Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contact a debilitating disease and suffer a loss of \(\$ 10,000\) a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 8.1 ) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) A fair policy covering the complete loss. (2) A fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first.

1. Calculate the expected loss: The risk-averse individual has a \(50-50\) chance of losing \(\$10,000\). The expected loss can be calculated as follows: $$ Expected\ Loss = Probability\ of\ Loss \times Loss\ Amount $$ $$ Expected\ Loss = 0.5 \times \$10,000 = \$5,000 $$ 2. Calculate the actuarially fair insurance premium: The actuarially fair insurance premium is equal to the expected loss, in this case, \(\$5,000\). 3. Use a utility-of-wealth graph to show preference: To show preference using a utility-of-wealth graph, plot the individual's utility function with wealth on the x-axis and utility on the y-axis. Since the individual is risk-averse, the utility function will be concave. 4. Position of fair insurance and gamble uninsured: To show preference, locate the points representing the individual's wealth when buying fair insurance, accepting the gamble uninsured, and suffering the loss on the graph. As the person is risk-averse, fair insurance will have a higher utility compared to accepting the gamble uninsured.

1. Calculate the cost of the second policy: The second policy covers only half of any loss incurred, i.e., \(\$5,000\). The fair insurance premium for the second policy is equal to half of the expected loss: $$ Cost\ of\ Second\ Policy = 0.5 \times \$5,000 = \$2,500 ``` 2. Compare both policies: To show that the second policy is inferior, compare the utility of wealth for both insurance policies. Since the second policy only covers half of the loss, the person would lose more of their wealth in the case of the debilitating disease. In this case, the utility of the second policy is less than the utility of the first policy. 3. Conclusion: Overall, the risk-averse individual will prefer the fair policy covering the complete loss (first policy) to a fair policy covering only half of any loss incurred (second policy). The cost of the second policy is less, but the individual's utility is higher with the first policy as it covers the complete loss.