Aggregation Substitut Ability And The Factor Payments Problem

In 1973-4 there was an oil crisis and an embargo. Oil prices in particular and energy prices in general rose rapidly. Economists wanted to assess the impact on economic growth. Dale Jorgenson and his colleagues introduced a four-factor version, known as KLEM (capital, labor, energy, materials). They also introduced a transcendental logarithmic production function of these four factors, expressed as prices rather than quantities (Jorgenson et al. Lau 1973; Jorgenson and Houthakker 1973; Jorgenson et al. 1982;

Jorgenson 1983, 1984). However, a simpler approach was to incorporate energy as a third factor E into a Cobb-Douglas production function (for example, Allen et al. 1976; Hannon and Joyce 1981). This approach retained the condition of constant returns to scale and the usual assumption that factor elasticities should correspond to factor payments' share in the national accounts.

What if a third factor (energy) is included? It is axiomatic that the sum of all money payments must exhaust the total output of the economy. The trouble is that payments to resource owners are not really payments 'to' energy. Indeed, energy is not a tangible commodity or substance. The term 'energy' is a conceptualization of physicists, which only evolved to the present level of understanding in the late 19th century (earlier names for this idea were 'phlogiston' and 'vis viva'.) Sunlight can be thought of as a pure form of energy but it cannot be owned or sold as such. Nor can the wind or the kinetic energy of tides or flowing water. What can be captured, whether by water or wind turbines, or in the products of photosynthesis (present or past), is value added to this non-substance by the investment of labor and man-made capital, plus some of the recovered energy (useful work) itself. There are no payments to or for energy per se, both because energy is a conserved quantity (that quantity which is consumed or used up is exergy) and because there is no entity with financial accounts to receive or disburse payments.

Economists have tried to get around this difficulty by assuming that energy (exergy) carriers (food, fuels) are equivalent to energy itself and that the owners of resources like land, coal and petroleum are the ones to be paid. Since Ricardo, these payments have been classified as 'rents'. But, from another perspective, land and mineral resources are really forms of capital (natural capital), whence useful energy ('useful work') is really a product of capital, both natural and man-made, plus labor. That seems reasonable, since both natural and man-made capital plus labor are obviously required for mining, drilling and agriculture. It is important to emphasize again, however, that capital and labor do not actually create the energy, which is either embodied in fossil fuels, flowing water, wind or sunlight, all of which are actually gifts of nature.

This seems reasonable at first sight. But then the energy (exergy) component of capital, which is not man-made, cannot be treated as a product of savings and investment. It is better to regard the extraction activities (agriculture, forestry, mining, drilling, etc.) as a distinct sector or sectors, whose inputs are man-made capital and labor and whose products are food, animal feed, wood, coal, petroleum, natural gas, etc. These products are derived directly from sunlight, movements of air or water or extracted from a stockpile of the above-mentioned gifts of nature. The costs of transforming them into marketed products are simply the payments to capital and labor used by the corresponding extraction and transformation sectors.

It is evident that, apart from rents to land or mineral resource owners, the extractive sectors per se account for a very small share of payments in the national accounts of an industrialized country. The more value is added to crude oil or coal in the chain of processes leading to a final service, the smaller the extractive component will be. Thus, as a number of influential economists have argued, if factor elasticities are equal to factor payments' share in the national accounts, factor price changes cannot make any significant difference in terms of explaining economic growth (for example, Denison 1979). But are output elasticities really equal to cost shares? Solow, Denison, Jorgenson et al. habitually thought in terms of a single-sector economy producing a single 'composite' all-purpose good. They also assumed equilibrium, perfect competition and homogeneous labor and capital. Finally, they assumed that all factors are mutually substitut-able. Actually the formal proof of equivalence between output elasticity and cost share also depends upon the assumptions of constant returns to scale, that firms are profit maximizers (and cost minimizers) in equilibrium (Appendix A). Needless to say these assumptions are all open to serious question.

What if the economy is more complex than a single sector producing a composite good? Does the famous factor share theorem (Appendix A) still hold for a two, three or multi-sector economy? Setting aside the question of substitutability among the factors (capital, labor, useful work), the answer is provisionally 'yes'. Consider a two-sector economy such that sector 1 consists of all the extractive and primary energy conversion sectors, such as electricity generation, and sector 2 consists of all the rest. The inputs to sector 1 are capital and labor, and gifts of nature. The output of this sector is energy services, or what we call 'useful work', some of which is utilized within sector 1 itself. Subject to the usual assumptions, the factor share theorem still holds, meaning that the costs of useful work consist of the capital and labor costs used to produce it.

Sector 2 consists of all the activities that convert the net output of useful work, plus additional capital and labor, into final products and services. Once again, the theorem holds. The costs of the final output (GDP) consist of the costs of all the capital and labor inputs to the second sector plus the cost of the useful work inputs from sector 1. But the costs of the latter are simply the cost of capital and labor utilized by sector 1. The total of all capital and labor for both sectors are now accounted for.

Now suppose 'useful work' were to be treated as an independent factor of production (provided by an exogenous source; presumably the same agency that leases capital equipment and provides workers as needed). For purposes of argument we can now pretend that it is not actually a product of capital and labor applied to natural resources. In this case we can also dispense with the two-sector approximation and revert to the single-sector model similar to that of Solow, but with three factors. The factor share theorem still applies, subject to the usual assumptions, but with three factors, as proved in Appendix A.

However, one of those questionable assumptions is that both or all three factors are mutually substitutable. In the real world this is a very strong condition, because it implies that GDP could be produced by any one of the three factors alone, without either of the others. In reality, there is at most a narrow range of substitutability in the neighborhood of the actual combination of factors that exists at any one time. Substitution does occur, of course, but only over time, accompanied by capital investment. But there is also a considerable degree of complementarity. Machines require workers to operate them. (Only in a distant and hard-to-imagine future populated by intelligent robots might it be possible to produce all the goods and services we now enjoy without human labor.) Labor requires tools to be productive. Both humans and animals require food or feed. Heat engines require exergy (fuel) to do useful work, and other machines require useful work (for example, electricity or high temperature heat) to function. In short, both labor and capital equipment also require exergy inputs. All three factors are essential and therefore not substitutable - except at the margin - in the economy as we know it.

Of course the real economy is a multi-sector system, consisting of many products that are not substitutable for each other in the short term, of which 'useful work' is one of the most important examples. But quite apart from energy services, food cannot be replaced by paper, plastics cannot be replaced by cement, steel cannot be replaced (except in special cases) by aluminum, and copper has no effective substitute, at least for electrical wiring. The multi-sector character of the economy is determined by these limits on substitutability. Of course the multi-sector economy is characterized by important intersectoral flows and interdependencies. Crude extractive inputs are converted first into finished fuels and materials, subsequently into components, then subsystems, then complex products and structures, and finally into transport, information, entertainment and other services.

To deal adequately with the sectoral non-substitutability problem, an input-output approach would seem to be appropriate. Up to now, however, serious practical difficulties have prevented significant progress in that area. This issue is addressed briefly in the next section.

It is not clear to us whether the mathematical relationship that is proved in Appendix A is still applicable to a multi-factor, multi-sector economy where the factors are not totally substitutable and the output of one sector is an input to some other sector. True, quite a number of economists, including Jorgenson and Allen (cited above), have tried to include a third factor, namely commercial energy E, while retaining the single sector 'composite product' assumption. But this is conceptually dubious. There seems to be a fundamental contradiction in using a model that assumes perfect substitutability while assigning the cost of energy as the payments to primary resource extractive industries, thus identifying those industries as a separate sector.

Farm products are not made entirely by farmers, and coal, oil and gas are not created by the firms that extract them. Moreover, the energy carriers produced by these sectors are subsequently refined and converted into useful work or useful heat by other downstream sectors, from food-processing to petroleum-refining to electric-power generation, and further down the line by the engines in motor vehicles and the furnaces in industrial plants or the heating plants in commercial and residential buildings, not to mention the cooking stoves in kitchens. A moment's thought suggests that treating the fossil fuel extractive industries as the 'source' of energy is not only wrong in logic, but that doing so implicitly treats the economy as a two (if not multi)-sector system. From this perspective, it becomes clear that the real, but indirect (downstream) elasticity of useful work is far greater than the cost share of the extractive industries at the beginning of the chain.

The assumption that the elasticity of exergy service output should correspond to the payments to primary exergy extraction in the national accounts is still so widespread (despite being very dubious) that a few more paragraphs can justifiably be devoted to the subject. Here is a quotation from a 2005 Nobel Laureate, taken from the Concise Encyclopedia of Economics, available on the internet (Schelling 2002):

Today, little of our gross domestic product is produced outdoors, and therefore, little is susceptible to climate. Agriculture and forestry are less than 3 percent of total output, and little else is much affected. Even if agricultural productivity declined by a third over the next half-century, the per capita GNP we might have achieved by 2050 we would still achieve in 2051.

This particular article is entitled 'Greenhouse Effect', which explains the context. But it is clear that Schelling assumes that a radical cut in agricultural production would affect only the agriculture sector. In other words, he ignores the chain of downstream impacts on food processing, agricultural chemicals, tractor sales, rail transport, wholesale and retail trade, hotels and restaurants, etc.8 In effect, Schelling assumes that all the downstream sectors that consume agricultural products will easily find substitutes.

Yet it is perfectly clear that they will not. Suppose (with Schelling) that physical agricultural output (harvested crops) were cut in half, and suppose - for simplicity - that this cut applied to every crop. Based on existing intersectoral relationships, the immediate result would be that animal feed, other than grass, would be eliminated almost completely since humans would need all the grain, and the output of chickens, turkeys, pigs and grain-fed beef would fall to near zero. Only lamb and (some) veal would remain on the market, and the meat-packing industry would virtually disappear, as would most of the butchers. The dairy industry would also have to reduce its output substantially. Transport requirements for grain, potatoes and other bulky crops would be cut in half. Alcoholic beverages (which depend on grain) would also be cut sharply, as would leather goods, tobacco products and so on.

To be sure, these consequences would be modified by prices. Steak lovers and beer/whiskey drinkers would bid up the prices of beef and alcoholic beverages, thus reducing the amount of grain products available to millers and bakers to make bread and cereal products. The prices of those products would consequently increase dramatically, and the poor would have less to eat. Meanwhile many workers in agriculture, food processing and transportation, not to mention retail trade, would also lose their jobs, to the extent that they are related to the processing and movement of physical quantities. Clearly an input-output approach is needed to assess the real economic impact of a cut in agricultural production.

According to Schelling's argument, it would seem to follow that a sudden 50 percent cut in US energy supplies, which account for about 4 percent of GDP, would only result in a 2 percent reduction in US GDP. Virtually everybody in touch with reality knows this to be absurd. The transport sector, the construction sector, the chemical sector and even agriculture would be devastated by such a cut, precisely because there is no substitute for energy. The downstream and indirect impacts will have a multiplier effect several times greater than the primary cut. In short, the output elasticity of energy (exergy) services must be significantly greater than the cost share.

To take a more extreme, but equally pertinent, example, consider the sector that delivers water and sewer services to cities (SIC 680302). The total value added by this sector may be only in the tens or low hundreds of billions of dollars, which is insignificant in terms of the whole US GDP. But if these services were eliminated, the economy would collapse utterly. The multiplier effect in this case might be 100 or more. The point is that the real economy is not a homogeneous entity producing a single 'composite'

good, as many simple models assume (and Schelling assumed in his encyclopedia article). The reality is that we have a diverse multi-sector economy in which most sectors are dependent on inputs from others, and some intermediates - like water, food and electricity - are essential and non-substitutable.

In a multi-sector world the cost-minimizing strategy for the firm is not determined only by the elasticities of labor or of capital goods per se, but by a combination of labor, capital goods and other intermediates purchased from other sectors. Mankiw's textbook example was of bakers producing bread, but only from capital and labor (Mankiw 1997). Real products like bread cannot be produced from abstractions. In the real multi-sector world, the bakers must also purchase flour and yeast from a food-processing sector that buys grain from farmers. They must also purchase ovens from a manufacturer and fuel for their ovens from a gas distributor. Each of those sectors purchases from others.

Therefore, the idealized single-sector model of firms utilizing only labor and durable capital goods cannot be generalized to the economy as a whole. The cost-minimizing process at the firm level leaves its imprint on the overall picture, like the grin of the Cheshire cat.

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