Chapter 3 Continued
A Commodity by Industry Approach
To accommodate for this new structure of economic activities, a technical matrix that describes distributed multiple output and input activities in an enhanced SIM model is presented. Seminal work by Stone (1961, 1966) and Victor (1972) are used as starting points to fully describe the production structure of Life-cycle SIM.
As discussed in Chapter 2, Victor (1972) utilized a Commodity-by-Industry model based on Stone (1961, 1966) to create a static ecological model schema in a Leontief framework. The Victor ecological static model, shown in Figure 2–8, contains two concepts necessary to represent Life-Cycle SIM. These include:
1. Multiple output in the form of secondary products and by-products that can be attributed to a single industry,
2. The ecological input and output integrated into the model consistent with environmental life-cycle assessment.
It is then necessary to convert the Victor model into a distributed activities model to create life-cycle SIM. This is accomplished by utilizing the make and use matrix constructs to create a distributed activities direct requirements matrix. From this direct requirements matrix, a representation of total production and emissions output distributed in time can be formulated.
The commodity-by-industry model can then be used in several configurations to develop total requirements matrices. For example, data may be available regarding final demand for commodities, however total impacts are to be analyzed as total industry output, therefore an industry-by-commodity total requirements matrix configuration would be desired. Alternatively, if impacts were to be analyzed as a function of total commodities a commodity-by-commodity or a commodity-by-industry total requirement matrix would be used.
The basic premise of the Victor model is the commodity balance equation. The commodity balance equation represents the total production of a commodity that in turn is the sum of all the amounts of that commodity consumed by industries in the economy plus any sales of that commodity to final customers:
(3-14)
In matrix terms this becomes:
(3-15)
where i is a column vector of ones. The direct requirements in commodity-by-industry terms is:
(3-16)
where bij is the dollars' worth of commodity i's output required to produce one dollar's worth of industry j's output. This is analogous to the traditional Leontief definition of direct requirements of the A matrix, where
(3-17)
In matrix form (3-16) becomes
(3-18)
where
is a diagonal matrix of the column vector X.
Solving for U:
(3-19)
and substituting into (3-15) yields:
(3-20)
which reduces to
(3-21)
Equation (3-21) represents the general concept that total commodity output is equivalent to intermediate commodity production plus commodity deliveries to final demand, where intermediate commodity production is found by multiplying commodity-by-industry direct requirements by total industry output. The calculation of total requirements at this point can be done using commodity-based technology, which will result in a commodity-by-commodity total requirements matrix. Alternatively one could use an industry-based technology assumption, which would result in an industry-by-industry total requirements matrix (Miller and Blair 1985).
The commodity-based technology convention is based on the assumption that an industry's total output is made up of commodities in fixed proportions. This is done by determining the fraction of total production of industry i that is attributable to production of commodity j. Mechanically this accomplished by dividing an element of the make matrix, vij, by the total output of industry i:
(3-22)
where cij is referred to as the industry output proportion. In matrix form this becomes:
(3-23)
A simple example of a commodity-based assumption would be to consider that the cosmetic industry produces two commodities, lipstick and mouthwash. As shown in Figure 3-14, if the production of the cosmetic industry varies, the total commodity output of lipstick and mouthwash will vary in fixed proportions.
Figure 3-14. Commodity-based assumption.
For the industry-based technology convention it is assumed that a commodity is provided by industries in fixed proportions. This is done by determining the fraction of total production of commodity j that is attributable to production by industry i. Mechanically this accomplished by dividing an element of the make matrix, vij, by the total output of commodity j:
(3-24)
where dij is referred to as the commodity output proportion. In matrix form this becomes:
(3-25)
An example of an industry-based assumption is as follows. Using the cosmetic industry example, if we were to examine the commodity, mouth-wash, those industries that produce mouthwash would be fixed. In this example mouthwash is produced by the cosmetic industry and the agriculture industry. If an increase in production of the commodity of mouthwash were to occur, the percentage of production by the cosmetic industry and the agriculture industry would be remain fixed as shown in Figure 3-15.
Figure 3-15: Industry-based assumption.
Victor (1972) describes several reasons for choosing one method versus the other, foremost is that most of the data available estimates industry waste production at the industry level not at the commodity level, hence an industry-based assumption is more appropriate. Hannon et al. (1983) demonstrates conditions under which the commodity-based and industry-based technology assumptions give equivalent results in specifying ecosystem models. This discussion will proceed with an industry-based assumption as a method to facilitate the illustration of the model. However, both methods as well as a combination of the two methods are possible and can be implemented based on resources and requirements of the analysis at hand (Miller and Blair 1985).
As described previously, the commodity output proportion for an industry-based assumption is defined as:
(3-25)'
This leads to the following expression:
(3-26)
where from Figure 2-8 it is readily observed that:
(3-27)
where i is a column vector of 1's. Substituting (3-26) into (3-27) results in:
(3-28)
We apply the commodity balance equation:
(3-21)'
where from (3-28):
(3-29)
and it follows:
(3-30)
Equation (3-30) represents an industry-by-commodity formulation with an industry-based technology assumption. One could also create an industry-by-industry formulation, where the final demand for commodities is converted to final demand for industries. This conversion is based on each element of D representing the proportion of total production of commodity i produced by industry j. Thus:
(3-31)
therefore, substituting into (3-30) leads to:
(3-32)
where DB represents the direct requirements matrix, AI, in the industry-by-industry formulation. This term can then be expressed as a function of the make and use tables:
(3-33)
(3-34)
where
(3-35)
Distributed
Make and Use Tables
Make and use tables are the primary data structures used to define the interindustry relationships in time for life-cycle SIM. For this distributed activities model they are respectively defined as:
(3-36)
(3-37)
Here, for the distributed make
matrix, 
, each element, 
, represents the amount of commodity j produced by industry i;
at time t for primary production
output time, s. This make table is an amalgamation of all
production techniques by all industries using an industry-based assumption,
that is technologies represent production techniques by industries that provide
commodities in fixed proportions.
Similar to enhanced SIM, production techniques for various commodities in life-cycle SIM are represented by mtip. A production technique, mtip is applied at time t by input industry i, where p = 1 … lp, where lp is the numeration of possible production techniques of industry i.
The use matrix describes the
commodity input to an industrial production process and captures the
destination of commodities. Each
element of the distributed use matrix, 
, is represented by 
. Thus each element of
the use matrix represents the amount of commodity i used by industry j at
time t for primary production output
time, s. U
is of dimension m ´ n.
Again, similar to enhanced SIM, production techniques for various
commodities in life-cycle SIM are
represented by mtjp. A production technique, mtjp
is applied at time t by input
industry j, where p = 1 …lp, where lp
is the numeration of possible production techniques of industry j.
From the definitions of a distributed make matrix and distributed use matrix it follows that a general matrix expression for the distributed technical matrix in an industry-based assumption is:
(3-38)
Here, the function on the right-hand side of equation (3-38) can be represented by various configurations - commodity-by-commodity, industry-by-commodity, commodity-by-industry and industry-by-industry. For example, an industry-by-industry configuration would use a technology matrix defined as shown by equation (3-35). The technical matrix in this example can be formulated exclusively in terms of the make and use tables.
Returning to the bread-making example of the slow rise technique, the distributed use and make matrices are illustrated. As shown Figure 3-16, there are two products in this example, bread dough and finished bread. Both products are associated with a single production technique. Bread is produced at time s, the time-stamp of all associated activities. The intermediary production of bread dough occurs at time interval s - 3. Thus, at time interval s - 3, bread dough can be purchased by another industry, such as bread dough used to aid yeast production or directly to the consumer.
Figure 3-16.
Multiple input and output of the slow rise technique.
In general the use table for baking bread at the final output stage, s, would have the following structure as shown in Figure 3-17. Figure 3-17 represents the use of wood to produce the final product bread for output time s.
Figure 3-17. The
Use table at time s.
A "stack" of use tables in time would then represent all of the inputs sequenced in time necessary to produce the bread as shown in Figure 3-18. In this example the use requirements to produce the other commodities present are not shown, only those to produce bread are depicted.
Figure 3-18. Stack
of Use tables to produce bread at time s.
The make tables associated with baking bread are similar to the use tables. For the make tables, rows are defined by the industries and the columns defined by the commodities. Note in this example, that the matrix is not square. Commodities outnumber industries by one. This is due to the production of the commodities of bread and dough by the bread industry.
The make tables for this example have the following structure as shown in Figure 3-19. In this example, the commodity bread is produced by the bread industry at time s. Moreover, it is assumed that all other commodities are output at time s.
Figure 3-19. The
final output Make table at time s.
For illustrative purposes the stack of Make tables for baking bread would be as shown in Figure 3-20. Note the output of dough at time s - 3 and its explicit association with the final output of bread at time s.
Figure 3-20. The
stack of Make tables for bread production.
The significance of the make and use tables depicted in Figures 3-18 and 3-20 is their representation of data stores of all possible industry and commodity activities. Likewise, there is also a temporal link between the make and use matrices. That is, the set of distributed make and use tables as pairs represent all the recipes for production, both input and output that transform final demand stimulus into economic activity.
The illustration is now expanded further, where there are two techniques of production, the slow rise technique and the quick rise technique. The sequence of production activities associated with the quick rise technique involves only one time interval for dough rising as depicted in Figure 3-21.
Figure 3-21.
Sequence of quick rise technique.
The use and make tables in this expanded example would include both techniques for producing bread. An example of the use tables is shown in Figure 3-22. The convention of using s, the time of production output is illustrated in this example, where output by both techniques occurs at interval s. However, note that the slow rise technique is commenced at interval s - 4 and the quick rise technique is commenced at interval s - 3.[7]
Figure 3-22. Use tables for both techniques of baking bread.
A formal presentation of the life-cycle SIM model is now presented based on the concepts illustrated thus far.
Life-cycle SIM mathematical model
In the SIM production model, the interindustry transfer of products from industry i to industry j is defined as xij, and is described by two temporal parameters. In life-cycle SIM, interindustry transfers will be defined in a similar way with the added distinction that the time of output is the time of output of primary product. That is, the time of output of primary product is when a product leaves the production gate in its final form, e.g. the sale of a completed automobile to the consumer. This parameter becomes a universal time-stamp that relates a primary product and any associated past or future input or output, such as by-products, to the interval in which it was produced. The advantage of utilizing this action as a time reference point, is the precedent of traditional accounting information that captures this transaction.
The interindustry transfer parameters are defined as follows:
(3-39)
The output by industry j, is defined as xj, and is also described by two temporal parameters, time of output, and again, time of output of final product.
(3-40)
The technical coefficient representing the ratio of the total input necessary to produce a given output within an interval of time is:
(3-41)
and, it follows that interindustry requirements can be described by:
. (3-42)
All intermediate input at interval t, then are:
(3-43)
Substituting, (3-42) into (3-43) yields:
(3-44)
and it follows from (3-9) on page 90:
(3-45)
Similar to the modified SIM presented, Life-cycle SIM includes parameters to model technological change.
From equation (3-11), equation (3-44) becomes:
(3-46)
And, from equation (3-12), total output from industry i at interval t is:
(3-47)
and substituting using expression (3-31) to create an expression in terms of commodity based final demand, it follows:
. (3-48)
Further,
(3-49)
represents the technical matrix formulation as a function of the distributed activities make and use tables.
Final Demand Structure
Typical ecological I-O analysis examines the interindustry activities associated with the production of a commodity. Social accounting expands the analysis by examining the various categories of consumers – by geography, income, behavior, age, or gender. A more appropriate analysis in this application of environmental LCA, is the clustering of the production and consumption of commodities by the associated activities to fulfill a function, or in LCA terminology a defined functional unit.
For example, consider a LCA study on a "typical" home computer. The traditional LCA would examine the components that make up the computer, the materials used to produce the computer, the energy used during the life of the computer, and its end disposition - parts recycled and those that are land filled. The LCA of the "typical" home computer in functional unit terms would be defined by the functions the computer performs. The LCA would then be bounded by the activities associated with accessing the internet, shopping, sending email, banking, and playing games. This view of the computer into a function expands what is to be included in the LCA, such as: production of hard copy requires a printer and printer paper; loading and saving programs and documents requires virtual memory devices such as removable disks and backup tapes; accessing the internet requires a modem and a telephone line; even the desk and chair and the space occupied to use the computer may also be considered.
By examining functional units, one can realize the pertinent alternative activities. This results in more robust comparisons of these relevant alternatives. For example, consider again, the "typical" home computer. One function performed on the computer is personal banking - balancing a checking account, paying bills, transferring funds, tracking assets, buying and selling securities, and accessing personal finance information on the internet. Consider typical methods to execute the function of just paying bills. Using the computer involves:
· the computer itself,
· the electricity to run the computer,
· the software to perform the function of paying the bill,
· the software that creates the electronic checkbook,
· the media to load the software (a CD-rom or over the internet),
· the telephone line or cable wire using a modem,
· some fraction of the infrastructure outside the home to support internet access,
· electronic messaging between you, your bank and the billing party's bank, and,
· a hardcopy from your bank in the form of a monthly statement sent via the via the U.S. Postal Service (USPS).
Comparatively, the activities necessary to perform this function without the aid of computer are as follows:
· The balancing of a checkbook to pay the bills using checkbook and pen,
· writing checks printed specifically for this task,
· sending to the billing party in an envelope with the proper postage stamp (USPS) (This may require a simple pick-up at one's home or a trip to the local post-office),
· the USPS delivers the check to the billing party, and,
· billing party, records receiving the check and then physically sends to their bank, which in turn sends it to your bank, that then sends it to you at your home included in a billing statement.
·
This example illustrates how one function pertaining to what appears to be a simple action can involve a complex web of activities. Certainly in this small example much detail has been left out. Depending on the intent of the analysis one would want to limit the boundaries of the analysis. However, it is important that the context of that LCA identify not only what is to be included but as important, what is omitted. For example, the emphasis of the LCA study of the typical home computer may be limited to just the computer. The infrastructure that involves analyzing the entirety of the currently evolving internet may be deemed too complex to include in the study. Acknowledgement of this boundary limitation then constrains the conclusions identified in the interpretation phase of the LCA. In this case, interpretations of the results of the LCA study would be limited to functions of the computer as a stand-alone device. A comparative assessment that involves functions that use the internet would not be appropriate.
Moreover, the importance of normalization of the LCA study should be mentioned. Normalization in the context of LCA involves deciding the sphere of influence of the functional unit. In the bill-paying example of the typical home computer, the assessment is greatly influenced by the definition of the exact functional unit. Examples of variations of how the functional unit can be normalized in increasing orders of spheres of influence include:
· the "typical" activities of one bill payment of one household,
· the average bills paid by one household over a month, or,
· the bills paid by the entire population of 150 million U.S. residences over a year.
Thus, variations of quantities, space and time via normalization are defined such that the results of the study are relevant to the specific goals of the LCA study. In summary, the functional unit, leads to robust analysis of alternatives and recognition of what is being omitted in the context of the study. Further, the goals of the study are met through proper normalization of the function unit.
In terms of economic input-output analysis, the functional unit will be predominately described by final demand. For example, instead of a LCA of an automobile that is defined by demand solely for automobiles, we conduct an analysis of providing personal transportation. This more general view allows for the inclusion of all activities associated with the life-cycle of the automobile. As shown in Figure 3-23, the functional unit of personal transportation is satisfied by the purchase of an automobile that produced in interval s. To satisfy the demand for personal transportation associated with use of the purchased automobile, each interval of time in the future will be represented by a set of purchases of consumables or consumption items, e.g. gasoline, oil, tires, replacement parts and the final purchase, its disposition or EOL status. Figure 3-23 represents a distribution of final demand associated with an automobile over its life span. In this example, the vehicle has an expected value or mean life span of nine years. The associated commodities that represent the cluster of final demand for personal transportation include:
· The automobile itself and all of the intermediate production associated with its completion as a final product,
· The consumption of gasoline,
· Periodic purchases of oil, tires, and replacement parts, and,
· the final disposition of the vehicle – scrap that ends up as reused parts, recycled materials, incinerated materials or materials destined for land-filling.
Figure 3-23. Final Demand defined by the function unit - "personal transportation".
The importance of the functional unit perspective is the representation of alternatives. The LCA is not just of an automobile, it is an LCA study of personal travel. This function can be accomplished not only by an ICEV, but alternatively by an EV, a hybrid vehicle, shared vehicle (i.e. car pooling), bicycle or one's own propulsion by simply walking. Figure 3-24 shows the life spans of two commodities that satisfy personal transportation, ICEVs and EVs. This representation of alternatives allows for the comparison and ultimately the optimization of the system that is represented by a specific life-cycle SIM system.
Figure 3-24. ICEVs and EVs as two methods to satisfy the functional unit of personal transportation.
In addition to the clustering demand for various commodities into a functional unit, these clusters are distributed in time with associated stochastic characteristics. In this form of SIM, emphasis on the production model has been expanded to the entire life-cycle of the economic system. In this new schema, parameters are introduced to capture this larger system boundary. As an illustration of the additional parameters, we return to the bread-baking example, see Figure 3-25. Here, the production process of baking bread is expanded to include the packaging, transport, consumption and disposal.
Figure 3-25. Life-Cycle SIM parameters.
A distinction between production and consumption application times is established. The production application time remains as the term hij, and the consumption application time is described by gij. Similarly, the pass-through time of production is distinguished between production and consumption where, hj defines the production pass-through time and gj represents the consumption pass-through time. Moreover, the life-cycle pass-through time is then the sum of the production pass-through and the consumption pass-through time as shown in equation.
(3-50)
Final demand for commodities, ei(t), is now associated with a distribution of consumption activities, ltic, analogous to the production techniques described by mtip. Thus we have a numeration of various consumption behaviors, described by ltic for a particular commodity over its consumption pass-through time, gj. This distinction allows for the description of variations of input and output based on variations of consumption behavior. For example, given a particular consumption behavior, such as ltic that describes automobile use that is predominately urban, an automobile that was produced in 1985 would have different input and output activities than that of a relatively new automobile produced in the year 2000. Examples of other consumption variations for the automobile could be: climate variation, elevation variation, highway driving conditions, cold starts, warm starts, short trips, long trips, and consumer behavior patterns (jack-rabbit driving habits versus economical driving habits).
Final demand is therefore defined with the following parameters:
(for commodity
based final demand) (3-51)
(for industry
based final demand) (3-52)
where,
ltic – numerates consumption behavior for a commodity or industry output, i, from c = 1…lc, the number of distinct consumption behaviors.
t – is the interval of application
s – is the time interval of production.
The model presented thus far represents
deterministic events. For example, the
vehicles depicted in the life-cycle SIM schema of Figure 3‑12, have an
expected life span of precisely nine years. However, not all vehicles have
precisely the same exact useful life-span.
Some are retired prematurely due to accidents or failure of major
components. Others may last far beyond
the expected life due to meticulous care and a little bit of luck. Thus a more appropriate description of the
end-of-life of a vehicle is a distribution of the likely retirement of the
population of particular model year vehicle as shown in Figure 3‑26.
Figure 3-26.
General stochastics for the life-span of a model year of vehicles.
Moreover, a more accurate Life-cycle SIM schema of the vehicles is shown by Figure 3-27, where the entire life-span for a population of vehicles for a specific model year is nearly twenty years. In this example, for each model year there are two intervals of production and twenty intervals of consumption/disposition status. That is consumption intervals and disposition intervals coexist over twenty intervals. Within a single interval certain automobiles continue to be used while others reach their EOL disposition state. Over the life span of a population of vehicles of a model year the ratio of the automobiles that continue as part of the consumption activities to the ones that have reached EOL slowly declines.
For example, when model year 2000 is at interval CD2 which is the year 2001, the second year of consumption activities, the majority of model year 2000 vehicles will still be operational, with very few vehicles at their EOL. By interval CD9, year 2008, half of the vehicles would be in operation and the other half would have reached EOL. By interval CD20, year 2019, the majority of model year 2000 vehicles would have reached EOL, with virtually zero vehicles in operation.
Figure 3–27. Stochastics of automobile life-spans.
By
incorporating the clustering of final demand and its associated stochastics,
from equation (3-47), total output
from industry i at interval t is now fully described by:
(3-53)
and from equation (3-48) for industry final demand:
(3-54)
for commodity final demand where for both (3-53) and (3-54) the technical matrix is a function of the make and use tables distributed in time:
(3-49)'
Empirical implementation of
Life-cycle SIM
The model presented thus far relies on detailed information of production sequences and their dynamics. However, data regarding these details are not collected and aggregated for national data sets. This section will introduce one method to approximate the temporal details incorporated into life-cycle SIM, facilitating the use of existing static and comparative static data sets.
In general, the anticipatory model represents a system where production occurs based on estimates of future demand. Similar to the Leontief model, for SIM:
Total output = intermediate demand + final demand, or
(3-55)
And here, time is divided into discrete industry intervals delimited by k:
(3-56)
For the time invariant anticipatory model, intermediate yield is expressed by:
(3-57)
The resulting total requirements responsive model is defined by:
(3-58)
Using a Z transform where r numerates future industry intervals:
(3-59)
The anticipatory model output, that is, total systems production, is expressed as a power series of past final demand stimuli weighted by powers of the A matrix (Romanoff and Levine 1981). From equation (3-59) we choose an approximation limit term, t, defined as the number of intermediate intervals or production rounds to estimate total production. This is analogous to setting a pass-through time of t for all industries. The time invariant model then expands to:
(3-60)
And simplifies to:
(3-61)
For the time-varying anticipatory model we define a time-varying technical matrix At. Thus it follows from (3-61), the total production for interval t is:
(3-62)
This simplifies to the following expression:
(3-63)
Here, each round of production that contributes to intermediate production utilizes the technology and price composition that would be in place for final demand in future intervals. In this interpretation, several technical matrices are used to define production activities to meet final demand and intermediate demand within the same interval. This is analogous to utilizing multiple technologies simultaneously to produce the same commodity. [8]
From (3-62) the general solution of (3-53) becomes:
(3-64)
where t is the number of intermediate production rounds that are computed to account for total production. Expression (3-64) can also be written in terms of commodity final demand and is shown below in (3-65).
(3-65)
Section 3.4 - Emissions to the Environment
In the previous section, the formulation of model that describes production activities and consumption activities based on present and future consumer demand activities. This sequence of production and consumption or use activities is then the foundation for estimating resources consumed and emissions generated. In this section a method to relate economic activity described in the previous section to resources used and emissions released to the environment will be presented.
The life-cycle SIM schema is a distributed activities model where interindustry relationships are stimulated exogenously by final demand. The life-cycle analysis is then a formulation to identify and quantify ecological commodities consumed and/or released to the environment based on production and final demand activities described by life-cycle SIM. Ecological commodities are those commodities that have yet to enter the economy as well as those that have been released. Ecological commodities include for example: unmined minerals, water, oxygen and CO2. This quantification involves a pre-multiplication of production and final demand by ecological commodity "intensity" matrices. The intensity matrices represent the assumption that the use and release of ecological commodities from/to the environment are proportional to total output and final demand. For example, for a given total industry output by the vehicle manufacturing industry, the industry releases a proportional amount of CO2. These fixed proportions are best utilized for small variations from their basis. The inaccuracies of fixed proportions can be partially mitigated by temporal detail. Moreover, gains in precision can be further accomplished by non-linear relationships that account for economies of scale factors. For example, if there was a two-fold increase in vehicle production, emissions of CO2 would increase an amount less than 200%. Linear relationships of proportionality will be used to simplify the implementation of this model, however their contribution to more precise results is recognized.
In this schema, the ecological commodities used and ecological commodities released to the environment due to production and consumption that is associated with final demand are accounted. For example, the demand for vehicles results in production activities. In the case of this model, production occurs in anticipation of demand. Therefore emissions to the environment occur prior to orders representing demand. Once final demand has been met for this commodity, moving forward in time, there are associated consumption activities: driving, periodic maintenance, end-of-life disposal, etc. These consumption activities generate demand for gasoline, oil, spare-parts, recycling services, and landfill space. Moreover, the purchased commodities to support the use of the primary commodity, a vehicle, have associated anticipatory production activities themselves. There are ecological commodities that are used and emitted due to their production, such as, resources used and emissions generated to produce gasoline, spare parts, recycle old parts and the resources final disposition.
The
formulation of the ecological portion of the model that is a total of all
ecological commodities used and release to the environment is as follows. The total industry output vector, X(t), is pre-multiplied by the transpose
of a production-side resources and residuals matrix 
. This product is
summed with the product of the final demand vector for industry Y(t) pre-multiplied with the transpose of
demand-side consumption resources and residuals matrix 
. One could also
define residual matrices 
, 
for commodity based
production and consumption residual calculations. The result is a temporally detailed ecological activities matrix
as shown in equation (3-66) [9]:
, (3-66)
where 
is a lr by 1 vector where lr is the number of resources
and residuals tabulated for each industry output. To indicate each resource and residual used disaggregated by
industry, the diagonal matrix (^) of X
and Y can be used:
, (3-67)
where 
is a lr by n matrix where lr is the number of resources
and residuals tabulated and n is the
number of industries examined.
Final demand can also be represented by demand for commodities. Using an industry-based assumption, the formulation for total resource use and residuals released becomes:
, (3-68)
Similar to the distributed
activities schema of life-cycle SIM, the ecological commodity resource and
residuals matrices, 
and 
, are defined as time-varying. The time-varying 
and 
are based on the age
of a commodity (its production output time, s),
the technology used to produce the commodity (technique m)
and the associated consumption chronology (consumption behavior l). The
matrices presented in equation (3-68)
are defined as follows:
– is the matrix
of ecological commodity resources used and residuals generated per industry
production output; 
is the amount of ecological commodity k discharged at time t
per unit of total output of industry i
for production that occurs at time s
using production technique mtip;
for lr ecological
commodities and n industries, RP is of dimension n ´ lr.
– is the matrix
of ecological commodity resources used and residuals generated per the
consumption of final demand by industry output; 
is the amount of
ecological commodity k discharged at
time t per unit of final demand for
output of industry i that occurs at
time s using commodity
behavior ltic; for lr ecological commodities and
n economic industry outputs, RC is of dimension n ´ lr.
– is a
column vector of total ecological commodity resources used and residuals
generated for total production and final demand by industry; 
is the amount of
ecological commodity k discharged at
time t for total production output of
industry i and final demand
activities of commodity j; RT is of dimension lr ´ 1.
Section 3.5 - Summary of Life-cycle SIM
The following is a final summary of the model parameters and defining equations for Life-cycle SIM.
Model Parameters:
General Indices
i – index referring to the rows of the system model,
j – index referring to the columns of the system model,
k – index referring to the ecological commodity used or released to the environment,
lr – the number of ecological commodities,
m – the number of economic commodities,
n – the number of industries,
Temporal Indices
t – time interval of input application,
s – time interval of output or product completion,
p – time period representing the sum of all time intervals,
fj – the life-cycle
pass-through time; the sum of the production pass-through and the consumption
pass-through time for industry j:
.
gij – consumption application time of an input from industry i used by industry j, referenced from the initial application interval, t, to product completion s,
gj – consumption pass-through time, that is, the consumption period of industry j, equivalent to the longest application period, hij,
hij – production application time of an input from industry i used by industry j, referenced from the initial application interval, t, to product completion s,
hj – production pass-through time, that is, the production period of industry j, equivalent to the longest application period, hij,
mtip – production technique applied at time t by input industry i, where p = 1…lp, where lp is the numeration of possible production techniques of industry i,
ltic – consumption behavior applied at time t for a commodity or industry output, i, from c = 1…lc, where lc is the number of distinct consumption behavior categories specific to a commodity or industry output.
Economic Parameters
X(t) – is the vector of industry total output, where xi(t) represents the total output of industry i time t; X is of dimension n ´ 1.
X(t,s) – is the vector of industry final product output where each element, xj(t,s) represents the final production output from industry j with input applications at interval t and production output at interval s.
– the make
matrix, where each element, 
, represents the amount of commodity j produced by industry i;
using production technique mtip
at time t for primary production
output time, s. V is of dimension n ´ m.
– the use matrix, where each element 
, represents the amount of commodity i used by industry j
using production technique mtip
at time t for primary production
output time, s. U
is of dimension m ´ n.
A(mtip; t,s) – technical coefficient matrix, where each element aij(mtip; t,s) describes the ratio of input required by industry j from industry i at time interval t using production technique mtip to produce output by industry j at time interval s. The technical coefficients are defined by:
(3-38)'
where A is of dimension n ´ n for the industry-by-industry formulation.
– is the vector
of commodity deliveries to final demand, where each element, 
represents demand for each commodity i; E is of dimension
1 ´ m.
– is the vector
of industry deliveries to final demand, where each element, 
represents demand for each industry output i; Y
is of dimension n ´ 1,
Q(t) – is the vector of commodity total output, where qi(t) represents the total output of commodities at time t; Q is of dimension 1 ´ m.
Environmental Parameters
– is the matrix
of ecological commodity resources used and residuals generated per industry
production output; 
is the amount of ecological commodity k discharged at time t
per unit of total output of industry i
for production that occurs at time s
using production technique mtip;
for lr ecological
commodities and n industries, RP is of dimension n ´ lr.
– is the matrix
of ecological commodity resources used and residuals generated per the
consumption of final demand by industry output; 
is the amount of
ecological commodity k discharged at
time t per unit of final demand for
output of industry i that occurs at
time s using commodity
behavior ltic; for lr ecological commodities and
n economic industry outputs, RC is of dimension n ´ lr.
–
is a column vector of total ecological commodity resources used and residuals
generated for total production and final demand by industry; 
is the amount of
ecological commodity k discharged at
time t for total production output of
industry i and final demand
activities of commodity j; RT is of dimension lr ´ 1.
To solve for total economic activity in an industry-by-industry configuration equation (3‑64) is utilized:
(3-64)'
Where the approximation limit term, t, is defined as the number of intermediate intervals or production rounds to estimate total production.
The use and release of ecological commodities are then defined by equation (3-66):
, (3-66)'