1) One's value scale is totally (linearly) ordered: i) Transitive; p q and q r imply p r ii) Reflexive; p p   iii) Anti-Symmetric; p q and q p imply p = q   iv) Total; p q or q p   2) Marginal (diminishing) utility, u(s), is such that: i) It is independent of first-unit demand. ii) It is negative monotonic; that is, u'(s) < 0.   iii) The integral of u(s) from zero to infinity is finite.

3) First-unit demand conforms to proportionate effect: i) Value changes each day by a proportion (called 1+j, with j denoting the day), of the previous day's value. ii) In the long run, the j's may be considered random as they are not directly related to each other nor are they uniquely a function of value.   iii) The j's are taken from an unspecified distribution with a finite mean and a non-zero, finite variance.

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 really 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, ε_j. (Note: epsilon, ε, is a mathematical symbol denoting a small impulse, negligible by itself but influential in quantity.)

My theory describes society, so my demand distribution, c(m), can be thought of 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." There are many examples in prediction literature, especially of investments, where the log-normal distribution is used to describe the possible future outcomes of an action one is considering. But my theory is descriptive, not predictive.

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. Oh well – ferocity is meaningless if it is not accompanied by effectiveness.

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:

1. 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.



2. 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.



3. 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.



4. 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.



5. If one is concerned about current events, particularly the financial crisis, the next step is to scroll down the home page to where “Critiques of Paulism,” “Critiques of Socialism” and “Critiques of Fascism” are listed. Here, one will find several short, non-mathematical papers about the Ron Paul movement, socialism and fascism. Their titles are fairly self-explanatory and they can be read in any order. Also, my Devil’s Dictionary of Economics makes many brief comments about prominent politicians and other newsmakers.