The Liberal Party of Canada is proposing to increase the federal tax rate on taxable income above $200,000 from the current rate of 29% to 33%. According to the LPC, this measure will generate $3b in revenues:

[An earlier version made a stupid mistake; I did everything under tha assumption that the increase was 3 percentage points, and not 4. Gah.]

[I started writing this post in May, but I stopped when it looked like the Liberals were not going to be in a position to implement this measure. Happily, I didn't actually delete it.]

In this post, I'm going to see if I can reproduce that estimate using publicly-available data. If you detect trace elements of snark in what follows, they are due to my irritation at having to be obliged to go through all this. In 2015, it shouldn't be enough to plop a number like that into public debate without supporting documentation.

I'm writing this as I work through the math. At this point, I don't know what sort of answer I'll get.

There are two questions that have to be addressed:

How big is the tax base upon which this new rate will be applied? Knowing what we know about how the tax base reacts to changes in the tax rate, how does a higher rate translate into higher revenues?

My strategy for dealing with the first question is to estimate the tax base in the most recent year available, and then assume that it has grown and will grow in line with nominal GDP.

The most recent publicly-available for data on taxable incomes is from 2012:

Top incomes: 2012 Total incomes between $150k and $250k Total incomes above $250k Total incomes above $150k Number of files 443,160 215,350 658,510 Average total income (Thousands) $185.287 $540.585 $301.497 Total income (Thousands) $82,111,674 $116,415,030 $198,526,704 Taxable income (Thousands) $72,288,146 $104,293,230 $176,581,376 Taxable income: percent of total income 88.0% 89.6% 88.9%

And as a check, here are the data from 2011:

Top incomes: 2011 Total incomes between $150k and $250k Total incomes above $250k Total incomes above $150k Number of files 414,890 212,450 627,340 Total income (Thousands) $76,926,292 $118,667,577 $195,593,869 Average total income (Thousands) $185.414 $558.671 $311.783 Taxable income (Thousands) $67,541,529 $105,586,721 $173,128,250 Taxable income: percent of total income 87.8% 89.0% 88.5%

Unfortunately, these data are tabulated for total incomes, and not taxable incomes. However, it looks as though the ratio of taxable to total income is fairly stable, between 88%-89%. So I'm going to assume that this ratio will hold throughout. People who have $200,000 in taxable incomes will have, on average, somewhere between $224,000 and $228,000 in total income. Let's split the difference and set the threshold at $226,000 and suppose that taxable income in this income range is 88.5% of total income.

Inconveniently, the CRA doesn't provide data for this threshold, so we're going to have to do some interpolation. What I want to interpolate is the interval between $226k and $250k.

People working in this field seem to have concluded that the Pareto distribution is well suited for modelling incomes in the far right tail of the income distribution, so I'm going to go along with that here.

Here are the densities and cumulative distribution functions of the Pareto distribution:

The β parameter is the lower bound, and the α parameter captures the skewness towards the top of the distribution. Small values of α indicate a higher concentration in the right-hand tail. If α≤1, the tail concentration is so strong that the mean of the Pareto doesn't even exist:

You can get a method of moments estimator for α from the lower bound and the mean, which is convenient. Here's what you get if you use the information in the tables above to recover α:

Method of moments estimates for α Total incomes above $150k Total incomes above $250k 2011 1.80 1.52 2012 1.99 1.86

This is the usual approach for estimating α in this literature, but I don't think it's appropriate for this particular interpolation exercise. In 2012, there were 658,510 tax files with total incomes above 150,00$, and 215,350 with total incomes above $250,000. In other words, 67.3% of the tax files with total incomes above $150,000 had incomes less than $250,000.

This is a larger fraction than would be suggested by values of α between (say) 1.5 and 1.9. If α = 1.9, the probability associated with the $150k-$250k interval would be 0.621. We're interested in values of total incomes not far from $250,000, and I think that this is the feature of the data we should be reproducing as closely as possible. So I'm going to use the value of α that reproduces the actual frequency observed in the tax data:

So for α = 2.19, the the conditional distribution for the interval between some lower bound b (b = $226k) and $250k is

The average total income for b < x < 250 is

Happily, this is one of those integrals that's easy to work out by hand - which no doubt explains some of the popularity of the Pareto distribution here. For α = 2.19, β=150, we have F(250) = 0.673, F(226) = 0.593 and F(250)-F(226) = 0.0808. This last number is simply the proportion of files in the $150k and over group with total incomes between $226 and $250k. The number of files in this interval is the share times the total: 0.0808 x 658,510 = 53,213.

The average income in this interval is therefore

Multiplying this average income by the number of files in this interval yields and estimate for total income; multiplying this by 0.885 is my estimate for taxable income.

These estimates can now be added to the tax files with total incomes above $250k:

Top incomes: 2012 Total incomes between $226k and $250k Total incomes above $250k Total: both groups Number of files 53,213 215,350 268,563 Average total income (Thousands) $237.357 $540.585 $480.504 Total income (Thousands) $12,630,582 $116,415,030 $129,045,612 Taxable income (Thousands) $11,178,066 $104,293,230 $115,471,303 Taxable income above $200k threshold (Thousands) 535,381 61,423,230 61,958,611

The relevant tax base for the Liberal proposal is the taxable income above $200k; to get this, we take the total number of eligible files, multiply it by $200k, and subtract it from the total taxable income. In 2012, this worked out to $62b, and four extra percentage points would have generated applied $2.5b in extra revenues. That's a bit below $3b, but then again, those are data from 2012. You'd expect these numbers to increase over time. All we have to do is inflate those numbers to account for growth in nominal income since 2012.

Then again, maybe not. Here is what you get if you apply the same procedure (including a different estimate for α) to the data from 2011:

Top incomes: 2011 Total incomes between $226k and $250k Total incomes above $250k Total: both groups Number of files 50,677 212,450 551,240 Average total income (Thousands) $237.371 $558.671 $338.542 Total income (Thousands) $12,029,216 $118,667,577 $186,180,077 Taxable income (Thousands) $10,645,856 $105,586,721 $165,237,512 Taxable income above $200k threshold (Thousands) 510,480 63,096,721 63,607,201

According to the national accounts, nominal GDP increased by 3.5% between 2011 and 2012. But the tax base for the new tax bracket fell by 2.5% from $63.6b in 2011 to $62b in 2012. This is perhaps not as surprising as it sounds: we know that the top income share declined between 2011 and 2012 (the last year for which data are available).

So any estimate for high end tax revenues has to be based on a conjecture for the evolution of top-end income concentration since 2012, and this conjecture has to be put into the context of a steady decline in the income share of the top one per cent since 2006. I'm going to assume here that this decline leveled out after 2012 and that the top-end tax base has since increased in line with nominal GDP. NGDP increased by 7.8% between 2012 and 2014, so that would have brought the top-end tax base up to $66.8b, and the extra tax bracket would have brought in $2b.

Now we want to get from 2014 to 2016. According to the Bank of Canada's July Monetary Policy Report, real GDP in 2016 is expected to be 3.5% higher than it was in 2014. The Bank of Canada doesn't provide forecasts for the GDP deflator, but the PBO has given it a shot: they have the deflator increasing by 3.1% over this time. This brings the total increase in nominal GDP over 2014-16 to 6.6%.

If we go on to make the - possibly heroic - assumption that the top-end tax base has tracked nominal GDP since 2012, this would bring it up to $71.2b in 2016. Applying the 4 percentage point increase to this base yields a revenue increase of $2.9b.

But of course, this assumes that the tax base stays constant in the face of the tax increase - that is to say, it's a static analysis. But we know that's unlikely to happen. When we estimate expressions like

log[tax base] = e * log[1 - tax rate] + controls

the value of the elasticity parameter e is significantly positive. An increase in the rax rate - that is, a decrease in the net-of-tax rate - reduces the tax base. Available estimates for e in Canada are between (say) 0.2 and (say) 0.7. According to the OECD tax database, the current top rate (a weighted average of the top rates in each province) is 0.495. Increasing this to 0.535 reduces the log of the top net-of-tax rate by 0.08. This implies a reduction in the tax base somewhere between 1.7% and 5.8%.

Applying these reductions to an initial tax base of $71.2b brings it down to somewhere between $67.1 and $70.0b.

Revenues from taxable incomes above $200,000: 2016 e = 0.2 e = 0.7 29% tax rate $71.2 * 0.29 = $20.6b $71.2 * 0.29 = $20.6b 33% tax rate $70.0 * 0.33 = $23.1b $67.1 * 0.33 = $22.1b Change in revenues $2.5 $1.5

which implies that new revenues from the increased tax rate would be somewhere around $2.0b, give or take a half-billion or so.

And I'm done. I got something that is a bit below the Liberals' estimate of $3b, but it's close enough.

[Update: It turns out that the Liberals' costing document - which I didn't have in May - cites a revenue of estimate of $2.8b.]

[Update2: I added the last table.]