utility value2

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:

car depreciation 3This 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.

If you liked this post you might also like:

Benford’s Law – Catching Fraudsters – how mathematics can help solve crimes

Game Theory and Evolution – how understanding mathematics helps us understand human behaviour

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.