I'm going to work through the math here, and I'm going to take it very slowly - partly for the benefit of people who are seeing these sorts of manipulations for the first time, but mostly so I don't screw it up.

In my recent National Post column , I make reference to some back-of-envelope calculations to the effect that replacing the fiscal anchor of balanced budgets to one of a fixed debt-GDP ratio allows the federal government to increase spending by 1.2 percentage points of GDP, or by about $25 billion.

Some notation first:

G is government spending

T is tax revenue

B (for bonds) is government debt

i is the interest rate

Balanced budget case

If the budget is balanced, then government spending must be equal to the revenues left over after serving the debt. Debt service payments are calculated by multiplying the interest rate i by the size of the debt B:

G = T - iB

Divide everything by GDP – denoted by Y – to express everything in ratios to GDP:

G/Y = T/Y - iB/Y

Note that B/Y is the debt-to-GDP ratio, denoted by DR (for debt ratio), so write this as

G/Y = T/Y - iDR

Fixed debt ratio case

We have to compare across years, so we’ll add subscripts to denote which year we’re talking about. If (say) debt in a given year is B t then debt in the following year is B t+1 . Next year’s debt B t+1 is the previous year’s debt B t plus the deficit accumulated from the previous year (G t + iB t – T t ). If there’s a surplus, then the ‘deficit’ is negative, and debt falls.

B t+1 = B t + iB t + G t - T t

Collecting terms:

B t+1 = (1 + i)B t + G t - T t

Again, divide through by GDP to express everything in ratios to GDP. For reasons that will become apparent in a minute, let’s divide by Y t+1 , GDP in the next year:

B t+1 /Y t+1 = (1 + i)B t /Y t+1 + G t /Y t+1 - T t /Y t+1

Let’s denote the growth rate of GDP by g, so that Y t+1 /Y t = (1+g), or, equivalently, Y t+1 = (1+g)Y t . Substitute [(1+g)Y t ] for Y t+1 on the right-hand side, so that all terms are expressed as a ratio of the GDP for the same year. This shows how the debt ratios in years t and t+1 are related:

B t+1 /Y t+1 = [(1 + i)/(1+g)]B t /Y t + [1/(1+g)]G t /Y t - [1/(1+g)] T t /Y t

Recall that the ratio B/Y is the debt ratio (DR) for a given year. Let’s say that the government sets B/Y at a constant ratio in both years, so we can write them as DR without the year subscript:

DR = [(1 + i)/(1+g)]DR+ [1/(1+g)]G t /Y t - [1/(1+g)] T t /Y t

Move the right-hand side term with DR to the left-hand side and collect the DR terms:

(1 - [(1 + i)/(1+g)])DR = [1/(1+g)]G t /Y t - [1/(1+g)] T t /Y t

Common denominator for the term in brackets on the left-hand side:

[(1+g) - (1 + i)]/(1+g)DR = [1/(1+g)]G t /Y t - [1/(1+g)] T t /Y t

Multiply through by (1+g):

[(1+g) - (1 + i)]DR = G t /Y t - T t /Y t

The left-hand side simplifies a bit:

(g - i)DR = G t /Y t - T t /Y t

A little bit of re-arranging so that G/Y is on the left-hand side and to get an expression we can compare easily with the balanced budget case:

G t /Y t = T t /Y t - iDR + gDR

We don’t need to distinguish between years anymore, so write this as

G/Y = T/Y - iDR + gDR

Comparing spending with the two anchors

Let’s compare the two expressions for spending-to-GDP ratios. Here, once again, is spending under a balanced budget:

G/Y = T/Y - iDR

And here is that ratio when the government follows a fixed debt ratio rule:

G/Y = T/Y - iDR + gDR

You can see that the spending-to-GDP ratio is higher under the debt ratio rule, and the difference is equal to the GDP growth rate g times the targeted debt ratio. That term gDR is also the size of the deficit, expressed as a ratio to GDP.

It’s interesting that the interest rate i doesn’t show up here: the increase in spending comes entirely from the anticipated growth in GDP, not whether or not interest rates are low or high.

Some back-of-envelope calculations

According to the October 2017 Fiscal Update, nominal GDP is expected to grow by about 4% between 2017 and 2018, so let’s set g = 0.04. Multiply this by the current debt ratio of 30%, and you get 0.04 x 0.30 = 0.012. In other words, going from a balanced budget anchor to a debt ratio anchor increases government spending by about 1.2% of GDP.

GDP is projected to be around $2.1 trillion in 2017, so 1.2% of GDP is roughly $25 billion.