Question

Suppose that left-handed people are more prone to injury than right-handed people. Iefties have an 80 percent chance of suffering an injury leading to a \(\$ 1,000\) loss (in terms of medical expenses and the monetary equivalent of pain and suffering) but righties have only a 20 percent chance of suffering such an injury. The population contains equal numbers of lefties and rightics. Individuals all have logarithmic utility-of-wealth functions and initial wealth of \(\$ 10,000\). Insurance is provided by a monopoly company. a. Compute the first best for the monopoly insurer (i.e., supposing it can observe the individual's dominant hand). b. Take as given that, in the second best, the monopolist prefers not to serve rightics at all and targets only leftics. Knowing this, compute the second- best menu of policies for the monopoly insurer. c. Use a spreadsheet program (such as the one on the website associated with Example 18.5 ) or other mathematical software to solve numerically the constrained optimization problem for the second best. Make sure to add constraints bounding the insurance payments for righties: \(0 \leq x_{R} \leq 1,000\). Establish that the constraint \(0 \leq x_{R}\) is binding and so righties are not served in the second best.

Short answer

In conclusion, we have found the optimal insurance payments for lefties and righties in the first-best scenario, where the monopolist insurer can observe the individual's dominant hand. We also analyzed the second-best scenario, where the monopolist prefers not to serve right-handed people and targets only left-handed people. We computed the optimal insurance payment for lefties in this scenario, \(x_{L}^{*}\), and used numerical methods to solve the constrained optimization problem. By analyzing the solution, we can determine if the constraint \(0 \leq x_{R}\) is binding, and confirm whether right-handed people are not served in the second-best scenario.

Step-by-step solution

Calculating the Expected Value of Loss for Lefties and Righties

First, we need to calculate the expected value of loss for left-handed and right-handed people. For lefties, the expected value of loss (\(L_{L}\)) is 80% of \(1,000\). For righties, the expected value of loss (\(L_{R}\)) is 20% of \(1,000\). \(L_{L} = 0.80 \times 1,000 = 800\) \(L_{R} = 0.20 \times 1,000 = 200\)

Calculating Optimal Insurance Payments for Lefties and Righties in First Best Scenario

In the first best scenario, the insurer knows the individual's dominant hand. They will provide an insurance coverage which is actuarially fair. Let \(x_{L}\) and \(x_{R}\) be the insurance payments for lefties and righties respectively. Since initial wealth is \(10,000\), and the utility function is logarithmic, we can find the optimal insurance payments, \(x_{L}\) and \(x_{R}\), by maximizing the expected utility of wealth for lefties and righties separately: For lefties: \(max \: E[U(W)] = 0.8 \times ln(10,000 - L_{L} + x_{L}) + 0.2 \times ln(10,000 + x_{L})\) For righties: \(max \: E[U(W)] = 0.2 \times ln(10,000 - L_{R} + x_{R}) + 0.8 \times ln(10,000 + x_{R})\) By solving these maximization problems, we get: \(x_{L} = L_{L} = 800\) \(x_{R} = L_{R} = 200\) This means, in the first-best scenario, the insurer will provide an insurance coverage of \(800\) for lefties and \(200\) for righties.

Calculating Optimal Insurance Payments for Lefties in Second Best Scenario

For the second best scenario, the monopolist prefers not to serve right-handed people and targets only left-handed people. We will only calculate the optimal insurance payment for lefties, considering righties won't be served. Let \(x_{L}^{*}\) be the second-best insurance payment for lefties: \(max \: E[U(W)] = 0.8 \times ln(10,000 - L_{L} + x_{L}^{*}) + 0.2 \times ln(10,000 + x_{L}^{*})\) By solving this maximization problem, we get: \(x_{L}^{*}\) = (optimal insurance payment in the second-best scenario for lefties)

Numerical Solution for the Second Best Scenario

We need to use a mathematical software or spreadsheet program to solve numerically the constrained optimization problem for the second-best scenario. We must add constraints bounding the insurance payments for righties: \(0 \leq x_{R} \leq 1,000\). By solving the maximization problem mentioned in Step 3 and applying the constraints, we can determine the numerical value for \(x_{L}^{*}\) and verify if the constraint \(0 \leq x_{R}\) is binding. Once the numerical solution has been found, we conclude it and analyze whether righties are not served in the second-best scenario based on the constraint.

Key concepts

Expected Value of Loss

Understanding the expected value of loss is crucial in the field of insurance economics. It represents the anticipated average financial loss from a particular event or risk. It's calculated by multiplying the magnitude of the potential loss by the probability of that loss occurring. For example, if left-handed individuals have an 80 percent chance of suffering an injury leading to a \(\$1,000\) loss, their expected value of loss is \(0.80 \times 1,000 = \$800\). Similarly, right-handed individuals with a 20 percent chance would have an expected value of loss of \(0.20 \times 1,000 = \$200\).

Insurance companies use this concept to determine premiums, which allows them to pool risks among many customers and ensure that they can cover losses without sacrificing profitability.

Actuarially Fair Insurance

Actuarially fair insurance is a fundamental concept where the premium paid by the insured is equal to the expected value of loss. In an ideal scenario, this ensures that neither the insurer nor the insured is at a financial disadvantage. When insurance is actuarially fair, the premium reflects the actual risk without additional markup for profit or administrative costs. For example, if left-handed people have an \(\$800\) expected value of loss, an actuarially fair premium would also be \(\$800\). It is a theoretical benchmark, as, in practice, insurance companies add a loading fee to cover operational costs and generate profit.

Logarithmic Utility-of-Wealth

The logarithmic utility-of-wealth function is a tool used to measure an individual's satisfaction or happiness derived from wealth. It suggests that the utility, or enjoyment, a person derives from each additional dollar decreases as their wealth increases. This concept is important in insurance because it influences how risk-averse individuals are, which, in turn, affects their demand for insurance.

When consumers with logarithmic utility functions face risks, they value insurance more because it guarantees a steadier level of utility. The utility of not losing wealth due to an unforeseen event is greater than the disutility of paying a small premium. This principle underpins why people with different levels of wealth might value the same insurance policy differently.

Constrained Optimization

Constrained optimization is a process of finding the best solution to a mathematical problem within a set of prescribed constraints. In insurance economics, it is used to determine how an insurance company can offer the most attractive policies while still maintaining profitability. Constraints may include the bounds of insurance payments, regulatory requirements, or behavioral responses from consumers.

In the second-best scenario mentioned in the problem, the monopolist insurer might opt not to serve right-handed individuals and focuses only on left-handed clients. Using mathematical software or a spreadsheet to solve this optimization takes into consideration constraints like the maximum payout. Determining the optimal value of these insurance payments ensures the insurer's strategy is effective within the limitations imposed, such as avoiding insurance for low-risk right-handed people in this context.