Growth Rates and the Rule of 70 - Online Tutor, Practice Problems & Exam Prep

Understanding growth rates is essential in analyzing economic performance, particularly GDP growth. The annual growth rate is calculated as the current GDP minus the prior year's GDP, divided by the prior year's GDP. To find the average annual growth rate, sum all yearly growth rates and divide by the number of years. The Rule of 70 estimates the time to double GDP by dividing 70 by the growth rate. For example, at a 2% growth rate, it takes approximately 35 years to double, while at 6%, it takes about 11.67 years, highlighting the significant impact of small growth rate changes on economic outcomes.

Growth Rates and the Rule of 70

Alright. Now, let's discuss growth rates a little more and how we analyze them as well as the rule of 70. So, there are several important calculations we're going to talk about here, and when we talk about economic growth, we're generally talking about growth in GDP. We're focused on the growth in GDP, and we're going to use a few different calculations we can use when we talk about what that growth is. The first one is just what is the annual growth rate? And this is basically just a simple percentage change calculation. A percentage change calculation where we take the annual growth rate being the current GDP minus the prior year's GDP, divided by the prior year's GDP, and remember, whenever we talk about percentage change, we talk about that new minus old divided by old and that's exactly what's going on here. The growth rate is going to be current GDP minus old GDP divided by old GDP. That'll give us the percentage change. Okay? So that's our annual growth rate.

We can also talk about an average annual growth rate. We've got growth rates for every year, maybe 2 years ago it was 2% and now it's 4% and then 5%, right? How do we find the average annual growth? Well, guess what? That's just like finding an average. To find your average in a class, you would add up all your grades on the exam divided by how many exams you took assuming they are all weighted the same, right? And we would find out the average. So what we're going to do is we're going to take all the growth rates. Let's say there was growth rate 1 this year, growth rate 2 in the next year until the final year, and then you'll divide it by the number of years, right? The number of years that you added growth rates for. So it's just a simple average calculation, and that will give you the average annual growth, right? Because maybe some years you have a lot of growth, some years you have a little, maybe 1 year you didn't grow at all. Well, we'll find the average of all those years together with that formula. Just add up all the growths and divide by the number, right? That's just a simple average formula there.

Finally, the rule of 70. They love to talk about this rule of 70 because it talks about how long it would take for a value to double. So how long will it take for GDP to double if this year GDP is $1,000,000,000? How long will it take for GDP to be $2,000,000,000? Well, that depends on the growth rate. So, with the rule of 70, it gives us an approximation of how long it is going to take to double and all we have to do with the rule of 70, the number of years to double is 70 divided by the growth rate. A lot of times, this will be the average growth rate. Generally, we'll talk about the average growth rate here. And when you do this calculation, the growth rate should be put in as a whole number. So 2% would be 2. We wouldn't put it as 0.02, okay? So that's the little trick here is that we put 2% as 2, not 0.02; that would give us a crazy number. So let's go ahead and do a little bit of practice here with the rule of 70 and then you guys can try as well. Let's do this first example.

Use the rule of 70 to analyze the following growth rates. Let's go ahead and see what happens. If the growth rate is 2%, 4%, or 6%. So in which case do you think we would have the fastest time till double? Which one is going to double the quickest? 6%, right? Because we're growing faster. If we're growing 6% every year, we would assume that we would double faster. Okay. So let me open up my calculator and let's do these calculations.

How long would it take us to double at a growth of 2%? What we would do is 70 divided by 2. The percentage there, 2 and that gives us approximately 35 years. How about when it's 4%? 70 divided by 4. So 70 divided by 4, that's going to give us 17.5 years. Now what about when it's 6%, 70 divided by 6. 70/6=11.67 years to double, right? So this is telling us how long it would take to double.

So, what do we have here? So notice that small differences. The point here is that small differences in growth rates can have large impacts on the standard of living in a country, right? So notice when it's just at 2%, it's going to take us 35 years to double. But just by going from 2% to 4%, even though that's double the amount but 4%, that still doesn't sound like that much growth, right? It means that the economy is going to double in 17.5 years, it's going to double twice as fast there, and then 6%, 17.67 years. So these small changes, these small percentage changes are having huge impacts in the long run of how long the economy is going to take to double there, okay? So that's the key here is that even small changes in economic growth can lead to large differences. Cool? Alright. Let's pause here and do this example in the next video.

Growth Rates and the Rule of 70

The country of Growtopia has a real GDP in the previous year of 1,450,000,000 dollars. The current year real GDP was 1,510,000,000 dollars. Based on this information, approximately how long would it take for Growtopia's real GDP to double if it continues to grow at a constant rate? Note, they don't mention the rule of 70 here, but the key is that they ask how long it will take to double if it continues to grow at a constant rate. The first thing we need to know is the growth rate, and then we can use the rule of 70 to estimate how long it would take to double. First, we need to calculate the growth rate using our percentage change formula.

To calculate the growth rate, use the current year GDP of 1,510,000,000 minus the previous year GDP of 1,450,000,000. The formula used is New minus Old divided by Old. Let's calculate it:
√ 1.51 − 1.45 divided by 1.45, which yields a percentage change. Calculating this, 1.51 − 1.45 divided by 1.45, gives us approximately 4.1379%.

It's important to use many decimals to get a precise answer. We typically don't want to round until the last step in any calculation to ensure accuracy. With our growth rate now identified, we can use the rule of 70 to determine how long it would take to double Growtopia's GDP:
√ 70 divided by 4.1379, which results in approximately 16.9192 years.

This calculation shows the importance of precision in the initial steps to ensure the final answer is accurate. With this process, we estimated it would take around 17 years for the GDP of Growtopia to double, assuming a constant growth rate.

Now, try a practice problem to apply what we've discussed.

Use the information in the table to calculate Growtopia's average annual growth rate for real GDP and the approximate amount of time it would take for Growtopia's real GDP to double.

• A. Average Growth:2.4% Time to Double:29.2 years
• B. Average Growth:3.8% Time to Double:18.4 years
• C. Average Growth:5.2% Time to Double:13.5 years
• D. Average Growth:7.6% Time to Double:9.2 years

Here’s what students ask on this topic:

What is the annual growth rate and how is it calculated?

The annual growth rate measures the percentage change in GDP from one year to the next. It is calculated using the formula:

$\frac{\mathrm{GDP}-\mathrm{GDP}{t}_{-}}{\mathrm{GDP}{t}_{-}}$

where $\mathrm{GDP}$ is the current year's GDP and $\mathrm{GDP}{t}_{-}$ is the previous year's GDP. This formula gives the percentage change, indicating how much the economy has grown or shrunk over the year.

How do you calculate the average annual growth rate?

The average annual growth rate is calculated by summing all the yearly growth rates and dividing by the number of years. The formula is:

$\frac{\sum \mathrm{Growth}\mathrm{Rates}}{\mathrm{Number}\mathrm{of}\mathrm{Years}}$

This method provides an average growth rate over a specified period, smoothing out the fluctuations of individual years.

What is the Rule of 70 and how is it used to estimate the time to double GDP?

The Rule of 70 is a simple way to estimate the number of years it will take for a value, such as GDP, to double given a constant annual growth rate. The formula is:

$\frac{70}{\mathrm{Growth}}$

For example, if the growth rate is 2%, it will take approximately 35 years to double (70/2 = 35). This rule highlights how small changes in growth rates can significantly impact the time required for economic doubling.

Why are small differences in growth rates significant for economic outcomes?

Small differences in growth rates can have large impacts on economic outcomes over time. For instance, a growth rate of 2% means GDP will double in 35 years, while a 4% growth rate means it will double in 17.5 years. This difference can significantly affect the standard of living, investment decisions, and long-term economic planning. Even a seemingly minor increase in the growth rate can lead to substantial improvements in economic prosperity over the long run.

How does the Rule of 70 apply to different growth rates?

The Rule of 70 can be applied to various growth rates to estimate the time required for GDP to double. For example, at a 2% growth rate, it takes approximately 35 years to double (70/2 = 35). At a 4% growth rate, it takes about 17.5 years (70/4 = 17.5), and at a 6% growth rate, it takes roughly 11.67 years (70/6 = 11.67). This demonstrates that higher growth rates lead to faster doubling times, emphasizing the importance of even small increases in growth rates.