# Determining the actions of a profit-maximising firm

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.

• 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 Bot
Jan 15 at 13:15

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