The Residual Shrinks

I took about half a dozen economics books, the big fat ones like Samuelson's, and looked up in the index: do the words 'energy,' 'entropy,' or 'thermodynamics' ever occur? Not once in any of them. Energy! You can't even have a dream at night without energy.
— Geoffrey West, in Complexity Economics (2020)

The economy is not a closed loop.

Walk through any operating data center in Loudoun County, Virginia, and the fact is palpable. This corridor hosts the world's largest concentration of data centers, drawing nearly five gigawatts from a grid that Dominion Energy has reinforced repeatedly to keep pace with demand that, as of late 2024, stands at roughly forty gigawatts in various stages of contracting. Electricity enters through high-voltage feeds, measured in megawatts. It drives processors that run inference on language, images, and code. And it exits as heat — enormous quantities of heat — carried away by chilled water, by industrial fans, by cooling towers that discharge it into the atmosphere. The electricity bill is the largest operating cost. The cooling bill is the second-largest. What enters as ordered energy leaves as disorder, and the useful computation happens in between, paid for by the irreversible degradation of the energy that drove it.

The circular-flow diagram in an economics textbook would show a data center buying electricity (an expenditure) and selling inference (an income), and the diagram would close. It would say nothing about the megawatts that entered, the useful work they performed, or the waste heat they became.

Standard theory treats production as a circular flow in which income becomes expenditure becomes income, and the diagram closes neatly. The abstraction is elegant but incomplete.(Georgescu-Roegen 1971, Part I) Nicholas Georgescu-Roegen, trained in statistical mechanics before he came to economics, saw the omission clearly. The textbook diagram shows money and goods circulating between households and firms as if the whole arrangement were self-sustaining. It omits both ends of the actual process.

At one end stand mine mouths, wellheads, photosynthetic capture, all drawing low-entropy energy and concentrated materials from sources that took geological time to accumulate. At the other end stand smokestacks, exhaust pipes, landfills, warm water discharge, returning high-entropy waste to a biosphere that can absorb only so much before the sinks fill. Between those endpoints runs a one-way transformation of order into disorder, the same transformation that powers every engine and sets the maintenance bill for every machine.

Useful Work

The difference between energy and useful work is visible in any aluminum smelter. At the Alcoa refinery in Point Henry, Australia, before its closure in 2014, roughly fifteen megawatt-hours of electricity entered the Hall-Héroult cells for every tonne of aluminum produced. The electricity had already been converted from coal at about thirty-five percent efficiency at the Anglesea power station, meaning that for every unit of useful work in the smelter, nearly two units had already been lost as waste heat at the power plant. The aluminum that emerged — light, strong, oxidation-resistant — embodied a chain of conversions, each with a measurable efficiency, each losing energy to entropy. The energy that entered the mine mouth as chemical bonds in coal was not the same as the energy that emerged as a billet of metal. The difference is what exergy measures, and it is what the Solow residual absorbs.

What if the Solow residual is not a measure of "technology" but an artifact of leaving energy quality out of the production function? Physicists and ecological economists have advanced this critique for decades. Most growth economists have treated it as peripheral: interesting in principle but difficult to operationalize. Energy enters the economy in many forms, at many stages, and its contribution to output is not easily captured by a single number. Measuring energy is one thing. Measuring what energy does is harder.

Robert Ayres, a physicist and industrial ecologist, and Benjamin Warr, an economist trained in systems dynamics, spent years constructing exactly that measurement.(Warr 2009, ch. 3–5) Their central move was to replace raw energy with useful work, an aggregate derived from exergy (the portion of energy that can, in principle, be converted into work) once real-world conversion efficiencies are accounted for. Unlike energy, exergy is not conserved. It is degraded in every real transformation. When a power plant burns coal to generate electricity, the energy in the coal is redistributed among electricity, waste heat, and exhaust gases, but the exergy is partially degraded. What arrives at the factory is less than what left the mine.

Ayres and Warr argued that this distinction is economically consequential. What matters for production is not how much fuel an economy burns but how much useful work it extracts from that fuel. A country that burns the same amount of coal as its neighbor but converts it more efficiently into motion, heat, and light will produce more output per unit of measured energy input. That efficiency gain shows up in standard growth accounting as higher TFP, because the model does not track conversion efficiency directly. The residual absorbs what the model does not measure.

The Empirical Record

To test the idea, Ayres and Warr constructed a production function in which useful work, defined as the product of primary energy and aggregate conversion efficiency, replaced the conventional energy input. They fitted the model to long-run data for the United States, the United Kingdom, and Japan.(Warr 2009, ch. 6–8) In their preferred specification, a function they called LINEX for its log-linear and exponential structure, the adjusted R² values exceeded 0.99 in some samples. When useful work entered as an explicit input, the residual contracted to a fraction of its former size. A series of researchers asked the same question with different data and got the same answer.

Reiner Kümmel and Dietmar Lindenberger reached similar conclusions with German and Japanese data using a LINEX production function of their own.(Lindenberger 2008) (Kümmel 2011, ch. 4–6) The functional form differed from Ayres and Warr's. The countries differed. The finding held. Kümmel's data extends through 2013. Independent verification with a different specification strengthens the case that the result is not an artifact of one team's modeling choices.

A 2022 panel study extended the finding across ten large economies.(Bergh 2022) Capital, labor, and exergy remained significant in the regression. The cross-country extension suggests the mechanism is not peculiar to the United States or to the OECD sample that anchored the original work.

The honest assessment cuts the other way. This literature has not been replicated with post-2020 data. Of roughly 2,975 citations to the Ayres-Warr book, only 11 originate from the top five economics journals. Mainstream growth economics has mostly ignored the finding rather than refuted it. That may reflect disciplinary conservatism: the profession's sunk cost in Solow-style models runs deep. It may also reflect genuine methodological concerns. Endogeneity plagues energy-output regressions. Energy demand rises with income. Output rises with energy use. Untangling cause from effect requires instruments or natural experiments that the useful-work literature has not fully supplied. The critique has not been answered. Neither has it been absorbed.

David Stern's work on energy and growth supplies additional perspective.(Kander 2012) With Astrid Kander, he traced energy and energy intensity through two centuries of Swedish data. The methods differ from the useful-work approach. The conclusion converges: energy and its efficient deployment explain a substantial share of productivity growth that standard models attribute to an exogenous residual. Stern's contribution reinforces the pattern without resolving the methodological disputes that keep the mainstream at arm's length.

The efficiency gains traced in the previous section, from Newcomen's single-digit thermal conversion to modern turbines exceeding sixty percent, are precisely what shows up in the data when useful work replaces raw energy as the input variable. Each improvement in conversion efficiency appears in the economy as an ability to do more with less, and that ability is exactly what productivity is supposed to measure. The Solow residual, which is supposed to capture "technology" or "ideas," is in part capturing the accumulated gains from better boilers, turbines, motors, insulation, and process controls. Those gains have physical explanations and physical limits.

The productivity slowdown in the United States after 1973 coincided with a period of energy price shocks and slower efficiency gains; the partial recovery in the late 1990s coincided with a shift toward more efficient information technology and a restructuring of energy-intensive industries.(Jorgenson 2001) The timing is consistent with the hypothesis that useful work is doing explanatory labor that standard models leave implicit. If energy were truly a marginal input with a few-percent cost share, it is harder to reconcile that co-movement with the idea that energy is merely a small intermediate input.

Neoclassical growth models treat capital and labor as the primary inputs, with technology as an external shifter that makes those inputs more productive over time. Energy, if it appears at all, is treated as an intermediate good, something purchased and used up in production like any other material input, with a cost share that is small and a contribution to output that is correspondingly modest. Standard cost-share accounting assigns to each input a weight proportional to its share of total costs, and since energy costs are typically only a few percent of GDP, energy's estimated contribution to growth is small.

Ayres, Warr, Kümmel, and others argue that this accounting is misleading. Low cost share does not imply low importance; it implies low marginal scarcity under existing infrastructure. Energy is cheap because it is abundant and because the infrastructure to produce and distribute it has been built up over more than a century at enormous capital expense. If energy services were suddenly unavailable at scale, output would collapse.

Cost share reflects substitutability at the margin, not necessity at the foundation. Energy appears in production functions as a residual precisely because the infrastructure that delivers it has already been paid for.

The 2021 Texas winter storm made the paradox empirical. In normal years, energy costs constituted roughly two to three percent of Texas GDP. When the grid failed for four days in February, the state suffered losses estimated at a hundred and thirty billion dollars.(Webber 2021) The cost share was small; the output dependence was total. Low marginal cost does not imply low systemic importance. It implies that the infrastructure delivering the input has already been capitalized. Remove the infrastructure, and the economy does not adjust at the margin. It stops.

This claim is empirically testable, and the tests reviewed above have produced results that are at least consistent with it. In the LINEX framework, the output elasticity of useful work is not a fixed constant but a time-varying quantity that rises as economies become more energy-intensive; by mid-century it significantly exceeds the output elasticity of labor. The output elasticities estimated by Kümmel, Ayres, and Stern are an order of magnitude above what cost-share accounting would assign. If those estimates are even approximately right, they explain why the residual is so large in standard models: the models are systematically underweighting an input whose productive importance dwarfs its market price. The interpretation remains contested. The pattern is not.

The Throughput Narrative

None of this means that energy is the only thing that matters, or that the entire residual can be explained by conversion efficiency. Human capital, institutions, innovation, management, and other factors all contribute to productivity growth, and disentangling their effects is a project that has occupied economists for decades without consensus. What the useful-work literature establishes is that one important channel, the thermodynamic channel, has been systematically under-modeled.

The standard narrative treats growth as a triumph of ideas, with energy as a supporting input that can be substituted or economized as prices and technologies change. The alternative narrative treats growth as a story about throughput, with ideas mattering insofar as they improve the efficiency with which energy is captured, converted, and applied. The two narratives are not mutually exclusive, for ideas about energy conversion are still ideas, but they have different implications for what binds and what relaxes, for where the bottlenecks are likely to appear, and for whether growth can be sustained when the physical substrate shifts. The throughput narrative does not dismiss innovation. It embeds innovation in a material context. Ideas matter; the question is whether they can substitute indefinitely for joules.

The English Industrial Revolution remains the template for how economists think about sustained growth, and it is precisely the case where energy and thermodynamics should be hardest to ignore.

Useful Work

The Empirical Record

Cost Share and Systemic Importance

The Throughput Narrative