A New Perspective on Entry in Horizontal Merger Analysis

Abstract

I analyze horizontal mergers in procurement settings in which sellers incur costs to participate. Considering existing sellers’ contest-level entry differs from antitrust authorities’ typical emphasis on new sellers’ market-level entry to counteract a merger’s anticompetitive harm. I show that profitable mergers can increase consumer and total surplus by inducing more and stronger contest-level entry by the merged seller, which echoes common claims from merging parties that their merger is beneficial because it creates a stronger competitor. This finding suggests caution by antitrust authorities: when contest-level entry costs matter, standard models that ignore those costs prescribe blocking procompetitive mergers.

Appendix

For each of the proofs to follow, consider a set A that does not include the merging sellers or the merged seller (sellers i, j, or M). For each scenario I can show that seller M’s expected profit versus the set A of active sellers exceeds the sum of seller i’s and seller j’s expected profits.

Nature’s move that selects active sellers’ production costs occurs after the sellers’ simultaneous entry decisions, so that the price competition stage is a subgame for each configuration of entry choices.

Proof of Lemma  1

Consider a subgame in which sellers i and j both are active pre-merger, facing a (potentially empty) set A of active sellers that does not include the merging sellers or the merged seller (sellers i, j, or M).

To show that the merger strictly increases the merging sellers’ expected profits, one must show that ${\overline{\pi }}_{i}^{A\cup \{i,j\}}+\overline{ \pi }_{j}^{A\cup \{i,j\}}<{\overline{\pi }}_{M}^{A\cup \{M\}}$. Using the formula for an active seller’s expected profit,

$$\begin{aligned} {\overline{\pi }}_{i}^{A\cup \{i,j\}}+{\overline{\pi }}_{j}^{A\cup \{i,j\}}= & {}\, \left( \int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A\cup \{j\}}\left[ 1-F_{k}(c)\right] \right) F_{i}(c)dc-e\right) \\&+\left( \int _{ {\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A\cup \{i\}}\left[ 1-F_{k}(c)\right] \right) F_{j}(c)dc-e\right) \\= & {} \int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c)\right] \right) \left( F_{i}(c)\left[ 1-F_{j}(c)\right] +F_{j}(c) \left[ 1-F_{i}(c)\right] \right) dc-2e \\= & {} \int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c)\right] \right) \left( F_{i}(c)+F_{j}(c)-2F_{i}(c)F_{j}(c)\right) dc-2e \\&<\int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c)\right] \right) \left( F_{i}(c)+F_{j}(c)-F_{i}(c)F_{j}(c)\right) dc-e \\= & {} \int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c)\right] \right) F_{M}(c)dc-e \\= & {} \,{\overline{\pi }}_{M}^{A\cup \{M\}}. \end{aligned}$$

The preceding argument is the same as one would make in the model without contest-level entry costs, plus the cost savings of e further increase seller M’s expected profit.

To show that the merger leaves rivals’ expected profits unchanged, one must show that ${\overline{\pi }}_{h}^{A\cup \{i,j\}}={\overline{\pi }}_{h}^{A\cup \{M\}}$ for all $h\in A$. Using the formula for an active seller’s expected profit,

$$\begin{aligned} {\overline{\pi }}_{h}^{A\cup \{i,j\}}&= {} \int _{{\underline{c}}}^{{\overline{c}} }\left( \prod \limits _{k\in A\backslash h\cup \{i,j\}}\left[ 1-F_{k}(c)\right] \right) F_{h}(c)dc-e \\&= {}\, \int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A\backslash h\cup \{M\}}\left[ 1-F_{k}(c)\right] \right) F_{h}(c)dc-e \\&= {} {\overline{\pi }}_{h}^{A\cup \{M\}}. \end{aligned}$$

The equality between the first and second lines of the preceding expressions arises because the merged seller’s cost distribution is $F_{M}(c)=1-\left[ 1-F_{i}(c)\right] \left[ 1-F_{j}(c)\right]$, so $\left[ 1-F_{i}(c)\right] \left[ 1-F_{j}(c)\right] =1-F_{M}(c)$. From the perspective of non-merging seller h, the merger does not change the distribution of the lowest of its rivals’ costs.

To show that the merger strictly increases the expected price, one must show that ${\overline{P}}^{A\cup \{i,j\}}<{\overline{P}}^{A\cup \{M\}}$. If the set A of rivals is non-empty, then using the formula for the expected price,

$$\begin{aligned} {\overline{P}}^{A\cup \{i,j\}}&= {} \int _{{\underline{c}}}^{{\overline{c}} }cG_{\left( 2:A\cup \{i,j\}\right) }^{\prime }(c)dc \\&= {} {\overline{c}}-{\underline{c}}-\int _{{\underline{c}}}^{{\overline{c}}}G_{\left( 2:A\cup \{i,j\}\right) }(c)dc \\&<{\overline{c}}-{\underline{c}}-\int _{{\underline{c}}}^{{\overline{c}}}G_{\left( 2:A\cup \{M\}\right) }(c)dc \\&= {} {\overline{P}}^{A\cup \{M\}}. \end{aligned}$$

The equality between the first and second lines of the preceding expressions follows from integration by parts, while the inequality between the second and third lines follows from the definitions of $G_{\left( 2:A\right) }(c)$ and $F_{M}(c)$. The merger increases the price in those instances in which sellers i and j would have had the lowest and second-lowest costs pre-merger, thereby increasing price post-merger to what would have been the third-lowest cost pre-merger. If the set A of rivals is empty, then a similar argument holds.

The merger strictly decreases the buyer’s expected profit because the merger strictly increases the expected price.

Finally, to show that the merger strictly increases expected total surplus, one must show that ${\overline{TS}}^{A\cup \{i,j\}}<{\overline{TS}}^{A\cup \{M\}}$. Using the formula for expected total surplus,

$$\begin{aligned} {\overline{TS}}^{A\cup \{i,j\}}&= {} \int _{{\underline{c}}}^{{\overline{c}}}\left( v-c\right) G_{\left( 1:A\cup \{i,j\}\right) }^{\prime }(c)dc-\left| A+2\right| e \\&= {} \int _{{\underline{c}}}^{{\overline{c}}}\left( v-c\right) G_{\left( 1:A\cup \{M\}\right) }^{\prime }(c)dc-\left| A+2\right| e \\&<\int _{{\underline{c}}}^{{\overline{c}}}\left( v-c\right) G_{\left( 1:A\cup \{M\}\right) }^{\prime }(c)dc-\left| A+1\right| e \\&= {} {\overline{TS}}^{A\cup \{M\}}. \end{aligned}$$

The equality between the first and second lines of the preceding expressions follows from the definitions of $G_{\left( 1:A\right) }(c)$ and $F_{M}(c)$. $\square$

Proof of Proposition  1

If the contest’s pre-merger equilibrium has sellers i and j both entering against the set A of active rivals, then it must be the case that ${\overline{\pi }}_{i}^{A\cup \{i,j\}}\ge 0$ and ${\overline{\pi }}_{j}^{A\cup \{i,j\}}\ge 0$. By Lemma 1, ${\overline{\pi }}_{M}^{A\cup \{M\}}>{\overline{\pi }}_{i}^{A\cup \{i,j\}}+ {\overline{\pi }}_{j}^{A\cup \{i,j\}}$, so seller M enters against the same set A of active rivals. With seller M’s entry, the merger’s effects on expected profits, the expected price, and expected total surplus all follow from Lemma 1. $\square$

Proof of Lemma  2

Consider a subgame in which seller i is active pre-merger, but seller j is not, facing a (potentially empty) set A of active sellers that does not include the merging sellers or the merged seller (sellers i, j, or M).

To show that the merger strictly increases the merging sellers’ expected profits, one must show that ${\overline{\pi }}_{i}^{A\cup \{i\}}+\overline{\pi }_{j}^{A\cup \{i\}}<{\overline{\pi }}_{M}^{A\cup \{M\}}$. Using the formula for an active seller’s expected profit, and recalling that inactive seller j’s expected profit is 0,

$$\begin{aligned} {\overline{\pi }}_{i}^{A\cup \{i\}}+{\overline{\pi }}_{j}^{A\cup \{i\}}&= {} \int _{ {\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c) \right] \right) F_{i}(c)dc-e \\&<\int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A}\left[ 1-F_{k}(c)\right] \right) F_{M}(c)dc-e \\&= {} {\overline{\pi }}_{M}^{A\cup \{M\}}. \end{aligned}$$

The inequality between the first and second lines of the preceding expressions follows from the definition of $F_{M}(c)$. In this instance the merger merely improves seller i’s cost distribution, so seller M earns the same expected profit for each cost draw as seller i did pre-merger, plus seller M’s average cost draw is lower.

To show that the merger strictly decreases active rivals’ expected profits, one must show that ${\overline{\pi }}_{h}^{A\cup \{i\}}>{\overline{\pi }} _{h}^{A\cup \{M\}}$ for all $h\in A$. Using the formula for an active seller’s expected profit,

$$\begin{aligned} {\overline{\pi }}_{h}^{A\cup \{i\}}&= {} \int _{{\underline{c}}}^{{\overline{c}} }\left( \prod \limits _{k\in A\backslash h\cup \{i\}}\left[ 1-F_{k}(c)\right] \right) F_{h}(c)dc-e \\&>\int _{{\underline{c}}}^{{\overline{c}}}\left( \prod \limits _{k\in A\backslash h\cup \{M\}}\left[ 1-F_{k}(c)\right] \right) F_{h}(c)dc-e \\&= {} {\overline{\pi }}_{h}^{A\cup \{M\}}. \end{aligned}$$

The inequality between the first and second lines of the preceding expressions arises because the merged seller’s cost distribution is $F_{M}(c)=1-\left[ 1-F_{i}(c)\right] \left[ 1-F_{j}(c)\right]$, so $\left[ 1-F_{i}(c)\right] >\left[ 1-F_{M}(c)\right]$. From the perspective of non-merging active seller h, the merger changes the distribution of the lowest of its rivals’ costs to put more weight on lower cost draws.

To show that the merger strictly increases expected total surplus, one must show that ${\overline{TS}}^{A\cup \{i\}}<{\overline{TS}}^{A\cup \{M\}}$. Using the formula for expected total surplus,

$$\begin{aligned} {\overline{TS}}^{A\cup \{i\}}&= {} \int _{{\underline{c}}}^{{\overline{c}}}\left( v-c\right) G_{\left( 1:A\cup \{i\}\right) }^{\prime }(c)dc-\left| A+1\right| e \\&<\int _{{\underline{c}}}^{{\overline{c}}}\left( v-c\right) G_{\left( 1:A\cup \{M\}\right) }^{\prime }(c)dc-\left| A+1\right| e \\&= {} {\overline{TS}}^{A\cup \{M\}}. \end{aligned}$$

The inequality between the first and second lines of the preceding expressions follows from the definitions of $G_{\left( 1:A\right) }(c)$ and $F_{M}(c)$.

To show that the merger strictly decreases the expected price, one must show that ${\overline{P}}^{A\cup \{i\}}>{\overline{P}}^{A\cup \{M\}}$. If the set A of active rivals is non-empty, then using the formula for the expected price,

$$\begin{aligned} {\overline{P}}^{A\cup \{i\}}&= {} \int _{{\underline{c}}}^{{\overline{c}}}cG_{\left( 2:A\cup \{i\}\right) }^{\prime }(c)dc \\&= {} {\overline{c}}-{\underline{c}}-\int _{{\underline{c}}}^{{\overline{c}}}G_{\left( 2:A\cup \{i\}\right) }(c)dc \\&>{\overline{c}}-{\underline{c}}-\int _{{\underline{c}}}^{{\overline{c}}}G_{\left( 2:A\cup \{M\}\right) }(c)dc \\&= {} {\overline{P}}^{A\cup \{M\}}. \end{aligned}$$

The equality between the first and second lines of the preceding expressions follows from integration by parts, while the inequality between the second and third lines follows from the definitions of $G_{\left( 2:A\right) }(c)$ and $F_{M}(c)$. The merger decreases the price in those instances in which seller i would have had the second-lowest cost or higher pre-merger, whereas post-merger the additional draw associated with seller j’s distribution causes seller M to have the second-lowest cost or lower post-merger. If the set A of active rivals is empty, then the expected price is v both pre-merger and post-merger.

If the set A of active rivals is non-empty, then the merger strictly increases the buyer’s expected profit because the merger strictly decreases the expected price. If A is empty, then there is no change in the buyer’s expected profit, because the expected price is the same pre-merger and post-merger. $\square$

Proof of Proposition  2

If the contest’s pre-merger equilibrium has seller i entering, but not seller j, against the set A of active rivals, then it must be the case that ${\overline{\pi }}_{i}^{A\cup \{i\}}\ge 0$ and ${\overline{\pi }}_{j}^{A\cup \{i\}}=0$. By Lemma 2, ${\overline{\pi }}_{M}^{A\cup \{M\}}>{\overline{\pi }}_{i}^{A\cup \{i\}}+ {\overline{\pi }}_{j}^{A\cup \{i\}}$, so seller M enters against the same set A of active rivals. With seller M’s entry, the merger’s effects on expected profits, the expected price, and expected total surplus all follow from Lemma 2.□

Proof of Proposition  3

If the contest’s pre-merger equilibrium has neither seller i nor seller j entering against the set A of active rivals, then it must be the case that ${\overline{\pi }} _{i}^{A\cup \{i\}}\le 0$, ${\overline{\pi }}_{j}^{A\cup \{j\}}\le 0$, ${\overline{\pi }}_{i}^{A\cup \{i,j\}}\le 0$, and ${\overline{\pi }}_{j}^{A\cup \{i,j\}}\le 0$. Calculations similar to those in the proofs of Lemmas 1 and 2 show that seller M’s expected profit versus the set A of active rivals strictly exceeds the expected profits of seller i, seller j, or both against the set A of active rivals. Therefore, seller M’s expected profit against the set A of active rivals might be strictly positive. If so, then seller M will enter against the set A of active rivals. If seller M enters, then the merger’s effects on active rivals’ expected profits, the buyer’s expected profit, the expected price, and expected total surplus follow from calculations similar to those in the proofs of Lemmas 1 and 2. $\square$