William Has a Utility Function Given By $U(x)=\ln (\mathrm{x})$ He Faces a Gamble That Pays 10 with Probability

Essay

William has a utility function given by $U(x)=\ln (\mathrm{x})$ . He faces a gamble that pays 10 with probability .5 and 15 with probability .5 . What is William's certainty equivalent - that is, what sure amount must he receive in order to be indifferent between the gamble and this sure amount?
a. Find $x_{C E}$ if William's utility was given by $U(x)=x$
b. Find $x_{C E}$ if William's utility was given by $U(x)=x^{2}$ .