Because of the essentiality (non-substitutability) condition noted several paragraphs above, we conceptualize the economic system as a multi-sector chain of linked processing stages, starting with resource extraction, reduction, refining, conversion, production of finished goods and services, including capital goods, final consumption (and disposal of wastes). Each stage has physical inputs and physical outputs that pass to the next stage. At each stage of processing, value is added and useful information is embodied in the products, while low value, high entropy, low information wastes are separated and disposed of.3 Global entropy increases at every step, of course, but the value-added process tends to reduce the entropy of useful products, while increasing the entropy of the wastes. An adequate description of the economic system, viewed in this way, must include all materials and energy flows, and information flows, as well as money flows. These flows and conversion processes between them are governed by the first and second laws of thermodynamics, as well as by monetary accounting balances.
It is evident that there are also feedbacks - reverse flows - along the process chain. For instance, capital goods are manufactured products that are important inputs to all stages, including the extraction and processing stages. Electric power and liquid motor fuels are intermediate products that are utilized in all sectors, including the extraction sectors. Information services, including financial services, produced near the end of the chain are also utilized by all sectors. This feedback is the fundamental idea behind Leontief's input-output model (Leontief 1936). When monetary flows are considered, the feedbacks are significant. Certainly they cannot be ignored. However, for the present, we are less concerned with monetary flows than with flows of mass/exergy (or useful work). From this perspective, the reverse flows are quantitatively small compared to the main mass/exergy flows in the forward (downstream) direction.
The next step must be to justify the use of a so-called aggregate production function in a situation where an input-output (I-O) model with fixed proportions might seem to be more appropriate, at least for short-run analysis. However, in the longer term, substitution between factors does occur - in conjunction with investment - whence the Leontief model with fixed coefficients is inappropriate.4 We expect to show that the relative importance of capital and energy (as useful work) have increased significantly over time vis-à-vis labor. This change reflects the long-term substitution of machines (in the most general sense) driven by exogenous energy sources mainly fossil fuels, for human and animal muscles, and human brains.
In the standard theory of productivity growth, beginning with Solow, firms produce goods and services - actually, a single composite product
- while households produce labor. Firms are assumed to be very small profit-maximizing price-takers, subject to constant returns to scale, producing a single composite good, and using capital and labor to the extent justified by marginal productivity of these factors. Consumers (households) sell labor and own capital, while firms may also own capital. In this idealized case, the cost shares for capital and labor in the national accounts would be equal to the corresponding output elasticities. We could, of course, generalize the Solow model by adding energy flows or useful work flows, provided by an exogenous utility. Each firm would purchase the amount of useful work justified by its marginal productivity. The question remains: what is the marginal productivity of useful work and what is its cost share? The latter question is particularly vexing. It can best be approached by means of an input-output model, as noted in the last chapter (and again later).
However, (in the spirit of evolutionary models) we do not assume that firms must operate on or move along the 'frontier' of a region in factor-space, as they would have to do if they were profit-maximizers with perfect information in a perfectly competitive market. On the contrary, we postulate (in the spirit of Milton Friedman (1953)) that if an assumed relationship explains (that is, reproduces) the empirical observations, one need not worry too much about the realism of every one of the underlying assump-tions.5 We also concede, in common with most neoclassical theorists, that the notion of a 'frontier', where all firms exist at all times, is quite a stretch from reality.6 In reality, the collection of firms in factor-space constitutes a sort of turbulent cloud (Figure 6.2). The 'frontier' idea is useful only to the extent that it describes the average of an ensemble.
We also recognize that the economy is really multi-sectoral. Firms operate in sectors where they compete with others within the sector, but not with firms in other sectors. This assumption reflects intersectoral non-substitutability, as mentioned above, but does not exclude the possibility that generic inputs (capital, labor and energy services as useful work) may substitute for each other even in the short run, within some small range.
In short, we argue that a postulated functional relationship among aggregates (capital, labor and mass/exergy - or useful work) flows is an adequate representation of the real world, at least for the purposes of explaining economic growth. Almost all firms are operating at some distance from this fictitious frontier, either inside it and outside it. The only further assumption needed to account for this picture is that firms do not have perfect knowledge or foresight, and that competition is not perfectly efficient. A firm too far inside the cloudy frontier is likely to be unprofitable and risks being selected out, in time, if it does not change its strategic behavior. On the other hand, a firm on the outside is likely to be above average in profitability, and may grow at the expense of its competitors.
Figure 6.2 The production frontier as a turbulent cloud
Figure 6.2 The production frontier as a turbulent cloud
If the condition of constant returns to scale is retained, it can be shown without difficulty (below) that adding a third term for materials and energy (exergy) resource inputs in a conventional Cobb-Douglas function for a single sector, while retaining the interpretation of output elasticity as share of payments in the national accounts, does not explain past economic growth any better than the original Solow model without a multiplier A(t). It is also inconsistent with the usual assumption that the economy is a single sector with a single composite output, as noted in Chapter 5. In other words, an exogenous time-dependent multiplier to reflect technical progress or total factor productivity is still required in this case.
However, if only for historical purposes, we start with the old Cobb-Douglas function.
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