Consumption, Production, Welfare B:Choice under Uncertainty (c‘td) Univ. Prof. dr. Maarten Janssen University of Vienna Winter semester 2013
Risk aversion • Expected utility does not mean we only consider expected value of a lottery • Risk aversion: u(w) is concave, where w is wealth • Risk neutral: u(w) is linear and only expected value matters • Risk loving: u(w) is convex • Figures • Risk aversion: expected utility of the lottery is smaller than the utility of the expected value: +) • Degree of absolute risk aversion measured by Arrow-Pratt coefficient: • Constant absolute risk aversion utility function: • Degree of relative risk aversion measured by Arrow-Pratt coefficient: • Constant relative risk aversion utility function:
Risk premium • F(.) is the cumulative distribution function of the lottery over wealth • Certainty equivalence c(F,u) is defined as • Risk premium (bold red line segment) is the amount of money you are willing to give up to exchange a lottery with an expected value of for the certainty equivalence of that lottery: u c(F,u) w
How much are people willing to pay for full Insurance? • What is the expected utility of the situation without insurance? • Full insurance takes all risks away. You are indifferent between risky situation and insured situation if after insurance you have x such that • But this is the certainty equivalent • Maximal willingness to pay is the risk premium! • How does this depend on degree of risk aversion? • If an individual is more risk averse than another, what can we say about the certainty equivalent? • What is the risk premium if you are risk neutral? • Fair bet is one that leaves expected income unchanged (fair price is what a risk neutral person is just willing to pay for fair bet) • Why is insurance (some demand, others provide) possible?
Probability premium • For any utility function with risk aversion, we have that for any w and any ε, • How much should probabilities be changed away from ½ such that equality is restored? Probability premium ) such that • Figure
Small experiment: What would you choose? • A1: 1 M euro for sure • Or • A2: 10% chance of 5M euro, 89% chance of 1 M euro and 1% chance of 0 • ============= • A3: 10% chance of 5M euro, 90% chance of 0 • A4: 11% chance of 1 M euro and 89% chance of 0
What does expected utility theory say? • Triangle representation (see article by Mark Machina) • Three possible outcomes • x: 0 • y: 1 M • z: 5 M • Choice between A1 and A2? Depends on risk aversion • Choice between A3 and A4? Also Depends on risk aversion • But consistency between two choices.
Indifference curves in triangle • or • Straight, upward sloping, parallel lines • Does not depend on degree of risk aversion • The larger the slope, the stronger the risk aversion (red more risk averse than green)
Allais’ Paradox • Many people do not make consistent choices (according to expected utility) • Choose A1 and A3 • Can only be if indifference curves are not parallel straight lines, as line through A1 and A2 has slope of 10 (10% more of z and 1% more of x) and line through A3 and A4 as well • Fanning out: when you are better off, you are more risk averse than when you are not
Alternatives to EUT • Transformations of probabilities, e.g., • Kahnemann, Tversky (1979) • People tend to attach more weight to low probability events than probability suggests
Insurance: anotherlook • Two statesoftheworld on axis (accidentdid happen, or not) • Initial endowment: point without insurance (blue dot) (w,0) • Indifferencecurve • Why not straightline? • Interpretation • Fair insurance: , where p is prob ofaccident, w isdamageand f isinsurancefee • Competitioninterpretation Wealthaccident Wealthnoaccident
Insurance: optimal choice • Fair insurance: • Fullinsurancepoint (w-f,w-f) on 45 line • Slope fair insurancelineis • Indifferencecurve.At 45lineslopeis • Ifriskaverseconsumersareoffered fair insurancethey will alwaysbuyfullinsurance. Wealthaccident Wealthnoaccident
Measures of Risk • First-order Stochastic dominance: • Distribution has first-order stochastic dominance over if • Define • Second-order stochastic dominance: • Distribution has second-order stochastic dominance over if for all y. • If is a mean-preserving spread (same mean, larger variance) of , then has second-order stochastic dominance over • Figures
Why do people buy lottery tickets? • Risk loving (obvious answer), • but most people seem to be rather risk averse (buy insurance) • Risk averse in the “normal domain of everyday life”, but like to enjoy increase in status (Friedman and Savage, 1948) • Figure with risk aversion in normal areas of income, risk loving in much higher areas of income • Overestimate likelihood of winning price • Kahnemann, Tversky (1979) • Prize winners are advertised, get attention.