####################################################################### # # Efficiency of Increasing the Real US GDP Per Capita: # # (Assuming an increase in the real 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 Real US GDP Per Capita: # # From: http://www.johncon.com/john/private/usgdp/uspopulation/us.real.gdp.distribution.2010 # # Total: 13087999999999.558594 # Population: 310108612.915182 # # Mode = 10798.485975 # Median = 29498.028764 # Mean = 42204.56786725644937622212 # # From: http://www.johncon.com/john/private/usgdp/uspopulation/us.real.gdp.capita.1790-2010 # # Note: The file, "us.real.gdp.capita.1790-2010", is the time series # of the real mean GDP per capita, i.e., the real annual GDP divided # by the population count. P(avg,rms) and G(avg,rms) are for the real # median GDP per capita. # # tslsq -e -p us.real.gdp.capita.1790-2010 # e^(-23.838304 + 0.017100t) # # exp (-23.838304 + (0.017100 * 2010)) # 37522.49879813671622348538 # # tsmath -l us.real.gdp.capita.1790-2010 | tslsq -p # 6.769825 + 0.017100t # # Note: These values are of the real mean GDP per capita, i.e., the # annual GDP divided by the population count, and finding the values # for the real median GDP per capita: # # r(2010) = sqrt ((2 / 3) * ln (mean(2010) / mode(2010))) # = sqrt ((2 / 3) * ln (42204.56786725644937622212 / 10798.485975)) # = 0.95328293165635798069 # # srms = r(2010) / sqrt (2010) # = 0.95328293165635798069 / sqrt (2010) # = 0.02126296324279258349 # # u(t) = a + (b * t) # mean(t) = e^(a + ((b + (srms^2 / 2)) * t)) # a = -23.838304 # b = 0.017100 - (srms^2 / 2) # = 0.017100 - (0.02126296324279258349^2 / 2) # = 0.01687394319706782575 # # median(t) = e^u(t) = e^(a + (b * t)) # = e^(-23.838304 + (0.01687394319706782575 * t)) # # From the calculate.avg.calc program, (srms = 0.02126296324279258349, # g = exp (0.01687394319706782575) = 1.0170171123188325426: # # avg = 0.01724451706865199472 # rms = 0.02737675975705266846 # G(avg,rms) = 1.01701711231883254259 # # Median Annual Real US GDP Per Capita: # # P(0.01724451706865199472,0.02737675975705266846) = # 0.81494810236279963015 # # (2 * P(0.01724451706865199472,0.02737675975705266846)) - 1 = # 0.62989620472559926029; Optimal annual Kelly criteria for # real median GDP per capita. # # 0.02737675975705266846 / 0.62989620472559926029 = # 0.04346233482860682624; Efficiency of annual "wagering" per # capita. # # G(0.62989620472559926029^2,0.62989620472559926029) = # 1.23884450571846976839; Maxium annual growth in the real median # GDP per capita. # # (1.01701711231883254259 - 1) / (1.23884450571846976839 - 1) = # 0.07124766076424179096; Efficiency of real annual median GDP per # capita growth. # # 1 - (0.02737675975705266846^2 / 0.01724451706865199472) = # 0.95653766517139317376; Optimum annual leverage. # ####################################################################### # # Effects of Governance on the Annual Mean Real US GDP Per Capita: # # The mean real 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.real.gdp.capita.1790-2010 | tsavg -p # 0.017967 # # tsfraction us.real.gdp.capita.1790-2010 | tsrms -p # 0.046864 # # srms = sqrt (0.046864^2 - 0.017967^2) # = 0.04328303832911917934 # # P(0.017967,0.046864) = # 0.69169298395356777057 # # G(0.034815,0.085756) = # 1.01702413233740541753 # # (2 * P(0.017967,0.046864)) - 1 = # 0.38338596790713554114; Optimal annual Kelly criteria for # real mean GDP per capita. # # 0.046864 / 0.38338596790713554114 = # 0.12223712895864640730; Efficiency of annual "wagering" per # capita. # # G(0.38338596790713554114^2,0.38338596790713554114) = # 1.07832381446867656866; Maxium annual growth in the real mean # GDP per capita. # # (1.01702413233740541753 - 1) / (1.07832381446867656866 - 1) = # 0.21735576149976385282; Efficiency of real annual mean GDP per # capita growth. # # 1 - (0.046864^2 / 0.017967) = # 0.87776287104135359270; 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 real US GDP per capita, # (i.e., real standard of living,) is dominated by issues of # governance by about a factor of two. In the case of the annual mean # real US GDP per capita, the wagering on innovation risk is about # 2.7%, (which is about 4.3% of optimal,) but the equivalent wagering # on governance risk is 4.7%, (which is about 12.2% of optimal,) # nearing twice as much, (if it is assumed that the affects of # governance risk affect the mean and median of the mean real 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 real US GDP per capita: # # sqrt (0.046864^2 - 0.017967^2) = # 0.04328303832911917934 # # sqrt (0.02737675975705266846^2 - 0.01724451706865199472^2) = # 0.02126296324279258349 # # sqrt (0.02126296324279258349^2 + 0.04328303832911917934^2) = # 0.0482238013108086337 # # sqrt (0.01724451706865199472^2 + 0.0482238013108086337^2) = # 0.05121433765846609563 # # avg = 0.01724451706865199472 # rms = 0.05121433765846609563 # # P(0.01724451706865199472,0.05121433765846609563) = # 0.66835634176947471409 # # G(0.01724451706865199472,0.05121433765846609563) = # 1.01607426367517698294 # # (2 * P(0.01724451706865199472,0.05121433765846609563)) -1 = # 0.33671268353894942818; Optimal annual Kelly criteria for # real mean GDP per capita. # # 0.05121433765846609563 / 0.33671268353894942818 = # 0.15210100528494599595; Efficiency of annual "wagering" per # capita. # # G(0.33671268353894942818^2,0.33671268353894942818) = # 1.05951434178000308367; Maxium annual growth in the real mean # GDP per capita. # # (1.01607426367517698294 - 1) / (1.05951434178000308367 - 1) = # 0.27009058983792645870; Efficiency of real annual mean GDP per # capita growth. # # Meaning that the individuals who are expanding the annual mean real # US GDP per capita, at an annual exponential rate of 1.016, are doing # so at a 27.0% efficiency, relative to the limitations of the Kelly # criteria, which places a maximum annual exponential rate limit of # 1.060 on the growth of the mean real US GDP per capita. # ####################################################################### # # Appendix: # ####################################################################### # # On a per worker bases, (i.e., those that are actually employed # generating the annual median real 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 * 0.02126296324279258349 # = 0.04252592648558516698 # # avg = 0.01724451706865199472 * 2 # = 0.03448903413730398944 # # rms = sqrt (0.04252592648558516698^2 + 0.03448903413730398944^2) = # = 0.05475351951410533692 # # G(0.03448903413730398944,0.05475351951410533692) # 1.03357362523003502235 # # P(0.03448903413730398944,0.05475351951410533692) # 0.81494810236279963015 # # (2 * P(0.03448903413730398944,0.05475351951410533692)) - 1 = # 0.62989620472559926029; Optimal annual Kelly criteria for real # median GDP per worker. # # 0.05475351951410533692 / 0.62989620472559926029 = # 0.08692466965721365248; Efficiency of annual "wagering" per # worker. # # G(0.62989620472559926029^2,0.62989620472559926029) = # 1.23884450571846976839; Maxium annual growth in the real median # GDP per worker. # # (1.03357362523003502235 - 1) / (1.23884450571846976839 - 1) = # 0.14056687269837743966; Efficiency of real annual median GDP per # worker growth. # # 1 - 0.05475351951410533692^2 / 0.03448903413730398944 = # 0.91307533034278634752; Optimum annual leverage. # ####################################################################### # # The mean real 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 (0.09572118866508811795^2 + 0.085756^2) = # 0.09572118866508811795 # # rms = sqrt (0.03448903413730398944^2 + 0.09572118866508811795^2) = # 0.1017449725302509182 # # G(0.03448903413730398944,0.1017449725302509182) = # 1.02984241173661079568 # # P(0.03448903413730398944,0.1017449725302509182) = # 0.66948765761890433902 # # (2 * P(0.03448903413730398944,0.1017449725302509182)) - 1 = # 0.33897531523780867803; Optimal annual Kelly criteria for real # median GDP per worker. # # 0.1017449725302509182 / 0.33897531523780867803 = # 0.30015451850489929908; Efficiency of annual "wagering" per # worker. # # G(0.33897531523780867803^2,0.33897531523780867803) = # 1.06035770111790436966; Maxium annual growth in the real median # GDP per worker. # # (1.02984241173661079568 - 1) / (1.06035770111790436966 - 1) = # 0.49442591722166189806; Efficiency of real annual median GDP per # worker growth. # # 1 - 0.1017449725302509182^2 / 0.03448903413730398944 = # 0.69984548149510070092; Optimum annual leverage. # ####################################################################### # # On a per worker bases, (i.e., those that are actually employed # generating the annual median real US GDP per capita,) and, for # 250 work days in a calendar year, the daily values of the variables: # # srms = 0.09572118866508811795 / sqrt (250) = # 0.00605393953040741694 # # avg = 0.03448903413730398944 / 250 = # 0.00013795613654921596 # # rms = sqrt (0.00013795613654921596^2 + 0.00605393953040741694^2) = # 0.00605551118679844913 # # P(0.00013795613654921596,0.00605551118679844913) = # 0.51139095712100842616 # # G(0.00013795613654921596,0.00605551118679844913) = # 1.00011963003390317854 # # (2 * P(0.00013795613654921596,0.00605551118679844913)) - 1 = # 0.02278191424201685231; Optimal daily Kelly criteria for # real median GDP per worker. # # 0.00605551118679844913 / 0.02278191424201685231 = # 0.26580344050414452122; Efficiency of daily "wagering" per # worker. # # G(0.02278191424201685231^2,0.02278191424201685231) = # 1.00025956394191968598; Maxium daily growth in the real # median GDP per worker. # # (1.00011963003390317854 - 1) / (1.00025956394191968598 - 1) = # 0.46088849251717076945; Efficiency of real daily median GDP # per worker growth. # # 1 - 0.00605551118679844913^2 / 0.00013795613654921596 = # 0.73419655949585547884; 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 real 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 = 29498.028764 # Mean = 42204.56786725644937622212 # # Median / Mean = 29498.028764 / 42204.56786725644937622212 = # 0.69892976648353369805 # # 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 # # Then, deflating the growth in the S & P 500 to 2005 chained dollars, # (from ../usdeflator/us.deflator.1790-2010, 10.66% of the growth # since 2005 was inflation-to obtain the real annual growth in the S & # P 500, through 2010): # # 1 + (0.04410547888399337599 / 1.1066) = # 1.03985674939815052954 # # For comparison, the growth in the per worker daily bases, (i.e., # those that are actually employed generating the annual median # real US GDP per capita, 1790 through 2010): # # G(0.00013795613654921596,0.00605551118679844913) = # 1.00011963003390317854 # # Annually, 1.00011963003390317854^250 = 1.03035738697085221345 # # Which are in agreement to: # # (0.03985674939815052954 - 0.03035738697085221345) / # 0.03985674939815052954 = # 0.23833761083735329557 # # Or, about 23.8%, which is somewhat reasonable, under the # assumptions, (and should be near the twice the growth rate in the # real mean GDP per capita, G(avg,rms) = 1.01701711231883254259, since # everyone works in the S & P 500, and about half do in the real 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. # #######################################################################