Confusions about calculations in Utility Theory Paper

Question

Recently I was improving my knowledge of Utility Functions in finances. I stumped upon very nice (yet quite old) article: "An Introduction to Utility Theory" by John Norstad (1999). What drove my attention was charts in section 4. They represent utility functions of wealth:

• First chart (figure 2): $- \dfrac{10^6}{w^3}$
• Second chart(figure 3): $-\dfrac{10^{10}}{w^5}$

(So basically $-\dfrac{10^{2c}}{w^c}$) with c as chosen constant)

Purely for learning purpose I was trying to recreate his example (by using Excel/R) from section 4 (details on page 7). In same page author stated that expected utility (of 105) will be equal to -0.98. It is also shown under figure 2 in table:

  Wealth   Utility
 $90   -1.37   Bad outcome
 $100   -1.00   Current wealth
 $105   -0.98   Expected outcome
 $120   -0.58   Good outcome

I run test and with those values and for \$105 it does not match. Using $- \frac{10^6}{105^3}$ I get -0.86. What is surprising other stated values are calculated correctly (90, 100, 120). At first I though it is nothing more than some kind of typo, but when I checked the second formula used to draft function in figure 3 the same situation occurred. For table:

Wealth   Utility
 $90   -1.69   Bad outcome
 $100   -1.00   Current wealth
 $105   -1.05   Expected outcome
 $120   -0.40   Good outcome

when I put value 105 to formula ($-\frac{10^{10}}{105^5}$) it seems to be incorrect (should be -0.78) while rest of them are right. This quite undermines my typo theory. Additionally this contradicts further conclusions (page 9 under the chart) made by author.

So my question is: did I overlooked something? Or did I just find a mistake? Or I am the one who is mistaken?

Answer 1

The notation is a bit confusing but what the author is computing is the expected utility of the investment, not the utility of the expected outcome.

Since the investment results in a wealth of \$120 with probability 0.5 and a wealth of \$90 with probability 0.5, the expected utility equals \begin{align*} \mathbb{E}U & = 0.5 \times (-\dfrac{10^6}{90^3}) + 0.5 \times (-\dfrac{10^6}{120^3}) \\ & \sim -0.98 \end{align*}

Notice that this value is lower than the utility of the expected outcome, which is approximately equal to $-0.86$ as you wrote. This is because the agent is risk-averse and must therefore be compensated for the risk.