Abstract
In this paper, I investigate the impact of a horizontal merger between firms that use price promotions. I find that after the merger, the merged firms increase their prices, but coordinate the promotions by never discounting their products simultaneously. The non-merged firm responds with a more aggressive pricing strategy, offering deeper and more frequent discounts. The effects of a merger on the firms’ profits and the consumer surplus are very small relative to the case with no promotions. These conclusions are not affected by the size of the change in market concentration or by the degree of substitutability between the merging products. Thus, the use of price promotions by the merging firms can be viewed as a mitigating factor by the antitrust authorities.
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Notes
In order to isolate the competitive implications of a merger, I assume it produces no efficiency gains.
This reaction of a firm to a decrease in the number of rivals competing for the switchers is similar to the effect of a decrease in the number of firms in pricing models with mixed strategy equilibria (Rosenthal 1980; Stahl 1989). Since these models do not consider price coordination, the prices of all firms go down as the number of firms decreases, whereas in my model, the merged firms increase their prices.
The findings for other non-merged firms were mixed. For Kellogg’s, prices of four brands went down while prices of five brands went up. All three Ralston brands increased their prices, and both Quaker brands decreased theirs.
To prevent firms from earning infinite profits from their loyals, it is standard in the theoretical literature to set an exogenous reservation price for this segment. The price of an outside option serves this role in my model. To allow for different strength of preferences for their preferred brand, I introduce heterogeneity among the loyals.
If there are no loyal consumers, i.e., α = 0, then the equilibrium is always in pure strategies and price promotions are not offered. In that case, all profits increase and the consumer surplus goes down after a merger (Werden and Froeb 1994).
This is in contrast to the atomless mixed strategies in Varian (1980) and subsequent papers that work with discontinuous demands.
The Matlab programs used to compute equilibria in this paper are available at https://sites.google.com/view/mergersandpromotions/home
The robustness checked were performed for the values of α between 0.05 and 0.2, the values of μl between 0.01 and 0.5, and the values of p0 between 0.5 and 1.5. Parameter δi was varied in the first extension. Qualitative findings of the paper were not affected.
Throughout the paper, I use the term “consumer heterogeneity” to refer to μ – the level of heterogeneity of the switching cohort.
For example, when μ = 0.06, the firms use high price p3 = 0.73 charged with probability 0.14, shallow promotion p2 = 0.54 charged with probability 0.17, and deep promotion p1 = 0.17 charged with probability 0.69.
The size of each bubble is proportional to the probability with which a corresponding price or a price pair is charged.
With a small probability, the merged firms charge the high price for both products.
The kinks and changes in monotonicity in this and the following figures happen when there is a change in the equilibrium structure due to firms adding more prices to the support of their price distributions.
The expected consumer surplus is computed by taking a weighted average of the values of consumer surplus at all possible outcomes of the mixed strategies. For any price combination, the consumer surplus of the switchers is \( {CS}_s=\mu \ln \left({\sum}_{j=0}^3{e}^{U_{sj}/\mu}\right) \) while the consumer surplus of the segment loyal to product i is \( {CS}_{li}={\mu}_l\ln \left({\sum}_{j\in \left\{0;i\right\}}{e}^{U_{lj}/{\mu}_l}\right) \) (Small and Rosen 1981). The total consumer surplus is \( {CS}_s+{\sum}_{i=1}^3{CS}_{li} \).
Formally, \( {CS}_s=\mu \ln \left({\sum}_{j=0}^3{e}^{U_{sj}/\mu}\right) \). Since ln is a concave function, an increase in CSs from product j going on sale (and corresponding increase in Usj) is larger when there are fewer other products on sale.
Similar to the results in the previous section, since these two firms are symmetric, if in the equilibrium, they use price pair (pi, pj) with some probability, a mixed strategy that splits this probability in any way between (pi, pj) and (pj, pi) is also an equilibrium strategy.
To facilitate comparison between the strategies after two different mergers, in the symmetric case of the merger between firms 1 and 2, if the firms use price pair (pM1, pM2) with a certain probability, I split this probability equally between (pM1, pM2) and (pM2, pM1).
When μ1 ≥ 0.136, firm 1 benefits more from the merger of firms 2 and 3. This is consistent with existing research that shows that a non-merged firm benefits more from a merger than the merged firms.
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Acknowledgements
The author thanks the co-editor Thomas Otter and two anonymous reviewers for their guidance in improving the paper. The author also benefited from the helpful comments of Ted Brandewie, Ryan Kasprzak, and Joel Sobel.
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Sinitsyn, M. Evaluating horizontal mergers in the presence of price promotions. Quant Mark Econ 18, 39–60 (2020). https://doi.org/10.1007/s11129-019-09213-7
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DOI: https://doi.org/10.1007/s11129-019-09213-7