Part I · The Hidden Variable

I.C — The Residual Shrinks

8 min read · 1,459 words

If Abramovitz’s “measure of our ignorance” is, in significant part, a measure of our inattention to thermodynamics, then the ignorance ought to shrink when thermodynamics is put back into the model. That is not a new idea; physicists and ecological economists have been saying something like it for decades. Most growth economists have treated the suggestion as peripheral: interesting in principle but difficult to operationalize. The difficulty is real. 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 who spent much of his career at INSEAD, and Benjamin Warr, an economist trained in systems dynamics, spent years trying to do exactly that.1 Their central move was to replace “energy” with “useful work”—exergy, in the thermodynamic vocabulary—as the relevant input variable. The distinction matters. Exergy is the portion of energy that can actually be converted into work under real conditions, and unlike energy, it is not conserved: every transformation dissipates some of it as waste heat, friction, or disorder. 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—the capacity to do useful work—is partially destroyed. What arrives at the factory is less than what left the mine, and the difference measures the conversion losses along the way.

Ayres and Warr argued that this distinction is economically consequential. What matters for production is not how much fuel you burn but how much useful work you extract from it. 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.

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, and asked how much of the residual could be explained by movements in useful work over time.1 The results were striking. 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. The residual shrank dramatically when useful work entered as an explicit input.

Those numbers require careful interpretation. A high R² does not prove causation, only fit; smooth, monotonically increasing variables will correlate strongly with output over long horizons; and the LINEX specification imposes parameter restrictions that are theoretically motivated but not universally accepted. There is a legitimate debate about whether the model is identifying a missing input or fitting curves to data that other variables would explain equally well.

What is harder to dismiss is the pattern across multiple studies. Reiner Kümmel, a German physicist who developed the LINEX approach independently in the 1980s, found similar results for Germany and other European economies.2 David Stern and Astrid Kander, using different methods and different data, found that energy and energy intensity explain a substantial share of productivity growth in Sweden over two centuries.3 The specifics vary—coefficient estimates differ, time periods differ, econometric techniques differ—but the direction of the finding is consistent: when energy and conversion efficiency are explicitly included in growth models, the unexplained residual shrinks, often substantially. The sign is stable even when the magnitudes are contested.

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 shows up 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—gains that have physical explanations and physical limits.

The converse is also suggestive. 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, among other things, a shift toward more efficient information technology and a restructuring of energy-intensive industries.4 The timing is consistent with the hypothesis that energy and efficiency are doing explanatory work that standard models leave implicit—though disentangling the causal channels remains difficult. What matters is that the correlation exists at all: if energy were truly a marginal input with a few-percent cost share, its fluctuations would not track productivity fluctuations so closely.

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 in part because it is abundant and in part because the infrastructure to produce and distribute it has been built up over more than a century at enormous capital expense. The low price does not mean that energy is unimportant; it means that the economy has invested heavily in making energy available at low marginal cost. If energy were suddenly unavailable, output would not fall by a few percent—it would collapse. The output elasticity of energy, they argue, is far higher than its cost share suggests, and the discrepancy is one reason why standard models underestimate energy’s role.

This claim is empirically testable, and the tests have produced results that are at least consistent with it. In the LINEX and related models, the estimated output elasticity of useful work is often an order of magnitude higher than energy’s cost share—twenty or thirty percent rather than two or three percent. If real, that discrepancy would explain why the residual is so large in standard models: the models are systematically underweighting an input that matters a great deal. The interpretation remains contested, and reasonable economists disagree about how to read the evidence. But the hypothesis has enough support to warrant serious attention—not as a fringe view but as a structured alternative to the standard framework.

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 suggests 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—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. The question is not whether ideas matter but whether ideas can substitute indefinitely for joules, or whether they eventually run up against constraints that no amount of cleverness can dissolve.

The next section turns from accounting to history. 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.