I am trying to understand Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion.
I am struggling to completely understand the following:
Suppose you have initial wealth of $W$, and that you reject a 50-50 lose 10/gain 11 gamble because of diminishing marginal utility of wealth. Then it must be that $U(W+11) - U(W) ≤ U(W) - U(W-10)$. Hence, on average you value each of the dollars between $W$ and $W+11$ by at most $\frac{10}{11}$ as much as you on average value each of the dollars between $W$ and $W-10$.
By concavity, this implies that you value the dollar $W+11$ at most $\frac{10}{11}$ as much as you value the dollar $W-10$.
Iterating this observation, if you have the same aversion to the lose 10/gain 11 bet at wealth level $W+21$, then you value dollar $W+21+11 = W+32$ by at most $\frac{10}{11}$ as you value dollar $W+21-10 = W+11$, which means you value dollar $W+32$ by at most $\frac{10}{11}\times \frac{10}{11} \approx \frac{5}{6}$ as much as dollar $W-10$.
May someone please provide the mathematics to the bolded italic part in the quote above?
Thank you very much
Gus