CHAPTER 2 – Background

This chapter is a presentation of the background knowledge relevant to the application of dynamic environmental life-cycle assessment in a structural economics context. The discussion begins with an introduction to static and dynamic economic models, then more specifically a discussion of static and dynamic I–O models. Central to the discussion of I–O modeling is the concept of equilibrium. I–O modeling is an empirical model based on Walrasian General Equilibrium constructs, thus, a brief discussion of its importance is included. The basis of dynamic I–O modeling is structural change. An introduction to this topic as it relates to the model as well as changing technologies and prices. The bridge between LCA and I–O is then discussed in an historical account of previous studies utilizing I–O for environmental analyses. Finally a discussion of how dynamic I–O will be used as the basis of dynamic environmental LCA. Dynamic LCA will be defined and described within this context.

Section 2.1 – Static versus Dynamic Economic Models General

Economists model how society meets its needs with given resources and technologies. There are essentially four general categories of temporal distinctions to the models that economists use: static, comparative static, dynamic and comparative dynamic.

The first temporal distinction, static models, represents phenomena at a single interval of time. A static model would be able answer the question: what was the level of employment in the United States on January 1 1998? In this context, static models are not precursors to dynamic models, they have an independent status of their own. There exist applied problems, which, by their nature, are purely static. This type of problem is one that involves the state of the economy and not process of change. Questions such as this are static questions and they can be readily solved within the realm of static modeling theory (Hicks 1985) (Ruth and Hannon 1997).

The second temporal distinction, comparative static models, compare phenomena at several instances in time, that is, they are comprised of a sequence of static models. Hence, comparative static models are often confused with dynamic models (Ruth and Hannon 1997). For example, a comparative static model is appropriate to answer the question: what was the annual trend of employment in the computer industry from 1975 to 1995? In this example, the comparative static model is a collection of similar static in this case, two models to compare employment in 1975 to employment in 1995.

Typically, static models are assumed to be in equilibrium. Hence, a comparative static analysis can be thought of as examining a system in equilibrium, before and after a change (Fisher 1983) (Baumol 1970). Although, comparative static models are ‘static’ in nature, there is still an important element of time involved with their formulation. “If the predictions of comparative statics are to be interesting in a world in which conditions change, convergence to equilibrium must be sufficiently rapid that the system, reacting to a given parameter shift, gets close to the predicted new equilibrium before parameters shift once more” (Fisher 1983, 3). Hence, in order for comparative static models to be useful, the parameters that remain static in the model within the period of study, must also remain relatively static in the real world.

The third temporal distinction, dynamic models, are models that try to reflect changes in time and take into account model components that are constantly changing as a result of previous actions or future expectations (Ruth and Hannon 1997). Here, a dynamic model would be appropriate to answer the question, what are the impacts of employment in the computer industry now and into the next century? It is important to note the distinction of comparative-statics to dynamic models. Although comparative static model may appear dynamic, they are not sufficient to answer issues concerning planning and are limited to historical accounts (Hicks 1985).

Therefore, a primary distinction of a dynamic model versus a static model is the scope of examining intra-period relationships. Within a period we define a sequence of intervals. Static models do not consider intra-period relationships of the intervals and dynamic models do. Dynamic economic modeling involves an understanding of how phenomena within an interval are related to activities outside the interval yet within the period of study.

Hicks (1985) eloquently points out the distinction of the inter-period study within static and dynamic models:

“Proper dynamic theory, even at its single-period stage, must take account of the fact that many activities that go on within the period are oriented outside the period; so that what goes on, even within the period, is not only a matter of tastes and resources, but also of plans and expectations. In statics there is no planning: mere repetition of what has been done before does not need to be planned. It is accordingly possible, in static theory, to treat the single period as a closed system, the working of which can be examined without reference to anything that goes on outside it (in the temporal sense). But this is not possible in dynamics. Even at the single-period stage the links which relate the single period to the rest of the dynamic process cannot be neglected” (Hicks 1985, 25).

Both static and dynamic models are concerned fundamentally with the structure of the interrelationships or interdependencies among variables and data of models (Kuenne 1963)(Kuznets 1953). However, a fundamental difference between models is that a specific solution to a static system yields a single solution vector, whereas a specific solution to a dynamic system is “a set of such vectors linked in a path through time” (Kuenne 1963, 14). A dynamic model may have more than one path converging to the same or different positions of rest (Kuenne 1963).

Incorporating the notion of inter-period relationships, “a dynamic model contains the potential for the derivation of theorems concerning the values of the variables, or changes in those values, before the position of rest, or equilibrium has been attained” (Kuenne 1963, 14). In contrast, a static system can yield theorems about “the values of the variables only in a state of rest, or theorems about changes in the values of the variables only between two states of rest” (Kuenne 1963, 14). Further, the distinction between static and dynamic models is not simply the existence of time in a dynamic system and its absence in a static model. The use of a static model must involve the interpretation of its solution “against time as a backdrop” (Kuenne 1963, 15). The difference between the models is more accurately reflected in the distinction between potentially derivable relationships (Kuenne 1963).

An intuitive presentation of the difference is presented by Kuenne (1963):

“A static model is one whose structural relationships do not contain time in any analytically meaningful way. By contrast, dynamic systems are those which do contain time-relationships among the relations of the variables in meaningful ways, i.e., in ways which could not be eliminated without affecting the solution to the system or eliminating the possibility of the solution” (Kuenne 1963, 457).

Essentially, a dynamic model is one where the incorporation of temporal information makes a difference in the ultimate solution of the system modeled.

The fourth distinction, comparative dynamics, is similar in concept to comparative statics. In the method of comparative static analysis, solutions to two sets of simultaneous equations are compared, while in comparative dynamics two sets of mutually exclusive differential equations are compared. For example, a dynamic system to determine quantity, q, consumed for the present time period is given by the differential equation of a consumption function F(x):

(2-1)

and by a second set of differential equations for the next time period:

(2-2)

The effective quantities for the first period are given by the values which solutions to (2-1) take on at t = ¥. Similarly, the values for effective quantities for the second period are given by the solutions described by (2-2) at t = ¥. By creating an imaginary system (2-2)´ (where (2-2) is the actual system in the next period but (2-2)´ is an imaginary one in the present), the analysis is then reduced from the comparison of effective quantities in two consecutive time periods to a comparison of two mutually exclusive dynamic systems (Morishima 1996, Ch. 5).

Baumol (1970) presents the following analogy: Here, an analogy to the study of a photograph can be made. A “still” photograph of a system in motion can be used to examine the position of various parts and the way they fit together. The “still” must include a reference to time, but the analysis of the still can be static (Baumol 1970, 5). One can expand the analogy that a motion picture is a succession of “still” photographs representative of comparative statics. And further, that the rules governing “continuity” such as in the art of cinematography, i.e., the relationship of a single frame to its preceding frame and its succeeding frame, is analogous to a dynamic study. It then follows that a comparative dynamic analysis would then be the comparison of the continuity of two segments of a motion picture.

In summary, both static and dynamic models involve some element of time. Static models are confined to a point in time or a single interval of time, and in general are concerned with changes in social behavior, such as price, wages, and demand. Further, the general assumption is that static models are at rest, or in a state of equilibrium. In contrast, dynamic models are expanded to include several intervals of time. They include inter-period relationships that go beyond social phenomena, such as the accumulation of capital and technological changes. In general, dynamic models describe time-based relationships that ultimately are reflected in the solutions obtained. Comparative static models are examinations of specific equilibrium points in time and comparative dynamic models are examinations of one system of inter-period relationships to another.

Section 2.2 – Static and Dynamic I-O Models

The distinction between static and dynamic I-O models is a complex topic that is interwoven with the mathematics of dynamic linear programming. The following discussion will introduce static I-O economics and the introduction of dynamic I–O in the mid–20^th century.

What is an input-output model? Economic I–O analysis is a method to systematically quantify the interrelationships among various sectors of an economic system. The economic system may be as large as a nation or even the entire world economy, or as small as the economy of a metropolitan area (Leontief 1985a). The I-O model is a formalization of the basic concepts published in 1758, by French economist, François Quesnay, in his "Tableau Economique." More than 100 years after Quesnay's work, León Walras developed a theory of general equilibrium that utilizes a set of production coefficients that related the quantities of factors required to produce a unit of particular product to levels of total production of that product. Wassily Leontief's model is a linear approximation of the Walrasian model, constructed in a similar fashion as the Tableau Economique, that allows the general theory of equilibrium to be applied (Miller and Blair 1985).

The approach to economic I–O analysis is as follows. The structure of each industry’s production process is represented by an appropriately defined vector of structural coefficients that describes in quantitative terms the relationship between the inputs it absorbs and the output it produces. The interdependence among the sectors of the given economy is described by "a set of linear equations expressing the balances between the total input and the aggregate output of each commodity and service produced and used in the course of one or several periods of time" (Leontief 1985a).

Thus if the economy is divided into n sectors, and if we denote X_i the total output (production) of sector i and by Y_i the final demand for sector i's product, we have:

(2-3)

for , and, .

The z terms on the right-hand side represent the interindustry sales by sector i to sector j. Thus, the entire right-hand side represents all of the interindustry sales, z_ij to final demand of sector i, Y_i, hence the total output of sector i. We then construct the system of equations for n sectors and we have:

(2-4)

If we then look at the jth column of z's we then have the following column vector:

(2-5)

These elements are the sales or inputs to sector j representing the intermediate input components necessary for the production of output by sector j. In addition to material inputs, there are labor activities (value added) that are also associated with production. We now have the basis of the Leontief table illustrated below:

Table 2–1. Input–Output Table of Interindustry Flows (Miller and Blair 1985).

				Purchasing Sector
		1	2		…	j	…	n
	1	z₁₁	z₁₂		…	z_1j	…	z_1n
	2	z₂₁	z₂₂			z_2j		z_2n
Selling Sector
	i	z_i₁	z_i₂			z_ij		z_in

	n	z_n₁	z_n₂		…	z_nj	…	z_nn

In input-output work, a fundamental assumption is that the interindustry flows from i to j depend entirely on the total output of sector j. That is, if sector j represents vacuum cleaners, we assume that if there is an increase in the sales of vacuum cleaners, there will be a corresponding increase in the sales of electric motors that are used in vacuum cleaners. From this concept we then formulate a ratio of input to output termed a technical coefficient. That is for a z_ij, the flow of input from i to j, and X_j, the total output of j, forms the technical coefficient denoted a_ij:

(2-6)

The a_ij's are fixed relationships between a sector's output and its inputs. Thus, there is an explicit definition of a linear relationship between input and output and there are no economies of scale, rather the Leontief model represents constant returns to scale. Thus, doubling inputs will double outputs, reducing inputs by half will reduce outputs by half. In essence, the coefficients represent the trade from industry i to industry j. By accepting the notion of fixed coefficients we can rewrite equations (2-4), replacing each z_ij, by a_ijX_j.

(2-7)

For input-output analysis, a common question to be asked is: given a forecasted demand (the Y_i's), how much output from each of the sectors would be necessary to supply these final demands? The Y_i's and the technical coefficients, a_ij are known and we solve simply solve for the outputs of each sector, the X_i's. Therefore, for a given set of Y's, we solve for n unknowns, X₁, X₂, …, X_n.

In matrix terms, we define:

, , (2-8)

Thus in matrix form the complete n × n system is:

(2-9)

The matrix A is known as the matrix of technical, input-output, or direct input coefficients. If , then can be found and a unique solution is given by:

(2-10)

where is commonly referred to as the Leontief Inverse. For a more detailed derivation of the input-output methodology, please refer to Miller and Blair (1985) and Leontief (1985a).

The following is a simple example using the input-output methodology. We are given data regarding two industrial sectors as shown in Table 2-2. This table quantifies the transactions between the two sectors, their final demand, and the row element of payments, representing the value added.

Table 2–2. Flows from two hypothetical sectors (Miller and Blair 1985).

		To Processing Sectors		Final Demand	Total Output
		1	2	(Y_i)	(X_i)
From Processing	1	150	500	350	1000
Sectors	2	200	100	1700	2000
Payments (value added)		650	1400
Total Outlays	(X_i)	1000	2000

The corresponding table of technical coefficients is found by dividing each flow in a particular column of the processing sectors in Table 2–2 by the total output. Thus a₁₁ = 150/1000 = 0.15. The resulting technical coefficient matrix, commonly referred to as the "A" matrix is shown in Table 2–3.

Table 2–3. Example technical coefficient matrix (Miller and Blair 1985).

		To Sectors
From		1	2
Sectors	1	0.15	0.25
	2	0.20	0.05

For this example, the technical coefficient matrix of Table 2–3, can be used to construct the recipe of commodities in each of our sectors. For sector 1, we need a interdependency of 15 cents of sector 1 and 20 cents of sector 2 to produce a dollars worth of commodity for sector 1. A typical question for this input-output analysis table is to find the resulting production as a result of increased demand, or output. For our example the initial output was $350 for sector 1 and $1700 for sector 2. It is important to note that for the year of observed data , and , thus the interindustry demands were greater than the final output.

For our analysis example we increase final demand to $400 for sector 1 and decrease demand for sector 2 to $1600, thus and . We then use equation (2-9) to find the new production levels.

, and .

The needed output are then found from equation (2-10):

The values of DX₁ and DX₂ then represent the impact to the economy as a result of changes in demand. For this case, DY₁ = $50 and DY₂ = -$100. This change in demand resulted in an increase of sector 1 activity of $29.7 and a decrease in sector 2 activity of $99.0 (Miller and Blair 1985).

Referring to the simple 2 × 2 model presented in Table 2-3, conceptually the simple model can be thought of as a nested feedback. The feedback is a relationship of the demand for two commodities (Y₁ and Y₂), and the two production activities that supply two commodities as shown in the digraph of Figure 2-1.

Figure 2–1. Simple Two-Sector Digraph.

In our example, the demand vector is , and we use the technical coefficient matrix presented in Table 2–3. For the first round, the amount of sector 1 and sector 2 production will be respectively, $400 and $1600 to meet, as a minimum, final demand. However, we know that from Table 2–3, that an additional (0.15)($400) = $60 sector 1 and (0.20)($400) = $80 from sector 2 will be necessary to produce $400 worth of sector 1 commodity. Also, for sector 2, we will need to produce an additional (0.25)($1600) = $400 from sector 1 and (0.05)($1600) = $80 from itself to produce the $1600 worth of sector 2 commodity. Therefore, for this "round", the total economic activity will be $60+$400 = $460 for sector 1, and $80 + $80 = $160 for sector 2. The next "Round-By-Round" impacts are then calculated for an intermediate demand of Y₁' = $460 and Y₂' = $160. The total "Round-By-Round" are summarized in Table 2–4:

Table 2-4: Round-By-Round Impacts (in Dollars) of (Miller and Blair 1985).

Round	1	2	3	4	5	6	7	³ 8
Sector
1	400	460	109	41.4	12.9	4.34	1.42	0.64
2	1600	160	100	26.8	9.61	3.06	1.02	0.51

Thus, the right hand side of equation (2–10) can be thought of the algebraic equivalent of:

(2–11)

Or removing the parentheses,

. (2–12)

Thus, the direct impacts can be considered: