“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.

“Rationality” in the Theory of the Firm... Part 1

Previously: Introduction

Turning now to Standish and Keen’s response, they first argue that I

completely failed to discuss... that the so-called profit-maximizing formula for an individual firm– of equating marginal cost and marginal revenue– provably does not maximize profits in any industry structure apart from monopoly.”

This is, to put it mildly, a not true. As I wrote in Section 5 (bold added)

There is no dispute as to whether or not profits would be higher at, say, the collusive result. Objectively, profits would be higher at that level than Cournot-Nash, and firms would be better off producing at that level. The neoclassical argument is that collusion is rational, but firms competing for the greatest profits will not forgo opportunities to increase their individual profits and so will over-produce (relative to the collusive level) even if that would result in lower profits for the industry on the whole. That is, competition hurts profits.

Far from discussing it, I address the specifics of this topic throughout my Comment. Each thread mustered in support this argument fails.

Returning to their response, they hide conceptual errors in their math, arguing

“Rationality” in the Theory of the Firm... Introduction

I have a Comment in the latest issue of World Economic Review. Please take a look.

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A little background. Early last year, I was asked to review a piece by Russell Standish and Steve Keen. Their piece was aimed at salvaging one of their critiques of neoclassical economics. Most significantly, Standish and Keen have made claims (my summary, quoted phrases theirs)

that the textbook model of perfect competition is “strictly false” in assuming the demand function has “dual [contradictory] properties” and thus the model contains a “fundamental flaw.”

Over the course of nearly a year and a half, correspondence with the authors led me to conclude that their critique is flawed beyond all repair. Though I made my concerns known to the editors, WER nevertheless deemed their paper worthy of publication without clarification– let alone correction of errors of fact. I am grateful that the editors permitted me a Comment to run alongside their paper, but it is unfortunate that they also allowed the authors to repeat again their nonsense without meaningfully addressing the concerns I lay out in the Comment. Indeed, the authors respond in part by asserting that I agree with them regarding a point upon which I spent considerable Comment space disagreeing.

I would, however, like to thank the editors of WER for attempting to facilitate a proper exchange. In particular, I thank John Harvey and Norbert Häring for their time. And finally, I thank Russell Standish for his own responsiveness. Standish left me an impression of being a solid fellow interested in dialogue despite the fact that at times I exposed to him my rawest frustration.

Before coming to the point, however, I would like to be up front regarding a very personal opinion. To put Steve Keen forward as a leading light and fail to address the fact that he either misunderstands or misrepresents first-semester undergraduate microeconomics an embarrassment to anyone who would make valid criticisms of the neoclassical orthodoxy. I believe that so long as this is so, it becomes all the more difficult to convince the mainstream that the heterodox community has anything important to say. And if it turns out that I have it all backwards and Keen is correct and I am wrong, then the shame is all the more mine. However, after the better part of a year and a half of trying to engage productively on these issues, I remain convinced of my position.

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I am not going to try here to summarize the state of the entire debate, except to reiterate that Keen misunderstands or misstates the theory he is critiquing. Specifically, he substitutes the generally-accepted definition of price-taking for his own. According to Keen, a price-taker accepts whatever price clears the market, given the industry level of production. This gives every firm at least some market power– by adjusting its own output, each firm has the ability to change (albeit indirectly) the price it receives for its production. Such market power runs contrary to the textbook model of perfect competition, in which firms are assumed to have no such market power. By the time firms decide how much to produce, the price they will receive is assumed to be known. Contra Keen, the price is at that point not a function of anything. Thus, Keen’s competitors become entirely indistinguishable from Cournot oligopolists. Little wonder, then, such imperfect competitors do not act like perfect competitors. In short, he uses a model of imperfect competition to argue that perfect competition is mathematically unsound. To paraphrase a certain irate Justice of the U.S. Supreme Court, this is the purest applesauce.

I am not going to lie. Further discussion can get very technical. Though I have already prepared a series of posts responding to their ongoing stream of nonsense, I am not sure what the level of my audience may turn out to be. That said, I will begin to address their response in Part 1.

Before I do, I would like to give shout-outs to folks who have tried to tackle this disaster before me. I think of folks like Donald Katzner and Paul Anglin and– surely– a veritable host of referees.

And of course, thanks to the woman who married me in the midst of all this and its many lost nights.

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

A Bit of Fun

This is a new one on me. Rogue started stalking my wife’s broccoli with garlic sauce. Nom?



Yes, nom. Weirdo.

Spreading Imports Thin Does Not Mean Exchange Rates Do Not Matter

I am in the middle of a Twitter debate with J.W. Mason. The starting point for the debate is an empirical paper suggesting that real currency depreciation does not increase real exports, but a real currency appreciation decreases real exports (pdf). Let us see if I can clarify my position that there is an implication that real currency depreciation leads to lower real imports.

Let us suppose there are 101 countries and everyone imports \$100 worth of goods from each of the 100 different partner countries, so that each country imports a total of \$10,000 worth of goods. Now suppose that my current depreciates, say, 10% so that everyone else’s currency appreciates 1% 0.1%. Suppose further that a 1% 0.1% appreciation reduces exports by 0.05%.

At first blush it seems that this implies a 0.05% reduction in my imports. After all, if every partner country reduces their exports to every other country by 0.05%, then my imports must fall by 0.05%– almost too small to measure.

But this ignores the fact that my partner countries exchange rates did not appreciate with each other. When each partner loses 0.05% of exports, that partner reduces exports to me by \$5 and exports to the rest of the world by \$0. Thus, my total imports from all countries falls by \$500, or 5% of my initial imports.

Which is to say, I do not understand the position that the reduction in real imports due to depreciation is “formally correct but practically and empirically irrelevant.”

Glossip v. Gross: Getting the Logic Right?

If we agreed that “no woman shall ever launch a nuclear weapon” does it then follow that it is okay for a man to do so? It seems that Scalia answers in the affirmative.

Scalia (pdf):

Mind you, not once in the history of the American Republic has this Court ever suggested the death penalty is categorically impermissible. The reason is obvious: It is impossible to hold unconstitutional that which the Constitution explicitly contemplates. The Fifth Amendment provides that “[n]o person shall be held to answer for a capital… crime, unless on a presentment or indictment of a Grand Jury,” and that no person shall be “deprived of life… without due process of law.”

He argues that because the Constitution says the government cannot execute unless if meets certain conditions, that the Constitution also permits it to execute so long as those conditions are met. I am willing to concede that the Founders thought that the death penalty was constitutional. And it would be one thing if the Constitution enumerated the power (”The government shall have the power to deprive a person of life withstanding due process of law.“

However, Scalia argues that the Constitution permits it to execute even if the government can find no way around some other constitutional barrier. There must always be a constitutional way for it to kill. From the Opinion of the Court (emphasis added):

Our decisions in this area have been animated in part by the recognition that because it is settled that capital punishment is constitutional, “[i]t necessarily follows that there must be a [constitutional] means of carrying it out.” Id., at 47. And because some risk of pain is inherent in any method of execution, we have held that the Constitution does not require the avoidance of all risk of pain. Ibid. After all, while most humans wish to die a painless death, many do not have that good fortune. Holding that the Eighth Amendment demands the elimination of essentially all risk of pain would effectively outlaw the death penalty altogether.

From Breyer’s dissent:

The relevant legal standard is the standard set forth in the Eighth Amendment. The Constitution there forbids the “inflict[ion]” of “cruel and unusual punishments.” Amdt. 8. The Court has recognized that a “claim that punishment is excessive is judged not by the standards that prevailed in 1685 when Lord Jeffreys presided over the ‘Bloody Assizes’ or when the Bill of Rights was adopted, but rather by those that currently prevail.” Atkins v. Virginia, 536 U. S. 304, 311 (2002). Indeed, the Constitu­tion prohibits various gruesome punishments that were common in Blackstone’s day. See 4 W. Blackstone, Com­mentaries on the Laws of England 369–370 (1769) (listing mutilation and dismembering, among other punishments).

Notice the conflicting arguments:

• Scalia: It is constitutional, therefore there must be a constitutional means of execution.
• Breyer: If there is no constitutional means, then it is unconstitutional.

Need I say that I find Breyer’s the more compelling argument? Regardless, if Scalia had not assumed his answer, he might have argued “If it is constitutional, then there must be a constitutional means.” But that would not differ in logic from Breyer’s argument.

I am not prepared to argue that the government is barred from exerting any inferred power which may come into conflict with any part of the Constitution. There are always questions of power and rights, but I find Scalia’s argument very strange.

What is the Conflict Between Uninformed Priors and Gambling?

Via Noah Smith at Twitter, Deborah G. Mayo asks “What do hard-nosed Bayesians like Gelman really mean by posterior probability?” In reading the piece, I grew concerned that Mayo left out something important to Andrew Gelman.

The following example pretty well describes the issue at hand.

Suppose I ask you show me a coin from your pocket and then ask you to flip it once. If it comes up heads, are you now willing to stake \$3 for a chance to win \$4 if among the next 1,000 flips at least 500 are heads? Or do you still think the coin is fair and you expect this bet on average would lose you \$1?

This illustrates the difference between an uninformed prior and a strong prior. If you began with a very weak prior– that is, prior to flipping the coin you believed the bias of the coin was equally likely to be always coming up heads as always coming up tails as anywhere in between– then you might take the bet. If on the other hand your experience is such that you do not frequently wind up with heavily biased coins in your own pocket then you might not think much of that first coin flip.

I think Gelman’s point is that however useful you may find it to employ an uninformative prior to communicate science, that does not mean your personal prior is necessarily uninformed. Therefore you might not make personal decisions based on a posterior derived from an uniformed prior– or, presumably, any prior much different than your own.

I am of two minds about the implications for reporting. My instinct is that biases should be made clear and so perhaps personal priors should be used in place of uniformed. On the other hand, a strong prior may greatly reduce the power of the study. If I believe that 999 of 1,000 coins yield tails 99 times out of 100, then a single flip of heads will do little to convince me that the coin is not biased toward tails. Is it really useful for me to report that I believe the coin is almost certainly biased toward tails? Is that science or opinion? It seems therefore that the likelihood– rather than the posterior– is the important scientific result of the study. In this case, the likelihood is identical to the posterior derived from the uniform prior, so there is no choice between these two.

To take an example from my own research, consider the audit of the April 14, 2013 Venezuelan election. There, a very extensive audit– 53 percent of more than 39,000 voting machines– turned up zero discrepancies between the numbers on the machines and paper ballots counted by hand. What conclusions may be drawn from this result? If you have uninformed priors, this seems overwhelmingly to suggest that the election was free of any meaningful audit-detectable fraud. If you believe strongly that the election results were fraudulent in a manner detectable by the audit, then it seems more sensible that the audit itself was a fraud. That is, your conclusion respecting the election result depends on your priors regarding possible fraud in the audit as well as the election. Scientifically, the important conclusion is that the audit results were not consistent with detectable fraud.¹

Coming back to the original posts, then, it seems to me that Mayo missed Gelman’s point that priors do matter. I think Gelman is suggesting that relatively uninformed priors are reasonable for the basis of scientific reporting; in going beyond a study the reader must construct their own posteriors. To whatever extent possible, apply your Bayesian inference to your own priors rather than allowing someone else to insert their own.

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1 To the point that a full audit of all voting machines would change nobody’s conclusions respecting the election results.^↩