What Is an Input-Output Table?

The economy as a grid

Every industry both buys and sells. A car manufacturer buys steel, electricity, and software; it sells vehicles to households, to the government, and to export markets. An input-output (I-O) table records all of these transactions — for every industry, in the same year, in a single rectangular grid.

Leontief built the first comprehensive table for the U.S. economy in the 1930s ^[1][2]. The idea was simple but powerful: if you know who sells to whom, you can trace a spending shock — a new highway project, an export boom — as it ripples across the whole economy. Leontief received the 1973 Nobel Prize in Economic Sciences for this framework ^[3].

The Leontief site carries 28 years of annual U.S. I-O data (1997–2024) at the BEA Summary level of 71 sectors — everything from Farms (sector 1) to Federal Reserve Banks to Computer Systems Design.

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The basic structure: a toy example

Imagine a three-sector economy: Farms, Food Processing, and Households (as final buyers). All values are in millions of dollars.

Selling \ Buying	Food Processing	Households (final demand)	Total output
Farms	60	40	100
Food Processing	0	120	120
Value Added	60	—	—
Total input	120	—	—

Reading across a row: Farms sells $60m of grain to Food Processing, and $40m of fresh produce directly to Households. Total farm output = $100m.

Reading down a column: Food Processing buys $60m from Farms and generates $60m of value added (wages + profits). Total inputs = $120m = total output. Columns must balance — every dollar of output came from somewhere.

This double-entry logic holds for all 71 BEA sectors. The table is essentially a national accounting identity expressed as a matrix.

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From transactions to "recipes": the A matrix

The raw transactions table $Z$ tells us how much each industry bought from each other. But we want to know the technology — the proportional recipe each industry uses. We divide each cell by the purchasing industry's total output:

$$a_{ij} = \frac{z_{ij}}{x_j}$$

Here $z_{ij}$ is the dollar flow from sector $i$ to sector $j$, and $x_j$ is sector $j$'s total output. The resulting matrix $A$ is the direct requirements matrix (glossary: direct-requirements): each column tells you how many cents of each input are needed to produce one dollar of output in that industry.

In the toy example, Food Processing spends $60m on Farms out of $120m total output, so $a_{\text{Farms},\text{FoodProc}} = 0.50$. Half of every dollar of processed food comes (directly) from the farm sector.

The live heatmap below shows the A matrix for 2024, aggregated to 15 sectors. Darker cells mean a stronger direct input link. Notice the bright diagonal block for Manufacturing — manufacturers buy heavily from other manufacturers — and the relative sparseness of service sectors, which draw more from labor (value added) than from intermediate goods.

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Going deeper: total requirements and the Leontief inverse

Direct requirements miss the cascade. Food Processing buys from Farms; Farms buys tractors from Manufacturing; Manufacturing buys fuel from Petroleum Refining; Petroleum Refining buys from Mining. A demand shock propagates through all these layers simultaneously.

Leontief showed that the total (direct + all indirect) requirements can be computed in one matrix operation. If $f$ is the vector of final demand — what households, government, and exporters want — then total output required is:

$$x = (I - A)^{-1} f = L f$$

The matrix $L = (I - A)^{-1}$ is the Leontief inverse (glossary: leontief-inverse), also called the total requirements matrix. Each entry $l_{ij}$ tells you: to deliver one additional dollar of final demand for sector $j$, how much total output must sector $i$ produce?

The power series $L = I + A + A^2 + A^3 + \cdots$ reveals the intuition: the first term is the direct delivery ($I$), the next is the first round of inputs ($A$), then inputs to those inputs ($A^2$), and so on ^[4].

The interactive table below shows the 2024 Leontief inverse $L$, aggregated to 15 sectors. Values on the diagonal are always ≥ 1 (an industry needs at least its own direct output to satisfy demand), and off-diagonal entries capture the cross-sector support each industry provides.

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Multipliers: what the inverse tells us

Sum down a column of $L$ and you get the output multiplier for that sector: the total dollars of production required economy-wide for each dollar of final demand delivered by that industry. Across the 71 BEA sectors, these multipliers ranged from 1.17 to 2.77 in 2024 (mean 1.87). They peaked around 2008 (mean 2.02) when energy prices inflated cross-industry costs, and compressed during the 2020 pandemic disruption (mean 1.88).

High-multiplier sectors — think Petroleum Refining or Construction — trigger long input chains. Low-multiplier sectors — Real Estate, some Finance — buy relatively little from other industries and instead generate value added almost entirely from labor and capital.

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Try it: load a year and compute output multipliers

import pandas as pd

# Download from the Leontief API or export button
L = pd.read_csv("2024_L.csv", index_col=0)

# Output multiplier = column sum of L
multipliers = L.sum(axis=0).rename("output_multiplier")
print(multipliers.sort_values(ascending=False).head(10))

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Where next

• Next tutorial: Reading Supply and Use Tables — how BEA actually constructs the I-O accounts from two separate tables, and what the value-added and final-demand rows contain.
• Dig into the data: Key Sectors: Backward and Forward Linkages — which of the 71 BEA sectors sit at the center of the U.S. production network, and how that has shifted from 1997 to 2024.

Further reading

• Miller & Blair (2009), ch. 2 — the foundations: notation, the transactions table, and the derivation of the Leontief inverse. ^[4]
• Leontief (1986), Input-Output Economics — the framework in its originator's own words. ^[5]
• ten Raa (2005), The Economics of Input-Output Analysis — a compact, rigorous modern treatment. ^[6]