# 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.

• – Kitsune Cavalry Jun 1 '16 at 16:59
• It is not true that maximising the sum of utilities yields a Pareto optimum only when utility is quasi-linear. Indeed, suppose we found an outcome that maximises the sum of everyone's utility. Then, by definition, any alternative choice must yield a weakly lower $\sum u$—meaning there must be a lower $u$ for at least one agent. It follows that the alternative does not Pareto dominate the original, utility-sum-maximising choice. What is true is that if utility is quasi-linear then every Pareto optimum maximises the sum of utilities. – Ubiquitous Jul 31 '16 at 15:26
• @Ubiquitous - I don't think it is true that if utility is quasi-linear then every Pareto optimum maximizes the sum of utilities. Refer to my answer for the counter example: economics.stackexchange.com/a/15582/11824 – Amit Feb 27 '17 at 3:09

## 2 Answers

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.

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

• Quasi-linear utility implies that utility is transferable. So in fact, $a_2$ can be a Pareto improvement from $a_1$ if individual 2 agrees to switch to $a_2$ and transfers $1$ unit of $x$ to individual 1. This way, individual 1 is no worse than in $a_1$ (he gets $4+1=5$), but individual 2 is strictly better off even after the transfer (he gets $4-1=3>2$).. – Herr K. Mar 29 '17 at 3:54
• Individual 2 does not have any private good to transfer to individual 1 at $a_1$. – Amit Mar 29 '17 at 5:20
• Yeah, you're right. I missed that, and I assumed utility were transferable. – Herr K. Mar 29 '17 at 5:35

The Lindahl Equilibrium $y^*$ with quasi-linear preferences is uniquely determined. That is, $y^*$ is independent of individual consumption levels of $x$

• Is there a way of understanding this without using the Lindahl equilibrium approach? (If not I will just have to learn what it is.) – hk39 Jun 1 '16 at 15:28