It’s driver’s licence renewal time in our household, and the State Government of South Australia helpfully allows her subjects to choose renewal periods ranging from one year for $42 up to ten years for $285. The renewal periods and their costs are summarised below:
Table 1: SA licence renewal options, total cost & annualised
It’s nice that my government offers so much flexibility in the way she collects her taxes, but I almost feel paralysed by choice here. Which option is best? Obviously renewing once a year, every single year, is a bad strategy (and in my view unfairly penalises people on low incomes). On the other hand, $285 for ten years, which does offer the best per annum rate over time, is quite a lot to fork over all at once. That money could potentially be better spent elsewhere. And by “better spent”, I mean on shiny toys like new iPhones obviously.
Luckily mathematics is there to sort it all out.
The annualised cost per year for a driver’s licence renewal is described by the formula:
annualised cost = $15/year + $27
which graphically looks a bit like this:
The turning point of the above function that I’m looking for occurs where the tangent (given by the first derivative) is equal to -1 (i.e. a negative slope on a 45 degree angle).
That is -15/year2 = -1
re-arranging yields when year = sqrt = 3.9
So calculus tells us that renewing for four years strikes a good balance between savings realised from an extended renewal period and minimising capital outlay.