TD3

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**b. On a balanced growth path, all variables grow at the same rate.**

*Answer***False**: Recall the definition of Balanced growth path at page 19 on you lecture notes:*Definition: A balanced growth path is a trajectory such that all variables grow at a constant rate*.Translated in mathematical terms, we have that all the variables

in our model must have , where is a constant. However, it is not specified that all must be equal! Each variable can grow at its own, constant rate. We have an example in the exercise below, where different variables of interest grows at different rates.

**c. The Solow model needs to assume technological change to check the stylised Kaldor facts of growth.**

*Answer***True**: Consider as an example Kaldor fact 1:*Kaldor fact 1: Labour productivity has grown at a sustained rate.*If we do not have technology, labour productivity does not grow in the steady state. In fact, if we do not have technology and we are in a steady state then

, since , if does not grow then also will not grow. Since , it is a measure of labour productivity (i.e. production per capita). You can check this on page 19 of your lecture notes, but the next exercises constitute a clear example of why this is true. Introducing a technological shift which grows at rate makes growth positive.*Question*: Try to argue the same thing by considering Kaldor fact 2 about capital per worker.

**d. The Solow model predicts convergence of all economies in the world to the same GDP per capita.**

*Answer***False**: The Solow model can be interpreted as a machine that takes as an input exogenous parameter and tells you what happens to the economy. To a different set of exogeneous parameters we obtain a different prediction. Consider as an example the economies at points a. and b. of exercise 3 of this TD(1), the growth predictions are completely different.

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The aim of this exercise is to get you used with growth rates calculations. It is in some sense less interesting from an intuitive point of view, but we will be able to link it to exercise 2 in TD2.

We have quite a lot of data. The production here is affected by three variables, capital

The law of motion of capital is the standard one

Throughout the problem we will use the following useful approximation:

Remember that

*Question*: Check the graph below. Do you think this approximation always works?

Graph1: Logarithmic approximation.

**a. In this problem use **

Let's use the definition and the approximation we are given. We start from

We can perform the same calculations to see that *Question*: Try to find

**b. Compute **

This seems like a daunting task, so let's divide this computation by steps.

First, we must identify the variable of which we want to compute the growth rate. In this case we have from the text

Second, we use the explicit expression of growth rates to understand how its growth rate is composed. Since we have that

Third, we take logs, so that we have a direct expression for

**Remark**: Remember that

We have exactly

**c. Compute **

Exactly as before, we exploit the definition of growth rate and what we know about

We elaborate a little bit on this expression to put it in a form that is convenient to us. First, we divide by

We managed to find an expression of

**d. Argue why, along a balanced growth path, **

Recall the definition of a balance growth path: all the variables must grow at a constant rate! This means, in order, that

As for the second question, the answer is only one step ahead of the previous reasoning. In order for the ratio

Which is what we wanted to prove.

**e. Using your answers to earlier parts of the problem, solve for **

From point **b.** we have that **d.** we know that along a balanced growth path

Which gives us

**f. What is the condition for **

To answer this question we have first to compute the quantity of interest. The rule is always the same:

Now we are ready to evaluate when this expression is positive. First, we know that the denominator is always positive, as

Before interpreting this result, we must understand what

The first two increases product per capita, while the third one decreases it. Therefore, growth will be positive when the sum of the first two is higher than the third.

*Question*: Can you guess what is the role of

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This problem is tightly related to the previous one but it has less computations and more intuition. Its focus is to study the employment of renewable and non-renewable resources and its sustainability.

We have the same production function and growth rates as before.

However, here we specify how

**a. Assume that initially, the economy is in a balanced growth path (BGP) where the stock of renewable resources is stable and where there are no non-renewable resources at all. What is the growth rate of **

We already have the growth rate of

However, in this case there are no non-renewable resources

The evaluation of its sign is the same as before. In particular, we know that

We have

captures the relative importance of capital in the production function. Intuitively, if capital is relatively more important there are more chances that per capita growth is positive, as it is directly affected by technological progress; is the rate of growth of technology. Of course, the more technology improves the more likely is that growth per capita increases; is counterpart for natural resources, it measures its relative importance in the production. Since these do not grow, if they are less important than growth will be positive despite the fact that the stock is fixed; represents the growth rate of population. Intuitively, if population increases the growth per capita decreases, as there are more mouths to feed.

Since we know that