No, not quickly, on the contrary, there's an inverse relationship:

"...the United States and United Kingdom had the lowest intergenerational vertical social mobility ..."

"...Studies have also found "a clear negative relationship" between income inequality and intergenerational mobility.[27] Countries with low levels of inequality such as Denmark, Norway and Finland had some of the greatest mobility, while the two countries with the high level of inequality -- Chile and Brazil—had some of the lowest mobility. A 2012 graph plotting the relationship between inequality and mobility in the United States and twelve other developed countries has been dubbed "The Great Gatsby Curve"[27][28][29] http://en.wikipedia.org/wiki/Social_mobility#Country_compari...

PEW report: http://nextbigfuture.com/2013/12/selling-out-over-long-term-...

U. Bekerley.edu / NYTimes feature: http://www.irle.berkeley.edu/features/20130726_upward_mobili...