Expected Utility showed a person who would take a sure \$156.25 instead of a lottery worth \$175 on average. That gap measures how much she dislikes risk, and its source is the shape of $u$, which is about Concavity and the Second Derivative Test.
Key idea
A person is risk averse when her utility of wealth is concave ($u'' < 0$). Then the certainty equivalent $CE$, the sure wealth with $u(CE) = E[u(W)]$, is below $E[W]$, and the risk premium $E[W] - CE$ is positive. The Arrow-Pratt measure $A(W) = -\dfrac{u''(W)}{u'(W)}$ gauges how risk averse she is at wealth $W$.
Why it works
For a concave function, the chord between two points of the graph lies below the graph (Jensen's inequality): $$p\,u(w_1) + (1 - p)\,u(w_2) < u(p\,w_1 + (1 - p)\,w_2)$$ The left side is the expected utility, the right side is the utility of the expected wealth. So the sure amount that matches the gamble's expected utility must be less than its expected value. Concavity also means diminishing marginal utility of wealth: the dollars lost in a bad outcome hurt more than the same dollars won in a good outcome help.
Worked example: Certainty equivalent and risk premium
$u(W) = \sqrt{W}$; wealth is \$400 or \$100 with equal chances.
- $E[u] = \tfrac{1}{2}(20) + \tfrac{1}{2}(10) = 15$.
- $\sqrt{CE} = 15$, so $CE = 225$.
- $E[W] = 250$, so the risk premium is $250 - 225 = 25$.
Answer: $CE = 225$, risk premium \$25.
Worked example: Log utility
$u(W) = \ln W$; wealth is 64 or 16 (thousand dollars) with equal chances.
- $$E[u] = \tfrac{1}{2}\ln 64 + \tfrac{1}{2}\ln 16 = \ln\sqrt{64 \cdot 16} = \ln 32$$
- So $CE = 32$, the geometric mean, while $E[W] = 40$.
- Risk premium: 8 thousand dollars.
Answer: $CE = 32$ thousand.
Worked example: Taking a good bet
With wealth \$100 and $u = \sqrt{W}$, should she accept a coin flip that wins \$44 or loses \$19?
- Accepting: $E[u] = \tfrac{1}{2}\sqrt{144} + \tfrac{1}{2}\sqrt{81} = \tfrac{1}{2}(12 + 9) = 10.5$.
- Refusing: $\sqrt{100} = 10$.
- Accept. The expected gain, \$12.50, is more than enough to pay for the risk.
Answer: Accept.
Common mistake
Thinking a risk-averse person refuses every gamble. Risk aversion means she needs to be paid for bearing risk, not that she never bears it: she takes a gamble when its expected gain is large enough to cover her risk premium, as in the last example. Check: compare expected utilities, not "is there risk?"
For $u = \sqrt{W}$, $A(W) = \dfrac{1}{2W}$; for $u = \ln W$, $A(W) = \dfrac{1}{W}$. Both fall as wealth rises, which fits the common observation that richer people take a given dollar gamble more calmly.