<p>Maybe an example will help you. Imagine you start working at a company today and they offer you a salary of <span class="math-container">$50,000$</span> USD with a promise of raising it to <span class="math-container">$55,000$</span> USD next year. At the same date your boss is also hired with a salary of <span class="math-container">$100,000$</span> USD which will be raised to <span class="math-container">$110,000$</span> 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</p>

<blockquote>
  <p>For you</p>
</blockquote>

<p><span class="math-container">$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{55,000 -50,000}{50,000} = 0.1$$</span></p>

<blockquote>
  <p>For your boss</p>
</blockquote>

<p><span class="math-container">$$
\frac{\text{Salary}(\text{today} + 1\text{yr}) - \text{Salary}(\text{today})}{\text{Salary}(\text{today})} = \frac{110,000 -100,000}{100,000} = 0.1$$</span></p>

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

<p><span class="math-container">$$
\frac{S(t + \Delta t) - S(t)}{S(t)} = r \Delta t
$$</span></p>

<p>where <span class="math-container">$r$</span> is the example above just means an increment of <span class="math-container">$10\%/{\rm yr}$</span>. Now let's organize this a bit</p>

<p><span class="math-container">$$
\frac{1}{S(t)} \frac{S(t + \Delta t) - S(t)}{\Delta t} = r 
$$</span></p>

<p>And from this, you probably will recognize the notion of derivative. If you take <span class="math-container">$\Delta t$</span> a <em>very</em> small number you will get</p>

<p><span class="math-container">$$
\frac{1}{S}\frac{{\rm d}S}{{\rm d}t} = r \equiv \frac{\dot{S}}{S}
$$</span></p>

<p>Note that this is just fancy notation to represent a fractional change of the quantity <span class="math-container">$S$</span>. In other words: how much does the quantity <span class="math-container">$S$</span> change in a moment, compared with this current value.</p>

<p>In most cases the quantity <span class="math-container">$r$</span> 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 <span class="math-container">$c$</span> is affected by other factors. For example <span class="math-container">$r$</span> will make it grow, but <span class="math-container">$n$</span> will make it decrease (because it is substracting)</p>

CC BY-SA 4.0
2019-01-17T23:26:22.850Z
2019-01-17T23:26:22.850Z
caverac
<p><a href="https://i.sstatic.net/HFEaa.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HFEaa.png" alt="enter image description here"></a></p>

<p>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.</p>

<p><a href="https://i.sstatic.net/ziNpk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ziNpk.png" alt="enter image description here"></a></p>

<p>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?</p>

<macroeconomics><mathematical-economics><economic-growth>
Dot over a variable, Neoclassical growth model?
Niko
# Dot over a variable, Neoclassical growth model?

(Asked by: username_0 on 2019-01-17)

[![enter image description here](https://i.sstatic.net/HFEaa.png)](https://i.sstatic.net/HFEaa.png)

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](https://i.sstatic.net/ziNpk.png)](https://i.sstatic.net/ziNpk.png)

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?

**Tags:** macroeconomics, mathematical-economics, economic-growth


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## 1 answers

### Answer by username_1 on 2019-01-17. Score: 2

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)


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