8. Annex 2: Inequality metrics

This annex presents the key metrics used to perform the analysis of income inequalities for the Nordic countries.

8.1. Global inequality indicators

8.1.1. Lorenz curves

The Lorenz curve of income typically shows the percentage of total income earned by the cumulative percentage of the population. In this case, the curve illustrates the percentage of average household income “earned” by a cumulative percentage of municipalities.

Under a perfectly equal distribution of household income across the territory, the “poorest” 25% of municipalities would receive 25% of the total average income, the “poorest” 50% would earn 50% of the total income and so on. In that scenario, the Lorenz curve would be a straight line following the 45° line of equality. As territorial inequality increases, the Lorenz curve deviates from the line of equality. For instance, the “poorest” 25% of municipalities might earn only 10% of the total household average income, while the “poorest” 50% might represent 20% of the mean household income and so forth.

Figure 20: Lorenz curve of household income (2022)

Figure 20 illustrates a Lorenz curve for all Nordic municipalities in 2022, calculated based on mean equivalised household income. The Lorenz curve shows a fairly balanced distribution of the indicator, reflecting a relatively equitable territorial structure. On average, mean equivalised household income at the municipal level appears to be evenly distributed across the territory. However, it is important to highlight that this metric can conceal significant intra-municipal disparities between income groups. Nevertheless, the Lorenz curve remains a fundamental metric for calculating many of the inequality measures described below.

8.1.2. An overview of widely used inequality indices

There are literally hundreds of indicators of inequality. In this section we review a selection of those that are most commonly used in social research:

Gini coefficient is the average difference between all possible pairs of incomes in the distribution, expressed as a proportion of total income. The Gini coefficient can be represented using the Lorenz curve; the coefficient is equal to the area below the line of perfect equality, formally:

$$ G = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} \left| x_i - x_j \right|}{2n^2 \bar{x}} $$

(2)

Ricci-Schutz coefficient (also called Pietra ratio, Hoover index or “Robin Hood index”) is the relative mean deviation of the population. It can be interpreted as the proportion of income that has to be transferred from those above the mean to those below the mean in order to achieve an equal distribution. Graphically, the Ricci-Schutz coefficient is equivalent to the maximum vertical distance between the Lorenz curve and the line of equal incomes, formally:

$$ RS = \frac{1}{2} \frac{\sum_{i} |x_i - \bar{x}|}{\sum_{i} x_i}$$$

(3)

Squared coefficient of variation: The coefficient of variation is a standardised measure of dispersion of a probability or frequency distribution. It shows the extent of variability of data in a sample in relation to the mean of the population. When applied to income distribution, the CV is smaller when income is distributed more equally between individuals. The coefficient of variation is a highly consistent and mathematically traceable indicator. However, unlike the Gini coefficient, it does not have an upper bound, making interpretation and comparison somewhat more difficult. Moreover, the two components of the coefficient of variation (the mean and the standard deviation) may be greatly influenced by extreme low or high income values.

Formally, the squared coefficient of variation can be expressed as:

$$ SCV = \sqrt{\frac{\sigma}{\mu}} $$

(4)

where

\sigma

is the standard deviation and

\mu

is the mean.

Atkinson index: The Atkinson index aims to overcome the main weakness of Lorenz curves and related inequality indicators, namely their inability to differentiate between various kinds of inequality distributions. In fact, contrasting patterns of income distribution may ultimately result in very similar Lorenz curves. The Atkinson index allows for varying sensitivity to inequalities in different parts of the income distribution by incorporating a sensitivity parameter

\left(\epsilon\right)

. Intuitively, Atkinson values can be used to calculate the proportion of total income that would be required to achieve an equal level of social (or territorial) welfare as at present if incomes were perfectly distributed. For example, an Atkinson index value of 0.20 suggests that we could achieve the same level of social welfare with only 80% of income. The theoretical range of Atkinson values is 0 to 1, with 0 being a state of equal distribution. The Atkinson index can be defined as:

A_{\varepsilon}(y_1,\ldots,y_N)=\begin{cases}1-\frac{1}{\mu}\left(\frac{1}{N}\sum_{i=1}^Ny_i^{1-\varepsilon}\right)^{1/\left(1-\varepsilon\right)} & \text{for }0\leq\varepsilon\neq1 \\ 1-\frac{1}{\mu}\left(\prod_{i=1}^Ny_i\right)^{1/N} & \text{for }\varepsilon=1\end{cases}

(5)

where

y_i

is individual income

\left(i=1,2,\ldots N\right),\mu

is the mean income and

\epsilon

is the sensitivity parameter. This parameter can be seen as a way to include the concept of social justice in the measurement of income inequality. The higher the value, the more sensitive the Atkinson index becomes to inequalities at the bottom of the income distribution. Even if

\epsilon

can be set to any positive value in theory, in practice

\epsilon

values of 0.5, 1, 1.5 or 2 are typically used. In this study we have opted for the conservative value of 0.5 for

\epsilon

Theil entropy index: The Theil index is an entropy-based measure of inequality. It is related to other generalised entropy indices

\left(E_{\theta}\right)

, developed by information theory. The Theil index represents the maximum possible entropy of the data minus the observed entropy in the sample. The theoretical range of

$$ \left(E_{\theta}\right) $$

, values is 0 to infinity, with 0 being a state of equal distribution and greater values representing increasing levels of inequality. If everyone had the same income, then

would equal 0. If one person has all the income available, then

gives the result

$$ \log N $$

, which is maximum inequality.

Like the Atkinson index, the Theil index incorporates a sensitivity parameter

\alpha

that varies in the weight given to inequalities in differing parts of the income spectrum. The general formula is given by:

E_{\theta}(\alpha) = \frac{1}{\alpha(\alpha - 1)} \left[ \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i}{\bar{x}} \right)^{\alpha} - 1 \right]

(6)

where

x_i

is the income of the

th person or subgroup and

\overline{x}

is the mean income in a population of

individuals or subgroups and the parameter

\alpha

in the

E_{\theta}

class represents the weight given to distances between incomes at different parts of the income distribution and can take any real value. The measure is sensitive to changes at the lower end of the distribution with an

\alpha

parameter value close to zero and is equally sensitive to changes at the higher end of the distribution for higher values of

\alpha

So, with

\alpha=1

, the Theil

T_T

index is defined as:

T_T = T_{\alpha=1} = \frac{1}{N} \sum_{i=1}^{N} \left( \frac{x_i}{\bar{x}} \cdot \ln \frac{x_i}{\bar{x}} \right)

(7)

where

x_i

, is the income of the

th person or subgroup and

\overline{x}

is the mean income in a population of

individuals or subgroups.

Given that in our calculations we applied

\alpha=1

, we assumed a relative concept of distance between income pairs. In other words, income levels

x_1

and

x_2

are the same distance apart as income levels

x_3

and

x_4

would be, if the ratios

x_1/x_2

and

x_3/x_4

are equal.

With

\alpha

, we have:

T_L = T_{\alpha=0} = MLD = \frac{1}{N} \sum_{i=1}^{N} \left( \ln \frac{\bar{x}}{x_i} \right)

(8)

T_L

is also known as the mean log deviation because it gives the standard deviation of

\ln\left(x\right)

T_L

is derivable from

T_T

T_L(x)=T_T\left(\frac{1}{x}\right)

(9)

T_L

is generally preferred over

T_T

when changes in lower incomes are more important.

8.1.3. Statistical properties of inequality indices

The metrics described above offer a wide range of statistical approaches to measure inequality. To determine the most suitable approach, five basic properties, or axioms, of inequality measures should be considered (Cowell, 2011):

Anonymity or symmetry: The index algorithm treats individual incomes anonymously, i.e. the inequality score does not depend on the particular assignment of labels to members of the population. This property distinguishes the concept of inequality from that of fairness, where who owns a particular level of income and how it has been acquired is of central importance. The anonymity criterion is met by all the indices considered here.

The Transfer principle (also called Distance principle): An inequality measure satisfies this principle if a transfer of a positive amount of income from a richer to a poorer group leads to a reduction in inequality (strong form) or at least overall inequality does not increase (weak form), no matter what amount is transferred. This quality is satisfied by all entropy-based indices, including
T_T
and
T_L
. The Gini, Atkinson and squared coefficient of variation indices have a weak performance in terms of this principle (e.g. the inequality value decreases more if the redistribution from richer to poorer involves extremes), whereas the relative mean deviation approach ( Ricci-Schutz index) only meets the weakest form of the axiom (i.e. a rich-to-poor transfer may not lead to a reduction in inequality score at all).

Scale independence: Simply put, this means that the measured inequality of the slices of the cake should not depend on the size of the cake. This property is shared by all the inequality measures we have examined.

Population independence: The value of the metric should not depend on population size either. This property is shared by all the inequality measures discussed here.

Decomposability: This property implies that there should be a coherent relationship between inequality in the whole of society and inequality in its constituent parts. Decomposability is found to be an important quality of income indicators, particularly when internal (e.g. social or territorial) disparities within the population might occur (Novotný, 2007). It turns out that it is possible for some inequality measures, including the Gini coefficient, “to register an increase in inequality in every subgroup of the population at the same time as a decrease in inequality overall” (Cowell, 2011). The criterion is met by all the indices, with the exception of the Ricci-Schutz and Gini coefficients, which are not decomposable.

It emerges that the Theil’s entropy index

T_T

is the only metric in the previous list to meet all the aforementioned criteria.

8.2. Convergence measures

This section describes the metrics used to analyse the evolution of income inequalities over time, focusing on their territorial expression.

8.2.1. Beta convergence

The beta-convergence hypothesis is tested by estimating the following regression:

\ln(\Delta y_{i,t})=\alpha+\beta\ln(y_{i,t-1}+\epsilon_{i,t})

(10)

where

y_{i,t}

is the level of indicator

in region

at time

t,\Delta y_{i,t}

, is the growth rate of indicator

in region

’at time

\alpha

, and

\beta

are the parameters to be estimated and

\epsilon_{i,t}

is the error term.

The beta-convergence hypothesis is verified when the partial correlation between growth in the income variable over time and its initial level are negative.

8.2.2. Sigma convergence

The sigma-convergence hypothesis is verified using the following equation:

\sigma_{i,t} = \left( \frac{1}{n} \right) \sum_{i=1}^{N} \left[ \left( \ln(y_{it}) - \mu_t \right)^2 \right]

(11)

where

\upsilon_i

is the sample mean of (log) income at time

in region

The hypothesis is confirmed when the dispersion of

\sigma

lessens over time (Furceri, 2005; Goecke and Hüther, 2016).

8.2.3. Club convergence

The club convergence method, introduced by Phillips and Sul, is formulated as a non-linear time-varying factor model that allows for different time paths and individual heterogeneity (Phillips and Sul, 2007). Unlike other methods where economies are grouped a priori, this method enables the endogenous determination of convergence clubs. The formal equation of their model is:

\log(y_{it}) = \alpha + \beta \cdot \log(y_{it-1}) + u_{it}

(12)

where

y_{it}

is the variable of interest for unit (i) at time (t),

\alpha

and

\beta

are coefficients – time-varying parameters –,

\left(y_{it}^{\ast}\right)

is the common component and

\left(e_{it}\right)

is the error term.