People and the environments they can inhabit are modeled as statistical distributions, each with their own population. They appear to be normal distributions. In the graph, Z is in sigma (standard deviations) centered on the people distribution, and population is the area under each curve. The environments we inhabit are part of a larger, maximum set of environments. As our population grows, the number of inhabited environments grows until it reaches the maximum number available. Over any given range of Z, the happiness (H) of the people (P) within it is the fraction of overlap between people and the environments (E) within the range:
Where the number of maximum environments (Emax) is greater than the number of inhabited environments (Epop), the average happiness (h) is the ratio of total population (Ppop) to inhabited environments:


Based on data from the New Economy Foundation, which tracks happiness along with ecological footprint and other variables, happiness and life expectancy vary with ecological footprint F in a predictable way. Using world data over time to find the best matches to these equations:
where F is in Earths/person/year, h is a fraction, and L is life expectancy in years. These relationships show that both h and L asymptotically approach a maximum value as more ecological resources are expended. For h, this value is hmax = 0.82; for L it is Lmax = 358 years.


World (people) population varies in a similar way with happiness, as seen when happiness is projected over history based on the derived F/h and F/L relationships. Following is the population error for this curve fit over the years 12013, which mostly represents the differences in the methods used to calculate happiness: 

For a number of reasons, life expectancy can be used more easily than happiness in making projections into the future. Among those reasons, it is a larger and more observed variable and is linearly related to happiness. Using polynomial curve fits to life expectancy over the period where footprint and life expectancy data is available, three scenarios have been chosen using coefficients of a thirdorder polynomial. Where t is the year:
The Expected case is a PERT estimate using leastmeansquares linear, secondorder, and thirdorder curve fits. The Worst case is the thirdorder curve fit; and the Best case uses the slope of the linear curve fit, with the slope (6.47E+02) adjusted to match the actual value for 2013. NOTE: The coefficients shown above are rounded. Higher precision values are used to generate the graphs.


As with life epectancy, the Worst and Expected cases for world population technically reach a maximum ("peak") within the next decade, and drop to zero by early in the next century. The Best case population continues to increase until 2821, reaching a maximum of 21.6 billion people as the happiness maximum is approached.


Projecting total ecological footprint (footprint per person times population) over time, the cost of increasing happiness is clearly prohibitive. The Worst and Expected cases peak at 1.79 and 1.88 Earths, respectively. In 2014, the total footprint is projected by the Worst case to be effectively at its peak. The Best case requires over 84,000 Earths at the beginning of the year (2821) that it gets closest to the maximum.


While the maximum happiness, approached in the Best case, is 0.82 (or "hmax"), the Worst and Expected cases both peak at nearly the same value (0.66).


Following are graphs of error over the time range of the curve fits to life expectancy, with thirdorder curve fits to the Best and Worst cases:  
The following sources were used to construct historical data:


Related pages:PopulationConsumption v4 Main Page Worst Case Technical Discussion Best Case Technical Discussion 
© Copyright 2014 Bradley Jarvis. All rights reserved.