#######################################################################
#
# Efficiency of Increasing the Nominal US GDP Per Capita:
#
# (Assuming an increase in the nominal US GDP per capita is a metric
# of increasing the US standard of living.)
#
# See:
#
#     http://www.johncon.com/ntropix/,
#     http://www.johncon.com/ndustrix/,
#     http://www.johncon.com/john/private/usgdp/uspopulation/tsinvestsim-lognormal/index.html,
#     http://www.johncon.com/john/private/usgdp/uspopulation/us.nominal.gdp.capita.1790-2010.README,
#     http://www.johncon.com/john/private/usgdp/uspopulation/us.real.gdp.capita.1790-2010.README
#
# for theoretical details.
#
#######################################################################
#
# Median Nominal US GDP Per Capita:
#
# From: http://www.johncon.com/john/private/usgdp/uspopulation/us.nominal.gdp.distribution.2010
#
# Total: 14526499999984.218750
# Population: 310108612.915182
#
# Mode = 11985.345852
# Median = 32740.152418
# Mean = 46843.26521416988446862474
#
# From: http://www.johncon.com/john/private/usgdp/uspopulation/us.nominal.gdp.capita.1790-2010
#
# Note: The file, "us.nominal.gdp.capita.1790-2010", is the time
# series of the nominal mean GDP per capita, i.e., the nominal annual
# GDP divided by the population count. P(avg,rms) and G(avg,rms) are
# for the nominal median GDP per capita.
#
# tslsq -e -p us.nominal.gdp.capita.1790-2010
# e^(-49.864050 + 0.029619t)
#
# exp (-49.864050 + (0.029619 * 2010))
# 15837.56612757001819720934
#
# tsmath -l us.nominal.gdp.capita.1790-2010 | tslsq -p
# 3.154062 + 0.029619t
#
# Note: These values are of the nominal mean GDP per capita, i.e., the
# annual GDP divided by the population count, and finding the values
# for the nominal median GDP per capita:
#
# r(2010) = sqrt ((2 / 3) * ln (mean(2010) / mode(2010)))
#         = sqrt ((2 / 3) * ln (46843.26521416988446862474 / 11985.345852))
#         = 0.95328293166682675904
#
# srms = r(2010) / sqrt (2010)
#      = 0.95328293166682675904 / sqrt (2010)
#      = 0.02126296324302608945
#
# u(t) = a + (b * t)
# mean(t) = e^(a + ((b + (srms^2 / 2)) * t))
# a = -49.864050
# b = 0.029619 - (srms^2 / 2)
#   = 0.029619 - (0.02126296324302608945^2 / 2)
#   = 0.02939294319706286072
#
# median(t) = e^u(t) = e^(a + (b * t))
#           = e^(-49.864050 + (0.02939294319706286072 * t))
#
# From the calculate.avg.calc program, (srms = 0.02126296324302608945,
# g = exp (0.02939294319706286072) = 1.02982917935065044642:
#
#     avg = 0.03005759655532852687
#     rms = 0.03681810310916595658
#     G(avg,rms) = 1.02982917935065044643
#
# Median Annual Nominal US GDP Per Capita:
#
# P(0.03005759655532852687,0.03681810310916595658) =
#     0.90819045546979326710
#
# (2 * P(0.03005759655532852687,0.03681810310916595658)) - 1 =
#     0.81638091093958653420; Optimal annual Kelly criteria for
#     nominal median GDP per capita.
#
# 0.03681810310916595658 / 0.81638091093958653420 =
#     0.04509917198675233829; Efficiency of annual "wagering" per
#     capita.
#
# G(0.81638091093958653420^2,0.81638091093958653420) =
#     1.47173592150816723573; Maxium annual growth in the nominal
#     median GDP per capita.
#
# (1.02982917935065044643 - 1) / (1.47173592150816723573 - 1) =
#     0.06323279188764091037; Efficiency of nominal annual median GDP
#     per capita growth.
#
# 1 - (0.03681810310916595658^2 / 0.03005759655532852687) =
#     0.95490082801324766171; Optimum annual leverage.
#
#######################################################################
#
# Effects of Governance on the Annual Mean Nominal US GDP Per Capita:
#
# The mean nominal US GDP per capita, with a population of 310
# million, (of which about half work,) should be a nearly perfect
# rising exponential. The deviation from that exponential is the
# result of governance related issues that affect, broadly, all, or
# nearly all, of the population. By definition, anything that affects
# all, or nearly all, of the population is an issue of governance:
#
# tsfraction us.nominal.gdp.capita.1790-2010 | tsavg -p
# 0.034815
#
# tsfraction us.nominal.gdp.capita.1790-2010 | tsrms -p
# 0.085756
#
# srms = sqrt (0.085756^2 - 0.034815^2)
#      = 0.07837095961515336709
#
# P(0.034815,0.085756) =
#     0.70298871216008209338
#
# G(0.034815,0.085756) =
#     1.03170223114205159087
#
# (2 * P(0.034815,0.085756)) - 1 =
#     0.40597742432016418676; Optimal annual Kelly criteria for
#     nominal mean GDP per capita.
#
# 0.085756 / 0.40597742432016418676 =
#     0.21123342053712480253; Efficiency of annual "wagering" per
#     capita.
#
# G(0.40597742432016418676^2,0.40597742432016418676) =
#     1.08853909360857487045; Maxium annual growth in the nominal
#     mean GDP per capita.
#
# (1.03170223114205159087 - 1) / (1.08853909360857487045 - 1) =
#     0.35805913354167520173; Efficiency of nominal annual mean GDP
#     per capita growth.
#
# 1 - (0.085756^2 / 0.034815) =
#     0.78876657946287519747; Optimum annual leverage.
#
#######################################################################
#
# Conclusion:
#
# Every individual in the population has to make decisions concerning
# increasing their personal standard of living. These decisions are
# largely gambling. For an individual, P is a metric on a individual's
# ability to synthesize innovation based on new information,
# (original, secret, or public,) and f = rms is a metric on the amount
# of the individual's wager on the innovation, (largely deduced
# intuitively,) with avg being the return, (positive or negative,)
# generated by the innovation, i.e., metrics on how smart an
# individual is, and how good a gambler.
#
# An individual's risk assessment to the mean nominal US GDP per
# capita, (i.e., nominal standard of living,) is dominated by issues
# of governance by about a factor of two.  In the case of the annual
# mean nominal US GDP per capita, the wagering on innovation risk is
# about 3.7%, (which is about 4.5% of optimal,) but the equivalent
# governance risk is 8.6%, (which is about 21.1% of optimal,) slightly
# more than twice as much, (if it is assumed that the affects of
# governance risk affect the mean and median of the mean nominal US
# GDP per capita about the same, since governance risk affects all, or
# nearly all, of the population.)
#
# Assuming the innovation and governance risks are IID, the
# perspective of the individual's, who are taking the risks in
# expanding the mean nominal US GDP per capita:
#
# sqrt (0.085756^2 - 0.034815^2) =
#     0.07837095961515336709
#
# sqrt (0.03681810310916595658^2 - 0.03005759655532852687^2) =
#     0.02126296324302608944
#
# sqrt (0.02126296324302608944^2 + 0.07837095961515336709^2) =
#     0.08120419272965084132
#
# sqrt (0.03005759655532852687^2 + 0.08120419272965084132^2) =
#     0.08658856753381000401
#
# avg = 0.03005759655532852687
# rms = 0.08658856753381000401
#
# P(0.03005759655532852687,0.08658856753381000401) =
#     0.67356561848417239583
#
# G(0.03005759655532852687,0.08658856753381000401) =
#     1.02672091064809493338
#
# (2 * P(0.03005759655532852687,0.08658856753381000401)) -1 =
#     0.34713123696834479166; Optimal annual Kelly criteria for
#     nominal mean GDP per capita.
#
# 0.08658856753381000401 / 0.34713123696834479166 =
#     0.24944043725372399436; Efficiency of annual "wagering" per
#     capita.
#
# G(0.34713123696834479166^2,0.34713123696834479166) =
#     1.06345438827470382832; Maxium annual growth in the nominal
#     mean GDP per capita.
#
# (1.02672091064809493338 - 1) / (1.06345438827470382832 - 1) =
#     0.42110421949725513328; Efficiency of nominal annual mean GDP
#     per capita growth.
#
# Meaning that the individuals who are expanding the annual mean
# nominal US GDP per capita, at an annual exponential rate of 1.027,
# are doing so at a 42.1% efficiency, relative to the limitations of
# the Kelly criteria, which places a maximum annual exponential rate
# limit of 1.063 on the growth of the mean nominal US GDP per capita.
#
#######################################################################
#
# Appendix:
#
#######################################################################
#
# On a per worker bases, (i.e., those that are actually employed
# generating the annual median nominal US GDP per capita,) for the
# log-normal distribution of productivity: the population decreases by
# a factor of about 2, but the productivity in each "bin" increases by
# a factor of about 2; also, srms increases by a factor of 2; and avg
# increases by a factor of about 2:
#
# srms = 2 * 0.02126296324302608945
#      = 0.0425259264860521789
#
# avg = 0.03005759655532852687 * 2
#     = 0.06011519311065705374
#
# rms = sqrt (0.0425259264860521789^2 + 0.06011519311065705374^2) =
#     = 0.07363620621833191317
#
# G(0.06011519311065705374,0.07363620621833191317)
#     1.05919128639619883616
#
# P(0.06011519311065705374,0.07363620621833191317) =
#     0.78859336093743695067
#
# (2 * P(0.06011519311065705374,0.07363620621833191317)) - 1 =
#     0.90819045546979326705; Optimal annual Kelly criteria for
#     nominal median GDP per worker.
#
# 0.07363620621833191317 / 0.90819045546979326705 =
#     0.08108013663306008968; Efficiency of annual "wagering" per
#     worker.
#
# G(0.90819045546979326705^2,0.90819045546979326705) =
#     1.66008928054132611359; Maxium annual growth in the nominal
#     median GDP per worker.
#
# (1.05919128639619883616 - 1) / (1.66008928054132611359 - 1) =
#     0.08967163706651505894; Efficiency of nominal annual median GDP
#     per worker growth.
#
# 1 - 0.07363620621833191317^2 / 0.06011519311065705374 =
#     0.90980165602649532340; Optimum annual leverage.
#
#######################################################################
#
# The mean nominal US GDP per capita, with a population of 310
# million, (of which about half work,) should be a nearly perfect
# rising exponential. The deviation from that exponential is the
# result of governance related issues that affect, broadly, all, or
# nearly all, of the population. By definition, anything that affects
# all, or nearly all, of the population is an issue of governance:
#
# srms = sqrt (0.07363620621833191317^2 + 0.085756^2) =
#     0.11303266077655919499
#
# rms = sqrt (0.06011519311065705374^2 + 0.11303266077655919499^2) =
#     0.12802429005841154366
#
# G(0.06011519311065705374,0.12802429005841154366) =
#     1.05356945797450389460
#
# P(0.06011519311065705374,0.12802429005841154366) =
#     0.73478041972827687724
#
# (2 * P(0.06011519311065705374,0.12802429005841154366)) - 1 =
#     0.46956083945655375449; Optimal annual Kelly criteria for
#     nominal median GDP per worker.
#
# 0.12802429005841154366 / 0.46956083945655375449 =
#     0.27264686341088506855; Efficiency of annual "wagering" per
#     worker.
#
# G(0.46956083945655375449^2,0.46956083945655375449) =
#     1.12153832050594494739; Maxium annual growth in the nominal
#     median GDP per worker.
#
# (1.05356945797450389460 - 1) / (1.12153832050594494739 - 1) =
#     0.44076187453885039035; Efficiency of nominal annual median GDP
#     per worker growth.
#
# 1 - 0.12802429005841154366^2 / 0.06011519311065705374 =
#     0.72735313658911493145; Optimum annual leverage.
#
#######################################################################
#
# On a per worker bases, (i.e., those that are actually employed
# generating the annual median nominal US GDP per capita,) and, for
# 250 work days in a calendar year, the daily values of the variables:
#
# srms = 0.11303266077655919499 / sqrt (250) =
#     0.00714881316086207516
#
# avg = 0.06011519311065705374 / 250 =
#     0.00024046077244262821
#
# rms = sqrt (0.00024046077244262821^2 + 0.00714881316086207516^2) =
#     0.0071528561422692208
#
# P(0.00024046077244262821,0.0071528561422692208) =
#     0.51680872421169250559
#
# G(0.00024046077244262821,0.0071528561422692208) =
#     1.00021490563246191391
#
# (2 * P(0.00024046077244262821,0.0071528561422692208)) - 1 =
#     0.03361744842338501118; Optimal daily Kelly criteria for
#     nominal median GDP per worker.
#
# 0.0071528561422692208 / 0.03361744842338501118 =
#     0.21277213107267064731; Efficiency of daily "wagering" per
#     worker.
#
# G(0.03361744842338501118^2,0.03361744842338501118) =
#     1.00056533264104095705; Maxium daily growth in the nominal
#     median GDP per worker.
#
# (1.00021490563246191391 - 1) / (1.00056533264104095705 - 1) =
#     0.38014014557200225601; Efficiency of nominal daily median GDP
#     per worker growth.
#
# 1 - 0.0071528561422692208^2 / 0.00024046077244262821 =
#     0.78722786892732935268; Optimum daily leverage.
#
#######################################################################
#
# Notes and Asides:
#
#######################################################################
#
# Note that, as a confidence estimate in the analysis, comparing the
# growth in the daily returns of the S & P 500, (a broad base market
# index, which is, therefore, in nominal dollars,) 1928 through 2010,
# with the growth in the per worker daily bases, (i.e., those that are
# actually employed generating the annual median nominal US GDP per
# capita, 1790 through 2010):
#
# And, converting the 2010 S & P 500 from the mean value, (since it is
# the sum aggregate of the productivities of the 500 companies, or
# more correctly, the market estimate thereof, plus a speculative
# variance,) to the median value, (assuming the ratio of the mean and
# median of S & P 500 is similar to the per worker daily basis):
#
# Median = 32740.152418
# Mean = 46843.26521416988446862474
#
# Median / Mean = 32740.152418 / 46843.26521416988446862474 =
#     0.69892976649493353331
#
# cut -f2 ../sp500/sp1928 | tslsq -e -p
# e^(1.715575 + 0.000247t)
#
# exp (0.000247) = 1.00024703050701169226
#
# And, for the last days in 2010, the value of the daily gain of the
# median S & P 500, (1928 through 2010):
#
# 1 + (0.00024703050701169226 * 0.69892976649493353331) =
#     1.00017265697458280711
#
# Annually, 1.00017265697458280711^250 = 1.04410547888399337599
#
# For comparison, the growth in the per worker daily bases, (i.e.,
# those that are actually employed generating the annual median
# nominal US GDP per capita, 1790 through 2010):
#
# G(0.00024046077244262821,0.0071528561422692208) =
#     1.00021490563246191391
#
# Annually, 1.00021490563246191391^250 = 1.05518977882054427392
#
# Which are in agreement to:
#
# (0.04410547888399337599 - 0.05518977882054427392) /
#     0.04410547888399337599 =
#     -0.25131344715029446800
#
# Or, about -25.1%, which is somewhat reasonable, under the
# assumptions, (and should be near the twice the growth rate in the
# nominal mean GDP per capita, G(avg,rms) = 1.02982917935065044643,
# since everyone works in the S & P 500, and about half do in the
# nominal mean GDP per capita.)
#
#######################################################################
#
# Note that the fraction of the U.S. population that works in the
# workforce is about 50%:
#
#    From:
#
#         http://quickfacts.census.gov/qfd/states/00000.html
#
#             23.5% of the population is Persons under 18 years,
#             percent, 2012.
#
#             23.5% * 16 / 17 = 22.11764705882352941176%
#
#         http://data.bls.gov/timeseries/LNS11300000?years_option=specific_years&include_graphs=true&to_year=2010&from_year=1948
#
#             Labor Force Participation Rate, 16 years and over, is
#             64.70833333333333333333%.
#
# Or, the fraction of the U.S. population working is (1 -
# 0.2211764705882352941176) * 0.6470833333333333333333 =
# 0.50396372549019607843, (although other estimates range from 40% to
# 60%.)
#
#######################################################################
#
# Note that the World's real mean GDP annual growth per capita has not
# changed since the early 1800's, (measured in 1990 International
# Dollars, records prior to 1850 removed,) either:
#
# tslsq -e -p ~/../historical/world.gdp.capita.1850-1950
# e^(-30.963479 + 0.016111t)
#
# exp (0.016111)
#      1.01624148195026728597
#
# And, for the real mean US GDP annual growth per capita, 1790 through
# 2010:
#
# tslsq -e -p us.real.gdp.capita.1790-2010
# e^(-23.838304 + 0.017100t)
#
# is 130 at about t = 1650, which is about the same time, (see:
# ../historical/world.gdp.capita.10000BCE-2010,) the modern era of
# civilization started; the world GDP per capita was nearly constant
# from the age of flint until 1600, when the rate of annual increases
# in the mean world real GDP per capita increased from unity to about
# 1.017, and has remained constant since the 1600's. Similarly, for
# the mean real US GDP per capita.
#
#######################################################################
#
# Note that the interval of 1800 (or even as far back as the early
# 1600's,) to 2000 in
# ~/us.energy.consumption/us.total.energy.consumption.1635-2000 and
# ~/us.energy.consumption/us.real.gdp.capita.1790-2010, the
# "mechanism" of an industrialized society is to increase energy
# usage, (perhaps through new product innovation, for example,) and
# multiply the cost of that increase in energy by a factor of about
# two, to get the increase in the mean US real GDP created by the
# innovation.
#
#######################################################################