# W1: Expected Value

An individual starts with a wealth of $100,000. With probability 0.3, they will get sick and incur a cost of $40,000.

## 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. Is this utility higher or lower than what you found in part (2)?**

The expected cost of illness is 12,000, so the expected wealth is 88,000. We just need to calculate the utility at this expected monetary value, \(u=(88,000)^{0.2}=\) 9.7476. As should be the case, this is higher than the expected utility from part (2) because this envisions a risk-less scenario whereas the expected utility in part (2) envisioned a risky scenario.