Leontief Input-output Model for Two Variables

In a real economy, sectors rarely work in isolation — an automobile sector might rely on steel, electricity and rubber, while those very sectors also consume some of what the automobile sector produces. The Leontief input-output model uses matrices to check whether such an interlinked, resource-sharing economy is viable: can it produce enough to satisfy both its own internal needs and the demand placed on it from outside?

Start by recording, for each sector, what fraction of its total output it uses on itself versus supplies to other sectors. Dividing each sector's resource-use figures by its own total output turns these figures into per-unit shares, assembled into a square matrix called the input-output coefficient matrix (or technology matrix), usually denoted $A$. Alongside it sits a demand vector $D$, recording how much external demand each sector must additionally meet, and an output vector $X$, the total output each sector needs to produce to cover both internal use and external demand.

These three are linked by a simple balance: each sector's total output must equal what gets used internally (given by $AX$) plus what's demanded externally ($D$):

Rearranging this equation, $D = X - AX = (I - A)X$, gives the requirement output directly in terms of demand:

This formula only makes sense — and only describes a realistic economy — if two conditions, known as the Hawkins–Simon conditions, are satisfied: