In 1957, Robert Solow published a short paper in the Review of Economics and Statistics. The paper did some subtraction. Solow took the growth of output in the American nonfarm economy between 1909 and 1949, subtracted the growth of the labor force, subtracted the growth of the capital stock, weighted each input by its share of national income, and looked at what was left. What was left was most of the story. Output per hour roughly doubled over those forty years. By his arithmetic, about one-eighth of the doubling came from workers having more machines behind them; seven-eighths came from the remainder — the line he labeled “technical change in the broadest sense” and that everyone afterward called the Solow residual.

The residual became the most quoted number in growth economics, and the least understood. Moses Abramovitz, running the same kind of accounting a year earlier on the period since 1870, had already given the number its honest name: a measure of our ignorance. Not of technology. Of everything the accounts had failed to write down.

What follows is the arithmetic inside that remainder — how the number is built, what it quietly absorbs, and why seventy years of attempts to shrink it have mostly succeeded in relocating the ignorance.

What the Solow Residual Actually Measures

The residual is whatever an accounting identity leaves behind. An economy produces output from a stock of equipment and structures and a flow of labor hours. Write output as a function of the two, let the function drift upward over time, and take year-over-year differences. The growth of output equals the payment-weighted growth of the inputs — labor’s wage share times the growth of hours, capital’s share times the growth of capital — plus a drift term. The drift term is the residual. The weights come from the income side of the accounts, which is the one place where the construction carries real information: it records what the market actually pays each factor at the margin, and it is why the arithmetic counts as economics rather than tautology.

Nothing in the identity says what the drift is made of. The production function Solow used descends from the one Cobb and Douglas fitted by hand in 1928, but the drift term inherits nothing from that lineage except a name. Solow called it technical change “in the broadest sense,” and broad is the operative word. Technology lives in the residual. So do organization, education, markup changes, capacity swings, weather, and every error committed by the price indexes used to deflate output and inputs alike.

The right mental model is a tide gauge. The old mechanical gauges float a pen on the water surface inside a stilling well — a vertical tube with a narrow opening near the bottom. The opening passes the slow rise and fall of the tide but damps the wind chop, so the record shows the tide and not the waves. Growth accounting is the economy’s stilling well: subtract the chop of input growth, keep the slow drift of technical change. The engineering is sound, and so is the failure mode. A stilling well cannot tell you which oscillations were noise. If some of the chop was signal — quality improving, machines idling, hours miscounted — the well damps that too, and the tide line quietly absorbs it.

How the Inputs Are Measured, and Where the Cracks Are

Capital enters the identity as a number assembled by the perpetual inventory method: each year’s investment, deflated by a price index, added to a stock that decays on a schedule. When a lathe joins the capital stock, the statistician records what it cost, not its serial number. The machine’s identity is gone at the door; only deflated dollars remain. A 1948 lathe and a 1958 lathe, run through the same machinery index, come out as equal quantities of capital, even if the newer machine holds tolerances the older one never could. If the index fails to price that improvement, the difference has to land somewhere, and where it lands is the residual.

Labor enters as hours. An hour is an hour. The accounts treat a veteran toolmaker’s hour and a first-week apprentice’s hour as the same unit, and the distance between them — training, judgment, the ability to read a worn die — feeds the drift term as well. Two brick walls built from the same number of bricks are not the same wall. The bond pattern, which bricks bear load and which tie the wythes together, does work the count cannot see. Growth accounting counts bricks.

What Happened When Economists Tried to Shrink It

In 1967, Dale Jorgenson and Zvi Griliches attacked the residual at the root. Their claim was simple: the residual is a measurement problem, and better instruments should abolish it. They rebuilt capital as a service flow rather than a stock, adjusted hours for composition and education, refined the deflators, and ran the accounting again. For the postwar decades, the unexplained portion fell to a fraction of a percent a year. For a moment the residual looked like a bookkeeping artifact, and bookkeeping artifacts can be abolished. It held for six years.

Then 1973 arrived. Output per hour, which had grown around 2.8 percent a year since 1948, fell to roughly half that and stayed there for two decades. Nothing visible had broken — no war, no destruction of capital, no collapse of the schools. The same instruments that had explained the postwar boom failed to explain the postwar stall, and they failed in the opposite direction: the residual came out too small. The Jorgenson-Griliches program had not abolished the ignorance. It had located it.

Griliches spent the rest of his career on the problem, and his 1994 presidential address to the American Economic Association — “Productivity, R&D, and the Data Constraint” — reads now as a concession. The residual-shrinking program had stalled, he argued, partly because the economy had drifted toward precisely the sectors where the statistics are weakest: services, trade, finance, health care. The data constraint had not loosened. The economy had grown into it.

Why Computers Made the Residual Worse Before They Made It Better

In July 1987, Solow published a one-line verdict in a book review: “You can see the computer age everywhere but in the productivity statistics.” Three months later, Stockholm called with the Nobel Prize. The quip stayed accurate for almost exactly eight more years.

From 1995 to 2004, output per hour grew near 3 percent a year, and the residual rose with it. Part of the acceleration was real — logistics redrawn around cheap tracking, retail reorganized around inventory data, plants rescheduled around demand signals that used to arrive by fax. Part of it was a change in the instruments. Through the 1990s the statistical agencies rebuilt their price indexes for computing on hedonic foundations, holding quality fixed by regression, and the measured price of a unit of computation fell by double digits year after year. A 1997 laptop and a 1985 machine are not the same good, and the semiconductor deflator is the piece of machinery that says so. Some of the residual’s late-nineties rise was a real machine the deflators had finally learned to see.

Then, around 2004, the residual fell back, and this time it stayed down. Output per hour settled near 1.4 percent a year — a number that looks uncomfortably like 1973.

Is the Slowdown Just Bad Measurement?

The tempting explanation for the post-2004 residual is that the statistics now miss most of what the economy produces. A search result, a mapping app, a video call carry no invoice. GDP counts production at market prices, so a good priced at zero contributes zero, however many hours of attention it absorbs. The consumer surplus is real — large, probably — and none of it is in the accounts.

Chad Syverson tested the mismeasurement explanation and found it wanting. His review in the Journal of Economic Perspectives runs the arithmetic several ways, and each way comes out against the hypothesis. For mismeasurement to explain the slowdown, the gains the accounts now miss would have to be several times larger than the ones they were already missing in the faster years of 1995 to 2004 — a jump in unmeasured value arriving exactly on the day measured growth fell. The slowdown also shows up in industries with clean, physical output, and in countries with very different levels of digital adoption. Mismeasurement exists, and always has. It does not appear to have changed sides on schedule.

There is also the older, deeper problem. In the late 1990s Susanto Basu and John Fernald showed that the measured residual moves with capacity utilization and with markups: a firm running its machines harder records a productivity gain no blueprint produced, and a sector with pricing power records productivity that is partly rent. This is the stilling well’s failure mode, measured. The gauge reads the tide, the chop, and a certain amount of water pushed around by pumps upstream, and the record does not say which is which.

Why Keep the Residual at All

Because it is a discipline before it is a measurement. The residual is the number that says: whatever you believed explained growth, most of growth is elsewhere. It has survived every attempt at elimination by absorbing each one — quality adjustment, labor composition, utilization correction — and every absorption has made the accounts more honest without making them complete. Abramovitz’s name still fits.

Seventy years on, the residual is either the most important number in economics or the most embarrassing one, and the honest position is that it is both. It measures our ignorance, and it measures the fact that we keep track of our ignorance, which is not nothing. What it has never done is shrink to a size anyone would call comfortable. There is no obvious reason it will.

Frequently Asked Questions

What is the Solow residual in simple terms?

It is the part of output growth left over after subtracting the growth of measured inputs — hours worked and the capital stock — each weighted by its share of income. Solow’s 1957 calculation attributed roughly seven-eighths of the doubling of American output per hour between 1909 and 1949 to this remainder.

Is the Solow residual the same as total factor productivity?

Nearly. The residual is the annual change; total factor productivity is the index built by compounding those changes. Practitioners use the two terms loosely and interchangeably, though “residual” carries the useful reminder that the quantity is defined by what is left over, not by what it contains.

Why can’t the residual simply be called technology?

Because it also absorbs measurement error, quality change, utilization swings, markup changes, education, and organizational learning. Solow flagged the problem in the original paper, calling the term technical change “in the broadest sense.” The broadness is the problem.

Was the post-2005 productivity slowdown caused by mismeasurement?

The evidence leans against it. Syverson’s review argues that unmeasured gains large enough to explain the slowdown would be several times bigger than the ones missed during the faster 1995-2004 period, and that the slowdown appears in industries with measurable output and across many countries at once. Mismeasurement is real; it just does not appear to have changed on cue.