ACCOUNTING FOR GROWTH: THE ROLE OF PHYSICAL WORKRobert U. Ayres and Benjamin Warr*ABSTRACT       We test several related h...
Ayres and Warr                           Accounting for growth                                Page 2of goods and services ...
Ayres and Warr                           Accounting for growth                                 Page 3as a factor.1 It is i...
Ayres and Warr                            Accounting for growth                               Page 4questionable assumptio...
Ayres and Warr                           Accounting for growth                                  Page 5animal) labor, resul...
Ayres and Warr                            Accounting for growth                                Page 6should be treated muc...
Ayres and Warr                           Accounting for growth                                Page 7scarce or not economic...
Ayres and Warr                           Accounting for growth                                 Page 8         In the simpl...
Ayres and Warr                           Accounting for growth                                Page 9II. PRODUCTION FUNCTIO...
Ayres and Warr                           Accounting for growth                              Page 10approximately by some f...
Ayres and Warr                          Accounting for growth                              Page 11III. CALCULATION OF PHYS...
Ayres and Warr                          Accounting for growth                               Page 12reconverted back into m...
Ayres and Warr                           Accounting for growth                              Page 13         The agricultur...
Ayres and Warr                           Accounting for growth                                Page 14(2a)             UB =...
Ayres and Warr                           Accounting for growth                              Page 15omitted) must be a homo...
Ayres and Warr                           Accounting for growth                                  Page 16where A is a multip...
Ayres and Warr                           Accounting for growth                                Page 17values of the variabl...
Ayres and Warr                           Accounting for growth                              Page 18A(t). Based on the resu...
Ayres and Warr                          Accounting for growth                               Page 19airlines by increasing ...
Ayres and Warr                             Accounting for growth                               Page 20         Turning to ...
Ayres and Warr                          Accounting for growth                              Page 21The MSE is a measure of ...
Ayres and Warr                        Accounting for growth                           Page 22Ayres, Robert. U; Ayres, Lesl...
Ayres and Warr                        Accounting for growth                             Page 23Fabricant, Solomon. “Econom...
Ayres and Warr                           Accounting for growth                        Page 24Kaufmann, Robert K. “A Biophy...
Ayres and Warr                        Accounting for growth                        Page 25Potter, Neal and Christy, Franci...
Ayres and Warr                         Accounting for growth                        Page 26Solow, Robert M. “A Contributio...
Ayres and Warr                           Accounting for growth                            Page 27FOOTNOTES*        Robert ...
Ayres and Warr                           Accounting for growth                                 Page 281.       The new A-K...
Ayres and Warr                            Accounting for growth                              Page 295.       There are sta...
Ayres and Warr                         Accounting for growth                           Page 309.       Kümmel introduced t...
Ayres and Warr                        Accounting for growth                             Page 31    Table A1. Statistical m...
Ayres and Warr                          Accounting for growth                             Page 32FIGURE HEADINGSFigure 1a....
12                                            K                                            L                              ...
20             Y GDP (actual)                           Y = Ka Lb Eg                           A(t), Solow Residual       ...
100                         TOTAL FOSSIL FUEL                         PHYTOMASS      90                 OTHER             ...
1.6                                                TOTAL EXERGY (incl. Phytomass) /                                       ...
40%                             High Temperature Industrial                             Heat               35%           M...
90%                                                  HEAT                                                  LIGHT80%       ...
40%                                                          800                          % Electric Power Generation and ...
0.30                     calculated using B and UB                     calculated using E and Ue        0.25        0.20ra...
40                                                    2.5                            Primary work (left scale)            ...
20              empirical GDP                      Theoretical estimates:                              using UE    - MSE =...
12              10              8% of growth              6              4              2              0                  ...
1.0             Capital - K             Labor - L             Work – UE0.80.60.40.20.0      1900    1920         1940     ...
1.0             Capital - K             Labor - L             Work - UB0.80.60.40.20.0      1900    1920         1940     ...
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Accounting for Growth, Robert Ayres & Benjamin Warr

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A seminal paper describing the use of useful work instead of raw energy as a factor of production used in explaining past economic growth.

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Accounting for Growth, Robert Ayres & Benjamin Warr

  1. 1. ACCOUNTING FOR GROWTH: THE ROLE OF PHYSICAL WORKRobert U. Ayres and Benjamin Warr*ABSTRACT We test several related hypothesis for explaining US economic growth since 1900.Introducing physical work, instead of energy, as a factor of production the historical growthpath is reproduced with high accuracy from 1900 until the mid 1970s. In effect, the Solowresidual is explained as increasing energy-conversion (to work) efficiency. The remainingunexplained residual amounts to only about 12 percent of total growth since that time.Information technology may be responsible for the unexplained growth.Keywords: Economic growth; Exergy; Productivity; Resources; Solow Residual; Work(JEL O39; alternate JEL O47) In the 1950s, it was discovered that the growth in capital stock could only account fora small fraction (about one eighth) of the historical US growth in economic output per worker(Moses Abramovitz 1952, 1956; Solomon Fabricant 1954). Economic growth theory wassubsequently formulated in its current production function form by Robert Solow and TrevorSwan (Robert M. Solow 1956, 1957; Trevor Swan 1956). The theory assumes that production
  2. 2. Ayres and Warr Accounting for growth Page 2of goods and services (in monetary terms) can be expressed as a function of capital and labor.But the major contribution to growth had to be attributed to something else, namely“technological progress”or just “technical progress”. Absent any fundamental economictheory of technical progress, or any convincing independent measure of it, technical progresshas been treated as an unexplained residual. This is another way of saying that it is anexogenous multiplier, either of labor or capital or both equally (i.e. of the whole productionfunction, usually taken to be Cobb-Douglas or CES in form). As it happens, however, the Solow model makes two fundamental predictions that donot correspond to historical experience over the last half century. One prediction is that therate of growth of an economy will decline as the capital stock grows, due to decliningmarginal productivity of capital (and the need to replace depreciation). The other prediction(known as “convergence”) is that poor countries, with smaller capital stocks, will grow fasterthan rich countries. The most popular “fix” is to dispense with the notion that capitaldepreciates. Another way of putting it is to assume that decreasing marginal productivity ofcapital is compensated by another effect, namely increasing returns to knowledge, arisingfrom positive externalities (“spillovers”). This notion, first incorporated in the standardtheoretical framework by Romer (Paul M. Romer 1986), has prompted an explosion of so-called “endogenous growth” theories (e.g. Romer 1986, 1987, 1990; Robert E. Lucas, Jr.1988; Gene M. Grossman and Elhanen Helpman 1991; Philippe Aghion and Peter Howitt1998). There is a lot of interest among theorists, at present, in the phenomenon of increasingreturns to scale (as exemplified by network systems of all kinds). Most of the recent endogenous growth theories are so-called A-K models, where K isgeneralized capital, defined to include knowledge and skills. The Solow multiplier A in thesemodels is supposed to be a constant, independent of time. Labor no longer appears explicitly
  3. 3. Ayres and Warr Accounting for growth Page 3as a factor.1 It is interesting to note that Solow’s argument for choosing capital and laborproductivities in the Cobb-Douglas production function (on the basis of shares in the nationalincome accounts) no longer applies to the A-K theories. The major difficulty with the “new”endogenous growth theories is that there is no independent empirical basis for determiningwhat the generalized capital K should be. In this paper we reconsider the possible role of resource inputs as factors ofproduction, and whether this approach offers a possible alternative to the AK theoriesmentioned above.I. THE ROLE OF NATURAL RESOURCES The possible contribution of natural resource inputs to growth (or to technicalprogress), was not considered seriously by most economists until the 1970s (mainly inresponse to the Club of Rome and “Limits to Growth” (Donella H. Meadows et al 1972)).The primary focus of the early economic responses to “Limits” was on the problem ofresource exhaustion (e.g. Partha Dasgupta and Geoffrey Heal 1974; Solow 1974; JosephStiglitz 1974). Despite the very small share of national income that could reasonably beattributed to resources, some economists, notably Jorgenson and his colleagues, introducedtranscendental logarithmic production functions and multi-sector (KLEMS) models withcapital, labor, energy and materials as factors (Dale W. Jorgenson et al 1973). Jorgensonsubsequently tried to calculate factor productivities empirically in a large multi-sector model(Jorgenson 1984). The results were not sufficiently unambiguous to prompt many imitators.In most more recent large model applications resource consumption has been implicitlytreated as a consequence of growth and not as a factor of production. This simplistic. and
  4. 4. Ayres and Warr Accounting for growth Page 4questionable assumption is built into most of the large-scale models used for policy guidanceby governments. An important “engine of growth” since the first industrial revolution has been thecontinuously declining real price of physical resources, especially energy (and power)delivered at a point of use.2 The increasing availability of energy from fossil fuels, and powerfrom heat engines, has clearly played a fundamental role in growth. Machines powered byfossil energy have gradually displaced animals, wind power, water power and human musclesand thus made human workers vastly more productive than they would otherwise have been.The term energy as used above, and in most discussions (including the economics literature)is technically incorrect, since energy is conserved and therefore cannot be “used up”. Thecorrect term in this context is exergy3, which is sometimes called “available energy” or“useful energy”.We use this term hereafter because of its generality. The generic exergy-power feedback cycle works as follows. Technological progresssuch as discoveries, inventions, economies of scale, and accumulated experience – orlearning-by-doing – result in cheaper exergy and power at the point of use. Moreover,inventions (such as the steam engine) enable the substitution of machines for human oranimal muscles, thus delivering more power at lower cost. This, in turn, enables tangiblegoods and intangible services to be produced and delivered to consumers at ever lower prices.Factories and mines also benefit from these price reductions. Lower consumer prices encourage higher demand, thanks to price elasticity ofdemand. Since demand for final goods and services (including investment) necessarilycorresponds to the sum of factor payments, most of which go back to labor as wages andsalaries, it follows that wages of labor tend to increase as output rises.4 This, in turn,stimulates the further substitution of fossil exergy and mechanical lower for human (and
  5. 5. Ayres and Warr Accounting for growth Page 5animal) labor, resulting in further increases in scale, etc. and still lower costs. In moderntimes the use of electric power has been particularly productive, not least because it hasstimulated the development of so many new industries. Based on both qualitative and quantitative evidence, the existence of the positivefeedback cycle sketched above implies that physical resource (exergy) flows are a majorfactor of production. This observation has tempted many economists to try to quantify therelationship. Indeed, including a fossil exergy flow proxy in a 3 (or 4) factor Cobb-Douglas orsimilar production function has apparently successfully accounted for economic growth quiteaccurately, at least for limited time periods, without any exogenous time-dependent term.Examples of such studies include (Bruce M. Hannon and John Joyce 1981; Reiner Kümmel1982; Cutler Cleveland et al 1984; Kümmel et al 1985; Cleveland 1992; Robert K.Kaufmann 1992; Bernard C. Beaudreau 1998; Cleveland et al 2000; Kümmel et al 2000). However, even a high degree of correlation (exhibited by some of these studies) doesnot necessarily imply causation. In other words, the fact that economic growth tends to bevery closely correlated with energy or exergy consumption for some period of time – a factdemonstrated in numerous studies – does not a priori mean that energy (or exergy)consumption is the cause of the growth. Indeed, most economic models assume the opposite:that economic growth is responsible for increasing energy consumption. This automaticallyguarantees correlation. It is also conceivable that both consumption and growth aresimultaneously caused by some third factor. The direction of causality must evidently bedetermined empirically by other means.5 A deeper question is: why should capital services be treated as a “factor ofproduction” while the role of energy (exergy) services – not to mention other environmentalservices – is neglected or minimized? The naive view would seem to be that the two factors
  6. 6. Ayres and Warr Accounting for growth Page 6should be treated much the same way. Yet, among many theorists, strong doubts remain. Itappears that there are two reasons. The first and most important is that national accounts areset up to reflect payments to labor (wages, salaries) and capital owners (rents, royalties,interest, dividends). In fact, GNP is the sum of all such payments and NNP is the sum of allsuch payments to individuals. There is no category for payments to resources. If labor and capital are the only two factors, neoclassical theory implies that theproductivity of a factor of production must be proportional to the share of that factor in thenational income. This proposition is quite easy to prove in a hypothetical single sectoreconomy consisting of a large number of producers manufacturing a single good using onlylabor and capital services (as taught in elementary economics texts.) Moreover, the supposedlink between factor payments and factor productivities gives the national accounts afundamental role in production theory. This is intuitively very attractive. Labor gets the lion’sshare of payments in the US national accounts, around 70 percent. Capital (defined asinterest, dividends, rents and royalties) gets all of the rest. The figures vary slightly from yearto year, but they have been relatively stable for the past century or more. Land rents arenegligible. Payments for fossil fuels (even in “finished” form, including electric power)altogether amount to only a few percent of the total GDP. It follows, according to the received theory, that energy (exergy) is not a significantfactor of production, or that it can be subsumed in capital, and can be safely ignored.However, there is a flaw in this argument. Suppose there exists an unpaid factor, e.g.environmental services. Since there are few economic agents (persons or firms) who receiveincome in exchange for providing or protecting environmental services, there are no explicitpayments for such services in the national accounts. Absent such payments, it would seem tofollow from the logic of the preceding two paragraphs that environmental services are not
  7. 7. Ayres and Warr Accounting for growth Page 7scarce or not economically productive. This implication pervades neoclassical economictheory. But it seems quite unreasonable to argue that environmental services areunproductive, given their central role in agriculture and forestry not to mention providing thewelfare benefits of equable climate, fresh water, clean air, waste disposal and so forth. True,there are no markets for these services, but that is primarily due to indivisibility and lack ofproperty rights. Free goods can still be scarce Just as environmental assets and services may be underpriced, so may exergy andexergy services. Many environmental economists note to the enormous direct and (mostly)indirect subsidies to energy use, especially in the US. Motor vehicles are the most obviouscase-in-point. Motor vehicles create very large public costs, not only for highwayconstruction and maintenance, but from air pollution damage, from uninsured accidents, fromthe military expenditures to protect the sea-routes from the Middle East and by the removal oflarge urban land areas from other public or private uses. Many studies have been done toquantify these costs, but the point is that if these damage costs were added to the price ofpetroleum products, as theory suggests, the price of fuel would soar – and the petroleum shareof national income would soar also. There is little doubt that all fossil fuels, as well as nuclearenergy, are significantly underpriced. This suggests that the energy share of the nationalaccounts could be too small by a factor of two or three, or even more. Quite apart from the question of under-pricing, there is another reason why factorpayments directly attributable to consumption of physical resources – especially exergy – aremuch smaller than the real productivity of exergy to the economy. In reality, the economy isnot a single sector producing a single good – say, bread – from capital and labor.6 On thecontrary it is a complex network of interacting sectors. Most sectors produce products (orservices) by adding value to inputs purchased directly from other sectors.
  8. 8. Ayres and Warr Accounting for growth Page 8 In the simple single sector model used to “prove” the neoclassical relationshipbetween factor productivity and factor payments, this crucial fact is disregarded, resulting in amajor distortion. If the economy were a one-sector model with only one product and one sortof producer (e.g. bakers of bread), the payments to each factor input would indeed reflect themarginal productivity of that factor. In a multi-sector economy, however most sectors buyfrom others (and sell to others), whence purchased intermediates constitute a significant shareof the factors of production for each sector. Only the extractive sectors consume rawmaterials (exergy) extracted directly from nature. The intermediate sectors utilize labor andcapital plus semi-finished materials and products, plus utilities (water, gas, electricity) and avariety of services (finance, insurance, transport, etc.). The intermediates are, in effect,“processed exergy”. If each sector is considered as a producer, with its own production function, itsfactors of production necessarily include exergy consumed in the production of purchasedintermediate goods, including utilities and services. Allowing for the exergy transmitted viaintermediates, the simple picture derived from the 1-sector model changes completely.Whereas, exergy extracted from nature accounts for a very small share of the national incomedirectly, “processed exergy” can (and does) contribute a much larger effective share of thevalue of aggregate production, indirectly (Robert U. Ayres 2001). So much for the anomalously small share of exergy in the national accounts or(looking at it the other way) the anomalously large share of exergy as a factor of productionand driver of growth.
  9. 9. Ayres and Warr Accounting for growth Page 9II. PRODUCTION FUNCTIONS It is now convenient to postulate two possible relationships of the form:(1a) Y = fEgEE or Y = fBgBB(1b) Y = gEUE or Y = gBUBwhere Y is GDP, measured in dollars, E is a measure of commercial energy (mainly fossilfuels), B is a measure of all “raw” physical resource inputs (technically, exergy), includingfuels, minerals and agricultural and forest products. Then f is the ratio of “useful work” Udone by the economy as a whole to “raw” exergy input (defined below), and g is the ratio ofeconomic output in value terms to work input. All the variables have implicit subscripts B orE, depending on whether or not we are considering only standard measures of energy (E) ortotal exergy (B). We neglect the subscripts hereafter where the choice is obvious. Since work appears implicitly in both numerator and denominator of (1), its definitiondepends on whether we choose B or E. (Note that equation (1 a,b) is merely a definition ofthe factors f, g. There is no theory or approximation involved). However we note that theexpressions (1a) or (1b) can be interpreted as a production function in either of two cases.The first possibility is that E or B is a factor of production and the product fE gE or fBgB can beapproximated by some first order homogeneous function of the three factors: labor L , capitalK and energy consumption E or exergy consumption B. The second possibility is that work Uis a factor of production (instead of E or B) and the function g can be expressed
  10. 10. Ayres and Warr Accounting for growth Page 10approximately by some first order homogeneous function of K, L and U. (Subscripts omitted.)We test these possibilities empirically hereafter. The traditional variables capital K, and labor L, as usually defined for purposes ofeconomic analysis, are plotted along with commerical energy (fossil fuels plus hydro andnuclear power) in Figure 1a from 1900 to 1998; deflated GDP and a traditional Cobb-Douglas production function of K,L,E are shown in Figure 1b. It is important to note thatGDP increases faster than any of the three contributory factors or (by extension) any firstorder homogeneous function of them. The need for a time-dependent factor representingtechnical progress (the Solow residual) is evident. It is plotted in the figure. The ratio E/GDP is the so-called Kuznets curve. It is often observed that, for manyindustrialized countries, the E/GDP (or E/Y) ratio appears to have a characteristic inverted“U-shape”, at least if E is restricted to commercial fuels. (Not to be confused with ourvariable U). However, when the exergy embodied in firewood is included the supposedlycharacteristic inverted U-shape is much less pronounced. When non-fuel and mineralresources, especially agricultural phytomass are included, fuel exergy E is replaced by totalexergy B, and the inverted U form is no longer evident. Figure 2 shows the various exergyinputs, plotted from 1900 to 1998. Figure 3 displays the three versions of the Kuznets curve.The top curve is the classical version, namely the ratio of commercial (mostly fossil fuel)exergy to GDP. The middle curve is the ratio of all fuels, including firewood, to GDP. Thepeak is still visible but much less pronounced. The third and lowest curve is the ratio of totalexergy inputs, including non-fuel exergy, especially agricultural phytomass, to GDP. Thepronounced inverted U in the top curve apparently reflects the substitution of commercialfuels for non-commercial fuels (wood) during early stages of industrialization.
  11. 11. Ayres and Warr Accounting for growth Page 11III. CALCULATION OF PHYSICAL WORK As noted earlier, the technical definition of exergy is the maximum work that a systemcan do as it approaches thermodynamic equilibrium (reversibly) with its surroundings. It isalso measured in energy units. Not surprisingly, exergy values are very nearly the same asenthalpy (heat values) for ordinary fuels. So, what most people mean when they speak of“energy”, is really “exergy, except that exergy is also definable for non-fuel materials. Wehave done the appropriate calculations in detail in other publications.) However, technically, exergy is also equal to maximum potential work. For non-engineers, mechanical work can be exemplified in a variety of ways, such as pulling a plow,lifting a weight against gravity or compressing a fluid. The term horsepower was introducedin the context of horses pumping water from flooded 18th century British coal or tin mines. Amore general definition of work is movement against a potential gradient (or resistance) ofsome sort. A heat engine is a mechanical device to perform work from heat, though not allwork is performed by engines as we will point out later. With this in mind, we can subdividework into three broad categories, namely work done by animal (or human) muscles7, workdone by heat engines (i.e. mechanical work) and work done in other ways (e.g. thermal orchemical work). Mechanical work can be further subdivided into work done to generate electric powerand work done to provide motive power (e.g. to drive motor vehicles.) The power sources inboth cases are called “prime movers”, including all kinds of internal and external combustionengines, from steam turbines to jet engines. So called “renewables”, including hydraulic,nuclear, wind and solar power sources for electric power generation are conventionallyincluded in energy statistics. Electricity can be thought of as “pure” work, since it can be
  12. 12. Ayres and Warr Accounting for growth Page 12reconverted back into mechanical work, chemical work or thermal work with little or no loss.However electric motors are not prime movers, because electricity is generated by a priorprime mover, usually a steam or gas turbine. Chemical work is exemplified by the reduction of metal ores to obtain the pure metal,or indeed any endothermic (heat consuming) chemical process, such as ammonia synthesis.Thermal work is exemplified by the transfer of heat from its point of origin (e.g. a furnace inthe basement) to its point of use, such as a living-room, via a heat-exchanger (e.g. a radiator). So much for definitions. To measure the work done U, by the economy as a whole, itis helpful to classify fuels by use. The first use-category is fuel used by prime movers to domechanical work. This consists of fuel used by electric power generation equipment and fuelused by mobile power sources such as motor vehicles, aircraft and so on. As regards mobilepower sources, we choose to define efficiency in terms of the whole vehicle, not just theengine itself. Thus the efficiency of an automobile is the ratio of work done by the wheels onthe road to the total potential work (exergy content) of the fuel. Data are available on fuelconsumed by electric power generating plants (known as the “heat rate”) but the work outputis also measured directly as kilowatt hours of electric power produced. The second broadcategory is fuel used to generate heat as such, either for industry (process heat to do chemicalwork) or space heat and domestic uses such as washing and cooking. Lighting can be thoughtof as third category, which is almost a special case. So far we have only considered exergy inputs. The inputs for animal work are, ofcourse, feedstuffs. Horses and mules, which accounted for most animal work on US farmsand urban transport, have not changed biologically since their heyday. The efficiency withwhich animals convert feed energy to work is generally reckoned at about 4 percent. Theuncertainty is unimportant for our purposes.
  13. 13. Ayres and Warr Accounting for growth Page 13 The agricultural phytomass that is converted into human food (as well as petfood,cotton, tobacco, and soap) contribute to the economy in the same way as other industrialmaterials. The recent phytomass-to-food conversion efficiency for North America (via meat,eggs and milk) has been estimated as 5.5 percent (Stefan Wirsenius 2000). In 1900 asignificant share – about 20 percent – went to feed horses and mules, but as people consumeever more animal products, the efficiency of conversion is unlikely to be increasing. On the other hand, the efficiency of heat engines, domestic and commercial heatingsystems and industrial thermal processes has changed significantly over the past 100 years.We have plotted these increasing conversion efficiencies, from 1900 to 1998 in Figure 4.(Detailed derivations of these curves can be found in another publication (Ayres et al 2002)).Work by horses and mules (mostly on farms) has been estimated from the work/feed ratio,together with the horse-mule population. Exergy allocations to different categories of workare shown in Figure 5. Animal work was still significant in 1900 but mechanical andelectrical work have since become far more important. Electrification has been perhaps the single most important source of work and (as willbe seen later) the most powerful driver of economic growth. Electrical work need not becomputed from fuel inputs, since it is measured directly in kilowatt-hours (kwh) generated.The fuel required to generate a kilowatt-hour of electric power has decreased by a factor ofnine during the past century. On the other hand, the consumption of electricity in the US hasincreased over the same period by a factor of more than 1300, as shown in Figure 6. (Thisexemplifies the positive feedback economic “growth engine” discussed earlier.) Effectively there are two definitions of work to be considered hereafter, namely
  14. 14. Ayres and Warr Accounting for growth Page 14(2a) UB = fBB(2b) UE = fEEThe ratios fB and fE are, effectively, composite conversion efficiencies. The former takes intoaccount animal work and all agricultural products, including animal feed. The latter neglectsanimal work and also agricultural production. These two efficiency trends are plotted from1900 to 1998 in Figure 7. It seems likely that, if the trend in f is fairly steadily upwardthroughout a long period (such as a century) it would seem reasonably safe to project thistrend curve into the future for some decades. The trends in physical work done by the economy, together with the ratio of physicalwork to deflated GDP, are shown in Figure 8. The total quantity of physical work inputs intothe economy rise steadily over the entire period and it seems likely that they will continueincreasing at a similar rate. Surprisingly, however, the slope of the ratio of work input to GDPchanges dramatically around 1970. Prior to this date the GDP output per unit of work inputwas decreasing. After this date the trend unaccountably reversed. We think it important tonote that this could not be the result of a business cycle or any other short-term phenomenon.IV. ELIMINATING THE SOLOW RESIDUAL: A NEARLY ENDOGENOUSPRODUCTION FUNCTION There are two important conditions to be satisfied for either version of the expression(1a,b) to be a production function. One of them is the Euler condition for constant returns toscale, which means that for (1a) to be a production function the product fg (subscripts
  15. 15. Ayres and Warr Accounting for growth Page 15omitted) must be a homogeneous first order function of three independent variables, K,L,E orK,L,B. Similarly for (1b) to be a production function g must be a function of K,L,UE, orK,L,UB . The other condition is that the marginal productivities of the three factors be non-negative at all times, at least over a rolling average. (The marginal productivities, logarithmicderivatives of output with respect to each of the factors, need not be constant in time. In factthere is no theoretical reason why marginal productivities should be constant.) It is already evident from Figure 1 that the Cobb-Douglas function cannot explain USeconomic growth since 1900. Of course Cobb-Douglas is the special case with constantproductivities. Of course, there are other functional forms combining the factors K, L, E (orB)that do permit variable marginal productivities and thus may provide slightly better fits thanCobb-Douglas, especially over moderate time periods. However, over the very long-term it iseasy to show that the constant returns (Euler) condition rules out any function of K, L, E or K,L, B, since a homogeneous first order function cannot explain observed growth because sucha function cannot increase faster than any of its arguments and none of the three argumentsincreases fast enough. In particular, exergy inputs – however defined – do not increase fastenough to compensate for the slow growth of labor and capital. Hence no such functionalform can eliminate the need for a time dependent multiplier (Solow residual). This essentiallyrules out (1a) as a viable production function choice.8 We are left with (1b) and, of course, either (2a) or (2b). Over the long time period ofour data base, we want to allow for the possibility of non-constant productivities. It happensthat a convenient functional form (the so-called LINEX function) has been suggested byKümmel (Kümmel 1982)9, namely(3) Y = A U exp{aL/U - b(U+L)/K}
  16. 16. Ayres and Warr Accounting for growth Page 16where A is a multiplier that should (in principle) be independent of time, while a, b areparameters to be determined econometrically. It can be verified without difficulty that thisfunction satisfies the Euler condition for constant returns to scale. It can also be shown thatthe requirement of non-negative marginal productivities can be met. The three factorproductivities are as follows:(4a) M lnY/M lnK = (MY/ MK)(K/Y) = b(L/K) > 0(4b M lnY/M lnL = (MY/ML)(L/Y) = a(L/U) - b(L/K) > 0(4c) M lnY/M lnU = (MY/MUE)(U/Y) = 1 - a(L/U) - b(U/K) > 0It follows from straightforward algebra that the requirement of non-negativity is equivalent tothe following three inequalities:(5a) b >0(5b) a > b(U/K)(5c) 1 > a(L/U) + b(U/K)The variable U in the above expressions (4,5) can, of course be interpreted in either of thetwo ways indicated in (2) without affecting the results. The first condition (5a) is trivial.However the second and third conditions are not automatically satisfied for all possible
  17. 17. Ayres and Warr Accounting for growth Page 17values of the variables. It is therefore necessary to do the fitting by constrained non-linearoptimization. The statistical procedures and quality measures are discussed in the Appendix. The two curves in Figure 9 show the LINEX fits, where we have tested bothdefinitions of work UE and UB respectively as factors of production. The best fit is obtainedby using UB, recalling that total exergy B includes all phytomass inputs and UB includesanimal work. The unexplained residual has essentially disappeared, prior to 1975 and remainssmall thereafter. In short, “technical progress” as defined by the Solow residual is almostentirely explained by historical improvements in exergy conversion (to physical work), assummarized in Figure 4, at least until recent times. The remaining unexplained residual,amounting to roughly 12 percent of recent economic growth, is shown in Figure 10. The marginal productivities of the factors can be calculated directly from equations(4). The three marginal productivities for each of the two interesting cases are plotted inFigure 11a,b, respectively. The differences are surprisingly large, but the results for the bestfit case appear much smoother and more reasonable. There is a small but noticeabledirectional shift that roughly coincides with the two so-called oil crises, and may well havebeen triggered by the spike in energy (exergy) prices that occurred at that time. The shift canbe interpreted as a structural change in the economy, and conceivably, a change in the locusof technological progress from energy conversion efficiency towards systems optimization.However, interpretation of this shift – assuming it is real – will have to await further analysis.V. IMPLICATIONS: TOWARDS AN ENDOGENOUS GROWTH MODEL In the “standard” Solow model a forecast of GDP requires a forecast of labor L,capital stock K and a forecast of the technical change (or multi-factor productivity) multiplier
  18. 18. Ayres and Warr Accounting for growth Page 18A(t). Based on the results described above, the technical progress term can be decomposedinto contributions from improved exergy conversion-to-(primary) work efficiency and“other”. The years 1972 marked a distinct “turning point” in the economic history of the US.Prior to those years the marginal increase per unit of primary work input was generallyfalling; a kind of saturation phenomenon. After 1972, the trend was reversed. Figure 8 expresses this behaviour in terms ofprimary work/GDP10. Evidently growth of GDP since 1972 has slightly outstripped growth ofthe three input factors, capital (K), labor (L) and physical work (U). Clearly primary physicalwork is still by far the dominant driver of growth. However, this does not mean that humanlabor or capital are unimportant. The three factors are not really independent of each other.Increasing exergy conversion efficiency requires investments of capital and labor, while thecreation of capital is highly dependent on the productivity of physical work. An additionalsource of value added is involved. Part of the missing source of value added may have beendue to improvements in the efficiency of “secondary work” or work done by electricity.Improvements in the efficiency of electric lighting, electric motors, refrigeration, electrolyticprocesses and so forth have occurred since 1972, but are not reflected in our efficiency curves(Figure 4). However, preliminary analysis suggests that these efficiency improvements arenot sufficient to account for the “extra” growth since 1972 or so. In the spirit of some endogenous growth theories, it would be possible to interpret thisadditional productivity to some qualitative improvement in either capital or labor. It istempting to argue that the observed shift starting in the 1970s reflects the growing influenceof information technology. Certainly large scale systems optimization depends very stronglyon large data bases and information processing capability. The airline reservation systemsnow in use have achieved significant operational economies and productivity gains for
  19. 19. Ayres and Warr Accounting for growth Page 19airlines by increasing capacity utilization. Manufacturing firms have achieved comparablegains through computerized integration of different functions. We cannot, however, offer anyeconometric confirmation of these conjectures. One of the more important implications of results we have reported in this paper isthat some of the most dramatic and visible technological changes of the past century haveapparently not contributed significantly to overall economic growth. An example in point ismedical progress. While infant mortality has declined dramatically and life expectancy hasincreased very significantly since 1900, it its hard to see any direct impact on economicgrowth, at least up to the 1970s. Greater life expectancy (beyond the age of 60) has addedlittle to labor productivity, and may actually decrease it. (However, it is possible that some ofthe GDP gains since 1970 can be attributed to increased expenditure on health services.) Thegain from medical progress in recent decades, at least, has been primarily in quality of life,not quantity of output. Changes in telecommunications technology since 1900 may constitute anotherexample. Faster communication has clearly generated more communication – a kind orrebound effect – but it is not clear that greater efficiency has resulted. (The examples of faxand email are instructive in this regard.) New service industries, like moving pictures, radioand TV have been created, but if the net result is new forms of entertainment, the gains inemployment and output may have come largely at the expense of earlier forms of public newsand entertainment, such as the print media, live theater, circuses and vaudeville. While thechanges have been spectacular, as measured in terms of information transmitted, theproductivity gains may not have been especially large, at least until recently. Again, the netimpact may have been primarily on quality of life.
  20. 20. Ayres and Warr Accounting for growth Page 20 Turning to the problem of forecasting, since economic growth for the past century, atleast up to 1980, can be explained with considerable accuracy by three factors, K, L, UB, it isnot unreasonable to expect that future growth for some time to come will be explained quitewell by these three variables, plus a growing contribution from IT. From a long-termsustainability viewpoint, this conclusion carries a powerful implication. If economic growthis to continue without proportional increases in exergy (especially fossil fuel) consumption, itis vitally important to use exergy ever more efficiently as well as to develop ways ofreducing fossil fuel exergy inputs per unit of physical work output. But, as declining marginalreturns to investment in R&D limits the future efficiency gains that can be expected – assaturation approaches – we must deliberately exploit new ways of generating value addedwithout doing more physical work. In other words, energy (exergy) conservation is the mainkey to long term environmental sustainability.APPENDIX A For each definition of the LINEX function, the optimal parameters were obtained byusing a quasi-Newton non-linear optimization method, with box-constraints. The constraintson the possible values of the parameters of the LINEX model were required to ensure that thefactor marginal productivities were non-negative. To avoid the influence of local minima thestarting values were selected iteratively. A statistical measure of the precision of fit wasprovided by the Mean Square Error,(A1) MSE = [ t=1 3n (e2 )] (n - k)
  21. 21. Ayres and Warr Accounting for growth Page 21The MSE is a measure of the absolute deviation of the theoretical fit from the empiricalcurve, where n is the number of samples, k the number of parameters and e the residual fromthe fitted curve. We also calculated the correlation coefficients (r2) between log-transformedGDP and each log-transformed factor. We tested the significance of the correlations using a t-test with Welch modification for unequal variances (Table A1). The alternative hypothesis isthat the true difference in the means is not equal to zero at a 95 percent confidence level. Theresults indicate that only fits with UE and UB are significant, and that of UB by far the mostsignificant as indicated by the low estimated t-values.REFERENCESAbramovitz, Moses. “Economics of Growth,” Haley (ed), A survey of contemporaryeconomics (Volume 2). New York: Richard D. Irwin, Inc, 1952, pp. 132-78.______. “Resources and Output Trends in the United States since 1870.” American EconomicReview, May 1956, 46.Aghion, Philippe and Howitt, Peter. Endogenous growth theory. Cambridge MA: The MITPress, 1998.Ayres, Robert U. “The Minimum Complexity of Endogenous Growth Models: the Role ofPhysical Resource Flows.” Energy, September 2001, 26(9), pp. 817-38.
  22. 22. Ayres and Warr Accounting for growth Page 22Ayres, Robert. U; Ayres, Leslie W. and Warr, Benjamin. “Exergy, Power and Work in theUS Economy, 1900-1998.” Energy, (Forthcoming, 2002).Barnett, Harold J. and Morse, Chandler. Scarcity and Growth: The Economics ofResource Scarcity, Baltimore MD: Johns Hopkins University Press, 1962.Beaudreau, Bernard C. Energy and Organization: Growth and Distribution Reexamined.Westwood CT, Greenwood Press, 1998.Cleveland, Cutler J. “Energy Quality and Energy Surplus in the Extraction of Fossil Fuels inthe US.” Ecological Economics, 1992, 6, pp. 139-62.Cleveland, Cutler J.; Costanza, Robert; Hall, C. A. S. and Kaufmann, Robert K.“Energy and the US Economy: A Biophysical Perspective.” Science, 1984, 255, pp. 890-97.Cleveland, Cutler J.; Kaufmann, Robert K. and Stern, David I. “Aggregation and theRole of Energy in the Economy.” Ecological Economics, February 2000, 32(2), pp. 301-17.Dasgupta, Partha and Heal, Geoffrey. “The Optimal Depletion of Exhaustible Resources,”Symposium on the Economics of Exhaustible Resources, Review of Economic Studies, 1974.Domar, Evsey D. Essays in the Theory of Economic Growth, London: Oxford UniversityPress, 1957.
  23. 23. Ayres and Warr Accounting for growth Page 23Fabricant, Solomon. “Economic Progress and Economic Change,” 34th Annual Report:National Bureau of Economic Research, 1954.Granger, C. W. J. “Investigating Causal Relations by Econometric Models and Cross-spectral Methods.” Econometrica, 1969, 37, pp. 424-38.Grossman, Gene M. and Helpman, Elhanen. Innovation and Growth in the GlobalEconomy. Cambridge MA: MIT Press, 1991.Hannon, Bruce M. and Joyce, John. “Energy and Technical Progress.” Energy, 1981, 6, pp.187-95.Harrod, Roy F. Towards a Dynamic Economics (Out of Print), 1947.Jorgenson, Dale W. “The Role of Energy in Productivity Growth.” The Energy Journal,1984, 5(3), pp. 11-26.Jorgenson, Dale W.; Christensen, Lauritz R. and Lau, Lawrence J. “TranscendentalLogarithmic Production Frontiers.” Review of Economics and Statistics, February 1973,55(1), pp. 28-45.
  24. 24. Ayres and Warr Accounting for growth Page 24Kaufmann, Robert K. “A Biophysical Analysis of the Energy/Real GDP Ratio: Implicationsfor Substitution and Technical Change.” Ecological Economics,, 1992, 6, pp. 33-56.______.“The Economic Multiplier of Envirnmental Life Support: Can Capital Substitute for aDegraded Environment?.” Ecological Economics , January 1995, 12(1), pp. 67-80.Kümmel, Reiner. “Energy, Environment and Industrial Growth,” Economic Theory ofNatural Resources, Würzburg, Germany: Physica-Verlag, 1982.Kümmel, Reiner; Lindenberger, D. and Eichhorn, Wolfgang. “The Productive Power ofEnergy and Economic Evolution.” Indian Journal of Applied Economics, 2000, 8, pp. 231-62.Kümmel, Reiner; Strassl, Wolfgang; Gossner, Alfred and Eichhorn, Wolfgang.“Technical Progress and Energy Dependent Production Functions.” Journal of Economics,1985, 45(3), pp. 285-311.Lucas, Robert E. Jr. “On the Mechanics of Economic Development.” Journal of MonetaryEconomics, July 1988, 22(1), pp. 2-42.Mankiew, N. Gregory. Macroeconomics, Worth Publishing, 1997.Meadows, Donella H.; Meadows, Dennis L.; Randers, Jorgen and Behrens, William W.III. The Limits to Growth: A Report for the Club of Romes Project on the Predicament ofMankind. New York: Universe Books, 1972.
  25. 25. Ayres and Warr Accounting for growth Page 25Potter, Neal and Christy, Francis T. Jr. Trends in Natural Resource Commodities,Baltimore MD: Johns Hopkins University Press, 1968Romer, Paul M. “Endogenous Technological Change.” Journal of Political Economy,October 1990, 98(5), pp. S71-S102.______. “Growth Based on Increasing Returns Due to Specialization.” American EconomicReview, May 1987, 77(2), pp. 56-62.______. “Increasing Returns and Long-run Growth.” Journal of Political Economy, October1986, 94(5), pp. 1002-37.Sims, C. J. A. “Money, Income and Causality.” American Economic Review, 1972.Smith, H. “The Cumulative Energy Requirements of Some Final Products of the ChemicalIndustry.” Transactions of the World Energy Conference, 1969, p. 18(section E).
  26. 26. Ayres and Warr Accounting for growth Page 26Solow, Robert M. “A Contribution to the Theory of Economic Growth.” Quarterly Journalof Economics, 1956, 70, pp. 65-94.______. “Intergenerational Equity and Exhaustible Resources.” Review of Economic Studies,1974, 41, pp. 29-45.______. “Perspectives on Growth Theory.” Journal of Economic Perspectives, Winter 1994,8(1), pp. 45-54.______. “Technical Change and the Aggregate Production Function.” Review of Economicsand Statistics, August 1957, 39, pp. 312-20.Stern, David I. “Energy Use and Economic Growth in the USA: a Multivariate Approach.”Energy Economics, 1993, 15, pp. 137-50.Stiglitz, Joseph. “Growth with Exhaustible Natural Resources. Efficient and Optimal GrowthPaths.” Review of Economic Studies, 1974.Swan, Trevor. “Economic Growth and Capital Accumulation.” The Economic Record, 1956,32(68), pp. 334-361.Wirsenius, Stefan. “Human Use of Organic Materials,” Doctoral Thesis, Gothenburg,Sweden: Chalmers Institute of Technology and Gothenburg University, 2000.
  27. 27. Ayres and Warr Accounting for growth Page 27FOOTNOTES* Robert U. Ayres Center for the Management of Environmental Resources INSEAD 77305 Fontainebleau, FRANCE Tel: 33 (0) 1 60 72 40 11 Fax: 33 (0) 1 60 74 55 64 Email: robert.ayres@insead.edu Benjamin Warr Center for the Management of Environmental Resources INSEAD 77305 Fontainebleau, FRANCE Tel: 33 (0) 1 60 72 41 28 Fax: 33 (0) 1 60 74 55 64 Email: benjamin.warr@insead.edu Research supported by Institute for Advanced Study, UN University, Tokyo and The European Commission, TERRA project. We thank Christian Azar, Jonathan Cave, Barry Hughes, Jens Jesinghaus, Yuichi Kaya, Reinhard Kümmel, Jeroen van den Bergh, Chihiro Watanabe, Eric Williams, and Ulrich Witt for helpful discussions and suggestions. All errors are our own.
  28. 28. Ayres and Warr Accounting for growth Page 281. The new A-K models are essentially throwbacks to the pre-Solow Harrod-Domar growth models (Roy F. Harrod 1948; Evsey D. Domar 1957).2. The tendency of virtually all raw material and fuel costs to decline over time (lumber was the main exception) has been thoroughly documented, especially by economists at Resources For the Future (RFF) (Harold J. Barnett and Chandler Morse 1962; Neal Potter and Francis T. Christy, Jr. 1968; H. Smith 1969).3. The proper definition of exergy is the maximum work that can be done by a system as it approaches equilibrium with its surroundings. Thus delivered exergy is effectively equivalent to work. The distinction netween exergy and energy is theoretically important because energy is a conserved quantity (first law of thermodynamics). This means that energy is not “used up” in physical processes, merely transformed from available to less available forms, with concomitant entropy production. On the other hand, exergy is not conserved: it is used up. The directionality of all spontaneous transformations the direction of increasing entropy (second law of thermodynamics).4. Marx believed (with some justification) that the gains would flow mainly to owners of capital rather than to workers. Political developments have changed the balance of power since Marx’s time. However, in either case, returns to energy or to physical resources tend to decline as output grows. This can be interpreted as declining real price.
  29. 29. Ayres and Warr Accounting for growth Page 295. There are statistical approaches to addressing the causality issue. For instance, Granger and others have developed statistical tests that can provide some clues as to which is cause and which is effect (C. W. J. Granger 1969; C. J. A. Sims 1972). These tests have been applied to the question whether energy consumption is a cause or an effect of economic growth (David I. Stern 1993; Kaufmann 1995). In brief, the conclusions depend upon whether energy is measured in terms of heat value of all fuels (in which case the direction of causation is ambiguous) or whether the energy aggregate is adjusted to reflect the “quality” (or, more accurately, the price or productivity) of each fuel in the mix. In the latter case the econometric evidence seem to confirm that energy (exergy) consumption is a cause of growth. We comment on this result later.6. See the discussion of national income allocation in N. Gregory Mankiew’s new economics textbook “Macroeconomics” (N. Gregory Mankiew 1997).7. Human muscular work is insignificant by comparison with other sources of work, and we can neglect it as regards the US or any industrialized country. Human “labor” is something else entirely. It is a combination of sense-based supervision and coordination and brain work.8. The so-called endogenous growth theory gets around this difficulty by relaxing the Euler condition and allowing positive returns. However, as Solow has remarked, this seemingly simple change implies that infinite growth can occur in a finite period of time (Solow 1994).
  30. 30. Ayres and Warr Accounting for growth Page 309. Kümmel introduced the LINEX function with E as a variable, rather than U.10. This is analogous to the so-called Kuznets (inverted U) curve for E/GDP shown in Figure 3 (upper curve).
  31. 31. Ayres and Warr Accounting for growth Page 31 Table A1. Statistical measures of the quality and significance of fitted models Variable t-value Degrees of p-value Correlation used freedom coefficient (r2) B 5.74 158.4 4.5e-08 0.98 E 3.09 174.3 0.002 0.97 UE 0.51 194.8 0.604 0.99 UB 0.19 195.9 0.845 0.99
  32. 32. Ayres and Warr Accounting for growth Page 32FIGURE HEADINGSFigure 1a. Traditional factors of production K, L, E, USA 1900-1998Figure 1b. Cobb Douglas production function and Solow Residual, USA 1900-1998Figure 2. Exergy inputs, USA 1900-1998Figure 3. The ratio of exergy inputs to GDP, USA 1900-1998Figure 4. Energy (exergy) conversion efficiencies, USA 1900-1998Figure 5. Allocation of exergy by types of work, USA 1900-1998Figure 6. Electricity production and conversion efficiencyFigure 7. Exergy conversion efficiency f, for two definitions of work and exergy, USA 1900- 1998Figure 8. Primary work and the primary work/GDP ratio, USA 1900-1998Figure 9. LINEX production function fits with different ‘energy’ factor inputs, USA 1900- 1998Figure 10. The percentage of observed growth unexplained by the LINEX fit with ‘work’ (UB)Figure 11a. Marginal productivities (elasticities) of each factor of production using UE in LINEX, USA 1900-1998Figure 11b. Marginal productivities (elasticities) of each factor of production using UB in LINEX, USA 1900-1998
  33. 33. 12 K L E 10standardised value (1900 = 1) 8 6 4 2 1900 1920 1940 1960 1980 2000 year
  34. 34. 20 Y GDP (actual) Y = Ka Lb Eg A(t), Solow Residual a = 0.28 b = 0.68 15 g = 0.04 MSE = 28estimates 10 5 0 1900 1920 1940 1960 1980 2000 year Base Year 1900 = 1992 $ 354 billion
  35. 35. 100 TOTAL FOSSIL FUEL PHYTOMASS 90 OTHER MINERALS & METALS RENEWABLES 80 70 60eJ 50 40 30 20 10 0 1908 1918 1928 1938 1948 1958 1968 1978 1988 1998 year
  36. 36. 1.6 TOTAL EXERGY (incl. Phytomass) / GDP RATIO TOTAL EXERGY (excl. Phytomass) / 1.4 GDP RATIO FOSSIL FUEL EXERGY / GDP RATIO 1.2 1 Greatratio Depression 0.8 WW I 0.6 0.4 WW II Peak US 0.2 Domestic Petroleum Production 0 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 year
  37. 37. 40% High Temperature Industrial Heat 35% Medium Temperature Industrial Heat Low Temperature Space Heat 30% Electric Power Generation and Distribution Other Mechanical 25% WorkFraction (%) 20% 15% 10% 5% 0% 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 year
  38. 38. 90% HEAT LIGHT80% ELECTRICITY OTHER PRIME70% MOVERS NON-FUEL60%50%40%30%20%10%0% 1908 1918 1928 1938 1948 1958 1968 1978 1988 1998 year
  39. 39. 40% 800 % Electric Power Generation and Distribution (left scale) 35% Index of Electricity Production (right scale) 700 30% 600 index of output (1900=5000 gWh) 25% 500efficiency 20% 400 15% 300 10% 200 5% 100 0% 0 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 year
  40. 40. 0.30 calculated using B and UB calculated using E and Ue 0.25 0.20ratio 0.15 0.10 0.05 0.00 1908 1918 1928 1938 1948 1958 1968 1978 1988 1998 year
  41. 41. 40 2.5 Primary work (left scale) 35 Primary work / GDP ratio (right) 2.0 30 25 1.5Exergy (eJ) ratio 20 1.0 15 10 0.5 5 0 0.0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 year
  42. 42. 20 empirical GDP Theoretical estimates: using UE - MSE = 52 15 using UB - MSE = 7GDP (1992$) 10 5 1900 1920 1940 1960 1980 2000 year
  43. 43. 12 10 8% of growth 6 4 2 0 1975 1980 1985 1990 1995 year
  44. 44. 1.0 Capital - K Labor - L Work – UE0.80.60.40.20.0 1900 1920 1940 1960 1980 2000 year
  45. 45. 1.0 Capital - K Labor - L Work - UB0.80.60.40.20.0 1900 1920 1940 1960 1980 2000 year

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