The increasing interest in multidimensional poverty and well-being analysis has added complexity to the way these phenomena are conceptualized and measured. When multiple attributes are considered, a criterion determining the relative importance attached to the different dimensions has to be adopted. There has not been thus far in the literature a specific attempt to conceptualize the nature of the desired hierarchy among the selected poverty dimensions. The aim of this paper is to take the first step in this direction. We envisage two simple and highly intuitive ways in which such a hierarchical system can be understood, which we label restricted and unrestricted hierarchy. The analytical conditions allowing the incorporation of these into a poverty index are derived and their implications in terms of the understanding of poverty are discussed. An empirical application shows how the choice of the hierarchical scheme for poverty dimensions can lead to opposite conclusions on the poverty trend.