TD6

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This exercise make you compute Pigouvian taxes. These kind of taxes are classical in the economics literature. Their aim is to correct for externalities that affect the market outcome without passing trough the channel of prices. In the fishing model an increase of boat affects the growth of natural resources in a way that is not transmitted to the market with the price

**a. Show that the optimal number of boats could be obtained by a tax per boat **

There are two ways to answer this question. The first way, which is the one you will have in the professor's solution, is to ask "which is the value of

and realise that

The second way amounts to perform the same step you did in the class, but the profits are

Which is the result we wanted.

**b. Illustrate this tax on the graph of the revenue of the fishing industry.**

The change in marginal revenues due to the introduction of the tax affects the point in which this line intersect the marginal cost

Figure 1: Profits and marginal costs for

**c. Show that an ad valorem tax on fish sales of **

Exactly as before, we can solve this problem in two ways. The first one is to compute the

You have the detail of this method in the professor's solution.

The second method follows the same reasoning of the previous point. We just change the expression for profits and find the optimal number of boats in free markets

Which is again the solution we wanted.

*Question*: How does the graph looks like here?

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In this exercise, we just have to reason with the phase diagram to get the answers. First, let's derive again the fundamental equations. Here we work in a system in which there is both growth of a natural resource, as in the fishing model, and population growth. As we just saw in the review question, the net growth of the natural resource is given by its natural growth rate minus the harvest (exactly as in the fishing model).

As for population growth, we can easily recover it by using the fundamental parameters of the model. We have that people die at a rate

These are the two **laws of motion** of the variables of interest in our model,

We have a system with two variables and two growth rates, which means that each law of motion will have more than one couple of

This equation has two solutions: the first is

We must now repeat the same exercise with

We have two solutions here too: the law of motion is null when

Hence, the equations that give us the set of points

When our variables respect these conditions there is no movement of

Figure 2: Phase diagram with