real-world economics review, issue no. 106
subscribe for freePutting energy back into economics
Steve Keen[Institute for Strategy, Resilience and Security, UCL 1]
Copyright: Steve Keen, 2023

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Human society is energy blind. Like a fish in water, it takes for granted the existence of that without
which it could not survive.
,.
economists
Smith.

,

Neoclassical,Smith led economics astray on the vital issue of energy in the very first sentence of The Wealth of
Nations, when he stated that:THE annual labour of every nation is the fund which originally supplies it with all(Smith1776, p. 10. Emphasis added),,
Essay on Economic Theory, which was published two
decades before The Wealth of Nations:Landthe form for its production, and wealth in itself is nothing but the food,conveniences, and pleasures of life. (Cantillon 1755, p. 21. Emphasis added)
With that one word altered, economics took a terrible lurch away from realism and into fantasy.
let alone outside
was the source of the material wealth we generate.
substitution saw an action within the economy itself the work of the labourer as the source ofvalue, and the division of labour over time as the source of its growth.
,though the form wealth took
was shaped within it was correct, according to the incontrovertible Laws of Thermodynamics(Ulgiati and Bianciardi 2004; Eddington 1928, p. 37).
,
contemplated that the closed system of the economy could produce more outputs than inputs over.
The Wealth of Nations, when the Laws ofThermodynamics were developed, so Smith cannot be criticised for that mistake. But economists
today should not persist with models of production that violate the Laws of Thermodynamics.
1 www.isrs.org.uk

64real-world economics review, issue no. 106
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From the First Law, that energy is conserved, we know that there cannot be a surplus of outputs
over inputs. From the Second Law, that energy degrades when used to do work, we know thatorder declines over time in a closed system which the economy, considered in isolation from the
, .
,
,
,
2
considered in isolation from the environment, must degrade rather than grow. To explain theeconomy, we must start from a flow of energy from the environment into the economy, and end
with waste that must be dumped back into the environment, as a consequence, not merely of
growth, but of any economic activity whatsoever, whether the economy is expanding orcontracting.
Classical and Neoclassical economics developed in ignorance of these Laws, and therefore.
times in the foundational text for Neoclassical economics, his Principles of Economics (Marshall1890 [1920]), but always to describe human initiative and action, and not once in thethermodynamic sense. Energy, which should be front and centre in the economic analysis ofproduction, instead disappeared from view.

Neoclassical Economics

the Cobb Douglas Production Function

Cobb and Douglas, when they developed the now dominant Neoclassical model of production,
considered only Labour and Capital as inputs
look forward toward including the third factor of natural resources in our equations and of seeing(Cobb and Douglas 1928, p. 164). That was neverdone. Instead, after an initially rocky reception, the Cobb-Douglas Production Function (CDPF),
with only Labour and Capital as inputs, became the accepted model of production for Neoclassicaleconomists.3 The reason for its acceptance was neatly expressed by Robert Solow when hequipped to Franklin Fisher that:
had Douglas found labor's share to be 25 per cent and capital's 75 per cent insteadof the other way around, we would not now be discussing aggregate production
functions. (Fisher 1971, p. 305)
Cobb and Douglas found that result by fitting the function shown in Equation
Error! Reference source not found. to index number data, which they had laboriously assembled
from Census data and an established index of manufacturing output (see Error! Reference
source not found. in the Appendix). In Equation Error! Reference source not found., P stands
for manufacturing output, L and C for employment and capital respectively in manufacturing, andb is a constant:
2 Georgescu-

isolated thermodynamic system the available energy continuously and irrevocably degrades into an equal
quantity of unavailable energy, so that the total energy remains constant while the unavailable energy keeps. (Georgescu-Roegen 1993, p. 187)
3 Computable General Equilibrium (CGE) models are an obvious exception, with their input-output tables for

production, but over time these have become relics in Neoclassical modelling, with the Cobb-Douglas
Production Function (in raw or CESform) reigning supreme in the eraof DSGE models.

65

real-world economics review, issue no. 106
subscribe for freeP = b Lk C1 k

(1)Their regression returned the result shown in Equation Error! Reference source not found.:P ' = 1.01 L3/4 C1/4
They reported an extremely high correlation
Error! Reference source not found.,eliminatedcoefficient,

(2)
not

merelyfor

Equationwith trends

The coefficient of correlation between P and P with trends included is .97 and withtrends eliminated is .94. (Cobb and Douglas 1928, p. 154)
This implied a high level of robustness for their result, but this is not the case. The results and highcorrelations for the absolute value data are correct, but as Samuelson later observed, this waslargely due to the collinearity of the data (Samuelson 1979, pp. 929). However, their stated results-trending, which was to analyse
the three-year moving average. When annual changes are used, the results are disastrous: the2coefficient for a is negative (,,is much lower) see Error! Reference
source not found..
Table 1: Parameter values and R2 from the Cobb-Douglas index data, and annual fractional change
in the data
Economists, data & assumptions

FunctionsFitted values

R2

1. Cobb-Douglas original data

P = b. C a.L1-ab=1.02, a = 0.25

0.94

2. Cobb-Douglas change data

DP/P=a.DC/C+(1- a).DL/L

a=-0.15

0.66

The results are similarly bad when modern data is fitted see Error! Reference source notfound. in the Appendix for the Penn World Tables data for the USA from 1950 till 2019 (Feenstra,
Inklaar, and Timmer 2015) and the fractional annual rate of change. The results from fitting the
CDPF to this data are shown in Error! Reference source not found. and are similarly disastrous
for Neoclassical theory. A fit of the CDPF to aggregate data returns an of 1.24, which heavily,.
change data generates a value for
1,,
Neoclassical theory of income distribution: it attributes 71% of the change in output to Capital andonly 29% to Labour. This may in fact be more realistic, but it conflicts with distribution of income
data, and therefore with Neoclassical theory. As Solow said, had Cobb and Douglas returned,.

66real-world economics review, issue no. 106
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Table 2: CDPF fitted to PWT data for the USA from 1950 till 2019
Data

FunctionsFitted values

R2

3. PWT rgdpna, emp & rnna

P = b. C a.L1-ab= 0.013, a = 1.24

0.997

4. Annual change fraction PWT

DP/P=a.DC/C+(1- a).DL/L

a = 0.71

0.29

However, Neoclassical economists were saved the embarrassment of encountering these results.
,
achieve his objective he had to add two assumptions that the exponents in the CDPF were themarginal products of Labour and Capital, and that these were equivalent to income-share data:
The new wrinkle I want to describe is an elementary way of segregating variations
in output per head due to technical change from those due to changes in theavailability of capital per head. Naturally, every additional bit of information has its
price. In this case the price consists of one new required time series, the share oflabor or property in total income, and one new assumption, that factors are paid
their marginal products. Since the former is probably more respectable than theother data I shall use, and since the latter is an assumption often made, the price
may not be unreasonably high. (Solow 1957, p. 312. Emphasis added),,
assume that their theory of production and of income distribution were both correct, and consistent
with each other. They could then derive the contribution of change in technology from thedifference between change in GDP and change in the two income-distribution.
,,
but were simply assumed to be correct, and equal to the shares of Labour and Capital in income
distribution data roughly 1/3rd for Capital and 2/3rds for Labour (Solow 1957, Table 1, p. 315).
.
,,
87.5% of growth was attributed to technical change, and only 12.5% to changes in the factor
proportions of Labour and Capital, was only moderately embarrassing. Subsequently, Neoclassicaleconomists have since simply assumed that their models of production and distribution are correct,
and the coefficients of the CDPF have altered from flawed empirical findings to unquestioned
theoretical assumptions.
All of this was without considering energy: to this day, the vast majority of Neoclassical models of
production consider only Labour and Capital as inputs. But when energy was considered by someNeoclassicals, it was accorded the same treatment: its exponent was set by its share in GDP, andthis was assumed to be equal to its marginal productivity.

67

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The Power(lessness) of Energy?
Two of the very few Neoclassical papers that include energy in a production function and ascribe
a numerical value to it 4 are (Engström and Gars 2016) and (Bachmann et al. 2022). The former
uses an exponent of 0.03 and the latter of 0.04, in production functions of the form shown inEquation Error! Reference source not found.:F ( K , L, E ) = K

L1

E

(3)5

productivity of Energy. This led Bachmann et al. to comment that:
Therefore, for example, a drop in energy supply of Dlog E = -10% reduces
production by DlogY = 0.04 x 0.1 = 0.004 =0.4%
production is quite insensitive to energy E as expected (Bachmann et al. 2022,
Appendix, p. 5. Emphasis added).
The data begs to differ. Error! Reference source not found. and Error! Reference source notfound. to Error! Reference source not found. show Gross World Product 6 against Primary
Energy Supply 7
1 1 201 .
,Y /YE / E is 0.97, rather

as assumed by Neoclassical economists, the empirically derived value ofthan the 0.03-0.04.
,,
is Energy.

Error! Reference source not found. shows the coefficients for regressing GDP and change in
GDP |(DY/Y) against linear equations for Energy and change in Energy (DE/E).
Table 3: Regression of Energy against Gross World Product
Data

FunctionsFitted values

R2

5. OECD Energy & World Bank GWP

P=a+b.E

a=3510, b = 0.14

0.99

DP/P=a+b. DE/E

a=-0.01, b = 0.97

0.7

6. Annual change fraction

4 (Solow 1974b; Solow 1974a; Stiglitz 1974b; Stiglitz 1974a) include energy or resources, but do not provide

numerical values for the exponents.
5 Equation 4c in (Engström and Gars 2016, p. 546)
6 https://data.worldbank.org/indicator/NY.GDP.MKTP.CD.
7 https://data.oecd.org/energy/primary-energy-supply.htm.

68real-world economics review, issue no. 106
subscribe for freeFigure 1: Gross World Product and Energy Consumption over time

Figure 2: Energy vs GWP

69

real-world economics review, issue no. 106
subscribe for freeFigure 3: Ratio of GWP in US$2015 billion to Energy in MTOE from 1971 till 2019

Figure 4: Change in GWP and Change in Energy in Percent p.a.

70

real-world economics review, issue no. 106
subscribe for freeFigure 5: Correlation of change in Energy and change in GWP in Percent p.a.

This empirical data, as Bachmann et al. unintentionally show, is an effective refutation of the
Neoclassical theories of production and income distribution, and confirmation of the PostKeynesian theories.
They compare the polar opposites of the Cobb-Douglas and the Leontief in a CES production
function, where the elasticity of substitution between inputs s for Cobb-Douglas equals 1 and thatfor Leontief equals 0. They correctly lay out the implications of the Leontief case, that:log Y =,
of substitution is exactly zero, production Y drops one-for-one with energy supply
Intuitively, the Leontief assumption means that energy is an extreme
bottleneck in production: when energy supply falls by 10%, the same fraction 10%of the other factors of production X lose all their value (their marginal product drops
to zero) and hence production Y falls by 10%. (Bachmann et al. 2022, Appendix,
pp. 5-6. Emphasis added)
They plotted the theoretical relationships between energy input and GDP output for different valuesof the substitution parameter s in their Figure 1 (reproduced as Error! Reference source notfound. here).

71real-world economics review, issue no. 106
subscribe for freeFigure 6: Bachmann et al.'s theoretical predictions of change in output for a change in energy

They then rejected the Leontief function, on the grounds that its prediction of a 1:1 fall in production
for a fall in energy leads to nonsensical results in terms of Neoclassical theory:
Extreme scenarios with low elasticities of substitution and why Leontief
production at the macro level is nonsensical1 showed that output falls one-forthe marginal product of energy jumps to 1/ [their exponent for energy] while the. If … factor prices equalmarginal products, this then implies that similarly the price of energy jumps to1/
expenditure share on energy jumps to 100% whereas the expenditure share onother factors falls to 0%. We consider these predictions to be economicallynonsensical. (Bachmann et al. 2022, p. 15. Italicised emphasis added)
These predictions are nonsensical, but at the same time, the Leontief case fits the empirical data
(
,
,
).
,
.
,
,
,
respectively. The Neoclassical Cobb-Douglas model of production is false, 8 and the PostKeynesian Leontief model of production is correct. The question now remains as to why the
Leontief model is correct.
8 This empirical critique adds tothe logical critique made by Shaikh (Shaikh 2005, 1987, 1980, 1974) and
McCombie and Felipe (Felipe and McCombie 2020; Felipe and McCombie 2014; Felipe and McCombie 2011;
Felipe and McCombie 2007; McCombie 2000) that the Cobb-Douglas function is simply a nonlinear mappingfrom the income identity that Wages plus Profits equals Income, given the empirically realistic assumption ofrelatively slow changes in the distribution of income.

72

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From Empirical Regularity to the Role of Energy in Production
The Leontief Production Function began as an empirical regularity between GDP, however
measured, and Capital, however measured. The ratio was relatively constant over time andshowed no trend see Error! Reference source not found. for Capital to Output ratios derived
from the Penn World Tables database.
Figure 7: Capital to Output Ratios are reasonably constant over time

This led to the pragmatic Post-Keynesian school adopting the capital to output ratio as its,,
real explanation as to why it was found. Leaving aside the minimum form in which the LPF is often
expressed but seldom used, we have, as in the Goodwin model (Goodwin 1967):

Y=

K
v

(4)

Here v is the capital to output ratio. With K having the dimension of Widgets, and Y of Widgets per
year, for dimensional accuracy, v must be a time constant denominated in Years.
The empirical regularity behind the LPF can be explained by the aphorism that Ayres, Standish(Keen, Ayres, and Standish 2019),
that:labour without energy is a corpse, while capital without energy is a sculpture.
(Keen, Ayres, and Standish 2019):
This suggested that the inputs of Labour and Capital assumed by both the CDPF and the LPF
should be replaced by the energy inputs to both Labour and Capital, via the substitution shown inEquation Error! Reference source not found.:73

real-world economics review, issue no. 106
subscribe for freeL

L EL eLK

K EK eK(5)

Here respectively L and K stand for units of Labour and Capital, 9 EL and EK represent the energy
consumed by a unit of Labour and a unit of Capital, and eL and eK are time constants (dimensioned
by 1/Year) representing the proportion of these inputs that are turned into useful work over a Year.
This then suggests a way to derive the LPF from the dimensionality of the substitution proposedin Equation (5). In the standard single commodity CDPF and LPF, Y and K are denominated inunits of a universal commodity that can be used for
either investment or consumption:

Widget
Year

Y

(6)

The substitution in (5) on the other hand has the dimensionality of units of Energy per year:

K EK eK

WidgetEnergy Scalar
Widget Year

EnergyScalarYear(7)

Call this Q, denominated in units of Energy per year, to distinguish it from Y, denominated in unitsof widgets per year:

Q = K EK eK(8)

Y is therefore equal to Q divided by EK:

Y=
=

Q
EK
K EK eK
EK= K eK(9)Equating Equation (9) with Equation (4) shows that the empirically derived capital to output ratio v
is in fact the inverse of the proportion of inputted energy that machinery turns into useful work:K= K eK
v
1
eK =
v

(10)

9 Units of Capital raise all the issues in the Capital Controversies (McCombie 2001; Harcourt 1972), but thisformulation also enables an empirically sound way around them.

74real-world economics review, issue no. 106
subscribe for freeThis provides a physical explanation for the empirical regularity on which the Post-Keynesianmodel of production is based: it is due to the role of machinery in turning energy predominantly
fossil fuel energy into useful work. This model therefore ties Post-Keynesian theory to the initialaccurate insights of the Physiocrats, that Nature is the source of wealth, and that what human
this superfluity that nature accords him as a pure gift
(Turgot 1774, p. 9) into useful work. Given the close relationship between GDP and Energy shownin Error! Reference source not found. to Error! Reference source not found., at a first
approximation, GDP is useful energy. Equation (9) can therefore be used in place of Equation (4)in Post-Keynesian models.
Increased living standards over time can now be explained by the increasing quantity of energy EK().
an extreme comparison, Energy per machine has risen from the roughly 30 tons of coal per day
1
,
1,000
methane consumed by. 10 This increasing energy
consumption per head is the true basis of the increase in the Wealth of Nations.
The Post-Keynesian model is also consistent with the Laws of Thermodynamics, including theSecond Law (which the Physiocrats did not realise) that doing work generates waste as well as
desired output. With the capital to output ratio averaging 4 globally, and ranging between 3 and 5
for developed nations, the magnitude of eK is of the order of 0.2-0.33. This then quantifies thewaste generated in production as being of the order of 0.67-0.8: humanity generates more waste
than output. The constancy of the capital to output ratio, much criticised by Neoclassicaleconomists, is in fact due to the impossibility of substituting any other input for energy, and intrinsic
limits to the efficiency of conversion of energy into useful work given by the Second Law ofThermodynamics. 11

Conclusion
The Neoclassical Cobb-Douglas Production Function, with its exponents assumed to be equal tothe income shares of factor inputs, and also equal to the marginal product of those inputs, cannot
be reconciled with energy data, or with the Laws of Thermodynamics, and it is therefore wrong.
The Post-Keynesian Leontief Production Function, on the other hand, is not only empiricallyaccurate, but is also consistent with the Laws of Thermodynamics. Though the construction of auniversal commodity in aggregate production functions has always been a convenience, the factthat GDP and Energy are so tightly coupled means that the LPF is a reasonable first approximation.
(Solow 1957, p. 213) is correct, but the only aggregate production
function that fits the bill is the Leontief Production Function.
10See
https://www.energy-cg.com/NorthAmericanNatGasSupplyDemandFund/NaturalGasDemand_
MethaneFuelMuskStarship.html.
11

a technical nature (i.e., are due to the unavailability of tools or cylinders and pistons without friction or perfectly) it is a constraint imposed by Nature and not by the use of technically poor instruments. (Ulgiati
and Bianciardi 2004, pp. 112-13. Emphasis added).

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Appendix: Data Tables
Please see here: http://www.paecon.net/PAEReview/issue106/Keen Appendix106.pdf

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