Keen still keen on attacking standard micro (updated)

Steve Keen has a another paper out on Debunking the theory of the firm—a chronology. Its by Keen and Russell Standish and appeared in issue 53 of Real-world Economics Review (never heard of it).

Matt Nolan and myself have commented on Keen's analysis before, see here. Remember that Keen's claim is that the standard (textbook) 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. He argues that his results follow from standard textbook assumptions, and that all other economists have simply gotten the maths wrong (I'm not sure how likely this last bit is. Many of the economists who have gotten it wrong, starting from Cournot and Marshall, have been trained as mathematicians.).

Chris Auld has a new post up at ChrisAuld.com which notes that Steve Keen still butchering basic microeconomics. Chris writes,

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 in Keen’s paper), let denote that firm’s output, and let denote the total output of the rest of the the firms, which in general depends on . Then we have , and, as Keen says, price must fall as increases if we hold R constant, since P() 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 units, its revenues will be , and its profits will then be



where is the cost of producing units. What value of 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:




If we hold other firms outputs constant, as Keen claims to do, and the expression simplifies to



which is the textbook solution. “Marginal revenue” here means “how much does revenue change when 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, (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, , as if it’s a parameter that affects all firms’ outputs. Instead of we could use some other symbol to denote this variable to highlight that it’s not really total output, but I will stick with . Keen treats each firm’s output as depending on this parameter Q and on the output of all other firms, so we could write

,

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



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

.

That is Keen’s major result, equation (0.9). It differs from the textbook result, equation (2), in that the number of firms, , appears in the first term. That is, again, because as Q increases 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 firm will react to a change in by changing their own outputs exactly as 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 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

,

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



and differentiating with respect to q gives



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.

For a more advanced version of the argument see Chris's Debunking Debunking Economics.

Update: Tim Worstall comments on Keen's piece here and Nick Rowe comments here.

Auld keen on debunking Keen (updated)

Steven Keen is going around the country right now explaining the many, many evils, as he see them, of standard economics. That the majority of economists don't agree with Keen will not come as any surprise to most people. One person who has done a great service in debunking Keen's ideas on "Debunking Economics" is Professor Christopher Auld. Back in 2002 he wrote on article on "Debunking Debunking Economics". Unfortunately this article has not been available online for some time but now Professor Auld has kindly allowed me to make it available once more. A copy is available here for those interested.

For some local comments on Keen's ideas see Anti-Dismal here and here and TVHE here and here.

Update: Matt Nolan at the TVHE blog is Debunking Keen on Bernanke: The issue of debt deflation

Critiques of economics

In the comments section of a recent post at TVHE blog the question of Keen's critique of economics given in "Debunking Economics" is raised. Responses to Keen's argument have been made in a number of different places. As to local blogs, in the past, Matt Nolan has commended here and here and I have posted here, here and here. Canadian economist Chris Auld has a more detailed discussion in Auld, M.C., 2002. Debunking Debunking Economics. Working Paper, University of Calgary. Available at http://jerry.ss.ucalgary.ca/debunk.pdf.

An executive summary of a standard reply to Keen's arguments about models of firm behaviour would be that given by Schiffman (2004: 1909-1)

According to Auld, Keen is mistaken concerning the distinction between perfect competition and monopoly (or lack thereof—topic 3), and his perception that mainstream modeling ignores dynamics (topic 6). These errors, in Auld’s estimation, are caused by "either a lack of familiarity with the literature, conceptual errors, or both". As Auld shows, perfect competition can be rigorously derived as the limit of a model of imperfect competition (as the number of firms becomes large). Assume that each firm takes competitors’ outputs as given, but recognizes that it has some degree of market power (its own output influences the market price). The ratio of output under this form of imperfect competition to output under perfect competition is n/(n+1) (where n is the number of firms). When standard theory assumes that firms take prices as given, it is making an innocuous assumption; for example, an imperfectly competitive industry with 100 firms will produce slightly over 99% of the perfectly competitive output.

Another possible approach to rigorously deriving perfect competition is to assume there exists a continuum of firms. In such a situation , it is literally true that any firm can change its output without changing price, even when the market demand is smooth and downward-sloping. Aumann, R. (1964) ("Markets with a continuum of traders," Econornetrica 32:39-50) is a standard reference.

Overall Schiffman notes,

To summarize, Keen is correct that many issues that should be taught to students are not being taught. There is need for a book that introduces students to controversies in theory and methodology, on a level that is accessible to advanced undergraduates. Debunking Economics is, however, too biased to fulfill this need. If one wishes to advocate a reform of economics (and Keen may very well be correct that it is a necessity), one must provide a more nuanced, more accurate, and more up to date picture of its current state.

Deadweight losses 3 (updatred)

Over at The Standard Steve Keen responses to Matt Nolan's comment. In part Keen writes,

My critique of neoclassical pricing theory has been published in Physica A, long after that exchange with Auld. The maths passed the scrutiny of physicists, who leave economists in the dust when it comes to mathematical reasoning.

An appeal to authority argument. Most people will know what to do with this. But I'm not sure its even true given the number of economists who have graduate and PhD training in mathematics. Keen goes on,

Furthermore, though the Cournot-Nash is mathematically correct, the Nash equilibrium is meta-unstable: independent competitive behaviour will lead instrumental profit maximisers to diverge from it without collusion. The only way to maintain the equilibrium is to presume competitive firms have “perfect knowledge” of each other’s strategies, which makes a nonsense of the concept of competition to agree with.

Now I'm not sure what Keen means by "meta-unstable" but note that by definition a Cournot-Nash equilibrium is where the best response functions of the firms intersect, and thus no one has any incentive to change their behaviour given their (correct) beliefs about the other players behaviour. The collusive equilibrium, on the other hand, is off the best response functions of all players and thus all players have an incentive to change their behaviour.

As to the information requirement, players need to be able to form (correct in equilibrium) beliefs about the other players possible actions, that is, they know the other players best response functions. They do not need or have perfect knowledge about the actual strategy being played by the other players. Martin Osborne explains the belief formation:

On what basis can such a belief be formed? The assumption underlying the analysis in this chapter and the next two chapters is that each player's belief is derived from her past experience playing the game, and that this experience is sufficiently extensive that she knows how her opponents will behave. No one tells her the actions her opponents will choose, but her previous involvement in the game leads her to be sure of these actions.

(More details are provide in a later section of his book on the question of how a player's experience can lead them to the correct beliefs about the other players' actions.)

Note however that the information requirements are really no stronger than those needed for the perfect competition model or the monopoly model.

Keen continues in response to part of Matt Nolan's comment on my posting,

For those on this blog, what Matt wrote was:

“MR-MC = (n-1)/n * (P – MC)”

As perfect competition assumes “many firms” (read infinite) n-1 converges to n, implying that P=MR.”

This is the formula for an individual firm, not the economy as a whole. The convergence Matt notes applies because the firm output “q” in the MR formula for the single firm (MR(q)=P+q*dP/dQ) must go to zero if the number of firms in an industry goes to infinity.

But we are looking for the equilibrium output for a single firms thus q makes sense. The condition MR=MC is for a single firm, and industry output will be n times the individual firm output given that each firm is the same under perfect competition. Under a Cournot oligopoly the total equilibrium out will be Q^N= n/(n+1)(a-c/b). This goes to the competitive equilibrium as n gets large and if n=1 it equals the monopoly output. Inverse demand is P=a-bQ. If we think of the residual demand curve that a given firm faces after all other firms have produced their output, and let the firm act as a monopolist within this residual market, as in the Cournot model, the quantity they produce will decrease as more firms enter the market as the residual market will get smaller. But the firm will still be setting MR=MC no matter how small the residual market is and MR will not be zero. All seems to be working as it should.

Keen also says,

Perfect competition is and always has been a crock that has stopped economists from actually confronting the real world. Though I despair of ever getting neoclassical economists to realise this, I hope that non-believers can appreciate this and start to ignore the irrelevant theories of neoclassical economists.

Another possible approach to rigorously deriving perfect competition is to assume there exists a continuum of firms. In such a situation , it is literally true that any firm can change its output without changing price, even when the market demand is smooth and downward-sloping. Aumann, R. (1964) ("Markets with a continuum of traders," Econornetrica 32:39-50) is a standard reference.

Update: Matt Nolan discusses The basic frame of a firm: Cournot.

Deadweight losses 2

In a comment to my previous posting, Deadweight losses, Steve claims that Bastard's arguments are correct. Matt Nolan at TVHE replies to Steve and makes all the points I would have made. Thanks Matt.

Let me give you the executive summary of what I was going to say about Steve's more general arguments; actually the summary is due to Schiffman (2004: 1909-1)

According to Auld, Keen is mistaken concerning the distinction between perfect competition and monopoly (or lack thereof—topic 3), and his perception that mainstream modeling ignores dynamics (topic 6). These errors, in Auld’s estimation, are caused by "either a lack of familiarity with the literature, conceptual errors, or both". As Auld shows, perfect competition can be rigorously derived as the limit of a model of imperfect competition (as the number of firms becomes large). Assume that each firm takes competitors’ outputs as given, but recognizes that it has some degree of market power (its own output influences the market price). The ratio of output under this form of imperfect competition to output under perfect competition is n/(n+1) (where n is the number of firms). When standard theory assumes that firms take prices as given, it is making an innocuous assumption; for example, an imperfectly competitive industry with 100 firms will produce slightly over 99% of the perfectly competitive output.

The reference to Auld is Auld, M.C., 2002. Debunking Debunking Economics. Working Paper, University of Calgary. Available at http://jerry.ss.ucalgary.ca/debunk.pdf.

Deadweight losses (updated)

You do get some odd economic arguments from non-economist at times. The following comes from comments by Draco T Bastard on the posting Back to the future: electricity privatisation at The Standard:

The supposed dead weight loss that monopolies bring are actually brought about by people trying to maximize profit. This has been proved (google Steve Keen, economist). As the government doesn’t need to make a profit that dead weight loss doesn’t exist.

This is wrong for a number of reasons.

The reason for a deadweight loss is that output is below the perfectly competitive level of output, call it q(c). I am assuming that no one wants to produce at a level greater than q(c) to avoid outright losses. I will also assume here that the price is read off the demand curve so that all output is sold. The reason for output being below q(c) doesn’t matter. Whether the firm is maximizing profits or not there will still be a deadweight loss if output is below q(c). For example, assume that q(c) is 10 and the monopoly level of output, q(m), is 5. Then any level of output between 5 and 10 will not maximise profits for the monopolist but will result in a deadweight loss. So not maximising profits does not mean no deadweight loss. What results in no deadweight loss is producing the perfectly competitive level of output.

Note also that both monopolists and competitive firms maximise profits. But only one results in a deadweight loss. So having firms maximise profits doesn't mean there will be a deadweight loss.

Even state owned firms need to make (non-negative) profits, in that their revenues have to at least as large as their costs of production to avoid any subsidies, and their implied taxes.

Bastard goes on,

Monopolies are actually more efficient than competition due to several factors: Economies of scale and having only to deal with itself and the customer (rather than several independent competitors and the customer) being the most notable.

Now here I'm guessing (the "Economies of scale" bit) that Bastard is assuming a natural monopoly and thus you get the standard result that a single firm is the most efficient form of production. Note that the whole discussing is about electricity and I'm not sure that electricity production is a natural monopoly and thus the natural monopoly arguments don't apply. Natural oligopoly may be.

Update: Bastard goes on to say,

And my description of dead weight loss is spot on – it is profit.

No I'm not making this up!