Calculating growth rates

How to calculate the change and growth in GDP

Let's say that we have GDP for two different years: 2019 and 2020.

Logs

Let's start with a simple function that the value of some variable $z$ depends on time, $z(t)$. Take the total derivative of this function

$$d z(t) = z'(t) dt$$

or the absolute change in $z(t)$ depends on the derivative $z'(t)$ and the actual change in time. If you divide both sides of this by $z(t)$, you get

$$\frac{d z(t)/dt}{z(t)} = \frac{z'(t)}{z(t)}.$$

On the left is the growth rate of $z$; the change in $z$ for a given change in time divided by $z$ itself. We'll be concerned a lot with growth rates, so I'm going to introduce some new notation here to avoid writing out this whole fraction over and over again.

$$g_z \equiv \frac{d z(t)/dt}{z(t)}.$$

Any time you see $g_z$ anywhere, it means the growth rate of $z$ (or whatever variable is in the subscript). As we'll often implicitly assume that the change in time is $dt = 1$, I will also often refer to $d z(t)/z(t)$ as $g_z$. We'll sometimes use the following notation as well to save on notation in certain situations,

$$G_z \equiv 1 + g_z,$$

and $G_z$ is sometimes referred to as the "growth factor" of $z$.

That definition works fine, but we can establish a nice relationship between the (natural) log of $z(t)$ and the growth rate that will save us a lot of time in the work ahead. Start with $\ln z(t)$, and again take the total derivative,

$$d \ln z(t) = \frac{z'(t)}{z(t)}dt.$$

This expression means that

$$\frac{d \ln z(t)}{dt} = \frac{z'(t)}{z(t)} = g_z.$$

The derivative of \textit{log} $z(t)$ for a given change in time is equal to the growth rate of $z$. This is incredibly useful for us. It effectively says that the derivative of the log of something with respect to time is equal to its growth rate. As we'll often look at figures which plot the log of a variable against $t$, this gives us a simple visual way to evaluate the growth rate. The slope of $\ln z(t)$ with respect to time is equal to the growth rate.

Things get even more stark if we have exponential growth, meaning there is a constant growth rate. Let

$$z(t) = z(0) e^{bt}$$

where $z(0)$ is some initial value of $z$, and the value of $z$ at any time $t$ is $z(t)$. $b$ is a parameter and is equal to the growth rate, as we will see. Take logs and you get

$$\ln z(t) = \ln z(0) + bt$$

and now this is log-linear. If we take the derivative of this with respect to time we get $d \ln z(t)/dt = b$, or the growth rate is $g_z = b$.\marginnote{You can confirm this by finding $z'(t) = b z(0) e^{bt}$, and dividing by $z(t)$, so that $z'(t)/z(t) = b z(0) e^{bt}/z(0)e^{bt} = b$.} Moreover, note that this last equation is simply the equation for a straight line. Which can use this visually as well. If we see that the relationship of (log) $z(t)$ with time is linear, we know it has a constant growth rate. The intercept of this line tells us about the size of $z(0)$.

Finally, you should be familiar with the properties of growth rates of products, ratios, and powers, which all can be confirmed by taking logs and then differentiating with respect to time.

X = ZW &\Rightarrow& g_X = g_Z + g_W \\ \nonumber X = \frac{Z}{W} &\Rightarrow& g_X = g_Z - g_W \\ \nonumber X = Z^{\alpha} &\Rightarrow& g_X = \alpha g_Z \nonumber