Von Neumann Morgenstern Utility - Expected Value Choice Under Uncertainty

If Mr.D doesnt break the speed limit he earns a guaranteed income of £30000. If he breaks speed limits he is able to double his income to £60,000 per

year. If caught speeding he will be banned from driving and will only be able to earn

£10,000 per year in unskilled work. The probability of getting caught for speeding is

0.5.

The decision to speed (or not) is based on rational criteria, with all ethical and similar

issues being irrelevant in this regard, i.e. Mr D seeks to maximise expected utility. He

has a Von Neumann-Morgenstern utility function.

Where $\displaystyle U = {Y}^{\propto } and \propto = 0.2 $

To analyse whether Mr.D should speed or not, I need to calculate the expected utility values for Mr.D in both scenarios of when he speeds and when he doesn't.

Certain expected utility from not speeding = $\displaystyle {30000}^{ 0.2} = 7.86 $

Expected wealth from uncertain situation = $\displaystyle (0.5*10000)+(0.5*60000) = 35000 $

Utility of expected wealth =$\displaystyle {35000}^{0.2 } = 8.13 $

Expected Utility =$\displaystyle 0.5*({10000}^{0.2 })+0.5*({60000}^{0.2 })= 7.66$

Therefore the utility of expected wealth > expected utility.

As Mr.D seeks to maximise expected utilitu, the higest utility figure is 8.13, which is greater than 7.86, which was the utility of certain income. Would Mr.D still choose to speed in this instance, is it possible to compare the utilities of two different income levels. 35000 to the power of any value will always provide a higher value than 30000 to the same power, so would Mr.D always choose to speed ?

In this case Mr.D recieves more utility from a certain £35000 than from an uncertain one. However where i am struggling to interpret this is that, the certain expected utility (7.86) is less than utility of expected wealth (8.13) so would Mr D choose to speed in this instance?