TD2

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**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 classBy setting

, i.e. zero growthWhich 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?

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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

**a. Represent graphically in the **

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

*Question*: Why? Be sure to really understand this.

As for the production function, we perform the same changes by dividing with *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 *” 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

We already know that

By multiplying on the right and on the left by

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

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

**b. Same question for the **

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

Notice that, contrary to what you see in the lecture notes, the exercise asks you to draw the graph in the space

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

However, we are still in the

So, does

However, in this model, this can never be true as we have

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

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

*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 **

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

if . This condition ensures that our production is positive if we use a positive amount of capital; . This condition tells us that we have a*positive marginal product*, that is, increasing capital always increases production; . 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.*

. This imposes that you can not have a positive production by employing zero capital; . This assumption tells us that a little bit of capital is infinitely productive, as we go from production to positive production; . 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

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