# Efficiency of the Nash Equilibrium

Let us compare the Nash equilibrium and the Pareto optimum. The provision of public goods at the Nash equilibrium is smaller than at the Pareto optimum. At the Nash equilibrium, public goods are provided to a smaller extent than at the Pareto optimum. This is because each person chooses her or his provision by considering her or his own welfare.

At the Nash equilibrium point, the marginal benefit of each person, which is given as the marginal rate of substitution, is equal to the marginal cost of public goods, p. Thus,

If the benefit is limited to each person as with private goods, this is the socially optimal condition. Thus, the private market may attain the Pareto optimum allocation, as the fundamental theory of welfare economics implies.

However, with regard to public goods, it is necessary to include the spillover effect on other agents. The social marginal benefit is larger than the personal marginal benefit in a multiple person economy. The Samuelson rule implies that the sum of the personal marginal rate of substitution, which is given as the social marginal benefit of public goods, is equal to the marginal cost of the public goods. Thus,

In other words, at point N we have

Hence, public good Y is undersupplied at the Nash equilibrium point N.

In Fig. 11.5, we draw the Pareto optimal point P, where the indifference curves of both persons are tangent at point P. However, at point N, the indifference curves of both persons intersect, which means that point N is not Pareto efficient. I_{1 }denotes an indifference curve of person 1, while I_{2} denotes an indifference curve of person 2 at N.

Note that indifference curve I_{1} becomes tangent to the horizontal line on curve N (y_{2}), while indifference curve I_{2} becomes tangent to the vertical line on curve N(y_{1}). For person 1, the horizontal line is the budget line in Fig. 11.5 since she or he may choose any level of y_{1} at a given level of y_{2}. The optimal point is the point where the budget line is tangent to her or his indifference curve, which corresponds to the reaction curve.