At Wednesday night's CNBC debate, moderator John Harwood confronted Marco Rubio on his $4 trillion tax cut plan, noting that while Rubio's rhetoric has focused on expanding opportunity to poor and middle-class families, his tax proposal would overwhelmingly benefit the rich. Rubio shot back, insisting that Harwood got his facts wrong and that his plan would in fact help poor households more than the rich:

The consensus among conservatives was that Rubio nailed it, that Harwood was straight-up lying about the facts, and that the whole exchange illustrated left-wing media bias more than anything else. And since the point of primary debates is to curry favor with the base, the fact that the exchange let Rubio look like the victim of an unscrupulous mainstream media reporter was a big win for his campaign, and a major reason many observers thought he won the debate overall.

But if you look at the actual exchange, that narrative is all wrong. Harwood got all his facts right, and then Rubio pivoted to unrelated issues, issuing a number of misleading claims along the way.

What was actually said

It got a bit convoluted, so it helps to look at the transcript first:

HARWOOD: The Tax Foundation, which was alluded to earlier, scored your tax plan and concluded that you give nearly twice as much of a gain in after-tax income to the top 1 percent as to people in the middle of the income scale. Since you're the champion of Americans living paycheck-to- paycheck, don't you have that backward? RUBIO: No, that's — you're wrong. In fact, the largest after-tax gains is for the people at the lower end of the tax spectrum under my plan. And there's a bunch of things my tax plan does to help them. Number one, you have people in this country that... HARWOOD: The Tax Foundation — just to be clear, they said the... RUBIO: ...you wrote a story on it, and you had to go back and correct it. HARWOOD: No, I did not. RUBIO: You did. No, you did. (APPLAUSE) HARWOOD: Senator, the Tax Foundation said after-tax income for the top 1 percent under your plan would go up 27.9 percent. RUBIO: Well, you're talking about — yeah. HARWOOD: And people in the middle of the income spectrum, about 15 percent. RUBIO: Yeah, but that — because the math is, if you — 5 percent of a million is a lot more than 5 percent of a thousand. So yeah, someone who makes more money, numerically, it's gonna be higher. But the greatest gains, percentage-wise, for people, are gonna be at the lower end of our plan, and here's why: because in addition to a general personal exemption, we are increasing the per-child tax credit for working families.

This is a fairly muddled exchange, and it's worth taking the time to parse it out clearly:

Harwood is alleging that the middle class gets a much smaller benefit than the rich, according to Rubio's own preferred analysis of his tax plan. This is unambiguously true.

Rubio did not answer this point. Instead, he insisted that the poor would benefit even more than the rich — and noted that when Harwood claimed the contrary in an tweet earlier this month, he issued a correction (Rubio claimed that Harwood corrected an article, which is wrong). It's true that the Tax Foundation, the right-wing think tank he's citing, found that the poor gained the most. But that wasn't what Harwood's question was about. Harwood was asking about the middle class, and on that point he was completely right.

Separately, you shouldn't believe that the poor benefit the most from Rubio's plan. The Tax Foundation report Harwood and Rubio both cited was a bad analysis premised on a fundamental misunderstanding of how Rubio's plan works. It's doubtful the poor would benefit anywhere near as much as the analysis suggests.

Rubio further argued that even though the rich might get a "numerically" bigger tax cut than the poor, that doesn't mean it's actually bigger in percentage terms, and this is what matters. That's a questionable way to look at tax policy. If you get $1,000 and Bill Gates gets $10,000, then Gates has gotten more even though the percentage increase in his net worth is trivial.

What the Tax Foundation analysis says

Here is the Tax Foundation table at the center of the dispute:

The left column is the "static" analysis: that is, the one that doesn't take into account the possibility that the tax cut would boost economic growth. The second, much more optimistic-looking column is the "dynamic" analysis, which assumes that Rubio's tax plan would provoke massive economic growth. Keep in mind that very few reputable economists outside the Tax Foundation think Rubio's plan could have growth effects anywhere near that dramatic, including economists who in general believe tax reform can boost growth.

Two things are clear, looking at this table.

One is that Harwood was right: The 15.3 to 15.7 percent income boost that people in the middle of the income distribution would get according to the dynamic analysis is considerably lower than the 27.9 percent boost the top 1 percent would get.

The other is that Rubio was right: The 55.9 percent income boost that the bottom 10 percent would get is bigger than the boost for the top 1 percent.

How is Rubio helping the poor so much? Well, Rubio's plan would replace the standard deduction and personal exemption with a $2,000 credit ($4,000 for couples). That's a big increase. Currently, for a couple in the bottom, 10 percent tax bracket, the standard deduction and personal exemption is worth a maximum of $2,040. A $4,000 credit is a near doubling of that.

That helps people at the bottom end — but only somewhat. After all, many of the poorest Americans don't owe any income taxes in the first place. In 2015, 40.4 percent of tax units will have a negative or zero income tax burden. That's mostly the poorest Americans, and they're not helped at all by a plain old credit. But Rubio's proposal, as originally laid out, is not a plain old credit. It's a fully refundable credit. Think about that for a second. Rubio's original proposal would give any household in America $2,000 or $4,000, no questions asked. It was a basic income. It was a massive increase in the welfare state of a kind that no Democratic candidate, including Bernie Sanders, is proposing.

Even Rubio's campaign says the Tax Foundation's analysis is wrong

So it's perhaps no surprise that when I asked his team about this, they insisted that this was a mistake, and the credit was in fact much more limited. "Rules would be tailored to ensure that our reforms would not create payments for new, non-working filers," a Rubio aide told me in April.

It's unclear what exactly that means, especially since the aide insisted that the tax was nonetheless refundable. But one thing it definitely does mean is that millions of people who would've benefited from a simple $2,000 to $4,000 refundable credit won't benefit under Rubio's actual plan.

"A fully refundable personal credit would be a radical change," the Tax Policy Center's Len Burman explained in a blog post. "It would cause millions of Americans with no earnings and no other reason to file to become tax filers purely for purposes of claiming the credit." Rubio's more restricted plan, which would bar "new, non-working filers," wouldn't help those millions.

Rubio should really stop going around saying that the poor benefit the most. It's just not true.

Here's the problem, though: The Tax Foundation assumed that Rubio had proposed a basic income. "The Tax Foundation assumed the proposal would make the new personal credit ($2,000 for singles and $4,000 for married couples) fully refundable," Burman wrote. "This assumption helps explain why the group concluded the Lee-Rubio plan would be highly progressive."

The Tax Foundation, in other words, concluded that the poor benefit from Rubio's plan by assuming that Rubio will create a massive new welfare program. Given that Rubio will not, in fact, create a massive new welfare program, this finding is pretty dubious. The fact of the matter is there's been no distributional analysis of the actual contents of Rubio's plan, only one of a bizarro-world version of his plan.

More credible analysis suggests the Rubio plan does little for the poor

An earlier version of the proposal introduced by Rubio's collaborator Sen. Mike Lee (R-UT) did get a thorough analysis from the Tax Policy Center. That analysis found that the bottom quintile would get the smallest benefit of any group: only a 0.5 percent boost in after-tax income. The top 0.1 percent, by contrast, would see a 3.8 percent boost. And Lee's version was, if anything, more progressive than the new Rubio-Lee plan. The new plan eliminates all taxes on capital gains, taxes that are almost exclusively paid by the wealthy (the Tax Policy Center estimates that 70.9 percent of capital gains taxes are paid by millionaires).

So, to recap: If you analyze Rubio's plan by assuming that it includes a massive welfare program that it does not, in fact, include, then it turns out to look very good for the poor! But the closest thing we have to a rigorous analysis of the plan suggests it's regressive.

$600,000 is more than $7,250

The other element of the dispute was Rubio's insistence that a 5 percent tax cut for a millionaire is the same size as a 5 percent tax cut for a poor person.

"Five percent of a million is a lot more than 5 percent of a thousand," he noted. "So yeah, someone who makes more money, numerically, it's gonna be higher. But the greatest gains, percentage-wise, for people, are gonna be at the lower end of our plan."

It is true that this is how percentages work, but it's a strange framework for tax analysis.

The Tax Policy Center doesn't calculate the average income of the bottom 10 percent of taxpayers, but it does calculate it for the bottom 20 percent. This year, it's $12,939. Let's assume someone making that much gets a 55.9 percent tax break — which is what Rubio's claiming he's giving to the poorest 10 percent. That tax cut would amount to $7,232.90.

But the Tax Policy Center also finds that the average income of the top 1 percent was $2.1 million. A 27.9 percent break for that group — as predicted in Rubio's preferred analysis — would be worth $588,001.15.

Rubio wants to argue, given these facts, that the poor are benefiting the most even though their tax cut would be more than 81 times smaller than the tax cut the top 1 percent would get. This doesn't bother Rubio, so long as the percentage cut is smaller for the top 1 percent. But it's unclear why we should care about that. Tax cuts cost money. That money can either go to poor people or it can go to rich people. And under Rubio's plan, much, much more of it would go to rich people.