<p>To calculate the <em>socially</em> optimal level of <span class="math-container">$e$</span>, you should look at <span class="math-container">$\pi_W+\pi_H$</span> (presumably that's also how you come up with the socially optimal <span class="math-container">$x$</span>). At <span class="math-container">$x=20$</span>, <span class="math-container">$e=1$</span>. </p>

<p>The intuition is that the woodworker's production creates an externality of <span class="math-container">$-2a=-x$</span> on the hotel. Since the hotel has to right to ban noise pollution, the woodworker would have to compensate the hotel for its cost from putting up with the noise. At social optimum, the marginal compensation (<span class="math-container">$e$</span>) should exactly cover the marginal external cost (<span class="math-container">$-1$</span>). Hence <span class="math-container">$e=1$</span> at the social optimum.</p>

CC BY-SA 4.0
2019-02-13T22:23:34.763Z
2019-02-13T22:23:34.763Z
Herr K.
<p>Let's say that there is a hotel owner <span class="math-container">$(H)$</span> and a woodworker <span class="math-container">$(W)$</span> working in close proximity to one another. </p>

<p>The woodworker produces <span class="math-container">$x$</span> units to sell at market at <span class="math-container">$p_{x}=6,5$</span>. From the woodworking activities, noise pollution is created that is measured by <span class="math-container">$a = \frac{1}{2}x$</span>, and to protect workers' hearing <span class="math-container">$W$</span> incurs costs of <span class="math-container">$1$</span> unit of money per noise pollution unit <span class="math-container">$a$</span>. Other costs of <span class="math-container">$W$</span> are at <span class="math-container">$\frac{1}{8}x^{2}$</span>.</p>

<p>It follows that <span class="math-container">$\pi_{W}=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x$</span></p>

<p>and <span class="math-container">$\pi_{H}=7g-\frac{1}{4}g^2-2a$</span>, where <span class="math-container">$g$</span> is the number of guests. </p>

<p>Now, I have calculated that the social optimum at which <span class="math-container">$H,W$</span> produce are <span class="math-container">$x = 20$</span> and <span class="math-container">$g = 14$</span></p>

<p>Question: The noise pollution will only be allowed if, for every noise pollution unit <span class="math-container">$a$</span> an emissions certificate at price <span class="math-container">$e$</span> is purchased. <span class="math-container">$H$</span> is given permission to sell these, under the premise that he will sell them at a welfare-maximzing price. Calculate <span class="math-container">$e$</span> and determine <span class="math-container">$\pi_{H}^{\operatorname{new}}$</span>.</p>

<p>My idea:</p>

<p>Looking at <span class="math-container">$\pi_{W}=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x-e a=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x-e \frac{1}{2}x$</span></p>

<p><span class="math-container">$\frac{\partial \pi_{W}}{\partial x}=6-\frac{1}{4}x-\frac{e}{2}=0$</span> and since social optimum is <span class="math-container">$x = 20$</span>.</p>

<p><span class="math-container">$\Rightarrow e = 2$</span></p>

<p>But in the solution set, it states <span class="math-container">$e=1$</span>... What have I done wrong?</p>

<macroeconomics><microeconomics><optimization><social-welfare>
Have I found the correct Emission Price
MinaThuma
# Have I found the correct Emission Price

(Asked by: username_0 on 2019-02-13)

Let's say that there is a hotel owner $(H)$ and a woodworker $(W)$ working in close proximity to one another.

The woodworker produces $x$ units to sell at market at $p\_{x}=6,5$. From the woodworking activities, noise pollution is created that is measured by $a = \frac{1}{2}x$, and to protect workers' hearing $W$ incurs costs of $1$ unit of money per noise pollution unit $a$. Other costs of $W$ are at $\frac{1}{8}x^{2}$.

It follows that $\pi\_{W}=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x$

and $\pi\_{H}=7g-\frac{1}{4}g^2-2a$, where $g$ is the number of guests.

Now, I have calculated that the social optimum at which $H,W$ produce are $x = 20$ and $g = 14$

Question: The noise pollution will only be allowed if, for every noise pollution unit $a$ an emissions certificate at price $e$ is purchased. $H$ is given permission to sell these, under the premise that he will sell them at a welfare-maximzing price. Calculate $e$ and determine $\pi\_{H}^{\operatorname{new}}$.

My idea:

Looking at $\pi\_{W}=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x-e a=6,5x-\frac{1}{8}x^{2}-\frac{1}{2}x-e \frac{1}{2}x$

$\frac{\partial \pi\_{W}}{\partial x}=6-\frac{1}{4}x-\frac{e}{2}=0$ and since social optimum is $x = 20$.

$\Rightarrow e = 2$

But in the solution set, it states $e=1$... What have I done wrong?

**Tags:** macroeconomics, microeconomics, optimization, social-welfare

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## 1 answers

### Answer by username_1 on 2019-02-13. Score: 1 | Accepted answer

To calculate the *socially* optimal level of $e$, you should look at $\pi\_W+\pi\_H$ (presumably that's also how you come up with the socially optimal $x$). At $x=20$, $e=1$.

The intuition is that the woodworker's production creates an externality of $-2a=-x$ on the hotel. Since the hotel has to right to ban noise pollution, the woodworker would have to compensate the hotel for its cost from putting up with the noise. At social optimum, the marginal compensation ($e$) should exactly cover the marginal external cost ($-1$). Hence $e=1$ at the social optimum.


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