Wednesday, July 15, 2015

“Rationality” in the Theory of the Firm... Part 6

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5

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 strategies 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.

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.
Keen and Standish do, however, present a new set of strategies in their response. We turn to that new arrangement in Part 7.

Read my original Comment (including Technical Appendix) at World Economic Review.

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