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I would like your help to understand the Maximum sustainable yield model. This is what I understood with some questions.


We consider a population of individuals who are born, possibly reproduce themselves, and die.

We assume that the population does not grow indefinitely, but after a certain amount of time the population reaches an equilibrium size, $K$. Time and population size are thought as continuous variables. Time spans an infinite horizon.

Specifically, the evolution of the population size over time is assumed to follow a logistic function as in Figure A of the picture below, where $N(t)$ denotes population size and $t$ denotes time.

Consequently, one can derive the slope of the logistic curve at each point, which we denote by $\frac{d N(t)}{dt}$. When plotting $\frac{d N(t)}{dt}$ as a function of $N(t)$, one obtains Figure B of the picture below.

$\frac{d N(t)}{dt}$ can be thought of as the "increase of population size that is obtained by natural processes during an infinitesimal amount of time".

Suppose now that a firm wants to harvest individuals and that the firm's profit increases with the number of harvested individuals. Let $H$ be the harvesting rate, that is "amount of individuals removed from the population during an infinitesimal amount of time". Suppose that the firm wants to keep the harvesting rate constant over time. At a given point in time, $H$ cannot be bigger than $\frac{d N(t)}{dt}$ because otherwise the harvesting is not sustainable and the population is at risk for decline to extinction. $\frac{d N(t)}{dt}$ is highest when the population size is equal to $N^*$. Hence the optimal strategy is to wait until the population has reached size $N^*$ and then apply a constant harvesting rate equal to $H^*$ (which, in turn, keeps the population size constant at $N^*$ and does not allow it to grow until $K$).

Questions:

1) Is my summary correct? For the last sentence to be true, are we assuming that "waiting until the population has reached size $N^*$" has no cost for the firm (or that, somehow, costs are compensated after)?

2) I read somewhere that the harvesting activity produces an harvested stock equal to $N^*$. What does this mean? Since the time horizon is infinite, once harvesting starts it will never stop. Therefore, how can one compute the final harvested stock?

3) Which are other underlying assumption of the model that I have not reported in my description above?


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