How do subsidies for positive externalities relate to welfare economics, public goods, and deadweight loss in a market economy?

Subsidies for positive externalities are financial support provided by the government to encourage activities that have benefits for society, like education or renewable energy. In welfare economics, these subsidies aim to improve overall well-being by promoting goods that create positive effects, which might not be fully appreciated in a free market. Public goods, like clean air, are often underprovided because individuals do not pay directly for them; subsidies help to make these goods more accessible. Without subsidies, the market may experience deadweight loss, meaning that there are missed opportunities to improve social welfare because the benefits of these goods are not fully captured in their prices. For example, if a government subsidizes solar panels, more people may install them, leading to cleaner energy and benefiting everyone by reducing pollution.