## Input-Output Models in Economics

*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 (From an unpublished chapter in my father's PhD thesis.)

As that parenthetical suggests, I learned about this from my father the economist, who did his thesis in this area, and his thesis adviser did her thesis with Leontief, who basically invented it. I have always felt like I ought to actually read Leontief, but never really done so.

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. (The 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.
--- *Update*, 2021: This seems to be the key idea in Sraffa, to
the extent I can make sense of Sraffa.

- See also:
- Complex Networks
- Economics
- Graph Theory
- Markov Models
- Optimization
- Planned Economies
- Social Networks [an entry in the I/O matrix is a social relation, or rather an abstraction representing many concrete social relations...]

- Recommended:
- William H. Miernyk, The Elements of Input-Output Analysis
- Piero Sraffa, Production of Commodities by Means of Commodities

- To read:
- Erik Dietzenbacher and Michael L. Lahr (eds.), Wassily Leontief and Input-Output Economics
- Gale, Theory of Linear Economic Models
- Hiroshi Iyetomi, Yasuhiro Nakayama, Hideaki Aoyama, Yoshi Fujiwara, Yuichi Ikeda, Wataru Souma, "Fluctuation-Dissipation Theory of Input-Output Interindustrial Correlations", arxiv:0912.1985
- Tjalling Koopmans, Efficient Allocation of Resources
- Wassily Leontief, Input-Output Economics
- Charles W. McArthur, Operations analysis in the U.S. Army Eighth Air Force in World War II
- Ronald E. Miller and Peter D. Blair, Input-Output Analysis: Foundations and Extensions
- Z. M. Shalizi, Multiregional input-output multipliers and the partitioned matrix solution of the augmented MRIO model
- Thijs ten Raa, The Economics of Input-Output Analysis