Question

Write down the Samuelson’s condition for the optimal provision of a public good. Use an example to explain your understanding of it.

Answer 1

A feasible allocation (x, G) is Pareto optimal if there exists no other feasible allocation (x' ,G') such that;

and for some i {1,...., n},

That is, a feasible allocation (x, G) is Pareto optimal if there is no way of making an agent strictly better off without making someone else worse off. We can characterize the set of Pareto optimal allocations;

max u₁(u₁, G)

for i = 2,3,....,n (multiplier )

(multiplier )

f(z) - G 0 (multiplier )

where u are treated as parameters of the problem. Conditions on the utility function implies that the non-negativity constraints can be ignored. The necessary and sufficient Kuhn-Tucker conditions are;

(x_i : )  

(G : )  

(z : )  

where we have set = 1 by convention

For the first n equalities,

From the last equality, we get;

Plugging these n + 1 equalities into the the middle condition regarding G;

This condition is referred to as the Samuelson condition.

The left hand of equation is the sum of the marginal rates of substitutions of the n agents.

denotes the quantity of private good agent i is willing to give up for a small unit increase in the level of the public good. The right hand of equation is the amount of private good required to produce an additional unit of public good known as the marginal rate of transformation. Hence the Samuelson condition says the following: "Any optimal allocation is such that the sum of the quantity of private goods consumers would be willing to give up for an additional unit of public good must equal to the quantity of private good that is actually required to produce the additional unit of public good."

If there are more than one private goods, say k private goods; and the public good is produced according to;

f(z₁,....,z_k)

then the corresponding Samuelson condition for the optimal level of public goods is given by;

for all j = 1,....,k