Insurance

1. Insurance

2. What is insurance? • Consider a club • 100 members • About the same age, about the same lifestyles • About once a year one of the members gets sick and incurs expenses of $1,000. • Club collects $10 from each member each year. • Invests it somewhere to maintain or increase its value. • Pays it out to members who file claims.

3. What has happened? • Insured pay $10 per year, guaranteed, to avoid the possibility of having to pay $1,000. • Although outlays for an individual may be highly variable, • Outlays for a group are generally rather predictable. • The “Law of Large Numbers” suggests that as group size increases, the distribution of the average rate of illness will collapse around the “true” probability of the illness.

4. Insurance Terms • Premium - $X premium for $Y of coverage. • Coinsurance and Copayment - The insured person must pay the loss. • % paid is the coinsurance rate (varies from 0 to 100%). • amount paid is the copayment. • Deductible - Some amount may be deducted from the payment to the insured person, irrespective of coinsurance. • Why coinsurance and deductibles? Discuss.

5. Risk and Insurance • What is “expected value?” • What’s the expected value of a coin flip that pays $1 for heads and 0 for tails. • A> (Prob. of heads) * $1 + (Prob. of tails) * 0. • E = 0.5 * 1 + 0.5 * $0 = $0.5 • How much would you pay to play this kind of game? • Why do we care? • A> Because insurance is based on expected losses!

6. Marginal Utility of Wealth and Risk Aversion • Would you bet $50 on a coin flip that would give you either 0 or $100. • Would you bet $5000 on a coin flip that would give you either 0 or $10000? • Why? • Consider a utility function with wealth. Utility 5 10 15 20 Wealth

7. Marginal Utility of Wealth and Risk Aversion • Suppose wealth is 10 (thousand). It gives him U = 140 200 140 Utility • Suppose wealth is 20 (thousand). It gives him U = 200 5 15 10 20 Wealth

8. Cardinal Utility • Just about everywhere else in microeconomics we use “ordinal utility”. • Here we use “cardinal utility.” • Why?

9. Expected wealth is due to risk E (U) = (Prob. Healthy * Utility of Wealth if Healthy) + (Prob. Ill * Utility of Wealth if Ill) 200 140 Utility E (U) = Prob. Healthy * Utility of 20 + Prob. Ill * Utility of 10 5 15 10 20 We’re on the maroon line rather than the blue curve BECAUSE of RISK Wealth

10. Expected wealth is due to risk We see that total utility is increasing, but what about marginal utility 200 140 Utility We see that it is decreasing. Why? 5 10 15 20 In this example, losses bother us more than gains help us, so we lose utility due to risk. Wealth Mgl. Utility Wealth

11. Deal or No Deal? • Biggest Fail Ending (Not my term). • www.youtube.com/watch?v=H9CQscwXBt0 • What does utility function look like? Utility E(W) 1 500T 1M

12. Deal or No Deal? • They offer him 412T and he refuses. Is he risk loving? • Maybe not. • What could utility function look like? Utility U[E(500)] U(400) E(W) 400T 500T 1 1M

13. Deal or No Deal? • What if it looked like this? Utility U[E(500)] U(400) E(W) 400T 500T 1 1M

14. Insurance puts us on the Blue Line Suppose the probability of illness is 0.05. E (U) = Prob. Healthy * Utility of 20 + Prob. Ill * Utility of 10 E (U) = (0.95 * 200) + (0.05 * 140). E (U) = 197. 200 140 Utility 19.5 BUT, without risk, Utility would be 199. 5 10 15 20 Wealth Mgl. Utility Insurance  risk (you end up at wealth = 19.5 and U=199 whether you’re sick or not). Wealth

15. Does everyone buy insurance? • Depends on marginal utility of wealth. • Some people “love” risk. • How would we draw this? 200 140 Utility • How would we draw curves for someone who was “risk neutral.” 5 10 15 20 Wealth Mgl. Utility Wealth

16. How much insurance? E (U) = (Prob. Ill * Utility of Wealth if Ill) + (Prob. Healthy * Utility of Wealth if Well) (8.3) If insured, new wealth = Wealth - insurance premium new wealth = W - aq q is the value of the insurance purchased a is the fraction paid, called the premium Wealth if ill = [(W - L) + q] - aq = W - L + (1-a)q Wealth if well = W - aq

17. How much insurance? E (U) = (Prob. Ill * Utility of Wealth if Ill) + (Prob. Healthy * Utility of Wealth if Well) (8.3) Wealth if ill = [(W - L) + q] - aq = W - L + (1-a)q Wealth if well = W - aq So we put the maroon stuff into the blue equation to get. [1] E (U) = [Prob. Ill * Utility (W - L + (1-a)q] + [2] [Prob. Healthy * Utility (W - aq)]

18. How much insurance? [1] E (U) = [Prob. Ill * Utility (W - L + (1-a)q] + [2] [Prob. Healthy * Utility (W - aq)] A purchase of 1 extra dollar of insurance increases utility in [1]. When he is ill, it will increase wealth by (1-a). The extra utility is (1-a) * MU (wealth when ill) * p. (Since wealth is lower, this is a relatively large marginal benefit). When he is healthy, it will decrease wealth by a. “Loss” of utility is a * MU (wealth when well) * (1-p).

19. Amount of Insurance MC Mgl. Benefits, Mgl. Costs A purchase of 1 extra dollar of insurance increases utility in [1]. When he is ill, it will increase wealth by (1-a). The extra utility is (1-a) * MU (wealth when ill) * p. (Since wealth is lower, this is a relatively large marginal benefit). When he is healthy, it will decrease wealth by a. “Loss” of utility is a * MU (wealth when well) * (1-p). MB Amount of insurance

20. Amount of Insurance MC Mgl. Benefits, Mgl. Costs A purchase of 1 extra dollar of insurance increases utility in [1]. When he is ill, it will increase wealth by (1-a). The extra utility is (1-a) * MU (wealth when ill) * p. (Since wealth is lower, this is a relatively large marginal benefit). When he is healthy, it will decrease wealth by a. “Loss” of utility is a * MU (wealth when well) * (1-p). MB q* Amount of insurance