Consumption, Production, Welfare B: Choice under Uncertainty

1. Consumption, Production, Welfare B:Choice under Uncertainty Univ. Prof. dr. Maarten Janssen University of Vienna Winter semester 2013

2. Many applications • Why do people take insurance? • Why do people buy lottery tickets? • Why do they do both? • How do we make choices regarding health( life and dealth) • Should a particular expensive medicine (or treatment) be part of an insurance package? • If you have a limited funds (or time) how to spend it if outcomes are uncertain? • Cost, benefit health related decisions

3. Choice between alternative chance events • Example of insurance against theft • No insurance, two outcomes • 0 if item (money) is stolen, with probability p • w if no theft , with probability 1-p • Full insurance: w-f with certainty, where f is full insurance fee • Partial insurance , two outcomes • if item (money) is stolen, with probability p (wheref is partial insurance fee, is deductible) • if no theft , with probability 1-p • In general, lotteryover different statesoftheworld: L=(,...,) with

4. Preference: Independence Axiom • Preferences satisfy the independence axiom if for all L, L´ and L” and all we have L≥ L´if, andonlyif, • Why is this called independence axiom? • Is it a natural condition? • What is equivalent in standard choice situations?

5. Von Neumann-Morgenstern Utility • U(.) attaches to each possible lottery to be chosen a number • U(.) is of the expected utility form if there exist for any possible outcome i, a number such thatforanylotteryoverthepossibleoutcomestheutilitycanbewrittenas • If preferences are continuous and satisfy the independence axiom, then they can be represented by a utility function of the expected utility form: • L≥ L´ if and only if

6. Implications • Utility functions are linear • Only linear transformations of given utility function represent same preferences • Expected utility is a cardinal measure • Difference in utility levels has a meaning • Consider four lotteries: difference in utility levels between lotteries is larger than difference in utility levels between lotteries : • if and only if ) iff • Why is “normal” utility only an ordinal measure.

7. 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 risk aversion measured by Arrow-Pratt coefficient:

8. 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

9. 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? • What is the risk premium of you are risk neutral? • Why is insurance (some demand, others provide) possible?

10. 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

11. Small experiment: What would you choose? • A1: 1 Meuro 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

12. What does expected utility theory say? • Triangle representation • 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.

13. 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)

14. 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

15. Alternatives to EUT