# On risk aversion and validity of utility functions

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.

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 :)

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.