While middle-class wages haven’t grown much in inflation-adjusted terms over the past few years, that’s not the case for many lower-wage workers. Recent analysis by economist Elise Gould shows, for example, that while median pay was unchanged in 2017, low pay — the 10th-percentile wage, meaning 90 percent of workers earn more — rose at a strong clip of 3.7 percent. Follow-up work by Gould shows roughly similar results through the first half of this year.

There are (at least) two explanations for this pattern. First, when the economy sniffles, disadvantaged workers catch pneumonia. That is, lower-paid workers get disproportionately hurt by labor market slack and vice versa. High earners are far less sensitive to the ups and downs of the business cycle. Low unemployment provides low-wage earners the bargaining clout they lack in slack economies.

The second explanation is even simpler: minimum wage increases. I realize it’s a very big “duh” to point out that raising the pay of low-wage workers ends up … um … raising their pay, but this simple fact gets obscured in the debate over raising minimum wages.

If you follow this policy, you know that I’m of course talking about state- and city-level increases. The federal minimum wage has been stuck at a ridiculously low $7.25 for almost a decade. But since the 1990s, subnational entities have stepped up and raised the minimum wage. Some of the places are the usual suspects, such as California, New York, Massachusetts and the District. Some other places, such as Arizona and South Dakota, might surprise you (the pay floors in both states hit $10.50 and $8.85, respectively, this year). But at this point, the federal minimum wage is basically the Southern minimum wage.

In fact, the figure below shows how much faster low wages went up in states that raised their minimums. The differential for female workers is particularly large.

Recent analysis I’ve done suggests that both of these factors — higher minimum wages and low unemployment — combine for a potent one-two punch to knock out wage stagnation for low-wage workers.

Based on a data set of low wages by state over the past couple of decades, I built a simple statistical model that predicts real 10th-percentile wages, using just a few variables, including the unemployment rate and where your state minimum wage is relative to the federal rate. This approach lets us to test whether low-wage workers get bigger bumps in places with higher state-level minimums compared with the federal level.

The modeling reveals that this is, unsurprisingly, the case. In California, for example, low wages rose quickly from 2015 through 2017, up by 10 percent in real terms. My results find that their increase in the minimum wage, which went from $9 to $10 in 2016, explains about a third of those gains, while lower unemployment explains about 10 percent. The magnitude of these results is just about the same for Arizona, which significantly raised its minimum from about $8 to $10. Real low wages were up 10 percent there from 2015 through 2017, with the minimum-wage increase accounting for a third of the 10th-percentile wage gain.

On the other hand, even as unemployment fell in Texas last year to a low 4.3 percent, low wages also fell slightly, showing that low-wage workers can’t always count on falling unemployment alone to help them get ahead. But had Texas raised its minimum from, say, $7.25 to $8.25, that would have more than offset the Lone Star State wage decline. Low-wage workers in Texas need the belt-and-suspenders policy of tight labor markets and higher minimum wages.

So, with apologies for repetition, the recipe for boosting the pay of low-wage workers has at least two clear ingredients: low unemployment and higher minimum wages. That’s not the complete recipe, of course. Unions, job training/apprenticeships, work supports (wage subsidies, health care and housing), direct job creation in places where labor demand remains low, and an accommodative Federal Reserve all matter a lot, too. But of all of these, higher minimum wages are the most direct way to raise market wages for low-wage workers.

At this point, if not sooner, someone always raises the specter of minimum-wage increases leading to job losses among low-wage workers. There’s been extensive, careful research on this question, tapping precisely the kind of regional variation that drives the model cited above. Summarizing, minimum-wage increases of the magnitudes we’ve seen in recent years do not generate significant job-loss effects. Even in cases where some losses occur, the benefits of the wage hike far surpass its costs, meaning low-wage workers come out ahead (they may work fewer hours but at higher pay).

I can show this using my model as well. Using state-level data on total employment, restaurant employment and fast-food employment, I ran the model with the same control variables: unemployment and the state minimum relative to the federal. In every case, the correlation was a big negative for unemployment (obviously, as lower unemployment and faster job growth are inversely correlated) but zero for the minimum wage variable.

Intrepid nerds are welcomed to go to my blog for statistical details, but rest assured, these are all common findings. What’s far too uncommon is the willingness of policymakers to recognize that raising pay for low-wage workers is no mystery. They just have to shut their doors on the low-wage employer lobby and raise the minimum wage.