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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?

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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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