Happiness Approximation 
Using a curvefit to global data, average happiness h is approximated by the equation
where F is percapita ecological footprint in Earths/person/year. This type of relationship can be summarized as
where
This relationship can be derived from a generalization of two observations: historical consumption of ecological resources, and completion over time. In preparing the Worst Climate Case of global population and consumption, it was discovered that the speed of population and consumption over time varies with the amount consumed. This observation was used in a refined model that produced backcasting and associated futurecasting, and can be generalized as follows for a time t:
where Consumption Growth Rate is a fraction. Completion varies with time based on the fraction of the task that remains (1  Completion) and the fraction of that which can be achieved in one unit of time (Completion Efficiency):
With the task starting at a time determined by the initial completion, and that time used to determine the consumption, where there is an initial consumption at the start time, completion will become fixed when all resources are being consumed (Consumption = 1). This is illustrated in the following example which simulates happiness over a lifetime, where Time is in years: 
Note the following about this example:
The following graph compares this simulation with happiness and with historical changes in happiness and percapita footprint (including the changes to the limiting footprint based on population). Dotted lines are linear curvefits to the data. 
Historical completion and consumption over time is shown below, including an extrapolation of consumption that shows when the resource limit will be reached without selfsustained climate change. 
For most normal tasks, the bestcase completion of a task occurs with Consumption Growth Rate and Completion Efficiency both equal to one. Initial Consumption must be nonzero to achieve any growth in consumption; so for the task to be completed when resources are all consumed (Completion = Consumption = 1) and starting completion at t = 0, Initial Consumption must be 0.75. 
The mode (or most observed) case can be expected to have conditions equal to the middle of each variable's range (except Completion, which starts at zero). That is: Completion Efficiency = Consumption Growth Rate = Initial Consumption = 0.5 as shown below. 
The expected case is determined using the PERT approximation to a normal curve for each variable:
The result is shown below, where the high values are from the first simulation above. 
The following graph compares all cases, including expected, history, and happiness: 
"Optimum" performance of a task occurs at the maximum amount of completion where completion equals consumption (Consumption / Completion = 1). This typically, but not always, happens when all resources are being consumed and completion is at its peak. Below are results for a sample of 100 cases with random values between 1E10 and 1 for all variables (Consumption Growth Rate, Completion Efficiency, Initial Consumption, and Initial Completion), along with linear curvefits to the data. 
Based on these results (where the probability of each data point is 1%):

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