1. Geoff Willis Risk Manager

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”

10. 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

11. 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

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

27. 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 N o is the total sum of people (factor of 10 arises from bandwidth of data:£101- £110 etc)

28. 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

32. 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

33. 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

37. 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, 1 st 6 points 1992, 1 st 12 points 1997 & 2002

40. 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

44. 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

48. 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

52. 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

54. More Theory Mimkes & Willis – Boltzmann distribution Souma & Nirei – this conference Simple explanation for power law, Allows saving Requires exponential base

55. Modelling Chattarjee, Chakrabati, Manna, Das, Yarlagadda etc Have demonstrated agent models that: –give exponential results (no saving) –give power tails (saving allowed)

56. 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

57. Geoff Willis