# Axiomatic Economics

#### Grozny

I have a degree in mathematics. I wrote a book about economics (Axiomatic Theory of Economics, ISBN# 1-56072-296-7) in 1999 in which I propose a new theory of economics based on three axioms:

1) One's value scale is totally (linearly) ordered:
i) Transitive; p <= q and q <= r imply p <= r
ii) Reflexive; p <= p
iii) Antisymetric; 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+Ej, with j denoting the day), of the previous day's value. (E means epsilon)
ii) In the long run, the Ej's may be considered random as they are not directly related to each other nor are they uniquely a function of value.
iii) The Ej's are taken from an unspecified distribution with a finite mean and a non-zero, finite variance.

#### Grozny

Simplified Exposition of Axiomatic Economics

How much of the book requires the reader to know functions with more than one independent variable?
Link: Simplified Exposition of Axiomatic Economics

Until this proof, only one semester of calculus had been required of the reader. Theorems 12 and 13 are about functions of two variables, however, and are more difficult...

Readers with only one semester of calculus can obtain most of the mathematics they need by reading a textbook on multivariable calculus up to but not including Lagrange multipliers. This is generally considered the easy part of multivariable calculus and is the work of six or eight lecture hours. To read Axiomatic Theory of Economics (without the simplifying axiom of this article) also requires some knowledge of infinite series. Fortunately, the â€œhardâ€ part of multivariable calculus (multiple integrals and vector fields) is never used.
By "simplifying axiom" I mean replacing axiom #2 with the claim that people never need more than one of anything.

The purpose of this article is to give a simplified exposition which is not too mathematically demanding. This is accomplished by replacing an axiom to assume away the infinite summations so that readers need not be familiar with real analysis. The essential points remain intact, however, as the theorems apply as well to partial sums (including the 0â€™th partial sum) as to infinite ones. But the proofs are simple enough to facilitate a cursory reading.

#### CRGreathouse

Forum Staff
Very interesting! Perhaps you should post on the Economics forum where you might get more attention.

(FWIW this book is relevant to my interests, as I studied math/econ in college and have maintained an interest in axiomatic economics, e.g. Arrow's Theorem and its ilk.)

I went to your site and started to read some of the material there, though I only finished a few pages.

Your first two axioms, though not completely obvious, are at least fairly reasonable and intuitive. I'm not so sure about the last one. Do you have any thoughts on why we should accept this axiom?

#### Grozny

Your first two axioms, though not completely obvious, are at least fairly reasonable and intuitive. I'm not so sure about the last one. Do you have any thoughts on why we should accept this axiom?
From my Simplified Exposition of Axiomatic Economics:

That first-unit demand conforms to the characteristics of proportionate effect must be regarded as an axiom. A plausibility argument is provided here. Let mj = Ï†(mj-1) with mj the number of monetary units to which one is indifferent relative to the first unit of a phenomenon on the j'th day of that person's life. We want to show that Ï†(mj-1) = (1+Îµj)mj-1. Consider a man who wants to take out a loan at interest. He must think he will have more money in the future than he does now. (More money holdings, not necessarily more wealth.) If he does, the value of individual monetary units will tend to decrease over time relative to other phenomena; that is, Ï† is a positive function when averaged over all phenomena. To determine how much interest he is willing to pay, the man must specify this average Ï†. For him to calculate the interest owed per unit of time as a percentage of the principle is equivalent to specifying Ï†(mj-1) = (1+Îµ)mj-1 with Îµ > 0 fixed. Fixing Îµ is a special case of Îµj being a random variable. Here, the probability density function is unity at Îµ and zero elsewhere. Thus, the axiom that first-unit demand conforms to the characteristics of proportionate effect is a generalization of calculating interest as a percentage of the amount owed. In fact, this is how people have calculated interest throughout recorded history, although economics having always been a soft science, they never asked for proof. Perhaps the value of money decays harmonically over time or in another way besides exponentially? This question is addressed in Axiomatic Theory of Economics, but for now let us proceed to investigate the consequences of people's points of indifference for their first unit of each phenomenon being lognormally distributed.