Part I · The Hidden Variable

I.B — The Billion-Dollar Error Term

8 min read · 1,460 words

If the ledger-and-law distinction established one fault line—between physical wealth and the claims written against it—then the history of growth accounting reveals another: between output as a number and output as a physical process. The residual that dominates conventional growth models is, in significant part, the trace of energy conversion that the framework was never designed to see.

In 1956, Moses Abramovitz set out to decompose a century of American economic growth into its constituent sources. The accounting framework he used was straightforward: add up the contributions of labor and capital, and whatever output remains unexplained is attributed to a residual—the portion of growth that measured inputs do not explain.1 What Abramovitz found was that the residual was enormous. The measurable inputs—more workers, more machines, more hours—explained less than half of the growth in output. The rest, the majority, came from somewhere else. He called it, with characteristic dryness, “a measure of our ignorance.”1

The phrase was honest, but the discipline kept the residual because it traveled well. It allowed comparison without inventorying the physical particulars of each economy—its engines, grids, fuels, machines, and constraints. The cost of that portability was explanatory blur: the residual became the place where mechanism could be implied rather than specified.

Robert Solow formalized the decomposition a year later, in 1957, with a paper that became one of the most cited in economics.2 Solow derived growth accounting from a neoclassical production function and showed how to extract the residual from aggregate data. The technique was elegant, and it gave the residual a respectable mathematical pedigree. It also gave it a name: “total factor productivity,” or TFP—a name for the gap between output and the inputs the model has chosen to count.

The Solow residual became the workhorse of growth economics. Cross-country comparisons, policy evaluations, and long-run projections all depended on it. When a country grew faster than its accumulation of labor and capital could explain, the difference was attributed to TFP growth. When a country stagnated despite investing heavily, the difference was attributed to TFP decline. The residual was versatile: it could absorb good news and bad news alike, and it did not require the analyst to specify what, physically, was changing.

To be fair, economists have never believed the residual is a single substance. It contains measurement error, changing utilization, imperfect price deflators, shifts in labor quality, sectoral reallocation, management practices, learning, and innovation. Entire literatures attempt to unpack it. But precisely because it is a bundle, it allows the most physical part of the story—the energy and material throughput that keeps the production function from becoming pure algebra—to disappear into the label.

How large is the residual in practice? In most studies of industrial-era growth, TFP accounts for somewhere between half and three-quarters of the increase in output per worker—often the majority of measured growth.3 Between 1948 and 1973, real GDP per hour worked in the United States roughly doubled. If measured inputs of capital and labor explained only 40 percent of that increase, then 60 percent—trillions of dollars of cumulative output—flowed from the residual. When a framework’s error term contains the majority of the phenomenon it claims to explain, something important is being left out of the ledger.

The Solow framework achieved its reach through abstraction. Economists wanted a model that could handle many countries and many periods without requiring detailed knowledge of each country’s physical infrastructure, and the residual delivered exactly that. By treating technology as a residual, the model remained agnostic about the mechanisms of improvement, which made it portable across contexts. But portability came at a cost. The production function at the heart of growth theory treats capital and labor as abstract inputs, denominated in value terms, without specifying what kinds of machines the capital purchased or what kinds of work the labor performed. A power station and a shopping mall are both “capital” in this framework; a coal miner and a management consultant are both “labor.” The aggregation is convenient, but it erases the distinctions that matter if you want to understand how economies actually produce output—and how physical constraints shape the possibilities.

Consider what the residual might be absorbing.

A factory installs a more efficient boiler. It can now produce the same output with less fuel, or more output with the same fuel. That improvement will show up in the data as higher productivity: more output per unit of measured input. But the improvement is not magic, and it is not “ideas” in some disembodied sense. It is a physical change in the conversion efficiency of a machine—the fraction of the energy in the fuel that ends up doing useful work rather than escaping as waste heat.

The resistance in the wires is lower. The combustion is more complete. The turbine blades are shaped to extract more momentum from the steam. The heat exchanger transfers more energy before the exhaust escapes. These are engineering facts, measurable in joules and watts. Newcomen’s first atmospheric engine converted perhaps 1 percent of the coal’s energy into useful work; Watt’s improved design reached 3 percent; a modern combined-cycle gas turbine converts over 60 percent.4 The curve is not smooth—it moved in steps, punctuated by decades of stagnation and bursts of improvement—but the direction is unmistakable. That trajectory sits inside the Solow residual, unnamed.

The same logic applies at larger scales. When a country shifts from burning wood to burning coal, it gains access to energy that is denser, more storable, and more transportable. When it shifts from coal to oil, it gains energy that is liquid and therefore easier to move through pipes and tanks. When it electrifies its factories, it gains the ability to transmit power over long distances and to apply it precisely where and when it is needed. Each transition required new extraction, transport, and conversion systems—mines, rail lines, pipelines, refineries, transmission grids—built over decades, financed and maintained like any other capital stock, but with constraints and failure modes that are not well summarized by the word “capital.” Each of these transitions shows up in growth accounting as a rise in TFP, because output increases faster than the measured inputs of labor and capital. But the transitions are not mysterious. They are changes in the energy substrate, and they have physical explanations.

The modern habit is to treat growth as a triumph of ideas, and the habit is not entirely wrong: ideas do matter, and the diffusion of useful knowledge has been one of the great engines of prosperity. But the record reads more like a succession of energy bottlenecks relieved than like a smooth accumulation of disembodied insights. The transitions that show up as discontinuities in the growth data—the British Industrial Revolution, the electrification of manufacturing, the postwar boom in the United States—are also transitions in how societies captured, converted, and deployed energy. The correlation is not accidental. Ideas about how to do things more efficiently are, in practice, often ideas about how to extract more useful work from a given quantity of fuel. And the value of those ideas depends on having the fuel to apply them to.

A steam engine is a technology, but it is also a device for turning the chemical energy in coal into mechanical motion, with a measurable efficiency that improved from Newcomen to Watt to Corliss. An integrated circuit is a technology, but it is also a device for switching electrical states at a certain energy cost per operation, and the progress of computing has been, among other things, a story of reducing that cost by many orders of magnitude. When we say that technology explains growth, we are often saying, without quite realizing it, that improvements in energy conversion explain a substantial share of growth.

Energy conversion is not the only channel inside the residual, but it is a dominant one, and it is systematically under-modeled. The Solow residual is, in significant part, a thermodynamic residual—the accumulated gains from doing more useful work with the same fuel, or the same useful work with less.

The residual is large because the framework is missing a physical input. What Abramovitz called “a measure of our ignorance” is, in significant part, a measure of our inattention to thermodynamics. The question—taken up in the section that follows—is what happens when that input is explicitly included: when energy, measured not merely as an expenditure but as a flow of useful work, enters the production function as an argument in its own right.