Suppose a family has saved enough for a 10 day vacation (the only one they will be able to take for 10 years) and has a utility function U = V1/2 (where V is the number of healthy vacation days they experience). Suppose they are not a particularly healthy family and the probability that someone will have a vacation-ruining illness (V = 0) is 20%. What is the expected value of V?

2 Days

Explanation:

First, there is the need to rewrite the utility function for clarity

U=V^{1/2}

1. The Probability of Falling ill by someone in the family is given as 20%

2. If someone should fall ill, the total number of days that would be spoiled is calculated as:

Total number of vacation = 10 days x Probability to fall ill = 20%

= 10 x 0.2 = 2 days

This means if someone should fall ill based on the probability, then 2 out of the total 10 days can be ruined

3. The number of days for vacation days to enjoy is 10-2 = 8 days

This means if the family gives up 2 days of probable illness, they can still enjoy their vacation.

V = 2 days