In Part 5, we saw that for $$ \frac{d{\bf q}}{dt}={\bf F}\!\left({\bf q}\right) $$ the choice of $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}a-2q_1-q_2\\a-q_1-2q_2\end{array}\right] $$ led to a globally stable $$ \frac{d{\bf q}\!\left({a}/{3},{a}/{3}\right)}{dt}=0. $$ We could just as easily have chosen $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}0\\a-q_1-2q_2\end{array}\right] $$ with eigenvalues determined by $$ 0=\left|\begin{array}{cc}\lambda&0\\1&\lambda+2\end{array}\right|=\lambda\left(\lambda+2\right) $$ so that $\lambda\in\!\left\{-2,0\right\}$. The zero eigenvalue here means Jacobian alone does not guarantee a stable ${\bf q}$ in the sense that a small change in initial $q_1$ assures convergence to a

*different*${\bf q}$. In fact there are a family of stable output pairs at ${\bf q}=\left(q_1,{\left(a-q_1\right)}/{2}\right)$. If this leads– as outlined in the previous Part– to a price $P=a-q_1-{\left(a-q_1\right)}/{2}={\left(a-q_1\right)}/{2}$ and profits $$ \boldsymbol{\pi}=\frac{a-q_1}{2}\left(q_1,\frac{a-q_1}{2}\right) $$ then firm 1’s ideal staring point is $q_1={a}/{2}$. That yields a price $P={a}/{4}$ and profits of $\pi_1={a^2}/{8}$ where the old strategy yielded a price $P={a}/{3}$ and profits of only ${a^2}/{9}$.

The point is, even though both sets of strategies lead ${\bf q}$ to converge to a fixed point, the first pair of

*strategies*are not stable. If either firm follows the initial strategy, the other has incentive to respond by fixing production at $q={a}/{2}$. The

**do not form a competitive equilibrium. Firm 2 may now profit from a change of strategy, but that is well and truly beside the point that the initial pair of strategies are not competitive.**

*strategies*On the other hand, if both firms employ the strategy of fixing production at $q={a}/{3}$, then neither has incentive to pursue an alternative strategy. If firm 1 fixes production at $q_1={a}/{3}$ there is nothing firm 2 can do in any period to generate greater profits than it would receive by fixing its own production at $q_2={a}/{3}$. No matter how much weight is given to any period in calculating the firm’s objective function, it loses. Therefore, given the strategy of firm 1, firm 2 must reciprocate. Such a (symmetric!) pair of strategies is both competitive and stable– dynamically and strategically.

In this way, a firm’s strategy for profit-maximization depends critically on the strategies employed by its competition. If no firm has incentive to change strategy, only then can the set of strategies form a competitive equilibrium. Importantly, as my Comment and Technical Appendix demonstrate, the strategy laid out by Keen and Standish cannot form the basis for a competitive equilibrium. There must be firms with incentive to pursue higher profits and so there are only two possibilities. Either

- all firms collude by agreeing not to pursue any individually-superior strategy, or
- the proposed equilibrium must fall apart.

Read my original Comment (including Technical Appendix) at

*World Economic Review*.

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