Understanding Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion

Question

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

Answer 1

From $U(W+11)−U(W)\le U(W)−U(W−10)$ we get that $\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}$, which is what the sentence before the bolded italic part says.

Now by concavity of $U(.)$ we know that $MU(W-10)\ge\frac{U(W)-U(W-10)}{10}$, and also that $MU(W+11)\le\frac{U(W+11)-U(W)}{11}$.

Therefore $MU(W+11)\le\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}\le\frac{10}{11}MU(W-10)$.

So $MU(W+11)\le\frac{10}{11}MU(W-10)$, wich is just the bolded italic statement.