Doomer literature is replete with stories of exponential growth and then collapse. It's one of their favorite themes. Some doomer sources (like Albert Bartlett, or the 7th fold) focus on exponential growth as their primary topic. In fact Albert Bartlett has famously claimed (famously in doomer circles anyway) that the inability of most people to understand exponential growth is one of the great problems facing the world today.
The reason exponential growth is a topic in doomer circles, is because exponential growth of almost any important quantity will lead to disaster before very long. For example, exponential growth of population, by 4% per year, would lead to 6.5 million times as many people in the world in 400 years, necessitating a massive die-off. And exponential growth of energy usage, by a few percent per year, would lead to the oceans boiling in only a few centuries. In most cases, sustained exponential growth leads to die-off or collapse before very long. Just look at colonies of bacteria or yeast, which (as doomers frequently point out) grow exponentially until they suffocate in their own waste or exhaust their food supply.
There is only one big problem, with all this talk of exponential growth and then collapse. The problem is: no important quantities are growing exponentially.
Take energy consumption as an example. Energy consumption is not growing exponentially. In fact, energy consumption per capita isn't growing at all, not even linearly, and hasn't grown much for decades. This point should be obvious to anyone over age 40, by just looking around and then remembering how things used to be. Do you use exponentially more electricity than in 1970? Do you set your thermostat 3% higher every year? Do you use exponentially more gasoline than in 1970? In fact, you probably use less gasoline, if your car in 1970 was anything like my parents' cars, which got about 10 mpg.
In fact, energy usage per capita has been essentially flat for the last 5 years or so, and was growing only very slowly in the 2 decades before that, and not at an exponential rate.
Well, even if energy usage per capita isn't growing exponentially, isn't the world population growing exponentially? Don't we face exponentially more people, and so exponentially more resource usage even if resource usage per capita remains the same?
No, the population of the world is not growing exponentially. In fact, the population is growing, but at a declining rate. Already, the rate of population growth has reached 0% in many industrialized countries, and is declining rapidly in developing countries. Furthermore, there is every reason to believe that the rate of population growth in developing countries will also reach 0%, since they're basically following the same pattern which industrialized countries have laid down. As a result, most professional demographers believe that world population will level off at between 10 and 12 billion people.
Well, isn't the economy growing exponentially? We see figures like "3% growth this year" which seems to imply it's growing exponentially, right? And, since we need energy for economic growth, doesn't that mean that the rate of energy usage is also growing at 3%?
First, economic growth isn't the same thing as growth in energy consumption. (This mistake is very common). The term "economic growth" refers to growth in production of things you can buy. If someone invents a new drug, he has caused the economy to grow, even if manufacturing that drug takes less energy than manufacturing the older equivalent.
Second, the economy is not growing exponentially. The rate of economic growth is delining everywhere. As economies mature, their rates of growth decline. This has already happened in all first-world countries, which do not enjoy anyhwere near the rates of growth of (say) China or India.
Thus, essentially no important physical quantity is growing exponentially1. Population is not growing exponentially. Energy usage is not growing exponentially. Industrial output is not growing exponentially. The economy is not growing exponentially. Food production is not growing exponentially.
Since these quantities are not growing exponentially, it would be a basic mathematical error to use exponential functions to model them.
Both population and energy usage will grow and then level off before very long. There is ample historical precedent for this.
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