The famous St. Petersberg Game. In that game, Daniel Bernoulli proposed the utility-of wealth function u0(w) =w. However, what if people had a different utility function?

(a)Suppose if instead of using the function, we used the function 1(w) = log(w). What would a person with this utility function be willing to pay to play the St. Petersberg Game?Show your work. (Hint:You can either figure this out using some calculus, or you can calculate it term-by-term using a calculator or a program like Excel.)

(b)Now, suppose another person had the utility function 2(w) = e2 log(w). Repeat (a), under this assumption.

(c)How do the results for u0, u1, and u2 compare to one another?What does this say about the preferences that create these three utility functions?

(d)An important property of utility functions is that they are equivalent under positive affine transformations (that is, being transformed by equations of the type y = mx + b where m > 0).What is the relationship between u1 and u2? How does it related to your answers above?

(e) Is the utility function representation of an individual's preferences unique? Explain why or why not, with reference to the above questions.