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 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 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 lefthand 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 (n1) firms other than firm 1 and they all increase their output by one unit too, . The term in square brackets is then equal to (n1) + 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, profitmaximizing 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 lefthand side) with marginal cost (the right). Keen appears to think that marginal revenue is defined as the expression “,” 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 oldschool 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 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 (n1) firms, R, changes by (n1) units. Consider equation (2) again, but set 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. 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 secondyear 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 pigeonholed 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 nontechnical 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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I don’t think Keen explains himself well, but he often raises some good points, even amongst some mistakes or contradictions (which is true of economists generally). My take away from his work is that the MR of a competitive firms is also downward sloping so that there are no conditions for getting output beyond the monopoly solution unless you invoke the price taking assumption – which is nonsense because if any single firm changes its output the total output changes. That’s been his message about chopping up downward sloping lines and getting flat lines.
Anyway, my questions for you are
1. Doesn’t your equation (3) give the monopolist solution anyway?
2. You seem to agree that the assumption of pricetaking is nonsense – does that mean you agree that a finite number of firms with the same cost structure gets to the monopoly outcome?
3. Where does that leave the competitive market model as an analytical tool?
4. Should anyone draw welfare implications from a model with pricetaking assumptions?
5. Why is the competitive model taught as the fundamental economic model to most students? Indeed, it is a fundamental tool for assessing the impacts of taxes, quotas, subsidies and other market interference, even though it is explicitly a shortterm (some input fixed) model.
Hi Cameron,
1. No, the monopolist would set industry marginal revenue (P(Q)Q + P) equal to marginal cost, not firm 1’s marginal revenue. This is not a subtle difference. Please read the paragraph in the post above starting “Assuming conjectural variations…” or have a read of the little piece I wrote a long time ago (Debunking).
2. I absolutely do not agree that the assumption of price taking is nonsense. More precisely, I think there are some situations in which clearly we should ignore strategic interactions when modeling some phenomenon—that is, assume pricetaking behavior—and some situations in which it would be crazy to assume pricetaking behavior. It depends on what the analyst is modeling.3. The competitive model is an extremely useful tool. There are, however, many reasons why one would not want to use this model. None of them are the ridiculous and innumerate objection Keen invented.
4. Yes.
5. The competitive model is a cornerstone of economic analysis. Ignoring strategic interactions is where we should start when explaining basic concepts, including those such as you discuss along with interventions to address negative externalities, monopoly power, and other market failures. Even in beginning undergrad courses, students will shortly go on to discuss a variety of oligopolistic models and many other models which relax or otherwise change the assumptions of the Econ 101 competitive model.

1. No
2. No, it gives the Cournot solution.
3. As a useful approximation of a market with a large number of (relatively small) firms. In the most basic version of the model (linear demand curve, all firms have the same marginal costs) the approximation error is 1/n where n is the number of firms.
4. Sure, but we should remember to regard them as an approximation.
5. For exactly the same reason that when students first learn chemistry they learn that electrons have fixed orbits – it is a (relatively) harmless approximation that is much easier to understand and work with. I agree that this isn’t an ideal situation, but until our first year students have a better math background I don’t think that there is much that we can do about this. [I know of at least one reasonably prominent economist who thinks that we shouldn’t teach any economics in 1st year courses – students should have to take a year of math before they start econ]. 
Chris, I think you are missing one of Keen’s important points. If R is fixed, then q_i is essentially Q (or at least a change in q_i changes Q by the same amount). When q_i increases by 1, then Q increases by one. Only one firm can be the marginal firm that will produce the next unit of output, but this one is actually whichever firm changes output. Thus you have to imagine a bunch of firms looking at the market’s downward sloping demand curve and deciding to increase or decrease output by one unit from the competitive solution. That is what Keen’s simulations do (as far as I know).
And it makes perfect sense. At any point in time, a firm can decrease output and increase profits if all other firms do nothing. Correct?
What would you do if this was your business? Produce one more unit at rising costs and sell all your output at a lower price, or produce one less at lower costs and sell all your output at a higher price, even if it by chance that decision means more profits for your competitors as well?
Take an example: P(Q) = 10002Q, P'(Q)=2, Q=R+q_i
The firm’s MR in your equation (3) is 10004R4q_i – a marginal revenue curve with twice the slope of the market demand curve. If R is fixed, each firm would be independently better off shifting away from the competitive equilibrium. And there is no condition that would get them to increase output again.
Remember, each firm faces this same problem because whichever firm changes output become the marginal firm.

Cameron, it is true that, at the competitive output, any firm deviating by producing less would increase its own profits when the number of firms of finite. Evaluated at the Cournot equilibrium, however, any firm which changed its own output would decrease its profits. The Cournot outcome is not the same as the monopoly outcome, but converges to the competitive outcome as the number of firms grows without bound.
Work through your own example. Suppose there are n=2 firms. They have zero costs. What are equilibrium outputs and prices in the competitive case? The Cournot case? The collusive case? Are the latter two identical, as Keen claims? At the collusive outcome, do firms have incentive to “cheat” and produce more? It’s a little trickier, but then proceed to figure out what happens with n firms rather than two. The conclusion I present above, which is the same as countless thousands of textbooks and journal articles, is mathematically correct. The conclusion Keen comes to is simply wrong, as a matter of mathematics.
This is all very basic stuff that second or third year undergrads work through in gory detail. Please work through my post, or the 2002 article, if you’re still not clear on where Keen goes wrong.

You have to be very careful of those who claim that their presentation is; “the same as countless thousands of textbooks” but yet cite not one.
Obviously you can set up a profit curve, revenues – costs, and find its maximum by differentiating. The argument seems more to do with how the curve is constructed.
But in real life firms profits are determined not by a profit curve but by commercial and financial activity including exploiting copyright, gaining subsidies, favoured locations, greater inputs from lowwage economies, different levels of borrowing, etc etc. Within a society, the average may come close to a mathematical curve, but this may only apply to 2 or 3 firms. The rest, the vast majority, are behaving completely differently and consequently reporting all manner of profits results.

Keen claims that his results follow from textbook assumptions, and everyone but him has the math wrong.
OK. But if you gave maximum charity (rephrase Keen’s arguments to give him the best shot) then do you think he would be getting somewhere? “Tyrone Cowen” also thinks dynamics are more important http://marginalrevolution.com/marginalrevolution/2006/03/my_good_friend_.html (let’s ignore whether they are treated in the best economic literature and focus instead on what the average business or poli sci student comes out of the university thinking … or what a typical elected politician thinks … etc.)

isomorphismes: I don’t think Keen has any valid point regardless of how charitably his arguments are interpreted. His claims about both the model discussed above and the content of modern economic thought more generally are very wrong, certainly including his shockingly ignorant claim that economists don’t use dynamic models.
I don’t know why we would focus on what the average poli sci student or elected politician thinks about research methods in economics. Keen claims to “debunk” methods in contemporary economic research, not the beliefs of poli sci undergrads, etc.

Chris, when you’re saying that almost all modern macroeconomics is based on dynamic models, I guess you’re talking about DSGE?

Well, DSGE models are not dynamic models, it’s comparative statics. Imagine that I’m saying that Antient Roman citizens didn’t knew how to drive a car. Someone step up and say I’m an ignorant and that “cars” were very common in Ancient Roman Empire. Well, the guy is talking about a chariot and calls it a car but it’s not that I’m ignorant, that’s simply not what I meant by car and anyone who live in our time would understand what I was talking about. It seems that a lot of economists are just completely ignorant of system dynamic modelling and think they are building dynamic models when they are into intertemporal maximization. Here’s an economist who looks like a guy looking at a car and saying it’s absurd because there’s no horse to propel it! :
More sophisticated economists, like Paul Krugman for instance, seem to be aware of the issue :
“New Keynesian models are intertemporal maximization modified with sticky prices and a few other deviations (such as balancesheet constraints). Even ISLM loosely appeals to maximazation arguments to derive the slopes of the curves, while analyzing outcomes by comparing equilibria.
Why do things this way? Simplicity and clarity. In the fairly rational and more or less selfinterested; the qualifiers are complicated to model, so it makes sense to see what you can learn by dropping them. And dynamics are hard, whereas looking at the presumed end state of a dynamic process – an equilibrium – may tell you much of what you want to know.
…
What would a truly nonneoclassical economics look like? It would involve rejecting both the simplification of maximizing behavior, going for full behavioral, and rejecting the simplication of equilibrium, going for a dynamic story with no end state.”
http://krugman.blogs.nytimes.com/2012/08/28/neofightsslightlywonkishandvague/
And more Krugman :
“Will we then be ready to start making recordings for the Time Vault? Actually, no – and I think never. If there eventually is a true, integrated social science, it will still be a science of complex, nonlinear systems – systems that are chaotic in the technical sense, and hence not susceptible to detailed longrun forecasts. Think of weather forecasting: no matter how good the models get, we’re never going to be able to predict that a particular storm will hit Philadelphia in a particular week 20 years from now.”
http://www.guardian.co.uk/books/2012/dec/04/paulkrugmanasimoveconomics
It happens that it’s just what dynamic modelling is about…

Mathieu, a “dynamic model” is simply one that models how something changes over time. None of your quotes from Paul Krugman are in conflict with that definition. You are referring a particular type of dynamic model—apparently ones that exhibit nonlinear or endogenous disequilibrium dynamics—but not all dynamic models are of that type. There are many valid criticisms of DSGE models, but “they’re not dynamic models” is not among them.
Moreover, Steve Keen’s extremely ignorant claim that economists do not use dynamic models cannot be rescued by semantics. Look again at the piece I wrote about his paper a decade ago (http://chrisaulddotcom.files.wordpress.com/2012/04/debunk.pdf). Skip to the part addressing Keen’s claim economists don’t use dynamic models. In his paper, Keen proceeds to discuss what he views as the kind of dynamic model which economists ought to use. What he presents is just a standard intertemporal profit maximization problem, not some complex dynamical system. (He then proceeds to get the solution conceptually wrong, and does the math wrong even given his confused solution concept, but I digress.)

For people used to system dynamics, dynamics means more than simply something that changes over time, it rhymes with nonlinearities and endogenous disequilibrium. You may complain about words that mean different things for different people with different backgrounds but when you see people saying that economist don’t use dynamic models, it’s not a matter of ignorance, they simply look at a DSGE model and don’t see any dynamics in there, given their own vision of dynamics. I think Krugman understand this, as he says that dynamics is hard so we look at an equilibrium instead.
The model Keen present in his paper is meant to criticize the marshallian model of perfect competition and is uninteresting. The interesting part of Keen’s work is his macro modeling, which exhibit chaotic dynamics and altought still at an embryonic stage, is a good starting point to build genuine dynamic models. If you want to see what he thinks economist should do, that’s what it would be.

Mathieu, you ignored my point: when Steve Keen claimed that economists do not use dynamics, he then specifically used an example of a standard intertemporal optimization problem of the sort commonly taught to undergraduate economic students to illustrate the sort of dynamic model he asserted economists ought to be using. It is not tenable to claim he had some other meaning of “dynamic” in mind, he was quite explicit. It is a matter of ignorance, and it is usually a matter of ignorance when noneconomists make that claim. Many critics talk as if all of economic thought is contained in the intro chapters of 101 textbooks, which typically do not feature dynamic models.
If “dynamic” implied “endogenous disequilibrium” we wouldn’t need that phrase, and there are many canonical models across the sciences called “dynamic” which are not of sort to which you refer. Paul Krugman did not say or imply otherwise: “going for a dynamic story with no end state” does not imply that “stories with end states” are, therefore, static. Many of Krugman’s most famous papers study dynamic, equilibrium (and usually deterministic) models, explicitly described as “dynamic” models.

Chris, it sounds like you’re distorting what Keen has said. He’s been quite explicit that he regards DGSE modeling as nothing more than competitive statics. Using his “debunking the theory of the firm” paper to invalidate his criticisms of modeling using comparative statics makes it seem like you’ve got something ideological against him. And, for the record, quite a few of Keen’s criticisms are aimed at the way things are taught at the classroom level.

Kyle, when I say that Keen claims economists don’t use dynamic models, I am referring to Keen’s explicit and ridiculous claims that economists don’t use dynamic models. This has nothing to do with the mistaken claim that DSGE models are not dynamic—again, I am not a macroeconomist and I am not a fan of DSGE models, but DSGE models are in fact dynamic models. And yet again, after making the ridiculous claim that economists ignore dynamics, Keen went on to give an example of what he considers a dynamic model, which turns out to be a bogstandard intertemporal profit maximization problem of the sort routinely discussed in undergraduate economics classes. Here is one version of his claim:
It is feasible to see Sraffa’s critique as simply an attempt to take seriously the limitations which Jevons, Walras, Marshall and Clarke all acknowledged were endemic in using static methods to analyse what are clearly dynamic problems. Their defence for the use of static tools was the inherent difficulty of dynamic analysis, and the absence of suitable tools. No such defence is available to modern economics, since dynamics is now a far more developed field of analysis (in sciences other than economics), and so many tools exist to analyse dynamic systems dynamically. We can begin this process of recasting economics as a dynamic science by taking Sraffa’s critique to heart and drop altogether the neoclassical treatment of time. Our first step here should be to take time seriously by treating revenue and costs as functions of not just quantity, but also time.
Notice that is not a claim that DSGE models are not “really” dynamic, or something similar. It’s a sweeping claim that economists ignore dynamic modeling. In fact, explicitly dynamic methods to study dynamic problems have been very common in economics since the development of suitable analytic methods in the 1950s and 1960s, and much of modern economic theory and empirics is built on such dynamic models. The strained, semantic defence that Keen has something different in mind by the use of the word “dynamic” doesn’t fly, because the example he proceeds to give of a “dynamic” model is in fact a standard neoclassical model.
If Keen limited his attention to claims that Econ 101 is not taught the way he would like, I would probably have never written a word about his criticism. He claims to be destroying the foundations of modern mainstream research. I am not distorting what he claims. He is, on the other hand, horribly misrepresenting methods and conclusions in modern economic thought.

You are criticising Prof Steve Keen on the dynamic modelling issue, but like your mainstream peers, you do not really know about what a dynamic model means, since your understanding of a dynamic model corresponds to those useless macro models, (be it a DSGE model, endogenous growth model, OLG model, banking model, computable general equilibrium model) which are solved for one steady state using some parameters, then subjected to a productivity shock(!) or a monetary shock(!) and either solved for a second steady state, and the transition from one steady state to another is determined (if the model is deterministic and therefore the shock changes a parameter) or impulse response functions to temporary or permanent shocks to distributions or timeseries processes are drawn (if the model is stochastic). To mainstream macroeconomists and, as far as I can see, to you, this is a dynamic model.
If you ever speak to a physicist or a mathematician, I recommend you avoid calling such a set up dynamic, as they will probably laugh at you. This is transition from one static equilibrium to another one disguised as dynamic.
A truly dynamic model does not require exogenous shocks in order to generate psuedo dynamics from one steady state to another, as in mainstream macro models. A truly dynamic model for instance can be a system of differential equations (as in Prof Keen’s papers), which generates dataconsistent business cycle behaviour of an economy (troughs and peaks) and generates this business cycle behaviour internally beginning from a dataconsistent set of initial points and evolving with respect to that set of differential equations (like weather predictions or systems of differential equations simulated by engineers, which are all very sensitive to initial points), without some unexplained unknown “shock” which makes the economy to move from one steady state to another. Therefore, he is right, there is NO macro model I have come across which are truly dynamic. If you have a quick glance to Prof. Keen’s macro models, you will see that is exactly what his models do. You may like his models or not, but they are truly dynamic. So your mistake lies with the illdefinition of “dynamic” in economic terminology.

There seem to be a lot of anonymous commentators insistent on making the same wrong, desperate, purely semantic point, studiously refusing to acknowledge that point has been repeatedly answered.
Sakir, read my last response, to Kyle. You’ll note that, even if you ignore everything preceding it above, it addresses your claims directly.
Note: I won’t be approving more hostile posts repeating the same nonsense points.

Chris I much appreciate your work of explication. I have a question. I had assumed that Keene wrote dQ/dq= 1 since this was true by definition, insofar as the industry demand curve is the same as the demand curve facing each individual producer. This would also make dq/dQ=1 by definition rather than as a result of some behavioral assumption. Where am I going wrong?

Hi Larry. Remember Q is not, despite how Keen treats it, a parameter. It is the sum of all firms’ output, and each firm’s output is an endogenous outcome.
Consider the simplest interesting case, n=2. Then we can write,
dQ / dq1 = d(q1 + q2) / dq1 = 1 + 0 = 1.
This is not true by definition, it follows from the behavioral assumption that each each firm takes other firm’s outputs as fixed when making its own choice.
But what is this object:
d q_1 / d( q1 + q2)?
It’s undefined, since q1 is not a function of (q1+q2).
Paul Anglin has pointed out to me that Keen now has a working paper kicking around admitting this is an error (http://arxiv.org/abs/1101.3409, see section 4), yet he somehow manages to come to the same wrong conclusion he’s been foisting for 15 years, by clumsily yet again deriving what he calls the “Keen equilibrium,” or what everyone else has referred to as the “collusive equilibrium” for ages, and insisting that that is not what he’s doing.

quote
“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, ”is this true in the real world
I would assert that in low volume industrys, eg capital goods that have a cost >1,000 dollars us, and about 1,000 units/yr or less, that, n fact, the price you charge goes up with volume because you are changing and improving hte product continuouslymaybe this represents a small, special case that is roundoff error; I don’t know

I wonder whether Fig 3.1 (a valid market demand curve) in Keen’s Debunking Economics (Chapter 3, revised ed. 2011) is wrong or not. I understand that his arguments on the SonnenscheinMantelDebreu Theorem (SMDT) refer to the aggregate excess demand –Z(p). However, he is showing a graph of an aggregate demand function –D(p), in Fig 3.1. I argue that showing a graph of D(p) while discussing the features of Z(p) is wrong but my colleagues in the Economics Graduate program at UNAM (University of Mexico) insist that, given the SMDT result, graph 3.1 is
indeed right. Could you please have a hint on this, please? I teach
microeconomics at a graduate level and I just want to give an objective and
robust lecture to my students on this puzzle. Thanks. 
Would not Keen’s result (if true), also imply that the (many) consumers of a pure “public” good P1, would behave the same as would a single buyer of a “public” good P1 (who would rationally pay for the optimum amount of P1, since he only takes into account his own and the supplier’s reactions)? By extension, Keen’s result (if valid), must also apply to the case of an upwardsloping MES (Marginal expense of input) curve, faced equally by a single monopsonist or by a competitive set of suppliers seeking to buy the good X1, who can NEVER achieve a horizontal MES=S curve as idealized by Samuelson, et al. Then, by extension, the many buyers of a pure “public” good, P1, MUST behave the same as would a single purchaser of P1, etc. Hence, following Keen (2001), markets can supply “public” goods as well as private ones! QED.

As a noneconomist whose background is in physics, Keen actually gets a lot more right here than you’re giving him credit for. With respect, I think you’ve missed importantly the difference between a derivative and a partial derivative, and really that is the key to making this all make sense. You can also see more about the “total derivative,” which Keen actually displays a very good grasp of, here: https://en.wikipedia.org/wiki/Total_derivative
Also, if you’re wondering why Keen takes the derivative of profits with respect to Q rather than q_i, it’s because it stands as a marker for the entire system. In other words, it describes what happens to the profits at firm_i if it and/or any other firm changes its output, whereas the expression usually used here dpi/dq_i (which should really be a partial derivative) only describes what happens to profits at firm_i if only its own output changes and other outputs is held constant, an unrealistic situation. A firm is doing well if it’s maximizing profits with respect to changes in its own output, but it’s doing even better if it’s maximizing profits with respect to changes in the outputs of all the firms. Notice that this doesn’t have to imply that firms are either colluding or reacting to each other, just that they’re allowed to change their outputs.
While the traditional approach is to examine what one business should do if all other businesses do nothing, Keen’s approach is to examine what one business should do if it has a reasonable expectation that other businesses are doing something. My instinct, without having run the math, is that for markets where outputs change infrequently, we would expect to see results closer to the traditional approach, while for markets with frequent output changes, it should be closer to Keen’s equation. In the former, the assumption that other businesses are leaving their outputs constant is reasonably good. As a noneconomist, I couldn’t answer the question as to whether such things actually exist.
Also, your language in this post seems to imply that you think a derivative implies a relationship where the variable in the denominator is independent and the variable in the numerator is dependent, but that’s really not how derivatives work at all. And the part where you say that the derivative of dpi/dQ is undefined is very much incorrect: you can take derivatives with respect to any sufficiently wellbehaved variable, either partial derivatives or total derivatives. The answer may be a function, including of variables besides just the two mentioned, or it may be zero if there is no relationship between the two. There certainly is a relationship between Q and pi_i, although it’s not obvious at first glance what it might be until Keen walks us through it.

Sam,
You seem to have taken Keen at his word that there is a “traditional approach” which assumes “firms do nothing” in response to the actions of other firms and that approach is to be contrasted with a new approach invented by Steve Keen which is more realistic. That is simply not true, as you can easily verify for yourself.
Keen’s solution is a very confused exposition of a special case of a very traditional, in fact, archaic, approach (again, look up “conjectural variations”). And much like in other cases in which he claims to have invented something that is actually wellknown, Keen proceeds to try to solve his new invention himself and doesn’t get the right answer.With regard to your math: Of course I do not think and in no place did I imply that “a derivative implies a relationship where the variable in the denominator is independent and the variable in the numerator is dependent.” Taking derivatives with respect to Q is an illdefined operation because “what happens to the profits at firm_i if it and/or any other firm changes its output” depends on the distribution if that change in output, not just its magnitude (it’s not a “sufficiently wellbehaved variable”). Suppose profits_1 = [ q_1^2 – ln(q_2) ] and Q=(q_1 + q_2). What is d(profits_1)/dQ?
Differentiating with respect to Q does not “describe what happens to the profits at firm_i if it and/or any other firm changes its output.” If firm i changes its own output (and that’s what we should be modeling, as ownoutput is the firm’s choice variable) we want to evaluate d(profits_i)/dq_i. If we seek a Nash equilibrium, we set dq_j/dq_i = 0 forall j ~i when evaluating that expression, and then solve the resulting system of equations for an equilibrium. That is not the same as assuming that firms believe their competitors won’t make strategic responses (look up the concept of a “reaction function”). If we wanted to “describe what happens to the profits of firm_i if it and/or any other firm changes its output” we would write an expression for d(profits_i) letting the entire vector of outputs change. That is not the same as the derivative of firm i’s profits with respect to Q. The standard Cournot (1838, yes, that’s the year of publication) model, which is what this is despite Keen’s claims to have invented it, generates a position of the *system* which is an equilibrium in a formallydefined sense.
Finally, as I said above, none of this is new. You can find countless expositions of the conjectural variations model Keen claims to have invented in textbooks written 50 to 70 years ago, and many mathematically rigorous expositions of standard game theoretic representations of the modern model in any modern IO textbook (along with much more sophisticated dynamic and stochastic models, and with extensive empirical evidence). For example, consider this textbook’s exposition of these basic models, along with some pointers to some modern extensions and empirics:
http://www.econ.yale.edu/~steveb/Econ600/chapter2.pdf
Generally, I encourage you to go look this stuff up in credible sources rather than trying to learn economics from Steve Keen.






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