the post-merger first order conditions. Farrell and Shapiro (2010)

refer to these opportunity costs as the UPP due to the merger. They propose UPP as

an initial screen in merger investigations, on the basis that higher marginal costs tend be

associated with higher prices.

2.2 UPP as a price predictor

The equilibrium post-merger price effects in this model depend upon how pricing pressure

is passed through to consumers. In merger simulation models, pass-through behavior is

determined by the demand system, and there is existing research that explores how functional

form restrictions on demand affect the accuracy of simulation (e.g., Crooke, Froeb, Tschantz

and Werden (1999); Miller, Remer, Ryan and Sheu (2016)). A somewhat more general

solution to calculating merger price effects is provided by Jaffe and Weyl (2013). If passthrough is observed (or can be estimated from data) then a first order approximation to the

4
price change can be obtained as

∆P = −



∂h(P)

∂P −1











P =P0

g(P

0

) (4)

where P

0

is the vector of pre-merger prices. The first order approximation equals UPP premultiplied by the opposite inverse Jacobian of h(P), which Jaffe and Weyl refer to as the

merger pass-through matrix. By inspection, merger pass-through depends on the first and

second derivatives of demand, while omitting higher order terms. Miller, Remer, Ryan and

Sheu (2016) provide Monte Carlo evidence that the first order approximation is an accurate

predictor of true price effects.

With this foundation in place, it follows that UPP itself may provide a useful prediction

of the price effect, insofar as the identity matrix can accurately proxy for the merger passthrough matrix. We provide a simple numerical example to fix ideas. Consider three firms,

each of which has a margin of 0.50 and a 30% market share (the outside good has a 10%

share). Consumer behavior is given by the logit demand system. With a merger between

the first two firms, equation (4) becomes







0.204

0.204

0.052





 =







0.771 0.180 0.297

0.180 0.771 0.297

0.122 0.122 0.776













.214

.214

0





 (5)

Here the value of UPP (0.214) nearly equals the first order approximation (0.204) for the

merging firms. This happens because the diagonal elements of the merger pass-through

matrix are somewhat below one, while the off-diagonal elements are positive. Thus, using

an identify matrix to proxy merger pass-through overstates some effects and understates

others; the balance is a prediction close to the first order approximation. Further, again in

this example, it is worth noting that both UPP and the first order approximation are close

to the true price effects (0.190 for the merging firms).

This idea extends beyond the simple example provided. Countervailing biases arise

provided that (i) the diagonal elements of the merger pass-through matrix are below unity,

and (ii) prices are strategic complements so the off-diagonal elements are positive. In such

settings, UPP may provide a reasonable approximation to the true price effects. The two

demand systems we consider that exhibit log-concavity (linear and logit) always satisfy both

conditions.6If instead demand exhibits greater convexity, then the diagonal elements of the

6Log-concavity is sufficient to ensure incomplete cost pass-through (e.g., Bulow and Pfleiderer (1983)),

5
merger pass-through matrix can exceed one, and UPP should understate price effects. As

we develop below, this typically is the case with almost ideal and log-linear demand.

The limited informational requirements of UPP could make it a valuable tool for antitrust authorities. UPP can be calculated with diversion and markups for only the merging

parties, and such information often becomes available during the course of merger investigations. By contrast, merger simulation models typically require a full set of demand

elasticities, encompassing consumer responses to the prices of all firms in the model. FOA

requires these demand elasticities along with pass-through. To the extent that UPP provides

accurate price predictions, the importance of obtaining elasticities and pass-through would

be diminished. This motivates the Monte Carlo experiments developed below.

2.3 Market shares and HHI

The economic theory outlined above demonstrates that competition between merging firms,

as characterized by diversion ratios and markups, is directly related to unilateral price effects

in markets with differentiated products. Except in a special case, no such direct theoretical

connection exists between unilateral price effects and market concentration. Nevertheless,

the 2010 Guidelines maintain that the HHI is useful in informing competitive effects, at

least in the broad sense of “identify[ing] some mergers unlikely to raise competitive concerns

and some others for which it is particularly important to examine whether other competitive factors confirm, reinforce, or counteract the potentially harmful effects of increased

concentration.”7 We calculate HHI as the sum of squared market shares:

HHI =

X

i

s

2

i

(6)

where siis between 0 and 100 and represents the market share of firm i. The change in HHI

due to a merger between firms j and k requires only the merging parties’ market shares:

∆HHI = 2sjsk (7)

The Monte Carlo experiments allow us to evaluate the accuracy of the HHI statistics as a

screening device. Because HHI and ∆HHI are merger-specific statistics (unlike UPP which

is firm-specific), we compare them to the average price change of the merging firms.

The direct theoretical connection between unilateral price effects and market concenbut cost pass-through differs somewhat from merger pass-through. See Jaffe and Weyl (2013) for details.

7See the 2010 Horizontal Merger Guidelines, Section 5.3.

6
tration arises if consumer diversion is proportional to market share. Then diversion from

product j to product k equals sk/(1−sj ), and can be approximated by sk(1+sj ) for small sj.

Diversion from k to j is analogous, meaning that the sum of the approximate diversion ratios

is sj + sk + 2sjsk or, equivalently, sj + sk + ∆HHI.8 This provides a theoretical foundation

for the many empirical studies that relate merger price effects to the predicted change in

HHI (e.g., Dafny, Duggan and Ramanarayanan (2012); Ashenfelter, Hosken and Weinberg

(2015)). Even in this special case, the correlation between ∆HHI and the price effects may

be weak because pricing pressure is determined through the interaction of diversion and

markups. Lastly, we note that the level of HHI often is influenced greatly by the shares of

non-merging firms. While the strategic reactions of rivals affect post-merger equilibrium,

typically they are of secondary importance.9

3 Design of the Monte Carlo Experiments

3.1 Data generation

We generate data that are consistent with the theoretical model outlined in the previous

section. Each draw of simulated data is independent and characterizes the pre-merger equilibrium conditions of a single market. Together, the data cover a wide range of competitive

conditions. The markets contain six firms that produce differentiated products at constant

marginal cost. Firms compete in prices and equilibrium is Nash-Bertrand. All pre-merger

prices are normalized to one, which results in price effects that are identical in levels and

percentages. The specific data generating process is as follows:

1. Randomly draw (i) market shares for six firms and an outside good, and (ii) the first

firm’s margin based on a uniform distribution bounded between 0.20 and 0.80.

2. Calibrate the parameters of a logit demand system based on the margin and market

shares, and calculate the demand elasticities that arise in the pre-merger equilibrium.

This entails selecting demand parameters that rationalize the random data. The parameters are exactly identified given market shares, prices, and a single margin.

3. Calibrate linear, almost ideal, and log-linear demand systems based on the logit demand elasticities. The parameters of these systems are exactly identified given market

8We first encountered these mathematics in Shapiro (2010).

9The market shares of competitors is more relevant for the likelihood and repercussions of post-merger coordinated effects. We confine the analysis to unilateral effects by maintaining the Nash-Bertrand equilibrium

concept throughout the paper.

7
shares, prices, and the logit elasticities.10

4. Calculate UPP for a merger of the first and second firms in each market, based on the

elasticities and margins for each draw of data. UPP is invariant to the demand system.

Simulate the effect of the merger under each of the demand systems.

5. Repeat steps (1) - (4) until 4,500 draws of data are obtained.

The algorithm generates 18,000 mergers to be examined, each defined by a draw of data and

a demand system. See Appendix A for mathematical details on the calibration process.

The data generating process allows us to assess the accuracy of UPP both in absolute

terms and relative to merger simulation conducted with a functional form misspecification

or with imprecise demand elasticities. To develop the first comparison, we note that the

elasticities of each demand system are identical in the pre-merger equilibrium, for a given

draw of data, so that differences in price effects arise solely due to functional form. Suppose

the true demand system is logit. The prediction error of UPP can be compared against

simulation results using almost ideal, linear, and log-linear demand.

For the second comparison, we incorporate imprecision into the observed demand elasticities, and evaluate how the predictive accuracy of UPP and simulation degrade. Specifically, we add a uniformly distributed error to each product’s own-price elasticity of demand.

To do so in a manner that preserves the property that own-price elasticities are less than

negative one, we define the observed own-price elasticities to be

ekk = kk + ν where ν ∼ U(−t(kk + 1), t(kk + 1)) (8)

The support of the error is element-specific and depends on t ∈ [0, 1]. We examine three levels of error: t = (0.2, 0.5, 0.8). We then scale each product’s cross-price elasticity according

to the percent error of that products’ own-price elasticity, i.e. ejk = jk

ekk

kk

. This restriction

eliminates economically unlikely scenarios in which a substitution away from a given product (in response to a price increase) is exceeded by substitution to other products. When

we generate data without such a restriction, even modest amounts of error often result in

negative price predictions. We focus the second comparison on the linear, almost-ideal, and

log-linear demand systems because these can accommodate changes in the elasticity matrix

10In the pre-merger equilibrium, consumer substitution between products is proportional to market share

because all the systems are calibrated based on logit elasticities. This reduces the dimensionality of the

random data that must be drawn. The diversion-by-share property is retained away from the pre-merger

equilibrium only for logit demand.

8
with a fixed set of market shares.

3.2 Summary statistics

Table 1 summarizes the empirical distributions of the data. The distribution of firm 1’s share

is centered around 14 percent, which reflects that shares are allocated among six products

and the outside good. The margin distribution is determined by the uniform draws with

support over (0.20, 0.80). The own-price elasticity of demand, which equals the inverse

margin, has a distribution centered around two. 90 percent of the own-price elasticities fall

between 1.31 and 4.37. The diversion ratios have a distribution centered at .17 , and 90

percent of diversion ratios fall between .02 and .32.

Market concentration is based on pre-merger market shares, following equations (6)

and (7). The median pre- and post-merger HHI are 1562 and 1931, respectively, and the

median change is 317. The simulated merger results in an HHI increase of greater than 200

in 65 percent of markets, an increase of between 100 and 200 in 15 percent of markets, and

an increase of less than 100 in 19 percent of markets. The median UPP is 0.07, and 90

percent of UPP values fall between 0.01 and 0.22. All of the above statistics are invariant

to the posited demand system, because the demand systems are calibrated to produce the

same first-order characteristics in the pre-merger equilibria.11

The median merger price effects are 0.06, 0.11, 0.05, and 0.18 for the logit, almost

ideal, linear, and log-linear demand systems, respectively. Because pre-merger prices are

normalized to one, these statistics reflect both the median level change and median percentage change. Dispersion across demand systems reflects the specific pass-through properties

of the systems, with greater own pass-through associated with larger price effects. This relationship, first observed in Froeb, Tschantz and Werden (2005), is explained by the theoretical

results of Jaffe and Weyl (2013). Dispersion within demand systems mainly reflects the range

of market conditions that arise from the data generating process, which is demonstrated in

part by the range of UPP.

11In calculating HHI, we use the shares of the six firms that strategically react to the merger, and ignore

the share of the outside good. This is equivalent to an assumption that the outside good is sold by an infinite

number of atomistic firms, and leads to a conservative estimate of the HHI level.

9
Table 1: Order Statistics

Median 5%. 10% 25% 75% 90% 95%

Market Conditions

Market share 0.15 0.02 0.04 0.08 0.20 0.25 0.28

Margin 0.49 0.23 0.26 0.35 0.64 0.73 0.76

Elasticity 2.03 1.31 1.37 1.56 2.89 3.87 4.37

Diversion 0.17 0.02 0.04 0.09 0.23 0.29 0.32

Herfindahl–Hirschman Index (HHI)

Pre-Merger 1562 1163 1231 1361 1816 2073 2255

Post-Merger 1931 1346 1452 1642 2277 2667 2937

Change 317 22 48 139 555 833 968

Upward Pricing Pressure

UPP 0.07 0.01 0.01 0.04 0.12 0.18 0.21

Merger Price Effects

Logit 0.06 0.01 0.01 0.03 0.11 0.16 0.20

AIDS 0.11 0.01 0.02 0.05 0.28 0.69 1.14

Linear 0.05 0.00 0.01 0.02 0.08 0.12 0.15

Log-Linear 0.18 0.01 0.03 0.08 0.46 1.18 2.13

Notes: Summary statistics are based on 4,500 randomly-drawn sets of data

on the pre-merger equilibria. The market share, margin and elasticity are for

the first firm. The cross-diversion is diversion from firm 1 to firm 2 (the two

merging firms). Market share and margin are drawn randomly in the data

generating process. The elasticity is the own-price elasticity of demand and

equals the inverse margin. The merger price effects are the change in firm 1’s

equilibrium price.

10
Figure 1: Graphical Illustration of UPP as a Price Predictor

Notes: The scatter plots characterize the accuracy of UPP as a price prediction when the underlying demand

system is logit, almost ideal, linear, and log-linear. Each dot represents the first firm’s predicted and actual

post-merger prices for a given draw of data.

4 Results

4.1 Graphical analysis

We begin by plotting the data. Figure 1 depicts the accuracy of UPP in predicting postmerger price increases under each of the demand systems considered. Each dot represents

the predicted and true changes in firm 1’s price for a given draw of data; its vertical position

is the prediction of simulation and its horizontal position is the true price effect. Dots that

fall along the 45-degree line represent exact predictions while dots that fall above (below)

the line represent over (under) predictions.

UPP appears to be quite accurate, albeit somewhat larger in magnitude than the true

price effect, if the underlying demand is logit or linear. These systems are log-concave so the

two biases developed in Section 2 are countervailing. UPP exceeds the actual price increases

because using ones to proxy the diagonal pass-through elements (which amplifies predictions

with incomplete pass-through) has a somewhat larger affect on results than suppressing the

cross terms (which damps predictions with strategic complements). UPP understates price

increases with almost ideal and log-linear demand, both of which exhibit greater convexity.

Because the diagonal elements of merger pass-through with these systems typically exceed

unity, the biases caused by using an identity matrix as a proxy are not countervailing and

11
Figure 2: Prediction Error from Standard Merger Simulations

Notes: The scatter plots characterize the accuracy of merger simulations when the underlying demand system

is logit (column 1), almost ideal (column 2), linear (column 3), and log-linear (column 4). Merger simulations

are conducted assuming demand is logit (row 1), almost ideal (row 2), linear (row 3), and log-linear (row 4).

Each dot represents the first firm’s predicted and actual post-merger prices for a given draw of data.

instead tend to lower UPP relative to the actual price increases.

To put this in context, Figure 2 depicts the accuracy of merger simulation conducted

with incorrect functional form assumptions. The scatter plots show data sorted by the

underlying demand system: logit (column 1), almost ideal (column 2), linear (column 3),

and log-linear (column 4), and by the merger simulation model: logit (row 1), almost ideal

(row 2), linear (row 3), and log-linear (row 4). In many instances, the prediction error that

arise due to functional form misspecification visibly exceeds the prediction error of UPP.

This sensitivity of merger simulation to functional form assumptions is well known (e.g.,

Crooke, Froeb, Tschantz and Werden (1999); Miller, Remer, Ryan and Sheu (2016)) and, in

antitrust settings, it is standard practice to generate predictions under multiple assumptions

as a way to evaluate the scope of potential price changes.

For a second comparison, Figure 3 depicts the accuracy of merger simulation conducted

with imprecisely measured demand elasticities. We plot only part of the generated data, for

12
Figure 3: Prediction Error from Standard Merger Simulations

Notes: The scatter plots characterize the accuracy of merger simulations when there is error in the observed

elasticities of demand. Merger simulations are conducted assuming the true demand system is known: almost

ideal (column 1), linear (column 2), and log-linear (column 3). Each dot represents the first firm’s predicted

and actual post-merger prices for a given draw of data.

the sake of brevity, showing the cases in which the elasticities of demand are observed with

20% error (row 1), 50% error (row 2), 80% error (row 3). As one might expect, predictions are

centered around the true effects but prediction error increases as elasticities lose precision.

Interestingly, predictions are relatively robust to imprecision in the elasticities with linear

demand; we suspect this is because pass-through with linear demand does not change much

with the elasticities.12 We explore the relationship between these predictors and UPP in

greater detail next.

4.2 Numerical analysis

Table 2 presents the median absolute prediction error (“MAPE”) of UPP when the true

underlying demand system is logit, almost ideal, linear, or log-linear. UPP is quite accurate if demand is logit; the MAPE is roughly 10% of the median price effect. UPP loses

12See the mathematics in Miller, Remer and Sheu (2013).

13
Table 2: Median Absolute Prediction Error

Underlying Demand System:

Logit AIDS Linear Log-Linear

UPP 0.006 0.042 0.022 0.110

Logit Simulation 0.000 0.049 0.014 0.117

AIDS Simulation 0.050 0.000 0.068 0.065

Linear Simulation 0.014 0.066 0.000 0.132

Log-Linear Simulation 0.123 0.065 0.139 0.000

Notes: The table provides the median absolute prediction error of

UPP and standard simulations when the true underlying demand

system is logit, almost ideal, linear, and log-linear.

Table 3: Frequency with Which UPP Improves Accuracy

Underlying Demand System:

Logit AIDS Linear Log-Linear

Logit Simulation · 92.2% 3.2% 100%

AIDS Simulation 95.1% · 90.8% 10.6%

Linear Simulation 69.0% 98.5% · 99.0%

Log-Lin Simulation 100% 74.6% 100% ·

Notes: This table shows the fraction of mergers for which UPP has a

smaller absolute prediction error than standard merger simulations

in predicting the price change.

some accuracy if demand is linear or almost ideal, but prediction error remains small once

benchmarked against median price effects. With log-linear demand, the MAPE of UPP is

more than 50% of the median price effect. The table also shows the MAPEs that arise with

misspecified merger simulations. Looking across the four demand systems, the MAPE of

UPP is always smaller than at least two of the three misspecified simulations.

Table 3 shows the frequency with which UPP provides a more accurate prediction than

misspecified simulation. Among the twelve comparison groups the data generating process

allows, UPP dominates misspecified in all but two instances (the exceptions being logit simulation with linear demand and almost ideal simulation with log-linear demand). We interpret

the results summarized in Tables 2 and 3 as suggestive that the inaccuracies associated with

using UPP as a price predictor often are somewhat smaller than the inaccuracies that can

arise due to functional form misspecification in merger simulation models.

14
Table 4: MAPE with Imprecise Demand Elasticities

Panel A: Simulation

Underlying Demand System:

AIDS Linear Log-Linear

20% Error 0.007 0.001 0.012

50% Error 0.019 0.004 0.030

80% Error 0.031 0.006 0.049

Panel B: Upward Pricing Pressure

Underlying Demand System:

AIDS Linear Log-Linear

20% Error 0.042 0.022 0.110

50% Error 0.041 0.022 0.108

80% Error 0.045 0.023 0.104

Notes: Panel A shows the median absolute prediction error of

simulation when elasticities are observed with 20%, 50%, and

80% error and the true demand system is known. Panel B shows

the median absolute prediction error of UPP when elasticities

are observed with 20%, 50%, and 80% error.

We next consider merger simulation conducted with the correct demand system but

imprecise demand elasticities. In this exercise, we recalculate UPP based on the imprecise

elasticities in order to help facilitate an “apples-to-apples” comparison. Panel A of Table 4

summarizes the prediction errors that arise with merger simulation. The MAPEs increase

significantly with the amount of imprecisions, but remain small relative to the median price

prediction for each underlying demand system. Panel B shows that the MAPEs of UPP, by

contrast, do not increase much with imprecision. We suspect the reason is that effects of

imprecision are partially “canceled out” when the own and cross elasticities are converted into

diversion. Despite this robustness, the MAPEs of UPP exceed those of merger simulation

even with a substantial amount of measurement error. Consistent with these results, UPP

is more accurate than simulation in a minority of the mergers, at each level of imprecision

considered.13

13The analog to Table 3 is available upon request.

15
Table 5: UPP as a Screen

Underlying Demand System:

Logit AIDS Linear Log-Linear

False Positives (Type I Error) 5.0% 0.2% 18.4% 0.0%

False Negatives (Type II Error) 0.0% 22.4% 0.0% 36.6%

Notes: Panel A shows the fraction of mergers for which the price change is less

than 10% but UPP exceeds 10%. Panel B the fraction of mergers for which the

price change exceeds 10% but UPP is less than 10%.

4.3 Preliminary screens in merger analysis

4.3.1 Upward Pricing Pressure

The Monte Carlo experiments also allow us to assess the properties of UPP as a preliminary

screen in merger analysis. The evidence shown thus far – most visibly in Figure 1 – demonstrates that UPP is strongly correlated with price effects, a property that is highly desirable

in a screen. Indeed, UPP is almost perfectly correlated with price changes under logit and

linear demand, and highly correlated with price changes under almost ideal and log-linear

demand. The correlation coefficients are 0.996, 0.955, 0.857 and 0.895, respectively.

Consider the following thought experiment: the objective of the antitrust authority

is to block mergers that increase price more than 10%, and employs a screen in which it

investigates if and only if UPP exceeds 10%. How well would the antitrust authority sort

mergers?14 Table 5 provides the frequency of the two possible errors: “false positives” (benign

mergers that are investigated) and “false negatives” (anticompetitive mergers that are not

investigated). Results are broken out by the true underlying demand system. If demand is

log-concave (i.e., linear or logit) then false positives are much more likely than false negatives.

From a policy standpoint, this probably is desirable to the extent that false positives can be

identified and cleared in the subsequent investigation, while no such correction is available

with false negatives. If demand instead exhibits considerable convexity then false negatives

far exceed false positives. This is problematic from a policy standpoint, by the same logic.

Taken together, the results support the notion that it is prudent to employ an inclusive UPP

screen, at least to the extent that substantial convexity in demand is deemed realistic.

14We select 10% solely based on the empirical distribution of price changes in the data. We have examined

other thresholds, and the qualitative results are unaffected.

16
4.3.2 Herfindahl-Hirschman Index

The 2010 Horizontal Merger Guidelines define a set of HHI levels and changes to help stratify

mergers into those that are unlikely to pose a problem and those that warrant a close

investigation. Five categories are defined as follows:

(i) Post-merger HHI>2500 and ∆HHI>200.

(ii) Post-merger HHI>2500 and ∆HHI∈(100,200].

(iii) Post-merger HHI∈(1500,2500] and ∆HHI>100.

(iv) Post-merger HHI≤1500.

(v) ∆HHI<100.

The Guidelines state that categories (i)-(iii) are likely to raise competitive concern and lead

to further investigation, while mergers in categories (iv) and (v) are unlikely to create competitive problems. Many in the antitrust community view the latter categories as providing

safe harbors, although this is not specifically endorsed in the Guidelines. Our Monte Carlo

experiments allow us to evaluate these HHI thresholds. Because the theoretical connection

between ∆HHI and unilateral effects is strongest with proportional consumer substitution,

the results can be interpreted from a best-case scenario for the concentration measures.

Table 6 shows the fraction of mergers that result in prices increases of at least 5%

(Panel A) and 10% (Panel B), sorted by HHI category. Mergers in categories (i) and (iii)

frequently produce substantial price increases. This is especially true of mergers in category

(i), which generate price elevations above 5% in around 90% of the mergers with logit

and linear demand, and in more than 95% of the mergers with almost ideal and log-linear

demand. Perhaps more surprising, mergers in category (ii) appear relatively benign, and

never produce a 5% price increase with log-concave demand. The results for category (iv)

indicate that a nontrivial minority of mergers in markets that are not deemed “moderately

concentrated” nonetheless result in price increases of 5% or higher. By contrast, virtually

no mergers in category (v) result in such price increases if demand is log-concave.

These result reinforce that ∆HHI is more directly connected to unilateral effects theory

than the post-merger HHI (e.g., see Section 2). Accordingly, we investigate whether the

∆HHI could be used effectively as a screen. The data indicate a strong correlation between

∆HHI and the price change. The correlation coefficients range between 0.519 and 0.846

depending on the demand system. Table 7 shows the fraction of mergers that result in

17
Table 6: HHI Category Screens

Panel A: Frequency of 5% Price Increase

Underlying Demand System:

Logit AIDS Linear Log-Linear

Category (i) 90.9% 95.7% 86.7% 98.8%

Category (ii) 0.0% 33.3% 0.0% 61.9%

Category (iii) 63.8% 79.8% 45.1% 92.4%

Category (iv) 19.3% 46.6% 6.0% 61.5%

Category (v) 0.2% 20.7% 0.0% 30.2%

Panel B: Frequency of 10% Price Increase

Underlying Demand System:

Logit AIDS Linear Log-Linear

Category (i) 53.2% 80.2% 49.5% 93.6%

Category (ii) 0.0% 9.5% 0.0% 33.3%

Category (iii) 17.7% 57.1% 8.0% 73.9%

Category (iv) 0.7% 27.0% 0.0% 36.7%

Category (v) 0.0% 5.8% 0.0% 9.9%

Notes: Panel A shows the fraction of mergers in each HHI

category for which the weighted-average change in the merging firms’ prices is greater than 5%. Panel B shows the the

same statistic for a price increase greater than 10%. Category (i): Post-merger HHI>2500 and ∆HHI>200. Category

(ii): Post-merger HHI>2500 and ∆HHI∈(100,200]. Category (iii): Post-merger HHI∈(1500,2500] and ∆HHI>100.

Category (iv): Post-merger HHI≤1500. Category (v):

∆HHI<100.

18
Table 7: Screens Based on ∆HHI

Panel A: Frequency of 5% Price Increase

Underlying Demand System:

Logit AIDS Linear Log-Linear

∆HHI >200 76.1% 87.0% 60.3% 96.9%

∆HHI∈(100,200) 20.2% 53.4% 0.3% 71.7%

∆HHI<100 0.2% 20.7% 0.0% 30.2%

Panel B: Frequency of 10% Price Increase

Underlying Demand System:

Logit AIDS Linear Log-Linear

∆HHI>200 27.4% 65.8% 17.6% 81.9%

∆HHI∈(100,200) 0.0% 30.9% 0.0% 45.5%

∆HHI<100 0.0% 5.8% 0.0% 9.9%

Notes: Panel A shows the fraction of mergers in each ∆HHI

category for which the weighted-average change in the merging firms’ prices is greater than 5%. Panel B shows the same

statistic for a price increase greater than 10%.

prices increases of at least 5% (Panel A) and 10% (Panel B), this time sorted by ∆HHI.

Results are provided for (i) change in HHI greater than 200, (ii) change in HHI between

100 and 200 and (iii) change in HHI less than 100. Most mergers with ∆HHI>200 produce

substantial price increases, as do a nontrivial minority of mergers with ∆HHI∈(100,200).

Mergers with ∆HHI<100 also cause substantial price increases – notice that this corresponds

with category (v) above. Virtually no mergers with ∆HHI<100 result in substantial price

increases if demand is log-concave (this corresponds to category (v) above).

5 Conclusion

This research evaluates the accuracy of UPP in predicting post-merger price changes, using

a large-scale Monte Carlo experiment. The results are supportive overall: we find that UPP

is quite accurate with standard log-concave demand systems, but understates price effects if

demand exhibits greater convexity. Prediction error does not systematically exceed that of

misspecified simulation models, nor is it much greater than that of correctly-specified models

simulated with imprecise demand elasticities. We provide a theoretical basis for these results

19
by observing that UPP is a restricted version of the first order approximation derived in Jaffe

and Weyl (2013). We conclude that UPP has greater utility than currently is recognized.

That UPP often outperforms simulation models in our Monte Carlo experiments raises

a question about the appropriate scope of application. In our view, the value of UPP as

a price predictor is greatest in merger investigations and similar policy endeavors, due to

its expediency and simplicity. The academic literature provides an array of methodologies

that are capable of both limiting functional-form misspecification in simulation models and

reducing standard errors in structural estimation. These methodologies (which we do not

examine in the Monte Carlo experiments) may well allow simulation to produce more robust

and accurate predictions than are available from UPP. Thus, we are skeptical that our results

have significant bearing on empirical industrial economics. By contrast, because state-of-art

academic methodologies often may be too time-consuming to be used within the constraints

of merger investigations, our results are immediately relevant for antitrust practice.

20
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23
Appendix

A Mathematical Details of the Calibration Process

We provide mathematical details on the calibration process in this appendix. To distinguish

the notation from that of Section 2, we move to lower cases and let, for example, si and pi

be the market share and price of firm i’s product, respectively.15 Recall that in the data

generating process we randomly assign market shares among the four single-product firms

and the outside good, draw the price-cost margin of the first firm’s product from a uniform

distribution with support over (0.2, 0.8), and normalize all prices to unity. The calibration

process then obtains parameters for the logit, almost ideal, linear and log-linear demand

systems that reproduce these draws of data.

Calibration starts with multinomial logit demand, the basic workhorse model of the

discrete choice literature. The system is defined by the share equation

si =

e

(δi−αpi)

1 + PN

j=1 e

(δj−αpj )

. (A.1)

The parameters to be calibrated include the price coefficient α and the product-specific

quality terms δi. We recover the price coefficient by combining the data with the first order

conditions of the first firm. Under the assumption of Nash-Bertrand competition this yields:

α =

1

m1p1(1 − s1)

(A.2)

where m1 is the price-cost margin of firm 1. We then identify the quality terms that reproduce

the market shares:

δi = log(si) − log(s0) + αpi, (A.3)

for i = 1 . . . N. We follow convention with the normalization δ0 = 0. Occasionally, a set

of randomly-drawn data cannot be rationalized with logit demand and we replace it with a

set that can be rationalized. This tends to occurs when the first firm has both an unusually

small market share and an unusually high price-cost margin.

The logit demand system sometimes is criticized for its inflexible demand elasticities.

Here, the restrictions on substitution are advantageous and allow us to obtain a full matrix

of elasticities with a tractable amount of randomly drawn data. The derivatives of demand

15We define market share si = qi/

PN

j=1 qj , where qi represents unit sales.

24
with respect to prices, as is well known, take the form

∂qi

∂pj

=

(

αsi(1 − si) if i = j

−αsisjif i 6= j

(A.4)

We use the logit derivatives to calibrate the more flexible almost ideal, linear and log-linear

demand systems. This ensures that each demand system has the same first order properties

in the pre-merger equilibrium, for a given draw of data.

The AIDS is written in terms of expenditure shares instead of quantity shares Deaton

and Muellbauer (1980). The expenditure share of product i takes the form

wi = αi +

X

N

j=0

γij log pj + βilog(x/P) (A.5)

where x is total expenditure and P is a price index. We incorporate the outside good

as product i = 0 and normalize its price to one; this reduces to N2the number of price

coefficients in the system that must be identified (i.e., γij for i, j 6= 0). We further set βi = 0

for all i, a restriction that imposes in income elasticity of unity. Under this restriction, total

expenditures are given by

log(x) = (αe + uβe) +X

N

k=1

αk log(pk) + 1

2

X

N

k=1

X

N

j=1

γkj log(pk) log(pj ) (A.6)

for some utility u. We identify the sum αe + uβe rather than α, u e and βe individually.16 Given

this structure, product i’s unit sales are given by qi = xwi/pi and the first derivatives of

demand take the form

∂qi

∂pj

=

( x

p

2

i

(γii − wi + w

2

i

) if i = j

x

pipj

(γij + wiwj ) if i 6= j

(A.7)

The calibration process for the AIDS then takes the following four steps:

1. Calculate x and wifrom the randomly drawn data on market shares, using a market

size of one to translate market shares into quantities.

2. Recover the price coefficients γij for i, j 6= 0 that equate the AIDS derivatives given in

16The price index P is defined implicitly by equation (A.6) as the combination of prices that obtains utility

u given expenditure x. A formulation is provided in Deaton and Muellbauer (1980).

25
equation (A.7) and the logit derivatives given in equation (A.4). Symmetry is satisfied

because consumer substitution is proportional to share in the logit model. The outside

good price coefficients, γi0 and γ0ifor all i, are not identified and do not affect outcomes

under the normalization the p0 = 1. Nonetheless, they can be conceptualized as taking

values such that the adding up restrictions PN

i=0 γij = 0 hold for all j.

3. Recover the expenditure share intercepts αifrom equation (A.5), leveraging the normalization that βi = 0. The outside good intercept α0 is not identified and does not

affect outcomes, but can be conceptualized as taking a value such that the adding up

restriction PN

i=0 αi = 1 holds.

4. Recover the composite term (αe + uβe) from equation (A.6).

This process creates an AIDS that, for any given set of data, has quantities and elasticities

that are identical in the pre-merger equilibrium to those that arise under logit demand. The

system possess all the desirable properties defined in Deaton and Muellbauer (1980). Our

approach to calibration differs from Epstein and Rubinfeld (2001), which does not model the

price index as a function of the parameters, and from Crooke, Froeb, Tschantz and Werden

(1999), which assumes total expenditures are fixed.

We turn now to the linear and log-linear demand systems. Linear demand takes the

form

qi = αi +

X

j

βijpj (A.8)

The parameters to be calibrated include the firm specific intercepts αi and the price coefficients βij . We recover the price coefficients directly from the logit derivatives in equation

(A.4). We then recover the intercepts to equate the implied quantities in equation (A.8)

with the randomly drawn market shares, again using a market size of one. Of similar form

is the log-linear demand system:

log(qi) = γi +

X

j

ij log pj (A.9)

where the parameters to be calibrated are the intercepts γi and the price coefficients ij .

Again we recover the price coefficients from the logit derivatives (converting first the derivatives into elasticities). We then recover the intercepts to equate the implied quantities with

the market share data. This process creates linear and log-linear demand systems that,

for any given set of data, has quantities and elasticities that are identical to those of the

calibrated logit and almost ideal demand systems, in the pre-merger equilibrium.

26