SM5005 Chap.6 Environmental Extensions and Footprints
Environmental Extensions and Footprints
Attach environmental pressure to economic output
The input-output core explains how production responds to final demand. An environmental extension adds direct emissions or another pressure by sector. Dividing direct emissions by sector output gives the intensity vector q: environmental pressure per unit of sector output.
Units must be explicit because monetary output may be expressed in a particular currency, price year and scale.
Upstream multipliers are qL
Multiplying the direct intensity row vector q by the Leontief inverse L gives m = qL. Each entry of m is an environmental multiplier containing the direct and upstream pressure associated with one compatible unit of final demand for the corresponding sector.
Applying m to a final-demand vector gives the total footprint attributable to that demand.
Keep three quantities separate
Direct emissions, direct intensity and footprint multiplier are not synonyms. Direct emissions are the extension totals observed for producing sectors. Direct intensity normalizes those totals by output. The footprint multiplier propagates the intensities through total production requirements.
Confusing the last two drops or duplicates supply-chain effects.
Audit dimensions and units
If q is environmental units per monetary output, L is monetary output per monetary final demand, and Y is monetary final demand, then qLY returns environmental units. This dimensional chain is a powerful error check.
The sector order and region coverage of the extension must match L, and the monetary basis of Y must match the factor basis before multiplication.
What this chapter covers
- 01
Environmental satellite accounts
- 02
Direct intensity q
- 03
Total-requirement matrix L
- 04
Multiplier equation m equals qL
- 05
Footprint equation qLY
- 06
Unit and sector-order checks
Worked example · free
From direct intensities to a footprint multiplier
- 2Define q and its units.
- 2Define L and its interpretation.
- 2Define m equals qL.
- 2Show why multiplying by Y returns environmental units.
Key terms
- Environmental Extension
- Sector-level environmental totals attached to the economic accounts.
- Direct Intensity
- Direct environmental pressure divided by sector output.
- Footprint Multiplier
- Direct and upstream pressure per compatible unit of final demand.
- Satellite Account
- Additional information linked to the core economic accounts by sector.
- Dimensional Analysis
- Checking a calculation by tracking the units entering and leaving each operation.
Environmental Extensions and Footprints FAQ
Why is q not already a footprint multiplier?
q contains only the pressure released directly by each producing sector per output unit. It has not followed the inputs that sector purchases from suppliers. Multiplication by L adds the direct and successive upstream production requirements.
What does one entry of m mean?
It is the environmental pressure across the represented supply chain associated with one unit of final demand for the corresponding sector. The exact unit and boundary depend on the extension, currency, year, valuation basis, regions and sectors used.
What alignment error is most dangerous?
A vector can have the correct length but a different sector order. The multiplication will run and silently attach each intensity or demand value to the wrong sector. Preserve identifiers and test named entries rather than relying on position alone.
Assessment move
Make a four-row ledger for q, L, m and Y: meaning, dimensions, units and sector order. Reconstruct q from direct emissions and output for two sectors, keeping tonnes, kilograms, dollars and millions of dollars visibly aligned. Multiply one column of L by q as a contribution sum before trusting the full matrix product.
Explain why q and m can display the same physical unit per currency while covering different parts of the supply chain. Reconstruct m for a tiny model and explain why the output changes from money to environmental pressure only after the extension is applied. Choose one service sector with a low direct rate and list the upstream sectors that could make its multiplier larger.
Deliberately apply q directly to final demand, identify the missing coverage and write the repair. Preserve stable sector identifiers when joining the extension to the inverse, because equal vector lengths do not prove equal ordering. End every calculation with a dimensional line showing how environmental pressure per output, output per final demand and final-demand currency cancel to the claimed total.
Build one comparison table with direct emissions, q, m and the applied footprint for the same sector. Explaining why the ranking can change across those columns is a stronger test than reproducing the formulas alone. Check the same comparison after changing final demand so intensity and activity effects stay conceptually separate.
