Here's a problem for the protesters demonstrating at fast-food restaurants across the country to demand $15 an hour: Whom are they talking to, exactly? A global megacorporation or a locally-owned small business?

The answer is both -- and each of them can use the other to say no.

The minimum wage, which has long lagged behind the cost of living and even the rate of inflation, is having a moment. After nearly a year of timidly asking for an increase to $9 an hour, President Obama appealed again for Congress to take action in a speech on inequality this week, and polling that suggests the idea is gaining widespread support.

Low-wage employers, needless to say, are not very happy. And the most vocal among them are franchises, the affiliates of giant corporations at which protesters have directed their ire. On Thursday, the International Franchising Association and the Chamber of Commerce released a survey of their members on whether they would "make personnel decisions in order to adjust" to a $9 and $15 hourly minimum wage. By a substantial margin, more franchises said they'd have to reduce staff or cut hours:

(Public Opinion Strategies)

Why are franchises so much more likely to react negatively to a higher minimum wage?

The International Franchising Association says it's in part because their members have already been disproportionately squeezed by new mandates, and so higher wages are just one more thing to deal with. Many own several stores in the same market, and since they qualify as one business under the Affordable Care Act, they fall above the 50-employee threshold that requires them to offer health insurance. But they don't have the resources of their parent companies to absorb the hit (they've been warning about this effect since 2011, and have continued to push for changes).

"Franchisees, at the end of the day, are bearing the brunt of the cost associated with things like Obamacare and higher tax rates, because in most cases they are individuals filing as small businesses," says IFA spokesman Matthew Haller. "And so, when unions and other protesters are saying that corporate America is greedy, what they're missing is that in this situation, these are locally-owned small businesses that are contributing to their communities."

But the other thing that's going on here? Franchises already pay their workers the least. Sixty-five percent of the non-franchise executives said they had no employees at all who made minimum wage, compared with 38 percent of franchise respondents. Put another way, 38 percent of franchise respondents said more than 11 percent of their employees were paid the minimum wage compared with only 19 percent of non-franchise respondents.

A lot of that has to do with the fact that franchises are heavily concentrated in the fast-food industry, which has lower profit margins than most. But there's also research to suggest that franchises pay less than even company-owned stores within the same chain.

Back in 1991, Alan Krueger -- the policy brain behind Obama's minimum wage proposal -- found that managers at company-owned stores were paid nine percent more than managers at franchises and that their wages rose faster over time. He figured that was because it's more difficult to monitor employee performance when the owner is more remote, so the parent corporation builds in financial incentives for good behavior instead.

More recently, Boston University researcher David Weil -- who's since been nominated to lead the wage and hour division at the Department of Labor -- calculated that franchises had significantly more wage and hour violations under the Fair Labor Standards Act than did company-owned stores, as measured by the amount of back wages paid. He chalked it up to the fact that since franchisees have to pay royalties to the parent company as a percentage of revenues, not profits, labor costs are the only way they can control the amount of money that goes into their own pockets. In addition, they're less concerned about the reputational damage that the restaurant brand might sustain as a result of treating workers poorly.

The common thread here is a growing feature of the labor market that Weil calls "fissured employment": While workers generally used to be employed directly by companies, they're increasingly kept separated through arrangements with technically independent contractors, whose workers can be paid far less. It's a huge concern for the labor movement, since even though the client company ultimately calls the shots, it's able to abdicate responsibility for wages and working conditions and is insulated from pressure from employees. Think of Amazon and its warehouses, Walmart and its suppliers, Apple and its factories, and of course Burger King and its franchisees -- all of them to a certain extent have missed or turned a blind eye to workplace abuses.

Damon Silvers, policy director of the AFL-CIO, named that divide as one of the fundamental legal obstacles in the way of effective worker representation. "What if the employer isn't really the employer?" he put it to me in an interview a few weeks ago. "What if the employer is a front?"

Federal legislation may yet help fix the problem, by making it so that paying more isn't a competitive disadvantage. But the more structural challenge is a labor market that's fractured to the point where employers are able to essentially say: "That worker's wages are not my problem, because she's technically not my employee, anyway."