Optimal Health Insurance and Trade‐Off between Health and Wealth

Health insurance is considered to be a special type of nonlife insurance with two important features. First, compared with property insurance, health insurance provides valuable hedge against unpredictable shocks to health status, instead of loss on property. Therefore, a modified utility function that describes the trade‐off between health and wealth should be applied in optimal indemnity design. Second, in the case that the insured is severely or critically ill, with necessary medical treatment, the insured may not fully recover from an illness or an injury. The doctor usually communicates with the patient to set up a personalized treatment plan and explains clearly about the expected outcome beforehand. Hence, there is some probability that health insurance helps to rescue the insured from disastrous financial burden, but it still yields a lower utility of health. By taking these special features into account, we formulate the optimization problem and characterize the optimal solutions via the Lagrange multiplier method and optimal control technique. Finally, we examine our optimal contracts by numerical illustration. Our research work gives new insights into health insurance design.