When you compare countries with each other, higher income inequality correlates with lower intergenerational mobility. For instance, in countries with higher income inequality, kids from low-income families tend to stay low-income as adults. Is there something about income inequality that reduces economic opportunity for children of the poor?
Seems so, but then account for the fact that when we look at historical trends within countries, income inequality is not correlated with intergenerational mobility. Sometimes periods of high income inequality are periods of high intergenerational mobility. Sometimes not.
Correlation is not causation. However, lack of correlation doesn't mean lack of a cause-and-effect relationship. It may be that the route from cause to effect is circuitous and depends on the presence of other factors. It may be under certain conditions, high income inequality reduces social mobility across generations and in other conditions, it doesn't. How to rise above this morass of conditionals to discern the causal pathways?
I'd start with the endpoint - social mobility - and then work backwards. What fosters social mobility? That is, what helps individuals move up the economic ladder? Mostly job skills and connections. In developed countries, job skills matter more than ever. Economic opportunity beckons those with the right skill set.
So I'm going to simplify this exploration: what leads to skill inequality? Does income inequality undermine skill development?
Next: Why do some kids grow up to be highly skilled adults and others don't?
Reference:
Perez-Arce, F., E. F. L. Amaral, H. Huang, and C. C. Price. 2016. Inequality and Opportunity: The Relationship between Income Inequality and Intergenerational Transmission of Income. Santa Monica, CA: RAND. http://www.jstor.org/stable/10.7249/j.ctt1d41dcd