Workers at In-N-Out, whose food is pictured here, make at least $10.50 an hour. (vagueonthehow/Flickr)

About 20 workers were arrested this morning for arriving at the busy Times Square McDonalds where they work, sitting down, and refusing to move. It was one of about 100 similar sit-ins that are planned across the country as fast-food workers demand a pay raise to $15 an hour, about $6 more than they currently make.

This will be the seventh such one-day, fast-food strike since 2012. The wage increase would likely be good for workers: Currently, they'd have to be making about $13 an hour just to afford a one-bedroom apartment on much of the east coast. Though labor groups support (and underwrite) the workers' efforts, conservative groups argue it would be bad for fast-food franchise owners, who they say wouldn't be able to cope with higher labor costs.

Indeed, several experts suggest that a wage increase of this magnitude would raise the price of fast-food menu items by about 10 to 20 percent. Using the higher estimate, that means a Big Mac would go for $4.79 (compared to about $3.99 today), while a Wendy's Spicy Chicken Sandwich would set you back $5.27.

But there's yet another potential side effect in the fight over worker pay: If fast-food wages rise, and fast-food prices do too, would Americans eat less of it? In other words, as KFC cashiers' paychecks grow, would Americans' collective waistlines shrink?

This is important because eating junk food is widely thought to be correlated with obesity, and tinkering with its price (think soda taxes) is one of policymakers' favorite ways of trying to "nudge" people toward health. Wouldn't it be great if we could help restaurant workers while helping ourselves?

The first thing to determine is whether people buy less fast food as prices rise. The answer seems to be a "yes, but not by much."

In 2014, the agribusiness professor Timothy J. Richards and the economist Lisa Mancino looked at how eating patterns shifted as the prices at various types of restaurants changed. From a sample of about 4,800 U.S. households, they found that expensive restaurants were the most "price elastic"—their customers reacted most strongly to fluctuations in menu price—but fast-food joints were the least elastic. People wanted their Doritos Locos Tacos even if they had to shell out a few more quarters for them.

The drop in demand was certainly there for fast food—each 10 percent increase in price led to a 7.4 percent drop in sales—but it wasn't much. Most other studies have similarly found that a 10 percent rise in fast-food prices cuts sales by roughly the same amount.

"When prices are rising, consumers prefer to change the type of restaurant they visit, rather than forego the experience entirely," Richards and Mancino wrote. They might head to Arby's rather than Applebee's, but they won't be cooking for themselves either way.

What's more, there are different ways to think about the "price" of a meal, and, in the case of fast food, time is money. A 2001 study found that the proliferation of fast-food places reduced the costs associated with eating it. A greater density of the restaurants meant it took less gas money and less time away from work to get there. Thus, consumption increased. So even if your burger cost $1 more, you might still buy it because there's just no way to drive through a Trader Joe's produce section.

The second part of the puzzle is whether people will substitute their foregone McNuggets for a homemade kale salad or grocery-store frozen chicken nuggets. Richards and Mancino found that people are actually more willing to economize on food they eat at home than on fast food. However, another paper published in 2010 found that customers were more sensitive to changes in the prices of restaurant foods than to those of fresh eggs, poultry, fruits, and vegetables. But even so, getting people to switch from eating fast food to eating healthy food at home would mean fresh meat and produce would have to become a lot cheaper. And that brings us back to a vexing problem at the heart of the obesity epidemic: Eating good-for-you food is more expensive, by about $1.50 a day.

The takeaway here might be that the demand for a cheap, convenient meal seems ingrained and relatively stable. Fast-food chains always seem to find a way to get hungry customers in the door, price hikes or no.

Case in point: In 1997, McDonald's tried cutting the price of a Big Mac from $1.90 to 55 cents (if purchased with fries and a drink) in an attempt to drive up sales. The promotion was a failure—franchise owners were throwing away all the extra Big Mac buns they had ordered in anticipation of a rush. At the same time, they had trouble keeping up with all of the orders for Happy Meals because of another faddish promotion that was going on. That's the year each Happy Meal came packaged with a tiny plush toy called a Beanie Baby.