## Saturday, July 11, 2015

### “Rationality” in the Theory of the Firm... Part 2

Previously: Introduction; Part 1

Keen and Standish asserted that “regardless of market structure” the “Neoclassical pedagogy” holds that profit maximization requires firms to zero out $$\frac{\partial\pi_i}{\partial q_i}=\frac{\partial\!\left(P\!\left(Q\right)\cdot q_i\right)}{\partial q_i}-\frac{\partial\mathrm{TC}_i\!\left(q_i\right)}{\partial q_i}\label{eq:sk1}$$ where $P\!\left(Q\right)$ is inverse demand evaluated at quantity supplied. As discussed in the previous post, this is not generally true. It is, however, true for Cournot oligopolists.

The authors then respond
[I]n the interests of illustrating the crucial point Rosnick ignores, we provide a simple comparison of the standard “Neo-classical profit maximization” formula and the actual profit maximization formula in the case of $n$ identical firms in an industry.

Consider a linear demand curve $P\!\left(Q\right)=a-bQ$ and an industry with $n$ identical firms, where each firm has the identical cost function $\mathrm{TC}\!\left(q\right)=k+cq+dq^2/2$. Then the total revenue for an individual firm will be $\mathrm{TR}\!\left(q\right)=P\!\left(Q\right)q=aq-bnq^2$ and profit will be: $$\pi\!\left(q\right)=a\cdot q-b\cdot n\cdot q^2-\left(k+c\cdot q+\frac{1}{2}\cdot d\cdot q^2\right)$$
Obviously, this formula for profits is maximized when $q$ is chosen at the collusive level of output– a collusive oligopoly recognizing that if all firms produce the same amount then the single best choice of $q$ sets to zero $$a-2bnq-\left(c+dq\right)$$ or $$q=\frac{a-c}{2bn-d}$$ By contrast, the Cournot-Nash level of output is larger and leads to lower profits. To Keen and Standish, this represents a flaw in textbook theory. After all, if firms are assumed to “maximize profits” why do they fail to maximize profits? What Keen and Standish fail to grasp is the distinction between the perfectly rational strategy on the part of firms to maximize profits by colluding and the outcome of firms competing for the greatest individual profits.

The authors overarching claim is that firms will— contrary to Cournot— find the collusive level of output even while competing for the greatest individual profits. But their logic is flawed.

A Cournot oligopolist reasons,
If the rest of the industry happens to produce $Q^r$, I should produce $q$ such that $P\!\left(Q^r+q\right)$ yields a price-output combination that gives me the greatest profits.
The $n$-firm symmetric Cournot equilibrium is achieved when all firms happen find their optimal $q$ happens to be ${Q^r}/{\left(n-1\right)}$. In this, the Cournot oligopolist is prepared for anything, but finds itself operating at the same level as its competitors.

Keen and Standish turn this thinking around. Their firm reasons
If we are all symmetric, then whatever level of output I find optimal, so will my competitors. Therefore, I should select $q$ so that $P\!\left(nq\right)$ yields a price-output combination that gives me the greatest profits.
This results in the collusive level of production and higher profits, so it must be a superior strategy, right? The problem with this new line of thought is that Keen and Standish effectively end it there. This is not the end of the story. To see this, let us use their numerical example, but simplify by changing $k$, $c$, and $d$ all to zero. This reduces costs to zero, but everything still works out fine. Let us also make our lives a little easier by selecting $n=99$, $a=19,\!800$ and $b=0.5$. According to Keen and Standish, the firms divide the collusive level of output. This implies $q=200$, $Q=19,\!800$ and the price is $9,\!900$. Each firm receives $1,\!980,\!000$ in profits.

Now, every firm sees this coming, and according to Keen and Standish they all believe that their competitors will all select $q=200$. However, any firm which believes that their competitors will all select $q=200$ is doing itself a disservice by also selecting $q=200$. If a firm convinces itself utterly that the rest of the industry will produce $19,\!600$ units, then the firm ought to get the price down to $5,\!000$ by producing $10,\!000$ units all by itself. Such a firm would reap $50,\!000,\!000$ in profits.

Now, $50,\!000,\!000$ is quite a bit larger than $1,\!980,\!000$ so the firm has a tremendous opportunity for greater profits. Why then does the firm not pursue it? According to Keen and Standish, the firm reasons
If I see this amazing opportunity, then so must all my competitors. And if we each produce $10,\!000$ units then we will have to pay people to take the stuff off our hands and lose plenty of money. The logical thing to do is to close my eyes and pretend that the profit opportunity is not there and hope beyond all reason that nobody else decides to make a whole lot of money for themselves.
But the firm now again has convinced itself that its competitors will produce only $200$ unit each, which means that the profit opportunity still does very much exist. The firm must still choose to forgo this opportunity for higher profits.

It does not matter how the firm tries to reason it out. Once the firm believes that its 98 competitors will produce 19,600 units, then it follows that the rational firm believes it will maximize its own profits by producing 10,000 units.

Thus, the collusive outcome is not a competitive equilibrium. On the other hand, if each firm believes that all of its competitors will produce $q=396$ units each, then the firm‘s best option is to produce $396$ as well. Because every firm thus realizes that no firm has incentive to produce at anything other than $396$ units, the result is stable. This is, of course, the Cournot result.

To be continued in Part 3.

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