The IO model: overview

Preparing the IO tables¶

Every country enters the model as a national IO table, but countries don’t all start from the same kind of table. The first step is to collect the best available table for each country and record where it came from: the source, version, year, valuation, and sector detail. Which table we start from depends on the country’s tier: the U.S. starts from Make and Use tables for 2024 that Cornerstone produces using BEA’s methodology, after redefinitions and following BEA’s 2017 detailed schema, while other countries start from national statistical agency tables, FIGARO or ICIO, or a template. See IO sources by country tier for what each tier starts from.

The detailed U.S. structure sets the model’s common sector list, and it is used to split other countries’ coarser sectors (see ABSR). Sector lists and currencies are reconciled in the later steps: mapping to the common sector list, linking with trade data, and balancing.

What’s in the tables¶

An economy’s IO accounts come as a pair of tables.

Supply (V) records the production of each commodity by each domestic industry, plus imports of each commodity. The Make table, used for the U.S., is a variant: it has the same production block but no imports row. Imports appear instead in the Use table’s final demand, as a negative column. Either way, the production block gives two output totals:

• Industry output x: the sum across an industry’s row, which is everything the industry makes, including secondary products.

• Commodity output q: the sum down a commodity’s column, which is everything domestic industries make of that commodity. Adding imports gives the total supply of the commodity.

Use (U and y) records who buys each commodity:

• Intermediate use (U): the commodities each industry buys as inputs.

• Final demand (y): the commodities bought by final users, namely households, government, investment, and exports.

• Value added: below each industry’s inputs, its wages, taxes on production, and profits.

The two tables balance. For each commodity, total supply equals total use. For each industry, inputs plus value added equal the industry’s output x.

Domestic and imported use¶

A purchase can come from a domestic producer or from abroad, and the two lead to different supply chains. So the Use table is split in two:

• Domestic use (Udom, ydom): commodities bought from domestic producers.

• Imported use (Uimp, yimp): commodities bought from producers in other countries.

U = Udom + Uimp. Final demand splits the same way, with exports on the domestic side: y = ydom + exports + yimp. Imported use is later traced back to the countries that made the imports, using trade data. How each country’s tables arrive at this split is described in IO sources by country tier.

From tables to direct requirements (A)¶

The direct requirements tables turn the Supply and Use tables into recipes: the inputs needed to make $1 of each commodity. They take three steps.

1. Inputs per dollar of industry output. Divide each industry’s column of domestic use by its output x.

2. Market shares. Divide each commodity’s column of the Supply table by its output q. This gives the share of each commodity made by each industry.

3. Combine. Multiply the two to get inputs per dollar of each commodity. Doing this with domestic use gives Adom; doing it with imported use gives Aimp.

For more details on the math behind it, contact team@cornerstonedata.org.

Adom and Aimp add up to the total direct requirements A. A column of A is the input recipe for $1 of that commodity, split by where the inputs come from.

Tables that are already symmetric, with the same sectors on rows and columns, skip step 2: their domestic and imported use are divided by output directly. This applies to the Tier 1 and Tier 2 tables.

Final demand for the national accounting balance¶

Final demand in the Use table includes purchases of imports. To match domestic production, we need final demand for domestic products only. We call this ynab, the final demand for the national accounting balance: domestic final demand for domestic products (ydom) plus exports.

Here ydom is what households, government, and investment buy from domestic producers, and yimp is what they buy from abroad. Imports not bought by industries go to final users, so yimp is total imports less the row sums of Uimp. Exports are the country’s own production, so they count toward ynab in full.

With ynab, the model reproduces the country’s commodity output: running ynab through the domestic supply chain (the Leontief inverse of Adom) gives back q. This is a check that the tables, Adom, and ynab are consistent.

From single-country tables to the multi-region model¶

Each country’s tables say how much of its inputs are imported, but not where they come from. The multi-region model links the countries so that a supply chain can be followed across borders.

1. Put every country in one table. The 65 countries in Tiers 0 to 2 each get their own block, so A is a 65 × 65 grid of country blocks, each 405 × 405 sectors. Each country’s Adom sits on the diagonal. A country’s column holds everything its industries buy; the rows say which country sold it.

2. Split imports by origin. Each country’s Aimp is spread down its column using bilateral trade data (BACI): for each product, the share the country imports from each origin. Products the trade data don’t cover, mostly services, use origin shares from FIGARO or ICIO. Imported final demand (yimp) is split the same way.

3. Add the rest of the world. Imports from countries outside the table come from a rest-of-world region. Its domestic input structure is pooled from the tables of Colombia, Mexico, Türkiye, and Chile. It is added as one more block, making A a 66 × 66 grid of country blocks. This region is separate from the template used for Tier 3 countries’ own factors (see IO sources by country tier).

4. Balance. The table is balanced with RAS, so that each country’s sales to all users, at home and abroad, add up to its output, without changing it’s domestic recipe. This makes small adjustments to the import coefficients.

5. Solve. The total requirements (L) are calculated for the whole table at once, so supply chains that cross several borders are captured in full (see below).

The other 83 of the model’s 148 countries (Tier 3) are linked to this table more simply; see IO sources by country tier.

Total requirements (L)¶

A shows only a sector’s direct suppliers. Take $1 of furniture. Its column of A lists what the furniture maker buys to make it: lumber, hardware, electricity, transport. These are the first tier of the supply chain.

Each of those suppliers has its own column in A. The sawmill buys logs, fuel, and machinery; the power plant buys fuel and equipment. These purchases are the second tier. Each supplier’s column is scaled by how much of it the first tier needs: if the furniture maker buys 20 cents of lumber for each $1 of furniture, the sawmill’s whole recipe is multiplied by 0.20. The second-tier suppliers have their own suppliers, the third tier, scaled again by what the second tier needs, and so on, without end. In the multi-region model the trace crosses borders: a second-tier supplier may be a steel mill in another country.

Each tier is smaller than the one before, because every dollar of output pays for value added (wages, taxes, profits) as well as purchases. So the tiers add up to a finite total: everything produced, anywhere in the supply chain, to deliver $1 of each commodity to a final user. Written out, that total is I + A + A² + A³ + …, where I is the $1 of the commodity itself, A the first tier, A² the second, and so on.

There is a simple way to calculate this infinite sum all at once: a special inverse of A, (I − A)⁻¹, called the Leontief inverse after the economist Wassily Leontief. That is why the result is called L. A column of L shows the total output needed from every sector, in every country, for $1 of final demand for that commodity.

Learn more¶

• Supply, Use, and IO tables in national accounts - the United Nations Handbook United Nations, 2018

• The standard textbook on IO analysis Miller & Blair, 2022;

• How BEA builds the U.S. IO accounts BEA, 2009