Upstream horizontal mergers involving a vertically integrated firm

Abstract

We study upstream horizontal mergers when one of the merging parties is vertically integrated. Under observable contracting in the pre-merger case, we show that such type of mergers always harm consumers. However, under unobservable contracting in the pre-merger case, the input price may decrease and consumer surplus may increase as a result of the merger even in the absence of exogenous cost-synergies between merging firms. A necessary condition for this finding is that the unintegrated downstream firm is more cost-efficient than the downstream division of the integrated firm.

Appendix

Derivation of the properties described in ( 2 )

The last-stage subgame final-good outputs and prices as functions of input price are $\hat{q}_{i} = q_{i} \left( w \right)$, $\hat{p}_{i} = p_{i} \left( {\hat{q}_{i} ,\hat{q}_{j} } \right)$. As noted in Sect. 2.3, these equilibrium outcomes are the same irrespective of whether the merger occurs or not. We derive here their properties described in (2).

Note first that $\hat{q}_{1}$ depends on the input price only indirectly through $\hat{q}_{2}$, thus $\hat{q}_{1} = q_{1} \left( {\hat{q}_{2} } \right)$ and $d\hat{q}_{1} /dw = \left( {dq_{1} /dq_{2} } \right)\left( {d\hat{q}_{2} /dw} \right)$. Given strategic substitutability, it holds that $dq_{1} /dq_{2} < 0$. It is then straightforward that $d\hat{q}_{1} /dw$ and $d\hat{q}_{2} /dw$ have opposite signs. We show that $d\hat{q}_{2} /dw < 0$.

The last-stage subgame equilibrium final-good outputs $\hat{q}_{1}$ and $\hat{q}_{2}$ must satisfy the F.O.Cs in the downstream market, therefore (1) can be written as: $\hat{p}_{2} + \hat{q}_{2} \left( {\partial p_{2} /\partial q_{2} } \right) - w - c_{D2} = 0$. Using the implicit function theorem in the above expression, we obtain $d\hat{q}_{2} /dw = 1/\left( {\partial^{2} \pi_{D2} /\partial q_{2}^{2} } \right) < 0$, where the denominator is negative due to the assumption that S.O.Cs are satisfied. Thus, it holds that $d\hat{q}_{2} /dw < 0$ and $d\hat{q}_{1} /dw > 0$. Moreover, given that $\left| {dq_{1} /dq_{2} } \right| < 1$, it also holds that $d\hat{q}_{1} /dw < \left| {d\hat{q}_{2} /dw} \right|$. The last inequality implies that an increase in the input price decreases the total quantity supplied in the downstream market, i.e., $d\hat{Q}/dw < 0$.

Regarding the effect of $w$ on $\hat{p}_{2}$, we have that,

$$ \frac{{d\hat{p}_{2} }}{dw} = \frac{{\partial p_{2} }}{{\partial q_{2} }}\frac{{d\hat{q}_{2} }}{dw} + \frac{{\partial p_{2} }}{{\partial q_{1} }}\frac{{d\hat{q}_{1} }}{dw} = \frac{{d\hat{q}_{2} }}{dw}\left[ {\frac{{\partial p_{2} }}{{\partial q_{2} }} + \frac{{\partial p_{2} }}{{\partial q_{1} }}\frac{{dq_{1} }}{{dq_{2} }}} \right] > 0, $$

where the bracketed term in the last inequality is negative since $\left| {\partial p_{2} /\partial q_{2} } \right| > \left| {\partial p_{2} /\partial q_{1} } \right|$ and $\left| {dq_{1} /dq_{2} } \right| < 1$. Finally, regarding the effect of $w$ on $\hat{p}_{1}$, we have that,

$$ \frac{{d\hat{p}_{1} }}{dw} = \frac{{d\hat{q}_{2} }}{dw}\left[ {\frac{{\partial p_{1} }}{{\partial q_{2} }} + \frac{{\partial p_{1} }}{{\partial q_{1} }}\frac{{dq_{1} }}{{dq_{2} }}} \right] > 0. $$

An increase in $w$ affects indirectly $\hat{p}_{1}$ through $\hat{q}_{2}$ in two ways: On the one hand, a decrease in $\hat{q}_{2}$ increases $\hat{p}_{1}$—a second-order effect. On the other hand, a decrease in $\hat{q}_{2}$ leads to an increase in $\hat{q}_{1}$ which in turn decreases $\hat{p}_{1}$—a third order effect. It is natural to assume that the second-order effect is of greater importance than the third-order effect, implying $\hat{p}_{1}$ increases with $w$.□

Proof of Lemma 1

By making use of $d\hat{q}_{1} /dw = \left( {dq_{1} /dq_{2} } \right)\left( {d\hat{q}_{2} /dw} \right)$, the expression in (4) can be rewritten as:

$$ \frac{{d\hat{q}_{2} }}{dw}\left[ {(w - c_{U} ) + \hat{q}_{2} \frac{{\partial p_{2} }}{{\partial q_{1} }}\frac{{dq_{1} }}{{dq_{2} }}} \right] = 0. $$

Given that $d\hat{q}_{2} /dw \ne 0$, the bracketed term in the above expression must be equal to zero, which gives the expression in (5). It is then straightforward that the input price is always lower than upstream marginal cost. □

Proof of Lemma 2

By making use of $d\hat{q}_{1} /dw = \left( {dq_{1} /dq_{2} } \right)\left( {d\hat{q}_{2} /dw} \right)$ and $\partial p_{i} /\partial q_{j} = \partial p_{j} /\partial q_{i}$, the expression in (7) can be rewritten as:

$$ \frac{{d\hat{q}_{2} }}{dw}\left[ {(w - c_{U} ) + \frac{{\partial p_{i} }}{{\partial q_{j} }}\left( {\hat{q}_{1} + \hat{q}_{2} \frac{{dq_{1} }}{{dq_{2} }}} \right)} \right] = 0. $$

Given that $d\hat{q}_{2} /dw \ne 0$, the bracketed term in the above expression must be equal to zero, which gives the expression in (8). It is then straightforward that in order for the input price to be lower than upstream marginal cost, it must hold that

$$ \hat{q}_{1} + \hat{q}_{2} \frac{{dq_{1} }}{{dq_{2} }} < 0 \Rightarrow \frac{{\hat{q}_{1} }}{{\hat{q}_{2} }} < \left| {\frac{{dq_{1} }}{{dq_{2} }}} \right|. $$

Given that $\left| {dq_{1} /dq_{2} } \right| < 1$, in order for the above condition to be satisfied it must hold that $\hat{q}_{1} < \hat{q}_{2}$. For any given input price, and given that the downstream division of the merged firm obtains the input at marginal cost, the latter can be true only if $c_{D1} > c_{D2}$. □