Incorporating energy into production functions

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In my last post on my Debt­watch blog, I fin­ished by say­ing that the Phys­iocrats were the only School of eco­nom­ics to prop­er­ly con­sid­er the role of ener­gy in pro­duc­tion. They ascribed it sole­ly to agri­cul­ture exploit­ing the free ener­gy of the Sun, and specif­i­cal­ly to land, which absorbed this free ener­gy and stored it in agri­cul­tur­al prod­ucts. As Richard Can­til­lon put it in 1730:

The Land is the Source or Mat­ter from whence all Wealth is pro­duced. The Labour of man is the Form which pro­duces it: and Wealth in itself is noth­ing but the Main­te­nance, Con­ve­nien­cies, and Super­fluities of Life. (Can­til­lon, Essai sur la Nature du Com­merce in Général (Essay on the Nature of Trade in Gen­er­al)

Ques­nay’s famous but neglect­ed “Tableau Economique” there­fore described the agri­cul­tur­al sec­tor as “the pro­duc­tive sec­tor” and man­u­fac­tur­ing as “sterile”—see Fig­ure 1.

Fig­ure 1: Ques­nay’s “Tableau Economique”, first draft­ed in 1759, two decades before Wat­t’s steam engine

This was a jus­ti­fied asser­tion at the time, giv­en that the Phys­iocrats wrote before the Indus­tri­al Revolution—and in par­tic­u­lar the wide­spread exploita­tion in man­u­fac­tur­ing of stored solar ener­gy in fos­sil fuels– and orig­i­nat­ed in France, which was then over­whelm­ing­ly a rur­al nation.

Smith, who was influ­enced by the Phys­iocrats and wrote in Britain when indus­try was start­ing to exploit fos­sil fuels (specif­i­cal­ly coal) on a grand scale, could have cor­rect­ed this over­sight. But rather than fol­low­ing the Phys­iocrats’ lead on ener­gy, Smith instead saw labour—not energy—as the font of wealth (which he described in the same terms as Can­til­lon: the “con­ve­nien­cies of life”), and ascribed the increase in pro­duc­tiv­i­ty over time to “the divi­sion of labour”:

The annu­al labour of every nation is the fund which orig­i­nal­ly sup­plies it with all the nec­es­saries and con­ve­nien­cies of life which it annu­al­ly con­sumes, and which con­sist always either in the imme­di­ate pro­duce of that labour, or in what is pur­chased with that pro­duce from oth­er nations…

The great­est improve­ment in the pro­duc­tive pow­ers of labour, and the greater part of the skill, dex­ter­i­ty, and judg­ment with which it is any­where direct­ed, or applied, seem to have been the effects of the divi­sion of labour. (Smith 1776, An Inquiry into the Nature and Caus­es of the Wealth of Nations)

Eco­nom­ics thus lost the Phys­iocrats’ focus on ener­gy, and instead descend­ed first into the “Labour the­o­ry of val­ue” and then into the Neo­clas­si­cal (and Post Key­ne­sian) notions of “pro­duc­tion func­tions” in which ener­gy played no role at all.

The abid­ing weak­ness of all schools of eco­nom­ics, ever since the Classicals—including today’s Neo­clas­si­cal and Post Key­ne­sian schools, which are nor­mal­ly at pains to point out how supe­ri­or one is to the other—is this fail­ure to acknowl­edge the key role of ener­gy in pro­duc­tion. In this brief note, I want to record some spec­u­la­tions about how mod­ern math­e­mat­i­cal­ly-inclined eco­nom­ics, with its use of pro­duc­tion func­tions, might be made as ener­gy aware as the Phys­iocrats were two and a half cen­turies ago. For the sake of non-math­e­mat­i­cal read­ers, I’ve put the equa­tions in an appen­dix at the end of this post.

Post Key­ne­sian mod­els typ­i­cal­ly see Out­put (Y) as a func­tion of Cap­i­tal (K) with a fixed ratio (v) between Cap­i­tal and Out­put, and a fixed ratio (a) between Out­put and Labour (L) (see Equa­tion 1).

When they go fur­ther than “corn econ­o­my” mod­els, they employ a so-called “Leon­tief pro­duc­tion func­tion”, in which the ratio between cap­i­tal and labour is fixed in each indus­try, though it varies between indus­tries.

Neo­clas­si­cal econ­o­mists crit­i­cise this approach because it ignores the sub­sti­tutabil­i­ty of cap­i­tal and labour in pro­duc­tion, which they embody in their core con­cept of an “iso­quant” which alleges that the same lev­el of out­put can be pro­duced by very dif­fer­ent com­bi­na­tions of labour and cap­i­tal. Post Key­ne­sians nor­mal­ly respond that this sub­sti­tutabil­i­ty is a chimera, and con­tin­ue using this “fixed coef­fi­cients” mod­el of pro­duc­tion any­way.

Neo­clas­si­cals typ­i­cal­ly see out­put as a func­tion of cap­i­tal and labour where one can be smooth­ly sub­sti­tut­ed for the oth­er. Their canon­i­cal mod­el is the Cobb-Dou­glas Pro­duc­tion Func­tion with con­stant returns to scale (Equa­tion 2).

Neo­clas­si­cals tout this mod­el’s fit to empir­i­cal data as a strength; Post Key­ne­sians note that this is sim­ply the result of account­ing iden­ti­ties, since this “pro­duc­tion func­tion” can be derived by manip­u­lat­ing the iden­ti­ty that Out­put equals Wages plus Prof­its under con­di­tions of a rel­a­tive­ly con­stant (or slow­ly vary­ing) dis­tri­b­u­tion of income (see Anwar Shaikh’s bril­liant paper “The Hum­bug Pro­duc­tion Func­tion”).

In this “he said/she said” bat­tle, both sides ignore the shared weak­ness that their mod­els of pro­duc­tion imply that out­put can be pro­duced with­out using energy—or that ener­gy can be treat­ed as just a form of cap­i­tal. Both state­ments are cat­e­gor­i­cal­ly false accord­ing to the Laws of Ther­mo­dy­nam­ics, which—in strong con­trast to so-called “Eco­nom­ic Laws” like the “Law of One Price” and the “Law of Demand”—can­not be bro­ken. As Sir Arthur Edding­ton once put it:

The law that entropy always increas­es holds, I think, the supreme posi­tion among the laws of Nature. If some­one points out to you that your pet the­o­ry of the uni­verse is in dis­agree­ment with Maxwell’s equa­tions — then so much the worse for Maxwell’s equa­tions. If it is found to be con­tra­dict­ed by obser­va­tion — well, these exper­i­men­tal­ists do bun­gle things some­times. But if your the­o­ry is found to be against the sec­ond law of ther­mo­dy­nam­ics I can give you no hope; there is noth­ing for it but to col­lapse in deep­est humil­i­a­tion. (Sir Arthur Stan­ley Edding­tonThe Nature of the Phys­i­cal World (1915), chap­ter 4)

Arguably there­fore, the pro­duc­tion func­tions used in eco­nom­ic theory—whether spout­ed by main­stream Neo­clas­si­cal or non-ortho­dox Post Keynesians—deserve to “col­lapse in deep­est humil­i­a­tion”.

This unac­cept­able state of affairs has inspired a num­ber of eco­log­i­cal­ly ori­ent­ed econ­o­mists to attempt to come up with pro­duc­tion func­tions in which ener­gy plays a cru­cial role. One such mod­el is the LINEX (“LIN­ear-Expo­nen­tial”) or KLEC (“cap­i­tal-labor-ener­gy-cre­ativ­i­ty”) mod­el used by Kum­mel, Lin­den­berg­er, Ayres and col­leagues. At a basic lev­el, this treats out­put as a func­tion of labour, cap­i­tal, ener­gy and time—effectively adding Ener­gy as a third input to the Cobb-Dou­glas mod­el (Equa­tion 3).

While this is an improve­ment on the basic Cobb-Dou­glas mod­el, it still implies log­i­cal­ly that the con­tri­bu­tion of ener­gy to pro­duc­tion could be nil: just set its expo­nent ? to zero. This is still in vio­la­tion of the Laws of Ther­mo­dy­nam­ics: we need a pro­duc­tion func­tion in which ener­gy plays an essen­tial and irre­ducible role.

A poten­tial way to achieve this is to accept that the whole idea of “labour” and “cap­i­tal” with­out ener­gy is a farce: labour with­out ener­gy is a corpse, and cap­i­tal with­out ener­gy is a sculp­ture.

Why not acknowl­edge this by, as an ini­tial abstrac­tion, treat­ing labour and cap­i­tal as both being means to har­ness ener­gy to do work, and treat­ing out­put (Y) itself as work? Then we start from treat­ing Labour and Cap­i­tal as means to har­ness the ener­gy they con­tain: EL for the ener­gy flow that a work­er can har­ness in a day, and EK for the ener­gy flow that a machine can har­ness in a day (Equa­tion 4).

The type and amount of ener­gy that a work­er or a machine can embody (as flows of ener­gy at a point in time) are of course vast­ly dif­fer­ent: the for­mer is lim­it­ed to food, and has a bio­phys­i­cal max­i­mum (say, 5000 calo­ries per day) where­as the lat­ter can be the Sun itself direct­ly, agri­cul­tur­al prod­ucts, fos­sil fuels, or nuclear ener­gy, and has risen from triv­ial lev­els before the Indus­tri­al Rev­o­lu­tion to tru­ly astro­nom­i­cal lev­els today.

Unpack­ing this fur­ther, we also need to acknowl­edge that not all the ener­gy embod­ied in labour or cap­i­tal can be used for work. Ener­gy avail­able to do work (these days called “exer­gy”) is the rel­e­vant fac­tor, rather than the total ener­gy embod­ied in labour or a machine; the effi­cien­cy with which that avail­able ener­gy is har­nessed is also a vital ingre­di­ent.

So rather than sim­ply show­ing the ener­gy embod­ied in labour and cap­i­tal, we need to mul­ti­ply it by the ratio of avail­able ener­gy (exer­gy, with ExL for labour and ExK for machin­ery) to ener­gy, and by the effi­cien­cy with which that exer­gy is har­nessed . Then, using L to sig­ni­fy the num­ber of work­ers and K (how­ev­er imper­fect­ly) to sig­ni­fy the num­ber of machines, we get Equa­tion 5, which treats out­put as a func­tion of labour, cap­i­tal and ener­gy.

Rear­rang­ing Equa­tion 5, we can derive the basic Cobb-Dou­glas for­mu­la­tion for labour and cap­i­tal, times ener­gy inputs. This is super­fi­cial­ly like the Kummel/Ayres LINEX for­mu­la­tion, but it has the advan­tage that the ener­gy con­tri­bu­tion of either labour or cap­i­tal can­not be set to zero with­out set­ting the con­tri­bu­tion of the relat­ed “fac­tor of pro­duc­tion” also to zero, since they have the same expo­nents (see Equa­tion 6). Ener­gy there­fore plays a cru­cial role in pro­duc­tion using this for­mu­la­tion: if the ener­gy input is zero, then so is out­put.

It may also tran­spire that the avail­able ener­gy embod­ied in machin­ery, and the effi­cien­cy of its exploita­tion, is the major expla­na­tion for the “Solow Residual”—the appar­ent para­dox that, despite econ­o­mists see­ing out­put at any point in time as a func­tion of labour and cap­i­tal, the vast major­i­ty of the change in out­put over time comes not from an increase in the amount of Labour or Cap­i­tal employed, but from the rel­a­tive­ly unspec­i­fied A(t) term in the stan­dard Cobb-Dou­glas func­tion.

With this term replaced by two ener­gy relat­ed terms—one of which has a def­i­nite max­i­mum (since there is only so much exer­gy that a human body can exert in a day, and this can com­fort­ably be treat­ed as a con­stant), the oth­er of which has gone from that of a water wheel in pre-Indus­tri­al times to that har­nessed by the Spacex Fal­con Heavy today—the “Solow Resid­ual” may turn out to be the expo­nen­tial increase in ener­gy and exer­gy input into pro­duc­tion over time (see Fig­ure 2).

Fig­ure 2: US Ener­gy con­sump­tion over time

This ener­gy-aware mod­el of pro­duc­tion is just a first step in prop­er­ly inte­grat­ing ener­gy and the eco­log­i­cal effects of using ener­gy into eco­nom­ics. It does not as yet con­sid­er the dif­fer­ent types of ener­gy resources, the impact of ener­gy use upon ener­gy resources and the ecol­o­gy, or the min­ing (not “pro­duc­tion”) costs of ener­gy in terms of ener­gy itself (the “Ener­gy Return on Ener­gy Invest­ed” or EROEI). But it is nec­es­sary to have a mod­el of pro­duc­tion in which ener­gy plays an essen­tial role to be able to con­sid­er these issues about the via­bil­i­ty of our ener­gy usage prop­er­ly.


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About Steve Keen

I am Professor of Economics and Head of Economics, History and Politics at Kingston University London, and a long time critic of conventional economic thought. As well as attacking mainstream thought in Debunking Economics, I am also developing an alternative dynamic approach to economic modelling. The key issue I am tackling here is the prospect for a debt-deflation on the back of the enormous private debts accumulated globally, and our very low rate of inflation.