Simplified Exposition of Axiomatic Economics
Section III: The Law of Price Adjustment
Theorems are numbered analogous to those in Axiomatic Theory of Economics.
Theorem 4: \(\lim_{m->0^+}c_0(m)=0 \)
Proof: \(c_0(m) > 0 \) for all m> 0 Thus, by the Squeezing Theorem, if c0(m) is less than some function for all m > 0 and that function is continuous and equals zero at zero, then \(\lim_{m\rightarrow 0^+}c_0(m)=0 \)
Consider hm with h a finite constant. Since hm vanishes at zero, it is sufficient to show that hm > \(\frac{e^{-\frac{1}{2}(\frac{ln(m)-1}{\sigma })^2}}{\sigma m} \) for all m > 0. By making the substitution y = ln(m), this is equivalent to \(y^2 + (4\sigma ^2 - 2μ)y - 2\sigma ^2ln(\sigma h) + μ^2 > 0 \) for all real y. By the Quadratic Theorem, this is true for \(\frac{e^{2(\sigma ^2 - μ)}}{\sigma } \) < h < \(\infty \). Thus, the demand distribution is equal to zero at zero. 
Alternate proof: Make the substitution y = ln(m) so
\(\lim_{m \to 0^+}c_0(m) = \lim_{y \to \infty }\frac{e^-\frac{1}{2}(\frac{y-μ}{\sigma })^2}{\sigma e^y} \)
= \(\frac{1}{\sigma }\lim_{y \to -\infty }e^{-\frac{(y-μ+\sigma ^2 )^2) + \sigma^2(\sigma ^2 - 2μ)}{2\sigma ^2}} \)
= \(\frac{1}{\sigma }\lim_{y \to -\infty }e^{-\frac{(y-μ+σ^2)^2) + 2\sigma ^2y}{2\sigma ^2}} \)
= 0
The former was chosen as the main proof because the Squeezing Theorem and the Quadratic Theorem can be visualized and are (hopefully) more intuitive than a purely algebraic proof.
Theorem 7: Stock is finite.
Proof: Make the substitutions \(\frac{lt(t) - μ}{\sigma } \) and dy = \(\frac{dt}{\sigma t} \) so
S(m) = \(\int_{z}^{\infty }e^{-\frac{y^2}{2}}dy \) with z = \( \frac{ln(m)-μ}{\sigma } \)
The integral is the standard normal distribution, which is tabulated as \(\alpha (z) = 1- \phi (z) \) in the back of any statistics text, though multiplied by the constant \(\frac{1}{(\sqrt{2π} )} \) so that the total area under the integrand is unity, a step which is omitted here since the integrand is not being used as a probability density function. However, since this integral never exceeds \(\sqrt{2π} \), we have the following inequality:
S(m) < \(\sqrt{2π} \).
Aggregate utility is defined as price multiplied by stock. This is because money is the measure of utility and everyone who possesses a unit of stock values it only as highly as its replacement cost, for that is all that one risks. Stock and price, however, are inversely related, so increasing one or the other does not necessarily increase aggregate utility. Aggregate utility being the common goal of people dealing in a phenomenon, they are interested in maximizing it. As stock increases, aggregate utility also increases up to saturation, where any further increases in stock reduce aggregate utility by driving the price down. That part of the demand distribution to the right of saturation (the high end), where increases in stock increase aggregate utility, is unsaturated and that part to the left (the low end) is saturated. At a constant stock, there is a zone of indeterminacy between the marginal pair within which the price may fluctuate. Such fluctuations appear to be of a saturated market whether the stock has reached saturation or not. Most markets are large enough, however, that the zone of indeterminacy is too narrow to be of practical concern.
Because the actions appropriate in an unsaturated market (increasing stock) are not those appropriate in a saturated market (decreasing stock), it is important to determine the point of saturation. Aggregate utility, mS(m), is at a relative maxima where its first derivative, S(m) - mc(m), equals zero. Thus, saturation is a price and stock such that S(m) = mc(m). Graphically, S(m) is represented by the area between the horizontal axis and the graph of the demand distribution from m to \(\infty \). mc(m) is represented by the area of the rectangle formed by the two axes and horizontal and vertical lines extending from the point m,c(m).
Theorem 10 (existence): The absolute maximum of aggregate utility is at a finite critical point.
Proof: By Theorem 4, the limit of c0(m) at zero is zero. Thus, stock is finite even if it is free, and aggregate utility goes to zero as price approaches zero. Since aggregate utility is always positive, it is sufficient to show that it also goes to zero as price approaches infinity to prove the existence of a relative maxima. One makes the substitutions \(y = \frac{ln(t)-μ}{\sigma } \)and \(dy = \frac{dt}{\sigma t} \) so
0 < mS(m) = m \(\int_{\frac{ln(m)-μ}{\sigma }}^{\infty }e^{-\frac{y^2}{2}}dy \)
= \(\leq \) m \(\int_{\frac{ln(m)-μ}{\sigma }}^{\infty }ye^{-\frac{y^2}{2}}dy \) if m ≥ \(e^{μ + \sigma } \)
= -m[\(\left.\begin{matrix}
e^{-\frac{y^2}{2}} &
\end{matrix}\right|_{\frac{ln(m)-μ}{\sigma }}^{\infty } \)
= \(me^{-\frac{1}{2}(\frac{ln(m)-1}{\sigma })^2} \)
= \(me^{-\frac{ln^2(m)-2μln(m)+μ^2}{2\sigma ^2}} \)
= \(\frac{Bme^{-\frac{ln^2(m)}{2\sigma ^2}}}{m^{-\frac{1}{\sigma ^2}}} \)
With B = \(e^{-\frac{μ^2}{2\sigma ^2}} \)
= \(\frac{Bm}{m^{\frac{ln(m)-21}{2\sigma ^2}}} \)
= \(\leq \) \(\frac{B}{m} \) if m \(\geq \) \(e^{2(μ + 2\sigma ^2)} \)
B is a constant, so \(\lim_{m \to \infty }B/m = 0 \) and, by the Squeezing Theorem, \(\lim_{m \to \infty }mS(m) = 0 \) Thus, there exists a finite price where aggregate utility is at a maximum.
This only proves the existence of a relative maxima and identifies it with the absolute maximum. There may be more than one relative maxima, in which case the largest of them is the absolute maximum. However, by the following proof there is only one relative maxima and it is the absolute maximum of aggregate utility. This justifies the use of the word "the" when referring to the saturation point.
Theorem 11 (uniqueness): Aggregate utility has only one relative maxima.
Proof: Because aggregate utility is always positive and it approaches zero at both ends of its domain, (0, \(\infty \) ), there is either a single relative maxima, or relative maximas and minimas alternate with the largest and smallest being relative maximas. The second derivative of aggregate utility is \(c_0m(\frac{ln(m)-μ}{\sigma ^2} - 1) \). It is positive at relative minimas and negative at relative maximas. Therefore, if there is more than one relative maxima, there are two disjoint intervals in (0,\(\infty \) ) where the second derivative is negative and they are separated by an interval where the second derivative is positive.
We wish to show where the second derivative is strictly negative. c0(m) > 0 for all m, so we only have to examine \(\frac{ln(m)-μ}{\sigma ^2} - 1 \). This is negative for all 0 < m < \(e^{μ+σ^2} \) and positive for all m > \(e^{μ+σ^2} \). Recalling that relative maximas and minimas alternate with the largest and smallest being relative maximas, there can only be one price such that S(m) = mc0(m) and it is a relative maxima.
It is an easy corollary that the saturation price is less than \(e^{μ+σ^2} \).
μ and σ change over time for a variety of reasons, each change necessitating a recalculation of the saturation point. It is the business of entrepreneurs to anticipate these changes and to adjust stocks accordingly. While most shifts in a demand distribution are of only local concern, one is of particular interest to economics. If some of the people represented by the demand distribution for a phenomenon receive money from the government, how does the saturation point change? Whether these people receive a grant, a low interest loan, or are doing contract work for the government, they are more liquid than they want to be. Knowing the negative effect of a loose monetary policy on the value of money, they are not going to hoard it. Relative to money, the importance of phenomena has increased. How are prices and stocks affected and which adjusts more dramatically to the increase in μ ?
It is an old adage that people get more out of something the more they put into it and, money being the measure of utility, one expects increases in the importance of a phenomenon relative to money to increase the phenomenon's price in proportion to the price that it has already attained. Mathematically, p = p_0e^μ, with p0 the price at saturation with no importance relative to money and p the price such that f(μ,m) = S(μ,m) - mc(μ,m) = 0. Notice that p is the particular price which satisfies the condition f(μ,m) = 0 while m denotes an arbitrary price. Variables included in the functional notation are allowed to vary while others which appear in a function but are not listed in the parenthesis of the function are assumed to be constant. Here, we are discussing changes in both price and importance where before only price was allowed to vary.
Theorem 12: The price at saturation increases exponentially in response to an increase in the importance of a phenomenon relative to money; that is, \(\frac{dp}{dμ} = p \)
Proof: f(μ,m) = S(μ,m) - mc0(μ,m) = 0 implicitly defines a level set in the μ,m plane. Let that level set be parametized by [ μ(t) - m(t)]. By the chain rule, the derivative of f(μ,m) = 0 is \(\frac{\partial f}{\partial μ}\frac{dμ}{dt} \) + \(\frac{\partial f}{\partial m}\frac{dm}{dt} \) = 0 or \(\begin{bmatrix}
\\
\frac{\partial f}{\partial μ}
\frac{\partial f}{\partial m}
\end{bmatrix} \)
\(\begin{bmatrix}
\\
\frac{\partial μ}{\partial t}
\frac{\partial m}{\partial t}
\end{bmatrix} \) The latter vector is the derivative (tangent) of the parametized level set, so \(\begin{bmatrix}
\\
\frac{\partial f}{\partial μ}
\frac{\partial f}{\partial m}
\end{bmatrix} \) is perpendicular to the level set which passes through any μ,m where it is evaluated. From the definition of saturation, this is downward (toward smaller m), so a 90 degree counter-clockwise rotation of \(\begin{bmatrix}
\\
\frac{\partial f}{\partial μ}
\frac{\partial f}{\partial m}
\end{bmatrix} \) is tangent to the level set of all μ,m combinations with f(μ,m) constant. Dividing its vertical component by its horizontal component gives the desired rate of change in price:
\(\frac{dm}{dμ} \) = - \(\frac{\frac{\partial f}{\partial μ}}{\frac{\partial f}{\partial m}} \) = \(\frac{m\frac{\partial f}{\partial m}}{\frac{\partial f}{\partial m}} \) = m with \(\frac{\partial f}{\partial m} \) = \(c_0(m)(\frac{ln(m)-μ}{\sigma } - 1) \)
This relation is true regarding the level set which passes through any point μ,m. Choosing only points along the level set f(μ,m) = 0 (rather than another constant) yields \(\frac{dp}{dμ} = p \)
Notice that f(m) in the above proof may be expressed as
f(m) = \(-\int_{m}^{\infty }f'(t)dt \) = \(\int_{0}^{\infty }c_0(t)(1 - \frac{ln(t)-μ}{\sigma ^2})dt \)
Also, the evaluation of \(\frac{\partial f}{\partial μ} \) requires an application of Leibnitz' Rule, justification of which is given in Axiomatic Theory of Economics. Incidentally, it does not matter that the rotation is counter-clockwise since a clockwise rotation also switches the components but negates \(\frac{\partial f}{\partial μ} \) instead of \(\frac{\partial f}{\partial m}\) Because the sign comes out front after the division, it is immaterial which way \(\begin{bmatrix}
\\
\frac{\partial f}{\partial μ}
\frac{\partial f}{\partial m}
\end{bmatrix} \)is rotated.
An alternative proof uses the chain rule to differentiate f(μ,g(μ,m)) = 0 with p = g(μ,m) to get
fμ(μ,g(μ,m)) + fm(μ,g(μ,m))gμ(μ,m) = 0
This equation is solved for \(\frac{dp}{dμ} = g_μ(μm) \) Notice that, by the uniqueness of saturation, p = g(μ) is a function; that is, a unique price is associated with every μ, though in general this is not required for gµ(μ,m) to be determined explicitly. In other words, not every gµ(,m) has an anti-derivative, g(μ). By the construction of gµ(μ,m), g(μ,m) is proven to be smooth and continuous, which is all that is required of it.
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 may find the alternative proof of Theorem 12 easier than the main proof if they are familiar with implicit differentiation. However, many students who have been introduced to calculus of several variables readily grasp the concept of level sets because of their familiarity with contour maps. Thus, for \(£ \Re ^{1+1} \rightarrow \Re ^1 \) recourse to the tangent seems more intuitive than a purely algebraic proof and the former was chosen as the main proof. 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 pamphlet) also requires some knowledge of infinite series. Fortunately, the “hard” part of multivariable calculus (multiple integrals and vector fields) is never used. Axiomatic Theory of Economics is similar to probability. Indeed, I see my book following in the tradition of Kolmogorov’s Foundations of Probability more than in any work of an economist. People who have worked with probability distributions are encouraged to read Axiomatic Theory of Economics even if they are only vaguely familiar with multivariable calculus.
By Theorem 12, the price at saturation increases exponentially in response to an increase in the importance of a phenomenon relative to money. What about stock? Intuitively, one expects stock to remain constant since, effectively, all the government does by issuing money is to change the figures in which prices are quoted and that should not affect the stock of phenomena that people keep in existence. Most economists would agree that this is true in the long run but would argue that, because of the uneven diffusion of fresh issues of money, the stock of phenomena is temporarily affected. Money diffuses unevenly from a central bank and that is the principal motivation for issuing it (otherwise those close to a government would not profit from their connections), but I assert that this does not provide any incentive for the stock of phenomena to increase.
Theorem 13: The stock at saturation remains constant in response to an increase in the importance of a phenomenon relative to money; that is, \(\frac{dS_p}{dμ} = 0 \)
Here, the subscript on stock denotes that it is the stock associated with the saturation price, p.
Proof: We are interested in the change in stock along the level set implicitly defined in the μ,m plane by the relation f(μ,m) = 0. As noted in the preceding proof, the tangent to this curve is [ 1 m ]. Normalizing this vector and taking the inner-product with the derivative of S(μ,m) gives the desired rate of change in stock. Since we are interested in proving that this change is always zero, it is sufficient to show that the numerator is always zero and we may omit normalizing the directional vector. The inner product of this with the derivative of stock, [ mc0(μ,m) -c0(μ,m) ], is zero.
Together, the two preceding theorems will be referred to as the Law of Price Adjustment. Because Theorem 13 is a corollary of Theorem 12, the term "Law of Price Adjustment" is used to denote both theorems. From a practical point of view, however, the assertion that the stock of phenomena is unaffected by depreciating a currency is more important because, by definition, economics is concerned with the wealth of a nation. Of course, the wealth of an individual can always be increased at the expense of other people by printing and spending money, but theoretical economics (hopefully) addresses more lofty aims.
It is important that the Law of Price Adjustment does not place any restrictions on marginal utility, on the importance of a phenomenon relative to money, or on the difficulty of substituting other phenomena. Within my economic theory, these three characteristics are all that distinguish phenomena from one another; that is, phenomena with the same u(s), μ, and σ are isomorphic. Thus, it is impossible to argue that my theory is inapplicable in certain situations because it has been proven to apply to all possible situations; that is, it applies to phenomena at every point in u(s),μ,σ space. Since any mathematician will confirm the deduction of the Law of Price Adjustment from the three axioms, for an economist to accept or reject the Law of Price Adjustment is equivalent to his acceptance or rejection of the three axioms, respectively. Attempts to divert the argument away from the acceptance or rejection of the theory's axioms should be discouraged.
The implications of the Law of Price Adjustment should be obvious to anyone who has studied mainstream economics; stickiness of prices is the cornerstone of Keynesian Economics. Even for those who do not follow the mathematics, common sense alone is sufficient to refute the Keynesian premise. Considering that a government can print money for itself within a day's notice, if the adjustment process could not be done in equal time, the whole system of indirect exchange would have collapsed long ago. Prices can be changed with a word, but the stock of phenomena can only be changed after considerable toil. It is obvious which is adjusted and which left constant. The average level of prices is "sticky" because it takes time for money to diffuse through a community and if one is averaging all prices, it is some time before one notices a change in one's statistics. This average is also meaningless for the same reason. The effect of issuing money is to redistribute wealth to the people who receive the new money first and that is only possible because of the slow diffusion of money through an economy.
Having arrived at a position so fundamentally opposed to mainstream economics, it is important to realize exactly where we parted company. The difference is that my theory is concerned with the price and stock of phenomena while mainstream economics is concerned with the price and supply of phenomena. I assert that the stock of phenomena is more important than the supply because all of the decisions made regarding a phenomenon are based on its stock (how much of it is in existence), and not on how much of it happened to be produced in some arbitrary time period. Phenomena are the same whether they are produced in one time period or another. Most people do not know and none care what the supply of phenomena is, they are concerned with the stock; this week's or month's supply is only a small part of the available stock. Even if a factory is temporarily closed for a week or a month, the price of its product is hardly affected because the total amount of phenomena in existence is hardly affected. Yet during that week or month the supply is zero. Mainstream economics, which relates price to supply, is unable to explain why the price does not increase dramatically as inspection of the supply and demand curves predicts that it should.
Parking on campus has a price, so mainstream economists must believe that there is a supply, that is, an influx, of parking spaces. Yet none are being produced. Clearly, it is the stock, the absolute quantity of them, that determines price. Supply never means anything in economics, though sometimes (for non-durable phenomena) it can pass for stock. There are three principle mistakes of mainstream economics, but addressing supply and demand instead of price and stock is the most egregious. The other two are assuming that all short term credit instruments function as money and believing that the average price level is a meaningful statistic and, hence, that prices are “sticky.”
