# A math/programming view of Canada GST/HST credit calculation

## Introduction

As a programmer and mathematician, I like to go beneath the surface of everyday activities to find and understand the math and logic hidden inside. When I was preparing my Canadian income tax returns, I couldn’t help but notice that the calculations in the tax forms were a form of math and programming expressed in a different notation. So I want to explore what would happen if I recast these calculations in math notation and code, and from there on apply standard techniques for manipulating and analyzing the logic (e.g. algebra and graphing).

In this article, I will illustrate using the real-life example of the Canadian GST/HST credit calculation form, for tax year 2012 (payable in mid-2013 to mid-2014), focusing on chart 2 (for an unmarried person with no children, which has the simplest calculation). The official version of the form, with copious instructions and definitions, are found here:

## Expressed as program code

As a side note, we could take the form and re-implement it in a spreadsheet (e.g. Excel), and it would look and behave identically to the original form. (This is left as an exercise for the reader.) The advantage is that you can execute the logic automatically on a computer and be able to try different input values with ease. But this technique is not so interesting to me, so we’ll skip it and start with a real programming language.

If we work through the form line by line and write Java code as literally as possible, this is what we get (31 lines):

double calculateGstHstCreditChart2(

double line1 = 265.00;

double line7;
if (netIncome > 8608.00) {
double line2 = netIncome;
double line3 = 8608.00;
double line4 = line2 - line3;
double line5 = 0.02;
double line6 = line4 * line5;
line7 = Math.min(line6, 139.00);
} else {
line7 = 0.00;
}
double line8 = line1 + line7;

double line13;
double line10 = 34561.00;
double line11 = line9 - line10;
double line12 = 0.05;
line13 = line11 * line12;
} else {
line13 = 0.00;
}
double line14 = line8 - line13;

return line14;
}

By translating from instructions written in English (a human natural language) to code in a real programming language, we immediately gain two useful properties. The instructions in the natural language are written in an ad hoc way, using a variety of wording and logic constructs. It’s difficult to ensure that every calculation form is understandable and unambiguous (see the notes for an example of ambiguity). Contrast this with the program code, where every operation has a clear definition, leaving no room for ambiguity.

Obviously, we can run the code on a computer to test its behavior and check its results. Computers are very fast, and we can run this calculation on millions of cases per second if we so desired. (In fact, if we had a database of the income of every person in Canada (about 35 million), we can easily run this function on every person, taking only a few seconds in total. That’s right, it’s entirely feasible to process an entire nation’s worth of data on a single personal computer – though it depends on the type of calculation.)

A standard code simplification technique is that if a variable is used only once, then its definition can be inlined into the place where it is used. Here we have numerous opportunities to do inlining, and it would significantly shorten the code. For example, in the first if-statement, the 6 lines of the body can be condensed to simply: line7 = Math.min((netIncome - 8608.00) * 0.02, 139.00);. For typical programmers, a line of code with this amount of complexity is actually considered quite simple, taking not much mental effort to process. It could be argued that the calculation can be split into 2 lines, but definitely any more than 2 would be too verbose and harmful for readability. After fully inlining all the variables and doing small tweaks to the logic, this is the much simplified code that we get:

double calculateGstHstCreditChart2(

double total = 265.00;
if (netIncome > 8608.00)
total += Math.min((netIncome - 8608.00) * 0.02, 139.00);
total -= (adjustedNetIncome - 34561.00) * 0.05;
}

We can clearly see the input-output relationship in the function: It takes two input numbers named netIncome and adjustedNetIncome, and it calculates one output number. This is useful to know because the information is complete, so we can draw conclusions about the non-relationships: for example, this calculation does not depend on the number of children (input), does not determine your provincial tax credit (output), does not involve your RRSP amounts, etc. Being able to identify the scope of the calculation helps to clarify the big picture.

## Expressed as a math formula

For the notation, line i is denoted as $$L_i$$, and input variables are written as acronyms (namely $$NI$$ and $$ANI$$).

First, we need to get rid of those unnatural if-statements. Observe that that we can rewrite $$L_7$$ and $$L_{13}$$ to use $$\max(\dots, 0)$$ instead: For example, regarding $$NI$$ and $$L_7$$, if $$NI < 8608$$ then we can pretend $$NI = 8608$$, so the long computation will still yield $$L_7 = 0$$ as wanted. The same reasoning goes for $$ANI$$ and $$L_{13}$$. Other than this consideration, we simply do repeated substitution of variables until the formula depends only on the input variables.

$$L_{14} = L_8 - L_{13} \\ = (L_1 + L_7) - (L_{11} \times L_{12}) \\ = (265 + \min(L_6, 139)) - ((L_9 - L_{10}) \times 0.05) \\ = (265 + \min(L_4 \times L_5, 139)) - ((\max(ANI, 34561) - 34561) \times 0.05) \\ = (265 + \min((L_2 - L_3) \times 0.02, 139) - ((\max(ANI, 34561) - 34561) \times 0.05) \\ = (265 + \min((\max(NI, 8608) - 8608) \times 0.02, 139) - ((\max(ANI, 34561) - 34561) \times 0.05) \\ = 265 + \min(0.02(\max(NI, 8608) - 8608), 139) - 0.05(\max(ANI, 34561) - 34561).$$

We have thus condensed the entire form into a one-line formula:

$$GHC(NI, ANI) = 265 + \min(0.02(\max(NI, 8608) - 8608), 139) - 0.05(\max(ANI, 34561) - 34561)$$,

where $$GHC$$ is the GST/HST credit, $$NI$$ is the net income, $$ANI$$ is the adjusted net income, and all variables denote monetary amounts in Canadian dollars.

## Mathematical analysis

For the purposes of analysis, let’s simplify the problem by assuming that $$ANI = NI$$. (The actual ANI requires a couple of calculations.) This may or may not introduce significant inaccuracies, but we just want to get a rough idea for how the GST/HST credit varies with a person’s net income.

Taking the function in the previous section, setting $$ANI = NI$$, renaming $$NI$$ to $$x$$, and renaming $$GHC$$ to $$f$$, we get:

$$f(x) = 265 + \min(0.02(\max(x, 8608) - 8608), 139) - 0.05(\max(x, 34561) - 34561)$$.

Graphing

We can plot the function with a suitable tool like Wolfram|Alpha, a computer algebra system like Mathematica/Maple/Maxima, a graphing calculator, etc.

The original form offered little insight about how income is related to the credit, but now the graph shows these facts as clear as day:

• The maximum credit is about $400 (in fact, exactly$404), which is attained for incomes of about $15 000 to$35 000.