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					Geoff Willis
 Risk Manager
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 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*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
  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 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 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 tail
         Data drops down, not up

Believed to be detailed comparison of
 manufacturing income versus
 services 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

				
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posted:6/4/2013
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