Heterogeneity and using a Multiplicative Index

[LeosonH]

Re: PNAS paper, multiplicative form of Sustainable Development Index Xi

“While, from a purely accounting perspective, an additive form provides a possible definition of a sustainable development index, X, such a choice has two major disadvantages: i) it must be renormalized by the number of objectives every time a new dimension is added, and ii) it conveys a sense that different dimensions are strict substitutes for each other (e.g. electricity for sanitation), which is incorrect. A multiplicative index, instead, emphasizes the simultaneous importance of all its dimensions.”

I really like the formulation of Xi as multiplicative and agree with its main advantage addressed in the paper - the non-substitutability of each dimension. This takes into account the factor heterogeneity of the existing infrastructure (e.g. lots of good plumbing, but a very poor power grid, would likely yield a poorer score on the index). Moreover, the index could potentially be a powerful tool for measuring and optimizing policy and intervention effectiveness – a “one size fits all” policy package will likely have a lower effect on Xi than a package that targets the most dire infrastructural needs of a municipality.

I have two questions that follow from the above:

1) Given the multiple advantages and potential uses of the multiplicative form, in what cases (if any) might it be useful/more advantageous to use additive means instead of geometric means in this context?

2) This is somewhat related to Jo’s question(#1) regarding the inclusion of more dimensions in the index: As the measure expands to include more components and a greater number of differing geographical areas, how might one manage the possible growing interdependency between the individual components (e.g. an environmental quality component may be dependent on other components such access to water, sanitation, health services, and clean energy)?

[lmbett]

Good questions: the interdependence needs to be analyzed, and I think that things (problems) will come together in correlated ways. Additive indices imply substitution (say toilet for electricity) so they may be appropriate in some cases when that applies.