Chasing the Exponent

Today is fun with LaTeX!

No, not rubber gloves, you naughty person. LaTeX, the mathematics formatting program.

But we can’t do math without a math problem, right? So let’s ask a math question: what does 2% annual growth of the GDP (Gross Domestic Product) mean? Is this a good growth rate, or is it bad? Is it better than 1% growth? Is it worse than 5% growth?

As it turns out, all of these rates are catastrophic. But not in the way most would think.

Two percent growth means that at the end of the year, you have two percent more than you started with, or 102% of what you started with. “Percent” stands for “per centum” which is Latin for “in every one hundred.” So one percent is “one in every one hundred,” while two percent is “two in every one hundred.” So:

102\% = \frac{102}{100} = 1.02

Percentage growth is also called “proportional” growth, because the growth is a proportion of what you started with. You can calculate it by simply multiplying the original amount by the proportion of growth, as follows:

GDP_1 = GDP_0 \times 1.02

If you keep growing at two percent each year, then at the end of the second year you have two percent more than you had at the end of the first year, which was already two percent more than you started with. So it looks like this:

GDP_2 = GDP_1 \times 1.02 = (GDP_0 \times 1.02) \times 1.02 = GDP_0 \times (1.02 \times 1.02)

At the end of the third year, you have:

GDP_3 = ((GDP_0 \times 1.02) \times 1.02) \times 1.02 = GDP_0 \times (1.02 \times 1.02 \times 1.02)

At the end of ten years, you have:

GDP_{10} = GDP_0 \times (1.02 \times 1.02 \times \cdots \times 1.02) = GDP_0 \times (1.02)^{10}

This notation, the “exponential” notation, indicates that you multiply 1.02 times itself ten times. This is why proportional growth is also known as exponential growth.

You can do this pretty easily on a calculator by entering 1.02, then hitting the times button twice. Now, every time you hit the equals button, it multiplies the result by 1.02. The result on my little four-banger after hitting the equals button nine times is 1.21899…, or about 1.22 (note that you hit equals once less than the number of years you want, because the first time you hit equals, you get the total at the end of the second year.) That means after ten years, this modest little growth of two percent every year has become twenty-two percent total growth.

We can play with this a little more to bring out an interesting feature of exponential growth. First, I need to point out a property of logarithms, which I won’t bother to prove, as follows:

\ln(a^b) = b \times \ln(a)

This turns out to be very useful in this next bit of reasoning.

IF

x = (1.02)^{n}

THEN

\ln(x) = \ln((1.02)^{n}) = n \times \ln(1.02) = n \times \ln(1.02) \times \frac{\ln(2)}{\ln(2)} \\ \medskip \hspace*{0.6in} = n \times \frac{\ln(1.02)}{\ln(2)} \times \ln(2) = \left( \frac{\ln(1.02)}{\ln(2)} \times n \right ) \times \ln(2) \\ \medskip \hspace*{0.6in} = \ln \left ( 2^{\frac{\ln(1.02)}{\ln(2)} n} \right )

THEREFORE

x = (1.02)^n = (2)^{\frac{\ln(1.02)}{\ln(2)} n}

What does this nonsense mean? It means that raising a small percentage growth (like two percent) to some exponent is exactly the same as raising the value two to some different exponent. And two means doubling. In other words:

GDP \times (1.02)^{n} = GDP \times (2)^{\frac{ln(1.02)}{\ln(2)}n}

The magic value of ln(2)/ln(1.02) is the doubling time for two percent annual growth. When n (in years) reaches the value of ln(2)/ln(1.02) = 35.003, the GDP will double:

GDP_{35} = GDP_0 \times (2)^{\frac{\ln(1.02)}{\ln(2)} \times \frac{\ln(2)}{\ln(1.02)}} = GDP_0 \times (2)^{1} = GDP_0 \times 2

A two percent annual growth means the GDP doubles every thirty-five years. In seventy years, it will double again, and will be four times its original value. In a little over a century, it will be eight times its original value.

proportional growth = exponential growth = doubling growth

This is simply mathematics.

Let’s do a little more advanced exploration of this exponential growth concept. There’s a common technique used in math called functional expansion. It lets you express a complicated function as a collection of simpler functions, typically a sum of simple functions. In particular, let’s do a conversion as follows, to make this a little easier:

x = (1.02)^{n} = (2)^{\frac{\ln(1.02)}{\ln(2)}n} = e^{\frac{\ln(1.02)}{\ln(e)}n} = e^{\ln(1.02)n} = e^{kn}

The letter e represents a magic number, like π. It has a value approximately equal to 2.71828…. It’s a useful number, like π, which I won’t go into right now. This is the so-called standard exponential function, and k is just a constant number based on the growth rate of two percent, while n is the number of years. Let’s do a functional expansion of the standard exponential function using something called a Taylor expansion:

e^{kn} = \sum_{i=0}^{\infty }\frac{(kn)^i}{i!} = 1 + \frac{(kn)}{1} + \frac{(kn)^2}{2} + \frac{(kn)^3}{6} + \frac{(kn)^4}{24} + \cdots

Why is this important?

It has to do with how fast things grow. We refer to the speed of growth as the order of the equation, such as O(logN) or O(N) or O(N^2). It’s traditional to use a capital N in this expression, but it means the same as the small n in this case. The reason this is important is that as n (or N) gets larger — as time passes — the maximum order of the growth starts to dominate everything. For instance, some quantity might grow like this:

Growth = a + bn + cn^2 + dn^3

where a, b, c, and d are constant values, and n is the number of years. Even if a, b, and c are very large, and d is very small, after enough time has passed, the dn3 starts to take over. This is because ngrows much faster than n2 or n. So we would call this O(N^3) growth, or cubic growth, even though there are some other things going on.

In fact, a, b, and c could all be negative numbers: they could represent shrinkage, rather than growth. But if d is positive, no matter how small, this equation will eventually show cubic growth, and all of the shrinkage will become negligible.

Going back to our original economic question, let’s say we have a fixed amount of wealth that we keep in a cave, like a dragon’s hoard. This is proportional to the first term in our standard exponential equation:

1

That is, it’s a fixed constant. It may not be one, but it never grows, and it never shrinks, no matter how many years we keep it in the cave.

Now let’s assume we decide to walk around and pick up things we find along the path — shiny rocks, seashells, fruit that has fallen from the trees, that sort of thing. We can only walk so far in a given day, and can only pick up things that are within reach of whatever path we walk, so we can only add to our store of wealth a certain fixed amount — on the average — every year. That’s proportional to the second term in our equation:

\frac{(kn)}{1}

Now let’s assume that we begin to have children, and that we start expanding across the surface of the earth at a steady rate from our original village. Let’s say we raise just exactly enough children to keep the population density constant as we take up more and more land. Every person walks about the same distance each year, but every year we have more people to cover more area, so our wealth increases as the size of the area. Areas are proportional to the square of the distance from the center, so this is proportional to the third term in our equation:

\frac{(kn)^2}{2}

Obviously, we’re going to face a problem once we’ve covered the surface of the earth and can’t expand any more. But by then, they tell us, we’ll have space travel! So now we can expand into the third dimension, sending out space ships that move at a steady rate away from our original Earth. Every year we have more people colonizing more worlds and walking new paths within the volume of space we fill, which is proportional to the cube of the distance from the center. Thus, this is proportional to the fourth term in our equation:

\frac{(kn)^3}{6}

I think the problem is pretty clear. To sustain exponential growth of wealth, we still have an infinite number of terms left to cover, and I’m pretty well out of ideas as to how we’d get to even the fifth term, let alone the thirty-seventh. After the thirty-seventh, there is still an infinite number of terms left.

Exponential growth is O(expN), which is faster — much faster — than populating outer space at the speed of light, which is only a paltry O(N^3).

There’s one more important characteristic of exponential growth. Chris Martenson over on his Crash Course in Peak Economics site gave a wonderful visual image of how exponential growth behaves, involving filling Fenway Stadium in Boston with water. To give a brief reprise, you find yourself handcuffed to the railing in the top row of seats at Fenway Stadium. A water main breaks right underneath the pitcher’s mound. It starts out leaking just one drop of water per minute, but every minute, that rate doubles: two drops, four drops, eight drops…. The two questions are:

  1. How long does it take Fenway Stadium to fill completely with water, drowning you?
  2. From the moment you notice that the stadium is beginning to flood, how much time do you have to get out of the handcuffs and escape to safety?

The answer (as I recall, and I haven’t checked his numbers) is that it takes something like 24 hours for the stadium to fill, but you won’t even be able to see the water from the top row until the last forty-five minutes.

Throughout his course he refers to the “hockey stick” shape of an exponential curve, meaning that it creeps along slowly and then — BANG — shoots through the roof. As it turns out, this is mathematically not quite correct, but it does give voice to something important about exponential curves. It has to do with how these higher-order terms start to dominate the equations. This is easier to show than to describe.

exponents1

Here is an exponential curve seen from time 0.0 to time 1.0. At time zero, the function has a value of 1.00, and at time one, it has a value of about 2.72. In this range, we see only a gentle bend to an otherwise straight line.

 

exponents5

Here’s that same curve seen from time 0.0 to time 5.0. At time zero, it has a value of 1.00, as before, while at time five, it has a value of about 148. Notice the small box showing the curve from 0.0 to 1.0 from the previous chart, and how sharply the curve bends upward after that.

 

exponents10Here’s that same curve seen from time 0.0 to time 10.0. At time zero, it has a value of 1.00, as before, while at time ten, it has a value of around 22,000. Notice again how much sharper the curve becomes after time 5.0.  The higher-order terms are beginning to dominate strongly now, and the curve is getting very steep indeed.

The point here is that exponential curves start out innocently. They look a lot like linear growth, as we see in the first chart. But as time passes, they accelerate. And accelerate. And accelerate. The acceleration never stops, and the curve grows impossibly steep.

But this idea of a “hockey stick” shape actually obscures the more important feature of exponential curves, illustrated below:

exponentsSim

{y = \left ( e^{x+0} - e^0\right ) \left( \frac{e^1 - e^0}{e^1 - e^0}\right ) + 0.00} (red)

{y = \left ( e^{x+1} - e^1\right ) \left (\frac{e^1 - e^0}{e^2 - e^1} \right )+ 0.01} (blue)

{y = \left ( e^{x+2} - e^2 \right ) \left ( \frac{e^1 - e^0}{e^3 - e^2} \right ) + 0.02} (green)

So what is this all about? Well, let’s take it in pieces.

First, note that the X-axis runs from zero to one for all three curves.

So the first equation is simply ex from 0.0 to 1.0 — this is the very first chart we saw above, the one that looks almost like a straight line. We’ve subtracted its starting value from the result to shift it down so it starts at (0,0). We’ve then multiplied that by a constant which, if you look at it closely, is equal to one. Then we’ve added zero to it. Whoopee.

The second equation is a bit more interesting. The function is still ex, but because we’re adding one to x, this is ex  seen from 1.0 to 2.0: it’s where we see the curve getting steeper in the second graph. In this case, we again subtract its starting value (e1), divide it by its total height (e2 – e1), and then multiply by the total height from time 0.0 to 1.0 (e– e0), so that this new function fits on the y-axis of the old graph. Finally, we’ve added a smidge (0.01) to push it up a bit, so we can see the curve.

Just to make sure, we did this one more time, this time looking at ex from 2.0 to 3.0, again scaling it so it fits on the same chart with the original curve, and adding a smidge more to separate the lines.

The curves are identical in shape.

What does this mean? It means that over any fixed range of time (in this case, one unit of time), regardless of where it is located in the curve, the curve is “scale-invariant” — it is exactly the same curve, just viewed at a different scale. The “hockey stick” shape is an illusion of scale.

This is what really underlies what is happening in the Fenway Stadium Drowning Incident.

Exponential curves move through different scales of magnitude at a constant rate. When the water main leak starts, it is of concern only to microbes handcuffed to the top row of a thimble. It is far beneath our human “scale of relevance.” It is unimportant to us.

But the problem doesn’t merely get bigger, like a single dripping faucet that eventually fills the basement. The problem keeps changing scale, and then doing exactly the same thing all over again at the new scale: that is precisely what “scale-invariance” means. After it has drowned the microbes, it starts to drown larger creatures, like ants, in exactly the same way. Then rats. Then cats. Then big dogs. Then people. Then elephants. Then giants. Then mountains. Then continents. Then planets.

An exponential water leak in Fenway Park, could its growth be sustained, would drown the entire universe in a finite amount of time. And at that point it would only be getting warmed up for the real work.

As humans, we have a fixed scale of relevance. The flooding of an anthill is of little concern to us. The flooding of a continent — or a solar system — is too big for us to cope with, almost too big to imagine. There’s just this narrow range of scales that’s important to us. Something on the order of yards, and pounds, and gallons, and years. That’s the human scale.

Exponential curves aren’t like that at all. All scales are relevant to them, sooner or later. They methodically plod through them all, taking successively bigger and bigger steps.

So the issue we have with exponential curves as humans is that it takes astonishingly little time for them to pass straight through our scale of relevance, from “negligibly small” to “intractably huge.” The reason we have only forty-five minutes to get out of our handcuffs, rather than the full twenty-four hours after the problem started, is because the exponential growth of water didn’t intrude on our scale of relevance for twenty-three hours and fifteen minutes. And then, in forty-five minutes, it ripped right through our entire scale of concern, from “puddle” to “catastrophe.”

Here’s the part that Chris didn’t talk about. Should the leak continue to grow at an exponential rate, it won’t matter whether you get out of your handcuffs. In a matter of minutes after the stadium floods, all of Boston will be awash in a tidal gusher fountaining from the pitcher’s mound. Some minutes after that, the entire United States will be underwater, with a gravity-distorting tidal wave of water rushing toward Europe and Asia. Within a few hours or days, water will fill the entire Solar System and will put out the sun.

By the time this exponentially growing watery menace intrudes on the scale of relevance for the Galactic Federation of Planets, they will have about forty-five minutes to save the entire galaxy.

Now, if this seems to be getting silly, it is.

It’s silly because nothing in nature can sustain exponential growth.

In nature you’ll see brief spurts of exponential growth, as when yeast cells divide in a vat of beer wort, or as when a new product is first introduced to the marketplace. After that, the exponential growth scales back until it reaches no-growth. Then it reverses, declines, and is recycled to make room for the next wave of growth for something else.

Nature may abhor a vacuum, but it simply will not tolerate sustained exponential growth.

So let’s go back, now, to that original economic question. What is the appropriate growth rate for the GDP? Five percent? Two percent? One percent?

The correct answer is zero percent. The economy cannot sustain exponential growth at any rate.


At this point, you may be suffering a little bit of cognitive dissonance. After all, the news is constantly yammering about how the GDP grew two percent this year, or a disappointing one percent, and how growth is necessary for a “healthy economy.” And what about investment income? People get a percentage return on CDs, and savings accounts, to say nothing of stocks, bonds, and futures. It’s how the economy works.

Right?

Wrong. This takes us right into the heart of the shell game that is our modern economy.

Rather than trying to summarize this, I’ll refer you to Chris’s excellent Crash Course, as well as John William’s Shadowstats site.

Physical reality trumps economic theory. Nothing in nature can sustain exponential growth. So if economic theory is reporting exponential growth, there is a big mistake in the theory.

The presenting symptom of this big mistake is inflation. Chronic inflation is a symptom of faulty economics and a broken economy.

Monetary inflation is simply a matter of the money supply increasing in excess of the real economy the money is used to facilitate.

If there’s one chicken left in a village, there will be a bidding war for the chicken, and whoever has the most money will get the chicken. Let’s say that’s $10. Now, let’s give everyone in the village an extra $10. There’s still only one chicken, so now the bidding war will end at $20 instead of $10, and the same person goes home with the chicken.

The chicken is the real economy. Adding money to the village doesn’t change anything but the price of the chicken. The same is true of any other good or service in the village: all the prices rise as you pour in money, because everyone has more money, but no more goods or services to trade. This is monetary inflation.

The proper reason to increase the money supply is to match a growing real economy, to keep prices stable. Otherwise the money itself grows scarce, and you see monetary deflation.

Consider doubling the size of the village, and the number of chickens (and everything else) but leaving the amount of money the same. The original inhabitants all have money, but the newcomers are flat broke and can buy nothing. Over time, that money will disperse to everyone, but on the average, people will have only half as much money — the same total amount, spread out over twice as many people. If they eat all the chickens but one, the bidding war will only be able to go up to $5. Prices fall.

When money deflates, people start to hoard it as precious in its own right, and then trade breaks down. So it’s important to pump more money into the village so that prices go back up to $10 and stay there as the real economy grows.

But if you pump too much money into the village, prices will rise above $10 through monetary inflation. When inflation keeps happening, year after year, you have a chronic problem with your money supply. When the inflation rate is exponential, you have an exponential problem.

If you browse the Internet for old product catalogues, it doesn’t take long to realize how far things have gone. In 1960, a new sedan, straight off the showroom floor, sold for about $2500. A new pair of jeans cost about $1.50. A simple meal in a diner cost $1.00. My father bought a new house in the suburbs for $16,000.  Gasoline cost $0.35/gallon. Penny candy cost a penny.

Just shift that decimal point over, and you’re looking at today’s prices for the same kinds of goods, with remarkably few exceptions (like computers). When computers finally reach the end of their run with Moore’s Law — they’re getting close — you’ll start to see them subject to the same inflation as everything else. We’ve seen an average of about 5% per year between 1960 and 2006, just based on the 10-fold increase in prices over that forty-six-year period. The annual rates have varied — I remember it going up to around 13% in the late 1970’s, and some years have had lower inflation rates.

But overall, the US dollar is being exponentially eroded by inflation. And remember how exponential problems rip through our scale of relevance. What we see in our economic system right now is the puddle on the pitcher’s mound. The problem is changing scales even as I write.

Chris and John can take you on a tour of exactly how this all operates: GDP and CPI, hedonics and weighting and chaining and more tricks and traps for your edification and amusement. All a collection of desperate contortions to deny that there’s a problem, because we won’t (or can’t) get out of the handcuffs.

It’s quite a ride.

This entry was posted in General.

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