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The decision to buy health insurance is fundamentally an economic choice that involves weighing the costs of insurance premiums against the expected benefits of risk reduction. In the context of the social sector and environment, health insurance has broader implications, not only for individual welfare but also for public health and resource allocation. The analysis of optimality conditions for purchasing health insurance can be approached using utility theory, risk preferences, and the effects of free-riding behavior. In the absence of free riders, an individual’s decision to purchase health insurance is primarily guided by the trade-off between the certainty provided by insurance coverage and the risk of incurring high medical costs without insurance. Let $U(W)$ denote the utility derived from wealth $W$, and assume the individual faces a probabilistic health risk. If the individual remains uninsured, the expected utility is: $$ E[U] = p \cdot U(W - L) + (1 - p) \cdot U(W), $$ where $p$ is the probability of falling ill and incurring a medical cost $L$, and $(1 - p)$ is the probability of remaining healthy. The individual considers purchasing insurance at a premium $P$, which guarantees coverage for the loss $L$. With insurance, the utility becomes $U(W - P)$, which is certain. The optimality condition is determined by equating the expected utility of remaining uninsured with the utility under insurance: $$ U(W - ___ _________ _________ _____ ________ ___ _____ ______.
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