Understanding Exponential Growth: The Rule of 70 and Compound Interest | Lecture notes Economics

Productivity and Growth

The Elements of Productivity and Growth

The Rule of 70, Compounding, and Growth Page 1 of 2

Now, listen very closely to this; if you start at age 19 and you put 2,000 dollars a year into a savings account at 10%

interest from age 19 until age 26 and no more money into the savings account for the rest of your life, at age 65, when

you retire, you will have about 1 million dollars. On the other hand, if you don’t saving money until age 27, you could

put the same 2,000 dollars a year into that savings account every year until you're 65, and only wind up with about

800,000 dollars. What’s going on?

The answer is the power of compound interest. Compound interest is something that a lot of us understand, since our

parents take us to the bank when we’re little and open a savings account and we think about how interest

accumulates and how sums of money grow. In general, however, we can think about interest as a representative

growth rate. There are lots of other things that grow at compound rates and, in this discussion, we’re going to look at

how compounding causes sums to grow at an exponential rate and what the implications are for economic

development.

Let’s take a simple numerical example. If you have an interest rate or a growth rate of r, then you can calculate next

year’s sum as a function of this year’s sum and the interest rate. Say you put 10,000 dollars into the bank today. If

the interest rate is 10%, then 1 plus 10% equals 11,000, which is what your savings account will be worth next year if

you leave you 10,000 dollars in at 10% interest. If you let that 11,000 dollars stay in the bank for another year at 10%

interest, you’ll have 12,100 at the end of the second year. 12,100 is equal to the original 10,000 times 1 plus 10%,

raised to the second power. You raise 1 plus 10% to the second power, because the money stayed in the bank for 2

years. We call this our compounding factor; 1 plus 10%, or 1.1, raise it to the power of the number of years involved

in the story and you’ll wind up with the amount that’s left at the end of that number of years.

Now, suppose you want to know how long it will take for your money to double. If I put money in the bank and I leave

it there a number of years, how many years will have to pass before I have twice as much money as I start with? The

answer can be summarized by what we call the Rule of 70. Now, I’m about to say the word “logarithm”, so please,

don’t freak out or tune out when I do that. Just stay with me and you’ll get a payoff. And you don’t even have to

understand logarithms to follow this explanation, because you can just apply the rule at the end. But some of you are

interested in where it comes from and, in order to understand where it comes from, you really have to think about

logarithms. So, that having been said, let’s go forward.

Here’s our compounding term: 1 plus the growth rate. So 1 plus the growth rate raised to the n-th power tells you how

much money you’ve got in the bank. And you wonder, “If I put 1 dollar in at the beginning, how long would it take me

to have 2 dollars?” Or, in general, how may years do you have to leave it in the bank until the compounding term is

2? How long does it take for your sum of money to double? Well, if you take the logarithm of both sides, then you get

this: the n comes down as the coefficient in front of the logarithm of what’s left, 1 plus r equals the logarithm of 2. And

I’m using ln, these are natural logarithms. Well, here’s a little trick that will help you make this simpler. If you have 1

plus a small percentage, then the logarithm of 1 plus a small percentage is approximately equal to that small

percentage. So if this 10%, the logarithm of 1 plus 10% is about equal to 10%. So what we’ve done here is we take n

and leave it on the left-hand side of the equation and we divide by sides by the logarithm of 1 plus r, which is

approximately equal to r. And here you have it, what mathematicians call the Rule of 70, because the natural

logarithm of the number 2 is about equal to .70. So if you take an interest rate of 10%, which we’ve been writing as

0.10, and divide that into 0.70, you can see that it takes 7 years for your money to double if the interest rate is 10%.

In general, divide any interest rate into the number 70 and you find out how many years it would take for your money

to double at that interest rate. So if the interest rate is 7%, it takes 10 years for your money to double. If the interest

rate is 10%, it takes 7 years for your money to double. If the interest rate is 35%, it only takes about 2 years for your

money to double. This is what the Rule of 70 says. It tells you how long it takes for a sum to double at a given rate of

growth or a give interest rate.

Here’s a graphical representation of the phenomenon of exponential growth. A is the sum that you start with. 1 plus r

represents the rate of growth, with r being the interest rate or the growth rate, and 1 plus r being the compounding

term. n is the number of years of compounding. So here we’ve graphed, starting with A, the purple line has this slope

that shows the amount of money increasing at an increasing rate over time. That is, if we start with 10,000 dollars,

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Productivity and Growth The Elements of Productivity and Growth

The Rule of 70, Compounding, and Growth Page 1 of 2

Now, listen very closely to this; if you start at age 19 and you put 2,000 dollars a year into a savings account at 10% interest from age 19 until age 26 and no more money into the savings account for the rest of your life, at age 65, when you retire, you will have about 1 million dollars. On the other hand, if you don’t saving money until age 27, you could put the same 2,000 dollars a year into that savings account every year until you're 65, and only wind up with about 800,000 dollars. What’s going on?

The answer is the power of compound interest. Compound interest is something that a lot of us understand, since our parents take us to the bank when we’re little and open a savings account and we think about how interest accumulates and how sums of money grow. In general, however, we can think about interest as a representative growth rate. There are lots of other things that grow at compound rates and, in this discussion, we’re going to look at how compounding causes sums to grow at an exponential rate and what the implications are for economic development.

Let’s take a simple numerical example. If you have an interest rate or a growth rate of r, then you can calculate next year’s sum as a function of this year’s sum and the interest rate. Say you put 10,000 dollars into the bank today. If the interest rate is 10%, then 1 plus 10% equals 11,000, which is what your savings account will be worth next year if you leave you 10,000 dollars in at 10% interest. If you let that 11,000 dollars stay in the bank for another year at 10% interest, you’ll have 12,100 at the end of the second year. 12,100 is equal to the original 10,000 times 1 plus 10%, raised to the second power. You raise 1 plus 10% to the second power, because the money stayed in the bank for 2 years. We call this our compounding factor; 1 plus 10%, or 1.1, raise it to the power of the number of years involved in the story and you’ll wind up with the amount that’s left at the end of that number of years.

Now, suppose you want to know how long it will take for your money to double. If I put money in the bank and I leave it there a number of years, how many years will have to pass before I have twice as much money as I start with? The answer can be summarized by what we call the Rule of 70. Now, I’m about to say the word “logarithm”, so please, don’t freak out or tune out when I do that. Just stay with me and you’ll get a payoff. And you don’t even have to understand logarithms to follow this explanation, because you can just apply the rule at the end. But some of you are interested in where it comes from and, in order to understand where it comes from, you really have to think about logarithms. So, that having been said, let’s go forward.

Here’s our compounding term: 1 plus the growth rate. So 1 plus the growth rate raised to the n-th power tells you how much money you’ve got in the bank. And you wonder, “If I put 1 dollar in at the beginning, how long would it take me to have 2 dollars?” Or, in general, how may years do you have to leave it in the bank until the compounding term is 2? How long does it take for your sum of money to double? Well, if you take the logarithm of both sides, then you get this: the n comes down as the coefficient in front of the logarithm of what’s left, 1 plus r equals the logarithm of 2. And I’m using ln, these are natural logarithms. Well, here’s a little trick that will help you make this simpler. If you have 1 plus a small percentage, then the logarithm of 1 plus a small percentage is approximately equal to that small percentage. So if this 10%, the logarithm of 1 plus 10% is about equal to 10%. So what we’ve done here is we take n and leave it on the left-hand side of the equation and we divide by sides by the logarithm of 1 plus r, which is approximately equal to r. And here you have it, what mathematicians call the Rule of 70, because the natural logarithm of the number 2 is about equal to .70. So if you take an interest rate of 10%, which we’ve been writing as 0.10, and divide that into 0.70, you can see that it takes 7 years for your money to double if the interest rate is 10%. In general, divide any interest rate into the number 70 and you find out how many years it would take for your money to double at that interest rate. So if the interest rate is 7%, it takes 10 years for your money to double. If the interest rate is 10%, it takes 7 years for your money to double. If the interest rate is 35%, it only takes about 2 years for your money to double. This is what the Rule of 70 says. It tells you how long it takes for a sum to double at a given rate of growth or a give interest rate.

Here’s a graphical representation of the phenomenon of exponential growth. A is the sum that you start with. 1 plus r represents the rate of growth, with r being the interest rate or the growth rate, and 1 plus r being the compounding term. n is the number of years of compounding. So here we’ve graphed, starting with A, the purple line has this slope that shows the amount of money increasing at an increasing rate over time. That is, if we start with 10,000 dollars,

Productivity and Growth The Elements of Productivity and Growth

The Rule of 70, Compounding, and Growth Page 2 of 2

then after 1 year at 10% interest, we have 11,000, after 2 years 12,100, after 3 years 13,310, and so forth. Eventually, the rate of money begins to increase quite rapidly, due to the phenomenon of compound growth. Now, let’s suppose we started with the same amount of money, 10,000 dollars, but instead of growing at a 10% rate annually, we grew at a rate of 20%. What that would do is give us a curve that is steeper at very point. We’d have exponential growth, but that exponential growth would take off faster and the curve would become steeper faster. So here we have now a formula that could be represented A 0 , the original amount, times 1 plus r 1 , a faster rate of growth, raised to the n-th power.

My point is this: if you are a developing country and you start from a lower base, but you’ve got a faster growth rate, eventually you can catch up with a larger country that’s growing at a slower rate. Here’s how you see it: let’s now start with a lower base, a lower initial amount. We’ll call this A 1 , and let’s start right here. But let’s suppose that we have this faster rate of growth, r 1 , so we have a curve then that looks like this. And here you have A 1 , the lower base, times 1 plus r 1 , the faster growth rate, raised to the n-th power. If we start at time zero, the beginning of our story, maybe this is the United States in 1920 and this is Japan in the same year. Japan has a smaller economy and much lower per capita income. However, Japan’s economy is growing more rapidly and, eventually at some point, because Japan’s growth rate is so much faster, the per capital income in Japan will catch up. That is, the average Japanese will be as well off as the average American after a period of time, simply because Japan’s economy is growing faster. And the time at which this happens we might call n*. The point is this: if you have a country that has a lower base or a lower starting rate, but a faster rate of growth, then after enough time has passed, the country that is growing faster will surpass the country that is growing slower, regardless of what their initial starting positions are. This is the magic of compound growth; that is, the exponential, the doubling and tripling and raising to higher powers eventually offsets any gap with which these two countries start. Whichever country is growing faster will, in the long-run, be richer.

Compound interest is a fastening thing and it kind of attracts us, because of its odd stories, like the million dollars that comes from 10,000 that you put away in your 20’s. The Rule of 70 helps you calculate how fast a sum of money is doubling. And this set of curves reminds you that, even if a country starts with a lower base, if it has a faster growth rate, eventually it will win the race.

Understanding Exponential Growth: The Rule of 70 and Compound Interest, Lecture notes of Economics

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