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Prime Minister Justin Trudeau speaks during a news conference in Ottawa, Dec. 9, 2015.CHRIS WATTIE/Reuters

Justin Trudeau's perfectly round numbers were a red flag.

On a hot and sunny May morning, Liberal supporters crammed into a family diner in the Gatineau neighbourhood of Aylmer, about 20 minutes from Parliament Hill, for the tightly orchestrated unveiling of the Liberal Leader's first major election plank.

Illustrated on red menus handed out at tables, the Liberal "Middle Class Tax Cut" promised $3-billion in personal income-tax cuts. They would be paid for by raising taxes on Canada's top 1 per cent, bringing in exactly $3-billion in new cash.

"The two tax changes will be revenue neutral to the federal government," the Liberals vowed.

Now in government, Prime Minister Justin Trudeau's tax plan no longer follows that simple math. This week's admission that both numbers are wrong and that the package will produce an annual fiscal shortfall of more than $1-billion is now raising a question as to how the Liberals arrived at their estimates. The answer relates to how the Liberals chose to measure the behavioural response of high-income Canadians to the new tax rate.

When the Liberals made the promise, they were still publicly in favour of running balanced budgets. Now, however, the tax changes threaten to add to a string of three deficits of up to $10-billion a year, promised during the election, which were supposed to be dedicated to infrastructure spending.

At a news conference on Wednesday, Mr. Trudeau was no longer committing to a firm deficit target. Instead, he insisted the Liberal government's two "anchor points" are to keep bringing down Canada's debt-to-GDP ratio and to balance the budget by 2019-2020. Questions remain, however, about what steps the government will have to take in response to the new fiscal landscape.

Mr. Trudeau added that regardless of the final numbers, he will stick with the "essence" of his commitment to raise taxes on high-income Canadians and lower them for members of the middle class.

Finance Minister Bill Morneau announced this week that cutting the second personal income-tax rate to 20.5 per cent from 22 per cent will cost $3.4-billion in lost revenue next year, rising to more than $4-billion a year by 2019-20.

Meanwhile, the new personal income-tax bracket of 33 per cent on income in excess of $200,000 will not in fact raise $3-billion a year. Instead, Finance says it will raise $2-billion next year, rising to about $2.5-billion a year in 2019-20.

That leaves a gap of about $1.5-billion in 2019-20. The government lowered this to $1.26-billion by including new revenue from the reversal of the previous government's changes to the contribution limits for tax-free savings accounts.

Predicting how Canadians will react to tax changes is an inexact science. Economists generally believe that high-income Canadians are far more likely to call up an accountant to find ways to reduce their tax bills in response to higher rates. This behaviour is assumed to increase as taxes rise. Economists call this the elasticity of taxable income, which is essentially a measure of the percentage change response to a tax change.

Finance Canada's new revenue forecast for the tax hike assumes an elasticity of 0.4. That's higher than the assumption in the Liberal platform, which primarily explains the difference.

But C.D. Howe Institute economist Alexandre Laurin questions Finance's latest figures. He notes that just five years ago, the department released a report estimating the elasticity response of the top 1 per cent at 0.62. That is the figure Mr. Laurin used in his recent analysis of the tax changes. Using that guide, the new tax hike would raise less than $1-billion a year. "I think they are taking a risk," said Mr. Laurin of the department's most recent numbers. The 0.4 figure was a new estimate produced by Finance as it moves to implement the new government's platform commitment.

The Liberals have argued they can reduce tax avoidance through tougher enforcement measures, but Mr. Laurin said he doubts there are large revenue gains to be made in that area. "Our system is one of the best," he said.

A Finance Canada spokesperson described the department's latest estimates as "reasonable" and said tax enforcement is now stronger than it was in 2010. The department also notes that there are fewer options to avoid changes to federal taxes.

Meanwhile, University of British Columbia economist Kevin Milligan, who was a member of the Liberal economic advisory team, says Finance's latest numbers may actually be low-balling how much revenue the new tax rate will bring in and the Liberal platform estimates may prove to be accurate in the end. The Liberals' original campaign estimate was based on a calculation provided to the party on request by the Library of Parliament.

"I'm not sure [the Liberals] got it entirely wrong," he said. "These new estimates by Finance, to my eye, look quite cautious and [are] building in enough room for a potential positive upside. … They may be making a judgment that it's better to take a small political hit now and look good three years from now, [rather] than the alternative."

Dr. Milligan has studied these issues in-depth and reviewed the Liberal numbers before they were made public. He told them that if they follow through with planned measures to crack down on tax avoidance, then the elasticity ratio would drop below 0.4 and raise the revenue they are counting on.

When the Liberal tax plan was announced in May, it was the first major attempt by the party to put concrete proposals in front of the electorate. Since winning the Liberal leadership a year earlier, Mr. Trudeau had campaigned on promises to help the middle class and offer a new, more positive politics, while refusing to unveil any details of his platform.

At the time, the party was in a funk, with Mr. Trudeau facing internal and external attacks for supporting the Conservative government's anti-terrorism legislation. While the promised "middle-class tax cut" received immediate play in the media, it quickly disappeared from the public consciousness as NDP Leader Rachel Notley won the Alberta election the following day. The victory helped to propel the federal New Democrats up the polls nationally, well ahead of the Liberals when the election was called on Aug. 2.

By voting day, Mr. Trudeau managed to position himself ahead of the NDP as the major agent of change with his promises to run deficits and change the tax brackets.

Now that the fiscal reform is proving to be more expensive than planned, and the economy more sluggish than predicted, Liberal officials caution that final figures on the cost of the changes will only be known after Canadians file their taxes.

"Finance has very prudent projections, which is appropriate," said spokesperson Kate Purchase.

As for Mr. Trudeau, he defended his party's projections Wednesday when asked how he got the tax numbers so wrong.

"There have always been, throughout the campaign, many different economists with very different analyses of how much it was going to cost, how much it was going to bring in," he said, arguing that the government is meeting the "essence" of the pledge. "We know that it's not just good for middle-class Canadians to get more money in their pockets every paycheque. It's also good for fighting against the income inequality that continues to be a problem for growth in Canada."