Utility function $U$ is defined over wealth or consumption.

Risk Aversion: An individual who prefers a certain amount of wealth over a risky gamble with the same expected monetary value.

Implies diminishing marginal utility of wealth ($U''(W) < 0$, so the utility function is concave).
Willing to pay a risk premium to avoid risk.
Certainty Equivalent: The amount of certain wealth that yields the same utility as a risky gamble.
Example 7.1: Coin Flip and Risk Aversion
- Calculates expected value of a gamble and expected utility for a risk-averse individual ($U(W) = \sqrt{W}$).
- Shows that expected utility is less than the utility of the expected wealth.
- Determines the risk premium.

Risk Neutrality: $U''(W)=0$ (linear utility). Indifferent between a fair gamble and its expected value.

Risk Loving: $U''(W)>0$ (convex utility). Prefers a fair gamble to its expected value.

Absolute Risk Aversion (Pratt): $r(W) = -\frac{U''(W)}{U'(W)}$. Measures risk aversion at a given wealth level.
- A higher $r(W)$ means greater risk aversion.
- Risk premium ($P$) for a small gamble with expected value 0 and variance $\sigma_h^2$ is approximately $P \approx \frac{1}{2} r(W) \sigma_h^2$.
Relative Risk Aversion: $rr(W) = W r(W) = -W \frac{U''(W)}{U'(W)}$. Measures risk aversion relative to wealth.
- Often assumed to be constant (CRRA).
Types of Utility Functions:
- CARA (Constant Absolute Risk Aversion): $U(W) = -e^{-AW}$ (for $A>0$). $r(W) = A$.
- CRRA (Constant Relative Risk Aversion): $U(W) = \frac{W^R}{R}$ (for $R \ne 0$) or $U(W) = \ln W$ (for $R=0$). $rr(W) = 1-R$.
  - If $R=0$, $U(W)=\ln W$, $rr(W)=1$.
  - If $R=1$, $U(W)=W$, $rr(W)=0$ (risk neutral).
- Example 7.2: Constant Relative Risk Aversion
  - Calculates $r(W)$ and $rr(W)$ for $U(W)=\ln W$ and $U(W)=\sqrt{W}$.
  - Shows that for $\ln W$, $rr(W)=1$. For $\sqrt{W}$, $rr(W)=1/2$.

Methods for Reducing Uncertainty/Risk:
- Insurance: Risk-averse individuals pay a premium to transfer risk to an insurer.
  - Fair insurance: Premium equals expected loss. Risk-averse individuals will fully insure.
- Diversification: Spreading investments across multiple independent assets to reduce overall risk without sacrificing expected return.
- Flexibility (Real Options): Maintaining the ability to adjust decisions as new information becomes available, which has value in uncertain environments. (e.g., delaying investment).
- Information: Acquiring knowledge to make better decisions in uncertain situations.
State-Preference Approach: Models choice under uncertainty as choices among contingent commodities (wealth in different "states of the world").
- Let $W_1$ be wealth in state 1 (e.g., "good times") and $W_2$ in state 2 ("bad times").
- Budget constraint: $p_1 W_1 + p_2 W_2 = I$, where $p_1, p_2$ are prices of contingent claims.
- If markets are "fair" (no transaction costs or