<p>From <span class="math-container">$U(W+11)−U(W)\le U(W)−U(W−10)$</span> we get that <span class="math-container">$\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}$</span>, which is what the sentence before the bolded italic part says.</p>

<p>Now by concavity of <span class="math-container">$U(.)$</span> we know that <span class="math-container">$MU(W-10)\ge\frac{U(W)-U(W-10)}{10}$</span>, and also that <span class="math-container">$MU(W+11)\le\frac{U(W+11)-U(W)}{11}$</span>.</p>

<p>Therefore <span class="math-container">$MU(W+11)\le\frac{U(W+11)-U(W)}{11}\le\frac{10}{11}\frac{U(W)-U(W-10)}{10}\le\frac{10}{11}MU(W-10)$</span>.</p>

<p>So <span class="math-container">$MU(W+11)\le\frac{10}{11}MU(W-10)$</span>, wich is just the bolded italic statement.</p>

CC BY-SA 4.0
2020-04-15T03:24:43.287Z
Sir - you are brilliant! Thank you very much.
Gustavo Louis G. Montańo
CC BY-SA 4.0
2020-04-15T00:36:09.877Z
2020-04-15T00:36:09.877Z
VARulle
<p>I am trying to understand Rabin's <em><a href="https://www.anderson.ucla.edu/faculty_pages/keith.chen/negot.%20papers/Rabin_EasyCalibrationTh00.pdf" rel="nofollow noreferrer">Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion</a></em>.</p>

<p>I am struggling to completely understand the following:</p>

<blockquote>
  <p>Suppose you have initial wealth of <span class="math-container">$W$</span>, and that you reject a 50-50 lose
  10/gain 11 gamble because of diminishing marginal utility of wealth. Then it must be that
  <span class="math-container">$U(W+11) - U(W) ≤ U(W) - U(W-10)$</span>. Hence, on average you value each of the dollars between <span class="math-container">$W$</span>
  and <span class="math-container">$W+11$</span> by at most <span class="math-container">$\frac{10}{11}$</span> as much as you on average value each of the dollars between <span class="math-container">$W$</span> and
  <span class="math-container">$W-10$</span>.</p>
  
  <p><strong><em>By concavity, this implies that you value the dollar <span class="math-container">$W+11$</span> at most <span class="math-container">$\frac{10}{11}$</span> as much as you value the dollar <span class="math-container">$W-10$</span>.</em></strong></p>
  
  <p>Iterating this observation, if you have the same aversion to the lose
  10/gain 11 bet at wealth level <span class="math-container">$W+21$</span>, then you value dollar <span class="math-container">$W+21+11 = W+32$</span> by at most
  <span class="math-container">$\frac{10}{11}$</span> as you value dollar <span class="math-container">$W+21-10 = W+11$</span>, which means you value dollar <span class="math-container">$W+32$</span> by at most
  <span class="math-container">$\frac{10}{11}\times \frac{10}{11} \approx \frac{5}{6}$</span> as much as dollar <span class="math-container">$W-10$</span>. </p>
</blockquote>

<p>May someone please provide the mathematics to the bolded italic part in the quote above?</p>

<p>Thank you very much</p>

<p>Gus</p>

<expected-utility><marginal>
Understanding Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion
Gustavo Louis G. Montańo
# Understanding Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion

(Asked by: username_0 on 2020-04-14)

I am trying to understand Rabin's *[Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion](https://www.anderson.ucla.edu/faculty_pages/keith.chen/negot.%20papers/Rabin_EasyCalibrationTh00.pdf)*.

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

**Tags:** expected-utility, marginal

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## 1 answers

### Answer by username_1 on 2020-04-15. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: Sir - you are brilliant! Thank you very much.  

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