# Simple General Equilibrium Model of Public Goods: Intuition?

• We assume a two person economy, denoted by person A and B. Good y is an ordinary private good, and each person begins with an allocation of this good given by $$y^{A*}$$ & $$y^{B*}$$ respectively.
• Each person may consume some y directly or devote some portion of it to the production of a single public good, x. The amounts contributed are given by $$y_{s}^{A}$$ and $$y_{s}^{B}$$, and the public good has production function defined by $$x=f\left(y_{s}^{A}+y_{s}^{B}\right)$$

• Resulting utilities for these two people are given: $$\begin{array}{c}{U^{A}\left(x, y^{A *}-y_{s}^{A}\right)} \\ {U^{B}\left(x, y^{B *}-y_{s}^{B}\right)}\end{array}$$

• The necessary conditions for efficient resource allocation in this problem consist of choosing the levels of public goods subscriptions ($$y^{A*}$$ & $$y^{B*}$$) that maximize, say, A’s utility for any given level of B’s utility. The Lagrangian expression is $$\mathscr{L}=U^{A}\left(x, y^{A *}-y_{s}^{A}\right)+\lambda\left[U^{B}\left(x, y^{B *}-y_{s}^{B}\right)-K\right]$$ where K is a constant level of B's utility. The first-order conditions for max are \begin{aligned} \frac{\partial \mathscr{L}}{\partial y_{s}^{A}} &=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}=0 \\ \frac{\partial \mathscr{L}}{\partial y_{s}^{B}} &=U_{1}^{A} f^{\prime}-\lambda U_{2}^{B}+\lambda U_{1}^{B} f^{\prime}=0 \end{aligned}
• A comparison of these two equations yields $$\lambda U_{2}^{B}=U_{2}^{A}$$ and when substituting into the preceding equation the end result is: $$\frac{U_{1}^{A}}{U_{2}^{A}}+\frac{\lambda U_{1}^{B}}{\lambda U_{2}^{B}}=\frac{1}{f^{\prime}}$$ or more simply, $$M R S^{A}+M R S^{B}=\frac{1}{f^{\prime}}$$

For such goods, the MRS in consumption must reﬂect the amount of y that all consumers would be willing to give up to get one more x, because everyone will obtain the beneﬁts of the extra x output. Hence it is the sum of each individual’s MRS that should be equated to dy/dx in production (here given by $$\frac{1}{f'}$$).

Q: I follow the order of operations mathematically but don't understand the bolded statement.

• I understand thank you. If I can ask, I seem to have confused myself with some of the math. In differentiating the Lagrangian function, how is this the result: $$\frac{\partial \mathscr{L}}{\partial y_{s}^{A}}=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}$$? The first term is a utility function composed of two variables: $$\left(x, y^{A *}-y_{s}^{A}\right)$$. But for example the first term is $$U_{1}^{A} f^{\prime}$$ Feb 19 '20 at 22:46