I read and watched with interest the Beveridge Lecture given by Professor Danny Dorling at the Royal Statistical Society in 2012.^{1,2} I was particularly interested in his comments on “the 9%” of the income distribution as an alternative statistic to the Gini coefficient. The bestoff 10% of people minus the bestoff 1% can be thought of as “the 9%” just below the bestoff 1%.^{1}
The Gini coefficient^{3 }is often used to summarise income inequalities. It is confusing because it has no parallel in real life. The ratio of the area under the Lorenz curve to the area of a triangle is not a very intuitive concept.^{1} Changes in the Gini coefficient are difficult to interpret over time. Assessing the significance of such changes is a rather abstract process.
The percentage share of income of “the 9%” is easier to understand. It is a good proxy statistic for the Gini coefficient.
The timeseries^{4} for the share of income by the top 0.1% of taxpayers in Ireland began in 1922 and ended in 1990, with a discontinuity of ten years from 1954 to 1963 and a missing value for 1974.
The decay model y = 0.67 + 11.09exp(–0.0435x) … … … (1) fits the available data from 1931 to 1990 with R^{2} = 97%. (See Diagram 1.)
A wave model can be fitted to the subset of the above data from1931 to 1953 by incorporating a sine function. The model (shown in Diagram 2) then is
y = 3.15 + 4.76exp(–0.093x) + 0.48[sin{2π(x–3.40)÷10.46}] + ε … … … (2)
which has R^{2} = 98%.
Further econometric analysis follows of the pattern in “the 9%” share of the overall income distribution in Ireland over the 35 years from 1975 to 2009^{5}. It shows how the share of incomes near the top is at a minimum every 12½ years.
The Celtic Tiger^{5} years between 1995 and 2008 were a period of rapid economic growth. Recession then set in.
Time series wavemodels can be fitted to the data^{6 }for the top 10% and the top 1% of earners.
(See footnote on the methodology.)
The model for the top 10% (shown in Diagram 3) is
y = 29.29 + 0.20x – 1.45[sin{2π(x+0.66)÷12.27}] + ε which has R^{2} = 91%. … … … (3)
The model for the top 1% (shown in Diagram 4) is
y = 4.87 + 0.17x + 0.92[sin{2π(x+3.12)÷25.86}] + ε which has R^{2} = 94%. … … … (4)
Subtract the r.h.s. of model (4) from the r.h.s. of model (3) to get
y = 24.41 + 0.03x – 1.45[sin{2π(x+0.66)÷12.27}] – 0.92[sin{2π(x+3.12)÷25.86}] + ε … … (5)
which has R^{2} = 79%.
The following improvement on model (5) yields a better fit to the data for “the 9%”:
y = 24.45 + 0.02x – 1.37[sin{2π(x+0.56)÷12.44}] – 1.11[sin{2π(x+7.57)÷35.36}] + ε … … (6)
which has R^{2} = 92%.
Diagram 5 shows how the share of all earnings for “the 9%” oscillates to a local minimum every 12½ years.
Diagram 6 shows Model (6) in the long run (assuming that the patterns since 1974 continue).
Data: See the Excel spreadsheet of calculations here.
Footnote on the methodology: The wave models of the time series from 1975 to 2009 are of the form y = A + Bx + C[sin{2π(x–D)÷E}] + ε where x = Year – 1974; y = income share (%); D = horizontal displacement from the yaxis; E = periodicity; and ε = residual error.
The parameters A, B, and C were estimated by the matrix formula for multiple regression coefficients.
Initial guesses (based on judgment of the periodicity by eye) were assigned to the parameters D and E, which were then adjusted by linear programming (using Excel Solver) to minimise further the sum of squares of the residual errors.
Changes in D and E automatically changed the values of A, B, and C in the regression formula.
References

Fairness and the changing fortunes of people in Britain 19702012 by Danny Dorling
(The Royal Statistical Society's 2012 Beveridge Lecture)
http://www.sasi.group.shef.ac.uk/publications/2013/dorling_jroyalstatsoc_176_1.pdf.
See also: The NoNonsense Guide to Equality by Danny Dorling (1998), ISBN 9781780260716, published by New Internationalist.  http://www.youtube.com/watch?v=wplYhygkiU
 Gini coefficient: http://en.wikipedia.org/wiki/Gini_coefficient

Alvaredo Facundo, Anthony B. Atkinson, Thomas Piketty and Emmanuel Saez
The World Top Incomes Database http://topincomes.gmond.parisschoolofeconomics.eu/ (accessed 28/01/2013)  The Celtic Tiger: http://en.wikipedia.org/wiki/Celtic_Tiger
 LongTerm Trends in Top Income Shares in Ireland, Nolan, Brian, 2007, in Top Incomes over the Twentieth Century: A Contrast between Continental European and EnglishSpeaking Countries, ed. Anthony B. Atkinson and Thomas Piketty, 501–30, Oxford University Press.