A study released in March by the Federal Reserve Bank of San Francisco reports that nearly half, 44%, of American adults who are in the bottom 20% in income were born to parents who were also in the bottom 20%; nearly half, 45%, of adults in the top 20% had parents who were also in the top 20%. Most Americans who were born in the middle 60% had parents who were also born in the middle 60%.

The cup of inter-generational mobility in the U.S. is thus about half full, and about half empty.
If you were born in the bottom 20%, your chances of ending up in the top 20% are about one in 20: 5%.
If you were born in the top 20%, your chances of ending up in the bottom 20% are about one in 20: 5%. 
It’s not entirely a hereditary aristocracy and hereditary serfs; but the circumstances, genes, and connections that a person is born with do have a marked impact in this country.   
Here’s the chart, from this study:
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Eric Zuesse