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