```
title: TD 1
statement-kinds:
proposition:
counter: self
definition:
counter: self
example:
counter: self
```

...

**a. Inequality of income across countries was already very wide at the end of the medieval period.**

*Answer***False:**It started widening after the Industrial Revolution. Before the revolution, growth was almost null around the world. Production was very close to the subsistence level and there were no investments. Technological progress was the catalyst of growth and, since it was heterogeneous across countries, induced an increase in inequality.

...

This problem aims to get you used to the basic logic of the Solow-Swan model. This exercise is a particular case in which we have a Cobb-Douglas production function, population growth and no technology. In the notation of the class, population growth is

*Question*: Can you get what is the original production function

*Question*: What is the interpretation of

**a. Express the steady-state level of consumption **

*Question*: Before starting, what is the difference between exogenous and endogenous?

First, we have to recall how to express consumption in this model. What is consumed is what is produced minus what is invested, therefore

Here we have

**b. Use the result to the previous question to find the optimum level of **

This question asks to maximise consumption with respect to capital. We want to answer the question: what is the level of

Remember that you can express this quantity differently by playing with the exponent!

*Question*: What about the second-order condition?

**c. Express the steady-state level of consumption **

We have to perform the same operation as before, but without including

We can now substitute

Here we have

**d. Use the result to the previous question to find the optimum level of the savings rate **

Again, same story as the previous point, but instead of maximising for

We are now ready to solve the maximisation problem (here I provided very detailed calculations, if you are comfortable with calculus you do not need to write everything as I do here). First, I recall the rules for deriving a product. In general, we have the following:

In our case the two functions are

Then, by applying the general rule:

By setting the derivative equal to zero, we can get rid of the constant (equivalent to dividing each side of the equality by the constant itself).

We are left with the following:

The

*Question*: Again, what about the second-order conditions for this problem?

**e. Comment the results obtained to questions **

Consider the expressions **b.** and **c.**

Next, consider the result **d.**

By combining

*Question*: There is another (probably many) way to answer this question, can you get it?

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