*This working paper is a slightly edited version of a paper that has been submitted to the*

**Journal of Ecological Economics**Steve Keen, University College London

Robert U. Ayres, INSEAD, France

Russell Standish, Kingston University London

### Abstract

We derive two energy-driven models of production from first principles, starting from the observation that energy cannot be treated as an independent input from labour and capital: instead labour and capital are complementary means to harness different types of energy flows to perform useful work. One (Equation ) uses Neoclassical assumptions about substitutability of factors of production, the other (Equation ) employs Post Keynesian assumptions about fixed proportions between Labour and Capital at a point in time.

### Introduction

Aggregation problems with production functions using labour and capital alone have been well explored in theory (Sraffa 1960; Samuelson 1966; Pasinetti 2000), but not well acknowledged in practice: aggregate production functions are the rule in macroeconomics, in both Neoclassical and Post Keynesian economic modelling.

The constant returns to scale Cobb-Douglas Production Function (henceforth *CDPF, *see Equation ) (Cobb and Douglas 1928), with output (*Q*) produced by labour (*L*), machinery (*K*) and "total factor productivity" (*A*), remains the dominant aggregate production function in economics, despite these and its own additional well-explored, but in practice ignored, problems of tautology (Solow 1957; Shaikh 1974; McCombie 2000) and incoherent dimensionality (Barnett 2004).

Post Keynesian modellers use a Leontief form, where output is the minimum of either labour productivity (*a*) times Labour (*L*) or machinery (*K*) divided by the capital-output ratio (*v*), where full capacity utilization of machinery sets the absolute maximum output level, and where labour productivity is assumed to grow exponentially over time:

In this paper, we show how both approaches can be replaced by energy-dependent production functions.

### Cobb-Douglas

As Barnett remarked, "It is probably no exaggeration to claim that the CD is the most widely used mathematical example in all of neoclassical economics" (Barnett 2004, p. 96). It is so dominant that it is simply taken as a given: the archetypical pre-crisis Neoclassical macroeconomic model used a *CDPF* without discussion (Smets and Wouters 2007, p. 12, Equation 5), as does the recent lead paper in a special issue of the *Oxford Review of Economic Policy* on "Rebuilding Macroeconomic Theory" after the crisis (Vines and Wills 2018, p. 32, Equation 3).

In this paper we focus upon three critical deficiencies of this Equation—the meaningless dimensions of the inputs *L*, *K* and especially *A*, the ignored problem of aggregating capital, and the absence of energy as an input to production—to derive dimensionally and aggregatively sound alternative equations, in which energy plays an essential role. We apply similar logic to make the slight modifications necessary to the Post-Keynesian/Leontief production to make it consistent with the essential role of energy in production.

### Dimensionality

Barnett notes that:

The consistent and correct use of dimensions is essential to scientific work involving mathematics… their existence … makes possible dimensional analysis, which can be a significant factor in avoiding error. In the equation y = f (•), if y should have dimensions then so also should f, and they should be identical to those of y. If y should not have them then neither should f have them. (Barnett 2004, p. 95)

As a scalar quantity in theoretical papers, *Q(t)* is the annual production in year *t* of an aggregate commodity which Barnett calls "widgets", and which used to be called "corn" (Skourtos 1991). *Q(t)* in Equation thus has the dimensions "widgets/year" (*wid/yr* in Barnett's notation), which means that the right hand side of must have the same dimensions. The inputs *L(t)* and *K(t)* have the dimensions of labour hours per year and machine hours per year respectively (*manhr/yr* and *caphr/yr* in Barnett's notation), and these are raised to non-integer powers (*1-*) and ** respectively. Prior to the specification of *A(t)*, the *CDPF* therefore has the dimensions shown in Equation

This necessarily means that "total factor productivity" *A* must have the dimensions:

This dimensionality is meaningful only if has the integer values of 0 or 1, but in either case it implies that output is produced either by labour alone (line 2 in Equation ), or by machinery alone (line 3 in Equation ):

If the empirically derived value of 0.3 is used (which is also the preferred value, given its congruence with the marginal productivity theory of distribution), Barnett asserts that the dimensions of *L* and *K* are meaningless, while that of A is, as Barnett puts it, "even more meaningless, if that is possible" (Barnett 2004, p. 95). The dimensionality of the conventionally calibrated *CDPF* is as shown in Equation :

### Aggregation

The key issue established in the Cambridge Controversies was that machinery cannot be aggregated in any unique way, because the amount of machinery measured in terms of the unique invariant measuring stick that Sraffa devised, the standard commodity, depends on the distribution of income. The measured magnitude of machinery, expressed in terms of aggregated dated labour time, therefore changes in a nonlinear way as the rate of profit varies (Samuelson 1966).

This made aggregate production functions non-viable, when they were expressed in terms of output being produced by Labour and Machinery alone. Labour can be aggregated, albeit poorly, in terms of homogeneous unskilled labour time. This is done in existing production functions, from the Cobb-Douglas and *CES* forms used by Neoclassical economists today, to the Leontief form used by Post Keynesians..

However, energy, for the sake of modeling, *can* be treated as homogeneous: though it originates in different forms (fossil fuel, renewable, nuclear), it is converted into electricity for most uses outside transportation, and is measured in consistent units across its various uses. This applies to both production and consumption, and at the disaggregated level as well as the aggregate: energy input and exergy output per industry can be measured, and indeed is measured by the *Global Climate and Energy Project* (*GCEP*) at Stanford University (https://gcep.stanford.edu/research/exergy/data.html), amongst others.

This brings us to problems on the *LHS* of the all existing production functions as well, and not merely the *CDPF*. Just as the concept of aggregated machinery is a fallacy, so is the concept of an aggregate commodity for *Q*. Just as there is no meaningful, income-distribution-independent way to derive an aggregate quantity of representative machines in production, there is no meaningful, income-distribution-independent way that the array of commodities produced in capitalism can be reduced to a representative aggregate commodity.

However, it is feasible, as we discuss below, to define output (Gross Domestic Product, GDP) in the more homogenized form of useful work. Then aggregate *Q* can be replaced by Q, a term expressing aggregate exergy, which has the units of joules (preferably gigajoules) per year (hereinafter GJ/Yr). From dimensional analysis, it is clear that the *RHS* of the production function must also be expressed in GJ/Yr, rather than the current dimensionality, and should represent the exergy of the inputs, also with the dimensions of GJ/Yr: as Barnett emphasises, both sides of the equation must have the same dimensionality.

### Energy

Attempts have been made to introduce energy and/or natural resources into production functions by economists from a range of intellectual traditions. Neoclassical authors such as Solow, Stiglitz and Weinstein (Solow 1974; Stiglitz 1974; Weinstein and Zeckhauser 1974), added natural resources as additional independent inputs alongside labour and machinery. Stiglitz (Stiglitz 1974, p. 141) proposed a modified *CDPF* including natural resources (*R*) and time (*t*) in addition to Labour and Machinery:

Kummel et al produced the *LinEx* production function (Kümmel, Ayres et al. 2010, pp. 166, 172) which "depends linearly on energy and exponentially on factor quotients" (Kümmel, Ayres et al. 2010, p. 166). Though these equations did not take *CDPF* form, a *CDPF* which they called the "energy-dependent Cobb–Douglas function" was an interim step in their derivation (in equation 1.7, *y*, *k*, *l* and *e* are machinery, labour and energy respectively normalized to dimensionless numbers by dividing by values at a base date) (Kümmel, Ayres et al. 2010, p. 162), to produce a production function expressed in dimensionless numbers:

They note that this equation "has been often used in quantitative analyses, where mainstream economics identifies and with the cost shares of capital, labour, and energy" (Kümmel, Ayres et al. 2010, p. 166, equation 50). While this equation is dimensionally correct, we argue that it makes the same fundamental error as in Solow, Stiglitz and Weinstein of treating energy as an independent input alongside Labour and Machinery.

In reality, energy cannot be delivered independently of labour and capital to produce output: that is akin to setting off an explosion in a factory and expecting finished products to result, rather than mayhem. Instead, Labour and Machinery should be seen as means by which energy inputs are transformed into useful work (exergy) under the direction of Labour and Capital.

We therefore define Q as real GDP expressed as exergy (in GJ/Yr), and L and K as the exergy outputs of Labour and Machinery respectively. We treat the exergy outputs as products of the number of units times the exergy per unit, with representing the exergy contribution in Gigajoules of a representative unskilled worker per year, and represents the exergy contribution of a representative machine per year (with aggregation problems for machinery which have an eventual empirical resolution, given the availability of data on aggregate and per industry energy consumption):

We can now make some simplifications by considering the total derivative of Q as a means to identifying the sources of growth.

These relationships are independent of any particular theoretical assumptions about the nature of production. We now use them to derive a Neoclassical energy-based production function.

### Neoclassical

The key feature distinguishing the Neoclassical approach from Post Keynesian is the assumption of the smooth substitutability of factors of production. We can derive this by assuming that the effects of labour and capital are scale independent:

Then we obtain

Integration yields two expressions for *ln(F)*:

where and respectively are terms not including and . Combining these we obtain the exergy-based Cobb-Douglas Production Function:

where *Q0* represents the level of output (in GJ/Yr) in a base year—say 1784 or 1950. Putting this in constant returns to scale form yields the exergy-corrected constant returns to scale Cobb Douglas Production Function:

This function correctly incorporates the role of energy in production. The dimensionally meaningless "total factor productivity" element is gone, though the fractional dimensions of L and K remain. Energy—in particular, the energy harnessed by machinery, which has risen exponentially over time since the development of the Watt steam engine—plays an essential role in production, as one would expect. This far exceeds its "cost share", which, using this function, can be shown not to hold, in a much more direct way than the critique attempted by Kummel et al (Kümmel, Ayres et al. 2010, p. 157-165).

### The cost-share theorem

One of the reasons for the initial and continuing appeal of the *CDPF* to Neoclassical economists is its congruence with the Neoclassical theory of income distribution. As is well known, a constant-returns-to-scale *CDPF* with Labour and Capital as inputs returns the result that the marginal product of labour and capital are equal to respectively the real wage and the rate of profit. The fact that Cobb and Douglas's regressions returned values for these which were consistent with income distribution data played a strong role in the *CDPF*'s rapid adoption by Neoclassical economists. As Douglas put it:

Under perfect competition with a production formula of this type we would expect a factor to receive as its share of the product, the proportion indicated by its exponent. From the income studies of theNational Bureau of Economic Research, we found that labour's share of the net value product of manufacturing during the decade 1909-1918, was estimated at 74.1 per cent, or almost precisely the value of the exponent for labour. (Douglas 1948, p. 7)

When energy (or Natural Resources as in (Stiglitz 1974)) was added as a third independent input, this result was extended to imply that energy's share in production also reflects its marginal product—which implies that energy is responsible for of the order of just 5% of the value of output. This result is sufficiently incongruous with our perception of the importance of energy in production to result in its ridicule, even by leading Neoclassical economists (Summers 2013), while several authors have tried to reconcile the obvious crucial role of energy in production with the *CDPF* (Kander and Stern 2014; Stern and Kander 2012). Nonetheless, the *CDPF* has continued to be used (normally without alteration for energy, even in the Stiglitz form), presumably because of its inherent appeal to Neoclassical theorists.

We outline the simple proof of the congruence of the standard *CDPF* including energy with the Neoclassical theory of income distribution in Equation to contrast it with our result shown in Equation . Define a *CDPF* with three independent inputs *L*, *K* and *E* (for Energy). Dropping time arguments for simplicity, Equation uses *w*, *r* and *e* represent respectively the real wage, the rate of profit and the rate of return to energy, and denotes the production measure of output by *QP*, and the income measure by *QD*:

Our exergy-modified *CDPF* defines the production measure of real *GDP**QP* as Q (dimensioned by GJ/Yr), while the income distribution measure is the same: the returns to *L*, *K*, and the exergy output of machinery *ExK* are respectively *w*, *r* and *e* as in Equation (though they are now in dimensionally correct units of GJ/Yr/[input unit, where this is respectively *L*, *K* and *E*]). It is easily shown that the production and distribution measures are no longer congruent, and in particular the distribution measure is false, if the return to factors is assumed to be their marginal product.

The cost-share theorem is therefore invalid when the relations of labour and capital to energy are properly considered. The returns to the factors of production L, K and *ExK* cannot be equal to their marginal products. The marginal product theory of income distribution is therefore false, and something other than marginal products must determine the distribution of income.

Alternately, it could be argued that the returns to factors should be restricted to Labour and Machinery (a.k.a. "Capital") alone; however, in this case, the cost share of energy can no longer be treated as indicating the respective contribution to production of the "factors of production" *L* and *K*, since neither factor can add to production in the absence of its associated exergy contribution *ExL* and *ExK* respectively. *ExL* can be effectively treated as a constant, while the exergy output of a "representative machine" can be treated, at a first pass, as having risen exponentially at the rate ** since the start of the Industrial Revolution in, say, 1880. From this perspective, the cost-share theorem remains a tautology, as argued by Shaikh and McCombie {McCombie, 2000 #3951;Shaikh, 1974 #1417}. But the specific contributions of Labour and Machinery to the change in exergy reduce to their vastly different capacities to harness energy:

While these are serious issues for Neoclassical economics, they are not problems for the Post Keynesian tradition.

### Post Keynesian

Post Keynesians reject the concept of diminishing marginal productivity on both theoretical (Sraffa 1926) and empirical grounds (Eiteman 1947; Eiteman and Guthrie 1952; Means 1972; Blinder 1998; Lee 1998), so their production functions have always been linear in terms of either Labour or utilized machinery. Their theory of distribution has also been based not on marginal products, but on the relative bargaining powers of workers and capitalists (Goodwin 1967)—and, when additional social classes are included, other factors such as the level of private debt and the interest rate as well (Keen 1995, p. 616; Keen 2017, pp. 16-20). The divergence between the theory of production and the theory of distribution outlined above for Neoclassical economics does not affect their approach: if anything, they strengthen the Post Keynesian approach.

Turning solely to the Post Keynesian approach to production, machinery and its utilization rate are the ultimate constraints on production, so the minimum function in can be replaced by an equality by introducing the dimensionless number for the level of utilization of machinery:

As noted earlier, the identity *a(t) *(dimensioned in widgets/manyear) is conventionally treated as a measure of labour productivity, and modelled as growing over time, so that the increase in *Q* is due to rising labour productivity. However, in exergy terms, the productivity of unskilled labour has a definite maximum, which is the amount of useful work that an unskilled labourer can do per year. So this cannot be the source of rising real output, whereas the rising exergy capacity of machinery over time certainly can. This implies that *a* should be treated, not as a measure of productivity, but as a function stating the number of workers needed per "representative machine" at a point in time:

It is easy to adapt this approach to one based on exergy as the measure of both *GDP *and inputs toproduction. Firstly, we replace *Q *(widgets/yr) with **Q**

(GJ/yr) and *K *(caphr/yr) with **K**

(GJ/yr). Secondly, we relate the actual exergy harnessed by machinery **K**

to the maximum exergy that machinery can harness KMax when the utilization rate is 100%:

This then yields an exergy-based Post Keynesian/Leontief production function, where v now represents an efficiency of conversion of exergy output from machinery (in GJ/yr) into useful work (in GJ/yr, embodied in both goods and services), and not a ratio between the flow of GDP (in wid/yr) and the stock of machinery:

Or more simply, to a first approximation at the aggregate level, useful work is a linear transformation of the exergy of machinery:

Secondly, we consider L rather than *L*, and treat it as being determined by Q, consonant with existing practice:

Thirdly, we treat the exergy output of unskilled labour as a constant **, so that we can express *L* as:

Dispensing with time arguments for convenience, this implies the following equations should be used to empirically test this model:

These tests will be undertaken in subsequent research.

### Afterthoughts and conjectures

Throughout this paper, we have persisted with the fiction of an aggregate level of output Q. Given the homogenous nature of the energy inputs to production, this is more acceptable than *Q* measured in wid/yr. However, there are issues with aggregation on both the production and consumption side that need to be addressed in future work.

Raw energy inputs to production are relatively limited in number (fossil fuels, renewables, nuclear and thermal power), and predominantly homogenized into either electricity or liquid fuels before use. Exergy outputs from production take all kinds of forms, and in their usage deliver exergy in many desired forms (such as warmth, light, movement) or in derivative products (clothing, glasses, vehicles) that in turn deliver exergy in a form desired by consumers. It may be feasible to represent this diversity in terms of exergy vectors whose aggregate sum could be derived by vector operations (See Voudouris, Ayres et al. 2015).

Aggregation may also be meaningful as a means to derive a quantification of the change in living standards over time. Comparisons could be made between utility today and utility at a base date using a standardized set of basic needs along the lines suggested by Lancaster's characteristics approach to utility (Lancaster 1966).

Finally, a puzzle remains regarding the exponents of the conventional Cobb-Douglas Production Function. As Shaikh, McCombie and Felipe have shown {Felipe, 2007 #3945;McCombie, 2000 #3951;Shaikh, 1974 #1417;Shaikh, 1987 #4653;Shaikh, 2005 #3954}, this model fits the data because it is tautological; however, why is it that its exponents, when fitted to data, are approximately those for the distribution of income between labour and capital? An as-yet-unpublished speculation by Tiago Domingos, based on empirical research using a database of exergy and production relations in Portugal derived by him and co-authors {Santosa, 2018 #5421}, is that this may be because the exponents, rather than representing factor shares and factor contributions to production, may unwittingly uncover the pivotal role of machine exergy in production. His team found that, with capital services derived from exergy data, there was an empirical relation between this definition of capital services and the exergy-labour ratio:

Domingos reasoned that this relation applies because exergy is captured partly by large machines (such as blast furnaces) and partly by primarily electrical energy applied by labour-operated machinery throughout the production process. This relation enables us to approximate the exergy harnessed by machinery in terms of capital and labour services:

These exponents are of the magnitude normally returned by fitting a standard CDPH to data using Q, K (measured as ) and L, which by coincidence are similar to the capital-labour shares found in the data (in Portugal's case, the average income distribution shares were 30.8% and 69.2% respectively {Santosa, 2018 #5421, p. 116}). When combined with our Equation , that posits a linear relationship between GDP and useful exergy in production, this generates the empirical regularity that:

The standard CDPH exponents may therefore uncover, not the marginal productivity of labour and capital implied by Neoclassical production and distribution theory, but the underlying role of exergy in production.

### Conclusion

We hope that this paper provides means by which energy can be properly incorporated into production theory in a manner which economists of all persuasions can accept. It may come with some undesired consequences for some, given the challenge to the marginal product theory of income distribution, but this challenge has already been laid by empirical research (Piketty 2014). And, as Samuelson observed decades ago when summing up the Cambridge Controversies over the nature of capital in economic theory:

If all this causes headaches for those nostalgic for the old time parables of neoclassical writing, we must remind ourselves that scholars are not born to live an easy existence. We must respect, and appraise, the facts of life. (Samuelson 1966, p. 583)

### References

Barnett II, W. (2004). "Dimensions and Economics: Some Problems." __Quarterly Journal of Austrian Economics__**7**(1): 95-104.

Blinder, A. S. (1998). __Asking about prices: a new approach to understanding price stickiness__. New York, Russell Sage Foundation.

Cobb, C. W. and P. H. Douglas (1928). "A Theory of Production." __The American Economic Review__**18**(1): 139-165.

Douglas, P. H. (1948). "Are There Laws of Production?" __The American Economic Review__**38**(1): i-41.

Eiteman, W. J. (1947). "Factors Determining the Location of the Least Cost Point." __The American Economic Review__**37**(5): 910-918.

Eiteman, W. J. and G. E. Guthrie (1952). "The Shape of the Average Cost Curve." __The American Economic Review__**42**(5): 832-838.

Felipe, J. and J. S. L. McCombie (2007). "Is a Theory of Total Factor Productivity Really Needed?" __Metroeconomica__ **58**(1): 195-229.

Goodwin, R. M. (1967). A growth cycle. __Socialism, Capitalism and Economic Growth__. C. H. Feinstein. Cambridge, Cambridge University Press**: **54-58.

Kander, A. and D. I. Stern (2014). "Economic growth and the transition from traditional to modern energy in Sweden." __Energy Economics__**46**: 56-65.

Keen, S. (1995). "Finance and Economic Breakdown: Modeling Minsky's 'Financial Instability Hypothesis.'." __Journal of Post Keynesian Economics__**17**(4): 607-635.

Keen, S. (2017). __Can We Avoid Another Financial Crisis? (The Future of Capitalism)__. London, Polity Press.

Kümmel, R., R. U. Ayres, et al. (2010). "Thermodynamic laws, economic methods and the productive power of energy." __Journal of Non-Equilibrium Thermodynamics__**35**: 145-179.

Lancaster, K. (1966). "A New Approach to Consumer Theory." __Journal of Political Economy__**74**(2): 132-157.

Lee, F. S. (1998). __Post Keynesian price theory__. Cambridge, Cambridge University Press.

McCombie, J. S. L. (2000). "The Solow Residual, Technical Change, and Aggregate Production Functions." __Journal of Post Keynesian Economics__**23**(2): 267-297.

Means, G. C. (1972). "The Administered-Price Thesis Reconfirmed." __The American Economic Review__**62**(3): 292-306.

Pasinetti, L. L. (2000). "Critique of the Neoclassical Theory of Growth and Distribution." __Banca Nazionale del Lavoro Quarterly Review__**53**(215): 383-431.

Piketty, T. (2014). __Capital in the Twenty-First Century__. Harvard, Harvard College.

Samuelson, P. A. (1966). "A Summing Up." __Quarterly Journal of Economics__**80**(4): 568-583.

Santosa, J., T. Domingosa, et al. (2018). "Useful Exergy Is Key in Obtaining Plausible Aggregate Production Functions and Recognizing the Role of Energy in Economic Growth: Portugal 1960–2009." __Ecological Economics__ **148**: 103-120.

Shaikh, A. (1974). "Laws of Production and Laws of Algebra: The Humbug Production Function." __Review of Economics and Statistics__**56**(1): 115-120.

Shaikh, A. (1987). Humbug production function. __The New Palgrave: A Dictionary of Economics__. J. Eatwell, M. Milgate and P. Newman, Palgrave Macmillan.

Shaikh, A. (2005). "Nonlinear Dynamics and Pseudo-Production Functions." __Eastern Economic Journal__ **31**(3): 447-466.

Skourtos, M. (1991). "Corn models in the classical tradition: P. Sraffa considered historically." __Cambridge Journal of Economics__**15**(2): 215-228.

Smets, F. and R. Wouters (2007). "Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach." __American Economic Review__**97**(3): 586-606.

Solow, R. M. (1957). "Technical change and the aggregate production function." __Review of Economics and Statistics__**39**(August): 312-320.

Solow, R. M. (1974). "The Economics of Resources or the Resources of Economics." __The American Economic Review__**64**(2): 1-14.

Sraffa, P. (1926). "The Laws of Returns under Competitive Conditions." __The Economic Journal__**36**(144): 535-550.

Sraffa, P. (1960). __Production of commodities by means of commodities: prelude to a critique of economic theory__. Cambridge, Cambridge University Press.

Stern, D. I. and A. Kander ( 2012). "The role of energy in the industrial revolution and modern economic growth." __The Energy Journal__**33**(3): 125-152.

Stiglitz, J. (1974). "Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths." __The Review of Economic Studies__**41**: 123-137.

Stiglitz, J. E. (1974). "Growth with Exhaustible Natural Resources: The Competitive Economy." __Review of Economic Studies__**41**(5): 139.

Summers, L. (2013). Larry Summers at IMF Economic Forum, Nov. 8.

Vines, D. and S. Wills (2018). "The rebuilding macroeconomic theory project: an analytical assessment." __Oxford Review of Economic Policy__**34**(1-2): 1-42.

Voudouris, V., R. Ayres, et al. (2015). "The economic growth enigma revisited: The EU-15 since the 1970s." __Energy Policy__**86**: 812-812.

Weinstein, M. C. and R. J. Zeckhauser (1974). "Use Patterns for Depletable and Recycleable Resources." __Review of Economic Studies__**41**(5): 67.