<p>If you own the lake then your profit is the sum of the profit of all boats, so
<span class="math-container">$$
\pi(b) = \sum_{i = 1}^b \left(p_f\frac{f(b)}{b} - \frac{p_f}{4}\right).
$$</span></p>

CC BY-SA 4.0
2022-01-15T15:47:42.060Z
2022-01-15T15:47:42.060Z
tdm
<blockquote>
<p>The main economic activity in Blue Lake is fishing. Anyone can send
out a fishing boat, but the cost is <span class="math-container">$\frac{p_f}{4}$</span> where <span class="math-container">$p_f$</span> is is
the price of a ton of fish. If <span class="math-container">$b$</span> boats are on the lake then <span class="math-container">$f(b)$</span>
fish will be caught in total (where <span class="math-container">$f(b)$</span> is a strictly concave
function). All fish caught will be distributed evenly amongst boats.</p>
</blockquote>
<p>The question first asked me to write a profit function for each boat as a function of the price of fish <span class="math-container">$p_f$</span> and the number of boats fishing <span class="math-container">$b$</span>. I wrote <span class="math-container">$\pi=p_f(\frac{f(b)}{b})-\frac{p_f}{4}$</span>, where <span class="math-container">$\pi$</span> is profit. It then asks to determine what a price-taking, profit-maximising firm that owned the lake would do (i.e. how many boats would it send fishing)?</p>
<p>I believe that to answer this question, I need to find the point at which <span class="math-container">$\pi'=0$</span>, the maximum profit possible. I think I need to take a partial differentiation of <span class="math-container">$\pi$</span> with respect to <span class="math-container">$b$</span>, holding all other variables constant. I am a little stuck though and need help.</p>

CC BY-SA 4.0
2022-01-15T13:15:50.367Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Community
<microeconomics>
Determining the actions of a profit-maximising firm
Alexis Huang
# Determining the actions of a profit-maximising firm

(Asked by: username_0 on 2022-01-15)

> The main economic activity in Blue Lake is fishing. Anyone can send
> out a fishing boat, but the cost is $\frac{p\_f}{4}$ where $p\_f$ is is
> the price of a ton of fish. If $b$ boats are on the lake then $f(b)$
> fish will be caught in total (where $f(b)$ is a strictly concave
> function). All fish caught will be distributed evenly amongst boats.

The question first asked me to write a profit function for each boat as a function of the price of fish $p\_f$ and the number of boats fishing $b$. I wrote $\pi=p\_f(\frac{f(b)}{b})-\frac{p\_f}{4}$, where $\pi$ is profit. It then asks to determine what a price-taking, profit-maximising firm that owned the lake would do (i.e. how many boats would it send fishing)?

I believe that to answer this question, I need to find the point at which $\pi'=0$, the maximum profit possible. I think I need to take a partial differentiation of $\pi$ with respect to $b$, holding all other variables constant. I am a little stuck though and need help.

**Tags:** microeconomics

> **Comments:**
> **username_1**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  

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## 1 answers

### Answer by username_2 on 2022-01-15. Score: 1

If you own the lake then your profit is the sum of the profit of all boats, so
$$
\pi(b) = \sum\_{i = 1}^b \left(p\_f\frac{f(b)}{b} - \frac{p\_f}{4}\right).
$$


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