<p>Set of Pareto efficient allocations is given by the following set:</p>
<p><span class="math-container">$\mathcal{P}=\{(m_s, m_f, d)\in\mathbb{R}^3_+|(d=50)\ \vee \ (0\leq d \leq 50 \ \wedge \ m_s=0) \ \vee \ (50\leq d \leq 100 \ \wedge \ m_f=0)\}$</span></p>
<p>In the example below, <span class="math-container">$m_s+m_f=60$</span> is taken just for the purpose of illustration. <span class="math-container">$\mathcal{P}$</span> is the set of efficient allocations, irrespective of the amount of total money <span class="math-container">$m_s+m_f$</span> in the economy.</p>
<p><a href="https://i.sstatic.net/EiSRJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EiSRJ.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-06-05T19:50:08.833Z
2023-06-06T03:16:07.533Z
2023-06-06T03:16:07.533Z
Amit
<p>Student S has one hobby of listening to music. The noise of the speaker can produce noise up to 100 decibels (measured by D). Her money is measured by M<sub>s</sub> . Her utility function is U<sub>s</sub> = 10D<sup>1/2</sup> + M<sub>s</sub></p>
<p>Student F has a utility function of U<sub>F</sub> = 10(100-D)<sup>1/2</sup> + M<sub>F</sub> . M<sub>F</sub> is Student F’s money and D is the decibels of Student S’s speaker.</p>
<p>What is a Pareto-Efficient Level? (We also learnt in the previous part that Student S has no money, we do not know about Student F)</p>
<p>I am aware that Pareto-Efficiency means we can’t make one student better off without making another student worse off, but since one likes loud music and one does not like any music. Isn’t any level of noise Pareto-Efficient?</p>

<pareto-efficiency>
Finding Pareto efficiency with negative externalities
alessioriccardi
# Finding Pareto efficiency with negative externalities

(Asked by: username_0 on 2020-12-04)

Student S has one hobby of listening to music. The noise of the speaker can produce noise up to 100 decibels (measured by D). Her money is measured by Ms . Her utility function is Us = 10D1/2 + Ms

Student F has a utility function of UF = 10(100-D)1/2 + MF . MF is Student F’s money and D is the decibels of Student S’s speaker.

What is a Pareto-Efficient Level? (We also learnt in the previous part that Student S has no money, we do not know about Student F)

I am aware that Pareto-Efficiency means we can’t make one student better off without making another student worse off, but since one likes loud music and one does not like any music. Isn’t any level of noise Pareto-Efficient?

**Tags:** pareto-efficiency

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## 1 answers

### Answer by username_1 on 2023-06-05. Score: 3

Set of Pareto efficient allocations is given by the following set:

$\mathcal{P}=\{(m\_s, m\_f, d)\in\mathbb{R}^3\_+|(d=50)\ \vee \ (0\leq d \leq 50 \ \wedge \ m\_s=0) \ \vee \ (50\leq d \leq 100 \ \wedge \ m\_f=0)\}$

In the example below, $m\_s+m\_f=60$ is taken just for the purpose of illustration. $\mathcal{P}$ is the set of efficient allocations, irrespective of the amount of total money $m\_s+m\_f$ in the economy.

[![enter image description here](https://i.sstatic.net/EiSRJ.png)](https://i.sstatic.net/EiSRJ.png)


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