On risk aversion and validity of utility functions

Question

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.

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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 :)

Answer 1

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.