Growth Rates and the Rule of 70

Alright now let's discuss growth rates a little more on how we analyze them as well as the rule of 70. So there are several several important calculations we're gonna talk about here and when we talk about economic growth were generally talking about growth in GDP. So we're generally focused on the growth in GDP and we're gonna use a few different calculations we can we can use when we talk about what that growth is. So 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 uh the annual growth rate being the current GDP minus the prior year's GDP divided by the prior year's GDP. And remember when whenever we talk about percentage change we talked 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 G. D. P. Divided by old GDP. That will 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 two 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. If you're gonna find your average in the 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 gonna do is we're gonna take all the growth rates, let's say there was growth rate, one in this year growth rate to in the next year dot dot dot until the final year. And then you'll divided 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 one year you didn't grow at all. What we'll find the average of all of those years together with that formula. So just add up all the growths and divided by the number, right? That's just a simple average formula there. And finally the rule of 70, they love to talk about this rule of 70 because it takes, it talks about how long it would take for a value to double. So this basically, how long will it take for GDP to double this year? GDP is say $1 billion. How long will it take for GDP to be $2 billion? Well, that depends on the growth rate. So with the rule of 70, it gives us an approximation of how long it's 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. So 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 This growth rate, we put it in as a number. So 2% would be too, we wouldn't put it as .02. Okay, so that's the little trick here, is that we would put 2% as to not .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. Uh So 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's going to double the quickest. 2%. 4% or 6%? 6%. Right. Because we're growing faster. If we're growing 6% every year we would assume that we would double um Double faster. Okay, so let me open up my calculator and let's do these calculations. So how long would it take us to double at? At a growth of 2%. What we would do 70 divided by to write the percentage there too. And that gives us 35 years approximately. Let me put in approximately equals to Okay, so this gives us an approximation of how long it takes us to double? 35 years when it's 2%. How about when it's 4% 70 divided by four. So 70 divided by four. That's gonna give us 17.5 years now. What about when it's 6% 70 divided by 6 70 divided by six equals 11. On these are approximates again. Right. These are both approximates 11.67 years to double. Right, so this is telling us how long it would take to double. So what what do we have here? Uh So notice that small differences. What were 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 we're just at 2% it's gonna take us 35 years to double, but just by going from 2% to 4% that I know that double the amount but 4%. That's still doesn't sound like that much growth. Right? 4%. Well, It means the economy is gonna double in 17.5 years, right? It's gonna double twice as fast there and then 6%. Right, 11.6, 7 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 their. Okay, so that's 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 let's do this example in the next video.

Alright, let's try this example. The country of grow Topia has Real G G D p in the previous year of 1.45 billion. The current year. Real GDP was 1.51 billion based on this information, approximately how long would it take for Grotto pia's Real GDP to double if it continues to grow at a constant rate, notice they don't mention the rule of 70 at all here. But the key is that they say that how long will it take to double If it continues to grow at a constant rate? So the first thing we need to know is what is the growth rate and then we can use the rule of 70 to see how long it would take to double. So the first thing we need to do is to calculate the growth rate using our percentage change formula. So we need to use the current year of 1.5, 1 billion - the previous year of 1.45 billion. Right, new minus old, divided by what's going to be on the bottom here. 1.5, -1.45 divided by 1.45. Right, new minus old, divided by old. Okay. And that's going to give us our percentage change. So let's go ahead and see what this is. 1.5, -1.45 divided by 1.45 that comes out to and I'm gonna put it as a, as a percentage here. four 1379%. Now the reason I use a lot of decimals is because we want to get a very as precise of an answer. You don't you really don't want to round until the last step in any calculation. You kind of want to save your rounding to the very end. So we'll go to four decim here and then we'll round it off. Notice they don't even have any decimals here, so we wanna make sure we get the correct answer. So now let's use our rule of 70. We're gonna take 70 divided by 4.1379. And let's see how long approximately it would take to double 70 divided by 4.1379. It gives us 16.919, 2 years, right? 9, 2 and notice it gave us 16 years and 17 years as a as a possible choice and that's why you don't want around too early because you want to make sure that you get the right answer. Let's say it had been four and we just rounded to 4 70 divided by four. It would come out to 17.5. So 17 would still be the best answer there, right? If we just rounded it to four. So that's what I like to do is I don't like to round too early just in case you never know uh if rounding is going to Bite you in the butt in the end of the day. All right. So that's how we use our percentage change formula and the rule of 70 there. Let's go ahead and try the next one. You guys do a practice problem.

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