Suppose there is a \(50-50\) chance that a risk-averse individual with a current wealth of \(\$ 20,000\) will contract 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 7.1 ) 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; and (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.

First, we need to find the expected loss from the debilitating disease. Given that there is a 50-50 chance of the individual contracting the disease and hence, suffering a loss of $10,000. The expected loss can be computed as follows: Expected Loss = Probability of contracting disease * Loss = 0.5 * $10,000 = $5,000

Actuarially fair insurance means that the insurance premium is equal to the expected loss. In this case, the cost of actuarially fair insurance can be calculated as: Cost of fair insurance = Expected Loss = $5,000

Since the individual is risk-averse, their utility function is concave. We can illustrate this situation with the following graph: - On the x-axis, plot wealth ranging from \(10,000 to \)20,000 - On the y-axis, plot the utility of wealth - Draw a concave utility function curve - Mark the initial wealth of the individual ($20,000) on the x-axis, and the corresponding utility on the curve - Also mark the utility points if the individual contracts the disease with and without insurance - The graph should show a higher utility of wealth for the individual when they have insurance compared to when they are uninsured Based on the utility graph, the risk-averse individual would prefer to purchase the fair insurance against the loss rather than accepting the gamble uninsured, as it provides them a higher utility of wealth. #Part B#

Since the individual already has a fair insurance policy covering the complete loss for a premium of $5,000 (from Part A), we can use this value to compare it with the second type of insurance policy. Cost of fair insurance covering complete loss = $5,000

Now, let's calculate the premium of the fair insurance policy covering only half of any loss incurred. To do this, we need to find the expected loss covered by the insurance policy: Expected Loss covered by the policy = 0.5 * $5,000 = $2,500 Since this is an actuarially fair insurance policy, the premium is equal to the expected loss covered by the policy: Cost of fair insurance covering half of the loss = Expected Loss covered by the policy = $2,500

Now, we can compare the two insurance policies: 1. Fair insurance policy covering the complete loss: Cost = $5,000 2. Fair insurance policy covering half of the loss: Cost = $2,500 Since the individual is risk-averse, they would prefer to minimize their exposure to the risk of loss. In this case, the first insurance policy (covering the complete loss) provides better protection for the individual compared to the second policy (covering only half of the loss). Therefore, the risk-averse individual would generally regard the first policy as superior to the second one.