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To derive the conditions of optimality for buying health insurance in the context of the social sector and environment, we must examine the economic rationale behind individuals’ decisions to purchase insurance, incorporating both the absence and presence of free riders. Health insurance is a mechanism to redistribute risk from the insured individual to the insurer, thereby providing financial protection against uncertain health expenditures. The optimality of purchasing health insurance depends on individual preferences, risk aversion, expected health costs, and the institutional setting, including whether free riders exist. 1. Optimal Health Insurance in the Absence of Free Riders In the absence of free riders, individuals are fully responsible for their insurance choices, and the market mechanism works without external interference. Let us consider an individual with income $Y$ and potential health expenditure $L$ in case of illness. Let the probability of falling ill be $p$ (where $0 < p < 1$) and the insurance premium be denoted by $P$. The utility function of the individual is $U(C)$, where $C$ is consumption, and it is assumed to be concave to reflect risk aversion. Without insurance, the expected utility of the individual is: $$ EU_{\text{no insurance}} = (1-p)U(Y) + pU(Y-L) $$ With insurance that covers the full loss $L$ and a premium $P$, the consumption in both states (healthy or ill) becomes $Y-P$. Hence, the expected utility with insurance is: $$ EU_{\text{insurance}} = U(Y-P) $$ The individual will purchase insurance if: $$ EU_{\text{insurance}} \geq EU_{\text{no insurance}} $$ Substituting the expressions: $$ U(Y-P) \geq (1-p)U(Y) + pU(Y-L) $$ This inequality represents the condition for optimal insurance purchase in the absence of free riders. It ensures that the individual’s utility after paying the insurance premium exceeds the expected utility without insurance. Derivation of Optimal Premium To find the optimal premium $P^*$, the individual maximizes expected utility by choosing $P$ such that the marginal benefit of reducing risk equals the marginal cost of ___ ____ _________ _____ ___ ___ ________.
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