GDP per Capita only started to rise in industrialized countries around 1800, but some historians believe that the Romans were close to having their own industrial revolution a little before 400 A.D. If they had managed it, and GDP per Capita there started to rise by 1% each year from about $1,800 per year at the time, today their average income would be around $15 billion per person. Is that a total fantasy, or will our descendants be so wealthy we can hardly imagine it?
Since economic growth got started (and even before), people have been predicting the inevitable collapse of our civilization. Pessimism is fashionable. But is it wise? Humanity faces many existential threats like planet-killing asteroids and super volcanoes. We also face enormous challenges like climate change, which could wreak havoc on our well-being and our economy. But humans have overcome challenges this big before. Perhaps the thing which should concern us the most is that, when we look out into the night sky, we don't see any evidence that these problems have been overcome by any kind of life on other planets. Where are all the aliens?
1. Rupert has decided to be cryogenically frozen at the time of his death so that he can be resurrected once medical science has advanced far enough to keep him alive. In preparation, Rupert places $100,000 into an investment account which earns an 8% rate of return per year. To his amazement, he is one day brought back to life. "How long has it been?" he asks. "210 years" the strangely dressed person replies.
How much money is sitting in Rupert's investment account?
We start with the growth equation:
Yend = Ystart(1 + r)t
In this case, the starting amount of money is $100,000, the rate of growth is 8% or 0.08, and the time period is 210 years. Plugging those numbers in, we can solve.
Yend = $100,000(1 + 0.08)210
Yend = $1,044,692,922,000
On a calculator, you may have seen a result like this:
1.044692922E12
That is called "Scientific Notation". The "E12" part means that they have moved the decimal place to the left 12 times. If you undo that, you would get the result I put above, which is 1 trillion, 44 billion, 692 million, 922 thousand dollars. Not bad.
2. Suppose that there is a 1% chance a planet will develop life, a 5% chance that the life will evolve into complex organisms, a 10% chance that complex organisms develop into advanced civilizations, and an X% chance that the advanced civilizations become intergalactic.
If there are 1 million planets, and we observe only 1 intergalactic civilization, what is the expected value of X?
Each of the probabilities given are the conditional probabilities of making it past the next filter. So, the overall probability that any given planet becomes an intergalactic civilization is:
1% * 5% * 10% * X%
The observational data provided suggests that these overall odds should be equivalent to 1 out of 1 million. That would be 1 / 1,000,000 which is 0.000001 or 0.0001%. So...
1% * 5% * 10% * X% = 0.0001%
Solving for X%, we would get:
0.005% * X% = 0.0001%
X% = (0.0001 / 0.005)%
X% = 2%
This means there is a 2% chance that an advanced civilization becomes an intergalactic civilization.
3. Referring to the previous question, how would your estimate for X change if we discovered simple life on another planet and increased our estimate for the probability that a planet develops life from 1% to 10%?
If we update our first probability, we can re-solve for X%:
10% * 5% * 10% * X% = 0.0001%
Solving for X%, we would get:
0.05% * X% = 0.0001%
X% = (0.0001 / 0.05)%
X% = 0.2%
This means there is a 0.2% change that an advanced civilization becomes an intergalactic civilization, instead of a 2% chance. Because we revised an early filter to be less difficult, we have to revise a later filter to be much more difficult!
Lao Tzu, an ancient Chinese philosopher, said, "He who knows that enough is enough will always have enough." At some point, earning higher incomes just isn't going to be doing much for our well-being, and most people will prefer to work less and have more time for themselves. We may choose not to grow our incomes, preferring leisure instead. There are also many who think that the low-hanging fruit has already been picked, and future growth will be harder to achieve. Society doesn't have to collapse, but perhaps we just stop growing. It might happen, but in all respect to Lao Tzu, I say, "Too much is always enough."
Maybe there is cause for optimism though. With so many promising technologies like virtual reality, self-driving cars, mRNA vaccines, bio-tech, and more, it seems like there is much more innovation and economic growth ahead of us. And while it is obvious that we cannot have infinite growth in production on a finite planet, economic growth has never been about how much we can produce, but rather how enriched we can become.
The hunter-gatherers of our ancestral past probably never imagined that one day their societies would be totally reorganized around farming, and they probably would have thought it sounded awful if we could have told them about it. Likewise, the farmers of our past never imagined that one day their societies would be totally reorganized around industry, and probably would have thought it sounded awful if we could have told them about it. So, it is natural to conclude that we folk in the industrial world are failing to imagine what the next big change will be. Let's try to correct this lack of foresight!
Deeper Thoughts and Extra Practice
Example Question
Suppose that average incomes today were $57,486 and are expected to grow at 2.7% each year for the next 110 years. The president is considering a policy which would increase the growth rate to 4.4%.
How much higher would the average income be in 110 years if this policy is enacted?
Round your final answer to two decimal places.