Quantitative Methods for Evaluating the Unilateral Effects of Mergers


We describe the quantitative modeling techniques that are used in horizontal merger review for the evaluation of unilateral effects, and discuss how the 2010 Horizontal Merger Guidelines helped legitimize these methods and motivate scholarly research. We cover markets that feature differentiated products pricing, auctions and negotiations, and homogeneous products, in turn. We also develop connections between quantitative modeling and market concentration screens that are based on the Herfindahl-Hirschman Index (HHI).

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  1. 1.

    For a formal definition of unilateral effects, see Werden (2008).

  2. 2.

    Mergers can also reduce innovation incentives, though we do not focus on that possibility here. See Federico et al. (2019) for analysis and a useful discussion of the literature.

  3. 3.

    Recent empirical research has made progress on the quantitative modeling of coordinated effects: Igami and Sugaya (2019), Miller et al. (2019).

  4. 4.

    We focus in this article on model-based methods, which contrast with other empirical work, such as the study of natural experiments, that is not tied to a structural framework.

  5. 5.

    Sometimes multiple models may be used in a single merger review, with each model designed to answer a particular question. At the risk of over-generalization, models that are designed for internal agency decision-making tend to be more sophisticated than those presented in litigation.

  6. 6.

    The DOJ and FTC also released a helpful discussion document in 2006: The Commentary on the Horizontal Merger Guidelines (henceforth, the “Commentary”).

  7. 7.

    Shapiro (2010, p. 712). The “Agencies” refers to the DOJ and FTC.

  8. 8.

    We use the “from hedgehog to fox” description of merger review throughout this article. This follows Shapiro (2010):

    “Isaiah Berlin’s famous allusion to the different ways in which the Hedgehog and the Fox view the world is a useful model for how to think about the evolution of the Merger Guidelines. The hedgehog knows one big thing. Likewise, the 1968 Guidelines were based on one big idea: horizontal mergers that increase market concentration inherently are likely to lessen competition.... [T]he fox knows many things. Likewise, merger enforcement in recent years has become increasingly eclectic, reflecting the enormous diversity of industries in which the Agencies review mergers and the improved economic toolkit available.”

  9. 9.

    The HHI equals the sum of squared market shares, with the shares being measured on a scale from zero to 100. The \(\Delta\)HHI is calculated as twice the product of the merging firms’ market shares.

  10. 10.

    We are not the first to propose that a screen that is based solely on the \(\Delta\)HHI would better align with unilateral effects theory. See, for example, Nocke and Whinston (2020), or the other articles that are cited herein.

  11. 11.

    Throughout this article, we use the superscript T to refer to the vector/matrix transpose operation.

  12. 12.

    See also Werden and Froeb (1998), p. 532: “A stylized fact of US industry is that marginal costs are typically constant....” We revisit this subject in our discussion of homogeneous products (Sect. 4).

  13. 13.

    Robert Willig, a primary author of the 1992 Guidelines, provides a description of the modeling framework in Willig (1991).

  14. 14.

    1992 Guidelines, §2.21.

  15. 15.

    2010 Guidelines, §6.1.

  16. 16.

    The 2010 Guidelines, §13, address partial acquisitions.

  17. 17.

    We define diversion explicitly in the next subsection.

  18. 18.

    See Miller et al. (2016) for details on the calibration of the linear, log-linear, Almost Ideal, and logit demand systems. Caradonna et al. (2020) show how to calibrate logit demand partially with (only) market shares; this is sufficient to obtain the percentage change in markups that is caused by a merger without efficiencies. As an extension, the academic literature has examined how merger simulation responds to the presence of consumer search costs or switching costs: e.g., Allen et al. (2013), MacKay and Remer (2019). To our knowledge, these refinements have yet to be applied in investigations.

  19. 19.

    See 2006 Commentary, §2.

  20. 20.

    See 2006 Commentary, §2. The commentary also mentions Bertrand merger simulation in two other DOJ cases: Interstate Bakeries/Continental (1995) and Vail Resorts/Ralston Resorts (1997). In the latter instance, demand was measured using survey data.

  21. 21.

    See the Memorandum Opinion at p. 76, which describes testimony by Frederick Warren-Boulton.

  22. 22.

    See Appendix C to the FCC Memorandum Opinion and Order.

  23. 23.

    See the demonstrative exhibit that was used by Aviv Nevo, at slides 63-66, available at https://www.justice.gov/atr/page/file/918706/download.

  24. 24.

    The upward pricing pressure approach of Farrell and Shapiro (2010) draws on preceding research on merger price effects: e.g., Werden (1996), O’Brien and Salop (2000), which we review below.

  25. 25.

    For the case of multi-product firms, see Eq. (8).

  26. 26.

    As best we can discern, the term “diversion ratio” was coined in Shapiro (1995, 1996), though the concept dates at least to Willig (1991). Empirical articles in industrial organization continue to focus more on demand elasticities; however, see Conlon and Mortimer (2019) for a detailed discussion of how diversion can be estimated from data on prices and shares.

  27. 27.

    For more on the development of upward pricing pressure indices, see Moresi (2010) and the other contributions that are cited in Farrell and Shapiro (2010, fn. 90).

  28. 28.

    2010 Guidelines, §6.1.

  29. 29.

    2010 Guidelines, §6.1. Notably, the Guidelines do not state that the Agencies rely on UPP-style analysis more than on the change in HHI for diagnosing unilateral effects. As we develop in Sect.  5, the change in HHI often can be quite informative of unilateral effects in differentiated products markets.

  30. 30.

    Mergers can also affect product quality or production costs, which matter for unilateral pricing incentives (Sect. 2.4). A host of other possibilities exist, but are beyond the scope of this article.

  31. 31.

    The 2010 Guidelines at §4.1.3 state that “The Agencies often estimate incremental costs, for example, using merging parties’ documents or data the merging parties use to make business decisions.” Fisher and McGowan (1983) discuss some ways that accounting data can be misleading.

  32. 32.

    Referring to UPP and concentration analysis, Farrell and Shapiro (2010, p. 3) state that “neither approach purports to quantify the likely equilibrium effect (e.g., the price change) of the merger....”

  33. 33.

    See also Miklos-Thal and Shaffer (2021).

  34. 34.

    We experienced this first-hand while attempting to estimate pass-through in the cement industry (Miller et al. 2014). For an exception that may prove the rule, see Muehlegger and Sweeney (2019). MacKay et al. (2014) provide a formal treatment of econometric bias in pass-through regressions.

  35. 35.

    Another concern is developed in the Conlon and Rao (2020) study of the distilled spirits market. There most prices end in 99 cents (e.g., $9.99), and price changes predominately occur in one dollar increments (e.g., $9.99 to $10.99). A retailer that operates under these constraints might not raise price at all in response to small cost increases but, in the event that a price rise is warranted, the magnitude of the price increase is likely to far exceed that of the cost increase. This can produce pass-through estimates that are implausibly small or implausibly large—depending on the sample.

  36. 36.

    See the slides that were used by Michael Whinston as part of his testimony, at page 59, available at https://www.justice.gov/atr/file/ge-px02015/download. The merger was abandoned before the trial concluded.

  37. 37.

    The exercise is similar in spirit to that of Miller et al. (2017), which also considers linear demand, the Almost Ideal Demand System, and log-linear demand.

  38. 38.

    Although it predates the formal development of UPP in the economics literature, the analysis that was presented by the FTC in Swedish Match/National Tobacco (2000) also included similar calculations. See the FTC Proposed Findings of Fact at pp. 125–126, which describe testimony by John Simpson, available at https://www.ftc.gov/sites/default/files/documents/cases/2000/06/swedishmatchpublic.pdf.

  39. 39.

    See the demonstrative exhibit used by Michael Whinston at p. 57, available at https://www.justice.gov/atr/file/ge-px02015/download.

  40. 40.

    Equation (12) applies to asymmetric single-product firms. With symmetry, a simplification yields \({\hat{c}}_j = (m/(1-m))(d/(1-d)).\) See Werden (1996) for the case of multi-product firms. Nocke and Whinston (2020) provide an analogous equation that depends only on the market shares of the merging firms, for the specific case of logit demand. Dutra and Sabarwal (2019) provide an adjustment to UPP that may improve accuracy in the presence of cost efficiencies.

  41. 41.

    See the economic appendix to the Staff Report at p. C-9 to C-10, available at https://docs.fcc.gov/public/attachments/DA-11-1955A2.pdf. The transaction was subsequently abandoned. Pittman and Li (2013) analyze the literature on the economics of density in mobile telephony, and conclude that large efficiencies probably were implausible.

  42. 42.

    An adjustment allows for cost and quality efficiencies to be evaluated together. Similarly, Nocke and Whinston (2020) provide the critical type efficiency—a nonlinear combination of quality and marginal cost improvements—that is necessary to counterbalance adverse unilateral effects exactly with logit demand. The critical type efficiency depends only on the market shares of the merging firms.

  43. 43.

    Notably, the responsiveness of demand to characteristics (\(\beta\)) differs from the responsiveness of demand to price (\(\alpha\)) because it cannot be inferred from a price-cost margin.

  44. 44.

    Because a more rigorous balancing requires an understanding of consumer heterogeneity, the random coefficients logit model of Berry et al. (1995) seems to have relatively more value for merger review when quality efficiencies are involved.

  45. 45.

    If the supplier has capacity constraints and is bidding on other procurements, then opportunity costs are relevant. This may be a substantial source of private information—particularly if firms have imperfect information about their competitors’ future plans.

  46. 46.

    1992 Guidelines, §2.21.

  47. 47.

    2010 Guidelines, §6.2.

  48. 48.

    See the Sysco/US Foods Memorandum Opinion at pp. 89–92, which discusses testimony by Mark Israel on behalf of the FTC; the Anthem/Cigna district-level Memorandum Opinion at pages 58–59 and 66–67, which discusses testimony by David Dranove on behalf of the DOJ; and the Wilhelmsen/Drew Marine Memorandum Opinion at pages 44–45, which discusses testimony by Aviv Nevo on behalf of the FTC.

  49. 49.

    2006 Commentary, §2.

  50. 50.

    See the demonstrative exhibits that were used by Kenneth Elzinga and Preston McAfee, available at https://www.justice.gov/atr/usdoj-antitrust-division-us-and-plaintiff-states-v-oracle-corporation.

  51. 51.

    See the Memorandum Opinion at paragraphs 267-273, discussing testimony by Carl Shapiro on behalf of the DOJ.

  52. 52.

    See the Memorandum Opinion at pp. 57–58, citing analysis done by Carl Shapiro on behalf of the FTC.

  53. 53.

    Relatedly, Chen and Schwartz (2016) show how the churn ratio may depart from diversion.

  54. 54.

    Diversion-according-to-share is given by \(d_{jk} = s_{k}/(1-s_j)\), where \(s_j\) is the market share of firm j.

  55. 55.

    See Shelanski et al. (2012).

  56. 56.

    See the demonstrative exhibit that was used by David Dranove at pp. 46–46, available at https://www.justice.gov/atr/page/file/914606/download.

  57. 57.

    2010 Guidelines, §8.

  58. 58.

    See the demonstrative exhibit that was used by David Dranove when testifying on behalf of the FTC, at slides 12–16, available at https://www.ftc.gov/system/files/documents/cases/131002stlukedemodranove.pdf.

  59. 59.

    See the discussion in the “Proposed Comcast-Time Warner Cable-Charter Transaction Economic Analysis Workshop,” the transcript of which is available at https://ecfsapi.fcc.gov/file/60001031131.pdf. In this case the merger was one between buyers, not suppliers. The transaction was subsequently abandoned.

  60. 60.

    It is telling that the Shapiro (2010) article on the drafting of the 2010 Guidelines mentions homogeneous products only in passing.

  61. 61.

    The first order conditions of Eq. (1) apply with the simplification that \(\partial q_i / \partial p_i = \partial Q/\partial p\), for all i, given market quantities \(Q \equiv \sum _i q_i\) and market price p.

  62. 62.

    2010 Guidelines, §6.3.

  63. 63.

    2010 Guidelines, §6.3.

  64. 64.

    The only consideration that does not emerge from a one-shot game of simultaneous production is (2), which can be an important consideration, especially in the presence of forward markets. A caveat to the Guidelines language is that merging firms have an incentive to reduce output commitments, such that forward markets can amplify rather than mitigate price effects ( Miller and Podwol 2020).

  65. 65.

    The code we use for the calibration and simulation of this model tracks the equations that are derived in McAfee and Williams (1992). See also Farrell and Shapiro (1990) and Werden (1991).

  66. 66.

    See Jaffe and Weyl (2013) for a generalized expression for the UPP formula that nests differentiatedproducts Cournot as a special case. As the amount of differentiation decreases and converges toward the homogeneous products case, UPP tends to zero but the pass-through of UPP to equilibrium prices tends to infinity. This mathematical difficulty makes the CMCR approach more useful than UPP for homogeneous-products markets.

  67. 67.

    See Froeb and Werden (1998). Nocke and Whinston (2020) provide an equivalent equation in which the change in the HHI—\(\Delta\)HHI\(=2s_j s_k\)—appears on the right-hand side.

  68. 68.

    See Greenfield et al. (2019) and the Memorandum Opinion at p. 33, discussing testimony by Nicholas Hill on behalf of the FTC. The FTC also applied a modeling framework that is designed to predict capacity reductions, which was developed in Hill (2008). The 2010 Guidelines, §6.3, point out that capacity reductions are one way to implement output suppression.

  69. 69.

    The observation that Cournot mergers with constant marginal cost and linear demand tend to be unprofitable was made originally in Salant et al. (1983). The result depends on particular modeling assumptions: Profitability is restored if capital is incorporated and the merging firms are sufficiently large (Perry and Porter 1985) or if demand is convex (Fauli-Oller 1997; Hennessy 2000).

  70. 70.

    The 2010 Guidelines, §6.3, describe homogeneous products as being “relatively undifferentiated.”

  71. 71.

    For an example of how these screens have been applied in litigation, see the demonstrative exhibit used by Aviv Nevo as part of the Aetna/Humana (2016) trial, slides 56-61, available at https://www.justice.gov/atr/page/file/918706/download.

  72. 72.

    The 2010 Guidelines define a relevant antitrust market as comprising a set of products for which a hypothetical monopolist would find it profitable to impose a small but significant price increase. There are straight-forward connections between the models that we have discussed for unilateral effects and those that are used to evaluate candidate markets. It is worth noting, however, that the finding of a significant GUPPI for products of the prospective merging firms can indicate that those products – by themselves – constitute a relevant market.

  73. 73.

    2010 Guidelines, §5.3.

  74. 74.

    Froeb and Werden (1998) consider the Cournot case that has already been discussed. Shapiro (2010) provides an approximation for the Bertrand logit context: Consider a merger that involves two products with pre-merger market shares \(s_j\) and \(s_k\). With logit demand, diversion from product j to product k equals \(s_k/(1-s_j)\) and can be approximated by \(s_k(1+s_j)\) for small \(s_j\). Diversion from k to j is analogous, so the sum of the approximate diversion ratios is \(s_j + s_k + 2s_j s_k\) or \(s_j + s_k + \Delta \text {HHI}\).

  75. 75.

    The results we obtain with the Bertrand and second-score models depend on a stochastic assumption that we discuss in the next subsection. We evaluate \(\Delta\)HHI at pre-merger market shares, with the shares calculated among the inside goods. This is a slight departure from Nocke and Schutz (2019), in which the shares are calculated among all goods, including the outside good. We exclude the outside good in the Cournot analysis.

  76. 76.

    Shapiro (2010, p. 707).

  77. 77.

    Shapiro (2010, p. 708).

  78. 78.

    A similar conclusion is reached by Nocke and Whinston (2020), who state, “[w]e show that there is both a theoretical and an empirical basis for focusing solely on the change in the Herfindahl index, and ignoring its level, in screening mergers for whether their unilateral effects will harm consumers.”

  79. 79.

    Efforts to link the HHI to coordinated effects date at least to Stigler (1964). For recent treatments that tie concentration and the number of firms to coordination, see Ivaldi et al. (2007) and Baker and Farrell (2020). Antitrust concerns about a lessening in potential competition could arise, for example, in situations that involve a dominant firm that acquires a promising smaller firm (Cunningham et al. 2020). See Werden and Limarzi (2011) for an insightful discussion of potential competition under current practice.

  80. 80.

    A full examination of such a new standard, across the myriad of mergers that are encountered in practice, is beyond the scope of this article.

  81. 81.

    In the second-score model of Miller (2014, 2017), consumers substitute to competitors in proportion to their market share if one firm’s value or cost changes. We refer to this as diversion for brevity.

  82. 82.

    Example 7 in the 2010 Guidelines, §4.1.1, makes the same point with respect to cars and motorcycles.

  83. 83.

    See the Memorandum Opinion for H&R Block/TaxACT (2011) at p. 76, or the demonstrative exhibit used by David Dranove during the Anthem/Cigna (2016) trial at p. 48, available at https://www.justice.gov/atr/page/file/914606/download, for example.

  84. 84.

    Memorandum Opinion, p. 78.

  85. 85.

    Memorandum Opinion, pp. 4–5.


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Correspondence to Gloria Sheu.

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The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the Board research staff or the Board of Governors. We thank David Benson, Breno De Castro Vieira, Joseph Farrell, Evan Gee, Arnd Klein, Alex MacKay, Russell Pittman, Carl Shapiro, Charles Taragin, and Nathan Wilson for helpful comments.

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Miller, N.H., Sheu, G. Quantitative Methods for Evaluating the Unilateral Effects of Mergers. Rev Ind Organ 58, 143–177 (2021). https://doi.org/10.1007/s11151-020-09805-8

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  • Mergers
  • Antitrust
  • Unilateral effects

JEL Classification

  • L13
  • L40
  • L41