## Digression Inputoutput Approaches

As mentioned in the previous section, the fact that substitution between inputs, or sectors, is limited in reality suggests that an input-output model with fixed coefficients might be an appropriate tool for analysis, at least in the short run. Input-output models were first introduced into economics by Wassily Leontief (1936, 1941). Fundamental to these models is the existence of an effective sectorization9 of the economy, normally represented as a square matrix with intersectoral transactions. The transactions table has rows that show fractional inputs to each sector from other sectors and columns that show fractional outputs from each sector to other sectors. This table is obviously dynamic; it changes from hour to hour, and certainly from year to year. But Leontief argued that the coefficients (matrix elements) represent technological relationships that do not change rapidly. By assuming fixed coefficients, the relationship in question becomes a model of the economy. The Leontief model assumes an exogenous final demand (including exports) vector Y which determines the vector of sectoral outputs X through a matrix relationship

where A is a matrix of coefficients, representing fractional inputs to each sector from other sectors. Hence it follows that

This famous relationship makes it possible to ascertain the sectoral impacts of an increase (or decrease) in some component of final demand. For instance, an increase in military spending will have implications for other sectors such as steel, aluminum and electronics. We note that the matrix inverse can be expressed as a power series where each term in the series after the first consists of a sum over all possible products of pairs, triples and n-tuples of coefficients. These n-tuples represent flows of products (or payments for products) from sector to sector through the economy. Of course, most sectors sell to only a few other sectors, so most of the possible products in the sums actually vanish, which is why the series converges. The non-vanishing combinations reflect actual flows of products (apart from some aggregation errors) or, in the case of interest to us, energy services, from any starting point - such as coal mining or petroleum and gas drilling to refineries, electric power generation and subsequently to other sectors.

For instance, one primary user of electric power (e-power) is the steel industry. A fraction of the output of the steel industry is produced in electric arc furnaces (including both recycled scrap and stainless steel). The recycled scrap is consumed mainly by the construction industry (for example, for concrete reinforcement), while stainless steel is consumed by a range of industries from kitchenware to plumbing products. These flows are represented by product terms of the form Ae-power,steelAsteel,construction and

Ae-power,steeiAsteeijpiumbing and so on. The extension to products of three or more terms is obvious.

It is tempting to assume that the sum of all such product terms from electric-power generation to others will be a measure of the economic importance of electric power in the economy, and hence of the impact of a price increase, or a supply scarcity. This is partly true. However, one of the other implicit assumptions underlying the Leontief model is that, in the event of a change in some element of final demand Y, all input requirements - including the factors of production (capital, labor and energy) - will be met automatically and instantaneously, or at least within the statistical year. This implies the existence of unused capacity and elastic factor supply curves (Giarratani 1976). That assumption is rarely justified in practice.

An alternative scheme is known as the supply-side I-O model (for example, Ghosh 1958). In this version, the final demand vector is regarded as endogenous, whereas the value-added (expenditure) vector, including expenditure for imports, is given exogenously. In this model, an increase (or decrease) in some element of the expenditure (supply) vector has implications for the outputs of all the other intermediate sectors, as well as final consumption. But again, this model assumes perfect substitutabil-ity of inputs at the sectoral level, and at the final demand level. Indeed, cost minimization implies that each sector will consume only the cheapest input, or that a single combination applies to every sector equally, which is clearly not realistic (Oosterhaven 1988; Gruver 1989). In any case, the supply-side I-O model is not satisfactory for assessing the impact of scarcity of an essential - non-substitutable - input such as petroleum or electric power.

It has been suggested by R.A. Stone that the problem can be addressed by a hybrid I-O model with some supply-constrained sectors and some unconstrained sectors (Stone 1961 p. 98). The idea is to fix the value-added in some sectors and the intermediate or final demand of others, exogenously. The procedure has been explained in detail by Miller and Blair (Miller and Blair 1985, p. 330 ff.), and it has been applied to several cases, primarily in the context of agriculture and limited land availability (for example, Hubacek and Sun 2001). On reflection, it is clear that a reduction in petroleum output by, say, 10 percent will necessarily cut automotive and air transportation activity by almost the same fraction, at least in the short run. Labor or capital cannot replace liquid fuel, so people will have to fly less. The cut in electricity production will have a similar impact on virtually every manufacturing sector, as well as final consumption, because there is virtually no substitute for electric power, at least in the short run, although the allocation of cuts among users might reduce the impact somewhat. Again, labor and capital cannot replace electricity in the near term. The point is that a cut in the availability of a primary fuel will have a downstream impact much larger than the impact on the primary production sector itself.

It is intuitively clear that, because of non-substitutability, the 'weight' of energy (and energy services or useful work) in the economy is much larger than its cost share. The magnitude of the multiplier can be calculated, in principle, from an I-O model. However, the multiplier is not the simple sum of value-added fractions attributable to the primary input, because some downstream substitution between sectors - for example, communication substituting for transportation - does occur. The question is: how much? Unfortunately, the supply-constrained model has not yet been applied, as far as we know, to the problem of constrained petroleum or exergy supplies. Such an application would obviously be desirable, but it is beyond the scope of this book.

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