CH 6: Decision-Making Under Uncertainty

Overview

• Topic: Decision-making under uncertainty using expected utility theory.
• Purpose: Apply economic tools to real-world choices involving risk, insurance, and lotteries.
• Key idea: People care about utility, not just expected monetary value; utility functions determine risk attitudes.

Expected Value vs Expected Utility

• Expected value (EV): sum of outcome values weighted by probabilities.
• Expected utility (EU): sum of utilities of outcomes weighted by probabilities.
• EV can mislead because utility is typically nonlinear in dollars.

Risk Attitudes (Definitions)

• Risk-averse: utility is concave; diminishing marginal utility of wealth.
  • Prefer certain outcome over gamble with same EV.
• Risk-neutral: utility is linear; care only about expected value.
• Risk-loving: utility is convex; prefer risky prospects, sometimes even negative EV.
• Loss aversion (behavioral): losses feel worse than equivalent gains feel good; a psychological bias beyond standard EU.

Utility Function Examples And Implications

• u(C) = sqrt(C): concave, risk-averse.
  • Example: start with C0 = $100. Coin flip: +$125 / -$100.
  • EV = 0.5(125) + 0.5(-100) = $12.50 positive.
  • EU = 0.5 u(225) + 0.5 u(0) = 7.5 < u(100)=10 → reject gamble.
  • Certainty equivalent: wealth giving same utility ≈ $56.25; risk premium ≈ $43.75.
• u(C) = 0.1 C: linear, risk-neutral.
  • Same starting point leads to EU > 10 → accept more-than-fair bet.
• u(C) = C^2 / 1000: convex, risk-loving.
  • Would accept risky or even unfair bets (negative EV) when EU increases.

Graphical Intuition

• Utility vs Wealth: concave curve for risk-averse.
• A two-outcome gamble maps to two points on the curve; EU is average of utilities, not utility at average wealth.
• For small stakes relative to wealth, utility is locally linear → people act more risk-neutral.

Key Formulas

• Expected Value: EV = Σ p_i * x_i
• Expected Utility: EU = Σ p_i * u(x_i)
• Risk premium = (expected monetary loss) difference between EV and certainty equivalent (how much one pays to avoid risk).

Behavioral Considerations

• Loss aversion: individuals demand more to give up an owned item than they'd pay to acquire it.
  • Example: people who own a $5 mug demand ~$7 to sell, while buyers pay ~$3.
• Loss aversion and risk aversion both reduce willingness to accept gambles.

Applications

Insurance

• Setup example:
  • Income = $40,000; probability of accident = 1% (0.01); loss = $30,000.
  • Expected monetary loss = $300/year.
  • With u(C) = sqrt(C) and no savings:
    • EU uninsured = 0.99 sqrt(40,000) + 0.01 sqrt(10,000) ≈ 199.
    • EU insured (pay premium X): sqrt(40,000 - X).
    • Indifference premium X* ≈ $399 > $300 → risk premium $99.
• Implications:
  • People pay more than expected loss to avoid risk → explains large insurance market.
  • Risk premium increases with larger losses relative to income.
  • Risk premium falls as income rises (wealth makes people more locally linear).
  • Small-value warranties may be refused by wealthy, accepted by poorer consumers.*

Lottery

• Lotteries have negative EV (e.g., $1 ticket yields $0.50 EV) yet remain popular.
• Four candidate explanations:
  1. Risk-loving people buy lotteries (contradicted by insurance purchases).
  2. Friedman–Savage: people are locally risk-averse, globally risk-loving (risk-loving for huge stakes). Predicts preference for big-jackpot tickets over small scratch-offs.
  3. Entertainment value: thrill or fun of playing adds utility, making tickets rational purchases.
  4. Ignorance: people misunderstand probabilities and EV; may explain heavy lottery spending in some low-income groups.
• Observed behavior: most spending is on small, frequent scratch-offs, which disfavors pure Friedman–Savage explanation.
• Policy implication: whether lotteries are welfare-improving depends on whether purchases reflect entertainment or ignorance.

Key Terms And Definitions

• Expected Value (EV): probability-weighted average monetary outcome.
• Expected Utility (EU): probability-weighted average utility outcome.
• Risk-averse: preferring certain wealth to risky prospect with same EV.
• Risk-neutral: indifferent between risk and certainty with same EV.
• Risk-loving: preferring risky prospect even with lower EV.
• Loss aversion: psychological bias where losses loom larger than gains.
• Risk premium: extra amount one pays above expected loss to avoid the gamble.
• Certainty equivalent: guaranteed wealth giving same utility as the gamble.

Action Items / Next Steps (for students)

• Practice computing EV, EU, certainty equivalents, and risk premiums for given utility functions.
• Compare decisions under different utility functional forms (concave, linear, convex).
• Reflect on behavioral biases (loss aversion) and how they modify standard EU predictions.
• Consider policy implications: how different explanations for lottery participation affect regulation and public finance.