Entry Deterrence, Concentration, and Merger Policy

^Abstract

In merger enforcement, entry is considered to be a factor that potentially can mitigate otherwise anti-competitive effects of a merger. The current framework for entry analysis evaluates whether potential entrants are likely to have the incentives and ability to enter the industry under the conditions of elevated profitability that are created by an anti-competitive merger. Missing from entry analysis is the notion that incumbent firms may proactively deter entry and how such incumbent incentives may change as a result of a merger. By modeling entry as the outcome of a game between incumbents and potential entrants, we show that a merger can reduce the likelihood of entry even at elevated profit levels by increasing incumbent incentives to invest in entry deterrence. The paper has two policy implications for merger enforcement: First, a merger that is benign by traditional measures may nonetheless have the effect of reducing future entry—entry that would have made the market more competitive relative to status quo. Second, evidence of recent historical entry—which is an important criterion that is used to assess the likelihood of post-merger entry—may be of less evidentiary value than is considered under the current merger enforcement policy.

Appendices

Appendix 1

Sufficient Condition in Proposition 2

In this appendix, we show that—both under Cournot competition (with linear demand) and under Bertrand competition (with logit demand)—for every firm $i$, the loss of profit suffered by incumbent $i$ due to entry is larger post-merger than pre-merger:

$$\left({\pi }_{post}^{i}\left(NE\right)-{\pi }_{post}^{i}\left(E\right)\right)>\left({\pi }_{pre}^{i}\left(NE\right)-{\pi }_{pre}^{i}\left(E\right)\right)$$

2.1 Cournot

We assume a Cournot model with $N$ firms, where: (i) each firm has constant marginal cost ${c}_{i}$; and (ii) demand is linear: $P=a-b\times Q$.

Firm $i$’s FOC is:

$$a-b\times Q-{c}_{i}-b\times {q}_{i}=0.$$

(15)

Summing over the FOC of all firms implies that:

$$N\times a-N\times b\times Q-N\times \overline{c }-b\times Q=0,$$

(16)

where $\overline{c }$ is the unweighted average cost across all firms: $\overline{c }=\frac{\sum_{i}{c}_{i}}{N}$. Therefore:

$$ Q = \frac{{N \times \left( {a - \overline{c}} \right)}}{{\left( {N + 1} \right) \times b}};\,{\text{and}} $$

(17)

$$ q_{i} = \frac{{a - c_{i} }}{b} - \frac{{N \times \left( {a - \overline{c}} \right)}}{{\left( {N + 1} \right) \times b}}. $$

(18)

Furthermore, (15) implies that $P-{c}_{i}=a-b\times Q-{c}_{i}=b\times {q}_{i}$, so that firm $i$’s profit is:

$${\pi }_{i}=b\times {\left({q}_{i}\right)}^{2}=\frac{1}{b}\times {\left(a-{c}_{i}-\frac{N}{\left(N+1\right)}\times \left(a-\overline{c }\right)\right)}^{2}.$$

(19)

We provide an analytic proof under the assumption that ${c}_{i}=\overline{c }$. We then run Monte Carlo simulations that do not impose this restriction.

If ${c}_{i}=\overline{c }$, then ${\pi }_{i}=\frac{1}{b}\times \frac{{\left(a-\overline{c }\right)}^{2}}{{\left(N+1\right)}^{2}}$. The condition $\left({\pi }_{post}^{i}\left(NE\right)-{\pi }_{post}^{i}\left(E\right)\right)>\left({\pi }_{pre}^{i}\left(NE\right)-{\pi }_{pre}^{i}\left(E\right)\right)$ translates to (without loss of generality, we normalize $b$ and ${\left(a-\overline{c }\right)}^{2}$ to 1):

$$\frac{1}{{\left(N\right)}^{2}}-\frac{1}{{\left(N+1\right)}^{2}}>\frac{1}{{\left(N+1\right)}^{2}}-\frac{1}{{\left(N+2\right)}^{2}}$$

To show this, it is sufficient to show that $\frac{1}{{\left(X\right)}^{2}}-\frac{1}{{\left(X+1\right)}^{2}}$ is decreasing in $X$, for $X\ge 2$. The derivative of the previous expression with respect to $X$ is $-\frac{2}{{\left(X\right)}^{3}}+\frac{2}{{\left(X+1\right)}^{3}}$, which is clearly negative.

Now we allow for firms to have different cost ${c}_{i}$. We construct each Monte Carlo experiment as follows²⁸:

• 1. Each firm’s cost is drawn from a uniform between 0 and 5.

• 2. $a$ is drawn from a uniform between 5 and 50.

• 3. The number of firms is between 2 and 10, drawn from a discrete uniform.

• 4. The slope $b$ is normalized to 1.

We ran 50,000 such Monte Carlo experiments. We used only parameter configurations where all firms have a positive quantity both pre-merger and post-merger. The Monte Carlo results confirm that a merger always increases each incumbent firm’s loss from entry.²⁹

2.2 Bertrand

We consider a model where demand is logit and single product firms are Bertrand competitors. Because this model does not have closed-form solution, we confirmed that $\left({\pi }_{post}^{i}\left(NE\right)-{\pi }_{post}^{i}\left(E\right)\right)>\left({\pi }_{pre}^{i}\left(NE\right)-{\pi }_{pre}^{i}\left(E\right)\right)$ by running Monte Carlo experiments as follows:

We assume that the latent utility that consumer $i$ derives from product $j$ is: ${U}_{ij}={d}_{j}-\alpha \times {p}_{j}+{e}_{ij}$, where ${e}_{ij}$ follow an extreme value distribution. It follows that the demand for product $j$ is: ${q}_{j}=\frac{\mathrm{exp}({d}_{j}-\alpha \times {p}_{j})}{1+\sum_{k}\mathrm{exp}({d}_{j}-\alpha \times {p}_{j})}$.

In our baseline simulations, we construct each Monte Carlo experiment as follows:

• 1. ${d}_{j}$ is drawn from a uniform between 0 and 2.

• 2. $\alpha$ is drawn from a uniform between 0 and 1.

• 3. Each firm’s cost is drawn from a uniform between 0 and 2.

• 4. The number of firms is between 2 and 10, drawn from a discrete uniform.

We ran 50,000 such Monte Carlo experiments. For each experiment, we find that³⁰:

• 1. For non-merging firms $\left({\pi }_{post}^{i}\left(NE\right)-{\pi }_{post}^{i}\left(E\right)\right)>\left({\pi }_{pre}^{i}\left(NE\right)-{\pi }_{pre}^{i}\left(E\right)\right)$: The loss from entry is larger post-merger than pre-merger.

• 2. This may not hold for each individual merging firm, but it holds for the sum of the merging firms’ profits, which is sufficient for our purposes.

Appendix 2

In this appendix we demonstrate analytically that, in the case of a symmetric Cournot model with linear demand and constant marginal cost, a merger always decreases the likelihood of entry when $D\left(\sum_{k}{x}_{k}\right)=\sum_{k}{x}_{k}$. We also implement Monte Carlo simulations that indicate that the same holds in the case of Bertrand competition with logit demand.

3.1 Cournot

As was derived in Appendix 1, the equilibrium quantity and profits are ${q}_{i}=\frac{\left(a-c\right)}{\left(N+1\right)\times b}, \pi \left(N\right)=\frac{{\left(a-c\right)}^{2}}{{\left(N+1\right)}^{2}\times b}$. In what follows, we treat $N$ as the number of incumbent firms pre-merger. A merger reduces the number of firms to $N-1$: Given symmetry, the merged firm will remove one of the pre-merger firms from the market. Entry increases the number of firms by 1.

3.1.1 Pre-merger

Entry increases the number of firms from $N$ to $N+1$. Therefore:

$${\Omega }_{pre}= \pi \left(N\right)-\pi \left(N+1\right)=\frac{{\left(a-c\right)}^{2}}{b}\left[\frac{1}{{\left(N+1\right)}^{2}}-\frac{1}{{\left(N+2\right)}^{2}}\right]=\frac{{\left(a-c\right)}^{2}}{b}\frac{2N+3}{{\left(N+1\right)}^{2}{\times \left(N+2\right)}^{2}} ;$$

$${\pi }_{pre}^{E}=\pi \left(N+1\right)=\frac{{\left(a-c\right)}^{2}}{{\left(N+2\right)}^{2}\times b} .$$

3.1.2 Post-merger

Entry increases the number of firms from $N-1$ to $N$. Therefore:

$${\Omega }_{post}= \pi \left(N-1\right)-\pi \left(N\right)=\frac{{\left(a-c\right)}^{2}}{b}\left[\frac{1}{{\left(N\right)}^{2}}-\frac{1}{{\left(N+1\right)}^{2}}\right]=\frac{{\left(a-c\right)}^{2}}{b}\frac{2N+1}{{\left(N\right)}^{2}{\times \left(N+1\right)}^{2}} ;$$

$${\pi }_{post}^{E}=\pi \left(N\right)=\frac{{\left(a-c\right)}^{2}}{{\left(N+1\right)}^{2}\times b} .$$

3.1.3 Comparison of Entry Probabilities

Assume that $D\left(\sum_{k}{x}_{k}\right)=\sum_{k}{x}_{k}$. Then, for $l=pre, post$, Eqs. (8) and (11) imply that:

$${X}^{l}={\left(\frac{{\Omega }_{l}\times {\pi }_{l}^{E}}{\alpha }\right)}^{1/2},$$

(20)

and the likelihood of entry is.

$${{Prob\_Entry}_{l}=\frac{{\pi }_{l}^{E}}{{X}^{l}}=\left(\frac{\alpha }{{\Omega }_{l}}\right)}^{1/2}{\left({\pi }_{l}^{E}\right)}^{1/2}.$$

(21)

Therefore;

$$\frac{{Prob\_Entry}_{post}}{{Prob\_Entry}_{pre}}=\frac{{\left(\frac{\alpha }{{\Omega }_{post}}\right)}^{1/2}{\left({\pi }_{post}^{E}\right)}^{1/2}}{{\left(\frac{\alpha }{{\Omega }_{pre}}\right)}^{1/2}{\left({\pi }_{pre}^{E}\right)}^{1/2}}={\left(\frac{{\Omega }_{pre}}{{\Omega }_{post}}\right)}^{1/2}\times {\left(\frac{{\pi }_{post}^{E}}{{\pi }_{pre}^{E}}\right)}^{1/2}$$

$$={\left(\frac{\frac{2N+3}{{\left(N+1\right)}^{2}{\times \left(N+2\right)}^{2}}}{\frac{2N+1}{{\left(N\right)}^{2}{\times \left(N+1\right)}^{2}}}\right)}^{1/2}\times {\left(\frac{{\left(N+2\right)}^{2}}{{\left(N+1\right)}^{2}}\right)}^{1/2}={\left(\frac{\left(2N+3\right)\times {\left(N\right)}^{2}}{{\left(2N+1\right)\times \left(N+1\right)}^{2}}\right)}^{1/2} .$$

The ratio, which depends only on $N$, is always less than 1, as Fig. 2 illustrates. For example, when there are four incumbent firms pre-merger, the ratio of the probabilities is 88.4%; consequently, if the pre-merger probability of entry is 40%, then the post-merger probability of entry is 35.4% (40% $\times$ 88.4%).

Fig. 2

Effect of the merger on the probability of entry—effect of the number of incumbents

3.2 Bertrand

For the Bertrand model, we implement Monte Carlo simulations (which were described in Appendix 1) that indicate that, when $D\left(\sum_{k}{x}_{k}\right)=\sum_{k}{x}_{k}$, the ratio of probabilities $\frac{{Prob\_Entry}_{post}}{{Prob\_Entry}_{pre}}$ is always less than 1: The entry probability always decreases.³¹

Figure 3 illustrates the relationship between $\frac{{Prob\_Entry}_{post}}{{Prob\_Entry}_{pre}}$ and number of incumbent firms. It shows that the negative effect on entry probability is greater when the number of incumbent firms is smaller.

Fig. 3

Effect of the merger on the probability of entry—effect of the number of incumbents

Table 1 illustrates the relationship between $\frac{{Prob\_Entry}_{post}}{{Prob\_Entry}_{pre}}$ and the share of the smaller of the two merging firms, conditional on the number of pre-merger incumbents. It shows that the reduction in entry probability is greater when the share of the smaller merging firm is smaller.

Table 1 Effect of the merger on the probability of entry