This lecture discusses public economics, focusing on the efficient allocation of resources in the presence of public goods and private goods. The analysis begins with a simple example of two roommates, Anne and Bruce, who have conflicting preferences over two goods: room temperature and playing cribbage. The lecture introduces the concept of Pareto optimality, where no one can be made better off without making someone else worse off. It explains that Pareto optimal points occur where the marginal rates of substitution between the two goods are equal for both individuals. The lecture then moves to a more complex example involving two individuals, Charles and Diana, who have different preferences over two goods: sherry and monuments. The analysis uses the method of Lagrange multipliers to find Pareto optimal allocations, leading to the Samuelson condition, which states that the sum of the marginal rates of substitution between public goods and private goods must equal the cost of an extra unit of the public good relative to an extra unit of the private good. The lecture also discusses the concept of transferable utility, where the utility function is linear in the private good, leading to a straight-line utility possibility frontier. It contrasts this with non-transferable utility, where the utility possibility frontier is not necessarily linear. The lecture then explores the role of social welfare functions in determining optimal allocations, noting that while a social welfare function is not always necessary, it can provide guidance in choosing among Pareto optimal points. It also discusses the importance of the Pareto criterion in policy-making, even in the absence of a social welfare function. Finally, the lecture addresses the issue of congestion, where the use of a public good (such as highways) leads to negative externalities. It discusses how tolls can be used to internalize these externalities and achieve efficient outcomes. The lecture concludes with an analysis of how to determine the optimal amount of public goods when individuals have different preferences and how to ensure that the total taxes collected are sufficient to cover the costs of public goods.This lecture discusses public economics, focusing on the efficient allocation of resources in the presence of public goods and private goods. The analysis begins with a simple example of two roommates, Anne and Bruce, who have conflicting preferences over two goods: room temperature and playing cribbage. The lecture introduces the concept of Pareto optimality, where no one can be made better off without making someone else worse off. It explains that Pareto optimal points occur where the marginal rates of substitution between the two goods are equal for both individuals. The lecture then moves to a more complex example involving two individuals, Charles and Diana, who have different preferences over two goods: sherry and monuments. The analysis uses the method of Lagrange multipliers to find Pareto optimal allocations, leading to the Samuelson condition, which states that the sum of the marginal rates of substitution between public goods and private goods must equal the cost of an extra unit of the public good relative to an extra unit of the private good. The lecture also discusses the concept of transferable utility, where the utility function is linear in the private good, leading to a straight-line utility possibility frontier. It contrasts this with non-transferable utility, where the utility possibility frontier is not necessarily linear. The lecture then explores the role of social welfare functions in determining optimal allocations, noting that while a social welfare function is not always necessary, it can provide guidance in choosing among Pareto optimal points. It also discusses the importance of the Pareto criterion in policy-making, even in the absence of a social welfare function. Finally, the lecture addresses the issue of congestion, where the use of a public good (such as highways) leads to negative externalities. It discusses how tolls can be used to internalize these externalities and achieve efficient outcomes. The lecture concludes with an analysis of how to determine the optimal amount of public goods when individuals have different preferences and how to ensure that the total taxes collected are sufficient to cover the costs of public goods.