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

The decision to buy health insurance involves several economic factors, including individual preferences, expected utility, and the presence of free riders. In the context of health economics, understanding the conditions of optimality for purchasing health insurance is crucial. These conditions depend on whether free riders exist in the market, as free riders can distort the equilibrium and affect the optimal decision-making process.

### Absence of Free Riders

In the absence of free riders, health insurance purchases are made by individuals who fully internalize the costs and benefits associated with insurance. The optimal decision to buy health insurance is based on maximizing expected utility, which is influenced by both the probability of incurring health costs and the financial protection provided by the insurance. 

1. Expected Utility Maximization: Individuals aim to maximize their expected utility, which is a combination of the utility derived from consumption (or health status) and the financial security provided by insurance. The expected utility \(U\) of an individual choosing health insurance can be written as:

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