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 best-off 10% of people minus the best-off 1% can be thought of as “the 9%” just below the best-off 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 time-series^{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(–0.0435 x) … … … (1) **fits the available data from 1931 to 1990 with

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(–0.093 x) + 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 wave-models 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

The model for the top 1% (shown in **Diagram 4**) is

**y**** = 4.87 + 0.17 x + 0.92[sin{2π(x+3.12)÷25.86}] + ε **which has

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

The following improvement on model (**5**) yields a better fit to the data for “the 9%”:

**y**** = 24.45 + 0.02 x – 1.37[sin{2π(x+0.56)÷12.44}] – 1.11[sin{2π(x+7.57)÷35.36}] + ε … … (6)**which has

**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 + B x + C[sin{2π(x–D)÷E}] + ε** where

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.

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## Comments

## Michael Mernagh

The above article has been mentioned in the blogosphere.

See http://www.irisheconomy.ie/index.php/2013/02/21/long-run-income-inequality-in-ireland/#comments

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## David O'Donnell

Much appreciated.

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