TD4

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**a. In an ecosystem, the natural growth of a renewable resource is an increasing function of the amount of this resource.**

*Answer***False**: As an example, in class, you considered a logistic growth, captured by the equation . As you can see, when then . Therefore, it is not true that if increases then its growth also increases.

**b. An improvement in extractive technology always increases fish production if fishing is free.**

*Answer***False**: In our model the total production of fish when there is free entry is given by the following:As you can see, we have an

at the denominator with a minus sign (positive effect on ), but we also have an at the denominator with a plus sign (negative effect on . Hence, the total effect is ambiguous.

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**b. Assume that at time **

Before answering this question we have to compute the growth rate of

By substituting the numbers we have we obtain the growth rate when the new non-renewable resource is discovered, at time

As for

For the last points, the professor run an analysis with particular values of parameters

**c. Which graph is **

Let's try to find a general way of answering these kind of questions. The variables involved are *C*, as there is a smooth evolution of the dashed line at time

To understand what happens here recall that

**d. Which graph is **

First step done, now we have to distinguish between *B*.

Here two things happen. First, as we saw before,

**e. Which graph is **

We are only left with one ratio and graph *A*, so the answer is easy here, but we must understand also what is going on.

The jump is always given by the steady increase of

“We are in the beginning of mass extinction, and all you can talk about is money and fairy tales of eternal economic growth”- Greta Thunberg at the United Nations Climate Action Summit, September 23, 2019

**f.** **There is little doubt that a mass extinction is going on. However, this widespread idea of sustained economic growth being a myth is open to debate. Expanding on the model, can we comment on it.**

This is more of a philosophical question, so fell free to answer whatever you think that is relevant and consistent with the model (it is true that many answer could be consistent with what we have here, but there are also many things that are not!). What this little exercise can tell us is that there is no need of non-renewable resources for an economy to grow, as long as other rates

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One of the problems that the fishing model has is that the only circumstance in which there is an extinction of fishes (or natural resources in general) is when the starting stock is equal to

**a. Find the values of **

As always, we first need to understand what the question is asking. When does the fish stock grows? When its growth rate is different then

Since

Hence, fishes will not grow when their stock is exactly equal to their maximum capacity

Figure 1: Graph of the growth rate of fishes for

**b. Of these values, which are stable, which are not?**

First of all, what does stable mean in this context? When speaking about steady states (growth rates equal to

Checking for stability seems a daunting task, but if you have the graph it becomes easier. Here I will show you to check for stability with both a graphical and a mathematical technique. Let's start from the graphical one. Consider the steady state

Figure 2: Stability of steady states.

If you don't like this graphical reasoning, there is also the math way. This amounts to taking the derivative of

We can now evaluate the derivative in the two points of interest. Recall that

This result confirms our graphical analysis. Since the derivative at

Which again goes in the same direction as the graphical intuition (

*Question*: Do you know an easier way to check the sign of the derivative from the graph?