Labor union members and supporters demonstrate in opposition to a right-to-work law in Lansing, Mich in 2012. (REUTERS/James Fassinger)

For all his talk about "middle class economics" in recent years, it might have seemed odd that President Obama didn't mention the "middle class" once during his nearly hour-long speech to Congress last night.

Instead, he talked about "working families." Those phrases aren't interchangeable anymore — so many working people have exited the middle class that it's no longer the demographic majority.

Now, here's another thing the president mentioned, even if indirectly: The power of unions to help. Working families, he said, "won’t get more opportunity or bigger paychecks by...allowing attacks on collective bargaining to go unanswered." Attacks, he might have said, like the case argued earlier this week at the Supreme Court that could bar unions from collecting fees from non-members for administration of the contracts that cover them, a prohibition that has tended to weaken their bargaining power and ability to push for middle class priorities in the political process.

That observation has some new analysis behind it, courtesy of a report from the Center for American Progress senior fellow David Madland and Harvard economist Richard Freeman. The headline number: The decline of union coverage is responsible for about 35 percent of the drop in the share of the workforce that falls within the middle class, the researchers found.

Of course, that isn't a new argument. Liberals have long touted the equalizing effects of unions, and have been doing so more stridently as inequality has risen and become central to the Democratic party's message. Building on earlier work, Madland and Freeman have been contributing some quantitative ammunition, first with a look at how children in more unionized regions and even those in union families have a better-than-average chance of climbing the economic ladder.

Here's what the math looks like:

Because people covered by union contracts both earn more and have more uniform earnings, they are more likely to be middle class, which the authors define as earning between 67 and 200 percent of median income, taking a cue from the Pew Research Center's parameters. Therefore, the decline in union coverage will disproportionately remove people from the middle class. That decline is responsible for 2.7 percentage points of the 7.6 point decrease in the share of workers included in the middle class between 1984 and 2014.

Meanwhile, the amount by which union workers are more likely to be middle class than non-union workers — the "union equality premium"— has also declined, since it's harder to maintain high wages with smaller membership bases. That effect, the authors calculate, takes another 1.8 percent off the share of workers in the middle class. The combination of those two effects actually cancel each other out a bit, reducing the impact the weakened labor movement — most of which can be attributed to the decline in union coverage — to 47 percent of the total decline in the share of the middle class.

So here's a question: How much of this effect is due to a simple shift in the composition of the U.S. economy, from highly-unionized manufacturing over to service occupations that were never organized at as high a rate? If it were just a consequence of the rapid offshoring of relatively highly-paid factory jobs over the past three decades, that might suggest unions don't necessarily have the ability to keep people in the middle class.

Madland says that it's not just a manufacturing phenomenon. Other professions that can't be offshored, such as construction, have also seen their their union density decline. Here's a graph that separates out those industries:

James Sherk, a legal fellow at the Heritage Foundation, is a professional skeptic of organized labor. He says that union members tend to earn better wages because organizing campaigns have historically targeted middle-income workplaces. Those who are highly motivated to earn lots of money tend to avoid them, he argues, pointing to a 1996 paper describing these selection effects.

"Economists expect to find more union members in the middle class, whether or not unions causally contribute to it," Sherk says. "This study makes no effort to distinguish that correlation from causation."

In response, Madland refers to a number of of other studies that do suggest a causal relationship. "What this study is especially good at is showing that because there are fewer union members, there’s fewer people earning a middle-class income," he says. "And as unions have weakened, they have been less able to pull people up into the middle class."

How unions ought to be rebuilt, both through policy and the efforts of unions themselves, is a big question that Madland and Freeman don't try to answer in their paper. But one thing's for sure: If the Supreme Court rules against public sector unions in the case it just heard, Friedrichs vs. the California Teachers Association, the labor movement will have to redouble its efforts to arrest its long decline.