Humans DO Understand The Exponential Function

I was reading through a mostly uninteresting book by William Ophuls called Immoderate Greatness—Why Civilizations Fail. Right there near the beginning, in the "Biophysical Limits" section, at the beginning of the "Exponential Growth" chapter, was physicist Al Bartlett's famous quote.

The greatest shortcoming of the human race is the inability to understand the exponential function.

NO, NO, NO! Humans (including economists) DO understand the exponential function.

The problem is they don't think it applies to them.

Allow me to explain. Lately I've been reading about the work of Thomas Piketty, whose new book on economic inequality is causing a stir in the economics community.

The first question we need to ask is where does growth come from?

Piketty, who is taking the very long view, answers that question in this interview with New York Times reporter Eduardo Porter.

Porter — Why are you confident that the economy will grow slower than returns on capital?

Piketty — Over the 1700-2012 period, world output [growth g, graph below] has grown at 1.6 percent per year on average, including 0.8 percent due to population growth and 0.8 percent due to per capita output growth.

This may seem small, but in actual fact this was sufficient to multiply the world population by more than 10 — from 600 million to seven billion inhabitants.

According to U.N. population forecasts, this seems unlikely to happen again in the coming decades and centuries. Indeed population has already started to stabilize or even decline in a number of European and Asian countries. Productivity growth can certainly continue forever, assuming we invent clean energy.

But in any case it will probably not be faster than 1 to 1.5 percent. It is only in exceptional periods, e.g. when countries are catching up with other countries, that productivity growth rates reach very high levels — say 4 to 5 percent or even higher.

From Piketty's book. Note that pure rate of return on capital r fell precipitously during the period 1913-1950, but is now returning to the presumably more normal level last seen in America's first Gilded Age in the late 19th century. The return on capital is contrasted with the growth rate of world output g. Note that growth continues right up through the year 2100, albeit at a slower rate than we had in the 20th century.

In contrast, rates of return on capital [r, graph above] can be 4 to 5 percent over centuries, or even higher for risky assets and high wealth portfolios.

Contrarily to what Karl Marx and other believed, there is no natural reason why rates of return should fall in the long run. According to Forbes’s global billionaires list, very top wealth holders have risen at 6 to 7 percent per year over the 1987-2013 period, i.e. more than three times faster than per capita wealth and income at the world level. Wealth concentration will probably stabilize at some point, but this can happen at a very high level.

To sum up—

• world output grew 1.6%/year (on average) in the period 1700-2012
• that expansion was due equally to 1) population growth and 2) productivity growth
• the rate of growth slows starting now, but growth continues up through the year 2100
• in so far as productivity growth "can certainly continue forever, assuming we invent clean energy," the decline in the growth rate is due entirely to a stable or declining global population

OK, I hope that's clear. Piketty cites the U.N. projections on population growth, noting that population has already stabilized (or is in decline) in some European and Asian countries (graph below).

Piketty is citing the "medium" case, as most people and all run-of-the-mill economists do. Source

And why does global population stabilize or begin to decline later in the 21st century?

As I explained in Humanity's Comeuppance — Population? Or Consumption?, fertility rates fall because the more wealth people have—wealth drives consumption, up to a point—the fewer babies they make.

ONE of the most significant phenomena of modern history is the demographic transition: as people get richer, they have smaller families.

This slowing of reproduction with economic development is the reason why Thomas Malthus’s prediction of disaster, caused by the human population outstripping its supply of food, is unlikely ever to come true. In the short term, Malthusian doom has been evaded by innovations that increased the food supply. But in the long term it is likely to be a ceiling on demand [consumption] that helps to save humanity. The world’s population, now some 7 billion, is expected to level out at a little over 10 billion towards the end of the century.

I don't particularly want to argue about any of this day. The expected outcome for economists generally is that everybody (including all those Asians, South Americans, etc.) will be living like kings later in this century. Consequently, all these nouveau riche won't feel the need to have so many babies, and the world population growth rate will slow (as it's already doing) or even decline. Which isn't good for long-run economic growth according to the usual story.

Thus, according to Piketty or any other mainstream economist you might name, global economic growth necessarily declines as population growth slows or declines. So I will repeat my initial point—humans (economists) understand exponential functions all too well. The problem is they don't think the limits to growth such functions dictate apply to Homo sapiens in the long run.

Assuming we invent clean energy, of course

Now, think about an Earth where there are 7-8 billion people living like Americans did in 1970...

You get the idea.

 

Comments

[Alexander Ač]

Dave, thanks for making my day (again)!

So I am happy, because:

* I don't care about impacts of climate change
* I don't care about ocean acidification
* I don't care about overfishing
* I don't care about deforestation
* I don't care about soil erozion
* I don't care about (rising) wealth and income inequality
* I don't care about peak oil and resource depletion
* I don't care about air pollution
* I don't care about water polution
* I don't care about impacts of wrecking the nitrogen cycle
* I don't care about overpopulation
* I don't care about relative and absolute greed
* I don't care about other idiots inhabiting this planet with me

and most of all:

* I DO care only about myself

-- NOTHING of that I don't care applies to me (otherwise, naturally, I would care), and I have nothing to do with it, therefore I am happy. And smiling at others. That is the best evidence there are no meaningful limits to growth and human ingenuity. Most of humanity is following me and I have many friends.

Best,

Alex