The results of Samuelson in the theory of public goods have provided the basis for most subsequent discussion of the optimum provision of public goods. Samuelson showed that a necessary condition for Pareto Optimality (and hence for maximizing a social welfare function which responds positively to individual utilities) is that the sum of the marginal rates of substitution (Σ MRS) between a public good and a private good be equal to the marginal rate of transformation (MRT). The sole constraint is that production is in the aggregate production set. This optimum can be achieved as a competitive equilibrium with the government supplying the public good up to the point where Σ MRS = MRT and financing its production by lump-sum taxation. Lump-sum transfers may also be employed to achieve the appropriate income distribution.