Measuring population decline through a composite fuzzy index: Evidence from Italian municipalities

The implications of population decline have been widely examined in the literature, with particular attention to rural, mountain, and peripheral areas. Most research in the field relies on approaches that measure depopulation by imposing fixed thresholds or by classifying geographical areas as “declining” versus “non-declining”. Such methods suffer from several shortcomings, including the arbitrariness of cut-off values, the treatment of depopulation as a dichotomous phenomenon, and the neglect of the timing and persistence of decline. To address these limitations, this study proposes the Composite Fuzzy Demographic Index (CFDI), defined as the convex combination of two components: one based on the log-differences of the demographic variable P between two subsequent time points, and the other on the persistence of the decreasing state of P. The computation of these components involves a time-weight vector that increases with time proximity and accounts for the variability of P across different time points. The proposed methodology is applied to census data on the resident population of Italian municipalities between 1951 and 2021. Results show that the CFDI produces spatial patterns consistent with established evidence on depopulation, confirming its empirical validity. At the same time, the proposed index uncovers novel insights that traditional threshold-based measures fail to capture.