<p>Almost correct. Setting <span class="math-container">$W(h)=0$</span> is wrong (but inconsequential for the solutions). Checking the SOC for completeness should be included, but is somewhat obvious here. Correctness of 3. holds only under the assumption that the consumer owns no shares of the firm, which seems to hold in this exercise.</p>

CC BY-SA 4.0
2021-04-07T10:28:35.647Z
2021-04-07T10:28:35.647Z
VARulle
<p>I am trying to solve the following question(s):</p>
<p>Let <span class="math-container">$h \geq 0$</span> represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility function and attaches a utility of <span class="math-container">$\phi(h) = -2h^2$</span> to the externality. The firm's profit function is <span class="math-container">$\pi(h)=120-2(h-10)^2 $</span>.</p>
<ol>
<li><p>Compute the pareto optimal level of <span class="math-container">$h$</span>, <span class="math-container">$h^o$</span>.</p>
</li>
<li><p>Compute the firm's optimal level of <span class="math-container">$h$</span>, <span class="math-container">$h^f$</span>.</p>
</li>
<li><p>Suppose the consumer rather than the firm can determine the specific externality levelh with which the firm operates. Compute the optimal level of <span class="math-container">$h$</span>, <span class="math-container">$h^c$</span>, from the consumer's perspective.</p>
</li>
</ol>
<p>My approach so far:</p>
<ol>
<li><p>To find <span class="math-container">$h^o$</span> I maximize the aggregate surplus/welfare of h. <span class="math-container">$$\max_{h \geq0} \pi(h) + \phi(h)$$</span>
<span class="math-container">$$ W(h)=120-2(h-10)^2 - 2h^2 = 0 $$</span>
<span class="math-container">$$ \frac{dW}{dh}= -4(h-10)-4h = -8h + 40 = 0 $$</span>
<span class="math-container">$$ 8h=40 \rightarrow  h^o=5$$</span></p>
</li>
<li><p>To find <span class="math-container">$h^f$</span>
<span class="math-container">$$FOC\ of\ \pi(h): \frac{d\pi}{dh} =-4(h-10) = -4h+40 = 0 \rightarrow h^f=10 $$</span></p>
</li>
<li><p>To find <span class="math-container">$h^c$</span>
<span class="math-container">$$ FOC\ of\ \phi(h): \frac{d\phi(h)}{dh} = -4h = 0 \rightarrow h^c=0 $$</span></p>
</li>
</ol>
<p>Is this correct?</p>

<microeconomics><optimization><externalities>
Calculating optimal level of negative externality
asd7
# Calculating optimal level of negative externality

(Asked by: username_0 on 2021-04-07)

I am trying to solve the following question(s):

Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility function and attaches a utility of $\phi(h) = -2h^2$ to the externality. The firm's profit function is $\pi(h)=120-2(h-10)^2 $.

1. Compute the pareto optimal level of $h$, $h^o$.
2. Compute the firm's optimal level of $h$, $h^f$.
3. Suppose the consumer rather than the firm can determine the specific externality levelh with which the firm operates. Compute the optimal level of $h$, $h^c$, from the consumer's perspective.

My approach so far:

1. To find $h^o$ I maximize the aggregate surplus/welfare of h. $$\max\_{h \geq0} \pi(h) + \phi(h)$$
   $$ W(h)=120-2(h-10)^2 - 2h^2 = 0 $$
   $$ \frac{dW}{dh}= -4(h-10)-4h = -8h + 40 = 0 $$
   $$ 8h=40 \rightarrow h^o=5$$
2. To find $h^f$
   $$FOC\ of\ \pi(h): \frac{d\pi}{dh} =-4(h-10) = -4h+40 = 0 \rightarrow h^f=10 $$
3. To find $h^c$
   $$ FOC\ of\ \phi(h): \frac{d\phi(h)}{dh} = -4h = 0 \rightarrow h^c=0 $$

Is this correct?

**Tags:** microeconomics, optimization, externalities

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## 1 answers

### Answer by username_1 on 2021-04-07. Score: 2 | Accepted answer

Almost correct. Setting $W(h)=0$ is wrong (but inconsequential for the solutions). Checking the SOC for completeness should be included, but is somewhat obvious here. Correctness of 3. holds only under the assumption that the consumer owns no shares of the firm, which seems to hold in this exercise.


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