## Thursday, July 16, 2015

### “Rationality” in the Theory of the Firm... Part 7

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6

In their response, Keen and Standish propose the following dynamics:
[A]round a Keen outcome ${\bf q}^K$, we get $$\frac{d{\bf q}}{dt}=\sum_j{\frac{\partial\pi_i}{\partial q_j}\left(q_j-q^K_j\right)}$$
Right off this math is confusing. Apparently, what they mean is $$\frac{dq_i}{dt}=\sum_j{\frac{\partial\pi_i}{\partial q_j}\left(q_j-q^K_j\right)}$$ This suggests that they have in mind $${\bf F}\!\left({\bf q}\right)\equiv\boldsymbol{\pi}\!\left({\bf q}\right)-\boldsymbol{\pi}\!\left({\bf q}^K\right)$$ which, by construction yields ${d{\bf q}}/{dt}=0$ when ${\bf q}={\bf q}^K$. This much is true no matter what ${\bf q}^K$ is chosen. It is also a very strange strategy for firms to employ. Firm $i$ expands only if it has profits in excess of its benchmark profits $\pi_i\!\left({\bf q}^K\right)$, and contracts only if it has profits below its benchmark. Clearly, the symmetric collusive oligopoly is not stable according to this dynamical system. If all firms start by underproduce by some tiny amount, then they will all reduce their outputs to zero.

If marginal costs are constant, then the eigenvalues are determined by the zeroes of $$\left|\begin{array}{cccc}\left(\lambda-P+\mathrm{MC}\right)-q_1P'&-q_1P'&\cdots&-q_1P'\\-q_2P'&\left(\lambda-P+\mathrm{MC}\right)-q_2P'&\cdots&-q_2P'\\\vdots&\vdots&\ddots&\vdots\\-q_nP'&-q_nP'&\cdots&\left(\lambda-P+\mathrm{MC}\right)-q_nP'\end{array}\right|$$ so $\lambda\in\left\{P-\mathrm{MC}+QP',P-\mathrm{MC}\right\}$. That is, profitability ($P>\mathrm{MC}$) implies instability.

Even if marginal costs are not constant, if any two firms have identical cost structures, there is no ${\bf q}$ which is both stable and symmetric. Stability is in general achieved when smaller firms are driven out of the industry and the market is monopolized by a single firm.

In Figure 1, we see 25 simulations of a duopoly ($n=2$) with costs and demand parameterized as in their earlier example. The initial output for each firm is randomly selected to be within 2 percent of the proposed equilibrium. Each simulation is seen as a line, with a marker at the end of each run– the ${\bf q}$ at time $t=0.1$ from simulation start.

Figure 1: Simulations of Keen-Standish Dynamics

Keen and Standish write
The condition describes not an equilibrium point, but rather an equilibrium manifold of constant total market production, which is stabilised by the agents ensuring that if any agent were to cause the system to stray from this manifold, then all agents will follow suit, causing that agent to not enjoy its advantage for long. Any rational agent will then return to the fold.
which is a really funny way of saying that they believe firms with higher initial production ruthlessly drive out the competition until the industry turns into a monopoly. Note that this is nothing like their earlier agent simulations in which equilibrium market shares are predictable without dependence on the initial firm choices. Keen and Standish may invoke (pdf) the movie War Games (“A strange game. The only winning move is not to play.”) but their response brings to mind Highlander. “There can be only one!”

However, the math in their response makes no sense to begin with, so the dynamical system Keen and Standish have in mind may be differ from the one analyzed above. Still, a Cournot-like strategy where firms move toward their best response to the current price such as $${\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}a-c\\a-c\end{array}\right]-\left[\begin{array}{cc}2b+d&b\\b&2b+d\end{array}\right]\left[\begin{array}{c}q_1\\q_2\end{array}\right]$$ shows globally stable dynamics. Figure 2 shows this strategy simulated with each firm’s output initialized randomly to within 50 percent of the Cournot-Nash result.

Figure 2: Simulation of a Cournot-like Dynamics

This alone does not imply that either dynamic system describes optimal firm behavior; such would require still an appropriately defined firm objective such as $$\Pi_i\!\left[{\bf q}\!\left(t\right)\right]=\int^\infty_0{\rho^t\pi_i\!\left[{\bf q}\!\left(t\right)\right]dt}$$ Rather, the examples demonstrate that Keen and Standish’s response respecting this thread is in fact math salad.

In Part 8, we will consider their response to my criticism of their computer simulations.

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