Driver’s Licence Renewal and Calculus

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

Renewal period
Licence fee
(total period)
Licence fee
1 $42 $42.00
2 $69 $34.50
3 $96 $32.00
4 $123 $30.75
5 $150 $30.00
6 $177 $29.50
7 $204 $29.14
8 $231 $28.88
9 $258 $28.67
$285 $28.50

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[15] = 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.