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

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

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.