<p>I don't think it is true in a standard public good economy the question is referring to. Consider the following counterexample:</p>
<p>Suppose <span class="math-container">$I  = \{1,2\}$</span> and utility of the individual <span class="math-container">$i$</span> depends on his consumption of public good <span class="math-container">$(G)$</span> and private good <span class="math-container">$x_i$</span>: <span class="math-container">$u_1(G, x_1) = 2\sqrt{G} + x_1$</span> and <span class="math-container">$u_2(G, x_2) = 2\sqrt{G} + x_2$</span>,</p>
<p>Also, the CRS technology used for production of public good uses private good as input:
<span class="math-container">$G = f(x_0) = x_0$</span>.</p>
<p>If the society has only 4 units of private good in the beginning, then the set of feasible allocations can be written as</p>
<p><span class="math-container">$\{(G, x_1, x_2)\in\mathbb{R}^3_+: G+x_1+x_2 = 4\}$</span>.</p>
<p>Notice that the allocation <span class="math-container">$a_1 = (G, x_1, x_2) = (1,3,0)$</span> is Pareto efficient, but does not maximize the sum of utilities. The reason is that allocation <span class="math-container">$a_2 = (G, x_1, x_2) = (4,0,0)$</span> yields the higher sum.</p>
<p><span class="math-container">$\color{blue}{u_1(1,3) + u_2(1,0)} = 5+2 =7 \color{blue}{&lt;} 8 = 4 + 4 = \color{blue}{u_1(4,0) + u_2(4,0)}$</span>.</p>
<p><em>Added later</em>: Here is the video of how we can find the set of all efficient allocations for a similar problem, the only difference is that the total endowment of private good is given 10 in this video:</p>
<p><a href="https://youtu.be/GPahrOHjUs4" rel="nofollow noreferrer">Representing set of all feasible allocations in a Dolbear Triangle</a></p>
<p><a href="https://youtu.be/g1gOVshgYGM" rel="nofollow noreferrer">Finding set of all efficient allocations</a></p>

CC BY-SA 3.0
2017-03-29T03:54:08.817Z
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.
CC BY-SA 3.0
2017-03-29T05:20:40.863Z
Individual 2 does not have any private good to transfer to individual 1 at $a_1$.
Amit
CC BY-SA 3.0
2017-03-29T05:35:32.683Z
Yeah, you're right. I missed that, and I assumed utility were transferable.
Herr K.
CC BY-SA 4.0
2017-02-26T23:24:58.167Z
2022-04-08T18:26:07.477Z
2022-04-08T18:26:07.477Z
Amit
<p>The Lindahl Equilibrium $y^*$ with quasi-linear preferences is uniquely determined. That is, $y^*$ is independent of individual consumption levels of $x$ </p>

CC BY-SA 3.0
2016-06-01T15:28:54.837Z
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
CC BY-SA 3.0
2016-06-01T13:04:41.190Z
2016-06-01T13:04:41.190Z
123
<p>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. </p>

CC BY-SA 3.0
2016-06-01T16:59:52.263Z
Related: http://economics.stackexchange.com/questions/11033/quasilinear-utility-pareto-optimality-implies-total-utility-maximization
Kitsune Cavalry
CC BY-SA 3.0
2016-07-31T15:26:57.970Z
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
CC BY-SA 3.0
2017-02-27T03:09:57.120Z
@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: http://economics.stackexchange.com/a/15582/11824
Amit
<microeconomics><public-goods>
Pareto efficient quantity of public good (quasi-linear preferences)
hk39
# Pareto efficient quantity of public good (quasi-linear preferences)

(Asked by: username_0 on 2016-06-01)

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.

**Tags:** microeconomics, public-goods
> **Comments:**
> **username_1**: Related: http://economics.stackexchange.com/questions/11033/quasilinear-utility-pareto-optimality-implies-total-utility-maximization  
> **username_2**: 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.  
> **username_3**: @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: http://economics.stackexchange.com/a/15582/11824  

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## 2 answers

### Answer by username_3 on 2017-02-26. Score: 3

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.

$\color{blue}{u\_1(1,3) + u\_2(1,0)} = 5+2 =7 \color{blue}{<} 8 = 4 + 4 = \color{blue}{u\_1(4,0) + u\_2(4,0)}$.

*Added later*: Here is the video of how we can find the set of all efficient allocations for a similar problem, the only difference is that the total endowment of private good is given 10 in this video:

[Representing set of all feasible allocations in a Dolbear Triangle](https://youtu.be/GPahrOHjUs4)

[Finding set of all efficient allocations](https://youtu.be/g1gOVshgYGM)

> **Comments:**
> **username_4**: 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 $4-1=3>2$)..  
> **username_3**: Individual 2 does not have any private good to transfer to individual 1 at $a_1$.  
> **username_4**: Yeah, you're right. I missed that, and I assumed utility were transferable.  

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### Answer by username_5 on 2016-06-01. Score: 0

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

> **Comments:**
> **username_0**: Is there a way of understanding this without using the Lindahl equilibrium approach? (If not I will just have to learn what it is.)  

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