Geoff Willis Risk Manager Geoff Willis Juergen Mimkes
![Geoff Willis Risk Manager Geoff Willis Risk Manager](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-1.jpg)
![Geoff Willis & Juergen Mimkes Evidence for the Independence of Waged and Unwaged Income, Geoff Willis & Juergen Mimkes Evidence for the Independence of Waged and Unwaged Income,](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-2.jpg)
![Income Distributions - History • Assumed log-normal - but not derived from economic theory Income Distributions - History • Assumed log-normal - but not derived from economic theory](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-3.jpg)
![Income Distributions - Alternatives • Proposed Exponential - Yakovenko & Dragelescu – US data Income Distributions - Alternatives • Proposed Exponential - Yakovenko & Dragelescu – US data](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-4.jpg)
![UK NES Data • • • ‘National Earnings Survey’ United Kingdom National Statistics Office UK NES Data • • • ‘National Earnings Survey’ United Kingdom National Statistics Office](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-5.jpg)
![UK NES Data • • • 11 Years analysed 1992 to 2002 inclusive 1% UK NES Data • • • 11 Years analysed 1992 to 2002 inclusive 1%](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-6.jpg)
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![Three Parameter Fits • Used Solver in Excel to fit two functions: • Log-normal Three Parameter Fits • Used Solver in Excel to fit two functions: • Log-normal](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-10.jpg)
![Three Parameter Fits • Used Solver in Excel to fit two functions: • Boltzmann Three Parameter Fits • Used Solver in Excel to fit two functions: • Boltzmann](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-11.jpg)
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![Reduced Data Sets • Deleted data above £ 800 • Deleted data below £ Reduced Data Sets • Deleted data above £ 800 • Deleted data below £](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-19.jpg)
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![Two Parameter Fits • Boltzmann function only • Reduced Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) It Two Parameter Fits • Boltzmann function only • Reduced Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) It](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-27.jpg)
![Two Parameter Fits • Boltzmann function, Red Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) B =10*No*P*P So: Two Parameter Fits • Boltzmann function, Red Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) B =10*No*P*P So:](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-28.jpg)
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![One Parameter Fits • Boltzmann function, Reduced Data Set F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G))) Parameters varied: P One Parameter Fits • Boltzmann function, Reduced Data Set F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G))) Parameters varied: P](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-32.jpg)
![One Parameter Fits • Boltzmann function analysed only • Fitted to Reduced Data Set One Parameter Fits • Boltzmann function analysed only • Fitted to Reduced Data Set](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-33.jpg)
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![Defined Fit can be calculated from the raw data • G is the offset Defined Fit can be calculated from the raw data • G is the offset](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-37.jpg)
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![Defined Fit • Used function: F(x) =10*No*(2/((Ko/No)-G))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) • Parameter No derived from raw data Defined Fit • Used function: F(x) =10*No*(2/((Ko/No)-G))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) • Parameter No derived from raw data](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-40.jpg)
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![US Income data • Ultimate source: US Department of Labor, Bureau of Statistics • US Income data • Ultimate source: US Department of Labor, Bureau of Statistics •](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-44.jpg)
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![US Income data • Note: No power tail Data drops down, not up Believed US Income data • Note: No power tail Data drops down, not up Believed](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-48.jpg)
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![Malleability of log-normal • Un-normalised log-normal F(x) = A*(EXP(-1*((LN(x)-M)))/(2*S*S)))/((x)*S*(2. 5066)) is a three parameter Malleability of log-normal • Un-normalised log-normal F(x) = A*(EXP(-1*((LN(x)-M)))/(2*S*S)))/((x)*S*(2. 5066)) is a three parameter](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-52.jpg)
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![More Theory • Mimkes & Willis – Boltzmann distribution • Souma & Nirei – More Theory • Mimkes & Willis – Boltzmann distribution • Souma & Nirei –](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-54.jpg)
![Modelling • Chattarjee, Chakrabati, Manna, Das, Yarlagadda etc • Have demonstrated agent models that: Modelling • Chattarjee, Chakrabati, Manna, Das, Yarlagadda etc • Have demonstrated agent models that:](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-55.jpg)
![Conclusions • Evidence supports: Boltzmann distribution low / medium income Power law high income Conclusions • Evidence supports: Boltzmann distribution low / medium income Power law high income](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-56.jpg)
![Geoff Willis Geoff Willis](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-57.jpg)
- Slides: 57
![Geoff Willis Risk Manager Geoff Willis Risk Manager](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-1.jpg)
Geoff Willis Risk Manager
![Geoff Willis Juergen Mimkes Evidence for the Independence of Waged and Unwaged Income Geoff Willis & Juergen Mimkes Evidence for the Independence of Waged and Unwaged Income,](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-2.jpg)
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 Assumed lognormal but not derived from economic theory Income Distributions - History • Assumed log-normal - but not derived from economic theory](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-3.jpg)
Income Distributions - History • Assumed log-normal - but not derived from economic theory • Known power tail – Pareto - 1896 - strongly demonstrated by Souma Japan data - 2001
![Income Distributions Alternatives Proposed Exponential Yakovenko Dragelescu US data Income Distributions - Alternatives • Proposed Exponential - Yakovenko & Dragelescu – US data](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-4.jpg)
Income Distributions - Alternatives • Proposed Exponential - Yakovenko & Dragelescu – US data • Proposed Boltzmann - Willis – 1993 – New Scientist letters • Proposed Boltzmann - Mimkes & Willis – Theortetical derivation - 2002
![UK NES Data National Earnings Survey United Kingdom National Statistics Office UK NES Data • • • ‘National Earnings Survey’ United Kingdom National Statistics Office](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-5.jpg)
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
![UK NES Data 11 Years analysed 1992 to 2002 inclusive 1 UK NES Data • • • 11 Years analysed 1992 to 2002 inclusive 1%](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-6.jpg)
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”
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![Three Parameter Fits Used Solver in Excel to fit two functions Lognormal Three Parameter Fits • Used Solver in Excel to fit two functions: • Log-normal](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-10.jpg)
Three Parameter Fits • Used Solver in Excel to fit two functions: • Log-normal F(x) = A*(EXP(-1*((LN(x)-M)))/(2*S*S)))/((x)*S*(2. 5066)) Parameters varied: A, S & M
![Three Parameter Fits Used Solver in Excel to fit two functions Boltzmann Three Parameter Fits • Used Solver in Excel to fit two functions: • Boltzmann](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-11.jpg)
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
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![Reduced Data Sets Deleted data above 800 Deleted data below Reduced Data Sets • Deleted data above £ 800 • Deleted data below £](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-19.jpg)
Reduced Data Sets • Deleted data above £ 800 • Deleted data below £ 130 • Repeated fitting of functions
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![Two Parameter Fits Boltzmann function only Reduced Data Set Fx BxGEXPPxG It Two Parameter Fits • Boltzmann function only • Reduced Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) It](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-27.jpg)
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)
![Two Parameter Fits Boltzmann function Red Data Set Fx BxGEXPPxG B 10NoPP So Two Parameter Fits • Boltzmann function, Red Data Set F(x) =B*(x-G)*(EXP(-P*(x-G))) B =10*No*P*P So:](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-28.jpg)
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
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![One Parameter Fits Boltzmann function Reduced Data Set Fx 10NoPPxGEXPPxG Parameters varied P One Parameter Fits • Boltzmann function, Reduced Data Set F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G))) Parameters varied: P](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-32.jpg)
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 Ko Will be overestimated due to extra wealth from power tail
![One Parameter Fits Boltzmann function analysed only Fitted to Reduced Data Set One Parameter Fits • Boltzmann function analysed only • Fitted to Reduced Data Set](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-33.jpg)
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))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) Parameter varied: G only
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![Defined Fit can be calculated from the raw data G is the offset Defined Fit can be calculated from the raw data • G is the offset](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-37.jpg)
Defined Fit 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, 1 st 6 points 1992, 1 st 12 points 1997 & 2002 • Ko & No
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![Defined Fit Used function Fx 10No2KoNoGxGEXP2KoPopGxG Parameter No derived from raw data Defined Fit • Used function: F(x) =10*No*(2/((Ko/No)-G))*(x-G)*(EXP(-(2/((Ko/Pop)-G))*(x-G))) • Parameter No derived from raw data](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-40.jpg)
Defined Fit • Used function: F(x) =10*No*(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
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![US Income data Ultimate source US Department of Labor Bureau of Statistics US Income data • Ultimate source: US Department of Labor, Bureau of Statistics •](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-44.jpg)
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
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![US Income data Note No power tail Data drops down not up Believed US Income data • Note: No power tail Data drops down, not up Believed](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-48.jpg)
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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![Malleability of lognormal Unnormalised lognormal Fx AEXP1LNxM2SSxS2 5066 is a three parameter Malleability of log-normal • Un-normalised log-normal F(x) = A*(EXP(-1*((LN(x)-M)))/(2*S*S)))/((x)*S*(2. 5066)) is a three parameter](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-52.jpg)
Malleability of log-normal • Un-normalised log-normal F(x) = A*(EXP(-1*((LN(x)-M)))/(2*S*S)))/((x)*S*(2. 5066)) is a three parameter function • A - size • M - offset • S - skew
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![More Theory Mimkes Willis Boltzmann distribution Souma Nirei More Theory • Mimkes & Willis – Boltzmann distribution • Souma & Nirei –](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-54.jpg)
More Theory • Mimkes & Willis – Boltzmann distribution • Souma & Nirei – this conference • Simple explanation for power law, Allows saving Requires exponential base
![Modelling Chattarjee Chakrabati Manna Das Yarlagadda etc Have demonstrated agent models that Modelling • Chattarjee, Chakrabati, Manna, Das, Yarlagadda etc • Have demonstrated agent models that:](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-55.jpg)
Modelling • Chattarjee, Chakrabati, Manna, Das, Yarlagadda etc • Have demonstrated agent models that: – give exponential results (no saving) – give power tails (saving allowed)
![Conclusions Evidence supports Boltzmann distribution low medium income Power law high income Conclusions • Evidence supports: Boltzmann distribution low / medium income Power law high income](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-56.jpg)
Conclusions • Evidence supports: Boltzmann distribution low / medium income Power law high income • Theory supports: Boltzmann distribution low / medium income Power law high income • Modelling supports: Boltzmann distribution low / medium income Power law high income
![Geoff Willis Geoff Willis](https://slidetodoc.com/presentation_image/3472e46726fd753fb192acfa12527cc6/image-57.jpg)
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