The Leontief System
MODEL LAB · LABORATORY 04
The Leontief
System.
Final demand is only the beginning. Solve the interdependent production system and reveal the direct and indirect output required throughout the economy to deliver goods and services to final users.
DIRECTROUND 2ROUND 3INDIRECT
FROM IDENTITY TO SOLUTION
Production serves itself
and final users.
Every sector must produce enough to meet final demand and the intermediate demand generated by all industries—including its own requirements.
x = Ax + f
Gross output supplies intermediate demand Ax and final demand f.
→
(I − A)x = f
Move intermediate requirements to the left side of the system.
→
x = (I − A)⁻¹f
Premultiply final demand by the total-requirements matrix.
WHY THE INVERSE MATTERS
One demand shock.
Successive production rounds.
Final demand
f
The economy must deliver the goods directly requested by final users.
Direct inputs
Af
Producing final goods requires the first round of intermediate inputs.
Upstream inputs
A²f
Suppliers themselves require inputs from other sectors.
Total requirements
(I−A)⁻¹f
All progressively smaller upstream rounds are accumulated.
LEONTIEF SOLVER
Solve the
interdependent economy.
Change the technology matrix and final demand. ECONORIA calculates the determinant, Leontief inverse, gross output and indirect production requirements.
SYSTEM READINGThe technology supports a finite non-negative solution. Final demand of €180m requires approximately €313m of gross production after all upstream rounds.
SCIENTIFIC DISCIPLINE
What the solution
does—and does not—say.
Total, not net
Gross output counts intermediate production repeatedly; it must not be confused with GDP or value added.
Demand driven
Basic IO assumes industries can expand supply to satisfy exogenous final demand at fixed coefficients.
No price response
The quantity model does not endogenously adjust prices, substitution, capacity or factor scarcity.
Check productivity
The inverse is economically meaningful only when the technology generates a finite non-negative solution.
SCIENTIFIC CHECKPOINT
What does an element
of L measure?
How should lᵢⱼ in the Leontief inverse be interpreted?
The inverse accumulates direct and all indirect production rounds.
