# A Two-Person Model

Consider the two-person economy. In this context, we formulate person 1’s optimization problem. Person 1’s utility function is given as

where U^{1} is her or his utility, x_{1} is her or his consumption of private goods, and Y is her or his consumption of public goods, which is equal to the total supply of public goods. For simplicity, preferences are identical between agents.

Person 1’s budget constraint is given as

where p is the price of public goods in terms of private goods and M is her or his income. For simplicity, two persons have the same income, and the marginal cost of public goods is fixed as p. y_{1} and y_{2} denote the private provision of public goods for each person respectively. Then, we have

From Eqs. (11.14) and (11.15), the budget constraint of person 1 is rewritten as

The right-hand side of Eq. (11.16) is called the effective income of person 1, which includes the benefit of the spillover effect of public goods provided by person 2. Person 1 effectively chooses x_{1} and Y so as to maximize welfare, Eq. (11.13), subject to budget constraint, Eq. (11.16), at given levels of M, p, and y_{2}.

This formulation assumes that person 1 regards others’ provision of public goods y_{2} as fixed when she or he chooses her or his own decision variables, x_{1} and Y. This

Fig. 11.4 **The Nash reaction function**

is called the Nash equilibrium approach, which is consistent with the definition of non-cooperative Nash equilibrium.

In Fig. 11.4, the vertical axis is private goods x and the horizontal axis is public goods Y. Line AB corresponds to the budget constraint, Eq. (11.16). AG corresponds to py_{2} and GO corresponds to M. The slope of the budget line is p. Person 1 may choose any point on line AB. Figure 11.4 draws her or his indifference curve. The indifference curve is a combination of private goods and public goods in order to fix person 1’s utility, which is concave toward the origin.

The optimal point, the highest utility point, on line AB is given by point E, where line AB is tangent to the indifference curve. Person 1 consumes public goods of OF. In other words, at the given level of OD = y_{2}, person 1’s optimal public provision is given as DF = y_{1}.

As shown in Fig. 11.4, the optimal levels of Y and y_{1}for person 1 are a function of y_{2}. This is the Nash reaction function of person 1. Thus,

У1 = N(y2). (11.17)

If person 2 provides more in the way of a contribution to public goods, person 1 reduces her or his own contribution and raises her or his consumption of private goods (N' < 0); however, she or he does not reduce her or his contribution of public goods to a greater extent than a reduction in the total supply (—1 < N'). In other words, if y_{2} increases, person 1’s effective income M + py_{2} increases. Because of the positive income effect, person 1 raises the consumption of private goods and public goods. Because her or his consumption of private goods increases, y_{1 }declines but the consumption of public goods, Y, increases.

Person 2’s optimizing behavior is similarly formulated. Her or his Nash response function is similarly given as

Fig. 11.5 **The Nash equilibrium**

The combination of yj and y_{2}, which satisfies Eqs. (11.17) and (11.18) at the same time, gives the Nash equilibrium.

Figure 11.5 explains the Nash equilibrium. The vertical axis is person 1’s provision, y_{1}, and the horizontal axis is person 2’s provision, y_{2}. Curve N(y_{2}) denotes person 1’s reaction curve and curve N(y_{1}) denotes person 2’s reaction curve. Since 0 > N' > —1, the slope of curve N(y_{2}) is negative and flatter than curve N(y_{1}). The intersection of both curves gives the Nash equilibrium point N.