Price index dispersion and utilitarian social evaluation

The living standard indicator in utilitarian social evaluation functions (USEF) is the ratio of a nominal living standard and a price index. We show that under weak association of price indices and nominal living standards and usual concavity conditions on utility functions, utilitarian social welfare increases with price index dispersion when the aggregate price level is superior to the arithmetic mean of price indices, and diminishes when it is inferior to the harmonic mean.