Decomposing Changes in Inequality and Welfare Between EU Regions: The Roles of Population Change, Re-Ranking and Income Growth

Abstract

This paper explores the changes in inequality and welfare between EU regions at the NUTS 3 level over the 2003–2011 period. Changes in absolute and relative inequalities are broken down into components explaining the effects of population change, re-ranking of regions and income growth between regional per capita incomes. Each component of inequality change is further decomposed by subgroup, revealing the contributions arising from changes within subgroups and from changes between subgroups. The decomposition of the change in absolute inequality is used to develop a decomposition of the change in welfare between EU regions.

Appendices

Appendix 1: Calculation of $\mathbf {A}_{h,t+1}$, $\mathbf {D}_{\mathbf{a},h}$, $\mathbf {A}_{gh,t+1}$, $\mathbf {D} _{\mathbf{a},gh}$

Let ${\mathbf {w}} _{h,t}$ be the $k\times 1$ vector with nonzero elements equal to 1 in the corresponding places of the t per capita incomes of the regions belonging to subgroup h (with $h=1,\ldots ,r$) in ${\mathbf {y}} _{t}$. The $k\times k$ matrix ${\mathbf {W}} _{h,t}={\mathbf {w}} _{h,t}{\mathbf {w}} ^{T}_{h,t}$ has its $\left( i, j\right)$-th element equal to 1 if and only if the $\left( i, j\right)$-th element of ${\mathbf {A}}_{t}$ is the difference between the per capita incomes of two regions belonging to subgroup h, otherwise the $\left( i,j\right)$-th element of ${\mathbf {W}} _{h,t}$ is equal to 0. By using the Hadamard product,²² the pairwise differences between the per capita incomes of the regions in subgroup h are drawn from ${\mathbf {A}}_{t}$:

$$\begin{aligned} {\mathbf {A}}_{h,t}={\mathbf {W}} _{h,t}\odot {\mathbf {A}}_{t}. \end{aligned}$$

(23)

The pairwise differences between the $t+1$ per capita incomes of subgroup h in ${\mathbf {A}}_{t+1\left| t\right. }$ occupy the same entries in which the pairwise differences between the t per capita incomes of subgroup h are arranged in ${\mathbf {A}}_{t}$. Thus, ${\mathbf {W}} _{h,t}$ can also be used to select the pairwise differences between the $t+1$ per capita incomes of subgroup h from ${\mathbf {A}}_{t+1\left| t\right. }$:

$$\begin{aligned} {\mathbf {A}}_{h,t+1\left| t\right. }={\mathbf {W}} _{h,t}\odot {\mathbf {A}}_{t+1\left| t\right. }. \end{aligned}$$

(24)

${\mathbf {w}} _{h,t+1}$ being the $k\times 1$ vector with nonzero elements equal to 1 in the corresponding places of the $t+1$ per capita incomes belonging to regions of subgroup h in ${\mathbf {y}} _{t+1}$, the $k\times k$ matrix ${\mathbf {W}} _{h,t+1}={\mathbf {w}} _{h,t+1}{\mathbf {w}} ^{T}_{h,t+1}$ selects the pairwise differences between the $t+1$ per capita incomes of regions within subgroup h from ${\mathbf {A}}_{t+1}$:

$$\begin{aligned} {\mathbf {A}}_{h,t+1}={\mathbf {W}} _{h,t+1}\odot {\mathbf {A}}_{t+1}. \end{aligned}$$

(25)

Since ${\mathbf {D}} _{\mathbf{a }}={\mathbf {A}}_{t}-{\mathbf {A}}_{t+1\left| t\right. }$, the Hadamard product between ${\mathbf {W}} _{h,t}$ and ${\mathbf {D}} _{\mathbf{a }}$ is the matrix whose nonzero elements are the elements of ${\mathbf {D}} _{\mathbf{a }}$ measuring the change in the absolute disparities between the per capita incomes of regions within subgroup h:

$$\begin{aligned} {\mathbf {D}} _{\mathbf{a},h}={\mathbf {A}}_{h,t} -{\mathbf {A}}_{h,t+1\left| t\right. }={\mathbf {W}} _{h,t} \odot \left( {\mathbf {A}}_{t}-{\mathbf {A}}_{t+1\left| t\right. }\right) ={\mathbf {W}} _{h,t}\odot {\mathbf {D}} _{\mathbf{a }}. \end{aligned}$$

(26)

The $k\times k$ matrix ${\mathbf {W}} _{gh,t}={\mathbf {w}} _{g,t}{\mathbf {w}} ^{T}_{h,t}+{\mathbf {w}} _{h,t}{\mathbf {w}} ^{T}_{g,t}$ shows nonzero elements equal to 1 in the entries corresponding to the pairwise differences between the per capita incomes of subgroup g and those of subgroup h in both ${\mathbf {A}}_{t}$ and ${\mathbf {A}}_{t+1\left| t\right. }$; hence, ${\mathbf {W}} _{gh,t}$ selects the between-group pairwise differences from both the matrices:

$$\begin{aligned} {\mathbf {A}}_{gh,t}={\mathbf {W}} _{gh,t}\odot {\mathbf {A}}_{t} \end{aligned}$$

(27)

and

$$\begin{aligned} {\mathbf {A}}_{gh,t+1\left| t\right. } ={\mathbf {W}} _{gh,t}\odot {\mathbf {A}}_{t+1\left| t\right. }. \end{aligned}$$

(28)

The Hadamard product between ${\mathbf {W}} _{gh,t}$ and ${\mathbf {D}} _{\mathbf{a }}$ selects the elements of ${\mathbf {D}} _{\mathbf{a }}$ measuring the change in the disparities between the per capita incomes of regions in subgroup g and the per capita incomes of regions in subgroup h:

$$\begin{aligned} {\mathbf {D}} _{\mathbf{a},gh} ={\mathbf {A}}_{gh,t}-{\mathbf {A}}_{gh,t+1 \left| t\right. }={\mathbf {W}} _{gh,t}\odot \left( {\mathbf {A}}_{t} -{\mathbf {A}}_{t+1\left| t\right. }\right) ={\mathbf {W}} _{gh,t}\odot {\mathbf {D}} _{\mathbf{a }}. \end{aligned}$$

(29)

The $k\times k$ matrix ${\mathbf {W}} _{gh,t+1}={\mathbf {w}} _{g,t+1}{\mathbf {w}} ^{T}_{h,t+1}+{\mathbf {w}} _{h,t+1}{\mathbf {w}} ^{T}_{g,t+1}$ has nonzero elements equal to 1 in the entries corresponding to the pairwise differences between the per capita incomes of subgroup g and the per capita incomes of subgroup h in ${\mathbf {A}}_{t+1}$. Thus, the matrix

$$\begin{aligned} {\mathbf {A}}_{gh,t+1}={\mathbf {W}} _{gh,t+1}\odot {\mathbf {A}}_{t+1} \end{aligned}$$

(30)

comprises the pairwise differences between the $t+1$ per capita incomes of subgroup g and those of subgroup h.

Appendix 2: Maps of EU Regions Divided in Two and Three Subgroups in 2007

Figure 3 shows the partition of regions in 2007 for $k=2$. Figure 4 shows the partition of regions in 2007 for $k=3$.

Fig. 3

Map of NUTS 3 regions divided in two subgroups in 2007

Fig. 4

Map of NUTS 3 regions divided in three subgroups in 2007