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Income Distributions - History Assumed log-normal - but not derived from economic theory Known power tail – Pareto strongly demonstrated by Souma Japan data
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Geoff Willis
Risk Manager
Geoff Willis & Juergen Mimkes
Evidence for the Independence of Waged and Unwaged Income,
Evidence for Boltzmann Distributionsin Waged Income,
and the Outlines of a Coherent Theoryof Income Distribution.
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
• 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• 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% 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”
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
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
Reduced Data Sets
• Deleted data above £800
• Deleted data below £130
• Repeated fitting of functions
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*Pwhere 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 SetF(x) =B*(x-G)*(EXP(-P*(x-G)))
B =10*No*P*PSo: F(x) =10*No*P*P*(x-G)*(EXP(-P*(x-G)))
Parameters varied: P & G only
One Parameter Fits
• Boltzmann function, Reduced Data SetF(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 overestimateddue to extra wealth from power tail
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
Defined Fit
• Ko & No can be calculatedfrom 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
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
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
US Income data
• Note: No power tailData drops down, not up
Believed to be detailed comparison of manufacturing income versusservices income
• Assumed that only waged income was used
Malleability of log-normal
• Un-normalised log-normal
F(x) = A*(EXP(-1*((LN(x)-M)*((LN(x)-M)))/(2*S*S)))/((x)*S*(2.5066))
is a three parameter function• A - size• M - offset• S - skew
More Theory
• Mimkes & Willis – Boltzmann distribution
• Souma & Nirei – this conference• Simple explanation for power law,
Allows savingRequires exponential base
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 incomePower law high income
• Theory supports:Boltzmann distribution low / medium income
Power law high income• Modelling supports:
Boltzmann distribution low / medium incomePower law high income
Geoff Willis
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