Income inequality in Ireland from 1922 to 2009

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 coefficient³ 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-series⁴ 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 R² = 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 R² = 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 2009⁵. It shows how the share of incomes near the top is at a minimum every 12½ years.

 

The Celtic Tiger⁵ 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 data⁶ 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 R² = 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 R² = 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 R² = 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 R² = 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 y-axis; 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