Approximations

The following table provides approximations for calculating world population, ecological footprint, the possible number of artificial environments that can provide total happiness for one person (happy environments), and their limits.

Population Growth Rate	Prate = 2.5% /year
Footprint Growth Rate	Frate = 0.5% /year
Total Resources	Femax = 3 Earths/year
Minimum Footprint	Fmin = 7^-12 = 1 / (7¹²) Earth/year/person
Minimum Population (with Minimum Footprint)	Pmin = 2E+07 = 20 million people
Resource Generators	a = 2

Footprint	f = F / Fmin
Footprint at time t (starting from f = 1 at t = 0)	fsim = (1 + Frate)^t
Population at time t	Psim = Pmin * (1 + Prate)^t
Maximum Population (f = 1)	Pmax = Femax / Fmin = Femax * 7¹²
Peak Population	Ppeak = Pmax / (f + a)
Peak Footprint (for population P)	fpeak = Pmax / P - a
Happiness	h = f / (1.22 * f + 1)
Happy Environments (for population P)	Env = h * P
Peak Happy Environments	Envpeak = h * Ppeak

The following graph shows how happiness changes with footprint, including historical values and projected values from the limit-seeking projected case.

The following graph shows peak happy environments and peak population for different values of a as footprint varies. For example, Envpeak(0) is Envpeak for a = 0.

The following graph shows how the peak values of population (Ppeak) and happy environments (Envpeak) change with footprint.

Approximations to historical growth of population (Psim) happy environments (Envsim) are also shown, along with values from the limit-seeking projected case (population Pproj and happy environments Envproj) which include historical data.