Competition, Mergers, and R&D Diversity

Abstract

This paper describes a model of research and development (R&D) investment in which firms can choose any number of R&D projects that have independent and identical probabilities of success. The measure of R&D diversity is the number of projects that are undertaken by the industry. Absent spillovers or profits at risk from innovation, mergers often—but not always—decrease R&D diversity; however, the incremental effects decline rapidly with the number of industry rivals. Mergers can have significant adverse effects if the merging firms have large profits that are at risk from an innovation. A merger can promote investment in R&D and increase expected consumer surplus if discoveries have sufficiently large information spillovers.

Appendix: Proofs of Propositions

Proposition 1

Part (i): If the only payoff occurs if a firm is the sole successful inventor, then from Eq. (5),

$$- {\ln (q)}q^{n} q^{{n\left( {R - 1} \right)}} \pi (1) = k$$

for all n and R. Hence the equilibrium total investment nR and the probability of discovery Y(R) are independent of the number of firms R. This proves the first part of the theorem.

Part (ii): Let n be each firm’s profit-maximizing number of R&D projects in an industry with R firms and$n^{\prime}$be the profit-maximizing number of R&D projects with R − 1 firms. If we assume that$\pi (j) = 0$for$R > \bar{R}$, Eq. (5) requires that

$$\mathop \sum \limits_{j = 0}^{{\bar{R}}} \frac{{\left( {R - 1} \right)!}}{{j!\left( {R - 1 - j} \right)!}}q^{{n\left( {R - j} \right)}} \left( {1 - q^{n} } \right)^{j} \pi \left( {j + 1} \right) = \mathop \sum \limits_{j = 0}^{{\bar{R}}} \frac{{\left( {R - 2} \right)!}}{{j!\left( {R - 2 - j} \right)!}}q^{{n^{\prime}\left( {R - 1 - j} \right)}} \left( {1 - q^{{n^{\prime}}} } \right)^{j} \pi \left( {j + 1} \right).$$

(12)

Suppose$n^{\prime}(R - 1) \to nR$and therefore$n^{\prime} \to n$as$R \to \infty$. Then in the limit equation (12) becomes

$$\mathop \sum \limits_{j = 0}^{{\bar{R}}} \left( {\frac{j}{R - 1 - j}} \right)\left[ {\frac{{\left( {R - 2} \right)!}}{{j!\left( {R - 2 - j} \right)!}}q^{{n\left( {R - j} \right)}} \left( {1 - q^{n} } \right)^{j} \pi \left( {j + 1} \right)} \right].$$

(13)

This must be zero in the limit as$R \to \infty$because the first term in brackets approaches zero and the sum of the second term in brackets is finite. Therefore$n^{\prime } (R - 1) \to nR$in the limit as$R \to \infty.$

Proposition 2

Consider a reduction in rivalry from R = 2 to R = 1. If$\pi (j) = 0$for$j \ge 2$and$q \in \left( {0,1} \right)$, Eq. (4) implies

$$q^{{n^{\prime}}} \pi (1) = q^{2n} \pi (1) + q^{n} \pi (2).$$

If$\pi (1) > \pi (2) > 0$, this can be satisfied only if$n^{\prime} < 2n$. Therefore, a merger to monopoly decreases the probability of discovery.

Proposition 3

Without loss of generality, let$\pi (1) = 1 \;{\text{and}}\; \pi(2)=\alpha$. If$\pi (j) = 0$for j > 2, Eq. (5) requires

$$q^{nR} \left[ {1 + \left( {R - 1} \right)\alpha h\left( n \right)} \right] = z,$$

(14)

where

$$h\left( n \right) = \frac{{1 - q^{n} }}{{q^{n} }},$$

$$z = - \frac{k}{ln(q)} ,$$

and n is the equilibrium number of projects chosen by each firm in an industry with R rivals. Furthermore,

$$q^{nR} \left[ {1 + \left( {R - 1} \right)\alpha h\left( n \right)} \right] = q^{{n^{\prime}\left( {R + 1} \right)}} \left[ {1 + R\alpha h\left( {n^{\prime}} \right)} \right] ,$$

(15)

where n′ is the equilibrium number of projects chosen by each firm in an industry with R+1 rivals. In the limit as$q \to 0$, Eq (14) implies

$$q^{n} = \left( {\frac{z/\alpha }{R - 1}} \right)^{{\frac{1}{{\left( {R - 1} \right)}}}} ,$$

(16)

and

$$q^{{n^{\prime}}} = \left( {\frac{z/\alpha }{R}} \right)^{{\frac{1}{R}}} .$$

(17)

Equation (15) with$\gamma \equiv n^{\prime}\left( {R + 1} \right) - nR$is

$$1 + \alpha \left( {R - 1} \right)h\left( n \right) = q^{\gamma } \left[ {1 + \alpha Rh\left( {n^{\prime } } \right)} \right].$$

A sufficient condition for $\gamma < 0$ (which corresponds to Y(R + 1) < Y(R)) is

$$\left( {R - 1} \right)h\left( n \right) > Rh\left( {n^{\prime } } \right).$$

Substituting Eqs. (16) and (17), this condition is

$$\left( {\frac{z}{\alpha R}} \right)^{{\frac{1}{{R\left( {R - 1} \right)}}}} - \frac{1}{R}\left( {\frac{z}{\alpha R}} \right)^{{\frac{1}{{\left( {R - 1} \right)}}}} < \left( {\frac{R - 1}{R}} \right)^{{\frac{R}{R - 1}}} .$$

(18)

Inequality (18) is satisfied if z is sufficiently small, which corresponds to$q < \tilde{q}(R)$with$\tilde{q}(R) > 0$. In that case,$\gamma$must be negative, which implies that$n^{\prime}(R + 1) < nR$and$Y(R + 1) < Y\left( R \right)$.