題組內容

Lastly, suppose that either Emma or Francesca can make extra efforts to improve Emma's consumption experiences. By making such efforts (at a cost \( I \)), the probability of a benefit of $800 becomes \( \frac{1}{2} + x \), and the probability of a benefit of $200 becomes \( \frac{1}{2} - x \), where \( 0 < x < \frac{1}{2} \).

8. _________ (3%) If Francesca can make the efforts, what is the maximal value of \( I \) such that Francesca is willing to do so?