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?