1、Why buy insurance?Demand for insurance driven by the fear of the unknown Hedge against risk-the possibility of bad outcomes Purchasing insurance means forfeiting income in good times to get money in bad timesIf bad times avoided,then money lost Ex:The individual who buys health insurance but never v
2、isits the hospital might have been better off spending that income elsewhere.Risk aversion Hence,risk aversion drives demand for insurance We can model risk aversion through utility from income U(I)Utility increases with income:U(I)0Marginal utility for income is declining:U(I)0Marginal utility decr
3、easing U(I)0Adding uncertainty to the modelAn individual does not know whether she will become sick,but she knows the probability of sickness is p between 0 and 1Probability of sickness is p Probability of staying healthy is 1-pIf she gets sick,medical bills and missed work will reduce her income IS
4、=income if she does get sickIH IS =income if she remains healthy Expected valueThe expected value of a random variable X,EX,is the sum of all the possible outcomes of X weighted by each outcomes probability If the outcomes are x1,x2,.,xn,and the probabilities for each outcome are p1,p2,.,pn respecti
5、vely,then:EX=p1 x1+p2 x2+pn xn In our individuals case,the formula for expected value of income EI:EI=p IS+(1-p)IH Example:expected valueSuppose we offer a starving graduate student a choice between two possible options,a lottery and a certain payout:A:a lottery that awards$500 with probability 0.5
6、and$0 with probability 0.5.B:a check for$250 with probability 1.The expected value of both the lottery and the certain payout is$250:EI=p IS+(1-p)IH EA=.5(500)+.5(0)=$250EB=1(250)=$250People prefer certain outcomesStudies find that most people prefer certain payouts over uncertain scenarios If a stu
7、dent says he prefers uncertain option,what does that imply about his utility function?To answer this question,we need to define expected utility for a lottery or uncertain outcome.Expected UtilityThe expected utility from a random payout X EU(X)is the sum of the utility from each of the possible out
8、comes,weighted by each outcomes probability.If the outcomes are x1,x2,.,xn,and the probabilities for each outcome are p1,p2,.,pn respectively,then:EU(X)=p1 U(x1)+p2 U(x2)+pn U(xn)ExampleThe students preference for option B over option A implies that his expected utility from B,is greater than his ex
9、pected utility from A:EU(B)EU(A)U($250)0.5 U($500)+0.5 U($0)In this case,even though the expected values of both options are equal,the student prefers the certain payout over the less certain one.This student is acting in a risk-averse manner over the choices available.Expected utility without insur
10、anceLottery scenario similar to case of insurance customerShe gains a high income IH if healthy,and low income IS if sick.Uncertainty about which outcome will happen,though she knows the probability of becoming sick is p Expected utility EU(I)is:EU(I)=p U(IS)+(1-p)U(IH)Consider a case where the pers
11、on is sick with certainty(p=1):EU=U(IS)equals the utility from certain income IS(Point S)Consider case where person has no chance of becoming sick(p=0):EU=U(IH)equals utility from certain income IH(Point H)EU(I)and probability of sicknessWhat if p lies between 0 and 1?For p between 0 and 1,expected
12、utility falls on a line segment between S and H Ex:p=0.25For p=0.25,persons expected income is:EI=0.25IS+(1-.25)IH Utility at that expected income is EU(I)(Point A)Expected utility and expected incomeCrucial distinction betweenExpected utility EU(I)Utility from expected income U(EI)For risk-averse p
13、eople,U(EI)EU(I)Risk-averse individuals Synonymous definitions of risk-aversion:Prefer certain outcomes to uncertain ones with the same expected income.Prefers the utility from expected income to the expected utility from uncertain income U(EI)EU(I)Concave utility functionU(I)0U(I)IH)and gains incom
14、e in the sick state(IS U(AP)U(A)The ideal insurance contractFor anyone risk-averse,actuarially fair&full insurance contract offers the most utilityHence,it is called the ideal insurance contractIdeal and non-ideal insurance contracts:Comparing non-ideal contractsAF Full but actuarially unfair contra
15、ctAP Partial but actuarially fair contractComparing non-ideal contractsIn this case,U(AF)U(AP)Even though AF is actuarially unfair,its relative fullness(i.e.higher payout)makes it more desirableBut notice if contract AF became more unfair,then expected income EI fallsIf too unfair,AF may generate le
16、ss utility than APSimilarly,AP may become more full by increasing its payoutUncertainty falls,so point AP movesAt some point,this consumer will be indifferent between the two contracts ConclusionDemand for insurance driven by risk aversionDesire to reduce uncertaintyDiminishing marginal utility from incomeU(I)is concave,so U(I)EU(I)Risk aversion can explain not only demand for insurance but can also help explainLarge family sizesPortfolio diversificationFarmers scattering their crops and land holdings
