# Dr. Albert Bartlett on Compounding

Dr. Albert Bartlett: Arithmetic, Population and Energy

*[Transcribed from a speech]*

It's a great pleasure to be here, and to have a chance just to share with you some very simple ideas about the problems we're facing. Some of these problems are local, some are national, some are global.

They're all tied together. They're tied together by arithmetic, and the arithmetic isn't very difficult. What I hope to do is, I hope to be able to convince you that the greatest shortcoming of the human race is our inability to understand the exponential function.

Well, you say, what's the exponential function?

This is a mathematical function that you'd write down if you're going to describe the size of anything that was growing steadily. If you had something growing 5% per year, you'd write the exponential function to show how large that growing quantity was, year after year. And so we're talking about a situation where the time that's required for the growing quantity to increase by a fixed fraction is a constant: 5% per year, the 5% is a fixed fraction, the “per year” is a fixed length of time. So that's what we want to talk about: its just ordinary steady growth.

Well, if it takes a fixed length of time to grow 5%, it follows it takes a longer fixed length of time to grow 100%. That longer time's called the doubling time and we need to know how you calculate the doubling time. It's easy.

You just take the number 70, divide it by the percent growth per unit time and that gives you the doubling time. So our example of 5% per year, you divide the 5 into 70, you find that growing quantity will double in size every 14 years.

Well, you might ask, where did the 70 come from? The answer is that it's approximately 100 multiplied by the natural logarithm of two. If you wanted the time to triple, you'd use the natural logarithm of three. So it's all very logical. But you don't have to remember where it came from, just remember 70.

I wish we could get every person to make this mental calculation every time we see a percent growth rate of anything in a news story. For example, if you saw a story that said things had been growing 7% per year for several recent years, you wouldn't bat an eyelash. But when you see a headline that says crime has doubled in a decade, you say “My heavens, what's happening?”

OK, what is happening? 7% growth per year: divide the seven into 70, the doubling time is ten years. But notice, if you want to write a headline to get people's attention, you'd never write “Crime Growing 7% Per Year,” nobody would know what it means. Now, do you know what 7% means?

Let's take an example, another example from Colorado. The cost of an all-day lift ticket to ski at Vail has been growing about 7% per year ever since Vail first opened in 1963. At that time you paid $5 for an all-day lift ticket. What's the doubling time for 7% growth? Ten years. So what was the cost ten years later in 1973? (showing slides of rapidly increasing prices) Ten years later in 1983? Ten years later in 1993? What was it last year in 2003, and what do we have to look forward to? (shows "2003: $80; 2013: $160; 2023: $320; audience laughter)

This is what 7% means. Most people don't have a clue. And how is Vail doing? They're pretty much on schedule.

So let's look at a generic graph of something that’s growing steadily. After one doubling time, the growing quantity is up to twice its initial size. Two doubling times, it's up to four times its initial size. Then it goes to 8, 16, 32, 64, 128, 256, 512, in ten doubling times it's a thousand times larger than when it started. You can see if you try to make a graph of that on ordinary graph paper, the graph’s gonna go right through the ceiling.

Now let me give you an example to show the enormous numbers you can get with just a modest number of doublings.

Legend has it that the game of chess was invented by a mathematician who worked for a king. The king was very pleased. He said, “I want to reward you.” The mathematician said “My needs are modest. Please take my new chess board and on the first square, place one grain of wheat. On the next square, double the one to make two. On the next square, double the two to make four. Just keep doubling till you've doubled for every square, that will be an adequate payment.” We can guess the king thought, “This foolish man. I was ready to give him a real reward; all he asked for was just a few grains of wheat.”

But let's see what is involved in this. We know there are eight grains on the fourth square. I can get this number, eight, by multiplying three twos together. It's 2x2x2, it's one 2 less than the number of the square. Now that continues in each case. So on the last square, I’d find the number of grains by multiplying 63 twos together.

Now let’s look at the way the totals build up. When we add one grain on the first square, the total on the board is one. We add two grains, that makes a total of three. We put on four grains, now the total is seven. Seven is a grain less than eight, it's a grain less than three twos multiplied together. Fifteen is a grain less than four twos multiplied together. That continues in each case, so when we’re done, the total number of grains will be one grain less than the number I get multiplying 64 twos together. My question is, how much wheat is that?

You know, would that be a nice pile here in the room? Would it fill the building? Would it cover the county to a depth of two meters? How much wheat are we talking about?

The answer is, it's roughly 400 times the 1990 worldwide harvest of wheat. That could be more wheat than humans have harvested in the entire history of the earth. You say, “How did you get such a big number?” and the answer is, it was simple. We just started with one grain, but we let the number grow steadily till it had doubled a mere 63 times.

Now there's something else that’s very important: the growth in any doubling time is greater than the total of all the preceding growth. For example, when I put eight grains on the 4th square, the eight is larger than the total of seven that were already there. I put 32 grains on the 6th square. The 32 is larger than the total of 31 that were already there. Every time the growing quantity doubles, it takes more than all you’d used in all the proceeding growth.

Well, let’s translate that into the energy crisis. Here’s an ad from the year 1975. It asks the question “Could America run out of electricity?” America depends on electricity. Our need for electricity actually doubles every 10 or 12 years. That's an accurate reflection of a very long history of steady growth of the electric industry in this country, growth at a rate of around 7% per year, which gives you doubling every 10 years.

Now, with all that history of growth, they just expected the growth would go on, forever. Fortunately it stopped, not because anyone understood arithmetic, it stopped for other reasons. Well, let's ask “What if?” Suppose the growth had continued? Then we would see here the thing we just saw with the chess board. In the ten years following the appearance of this ad, in that decade, the amount of electrical energy we would have consumed in this country would have been greater than the total of all of the electrical energy we had ever consumed in the entire proceeding history of the steady growth of that industry in this country.

Now, did you realize that anything as completely acceptable as 7% growth per year could give such an incredible consequence? That in just ten years you'd use more than the total of all that had been used in all the proceeding growth?

Well, that's exactly what President Carter was referring to in his speech on energy. One of his statements was this: he said, “In each of those decades (1950s and 1960s) more oil was consumed than in all of mankind's previous history.” By itself that's a stunning statement.

Now you can understand it. The president was telling us the simple consequence of the arithmetic of 7% growth each year in world oil consumption, and that was the historic figure up until the 1970s.

There's another beautiful consequence of this arithmetic. If you take 70 years as a period of time—and note that that's roughly one human lifetime—then any percent growth continued steadily for 70 years gives you an overall increase by a factor that's very easy to calculate. For example, 4% per year for 70 years, you find the factor by multiplying four twos together, it's a factor of 16.

A few years ago, one of the newspapers of my hometown of Boulder, Colorado, quizzed the nine members of the Boulder City Council and asked them, “What rate of growth of Boulder's population do you think it would be good to have in the coming years?” Well, the nine members of the Boulder City council gave answers ranging from a low of 1% per year. Now, that happens to match the present rate of growth of the population of the United States. We are not at zero population growth. Right now, the number of Americans increases every year by over three million people. No member of the council said Boulder should grow less rapidly than the United States is growing.

Now, the highest answer any council member gave was 5% per year. You know, I felt compelled, I had to write him a letter and say, “Did you know that 5% per year for just 70 … ” I can remember when 70 years used to seem like an awful long time, it just doesn't seem so long now. (audience laughter). Well, that means Boulder's population would increase by a factor of 32. That is, where today we have one overloaded sewer treatment plant, in 70 years, we'd need 32 overloaded sewer treatment plants.

Now did you realize that anything as completely all-American as 5% growth per year could give such an incredible consequence in such a modest period of time? Our city council people have zero understanding of this very simple arithmetic.

Well, a few years ago, I had a class of non-science students. We were interested in problems of science and society. We spent a lot of time learning to use semi-logarithmic graph paper. It's printed in such a way that these equal intervals on the vertical scale each represent an increase by a factor of 10. So you go from one thousand to ten thousand to a hundred thousand, and the reason you use this special paper is that on this paper, a straight line represents steady growth.

Now, we worked a lot of examples. I said to the students, “Let’s talk about inflation, let’s talk about 7% per year.” It wasn't this high when we did this, it's been higher since then, fortunately it's lower now. And I said to the students, as I can say to you, you have roughly sixty years life expectancy ahead of you. Let’s see what some common things will cost if we have 60 years of 7% annual inflation.

The students found that a 55-cent gallon of gasoline will cost $35.20; $2.50 for a movie will be $160; the $15 sack of groceries my mother used to buy for a dollar and a quarter, that will be $960; a $100 suit of clothes, $6,400; a $4000 automobile will cost a quarter of a million dollars; and a $45,000 home will cost nearly 3 million dollars.

Well, I gave the students these data (shows overhead). These came from a Blue Cross, Blue Shield ad. The ad appeared in Newsweek magazine and the ad gave these figures to show the cost escalation of gall bladder surgery in the years since 1950, when that surgery cost $361. I said, “Make a semi logarithmic plot, let’s see what's happening.” The students found that the first four points lined up on a straight line whose slope indicated inflation of about 6% per year, but the fourth, fifth, and sixth were on a steeper line, almost 10% inflation per year. Well, then I said to the students, “Run that steeper line on out to the year 2000, let’s get an idea of what gall bladder surgery might cost,” and this was, 2000 was four years ago—the answer is $25,000. The lesson there is awfully clear: if you're thinking about gall bladder surgery, do it now. (audience laughter)

In the summer of 1986, the news reports indicated that the world population had reached the number of five billion people growing at the rate of 1.7% per year. Well, your reaction to 1.7% might be to say “Well, that's so small, nothing bad could ever happen at 1.7% per year.” So you calculate the doubling time, you find it’s only 41 years. Now, that was back in 1986; more recently in 1999, we read that the world population had grown from five billion to six billion . The good news is that the growth rate had dropped from 1.7% to 1.3% per year. The bad news is that in spite of the drop in the growth rate, the world population today is increasing by about 75 million additional people every year.

Now, if this current modest 1.3% per year could continue, the world population would grow to a density of one person per square meter on the dry land surface of the earth in just 780 years, and the mass of people would equal the mass of the earth in just 2400 years. Well, we can smile at those, we know they couldn't happen. This one make for a cute cartoon; the caption says, “Excuse me sir, but I am prepared to make you a rather attractive offer for your square.”

There's a very profound lesson in that cartoon. The lesson is that zero population growth is going to happen. Now, we can debate whether we like zero population growth or don't like it, it’s going to happen. Whether we debate it or not, whether we like it or not, it’s absolutely certain. People could never live at that density on the dry land surface of the earth. Therefore, today’s high birth rates will drop; today’s low death rates will rise till they have exactly the same numerical value. That will certainly be in a time short compared to 780 years. So maybe you're wondering then, what options are available if we wanted to address the problem.

In the left hand column, I’ve listed some of those things that we should encourage if we want to raise the rate of growth of population and in so doing, make the problem worse. Just look at the list. Everything in the list is as sacred as motherhood. There's immigration, medicine, public health, sanitation. These are all devoted to the humane goals of lowering the death rate and that’s very important to me, if it’s my death they’re lowering. But then I’ve got to realise that anything that just lowers the death rate makes the population problem worse.

There’s peace, law and order; scientific agriculture has lowered the death rate due to famine—that just makes the population problem worse. It’s widely reported that the 55 mph speed limit saved thousands of lives—that just makes the population problem worse. Clean air makes it worse.

Now, in this column are some of the things we should encourage if we want to lower the rate of growth of population and in so doing, help solve the population problem. Well, there’s abstention, contraception, abortion, small families, stop immigration, disease, war, murder, famine, accidents. Now, smoking clearly raises the death rate; well, that helps solve the problem.

Remember our conclusion from the cartoon of one person per square meter; we concluded that zero population growth is going to happen. Let’s state that conclusion in other terms and say it’s obvious nature is going to choose from the right hand list and we don't have to do anything—except be prepared to live with whatever nature chooses from that right hand list. Or we can exercise the one option that’s open to us, and that option is to choose first from the right hand list. We’ve got to find something here we can go out and campaign for. Anyone here for promoting disease? (audience laughter)

We now have the capability of incredible war; would you like more murder, more famine, more accidents? Well, here we can see the human dilemma—everything we regard as good makes the population problem worse, everything we regard as bad helps solve the problem. There is a dilemma if ever there was one. The one remaining question is education: does it go in the left hand column or the right hand column? I’d have to say thus far in this country it’s been in the left hand column—it's done very little to reduce ignorance of the problem.

So where do we start? Well, let’s start in Boulder, Colorado. Here’s my home town. There’s the 1950 census figure, 1960, 1970—in that period of twenty years, the average growth rate of Boulder’s population was 6% per year. With big efforts, we’ve been able to slow the growth somewhat. There’s the 2000 census figure. I’d like to ask people: let’s start with that 2000 figure, go another 70 years—one human life time—and ask: what rate of growth would we need in Boulder’s population in the next 70 years so that at the end of 70 years, the population of Boulder would equal today’s population of your choice of major American cities?

Boulder in 70 years could be as big as Boston is today if we just grew 2.58% per year. Now, if we thought Detroit was a better model, we’ll have to shoot for 31?4% per year. Remember the historic figure on the preceding slide, 6% per year? If that could continue for one lifetime, the population of Boulder would be larger than the population of Los Angeles. Well, I’ll just tell you, you couldn’t put the population of Los Angles in the Boulder valley. Therefore it’s obvious, Boulder’s population growth is going to stop and the only question is, will we be able to stop it while there is still some open space, or will we wait until it’s wall-to-wall people and we’re all choking to death?

Now, every once in a while somebody says to me, “But you know, a bigger city might be a better city,” and I have to say, “Wait a minute, we’ve done that experiment!” We don’t need to wonder what will be the effect of growth on Boulder because Boulder tomorrow can be seen in Los Angeles today. And for the price of an airplane ticket, we can step 70 years into the future and see exactly what it’s like. What is it like? There’s an interesting headline from Los Angeles. (“…carcinogens in air…”) Maybe that has something to do with this headline from Los Angeles. (“Smog kills 1,600 annually…”)

So how are we doing in Colorado? Well, we’re the growth capital of the USA and proud of it. The Rocky Mountain News tells us to expect another million people in the Front Range in the next 20 years, and what are the consequences of all this? (“Denver's traffic…3rd worst in US…”) These are totally predictable, there are no surprises here, we know exactly what happens when you crowd more people into an area.

Well, as you can imagine, growth control is very controversial, and I treasure the letter from which these quotations are taken. Now, this letter was written to me by a leading citizen of our community. He’s a leading proponent of “controlled growth.” “Controlled growth” just means “growth.” This man writes, “I take no exception to your arguments regarding exponential growth.” “I don't believe the exponential argument is valid at the local level.”

So you see, arithmetic doesn't hold in Boulder. (audience laughs) I have to admit, that man has a degree from the University of Colorado. It’s not a degree in mathematics, in science, or in engineering. All right, let’s look now at what happens when we have this kind of steady growth in a finite environment.

Bacteria grow by doubling. One bacterium divides to become two, the two divide to become 4, the 4 become 8, 16 and so on. Suppose we had bacteria that doubled in number this way every minute. Suppose we put one of these bacteria into an empty bottle at 11:00 in the morning, and then observe that the bottle is full at 12:00 noon. There's our case of just ordinary steady growth: it has a doubling time of one minute, it’s in the finite environment of one bottle.

I want to ask you three questions. Number one: at what time was the bottle half full? Well, would you believe 11:59, one minute before 12:00? Because they double in number every minute.

And the second question: if you were an average bacterium in that bottle, at what time would you first realize you were running of space? Well, let’s just look at the last minutes in the bottle. At 12:00 noon, it’s full; one minute before, it’s half full; 2 minutes before, it’s a quarter full; then an 1?8th; then a 1?16th. Let me ask you, at 5 minutes before 12:00, when the bottle is only 3% full and is 97% open space just yearning for development, how many of you would realize there’s a problem?

Now, in the ongoing controversy over growth in Boulder, someone wrote to the newspaper some years ago and said “Look, there’s no problem with population growth in Boulder, because,” the writer said, “we have fifteen times as much open space as we've already used.” So let me ask you, what time was it in Boulder when the open space was fifteen times the amount of space we’d already used? The answer is, it was four minutes before 12:00 in Boulder Valley. Well, suppose that at 2 minutes before 12:00, some of the bacteria realize they’re running out of space, so they launch a great search for new bottles. They search offshore on the outer continental shelf and in the overthrust belt and in the Arctic, and they find three new bottles. Now that’s an incredible discovery, that’s three times the total amount of resource they ever knew about before. They now have four bottles, before their discovery, there was only one. Now surely this will give them a sustainable society, won’t it?

You know what the third question is: how long can the growth continue as a result of this magnificent discovery? Well, look at the score: at 12:00 noon, one bottle is filled, there are three to go; 12:01, two bottles are filled, there are two to go; and at 12:02, all four are filled and that’s the end of the line.

Now, you don't need any more arithmetic than this to evaluate the absolutely contradictory statements that we’ve all heard and read from experts who tell us in one breath we can go on increasing our rates of consumption of fossil fuels, in the next breath they say “Don't worry, we will always be able to make the discoveries of new resources that we need to meet the requirements of that growth.”

Well, a few years ago in Washington, our energy secretary observed that in the energy crisis, “we have a classic case of exponential growth against a finite source.” So let's look now at some of these finite sources. We turn to the work of the late Dr. M. King Hubbert. He’s plotted here a semi-logarithmic graph of world oil production. You can see the lines have been approximately straight for about 100 years, clear up here to 1970, average growth rate very close to 7% per year. So it’s logical to ask, well, how much longer could that 7% growth continue? That’s answered by the numbers in this table (shows slide). The numbers in the top line tell us that in the year 1973, world oil production was 20 billion barrels; the total production in all of history, 300 billion; the remaining reserves, 1700 billion.

Now, those are data. The rest of this table is just calculated out assuming the historic 7% growth continued in the years following 1973 exactly as it had been for the proceeding 100 years.

Now, in fact the growth stopped; it stopped because OPEC raised their oil prices. So we’re asking here, what if? Suppose we just decided to stay on that 7% growth curve? Let’s go back to 1981. By 1981 on the 7% curve, the total usage in all of history would add up to 500 billion barrels; the remaining reserves, 1500 billion. At that point, the remaining reserves are three times the total of everything we’d used in all of history. That’s an enormous reserve, but what time is it when the remaining reserve is three times the total of all you’ve used in all of history? The answer is, it’s two minutes before 12:00.

We know for 7% growth, the doubling time is 10 years. We go from 1981 to 1991. By 1991 on the 7% curve, the total usage in all of history would add up to 1000 billion barrels; there would be 1000 billion left. At that point, the remaining oil would be equal in quantity to the total of everything we’d used in the entire history of the oil industry on this earth, 130 years of oil consumption. You'd say, “That’s an enormous reserve.” But what time is it when the remaining reserve is equal to all you’ve used in all of history? And the answer is, it’s one minute before 12:00. So we go one more decade to the turn of the century—that’s like right now—that’s when 7% would finish using up the oil reserves of the earth.

So let's look at this in a very nice graphical way. Suppose the area of this tiny rectangle represents all the oil we used on this earth before 1940; then in the decade of the 40s, we used this much (uncovering part of chart): that's equal to all that had been used in all of history. In the decade of the 50s, we used this much (uncovering more of chart) : that's equal to all that had been used in all of history. In the decade of the 60s, we used this much (uncovering more of chart): again that's equal to the total of all the proceeding usage. Here we see graphically what President Carter told us. Now, if that 7% growth had continued through the 70s. 80s, and 90s, there's what we’d need (uncovering rest of chart) . But that's all the oil there is.

Now, there’s a widely held belief that if you throw enough money at holes in the ground, oil is sure to come up. Well, there will be discoveries in new oil; there may be major discoveries. But look: we would have to discover this much new oil if we would have that 7% growth continue ten more years.

Ask yourself: what do you think is the chance that oil discovered after the close of our meeting today will be in an amount equal to the total of all we’ve known about in all of history? And then realize if all that new oil could be found, that would be sufficient to let the historic 7% growth continue ten more years.

Well, it’s interesting to see what the experts say. Here’s from an interview in Time magazine, an interview with one of the most widely quoted oil experts in all of Texas. They asked him, “But haven’t many of our bigger fields been drilled nearly dry?” And he responds, saying “There’s still as much oil to be found in the US as has ever been produced.” Now, lets assume he’s right. What time is it? And the answer: one minute before 12:00. I’ve read several things this guy’s written; I don't think he has any understanding of this very simple arithmetic.

Well, in the energy crisis about thirty years ago, we saw ads such as this (shows slide). This is from the American Electric Power Company. It’s a bit reassuring, sort of saying, now, don’t worry too much, because “we’re sitting on half of the world’s known supply of coal, enough for over 500 years.” Well, where did that “500 year” figure come from? It may have had its origin in this report to the committee on Interior and Insular Affairs of the United States Senate, because in that report we find this sentence: “At current levels of output and recovery, these American coal reserves can be expected to last more than 500 years.”

There is one of the most dangerous statements in the literature. It’s dangerous because it’s true. It isn’t the truth that makes it dangerous, the danger lies in the fact that people take the sentence apart: they just say coal will last 500 years. They forget the caveat with which the sentence started. Now, what were those opening words? “At current levels.” What does that mean? That means if—and only if—we maintain zero growth of coal production.

So let’s look at a few numbers. We go to the Annual Energy Review, published by the Department of Energy. They give this (pointing) as the coal demonstrated reserve base in the United States. It has a footnote that says “about half the demonstrated reserve base… is estimated to be recoverable.” You cannot recover —get out of the ground and use—100% of the coal that’s there. So this number then, is ½ of this number (pointing). We’ll come back to those in just a moment. The report also tells us that in 1971, we were mining coal at this rate, twenty years later at this rate (pointing). Put those numbers together, the average growth rate of coal production in that twenty years: 2.86% per year. And so we have to ask, well, how long would a resource last if you have steady growth in the rate of consumption until the last bit of it is used?

I’ll show you the equation here for the expiration time. I’ll tell you it takes first year college calculus to derive that equation, so it can’t be very difficult. You know, I have a feeling there must be dozens of people in this country who’ve had first year college calculus, but let me suggest, I think that equation is probably the best-kept scientific secret of the century!

Now, let me show you why. If you use that equation to calculate the life expectancy of the reserve base, or of the ½ they think is recoverable, for different steady rates of growth, you find if the growth rate is zero, the small estimate would go about 240 years and the large one would go close to 500 years. So that report to the Congress was correct. But look what we get if we plug in steady growth. Back in the 1960s, it was our national goal to achieve growth of coal production up around 8% per year. If you could achieve that and continue it, coal would last between 37 and 46 years. President Carter cut that goal roughly in half, hoping to reach 4% per year. If that could continue, coal would last between 59 and 75 years. Here’s that 2.86%, the average for the recent period of twenty years. If that could continue, coal would last between 72 and 94 years. That’s within the life expectancy of children born today.

The only way you are going to get anywhere near this widely quoted 500 year figure, is to be able to do simultaneously two highly improbable things: number one, you’ve got to figure out how to use 100% of the coal that is in the ground; number two, y