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Simplified Exposition of Axiomatic Economics

Section II:  Second and Third Axiom

There is an upper bound to one's value of any stock of a phenomenon which will be denoted M. This includes one's need for saving phenomena for future use. Total utility is the marginal utility of a phenomenon when the unit is defined as the entire quantity possessed. It increases from zero up to its maximum point, M, as one's stock increases. Hence, total utility is a cumulative distribution function and marginal utility is the associated probability density function (after normalization), denoted U(s) and u(s), respectively. Since the utility of a given stock is measured by the quantity of money which stands beside it on one's value scale, U(s) is a mapping from the stock of a phenomenon one possesses to the money one associates with that stock. u(s) is its first derivative. u(s) must be negative monotonic because utility diminishes as one adds units to one's stock. The integral of u(s) must also converge, that is, \(\int_{0}^{\infty } u(s)ds < \infty \) This is because marginal utility is the probability density function of a cumulative distribution. Nothing else is known about u(s) and this is the first parameter (and the only function) used to distinguish phenomena from one another. Its characteristics must be regarded as an axiom. Later, two more parameters (both from \(\Re \)) will be introduced which will be sufficient to completely describe every phenomenon.

As will be shown shortly, we are only concerned with the ratio \(\frac{u(0)}{u(r)}\) for non-negative integers, r. This ratio is invariant under a rescaling of the vertical axis, so u(s) can be normalized by setting the upper bound on the distribution function, M, to unity. This makes u(s) a true probability density function as the total area under it is unity.

It should be noted here that the requirement that u(s) be negative monotonic does not imply that firms must be small, which is clearly not true because there are many large and successful corporations. Economists have used the term “marginal (or diminishing) utility” to denote both the first derivative of one’s total utility for some phenomenon and the assertion that firms receive less and less return on their investments as they grow bigger. Capital, like all phenomena, has diminishing utility because one quickly becomes sated on it. However, like most things on which one temporarily sates oneself, one is ready for more the next day and the day after that. Thus, while a firm cannot immediately make use of all the capital it might consider buying, it can start with a small capital project and use the profits from that to train the managers and laborers that will make an expansion feasible. In this way, firms can become global in scale without ever contradicting the assertion that u(s) is negative monotonic for capital. The large corporation embarking on another great expansion may have started out as a small mom-and-pop outfit, but it is not that little company anymore and it has a (very) different utility function now. Since Axiomatic Theory of Economics is about stock, not supply, the relative sizes of the firms supplying a phenomenon is of no concern.

I assert that the distribution of people's points of indifference for their first unit of a phenomenon relative to money, c₀(m), is lognormal; that is, the natural logarithm of the number of people who are indifferent at a particular price, m, is cumulatively (normally) distributed. The cumulative distribution is applicable to a variable that is subject to a process of change such that, at each step, a random quantity is added to the accumulated value of that variable. By the Central Limit Theorem, the distribution of the sum of a large number of independent, identically distributed random variables (from an unspecified distribution with a finite mean and a non-zero, finite variance) is approximately normal.

c₀(m), however, does not accumulate, rather it is analogous to the growth of the value of money through history: It conforms to the characteristics of proportionate effect. After the j'th day of a person's life, the change in the number of monetary units to which he is indifferent, relative to the first unit of a phenomenon, is a proportion of his indifference point the day before. That anthropometric variables (height, size of organs, tolerance to drugs, etc.) conform to the characteristics of proportionate effect is well established in the literature.

Theorem 1 (Law of Proportionate Effect): Phenomena which conform to the characteristics of proportionate effect are lognormally distributed.

Proof:

m_j - m_j-1 = ε_jm_j-1      The difference between each stepand the last one is the last one multiplied by a random quantity.

\(\frac{m_j - m_j-1}{m_j-1} = \epsilon _j \)      Devide through by m_j-1 to get ε_j,the change in m relative to its previous value, m_j-1

\(\sum_{j = 1}^{n}\frac{m_j - m_j-1 }{m_j-1} \) = \(\sum_{j = 1}^{n}\epsilon _j \)      Find the sum of all ε_j from the initiation of the process to its termi-nation after n steps.

\(\int_{m_0}^{m_n}\frac{dm}{m} \) = \(\sum_{j = 1}^{n}\epsilon _j \)     If each step is small, m_j - m_j-1 can be approximated by dm.

ln|m_n| - ln|m₀| = \(\sum_{j = 1}^{n}\epsilon _j \)     Integrate from m₀ to m_n.

ln|m_n| = ln|m₀| + ε₁ + .. +ε_n     solve for ln(m_n)

As can be seen from the last step, the natural logarithm of one's indifference point after the n'th day is a constant (the logarithm of its initial quantity) with a large number of random and identically distributed quantities accumulated onto it. Hence, after having lived through n days and having seen their point of indifference change by a small proportion each day, consumers of their first unit are normally distributed with regard to the variable ln(m) and, hence, are lognormally distributed with regard to the variable m.  

The absolute value operation may be dropped, since we are only interested in positive prices.

That first unit demand conforms to the characteristics of proportionate effect must be regarded as an axiom. A plausibility argument is provided here. Let \(m_{j } = ∅(m_{j-1 } )\) with m_j 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 \(∅(m_{j-1 } ) = (1+\varepsilon_{j} )(m_{j-1 } )\) 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, \(∅ \) 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 \(∅ \). For him to calculate the interest owed per unit of time as a percentage of the principle is equivalent to specifying \(∅(m_{j-1 } ) = (1+\varepsilon_{j} )(m_{j-1 } )\) with > 0 fixed. Fixing \(∅ \) is a special case of \(\varepsilon \)_j being a random variable. Here, the probability density function is unity at \(\varepsilon \) 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 an appendix of my book 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. I believe that this axiom is on solid intuitive ground and will not be criticized. Even if it is, it is unlikely that critics will succeed in convincing the banking industry to calculate interest with a different formula, so the weight of tradition will continue to support my choice of the lognormal distribution for first unit demand.

Before continuing, let us explicitly state our three axioms:

ln(m) is linearly transformed by \(\frac{ln(m)-μ}{\sigma } \) The location parameter, μ (mean), quantifies the importance of a phenomenon relative to money and the scale parameter, \(\sigma \) (standard deviation), quantifies the difficulty of substituting other phenomena for the one in question. Easily substituted phenomena have very little probability in the tail of their demand distribution; only the eccentric purchase a phenomenon at a high price when there are cheaper substitutes available. As substitution becomes more difficult, people must purchase the phenomenon even at high prices, and their distribution is less skewed. Both μ and \(\sigma \) must be positive. With u(s), μ and \(\sigma \) describe all phenomena. Thus, every phenomenon is associated with a point in u(s),μ,\(\sigma \) space where u(s) is a negative monotonic probability density function on \(μ \) and μ and \(\sigma \) are both from \(μ \). For the purpose of economics, nothing else distinguishes one phenomenon from another.

The equation for the distribution of first unit demand is c₀(m) = \(\frac{e^{-\frac{1}{2}(\frac{ln(m)-1}{\sigma })^2}}{\sigma m} \) This is the equation of the lognormal distribution, \(e^{-\frac{ln^2(m)}{2 }} \) multiplied by the derivative of the linear transformation which is substituted for ln(m). It need not be divided by its total area, \(\sqrt{2π} \), since it will not be used as a probability density function.

The number of people with a point of indifference at a particular price, m, for their first unit is c₀(m). Whoever's point of indifference for his first unit is \(\frac{u(0)}{u(1)} \) times greater than that price values his second unit equivalent to price m. Whoever's point of indifference for his first unit is \(\frac{u(0)}{u(2)} \)times greater than that price values his third unit equivalent to price m, and so on. To find c(m), all the people with a point of indifference at m are summed up, whether it is their first purchase or a later purchase. Recall the analogy of the demand distribution being 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. Now consider a person who wishes to possess more than one unit of the phenomenon; each of his agents appears behind a different point on the money line. If he himself appears in the column assembled behind m monetary units, the first person he sends to get another unit is directed to the column behind \(\frac{mu(1)}{u(0)} \) monetary units. His next agent is in the column behind \(\frac{mu(2)}{u(0)} \) monetary units, and so on. Hence, we have the following formula for the demand distribution which, unfortunately, is impossible to integrate in closed form, even with u(s) fixed.

c(m) = \(\sum_{r=0}^{\infty }c_0(x) \)      with x = \(\frac{mu(0)}{u(r)} \)

c₀(m) can be thought of as the 0'th partial sum of c(m) and, in general, c_n(m) denotes the
n'th partial sum of c(m). Thus,

c_n(m) = \(\sum_{r=0}^{n}c_0(x) \)      with x = \(\frac{mu(0)}{u(r)} \)

Most of the real analysis in Axiomatic Theory of Economics stems from the infinite summation, c(m). To simplify the proofs in this pamphlet, the second axiom is replaced with the assertion that people never need more than one of anything at a time. This assumption is neither accurate nor necessary, as all of the results of my theory can be (and are) proven in their full generality. However, some economists do not have the mathematical background necessary to read Axiomatic Theory of Economics, so, for expository purposes, simplified proofs are provided here. Also, before the theory becomes accepted, it will receive cursory reviews, perhaps at the end of courses on mainstream economics. In this case, if a professor is sympathetic to my theory, he may wish to prove some of its assertions, but he will not have time to prove them in their full generality. As long as he mentions that the complete proofs do exist, his students can get the essence of my theory from the simplified proofs. The important thing for them to understand is that this theory is deduced from axioms. So, for the remainder of this pamphlet, all of the theorems will be proven using the 0'th partial sum, c₀(m), rather than c(m). When S(m) appears in a proof, it will refer to \(S(m) = \int_{0}^{\infty }c_0(t)dt \) f(μ,m), which will be defined later, will also be defined in terms of c₀(m) rather than c(m).