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Axiomatic Economics Blog

### Introduction

In this blog, we will discuss the axiom set that I proposed in 1999:

1) One's value scale is totally (linearly) ordered:

 i) Transitive; p $$\leq$$ q and q $$\leq$$ r imply p $$\leq$$ r ii) Reflexive; p $$\leq$$ p iii) Anti-Symmetric; p $$\leq$$ q and q $$\leq$$ p imply p = q iv) Total; p $$\leq$$ q or q $$\leq$$ 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+ &epsilone;j, with j denoting the day), of the previous day's value. ii) In the long run, the &epsilone;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 &epsilone;j's are taken from an unspecified distribution with a finite mean and a non-zero, finite variance.

Comparison will be made to famous axiom sets in physics, specifically the laws of motion proposed by Isaac Newton. For reference, Newton’s axioms are three:

1. Law of Inertia. If the net force acting on a body is zero, it is possible to find a set of reference frames in which that body has no acceleration.

2. Force equals mass times acceleration. F=ma (Boldface denotes vectors in three-dimensional space.)

3. To every action there is always an equal and opposite reaction. FAB=-FBA (FAB is the force exerted on body A by body B; FBA is the force exerted on body B by body A.)

We will also consider the two axioms added by Einstein:

1. The laws of physics are the same for observers in all inertial reference frames. No frame is singled out as preferred.

2. The speed of light in free space has the same value c in all directions and in all inertial reference frames.

Einstein’s work provides a good example of how an axiomatic system may be amended as new data comes to light. Such an amendment is only acceptable if there is a smooth transition with previous results; in this case, as the speeds of particles are reduced to the point that relativistic effects become negligible, then Newton’s original laws of motion apply with negligible error. Similarly, the correspondence principle of quantum physics – that the greater the quantum number, the closer it approaches classical physics – also assures a smooth transition.

The issue of continuity is important because I have recently (2013) suggested a new axiom: The distribution of importance, μ, is exponential. This axiom explains why large fluctuations in prices have an inverse power law distribution. Just as Einstein did not disturb Newtonian physics as it applies to slow moving particles, this new axiom must leave the results of classical (1999) axiomatic economics undisturbed.

We will also consider the axiom set employed by Leonhard Euler to describe the trajectory of trench mortars. Both Euler and Newton are excellent role models for economists.

1. Constant atmospheric density from the ground to the apogee.

2. Drag is proportional everywhere to the square of the speed.

3. Gravity is everywhere pointed downwards; e.g. the Earth is flat.

This axiom set provides a good example of how axioms that are patently false (the Earth is actually a sphere) may be acceptable and quite useful for their stated purpose. Trench mortars simply do not go far enough for the curvature of the Earth to matter; taking curvature into consideration would introduce undue complication for no gain. But howitzers can fire on targets over the horizon, so, just as Einstein amended Newton’s axiom set, artillerists must amend the foundational work of Euler. My own fire control software does not amend the flat Earth axiom, though it does consider velocities above 240 m/s where drag is proportional to the cube of the speed. Also, it considers an apogee high enough for the air to be significantly thinner than at ground level.

I must ask readers to refrain from mentioning the axiom set of Gerard Debreu. Please stick to examples from the physical sciences (including chemistry and meteorology), which I feel are really the best role models for economics. There are two reasons for this:

1. Nobody cares about Debreu. Essentially all modern economists have rejected general equilibrium theory. A theory that is kept alive only for the sake of demonstrating one’s ability to take it apart turns it into a straw man. In this blog, we do not engage in straw man attacks.

2. Nobody knows what Debreu’s axioms are. Debreu changed his axiom set more often than he changed his underwear; every time a problem appeared he amended his axioms to “fix” the problem. Modern commentators are never on the same page regarding which patchwork they are criticizing.

I will begin with a discussion of the advancements in economic methodology over the last thirty years, if there are any. The axiomatic system that Euler developed to describe the trajectory of trench mortars is, in my opinion, the best role model available for economists to emulate; this is the topic of my next column. The failure of pre-Newtonian physics is demonstrated by David Hume’s floundering discussion of the game of billiards and, in the following column, I explain how Newton corrected Hume by proposing an axiomatic system. In the next column, I discuss Newton’s definition of limits in calculus and I point out that the reason why economics before 1999 was such a complete failure is because they were trying to use calculus without convergence. Following this, I will discuss quantum mechanics as a role model for economics.