The recent paper by Chetty et al. finds that the regression of kids' income rank on parents' income rank has a coefficient of 0.3. (See Figure 1.) That implies an R2 for the regression of 0.09. In other words, 91 percent of the variance is unexplained by parents' income.
I would be willing venture a guess, based on adoption studies, that a lot of that 9 percent is genetics rather than environment. That is, talented parents have talented kids partly because of good genes. Conservatively, let's say half is genetics. That leaves only 4.5 percent of the variance attributed directly to parents' income.
Now, if you let me play a bit fast and loose with the difference between income and income rank, these numbers suggest the following: If we had some perfect policy invention (such as universal super-duper pre-school) that completely neutralized the effect of parent's income, we would reduce the variance of kids' income to .955 of what it now is. This implies that the standard deviation of income would fall to 0.977 of what it now is.
The bottom line: Even a highly successful policy intervention that neutralized the effects of differing parental incomes would reduce the gap between rich and poor by only about 2 percent. [Bold in original]