Derive the conditions of optimality for buying health insurance in cases of absence/presence of free riders.

Health insurance is a critical component of the economics of the social sector, as it ensures financial protection against health risks and promotes overall welfare. The optimality conditions for buying health insurance depend on multiple factors, including risk aversion, asymmetric information, and the presence or absence of free riders. The decision to purchase health insurance is influenced by expected utility maximization, market equilibrium, and externalities.

### Optimality Conditions for Buying Health Insurance in the Absence of Free Riders

1. Expected Utility Maximization  
   An individual decides to purchase health insurance if the expected utility with insurance is greater than or equal to the expected utility without it. This is given by:  

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   p U(W - P) + (1 - p) U(W - P - L) \geq p U(W - L) + (1 - p) U(W)
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