<p>This study investigates the optimal demand for insurance and the interrelationship between insurance and prevention, such as self-insurance and self-protection, within a two-argument utility framework. We identify the conditions under which insurance is a normal good even under decreasing absolute risk aversion (DARA). These conditions depend on the correlation between income and the asset, specifically, whether the individual exhibits correlation-loving or correlation-averse preference. We also find that even when preferences exhibit prudence, the presence of background risk may not increase insurance demand, and the relative magnitudes of cross-prudence and prudence matter. Lastly, this study provides the conditions that self-insurance/self-protection and insurance are substitutes in terms of insurance premiums. This study has implications for government policies, suggesting that subsidies can increase the demand for insurance, whereas subsidies for different risk management instruments may crowd each other out.</p>