President Trump’s vow to raise the annual growth rate to 4 percent has been dismissed as outlandish by economists.

But what if the economy has already been growing at that rate — but no one realized it?

This isn’t a matter of rigged statistics, political spinning or fanciful alternate realities, but one of measurement.

As the economy has shifted from one that primarily produced things — refrigerators and cars, guns and shoes — to one that now deals largely in services and information, economists have grown more and more skeptical that the traditional measure of gross domestic product — the nation’s total output — is accurately capturing much of the economy’s innovation and improvements.

“I think the official data on real growth substantially underestimates the rate of growth,” said Martin Feldstein, an economist at Harvard.

These calculations may seem to be the esoteric domain of number crunchers, but the yearly growth figures — fairly or not — end up being used to answer several essential questions. Is the country getting richer? Are standards of living rising? Are businesses and workers more productive? Is the economy improving?

Indeed, the measure was initially created to back up President Herbert Hoover’s overly buoyant claim in 1930, based solely on sprinkled anecdotes of improvements, that “the Depression is over.”

Gross domestic product has always been an imperfect answer to such questions. It is designed to measure production and just production — not welfare or happiness.

Since the end of the recession in 2009, the government has estimated that G.D.P. has trudged forward at an average annual rate of roughly 2 percent. Last year, as the Commerce Department reported on Jan. 27, growth amounted to a disappointing 1.6 percent. That’s certainly better than the shrinkage that occurred during the recession, but nowhere close to the typical yearly gains of 3-plus percent or more that undergirded American prosperity for decades after World War II.

Such ho-hum performance has become both an emblem of and explanation for the anxiety that smears the economic outlook like a bathtub ring. And every day, decisions about government policy, investment, regulations, taxes, trade and more are based on such measurements.

At its most basic, G.D.P. is calculated by looking at prices — the price of materials, workers, overhead and so on that it costs to make a product and the price that consumers in turn pay for that product.

And while prices can be measured, they don’t necessarily reflect the value of quality and experience. As far as G.D.P. is concerned, a delectable $20 meal that would wow Julia Child is equal to a rubbery, tasteless one that costs the same amount.

The growing suspicion, however, is that in a digital world overflowing with free services like Facebook, Google and YouTube, price is an increasingly ill-suited proxy for value.

What is the worth of a free software update that protects against a nasty virus? Of the streaming service that enables you to watch shows on your computer instead of on a television? Of the hours and hours saved by looking up a fact on Wikipedia rather than having to go to a library? All have productive value but no price.

Trying to measure an economy as large and complicated as the United States’ is daunting even under the best of circumstances. Ever since the Nobel Prize winner Simon Kuznets helped create the government’s first estimate of the national income in 1934, the comprehensiveness and accuracy of the measure have been debated. Kuznets, for example, wanted to include the value provided by mothers taking care of their children and the home, which would cost a sizable amount if it were paid labor.

Kuznets lost that battle, but economists and statisticians have continually struggled to compensate for the measure’s shortcomings — like qualities not reflected in price and the constant cavalcade of new goods and services.

Mr. Feldstein and other critics, some of whom raised the issue at the recent meeting of the American Economic Association, are quick to point out that government economists who calculate a nation’s G.D.P. do an extremely difficult job well with the tools available.

They are constantly adjusting and fine-tuning for quality, trying to account not only for the better-built mousetrap, but also the new mousetrap app, its upgrades and sometimes even the version sold on the black market. (When the European Union decided to include recreational drugs and paid sex work in 2013, Britain’s G.D.P. grew by 0.7 percent.)

Economists seek to attach values to specific improvements, whether faster internet speed, less pollution or a medical treatment that leaves you healthy enough to water ski instead of stuck in a hospital bed. And they look at ad revenue that free services like Google and Facebook rack up and jobs they create.

Still, given how much of people’s time and activity has migrated online, there remains a sizable chunk of “digital dark matter” that is not counted. At the same time, measuring the increased value to consumers can be even more difficult when it comes to services and experience than it is with goods.

“It’s hard to evaluate consumer preferences,” said Richard Freeman, an economist at Harvard, and it may be even harder to figure for “things where knowledge is involved and for some kind of qualitative changes.”

The best biology textbook in the 1950s, for example, would still not have the accuracy, depth and complexity of one published in 2017. “A doctor who graduates today knows and learns many more techniques and medicines than someone who graduated years ago,” Mr. Freeman said. “And we don’t really build that in.”

Mr. Feldstein likes to illustrate his argument about G.D.P. by referring to the widespread use of statins, the cholesterol drugs that have reduced deaths from heart attacks. Between 2000 and 2007, he noted, the death rate from heart disease among those over 65 fell by one-third.

“This was a remarkable contribution to the public’s well-being over a relatively short number of years, and yet this part of the contribution of the new product is not reflected in real output or real growth of G.D.P.,” he said. He estimates — without hard evidence, he is careful to point out — that growth is understated by 2 percent or more a year.

The critical question, though, is whether growth and productivity figures are being underestimated more than ever before.

Last year’s annual report from the president’s Council of Economic Advisers pointed out that there have always been services — like electricity and indoor plumbing — that consumers value more than is reflected in market prices. The report concluded that while the mismeasurement might have worsened in recent years, the problem of digital dark matter was modest (one estimate puts it at 0.4 percent) compared with other hard-to-measure sectors like health care.

“Growth statistics have always missed some important quality improvements, and it’s not obvious that we’re missing more today than in the past,” said Jason Furman, the head of the council under President Barack Obama and now a senior fellow at the Washington-based Peterson Institute for International Economics.

Mr. Furman said that Americans were feeling a genuine slowdown in growth, but aggravating the impact was that the growth that had occurred “is less equally shared than before.”

For Mr. Feldstein, it is misleading measurements that are contributing to a public perception that real incomes — particularly for the middle class — aren’t rising very much. That, he said, “reduces people’s faith in the political and economic system.”

“I think it creates pessimism and a distrust of government,” leading Americans to worry that “their children are going to be stuck and won’t be able to enjoy upward mobility,” he said. “I think it’s important to understand this.”