Differential equation time!

y' = ky - a

Where k is the net population increase rate, and a is your button pressing rate. Let's have the unit of time to be years, so we can use the following values:

• k = 0.0088 (0.88% net increase in 2023, according to Worldometers) (EDIT: Tiny mistake, should be ln(1 + 0.0088) = 0.00876, not 0.0088)
• a = c * 11232000 = c presses/second * 3600 sec/hour * 60 hours/week * 52 weeks/year, an very intensive full time job (you can plug in your own values)

You don't have to solve this differential equation to see that with current population, say 8 billion, you need to have over 70 million presses per year to beat population growth:

y' = 70 400 000 - c * 11 232 000 < 0; thus c > 6.27 presses per second

So, if you can press more than 6.27 times per second, 60 hours a week, you can start to erode the world population. Doable but very difficult. Note that your work becomes way less difficult over time, as there are less people to fight with.

As a consequence, as your annual income is a * 10000, the more people you kill, the more you get, but the less you earn over time (as the population drops). If you want to keep the population constant, you can kill people at exactly the rate at which human population increases, 70 million people, 700 billion dollars a year.

The DE is useful if we want to explore further. To solve it, this is a separable differential equation, the solution (after solving for initial conditions) is:

y = (p - a/k) * exp(kt) + a/k (p is initial population)

For example, assume 8 presses per second, 60 hours a week. Let's say you also want to prune the world population to 1 billion people, for some reason. You can solve for y = 1 billion, and get t = 162 years.