In 1936, Wassily Leontief mapped how every sector of the American economy buys from and sells to every other. His input-output model reveals why a change in one place ripples everywhere.
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Consider a simplified economy with three sectors. Agriculture grows food and raw materials. Manufacturing turns them into goods. Services — finance, transport, healthcare — supports them all.
Each sector buys from the others. Agriculture needs manufactured equipment and financial services. Manufacturing needs raw materials and logistics. Services needs food and physical goods.
The arrows show the dollar flows between them — thicker means more trade.
Leontief organized these flows into a table. Each row shows where a sector's output goes — to other sectors as intermediate goods, or to consumers as final demand.
Each column shows where a sector gets its inputs.
Divide each column by its total output to get the technology matrix A. Entry aij tells you: to produce one dollar of sector j's output, you need aij dollars of sector i's output.
This matrix encodes the structure of the economy.
Total output x must equal intermediate demand Ax — what other sectors buy — plus final demand d — what consumers, government, and exporters buy.
The equation x = Ax + d ties everything together.
What if consumers want $10 billion more in manufactured goods? Manufacturing must increase output. But it can't do that alone — it needs more agricultural inputs and more services.
Agriculture and Services increase output to supply Manufacturing. But they in turn need more inputs from each other and from Manufacturing. Each round of expansion is smaller than the last, but they accumulate.
The total effect far exceeds the initial $10B.
The matrix (I − A)⁻¹ captures all rounds at once — direct effects, indirect effects, and the infinite chain of feedback.
Its column sums are the output multipliers. Manufacturing's is about 2.1× — every dollar of new demand produces two dollars of total output.
In our example, a $10 billion increase in demand for manufactured goods requires about $20.8 billion in total new output across all sectors. More than half of that total is indirect — the cascading needs of interconnected industries.
This is the multiplier effect: in a connected economy, nothing changes in isolation.
Each round of indirect effects is smaller than the last. The stacked bars show each round's contribution; the line shows the running total approaching the Leontief inverse solution.
Demand shock: +$10B to Manufacturing. Rounds computed as Aᵏ × Δd. Final total from (I − A)⁻¹ × Δd.
Leontief published his first input-output table for the United States in 1936, tracking 41 sectors. Today the Bureau of Economic Analysis publishes tables covering over 400 industries. Input-output analysis is used worldwide to study supply chain vulnerability, estimate the economic impact of disasters, evaluate climate policy, and trace environmental footprints embedded in trade.
For this work, Wassily Leontief received the Nobel Memorial Prize in Economic Sciences in 1973.
Adjust final demand for each sector and see how total output responds through the Leontief inverse. Then shock a sector to watch the multiplier in action.