First, what are axioms? One thing that took me by surprise when I published my book in 1999 was that almost nobody in the general public knows what the word “axiomatic” means. Mention of my book’s title was met with blank stares and open derision. The blank stares I could understand, but derision? How mean! Eventually I came to realize that people think “axiomatic” is an empty, feel-good adjective, like “super-duper.” They were derisive because they thought that I was calling my ideas, effectively, the “super-duper theory of economics.” For example, Mike Montagne writes, “While the title [Aguilar] gives his book is no more than a name if it pretends the content is self evident (axiomatic), egos which pretend such things only prevail so long as shills and dupery.” Admittedly, such pretense would be kind of lame; however, that is not what the word means.
The Random House College Dictionary offers three definitions: “axiom, n, 1. a self-evident truth. 2. A universally accepted principle or rule. 3. Logic, Math, a proposition that is assumed without proof for the sake of studying the consequences that follow from it.” I employ definition #3. The problem with #1 is that, lacking a “burning bush” experience, nothing ever appears sufficiently self-evident. The problem with #2 is the same one encountered when ordering pizza: Everybody will go hungry if they must wait for universal agreement on which toppings they want.
One’s axioms are introduced at the beginning of one’s book and then all of the subsequent theorems are derived from the axioms. For example, Euclid claimed that, given a line and a point not on the line, there exists a unique line that passes through the point and is parallel to the given line. Lobachevski asserted that there is more than one parallel and Reimann that there are none. Thus, there are three geometries and, similarly, there may be more than one economics. I am, however, the only economist to ever precisely state my axioms and to base an entire theory on exactly those axioms and on nothing else.
Axiom #1: One’s value scale is totally (linearly) ordered.
An ordering is a way of comparing any two items in a set. For instance, if you were a grade-school teacher, you could order all of the children in your class by height. Asking them to form a line from the tallest to the shortest is unambiguous. There may be some squabbles as to which of two boys is taller, but the issue can easily be settled with a tape measure. If it cannot, then they are of equal height and (conceptually, if not literally) they occupy the same point on the line.
Intelligence is a different matter. What if one child is good at math, another at chess and a third excels in music? Which one is more intelligent? There is no tape measure that can resolve this dispute. Thus, intelligence is not an ordering, while height is. I claim that value is an ordering, like height. Specifically, it is transitive, reflexive and anti-symmetric.
But what does “total” mean? It means that any two items can be compared. An example of an ordering that is not necessarily total is subset. It can be total: If one’s sets are defined as everybody less than or equal to 5’0’’, everybody less than or equal to 5’1’’, etc, then all of those sets can be ordered by the concept of subset just as the children were by height. But what if one set is girls and one set is blondes? Not all girls are blonde, but neither are all blondes, girls – boys can have yellow hair too. So subset can be, but is not necessarily, total. Height is a total ordering and, I claim, so is value.
Axiom #2: Marginal (diminishing) utility has three characteristics.
Marginal and diminishing utility are the same thing. Also, utility and value are synonymous; early economists came from many different lands and never settled on a common lexicon.
Diminishing utility refers to how the value of an item depends on how many of them one already owns. For instance, if my only means of transportation around town is a bicycle, then obtaining a car is very important to me. But, if I already own a car, getting another one is not particularly important. It may be nice to have both a sedan and a pickup but, realistically, most people get by with just one vehicle; it is possible to strap cargo to the roof of a car and it is possible to squeeze one’s entire family into the front seat of a pickup. Rich people may have a whole stable of cars but, for most people, three is about the limit.
So we see that, for cars, utility diminishes very rapidly down to almost nothing for the third unit. For guys, the value of shoes diminishes about as fast as it does for cars. For women, the value of shoes diminishes less quickly, dropping to zero only after a dozen or more pairs are obtained. Imelda Marcos found value in her thousandth pair of shoes. How rapidly utility diminishes is described mathematically by diminishing utility.
I claim that diminishing utility is independent of first-unit demand. For instance, the same mathematical function can describe how rapidly utility diminishes for a man’s vehicles and for his shoes; both drop to a negligible amount relative to the first unit’s value after about three are obtained. The former has the same diminishing utility as the latter, only with more zeroes on the numbers.
I claim that diminishing utility is negative monotonic, which means that it always diminishes. The only thing that could conceivably go back up in value as one obtained more of it is an addictive drug, like crack. However, my theory describes only one moment at a time and addiction takes months and years to develop. The addict is not the same person that he was before he took up the pipe, both literally and figuratively. So, at no point in time, was my theory ever inapplicable to his valuation of narcotics. For most phenomena, this claim is uncontroversial.
I claim that there is a limit to how slowly utility can diminish. Non-mathematical readers do not need to know what an integral from zero to infinity is, only that its being finite implies that utility must diminish at a good clip. There is no phenomenon that is still valued even after millions and billions of units have been obtained.
Axiom #3: First-unit demand conforms to proportionate effect.
First-unit demand is demand for the first unit (the first car or the first pair of shoes) so, by axiom #2, it is independent of diminishing utility. Proportionate effect describes things that gradually increase over time, but as a proportion (percentage) of their current value. After many individuals have independently experienced such growth for a period of time, they are log-normally distributed.
When describing the current state of society, "many individuals" means everybody in society, each of whom grew to their current state independently, but in the way described by proportionate effect. When predicting one's future, "many individuals" means the many paths that one may follow as one is buffeted about by each day’s epsilon, εj. (Note: epsilon, ε, is a mathematical symbol denoting a small impulse, negligible by itself but influential in quantity.)
As Wikipedia puts it, "A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independent factors which are positive and close to 1. For example the long-term return rate on a stock investment can be considered to be the product of the daily return rates."
Height conforms to proportionate effect. Throughout one’s childhood, there are good days when one gets plenty to eat and there are bad days when one does not. But the effect on one’s height is proportionate to what one has already obtained. Big kids eat more on the good days than little kids because they have bigger stomachs. Height is not distributed normally, that is, it does not have a bell-shaped curve. The graph of the distribution of height rises quickly to a peak and then tapers off gradually to a long tail. This long-tailed distribution is called log-normal, though non-mathematical readers do not have to know what this word means, only that I claim that first-unit demand is described by it.
There is an easy plausibility argument for axiom #3: Interest is calculated as a percentage of the amount owed. Fixing this percentage (using the same percentage throughout the calculation) is a special case of the percentage varying slightly every day. Thus, the fact that people have calculated interest in this way throughout history, and done so unquestioningly, implies that they should accept my axiom as self-evident. There are other conceivable ways of calculating interest (which I consider in the appendix of my book) but, as far as I know, nobody has ever thought to even ask this question. Thus, axiom #3 actually meets the dictionary’s more restrictive second definition of axiom, “a universally accepted principle or rule.”
Ludwig von Mises was not a mathematician – he actually despised us – but, in his own vague way, he made the same claim about money. Money is valued today because it had value yesterday and it had value yesterday because it had value the day before. In this way, Mises’ Regression Theorem traces the value of money back to when it had value primarily for its use. But that is as far as he got; not being a mathematician, he had no way of knowing that the distribution of people’s valuations of their first unit of the monetary unit is log-normally distributed.
I went beyond Mises by recognizing that his Regression Theorem is not a theorem but an axiom and that it applies to all phenomena, not just money. When I wrote a letter of introduction to Murray Rothbard in 1993, I fully expected the Austrians to recognize that I was extending Mises’ rudimentary work and applaud me for doing so. How naïve! The Austrians have enough hatred in them to feed seven hells. To this day I am appalled by the ferocity of their attacks on me. Oderunt dum metuant!.
The Demand Distribution
With these three axioms it is possible to derive the equation for the demand distribution, c(m). In my Simplified Exposition I write:
The graph of the distribution of points of indifference, c(m), can be pictured as an aerial view of the people who value a phenomenon assembled along a line marked "money," where they are asked to stand by the number of monetary units that are equal to a unit of that phenomenon. If more than one person has the same valuation, they stand behind the corresponding number. The stock of that phenomenon naturally tends toward the high end, as anyone who possesses a unit of it who sees his neighbor to the right without one will sell it to him.
My theory relates stock and price, not supply and demand. The stock associated with a particular price is the integral of the demand distribution (the area under the graph) from that price out to infinity.
The image of the demand distribution as an aerial view is helpful. Initially, the stock is scattered throughout the distribution, but it migrates to the right due to the people towards the right side valuing it more and buying it from those to their left. In the same way that the water in a glass migrates towards the bottom as water molecules push air molecules aside, the phenomena all migrate towards those who value them.
A sale is like a water molecule pushing an air molecule aside as gravity draws it downward. When the glass is quiescent, there are no instances of water molecules pushing air molecules aside because the water has all already settled on the bottom. The water has a smooth surface that defines a sharp boundary between the water and the air. It is only if the glass is shaken that water molecules find themselves above air molecules and push them aside in their effort to re-fill the bottom of the glass.
The computer printout is of a quiescent market where nobody is selling anything to anybody. If the market has been shaken, possibly because some or all of the participants have come to the realization that their previous position was foolish, then there are people who possess some of the stock who do not want it as much as others who have none. Like the water molecules that suddenly found themselves above air molecules, a sale is made so the have-guy and the have-not-guy reverse roles. When the market is quiescent again, all the stock will have found its way to the right side of the demand distribution just as all the water will eventually find its way to the bottom of the glass.
There are two ways that my theory can be used for prediction:
First, it tells one what will happen to the market after it gets shaken up. For instance, suppose that some of the market participants come to the realization that their position is foolish and they do something about it: They move leftward on the graph. So the shape of the demand distribution changes, becoming fatter on the left and thinner on the right. This is like if one has a plastic water bottle and suddenly gives the bottom a squeeze, splashing water up and leaving a new bottle shape, one that is fatter on top and thinner on the bottom.
An axiomatic economist is not a visionary. I cannot predict that these people will suddenly have a forehead-slapping moment and decide to change their position. What I can predict, during the resulting turmoil, is where the price and stock will settle, provided that I can estimate the changes in the parameters that have occurred. In the same way, I can predict the new water level in the squeezed bottle if I can estimate the changes to its shape.
Second, it is possible to prove theorems that are always true but are not intuitive.
For instance, ever since the first caveman constructed a bow and arrow, people have known that arrows follow a curved path through the air. But it was not then and it is still not intuitive that this path is a parabola. (For simplicity, we will ignore air resistance in this discussion; inclusion just requires another axiom, that drag is proportional to the square of velocity.)
Newton's innovation was to state three axioms, his three laws of motion, which are intuitive and then to prove, using those axioms alone, theorems like the one about the parabolic path, which are not intuitive. Using logic alone, he was able to go on to describe things completely outside our intuition, like the elliptic path of missiles that achieve orbit or the hyperbolic path of missiles that escape into outer space.
This is also a type of prediction. Based on a theory that had been worked out entirely in the form of algebraic equations, and which had been confirmed only in the easily observed case of parabolic trajectories, Victorian-era scientists were able to predict the path of missiles that would not be built for another hundred years, not until chemists came up with fuels powerful enough to propel ships into orbit or beyond.
Similarly, I have stated three axioms which are intuitive and, based only on those axioms, I have proven theorems that always hold true and yet are not at all intuitive. Specifically, I proved the Law of Price Adjustment. But readers will have to consult my Simplified Exposition to see what the Law of Price Adjustment is about. It is too complicated to explain in a page advertised as being non-mathematical.
On page 95 of Axiomatic Theory of Economics, I write:
250 years ago, when economics was first studied systematically, supply and demand made more sense. The only organized market was for agricultural products and, even there, the market did not operate continuously. Once a week, everybody came to town to attend church and to buy and sell food products. Without refrigeration, nothing lasted from one week to the next and so supply coincided with stock. Since the market was only held once a week, economists were not trying to be deceptive when they shortened “weekly quantity” to “quantity.” Perfect competition, as the term is used by mainstream economists, was fairly descriptive. After the industrial revolution, supply and demand no longer worked. Industrial products are durable and it is their stock, not their supply, that is related to price. Unfortunately, economists continued to use their theory and to label the horizontal axis of their graphs “quantity” (but to mean weekly quantity) even though markets now operated continuously and the week was an arbitrary time unit. Late into the industrial revolution, tiny villages surrounded by family farms were still very common. As long as economists chose their examples carefully, it was not immediately apparent that they had gone astray. Today, it is apparent to everybody who gives the subject some thought. It is truly appalling that, as I write these words, the 21st century approaches and supply and demand is still the undisputed theory of mainstream economics.
For instance, the price of eggs is based on the stock of eggs; that is, all the fresh eggs in existence at this moment. However, since all of these eggs will have either been eaten or spoiled by next week, the stock is actually the same thing as the weekly supply; that is, how many eggs are laid and delivered to the grocery stores every week.
Because Neoclassical Economics overlaps sound (axiomatic) theory in this one area, mainstream economists can maintain a semblance of respectability by carefully restricting all of their textbook examples to consumables, doing axiomatic economics and then just saying “supply” when they really mean stock. In this way, students can be convinced that the theory of supply and demand is still relevant 250 years after it became irrelevant.
But it should be clear to all intelligent observers that it is actually stock, not supply, that is important in the egg market. For instance, suppose that there is a power outage and all the housewives in the city must throw out the dozen or two of eggs that they possess, but the egg farmers are unaffected as they have backup generators for air conditioning their hen houses. Neoclassical economists would attempt to describe this situation by saying that demand has increased as all those housewives converge on the grocery stores to replenish their stocks of eggs. (The supply of ice is unaffected by the power outage, so the housewives can store their newly purchased eggs in ice chests.)
But, clearly, demand has not increased – people want the same number of eggs for their breakfasts whether the lights are on or not. Axiomatic economists count both the eggs in possession of professionals (farmers, grocers, etc.) as well as those in possession of consumers (housewives) toward their measure of stock. No distinction is made between an egg in my refrigerator and one in the grocer's refrigerator – both just count as one more egg in the stock. So the power outage is easy to describe: stock went down, demand remains the same.
When the Berlin Wall fell in 1989, General Equilibrium Theory (the basis of the Soviet Union's five-year plans) was discredited, which left a power vacuum in the economics establishment that the sociologists rushed in to fill. They immediately set out to purge economics departments of anyone interested in the axiomatic method, whom they denounced as autistic. Stating one's assumptions clearly before proving theorems based only on those assumptions was anathema to them. They did not want an axiomatic system with realistic assumptions, they just wanted to be rid of the axiomatic method. Example from 1992.
James Devine writes in his online biography:
My brand of political economy takes into account the large class differences in political power, so that those with large net worth have disproportionate clout in political decision-making. I see capitalism (which is much more than mere markets) as a human-created institution that will not last forever… I see the divisions between the social sciences as largely artificial… even narrowly-defined economics should be seen as a sub-set of sociology.
This is the clearest definition of the goals of the Post-Autistic Economics Network; basically, they wish to frame every question in terms of class warfare and to subsume economics within sociology. James Devine has also provided us with the clearest definition of the methods that the Post-Autistic Economics Network will use to attain these goals:
The original statements by the rebellious French economics students define autistic economics in terms of its one-sided and exclusionary interest in “imaginary worlds,” “uncontrolled use of mathematics” and the absence of pluralism of approaches in economics. The hard-core autistic walling off from the societal environment can be seen most strongly in the specific, highly abstract, axiomatic school that the students protested against.
Basically, James Devine and his ilk will start a whisper campaign against any economist who believes in or even studies capitalism, accusing him or her of being autistic and, hence, ineligible to publish in economics journals.
The sociologists were leftists themselves who, in their youth, had been gung-ho for the Soviet Union and its five-year plans. The communists have periodic purges and this attack on the axiomatic method was initially just a change in focus from central planning towards piecemeal government intervention. Milton Friedman famously claimed in 1953 that assumptions do not matter, which relegated supporters of the free market to the status of ideologues, not academics. When Arrow and Debreu introduced their rather faulty set of axioms in an Econometrica article a year later, academics were divided into axiomatists, who supported central planning, and sociologists, who supported piecemeal government intervention. Free-market economists were not considered academics and were known for TV shows like Free to Choose rather than for scholarly papers. In academic journals central planning dominated piecemeal intervention for a decade or two after Theory of Value was published in 1959 but was already well on its way out (because Debreu's axioms were unsound) when the Soviet Union fell in 1989, at which time the sociologists purged academia of axiomatists. The possibility of an axiomatic system describing the free market did not occur to them in 1989 but, ten years later when one was introduced, they already had their guns in place to shoot it down.
The sociologists set out to make economics journals a forum for descriptive essays, no two having any logical connection to each other, but each just describing some aspect of society, very much as anthropologists might describe a new tribe they had discovered in Borneo. Observe that their flagship document, Freakonomics, is not actually a treatise but a collection of essays, each brandishing different but equally dubious statistics while touting the author's pre-conceived opinions on this, that and the other sociology topic. Because the sociologists' understanding of statistics is so weak (all that math – yuck), defense of their proposals still relies mostly on denouncing their critics as autistic.
In 1992 I was a senior majoring in math and minoring in economics. I took
a graduate-level course in Non-Euclidean Geometry and was told by my
economics professors that this would preclude my ever publishing in an
economics journal. They could accept a statistician co-authoring an
economics paper, but a geometer the journal editors would hate and fear.
I thought they were joking, but they actually called that one correctly. I
published my book, Axiomatic Theory of Economics, in 1999 and, a few
months later, the Post-Autistic Economics Network was formally established
with the express purpose of blacklisting textbooks that they considered
toxic, that is, written by an autistic. James Devine had a nine-year-old
son who had been diagnosed (by real doctors, not economists) with autism
and this, apparently, gave James Devine the amazing ability to diagnose
mental illness in people he had never met in person nor read beyond the
title of their book. Needless to say, I have the dubious distinction of
writing the first “toxic textbook” to be blacklisted.
This is not a new phenomenon. In 1545, Gerolamo Cardano denounced the study of imaginary numbers as being “as subtle as it is useless.” Of course, today we know that all of modern electronics, from smart phones down to the common light bulb, could not exist without electrical engineer’s access to complex analysis. Similarly, quantum mechanics could not exist without functional analysis, which is an axiomatic system. And what is Einstein's claim that the speed of light in a vacuum is constant for all observers except an axiom? Constant means always the same, which cannot be an empirical observation because “always” is vastly stronger than “so far.” We have already seen how classical mechanics is based on the axiomatic system proposed by Newton, now commonly referred to as Newton's Laws of Motion. In my dialogue, Socrates and Hume at Billiards, I explain how the conservation laws of physics are an axiomatic system, as is any science based on the solving of simultaneous equations, such as IS-LM Analysis. A better-known example, made famous by Stephen Hawking's work, is black holes, which were deduced entirely from the fundamental axioms of physics a generation before astronomers found evidence of them.
A more recent example is the discovery of the Higgs boson, which confirms
predictions made in 1964 by the Standard Model of particle physics, which
is deduced from a few axioms that Lawrence Krauss boasts are so succinct
that they "can fit on a T-shirt."
So we see that everything that is good and useful in physics is the result of the axiomatic method. And, at every step of the way, people like Cardano stood ready to denounce the founders of modern physics for their lack of “realism.” Indeed, the term “imaginary numbers” was originally coined as an insult to those who would study “imaginary worlds.” The heroes of modern electrical engineering prevailed, as evidenced by the electronic gadgets that fill the stores today, though they never shed this insulting characterization of their work. But the Post-Autistic Economics Network has taken this one step further, denouncing anyone who clearly states his assumptions before proving theorems from them as insane, afflicted with autism, which is a neurological disorder. In recent history, only the Soviet Union has been so quick to condemn their ideological opponents to mental institutions.
Lawrence H. White agrees with me:
Autism is a neurological disorder. It is not a methodological approach, an attitude, or a worldview. It is not a variety of solipsism. Implying that the position of one's intellectual opponent is attributable to a neurological disorder is a pathetic debating tactic.
The best historical analogy to the Post-Autistic movement is the German Historical School, which dominated German economics departments a generation before the rise of Nazism. Historian Ludwig Mises writes,
Economics in the second German Reich, as represented by the government-appointed university professors, degenerated into an unsystematic, poorly assorted collection of various scraps of knowledge borrowed from history, geography, technology, jurisprudence, and party politics, larded with deprecatory remarks about the errors in the “abstractions” of the Classical School.
After 1866, the men who came into the academic career had only contempt for “bloodless abstractions.” They published historical studies, preferably such as dealt with labor conditions of the recent past. Many of them were firmly convinced that the foremost task of economists was to aid the “people” in the war of liberation they were waging against the “exploiters.”
This was the position Gustav Schmoller embraced with regard to economics. Again and again he blamed the economists for having prematurely made inferences from quantitatively insufficient material. What, in his opinion, was needed in order to substitute a realistic science of economics for the hasty generalizations of the British “armchair” economists was more statistics, more history, more collection of “material.” Out of the results of such research the economists of the future, he maintained, would one day develop new insights by “induction.”
What if there was an organization called the Post-Autistic Astronomy Network? They would denounce black holes as an example of “imaginary worlds” and anybody who mentioned them as “autistic.” Then, a few years later, they would change their name to Real World Astronomy to be more politically correct, though everybody would know what the “A” in their initials and in the URL of their website stood for.
Observatories would only be allowed to point their telescopes at “real worlds” like the moon. The founder of Post-Autistic Astronomy would create a blacklist of books that we are not allowed to read called Toxic Textbooks, though cravenly not printing his name anywhere on his website (Hint: It’s Edward Fullbrook).
Prediction in astronomy would consist of statements like, “five of the last nine craters we have observed were over 100 meters deep; therefore the next one probably will be too.” Discovery in astronomy would consist of finding a crater on the moon that nobody had written about yet, describing it and then naming it after oneself. The only difference between popular essays and refereed journal articles would be that the latter would include statistics, like the average depth of a crater measured at various points in its interior.
If Stephen Hawking asked to have a telescope pointed into deep space so that he could look for evidence of black holes, he would be banned for life from the observatory. Real-world astronomers would point at him behind his back, whisper the word “autistic” and make snarky comments about him on the internet.
Edit: On 16 May 2011, the Post-Autistic Economics Network changed its
name to the World Economics Association. Their leadership is unchanged
and they continue to maintain the same blacklist, though the economists on
the blacklist are no longer referred to as "autistic," apparently in
deference to real scientists who know that "autistic" is not just a bad
word to hurl at one's ideological opponents but a neurological disorder
that should be diagnosed by medical professionals, not economists.
In Section XIV of my Critique of Austrian Economics, I compare my axiomatic system to other axiomatic systems and do so in a relatively non-mathematical way. Readers who have made it all the way through this page and found it interesting would do well to read Section XIV of my Critique next. From there, the path branches, depending on one’s background and motivation:
If one is primarily interested in learning more about Axiomatic Economics, one should try my Simplified Exposition. It is considerably more mathematically intense than this paper but, as the proofs of the theorems are clearly delineated, one can tone down the intensity a bit by reading about the theorems while skipping over their proofs.
If one is interested in the methodology of economics, one should read Socrates and Hume at Billiards, which compares my axiomatic system to that of Isaac Newton and demonstrates that IS-LM Analysis can also be thought of as an axiomatic system, though it is not usually taught as such. My Review of Fuerle’s Pure Logic of Choice also discusses epistemology, touching on the theories of Euclid and Kant.
If one is from an Austrian background, the next step is to read my complete Critique of Austrian Economics. It is only slightly more mathematical than this paper. Robert Murphy of the Mises Institute responded and I replied; both papers are in my Rebuttals page. The Wreckage of Austrian Business Cycle Theory is a short summary of this debate.
If one is from a neo-classical (mainstream) background, the next step is to read, Cutting the Gordian Knot of GE Theory. This short, non-mathematical paper explains what I hope to accomplish by taking the rather drastic step of inventing my own axiomatic system, rather than just working within Debreu’s framework. If one is still with me, one should then try my Simplified Exposition.
If one is concerned about current events, particularly the financial crisis, the next step is to read my short paper, Is the Collapse of the Dollar inevitable? One may then scroll down the home page to where “Critiques of the Tea Party,” “Critiques of Socialism” and “Critiques of Fascism” are listed. Here, one will find several short, non-mathematical papers about the Tea Party, socialism and fascism. Their titles are fairly self-explanatory and they can be read in any order.