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*} $ &amp; $ 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*} $ &amp; $ 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
$$
\begin{aligned} \frac{\partial \mathscr{L}}{\partial y_{s}^{A}} &amp;=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}=0 \\ \frac{\partial \mathscr{L}}{\partial y_{s}^{B}} &amp;=U_{1}^{A} f^{\prime}-\lambda U_{2}^{B}+\lambda U_{1}^{B} f^{\prime}=0 \end{aligned}
$$
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.
You can take MRS as the willingness to pay (Take the other good in MRS as money). If the good is a private good that costs 3, your willingness to pay should be 3 because you are buying it for yourself. However, a public good is used by all payers, so we only need the sum of the willingnesses to pay to be equal to 3. (You are simply splitting the bill)