Utility Value – How Maths Can Make You Happier
The use of utility curves to make optimal decisions is something which is never really touched on in mathematics – even though they are a powerful tool for making good choices in life. “Utility” is used to represent personal benefit – and in this graph, maximum utility value has been set as 1.
For these types of concave utility graphs you have a rapid increase in utility value initially (as spending £2000 on a car brings a large increase in utility value than not having a car), but then it quite quickly levels off (as there is little added utility value in spending £22,000 rather than £20,000).
Obviously everyone’s personal utility curve will be different – perhaps someone who spends several hours every day in their car will really value that extra comfort and luxury that spending a few thousand pounds brings – and so their own curve will flatten out more slowly. Nevertheless they will always demonstrate the law of diminishing returns – where we ultimately end up spending more and more for ever reduced benefits.
The graph above is generated by the function f(x) = 1 + 2/(x+2). By adding a slider on Geogebra in the form f(x) = 1 + a/(x+a) you can amend the utility curve to have different levels of steepness – so students can easily generate their own utility curves.
The Khan Academy video on Marginal Utility is a really nice example of how we can use these calculations to optimise our happiness. In the video they discuss how best to spend $5 when faced with the choice of either fruit or chocolate.
Staying with cars, another factor to take into account is depreciation:
This graph is taken from What Car which allows you to enter any make and model of a car and see how its cost depreciates with time. In this particular case (with a Ford Focus), the price depreciates from £17,405 to £9,829 in just 12 months – that’s a stunning 44% decrease in value in one year. Or to think of it another way, you’ve lost £630 every single month.
So it is clear that buying a new car will lead to a massive yearly loss in your investment. The only strategies that make sense therefore are either buying a new car (with the peace of mind of long term warranties) and driving it for the next decade (thus averaging out the initial large loss in value), or buying a car that is already a year old – and avoiding the sharp depreciation completely. The absolute worst thing you can do is buy a new car and after 3-4 years sell it to buy another one – this really is just throwing money away!
So, understanding some simple mathematical concepts can help us make better decisions when it comes to investments and how we spend our money – and by helping to maximise our utility also has the potential to make us happier individuals.
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Benford’s Law – Catching Fraudsters – how mathematics can help solve crimes
Game Theory and Evolution – how understanding mathematics helps us understand human behaviour