SM5005 Chap.5 Tracing Supply Chains with Leontief
Tracing Supply Chains with Leontief
Final demand creates successive production rounds
A final-demand vector Y requires direct output Y. Producing that output requires intermediate inputs AY. Those suppliers require A-squared Y, and further rounds continue. The series I + A + A-squared and onward collects the direct and indirect requirements.
Under the model conditions, it equals the inverse of I minus A, called the Leontief inverse L.
The output solution is X = LY
Multiplying L by final demand produces the total sector output needed throughout the linked production system. It is not the footprint yet. It is an economic requirement vector containing direct output for the demanded products and indirect output from suppliers at every represented tier.
Orientation matters: Y must use the same sector order as A and L.
Read an inverse entry as a total requirement
Entry L_ij describes total output from sector i required per unit of final demand for sector j, within the model's units and boundaries. It includes the direct identity contribution where relevant and the accumulated upstream rounds.
This differs from A_ij, which describes only the direct input requirement of j from i.
Use series intuition to diagnose results
The inverse can feel like a black box. Expanding a small model into Y, AY and A-squared Y shows what is being accumulated and reveals a wrong multiplication order. Compare the matrix solution with several explicit rounds for a tiny example.
The partial sum should approach LY when the production structure is productive and the series converges.
What this chapter covers
- 01
Direct output equal to final demand
- 02
First upstream round AY
- 03
Further rounds through powers of A
- 04
Leontief inverse L equals inverse I minus A
- 05
Total output solution X equals LY
- 06
Inverse entries as total requirements
Worked example · free
Explaining the supplier-layer series
- 1Interpret Y.
- 2Interpret AY.
- 2Interpret A²Y.
- 3State and interpret the closed-form result.
Key terms
- Identity Matrix
- The matrix I that preserves a vector and represents the direct-demand term.
- Leontief Inverse
- The inverse of I minus A, collecting direct and successive indirect requirements.
- Supplier Layer
- One upstream production round generated by another multiplication by A.
- Output Requirement
- Sector production needed to support a specified final-demand vector.
- Convergence
- The condition under which successive upstream rounds approach a finite total.
Tracing Supply Chains with Leontief FAQ
Why does the identity matrix appear?
The identity term carries the direct output delivered to final demand before upstream inputs are added. Without it, the series would begin with suppliers and omit the demanded product itself. It also makes the algebraic expression for the inverse work.
What is the difference between A and L?
A contains direct input requirements per output unit. L contains total requirements after direct demand and every represented upstream round are accumulated. Treating an A entry as a footprint multiplier omits the rest of the supply chain and the environmental extension.
Does X equal emissions?
No. X is total sector output associated with final demand. Emissions require a compatible environmental intensity vector. Multiplying that intensity through the total requirements converts the economic production response into an environmental result.
Assessment move
For a two-sector example, compute Y, AY and A²Y by hand, label each supplier layer and compare their partial sum with LY. Add another round and notice whether its contribution becomes smaller, then explain the convergence in economic language. Form I minus A, calculate the inverse and substitute the resulting X into X = AX + Y. Treat that substitution as required evidence rather than optional arithmetic.
Interpret one diagonal and one off-diagonal entry of L without mentioning emissions: name the supplying row, demanded column and total-output meaning. Next change only one element of Y and identify which output requirements move and why. Always write the sector order beside the vector before multiplying, and deliberately transpose Y once to see how software may accept a shape that carries the wrong meaning.
Build a short error checklist covering determinant or inversion entry, vector orientation, sector permutation and accidental use of A where L is required. Finish by drawing the sequence of direct demand and supplier layers from memory before using the closed form. Compare an A entry and the matching L entry in one sentence: the first is a direct recipe coefficient, while the second gathers total requirements.
This verbal contrast prevents a plausible direct coefficient from being misreported as a supply-chain multiplier. Verify one off-diagonal entry after changing the demanded sector, then restate which index moved.
