# Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility... ## Question: Ms. Fogg is planning an around-the-world trip on which she plans to spend$10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by

U(Y) = In Y.

a. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip's expected utility b. Suppose that Ms. Fogg can buy insurance against losing the$1,000 (say, by purchasing traveler's checks) at an "actuarially fair" premium of $250. Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the$1,000 without insurance.

c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1,000 ## Risk Premium: Individuals who are risk averse will be willing to pay insurance to avoid a gamble. This means that they would prefer to receive an amount of money with certainty that is lower than the expected value of the gamble. The difference between the expected value of the gamble and the amount of money that, when received with certainty, produces the same level of utility as the gamble is known as the Markowitz risk premium. This is the maximum amount of money an individual would be willing to pay as insurance. ## Answer and Explanation: a. The probability Ms. Fogg has$10,000 to spend is 75%. The probability she only has $9,000 to spend is 25%. The expected utility of the trip is equal to the probability she doesn't lose any cash multiplied by the utility of spending$10,000 on the trip plus the probability she does lose $1,000 plus the utility derived from spending$9,000 on he trip.

ln(10,000) x 0.75 + ln(9,000) x 0.25 = 9.184000243.

b. If Ms. Fogg purchases the insurance for $250, she will have$9750 to spend on the trip with certainty. Her expected utility will therefore be equal to: ln(9750) = 9.185022564, which is higher than if she does not purchase the insurance.

c. We need to calculate Ms. Fogg's certainty equivalent - that is, the amount of money she would be willing to pay in order to avoid the gamble of going on holiday without insurance. From a., we know that the expected utility of the gamble is 9.184000243. We know that U(Y) = ln(Y). Therefore, {eq}Y = e^{(U(Y))} {/eq}

Y = $9740.04.$9740.04 with certainty would make Ms. Fogg ambivalent between taking the gamble and taking out insurance.

The difference between this certainty equivalent and $10,000 is$259.96.

\$259.96 is the maximum amount of money she is willing to pay for insurance.