## 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 (years) Licence fee (total period) Licence fee (annualised) 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 10 \$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.

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### 7 Responses

1. Nice article. Interesting and simple to understand 🙂 Although, just for the sake of easy numbers, I would probably end up renewing for 5 years.

• Thanks for the feedback. Yeah, after all that I’ll probably renew for 5 years as well! 😛

2. What is your assumed interest rate? In the case in which the full 10 years worth of license is purchased at once, the state wants their money now, whereas in all other scenarios, there is deferment of payment, which allows for return in whatever savings account or investment you like.

• Hi, Will. You raise some good points. I was going to include things like inflation, returns on investment and rises in government charges into the analysis, but decided it just added too much complexity to a lighthearted post that I wrote for a bit of fun.

3. Unbelievabe – only costs \$44.30 for a 10 year drivers licence in NZ which can be used in Australia

• Licence renewal (3 or 10 years) in Victoria
(car or motorcycle) fee: 3 years \$46.60 or 10 years \$158.60.

4. On the subject of licence renewal, why when doing the eye sight test do we have to cover one eye and then the other? I could not imagine anyone ever does this in any driving situation.