Stability of the merger-to-monopoly and a core concept for partition function games

Abstract

This paper is concerned with an old question: Will oligopolistic firms have incentives to merge to monopoly and will the monopoly, if the firms indeed merge, be stable? To answer this question, I motivate and introduce a new core concept for a general partition function game and prove stability of the merger-to-monopoly by applying the new core concept, labelled the strong-core, to Cournot oligopoly modelled as a partition function game. The paper shows that the Cournot oligopoly with any finite number of homogeneous firms without capacity constraints admits a non-empty strong-core and so does the Cournot oligopoly of not necessarily homogeneous firms with capacity constraints that are equal to their “historical” outputs. These results imply that oligopolistic firms will have incentives to merge to monopoly both in the long- and short-run and if the firms indeed merge to monopoly, the merger-to-monopoly will be stable.

Appendix

Proof of the Lemma

Since each ${\pi }_{i}\left(.\right)$ is concave and continuous in ${q}_{1},\dots ,{q}_{n}$ and each ${A}_{i}$ is compact and convex, the game $\left(N,A,\pi \right)$ admits a Nash equilibrium $({\bar{q}}_{1},\dots ,{\bar{q}}_{n})$. Suppose contrary to the assertion that the game has another Nash equilibrium, say $({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n})$, and $({\bar{q}}_{1},\dots ,{\bar{q}}_{n})\ne ({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n}).$ Without loss of generality, let $\bar{q}=\sum_{i\in N}{\bar{q}}_{i}\ge {\sum }_{i\in N}{\overline{\overline q}}_{i}=\overline{\overline q}.$ Since $({\bar{q}}_{1},\dots ,{\bar{q}}_{n})\ne ({\overline{\overline q}}_{1},\dots ,{\overline{\overline q}}_{n}),$ ${\bar{q}}_{i}>{\overline{\overline q}}_{i}$ for at least one $i.$ Furthermore, ${p}^{{\prime}}\left(\overline{\overline q}\right){\overline{\overline q}}_{i}+p\left(\overline{\overline q}\right)>{p}^{{\prime}}\left(\overline{\overline q}\right){\bar{q}}_{i}+p\left(\overline{\overline q}\right)\ge {p}^{{\prime}}\left(\bar{q}\right){\bar{q}}_{i}+p\left(\bar{q}\right)$, since $\bar{q}\ge \overline{\overline q}$ and by assumption the marginal revenue of each firm is non-increasing with total demand $q.$ From the first order conditions for a Nash equilibrium ${c}_{i}^{{\prime}}\left({\overline{\overline q}}_{i}\right)={p}^{{\prime}}\left(\overline{\overline q}\right){\overline{\overline q}}_{i}+p\left(\overline{\overline q}\right)>{p}^{{\prime}}\left(\bar{q}\right){\bar{q}}_{i}+p\left(\bar{q}\right)={c}_{i}^{{\prime}}({\bar{q}}_{i})$ implying ${\bar{q}}_{i}<{\overline{\overline q}}_{i},$ which is a contradiction. $\square$