A compact planet model: renewable resources, capital, pollution and population respond to four policy levers. Find the balance where people thrive without overshooting the planet.
Model and sources
Resource: dR/dt = r R (1 − R)(1 − 0.5 Z/(1+Z)) − q H with harvest H = e R P, after Brander and Taylor (Am. Econ. Rev. 1998, the Easter Island model). Capital: dK/dt = s (1 − u/2) Y − δK. Output Y = A K^0.35 H^0.65, reduced by pollution damage. Pollution stock: dZ/dt = ε Y (1 − u) − δZ Z.
Population follows a demographic transition: births fall as income per head rises and with education, deaths fall with income and rise with pollution and famine. Wellbeing is a geometric mean of income, life-expectancy and education indices as in the UNDP Human Development Index. The sustainable development index divides it by the overshoot of harvest over the maximum sustainable yield r/4 whenever that exceeds 1 (Hickel 2020, Lancet Planetary Health). Units are normalised and the model is illustrative, not a forecast.