Our goal: build tools (probability, expected value, utility) to understand how people make choices under risk.
To understand risk in this class, we need four building blocks:
Definition: The likelihood that a given outcome will occur
Examples:
- 10% chance of heart disease in 10 years → 100 of 1,000 people like me will get sick
- 5% chance of a car accident in a year → 50 of 1,000 drivers will file a claim
In this class: two outcomes only — sick (probability \(p_s\)) or healthy (probability \(p_h = 1 - p_s\)).
Definition: the monetary value of each possible outcome
Example:
- Start with $1,000 in wealth
- If sick → pay $500 for care → \(w_s = 500\)
- If healthy → pay nothing → \(w_h = 1000\)
Definition: The probability-weighted average of possible payoffs
For two outcomes, \(x_1\) and \(x_2\), with probabilities \(p_1\) and \(p_2\):
\[E[x] = p_1 x_1 + p_2 x_2\]
Example: With 90% chance of \(w_h = 1000\) and 10% chance of \(w_s = 500\):
\[E[w] = 0.9 \times 1000 + 0.1 \times 500 = 950.\]
What is my expected cost?
I will incur a cost of $100,000 with 10% probability. So my expected cost is just \(E[cost]=0.1*100,000 =\) 10,000.
Definition: How individuals value different outcomes, often with a utility function \(u(w)\)
Expected utility combines probabilities and utilities:
\[E[u(w)] = p_h u(w_h) + p_s u(w_s)\]
An individual starts with a wealth of $100,000. With probability 0.3, they will get sick and incur a cost of $40,000.
With probabilities, payoffs, expected values, and utilities/preferences, we can now measure preferences toward risk (i.e., how people feel about uncertain outcomes).
In economics, we usually assume people are risk averse.
Risk aversion follows from diminishing marginal utility.
\(u'(x_{1}) > u'(x_{2})\) for \(x_{1} < x_{2}\)
What does this mean in words?
Say your utility function is \(u(w)=\sqrt{w}\) and that you’re starting with \(w=\) $100. I propose a lottery in which I flip a coin…heads you win $20 and tails you lose $20.
Expected wealth is simply \(\frac{1}{2} \times 80 + \frac{1}{2} \times 120 =\) 100, which yields a utility of \(u(w)=\) 10
But your expected utility is \(\frac{1}{2} \times u(w_{heads}) + \frac{1}{2} \times u(w_{tails}) = \frac{1}{2} \times \sqrt{80} + \frac{1}{2} \times \sqrt{120} =\) 9.95.
Because expected utility < utility at expected wealth, the lottery is less attractive than a sure outcome of the same expected value.
This gap is what we’ll later call the risk premium — and it’s the reason people are willing to pay extra for insurance.