In 2002, I wrote a small piece noting that Steve Keen’s novel criticism of economics in his book *Debunking Economics* is simply wrong (Debunking Debunking Economics). Part of that novel criticism is Keen’s claim that the standard analysis of the competitive model is *mathematically* wrong, and if one does the math correctly, one finds that the competitive equilibrium and the collusive outcome are the same. Which is an extraordinary claim! Everyone has been just doing the math wrong for well over a century, and if we were to do the math correctly we’d find that all industry structures actually behave as if the industry were monopolized, under *textbook* assumptions. Again, it’s important to emphasize this isn’t an appeal to some more complex model, or to empirical evidence, or criticism of some unrealistic assumption in the standard model: Keen’s claim is that this theoretical result follows from textbook assumptions if one merely does the math correctly.

Over on Worthwhile Canadian Initiative, Nick Rowe made an attempt a few days ago to explain that an elaborated version of that argument which was published in the “Real World Economics Review” (paper) is wrong. The elaborated version includes an alleged proof of an assertion in the original paper. This post points out the conceptual and mathematical errors in that “proof.” These are errors in high school level mathematics and elementary microeconomics.

Looking back at what I wrote about Keen’s argument in 2002, I see I pitched it at too high a level. If you can follow my argument, you don’t need to read my piece to see for yourself that Steve Keen is just plain wrong. So I am going to attempt to write this post in such a way that anyone with a reasonable grasp of introductory calculus can follow along, even if you’ve never studied economics. I also think both Nick’s blog post and my previous piece make an error in possibly leaving the reader with the impression that Keen’s argument is correct if the competitive model is unrealistic or fails empirically, but that’s not the issue. Again, Keen claims that his results follow from textbook assumptions, and everyone but him has the *math* wrong.

A “competitive” firm in economic theory is one which takes prices as given, ignoring the effect of its own output on price. This is an *assumption*, not a result. Keen notes, correctly, that this assumption is false when there are a finite number of firms. Suppose demand is given by P(Q), where P is price and Q is the total output of all firms. Consider any one firm, which without loss of generality I will call firm 1 (same as firm \(i\) in Keen’s paper), let \(q_1\) denote that firm’s output, and let \(R(q_1)\) denote the total output of the rest of the the firms, which in general depends on \(q_1\). Then we have \(P(Q)=P(R+q_1)\), and, as Keen says, price must fall as \(q_1\) increases if we hold R constant, since \(P(\cdot)\) is by assumption decreasing in its argument.

Along with Keen, suppose firm 1 does not take price as given. Rather, firm 1 acts to maximize its own profits taking into account that it will fetch a lower price for each incremental unit it produces, holding constant the output of all other firms. If firm 1 produces \(q_1\) units, its revenues will be \(P(R(q_1)+q_1)q_1\), and its profits will then be

\(P(R(q_1) + q_1) q_1 – c(q_1), \>\>\> (1) \)

where \(c(q_1)\) is the cost of producing \(q_1\) units. What value of \(q_1\) maximizes firm 1’s profits? To find that, we find how much profits change as output changes, and find the maximum by setting that derivative to zero:

\(P'(R + q_1)[ R'(q_1) + 1]q_1 + P(R+q_1) – c'(q_1) = 0. \>\>\> (2)\)

If we hold other firms outputs constant, as Keen claims to do, \(R'(q_1)=0\) and the expression simplifies to

\(P'(Q)q_1 + P(Q) = c'(q_1), \>\>\> (3)\)

which is the textbook solution. “Marginal revenue” here means “how much does revenue change when \(q_1\) increases by one unit?” Note that the left-hand side is firm 1’s marginal revenue and the right is firm 1’s marginal cost, so the firm equates the two to maximize profits.

Steve Keen claims that that bit of math is wrong. He claims (page 62):

However, the individual firm’s profit is a function, not only of its own output, but of that of all other firms in the industry. This is true regardless of whether the firm reacts strategically to what other firms do, and regardless of whether it can control what other firms do. The objectively true profit maximum is therefore given by the zero of the total differential: the differential of the firm’s profit with respect to total industry output.

Let’s consider that claim. Yes, firm 1’s profits in equation (1) depend on firm 1’s own output and on the output of all other firms, R. No, that does not imply that we solve firm 1’s profit maximization problem by taking the derivative of equation (1) with respect to total output. And, no, the term “total derivative” does not mean “derivative with respect to a total.” This conceptual confusion then leads Keen to incoherent math: he takes the derivative of firm 1’s profits with respect to, in the notation here, \(Q = ( R + q_1 )\) (equation 0.4). That derivative isn’t defined because firm 1’s profits don’t depend solely on the sum of its own output and the output of all other firms.

The math Keen proceeds to do treats total output, \(Q\), as if it’s a parameter that affects all firms’ outputs. Instead of \(Q\) we could use some other symbol to denote this variable to highlight that it’s not really total output, but I will stick with \(Q\). Keen treats each firm’s output as depending on this parameter Q and on the output of all other firms, so we could write

\(q_1 = q_1( q_1(Q),…, q_n(Q), Q)\),

and likewise for all other firms’ outputs, to clarify what’s being assumed. Keen then asks what value of this parameter Q maximizes firm 1’s profits. Notice this problem has nothing to do with the problem we’re supposed to be considering: how does firm 1 set its own output to maximize its own profits?

The way Keen has set this up, as the parameter Q changes, a firm’s output changes for two reasons: there is a direct effect of Q on each firm’s output, and there is an indirect effect operating through the effect of Q on other firm’s outputs. Keen takes the derivative of firm 1’s profits with respect to this parameter Q. He claims to treat firms as atomistic, that is, they ignore the effect of their own outputs on other firm’s outputs, by setting the derivatives of all firms’ outputs with respect to all the other firms’ outputs to zero. But he sets the derivatives of all firms’ outputs with respect to the parameter Q to one. Since firm 1 is for some reason choosing this parameter Q, to increase its own output by one unit, it increases Q by one unit. When firm 1 increases Q by one unit, all other firms also increase their output by one unit. Keen claims repeatedly and explicitly that he assumes other firms do not respond to changes in firm 1’s output, but the math he actually does assumes otherwise.

Getting back to the problem Keen for some reason considers: How should firm 1 set Q to maximize its own profits? Take the derivative of firm 1’s profits (1) with respect to the parameter Q and set it to zero to find

\(P'( R + q_1 )[ dR/dQ + dq_1/dQ]q_1 + P(\cdot) – c'(q_1)dq_1/dQ=0.\)

Keen assumes that all firms including firm 1 increase their output by one unit when Q increases by one unit. Then trivially \(dq_1/dQ=1\), and since there are (n-1) firms other than firm 1 and they all increase their output by one unit too, \(dR/dQ = (n-1)\). The term in square brackets is then equal to (n-1) + 1 = n, and the equation above simplifies to

\(P'(Q)nq_1 + P(\cdot) = c'(q_1). \>\>\> (4) \)

That is Keen’s major result, equation (0.9). It differs from the textbook result, equation (2), in that the number of firms, \(n\), appears in the first term. That is, again, because as Q increases \(q_1\) and all other firms’ outputs increase at the same rate in the problem Keen solves. Firm 1 then must take into account that as it increases output, price will fall much more rapidly when all other firms respond by increasing their output than when all other firms’ outputs are fixed. Keen does not solve firm 1’s problem taking all other firm’s outputs as given.

Keen insists that, if we do the math correctly, profit-maximizing firms do not equate marginal revenue and marginal cost. But equation (4), which is, again, Keen’s solution, says that the firm sets Q to equate marginal revenue (the left-hand side) with marginal cost (the right). Keen appears to think that marginal revenue is defined as the expression “\(P'(Q)q_i + P\),” so whenever marginal revenue cannot be expressed in exactly that way, it’s not marginal revenue. All of the claims about marginal revenue not equalling marginal cost follow from that basic conceptual error. Generally, any optimization problem that can be expressed as maximizing (f(x) – g(x)) with respect to x has the property that f'(x)=g'(x) at an internal solution (assuming differentiability, etc, which Keen does), so marginal revenue equalling marginal cost is a very general condition. Keen thinks he’s arguing against the “neoclassical dogma” that equates marginal revenues and costs, but he’s actually arguing the sum rule of differentiation doesn’t hold.

We can also see that Keen implicitly assumes all firms react to changes in firm 1’s output by increasing their own output by the same amount by noting that that assumption is the same as an old-school approach to strategic interaction among firms called “conjectural variations” (Keen implies later in the paper, starting on page 74, that he invented this approach. It’s actually not just textbook, it’s outdated textbook, as it’s an approach which has been eclipsed). A “conjectural variation” of 1.0 means here that firm 1 assumes that all other firms will react to a change in \(q_1\) by changing their own outputs exactly as \(q_1\) changes: if firm 1 increases its output by one unit, it expects all other firms to also increase their output by one unit in response. So if \(q_1\) goes up by one unit, the output of the other (n-1) firms, R, changes by (n-1) units. Consider equation (2) again, but set \(R'(q_1) = (n-1)\) instead of zero to find

\(P'(Q)nq_1 + P(\cdot) – c'(q_1) = 0\),

which is exactly the same as equation (4), which, again, is the same as Keen’s equation 0.9.

Assuming conjectural variations of one is almost but not quite the same as simply assuming that firms collude. If firms collude, firm 1 would set its own output to maximize industry profits rather than its own profits, which entails setting industry marginal revenue rather than firm 1’s own marginal revenue equal to firm 1’s marginal cost. One sufficient condition for Keen’s problem to be exactly the same as assuming collusion is that we restrict attention to outcomes in which all firms produce the same amount. Call that amount q. Then firm 1’s profits can be expressed

\(P(nq)q – c(q),\)

and differentiating with respect to q gives

\(P'(nq)nq + P = P'(Q)Q + P = c'(q),\)

because total output Q is equal to nq. P'(Q)Q+P is industry marginal revenue, so this is exactly the same as simply finding the collusive outcome. Another way to see this is to note that if all firms produce the same output and have the same costs, then total profit is just n times the profit of any given firm, so maximizing any given firm’s profits is just maximizing (1/n) times total profits, so the solutions must be identical. This is just a clumsy way of solving the Econ 101 monopolist’s problem.

Steve Keen’s arguments are simply wrong. They cannot be rescued by any appeal to realism or empirical evidence, because he is arguing about math, not empirical implications, and he simply has the math wrong. It’s no surprise his paper was rejected at every reputable economics journal to which he sent it. And I am hardly the first person to point out that Keen seems to misunderstand very simple issues in basic mathematics and microeconomics. For example, here’s David Stern, writing in *Ecological Economics*—hardly a bastion of mainstream thought—about the versions of these arguments Keen puts forth to laymen in his book *Debunking Economics* (ungated .pdf),

However, despite containing much useful material the book is seriously flawed. Almost all the new criticisms of economics put forward by the author are wrong. While the author claims to know mathematics better than most economists, the mathematics in these arguments is incorrect. Some of these errors are glaring and will be apparent to anyone trained in basic calculus; others are more subtle and may not be picked up by people who have not taken advanced economics courses.

Steve Keen is offering just plain wrong arguments about very basic versions of very basic models taught to second-year undergraduates. I hope that people who take these arguments seriously attempt to reproduce Keen’s results, so they can demonstrate for themselves that Keen is wrong. If you can’t do basic calculus, consider this: it is either the case that Keen has made basic errors in basic math, or it is the case that hundreds of thousands of economists and mathematicians over many generations have all made basic errors in basic math. Which seems more likely?

I’ll close by noting that the professional literature on strategic interactions among firms, which falls under the field of economics called Industrial Organization, is highly technical and empirical. Mainstream economics, which has not been reasonably pigeon-holed as “neoclassical economics” for many decades, investigates the behavior of firms in uncertain, dynamic environments, typically using Bayesian game theory, often with a focus on when and how firms will be able to maintain implicit collusion (that is, keep prices high to make more money, at the expense of consumers). Over the last two decades researchers have developed structural microeconometric methods to deeply integrate these models with extensive empirical evidence on firms’ behavior (here’s a recent non-technical survey of this literature by Einav and Levin). Steve Keen does not mention any of this literature when he attacks mainstream economics. He limits attention to basic theory found in introductory textbooks, and his analysis of those models is just plain wrong.

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