In the context of economic models, how does a lump-sum tax differ from marginal tax rates in terms of efficiency and equity from various theoretical perspectives on taxation?

A lump-sum tax is a fixed amount that everyone pays, regardless of their income or wealth, while marginal tax rates increase as a person's income rises. From an efficiency perspective, lump-sum taxes are considered efficient because they do not change people's behavior; people will still work and earn money without worrying about higher taxes on extra income. However, when it comes to equity, lump-sum taxes can be seen as unfair because they take the same amount from everyone, which can be a heavier burden for those with lower incomes. For example, if everyone pays $1,000, a wealthy person might hardly notice it, while a low-income person could struggle to pay it. In contrast, marginal tax rates aim to distribute the tax burden more fairly, as higher earners pay a larger percentage, but they might discourage people from earning more, which can lead to inefficiencies.