Strong correlation between income inequality within nations [1] and a number of measures that measure 'wellbeing of society' (life expectancy, education, crime, health, trust and aggregate measures). Plotting these measures against the GDP per person (richness of nation) shows weak or no correlation.
Bottom line: More unequal societies have more social problems.
[1] ratio between wealth of top 20% and wealth of bottom 20%.
EDIT: economic problems -> social problems in bottom line