(A shout out to Claudio Tereso for hammering out some of the ideas below with me over a beer somewhere near Marinha Grande, Portugal in summer 2018)
A quick Google search will reveal that the Happiness Curve is “a thing”, and it’s roughly U-shaped, like this:
You can triangulate this with multiple sources, and there was even a book written about it recently:
There’s also the corresponding depression curve:
And of course the chicken of depression:
Thinking like a mathematician
Several questions spring to mind as I contemplate such graphs:
- What are the units of the y-axis?
- Is a quadratic function an appropriate model?
- What do the gradient and area functions represent, respectively?
- What factors change the curve’s shape?
- How can we use this model to optimise life happiness?
The basic happiness axiom
The simplest mathematical models that have been proposed for happiness are those involving either a quotient or difference of expectations and reality, for example:
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Obviously, these models imply that both reality and expectations are functions of time. In the case of the quotient model, this then implies that happiness is infinite at birth (as one has zero expectations). Such a model could still be made to fit the available data and is plausible (since we have no way of measuring the happiness of very young children, and perhaps they really do have crazy high happiness levels until we adults ruin it for them), but more thought needs to be given to how to create a function that bottoms out in midlife and then curves upward again – suggestions welcome!
Let’s assume though, that that we can model the curve with a sensible mathematical function that behaves as we want it to and is consistent with underlying first principles.
Integrating the happiness curve
Assuming such a model exists, I’m working on a theory about the best actions to take being whatever will result in the optimisation of the integral of the happiness curve (ie happiness versus time, or h(t), where h is an arbitrarily defined “happiness level”), over the length of one’s life (ie, the total area under the h(t) curve represents the total happiness in one’s life, and it should be maximized).
This is similar to an idea from Rebecca Dias that I found a few years ago (funnily enough when looking for inspiring things for my maths classroom wall):
I’m not sure how to interpret integrating h(t)/t as opposed to just h(t) (again, I welcome your suggestions in the comments), but I can see how it would give the correct units of happiness, whereas my approach would give an answer in units of “happiness years”.
The problem with optimising the integral is you don’t know when you’re going to die (or lose your memory). To get around this, you’d have to simultaneously optimise the rate of happiness increase (dh/dt) and the current value of h(t) at any given point in time, in addition to the integral.
Towards optimisation
At my current level of mathematical mastery (I teach a university preparation calculus course), I’m stumped on a possible analytical approach. But perhaps the principles of PID control could be applied here? For those unfamiliar with the idea, read the first two paragraphs of the Wikipedia entry.
Essentially what I’m proposing is that e(t) in the diagram below is the difference between the actual happiness level and some optimal happiness level:
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This then raises the question of what the “set point” should be – and whether happiness has an upper limit, or whether it can approach infinity.
Maybe I should take up a religion after all…
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