Horizontal mergers and innovation in concentrated industries

Abstract

It is an open question in antitrust economics whether allowing dominant firms to acquire smaller rivals is ultimately helpful or harmful to the long run rate of innovation and therefore long-term consumer welfare. I develop a framework to study this question in a dynamic oligopoly model where firms endogenously engage in investment, entry, exit and mergers. Firms produce vertically differentiated goods, compete by innovating on product quality, and can acquire rival firms to gain market power. In a benchmark model, mergers are modeled to be exclusively harmful to consumers in the short run by reducing competition and increasing prices. Despite this, under standard industry settings it is possible to show that the prospect of a buyout creates a powerful incentive for firms to preemptively enter the industry and invest to make themselves an attractive merger partner. The result is significantly higher rate of innovation with mergers than without and significantly higher long-run consumer welfare as well. Further results explore the circumstances under which this result is likely to hold. In order for the long run increase in innovation to outweigh the short run harm to consumers caused by mergers, entry costs must be low, entrants and incumbents must both have the ability to innovate rapidly, and the degree of horizontal product differentiation must be low. Alternatively, when dominant firms can directly incorporate the acquired firm’s innovation into their own product, mergers will typically benefit consumers in both the short run and long run.

Appendix: Algorithm for solving the model

This appendix describes the algorithm used to solve for an equilibrium to the model described in Section 3. This builds on the work of Pakes and McGuire (1994) and Fershtman and Pakes (2012) but adds a novel element from the reinforcement learning literature to improve convergence properties. It has been noted that the basic stochastic algorithm often performs badly on convergence. Indeed, without the changes described below the basic algorithm almost always fails to reach an equilibrium. Here I review some of the reasons for this and how they can be fixed. For greater detail on the basic algorithm see Pakes and McGuire (1994) and Fershtman and Pakes (2012).²⁰

The algorithm proceeds iteratively, simulating the dynamic game using a stored value function that firms use to generate policies regarding entry, exit, investment, and mergers. At each step of this simulation, the value function is updated with the payoffs realized for each action taken or not taken. The key components that are stored in memory are the current state of the industry at each iteration k, called Ω_k, the stored value functions defining payoffs for each action at that state: ${W^{I}_{k}}({\Omega }_{k}, x_{i})$, ${W^{E}_{k}}({\Omega }_{k}, xe_{i})$, and ${W^{M}_{k}}({\Omega }_{k}, i,j)$, and a counter that stores the number of prior visits to state Ω_k to that point: ${h^{I}_{k}}({\Omega }_{k}, x_{i})$, ${h^{E}_{k}}({\Omega }_{k}, xe_{i})$, and ${h^{M}_{k}}({\Omega }_{k}, i,j)$. If ${h^{I}_{k}}({\Omega }_{k}, x_{i})=0$, ${W^{I}_{k}}({\Omega }_{k}, x_{i})$ is empty. When the state (Ω,x_i) is reached for the first time, ${W^{I}_{k}}({\Omega }, x_{i})$ is set to an initial value and updated from there.

Profits π(Ω) are computed offline for all states. Each value function is initialized at some level that I discuss in more detail below. The timing is as follows, at each iteration k:

• 1. At state Ω_k draw from memory:${W^{I}_{k}}({\Omega }_{k}, x_{i})$, ${W^{E}_{k}}({\Omega }_{k}, xe_{i})$, ${W^{M}_{k}}({\Omega }_{k}, i,j)$, ${h^{I}_{k}}({\Omega }_{k}, x_{i})$, ${h^{E}_{k}}({\Omega }_{k}, xe_{i})$, and ${h^{M}_{k}}({\Omega }_{k}, i,j)$.

• 2. For all incumbent firms i and for the potential entrant, draw investment costs $\tilde {c_{i}}$

• 3. Incumbents solve:

  $$ \max_{x_{i}}\{-C(\tilde{c_{i}},x_{i})+{W^{I}_{k}}({\Omega}_{k},x_{i})\} $$

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  and exits if the max of this term is less than zero. Entrants solve:

  $$ \max_{xe_{i}}\{-C(\tilde{c_{i}},xe_{i})+{W^{E}_{k}}({\Omega}_{k}, xe_{i})\} $$

  and enter if the max of this term is greater than zero.

• 4. Randomly draw the industry-wide depreciation shock η.

• 5. Using η and the investment, entry and exit decisions of incumbents and the potential entrant, update the market state from Ω_k to ${\Omega }^{\prime }_{k}$.

• 6. Begin the merger stage by drawing a random ordering of firms to act as merger proposers.

• 7. Loop over all firms, for each solve for the best merger partner as $max_{j} {W^{M}_{k}}({\Omega }^{\prime }_{k}, i,j)$. If the value of merging for both firms is higher than the option value of letting the next firm in the merger order proceed, they agree to merge, τ_ij is calculated, ${\Omega }^{\prime }_{k}$ is updated to Ω_k+ 1, and the merger stage ends. During this stage, Ω_k+ 1 is necessarily rounded to the nearest integer value. Once a merger has occurred or all firms have had a chance to propose, the stage ends.

• 8. Profits for all firms are calculated as π(Ω_k+ 1).

• 9. Stored value functions are updated as:

  $$ \begin{array}{@{}rcl@{}} W^{I}_{k+1}({\Omega}_{k},x^{*}_{i}) &=& \alpha^{I}({\Omega}_{k}) \beta [\pi(\omega^{\prime}_{i}) - FC + \mathbb{E} {W^{I}_{k}}({\Omega}_{k+1})]\\ &&+ (1-\alpha^{I}({\Omega}_{k}) ) {W^{I}_{k}}({\Omega}_{k},x^{*}_{i}) \end{array} $$

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  $$ \begin{array}{@{}rcl@{}} W^{E}_{k+1}({\Omega}_{k},xe^{*}_{i})&=& \alpha^{E}({\Omega}_{k}) \beta [\pi(\omega^{\prime}_{i}) - FC +\mathbb{E} {W^{I}_{k}}({\Omega}_{k+1})]\\ &&+ (1-\alpha^{E}({\Omega}_{k}) ) {W^{E}_{k}}({\Omega}_{k},xe^{*}_{i}) \end{array} $$

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  If firm i is acquired by firm j:

  $$ W^{M}_{k+1}({\Omega}^{\prime}_{k}, i,j)= \alpha^{M}({\Omega}^{\prime}_{k},i,j) \tau_{ij} + (1-\alpha^{M}({\Omega}^{\prime}_{k},i,j) ) {W^{M}_{k}}({\Omega}^{\prime}_{k}, i,j) $$

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  If firm i acquires firm j:

  $$ \begin{array}{@{}rcl@{}} W^{M}_{k+1}({\Omega}^{\prime}_{k}, i,j)&=& \alpha^{M}({\Omega}^{\prime}_{k},i,j) [ \mathbb{E} {W^{I}_{k}}({\Omega}_{k+1})-\tau_{ij} ]\\ &&+ (1-\alpha^{M}({\Omega}^{\prime}_{k},i,j) ) {W^{M}_{k}}({\Omega}^{\prime}_{k}, i,j) \end{array} $$

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  where α^I(⋅), α^E(⋅), and α^M(⋅) are weighting functions to be described in detail below. In addition, counters ${h^{I}_{k}}\cdot )$, ${h^{E}_{k}}(\cdot )$, and ${h^{M}_{k}}(\cdot )$ are incremented by 1.

• 10. Return to step 1 at state Ω_k+ 1.

The algorithm is periodically paused to test for whether an equilibrium has been reached. This test follows Fershtman and Pakes (2012) and checks whether the value functions are consistent with equilibrium notions described in Section 3.2. This simulates a sample path of the model and keeps a separate memory of the distribution of outcomes reached at each state on this sample path. The mean squared difference between these outcomes and the value function stored in memory is used to calculate bias. This is done separately for ${W^{I}_{k}}(\cdot )$, ${W^{E}_{k}}(\cdot )$, and ${W^{M}_{k}}(\cdot )$ and the highest bias value of the three is compared to .001 to determine if an equilibrium has been reached.

In most cases, this equilibrium calculation is taken inside a larger loop over possible parameter values as part of the Simple Continuation Method. This method traces out the equilibrium outcomes of the model as parameters vary. In this case, when an equilibrium is reached the value functions ${W^{I}_{k}}(\cdot )$, ${W^{E}_{k}}(\cdot )$, and ${W^{M}_{k}}(\cdot )$ are saved but the counters $({h^{I}_{k}}\cdot )$, ${h^{E}_{k}}(\cdot )$, and ${h^{M}_{k}}(\cdot )$ are returned to 1 and the weighting functions described below are similarly updated

A key consideration in the algorithm is what function to use to weight realized payoffs in iteration k and how much to weight the current estimate of W_k(⋅). One alternative would be to simply use $\alpha ^{I}(\cdot ) = \frac {1}{{h^{I}_{k}}({\Omega }_{k})}$, ie the number of previous visits to that state. This ultimately would give value functions equal to the arithmetic mean of realized payoffs across all visits to that state. One problem with this approach is that if the initialized value functions are far from their true values, it could take a very long time for the algorithm to converge.

A second and more serious problem with the algorithm as described above is that for discrete choices such as entry and exit, it can get “stuck” at a suboptimal choice. For example, if the value functions for firm investment are set high in order to encourage exploration, firms will initially invest large amounts and rarely exit. At the same time as they make these choices, potential entrants are exploring entry strategies and updating their entry value function with the realized outcomes. In the case where incumbents invest highly entry is rarely profitable and so in certain states potential entrants may update all entry options as having negative value. Once they have done so, entry will cease at those states and as incumbent firms investment policies converge towards equilibrium the potential entrant will have stopped testing entry even though it may be profitable to do so.

I solve this second problem by implementing a strategy from the reinforcement learning literature known as 𝜖-greedy exploration. In this case, firms will take what they perceive as being the optimal action with probability 1 − 𝜖 and with probability 𝜖 they will choose a policy at random from their set of possible actions. The researcher sets the initial value of 𝜖 to encourage exploration and as the algorithm proceeds it declines slowly to 0. In states in which firms take suboptimal policies as a result of this process, they learn and update values for those policies but the other firms in the market do not update at those states.

This approach is simple but has the advantages that it ensures each action will be taken a large number of times and that policies ultimately converge to the optimal ones. In practice, without implementing this strategy the algorithm almost always converged to an outcome where a suboptimal discrete choice was taken and which was strongly rejected as an equilibrium outcome by the testing procedure. This would remain true regardless of how long it ran. Even if this extreme case of non-convergence were not possible, it is highly likely the 𝜖-greedy exploration improves the speed to convergence of even a simple model.

1.1 A.1 Computational details

Here I provide specific details on the implementation of the algorithm. For incumbent and entrant value functions, I initialize the values above the level of discounted profits if the state they entered were permanent. A high initialization is useful for ensuring firms explore their strategy space early on. That is:

$$ {W_{0}^{I}}({\Omega}) = \frac{1.1\pi({\Omega})}{1-\beta}. $$

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Merger value functions are set at a flat constant value of 5, which is high enough to encourage exploration. For all policies, 𝜖 is set initially at .1 and declines such that $\epsilon ^{\prime } = .9\epsilon$ every 200,000 iterations.

The weighting functions used are:

$$ \alpha^{I}({\Omega}_{k})=\frac{1}{min({h^{I}_{k}}({\Omega}_{k}, x_{i}), \bar{h}^{I})} $$

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where $\bar {h}$ is a cap on high h_k(⋅) for weighting purposes. This effectively places more weight on the more recent $\bar {h}$ observations. In practice, $\bar {h}$ begins at 100 and doubles every 1,000,000 iterations until it ceases to bind. In all cases, the test concluded that an algorithm had been reached before at most 350 million iterations and in some cases much sooner.