Measurement Error and Misspecification in DemandBased Diversion Ratios
Patrick Sun
August 3, 2022
OEA Working Paper 53
Office of Economics and Analytics
Federal Communications Commission
Washington, DC 20554
Abstract: When analyzing a horizontal merger it is often important to determine the extent to which the
products of the merging firms are close substitutes. A commonlyused measure to assess the closeness of
substitution is the diversion ratio: the fraction of demand leaving a product due to a price increase that
goes to a specific rival product. One method of estimating diversion ratios is to first estimate a demand
system and then calculate the implied diversion ratios. When estimating a demand system, two issues
arise. First, using incorrect data values leads to measurement error. Second, using an incorrect model of
demand leads to specification error. Through simulated datasets and using a known demand system, I
examine how these errors can bias the diversion ratio estimates and a related, preliminary estimate of
competitive harm, the Gross Upward Pricing Pressure Index (GUPPI). I find that even moderate
measurement error results in biases comparable to the biases in sharebased estimates of diversion that
underestimate diversion between similar products. Further, I find that model specification error can result
in substantial bias in the diversion and GUPPI estimates resulting in either overestimates or
underestimates, depending on the specific nature of the specification error. My results suggest that: (1)
measurement error is a serious concern when using demographic data to proxy for differences in price
sensitivity if that data do not very accurately represent the sample demographics; (2) practitioners should
prefer flexible random coefficient models to avoid specification error; (3) practitioners should consider
using observed markups for GUPPIs instead of estimating them; and (4) practitioners should more
carefully consider the use of demandbased diversion ratios relative to alternatives..
OEA Working Paper
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OEA Working Paper
Measurement Error and Misspecification in DemandBased Diversion Ratios
1. Introduction
When analyzing a horizontal merger it is often important to determine the extent to which the
products of the merging firms are close substitutes. A commonlyused measure to assess the closeness of
substitution is the diversion ratio which is emphasized in the 2010 Horizontal Merger Guidelines. 1 The
diversion ratio from an “origin” product to a “destination” product is defined as the share of quantity
demanded of the origin product that is lost due to an increase in its price that is captured by the
destination product. Diversion ratios, along with gross margin estimates, provide information on the
likely incentive of the merged firm to raise prices postmerger, because a higher diversion ratio means
that a higher percentage of lost sales caused by the price rise will go to the merging partner’s products.
While estimates of diversion ratios based on demand system estimates are increasingly used in practical
antitrust applications, little attention has been paid to the degree to which these estimates may be biased
by measurement error and demand system misspecification. I examine the effect of measurement error
and demand system misspecification using Monte Carlo experiments. I find that in certain situations both
can lead to diversion ratio bias that is comparable to or worse than market sharebased diversion ratios, ,
which are accurate for only very specific demand systems.
There are three main ways of estimating diversion ratios. First, diversion ratios can be directly
calculated from data on consumer switching between products. One drawback of this approach is that,
while firms often have data on gained or lost sales, they often do not have data on the specific products to
which or from which consumers switch. 2 Further, such data often do not include why the consumers
switched.3 If consumers switch for reasons other than a price change, the implied diversion would not
necessarily reflect what would happen after a price change, a major subject of interest in a merger
review. 4
Second, diversion ratios can be calculated as the market share of the destination product divided
by one minus the share of the origin product. This “market shareproportional” or “market sharebased”
diversion ratio is easy to calculate from readily available market share data. It suffers from the problem
that it is only a good estimate of diversion ratios if product switching happens in proportion to the product
1 Horizontal Merger Guidelines, U.S. Department of Justice and the Federal Trade Commission at § 6.1 (Aug. 19,
2010). See generally Carl Shapiro & Howard Shelanski, Judicial Response to the 2010 Horizontal Merger
Guidelines, 58 Rev. Indus. Org. 51 (2021) and Tommaso Valletti & Hans Zenger, Mergers with Differentiated
Products: Where Do We Stand? 58 Rev. Indus. Org. 179 (2021).
2 Consulting company Oxera notes that data for calculating diversion ratios directly “is rarely complete or available
in an appropriate form.” Oxera, Diversion Ratios: Why Does It Matter Where Customers Go if a Shop Is Closed?
Agenda: Advancing Economics in Business (Feb. 15. 2009), https://www.oxera.com/wp
content/uploads/2018/07/Diversionratiosupdated_11.pdf1.pdf. For issues regarding estimating diversion ratios
from survey data, see generally Kirsten Edwards, Estimating Diversion Ratios: Some Thoughts on Customer Survey
Design, in European. Competition Law Annual 2010: Merger Control in European and Global Perspective 31 (eds.
Philip Lowe and Mel Marquis 2013).
3 Knowing why consumers switched would require a survey in which a question was asked about why they
switched, or “experimental” variation, where switching occurred after change in price when no other factors
substantially changed.
4 See Yongmin Chen & Marius Schwartz, Churn Versus Diversion In Antitrust: An Illustrative Model, 83
Economica 564 (2016).
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choices of the market as a whole.5 An example of such a case would be a demand system in which
consumers choose products with certain probabilities, and these probabilities are the same for every single
consumer. In contrast, if the consumers have differing tastes causing probabilities to vary across
consumers, then market sharebased diversion ratios likely would be a poor fit.
Third, diversion ratios can be estimated by first estimating a demand system and then
mathematically deriving the implied diversion ratios. I refer to this as “demandbased diversion ratio
estimation.” Demandbased diversion ratios do not require switching data, and if demand is estimated
well, the diversion ratios will capture substitution patterns that cannot be captured by market sharebased
diversion.6 Some examples in merger review include the applicants’ analysis in AT&TDirecTV, 7 the
U.S. government’s analysis in AetnaHumana, 8 and the applicants’ analysis in TMobileSprint.9
Using Monte Carlo experiments, I document biases in demandbased diversion ratio estimates
derived for a variety of commonly used demand systems that suffer from either measurement error or
misspecification. Measurement error in the demand system naturally affects the estimated diversion
ratios. Error in the specification will not only result in inaccurate estimation of demand, but also
introduce errors in the formulas for implied diversion ratios. Although there is literature on the impact of
specification error on demand estimation10 and calibrated merger simulations,11 I have been unable to
find any paper that investigates how errors in the estimated demand system affect estimated diversion
ratios.12 Using a simulated set of demand data generated from a known demand system, I estimate
diversion ratios using different demand models that may plausibly be used in a merger review. I compare
the performance of these demand systems in diversion ratio estimation and in the estimation of the most
5 Robert D. Willig, Merger Analysis, Industrial Organization Theory, and Merger Guidelines, 1991 Brookings
Papers Econ. Activity: Microeconomics, 281, 30104.
6 Another potential benefit is that, if market definition is somewhat unclear, the demand system may be less
sensitive to including too many products, a problem that can be significant in the case of market sharebase
diversion ratios. If the demand estimation is precise and accurate, products with little to no relation to the market in
question will be estimated to have little substitution with the irrelevant products. However, this relies on
overcoming all the difficulties faced in demand estimation, which may be compounded by adding irrelevant
products. For example, Conlon & Mortimer (2013) find bias occurs when estimating demand on data in which
inventories are commonly exhausted but all products are always assumed available. Christopher T. Conlon & Julie
Mortimer, Demand Estimation Under Incomplete Product Availability, 5 Amer. Econ. J.: Microecon. 1 (2013).
7 Letter from Maureen Jeffreys, Counsel to AT&T Inc., to Marlene H. Dortch, Secretary, FCC, MB Docket No. 14
90, Attach. (files Jul. 17, 2014), https://ecfsapi.fcc.gov/file/7521680277.pdf.
8 Ari D. Gerstle, Helen C. Knudsen, June K. Lee, W. Robert Majure, & Dean V. Williamson, Economics at the
Antitrust Division 2016–2017: Healthcare, Nuclear Waste, and Agriculture, 51 Rev. Indus. Org. 515, 52223
(2017).
9 Letter from Nancy Victory, Counsel to TMobile, to Marlene H. Dortch, Secretary, FCC, WT Docket No. 18
197, Attach. A (filed Nov. 6, 2018) (TMobile/Sprint Expert Economic Analysis),
https://ecfsapi.fcc.gov/file/11060648404338/Nov.%206%20Public%20SuppResponse.pdf.
10 Dongling Huang, Christian Rojas, & Frank Bass, What Happens when Demand is Estimated with a Misspecified
Model? 56 J. Indus. Econ. 809 (2008).
11 Philip Crooke, Luke Froeb, Steven Tschantz, & Gregory J. Werden, Effects of Assumed Demand Form on
Simulated PostMerger Equilibria, 15(3) Rev. Indus. Org. 205 (1999).
12 One related paper is Rossi, Whitehouse & Moore (2019) which compares switching databased diversion ratios
estimates based on hospital referrals to demandbased diversion ratios estimates. However, the authors assume no
systematic estimation error in their demandbased diversion ratios. Instead, they use the demandbased diversion
ratios as a benchmark to evaluate the performance of the referralbased diversion ratios. Cecilia Rossi, Russell
Whitehouse & Alex Moore, Estimating Diversion Ratios In Hospital Mergers, 15 J. Competition Law Econ., 639
(2019).
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common measure of pricing pressure, the Gross Upward Pricing Pressure Index (GUPPI), which is the
product of a diversion ratio and a ratio of the destination product markup over the origin product price.13
I find that even moderate levels of measurement error in data proxying for differences in price
sensitivity can lead to significant biases. I use several specifications with different levels of measurement
error, where the data used to measure the consumer’s price sensitivity is only correlated with the true
price sensitivity. Reducing the correlation to 0.75 produces diversion ratios that are similar to market
sharebased diversion ratios that use no consumer specific data at all and underestimates diversion
between similar products. For comparison, consider the common practice of imputing an individual’s
data by using averages of their local area. If one imputes household income available in the 2019
American Community Survey (ACS) using the weighted median of the household’s local area, the true
income and the imputation have only a 0.32 correlation. 14 Further, a specification using data on five
ordered groupings of consumer price sensitivities rather than the exact price sensitivity can lead to a
numerically unstable estimate because some consumers’ estimated demand has a positive price effect. In
contrast, I find using ten groupings of price sensitivities yields more accurate and stable estimates, but this
higher level of granularity is uncommon in practice.
The results examining misspecification suggest that incorrectly assuming the nature of price
sensitivity leads to significant biases. For example, one specification is misspecified by having no
differences among consumers in their price sensitivities but captures differences in product value through
a term that is specific to consumer type, product and market. This specification simultaneously predicts
expected individual demand very well, while overestimating diversion originating from high price
products because it overestimates the price sensitivity of consumers of high price products. A random
coefficient specification, which takes account of variation in price sensitivity as a random variable, is
accurate when it assumes price sensitivity has its true distribution, but has similar biases to the five
ordered groupings specification when it uses a commonly assumed alternate distribution (i.e. assuming
the distribution is normal when it is actually lognormal). The commonly used nested logit specification
makes diversion reflect assigned product category shares. However, the estimated biases do not always
improve upon, and are sometime worse than, the biases from sharebased diversion.
These results suggest caution when using demandbased diversion ratio estimates. Unless the
consumer demographic data accurately represent the true sample demographics, such as is typical with
consumer panelist data or administrative data, measurement error will be a serious concern when using
such data to proxy for difference in price sensitivities.. Whether you have consumer demographics data
or not, misspecification is always a concern because matching the true variation in price sensitivity is
crucial. The estimates with the higher and more imprecise estimates comes from inaccurate predictions
that many consumers have positive price sensitivities, suggesting restriction of all price sensitivities to be
13 Steven C. Salop & Serge Moresi, Updating the Merger Guidelines: Comments, 2009 Georgetown Law Faculty
Publications and Other Works 1662,
https://scholarship.law.georgetown.edu/cgi/viewcontent.cgi?article=2675&context=facpub.
14 I use the 2019 OneYear ACS Public Use Microdata Sample (PUMS). PUMS is a subsample of ACS responses
which meant to representative of national demographics and represents about 1% of the U.S, population. I use
weighted medians of Public Use Microdata Areas (PUMAs), which are the smallest areas identified in the PUMS.
PUMAs are designed to have approximately 100,000 residents. American Community Survey Office, United States
Census Bureau, AMERICAN COMMUNITY SURVEY 2019 ACS 1YEAR PUMS FILES ReadMe 34 (2020),
https://www2.census.gov/programssurveys/acs/tech_docs/pums/ACS2019_PUMS_README.pdf. For access to
the data, use the “Explore Census Data” page. United States Census Bureau, Explore Census Data,
https://data.census.gov/cedsci/ (last visited Nov. 19, 2021).
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negative in the demand estimation. Practitioners could avoid these issues by using random coefficient
specifications where price sensitivity distribution is flexibly estimated, though their applicability is
limited due to increased complexity and data requirements. Measurement error and misspecification also
impact markups estimated from a demand system, so practitioners should consider using observed
markup data when available and trustworthy. In general, the challenges posed by errors in demand
estimation suggest that practitioners should carefully consider the use of demandbased diversion ratios
compared with alternative ways to estimate diversion ratios..
2. Diversion Ratios and Upward Pricing Pressure
Diversion as a tool for antitrust analysis appears as early as Willig (1991)15 and the 1992
Horizontal Merger Guidelines, 16 with the DOJ using the term “diversion ratio” by 1995. 17 Shapiro (1996)
presents an early treatment of diversion ratios as one of the U.S. antitrust authorities’ tools. 18 Werden
(1996) discusses how one can use diversion ratios to measure cost efficiencies necessary to offset merger
induced priceincreases; he views this approach as a substitute for merger simulations based on
parametric demand estimation, which he says are “vulnerable to attack” due to the need for functional
form assumptions. 19 A later set of papers develops pricing pressure indices, which included diversion
ratios as part of their formulas, and proposed these indices as an initial screen of likely anticompetitive
effects in merger reviews.20
To explain the importance of diversion ratios, let us assume a differentiated products market
with products. Firms charge all consumers the same price for product in market . Define the
𝑡𝑡
vector of all𝑡𝑡 prices in market as . I make no assumptions on the functional form for demand ( )
𝑗𝑗 ∈ 𝐽𝐽 𝑃𝑃𝑗𝑗𝑗𝑗 𝑗𝑗 𝑡𝑡
for each product, but for simplicity, I assume constant returns to scale costs for each product, where is
𝑡𝑡 𝑷𝑷𝒕𝒕 𝐷𝐷𝑗𝑗𝑗𝑗 𝑷𝑷𝒕𝒕
the per unit cost for product in market . 21
𝐶𝐶𝑗𝑗𝑗𝑗
𝑓𝑓𝑓𝑓 𝑡𝑡
𝑗𝑗 𝑡𝑡= 𝑓𝑓𝑡𝑡𝐽𝐽 ⊂ 𝐽𝐽 𝑓𝑓 ( ) . (1)
𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝒕𝒕
𝜋𝜋 �𝑓𝑓𝑓𝑓�𝑃𝑃 − 𝐶𝐶 �𝐷𝐷 𝑷𝑷
The first order condition with respect to the 𝑗𝑗price,∈𝐽𝐽 , is
𝑃𝑃𝑗𝑗𝑗𝑗
15 Willig, supra note 5, at 299305.
16 Horizontal Merger Guidelines, U.S. Department of Justice and the Federal Trade Commission at § 2.2 (April 2,
1992).
17 Carl Shapiro, The 2010 Horizontal Merger Guidelines: From Hedgehog to Fox in Forty Years, 77 Antitrust L.J.
701, 7134 (2010).
18 Carl Shapiro, Mergers with Differentiated Products, 10 Antitrust 23 (1996).
19 Gregory J. Werden, A Robust Test for Consumer Welfare Enhancing Mergers Among Sellers of Differentiated
Products, 44 J. Indus. Econ. 409 (1996).
20 Daniel P. O'Brien & Steven C. Salop, Competitive Effects of Partial Ownership: Financial Interest and Corporate
Control, 67 Antitrust L.J. 559 (1999); Salop & Moresi, supra note 13; and Joseph Farrell & Carl Shapiro, Antitrust
Evaluation of Horizontal Mergers: An Economic Alternative to Market Definition, 10 BE J. Theoretical Econ.
Article number: 0000102202193517041563 (2010).
21 The resulting equations are more complicated with nonconstant marginal costs, where the shape of the supply
curve becomes an important consideration in addition to diversion ratios. For most applications, researchers and
practitioners assume constant marginal costs, which is a good approximation for industries at scale.
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0 = ( ) + + ( ) . (2)
𝑗𝑗𝑗𝑗 𝑘𝑘𝑘𝑘
𝑗𝑗𝑗𝑗 𝒕𝒕 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝜕𝜕𝐷𝐷 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝜕𝜕𝐷𝐷
𝐷𝐷 𝑷𝑷 �𝑃𝑃 − 𝐶𝐶 � 𝑗𝑗𝑗𝑗 � 𝑓𝑓𝑓𝑓 𝑃𝑃 − 𝐶𝐶 𝑗𝑗𝑗𝑗
I define the priceinduced diversion ratio from product𝜕𝜕𝑃𝑃 to𝑘𝑘 ≠product𝑗𝑗∈𝐽𝐽 in market𝜕𝜕𝑃𝑃 as
𝑗𝑗 𝑘𝑘 𝑡𝑡
= / . (3)
𝑗𝑗𝑗𝑗 𝜕𝜕𝐷𝐷𝑘𝑘𝑘𝑘 𝜕𝜕𝐷𝐷𝑗𝑗𝑗𝑗
𝐷𝐷𝑅𝑅𝑡𝑡 −
Rearranging (2) we can find a formula of the markup𝜕𝜕 for𝑃𝑃𝑗𝑗𝑗𝑗 product𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 as a function of diversion ratios:
= ( )/ + ( 𝑗𝑗 ) . (4)
𝜕𝜕𝐷𝐷𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗
𝑃𝑃𝑗𝑗𝑗𝑗 − 𝐶𝐶𝑗𝑗𝑗𝑗 −𝐷𝐷𝑗𝑗𝑗𝑗 𝑷𝑷𝒕𝒕 � 𝑃𝑃𝑘𝑘𝑘𝑘 − 𝐶𝐶𝑘𝑘𝑘𝑘 𝐷𝐷𝑅𝑅𝑡𝑡
𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓
The term ( ) represents how a firm that owns𝑘𝑘≠𝑗𝑗∈ 𝐽𝐽product j benefits from acquiring product k.
A high diversion ratio from𝑗𝑗𝑗𝑗 j to k and a high mark up on k implies the merged firm will recapture a large
𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑡𝑡
amount of 𝑃𝑃lost −profits𝐶𝐶 𝐷𝐷 from𝑅𝑅 sales of j due to a price increase of j through additional sales of product k.
Thus, the higher is the diversion ratios between an owned product and an acquired product the greater is
the incentive (all else equal) to raise price postmerger.
In a horizontal merger between firm and a rival firm , the set of owned products of the merged
firm is larger than that of the individual firms. Accordingly, if firm sells a set of products in
𝑓𝑓 ℎ
market as well, then additional terms of ( ) for each product from firm will beℎ𝑡𝑡 added𝑡𝑡 to
the postmerger formula for (4): 𝑗𝑗𝑗𝑗 ℎ 𝐽𝐽 ⊂ 𝐽𝐽
𝑡𝑡 𝑃𝑃𝑘𝑘 − 𝐶𝐶𝑘𝑘 𝐷𝐷𝑅𝑅 ℎ
= ( )/ + ( ) . (5)
𝜕𝜕𝐷𝐷𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗
𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝒕𝒕 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 𝑡𝑡
𝑃𝑃 − 𝐶𝐶 −𝐷𝐷 𝑷𝑷 𝑗𝑗𝑗𝑗 �𝑓𝑓𝑓𝑓 ℎ𝑡𝑡 𝑃𝑃 − 𝐶𝐶 𝐷𝐷𝑅𝑅
This means that two firms with high diversion 𝜕𝜕ratios𝑃𝑃 will𝑘𝑘≠𝑗𝑗 ∈have𝐽𝐽 ∪ 𝐽𝐽a higher incentive to increase markups
after merging. As a result, the formula ( ) calculated with premerger markup and diversion
ratios has become an index of pricing pressure on product𝑗𝑗𝑗𝑗 due the addition of product in merger
𝑘𝑘 𝑘𝑘
review. “Upward Pricing Pressure,” or UPP𝑃𝑃 − has𝐶𝐶 been𝐷𝐷𝑅𝑅 defined as:22
𝑗𝑗 𝑘𝑘
= ( ) . (6)
𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗
Note that equation (5) is a function of equilibrium𝑈𝑈𝑈𝑈𝑃𝑃𝑡𝑡 𝑃𝑃prices,𝑘𝑘𝑘𝑘 − 𝐶𝐶 so𝑘𝑘𝑘𝑘 the𝐷𝐷𝑅𝑅 𝑡𝑡true additional postmerger markup
cannot be directly calculated with premerger markups or diversion ratios. One can also think of UPP as
a measure of the minimum required cost efficiencies necessary to maintain premerger prices,23 but again,
this is not exact because (5) uses premerger costs instead of postmerger costs.24 Even so, in practice
premerger markups and diversion ratios are used, with the understanding that this estimate of UPP is not
excessively biased when the postmerger equilibrium prices are not significantly different from the pre
merger prices. Moreover, a large UPP estimated using premerger data implies a large change in
equilibrium prices after the merger. In Monte Carlo experiments, Miller, Ryan, Miller and Sheu (2017)
22 O’Brien & Salop, supra note 20 and Farrell & Shapiro, supra note 20.
23 Farrell & Shapiro at 911, supra note 20.
24 A more exact measure of the required efficiencies to keep prices equal is Compensating Marginal Cost Reductions
(CMCR), which also uses diversion ratios. Werden, supra note 19.
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show the UPP performs well with logconcave demand systems and performs comparably well to merger
simulations with misspecified models.25
In practice, the scale of UPP is dependent on how high prices are in the industry under study – a
UPP of 10 is much less concerning if the average price of a good is $10,000 compared to when the
average price is $10. It is therefore common to calculate the “Gross Upward Pricing Pressure Index” or
GUPPI, which is the UPP divided by the price of the origin product: 26
( )
= . (7)
𝑗𝑗𝑗𝑗 𝑃𝑃𝑘𝑘𝑘𝑘 − 𝐶𝐶𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐼𝐼𝑡𝑡 𝐷𝐷𝑅𝑅𝑡𝑡
There is no hardandfast rule on what value of GUPPI𝑃𝑃 indicates𝑗𝑗𝑗𝑗 a harmful merger, but Salop, Moresi, and
Woodbury (2010) suggest less than 0.05 as presumptively not anticompetitive and greater than 0.1 as
presumptively anticompetitive.27 The literature has extended the GUPPI into new forms to study other
postmerger incentives. For example, the “vGUPPI” is used to evaluate vertical mergers. 28
3. Why Use Diversions Ratios if You Have Estimated a Demand System?
Hausman (2011) comments that it seems pointless to use an estimated demand system to estimate
diversion ratios for UPP, because estimated demand systems can deliver postmerger price changes
directly.29 As I have noted earlier, UPPs and GUPPIs calculated with premerger data are only indicative
of price changes rather than actual predictions of what those changes would be. In contrast, a fully
estimated demand system can be used to directly predict the postmerger prices by changing supplyside
assumptions, as in Nevo (2000).30 Given assumptions about the nature of competition both before and
after the merger, and the extent and nature of any merger efficiencies, an analyst can solve for the prices
that would be optimal for all competing firms given the demand under the new market structure.
There are a few reasons why calculating diversion ratios with a demand system may be desirable,
especially in policy and/or legal contexts. First, merger applicants have presented demandbased
diversion ratios in merger reviews before and will likely continue to do so. Understanding likely biases in
these measures is thus important regardless of alternative measures. Second, diversion ratios can be more
easily grasped by noneconomists and can be more indicative of close substitutes than crossprice
elasticities.31 This is especially helpful when communicating substitution patterns to regulatory or
25 Nathan H. Miller, Marc Remer, Conor Ryan, & Gloria Sheu, Upward Pricing Pressure as a Predictor of Merger
Price Effects, 52 Int’l J. Indus. Org. 216 (2017).
26 Salop & Moresi, supra note 13.
27 Steven C. Salop, Serge Moresi, & John R. Woodbury, Scoring Unilateral Effects with the GUPPI: The Approach
of the New Horizontal Merger Guidelines, CRA Competition Memo, Charles River Associates (2010).
28 Serge Moresi & Steven C. Salop, vGUPPI: Scoring Unilateral Pricing Incentives in Vertical Mergers, 79
Antitrust L.J. 185 (2013).
29 “Of course, if an econometric demand model had already been estimated, there seems little reason not to perform
a merger simulation rather than an upward pricing pressure calculation.” Jerry Hausman, 2010 Merger Guidelines:
Empirical Analysis, 2011 Working Paper, 4 & n. 12, https://economics.mit.edu/files/6603.
30 Aviv Nevo, Mergers with Differentiated Products: The Case of the ReadytoEat Cereal Industry, 31 RAND J.
Econ. 395 (2000).
31 Conlon and Mortimer (2021) provide the following example. Assume there are three substitute products , ’,
and . Product has a crossprice elasticity with of 0.03 and a market share of 0.1. Product ’ has a crossprice
elasticity with of 0.01 but a market share of 0.35. After a 1% price increase in , the number of switchers 𝑘𝑘to 𝑘𝑘’
𝑗𝑗 𝑘𝑘 𝑗𝑗 𝑘𝑘
𝑗𝑗 𝑗𝑗 𝑘𝑘
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judicial officials who review mergers, but otherwise do not regularly deal with economic issues. Such
officials are also becoming more familiar with diversion ratios as their use in merger reviews becomes
more common. Third, the nature of the supplyside may not be obvious due to industry complexity or
nonpublic business practices, so any merger simulation would be making strong supplyside
assumptions. If the nature of the demand side is more obvious, it may be more credible to present
demandbased diversion ratios and simply point out whether merging firms have strong substitutes or not.
Fourth, diversion ratios can also act as a transparent check on the performance of full merger simulations
based on the same demand system. As shown above, the formulas for simulated price changes are very
closely related to diversion ratios, so if the diversion ratios are wrong, demand simulations are also likely
wrong. Finally, as mentioned earlier, Miller, Remer, Ryan, and Sheu (2017) have shown that UPP
performs well compared to merger simulation under misspecification.32 However, this finding is
conditional on observing diversion ratios, so the contribution of the present paper is to characterize what
happens when diversion ratios are not perfectly observed.
4. Discrete Choice Demand
The most popular framework amongst economists for studying differentiated product demand is
the discrete choice random utility model (RUM).33 Generically, consumer in a market of population
has a utility for product with the bipartite form:
𝑖𝑖 𝑡𝑡
𝑁𝑁𝑡𝑡 𝑗𝑗 = + . (8)
is the product taste shock, a random variable𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖 that𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 represents𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖 the component of utility each consumer
has for product that cannot be explained by observable data. is the component of utility that
𝜖𝜖𝑖𝑖𝑖𝑖𝑖𝑖
consumer has for product that varies as a function of product 𝑖𝑖characteristics𝑖𝑖𝑖𝑖 and possibly consumer
characteristics. 𝑗𝑗Because consumer level variation in is generally𝛿𝛿 assumed to vary with product
characteristics,𝑖𝑖 individual differences𝑗𝑗 in correspond to taste variation in product characteristics. I will
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
follow most industrial organization applications and assume is indirect utility, so is a function of
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
price. Thus, an example of individualvarying sensitivity to a 𝑖𝑖product𝑖𝑖𝑖𝑖 characteristic is income𝑖𝑖𝑖𝑖𝑖𝑖 specific
price sensitivity: = , where represents a constant𝑈𝑈 nonprice utility each𝛿𝛿 consumer has for
product and measures individualspecific sensitivity to price that is a function of income.
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 𝛽𝛽𝑗𝑗 − 𝛼𝛼𝑖𝑖𝑃𝑃𝑗𝑗𝑗𝑗 𝛽𝛽𝑗𝑗
Each𝑗𝑗 consumer𝛼𝛼𝑖𝑖 chooses only the product with the highest𝑃𝑃 utility𝑗𝑗𝑗𝑗 – hence the “discrete choice”
moniker. Thus, expected individual demand (integrating over the product taste shocks ) is simply the
individual choice probability of choosing , [ ] = . Each consumer has a type ~ (possibly
𝜖𝜖𝑖𝑖𝑖𝑖
observed or degenerate) that is independent of the product taste shocks, . For a particular market , all
𝑖𝑖 𝑗𝑗 𝐸𝐸 𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖 𝐹𝐹
consumers face the same products so consumers’ indices vary only by . This implies that expected
𝜖𝜖𝑖𝑖𝑖𝑖 𝑡𝑡
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖
(~0.35% of the market) would be larger than the number of switchers to (~0.30% of the market), even though the
crossprice elasticity to k would be larger. Thus given large differences in market share, crossprice elasticities can
be misleading in terms of the closeness of substitution. Diversion ratios 𝑘𝑘do not suffer this issue as they directly
report the number of switchers. Christopher T. Conlon & Julie Holland Mortimer, Empirical Properties of
Diversion Ratios, (2021) RAND J. Econ., 697 & n. 8.
32 Miller, Remer, Ryan, & Sheu, supra note 25.
33 See generally Simon P. Anderson, Andre De Palma, & JacquesFrancois Thisse, Discrete Choice Theory of
Product Differentiation (1992); Kenneth E. Train, Discrete Choice Methods with Simulation, (2d ed. 2009); and
Steven T. Berry & Philip Haile, Foundations of Demand Estimation, in Handbook of Industrial Organization, Vol.
4, 1 (eds. Kate Ho, Ali Hortaçsu, & Alessandro Lizzeri 2021).
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market share, (integrating over the product taste shocks and consumer types ), is simply the
expected value of :
𝑆𝑆𝑗𝑗𝑗𝑗 𝜖𝜖𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 [ ( )]
= = [ ( )] ( ) = ( ). (9)
𝐸𝐸 𝐷𝐷𝑗𝑗𝑗𝑗 𝑷𝑷𝒕𝒕
𝑆𝑆𝑗𝑗𝑗𝑗 ∫ 𝐸𝐸 𝐷𝐷𝑗𝑗𝑗𝑗 𝑷𝑷𝒕𝒕 𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖 ∫ 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖
In principle, could be drawn𝑁𝑁 from any distribution. For example, if is multivariate normal
then choice probabilities𝑖𝑖𝑖𝑖 follow the multinomial probit, which has been used in notable𝑖𝑖𝑖𝑖 applications in
industrial organization𝜖𝜖 . 34 Still more popular is assuming that is distributed i.i.d.𝜖𝜖 according to the
35
Gumbel distribution, also known as the Type I Extreme Value distribution.𝑖𝑖𝑖𝑖 In contrast to probit, which
has no analytical form for the probabilities, the Gumbel distribution𝜖𝜖 assumption implies the “logit” or
“softmax” choice probability equations. 36 Assuming an outside option = 0 with utility normalized to 0,
the choice probabilities are 37
𝑗𝑗
exp
= . (10)
1 + exp( )
�𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖�
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
The expected diversion ratios of discrete choice∑𝑘𝑘∈𝐽𝐽 demand𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 can be written as weighted averages. 38
Using (9) to rearrange (3) shows that the expected diversion ratio is a function of the derivatives of the
expected market shares, which implies they are also functions of the derivatives of choice probabilities:
( )
( )
= 𝜕𝜕𝜕𝜕�𝐷𝐷𝑗𝑗𝑗𝑗 𝑷𝑷𝒕𝒕 � = 𝜕𝜕𝑆𝑆𝑘𝑘𝑘𝑘 = 𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 . (11)
( ) ∫ 𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖
𝑗𝑗𝑗𝑗 𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 ( )
𝑡𝑡
𝐸𝐸�𝐷𝐷𝑅𝑅 � − 𝑗𝑗𝑗𝑗 𝒕𝒕 − 𝑗𝑗𝑗𝑗 − 𝑖𝑖𝑖𝑖𝑖𝑖
𝜕𝜕𝜕𝜕�𝐷𝐷 𝑷𝑷 � 𝜕𝜕𝑆𝑆 𝜕𝜕𝑆𝑆 𝑖𝑖
𝑗𝑗𝑗𝑗 ∫ 𝑗𝑗𝑗𝑗 𝜕𝜕𝜕𝜕 𝑦𝑦
Given a large number of consumers in a market,𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 the expected𝜕𝜕𝑃𝑃 and realized𝜕𝜕𝑃𝑃 diversion ratios should be close
in value. For the remainder of this study, I will treat the “the expected diversion ratio” as a highly
accurate estimate of the realized “diversion ratio,” and I refer to them interchangeably unless specifically
noted.
/
Introducing the term inside the integral of the numerator of (11) allows the expected
𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑡𝑡 𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑡𝑡
marketlevel diversion ratio to𝜕𝜕 𝑃𝑃be𝑗𝑗𝑡𝑡 rewritten𝜕𝜕𝑃𝑃𝑗𝑗𝑡𝑡 as the weighted average of “individual diversion ratios,”
:
𝑗𝑗𝑗𝑗
𝐷𝐷𝑅𝑅𝑖𝑖 = ( ). (12)
𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗
𝐸𝐸�𝐷𝐷𝑅𝑅𝑡𝑡 � −∫ 𝜔𝜔𝑖𝑖𝑖𝑖𝑖𝑖𝐷𝐷𝑅𝑅𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖
34 E.g. Austan Goolsbee & Amil Petrin, The Consumer Gains from Direct Broadcast Satellites and the Competition
With Cable TV, 72 Econometrica 351 (2004).
35 The Gumbel distribution is singlepeaked, asymmetric and unbounded above and below. It has a cumulative
density function of ( ) = exp( exp( )). Train, supra note 33, at 34.
36 Id. at 3637.
37 The choice probabilities𝐹𝐹 𝜖𝜖 are the− expectation𝜖𝜖 of choice over the distribution of , but not consumer type . So
each consumer has their own value for their , and choice probabilities are based𝑖𝑖𝑖𝑖 on . 𝑖𝑖
38 𝜖𝜖 𝑦𝑦
The following discussion relies on the exposition𝑖𝑖𝑖𝑖 by Conlon and Mortimer (2021). Conlon𝑖𝑖𝑖𝑖 & Mortimer, supra
note 31, at 698706. 𝛿𝛿 𝛿𝛿
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Individual diversion ratios, , are equal to consumer ’s ratio of the change of choice
probability of product over the change𝑗𝑗 𝑗𝑗choice probability for product due to a change of product
𝑖𝑖𝑖𝑖
price: 𝐷𝐷𝑅𝑅 𝑖𝑖
𝑗𝑗 𝑘𝑘 𝑗𝑗′𝑠𝑠
= / . (13)
𝑗𝑗𝑗𝑗 𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
To avoid confusion between individual diversion𝑖𝑖𝑖𝑖 ratios and marketlevel diversion ratios, I will refer to
𝐷𝐷𝑅𝑅 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗
individual diversion ratios as “individual diversion ratios𝜕𝜕𝑃𝑃 ,” 𝜕𝜕while𝑃𝑃 “diversion ratios” will refer to market
level diversion ratios.
Weights measure how strongly each consumer is expected to substitute away from :
𝑗𝑗
= 𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 . (14)
𝑗𝑗𝑗𝑗 ( )
𝑖𝑖𝑖𝑖𝑖𝑖 𝜕𝜕𝑃𝑃
𝜔𝜔 𝑖𝑖𝑖𝑖𝑖𝑖
𝜕𝜕𝑆𝑆 𝑖𝑖
∫ 𝑗𝑗𝑗𝑗 𝜕𝜕𝜕𝜕 𝑦𝑦
(12), (13), and (14) show there are two channels𝜕𝜕𝑃𝑃 through which errors can bias diversion ratios –
either individual diversion ratios are wrong or the “weight,” measuring how much individual is expected
contributes to the total change in the market share of , , is wrong. 39 Individual diversion ratios and
weights are thus important diagnostic metrics for the rest of this paper. 𝑖𝑖
𝑗𝑗 𝜔𝜔𝑖𝑖𝑖𝑖𝑖𝑖
Logit individual diversion ratios and weights can be derived substituting the formula for choice
probabilities (10) into (13) and (14). The logit choice probabilities exhibit the Independence of Irrelevant
Alternatives (IIA) property as defined by Luce (1995): as long as the same two options are available in
otherwise different choice sets, the ratio between the choice probabilities for those two options is always
equal. 40 IIA makes the individual diversion ratio a function of only the choice probabilities:
= . (15)
1
𝑗𝑗𝑗𝑗 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
𝐷𝐷𝑅𝑅𝑖𝑖𝑖𝑖
Weights are a function of the choice probabilities and the− 𝑆𝑆 derivatives𝑖𝑖𝑖𝑖𝑖𝑖 of with respect to price, i.e., the
price sensitivity:
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
1
= 𝜕𝜕𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 . (16)
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖� − 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖�
1 𝜕𝜕𝑃𝑃 𝑗𝑗𝑗𝑗 ( )
𝜔𝜔𝑖𝑖𝑖𝑖𝑖𝑖
𝜕𝜕𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
∫ 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 − 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖
Thus marketlevel diversion of logit demand� systems� 𝜕𝜕𝑃𝑃𝑗𝑗 𝑗𝑗is not only a function of weights and
individual diversion ratios, but also of choice probabilities and price sensitivities. Further, (16) shows
39 An analogous derivation exists for the realized diversion rather than the expected diversion. In that case,
diversion ratios are still a weighted average with the form of (12), but realized individual demand replaces
choice probabilities in (13) and (14). The (13) and (14) analogues are not differentiable everywhere, because
𝐷𝐷𝑖𝑖𝑖𝑖
individual demand is 𝑖𝑖binary𝑖𝑖 for discrete choice demand. I therefore use the expected demand formulation in the
current application of𝑆𝑆 discrete choice; however, the realized demand formulation could be useful in in applications
with continuous demand.
40 R. Duncan Luce, Individual Choice Behavior: A Theoretical Analysis 9 (1959).
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that accurate diversion ratio estimation requires accurate estimation of the joint distribution choice
probabilities and the price sensitivities. If the marginal distributions of choice probabilities are correct,
then individual diversion ratios can be accurately estimated. However, if the marginal distribution of price
sensitivities or their joint distribution with choice probabilities are wrong, then the weights will not be
estimated correctly. Measurement error in information about consumer types thus can bias market
level diversion ratios by leading to biased choice probability estimates and biased estimates of price
𝑖𝑖
sensitivities. Specifications that do not have data on consumer types could work𝑦𝑦 if they manage to
somehow recreate the joint distribution of choice probabilities and price sensitivities, even if predictions
for individuals are wrong. However, misspecification not only can result in biased choice probabilities
but also may lead to direct misspecification of . This suggests that misspecification is a particular
𝜕𝜕𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
threat to unbiased estimation of weights. 𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗
One important case to consider is when has no variation across consumers, what I will call
41
the “simple logit” model. Choice probabilities and𝑖𝑖𝑖𝑖𝑖𝑖 weights are then constant across consumers in the
same market, and (15) simplifies to the market share𝛿𝛿 based diversion ratios:
= . (17)
1
𝑗𝑗𝑗𝑗 𝑆𝑆𝑘𝑘𝑘𝑘
𝐸𝐸�𝐷𝐷𝑅𝑅𝑡𝑡 �
As seen from the derivation, share proportional diversion− ratios𝑆𝑆𝑗𝑗𝑗𝑗 mean that consumers have no taste
variation for product characteristics. This seems implausible as basic economic theory implies consumers
should vary at least in price sensitivity by income. Further, because diversion is perfectly proportional to
market share, there is no way for two low share products to have high diversion with each other. This is
inconsistent with the plausible situation where a niche of similar products might be unpopular with the
general populace but the niche has a customer base that primarily substitutes within the niche.
For the remainder of this study, I will calculate market shares, diversion ratios and derived
statistics like GUPPIs by using their expectations – this assumes the samples of the Monte Carlo
experiments are large enough so that the expectation of the diversion ratio is a good approximation for the
diversion ratio itself.
5. Monte Carlo Experiments Set Up
I perform Monte Carlo experiments in the following steps:
1. Simulate Consumers: I draw 500 consumer types from the distribution F( ) and assign
these to consumers in a market . I do this 10 times for a total of 5,000 consumers distributed
𝑖𝑖 𝑖𝑖
across 10 markets. 500 consumers is a reasonable 𝑦𝑦number of consumers to observ𝑦𝑦 e per
market in a market research survey𝑡𝑡 42 and allows me to perform the computations in a
reasonable period of time given my limited computing resources. I experimented with using
more markets to create more variation in prices but I achieve reasonably precise estimates for
41 The literature often refers to this specification as simply the “logit” or “multinomial logit” model. This could be
confusing in this paper since all the models in this paper use logittype taste shock errors. I therefore use the term
“simple logit” to improve clarity, which I borrow from Train, supra note 33, at 50.
42 For example, Goolsbee and Petrin (2004) reports about 30,000 households over 317 markets in their data, which is
about 100 consumers per market. Goolsbee & Petrin, supra note 34, at 356.
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the true specification at 10 markets. I therefore keep the number of markets low for faster
estimation.
2. Simulate Consumer Product Choices: Given the assortment of products and the prices
in market , I calculate choice probabilities of selecting products assuming a true
𝐽𝐽𝑡𝑡 𝑃𝑃𝑗𝑗𝑗𝑗
demand system (explained later in this section)𝑗𝑗𝑗𝑗 . I then use these probabilities to divide up the
unit interval𝑡𝑡 into lengths equal to the choice𝑆𝑆 probabilities. To simulate𝑗𝑗 the choices, I then
draw from the uniform distribution, and assign the chosen product depending on what region
the draw is in. 43
3. Estimate Demand System Specifications: With this synthetic dataset of consumers, products,
markets, prices, consumers types , and choices, I estimate all my 11 demand specifications
using maximum likelihood estimation (or simulated maximum likelihood if the specification
𝑖𝑖
has random coefficients). The likelihood𝑦𝑦 function is based on choice probabilities of
observed choices.
4. Calculate Diversion Ratios and GUPPIs: I calculate the associated expected marketlevel
diversion ratios and GUPPIs for each demand system based on the parameter estimates for
Market 1. Aggregation to the market level requires an empirical analogue of integration over
consumer types. In the case of observed types, I simply sum over the sample of consumers in
each market. For specifications with random coefficients, I simulate 1,000 copies of each
consumer with a different price sensitivity drawn using the estimated distribution. I calculate
choice probabilities, individual diversion ratios and weights for each copy, and then average
over all 1,000 copies of the synthetic data to calculate marketlevel diversion and GUPPIs.
5. Repeat: I then repeat steps 14 100 times to make 100 synthetic dataset and 100 sets of
estimates for each of the 11 demand system specifications. I report bias, standard deviation
of errors, and the root mean square error (RMSE). These results allow me to measure the
expected sampling error in practical applications.
43 For example, if there were only product 0 and product 1 in market and = and = , then I would divide
1 2
0 0, 1 ( , 1] 𝑖𝑖0𝑡𝑡 𝑖𝑖1𝑡𝑡
the unit interval into region , , and region , . If I draw from𝑡𝑡 the𝑆𝑆 uniform3 and𝑆𝑆 the draw3 is less than , I
1 1 1
would assign product 0. If the draw� 3� is more than then3 I would assign product 1. 3
1
3
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Table 1: Assortments, Prices and Market Shares
Low Tier High Tier
Low Tier High Tier
Market Market
Market Assortment Price Price
Share Share
(1 & 2) (3 & 4)
(1 & 2) (3 & 4)
1 1, 2, 3, 4 1.34 2.17 45.7% 42.3%
2 1, 2, 3, 4 1.47 2.38 46.6% 39.2%
3 1, 2, 3, 4 1.67 2.71 47.3% 35.0%
4 1, 2, 3, 4 1.20 1.95 43.9% 46.2%
5 1, 2, 3, 4 1.00 1.63 41.2% 51.7%
6 1, 3, 4 1.34 2.17 35.2% 51.5%
7 1, 2, 3 1.34 2.17 55.6% 32.5%
8 1, 2 1.34 2.17  80.0%
9 3, 4 1.34 2.17 87.9% 
10 1, 3 1.34 2.17 44.6% 42.4%
I will provide further detail in the following sections. This set up is markedly different from most
of the related demand estimation literature in that it focuses on individual level choice data instead of
market share data, and there are no productmarket level unobservables. 44 The focus of the demand
estimation literature is different from mine in that the literature typically addresses demand estimation
under limited data and endogeneity so eschews other possible sources of error. Analogously, I assume a
generous data environment and no endogeneity to fully isolate the impact of measurement error and
misspecification. This present paper should be seen as complementary to the preexisting literature for
informing practitioners about demand estimation.
5.1 Markets, Assortments, Qualities, Prices
I define 10 different markets, which will vary by the assortment of products from which
consumers can choose and by the price levels of the products. Table 1 presents variation in products
across markets. Market 1 will be my baseline market – it will include all possible products and will have
my baseline prices. Markets 210 will be altered from Market 1 to aid in identifying parameters of the
various demand systems. If I only used Market 1, it would not be possible to distinguish variation in
utility due to price differences from variation due to quality differences. My specifications allow
differences in both quality and prices across products but not within markets, so I need multiple markets
with the same product to measure how price changes demand given the same quality.
Markets 1 to 5 will have 4 different products plus an Outside Option. There are two product
quality tiers: symmetric products 1 and 2 of the “Low Tier” have quality = 2.5, while symmetric
products 3 and 4 of the “High Tier” have = 3. Other markets vary by assortment to help with
𝑄𝑄𝑗𝑗
identification of the model parameters that govern substitution across products: comparing markets with a
𝑄𝑄𝑗𝑗
44 See generally Berry & Haile, supra note 33.
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product and markets without that product shows how consumers will substitute when forced to switch.45
Market 6 will not have Product 2; Market 7 will not have Product 4; Market 8 will not have High Tier
products; Market 9 will not have Low Tier Products; and Market 10 will not have Products 2 or 4.
For simplicity, I assume firms only produce one product. Firms produce with constant marginal
cost. Firm f’s profits from producing its one product are equal to the oneproduct version of equation
(1):
𝑗𝑗
= . (18)
In line with the traditional diversion ratios 𝜋𝜋assumption𝑓𝑓𝑓𝑓 �𝑃𝑃𝑗𝑗𝑗𝑗 − of𝐶𝐶 𝑗𝑗firm𝑗𝑗�𝐷𝐷𝑗𝑗 𝑗𝑗conduct and market structure, firms
compete as NashBertrand price setters in Market 1. Market 1 prices are thus the equilibrium strategies of
the NashBertrand price setting game of the firms selling to the entire Market 1 population of all 100
datasets. This is analogous to treating the full set of simulations as the “population” of the market and
each individual dataset as a sample.46 High Tier products cost more to produce than Low Tier Products:
in market 1, Low Tier products have marginal costs of = 1.0 and High Tier products have marginal
costs of = 1.5.
𝐶𝐶𝑗𝑗
I𝐶𝐶 𝑗𝑗directly assume the variation in the prices for Markets 2 to 5 to aid in identifying price
sensitivity parameters. This price variation can be thought as the result of cost variation across markets or
some regulatory intervention. I eschew modeling how exactly the supply side works in these other
markets because finding the equilibrium costs that would imply these exact prices can be complex.
Relative to prices in Market 1, Market 2 prices are 10% greater, Market 3 prices are 25% greater, Market
4 prices are 10% lower, and Market 5 prices are 25% lower. For the same reason, I keep the prices in
Markets 610 the same as Market 1 as the demand under different assortments can be directly compared
to demand in Market 1 without potentially confounding variation in prices.. The resulting prices can be
observed in Table 1.
Using all markets to calculate diversion and GUPPIs introduces composition effects in aggregate
diversion ratios and GUPPI estimates.47 I therefore use only Market 1 data when deriving diversion ratios
and GUPPIs for simpler interpretation of the results. I use data for Markets 210 only in demand
estimation.
5.2 The “True” Demand System
I will assume that consumers in our simulations have “true” discrete choice demand with logit
product taste shocks and that has the form
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
45 However, as the results will show, measurement error and misspecification may prevent a specification from
estimating why a consumer switched (i.e. based on their price sensitivity), and thus generate incorrect predictions for
diversion due to a price change.
46 There is an argument that it would have been more appropriate to construct our individual datasets via
bootstrapping from the entire population of the 100 synthetic datasets, but this would have only given a negligible
conceptual benefit at a significant increase in complexity.
47 For example, if I include both Market 1 and Market 6 in a single diversion ratio calculation, I would need to
account for the fact that half of consumers in Market 6 cannot divert to or from the missing Product 2. Inclusion of
a market with different prices like Market 2 is feasible, but then the resulting diversion ratios would be a blend of
diversion from two separate price levels.
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= . (19)
is quality; is market specific price;𝛿𝛿 𝑖𝑖and𝑖𝑖𝑖𝑖 𝜃𝜃 �is𝑄𝑄 an𝑗𝑗 − individual𝑦𝑦𝑖𝑖𝑃𝑃𝑗𝑗𝑗𝑗� specific consumer type that (up to the
factor𝑗𝑗 ) explains𝑗𝑗𝑗𝑗 price sensitivity. In many demand𝑖𝑖 models, a common example of such a characteristic
would𝑄𝑄 be income,𝑃𝑃 which reduces𝑡𝑡 price sensitivity𝑦𝑦 through the income effect. In (19), the price sensitivity
increases𝜃𝜃 in , so is more analogous to inverse or negative income than income itself.
Conditional𝑦𝑦𝑖𝑖 𝑦𝑦 𝑖𝑖on the Outside Option’s = 0 as a normalization, larger implies that taste shock
is a smaller component of utility than . That is, if is measured in dollars, then the standard
𝑖𝑖0𝑡𝑡
48 𝛿𝛿 𝜃𝜃
deviation𝑖𝑖𝑖𝑖 of is dollars. As approaches𝑖𝑖𝑖𝑖𝑖𝑖 to 0, the𝑗𝑗𝑗𝑗 model approaches a simple logit where
𝜖𝜖 𝜋𝜋 𝛿𝛿 𝑃𝑃
= 0 for all consumers, products and markets: products and the Outside Option are chosen at random
𝜖𝜖𝑖𝑖𝑖𝑖 √6 𝜃𝜃𝑦𝑦𝑖𝑖 𝜃𝜃
with𝑖𝑖𝑖𝑖𝑖𝑖 equal probability. As tends to infinity, the model approaches the “vertical model” of Shaked and
Sutton𝛿𝛿 (1982) and Bresnahan (1987), where all taste variation is explained by variation in price sensitivity
and there are no product taste𝜃𝜃 shocks.49
This demand system is intentionally very simple. While there are interesting questions about how
the interaction between price sensitivity and the taste for other product characteristics impacts diversion
ratios, using a simpler model will make the results more transparent. I therefore use only one random
coefficient and also absorb all nonprice characteristics into the single quality variable. Further, I assume
that there is no unobserved productmarket utility, i.e. price endogeneity is not an issue.
As the true model is still logitbased, (15) remains the formula for individual diversion ratios.
With linear , price sensitivity is = , so there is a more specific version of the (16) formula for
𝜕𝜕𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
weights: 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 𝜃𝜃𝑦𝑦𝑖𝑖
1
= . (20)
1 ( )
� − 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖�𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑦𝑦𝑖𝑖
𝜔𝜔𝑖𝑖𝑖𝑖𝑖𝑖
The formula for individual diversion ratios are∫ � typical− 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 for�𝑆𝑆𝑖𝑖 𝑖𝑖𝑖𝑖logit𝑦𝑦𝑖𝑖𝜕𝜕𝜕𝜕based𝑦𝑦𝑖𝑖 models, but directly impacts the
value of the weights. What is less obvious is the role of ; high mechanically makes all larger in
𝑦𝑦𝑖𝑖
magnitude, so choice probabilities (relative to the choice probability for the Outside Option)𝑖𝑖 𝑖𝑖𝑖𝑖are more
extreme for extreme values of . As choice probabilities𝜃𝜃 appear𝜃𝜃 in the weights, this can create𝛿𝛿 a more
lopsided weighted average.
𝑦𝑦𝑖𝑖
I will assume is lognormal distributed with a mean of 1 and skewness of 5. 50 I use the
lognormal distribution because this guarantees that price sensitivity will always be negative. I choose a
𝑖𝑖
mean of 1 for ease of exposition𝑦𝑦 : utility of 1 unit of price (i.e., money) to the mean consumer is . I
choose a high skewness because this accentuates the differentiation of consumers across their chosen
𝑃𝑃𝑗𝑗𝑗𝑗 𝜃𝜃
48 Relative to the units of the coefficients of an estimated multinomial logit, the standard deviation of taste shock
is . Train, supra note 33, at 4041.
𝑖𝑖𝑖𝑖
𝜋𝜋 𝜖𝜖
49 Avner Shaked & John Sutton, Relaxing Price Competition Through Product Differentiation, 49 Rev. of Econ.
√6
Stud. 3, (1982); and Timothy F. Bresnahan, Competition and Collusion in the American Automobile Industry: The
1955 Price War, 35 J. Indus. Econ. 457 (1987).
50 A lognormal random variable is the log of a normal random variable with mean and standard deviation . For
this paper, the parameters underlying the normal distribution seem less meaningful than the mean and skewness of
the lognormal distribution, but for interested readers, I am effectively assuming =𝜇𝜇 0.42 and = 0.92. 𝜎𝜎
14 𝜇𝜇 − 𝜎𝜎
OEA Working Paper
products. Diversion ratios will then be very different from the case in which all consumers have the same
price sensitivity, i.e., shareproportional diversion.
5.3 and GUPPIs
𝜽𝜽To limit the amount of information reported, I will report only one diversion within the Low Tier
(Product 1 to Product 2), one diversion from the Low Tier to the High Tier (Product 1 to Product 3), one
diversion from the High Tier to Low Tier (Product 3 to Product 1), and one diversion within the High
Tier (Product 3 to Product 4). As products within tiers are symmetric in quality and price, the presented
diversions are virtually identical (up to the simulation/estimation error) to the diversions I do not present.
For example, the diversion ratio of Product 1 to Product 3 is nearly identical to the diversion ratio of
Product 2 to Product 4 because they are both Low Tier to High Tier diversions.
Due to computational time, I focus on a single value of . However, it is instructive to show how
varying impacts the diversion and GUPPIs, as this informs the that I ultimately select. For the Market
1 subset of all 100 datasets, I calculate the overall diversion ratios𝜃𝜃 and GUPPIs for all consumers in all
datasets.𝜃𝜃 Figure 1 indicates that, as increases (i.e., making price𝜃𝜃 sensitivity variation more important
relative to product taste shocks), diversion within tiers goes up noticeably. However, Figure 2 indicates
that the GUPPI decreases as increases.𝜃𝜃 Recall that GUPPIs are the product of percent markups and
diversion. As increases, ownprice elasticity increases and thus markups decrease as seen in Figure 3.
Markups decrease more rapidly𝜃𝜃 than diversion increases, so the markup decrease dominates in their
resulting product.𝜃𝜃
For my purposes I choose = 5 as this implies a sizeable gap between true GUPPIs and share
proportional GUPPIs for both withintier diversions. Table 1 shows market shares and prices for Market
1. Low Tier products have market𝜃𝜃 shares of 23% and High Tier products have market shares of 21%.
Low Tier products have prices of 1.37 (markup is 27% of price) and High Tier products have prices of
2.17 (markup is 31% of price). Table 2 shows Market 1 diversion ratios: expected diversion is high
within tier: 59% in the Low Tier and 38% in the High Tier. This compares to the shareproportional
diversion of 29% within the Low Tier and 27% in the High Tier. Diversion between the High Tier and
the Outside Option is essentially 0, which means the model with its high skewness of price sensitivity is
close to the “vertical model” where products only substitute to the nextlowest and nexthighest products
in quality level. For comparison, Table 3 shows shareproportional diversion: all products beside the
Outside Option have roughly the same market share so diversion is close to symmetric.
Individual diversion ratios and weights differ greatly by products and . Figure 4 shows that
individual diversion is mostly dominated by the destination product of diversion. Diversion ratios to
𝑖𝑖
High Tier products are strongest amongst weakly price sensitive consumers with𝑦𝑦 low . Diversion ratios
to the Outside Option are strongest amongst strongly price sensitive consumers with high Diversion
𝑖𝑖
ratios to Low Tier products are strongest amongst moderate price sensitive consumers𝑦𝑦 with moderate .
𝑖𝑖
Figure 5 shows weights are also determined by price sensitivity. An extreme amount of weight𝑦𝑦 is put on
𝑖𝑖
highly price sensitive consumers with large for diversion from the Outside Option. Weight is highest𝑦𝑦
for low for the High Tier products, and the weight is highest for medium for the Low Tier goods. In
𝑖𝑖
summary, consumers with high individual diversion𝑦𝑦 ratios having high weights is what makes withintier
𝑖𝑖 𝑖𝑖
diversion𝑦𝑦 high in my baseline specification. 𝑦𝑦
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Figure 1: Diversion Ratios in Market 1 as Changes
𝜽𝜽
Evaluated at = {1, 2, 3, 4, 5 , 10 , 15, 20, 25, 30}.
Figure 2:𝜃𝜃 GUPPIs in Market 1 as Changes
𝜽𝜽
Evaluated at = {1, 2, 3, 4, 5 , 10 , 15, 20, 25, 30}.
𝜃𝜃
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Figure 3: Markups in Market 1 as Changes
𝜽𝜽
Evaluated at {1, 2, 3, 4, 5 , 10 , 15, 20, 25, 30}.
𝜃𝜃 ∈
Table 2: True Diversion Ratios (Market 1, = )
𝜽𝜽 𝟓𝟓 Diverting to…
0 1 2 3 4
0  0.496 0.496 0.004 0.004
1 0.115  0.591 0.147 0.147
Diverting
2 0.115 0.591  0.147 0.147
from…
3 0.002 0.311 0.311  0.376
4 0.002 0.311 0.311 0.376 
Diversion ratios calculated over all simulated consumers in Market 1 across all 100 synthetic datasets .
Table 3: Estimated ShareProportional Diversion Ratios (Market 1, = )
Diverting to… 𝜽𝜽 𝟓𝟓
0 1 2 3 4
0  0.261 0.258 0.240 0.241
1 0.155  0.295 0.274 0.276
Diverting
2 0.155 0.297  0.273 0.275
from…
3 0.152 0.291 0.288  0.269
4 0.152 0.291 0.289 0.268 
Diversion ratios calculated for all consumers in Market 1 across all 100 synthetic datasets.
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Figure 4: Individual Diversion Ratios by Percentile (Market 1, = )
𝒚𝒚𝒊𝒊 𝜽𝜽 𝟓𝟓
Individual diversion ratios calculated for all simulated consumers across all synthetic datasets in Market 1.
Figure 5: Individual Weights by Percentile ( = )
𝒚𝒚𝒊𝒊 𝜽𝜽 𝟓𝟓
Weights calculated for all simulated consumers in Market 1 across all synthetic datasets. Weights are normalized so that the mean weight is 1.
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Table 4: Prices Changes and GUPPIs (Market 1, = , )
Pre 𝜽𝜽 𝟓𝟓 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴Post𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 𝑷𝑷 𝑩𝑩Fraction𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩
Merger
Product Merger GUPPI Merger Price
Type
Price Price Change
1 1.34 0.15 1.50 0.12
Low & 2 1.34 0.15 1.50 0.12
Low 3 2.17  2.09 0.04
4 2.17  2.09 0.04
1 1.34 0.07 1.42 0.06
Low & 2 1.34  1.37 0.02
High 3 2.17 0.05 2.29 0.06
4 2.17  2.17 0.00
1 1.34  1.41 0.06
High & 2 1.34  1.41 0.06
High 3 2.17 0.12 2.72 0.26
4 2.17 0.12 2.72 0.26
An important feature of the model is that the expected weights for products drops off quickly as
approaches zero because weights are proportional to : less price sensitive consumers have lower
weights than price sensitive consumers because they respond less to price. The High Tier consumers
𝑖𝑖 𝑖𝑖
𝑦𝑦have disproportionally low so marginal increases in the𝑦𝑦 price of High Tier products result in more
muted increases in unit sales lost than when prices increase for Low Tier Products. Given the market
𝑖𝑖
shares are all about equal for𝑦𝑦 nonoutside option products, market sharebased diversion ratios between
these products are roughly similar (27%30%). Since the baseline of sharebased diversion ratios is
similar, when I compare the difference between true diversion and sharebased diversion, the differences
are more pronounced for the Low Tier goods that have more price sensitive consumers: the withinHigh
Tier diversion ratio is substantially greater than the sharebased diversion ratio (38% vs. 27%) but not as
high as the difference for withinLow Tier diversion (59% vs. 30%). This also means specifications that
do not properly account for heterogeneity will have a tendency to overestimate diversion originating
from higher tier products because they will overweight the individual diversion ratios of very low .
𝑦𝑦𝑖𝑖
Given these parameter assumptions, Market 1 is problematic for any merger. Table 4 shows𝑦𝑦𝑖𝑖
GUPPIs and postmerger price changes based on the true demand system. All possible mergers will result
in higher than 5% price increases for products of merging firms and, in the case of a merger of the High
Tier products, all products. 51 Using the 0.05 and 0.10 thresholds for GUPPIs explained in the earlier
section on GUPPIs 52, any merger within tier should be presumptively anticompetitive and a crosstier
merger should warrant further review: GUPPIs between the Low Tier products are 0.15 while GUPPIs
between the High Tier products are 0.12; and LowtoHigh GUPPIs are 0.7 while HightoLow GUPPIs
are 0.5. GUPPIs are similar in magnitude to the resulting price increases in LowLow and LowHigh
51 Interestingly, a merger between Low Tier products generates slightly negative pricing pressure: the Low Tier
price increases are so extreme that the residual demand for High Tier products becomes much more price sensitive
so that High Tier prices end up slightly lower despite the reduction in competition.
52 Salop, Moresi, & Woodbury, supra note 27.
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OEA Working Paper
mergers, but underestimate the 26% price increase in a HighHigh merger – the low price sensitivity of
High Tier consumers support higher price hikes and make the postmerger equilibrium different enough
that the GUPPI using premerger data are no longer accurate. This failure to perfectly replicate these
price changes reinforces that GUPPIs are not direct estimates of price changes themselves. 53
GUPPI calculations for the estimated demand specifications raise the issue of markup estimation.
Markups are often themselves estimated in practice because marginal costs are not generally observed.
One common estimation method for markups is to use accounting data, but accounting standards often
conflate fixed and variable costs, and only deal with measurable pecuniary costs. Further, accounting cost
data are often only available at the firm level, causing averaging errors for multiproduct firms. 54
In academic research with full demand estimation, margins are often estimated using the first
order condition of the profit function. In my setup of NashBertrand competition between single product
firms with discrete choice demand, expected markups are a version of equation (4) without
cannibalization effects:
( )
= . (21)
∫ 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝜕𝜕𝜕𝜕 𝑦𝑦(𝑖𝑖 )
𝐸𝐸�𝑃𝑃𝑗𝑗𝑗𝑗 − 𝐶𝐶𝑗𝑗𝑗𝑗� −
𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
∫ 𝜕𝜕𝜕𝜕 𝑦𝑦𝑖𝑖
While expected demand ( ) is likely to be estimated𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 well or realized demand may be available,
( )
, the denominator∫ 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝜕𝜕 𝜕𝜕of 𝑦𝑦(𝑖𝑖21) is also the denominator of the weight function in (14). Thus
𝜕𝜕𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
∫problems𝜕𝜕𝑃𝑃𝑗𝑗𝑗𝑗 ��𝜕𝜕𝜕 in𝑦𝑦 𝑖𝑖estimating the weights will also affect estimation of markups. When I report GUPPIs, I will
report both estimates using the true markups as well as those using estimated markups from the demand
system.
6. Demand Specifications
For each dataset, I will estimate demand using specifications with precedent in antitrust analysis
for comparison. I simulate product choices at the individual level, so I maximize the negative log
likelihood of the choices made by each simulated consumer. I report the mean and standard deviation of
the estimated coefficients over the 100 datasets to give a sense of the precision of the estimates.
Below are the details of the specifications I estimate. They vary either by how accurately data on
consumer type is observed, which reflects measurement error, or by the functional form of , which
reflects model misspecification.
𝑦𝑦𝑖𝑖 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
53 There is a lack of consensus on what UPPs and GUPPIs exactly represent. “UPP does not predict postmerger
prices, but only predicts the sign of changes in price.” Joseph Farrell & Carl Shapiro, Upward Pricing Pressure In
Horizontal Merger Analysis: Reply to Epstein and Rubinfeld, 10 BE J. Theoretical Econ., 3 (2010). “We do see
UPP as a simple and useful measure that is generally indicative of likely price effects.” Joseph Farrell & Carl
Shapiro, Upward Pricing Pressure and Critical Loss Analysis: Response, 2010 CPI Antitrust J., 4,
https://www.competitionpolicyinternational.com/assets/Uploads/ShapiroFarrellFEB10copy.pdf
54 See generally Seth B. Sacher & John Simpson. Estimating Incremental Margins for Diversion Analysis, 83
Antitrust L.J. 527 (2020).
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6.1 Simple Logit Specification
My benchmark for poor performance is a simple logit specification estimated on choice and price
data but no other individualspecific data. Recall that this is essentially assumed when an analyst uses
shareproportional diversion ratios. I assume the for this specification is
= 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 + . (22)
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 PURE 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃
Given the lack of individual heterogeneity,𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 it is notβj possible𝛼𝛼 for th𝑃𝑃𝑗𝑗is𝑗𝑗 specification to predict distribution of
the choice probabilities or the price sensitivities. Thus it is practically impossible for the specification to
generate the correct diversion ratios because the weighted average of individual diversion ratios (12)
would have to coincidentally equal the share proportional formula (15).
6.2 Correct Model and Data Specification
For this specification, I use the “correct” model of utility of equation (19) used to generate the
synthetic data, and assume is observed. This corresponds to the case where the econometrician has
access to individualspecific data on consumer characteristics (microdata) which is informative for
𝑖𝑖
product choice. In this case,𝑦𝑦 the microdata on is perfectly informative of price sensitivity as price
sensitivity is proportional to . I use a common transformation to include the impact of varying price
𝑖𝑖
sensitivity: I include both price and interactions 𝑦𝑦of price with a centered and denote the centered as
𝑖𝑖
. I use the following form 𝑦𝑦of the indices for estimation:
𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖
𝑦𝑦̇𝑖𝑖 = + + . (23)
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 TRUE 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
It follows from (19) and the assumption𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 sβ thatj =𝛼𝛼5 and 𝑃𝑃the𝑗𝑗𝑗𝑗 mean𝛾𝛾 of 𝑦𝑦̇𝑖𝑖 𝑃𝑃is𝑗𝑗𝑗𝑗 1 that this specification
estimates the true choices probabilities and price sensitivities if estimated parameters = 5 ,
𝜃𝜃 𝑦𝑦𝑖𝑖
= 5, = 5. This specification allows decomposition of the price effect𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 into its mean
β�𝑗𝑗 𝑄𝑄𝑗𝑗
plus𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 a varying term,𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 which is common when the demand researcher is unsure whether price sensitivity
varies.𝛼𝛼� − 𝛾𝛾� −
There are drawbacks to this specification. The first is that the additional parameter makes
estimation less efficient. The second is that for very negative (near zero ), this can imply positive
price sensitivity if > . The denominator of equation (20), the formula for weights, is a
𝑖𝑖 𝑖𝑖
function of choice probabilities𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 and price sensitivities. When 𝑦𝑦pricė sensitivities𝑦𝑦 are both positive and
negative, the denominator𝛼𝛼 of𝛾𝛾 weights can be small and/or be the opposite sign of its numerator. Thus,
some estimated weights can then be unrealistically large in magnitude and/or strongly negative.
Moreover, it could be profit maximizing to charge extremely high prices. While most consumers may no
longer buy such highly priced products, consumers with positive price sensitivities would continue to buy
no matter what the price is. It therefore may be a corner solution to charge an incredibly high price to just
these priceloving consumers rather than to sell to consumers more generally. 55 The diversion ratios
would then no longer be informative of the postmerger equilibrium, because pricing equation (5) is an
interior solution. In general, applications tend to ignore this issue, so I will simply assume that budget
55 Under the assumption that represents indirect utility, there are budget constraints so prices cannot be infinite.
𝑖𝑖𝑖𝑖𝑖𝑖
𝑈𝑈 21
OEA Working Paper
constraints are such that no local corner maximum yields greater profit than the interior solution.56 I will
document these positive price coefficients when they occur in the estimation result for this and other
specifications.
6.3 Correct Model but Mismeasured Data Specification
For three specifications, I use mismeasured data to explore measurement error. Mismeasured
data have the same marginal distribution of , but are only correlated with . 57 I use correlations of
𝑖𝑖
0.95, 0.85, and 0.75 to illustrate the decline in performance𝑦𝑦 as the data become less accurate. These
𝑖𝑖 𝑖𝑖
specifications not only correspond to when measured𝑦𝑦 has been contaminated𝑦𝑦 with errors, but also the
common case when is unobserved but some correlated proxy is used instead.
𝑦𝑦𝑖𝑖
I denote the 𝑦𝑦centered𝑖𝑖 version of mismeasured with correlation with as ( ). The
associated choice probabilities are determined by the :
𝑦𝑦𝑖𝑖 𝜌𝜌 𝑦𝑦𝑖𝑖 𝑦𝑦̇𝑖𝑖 𝜌𝜌
𝑖𝑖𝑖𝑖𝑖𝑖
= + 𝛿𝛿 + ( ). (24)
𝑀𝑀𝑀𝑀𝑀𝑀 MMρ 𝑀𝑀𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀
The resulting bias will depend𝛿𝛿 on𝑖𝑖𝑖𝑖𝑖𝑖 the degreeβj of inaccuracy,𝛼𝛼 𝑃𝑃𝑗𝑗𝑗𝑗 but𝛾𝛾 it is𝑃𝑃 𝑗𝑗an𝑗𝑗𝑦𝑦 ̇open𝑖𝑖 𝜌𝜌 question on how severe the
issue is in practice.
6.4 Quantile Coefficient Specification
In contrast to the case where mismeasured data have the correct marginal distribution, another
common form of measurement error is where is only observable up to discrete ranges, which I will call
“bins” indexed by . For example, such data are often produced by surveys where giving a range is
𝑖𝑖
easier than giving an exact number. Binning is𝑦𝑦 also used as a strategy of anonymizing data when
releasing to the public𝑏𝑏 ∈. 𝐵𝐵
To recreate this case, I estimate a model with bins designated by quantiles. The takes the
form
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
= + ( ) . (25)
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄
( ) is a binspecific coefficient𝛿𝛿𝑖𝑖𝑖𝑖 which𝛽𝛽𝑗𝑗 take𝛼𝛼𝑄𝑄 a𝑄𝑄 𝑄𝑄𝑄𝑄particular𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑦𝑦𝑖𝑖 value𝑃𝑃𝑗𝑗𝑗𝑗 based on what quantile is in.
Weights will take on the form
𝛼𝛼𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖
56 For example, Nevo (2000) estimates a distribution with 0.7% positive price sensitivities. Aviv Nevo, Measuring
Market Power in the Ready‐to‐Eat Cereal Industry, 69 Econometrica 307, 329 (2000).
57 I generate a with correlation with and the same marginal distribution, ( , ), as by drawing
a standard normal with correlation = (1 + ) with = ln ( ), and then setting = 2( + ).
Correlation 𝑍𝑍 between the standard𝜌𝜌 normal𝑌𝑌 implies correlation between their𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 exponentiation𝜇𝜇 𝜎𝜎 due to 𝑌𝑌following
𝑛𝑛
proof: Algebraic manipulation𝑧𝑧 of the 𝜌𝜌definition𝑙𝑙𝑙𝑙 of 𝜌𝜌covariance𝑦𝑦 and the𝑌𝑌 lognormal imply 𝑍𝑍 (𝑒𝑒𝑒𝑒𝑒𝑒, )𝜇𝜇= 𝜎𝜎[𝜎𝜎exp ( +
𝑛𝑛
)] [ ] 𝜌𝜌. and are both normal, so + is normal and exp𝜌𝜌 ( + ) is lognormal. Using the above covariance
formula, the2 formulas for the mean and standard deviation of a sum correlated normals,𝑐𝑐 the𝑐𝑐𝑐𝑐 definitions𝑌𝑌 𝑍𝑍 𝐸𝐸 of mean𝑦𝑦 and
𝑧𝑧standard− 𝐸𝐸 𝑌𝑌deviation𝑦𝑦 of𝑧𝑧 a lognormal, and the𝑦𝑦 definition𝑧𝑧 of correlation, 𝑦𝑦the implied𝑧𝑧 correlation between y and z is =
(1 + ).
𝑛𝑛
𝜌𝜌
𝑙𝑙𝑙𝑙 𝜌𝜌 22
OEA Working Paper
1 ( )
= 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 . (26)
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 1 � − 𝑆𝑆𝑖𝑖𝑖𝑖�𝛿𝛿𝑖𝑖𝑖𝑖 �� 𝑆𝑆𝑖𝑖𝑖𝑖�𝛿𝛿𝑖𝑖𝑖𝑖 �𝛼𝛼𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄(𝑄𝑄 𝑦𝑦)𝑖𝑖 ( )
𝜔𝜔𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄
𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑖𝑖 𝑖𝑖
I estimate two versions of th∫is� specification.− 𝑆𝑆 �𝛿𝛿𝑖𝑖𝑖𝑖 The first�� 𝑆𝑆 uses�𝛿𝛿 𝑖𝑖quintiles𝑖𝑖 �because𝛼𝛼 five income𝑦𝑦 𝜕𝜕𝜕𝜕 𝑦𝑦bins are
common for applications. 58 The second uses deciles, which will show how much estimate improve as the
binning becomes more granular.
6.5 ProductMarket Coefficients Specification
An alternative to the simple logit that avoids the difficulties in incorporating varying price
sensitivity but allows to capture taste variation due to price would be a model with productmarket
specific coefficients on . The associated index is
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
𝑦𝑦𝑖𝑖 = + + . (27)
𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃
A similar demand system was used 𝛿𝛿by𝑖𝑖𝑖𝑖𝑖𝑖 the applicantsβj 𝛼𝛼 in the𝑃𝑃 𝑗𝑗FCC’s𝑗𝑗 𝜓𝜓 𝑗𝑗T𝑡𝑡𝑦𝑦Mobile/Sprinṫ𝑖𝑖 merger review
process, though that had a more complicated set of covariates.59
An advantage of this specification is that it is possible to estimate the choice probabilities
perfectly – captures all the utility variation that is ignored by misspecifying the relation of price
sensitivity to . With strong variation in data on choice and , an econometrician should be able to
𝜓𝜓𝑗𝑗𝑗𝑗𝑦𝑦̇𝑖𝑖
estimate = 5 , = 5, and = 5 : then = . In fact, it would be possible
𝑦𝑦𝑖𝑖 𝑦𝑦𝑖𝑖
to estimate 𝑃𝑃choice𝑃𝑃𝑃𝑃 probabilities𝑃𝑃𝑃𝑃𝑃𝑃 without prices if there was also𝑃𝑃𝑃𝑃 a𝑃𝑃 product𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇market fixed effect. 60 I take
β�j 𝑄𝑄𝑗𝑗 𝛼𝛼� − 𝜓𝜓𝑗𝑗𝑗𝑗 − 𝑃𝑃𝑗𝑗𝑡𝑡 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
this specification to be a “best case” scenario for a misspecified model that does not allow variation in
price sensitivity – there exists an approximation to this implementation in which there is no direct data on
but instead data to proxy for the entire value of . 61
𝑦𝑦𝑖𝑖 The disadvantage of this specification is that𝜃𝜃𝑦𝑦 𝑖𝑖the𝑃𝑃𝑗𝑗𝑗𝑗 diversion ratios are biased. As the price
coefficient is now constant, it factors out in the implied weights:
1
= 𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃 . (28)
1 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 ( )
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 � − 𝑆𝑆 �𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 �� 𝑆𝑆 �𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 �
𝜔𝜔𝑖𝑖𝑖𝑖𝑖𝑖 𝑃𝑃𝑃𝑃𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃
Even if the choice probabilities are estimated∫ � −perfectly𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖�𝛿𝛿𝑖𝑖,𝑖𝑖𝑖𝑖 the� diversion�𝑆𝑆𝑖𝑖𝑖𝑖�𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 ratio�𝜕𝜕𝜕𝜕 will𝑦𝑦𝑖𝑖 still be biased because the
weights ignore . In particular, diversion from High Tier products will be overestimated as this formula
𝑦𝑦𝑖𝑖
58 Examples with five income groups in a demand system include Goolsbee & Petrin, supra note 34; Leemore
Dafny & David Dranove, Do Report Cards Tell Consumers Anything They Don't Already Know? The Case of
Medicare HMOs, 39 RAND J. Econ. 790 (2008); and Amil Petrin & Kenneth E. Train, A Control Function
Approach to Endogeneity in Consumer Choice Models, 47 J. Marketing Res. 3 (2010).
59 TMobile/Sprint Expert Economic Analysis at 2124, paras. 4858.
60 For example, the authors of the TMobile/Sprint demand model did not estimate price sensitivity directly in the
demand system but estimated it in an auxiliary procedure. This was because major U.S. mobile telephone services
use national pricing, so there is little to no market level price variation. Id. at 21, para. 50 & n. 45, and at 63, para.
63 & n. 54.
61 For example, machine learning techniques could incorporate variation from many variables correlated with
, or even more generally, to produce accurate predictions of .
𝑖𝑖 𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖
𝜃𝜃𝑦𝑦 𝑃𝑃 𝛿𝛿 23 𝑆𝑆
OEA Working Paper
gives the priceinsensitive consumers who disproportionately choose High Tier products significant
weight while in reality they should have little weight at all.
6.6 Random Coefficient Specification
If microdata on varying taste for price (or other product characteristics) are not available, it is
possible to estimate the distribution of by assuming a parametric distribution for it. In this case this
amounts to using
𝑦𝑦𝑖𝑖
= + . (29)
𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅
Simulations or numerical integrals of 𝛿𝛿are𝑖𝑖𝑖𝑖𝑖𝑖 usedβj to form𝛼𝛼𝑖𝑖 choice𝑃𝑃𝑗𝑗𝑗𝑗 probabilities for every guess of the
parameters, including those for the distribution𝑅𝑅𝑅𝑅 of . Random coefficient specifications like this can be
𝛼𝛼𝑖𝑖
computationally difficult and timeconsuming and are𝑅𝑅𝑅𝑅 rarely used in merger reviews.
𝛼𝛼𝑖𝑖
The specification can recreate the true choice probabilities if = 5 and is estimated to
be the negative of a lognormal distribution with = 0.42 + (5) and𝑅𝑅𝑅𝑅 = 0.92 (which𝑅𝑅𝑅𝑅 produces a
β�𝑗𝑗 𝑄𝑄𝑗𝑗 𝛼𝛼�𝑖𝑖
mean price sensitivity of 5 and skewness of 5). I estimate two versions of this specification: one
assuming the correct negative lognormal distribution,𝜇𝜇 − and the another𝑙𝑙𝑙𝑙 assuming𝜎𝜎 a normal distribution.
The latter will reflect the impact of misspecifying the strictly positive and asymmetric price sensitivity
distribution with a symmetric distribution over a full support.
To generate the likelihoods for estimation, I use simulation in which a finite number of draws of
are taken and resulting choice probabilities are averaged over the draws. 62 Integration for the
market𝑅𝑅𝑅𝑅 level diversion ratios and GUPPIs use simulated draws of using Halton draws, which have
𝛼𝛼�𝑖𝑖
good performance in random coefficients models. 63 𝑅𝑅𝑅𝑅
𝛼𝛼�𝑖𝑖
6.7 Nested Logit Specification
Nested logit is a popular formulation in discrete choice estimation because it admits more flexible
substitution patterns and is relatively simple to estimate. 64 Nested logit is often estimated as a first step in
analyzing a demand dataset: if certain estimated parameters are less than one then there is evidence true
model lacks IIA. 65 Estimating nested logits is much easier than estimating a random coefficient model
because there is no integration of coefficients required. The merger reviews of Aetna/Humana and
AT&T/DirecTV used nested logit demand systems for their diversion ratio estimates. Grigolon and
Verboeven (2014) compare random coefficient and nested logit estimates in Monte Carlo experiments
62 I use Stata’s asmixlogit function to estimate the random coefficient specifications and use the default settings.
This means 50 draws generated through Hammersley sequences. Stata, asmixlogit — AlternativeSpecific Mixed
Logit Regression, https://www.stata.com/manuals15/rasmixlogit.pdf (last visited Nov. 19, 2021).
63 Kenneth E. Train, Halton Sequences for Mixed Logit, 2000 UC Berkeley Working Paper No. E00278,
https://escholarship.org/uc/item/6zs694tp.
64 See generally Train, supra note 33, at 7786.
65 The parameters in question are the nesting parameters , which governs substitution patterns as explained later in
this subsection. The most wellknown formal test is from 𝑔𝑔Jerry Hausman & Daniel McFadden, Specification Tests
for the Multinomial Logit Model, 52 Econometrica 1219,𝜆𝜆 12261229 (1984).
24
OEA Working Paper
similar to those in this paper. Using automobile data, they find the nested logit to be a good
approximation of random coefficients in substitution patterns. 66
The nested logit diversion ratio does not share the weighted average formula for individual
diversion ratios or weights in (15) and (16) . The nested logit varies from the previous logit specifications
by assuming that product taste shocks are not i.i.d but are correlated for products in the same product
“nest,” , which is one of several mutually exclusive product groupings in the set of nests . Given the
nested logit distribution of taste shocks, the nested logit choice probabilities for a product in set are
𝑔𝑔 𝐺𝐺 𝑔𝑔
=  𝑗𝑗 𝐽𝐽 (30)
where 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖𝑆𝑆𝑖𝑖𝑖𝑖 𝑔𝑔𝑔𝑔
exp
𝜆𝜆𝑔𝑔
= 𝑔𝑔 𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 (31)
�∑𝑘𝑘∈𝐽𝐽 � ��
exp𝜆𝜆𝑔𝑔
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 𝜆𝜆ℎ
ℎ 𝑖𝑖𝑖𝑖𝑖𝑖
ℎ∈𝐺𝐺 𝑘𝑘∈𝐽𝐽 𝛿𝛿
∑ �∑ � ℎ ��
exp 𝜆𝜆
𝑖𝑖𝑖𝑖𝑖𝑖
 = 𝛿𝛿 . (32)
exp� 𝑔𝑔 �
𝑖𝑖𝑖𝑖 𝑔𝑔𝑔𝑔 𝜆𝜆
𝑆𝑆 𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖
𝑘𝑘∈𝐽𝐽 𝛿𝛿
∑ � 𝑔𝑔 �
can be interpreted as the choice probability for a nest and𝜆𝜆  can be interpreted as the choice j
probability𝑖𝑖𝑖𝑖𝑖𝑖 for a product conditional on the nest g being chosen.𝑖𝑖 𝑖𝑖 Thus𝑔𝑔𝑔𝑔 this model can be conceptualized
𝑆𝑆as a consumer choosing a nest first according to a logit over the𝑆𝑆 nestspecific “inclusive value,”
exp , and then limiting themselves to that nest when making a final decision of the product
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
𝑔𝑔 67
according∑𝑘𝑘∈𝐽𝐽 to� 𝜆𝜆a𝑔𝑔 nest� specific logit. The nestspecific parameters (“nesting parameters” or
“dissimilarity parameters”) govern the correlation of the product taste shocks within nests such that as
𝜆𝜆𝑔𝑔
approaches 0, the product taste shocks become more correlated. As approaches 1, the shocks become
𝜆𝜆𝑔𝑔
independent and the model reverts to a logit.
𝜆𝜆𝑔𝑔
Assuming no variation in consumer types, , expected marketlevel diversion ratios are equal to
constant individual diversion ratios:
𝑦𝑦𝑖𝑖
1 +
= = . (33)
𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗𝑗𝑗
1 𝑆𝑆 1𝑆𝑆 � Λ �
𝐸𝐸�𝐷𝐷𝐷𝐷𝑡𝑡 � 𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖 −
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖
� − � − 𝜆𝜆𝑔𝑔�𝑆𝑆𝑖𝑖𝑖𝑖 𝑔𝑔𝑔𝑔 − 𝜆𝜆𝑔𝑔𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖�
where varies based on whether j and k are in𝜆𝜆 𝑔𝑔the same nest g:
Λ𝑗𝑗𝑗𝑗
66 Laura Grigolon & Frank Verboven, Nested Logit or Random Coefficients Logit? A Comparison of Alternative
Discrete Choice Models of Product Differentiation, 96 Rev. Econ. Stat. 916 (2014).
67 It is possible to extend nesting by adding several layers of nests. Train, supra note 33, at 8688.
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OEA Working Paper
0,
= 1 .
,
−1 𝑗𝑗 𝑎𝑎𝑎𝑎𝑎𝑎 𝑘𝑘 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛
𝑔𝑔
Λ𝑗𝑗𝑗𝑗𝑗𝑗 �𝜆𝜆 −
𝑗𝑗 𝑎𝑎𝑎𝑎𝑎𝑎 𝑘𝑘 𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔
Thus unlike the general𝑆𝑆𝑔𝑔𝑔𝑔 model of Section IV where product taste shocks are independent, the
individual diversion ratios of the nested logit are not just . Two products can have high diversion
𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖

with low shares if is small – the diversion ratio then approaches1−𝑆𝑆𝑖𝑖𝑖𝑖𝑖𝑖 as approaches 1.
𝑆𝑆𝑖𝑖𝑖𝑖 𝑔𝑔𝑔𝑔
𝑔𝑔 𝑖𝑖𝑘𝑘 𝑔𝑔𝑔𝑔 𝑔𝑔
My nested𝜆𝜆 logit specification admits no individual level variation,1−𝑆𝑆 so t𝜆𝜆he is the same as the
Simple Logit specification: 68
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖
= + . (34)
One caveat is that some simulations estimate𝑁𝑁𝑁𝑁 to NLbe larger𝑁𝑁𝑁𝑁 than 1. BorschSüpan (1990) 69 shows that
𝛿𝛿𝑖𝑖𝑖𝑖𝑖𝑖 βj 𝛼𝛼 𝑃𝑃𝑗𝑗𝑗𝑗
this could be consistent with a utility maximizing𝑔𝑔 individual based on other parameters. Kling and
Herriges (1995)70 and Herriges and Kling (199𝜆𝜆6)71 find the scope for this possibility is limited in theory
and empirical practice, respectively. To sidestep these issues, I constrain to be at most 1, which I find
to be generally within the 95% confidence interval of estimate when it does violate the bound.72
𝜆𝜆𝑔𝑔
𝑔𝑔
7. Coefficient Estimates 𝜆𝜆
The mean coefficient estimates and the standard deviations for all models are reported in Table 5,
along with information on the frequency of positive estimated price sensitivities and McFadden’s ’s. 73
As expected, the Simple Logit without any individual data yields poor estimates – all coefficients are2
biased towards the origin and the variance explained by the model is low, as reflected in the very low𝑅𝑅
McFadden’s . Meanwhile using the correct model with correct data yields highly accurate estimates,
with bias of no2 more than 0.1 and with very small standard deviations. Unsurprisingly, the correct model
𝑅𝑅
68 When I incorporate individual level data with nested logit find that estimation correctly estimates nesting
parameters near 1 when the individual level data explains choices well. Thus, I focus on the case where nesting
might best serve as a substitute for when such identifying data is unavailable.
69 Axel BörschSüpan, On the Compatibility of Nested Logit Models with Utility Maximization, 43 J. Econometrics
373 (1990).
70 Catherine L. Kling & Joseph A. Herriges, An Empirical Investigation of The Consistency of Nested Logit Models
with Utility Maximization, 77 Amer. J. Agric. Econ. 875 (1995).
71 Joseph A. Herriges & Catherine L. Kling, Testing the Consistency of Nested Logit Models with Utility
Maximization, 50 Econ. Letters 33 (1996).
72 I use STATA’s nlogit command to estimate the nested logit demand system which does not admit inequality
constraints. When I find a > 1, I reestimate the nested logit with an equality constraint on the nesting parameter
that exceeds 1 and report the𝑔𝑔 corresponding set of estimates. Stata, nlogit — Nested Logit Regression,
https://www.stata.com/manuals/cmnlogit.pdf𝜆𝜆 (last visited Nov. 19, 2021).
73 McFadden’s is a test statistic for likelihood estimates analogous to the : it is 1 minus the ratio of the log
likelihood of the 2model over the log likelihood of a version of the model with2 only choicespecific intercepts as
covariates. It represents𝑅𝑅 how much unexplained variation in the simpler model𝑅𝑅 is explained by the more
complicated one. Daniel McFadden, Conditional Logit Analysis of Qualitative Choice Behavior, in Frontiers in
Econometrics 122 (ed. Paul Zarembka 1973).
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has an excellent fit with a McFadden’s of 0.35. 74 However, 34% of samples have some consumers
with positive price sensitivities, though in2 those samples they only make up 2.8% of the sample on
average. 𝑅𝑅
The mismeasurement of the microdata has a very striking effect – estimation performance
rapidly declines with modest drops in the correlation between the true and mismeasured data. Even the
0.95 Correlation specification has biases toward the origin of more than 1.0 for every parameter. The
0.75 Correlation specification has biases to toward the origin by a factor of at least 2, and the mean
McFadden’s of 0.15 is less than half of the mean for the correct model. Interestingly, the
mismeasured data2 specifications have a smaller prevalence of positive price sensitivities because the
attenuation is𝑅𝑅 stronger for the pricecentered interactions than for the price coefficient – positive price
coefficients happen for only 2.8% of the experiments for 0.95 correlation, and simply do not happen for
𝑖𝑖
0.85 or 0.75 correlation. 𝑦𝑦
The Quintile Coefficients specification demonstrates severe problems with positive price
sensitivities. While the fit is excellent with a mean McFadden’s of 0.28, 99% of the experiments
exhibit positive price sensitivities for some consumers, with either 2the lowest and sometimes the second
lowest quintile coefficients being positive. This is because the model𝑅𝑅 is not flexible enough to capture the
extreme price sensitivity of consumers with high . Instead, the model compensates by estimating lower
quality for every good. However, because High Tier products appear to have worse quality, the only way
𝑖𝑖
the specification can rationalize priceinsensitive 𝑦𝑦consumers choosing expensive High Tier products is to
assign them positive value for price. The implied price sensitivity results in an average price coefficient
over all simulations of 2.0 instead of the true 5.0. In contrast, the Decile Coefficients specification is
flexible enough so this problem is far less severe. While the quality estimates are still biased downwards,
only 34% of simulations− have some consumers− with positive price sensitivity, and in these simulations
these consumers usually only make up the lowest decile. Compared to the Quintile Coefficients
specification the average price coefficient over all simulations is still somewhat attenuated but greater at
𝑖𝑖
3.7. The specification has an excellent fit with a𝑦𝑦 mean McFadden’s of 0.33.
2
− The ProductMarket Coefficient specification estimates also shows𝑅𝑅 excellent fit. Moreover, the
price and quality coefficients are very close to the true values and on average the specification has a
slightly higher mean McFadden’s than the correct specification. This is not surprising, as the product
marketspecific coefficients are so2 flexible that they absorb some of the simulation error in choices, i.e.
this specification slightly overfits relative𝑅𝑅 to the true specification.
𝑦𝑦𝑖𝑖
74 Values of 0.2 to 0.4 represent “excellent fit.” Daniel McFadden, Quantitative Methods for Analyzing Travel
Behaviour on Individuals: Some Recent Developments, in Behavioural Travel Modelling 306 (eds. David Hensher &
Peter Stopher 1979).
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Table 5: Coefficient Estimates
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
Product Lognormal Normal
True 0.95 Corr. 0.85 Corr. 0.75 Corr. Quantile Decile Nested
Parameter Pure Correct Market Random Random
Values Coefficients Coefficients Logit
Coefficients Coefficient Coefficient
1.33 5.00 3.58𝒚𝒚𝒊𝒊 2.43𝒚𝒚𝒊𝒊 1.98𝒚𝒚𝒊𝒊 5.03 1.30
5
(0.18) (0.42) (0.33) (0.27) (0.24) (0.77) (0.18)
𝛼𝛼 4.99 3.29 1.89 1.30
5
(0.14) (0.11) (0.07) (0.06)
𝛾𝛾 1.20 1.49
1.19
(0.24) (0.40)
𝜇𝜇 0.93 1.35
0.92
(0.14) (0.61)
𝜎𝜎 0.63
1
(0.10)
𝐿𝐿𝐿𝐿𝐿𝐿
𝜆𝜆 0.58
1
(0.11)
𝐻𝐻𝐻𝐻𝐻𝐻ℎ
𝜆𝜆 2.64 12.50 8.62 5.57 4.38 7.35 10.63 12.61 13.11 4.17 2.81
12.5
(0.25) (0.60) (0.48) (0.39) (0.35) (0.48) (0.55) (1.09) (2.94) (1.27) (0.25)
1
𝑄𝑄 2.66 12.50 8.63 5.58 4.40 7.36 10.63 12.61 13.12 4.18 2.81
12.5
(0.25) (0.60) (0.49) (0.39) (0.35) (0.49) (0.55) (1.09) (2.93) (1.27) (0.25)
2
𝑄𝑄 3.67 15.00 10.55 7.01 5.65 8.25 12.66 15.13 15.70 5.15 3.84
15
(0.40) (0.89) (0.74) (0.60) (0.54) (0.76) (0.83) (1.67) (3.36) (1.47) (0.41)
3
𝑄𝑄 3.68 15.00 10.56 7.03 5.67 8.26 12.66 15.12 15.70 5.16 3.84
15
(0.40) (0.89) (0.74) (0.59) (0.54) (0.76) (0.82) (1.67) (3.34) (1.46) (0.40)
4
% Sims𝑄𝑄 w. + Price Sens. 0% 27% 11% 0% 0% 99% 34% 0% 0% 100% 0%
Mean % Sample w. + Price
. 2.8% 2.8%   26.8% 11.0%   12.9% 
Sens. If Any in Simulation
0.00 0.35 0.28 0.20 0.15 0.28 0.33 0.35 0.35 0.01 0.01
McFadden’s Pseudo
2 (0.00) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.00) (0.00)
Out of the 100 simulations,𝑅𝑅 means and standard deviations of coefficient estimates and McFadden’s Pseudo reported. Also reported is the percentage of simulations with some consumer with positive price
sensitivity and, conditional on being one of those simulations, the mean % of consumers with positive price2 sensitivity. Quantile price coefficients of specifications (6) and (7) and the productmarket
𝑅𝑅
coefficients of specification (8) are too numerous to report here but are available upon request. The and estimates of specification (10) are not directly comparable to that of (9) but are reported on the same𝑖𝑖
rows to conserve space. 𝑦𝑦
𝜇𝜇 𝜎𝜎
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The performance of the Random Coefficient specifications depends on the assumed distribution
of the random coefficients. When the correct lognormal distribution is assumed, the corresponding
distribution parameters mirrors the true distribution very closely and only slightly overestimates the
quality parameters. In contrast, if the normal distribution is assumed, the results are very similar to the
Quintile Coefficients specification. The Normal Random Coefficients specification cannot rationalize the
long tail of very pricesensitive consumers, and so compensates by assuming worse quality and a large
percentage of consumers with positive price sensitivity in every simulation: 12.9% on average. The
average of normal random price coefficient is also only 1.5, which is far lower than the true mean of
5.0. For both distributional assumptions, the mean McFadden’s is small with 0.007 for Lognormal
and 0.005 for Normal. This is a limitation of assuming −the coefficients2 are unobservable: each consumer
is− observationally identical aside from choice, so a random coefficient𝑅𝑅 choice probability is the average
over all draws of the random coefficient, i.e., the same number. Thus every consumer gets the same
choice probabilities, and so the likelihood is low because there is never a case where the choice
probability of an observed choice is especially high.
The Nested Logit specification has nesting parameters of around 0.6, representing significant
nesting. Markets without certain products provide ample evidence that consumer prefer products in the
same tier: under the true model consumers without access to one product in a tier disproportionately
choose the remaining product in that tier. However, with a constant price coefficient, the specification is
unable to infer that this diversion pattern is because of variance in price sensitivity and the difference in
tier price. As a result, the nonnesting parameters are biased towards the origin much like the Simple
Logit specification. Technically, (33) and (34) imply the effective values for these parameters should be
divided through by the appropriate nesting parameter, but even with this correction the coefficients are
less than half of the true coefficients. Like the Random Coefficients specification, the choice
probabilities are the same across all consumers, so the McFadden’s is small as well at 0.006.
2
8. Diversion Ratios Results 𝑅𝑅
The bias (mean error), the standard deviation of error, and the root mean square error (RSME) of
the expected marketlevel diversion ratios are reported in Table 6 for all models. In general, precision of
the estimates is relatively high with most standard deviations below 0.03. As a result, the RSME
corresponds mostly to bias when the bias is nonnegligible.
As it returns shareproportional diversion ratios, the Simple Logit specification underestimates
diversion within tier. Likewise, the Simple Logit overestimates the Low Tier to High Tier diversion
ratios, but actually underestimates the diversion from High Tier to Low Tier. HightoLow diversion is
displaced by the large overestimate of HightoOutside Option diversion (mean diversion ratio of 0.15
versus true diversion of essentially 0). In contrast, using the Correct Model specification results in biases
lower than 0.004 for every different case of diversion.
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OEA Working Paper
Table 6: Simulated Diversion Ratios
From From From From From From
Specification Statistic
1 to 0 1 to 2 1 to 3 3 to 0 3 to 1 3 to 4
Truth 0.12 0.59 0.15 0.00 0.31 0.38
Bias 0.04 0.30 0.13 0.15 0.02 0.11
(1) Simple Logit SD (0.02) (0.02) (0.02) (0.02) (0.02) (0.02)
RMSE [0.05] [0.30] [0.13] [0.15] [0.03] [0.11]
Bias 0.00 0.00 0.00 0.00 0.00 0.00
Correct
(2) SD (0.02) (0.02) (0.01) (0.00) (0.02) (0.03)
Model
RMSE [0.02] [0.02] [0.01] [0.00] [0.02] [0.03]
Bias 0.03 0.11 0.04 0.01 0.01 0.04
(3) 0.95 Corr. SD (0.02) (0.02) (0.01) (0.00) (0.01) (0.02)
RMSE [0.04] [0.11] [0.04] [0.01] [0.02] [0.05]
𝛾𝛾𝑖𝑖
Bias 0.06 0.21 0.07 0.04 0.01 0.06
(4) 0.85 Corr. SD (0.02) (0.01) (0.01) (0.01) (0.01) (0.01)
RMSE [0.07] [0.21] [0.07] [0.04] [0.01] [0.07]
𝛾𝛾𝑖𝑖
Bias 0.07 0.25 0.09 0.07 0.00 0.08
(5) 0.75 Corr. SD (0.02) (0.01) (0.01) (0.01) (0.01) (0.01)
RMSE [0.07] [0.25] [0.09] [0.07] [0.01] [0.08]
𝛾𝛾𝑖𝑖
Bias 0.27 0.14 0.07 0.03 0.12 0.29
Quintile
(6) SD (0.04) (0.02) (0.02) (0.30) (2.07) (4.38)
Coefficients
RMSE [0.27] [0.14] [0.07] [0.30] [2.06] [4.37]
Bias 0.01 0.00 0.01 0.00 0.03 0.06
Decile
(7) SD (0.02) (0.02) (0.01) (0.00) (0.03) (0.05)
Coefficients
RMSE [0.02] [0.02] [0.01] [0.00] [0.04] [0.08]
Product Bias 0.06 0.10 0.08 0.00 0.06 0.13
(8) Market SD (0.01) (0.02) (0.02) (0.00) (0.02) (0.03)
Coefficients RMSE [0.06] [0.10] [0.08] [0.00] [0.07] [0.13]
Lognormal Bias 0.00 0.01 0.00 0.00 0.00 0.01
(9) Random SD (0.02) (0.06) (0.02) (0.00) (0.02) (0.03)
Coefficient RMSE [0.02] [0.06] [0.02] [0.00] [0.02] [0.03]
Normal Bias 0.13 0.19 0.03 0.15 0.33 0.82
(10) Random SD (0.03) (0.05) (0.03) (0.09) (2.09) (4.23)
Coefficient RMSE [0.14] [0.20] [0.04] [0.17] [2.11] [4.29]
Bias 0.02 0.09 0.06 0.09 0.11 0.13
(11) Nested Logit SD (0.01) (0.06) (0.02) (0.01) (0.03) (0.07)
RMSE [0.02] [0.11] [0.06] [0.09] [0.11] [0.15]
Out of 100 simulations, the bias, the standard deviation (SD) of error and the root square error (RMSE) reported.
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Unsurprisingly, the measurement error specifications decline in performance as the correlation
declines. The 0.95 Correlation Specification moderately overestimates acrosstier expected diversion and
moderately underestimates withintier diversion. As the correlation becomes weaker, the specifications
yield more inaccurate diversion ratios. In particular, the mean withinLow Tier and within High Tier
diversion ratios of the 0.75 correlation specification are about half and three fourths of true diversion
ratios, respectively (0.34 vs 0.59; 0.31 vs. 0.38). In contrast, the mean LowtoHigh diversion ratio is
about 60% higher than under true expected diversion (0.24 vs. 0.15). Looking at Figures 6 and 7, the 0.75
Correlation specification too weakly differentiates consumers in their individual diversion ratios and
weights. The distributions of both individual diversion ratios and weights are too flat, and the estimates
𝑖𝑖
are quite noisy𝑦𝑦 on an individual basis. The 0.95 and 0.85 Correlated specifications exhibit similar
patterns but are less severe.
𝑦𝑦𝑖𝑖
The performance of the Quantile Coefficients specification depends on the number of quantiles.
The downward bias in the quality variables is severe enough that model biases down diversion from the
Low Tier ( 0.14 withinLow Tier and to 0.7 to High Tier) and biases up diversion to the Outside
Option (0.27 to the Outside Option). The bias of withinHigh Tier diversion ratio is high and negative
( 0.29) not− because the model underestimates− the magnitude of diversion, but because High Tier
consumption is predicted to increase when price goes up because of the prevalence of positive price
sensitivities− . However, the precision of this diversion ratio estimate is perhaps even more concerning
than the bias. The consumers with positive price sensitivities are more likely to choose the High Tier
Products (with high prices) so diverting consumers are mix of pricehating and priceloving consumers. In
many cases, positive and negative terms in the denominator for the weights described in (26) more or less
cancel out, resulting in very small denominators. Moreover, simulation error causes these small
denominators vary between positive and negative across simulations. Weights are thus sometimes
negative for consumers with positive price sensitivities in some simulations and often very large in
magnitude.75 Figure 9 shows that weights specific for High Tier products vary widely and can be over
200 times that of the average consumer depending on the simulation. 76 This translates into the standard
deviations of the withinHigh Tier and the HightoLow expected marketlevel diversion ratios being
more than 4.3 and 2.0, respectively.
In contrast, the Decile Coefficient specification estimates very accurate diversion ratios. Most
biases are below 0.03 in magnitude, with the least accurate diversion is being the withinHigh Tier
diversion ratio (bias of 0.06). Examination of individual diversion ratios and weights in Figures 10 and
11, respectively, reveal that ten discrete groups are nearly enough groups to approximate the true
distributions without making− the positive price sensitivities matter too much. The comparison with poor
𝑖𝑖
performance of the Quintile Coefficient 𝑦𝑦specification highlights how important it is to have many groups
for discretized data.
75 This is the same potential issue with positive price coefficients discussed for the Correct specification, though for
the Correct specification the number of positive price sensitivities estimated turn out to be negligible. Supra Section
6.2.
76 That is, in one of the simulated markets of 500 consumers, the magnitude of the weight would be more than
200 × = .
1 2
500 5
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Figure 6: Individual Diversion Ratios of 0.75 Correlated Specification
𝒚𝒚𝒊𝒊
Calculated for all Market 1 individuals across all 100 synthetic datasets.
Figure 7: Weights of 0.75 Correlated Specification
𝒚𝒚𝒊𝒊
Calculated for all Market 1 individuals across all 100 synthetic datasets. Weights are normalized so that the mean weight is 1 within a single
dataset.
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OEA Working Paper
Figure 8: Individual Diversion Ratios of Quintile Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets.
Figure 9: Weights of Quintile Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets. Weights are normalized so that the mean weight is 1 within a single
dataset.
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OEA Working Paper
Figure 10: Individual Diversion Ratios of Decile Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets.
Figure 11: Weights of Decile Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets. Weights are normalized so that the mean weight is 1 within a single
dataset.
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OEA Working Paper
The ProductMarket Coefficients specification produces biased marketlevel diversion ratios even
though the predicted demand fits “better” than the true model. The mean LowtoOutside Option
diversion ratio is only half as large as they should be (0.06 instead of 0.12). The mean withinLow Tier
diversion ratio is about one sixth too low (0.49 instead of 0.59). The mean LowtoHigh diversion ratio
is about twice as large (0.15 instead of 0.08). The mean HightoLow diversion ratio is about one sixth
too low (0.25 instead of 0.31). Finally, the mean withinHigh Tier diversion ratio is about one third too
large (0.51 instead of 0.38). Examination of the individual diversion ratios in Figure 10 reveals the
specification approximates the true individual diversion ratios almost perfectly. The individual diversion
ratios in discrete choice models are just functions of choice probabilities, so this is expected from the very
good fit of the choice model. The bias that does exist in the marketlevel diversion ratios comes entirely
from the weights, which are misspecified because they ignore heterogeneity in price sensitivity. Figure
13 shows this specification significantly overweights diversion from the High Tier, because the
specification overpredicts how responsive priceinsensitive consumers are to price.
The performance of the Random Coefficient specifications depends on the distribution assumed.
As noted before, the Random Coefficient specification cannot generate a high McFadden’s because
each estimated individual choice probability is the same. However, the estimated diversion ratios2 of
Lognormal Random Coefficient end up having little bias because it recreates the joint distribution𝑅𝑅
between price sensitivity and choice probabilities needed for the integration over types in (12). The
variance in expected marketlevel diversion ratios is only appreciably larger than using the Correct Model
specification for diversion within the Low Tier, where standard deviation is 0.06 instead of 0.02. In
contrast, assuming the wrong Normal distribution for the Random Coefficients specification leads to
problems similar to the Quintile Coefficients specification, which also estimate a significant amount of
positive price coefficients. Attenuated parameters lead to too little diversion within the Low Tier (bias of
0.19) and too much LowtoOutside option diversion (bias of 0.13). Positive price sensitivity causes
even more bias in expected marketlevel diversion within the High Tier compared to the Quintile
−Regression ( 0.88 vs. 0.29, respectively). 77 Similarly, bias for the HightoLow diversion ratio is even
higher than the Quintile Specification (0.36 vs. 0.12 respectively). These two estimated diversion ratios
likewise also− have similar− imprecision; the withinHigh Tier diversion ratio has a standard deviation of
4.8 and the HightoLow diversion ratio has a standard deviation of 2.4.
The Nested Logit specification yields inaccurate diversion ratios, but as demonstrated by the
Lognormal Random Coefficient specification this is not due to low McFadden’s . The estimated
diversion ratios are almost perfectly symmetric – within tier diversion ratios are about2 0.5, across tier
diversion ratios are about 0.2, and diversion ratios to the Outside Option are about𝑅𝑅 0.1. As the true
diversion ratios vary by tier, this is inaccurate. The biases stem from the specification’s estimate of the
nesting parameters in both nests to be about equal (about 0.6) and the nest shares also to be about equal
(about 45% for the Low Tier and 42% for the High Tier). Because the nest share and nesting parameters
are the only things that make the expected marketlevel diversion ratio vary from the Simple Logit share
proportional diversion ratios, diversion within nests ends up roughly equal as well. In contrast, true
diversion is less intense within the High Tier because 1) price sensitive consumers who happen to choose
the High Tier are more likely to switch to the Low Tier and 2) High Tier products attract price insensitive
77 In case of the Normal Random Coefficients, this is caused by very negative outliers, so the median bias of the
withinHigh Tier diversion ratio is actually only 0.22. In the case of the Quintile Regression, the median of the
withinHigh Tier diversion ratio is actually lower than the mean, with a bias of 0.34.
−
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OEA Working Paper
consumer who are less likely react to price changes at all (0.38 withinHigh Tier vs. 0.59 withinLow Tier
diversion ratios).
The Nested Logit results contrast with Grigolon and Verboeven (2014) in which nested logit and
random coefficients are good proxies for each other in Monte Carlo experiments.78 I speculate that this is
because the two major categories of true underlying demand that Grigolon and Verboeven (2014)
consider are 1) nestspecific random effects, which is similar to nested logit but not to my
“characteristics” setup, and 2) there is both a random coefficient and nesting, so including nesting is
always important.79 Thus the poor performance here of Nested Logit should be thought of as
emphasizing the importance of misspecification – with a different true demand system the Nested Logit
would perform much better.
Figure 12: Individual Diversion Ratios of ProductMarket Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets.
78 Grigolon & Verboven, supra note 66.
79 Id.
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Figure 13: Weights of ProductMarket Coefficients Specification
Calculated for all Market 1 individuals across all 100 synthetic datasets. Weights are normalized so that the mean weight is 1 within a single
dataset.
Figure 14: Individual Diversion Ratios of Normal Random Coefficient Specification
Estimated points calculated for 1,000 draws of from the estimated normal distribution for one Market 1 individuals per all 100 synthetic
datasets percentile is specific to each synthetic𝑖𝑖 dataset. True points calculated for all Market 1 individuals across all 100 synthetic datasets.
True points plotted over percentiles because 𝛼𝛼� = .
−𝛼𝛼𝑖𝑖
𝑦𝑦𝑖𝑖 −𝛼𝛼𝑖𝑖 𝜃𝜃𝑦𝑦𝑖𝑖
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Figure 15: Weights of Normal Random Coefficient Specification
Estimated points calculated for 1,000 draws of from the estimated normal distribution for one Market 1 individuals per all 100 synthetic
datasets percentile is specific to each synthetic𝑖𝑖 dataset. True points calculated for all Market 1 individuals across all 100 synthetic datasets.
𝛼𝛼� =
True points plotted𝑖𝑖 over percentiles because . Weights are normalized so that the mean weight is 1 within a single dataset.
Estimated− weights𝛼𝛼 for Product 3 censored at ± 50 which excludes less than 1% of estimated weights.
𝑦𝑦𝑖𝑖 −𝛼𝛼𝑖𝑖 𝜃𝜃𝑦𝑦𝑖𝑖
9. GUPPI Estimates Results
I report the bias, the standard deviation of errors, and the RSME in GUPPIs both calculated using
true markups and using markups implied by estimated demand in Table 7. As one might imagine,
specifications with accurate coefficients using both true and estimated markups are the ones that produce
good estimates of diversion. The Correct specification, the Decile Coefficients specifications, the
Lognormal Random Coefficient, and even the 0.95 Correlation specification have low biases. For the
Decile Coefficient and 95% Correlation , the bias is greater and positive when using the estimated
𝑖𝑖
markups. This is in line with the attenuation of estimated price𝑦𝑦sensitivities in these specifications: they
𝑖𝑖
underestimate price elasticities and overestimate𝑦𝑦 markups, so GUPPIs are larger.
This is even clearer in specifications that do not have good diversion ratios estimates. When
using true markups, the GUPPIs of the Simple Logit mirror the bias of sharebased diversion compared to
true diversion, with within tier GUPPIs underestimated because they are higher than what market share
would suggest. However, the Simple Logit GUPPIs with estimated markups are all overestimated,
sometime by more than double. This is because of the very attenuated parameter estimated (1.3 versus
the true 5.0) greatly inflates the markups. By sheer coincidence, the withinHigh Tier GUPPIs are about
right, because the low price parameter counteracts the underestimate of diversion ratios from shares. The
0.85 Correlated and the 0.75 Correlated specifications are similar, in which their GUPPIs biases
using observed markups reflect their diversion ratio biases, and the attenuation in price sensitivity leads to
𝑖𝑖 𝑖𝑖
large GUPPI upward𝑦𝑦 biases using estimated𝑦𝑦 markups.
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Table 7: Simulate GUPPIs
From 1 to 2 From 1 to 3 From 3 to 1 From 3 to 4
True Est. True Est. True Est. True Est.
Specification Statistic
Markup Markup Markup Markup Markup Markup Markup Markup
Truth 0.15 0.07 0.05 0.12
Bias 0.07 0.07 0.06 0.13 0.00 0.09 0.03 0.01
Simple
(1)
Logit SD (0.01) (0.03) (0.01) (0.04) (0.00) (0.02) (0.01) (0.02)
RMSE [0.08] [0.08] [0.06] [0.13] [0.00] [0.09] [0.03] [0.02]
Bias 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Correct
(2) Model SD (0.00) (0.02) (0.01) (0.01) (0.00) (0.01) (0.01) (0.02)
RMSE [0.00] [0.02] [0.01] [0.01] [0.00] [0.01] [0.01] [0.02]
Bias 0.03 0.02 0.02 0.03 0.00 0.02 0.01 0.00
0.95
(3) SD (0.00) (0.02) (0.01) (0.02) (0.00) (0.01) (0.01) (0.02)
Corr.
𝑖𝑖
𝛾𝛾 RMSE [0.03] [0.03] [0.02] [0.03] [0.00] [0.02] [0.01] [0.02]
Bias 0.05 0.03 0.04 0.07 0.00 0.04 0.02 0.01
0.85
(4)
Corr. SD (0.00) (0.02) (0.01) (0.02) (0.00) (0.01) (0.00) (0.02)
𝑖𝑖
𝛾𝛾 RMSE [0.05] [0.04] [0.04] [0.07] [0.00] [0.05] [0.02] [0.02]
Bias 0.06 0.04 0.04 0.09 0.00 0.06 0.02 0.01
0.75
(5) SD (0.00) (0.03) (0.01) (0.03) (0.00) (0.02) (0.00) (0.02)
Corr.
𝑖𝑖
𝛾𝛾 RMSE [0.06] [0.05] [0.04] [0.09] [0.00] [0.06] [0.02] [0.02]
Bias 0.04 0.05 0.03 0.09 0.02 0.06 0.09 19.56
Quintile
(6) Coefficient SD (0.00) (0.04) (0.01) (1.05) (0.32) (0.68) (1.35) (199.95)
s
RMSE [0.04] [0.06] [0.03] [1.05] [0.32] [0.68] [1.35] [199.91]
Bias 0.00 0.03 0.00 0.02 0.00 0.01 0.02 0.01
Decile
(7) Coefficient SD (0.00) (0.02) (0.01) (0.02) (0.00) (0.01) (0.02) (0.02)
s
RMSE [0.01] [0.04] [0.01] [0.03] [0.01] [0.02] [0.02] [0.03]
Product Bias 0.03 0.04 0.04 0.02 0.01 0.01 0.04 0.04
Market
SD (0.00) (0.02) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01)
(8) Coefficient
s RMSE [0.03] [0.04] [0.04] [0.02] [0.01] [0.02] [0.04] [0.05]
Bias 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Lognormal
(9) Random SD (0.02) (0.02) (0.01) (0.02) (0.00) (0.01) (0.01) (0.02)
Coefficient
RMSE [0.02] [0.02] [0.01] [0.02] [0.00] [0.01] [0.01] [0.02]
Normal Bias 0.05 0.09 0.01 0.22 0.05 0.15 0.25 7.50
Random
(10) SD (0.01) (0.05) (0.02) (0.55) (0.33) (0.37) (1.30) (68.34)
Coefficient
s RMSE [0.05] [0.11] [0.02] [0.55] [0.33] [0.39] [1.32] [68.41]
Bias 0.02 0.12 0.03 0.03 0.02 0.02 0.04 0.04
Nested
(11)
Logit SD (0.02) (0.04) (0.01) (0.03) (0.00) (0.02) (0.02) (0.02)
RMSE [0.03] [0.13] [0.03] [0.04] [0.02] [0.03] [0.05] [0.05]
Out of 100 simulations, the bias, the standard deviation (SD) of error and the root square error (RMSE) reported.
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The ProductMarkets Coefficients specification has the reverse bias at work: price insensitive
consumers are overly weighted so there is too little diversion to Low Tier products and too much
diversion to High Tier products. GUPPIs to Low Tier products are too low and GUPPIs to High Tier
products are too high using true markups. However, this bias has a strongly countervailing impact when
considering estimated markups – the specification overestimates the price sensitivity of the consumers
who do not choose the Outside Option, especially High Tier consumers. Thus, the implied markups are
too low, especially for the High Tier. While GUPPIs for diversions to Low Tier goods are only slightly
reduced compared to when using the true markups, the GUPPIs for diversions to High Tier goods drop by
50%, so much so that they change from overestimates to underestimates.
The substantial fraction of priceloving consumers estimated by the Quintile Coefficients and
Normal Random Coefficient specifications make their GUPPIs using estimated markups especially
volatile. While the GUPPIs using observed markups also are consistent with the biases from the
diversion ratios, their attenuated price sensitivities do not always lead to higher GUPPIs using estimated
markups. Recall in the case of the Quintile Coefficients specification, priceloving consumers mean that
the denominator of the weight formula in (26) is sometime very small and/or negative for the High Tier
goods. This value is also the denominator of (21), the formula for estimated markups, so estimated
markups can be very large and/or negative. Putting aside the fact negative markups or markups above the
High Tier price of 2.17 are nonsensical, these markups in combination with volatile expected marketlevel
diversion ratios leads to highly volatile GUPPIs for diversion to High Tier products. Using estimated
markups, the Low Tier to High Tier GUPPI has a standard deviation of bias of 1.35, while the within
High Tier GUPPIs has a standard deviation of almost 200. The Normal Random Coefficient
specification has similar issues: the Low Tier to High Tier GUPPI has a standard deviation of 1.3, while
the withinHigh Tier GUPPI has a standard deviation of nearly 70. The precision is somewhat higher
than the Quintile Coefficients specification because the Normal Random Coefficient specification
estimates fewer positive price sensitive consumers than the Quintile Coefficients specification. While
99% of simulation estimate at least 1 quintile of positive price sensitive consumers, the Normal Random
Coefficient specification only estimates 13% on average, so very small denominators of the markup
formula (21) are less frequent and not as small.
GUPPIs with true markup for the Nested Logit specification mirror the diversion ratios biases:
they underestimate diversion to Low Tier goods and overestimates diversion to High Tier goods. Like the
Simple Logit, the price coefficient estimate is attenuated, so markups are too high for Low Tier products
but about the right markups for High Tier products. As a result, the Nested Logit GUPPIs using
estimated markups are all overestimates of the true GUPPIs.
10. Discussion and Conclusion
This paper documents bias in demandbased diversion ratios that stems from measurement error
and misspecification of the demand system. I run Monte Carlo simulations, in which I estimate diversion
ratios and GUPPIs with incorrect demand specifications, and compare the results to the true values of the
correct demand specification. All of these specifications are commonly used or have been used in
practice, and they illustrate different aspects of measurement error and misspecification as sources of bias.
Measurement error can bias parameters of the demand system, which will carry over to values calculated
using that demand system, including diversion ratios. Misspecification can both lead to biased demand
parameters and incorrect formulas for diversion ratios.
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I find that even a modest amount of measurement error yields diversion ratios comparable to
those obtained from a model accounting for no consumer heterogeneity at all. I find small bias for all
diversions ratios in specifications for which the joint distribution of choice probabilities and weights were
accurately estimated. The ProductMarket Coefficients specification estimates choice probabilities very
well, but does not estimate diversion ratios well because it cannot replicate the distribution of weights
correctly because it underestimates price sensitivities of priceinsensitive consumers. This problem is
even worse in the Quintile Coefficients and Normal Random Coefficients specifications, where price
insensitive consumers have demand estimates with positive price effects. This results in numerically
unstable estimates. GUPPIs replicate these patterns when observed markups are used for calculation, but
are biased upwards by the use of estimated markups.
These results reinforce the need for practitioners to verify the robustness of their results through
the use of multiple specifications and to ensure data quality. They suggest a few key takeaways for
practitioners, especially in merger reviews.
The effectiveness of microdata on individual characteristics to proxy for differences in price
sensitivity can be limited by both measurement error and misspecification. Microdata used in practice,
whether from surveys, imputation or imperfect recordkeeping, will often have similar or more extreme
amounts of measurement error than the specification of 0.75 correlated data I used. It seems that using
consumer data that accurately represent the demographics of its subjects, for example consumer panelist
or administrative data, would be completely reliable to proxy for differences in price sensitivity. This
also implies a proxies for differences in price sensitivity with weak theoretical bases—for example, using
education instead of income because they are correlated—are additionally problematic. Further, even if
the data are correct, an incorrect specification like the ProductMarket Coefficients specification can
produce poor diversion ratio estimates. Such a problem may be hard to notice in practice because the
specification can also produce good estimates of demand.
The numerical instability of the Quantile Coefficients specifications suggest that practitioners
should use a large number of bins with discretized microdata to proxying for differences in price
sensitivity, but should not use a small number of bins. The Quintile Coefficients specification and
Normal Random Coefficient specification also indicate that if one estimates positive price coefficients for
some of the consumer sample, the specification may not be flexible enough and should be amended. 80 In
general, the instability from positive price sensitivities suggests that demand estimations should ex ante
restrict the price sensitivity to be negative.
The difficulties in estimating price sensitivities from data or under parametric assumptions
suggest that practitioner should prefer flexible random coefficient models. When the correct distribution
of random coefficients is assumed, diversion ratio bias is small and accuracy is high. If practitioners
could rely on random coefficient specifications without worrying about misspecification, practitioners
could avoid having to deal with the issues involving microdata entirely. The major caveat is that the
misspecification is a large problem with random coefficients: the Normal Random Coefficient
specification yielded biased and inaccurate results. Thus practitioners would need specifications that
estimate the distribution of the random coefficients or relax the distributional assumptions.
80 Positive price sensitivities can also be the result of unaddressed endogeneity. Steven T. Berry, Estimating
DiscreteChoice Models of Product Differentiation, 25 RAND J. Econ. 242, 243, 25758 (1994).
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OEA Working Paper
Several threads in the literature explore these options. The demand estimation procedure of Fox,
Kim, Ryan, and Bajari (2014) replaces estimating distribution parameters with estimating the frequency
of consumer types. 81 Dubé, Hitsch, and Rossi (2010) estimate a random coefficient model where the
random coefficient distribution is a flexible mixture of normal distributions.82 Brenkers and Verboeven
(2006) introduce the Random Coefficient Nested Logit used in the aforementioned Grigolon and
Verboeven (2014) which includes random coefficients in a nested logit.83 Compiani (2022) proposes a
nonparametric estimation approach which allows demand beyond standard discrete choice. 84 While these
approaches are promising and have prominent applications,85 they do have more intense data
requirements and are more computationally demanding. Such specification would also require addressing
common issues I ignored in this paper, including endogeneity and the use of aggregate data. Thus these
specifications may be of limited use in a merger review, when time and resources are limited.
Finally, the GUPPI results strongly suggest that practitioners should consider using observed
markups for GUPPIs. In my experiments, markups estimated from demand introduce bias that is often
more significant than the diversion ratio bias. When trustworthy markup data is available, the practitioner
can eliminate this potential channel for measurement error and misspecification bias by using that data
instead of estimates from the demand model.
All of the above challenges of properly using demand estimation as the basis of diversion ratio
estimates suggest practitioners should carefully consider the use of demandbased diversion ratios over
alternatives. As I have shown, measurement error and misspecification introduce potential errors in
potentially many different ways for demandbased diversion estimates. Insofar as diversion ratios are
indicative of the price predictions from merger simulations based on the same demand system, these
results suggest that those price predictions suffer similar biases. When faced with deciding how to
estimate diversion ratios, the practitioner should carefully weigh the pros and cons of the demandbased
method, especially given the complexity of demand estimation.
While the current study is quite suggestive, it is only done with a specific demand model with a
specific set of parameters. The level and direction of biases are specific to the true specification, the kind
of the measurement error or misspecification, and the identities of the destination and origin products. I
do not claim that the results will fully generalize to all settings. Moreover, I examined a relatively limited
set of specifications, and it is unclear what will happen to more complicated specifications. In particular,
I did not look at microdata proxying for sensitivity to product characteristics other than price. In
81 Jeremy T. Fox, Kyoo Il Kim, Stephen P. Ryan, & Patrick Bajari, A Simple Estimator for the Distribution of
Random Coefficients, 2 Quantitative Econ. 381 (2011).
82 Jean‐Pierre Dubé, Günter J. Hitsch, & Peter E. Rossi, State Dependence and Alternative Explanations For
Consumer Inertia, 41 RAND J. Econ. 417 (2010).
83 Randy Brenkers & Frank Verboven, Liberalizing a Distribution System: the European Car Market, 4 J. Eur.
Econ. Assoc. 216 (2006); and Grigolon & Verboven, supra note 66.
84 Giovanni Compiani, Market Counterfactuals and the Specification of MultiProduct Demand: A Nonparametric
Approach, 13(2) Quantitative Econ. 545 (2022).
85 Nevo, Turner and Williams (2016) examines residential broadband demand using a frequencybased estimation.
Aviv Nevo, John L. Turner, & Jonathan W. Williams, Usage‐Based Pricing and Demand for Residential
Broadband, 84 Econometrica 411 (2016). Random Coefficients Nested Logit is popular in alcoholic beverage
demand estimation, where there are widely recognized categories that can serve as nests. E.g. Nathan H. Miller &
Matthew C. Weinberg, Understanding the Price Effects of the MillerCoors Joint Venture, 85 Econometrica 1763
(2017); Eugenio J. Miravete, Katja Seim, & Jeff Thurk, Market Power and the Laffer Curve, 86 Econometrica 1651
(2018); and Christopher T. Conlon & Nirupama L. Rao, Discrete Prices and the Incidence and Efficiency of Excise
Taxes, 12 Amer. Econ. J.: Econ. Pol. 111 (2020).
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OEA Working Paper
addition, I assumed extremely favorable conditions for identification, so performance of the low bias
specifications may be worse in practical settings. I also do not use real data for my experiments; there
may be value in examining the performance of demand models with my data or in experiments with
subjects. I therefore look forward to future work that builds on this study.
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OEA Working Paper
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