A Measure of Well-Being Across the Italian Urban Areas: An Integrated DEA-Entropy Approach

Abstract

In recent years, there has been an increasing proliferation of initiatives focusing on the concept of quality of life and well-being. At the centre of these studies there is the recognizing that the GDP offers only a partial perspective of factors affecting people’s lives. Following this line of the research, this paper is aimed at computing the well-being efficiencies of a sample of Italian Province capital cities, using a methodological approach that combines data envelopment analysis (DEA) with Shannon’s entropy formula. To avoid subjectivity in choosing a representative set of variables that proxy the phenomenon under study, we rely on the theoretical framework adopted by the Italian National Institute of Statistics (ISTAT) within the equitable and sustainable well-being (BES) project. The dashboard of indicators included in the analysis are related to the Ur-BES initiative, promoted by ISTAT to implement the BES framework at cities level. In a first step of the analysis, an immediate focus on separate dimensions of urban well-being is obtained by summarizing the plurality of available indicators through the building of composite indices. Next, the adopted integrated DEA–Shannon entropy approach has permitted to increase the discriminatory power of DEA procedure and attain a more reliable profiling of Italian Province capital cities well-being efficiencies. The results show a marked duality between the Northern and Southern cities, highlighting important differences in many aspects of human and ecosystem well-being.

Appendix

1.1 Adjusted Mazziotta–Pareto Index (AMPI)

Mazziotta–Pareto Index (MPI) is a non-linear composite index method which transforms a set of individual indicators in standardized variables and summarizes them using an arithmetic mean, adjusted by a “penalty” coefficient related to the variability of each unit (Mazziotta and Pareto 2007; Mazziotta et al. 2010).

Two steps are involved in the construction of the MPI which require the normalization of individual indicators by “standardization” and the aggregation of the standardized indicators by arithmetic mean with penalty function based on “horizontal variability” (variability of standardized values for each unit). The penalty is based on the coefficient of variation and it can be added or subtracted, depending on the nature of phenomenon to be measured and hence on the direction of the individual indicators (De Muro et al. 2011).

In what follows, we describe how proceeds the construction of a variant of MPI, known as Adjusted Mazziotta–Pareto Index (AMPI) (Mazziotta and Pareto 2016).

For the AMPI it has been adopted a different procedure of data normalization to guarantee absolute comparisons over time. That data transformation requires a re-scaling of the elementary indicators respect two goalposts, that is respect to a minimum and maximum, which represent the range of each indicator over the given time period.

Let $X = \left\{ {x_{ij} } \right\}$ be the matrix with n rows (geographical units) and m columns (indicators), the normalized matrix $R = \left\{ {r_{ij} } \right\}$ is defined through a min–max transformation.

According to the original direction of the indicator is used min–max formula (5) or (6)

$$r_{ij} = \frac{{x_{ij} - Min_{xj} }}{{Max_{xj} - Min_{xj} }} 60 + 70$$

(5)

$$r_{ij} = \frac{{Max_{xj} - x_{ij} }}{{Max_{xj} - Min_{xj} }} 60 + 70$$

(6)

where $x_{ij}$ is the value of indicator j for the geographical unit i whereas $Min_{xj}$ and $Max_{xj}$ are the goalposts.

In our study, we deal with a min–max transformation in a continuous scale from 70 (minimum) to 130 (maximum).

To facilitate the interpretation of results, the “goalposts” can be chosen so that 100 represents a reference value (e.g., the average in a given year).

Let $Ref_{{x_{j} }}$ be the reference value for the indicator i, then the goalposts are defined as $Ref_{{x_{j} }} \pm\Delta$ where $\Delta = \frac{{(\sup x_{j} - \inf x_{j} )}}{2}$ and $\sup x_{j}$ and $\inf x_{j}$ are the minimum and maximum of indicator j across all units and all time periods considered.

The above formulas take into account the polarity of indicator, that is the sign of the relationship between the indicator and the phenomenon under study (+if the indicator represents a positive dimension and—if the indicator represents a negative dimension).

In our case, this data transformation assures a direct reading of values in terms of well-being: higher values reflect better performance.

Let $M_{ri}$ and $S_{ri}$ be the media and standard deviation, respectively, of the normalized values for the i-th unit. The composite index is defined as:

$$AMPI_{i}^{ + / - } = M_{ri} \pm S_{ri} cv_{ri}$$

(7)

where $cv_{ri} = S_{ri} /M_{ri}$ is the coefficient of variation for the ith unit and the sign $\pm$ depends on the kind of phenomenon to be measured.

This approach is characterized by the employment of a product ($S_{ri} cv_{ri}$) which penalizes the units showing unbalanced values of the indicators. Thus, the AMPI can be viewed as a combination of a “average effect” ($M_{ri}$) and a “penalty effect” ($S_{ri} cv_{ri}$) and indicate how each indicator is located compared to the goalposts.