Given a variable $x$, is its "growth rate" (or "rate of change") $\frac{\mathrm{d}x}{\mathrm{d}t}$ or $\frac{\frac{\mathrm{d}x}{\mathrm{d}t}}{x}$?

Question

I think it is (or should be) the latter (reasoning below). But many writers (e.g. Stewart, Calculus) use the former instead.

So I'm confused and hence this question. (Maybe this is just one of those phrases/terms that have different definitions according to the writer?)

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Say $x_t$ is population, $x_{2021}=100000$, and $x_{2022}=101000$. Then most people would say that the annual growth rate of population was $1\%$: $$g_x=\frac{\frac{x_{2022}-x_{2021}}{1\ \text{year}}}{x_{2021}}=\frac{\frac{\Delta x}{\Delta t}}{x}.$$

(I don't think many would say that the growth rate of population was $1000$.)

Now going from the above discrete case to the continuous case, shouldn't we similarly have that

$$g_x=\frac{\frac{\mathrm{d}x}{\mathrm{d}t}}{x}?$$

Answer 1

1. Many calculus textbooks define $\frac{\mathrm{d}x}{\mathrm{d}t}$ as the (absolute) growth rate and $\frac{\frac{\mathrm{d}x}{\mathrm{d}t}}{x}$ as the relative growth rate.

2. But many social scientists (economists, demographers) often simply call the latter the "growth rate".

So, growth rate is $\frac{\mathrm{d}x}{\mathrm{d}t}$ in math but $\frac{\frac{\mathrm{d}x}{\mathrm{d}t}}{x}$ in the social sciences.

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1. Stewart, Clegg, and Watson (Calculus: Early Transcendentals, 2021), pp. 230, 240:

growth rate ... $=\frac{dn}{dt}$ ...

$\frac{dP/dt}{P}$ ... is called the relative growth rate.

Briggs, Cochran, Gillett, and Schulz (Calculus: Early Transcendentals, 2019), p. 493:

If $y\left(t\right)$ represents a population, then $y^{\prime}\left(t\right)$ is the (absolute) growth rate of the population ... Another way to talk about growth rates is to use the relative growth rate, which is the growth rate divided by the current value of the quantity, or $y^{\prime}\left(t\right)/y\left(t\right)$.

Hass, Heil, and Weir (Thomas' Calculus, 2018), p. 541:

$\frac{dP/dt}{P}$ ... is called the relative growth rate.

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2. Abel, Blanchard, Bernanke, and Croushore (Macroeconomics, 2017):

Let $\Delta X/X$ and $\Delta Z/Z$ represent the growth rates

Our World in Data:

The global population growth rate has already slowed down considerably: it reached its peak at over 2% in the 1960s and has been falling since.

Mankiw (Principles of Economics, 2020):

With a growth rate of 2 percent per year, productivity and real wages double about every 35 years.

Answer 2

The growth rate is the rate at which a population is increasing (or decreasing) during a year. In this case, i think the rate is a positive rate of 1000 per year. However, this growth rate is usually expressed as a percentage of the base population. (1000 / 100000)*100 = 1% .

Answer 3

Say $x_t$ is population, $x_{2021}=100000$, and $x_{2022}=101000$. Then most people would say that the annual growth rate of population was $1\%$: $$g_x=\frac{\frac{x_{2022}-x_{2021}}{1\ \text{year}}}{x_{2021}}=\frac{\frac{\Delta x}{\Delta t}}{x}.$$

Your formula gives the relative average annual growth; more clearly: the average annual growth relative to the base case. Here, the time dimension is related to the averaging process rather than to the 'percentaging' process.

On the other hand, the average annual growth rate is actually $$\frac1n\sum_{i=1}^n(x_i-x_{i-1})$$ or $$\frac1{t_2-t_1}\int_{t_1}^{t_2}\frac{\mathrm dx}{\mathrm dt}\,\mathrm dt,$$ where the measurements are taken yearly and $t$ corresponds to years.