Professor Syverson notes that a slowdown has come to dozens of advanced economies, more or less at the same time, which indicates it is a general phenomenon. Furthermore, the countries with smaller tech sectors still have comparably sized productivity slowdowns, and that is not what we would expect if a lot of unmeasured productivity were hiding in the tech industry.

An additional problem for the optimistic interpretation is this: The productivity slowdown is too big in scale, relative to the size of the tech sector, to be plausibly compensated for by tech progress. Basically, under a conservative estimate, as outlined by Professor Syverson, the productivity slowdown has led to a cumulative loss of $2.7 trillion in gross domestic product since the end of 2004; that is how much more output would have been produced had the earlier rate of productivity growth been maintained. To make up for this difference, Professor Syverson estimates, consumer surplus (consumer benefits in excess of market price) would have to be five times as high as measured in the industries that produce and service information and communications technology. That seems implausibly large as a measurement gap.

Keep in mind that while Facebook is free to its users, much of the tech industry consists of services that are measured and at least partly incorporated into G.D.P. For instance, the free Uber app arranges paid transportation. Even Facebook, Google and Wikipedia have a lot of their value expressed in G.D.P. figures, albeit indirectly. Those services make people eager to pay for smartphones, broadband connections and iPads, and those purchases are recorded as part of G.D.P.

From this standpoint, the measurement problem isn’t all that serious. It’s like going to an all-you-can-eat restaurant buffet. The chicken may be free, but the presence of so many quality items induces customers to pay more for the overall bundle, and that registers as a sale, regardless of the price of the individual items you put on your plate. Also, some of the value of Google and Facebook comes from the advertisements, which induce many purchases, which again are recorded in the national statistics. Consumers pay for many apps as well.

Or look at it this way: The tech economy just isn’t big enough to account for the productivity gap. That gap has caused measured G.D.P. to be about 15 percent lower than it would have been otherwise, yet digital technology industries were only about 7.7 percent of G.D.P. in 2004. Even if the free component of the Internet has become more important since 2004, it’s hard to imagine that it is so much better now that it accounts for such a big proportion of G.D.P.