EXPECTED UTILITY AN D RISK AVERSION Consider an individual who faces two possible states of the world. With 20 percent probability he will face the bad state and have an income of $25, and with 80 percent probability he will face the good state and have an income of 5100. His utility is a function of income, U = W. a. Graph the utility function in the curve sheet below. Carefully, label the two possible outcomes mentioned above. b. Draw a line connecting the two possible outcomes on the graph. Calculate the expected utility in this example. Show this falls on the line. Label this expected utility on the graph. Calculate the certainty equivalent and risk premium. c. Now suppose the probability of the bad state increases to 50 percent, implying the probability of the good state is also 50 percent. Calculate the expected utility in this example. Mathematically show this falls on the same line. Label this expected utility in the graph. Calculate the certainty equivalent and risk premium. d. Suppose the probability of the bad state is 20 percent, implying the probability of the good state is 80 percent. However, now the bad state yields an income of So, while the good state still yields $100. Calculate the expected utility in this example. Draw a new line for finding expected utility given the new states of the world. Label this expected utility in the graph. Calculate the certainty equivalent and risk premium. e. Finally, return to the original example, except assume utility is U = W. Plot this new utility function and the expected utility line with the two possible outcomes. Calculate expected utility, the certainty equivalent and risk premium. f. Interpret the results from parts c, cl, and e. What did we learn about risk? INSURANCE DECISION Consider an individual who faces two possible states of the world. With 20 percent probability he will face a bad state ofthe world, in which he must pay out $100 in health care expenses, and in the good state of the world he has no health care expenses. In either state of the world he earns income of $100 and consumes his income less his health care expenses. He is offered an opportunity to pay $20 up front [thereby reducing his consumption by $20 in either state of the world} for a health insurance contract that promises to pay him $100 to cover health care expenses in the bad state of the world. He seeks to maximize expected utility, where the utility he would derive in any state of the world is given by U = IE, where C is his consumption in that state of the world. a. Calculate the expected value of his consumption, rst assuming that he does not purchase insurance and then assuming that he does. b. Calculate his expected utility, rst assuming that he does not purchase insurance and then assuming that he does. Would he choose to purchase the insurance? c. Now suppose that an insurer sells health insurance contracts to 10,000 individuals. Each individual faces a 20 percent chance of experiencing a health shock that requires payment of $100 in health care expenses. Their risks are idiosyncratic. This implies that in any one year approximately 20 percent of them will be hit by health shocks. Each insurance contract requires an up-front payment of $20 and pays out $100 if the buyer experiences a health shock. Approximately how much will it cost the insurer to fulfill its promise of paying out $100 to every insurance contract buyer who experiences a health shock? d. If the cost of administering the insurance arrangement is $1 per contract holder, approximately how large a premium would the insurer have to charge to cover its total costs {including payouts for medical expenses and administrative costs)? Would the contract holders be willing to pay this premium? e. If the contract holders were risk neutral and sought to maximize the expected value of consumption rather than expected utility, would they be willing to pay this premium