From: John Conover <john@email.johncon.com>

Subject: Quantitative Analysis of Non-Linear High Entropy Economic Systems II

Date: 14 Feb 2002 07:38:46 -0000

As mentioned in Section I, much of applied economics has to address non-linear high entropy systems-those systems characterized by random fluctuations over time-such as net wealth, equity prices, gross domestic product, industrial markets, etc.

The characteristic frequency distributions, over time, of non-linear high entropy systems evolve into log-normal distributions requiring suitable methodologies for analysis.

Note: the C source code to all programs used are available from the NtropiX Utilities page, or, the NdustriX Utilities page, and are distributed under License.

Figure I presents a normal/Gaussian frequency distribution made
with the

,
and, *tsgaussian*

programs. The distribution was integrated, using the *tsnormal*

program, and the X-Axis values modified using the *tsintegrate*

program such that both the distribution and its cumulative fit on the
plot. The root-mean-square, (i.e., the deviation,) of the distribution
is unity, and the mean equals the median, which is zero.*tsmath*

If the *difference* of the increments of a system are
characterized by a normal/Gaussian distribution-i.e., *Brownian
motion*, or a *random walk*-then the system is a cumulative
sum of a random variable,

, with a
normal/Gaussian distribution; an **a(t)***additive* construct:

**
V(n + 1) = V(n) + a(t) .............................(2.1)
**

However, the non-linear, high entropy economic model described in
Section
I, is a *multiplicative* construct-it is a *geometric
progression* of a random variable,

, with a normal/Gaussian
distribution:**a(t)**

**
V(n + 1) = V(n) * (1 + a(t)) .......................(2.2)
**

which has a long term evolutionary distribution that is log-normal,
(although the *marginal* increments have a normal/Gaussian
distribution.)

To make a log-normal distribution from a normal/Gaussian
distribution, the X-Axis is *exponentiated*, (i.e., for each
number on the X-Axis, use

to that
number, to make a new X-Axis for a log-normal plot.)**e**

Figure II is a plot of the normal/Gaussian frequency distribution,
shown in Figure I, and its cumulative, with the X-Axis exponentiated,
using the

program. Note that the *tsmath**mean*
diverges to infinity, (i.e., becomes undefined,) for the log-normal
distribution. The median of the log-normal distribution,

is:**medianLN**

**
meanN
medianLN = e ..................................(2.3)
**

where

is the mean of the
normal/Gaussian distribution. The one
deviation values, **meanN**

, and
**sdLN+**

, of the log-normal distribution
are:**sdLN-**

**
sdN
sdLN+ = e * medianLN ............................(2.4)
**

**
-sdN
sdLN- = e * medianLN ...........................(2.5)
**

where

is the deviation of the
normal/Gaussian distribution, and **sdN**

is the high side deviation for the log-normal distribution, and
**sdLN+**

, the lower side. Likewise, the two
deviation values for the log-normal distribution are
**sdLN-**

and
**e^2sdN * medianLN**

, and so on.**e^-2sdN * medianLN**

The term *deviation* in normal/Gaussian and log-normal
distributions has the same meaning; for the cumulative,
84.1344746068542954% of the time, the value will be less than one
deviation, 97.7249868051820868% of the time less than two deviations,
and so on.

As a methodology, we can *map* back and fourth between
log-normal and normal/Gaussian distributions. To convert a log-normal
frequency distribution to a normal/Gaussian distribution, take the log
of the X-Axis. To convert back to a log-normal distribution,
exponentiate the X-Axis. The median of the log-normal distribution is
the mean of the normal/Gaussian distribution, exponentiated. The
deviation values of the log-normal distribution are the median of the
log-normal distribution, multiplied by by both plus and minus the
deviation of the normal/Gaussian distribution, exponentiated.

As a side bar, it is frequently expedient-particularly when
analyzing the dynamics of a high entropy non-linear system with
log-normal characteristics-to work with the Brownian
motion/random walk equivalent of the system's time series. The
system's time series can be converted back and forth between the
two, for example, (using the time series,
and:
will convert a DJIA time series to its Brownian motion/random walk equivalent, and back. Different techniques are offered in Appendix I. |

The relationship between the median of the log-normal distribution,

, and the median of the Brownian
motion/random walk equivalent, **medianLN**

,
is:**meanN**

**
meanN
medianLN = e ..................................(2.3)
**

The relationship between the one deviation value of the Brownian
motion/random walk equivilent,

, and
the one deviation values, **sdN**

, and
**sdLN+**

, of the log-normal distribution
are:**sdLN-**

**
sdN
sdLN+ = e * medianLN ............................(2.4)
**

and:

**
-sdN
sdLN- = e * medianLN ...........................(2.5)
**

respectively.

Suppose a high entropy economic system has daily marginal
increments with a root-mean-square value of the deviation equal to
0.02, (i.e., 2% per day, which would be a *typical* value for
an equity price on the US markets, the US GDP, or an industrial market
growth/market share, etc.) At the end of a calendar year, consisting
of 253 business days, the estimate, (using the Brownian motion/random
walk model-as applied in classic Black-Scholes-Merton
methodology,) for the root-mean-square of the frequency distribution
of system's value, (i.e., the equity's increase in value after
investing a calendar year,) would be

; meaning that for 84.1344746068542954% of
the time, an annual investment would increase in value by less than
31.8119474%, (and 50% of the time, the investment would decrease in
value.)**0.02 * sqrt (253) =
0.318119474**

Figure III is a plot of the normal/Gaussian frequency distribution
of the investment's value at the end of a calendar year using a random
walk model, and the non-linear high entropy model described in Section
I which produces a log-normal distribution. The graphs were
shifted using the

program for comparison of the two distributions' deviations from their
median values.*tsmath*

For the log-normal distribution, the deviations are

, or a high
side deviation of 37.454047383%, and **e^0.318119474 = 1.37454047383**

, or a low side deviation of 27.2484136%,
compared with the random walk model's deviations of +31.8119474%, and
-31.8119474%; an accuracy within 10% for for one deviation.**e^-0.318119474 =
0.727515864**

However, if we extrapolate the random walk model's deviation to three deviations, (i.e., 3 * 0.318119474 = 0.95435842149,) and expect the investment's value to be less than that for 99.8650101968363646% of the time, we have committed a two order of magnitude error in the assessment of risk! (Restated approximately-instead of the investment's gain in value being lower than a factor of 2 in one calendar year, 0.1% of the time, its actually, 10% of the time.)

The median household net wealth in the US is about $40,000 from http://www.whitehouse.gov/fsbr/income.html, and there are about a hundred million households in the US. Bill Gates' family is the richest in the US, with an estimated net wealth of around fifty billion dollars. What are the chances that any household, in the 25 year period that it took Gates to acquire his wealth, could have become the richest in the US, using a typical value for the daily variance of wealth of 3%?

Assuming wealth distribution in a society is log-normal, and using
the methodology outlined above, on the normal/Gaussian distribution's
X-Axis scale, the log-normal median would become the mean,

, and Gates'
net wealth would be **ln (40000) = 10.5966347331**

. The deviation of the normal/Gaussian
distribution would be **ln (50000000000) =
24.6352888424**

after 25 years of 253 business days.**0.03 * sqrt (25 * 253) =
2.38589605809**

We now have to solve the equation

, where
**n = (24.6352888424
- 10.5966347331) / 2.38589605809**

is the number of deviations that
separate the Gates' family from the mean in the normal/Gaussian
distribution, or **n**

deviations.**n = 5.88401747918**

But we already knew that answer. What number of deviations is one
household in one hundred million?

.
**5.616**

With a US median wealth of about $40,000, in any 25 year period, it is a virtual certainty that one person will have a net worth in the many billions of dollars. Whether Bill Gates won the lottery of the non-linear high entropy economic system to get the tail point in the log-normal distribution, or is a genius depends on who is telling the story.

Of passing interest, the distribution of a country's wealth is
characterized by the *median*, and not the *arithmetic
average*; the calculation of the average is not stable, and
diverges to infinity in the evolution of log-normal distributions-the
term *average wealth* of a country makes no sense. The peak of
the log-normal bell curve lies below the median, also-meaning that the
largest frequency of wealth is below the median value.

In other words, without wealth re-distribution, a society's household wealth will evolve into a log-normal distribution; the rich get richer, and the poor get poorer-relative to the rich-with an ever widening gap between the two.

As a side bar, it would seem-at least intuitively-that the the rich provide the capital which fuels the expansion of the economy. But in general, that is not necessarily the case. To illustrate the point, consider a very simple economy,
where the participants increase wealth, through productivity, at
about
and from Section I, Equation (1.20),
for each household, without taxation or wealth re-distribution. Now, consider there is a
since the standard deviation,
And, Equation (1.20) from Section I becomes:
doubling the society's GDP expansion. Note that all industrialized countries have Now, consider that the king of the simple society is not so
benevolent, and has his The moral? Increasing tax rates is not necessarily detrimental to the growth of the GDP-and can even enhance it-provided the government is efficient, (and an efficient government is more important than the tax rate in determining the increase in prosperity, over time.) |

According to the March, 1999 article in Information Today, Microsoft had an 86 percent marketshare of the desktop computer market, and Linux, 2.5 percent. What are the chances of Linux becoming dominant in the desktop marketplace?

We can use the Brownian motion/random walk equivalent of the
non-linear high entropy desktop marketplace, and treat the problem as
a gambler's
ruin;

, and
**ln (0.82) = -0.198450939**

, and the
probability, **ln (0.025) = -3.68887945411**

, of Linux supplanting
Microsoft would be:**L**

**
-0.198450939
L = -------------- = 0.0537970788
-3.68887945411
**

or, about 5%, or about one in twenty.

Technically, the few percentage point chance is the chance that Microsoft will fall out of favor, (or go bust,) before Linux does-like the gambler's ruin, the chance of the house going broke before the player does is virtually zero-even if the game is fair.

But that does not mean Microsoft is invincible-it would take 20 attempts to cut Microsoft's chances of remaining on top to 50%. And how many attempts has their been? Corel, FreeBSD, Sun, Linux, Borland, Apple, Digital Research, DEC, SCO, UnixWare, AT&T, IBM OS/2, AIX, etc., all of which have/had a few percentage points of the desktop marketshare.

As a side bar, for the initiated, the technique used here is not inconsistent with Cournot-Nash equilibrium for the duopoly. It does assume, as a first order approximation, that the equilibrium has simple fractal characteristics. Interestingly, Cournot-Nash equilibrium is similar to the iterated prisoner's dilemma strategic game; probably the simplest of all games where the players have to make decisions under uncertainty do to conflict of interest and self-referentiality, which is why the characteristics are stochastic in nature. The concept of self-referential indeterminacy is ubiquitous in modern economics. It would be difficult to underestimate the contributions of the penetrating intellects of Gérard Debreu and Kenneth Arrow to modern economics. Note that Walrasian General Equilibrium Theory, the static equilibrium concepts popularized by Marshall and Keynes, is qualitatively different than Cournot-Nash equilibrium. The concept of static general equilibrium was dealt a fatal intellectual blow by Debreu Sonnenschein and Mantel, (DSM,) in the early 1970's, and has largely been discounted. |

We could, also, approximate the non-linear high entropy marketplace of the computer desktop with a Brownian motion fractal, in which case, the gambler's ruin becomes:

**
0.025
L = ----- = 0.0304878049
0.82
**

or, about 3%, or about 1 in 33, which is a reasonably good approximation for such small probabilities.

It is often expedient to convert a time series with log-normal characteristics, (for example, equity prices and index values, industrial market metrics, and GDPs,) to a Brownian motion/random walk equivalent-which is the simplest fractal, and has a large mathematical infrastructure dating back to the Sixteenth Century.

To explore the different techniques available for converting a time
series with log-normal characteristics to its Brownian motion/random
walk equivalent, the

program will be used to generate a time series with known log-normal
characteristics, and the simulation converted to its Brownian
motion/random walk equivalent using the different techniques. The
characteristics are defined in a single file,
*tsinvestsim*

, which contains the single
record, "*ts*

".**dummy, p = 0.51**

The log-normal characteristics of the simulation will have an
average and deviation of the marginal increments,

and **avg =
0.0004**

,
respectively, a marginal gain, **rms = 0.02**

, and a likelihood of an up movement,
**g =
1.0002**

, using Equation
(1.24), and, Equation
(1.20). The time series will have 100,000 time units. (For an
example of such a simulation, see the analysis of GE
Equity Price in Section
I.)**P = 0.51**

The simplest two ways of conversion:

**
***tsinvestsim* -n 1000 *ts* 100000 | *tsmath* -l > *tsinvestsim.tsmath-l*
*tsinvestsim* -n 1000 *ts* 100000 | *tsfraction* | \
*tsintegrate* > *tsinvestsim.tsfraction.tsintegrate*

And plotting:

Figure IV is a plot of the Brownian motion/random walk equivalent
of the simulated time series. The first technique just takes the
natural logarithm, using the

program, of each element in the time series-since a geometric
progression of a random variable has exponential characteristics,
taking the logarithm *tsmath**linearizes* the time series. The second
technique finds the marginal increments of each element, using the

program, which is the random sequence of fluctuations in the simulated
time series, and integrates it using the *tsfraction*

program-a Brownian motion/random walk is always the cumulative sum, or
integration, of a random process.*tsintegrate*

Note that the two graphs in Figure I are identical, except in
slope. The slope of the top line is

, (note that this is a technique for measuring
**avg =
0.0004**

,) and the slope of the bottom line
is **avg**

, (which
is a technique for measuring, **ln (g) = ln (1.0002) ~ 0.0002**

, too.)
Of passing interest is that the gain, **g**

,
is not the average, **g**

, of the marginal
increments of the time series of a non-linear high entropy system with
log-normal characteristics, (a subtle fact that has **avg***bitten*
many using regression analysis.)

Of interest is the straight line slope; in the case of taking the
natural logarithm of each element in the time series, it represents
the median value of the log-normally distributed time series. If the
value of the Brownian motion/random walk equivalent is above this
line, it is a positive *bubble*-likewise, if it is below the
line, it is a negative *bubble*. The distance the value is away
from the straight line slope is the *magnitude* of the
*bubble*, and the time between crossings of the straight line
slope, the *duration* of the *bubble*. Fortunately,
there is a large mathematical infrastructure for analyzing Brownian
motion/random walk *bubbles*, (which will be explored in Section
III, and, Section
IV.)

When analyzing the dynamics of non-linear high entropy economic
systems, i.e., the duration and magnitude of *bubbles*, it is
frequently advantageous to eliminate the slope in the conversion to a
Brownian motion/random walk equivalent fractal. There are two common
methods: subtract the average,

, from
the marginal increments of the time series; least squares fit the
slope of the Brownian motion/random walk equivalent. The first
subtracts a slope that is calculated using the **avg***average* of the
marginal increments, the latter subtracts a slope that is calculated
using the *variance* of the marginal increments. If the data
set size is sufficient, both will produce nearly identical
results.

There is a third method of removing the slope in the conversion of
a non-linear high entropy system's log-normal characteristics to a
Brownian motion/random walk equivalent, and that is to subtract a
value that will make the gain,

,
unity.**g**

From Section I:

**
avg
--- + 1
rms
P = ------- ........................................(1.24)
2
**

or:

**
avg
1 - ---
rms
1 - P = ------- ....................................(2.6)
2
**

and substituting into:

**
P (1 - P)
g = (1 + rms) (1 - rms) ....................(1.20)
**

after taking the natural logarithm of each side:

**
avg
2 * ln (g) = (1 + ---) * ln (1 + rms) +
rms
avg
(1 - ---) * ln (1 - rms) ..............(2.7)
rms
**

and manipulating for

:**avg**

**
avg
2 * ln (g) = ln (1 + rms) + --- * ln (1 + rms) +
rms
ln (1 - rms) -
avg
--- * ln (1 - rms) ....................(2.8)
rms
**

or:

**
avg avg
--- * ln (1 + rms) - --- * ln (1 - rms) =
rms rms
2 * ln (g) - ln (1 + rms) - ln (1 - rms) .......(2.9)
**

and if

, then:**g = 1**

**
avg * ln (1 + rms) - avg * ln (1 - rms) =
rms (- ln (1 + rms) - ln (1 - rms)) ............(2.10)
**

and, finally:

**
rms (- ln (1 + rms) - ln (1 - rms))
avg = ----------------------------------- ..........(2.11)
ln (1 + rms) - ln (1 - rms)
**

for

.**g = 1**

Of passing interest, making

will **avg = 0***not* make

; it will
make it negative.**g = 1**

So, there are several techniques available to remove the straight line slope in the conversion from a non-linear log-normal time series to a Brownian motion/random walk equivalent:

**
***tsinvestsim* -n 1000 *ts* 100000 | *tsmath* -l | *tslsq* -o > *tsinvestsim.tsmath-l.tslsq-o*
*tsinvestsim* -n 1000 *ts* 100000 | *tsfraction* | *tsintegrate* | \
*tslsq* -o > *tsinvestsim.tsfraction.tsintegrate.tslsq-o*
*tsinvestsim* -n 1000 *ts* 100000 | *tsfraction* | *tsmath* -s 0.0004 | \
*tsintegrate* > *tsinvestsim.tsfraction.tsmath-s0.0004.tsintegrate*
*tsinvestsim* -n 1000 *ts* 100000 | *tsfraction* | *tsmath* -s 0.0002 | *tsunfraction* | \
*tsmath* -l > *tsinvestsim.tsfraction.tsmath-s0.0002.tsunfraction.tsmath-l*

And plotting:

Figure V is a plot of the *detrended*, (i.e., with the
straight line slope removed,) Brownian motion/random walk equivalent
of the simulated non-linear log-normal time series. The first two
techniques just use the

program to least squares fit, and subtract, the slope of the lines in
Figure IV. The third technique subtracts the average,
*tslsq*

of the marginal increments, using
**avg**

and the *tsfraction*

programs, and integrates, using the *tsmath*

program to produce the Brownian motion/random walk equivalent. The
forth technique uses Equation
(2.11), and is similar to the third techique, but uses the
*tsintegrate*,

program to construct the Brownian motion/random walk equivalent.*tsunfraction*

Which technique should be used? Probably:

**
***tsmath* -l *ts* | *tslsq* -o > *ts.brownian.zero.mean*

is probably the easiest, and most reliable. But many will use one
of the

and **averaging**

techniques, both-in general, they converge
from opposite directions, and it is a handy thing to know if there is
a convergence problem, (like the data set size is too small.)**least
squares**

Note that the Y-Axis values in Figures IV and V are the Brownian motion/random walk equivalents. To convert back to non-linear log-normal characteristics, (using the previous example):

**
***tsmath* -l *ts* | *tslsq* -o | *tsmath* -e > *ts.log-normal.unity.median*

which will give the minimum variance, *exponentially
detrended*, non-linear log-normal characteristics of the time
series with a median of unity-and is a very useful technique.

-- John Conover, john@email.johncon.com, http://www.johncon.com/

Copyright © 2002-2004 John Conover, john@email.johncon.com. All Rights Reserved. Last modified: Mon Jan 8 18:17:29 PST 2001 $Id: 020217114704.27107.html,v 1.0 2004/05/26 05:56:13 conover Exp $