Geoff Willis
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Transcript of Geoff Willis
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 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
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 SetF(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 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
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 saving
Requires 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 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