e. At the steady state, investments are equal to what is lost to depreciation, population growth and technological progress.

• Answer

  TRUE: At the steady state, capital per unit of effective labour must not grow. From the results of the class

  By setting , i.e. zero growth

  Which exactly means that investments are equal to the loss in capital due to depreciation, population growth and technological progress.

f. The golden rule of savings states that in steady state, capital should be barely productive enough to compensate for depreciation, population growth and technological progress.

• Answer

  TRUE: The golden rule of savings tells us what is the optimal (or , in the previous TD we found a relation between these two problems) to maximise steady state consumption . To work it out we need to recall how to express consumption. We consume what we produce minus what we invest, therefore . We can exploit the expression of in steady state, which, by the result of the previous question, is the following:

  What is the capital that maximises ? We have to solve a standard optimisation problem.

  Which exactly means that the marginal productivity of capital must offset the loss due to depreciation, population growth and technological progress (remember that tells us how much production increases after an infinitesimal change in , namely the marginal product of effective capital).

  Question: What is the difference with what we got in the previous TD?

The aim of this exercise is to understand the role of assumptions in the Solow - Swan model. It may be tempting to read assumptions once and then forget about them, but they are of crucial importance in these and in all other theories in economics (science and reasoning in general).

Our production function for the first point is . The saving rate is , population growth rate is and capital depreciates at rate

a. Represent graphically in the plane the dynamics of the Solow model with population growth and no technological change.

The first step is to transform all the variables in per capita quantities. We perform this step because we are interested in the steady state of capital per worker , and not in its absolute value .

Question: Why? Be sure to really understand this.

As for the production function, we perform the same changes by dividing with and exploiting constant returns to scale (CRS).

Question: Can you prove that this production function satisfies constant returns to scale?

To check for the dynamics of the model we need the law of motion of capital, as capital is the principal state (endogenous) variable which determines what happens in the economy as time changes. The law of motion tells us how capital evolves over time. We ask ourselves the question: "if at time I have capital (per capita), how much capital do I have in the next period?” To answer this question we need to know what is the relation between these two variables.

On the one hand, we have saved resources that we can use in the next period , on the other hand, capital depreciates at rate , and therefore we lose . Moreover, in this formulation of the model, there is also population growth. An increase in population decreases the amount of capital per capita. These are the two relevant factors that affects the dynamics of capital. From this intuitive reasoning you should already see that , i.e. the difference between capital tomorrow and capital today is savings minus what is lost due to depreciation and population growth. However, we want to be rigorous and find the law of motion via rules of growth rates of products and ratios. Define , then

We already know that . Moreover, we know from the standard Solow model that (why?). Therefore,

By multiplying on the right and on the left by we get the final expression:

In the steady state variables do not change over time, therefore capital per capita will be stable, which means .

Investment exactly offsets the loss due to depreciation and population growth.

In our case , therefore the three elements of our graph are:

Graph1: Dynamics of the Solow Model with Population Growth.

Remark: Always, always, always put labels on axes when you draw graphs!

As always we have that , and , as shown in the graph.

b. Same question for the production function , assuming . Does exist in this case?

For this point, I offer a different path to reach the solution compared to what you will receive from the professor. I think my way is more in line with the standard method, but you choose which one you prefer.

To answer this question we proceed as we did in the previous point, but of course, we have to take into account the different production function and the technological change. First, let's express as a function of capital per capita. We perform the same computation as before.

Notice that, contrary to what you see in the lecture notes, the exercise asks you to draw the graph in the space and not , that's why I do not divide by units of effective labour .

Question: What if we had ? Answer: Due growth rates rules:

However, we are still in the space, so we are interested in the law of motion of capital per capita , which is the same we had in the previous point (except for the different production function). We have the following:

So, does exists in this case? The steady state condition is always the same:

However, in this model, this can never be true as we have ! So the answer is no, there exists no steady state capital (notice however that is a solution, in fact, the condition is respected in this case). The reason is also apparent from the graph. Inspired by a question of one of your classmates I also plotted a graph of the two economies we just studied in the space.

Graph 2 and 3: Dynamics of the Solow model with technological change and linear production function. Question: Can you guess what point the intersection between the curve and the -axis is?

The problem here is that the saving rate and the technological change outset the decrease of capital per capita due to depreciation and population growth. This is due to the fact that the coefficient of the (linear) savings function is greater than the coefficient of the depreciation . Therefore, the increase in capital will always be greater than its loss. Its growth will never stop, it will continue to increase in every time . This example shows why assumptions are a fundamental ingredient of the model and not something we use just for convenience and that we can forget by putting them below the carpet. We will discuss this in the following point.

Question: Do you know what would happen if ?

Question: Do you think the results would have been different if we checked capital per units of effective labour ?

c. Make a list of the properties that the function does not satisfy with regards to the Solow model. Which one explains the previous result?

In the Solow model, there are three assumptions on the production function and three extra assumptions that are dubbed Inada Conditions.

1. if . This condition ensures that our production is positive if we use a positive amount of capital;
2. . This condition tells us that we have a positive marginal product, that is, increasing capital always increases production;
3. . This condition is a crucial one in this exercise. It ensures that the marginal product is decreasing in . This means that the unit of capital will increase the production less than the one. We will show that this does not hold in the previous point.

Then we have the Inada Conditions.

1. . This imposes that you can not have a positive production by employing zero capital;
2. . This assumption tells us that a little bit of capital is infinitely productive, as we go from production to positive production;
3. . This assumption ensures that employing an infinite amount of capital is not convenient, as the marginal product will eventually reach so that using capital will not be productive at all and therefore will be wasted.

Let's check that does not satisfy assumption and Inada condition . and hence leads to no positive steady state level of capital per capita. We have

Therefore, as we noticed before, capital is too productive and its increase due to production always offset its loss due to depreciation and population growth.

Question: Check that the production function in the first part of the exercise indeed satisfies all these assumptions.

Question: Are you sure that satisfies all other assumptions except 3 and Inada condition 3?