# EIOLCA

Economic input-output life cycle assessment, or EIO-LCA involves use of aggregate sector-level data quantifying how much environmental impact can be directly attributed to each sector of the economy and how much each sector purchases from other sectors in producing its output. Combining such data sets can enable accounting for long chains (for example, building an automobile requires energy, but producing energy requires vehicles, and building those vehicles requires energy, etc.), which somewhat alleviates the scoping problem of traditional LCA. EIO-LCA analysis traces out the various economic transactions, resource requirements and environmental emissions (including all the various manufacturing, transportation, mining and related requirements) required for producing a particular product or service.

EIO-LCA relies on sector-level averages that may or may not be representative of the specific subset of the sector relevant to a particular product. To the extent that the good or service of interest is representative of a sector, EIOLCA can provide very fast estimates of full supply chain implications for that good or service.

## Background

Economic input-output analysis was developed by Wassily Leontief (who won a Nobel Prize in 1973). It quantifies the interrelationships among sectors of an economic system, enabling identification of direct and indirect economic inputs of purchases. This concept was extended by including data about environmental and energy analysis from each sector to account for supply chain environmental implications of economic activity.[1]

## Theory

Input-output transactions tables, which track flows of purchases between sectors, are collected by the federal government in the United States. EIO works as follows: If $X_{ij}$ represents the amount that sector $j$ purchased from sector $i$ in a given year and $y_i$ is the "final demand" for output from sector $i$ (i.e., the amount of output purchased for consumption, as opposed to purchased by other businesses as supplies for more production), then the total output $x_i$ from sector $i$ includes output to consumers plus output sold to other sectors:

$x_i = y_i + \sum_jX_{ij}$

If we define $A_{ij}$ as the normalized production for each sector, so that $A_{ij} = X_{ij}/x_j$, then

$x_i = y_i + \sum_jA_{ij}x_j$

In vector notation

$\mathbf{x} = \mathbf{y} + \mathbf{Ax}$

$\mathbf{y} = (\mathbf{I - A})\mathbf{x}$

$\mathbf{x} = (\mathbf{I - A})^{-1}\mathbf{y}$