The Residual Shrinks

A century of coal consumption was easier to recover than a century of uses for coal. In reconstructing the American energy economy, Robert Ayres, Leslie Ayres and Benjamin Warr encountered a difficulty that the national fuel total could not resolve. Coal had heated buildings, driven factory machinery and hauled trains; increasingly, it had generated electricity. The early statistics did not give them all these divisions. To estimate fuel used by different classes of engines, they allocated consumption in proportion to the horsepower recorded for each class.¹

This was an assumption about operation made from evidence about equipment. Horsepower could tell them something about the capacity installed. It could not tell them how long each engine had run, under what load, or how carefully it had been kept. Yet refusing the assumption would not produce a more faithful account of the work. It would leave the uses combined in a total that answered another question. The investigators had to construct a history precise enough to be useful from records that had not been kept for their inquiry.

The difficulty is familiar after the Centennial engine. There was a considerable distance between owning the machine and obtaining the work for which it had been bought. In a national series that distance must be followed through a succession of changing arrangements. Coal consumption alone cannot distinguish a more economical engine from one that wastes more fuel while running longer. Nor will an account of expenditure necessarily distinguish them: an advantageous coal contract can make the waste cheap. Before there is a question about how much growth energy explains, there is a question about what the measured input has been doing.

At the motor shaft

The distinction between energy and exergy gives this inquiry its physical basis. Energy is conserved through a transformation. The capacity to perform work is not. Exergy measures the maximum work obtainable as a system comes reversibly into equilibrium with a specified environment; the actual conversion loses some of that capacity. A quantity of heat near the surrounding temperature and the same quantity at a much higher temperature do not offer the same possibilities. An energy total can preserve the quantity while concealing the difference that matters to an engine.

Useful work takes the further step from available capacity to an estimated service obtained from it. In the Ayres reconstruction this meant allocating inputs among uses and applying conversion efficiencies to them, then bringing the resulting quantities into a common account. Motion, electrical work and thermal services entered through different calculations. The sum had a physical basis, but its construction required judgments about what counted as the service and where it should be measured. A joule provided a common unit. It did not choose the end of the process.

Electric motive power makes the choice unusually clear. A comparison with a steam-driven factory must follow the coal through generation, the distribution of electricity and the motor that turns it back into motion. An efficient motor cannot retroactively recover losses at the generating plant. Equally, an account that stops at generation cannot see every improvement achieved by the motor or by the machinery it serves. In their historical discussion, the researchers put a generating-and-distribution efficiency of nearly ten percent beside a motor efficiency of roughly eighty percent for the period around 1920. The combination yielded about eight percent, not eighty. The large motor percentage belonged to a later stage of the conversion.²

Obtaining electricity from fuel and delivering it belongs to a different history from putting electricity to work. They require different equipment, admit different improvements and can proceed at different speeds. A claim about the economy of electric drive must cross both. The motor's success is real, but it is not the whole undertaking.

The same care is needed when these operations enter a national aggregate. Electricity generated and the shaft work subsequently obtained from it cannot simply be added as though they were two independent services supplied by the original fuel. One can measure at the earlier stage, or follow the electricity into its uses, provided the boundary is kept consistent. The early Ayres–Warr growth account used a primary-work series in which electricity itself counted as work. A later account followed electrical output into further end uses. A better lamp or a better arrangement of a pump can accomplish more with the same electricity, while a series measured at the earlier boundary records no increase at all.³

One might wish to pursue the measurement until it reached the final purpose: the useful light, the delivered water, the successful journey. But there would still be a question about the purpose. Moving a heavier vehicle farther is a transport achievement on one measure; carrying the same passengers in a lighter vehicle may be an improvement on another. The researchers themselves confronted the difference between moving the vehicle and moving its payload. An account of work can become more informative as it approaches a use without becoming an account of everything for which that use is wanted. A hospital and a casino can both be well lit. Their equality in this measurement settles very little about them.

Across the series, changes in the amount and kind of service obtained from fuel can enter the history explicitly. Some improvement that otherwise appears only in the relation between inputs and output acquires an intermediate description: a conversion became less wasteful, a use shifted, a system delivered more work. There is now a place to look for the machines and practices that made the difference, and a basis for asking whether the estimated improvement agrees with their history. The national quantity is no longer required to carry an explanation it was never designed to contain.

Fitting the century

Ayres and Warr then asked whether this constructed series helped explain American output over the twentieth century. They compared capital, hours of labour and alternative measures of the energy input over 1900–1998. Their preferred measure included animal work as well as inanimate sources. This mattered for a history beginning when animals still supplied work that would later be supplied by engines. Leaving that earlier contribution outside the account would change the starting point against which the growth of mechanical provision was judged.

The striking result was a curve that could follow much of the historical rise in output without a separate multiplier continually increasing with time. In the authors' account, replacing raw energy with converted work greatly improved the fit of their chosen production function, particularly before the mid-1970s. The war years remained troublesome, and later output pulled ahead again. The residual shrank within this comparison: less of the observed path had to be supplied by an unexplained trend.⁴

The conversion assumptions refer to things that engineers and historians can investigate independently of the output curve. A presumed improvement in generating efficiency can be confronted with operating evidence; an allocation of fuel to engines can be challenged by a better account of their use. The interpretation has acquired obligations to a physical history. That is a substantial advance over giving the unexplained increase a name and leaving it there.

The fit is a different achievement, and it has to earn its authority separately. The parameters were selected against the historical series. A close retrospective curve is not yet a successful prediction. The statistical discussion in the inspected author manuscript also contains errors; its reported significance cannot be accepted on those explanations.⁵ The curve remains a reported result worth explaining and testing. It cannot carry the further assertion that a determinate share of economic growth has been caused by conversion efficiency. The people who improved the conversion had equipment, knowledge, finance and organizations to work with. Measuring the service they obtained does not subtract those conditions from its production.

Nor does moving a gain into an input series make the underlying advance less technological. Solow's residual was broad enough to contain changes in the quality of inputs that his available measures failed to capture. Hulten's account of disaggregation and intermediate production shows how much work has already gone into recovering such differences. The useful-work inquiry adds a particular physical reconstruction. To show what it adds to a modern economic account, one must compare it with an account that already takes the relevant services and industrial connections seriously. It is not enough to outperform an input total whose limitations have been stipulated in advance.⁶

There was more to investigate than the resemblance of two rising curves. Warr and Ayres's later study used American data for 1946–2000 to ask whether past movements in exergy or useful work helped account for subsequent movements in GDP, alongside capital and labour. Their tests allowed for a long-run relation among the series and for adjustment when they departed from it. They reported a direction running from the energy measures toward GDP; the useful-work result appeared in the long-run test, not the short-run one.⁷ This was an additional inquiry into temporal dependence, with a finding that a static fit could not supply. It still did not isolate what would happen under a particular intervention. The distinction matters when the question becomes whether to build a power station, improve a conversion or reorganize a use: these are different acts, although each may alter the measured service.

Other investigators pursued different questions. Robert Stresing, Dietmar Lindenberger and Reiner Kümmel examined cointegration with an energy-dependent Cobb–Douglas specification, not another version of the same useful-work model. A later ten-economy study tested exergy as an alternative or addition to labour alongside capital, and its country results differed.⁸ These studies make it harder to dismiss physical inputs as an unpromising subject of economic investigation. They do not establish one permanent coefficient with which to divide history among energy, labour and capital.

One can also ask how energy became cheap and dependable enough to be taken for granted.

When power becomes ordinary

Stern and Kander developed this possibility through the Swedish historical record. Their model allowed energy to constrain what capital and labour could accomplish, while also allowing the constraint to weaken as effective energy became more abundant. The term effective did important work. Their measure combined quantities, changes in the mix of energy sources and a fitted account of technological change. It was not the Ayres–Warr series under another name. They were investigating how the economy behaved as its opportunities for using energy changed.⁹

When energy services are scarce and difficult to replace, adding equipment encounters an obvious difficulty: more machines need more means of operating them. Investment can increase the demand against the constraint faster than it relieves the constraint itself. An improvement that supplies more usable power, or secures the same operation with less, then changes what further investment can accomplish. The enabling change need not retain the largest measured contribution forever. Once power is adequately supplied, another machine may wait for a skilled worker, a market or a better way of arranging production. These are possibilities created within a less constraining energy arrangement, not evidence that the arrangement has ceased to matter.

The Swedish model gave this argument a historical form. Energy-related expansion and improvement were especially important in its account of earlier growth; the technological improvement represented on the capital-and-labour side of their model became dominant in the second half of the twentieth century. Their later regime approached the behaviour of a Solow economy as energy became abundant. Solow was not an opponent left outside the model. He appeared within it, under conditions whose arrival had to be explained.

This is more demanding than declaring a neglected input indispensable. It requires an account of the transition from an obstacle that makes another investment unprofitable to a provision dependable enough for attention to move elsewhere. It also explains why a small routine payment need not resemble the importance assigned to energy in a history of industrialization. The payment is made after generations of investment and adaptation have made the service available. It cannot, by itself, tell us what those changes enabled. Conversely, their historical importance gives us no license to assign every additional unit supplied today the importance of the earlier escape.

The Swedish reconstruction does not settle how every such transition occurred. Its early data are partly estimated, and its technological trends are fitted rather than read from engineering records. But the mechanism it examines changes the question. Instead of asking which factor was secretly responsible for growth all along, we can ask what repeatedly prevented further production, how that impediment was reduced, and what became possible once it was. A fixed ranking of inputs would be an obstacle to this history. It would force successive difficulties to take the same form.

A limit without a timetable

There is an older argument here that deserves to survive the polemics with which it has sometimes travelled. Georgescu-Roegen insisted that production has a physical direction: the restoration of a financial balance does not restore the material conditions consumed in earning it. Yet he also insisted that thermodynamics does not give a process its complete path or timetable.¹⁰ Both judgments are needed. A physical limit is not an economic chronology, and a profitable chronology is not evidence that its physical conditions will renew themselves.

Improving a conversion, changing a product and finding a less demanding way to accomplish a purpose are all ways of altering the relation between a resource and a life made possible through it. The useful-work measure follows some of these changes better than others. The boundary that made the earlier history intelligible may cease to be the best boundary for the next one. That need not be a defeat for measurement. It is a reason to return to the operation, as the first investigators had to do when their fuel returns would not tell them how the coal was being used.

Abundance, meanwhile, is an arrangement that must continue to work. The Texas freeze of 2021 exposed feedback between gas supply and electricity: failures of power provision interfered with parts of the gas system, while inadequate gas supply compounded the difficulty of generating power. Busby and his coauthors traced the interaction through deficiencies in preparation and coordination.¹¹ The episode concerned the failure of a particular operating arrangement. It did not measure the ordinary marginal product of electricity. Its force is that provision which appears routine in one set of conditions can become the immediate constraint in another.

Making a constraint less oppressive is itself work. A national fuel total cannot reveal that work on its own; a growth curve cannot explain it merely by following the result. The conversions have to be reconstructed, their boundaries understood, and their changing place in production examined. Useful-work research gives part of that history a measure. It also leads beyond a single measure to the conditions under which the next expansion becomes possible.

For an economy drawing much of its power and materials from annual organic growth, those conditions included the claims made on the land. More feed for working animals, more wood for heat and more material for construction could compete with other uses of the same ground. A more economical machine might ease a claim without freeing production from that competition. Coal offered a different possibility, but its existence underground did not deliver the escape. The deposits had to be reached; the fuel had to be brought to uses that could exploit it. What matters next is how that possibility became an operating world, and whose land and labour were drawn into making it one.

Source notes

Footnotes

1. Robert U. Ayres, Leslie W. Ayres and Benjamin Warr, Exergy, Power and Work in the US Economy, 1900–1998, INSEAD working paper 2002/52/EPS/CMER, pp. 13–15, especially p. 14. The later article appeared in Energy 28 (2003), pp. 219–273, DOI 10.1016/S0360-5442(02)00089-0. The detailed account here follows the inspected INSEAD version, not uninspected published pagination. The authors distinguish locomotive, stationary-engine and generating uses; they state that early categories were not separately available and allocate by horsepower. The inference about what installed capacity alone cannot establish is this chapter's, not an additional claim about an individual engine's operation. ↩

2. Ayres, Ayres and Warr, INSEAD text, pp. 3–4 and 10–11 for exergy, work and the primary/secondary distinction; pp. 18–19 for the historical electric-drive comparison. The ten-percent and eighty-percent figures are their approximate reconstruction, not new measurements or an identified plant's performance. Generation and distribution are already combined in the first figure; multiplying by motor efficiency does not subtract network losses a second time. Eight percent follows from 0.10 × 0.80. No ratio to an unmatched steam-engine benchmark is asserted. ↩

3. Robert U. Ayres and Benjamin Warr, “Accounting for growth: the role of physical work,” author manuscript associated with the 2005 article, pp. 9–15, especially Figure 7, explicitly labelled primary work; the manuscript version is identified in the following note. Compare B. S. Warr and R. U. Ayres, “Evidence of causality between the quantity and quality of energy consumption and economic growth,” author manuscript dated 10 December 2009, pp. 9–10, on further allocation of electricity across uses. Its published version is Energy 35 (2010), pp. 1688–1693, DOI 10.1016/j.energy.2009.12.017. The author-posted text governs those internal locators. Ayres, Ayres and Warr's earlier discussion, pp. 43–45, already distinguishes electrical end-use improvement and vehicle movement from payload services. None of these physical aggregates is a welfare measure. The hospital/casino comparison is explanatory reasoning, not a reported comparison of facilities. ↩

4. Ayres and Warr, “Accounting for growth,” Structural Change and Economic Dynamics 16 (2005), pp. 181–209, DOI 10.1016/j.strueco.2003.10.003. Numerical details were inspected in the author manuscript uploaded by Warr, internally paginated pp. 1–37; these are not the journal's pages. Sections 4–6 specify deflated US GDP, capital, labour hours and alternative raw or converted inputs for 1900–1998. Equation (3) is a linear-exponential (LINEX) function with a time-independent multiplier and parameters chosen by nonlinear fitting. Figure 8, p. 17, reports mean-square errors of 22.9 for commercial energy, 3.6 for the corresponding work series and 0.9 for the broader work series including animal work. The broad series' R/B lettering differs between legend and prose. Appendix B defines the statistic as summed squared fitting errors divided by n−k. These are reported errors on an indexed-output scale, not percentages of growth. The full estimation data and fitted parameter values were not available here; this chapter does not claim to have reproduced the fit. The abstract of the published article supports the broad account of the improvement and the later shortfall, but its final numerical presentation has not been independently checked. ↩

5. In that author manuscript, Appendix B, pp. 36–37, large p-values are described as establishing significant correlations, and a Durbin–Watson value of 0.2387 is followed by acceptance of no serial correlation. Neither supports the stated conclusion. Equation (4a)'s printed capital elasticity also fails to differentiate the displayed equation (3) correctly. These may include typesetting errors, but no fitting code is available to resolve their consequences. The chapter therefore reports the historical curves without treating this manuscript's significance account or printed elasticities as verified. This criticism is restricted to the inspected version. It does not assert that an uninspected final typesetting contains the same errors, or that the underlying physical series was fitted to manufacture them. ↩

6. Solow, “Technical Change and the Aggregate Production Function” (1957), pp. 312–315; Charles R. Hulten, “Total Factor Productivity: A Short Biography,” published in New Developments in Productivity Analysis (2001), pp. 5–10, 13–15, 36–38. Editions and verification are those used in Chapter 2. Service reconstruction, disaggregation and a causal allocation of growth remain different achievements. The Ayres–Warr fitting comparison is not a head-to-head evaluation against a contemporary capital-services and industry-accounting system. ↩

7. Warr and Ayres, December 2009 manuscript, §§3.1–3.3, pp. 11–15, associated with the 2010 Energy article. Two vector error-correction models use GDP, capital, labour and alternatively exergy or useful work, with three lags and two cointegrating vectors. The reported short-run exergy result, absent short-run useful-work result and long-run findings agree with the published abstract. Granger causality concerns restrictions on prediction and adjustment within the selected variables and specification; it does not by itself identify the effect of an intervention. No numerical test statistic or production elasticity is taken from these models. ↩

8. Robert Stresing, Dietmar Lindenberger and Reiner Kümmel, “Cointegration of output, capital, labor, and energy,” European Physical Journal B 66 (2008), pp. 279–287, inspected in their EWI Working Paper 08.04, §§3–6, Tables 1–3. German, Japanese and US sector/total-economy samples have different periods and unequal strength of cointegration evidence. Their constrained Cobb–Douglas exercise is distinct from the related LINEX studies it discusses. Robert U. Ayres, Ivan Savin, Jeroen van den Bergh and Lu Hao, “Exergy versus labour in aggregate production functions: estimates for ten large economies,” International Journal of Exergy 38 (2022), pp. 320–332, especially pp. 323–328 and Table 2, inspected in the VU publisher PDF. All three coefficients are significant in the pooled model; in separate estimates, exergy and labour do not have the same pattern of significance. These are exergy regressions, not a replication of the converted-work mechanism. The empirical comparison record preserves periods, data definitions, imputations and country differences. ↩

9. David I. Stern and Astrid Kander, “The Role of Energy in the Industrial Revolution and Modern Economic Growth,” Energy Journal 33, no. 3 (2012), pp. 125–152, DOI 10.5547/01956574.33.3.5, especially pp. 129–132, 136–141 and 145–148. The published text combines a capital–labour Cobb–Douglas aggregate with energy in a nested CES function. Energy quality is measured through price-weighted changes in the fuel mix, and augmentation is estimated. Full-model estimation uses 1853–1998; the 1800–2000 simulation requires additional assumptions for the earlier years. Under their nesting, the parameter termed labour augmentation cannot be distinguished from capital augmentation; it is not an independently measured improvement attributable exclusively to workers. The later historical contrast is the authors' interpretation of their model, not a new causal estimate. Their discussion expressly limits the model to exploring a mechanism through which relative energy abundance constrains or enables growth. The 2011 working paper is a different edition; its page references are not used here. ↩

10. Nicholas Georgescu-Roegen, The Entropy Law and the Economic Process (Harvard University Press, 1971), introduction, pp. 1–3 and 10–13, especially p. 12, inspected in a university teaching scan of the original. Irreversibility and the absence of a complete physical timetable are both part of his argument. The chapter does not adopt his blanket criticism of economic models, equate circular-flow accounting with physical self-sufficiency, or derive economic value from entropy. ↩

11. Joshua W. Busby and coauthors, “Cascading risks: Understanding the 2021 winter blackout in Texas,” Energy Research & Social Science 77 (2021), article 102106, §§2–3, DOI 10.1016/j.erss.2021.102106. The gas/electricity feedback and deficiencies in winter preparation are reported in the authors' retrospective. No loss-to-expenditure ratio, claim of total dependence or ordinary output elasticity is inferred. The closing passage's organic-resource question is developed in Chapter 4; it does not assume that coal's availability alone accounts for industrialization. ↩