SALT LAKE CITY — Stephanie Pappas and her brothers built their roofing supply company in this fast-growing region by promising next-day delivery, but lately they’ve been forced to tell some customers that tomorrow is impossible.

Their company, Roofers Supply, employs 28 drivers across Utah, and Ms. Pappas said she would need at least 15 more to meet the exploding demand for shingles and tiles. The company has raised its starting wage by 10 percent since the beginning of the year to $17.50 an hour, but it’s not enough.

“We never want to have to say, ‘We can’t do it,’ but we need people,” Ms. Pappas said.

The Run-Up

The podcast that makes sense of the most delirious stretch of the 2016 campaign.

After eight years of steady growth, the main economic concern in Utah and a growing number of other states is no longer a lack of jobs, but a lack of workers. The unemployment rate here fell to 3.1 percent in March, among the lowest figures in the nation. Nearly a third of the 388 metropolitan areas tracked by the Bureau of Labor Statistics have an unemployment rate below 4 percent, well below the level that economists consider “full employment,” the normal churn of people quitting to find new jobs. The rate in some cities, like Ames, Iowa, and Boulder, Colo., is even lower, at 2 percent.

That’s good news for workers, who are reaping wage increases and moving to better jobs after years of stagnating pay that, for many, was stuck at a low level. Daniel Edlund, a 21-year-old call center worker in Provo, Utah, learned Monday that his hours were changing. On Wednesday, he had his first interview for a new job.

“I’m trying to find a company that treats you well,” he said.

But labor shortages are weighing on overall economic growth, slowing the pace of expansion in northern Utah and other fast-growing regions even as unemployment remains stubbornly high in Rust Belt cities like Cleveland and in regions still recovering from the 2008 recession, like inland California.

To Todd Bingham, the president of the Utah Manufacturers Association, “3.1 percent unemployment is fabulous unless you’re looking to hire people.”

“Our companies are saying, ‘We could grow faster, we could produce more product, if we had the workers,’” he said. “Is it holding the economy back? I think it definitely is.”

President Trump continues to promise that he will accelerate job growth by cutting taxes and regulations. But the accumulating evidence that workers are getting harder to find, and that wages are rising more quickly, has convinced many economists that significantly faster growth is unlikely. The Federal Reserve has cited the trend as its reason for moving to wind down its own economic stimulus campaign. The Fed may raise interest rates again at its next meeting in June.

Qualtrics, which conducts online market research, is a prime example of the rapid growth of the Utah economy — and the sense that Utah is straining at the limits of its growth potential. Scott Smith started the company with his son, Ryan, and a college classmate in his Provo home in 2002. Qualtrics now employs 1,300 people, including about 800 in a new headquarters building opened in August at the mouth of Provo Canyon. And it is bringing workers to Utah as fast as it can.

Each Monday, the company ties red balloons to the desks of that week’s batch of new employees. Last week, there were several dozen of those balloons. The parking lot outside the new headquarters building is already overstuffed, including many cars that still have out-of-state plates.

Ryan Smith, now the chief executive, said Qualtrics had hired about three dozen graduates from the University of Michigan alone last year. The company estimates that new arrivals bought 100 homes in Provo last year.

Utah’s tech scene is growing alongside the company. More local university students are studying engineering; more start-ups are popping up in the region, which boosters would very much like everyone to call “Silicon Slopes.” But by the end of the year, Mr. Smith said, he expects the company will have more employees outside Utah than in its home state. It is growing where it finds workers.

Companies in Utah, as in the rest of the country, were slow to raise wages in recent years. At first there were plenty of available workers. But by the end of 2015, a report by Utah’s Department of Workforce Services concluded that inadequate wages had become a key reason companies were struggling to find employees.

“It was as if employers hadn’t adjusted their approach to the labor market” as the economy recovered, said Carrie Mayne, the department’s chief economist.

Now there are signs the logjam is breaking. Adam Himoff, the president of Xemplar Skilled Workforce Solutions, a recruiting firm hired by Roofers Supply to find drivers, said he had seen an increase this year in the willingness of clients to raise wages.

“Labor has become the constraint on their growth goals, and they’re recognizing that they’re going to have to increase wages to achieve what they want to achieve,” he said.

Ms. Mayne said the state also saw signs of what she described as a broad-based acceleration in wages in the most recent data, through the end of last year.

But the share of Utah adults who have withdrawn from the labor force remains higher than before the recession. Last year, 31.7 percent of adults in Utah were neither working nor looking for work, up from 28.2 percent in 2006. That is part of a broad national trend.

And a 3.1 percent unemployment rate still means that about 50,000 people in Utah were trying to find jobs in March.

Some, like Monica Von Strahl, expect to find work quickly. Ms. Von Strahl, 44, moved to Utah from Oregon in April for family reasons. She left a job as a caregiver for adults with disabilities that paid $16 an hour; so far, the most she has been offered in Utah is $10 an hour. She plans to keep looking a little longer. (Scholars at M.I.T. estimate that a living wage in Utah for a single person is $10.71 an hour.)

But even in a red-hot market, some of the people who are looking for work struggle to find the right fit. Noel Nampijja, 42, left her job as a nurse’s aide two months ago because the work of moving patients was hurting her back. She just completed training as a phlebotomist, a medical assistant who draws blood.

“I’m hoping to find a job that won’t hurt as much,” she said.

In less lucrative industries, labor shortages may remain an intractable problem.

Ron Gibson, a fifth-generation dairy farmer, tends 1,500 cows on family land outside Ogden. Last month, he placed an ad in local papers seeking three workers at wages starting around $12 an hour. It did not draw any responses.

Mr. Gibson cannot afford to chase workers by raising wages. The price of milk, adjusted for inflation, is lower now than in the 1980s. Instead, he is producing less milk. Each cow is milked three times a day; only 15 percent get a fourth milking.

He also laughed at the idea that Americans might move from other states to milk cows in Utah. He relies primarily on immigrant labor, communicating with his two dozen workers in the Spanish he learned as a young Mormon missionary in Argentina. And since Mr. Trump’s election, he said, workers are harder to find.

“We are either going to import workers or we are going to import milk,” Mr. Gibson said.

The work “is dirty, stinky and hard,” he added. “It’s not what we teach our young people to do.”

But there is another solution on the horizon: automation. Last year, Mr. Gibson and his son visited a farm in upstate New York where robots milk cows. The cows learn to approach the machines when their udders are full.

Mr. Gibson is not yet ready to make the jump. Each machine costs half a million dollars, and the New York farmer spends about as much on mechanics as he spent on farmhands. But Mr. Gibson said he expected his children would use robots to milk cows.