Input-Output Networks¶
The previous subsection solved the equilibrium without using the language of networks. We can now reinterpret those same objects in a way that makes the propagation mechanism transparent.
Definition 17.1: Input-output matrix
For each pair of sectors \((s,k)\), define the expenditure share of sector \(s\) on input \(k\) by
The matrix \(\Omega=[\omega_{sk}]\) is the input-output matrix of the economy.
The value of \(\omega_{sk}\) captures how important sector \(k\) is for the production of sector \(s\).
In the Domar economy with Cobb-Douglas technology, \(\omega_{sk}=a_{sk}\) for every \(s,k\), so the technological coefficient matrix and the input-output matrix coincide with \(\Omega=A\).
Definition 17.2: Leontief inverse
The Leontief inverse is the matrix
we denote its \((s,k)\) element by \(\psi_{sk}\). When the Neumann series converges, this matrix admits the expansion
The element \(\psi_{sk}\) adds up all direct and indirect ways in which sector \(k\) enters the production of sector \(s\). The direct effect is \(\omega_{sk}\). The second-round effect is the contribution of \(k\) to the suppliers of \(s\), and so on. The Leontief inverse therefore summarizes all downstream linkages.
Directed graphs and centrality.¶
In graph-theoretic language, the production network is a weighted directed graph. The graph has one node per sector and a directed edge from \(k\) to \(s\) whenever sector \(s\) buys the output of sector \(k\) as an intermediate input. The edge weight is \(\omega_{sk}\).
A node is central when many important paths run through it. This is exactly what the column sums of the Leontief inverse measure.
Definition 17.3: Downstream centrality
For each sector \(k\), define its downstream centrality by
Equivalently, \(v_{k}\) is the sum of the entries in column \(k\) of the Leontief inverse \(\Psi\).
The measure \(v_{k}\) is a Bonacich-type centrality index: Sector \(k\) is central when it is an important supplier to sectors that are themselves important in production. We can prove this in general from the definitions of the input-output matrix \(\Omega\) and the Leontief inverse \(\Psi\).
Proposition 17.1: Domar weights are weighted downstream centrality measures
Consider a competitive economy with constant returns to scale. Let \(\Omega=[\omega_{sk}]\) denote the input-output matrix, \(\Psi=(I-\Omega)^{-1}\) the Leontief inverse, \(\beta_{s}\equiv\frac{p_{s}c_{s}}{Y}\) the final expenditure shares, and \(\lambda_{s}\equiv\frac{p_{s}y_{s}}{Y}\) the Domar weights. Then,
If final expenditure shares are uniform, \(\beta_{s}=1/S\) for all \(s\), then
Proof
For each sector \(s\), market clearing implies
Replace the expenditure in each input using input-output coefficients, \(p_{s}x_{js}=\omega_{js}\,p_{j}y_{j}\), so
Divide both sides by GDP, \(Y\), and express in terms of expenditure shares,
Stacking these equations across sectors yields
Rearranging gives the result.
Reinterpreting the Cobb-Douglas solution.¶
The sectoral formula
shows that productivity shocks propagate downstream through the network.
The aggregate formula
then says that the aggregate importance of each sector is summarized by a centrality object, its Domar weight. A sector matters for aggregate activity not only because households buy it directly but also because many other sectors buy it as an input.
It is useful to read the matrix \(\Psi\) in two different ways. Fix the row index \(s\). Then the coefficient \(\psi_{sk}\) measures how much sector \(s\) depends, directly and indirectly, on sector \(k\) as an upstream supplier. In this sense the row of \(\Psi\) attached to sector \(s\) describes where that sector's production process comes from (not just the direct inputs).
So, productivity shocks in sector \(k\) lower costs not only for its immediate customers but also for the customers of those customers, and so on. That is, productivity shocks propagate downstream: the sectors that respond are precisely the sectors that use the shocked producer, either directly or through a chain of intermediate-input linkages.
Now fix the column index \(k\). The column sum of \(\Psi\) tells us how important sector \(k\) is for the economy as a whole. Proposition 17.1 shows that the Domar weight of a sector captures this idea, since \(\lambda_{k}\) is a weighted sum of all the ways in which \(k\) enters goods that are ultimately absorbed in final demand.
A sector can have a large Domar weight either because households consume it directly, or because many important sectors use it as an input, or because it appears repeatedly along long production chains. In this way, goods such as electricity can be important even if their direct share in final expenditure is modest, because they are embedded in the production of many other goods.