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Choice Under Risk and Uncertainty
Decision-making under risk and uncertainty is a key topic in microeconomics and behavioral economics. It examines how individuals and firms make choices when outcomes are not certain.
🔹 Risk vs. Uncertainty:
- Risk: Probabilities of different outcomes are known (e.g., rolling a die).
- Uncertainty: Probabilities of outcomes are unknown (e.g., future stock market returns).
There are two main approaches to analyzing such decisions:
- Expected Utility Theory (EUT) – Used when probabilities are known.
- Subjective Expected Utility (SEU) – Used when probabilities are unknown.
1. Expected Utility Theory (EUT)
Basic Concept
When making decisions under risk, individuals maximize their expected utility, not expected monetary value. EU=p1U(X1)+p2U(X2)+…+pnU(Xn)EU = p_1 U(X_1) + p_2 U(X_2) + … + p_n U(X_n)
where:
- pip_i = Probability of outcome XiX_i
- U(Xi)U(X_i) = Utility from outcome XiX_i
🔹 Example: Suppose you are offered two bets:
- Bet A: 50% chance to win $100, 50% chance to win $0.
- Bet B: 100% chance to win $40.
The expected monetary value: E(A)=(0.5×100)+(0.5×0)=50E(A) = (0.5 \times 100) + (0.5 \times 0) = 50 E(B)=(1×40)=40E(B) = (1 \times 40) = 40
But a risk-averse person may prefer B if its expected utility is higher.
2. Attitudes Toward Risk
Consumers react to risk in different ways, categorized as:
| Risk Attitude | Utility Function | Example Behavior |
|---|---|---|
| Risk Averse | Concave U(X)U(X) | Prefers a sure thing over a risky bet (e.g., buying insurance). |
| Risk Neutral | Linear U(X)U(X) | Indifferent between sure thing and risky bet of equal expected value. |
| Risk Loving | Convex U(X)U(X) | Prefers risky bet over a sure thing (e.g., gambling). |
🔹 Risk Aversion and Certainty Equivalent
The certainty equivalent (CE) is the amount of money a person would accept instead of taking a gamble.
- A risk-averse person prefers CE < Expected Value (EV).
- Risk premium = EV−CEEV – CE (amount a person is willing to pay to avoid risk).
3. Measures of Risk Aversion
The Arrow-Pratt measure of risk aversion is given by: r(X)=−U′′(X)U′(X)r(X) = – \frac{U”(X)}{U'(X)}
- If r(X)>0r(X) > 0, the person is risk-averse.
- If r(X)=0r(X) = 0, the person is risk-neutral.
- If r(X)<0r(X) < 0, the person is risk-loving.
4. Choice Under Uncertainty: Subjective Expected Utility (SEU)
When probabilities are unknown, decision-makers assign subjective probabilities to outcomes.
🔹 Savage’s Axioms
- Consumers form subjective beliefs about probabilities.
- They maximize subjective expected utility (SEU): SEU=p1U(X1)+p2U(X2)+…+pnU(Xn)SEU = p_1 U(X_1) + p_2 U(X_2) + … + p_n U(X_n)
- Follows the axioms of rational choice, like completeness and transitivity.
🔹 Ellsberg Paradox (Violation of SEU)
- People prefer known risks over unknown risks, showing ambiguity aversion.
- Example: Given two urns, one with 50 red & 50 black balls, another with an unknown mix of red & black, most people prefer the first, even though probabilities are unknown in the second.
5. Alternative Decision Models
- Prospect Theory (Kahneman & Tversky)
- People perceive gains and losses differently.
- Losses hurt more than gains of the same magnitude (loss aversion).
- Uses value function rather than utility function.
- Maximin & Maximax (Decision Criteria for Uncertainty)
- Maximin: Choose the option with the best worst-case scenario (pessimistic).
- Maximax: Choose the option with the best possible outcome (optimistic).
- Minimax Regret Criterion
- Minimize the maximum regret from making a wrong decision.
6. Applications of Choice Under Risk and Uncertainty
- Insurance Markets: Risk-averse individuals buy insurance even when its expected value is negative.
- Investment Decisions: Investors balance expected returns vs. risk.
- Behavioral Finance: Explains market anomalies like overreaction and underreaction.
- Public Policy: Governments use uncertainty models in climate change and health policies.
7. Conclusion
- Expected Utility Theory helps explain decisions under risk.
- Subjective Expected Utility applies when probabilities are unknown.
- Prospect Theory and other models explain real-world deviations.