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.

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 , labour and a natural resource . We also have capital augmenting technology . The function is the following

The law of motion of capital is the standard one where . Technology, grows at an exogenously fixed rate . Also the stock of natural resources grows at an exogenous fixed rate . Labour also grows, as we already saw .

Throughout the problem we will use the following useful approximation:

Remember that , the growth rate.

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

Graph1: Logarithmic approximation.

a. In this problem use to denote the growth rate of the variable (for example ). From the definitions, write , and .

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

We can perform the same calculations to see that and . Question: Try to find and as an exercise.

b. Compute in terms of and .

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 , we first have to compute .

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

Remark: Remember that and .

We have exactly in terms of and .

c. Compute in terms of and .

Exactly as before, we exploit the definition of growth rate and what we know about . The law of motion of capital is always the same.

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

We managed to find an expression of in terms of and .

d. Argue why, along a balanced growth path, must be constant. Then argue why .

Recall the definition of a balance growth path: all the variables must grow at a constant rate! This means, in order, that must grow at a constant rate, that must be equal to a constant, and that must be constant. We know that and are indeed constant, but, if we are not on a balance growth path evolves with time. Therefore, must not change for to be constant, so that grows at a constant rate.

As for the second question, the answer is only one step ahead of the previous reasoning. In order for the ratio to be constant, the two variables must grow at the same rate in each time . If, as an example, grows quicker than , the ratio will not be constant in time, therefore . More precisely, if grows at a constant rate, it means that . By exploiting the rules of growth rates:

Which is what we wanted to prove.

e. Using your answers to earlier parts of the problem, solve for in terms of and (from now on I omit the index for simplicity).

From point b. we have that , while form point d. we know that along a balanced growth path . By substituting the second condition into the first on we obtain:

Which gives us in terms of and .

f. What is the condition for to be positive along a balanced growth path? Interpret.

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 . Therefore, the whole fraction is positive when the numerator is positive.

Before interpreting this result, we must understand what indicates. It is the growth rate of what we usually denote , production in per capita terms. So, asking when is positive is the same as asking: "when does production in per capita terms has a positive growth rate?". Hopefully this interpretation of the question helps us understand this condition. There are three factors that affect consumption per capita.

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 and exactly?

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 is composed. We can split natural resources in renewable and non-renewable . The rate of exploitation of is , therefore we have that .

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 ? Interpret in what conditions we get a positive rate of growth for . Knowing what we can anticipate about the rate of technology and population growth in the century, should we expect to grow or not in the coming decades ?

We already have the growth rate of on a balanced growth path from the previous exercise.

However, in this case there are no non-renewable resources and the stock of renewable resources is stable, which implies that it is not growing. Since and we have that here . The growth rate of was in the previous exercise, but since here does not grow, does not grow either, as it is composed by only. This translates into . The new growth rate is therefore:

The evaluation of its sign is the same as before. In particular, we know that is always positive, therefore the sign of the numerator is the significant one. The whole fraction is positive when the numerator is positive:

We have factors that affect this inequality:

1. 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;
2. is the rate of growth of technology. Of course, the more technology improves the more likely is that growth per capita increases;
3. 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;
4. 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 is quite low while is high, if the premises of this model are true then we would be sure to enjoy positive growth in the future.