The Pure Theory of Public Expenditure by Paul A Samuelson in English and Chinese TranslationTHE PURE THEORY OF PUBLIC EXPENDITUREPaul A Samuelson1 Assumptions Except for Sax WicksellLindahl Musgrave

s desired. This means that the analysis can be conducted in terms of either maximizing outputs or minimizing inputs. I shall choose the former for convenience.2. The Optimization Problem. Given a set of collective consumption goods G and a set of individuals, each with a utility function u (x_i, g), where x_i is individual i's private consumption, the problem is to determine the optimal level of public expenditure on each collective good g_i, subject to a budget constraint. The budget constraint is given by the total tax revenue collected from the individuals in the economy, which is assumed to be a fixed proportion of their income. Thus, the budget constraint can be expressed as follows:where y_i is individual i's income and t is the tax rate. The problem can then be stated as follows:where the first term in the objective function represents the sum of all individual utilities and the second term represents the total cost of public expenditure.3. The Solution. To find the optimal level of public expenditure on each collective good, we must first determine the marginal utility of each individual with respect to each collective good. Let M_i (g) denote the marginal utility of collective good g to individual i. Then we can express the total marginal utility of all individuals with respect to each collective good as follows:where N is the total number of individuals in the economy. The optimal level of public expenditure on each collective good is then given by the condition that the marginal cost of providing an additional unit of the collective good is equal to the sum of the marginal utility of all individuals with respect to that collective good:where MC (g_i) is the marginal cost of providing an additional unit of collective good g_i. This condition ensures that the additional unit of the collective good provides as much marginal benefit to society as it costs to produce.4. Conclusion. The pure theory of public expenditure provides a framework for analyzing the optimal level of public expenditure on collective goods. The key insight is that the optimal level of public expenditure is determined by the marginal utility of each individual with respect to each collective good, and that this marginal utility can be aggregated to determine the overall marginal benefit of the collective good to society