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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.

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  • $\begingroup$ 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. $\endgroup$
    – Community Bot
    Jan 15 at 13:15

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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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