Input-output Model

An input-output model (I/O model), sometimes instead called an input-output framework, is a model of production interactions.

Model

An economy is segmented into n distinct sectors. Every sector produces x units of a commodity. Some of that output is sold to consumers (final output or final demand) which is notated as d.

Most sectors also take inputs from other sectors in order to produce a unit of output. For any sector i, the quantity of their output needed to produce 1 unit of output in sector j is notated as a_ij. This creates a feedback loop where final demand for one sector creates intermediary demand for other sectors, creating intermediary demand for other sectors, and so on.

Altogether, the output of sector i is given as x_i = (Σ a_ijx_j) + d_i. The output of all sectors is notated as x = Ax + d. This is rewritten (I - A)x = d and, if (I - A) is invertible, this system can be solved.

Applications

Many national governments estimate the interaction terms from economic activity data and publish them as Input-Output Accounts tables. For example, the BEA publishes this data for the United States.

Given estimates for the interaction terms, this model can be used to estimate economic impact of an investment. Given a purchase on top of normal economic activity, the multiplicative effect on gross production (overall and in each sector) can be derived.

Assumptions

The interactions of sectors are taken as given and static. Dynamic effects, such as an investment into one sector crowding out investment in another, are not captured.

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