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The decision to purchase health insurance can be analyzed through the lens of microeconomic theory, particularly under the framework of expected utility and risk aversion. In an environment where individuals face uncertain health risks, the rationale for buying health insurance is fundamentally tied to the trade-off between the certain cost of premiums and the uncertain financial burden of potential medical expenditures. The optimality conditions for health insurance uptake differ significantly depending on whether there are free riders—individuals who benefit from others' insurance coverage without purchasing insurance themselves—or whether each individual acts independently in the absence of free riders. In the absence of free riders, the individual’s decision to buy health insurance can be modeled using expected utility theory. Let us denote the individual’s wealth as $W$, the cost of health insurance premiums as $P$, and the potential health expenditure in the event of illness as $L$. The probability of falling ill is $p$, and the individual derives utility $U(\cdot)$ from wealth, which is assumed to be strictly concave due to risk aversion. The expected utility without insurance is given by: $$ EU_{\text{no insurance}} = (1-p)U(W) + pU(W-L) $$ With full insurance coverage, the individual pays a premium $P$ and transfers the risk to the insurer. The expected utility with insurance is: $$ EU_{\text{insurance}} = U(W-P) $$ The optimality condition for purchasing insurance in this context is that the individual will choose to buy insurance if the expected utility with insurance exceeds the expected utility without insurance: $$ U(W-P) \ge (1-p)U(W) + pU(W-L) $$ This inequality represents a critical condition: it depends on the premium $P$, the level of risk $p$, the potential loss $L$, and the individual’s degree of risk ___ _________ _________ _________ _________ _______ _________ ___ ____ ________ ___ ___.
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