Worksheet 1: Expected Value

Question 1:

What is this person’s expected cost of illness?

The expected cost is the probability of being ill (0.3) times the cost of being ill (40,000),

$E[cost]=0.3\times 40,000=$ 12,000.

Question 2:

Assume this individual has a utility function of the form, $u(w)=w^{0.20}$. What is this person’s expected utility?

Expected utility works the same as any expectation…the “tricky” part is that we’re using the utility function to the find the values over which we form the expectation. In this case, we have two possible outcomes: a) healthy, which gives us a wealth of $100,000; or b) sick, in which case we incur the cost of illness and end up with $60,000. So to find the expected utility, we need to find the utility associated with each possible wealth value, and then we need to take the expectation over those utility values:

Step 1: Find utility values

• If healthy: $u(w){|}_{w=100,000}=100,{000}^{0.2}=$ 10

• If sick: $u(w){|}_{w=60,000}=60,{000}^{0.2}=$ 9.0288

Step 2: Take the expectation
Taking the expectation over these utility values yields: $E[u]=0.7\times$ 10 $+0.3\times$ 9.0288 $=$ 9.7086.

Question 3:

Calculate this person’s utility if they were to incur the expected cost of illness with certainty. Is this utility higher or lower than what you found in part (2)?

If they are definitely going to be sick (i.e., with probability, $p=1$), then we just need to plug in the wealth value in the sick state (60,000) into the utility function, $u=(60,000{)}^{0.2}=$ 9.0288. As should be the case, this is lower than our expected utility from part (2).