Welfare Maximization: Samuelson's Analysis with Public and Private Goods

Samuelson (1955) and Bator (1957) have provided geometric techniques for deriving or illustrating the conditions for Pareto-optimality in exchange and production. Bator's approach merely serves to illustrate what the distribution of private consumption and of consumer welfare must be if it is to make any arbitrarily chosen output combination Pareto-efficient. Samuelson. on the other hand, derives the Pareto-efficient output configuration, given a constraint on the welfare of one of two individuals. Even though Samuelson's approach makes the better use of the geometry, Bator's demonstration was the first to be extended from private to public goods (by McLure, 1968). The reverse extension of Samuelson's model, from public to private goods, is attempted in this note.