Tuesday, July 14, 2015

“Rationality” in the Theory of the Firm... Part 5

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

In their response, Keen and Standish continue to obfuscate by abusing mathematical notation. They write
It is true that we haven’t specified dynamical equations for the Marshall model as the model is not sufficiently detailed to specify it completely, but suppose the dynamical equations are: $$ \frac{d{\bf q}}{dt}={\bf F}\!\left(\boldsymbol{\pi}\!\left({\bf q}\right)\right) $$ where ${\bf q}=\left(q_1,q_2,\ldots,q_n\right)$, $\boldsymbol{\pi}=\left(\pi_1,\pi_2,\ldots,\pi_n\right)$ and generally we denote vector quantities in bold face. ${\bf F}$ has to be a function of the profit vector $\boldsymbol{\pi}$ as the agents’ behavior is entirely determined by their profit seeking rationality.

[The above equation] has equilibria where the derivative $D_{\bf q}F=0$, or equivalently where ${\partial F_i}/{\partial q_j}=0$, $\forall i,j$. The chain rule is $$ D_{\bf q}F=D_\boldsymbol{\pi}{\bf F}\cdot D_{\bf q}\boldsymbol{\pi} $$ where $\cdot$ is the usual matrix multiplication.

I have to admit, I have no good idea what they mean by “the [Marshall] model is not sufficiently detailed to specify [dynamical equations] completely.” This is very strange, as I thought Keen and Standish had just insisted the current topic to be infinitely-repeated Cournot-Nash. Indeed, they again fix the number of firms, suggesting ongoing conflation of different models. Nevertheless, let is accept that the above describes the dynamics of some model or another.

Now, $F$ (presumably a synonym for ${\bf F}$) is a function of $\boldsymbol{\pi}$ alone, and it is not clear exactly what the operator $D_{\bf q}$ represents or what its zero signifies. However, any effort at understanding is waylaid by the claim that equilibria lay where ${\partial F_i}/{\partial q_j}=0$, $\forall i,j$. Again, this is always true because they appear to define $F_i$ not as a function of $q_i$ but only $\boldsymbol{\pi}$. So yes, it is true that ${\partial F_i}/{\partial q_j}=0$, $\forall i,j$ for any equilibrium, but it is just as true out of equilibrium.

What is more, it is not clear why ${\bf F}$ should be limited to a function of $\boldsymbol{\pi}$ rather than, say, a function of ${\bf q}$ directly. After all, if the firm knows only its profits and the profits of each of its competitors, how does indicate how best to change production? Keen and Standish simply churn out meaningless math salad. A useful dynamic analysis– if one were interested– might start from $$ \frac{d{\bf q}}{dt}={\bf F}\!\left({\bf q}\right) $$ and search for a stable equilibrium where ${\bf F}\!\left({\bf q}^{eq}\right)=0$ such that there all eigenvalues of the Jacobian ${\bf J}_{\bf F}\!\left({\bf q}^{eq}\right)$ have negative real parts. However, this gets us nowhere until ${\bf F}$ is known– and ${\bf F}$ is derived from the firm strategies. Take for example the Cournot duopoly with zero costs and an industry (inverse) demand curve $P\!\left(Q^d\right)=a-Q^d$. Now, if we take $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}a-2q_1-q_2\\a-q_1-2q_2\end{array}\right] $$ then $$ 0=\left|\begin{array}{cc}\lambda+2&1\\1&\lambda+2\end{array}\right|=\left(\lambda+2\right)^2-1 $$ implies eigenvalues $\lambda\in\!\left\{-3,-1\right\}$ regardless of ${\bf q}$. Thus, all zeroes of ${\bf F}$ are stable. Of course, ${\bf F}\!\left({a}/{3},{a}/{3}\right)=0$, confirming that these choices of strategy lead to the Cournot equilibrium.

Yet, even this analysis is of relatively little interest. This merely tells us that ${\bf q}$ stabilizes at the Cournot outcome if the firms choose to evolve their outputs in the manner described. Discovering this did not even require us to note the firms were Cournot oligopolists or what the demand curve looked like– except to confirm that the result happened to match the Cournot equilibrium. It tells us nothing about whether– given the strategy of one firm– the other firm’s strategy is optimal. Part 6 will investigate this question in more detail.

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

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