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Geoff Willis

Risk Manager. Geoff Willis. Geoff Willis & Juergen Mimkes. Evidence for the Independence of Waged and Unwaged Income, Evidence for Boltzmann Distributions in Waged Income, and the Outlines of a Coherent Theory of Income Distribution. Income Distributions - History.

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Geoff Willis

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  1. Risk Manager Geoff Willis

  2. Geoff Willis & Juergen Mimkes Evidence for the Independence of Waged and Unwaged Income, Evidence for Boltzmann Distributions in Waged Income, and the Outlines of a Coherent Theory of Income Distribution.

  3. Income Distributions - History • Assumed log-normal - but not derived from economic theory • Known power tail – Pareto - 1896 - strongly demonstrated by Souma Japan data - 2001

  4. Income Distributions - Alternatives • Proposed Exponential - Yakovenko & Dragelescu – US data • Proposed Boltzmann - Willis – 1993 – New Scientist letters • Proposed Boltzmann - Mimkes & Willis – Theortetical derivation - 2002

  5. UK NES Data • ‘National Earnings Survey’ • United Kingdom National Statistics Office • Annual Survey • 1% Sample of all employees • 100,000 to 120,000 in yearly sample

  6. UK NES Data • 11 Years analysed 1992 to 2002 inclusive • 1% Sample of all employees • 100,000 to 120,000 in yearly sample • Wide – PAYE ‘Pay as you earn’ • Excludes unemployed, self-employed, private income & below tax threshold “unwaged”

  7. Three Parameter Fits • Used Solver in Excel to fit two functions: • Log-normal F(x) = A*(EXP(-1*((LN(x)-M)*((LN(x)-M)))/(2*S*S)))/((x)*S*(2.5066)) Parameters varied: A, S & M

  8. Three Parameter Fits • Used Solver in Excel to fit two functions: • Boltzmann F(x) = B*(x-G)*(EXP(-P*(x-G))) Parameters varied: B, P & G

  9. Reduced Data Sets • Deleted data above £800 • Deleted data below £130 • Repeated fitting of functions

  10. Two Parameter Fits • Boltzmann function only • Reduced Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) It can be shown that: B =10*No*P*P where No is the total sum of people (factor of 10 arises from bandwidth of data:£101-£110 etc)

  11. Two Parameter Fits • Boltzmann function, Red Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) B =10*No*P*P So: F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G))) Parameters varied: P & G only

  12. One Parameter Fits • Boltzmann function, Reduced Data Set F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G))) Parameters varied: P & G only • It can be further shown that: P =2 / (Ko/No – G) where Ko is the total sum of people in each population band multiplied by average income of the band • Note that KoWill be overestimated due to extra wealth from power tail

  13. One Parameter Fits • Boltzmann function analysed only • Fitted to Reduced Data Set F(x) = B*(x-G)*(EXP(-P*(x-G))) • Can be re-written as: F(x) =10*No*(2/((Ko/No)-G))*(2/((Ko/No)-G))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) Parameter varied: G only

  14. Defined Fit • Ko & No can be calculated from the raw data • G is the offset - can be derived from the raw data - by graphical interpolation Used solver for simple linear regression, 1st 6 points 1992, 1st 12 points 1997 & 2002

  15. Defined Fit • Used function: F(x) =10*No*(2/((Ko/No)-G))*(2/((Ko/No)-G))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) • Parameter No derived from raw data • Parameter Ko derived from raw data • Parameter G extrapolated from graph of raw data Inserted Parameter into function and plotted results

  16. US Income data • Ultimate source: US Department of Labor, Bureau of Statistics • Believed to be good provenance • Details of sample size not know • Details of sampling method not know

  17. US Income data • Note: No power tail Data drops down, not up Believed to be detailed comparison of manufacturing income versus services income • Assumed that only waged income was used

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