توزیع انرژی و رشد اقتصادی

This research examines the physical constraints on the growth process. In order to run, maintain and build capital energy is required to be distributed to geographically dispersed sites where investments are deemed profitable. We capture this aspect of physical reality by a network theory of electricity distribution. The model leads to a supply relation according to which feasible electricity consumption per capita rises with the size of the economy, as measured by capital per capita. Specifically, the relation is a simple power law with an exponent assigned to capital that is bounded between 1/2 and 3/4, depending on the efficiency of the network. Together with an energy conservation equation, capturing instantaneous aggregate demand for electricity, we are able to provide a metabolic-energetic founded law of motion for capital per capita that is mathematically isomorphic to the one emanating from the Solow growth model. Using data for the 50 US states 1960–2000, we examine the determination of growth in electricity consumption per capita and test the model structurally. The model fits the data well. The exponent in the power law connecting capital and electricity is 2/3.

The work of Domar (1946) and Solow (1956) marked the beginning of the formal analysis of the growth process where physical capital accumulation is seen a key growth engine. This body of research has unveiled fundamental structural characteristics which impinge on the determination of labor productivity in the long run: savings, population growth, technological change and more. These factors share the common feature that they importantly affect the ability of an economy to mobilize resources for the purpose of capital accumulation. At the same time, the basic neoclassical growth theory abstracts from the fact that it makes little sense to acquire a piece of machinery, at a particular time and place, unless the machine can be supplied with electricity and put to use. Surely, it is of first order importance that an economy is able to distribute electricity across the economy to the sites where investments are deemed profitable. 1 Omitting the physical preconditions for growth may be highly problematic as such factors can represent a binding constraint on economic growth. Hence, in the present study we provide an attempt to model electricity distribution, and take a first step towards examining its implications for the growth process. 2 Electricity networks are highly complex systems; too complex, one might think, for macroeconomic modeling. Fortunately, progress has been made in the natural sciences in describing and modeling the aggregate properties of similarly complex networks. The work of West et al., 1997 and West et al., 1999 and Banavar et al., 1999 and Banavar et al., 2002 is a case in point. By modeling biological organisms as an energy distributing networks these authors have been able to show, in keeping with the evidence, why the energy needs of an organism as a whole rises with the size of the organism in accordance with B = B0 · mb, where B is basal metabolism, m is body mass, B0 is a constant, and b = 3/4. 3 In the present context one might wonder if a similar sort of result could be derived in the case of an electricity network, thus linking electric power consumption and a measure of the “size” of an economy, such as the capital stock. At first it may seem far-fetched to believe that empirical laws and mathematical theories pertaining to biological organisms should have any sort of bearing on man-made networks. But upon reflection the link is perhaps not improbable, for three reasons. First, the cardiovascular system and the power grid share the feature of being energy distributing networks (of nutrients in the former case, electricity in the latter). Conceptually they are therefore highly related. Of course, the networks are very different at the more detailed level: in appearance and in terms of the matter being distributed. Nevertheless, and in spite of being developed for the investigation of biological systems, the inventors of the network theory that we employ to the study of electricity distribution below suggest themselves that their theory could be adapted to the case of electrical currents (Banavar et al., 1999, p. 132). Second, there is a good reason why biological networks and man-made networks would come to have similar aggregate properties. Both biological and man-made networks are developed over time through a process of gradual optimization. In the former case this occurs through natural selection; in the latter it is the result of deliberate decisions to rework, extend and improve the efficiency of the network in question. Undoubtedly, the process of natural selection has worked to produce efficient networks. Quite possibly, man-made networks have moved in a similar direction. As efficient networks have certain unique properties (e.g., minimal transmission losses), biological and man-made networks should have some aggregate characteristics in common. Third, physicists have already started to apply the empirical methods from the study of biological organisms to the case of artificial networks in an effort to uncover universal scaling laws with bearing on human societies (e.g., Bettencourt et al., 2007).4 This research strategy should be seen and appreciated through the lens of the remarks above. Biological networks and man-made networks have similar objectives, namely to distribute resources for final application (be it cell maintenance or the charging of an ipad) as efficiently as possible. Both networks are under constant selective pressure through which they edge towards optimality. In the end it would actually be rather surprising if biological networks and man-made networks did not have many common characteristics. Accordingly, in this study we begin by developing a model of an economy viewed as a transportation network for electricity. The model predicts that electricity consumption per capita (e) can, loosely speaking, be seen as the economic counterpart to metabolism, and capital per capita (k) as the counterpart to body size; the association between electricity and capital is concave, and log-linear as Kleiber’s law. The relevant interpretation of this power law association is as a supply relation: it captures the ability of an economy to make electricity available at geographically dispersed sites for the purpose of final use. The model delivers the above mentioned power law association between e and k and offers predictions regarding the size of the key elasticity linking capital and electricity consumption. Specifically, it is demonstrated that depending on the efficiency of the economy in the context of electricity distribution (in a sense to be made precise below), the elasticity should fall in a bounded interval ranging from 1/2 to 3/4; the more efficient the economy the larger the elasticity. By implication, economies that are more efficient in electricity distribution will be able to make more electricity available for final use. Ceteris paribus, such economies should be able to accumulate capital at a faster rate than less efficient economies. Needless to say, our characterization of electricity distribution neglects a lot of the details of actual power supply. It remains a crude aggregate representation of the network, for which we can provide some micro foundations. In this sense the equation bears some similarity to the aggregate production function. When macroeconomists write down a production function it is not because they believe this to be an accurate description of the full complexity of the myriads of production processes that ultimately generate GDP. It is viewed as a useful short cut, which is justified by the need to gain some understanding of how the economy operates. Our network representation should be viewed in a similar light, and is subject to similar limitations. With an equation summarizing power distribution in hand we subsequently add a representation of electricity demand. From an accounting perspective electricity can be viewed as being used for three basic purposes: running and maintaining existing capital and creating new capital. The notion of capital is broad, including both electricity consuming capital used by households (e.g., an air conditioner in a living room) as well as capital used by firms (e.g., an air conditioner in a factory hall).5 If a “characteristic” machine consumes a certain amount of electricity in use and requires a certain amount of electricity to be created, total electricity demand (at an instant in time) is simply the sum of the electricity requirements related to use, maintenance, and construction of capital. Assuming the demand for electricity (thus determined) equals supply (as reflected in the power law association) we can derive a simple first order differential equation governing the evolution of capital per capita over time. We can then proceed to study the implied dynamics and characterize the steady-state. The law of motion for capital is mathematically isomorphic to the one emanating from a Solow (1956) model, where the aggregate production function is assumed to be Cobb–Douglas. But in contrast to the Solow model the structure developed below holds predictions for the amount of capital (usable in consumption or production) that an economy can sustain in the long run from a physical perspective. This level of capital is determined by the efficiency of the electricity distributing network, as well as the energy cost associated with running, maintaining, and creating capital. Below we argue that technological change is the key driver behind changes in this physical limit to capital accumulation per capita in the long run. Hence, one should not view the steady-state derived below as an absolute boundary for capital but rather as a constraint which continually is being modified due to technological change. From an empirical perspective, however, the law of motion for capital is difficult to confront with data since our definition of capital necessarily is broader than the national accounts concept (cf. footnote 5). However, we demonstrate below that the central law of motion for capital per capita can also be restated in terms of electricity consumption per capita, which is a directly observable variable. If economies – on average perhaps – are operating at the boundary of physical feasibility, the model should be able to match the data on electricity consumption per capita. In order to examine whether this is the case or not, we examine the model’s implications using cross-state data for the 50 US States, 1960–2000. In terms of electricity consumption per capita the model holds several strong predictions. First, conditional on structural characteristics of an economy (notably population growth) one would expect to see β-convergence in electricity consumption per capita. Second, the model can be structurally estimated, which allows for the identification of the networks parameter for which we have a theoretical prior. Using cross-sectional data on electricity sales for the 50 US states we find strong support for the model. Over the period 1960–2000 there is marked tendency for conditional β-convergence in electricity consumption per capita. Moreover, the 95% confidence interval for the networks parameter conforms with the theoretical predictions: 1/2 to nearly 3/4. The point estimate is about 2/3. The remaining part of the paper proceeds as follows. Section 2 lays out the model of the economy, viewed as an electricity distributing network, and derives the power law association between electricity consumption and capital. We then proceed, in Section 3, to add electricity demand, and derive the law of motion for capital and electricity consumption, respectively. Section 4 discusses the model’s implications for long-run development and Section 5 contains the empirical analysis. Finally, Section 6 concludes.

The fundamental notion that economic growth originates from (and is limited by) energy has a long intellectual history, going back to Herbert Spencer’s (1862) First Principles. According to Spencer the evolution of societies depends on their ability to harness increasing amounts of energy for the purpose of production. Differences in stages of development can be accounted for by energy: the more energy a society consumes the more advanced it is. Chemist and Nobel prize winner Wilhelm Ostwald (1907) developed the Spencerian ideas further. Ostwald emphasized that it is not the sheer use of energy, but the degree of efficiency by which raw energy is made available for human purposes that defines the stage of economic (and according to Ostwald also cultural) development of society. 16 The theory developed above demonstrates that this notion of development, when given a modern network interpretation, is compatible with neoclassical growth theory. Indeed, it coincides with the structural form of the economist’s core model of economic growth, the Solow growth model. The central element of the theory, the network equation for electricity supply, receives support in US state-level data. The theory has bearing on the fundamental “limits to growth” debate. In particular, while conceding the importance of energy for growth, the theory also highlights the crucial importance of human ingenuity. As shown above, absent technological change, growth will come to a halt even with unlimited supplies of energy, since energy dissipation increases as the economic network (appliances and machines connected) becomes larger. This result therefore implies that technology, associated with the harnessing and use of energy, is as important for growth prospects as the supply of energy itself; energy and technology are equal partners in development. Indeed, as argued above, “major” innovations (which usually are referred to as GPTs) can be seen as rare instances of progress, which in a profound way improve the harnessing, transformation, and/or distribution of energy. Integrating the literature on endogenous technological change, with the present model of capital accumulation, would therefore seem like a useful topic for future research. The framework could also be adapted to the study of growth in the very long run. It seems widely conceded that human societies at large enjoy income and consumption levels of historically unprecedented magnitudes (e.g. Galor, 2005). A key implication of the model above is that such increases is inescapably linked to the ability of human societies to expand energy supply, which requires technological innovations. In particular then, such a long-run growth model would suggest that the recent harnessing of electricity during the 19th century should sow the seeds of a dramatic change in human societies. First, it is the period during which the modern day energy transport network is created. That is, this period represents the genesis of e(t) = ϵk(t)a. Second, as a result, these innovations allowed for investment growth, and thus income growth, of unprecedented scale, by removing the constraint on capital accumulation previously imposed by energy supply in ways of the metabolism of humans and animals. Accordingly, integrating the framework above with the unified growth literature also seems like a fruitful avenue for future research.