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## The real economy is never in equilibrium

from Philip George

What are vectors?

. . . we defined a vector as a quantity having both magnitude and direction and represented it by an arrow. This makes sense in Euclidean 3-dimensional space. But in higher dimensions the idea of direction is not intuitive and we need a more formal definition that is consistent with the definition in three dimensions. In mathematics, an object is defined as a vector if it is an element in a vector space. This seems a circular definition but the additional requirements make it clear why it is defined in this way. Thus, when a vector is multiplied by a scalar (a real number, for our purpose) the result must be an element of the vector space, i.e., another vector. And a vector added to another vector must also be a vector in that vector space.

Consider two 10-tuples of numbers, T1 = (t1, t2, t3, … , t10) and T1¢ = (t1¢, t2¢, t3¢,… , t10¢). Let these represent the temperatures at ten points along the lengths of two metal bars. Then it is obvious that these 10-tuples cannot be vectors because they fail the requirement of vector addition; it makes no sense to add t1 and t1¢ because, as shown in the section on temperature, adding temperatures is a meaningless operation.

Similarly, consider two 10-tuples of numbers P1 = (p1, p2, p3, … , p10) and P1¢ = (p1¢, p2¢, p3¢, … , p10¢). Let P1 be the prices of 10 goods which an individual consumes on Monday and P1¢ be the prices of the same goods which he consumes on Tuesday. Now it is meaningless to add the price of a good on Monday to the price of the good on Tuesday. Therefore, it is meaningless to add the elements of the two 10-tuples P1 and P1¢. Hence, the two 10-tuples cannot be vectors and, indeed, there cannot be such a thing as a price vector.

We are therefore forced to conclude that General Equilibrium Theory (GET), which in its modern version is nearly three quarters of a century old, is merely highfalutin nonsense.

A bit of history

Einstein described Gibbs as “the greatest mind in American history”. And later, when asked who were the most powerful thinkers he had known, Einstein said: “Lorenz”, adding, “I never met Willard Gibbs; perhaps had I done so, I might have placed him beside Lorenz.”

Gibbs’s influence on modern economics, especially on optimisation theory, is well known. That influence was through the effect on Samuelson, via Gibbs’s pupil, E.B. Wilson.

But Gibbs, along with Oliver Heaviside, was also the inventor of vectors, the subject matter of this paper. So, his influence runs through the other important strand of modern mathematical economics as well, mainly in proofs of general equilibrium, though the price vector has since ramified into other areas such as international trade theory.

Critics of mathematical economics would say that Gibbs was doubly unfortunate in his disciples, at one remove. But, of course, that was hardly his fault.

Samuelson must be blamed for erecting a huge mathematical edifice, without first ascertaining that the utility and profit functions are differentiable.

In GET, the error that Arrow and Debreu made was in blindly transferring mathematical ideas to the real world without first ascertaining that those ideas were transferable. It is significant that both of them were mathematicians who wandered into economics.

Conclusion

Debreu noted in his Nobel Prize lecture that the success of the mathematization of economic theory depended “on the fact that the commodity space has the structure of a real vector space”. We have shown that this is incorrect. The “price vector” is not a vector, and GET is therefore false. But we may go further and assert that not only was the proof incorrect, what was set out to be proved was not true in the first place. The real economy cannot be brought into equilibrium by adjusting prices. And indeed, the real economy is never in equilibrium.  http://www.paecon.net/PAEReview/issue101/George101.pdf

1. December 5, 2022 at 7:11 am

Vectors are defined coherently in GET, and addition and other algebraic operations are legitimately conducted. In classical and Marxian economics the practice is to pack them as rows and columns in matrices. The price vector is not an element of the commodity space. It is a non-negative vector that iterates as a member of a set until it settles at market clearing. Price vectors are not added. Differentiability of utility and profit functions and functions in other sciences, not excluding the physics of temperature, I daresay, is routinely assumed. On a personal note, I draw on the work of Jean-Pascal Bénassy, disciple of Debreu and active worker in the area of so-called disequilibrium economics as he responds to the “real economy” with books of startling slimness and elegance.

• December 6, 2022 at 2:18 am

In Debreu’s Nobel Prize lecture that I have quoted from there is a diagram in which the price vector is depicted as an arrow. So a “direction” is implied. One of the conditions an entity has to satisfy in order to be a vector is multiplication by a scalar including -1. The condition of multiplication by -1 is to ensure a reversal of direction. So something cannot be a vector if the possibility of its being negative is ruled out.

Yes, the differentiability of utility is assumed. That does not mean it is true. In previous papers on this site, Prof Jonathan Barzilai has shown why utility cannot be differentiated.

2. December 5, 2022 at 3:49 pm

3. December 5, 2022 at 9:33 pm

I have been opposed to General Equilibrium Theory (GET) since 1970’s. I have expressed this opinion several times in this blog page when it seemed opportune. But, I cannot approve the argument developed in Philip George’s paper. It is totally out of reasons.

I wonder why this paper has been accepted in Real-World Economics Review (RWER). Both referees and editors must be blamed. It is a great disgrace for RWER. I hope this was the worst paper ever published in RWER.

All physical quantities do not have the same property. There are two famous properties: intensive and extensive. (See “Intensive and extensive properties” in en.Wikipedia)

Mass and volume are extensive quantities. You can add those quantities (if they are of the same kind). Temperature, pressure, thermal or electric conductivity, and mass density are intensive quantities. Normally, we cannot add these quantities (even if they are quantities of the same kind).

N.B A special type of “addition” (parallel addition) is possible even for temperatures. See David Ellerman (2009) Series-Parallel Duality in the Mathematics of Appraisal. But I do not enter in this argument.

It is reasonable that Feynman was appalled by the question: “What is the total temperature of the stars seen by John and his father?” Physically speaking it was a meaningless question.

Philip George is right to claim that it is impossible to add temperatures. He is also right to claim that adding prices has no meaning. But this fact does no damage to the formulation of an equilibrium and to the proof for the existence of an equilibrium.

Philip George cannot know the difference of properties between baskets of goods and their prices or price vector. If we have two baskets of goods (say composed of ten kinds of goods), we can treat these baskets as vectors. We can add them. We can multiply them by a scalar.

Then, why do we not treat prices (or a set of prices) as a vector. Often, we speak of it as a vector. But, correctly speaking, it is not an ordinary vector, because it is only a representative of relative prices. Relative prices has a direction but no magnitude. Any set of prices that has the same direction (or lies in the half line that starts form the origin) is a representative of “relative prices”.

We often talk of price vector, but in fact we are talking of relative prices. Mathematically speaking, we are speaking of a point of projective space.

The fact that we are speaking of “relative prices” (or a half line) does not give any damage to the proof of existence of an equilibrium. Separation theorem may be easier to understand the situation. This is also a famous theorem that is often used to prove that an equilibrium point is Pareto optimum. The theorem is stated like this:

Given a closed, bounded, convex set C in a Euclidean N-space (or R^N) and a point P at the boundary, there exists a hyperplane H that passes the point P and one of two open half spaces that are separated by H contains no points of C. This hyperplane can be expressed as
H = { x | (x – P, v) > 0}
by using a vector v (a point of the dual space of R^N). One can replace v by β v for any positive β. One of these vector v is called normal vector to H or to C. In reality, v is a representative of the normal direction.

There is no trouble even if normal “vector” is not in fact a vector. The latter is only a representative of a normal “direction”. Even if Arrow or Debreu talk of equilibrium price “vector”, this kind of convention is commonly permitted, because it makes no difference for the substance of the proof (or mathematical formulations). If Philip George claims that he had discovered a fundamental flaw in Arrow and Debreu’s theorem, it only reveals he has no knowledge of mathematics.

As I have stated at the top of this post, this is a real scandal for RWER and for all heterodox economics. Geoffrey Hodgson once talked about the necessity of quality control for heterodox economics. The title of the book was “Is there a future for heterodox economics?” (Edward Elgar, 2019) I am very sad to know that the minimum level of quality control did not worked for RWER and for this blog site. If we stay at this level of arguments, there will be no future for heterodox economics. If I borrow the title of George’s paper, this is really a blunder to heterodox economics.

• December 8, 2022 at 1:51 pm

Let me change the last sentence. I was using the word “blunder” in a wrong way. I should have said that “This is really a giant damage to heterodox economics.”

4. December 6, 2022 at 2:28 am

A basket of goods is not a vector and hence cannot be treated as such. There is really nothing more to be said on the subject.

In my book (“Economics Redefined” https://www.amazon.com/dp/B098PYL8ZW) I have also showed that because it makes no sense to add the price of milk to the price of oil the application of fixed point theorems to economics is incorrect.

I also deal with the fallacy that if relative prices remain unchanged the economy is not affected.

• December 6, 2022 at 10:02 pm

Philip,
you are confusing the formal definition of vector and its conventional usage. A vector is defined as a point of a vector space that must satisfies a set of axioms. But it is customarily permitted to call an entity vector when the latter is an element of a subset of a vector space.

Take the case of baskets of goods. If there is a basket that contains 3 apples, 5 bananas, and 2 corns, we may write the basket as (3, 5, 2). If there is another basket that contains 2 apples, 2 bananas, and 3 corns, we write it (2, 2, 3). If we combine two baskets and make a third basket, it becomes basket of (5, 7, 5).

You cannot subtract basket (2, 2, 3) from basket (3, 5, 2), if we agree that a basket composed of (1, 2, -1) does not make sense (other interpretation is possible). However, it is the usual convention to call (3, 5, 2) a vector, because it is an element of the vector space
R^3 or the set of all triplets (x, y, z) where x, y, and z are real numbers. In other occasion, it is interpreted as a member of the vector space C^3, the set of all triplets (x, y, z) where x, y, and z are complex numbers.

When the unit price of an apple is 1 dollar, banana 50 pence, and corn 80 pence, it is permitted to write the situation as (1, 0.5, 0.8). This is often called a vector, but it does not mean it is an element of a vector space. It is simply used as a shorthand of “triplet of prices”.

These are all questions of mathematical conventions. The trouble of your argument is that you are claiming that you have discovered a “great blunder” to Arrow and Debreu’s theorem, i.e., a conceptual and mathematical error in the existence theorem of competitive equilibrium. You can argue as you have done in one of your past papers that Arrow and Debreu’s equilibrium state excludes involuntary unemployment by definition or by assumption. But what you are doing in your “blunder” paper (Real-World Economics Review, #101, 38-43, http://www.paecon.net/PAEReview/issue101/George.pdf) is the negation of a mathematical theorem (the existence of a competitive equilibrium). You argue that what is recollected as price vector in Arrow (or any others) is in reality not a vector. It is irrelevant to Arrow and Debreu’s theorem. It is sufficient for the theorem to prove that there exists a set of prices that satisfy the equilibrium conditions.

The proof of the existence of equilibrium point is logically perfect. It is as perfect as modern mathematics after 1950 are. (Do not say there may be some foundational problems as mathematics. They are different questions.) No economists do not add or subtract two equilibrium price vectors. Normally they are treated as a point of a half line and no more. I repeat. Almost all economists know that an equilibrium vector indicates “relative prices”. I know no case in which an economist of the first class of modern neoclassical economics has ever tried to add two “relative prices” as if they are vectors.

You are accusing neoclassical economics on a totally irrelevant and unjustified reason. It is an act of degrading all heterodox economics.

5. December 8, 2022 at 1:26 am

The proof of economic equilibrium applies to the real world and therefore cannot be a “mathematical” theorem. A mathematical theorem is true without any reference to the real world.

Debreu was a mathematician. If he intended to mean “sets” he would have used the word “sets”. Since he used the word “vector” it must be because he intended to mean “vectors”. In “Theory of Value” he defines both sets and vectors.

• December 8, 2022 at 2:06 am

6. December 8, 2022 at 3:59 am

I have posted a comment on December 7, 2022 at 1:04 am Reply. It is still waiting moderation. It is addressed to the editors of this Blog, of RWER and the referees. It is also addressed to all readers and supporters of this Real-World economics movement. As I claimed at the end of my comment above mentioned, George’s paper is really a great blunder, not to General Equilibrium Theory, but to Real-World Economics movement (and to heterodox economics in general). If we cannot treat this affair properly, the REW movement will be disgraced as a rabble of frustrated people.

7. December 8, 2022 at 9:49 am

It is easy to be overawed by the mathematics of General Equilibrium Theory and omit to use common sense.

What is general equilibrium? It is not equivalent to saying that everything that is bought is sold because that is a tautology. Something cannot be sold unless it is bought.

General equilibrium means that demand and supply can be made equal to each other by adjusting prices. But this works only in one direction.

The demand for a good can be reduced by raising prices. But a firm cannot cut the price of its goods until everything it produces is sold. There is a lower limit which is the cost of producing goods. It can sell its goods in a fire sale or junk them but thereafter it must reduce its output or go bankrupt. The quantity of goods sold is made equal to the quantity of goods bought by reducing the quantity of goods manufactured or ceasing manufacture altogether, not by adjusting prices. (The cost of production does not appear anywhere in general equilibrium. Also, as I have pointed out in http://www.paecon.net/PAEReview/issue99/George99.pdf it is not possible to calculate marginal costs unless fixed costs are ignored.)

There is also no guarantee that reducing the prices of goods will increase demand for them, as is obvious during recessions, a clear case of disequilibrium. When consumers have lost a large chunk of their net worth due to asset market crashes, their primary concern is to recover that net worth, and the rational way of doing so is by increasing saving ie. by reducing consumption. When firms cut the prices of their goods it does not stimulate consumption, as every recession shows.

• December 9, 2022 at 4:51 am

“Something cannot be sold unless it is bought.”

Do options markets relax this assumption, since you are buying and selling only the option to buy or sell some other good or service, and the price of the option fluctuates before that decision has to be made, allowing one to profit by buying and selling just the right to buy or sell other things?

“The demand for a good can be reduced by raising prices.”

Since you can buy fractional shares of Tesla, does this statement apply to stock markets?

“It can sell its goods in a fire sale or junk them but thereafter it must reduce its output or go bankrupt.”

Does this statement ignore financing options? If Lehman Brothers had gotten a loan from the Fed as AIG did, would either have gone bankrupt?

In other words, in a world where finance wags the real economy as a dog its tail, why ignore finance’s ability to relax the traditional constraints of mainstream economics?

8. December 8, 2022 at 1:44 pm

Philip,
you are changing your point of argument. Please withdraw what you claimed “a giant blunder” in the core of GET in your paper:
A giant blunder to the core of General Equilibrium Theory.

I have read and examined three other papers that you published in the RWER. Although I cannot say they contain full of insights and they are presented in a logically coherent way, they are permissible as along as your claims were not as exorbitant as the “Giant blunder” paper. Despite of many defects, they contained a bit of truth. The RWER can be as tolerant as to publish these papers to encourage posts and submissions of non-economists who have no academic training in economics. (If I were asked to write a referee report on your three papers, I must have been quite negative. I can point several flaws and distorted arguments, but I am not arguing them.)

To claim that you have discovered a giant blunder in the core of GET, you should doubt whether your “discovery” is really supportable and reasonable. You are now retreating your original contention in the “Giant blunder” paper. It only shows that you are feeling uneasy with your original argument. Do not try to cover up your own blunder.

9. December 9, 2022 at 1:01 am

Some people in RWER’s comment section take themselves (and their pet theories) far too seriously. Lars once posted nothing but a bit of humor, a comic titled “Economists Saving the World.” I read in a book long ago that humor often functions to lesson the shock of the unexpected impact of fact or truth, rigid underlying fact and ever-living flexible truth. Humor, for those who are blessed with its saving grace, helps them swiftly grasp–see the point and achieve insight–of the unexpected nature of a given situation be it fact or be it truth. The cartoon indeed is true to an important degree and reflects similar jokes about economists, like the one about the three scientists who were lost on a desert island with the only food being a bunch of cans that washed up on the beach. The engineer suggested they climb a tree and throw the cans down on the rocks of open them, while the physicist suggested the warm them in the fire until the heat opens the cans; alas the economists with perfect assurance of his assumptions said, “Let’s just assume we have a can opener.”
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But alas, for those like Shiozawa, lacking humor and full of self-confidence (to put it nicely), is simply unable to avail himself of the insight that such humor serves. Instead, he twists meanings and distorts what is being said for some self-serving purpose, often catastrophizing to turn a simple cartoon into something it is not while missing what it really is all about. Taking himself (and his pet theory) too seriously, he robs himself of basic insights into human nature and the reality we live within which others grasp quickly seeing the point being able to appreciate the insight, as one with a sense of humor so eloquently expressed:
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The cartoon is amusing and of course divisively so; but that surely tells us something doesn’t it? Anyone interested in economy for the purposes of human well being must be evidence oriented, and cannot work effectively without theory regarding how the world works. As Lars has pointed out many times, many economists lack adequate theory and have a reductive sense of what evidence is. We have a great deal of inadequate formal theory and (often) highly limited analytical statistics. Both tell us things about the world we don’t live in (in world x based on axioms a-z the following pertains; in restricted experiment f we observe y; in model T Y is dependent on… in exactly this place and time (maybe)). Argument and thus the sociology of knowledge are influenced by these and so “progress” is an odd term in economics… Sometimes the problem is maths and so more maths cannot be the answer, nor can mathematical argumentation be the appropriate medium… If we are dominated by our maths we are not thinking about its appropriateness… (Jamie, RWER, 1/19/2020)

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Nevertheless, Shiozawa went out of his way to distort the meaning of this eloquent and well-intentioned comment, eliding its intent and meaning altogether to twist words and distort meanings as he so frequently has done throughout his comments. He does this so frequently it is clearly a habitual way of communicating for him. It is narcissistic and lacks good will and fairness and that human decency and charity to accurately represent one’s fellows’ arguments. Of course, Jamie was kind enough to point out that ” Sometimes applies to both parts of the sentence,” while insightfully adding, “and you seem to be missing the point that maths is not always and everywhere the medium and motive force of knowledge, which is what is problematic regarding the discourse under consideration.” Something that a simple bit of humor teaches those who have two important traits: humility and sense of humor.
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Philip is just a new target of Shiozawa’s catastrophizing. I long ago concluded he does this to get attention. Who on earth does he think he is to demand Philip withdraw his paper or otherwise acquiesce his demands. And how judicious to believe or claim one paper on RWER threatens the entire future for heterodox economics. Someone needs a chill-pill and timeout, including me, for reading Shiozawa’s nonsense on stilts is a bit nauseating.
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So, it seems to me that it is time that RWER’s editors apply some degree of ethical standards to Shiozawa’s humorless and malicious accusations and catastrophizing that show a certain degree of contempt for the basic values of human decency when it comes to intellectual debate and discussion, the first rule of which is to accurately quote one’s fellow and don’t cut quotes off or twist their actual meanings, as he frequently does, to make something say another thing as pretext to his own self-serving arguments; and don’t willfully misrepresent or distort one’s fellow’s meanings by neglecting to accurately represent their argument.

• December 10, 2022 at 5:06 am

I cannot speak for Yoshinori Shiozawa’s (YS) sense of humor and so will confine myself to the topic at hand which is his repeated laborious tutorials directed at Philip George (PG). In a private communication, I advised him to give it up because PG has no understanding whatsoever of mathematical economics and GET and his reactions only underline that fact. I interpreted YS’s appeal to the editors of RWER as an expression of fraternal alarm and concern as a member of the heterodox economics community.

• December 10, 2022 at 7:19 am

I interpreted YS’s appeal to the editors of RWER as an expression of fraternal alarm and concern as a member of the heterodox economics community.

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Perhaps you are new to this blog and are not familiar with YS history of trolling Lars and attacking him personally. There is nothing fraternal in his ad hominem attacks on Lars (and others), and his catastrophizing while distorting the real meanings and substance of Lars arguments. I would play them all back here in quotes, but I doubt the editors would like it, and they are well aware, no doubt, already.
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Philip is just the latest target of Shiozawa’s “repeated laborious tutorials”

10. December 11, 2022 at 3:40 am

Philip, I managed to find your paper and read it. Are you familiar with Philip Mirowki’s “More Heat than Light”? I recommend reading chapter 5 if you are inclined. Your paper has led me to return to it. I will compare your book (which I have) with Mirowski, in due course. I am a low reader though …

11. December 15, 2022 at 1:57 am

I posted a long reply on December 7, 2022 at 1:04 am. It is a proposal to the editors of this blog page. It remains always hidden under “moderation”. Why isn’t it uncovered even after a week?

12. December 15, 2022 at 9:17 pm

Gentlemen, is this a kerfuffle or an imbroglio? That’s my attempt at humor and I hope Meta Capitalism gives me a little credit for that. I make this joke even though, or rather precisely because, I think the paper by Philip George is of real interest and deserves to be taken seriously. This is whether or not it offends certain research program sensibilities. Firstly, I don’t think heterodox economics should care about conventional economic sensibilities. I also don’t think Philip George’s argument can be easily dismissed.

Economics is a wicked problem. What are we dealing with when we deal with wicked problems? Modern Complex Systems science recognizes the seemingly irreducible complexity of the world. Here is an interesting view of the matter from the angle of the philosophy of physics and complex systems.

“One aspect of the variability (of complex systems theories and models) is the variety of complex systems phenomena engaged: in one application it may be counter-intuitive dynamics — such as
the survival of cooperation in a sea of cutthroat competition — in another, self-organisation — such as rhythmic entrainment among food-stressed slime mould amoebae — in still another the onset of chaos — such as in local climate fluctuations — and so on. Another aspect of the variability is that characterising complex system principles is often a ‘wicked’ problem where the key dynamics generating a phenomenon is itself a function of the application conditions.

To take a simple example, traffic jams on expressways may be caused by any of entry/exit rates, truck/car proportions, flow density, driver pathway correlations, etc. Moreover, the dynamics of jam formation for each of these conditions is significantly different. For instance, truck/car speed differential is important near lane-change originated jams but less important for high density braking-originated jams, and
unimportant for pathway convergence jams. So there is no usefully generalisable, detailed dynamical rules for traffic jam formation.” – “Introduction to philosophy of complex systems: A” by Cliff Hooker in the volume “Philosophy of Complex System” – Edited by Cliff Hooker.

If the formation of traffic jams is this complex how much more complex are economies, especially when we consider traffic jams, general congestion etc. affect economies along with a myriad of other factors? Note that we haven’t solved the problem of preventing traffic jams and their inefficiencies and is not conventional economics supposed to be about the efficient use of resources? That is to say we haven’t priced traffic jams correctly, probably because negative externalities are an endless set and at some point (usually very early because of special interest pleading) we stop pricing negative externalities.

To think that price theory alone and equilibriating with prices (market fundamentalism) can control for such features is an heroic assumption. But the greater heroic assumption is to think that an economy (in its entirety as a socioeconomy, in its aspect as a dissipative thermodynamic system out of equilibrium with its environment) can be described by an equilibrium model. It cannot be so described or modelled. When conventional economics began adopted GET or DSGE theory, even physics and certainly the biological sciences were abandoning the equilibrium paradigm outside the classical “simple” physics arena. Conventional economics and GET with their pretensions to scientific status are wholly called into doubt as a superseded paradigm.

“The study of simple physical systems of a few components and of many-component systems at or near stable equilibrium supported the idea that the paradigm of scientific understanding was linear causal analysis and reduction to linear causal mechanisms, with the real as what was invariant under symmetry groups (a formal stability) or invariant to small perturbations (dynamical stability). Paradigm cases
included 2-body solar system dynamics, engineering lever and circuit equations, equilibrium thermodynamics of gases.

“The philosophy of science evolved compatibly, focusing on determinism, universal a-temporal (hence condition-independent) causal laws, analysis into fundamental constituents then yielding bottom-up mechanical synthesis. To this was added a simple deductive model of explanation and prediction — deduction from theory plus initial conditions gives explanation after the event and prediction before it. Reduction to fundamental laws and separate contingent initial conditions became the basic explanatory requirement. This supports an ideal of scientific method as
logical inference: logical induction from the data to form the most probably correct theory, deduction from theory for prediction and explanation, and deduction from data that conflict with prediction to a falsification of the predicting theory, or other assumptions.” – Cliff Hooker op. cit.

“… by the late 1970’s it is clear in retrospect that science had begun
to pull together many of the major ideas and principles that would undermine the hegemony of the simple symmetry/ equilibrium orthodoxy. Instabilities were seen to play crucial roles in many real-life systems — they even conferred sometimes valuable properties on those systems, such as sensitivity to initial conditions and
structural lability in response. These instabilities broke symmetries and in doing so produced the only way to achieve more complex dynamical conditions. The phenomenon of deterministic chaos was not only surprising to many, to some extent it pulled apart determinism from analytic solutions, and so also from prediction, and
hence also pulled explanation apart from prediction. It also emphasised a principled, as opposed to a merely pragmatic, role for human finitude in understanding the world. The models of phase change especially, but also those of far-from equilibrium dynamical stability, created models of emergence with causal power (‘downward’ causality — see above) and hence difficulty for any straightforward
idea of reduction to components. And, although not appreciated until recently, they created an alternative paradigm for situation or condition-dependent, rather than universal, laws.” – Cliff Hooker op. cit.

Notice that hard science was just beginning to fully appreciate, in the 1970s, that “the hegemony of the simple symmetry/ equilibrium orthodoxy” was collapsing, just as the Arrow–Debreu model was erected assuming simple symmetry and equilibrium. The problem has to do with “axiomatisation” in science and in quasi-science and non-science. I can flesh out a simple discussion of this issue if I have piqued any interest.

But to finish here, I will state that I provisionally accept the likely correctness of the thesis in “The giant blunder at the heart of General Equilibrium Theory” by Philip George. The maths argument looks valid to me so far as I can understand it, which is not all the way I admit. I am not sure what Arrow–Debreu are doing in their theoretical dimensioned spaces. However, the Philosophy of Science (Complex Systems) considerations which I can understand from Hooker certainly do suggest Arrow–Debreu are in the wrong paradigm space.

• December 16, 2022 at 6:52 am

Nobody will disagree with you, Ikonoclast, least of all GET workers who have been appalled by the DSGE agenda. With giants like Werner Hildenbrand and Alan Kirman, the style has been to launch immanent and constructive critiques of GET resulting in the distribution of agents instead of the representative agent as a building block (WH), and emergence and novelty (AK). I mourned the passing away of Jacques Drèze a few months ago, GE pioneer of the T of nonclearing markets and the study of the inefficiencies of the equilibria.
Scholars, mainly in Europe, pursue the research programme unselfconsciously. Recently, I witnessed an exchange with a senior GE statesman, a contributor to chaos, Lionel McKenzie’s blue-eyed boy in his time. When confronted with the word heterodox, he asked, puzzled, “What is orthodox?”
Decades ago an entire issue of the Journal of Economic Theory was devoted to the themes you raise and more. All the authors had cut their teeth on Debreu’s Theory of Value. Economists were pushing the frontiers of nonlinear dynamics. A significant figure in the group was Jean-Michel Grandmont. His paper “On Endogenous Competitive Business Cycles” in Econometrica in 1985 was a faceoff with the best that Robert Lucas could offer. Alas, anti-science won. The advantage of mathematics, apart from the limited possibility of misunderstanding the language, is precision and rigor. ‘Complex’, for instance, can mean different things to different people. Thus, complex DYNAMICS from simple SYSTEMS.
All this might suggest dialogue between heterodoxy and orthodoxy. Roger Farmer has initiated a rapprochement with heterodox macroeconomics.

13. December 16, 2022 at 11:37 pm

“A qubit can be in a state of 1 or 0 or a superposition of both. Using linear algebra, the state of a qubit is described as a vector”

What’s the direction of a qubit?

• December 17, 2022 at 7:46 pm

Fools rush in, so I am rushing in to “answer” this. ;)

1. Isn’t it generally a big mistake to think that anything in the quantum world makes sense in terms of macro world analogies or homomorphisms?

2. Notwithstanding the above: Is the direction of a qubit its spin if it is an electron “holding” an information state? Is the direction of a qubit its wave orientation if it is a photon “holding” an information state?

As per:

“In quantum computing, a qubit (/ˈkjuːbɪt/) or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.” – Wikipedia.

• December 18, 2022 at 12:16 am

There is quantum macroeconomics pioneered by Bernard Schmitt and carried forward by Alvaro Cencini (His latest book should be out.), Claude Gnos, and other members of the ‘Dijon School’. I will be immodest and direct you to my article on the subject in the Review of Keynesian Studies, 2020, vol. 2.

14. December 17, 2022 at 10:33 am

As a “Capital as Power” theorist said of GET/DSGE, assuming there was a math error in it, “It’s like discovering a mathematical error when counting the number of angels sitting on a pinhead.”

Wikipedia tells us even orthodox economics is not monolithic on the issue of GET/ DSGE.

“Post-Keynesians reject the notions of macro-modelling typified by DSGE. They consider such attempts as “a chimera of authority,”… – Wikipedia.

15. December 19, 2022 at 7:10 pm

YS asked me to look at the article as a (somewhat pedantic) mathematician. There seems a possible interpretation of the conclusion that might be supportable, but I agree with YS in so far as it might be misleading if read without suitable clarification.

PG claims that Debreu thought that the ‘commodity space has the structure of a real vector space’. I do not know if this is a fair interpretation of Debreu and I haven’t checked GP’s critique, but it does seem important to note that the existence of a ‘price vector’ is questionable. Indeed, I can’t imagine what meaning I could give to a ‘price vector’ when the markets are closed or in free-fall, except where there is some kind of price stability, which rather begs the question.

The assertion that ‘the real economy cannot [ever? always?] be brought into equilibrium by adjusting prices’ seems a bit strong, but seems more credible to me than that one could always bring the economy into equilibrium simply by adjusting prices. More was done in 2008, and surely more needed to be done? (Even if what was actually done was not necessarily ‘optimal’.)

It is further claimed that ‘the real economy is never in equilibrium’. It seems to me that economists sometimes mean different things by ‘equilibrium’. It seem unlikely to me that any real economy is ever precisely at ‘the equilibrium point’ (whatever that might mean), but on the other hand there seems some merit in the view that it sometimes useful for specific purposes to imagine that an economy is ‘equilibriating’ and actually close to a (temporary not necessarily unique or ‘objective’) equilibrium.

What may have upset YS is the claim that ‘GTE is therefore false’. This is a clear non-sequitur: the most one could say is that the version of GTE as presented by PG is false. According to wikipedia there are many versions of GTE and PG has not debunked them all. But the key things is to characterise whatever equilibrium-like behaviour there may actually be, and to understand what may promote it or destabilise it.

My main concern in this is that economic theory seems in a muddle and would benefit from further clarity. I note some good points in the other comments, which I may ponder on, so PG’s article and YS’s reaction both have some merit in potentially helping to bring clarity. I do hope I am not confusing things too much.

• December 20, 2022 at 2:42 am

《the existence of a ‘price vector’ is questionable. Indeed, I can’t imagine what meaning I could give to a ‘price vector’ when the markets are closed or in free-fall, except where there is some kind of price stability, which rather begs the question.》

Why shouldn’t the same criticism apply to representing qubits as vectors, because a qubit can encode a superposition of all possible spin directions?

• December 22, 2022 at 3:20 pm

Qubit representations have more degrees of freedom, which is good, particularly if one wishes to ‘model’ the limitations of more conventional approaches. But any sort of state-space representation is questionable, and ultimately dependent on some sort of induction, which seems to preclude substantive innovation.

• December 20, 2022 at 10:01 am

Thank you, Dave Marsay! A wise man of Indian macro once advised us that the even basic math was best taught to economics students by mathematicians. General equilibrium is market clearing. Demand equals supply in all markets. The metaphor of the medieval fair was used with the entry of agents lugging their initial endowments of commodities and services and characterised by notional demands and supplies. You have unconsciously hit upon a well-known artifice in the account. Into this closed system, a deus ex machina had to be introduced to collate bids and offers. An auctioneer announced a series of price vectors, agents modified their plans accordingly, until the cry of a price vector at which transactions, purchases and sales, took place. The prices were relative prices or price ratios. One commodity was selected as a numéraire. That commodity was called money. While critics were quick to dismiss many aspects of the model including the putative tâtonnement process or “higgling” or “groping” in the market, as some have called it, that leads to equilibrium, diligent insiders set about silently, and obliviously, constructing non- tâtonnement models. Thus far for the existence of equilibrium.
Permit me to make a detour at this juncture. Debreu’s Theory of Value was published in 1959. Research lines have taken different directions since then. Mathematical sophistication has grown apace. Measure theory was introduced ages ago. As with all productive research programmes, progress was made through relaxing assumptions and extending results. I wonder, therefore, at the point of going for Debreu. Often, it seems to me, heterodox revolutionaries torch straw men.
I return to the theme, now with the stability of equilibrium. You, better than I, appreciate the distinction between local and global stability. When not at the intersection point of demand and supply functions, both quantities (Marshall) and prices (Walras) as you note adjust to restore equilibrium. The “free fall of prices” you refer to concern absolute prices or money prices and, especially in the case of the financial crisis of 2008, asset prices. The absence of money and finance in the DSGE model was exposed. Disciples were quick to respond with models with bank programs added to familiar maximands. The essentials in terms of production functions and preferences remained unchanged. Money and finance were superimpositions (not superpositions!) on the real.
It is here that the full force of heterodox theory of all stripes applies. The economy is a monetary production economy and banks, especially, are the medium through which agents solve for their demand and supply functions. I believe that PG and YS share this perspective.
You astutely mention other things. “Temporary equilibrium” is serious GET. Also, you suggest the multiplicity of equilibria and raise the issue of uniqueness but that is an engagement for another day.

• December 22, 2022 at 3:26 pm

As a non-economist, it seems to me that there there is much more scope for a ‘serious GET’ than ‘the received wisdom’. But first I think we need to establish some shared understanding of ‘mathematics’, ‘modelling’ etc., or else we are doomed to ‘talk past each other’.

• December 25, 2022 at 6:27 am

Why don’t Lars’s criticisms of economics apply to mathematics, as indicated by his recent series of combinatorics problems (on his site) whose correct answers Lars has modeled using implicit assumptions not always obvious to other puzzle solvers? Do “correct” math solutions similarly depend on an implied, fickle consensus on implicit and often inconsistent assumptions? In short how coercive do you require mathematics to be?

• December 26, 2022 at 2:36 pm

Thanks for drawing my attention to this. I have commented on the maths at https://djmarsay.wordpress.com/notes/puzzles/dining-puzzle/ .

If you are talking about some examples that are too often used in attempts to teach maths, then your remarks recall my own view as a school student. But ‘proper’ maths is not like this, and I think we would all prefer it if all examples were carefully vetted before use. But this is hard work, and even mathematicians are human, work under time-pressure etc.

Regards.

• December 27, 2022 at 12:24 am

Is there a Freudian slip in Dave’s linked blog entry? 《The answer is given as 54!2!=72.》Is 54! something like 2.3 x 10 to the 71st power? Writing about human errors, did Dave ironically make an error in quoting the math in Lars’s post?

Is “proper math” inconsistent and/or incomplete? Why doesn’t the Banach-Tarski paradox prove 1 = 2 and thus entail trivialism, due to explosion?

Why else is 0! = 1 unless you use sophistry to define away the inconsistent axiom that any number times zero equals zero?

How much coercion (social or otherwise) is required to cherry-pick away these objections to any claimed privileged status for “proper math” relative to applied math?

16. December 27, 2022 at 12:13 pm

1. Not a Freudian slip: Finger trouble, I’m afraid. (And maybe a new keyboard?)

2. I initially thought the https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox a bit of a red herring, as it is about physics rather than maths or economics. But there is a link. Many people have a common sense that every sub-set of space is measurable, which is much like the common sense view that every possibility has a measurable probability. So this an example of a paradox being simply a conflict between common sense and logic.

My own view is that if we are to apply logic to real-world problems we need to seek out all such paradoxes and seek to correct them, as Lars and you do. This cannot be done from mathematics alone but it seems to me that mathematics could help to identify such issues, rather then – as seems too often to be the case – to obfuscate them.

3. Proper applied math is proper math. The rest is what Keynes called pseudo-maths. But how to tell the difference? This matters in e.g. statistics, where practice too often seems to outrun logic.

I guess a key general point of Lars’ (following Boole and Keynes) is that probability theory is often applied improperly. There is a similar point to be made about equilibria, but in this case I just don’t see how anyone could think that economies were in equilibrium in anything other than a hand-wavy or very limited sense that would not justify the reliance that some economists seem to put on such a ‘fact’. But there is surely more that could be said.

17. January 4, 2023 at 5:43 am

Do vectors as used in natural language recognition have directions, or is direction just an artifact of representing physical forces with the more general vector? Is defining a vector as only that which can have direction, do you get bogged down in the mere physical too much?

“In one-hot encoding a symbol is represented by an array of mostly zeros, the same length of the vocabulary, with only a single element having a value of one. […] Heads-up, I’ll be using the terms “one-dimensional array” and “vector” interchangeably. Likewise with “two-dimensional array” and “matrix”.”

From https://e2eml.school/transformers.html “Transformers from Scratch”

Should natural language processing be forbidden from using vector math, because word vectors don’t have a meaningful direction?

• January 4, 2023 at 9:07 pm