People tended to choose personal goals for which they had relevant resources, and the degree of congruence of individuals’ goals with their resources predicted their SWB [Subjective Well-Being]. Diener and Suh  (1997) p 202

To the extent that we're making progress on our goals, we're happier emotionally and more satisfied with our lives. Timothy A Pychyl (2008)  

Does inequality interfere with our ability to live purposeful lives as valued members of a community? That's a pretty abstract question, so I'll make it simpler: does inequality undermine the quest for social status? 

Goals related to social status require a reference point: higher status in comparison to what? The answer is our old selves and other people. Not just any ol' other people, but people we know or who are like us. If I'm a roofer living in a working class neighborhood, I'm not measuring my status against super-rich CEOs or Hollywood superstars, I'm looking at how my relatives, neighbors, friends, associates, and fellow roofers are doing.

In the absence of favorable social comparisons, we at least want to feel movement from where we used to be to where we want to be - that is, we want to feel a sense of progress. A sense that we're going somewhere. Getting closer to where we want to be. 

In the absence of favorable social comparisons or a sense of progress, we feel miserable or adjust our goals.

When the prospects of social comparison are unfavorable, individuals often rearrange the priority of their goals and redirect their attention to tasks or situations that are more likely to result in a positive outcome for themselves Diener and Suh  (1997) p. 203

How does this relate to inequality? It doesn't, at least not directly. It relates to social mobility - the ability to move up the socioeconomic ladder within one's lifetime. Social mobility holds the promise of progress towards higher status. Social mobility makes for happier campers.

Right now in the US we have a situation of high inequality and stalled social mobility for a good chunk of the population. But is the relationship causal or merely correlational? Does high inequality somehow lead to lower social mobility? Or are levels of inequality and social mobility caused by something else, so that they tend to co-vary together ?

I don't know. But I'm pretty confident there are things we can do to increase social mobility for people who have stalled out. And thereby contribute to their happiness.

Next: How to increase social mobility for people who have become stuck.

References:

Diener, E. & Suh, E. (1997). Measuring quality of life: Economic, social, and subjective indicators. Social Indicators Research, 40, 189–216.

Timothy A Pychyl (2008) Goal Progress and Happiness. Accessed at: https://www.psychologytoday.com/blog/dont-delay/200806/goal-progress-and-happiness