Exponents are hard

It's the year 2030, and a group of scientists figure out time travel. Due to the particulars of how time travel works, they're unable to travel backwards in time and thus cannot change the past, but they're able to travel into the future and return to the present.

The scientists decide to run an experiment to send someone 200 years into the future to determine whether humanity survives the threats of overpopulation and poverty. They send a robot because they agree it's too dangerous for one of themselves to go. Archie, the robot they choose, sits in his time-traveling contraption and blinks in and out before the scientists' eyes. Though taking no time by the scientists' perceptions, Archie reports that he spent five years 200 years in the future.

All is well with humanity, Archie says. In the future, poverty has been solved for all, and the world's population has been brought down to one billion from the ten billion living in 2030. Furthermore, humans are thriving in bases on the Moon, Mars, and the asteroid belt. In short, the future is magnificent.

—So Asimov wrote over twenty years ago in one of his last robot stories, Robot Visions. The story continues with its unnamed narrator not believing Archie's report and not sharing the scientists' relief of humanity's good fortune. As the narrator figures:

I asked myself if population decreased from ten billion to one billion in the course of two centuries, why did it not decrease from ten billion to zero? There would be so little difference between the two alternatives.

Asimov was a smart guy, but how could he be so wrong about this point? As anyone knows who's had an infestation of cockroaches, ants, or any other pest, there's a big difference between most dying and all dying. In this particular case, for a population of ten billion to reduce to one billion over 200 years requires an annual growth rate of about -1%. That's barely negative and is the consequence of small changes in the mortality rate and the birth rate. Today in 2011, some industrialized countries, such as Japan, already have negative growth rates due only to a decrease in the birth rate.

Whereas, for a population to go extinct over 200 years, the annual growth rate must be about -10%. That's a big difference—like suffering an epidemic every year for two centuries. Or suffering continuous non-stop war as destructive as the world wars. While a -1% rate is historically precedented, a continued rate of -10% is not. Surely, Asimov knew this. I guess you ignore a few details when you average 1,700 published words per day for 40 years.