Pareto efficient quantity of public good (quasi-linear preferences) I am doing a question on finding the Pareto efficient quantity of a public good. Instead of using the condition $\sum MRS_i = c'(G)$ where $c(G)$ denotes the cost of the public good, it asks you to find the efficient quantity by maximising the sum of the agents' utilities. Apparently this is only valid if preferences are quasi-linear so whilst I can do the question I do not understand why this is a valid approach. Any help on this matter would be appreciated.
I don't think it is true in a standard public good economy the question is referring to. Consider the following counterexample:

Suppose $I  = \{1,2\}$ and utility of the individual $i$ depends on his consumption of public good $(G)$ and private good $x_i$: $u_1(G, x_1) = 2\sqrt{G} + x_1$ and $u_2(G, x_2) = 2\sqrt{G} + x_2$, 

Also, the CRS technology used for production of public good uses private good as input:
$G = f(x_0) = x_0$. 

If the society has only 4 units of private good in the beginning, then the set of feasible allocations can be written as

$\{(G, x_1, x_2)\in\mathbb{R}^3_+: G+x_1+x_2 = 4\}$.

Notice that the allocation $a_1 = (G, x_1, x_2) = (1,3,0)$ is Pareto efficient, but does not maximize the sum of utilities. The reason is that allocation $a_2 = (G, x_1, x_2) = (4,0,0)$ yields the higher sum.

$\color{blue}{u_1(1,3) + u_2(1,0)} = 5+2 =7 \color{blue}{&lt;} 8 = 4 + 4 = \color{blue}{u_1(4,0) + u_2(4,0)}$.