Math problems with the clean car discount

Clean Car Discount overview page provides some preliminary information on the rebate amount. Detailed information such as about rebate amount per gram/kilometre of CO2 emissions is missing. However, a graphic titled “Clean Car Programme 2022” is provided and we have shown that graphic as figure 1 here. In this blog post we explain the problems with this graphic.

Clean car discount

To be able to clearly see the problems in the Clean Car Discount graphic it helps to see a graphic that does not have the problems. See figure 2, titled “the proposed feebate scheme”. As I learnt in my year 12 maths class many years ago, you should always label your graphs and put the correct units on it. Figure 1, has (gCO2/km) on its horizontal axis. This should be (g/km) instead as it is a measure for mass (weight) over distance in units of grams per kilometre. The fact that it refers to carbon dioxide is correctly labelled as CO2 emissions.

Clean Car Discount overview
Figure 1: Clean Car Discount overview as provided by NZTA.
Proposed Feebate Scheme
Figure 2: Proposed feebate scheme with correct labels.

I chose to call figure 1 a graphic instead of a graph, with the distinction that a graphic is for visual purposes and need not be mathematically correct whereas a graph should show the correct mathematical relationship between quantities shown on the horizontal and vertical axes. If figure 1 is taken as a graph, it has a bigger problem, in that it makes no mathematical sense. The vertical axis has a label “Zero band ($0)” and a rectangular box with the text “Vehicles with emissions within the zero band do not receive a rebate or incur a fee”. On closer inspection this makes no mathematical sense because it is shown on the wrong axis. This text box should appear vertically on the horizontal axis because the band refers to a range on the horizontal axis which is 146 – 192 grams of CO2 per kilometre.

This leads us to the third mistake in the graphic. The asterisked text at the bottom right of figure 1 reads “Rebates end at 146 CO2 and fees begin at 192 CO2” but it should be changed to “Rebates end at 146 g/km and fees begin at 192 g/km”.

Now the 4th problematic aspect of the graphic is that it conveys a linear mathematical relationship. In year 12 maths students are taught that a linear mathematical relationship is graphically represented by a straight line with the general equation for such a relationship being y=mx + c. The convention is to graph the dependent y variable on the vertical axis and the independent x quantity on the horizontal axis. The variable m is the gradient of the graph and depicts the steepness of the graph. The variable c is a constant that labels when the line cuts the vertical axis. A linear relationship for new import rebates would then be R=-59.08E+8625, where R is rebate amount in NZD and E is CO2 emissions in units of (g/km).

Let’s check if our equation works by applying it to calculate the rebate for Mitsubishi Outlander that has E=44 (g/km) resulting in a rebate of $6025 which is higher than the $5750 for plugin hybrid electric vehicles stated on the NZTA website. Could this difference be because we did not take GST into account, we don’t know and this leads us to the final problem with the information about the clean car discount.

Perhaps the biggest problem with information about the clean car discount is that the information is not detailed and provides no reference to sources where details could be found. The information also incorrectly suggests that all vehicles qualify for the maximum rebate amounts listed, this is simply false advertising! There are websites that have created calculators for calculating the clean car discount, we can’t trust them because they don’t explain how they do their calculations or where they get their information from.


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