Input-output Model - Basic Derivation

Basic Derivation

Say that we have an economy with sectors. Each sector produces a single homogeneous good, . Assume that the th sector, in order to produce 1 unit, must use units from sector . Furthermore, assume that each sector sells some of its output to other sectors (intermediate output) and some of its output to consumers (final output, or final demand). Call final demand in the th sector . Then we might write

$x_i = a_{1i}x_1 + a_{2i}x_2 + \ldots + a_{ni}x_n + d_i,$

or total output equals intermediate output plus final output. If we let be the matrix of coefficients, be the vector of total output, and be the vector of final demand, then our expression for the economy becomes

$\vec{x} = A\vec{x} + \vec{d}$

which after re-writing becomes . If the matrix is invertible then this is a linear system of equations with a unique solution, and so given some final demand vector the required output can be found. Furthermore, if the principal minors of the matrix are all positive (known as the Hawkins-Simon Condition), the required output vector is non-negative.

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