# Dot over a variable, Neoclassical growth model?

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.

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?

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

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

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

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)