Here is the problem:
Commercial shrimp fishing is an important business off the coast of New Orleans. The daily output of shrimp depends on the number of shrimp-fishing boats and the production function is given by $$f(x) = 30x - 2x^2$$ where $x$ is the number of fishing boats and $f(x)$ is the amount in hundreds of tonnes. The amount of shrimp that an individual boat captures is on average $\frac{f(x)}{x}$. It can sell the shrimp at a price of \$100 per ton and has a fixed cost of \$200 per day for operating a boat with a crew.
a) What is the profit function of a boat?
b) How many boats would be out fishing per day if there are no restrictions on the number of fishing boats? That is, find the Nash equilibrium number of fishing boats when the number of fishing boats are not restricted.
c) Is the amount of fishing “efficient”? If not what is the efficient number of boats that should be allowed to fish each day?
My attempt:
a) $\pi (x) = 100 \frac{f(x)}{x} - 200 = 100(30-x) - 200 = 100[(30-2x)-200] = 2800-200x$
b) I suspect the profit function is set equal to $0$ and then solved for $x$, but I am not certain.
c) I am wondering if the answer to this one is actually the answer to b), and that b) has a different solution, or perhaps $f(x)$ or its derivative are set equal to $0$.