Malthus Was Right

His* logic is unflawed. The only question: are his premises consistent with reality?

The Argument

Let's take a look at a revised form of his argument.

• It is, in fact, true that unchecked population growth is geometric. Take a look at human population growth over the history of humanity: it's a J-shaped growth curve sharper than an exponential (Excel gives P = 8e7exp{0.0017t} as the closest exponential approximation, and it still falls short ). This is relatively unchecked growth; the only dip is slight, in the 1300s, because of the Black Plague in Europe. As another example, consider Fibonacci's famous rabbits. If they breed unchecked, the total number of rabbits will look something like this as time goes on: 1,1,2,3,5,8,13,21,36, .... The closed-form expression for the Fibonacci Sequence is F(n) = (p^n - (1-p)^n)/sqrt(5)), where p = (1+sqrt(5))/2 is the golden ratio. This is also geometric growth, albeit slightly more complicated: the "^n" gives it away.
• Is it true that food growth only increases linearly? Well, sort of. This initial assumption can be reforged and tempered by noting that "food growth" can be replaced by the more general energy harvesting. What does this mean? Well, to live, you must use energy, whether as electricity for appliances, food for eating, gas for your car, or some other form. In the US, for example, the energy use per capita is a good 2.4e11 (!) J. This is far, far more than what is strictly necessary for survival, but that's beside the point. The point, in fact, is that there is a finite amount of energy locked in this planet, the Earth, and it is replenished and added to at a linear rate from the Sun.

In fact, we can construct a model of the energy in the Earth:

where E(t) is the amount of energy at a time t, E_0 is the total energy in the Earth at some reference point t=0, and P** is the Sun's power input into the system of the Earth (calculated by taking the solar irradiance and multiplying it by half the Earth's surface area -- approximately 1366 W/m^2 * 5.1e14 m^2 = 7.0e17 W). And keep in mind that most of the energy, in fact, is unusable: how are we to tap the heat directly at the Earth's core, for example?

So, in fact, there is a linear increase in the amount of energy available for human consumption, while the amount of energy humans will actually consume increases faster than exponentially, directly with the population, as the technology to extract that energy becomes available.

Malthus Was Right

Malthus was right. If we do not expand beyond the Earth in the future, the reserve of consumable energy will be gone, and we will only be able to consume at less than the rate at which the Sun replenishes the Earth's supply of energy. Rate of consumption, going from increasing exponentially, will be forced to linearity. Since consumption itself is directly proportional to population size, we see that population will be constrained to follow a logistic curve along with consumption.

Mathematically, it will look something like this, I think:

[Note: I'm going to have to think a bit more about this part above. Something feels off.]

Malthus' prediction will have come true, regardless of the details of the math: the population will be constrained beneath a J-shaped growth curve (greater than exponential). Doom and gloom, etc.

Application to Now

Let's break down the usable energy sources in the Earth's system and see what sort of energy they provide our economies and lives, and what they correspond to in the equation E = E_0 + Pt.

• Fossil fuels: Petroleum, Natural Gas, Coal. These provide a majority of our electricity and a vast majority of our transportation network. Since the rate of replenishment is negligible, these fall squarely into the E_0 part of the equation: they'll eventually be used up, and their use will follow a Gaussian curve.
• Nuclear power: Uranium, Uranium derivatives. These almost solely provide electricity. They are not replenished at all, so they fall completely under the E_0 part of the equation. They will eventually be used up.
• Nuclear power: hydrogen and fusion. These will solely provide electricity, and, since hydrogen can be pulled from water, fall under the E_0 part of the equation. Moreover, since water is a vast resource, we could derive seemingly endless amounts of electrical energy from this. Relying on this is, however, dangerous and irrational: it is not a fully developed technology, and may never be. Commercial implementation i's fifty years away, but it's been fifty years away for more than fifty years.
• Solar power. This source is exactly the Pt part of the equation.
• Hydroelectric, wind power. These replenishable sources all provide electricity only, driven by the Sun's energy injection into the Earth's system (both weather patterns and the water cycle are driven by the Sun's energy). Therefore, they fall into the Pt category.
• Biomass: Wood and other life derivatives. These are replenishable, but also depend directly on the Sun's injection of energy into the atmosphere, so fall under the Pt category. Burning, they can be substituted directly for fossil fuels in a way electricity cannot.
• Geothermal: Tapping heat from the crust. We are only able to harness a minute fraction of the Earth's internal energy, which is an overwhelming majority of the E_0 energy content of the Earth. This largely generates electricity, as I understand, although it also generates some heat directly.

What can we draw from this? Consumption of fossil fuels and uranium will follow a strict logistic curve, since there's a finite amount in the Earth and replenishment is actually negligible (actually, natural uranium decay indicates that the Earth's supply of uranium is actually decreasing, but even in the long run, that's negligible: the half-life of U-238 is on the order of tens of billions of years). Therefore, production, the derivative of consumption, will follow a bell-shaped curve: will increase, peak, and decline.

Application to Economics

Our economy is based largely on fossil fuels. Our society is based largely on fossil fuels. Therefore, our economic growth and growth as a society, which has been largely exponential until now, will falter and follow Malthus' model for a period of time as fossil fuels peak and run out and we accomplish the transition to nuclear power, which is the only long-term sustainable alternative. In fact, because the US economy is a net importer of energy -- some 25% of our energy is imported*** -- the shock will be especially great: exporters will cut exports over a short period of time relative to the amount of time needed to absorb the shortages to deal with internal energy shortages. Because energy is so inelastic, price increases will simply be absorbed by the economy, causing damage. Therefore, demand will only cut back substantially when physical shortages occur. We can predict that the shocks will destroy our economy completely.

In the Long Run, We're All Dead

This application is analogous to the long-, long-run situation with respect to the rest of the non-renewable energy locked up in the Earth. We will need this energy to jump from the Earth and colonize the Solar System. If we can get our hands on fusion power, there is enough in the gas giants and Sun to sustain us for billions of years. If we can use that energy to launch us on to the rest of the stars, we'll have an entire galaxy to conquer, an entire galaxy's worth of energy to use. Once we do that, and especially if we can spread to other galaxies, I imagine that the closest thing we'll have to worry about as a species is proton decay or the heat death of the universe. See Figure 2 below for an illustration.

In the long run, we're all dead, but the human race isn't unless we let it.

Footnotes

* Do you not know who Thomas Malthus was? Well, you should; go look him up.

** P is not strictly input from the Sun; part of it also comes from work done by tides from the Sun and Moon, chiefly (other planets contribute, but negligibly: tidal forces decrease with the cube of distance). It is, regardless, constant, so linearity still stands. Interesting fact: the geothermal activity of Jupiter's moon Io, the most volcanically active body in the Solar System, is caused chiefly by Jupiter's tides.

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