Local Public Goods
To summarize, solutions of the first best problem can be sustained by a system of prices, rents and payments such that conditions (2.5.), (2.6.), (5.1.), (5.3.) and (5.4.) hold. Sustain ability is necessary for an allocation to solve the first best problem which, in this sense, is shown to allow for a second welfare theorem (see ,  for similar results). Unfortunately, a corresponding first welfare theorem does not generally hold. The above conditions of sustainability are not sufficient for an allocation to be a solution. Of course, any allocation (n*, x*, y*, z*, S*) sustained in the above sense must solve both the assignment and the Lindahl-Samuelson problem as follows from Theorems 1 and 4. In other words, it must be true that S*=s(u,n*,z*)~s(u,n,z*) for all assignments n as well as S* = s(u, n*, z*)
~s(u, n*, z) for any other supply z of public goods. But since s(u, n, z) is not, in general, concave as a function of nand z together, a different assignment and level of public spending could still exist which would lead to a higher level
of exports than S*. Allocations which are not first best may be sustainable.