Input-output tables: how a shock spreads across sectors and countries

01What an input-output table is

An input-output table is a large matrix of the economy. Each column is a sector as a buyer; each row is a sector as a seller. The cell at row i and column j says how much sector j bought from sector i in a year. The rows also show sales to final users (households, governments, investment, exports), and the columns show value added (wages, profits, taxes) on top of intermediate purchases.

The approach was developed by Wassily Leontief, who received the Nobel Memorial Prize in Economic Sciences in 1973 for it. National statistical offices publish national tables; international organisations combine them with trade data into inter-country tables, where the buyer and seller sectors are also split by country. In an inter-country table you can read, for example, how much the German motor vehicles sector buys from the Chinese basic metals sector.

02From transactions to coefficients

Dividing each column by the total output of the buying sector gives the technical coefficients: how many cents of input from sector i are needed per euro of output of sector j. Collected in a matrix A, they are the recipe of the economy.

a(i,j) = purchases of sector j from sector i / total output of sector j

Total output needed to deliver final demand y:
x = (I − A)⁻¹ · y          (the Leontief inverse)

Cost-push price model (shock to the price of inputs, passed on in full):
Δp = (I − Aᵀ)⁻¹ · Δv

The Leontief inverse adds up an infinite chain: the direct inputs, the inputs needed to make those inputs, and so on. Each round is smaller than the one before, so the series converges quickly.

03A small worked example

Take three sectors with simplified coefficients: steel, machinery and furniture. Machinery uses €0.20 of steel per euro of output. Furniture uses €0.10 of steel and €0.05 of machinery per euro of output. Steel prices rise by 10%, and every sector passes cost increases on in full.

In this toy case the indirect effect on furniture is only a tenth of the direct one. In real tables, with dozens of sectors and inputs such as energy and chemicals that everything uses, indirect effects are often as large as direct ones, and for services sectors they can be most of the effect.

04The OECD ICIO and TiVA databases

The OECD publishes free inter-country input-output tables (ICIO) and a derived database on Trade in Value Added (TiVA). The current release covers 76 economies plus a rest-of-the-world aggregate, 45 industries based on the ISIC Rev.4 classification, and annual data over two decades.

Simdots uses the OECD ICIO tables to estimate how a shock to one country and sector reaches the sectors and countries where your suppliers operate. Our interactive page on how a shock cascades shows the mechanism with real data.

05Using input-output data as a manufacturer

You do not need to build a model to get value from these tables. Three practical uses:

1. Indirect country exposure. Your purchase records show you buy only from EU suppliers. TiVA shows that a sizeable share of the value added in, say, EU electrical equipment comes from China. That is your indirect exposure to a China shock.
2. Indirect tariff and price effects. When estimating the effect of a tariff or a commodity price shock, the Leontief price model gives the increase expected in your domestic suppliers' sectors, which you can apply to your spend with them.
3. Prioritising supplier mapping. If a sector of your supply base has a high indirect dependency on a risky country, start tier-2 mapping there.

06Limits to keep in mind

• Aggregation. "Basic metals" mixes steel, aluminium and copper. A shock to one metal is diluted across the whole sector.
• Lag. Tables are built from national accounts and trade statistics and are published with a delay of a few years. Structural changes, such as a recent shift of sourcing, are not yet visible.
• Fixed coefficients. The basic model assumes the recipe does not change. In reality, companies substitute inputs when prices move, so long-run effects are smaller than the model suggests.
• Prices vs quantities. A price model assumes full pass-through. Real pass-through is partial and delayed.
• Capacity. Input-output models do not know about bottlenecks. When a sector is at capacity, a shortage can have effects much larger than its share of costs, as the semiconductor shortage of 2021 showed for car production.

These limits are why input-output models work best inside a broader simulation: they provide the structure of who depends on whom, while Monte Carlo simulation handles uncertainty about magnitudes and timing.