0
$\begingroup$

enter image description here

Looking at the Neoclassical growth model, I have hard time understanding what dot over a variable mean? I know it is short for taking the derivative with respect to time, I don't understand the meaning behind it.

enter image description here

The second question concerns the Euler equation for consumption. What does it tell us? And is c(t)dot/c(t) the growth rate of consumption?

$\endgroup$
2
$\begingroup$

Maybe an example will help you. Imagine you start working at a company today and they offer you a salary of $50,000$ USD with a promise of raising it to $55,000$ USD next year. At the same date your boss is also hired with a salary of $100,000$ USD which will be raised to $110,000$ next year. Surely you will think it is unfair your boss will get a lot more next year than you will. But let's look at it from another perspective

For you

$$ \frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{55,000 -50,000}{50,000} = 0.1$$

For your boss

$$ \frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{110,000 -100,000}{100,000} = 0.1$$

Which is exactly the same number, people usually call this a relative increment, which in this case is $10\%$. So from this point of view it is the pretty fair. To make this a bit more technical call $S$ the salary, $t$ today, and $\Delta t$ make it 1 year, so that the equations above becomes

$$ \frac{S(t + \Delta t) - S(t)}{S(t)} = r \Delta t $$

where $r$ is the example above just means an increment of $10\%/{\rm yr}$. Now let's organize this a bit

$$ \frac{1}{S(t)} \frac{S(t + \Delta t) - S(t)}{\Delta t} = r $$

And from this, you probably will recognize the notion of derivative. If you take $\Delta t$ a very small number you will get

$$ \frac{1}{S}\frac{{\rm d}S}{{\rm d}t} = r \equiv \frac{\dot{S}}{S} $$

Note that this is just fancy notation to represent a fractional change of the quantity $S$. In other words: how much does the quantity $S$ change in a moment, compared with this current value.

In most cases the quantity $r$ is not a constant, but depends on time. Such in your second expression, but do not loose track of the meaning there: It tells you how the relative change of quantity $c$ is affected by other factors. For example $r$ will make it grow, but $n$ will make it decrease (because it is substracting)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.