Merger policy in innovative industries

Abstract

We analyze optimal merger policy in R&D-intensive industries with product innovation aiming to improve the quality of products. Our results suggest that a permissive merger policy is rarely optimal in high-tech industries when the antitrust authority considers a welfare standard that balances the impact of mergers on consumers’ surplus and firms’ profits. In particular, relative to a benchmark where the effects from R&D are absent, we show that the optimal merger policy should not be substantially more permissive in the presence of those effects from R&D.

Appendix

1.1 Proof of Lemma 1

Part i): Note that ε(k,n,t) is strictly convex in k:

$$\frac{\partial^{2}\varepsilon (k,n,t)}{\partial k^{2}}= 6\frac{(t+1)^{2}}{\left( n-k+1\right)^{4}}>0. $$

Because ε(0,n,t)=0, the convexity of ε(k,n,t) with respect to k implies that there is, at most, a unique k ^∗(n,t)>0 satisfying Lemma 1.

Part ii): The critical k ^∗(n,t) is determined by the condition

$\varepsilon (k,n,t)\equiv (t+1)^{2}\left (\frac {1}{(n-k+1)^{2}}-(k+1)\frac {1 }{(n+1)^{2}}\right ) +k=0$. Therefore, $\frac {\partial \varepsilon }{\partial t}<0$ at k ^∗(n,t). Moreover, because ε(0,n,t)=0, the strict convexity of ε(k,n,t) which respect to k implies $\frac {\partial \varepsilon (k,n,t)}{\partial k}>0$ at k ^∗(n,t). Therefore, the proof is completed by noticing that $\frac {dk^{\ast }(n,t)}{dt }=$ $-\frac {\frac {\partial \varepsilon (k^{\ast },n,t)}{\partial t}}{\frac { \partial \varepsilon (k^{\ast },n,t,F)}{\partial k}}>0$.

1.2 Proof of Proposition 2 and calculation of β ⁰(n,t) in Eq. 15

For convenience, let us define the following monotonically increasing transformation of the original welfare function W:

$$ w(n,\gamma ,\beta )=\ln \left( W(n,\gamma ,\beta )\right)^{\frac{\gamma -2}{ \gamma }}. $$

(18)

The optimal value of n is given by

$$ \frac{\partial }{\partial n}\left( w(n,\gamma ,\beta )\right) =\frac{1}{n}- \frac{2}{n+1}+\frac{\gamma -2}{\gamma }\frac{\frac{\beta }{2}-\frac{2}{ \gamma }}{(\frac{\beta }{2}-\frac{2}{\gamma })n+1}=0. $$

(19)

This derivative is clearly increasing in β and γ. Moreover,

$$ \frac{\partial }{\partial n}\left( w(n,\gamma ,\beta )\right)\! =\!\frac{2\left( 4n-4n\gamma -n\beta \gamma +4n^{2}+\gamma^{2}-n\gamma^{2}+n\beta \gamma^{2}-n^{2}\beta \gamma \right) }{\left( 2\gamma -4n+n\beta \gamma \right) \left( n+1\right) \gamma n}, $$

(20)

whose sign is given by

$$ f(n,\gamma ,\beta )=4n-4n\gamma -n\beta \gamma +4n^{2}+\gamma^{2}-n\gamma^{2}+n\beta \gamma^{2}-n^{2}\beta \gamma . $$

(21)

This leads to

$$ \frac{\partial f(n,\gamma ,\beta )}{\partial n}=8n-4\gamma -\beta \gamma -2n\beta \gamma -\gamma^{2}+\beta \gamma^{2}+4. $$

(22)

Since this derivative is decreasing in n and f(0,γ,β) = γ ²>0, it follows that f(n,γ,β) is concave in n with just one positive root. Thus, the welfare function is increasing (decreasing) in n if n is lower (greater) than this root and, by the second-order condition, the optimal value of n is increasing in β.

Recall that the upper bound t of the number of active firms (the maximum number consistent with non-negative profits) is given by $t=\frac {\gamma }{2}$, which substituted into Eq. 20 gives:

$$ f(n,2t,\beta )=4n-8nt-2n\beta t+4n^{2}+4t^{2}-4nt^{2}+4n\beta t^{2}-2n^{2}\beta t=0, $$

(23)

and solving out for β gives rise to

$$ \beta^{0}(n,t)=\frac{2}{tn\left( 2t-n-1\right) }\left( 2nt-n-n^{2}-t^{2}+nt^{2}\right) , $$

(24)

where β ⁰(n,t) is defined as the critical value of β such that n is the optimal number of firms, given the upper bound t on the feasible number of firms.

According to our previous analysis, it turns out that β ⁰(n,t) increases with n. Furthermore, $\beta ^{0}(t,t)=2\frac {\left (t+1\right ) }{ t}\leq 3$ and $\beta ^{0}(t-1,t)=2\left (1-\frac {1}{t(t-1)}\right ) <2$. Therefore, if β≥2 and n<t then no merger can contribute to the welfare standard that arises from W.

1.3 Proof of Lemma 2

Part i): Note, first, that E(k,n,t) is strictly convex in k:

$$\begin{array}{@{}rcl@{}} \frac{\partial E(k,n,t)}{\partial k}&=&\frac{1}{\left( t-1\right) \left( n-k+1\right) }\left( \frac{2t(n-k)-1}{\left( n- k\right)^{2}}-1\right) \left( \frac{n-k}{\left( n-k+1\right)^{2}}\right)^{\frac{t}{t-1}}\\ &&-\frac{1}{ tn}\left( t-n\right) \left( \frac{n}{\left( n+1\right)^{2}}\right)^{\frac{t }{t-1}}, \end{array} $$

which is strictly increasing in k. To see this, note that $\left (\frac {2t(n-k)-1}{\left (n-k\right )^{2}}-1\right )$ is positive and strictly increasing in k, as well as $\frac {n-k}{\left (n-k+1\right )^{2}}$ and $\frac {1}{\left (t-1\right ) \left (n-k+1\right ) }$. Therefore, we have proved that E(k,n,t) is strictly convex in k. Because E(0,n,t)=0, the convexity of E(k,n,t) with respect to k implies that there is, at most, a unique k ^o(n,t)>0 satisfying Lemma 2.

Part ii): The critical k ^o(n,t) is determined by the condition E(k,n,t)=0, and the sign of E(k,n,t) is the same as that of $g(k,n,t)\equiv -\left (\frac {1}{\frac {k}{t-n}+1}\right ) \frac {(n-k)\left (k+1\right ) }{n}+\left (\frac {\left (n+1\right )^{2}\left (n-k\right ) }{n\left (n-k+1\right )^{2}}\right )^{\frac {t}{t-1}}$, which is strictly decreasing in t. Therefore, $\frac {\partial E(k,n,t)}{\partial t} <0$ at k ^o(n,t). Moreover, because E(0,n,t)=0, the strict convexity of E(k,n,t) which respect to k implies $\frac {\partial E(k,n,t)}{\partial k}>0$ at k ^o(n,t). Therefore, the proof is completed by noticing that $\frac {dk^{0}(n,t)}{dt}= -\frac {\frac {\partial E(k^{o},n,t)}{\partial t}}{\frac {\partial E(k^{o},n,t)}{\partial k}}>0$.

1.4 Critical β for the benchmark vs. the innovative industry

The critical β for the benchmark is:

$$\beta^{\ast }(n,t)=\frac{1}{n\left( t+1\right)^{2}}\left( \left( n+1\right)^{3}+\left( t+1\right)^{2}\left( n-1\right) \right) , $$

and for the innovative industry with a one-firm lag it is:

$$\beta^{0}(n-\!1,t)\,=\,\frac{2}{t(n\,-\,1)\left( 2t\,-\,n\right) }\left( 2t(n\,-\,1)-(n-1)-(n-1)^{2}-t^{2}+(n-1)t^{2}\right) . $$

By inspection of Tables 3 and 4 it is routinely checked that β ^∗(n,t)−β ⁰(n−1,t)>0 for t≤7. Therefore, if the welfare standard of the benchmark is used in the innovative industry, then the induced maximal discrepancy in the optimal number of firms is at most 1 firm. This confirms that merger policy should not be substantially more permissive in the innovative industry.