# 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 + 105.125\\ & = 50, \end{aligned} implying that $$d = -\frac {882} {326873}$$ and, for non-satiation, \begin{aligned} w & \leq -\frac 1 {2d}\\ & \approx 185. \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.