<p>The utility function under examination does reflect risk-aversion, being concave.
But there is an ambiguity in how the problem is formulated. It is said that the investor &quot;is non-satiated&quot;. And with the computed value of <span class="math-container">$d$</span>, we can determine that <em>currently</em>, with wealth of <span class="math-container">$100$</span>, the investor is in the rising curve of this utility function, so they are non-satiated.</p>
<p>Certainly, if it so happens that they sell all the boxes for <span class="math-container">$30$</span> they will land in the downward sloping  curve of this utility function.</p>
<p>But we are <em>not</em> told that &quot;the investor should/wants to <em>remain</em> non-satiated under the highest possible outcome&quot;. We are only told they &quot;they are non-satiated&quot;, which means <em>now</em>, not after the uncertainty is resolved.</p>
<p>On the other hand, one could argue as follows: &quot;It should be self-understood that the investor would have no incentive to engage in a certain economic activity if it were the case that there was non-zero probability that they would become satiated after the conclusion of said economic activity. Therefore, the utility function must be such that the investor will remain non-satiated even after the highest possible outcome&quot;.</p>
<p>Contemplate.</p>

CC BY-SA 4.0
2022-08-16T14:43:39.037Z
2022-08-16T14:43:39.037Z
Alecos Papadopoulos
<p><strong>Question</strong></p>
<p>A risk-averse, non-satiated investor has decided to use the utility function <span class="math-container">$$U(w) = w + dw^2,$$</span> where <span class="math-container">$$d \leq 0$$</span> is a constant, to describe his preferences.</p>
<p>The investor has a current wealth of <span class="math-container">$\\\$100$</span> and buys seven boxes of vegetables for <span class="math-container">$\\\$10$</span> per box. He knows that he can sell them for <span class="math-container">$\\\$30$</span>, <span class="math-container">$\\\$12$</span>, <span class="math-container">$\\\$10$</span> or <span class="math-container">$\\\$0.5$</span> per box with equal probability. All boxes of vegetables sold at a given time will be sold for the same price. After selling them, his expected utility of wealth will be <span class="math-container">$50$</span>.</p>
<p>Discuss whether <span class="math-container">$U(w)$</span> is appropriate for the investor.</p>
<hr />
<p><strong>My answer</strong></p>
<p>First, we find <span class="math-container">$d$</span> by simply solving
<span class="math-container">$$\begin{aligned}
\frac 1 4 \{U[100 + 7(30 - 10)] + U[100 + 7(12 - 10)] + U[100 + 7(10 - 10)] + U[100 + 7(0.5 - 10)]\} &amp; = \frac 1 4 [U(240) + U(114) + U(100) + U(33.5)]\\
&amp; = \frac 1 4 (240 + 57600d + 114 + 12996d + 100 + 10000d + 33.5 + 1122.25d)\\
&amp; = 20429.5625d + 121.875\\
&amp; = 50,
\end{aligned}$$</span>
implying that
<span class="math-container">$$d \approx -0.0035$$</span>
and, for non-satiation,
<span class="math-container">$$\begin{aligned}
w &amp; \leq -\frac 1 {2d}\\
&amp; \approx 142.
\end{aligned}$$</span></p>
<p>Now, in the scenario that all seven boxes of vegetables are sold at <span class="math-container">$\\\$30$</span> each, the investor will end up with a wealth of <span class="math-container">$240$</span>, which exceeds the proposed upper bound. Thus, <span class="math-container">$U(w)$</span> is inappropriate for the investor.</p>
<hr />
<p>As I have just covered risk-aversion and utility functions, I would like to know whether I have approached the problem correctly and whether my answer is correct. If not, then any intuitive explanations will be greatly appreciated :)</p>

<utility><preferences><expected-utility><risk-aversion>
On risk aversion and validity of utility functions
Ethan Mark
# On risk aversion and validity of utility functions

(Asked by: username_0 on 2022-08-02)

**Question**

A risk-averse, non-satiated investor has decided to use the utility function $$U(w) = w + dw^2,$$ where $$d \leq 0$$ is a constant, to describe his preferences.

The investor has a current wealth of $\\\$100$ and buys seven boxes of vegetables for $\\\$10$ per box. He knows that he can sell them for $\\\$30$, $\\\$12$, $\\\$10$ or $\\\$0.5$ per box with equal probability. All boxes of vegetables sold at a given time will be sold for the same price. After selling them, his expected utility of wealth will be $50$.

Discuss whether $U(w)$ is appropriate for the investor.

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**My answer**

First, we find $d$ by simply solving
$$\begin{aligned}
\frac 1 4 \{U[100 + 7(30 - 10)] + U[100 + 7(12 - 10)] + U[100 + 7(10 - 10)] + U[100 + 7(0.5 - 10)]\} & = \frac 1 4 [U(240) + U(114) + U(100) + U(33.5)]\\
& = \frac 1 4 (240 + 57600d + 114 + 12996d + 100 + 10000d + 33.5 + 1122.25d)\\
& = 20429.5625d + 121.875\\
& = 50,
\end{aligned}$$
implying that
$$d \approx -0.0035$$
and, for non-satiation,
$$\begin{aligned}
w & \leq -\frac 1 {2d}\\
& \approx 142.
\end{aligned}$$

Now, in the scenario that all seven boxes of vegetables are sold at $\\\$30$ each, the investor will end up with a wealth of $240$, which exceeds the proposed upper bound. Thus, $U(w)$ is inappropriate for the investor.

---

As I have just covered risk-aversion and utility functions, I would like to know whether I have approached the problem correctly and whether my answer is correct. If not, then any intuitive explanations will be greatly appreciated :)

**Tags:** utility, preferences, expected-utility, risk-aversion

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## 1 answers

### Answer by username_1 on 2022-08-16. Score: 1 | Accepted answer

The utility function under examination does reflect risk-aversion, being concave.
But there is an ambiguity in how the problem is formulated. It is said that the investor "is non-satiated". And with the computed value of $d$, we can determine that *currently*, with wealth of $100$, the investor is in the rising curve of this utility function, so they are non-satiated.

Certainly, if it so happens that they sell all the boxes for $30$ they will land in the downward sloping curve of this utility function.

But we are *not* told that "the investor should/wants to *remain* non-satiated under the highest possible outcome". We are only told they "they are non-satiated", which means *now*, not after the uncertainty is resolved.

On the other hand, one could argue as follows: "It should be self-understood that the investor would have no incentive to engage in a certain economic activity if it were the case that there was non-zero probability that they would become satiated after the conclusion of said economic activity. Therefore, the utility function must be such that the investor will remain non-satiated even after the highest possible outcome".

Contemplate.


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