# Vertical vs Horizontal Integration: Pre-emptive merging: A comment

Eric Avenel

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CAHIERS ECO & MATHS

96.53

VERTICAL VS. HORIZONTAL INTEGRATION
AND PRE-EMPTIVE MERGING : A COMMENT

Eric AVENEL

CEME
VERTICAL VS. HORIZONTAL
INTEGRATION :
PRE-EMPTIVE MERGING :
A COMMENT

Eric Avenel

Université de Paris 1 - C.E.M.E.

12, Place du Panthéon - 75231 Paris Cedex 05

e-mail avenel@univ-paris 1.fr

June 1996
Abstract

In a recent issue of The Journal of Industrial Economics, G. Colangelo (1995) develops the concept of pre-emptive merging. In a successive markets framework, with both upstream and downstream firms bidding to integrate with a target firm, a downstream (upstream) firm may merge with an upstream (downstream) firm, so that vertical integration occurs, in order to prevent (pre-empt) a detrimental horizontal merger. Colangelo claims that it is a quite general outcome.

We propose here a new resolution of the model, leading to more restrictive emergence conditions. Pre-emptive merging may indeed take place in equilibrium for an upstream monopoly and a downstream duopoly (the bidder and the target), as long as goods are not close complements. However, with an upstream duopoly, pre-emptive vertical integration never occurs, no matter if the target firm is upstream or downstream.

**keywords** : pre-emption, vertical integration, horizontal integration, foreclosure.

JEL classification : L12, L22.

Résumé

Dans un article récemment paru dans le Journal of Industrial Economics, G. Colangelo(1995) développe le concept d'intégration verticale pré-emptive. Dans un contexte de structure verticale, l'auteur montre que l'intégration verticale d'une des deux firmes aval avec la firme amont répond à un objectif stratégique consistant à empêcher le rachat par l'autre firme aval. Selon Colangelo, ce phénomène apparaîtrait à l'équilibre de façon relativement générale.

Nous proposons ici un réexamen des différents modèles présentés. Il en résulte que l'intégration verticale pré-emptive est un résultat beaucoup moins général que ne le prétend Colangelo. Elle n'intervient à l'équilibre que dans le cas d'un monopole amont, lorsque les biens ne sont pas trop complémentaires.

**mots clés** : Pré-emption, intégration verticale, intégration horizontale, foreclosure.

Classification JEL : L12, L22.
# 1. Introduction

In the September 1995 issue of this Journal, G. Colangelo develops the concept of pre-emptive merging. He claims that in a context of successive duopolies where both vertical and horizontal integration are feasible, vertical integration occurs in equilibrium for a large range of the parameter values, not because it is per se more profitable than a horizontal merger, but because it allows the merging firms to avoid facing a detrimental horizontal merger. He derives conditions for the emergence of vertical integration in equilibrium : these are propositions 1, 2 and 3 of Colangelo's paper.

In the three following sections of this comment, we show that each of these three propositions are false and we determine the correct emergence conditions. Furthermore, we show that vertical integration doesn't necessary have a pre-emptive nature. In the final section, we give some concluding comments.

# 2. First case : Upstream monopolist and downstream target

## 2.1. Structure of the game

An upstream monopolist, $U_1$, produces an intermediate good which is transformed on a one-for-one basis by two differentiated downstream firms, $D_1$ and $D_2$. Product differentiation is introduced through demand functions :

$$q_i = \alpha - p_i + \gamma p_j; i = 1, 2; i \neq j; \alpha > 0; -1 < \gamma < 1.$$

where $p_k$ is the price charged by $D_k$. For $\gamma > 0$, products are substitutes, while they are complements for $\gamma < 0$ and independent for $\gamma = 0$.

In the first stage of the game, $U_1$ and $D_2$ simultaneously bid for $D_1$. We denote the three possible industry structures after stage 1 by :

* NI (No Integration) : neither $U_1$ nor $D_2$ bids,

* VI (Vertical Integration) : $U_1$ bids more than $D_2$,

* HI (Horizontal Integration) : $D_2$ bids more than $U_1$.

In the second stage, $U_1$ sets the input prices.

In the third stage, $D_1$ and $D_2$ compete on prices. The costs of production are equal to 0.
## 2.2. Profits expressions

Solving stages 2 and 3, Colangelo determines the profits after stage 1 for the different structures :

$$NI : \pi_{U_1}(NI) = \frac{\alpha^2}{2(2-\gamma)(1-\gamma)}; \pi_{D_1}(NI) = \pi_{D_2}(NI) = \frac{\alpha^2}{4(2-\gamma)^2}$$

$$VI : \pi_{U_1-D_1}(VI) = \frac{\alpha^2(2+\gamma)(6-\gamma+\gamma^2)}{4(1-\gamma)(8+\gamma^2)}; \pi_{D_2}(VI) = \frac{\alpha^2(2+\gamma^2)^2}{(8+\gamma^2)^2}$$

$$HI : \pi_{U_1}(HI) = \frac{\alpha^2}{4(1-\gamma)}; \pi_{D_1-D_2}(HI) = \frac{\alpha^2}{8(1-\gamma)}.$$

## 2.3. Colangelo's result

For every $\gamma \in ] -1; 1[$, Colangelo establishes the following inequality :

$$\pi_{U_1-D_1}(VI) - \pi_{U_1}(HI) > \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI) \quad (1).$$

From (1), Colangelo concludes that $U_1$ is ready to bid more for $D_1$ than $D_2$ and, consequently, integrates with $D_1$. Vertical integration takes place in equilibrium. This is proposition 1.

Plotting the profit functions, one can easily check that, for $\gamma < -0,83$, both $\pi_{U_1-D_1}(VI) - \pi_{U_1}(HI)$ and $\pi_{D_1-D_2}(HI) - \pi_{D_2}(VI)$ are negative. It is therefore doubtful that Colangelo's argument is correct, at least for this range of the differentiation parameter.

We propose a complete resolution of the game and show that Colangelo's proposition is false. Both vertical and horizontal integration can occur in equilibrium for $\gamma < -0,83$.

## 2.4. Resolution of the game

Considering that the winner pays its bid, it is straightforward to get the payoffs of $U_1$ and $D_2$ for the bidding game. For example, if $b(U_1) > b(D_2) \ge 0$, where $b(x)$ is the bid submitted by $x$, $U_1$'s profit is given by $\pi_{U_1-D_1}(VI) - b(U_1)$.

We define six regions and characterize the existence of an equilibrium in each of these regions :

$$\text{region 1 : } b(U_1) > b(D_2) \ge 0$$

There exists an equilibrium $(b(U_1), b(D_2))$ in this domain iff neither $U_1$ nor $D_2$ has an incentive to deviate from its bid.
The only way $D_2$ can modify its payoff is to bid more than $U_1$. This is not (strictly) profitable iff $D_2$ prefers facing an integrated rival to integrating himself and paying ($\varepsilon$ more than) $b(U_1)$, that is $\pi_{D_2}(VI) \ge \pi_{D_1-D_2}(HI) - b(U_1)$.

As long as $b(U_1) > b(D_2)$, $U_1$ can increase its profits by lowering its bid to $b(U_1) - \varepsilon > b(D_2)$. However, as long as it prefers winning, bidding $b(D_2)$ lowers its profit because it loses the bidding game with some probability<fnref n="1" />. This is a classical "openness problem". This is only technical and we will ignore it, considering that $U_1$ can bid $b(D_2)$ and win the bidding game. So we need $b(U_1) = b(D_2)$.

Clearly, $U_1$ has no interest in bidding more than $b(U_1)$.

Finally, $U_1$ may bid less than $b(D_2)$. This is not profitable iff $\pi_{U_1-D_1}(VI) - b(U_1) \ge \pi_{U_1}(HI)$.

We can now characterize the existence of an equilibrium in region 1.

There exists an equilibrium in region 1 iff

$$ \exists b \ge 0 \text{ such that } b \le \pi_{U_1-D_1}(VI) - \pi_{U_1}(HI) \text{ and } b \ge \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI). $$

region 2: $b(U_1) > b(D_2) = -\infty$, where bidding $-\infty$ means not bidding.

This is the same as for region 1, except that $U_1$ cannot bid less than $D_2$. It just can stop bidding, in which case no integration takes place. What is more, the openness problem disappears. Bidding 0, $U_1$ wins the bidding game and cannot increase its profits by bidding less unless it stops bidding. It chooses between bidding 0 and not bidding. So we have:

There exists an equilibrium in region 2 iff

$$ 0 \le \pi_{U_1-D_1}(VI) - \pi_{U_1}(NI) \text{ and } 0 \ge \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI). $$

region 3: $b(D_2) > b(U_1) \ge 0$

As in region 1, we have:

There exists an equilibrium in region 3 iff

$$ \exists b \ge 0 \text{ such that } b \le \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI) \text{ and } b \ge \pi_{U_1-D_1}(VI) - \pi_{U_1}(HI). $$

region 4: $b(D_2) > b(U_1) = -\infty$

There exists an equilibrium in region 4 iff

<footnote n="1">¹For equal non-negative bids, each firm wins with probability 1/2.</footnote>
$$0 \geq \pi_{U_1-D_1}(VI) - \pi_{U_1}(HI) \text{ and } 0 \leq \pi_{D_1-D_2}(HI) - \pi_{D_2}(NI).$$

$$\text{region 5 : } b(D_2) = b(U_1) \geq 0$$

This is an equilibrium iff both bidders are indifferent between losing the bidding game and winning (and paying the bid).

There exists an equilibrium in region 5 iff

$$\exists b \geq 0, b = \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI) = \pi_{U_1-D_1}(VI) - \pi_{U_1}(HI)$$

$$\text{region 6 : } b(D_2) = b(U_1) = -\infty$$

This is an equilibrium iff both bidders prefer no integration to integrating, even paying 0.

There exists an equilibrium in region 6 iff

$$0 \geq \pi_{U_1-D_1}(VI) - \pi_{U_1}(NI) \text{ and } 0 \leq \pi_{D_1-D_2}(HI) - \pi_{D_2}(NI).$$

The resolution of the game just requires considering four functions of $\gamma$. Let us consider the existence of an equilibrium in any of the six regions.

According to inequality 1, there exists an equilibrium in region 1 iff

$$\pi_{VI}(U_1 - D_1) - \pi_{HI}(U_1) \geq 0.$$

This is true iff $\gamma \geq -0,83$.

It is easy to show that inequality (2) is satisfied :

$$\forall \gamma \in ]-1; 1[, 0 \leq \pi_{U_1-D_1}(VI) - \pi_{U_1}(NI) (2),$$

so that there exists an equilibrium in region 2 iff

$$0 \geq \pi_{D_1-D_2}(HI) - \pi_{D_2}(VI).$$

This is true iff $\gamma \leq -0,58$.

From inequality 1, there is no equilibrium in this region 3.

The following inequality (3) is easily established :

$$\forall \gamma \in ]-1; 1[, 0 \leq \pi_{D_1-D_2}(HI) - \pi_{D_2}(NI) (3),$$

so there exists an equilibrium in region 4 iff $\gamma \leq -0,83$.

From inequality 1, there is no equilibrium in region 5.

Finally, from inequality (2), there is no equilibrium in region 6.
## 2.5. Conclusion

To sum up the results for the first case, proposition 1 in Colangelo's paper has to be corrected the following way: vertical integration occurs in equilibrium for $\gamma > -0.83$, while for $\gamma \le -0.83$, both vertical and horizontal integration can take place in equilibrium.

For $\gamma > -0.58$, the merger game does include the two firms, because both prefer winning to losing. Vertical integration occurs both because it is per se profitable and because it pre-empts horizontal integration. Even more, vertical integration occurs although it is per se less profitable than horizontal integration. Over this range, Colangelo's results are correct.

For $-0.83 < \gamma < -0.58$, the merger game is trivial. $U_1$ prefers integrating to losing or to no integration, while $D_2$ prefers losing to winning or to no integration. There is no pre-emptive motive for vertical integration, because $D_2$ has no incentive to merge.

For $\gamma < -0.83$, the merger game is a war of attrition. Both firms prefer losing to winning. However, they prefer integrating to staying at the statu quo, so that $U_1$ and $D_2$ are both ready to bid if their rival does not bid. There is no pre-emptive motive for vertical integration. On the contrary, vertical integration is less profitable for $U_1$ than horizontal integration by its rival.

# 3. Second case : Upstream duopoly and upstream target

Colangelo considers two models. In the first one, the bid loser must remain unintegrated. In the second one, it can make an offer to the fourth firm and merge with it, as long as this firm accepts the offer. We will consider here both games.

## 3.1. The bid loser must remain unintegrated

### 3.1.1. An initial remark on vertical foreclosure

Suppose $D_1$ wins the bidding game and integrates with $U_1$. One could think that $U_1$ will stop selling on the intermediate market because the integrated firm as an incentive to foreclose its downstream rival, $D_2$. This is what Colangelo supposes. However, as Hart&Tirole(1990) shows, this is not correct. In this (symmetrical) Bertrand framework, the integrated firm will not foreclose its rival. As a consequence, the equilibrium on the intermediate market and therefore on
the final market will remain unchanged. Vertical integration has no effect, except that it pre-empts horizontal integration.

Why doesn't the integrated firm foreclose its rival? Consider that it does. $U_2$ will sell to $D_2$ a quantity $q_{22}$ at price $p_2$. Now, the integrated firm $U_1 - D_1$ can increase its profit by offering $D_2$ the same quantity at a price slightly less than $p_2$. The integrated firm has an incentive to cut its rival's price, even if it finally leads to a lower profit. This is because, considering a possible deviation from foreclosure, the firm doesn't consider the final profit, as Colangelo does, but the immediate profit increase.

### 3.1.2. Resolution of the game

Calculating the profits, we find:

$$NI: \pi_{U_1}(NI) = \pi_{U_2}(NI) = 0; \pi_{D_1}(NI) = \pi_{D_2}(NI) = \frac{\alpha^2}{(2-\gamma)^2}$$

$$VI: \pi_{U_1-D_1}(VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{D_2}(VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{U_2}(VI) = 0$$

$$HI: \pi_{D_1}(HI) = \pi_{D_2}(HI) = \frac{\alpha^2}{4(2-\gamma)^2}; \pi_{U_1-U_2}(HI) = \frac{\alpha^2}{2(2-\gamma)(1-\gamma)}$$

The resolution of the game is similar to the resolution in the preceding section. One just has to change the expressions of profits and verify the existence conditions.

There exists equilibria such that $b(U_2) > b(D_1) \ge 0$ and only such equilibria, that is $U_2$ wins the bidding game, which is competitive for every $\gamma$.

### 3.1.3. Conclusion

No vertical integration takes place in equilibrium, the pre-emptive incitation is not strong enough to compensate the difference between the direct gains from horizontal and vertical integration. This is in contrast with Colangelo's result, which is not surprising because he considers vertical foreclosure, so that vertical integration profit is higher.

## 3.2. The bid loser can integrate with $D_2$

### 3.2.1. Post-merger reactive merging

Clearly, as vertical integration drives no benefit in itself, $U_2$ has no incentive to merge with $D_2$ after a $U_1 - D_1$ merger.
Suppose now that $D_1$ loses the bidding game. Staying unintegrated, it gains $\pi_{D_1}(HI)$. A $D_1-D_2$ merger would gain a profit $\pi_{D_1-D_2}(2HI)$ given by $\pi_{D_1-D_2}(2HI) = \frac{\alpha_2^2}{8(1-\gamma)}$. As $D_1$ must offer $D_2$ at least $\pi_{D_2}(HI)$ for it to accept being absorbed by $D_1$, integration drives for $D_1$ the net profit $\pi_{D_1-D_2}(2HI) - \pi_{D_2}(HI)$, which is positive for every $\gamma \in ]-1; 1[$. As a conclusion, $D_1$ always merges with $D_2$ if it loses the bidding game.

### 3.2.2. Resolution of the game

Determining the profits, we find :

$$NI : \pi_{U_2}(NI) = 0; \pi_{D_1}(NI) = \frac{\alpha^2}{(2-\gamma)^2}$$

$$2VI : \pi_{D_1}(2VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{U_2}(2VI) = 0$$

$$2HI : \pi_{D_1}(2HI) = \frac{\alpha^2}{8(1-\gamma)} - \frac{\alpha^2}{4(2-\gamma)^2}; \pi_{U_2}(2HI) = \frac{\alpha^2}{4(1-\gamma)}.$$

The opportunity for $D_1$ to merge with $D_2$ after $U_2$ integrates with $U_1$ reduces $U_2$'s gains from integration, but it also reduces the decrease in $D_1$'s profit induced by the merger, so that both $U_2$ and $D_1$ have less incentives to integrate with $U_1$ if the bid loser can merge with $D_2$.

Solving the game, we find that $U_2$ wins the bidding game, so that double horizontal integration occurs in equilibrium. For $\gamma > \sqrt{15}-3$, the game is trivial. For $\gamma < \sqrt{15}-3$, $D_1$ has an incentive to pre-empt horizontal integration, but this incentive is not strong enough to win the bidding game.

## 3.3. Conclusion

No vertical integration occurs in equilibrium in these two models.

## 4. Third case : Upstream duopoly and downstream target

We will not give here any details, as the resolution is similar to the precedent ones.
### 4.1. The bid loser must remain unintegrated

Profits are given by :

$$NI: \pi_{U_1}(NI) = \pi_{U_2}(NI) = 0; \pi_{D_1}(NI) = \pi_{D_2}(NI) = \frac{\alpha^2}{(2-\gamma)^2}$$

$$VI: \pi_{U_1-D_1}(VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{D_2}(VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{U_2}(VI) = 0$$

$$HI: \pi_{D_1-D_2}(HI) = \frac{\alpha^2}{4(1-\gamma)}; \pi_{U_1}(HI) = \pi_{U_2}(HI) = 0.$$

Note that $U_1$ is indifferent between no integration and horizontal integration. Thus, it has no pre-emptive incentive. Furthermore, $D_2$ is indifferent between no integration and vertical integration, because there is no foreclosure, so that neither $U_1$ nor $D_2$ has a pre-emptive incentive. The solution of the bidding game depends only on the direct gain from integration. For $\gamma < 2(\sqrt{2}-1) \approx 0,83$, $U_1$ gains more from integration than $D_2$, so that vertical integration occurs in equilibrium, while the reverse is true for $\gamma > 2(\sqrt{2}-1)$. Competition for integration does always occur, except for $\gamma = 0$, where $D_2$ is indifferent between no integration, vertical integration and horizontal integration.

### 4.2. The bid loser can integrate with $D_2$

If $U_1$ loses the initial bidding game, it will merge with $U_2$, making a strictly positive profit, while separate upstream firms make 0 profit. For this reason, there is no need to compensate $U_2$.

Profits are :

$$NI: \pi_{U_1}(NI) = 0; \pi_{D_2}(NI) = \frac{\alpha^2}{(2-\gamma)^2}$$

$$VI: \pi_{U_1}(VI) = \frac{\alpha^2}{(2-\gamma)^2}; \pi_{D_2}(VI) = \frac{\alpha^2}{(2-\gamma)^2}$$

$$HI: \pi_{U_1}(HI) = \frac{\alpha^2}{4(1-\gamma)}; \pi_{D_2}(HI) = \frac{\alpha^2}{8(1-\gamma)}.$$

$U_1$ always prefers losing the bidding game to winning. There is no pre-emptive incentive to vertical integration. $D_2$ prefers integrating to losing iff $\gamma > 2(\sqrt{2}-1)$. In equilibrium, double horizontal integration takes place over this range. The game is trivial. For $\gamma < 2(\sqrt{2}-1)$, $D_2$ prefers any structure to horizontal integration. As $U_1$ prefers integrating to staying at the statu quo, double vertical integration takes place in equilibrium. The game is a war of attrition.
4.3. Conclusion

Although vertical integration can occur in equilibrium, it never has a pre-emptive
nature.

5. Conclusion

Our conclusion is twofold. First, our conditions for equilibrium vertical integration differ from Colangelo's. This is because he ignores the possibility that a firm may prefer, over some range of γ, losing the bidding game to winning it and because Colangelo considers that a vertically integrated firm forecloses its downstream rival, which is not an equilibrium strategy.

Second, vertical integration doesn't necessary have a pre-emptive nature. On the contrary, in many situations, vertical integration occurs while the competitor has no incentive to merge (horizontally), so that there is no pre-emptive incentive to vertical integration.

Finally, pre-emptive vertical integration occurs only with an upstream monopoly, as long as goods are not too much complementary. With an upstream duopoly, pre-emptive vertical integration is never an equilibrium.

References

[1] Colangelo, G., 1995, "Vertical vs. horizontal integration : pre-emptive merg-
ing", Journal of Industrial Economics, Vol. XLIII, No. 3, pp. 323-337.

[2] Hart, O. and J. Tirole, 1990, "Vertical integration and market foreclosure", Brookings Papers on Economic Activity, Microeconomics, pp. 205-276.