## Abstract

This article derives simple and intuitive formulas for calculating the effects that mergers that enhance product quality have on nominal and hedonic prices. If the value of quality enhancements can be estimated, the formulas proposed here can be applied using data realistically available in the merger investigations context and should therefore be valuable additions to the practitioner’s toolbox. This article also explores strategic effects between the merged entity and non-merging rivals in the presence of merger-specific quality enhancements, and it highlights the importance in ex post merger assessments of paying close attention to non-merging rivals’ price responses to the merger.

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## Notes

This was followed in 2018 by two other agricultural mergers: Bayer/Monsanto, and Bayer/BASF.

This recent literature shows that the classic results of Arrow (1962), Schumpeter (1942), and Aghion et al. (2005) do not readily carry over to mergers and that an important aspect of competition and innovation in the merger context is that mergers can create downward innovation pressure by internalising the negative externality of innovation between the merging firms. See Valletti and Zenger (2021) for a recent discussion.

Farrell and Shapiro (2010) already proposed applying pass-through rates to UPP as a rough indication of price effects.

Willig notes in the context of his model that “the impacts of a merger on product quality and on marginal costs are additive in their influences on pricing pressure”. Motta and Tarantino (2013) apply a similar insight in the ‘mergers and innovation’ context of their paper (see their Sect. 3.2.1).

Valletti and Zenger (2021) show independently from this article that CMCRs apply in a model with quality improvements only. The present analysis effectively combines this with Werden’s analysis by considering both quality improvements and marginal cost reductions. It also drops the unnecessary assumption that marginal costs are constant.

For brevity these will be referred to simply as quality enhancements.

For alternative demand models that can be used to carry out a similar analysis, see Motta and Tarantino (2018, Sec. 3.2.1).

Cf. Werden (1998). Some economists have called \({D}_{ij}\) a ‘revenue diversion ratio’, but ‘adjusted diversion ratio’ is more appropriate.

The analysis in this section extends previous work by adding quality enhancements. Moreover, it uses a more general demand, and the results are slightly simpler than in Neurohr (2019) and significantly more intuitive than in Hausman et al. (2011). Jaffe and Weyl (2013) is more general but more complex and thus less practitioner-friendly; and, as discussed in Neurohr (2019), the presentation of their results mischaracterises the underlying economic intuition.

Hedonic prices with linear demand can be motivated by the quasilinear quadratic utility function \( U\left( {\mathbf{q}_{0}} ,{\mathbf{q}} \right) = q_{0} + \left( {\mathbf{a}} + {\mathbf{v}} \right)^{\mathbf{\prime}} {\mathbf{q}} - \mathbf{\frac{1}{2}q^{\prime}Bq} \), based on the non-hedonic version in Choné and Linnemer (2020), Eq. 3 and Eq. A.1, where: \({q}_{0}\) is the quantity of the numeraire good; \(\boldsymbol{q}\) is the vector of all other quantities; \(\boldsymbol{a}\) and \(\boldsymbol{B}\) are, respectively, a vector and a positive definite matrix of parameters; and \(\boldsymbol{v}\) is the vector of quality enhancements. This can be shown to satisfy the conditions in Willig (1978, Theorem 3). Cf. Willig (2011, fn. 23) and Israel et al. (2014, fn. 47).

Given the strategic complementarity of hedonic prices, price responses of competitors reinforce the merger effects that are related to the merged entity. Thus if the merged entity’s hedonic prices rise (fall), this would be reinforced by competitors’ (hedonic) prices rising (falling) as well in response. Ignoring competitor responses thus means weakening the overall effect that is calculated for the merger but not changing its directionality.

Whereas Werden’s CMCR test needs to hold precisely for both products to be conclusive, here the

*extent*to which it does not hold for one product or another determines the extent of the price effects.As was discussed in Hausman et al. (2011), this assumption will be reasonable in many cases: either because the merging firms are producing intermediate goods that they sell to downstream firms rather than to end-consumers, in which case the assumed symmetry follows from cost minimisation and constant marginal costs for the downstream firms; or, absent constant marginal costs, symmetry holds approximately if the inputs are only a small proportion of variable costs; or for increasing marginal costs if the production function is homothetic. Or, if the merging firms sell final consumer goods, the assumed symmetry (roughly) follows from Slutsky symmetry and the absence of (significant) income effects.

In the case of (10) this simply means changing the sign in front of \(\frac{{v}_{i}}{{p}_{i}}\) from minus to plus, which nicely illustrates the fact that quality enhancements increase the observable nominal price that is relevant for producer surplus while reducing the unobservable hedonic price that is relevant for consumer welfare. Cf. Willig (2011, fn. 24), which states that with linear demand and constant marginal costs a quality (or unit value) increase of \({v}_{i}\) leads to an increase of \({v}_{i}/2\) in the nominal price \({p}_{i}\) and a decrease of \({v}_{i}/2\) in the quality-adjusted price \({h}_{i}\). Thus in this case the unit-value gain is shared equally by producers and consumers.

\(1/2\) is the standard pass-through rate of a single-product linear-demand monopolist. For reasons that will be given below, it also applies to the merged entity, which is a two-product firm. To be more precise: The post-merger pass-through matrix in the two-firm case is given by \(\frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\): \(\frac{\partial {p}_{j}}{\partial {c}_{i}}\) is zero for \(i\ne j\).

Note that the price change no longer depends on the efficiencies of the other product.

Algebraically, consider the conditions that obtain from totally differentiating the two post-merger first-order conditions with respect to \({c}_{i}\): \(\frac{{\partial }^{2}\pi }{\partial {p}_{i}\partial {c}_{i}}=\frac{\partial {q}_{i}}{\partial {p}_{i}}\frac{\partial {p}_{i}}{\partial {c}_{i}}+\frac{\partial {q}_{i}}{\partial {p}_{j}}\frac{\partial {p}_{j}}{\partial {c}_{i}}+\left(\frac{\partial {p}_{i}}{\partial {c}_{i}}-1\right)\frac{\partial {q}_{i}}{\partial {p}_{i}}+\frac{\partial {p}_{j}}{\partial {c}_{i}}\frac{\partial {q}_{j}}{\partial {p}_{i}}=0\) and \(\frac{{\partial }^{2}\pi }{\partial {p}_{j}\partial {c}_{i}}=\frac{\partial {q}_{j}}{\partial {p}_{i}}\frac{\partial {p}_{i}}{\partial {c}_{i}}+\frac{\partial {q}_{j}}{\partial {p}_{j}}\frac{\partial {p}_{j}}{\partial {c}_{i}}+\frac{\partial {p}_{j}}{\partial {c}_{i}}\frac{\partial {q}_{j}}{\partial {p}_{j}}+\left(\frac{\partial {p}_{i}}{\partial {c}_{i}}-1\right)\frac{\partial {q}_{i}}{\partial {p}_{j}}=0\), where we have made use of the linearity of demand. If \(\frac{\partial {p}_{i}}{\partial {c}_{i}}=\frac{1}{2}\) and \(\frac{\partial {p}_{j}}{\partial {c}_{i}}=0\), then the condition \(\frac{{\partial }^{2}\pi }{\partial {p}_{i}\partial {c}_{i}}=0\) is satisfied. But in \(\frac{{\partial }^{2}\pi }{\partial {p}_{j}\partial {c}_{i}}\), although the two middle terms are then zero, the first term—which gives the upward pressure on \({p}_{j}\) as a result of the increase in \({c}_{i}\) (and the concomitant increase in \({p}_{i}\)) causing sales to divert to product \(j\)—becomes \(\frac{1}{2}\frac{\partial {q}_{j}}{\partial {p}_{i}}\), and the last term—which gives the reduction in UPP for product \(j\) due to the reduced margin for product \(i\)—becomes \(-\frac{1}{2}\frac{\partial {q}_{i}}{\partial {p}_{j}}\). These two effects perfectly offset each other if the cross-price derivatives of demand are symmetric. The condition \(\frac{{\partial }^{2}\pi }{\partial {p}_{j}\partial {c}_{i}}=0\) is then also satisfied, and therefore also the condition \(\frac{\partial {p}_{j}}{\partial {c}_{i}}=0\).

It is for this reason that efficiencies for one product do not affect the pricing of the other in the case with symmetric cross-price derivatives of demand.

The matrix product that precedes the net UPP vector with entries \(\left({y}_{i}^{*}-{y}_{i}\right)/{p}_{i}\) is very closely related to the Jacobian of the price vector with respect to marginal costs: the matrix of pass-through rates, which is given simply by multiplying the adjusted diversion ratios \({D}_{ij}\) in this matrix by \({p}_{i}/{p}_{j}\) to get the standard diversion ratios. With and without adjusted diversion ratios, it can be shown under reasonable assumptions that all entries of this matrix are positive and therefore that net UPP on one product tends to raise all other prices in the market. This is not necessarily the case when cross-price derivatives of demand are asymmetric, as can be shown from (9). A proof for the \(3\times 3\) case is available from the author upon request.

The relevant diversion ratios for calculating pass-through rates (and

*absolute*price changes) are the standard, unadjusted diversion ratios. The matrix of pass-through rates in the present example is \(\left[ {\begin{array}{*{20}c} 2 & { - 2\gamma } & { - \mu } \\ { - 2\gamma } & 2 & { - \mu } \\ { - \mu } & { - \mu } & 2 \\ \end{array} } \right]^{{ - 1}} \left[ {\begin{array}{*{20}c} 1 & { - \gamma } & 0 \\ { - \gamma } & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] = \frac{1}{{4\left( {2\left( {1 - \gamma } \right) - \mu ^{2} } \right)}}\left[ {\begin{array}{*{20}c} {4\left( {1 - \gamma } \right) - \mu ^{2} } & {\mu ^{2} } & {2\mu } \\ {\mu ^{2} } & {4\left( {1 - \gamma } \right) - \mu ^{2} } & {2\mu } \\ {2\mu \left( {1 - \gamma } \right)} & {2\mu \left( {1 - \gamma } \right)} & {4\left( {1 - \gamma } \right)} \\ \end{array} } \right]\).For the reasons that were given above, and as seen from setting \(\mu =0\), the merging partner does not change its price in the absence of a non-merging rival. Thus, both the rival’s and the merging partner’s responses are characterised by the strategic complementarity of hedonic prices.

Absent a non-merging rival (\(\mu =0\)), this pass-through rate is \(1/2\). Cf. footnotes 18 and 19.

There may of course be other factors to consider, including other types of conduct and product repositioning.

The author would like to thank Bobby Willig for drawing his attention to this point. It relates closely to the statement from Willig (2011) that “…an empirical pattern of falling nominal prices of rivals, increased nominal price of the possibly improved product of the merged firm, and increased output of the merged firm would be consistent with and possibly indicative of a merger that improved product quality enough to lower the firm’s quality-adjusted price”

*.*An alternative ex post test might thus focus on the merged entity’s output and whether this has increased despite its charging higher prices. As noted in Willig (2011) and Israel et al. (2013), Kwoka and Shumilkina (2010) find but apparently misinterpret such an “empirical pattern” in their study of the 1987 USAir/Piedmont merger.

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The views and opinions expressed here are those of the author. Thanks are due to Bobby Willig for detailed discussions and great generosity with his time and to Larry White, an anonymous referee, and my colleagues at Oxera for helpful comments.

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Neurohr, B. Unilateral Effects of Mergers that Enhance Product Quality.
*Rev Ind Organ* **60**, 587–596 (2022). https://doi.org/10.1007/s11151-022-09859-w

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DOI: https://doi.org/10.1007/s11151-022-09859-w