Putting It Together: Growth Models
Growth models can help us to understand the world around us. In this module, we have learned to describe and compare growth by understanding a little bit about different types of growth. We saw that linear growth is characterized by having a constant rate of change. When something is increasing linearly, it increases by the same amount every year (or every month, every day, etc.). Because of this, graphs which model linear growth are straight lines. Exponential growth is similar, except that instead of increasing by the same amount every year, quantities increase by the same percent each year. The constant percent change is called the growth rate. What this means for exponential growth, is that every year the increase will be by a bigger amount than it was the previous year (in the next module this will have an importance consequence, called compounding). This implies that graphs which model exponential growth look like:![](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/5362/2016/12/25224737/Exponential.png)
![](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/5362/2016/12/25225426/Logistic.png)
\[ P_n = P_0 +dn,\]
where is the amount present after years/days/months, is the initial amount, and is the constant rate of change. For exponential growth, our explicit formula is: \[P_n = P_0 \left(1+r\right)^n,\] where is the growth rate (written as a decimal). (For those who are curious, there is also an explicit formula for logistic growth. Understanding where the formula comes from typically involves calculus though, so we won't use it in this class.) When making predictions, it is important to keep in mind that some models may exhibit linear or exponential growth in the short-run but the models might change in the future. Because of this we should always be careful whenever we extrapolate (i.e.: extend our model or graph by inferring unknown values from trends in the known data).