TD5

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**c. An improvement in extractive technology always increases fish production if fishing is socially optimal.**

*Answer***True**: From you notes (page ), the expression for the total harvest in the social planner solution is . You can see that an increase in will augment total harvest even without taking derivatives.*Question*: Don’t you think the comparison between this question and the solution to point**b.**is interesting?

**d. An improvement in extractive technology is always a bad thing from an environmental point of view.**

*Answer***True**: We can check from the equilibrium expression of the stock of natural resources . If increase then the stock of natural resources decreases.

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**c. What is the natural growth of the fish population at **

To answer this question we just need to evaluate the growth rate in the point

We should have expected this result, as we know that

Figure 1: Graph of the growth rate of fishes for

*Question*: Is this new steady state stable?

**d. What is the maximum number of fish that can be caught per unit of time such that the fish population is constant? This is also called the maximum sustained yield. What is the fish stock **

To answer this question we must ask when the growth rate of fishes is the highest. This would allow us to capture the maximum number of fishes every time

Now that we have the stock of fishes that maximises growth, we can ask by how much fish grows for this value of the stock. Of course, to answer this question we just need to plug the value we just found in the growth rate.

This expression tells us by how much fish we can catch for growth to always be at its maximum.

**e. Graph the fish growth function **

We already did a big part of the graph, the one below has also the answers to the last question.

Figure 2: Graph of the growth rate of fishes for

**f. If there are **

First, notice that

This is a second order degree equation of which we have to find the roots by the usual formula. There are two solutions that we label

These are the two steady state populations of fish.

**g. Graph the dynamics of the stock with resource extraction and identify the equilibrium population(s) of fish. Show with arrows how population dynamics pushes **

Here the picture where I added the solutions computed in the previous point. Notice that here the growth rate is given by the difference

Figure 3: Same graph as before with

**h. Is there an intensity of fishing **

To answer this question it is enough to notice that if you increase

If you perform the same operation in the plus (+) case you will find that

*Question*: Can you guess what happens if

*Question*: Can you think about other methods to do this point?

**i. The profit from a boat is **

If boats will continue to enter as long as profits are positive, then they will stop when profits are

However, notice that in class we had