Question

Marco's cousin Polo is also risk averse, but less so than Marco. Let CEMarco (Y) be Marco's certainty equivalent for Y, and CEpolo (Y) be Polo's. Which of the following statements is FALSE? a. CEPolo(Y) < EV(Y) b. CEMarco(Y) < CEpolo (Y) c. CEMarco(Y) < CEpolo (Y) < EV(Y) d. CEMarco (Y) < EV(Y) < CEPolo (Y) Marco's cousin Polo is also risk averse, but less so than Marco. Let EUMarco(Y) be Marco's expected utility from Y and EUpolo (Y) be Polo's. Which of the following statements is necessarily EALSE2 a. U[EV(Y)] > EUMarco (Y)

Answer 1

1.

Marco's cousin Polo is also risk-averse (a person who refuses a fair bet is risk averse), but less so than Marco. A person is risk-averse if their certainty equivalent of a gamble is less than the gamble's expected monetary value. In this case, expected value of Y is denoted as EV(Y). Thus, Marco's CE(Y) and Polo's CE(Y) must be less than the expected value of Y: EV(Y).
Thus, we can say that
Marco's CE(Y) < EV(Y) and
Polo's CE(Y) < EV(Y)  

The certainty equivalent CE(Y) is an indicator of how risk-averse the person is. A higher certainty equivalent implies that the person is less risk-averse. Thus, is Polo is less risk-averse than Marco, the CE(Y) for Polo will be greater than Marco's CE(Y).
Thus we can say that
Marco's CE(Y) < Polo's CE(Y)

Using the relations with obtained above, we can see that the statement (d) is clearly false, as it states that Polo's CE(Y) is greater than EV(Y), which is not possible if Polo is risk-averse.

2.

This question requires a basic understanding of utility theory. For a risk-averse person, a gamble with high risk has less utility than a gamble with high risk. Again, Polo is less risk-averse than Marco.
Remember that, U(CE) = EU
Thus, expected utility of Marco/Polo is determined by applying a simple operator (which does not change signs and maintains the inequality) to their CE.
EV(Y) > CE(Y) for both Marco and Polo
thus, U(EV(Y)) > EU(Y) for both Marco and Polo

In the first question, we derived that Marco's CE(Y) < Polo's CE(Y)
If the inverse of U operator is applied,
U^-1[Marco's CE(Y)] < U^-1[Polo's CE(Y)] -> notice there is no change in inequality

Thus, the statement c is clearly false.

Hope this helps!