Notebooks

## Input-Output Models

03 Oct 1994 12:01

The basic idea is that the outputs of some industries are the inputs of others, and you can keep track of this with a matrix. This can then be used to study the structure of industry, do planning, etc. The Air Force used in in WWII to decide which German factories to bomb, by seeing which industries were most vital --- the answer seems to have been ball-bearings. (From an unpublished chapter in my father's PhD thesis.)

Non-linear variants would be interesting.

Dynamics: If the input-output relations were truly linear, and the matrix time-invariant, then the economy should converge on a particular long-run distribution of the relative proportions of different goods. T(he argument would be the same one, from the Perron-Frobenius theorem, as that used in the theory of Markov chains, and similarly I'd guess that this long-run distribution should be unique, since otherwise there'd have to be at least two "connected components" of goods, none of which were used to produce the other.) However, whether this long-run economy was growing, in a steady state or shrinking would depend on whether the leading eigenvalue of the matrix was greater than, equal to, or less than one... This is all elementary, so I presume people have already worked it out and gone far beyond this.