Part 5: Input-Output Networks and Economic Activity Section 18

Examples of Input-Output Networks

Horizontal economy.

flowchart TD
L((N))
s1((1))
ss((s))
sS((S))
H[HH]
L --> s1
L --> ss
L --> sS
s1 --> H
ss --> H
sS --> H

Consider an economy with no intermediate inputs. Instead there are \(S\) sectors, each producing a final output directly consumed by the representative household. With only one input of production and constant returns to scale we have linear output:

\[ y_{s}\;=\;z_{s}n_{s}. \]

In terms of the Domar economy developed above, this implies,

\[ a_{sk}=0\qquad\text{for all }s,k,\qquad\alpha_{s}=1. \]

Hence

\[ \Omega=A=0,\qquad\Psi=I,\qquad\lambda_{s}=\beta_{s}. \]

There is no propagation across sectors. The sectoral formula collapses to

\[ d\log y_{s}=d\log z_{s}, \]

and the aggregation formula becomes

\[ d\log Y=\sum_{s=1}^{S}\beta_{s}\,d\log z_{s}. \]

This is the natural benchmark in which aggregate productivity is just a weighted average of sectoral productivity. The proper weights are now value added weights because each sector only produces final goods.

Economy with a basic input.

flowchart TD
L((N))
s0((0))
s1((1))
ss((s))
sS((S))
H[HH]
L --> s0
L --> s1
L --> ss
L --> sS
s0 --> s1
s0 --> ss
s0 --> sS
s1 --> H
ss --> H
sS --> H

Now consider an economy with one additional sector, sector \(0\), that produces a basic input using only labor with a linear production technology. Households do not consume good \(0\) directly, but every sector \(s=1,\dots,S\) uses it in production alongside labor with a Cobb-Douglas technology. The production functions are

\[ y_{0}\,=\,z_{0}n_{0},\quad\text{and}\quad y_{s}\,=\,z_{s}n_{s}^{\alpha_{s}}x_{s0}^{\,1-\alpha_{s}}\quad\text{for}\quad s\,=\,1,\dots,S. \]

Household preferences are also Cobb-Douglas. Since good \(0\) is not demanded by consumers, we have

\[ \beta_{0}\,=\,0,\qquad\sum_{s=1}^{S}\beta_{s}=1. \]

In terms of the input-output coefficients,

\[ \alpha_{0}\,=\,1,\qquad a_{s0}\,=\,1-\alpha_{s}\quad\text{for }\;s=1,\dots,S,\qquad a_{sk}\,=\,0\ \text{otherwise}. \]

Hence, ordering sectors as \((0,1,\dots,S)\),

\[ \Omega\,=\,\begin{bmatrix}0 & 0 & 0 & \cdots & 0\\ 1-\alpha_{1} & 0 & 0 & \cdots & 0\\ 1-\alpha_{2} & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1-\alpha_{S} & 0 & 0 & \cdots & 0 \end{bmatrix}. \]

There are no input chains longer than one step, because sector \(0\) itself uses only labor. Therefore \(\Omega^{2}=0\), so the Leontief inverse collapses to

\[ \Psi\,=\,(I-\Omega)^{-1}\,=\,I+\Omega\,=\,\begin{bmatrix}1 & 0 & 0 & \cdots & 0\\ 1-\alpha_{1} & 1 & 0 & \cdots & 0\\ 1-\alpha_{2} & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1-\alpha_{S} & 0 & 0 & \cdots & 1 \end{bmatrix}. \]

The basic input affects every final sector, but there is no further propagation beyond that first round.

Using \(\lambda\,=\,(I-\Omega')^{-1}\beta\), we obtain the equilibrium Domar weights

\[ \lambda_{0}\,=\,\sum_{s=1}^{S}(1-\alpha_{s})\beta_{s},\qquad\lambda_{s}\,=\,\beta_{s},\qquad s\,=\,1,\dots,S. \]

So the Domar weight of the basic-input sector is its average cost share across the sectors that are ultimately absorbed in final demand. The sectoral solutions are

\[ d\log y_{0}\,=\,d\log z_{0}, \]

and, for \(s\,=\,1,\dots,S\),

\[ d\log y_{s}\,=\,d\log z_{s}+(1-\alpha_{s})\,d\log z_{0}. \]

Aggregating,

\[ d\log Y\;=\;\lambda_{0}\,d\log z_{0}\,+\,\sum_{s=1}^{S}\lambda_{s}\,d\log z_{s}\;=\;\left(\sum_{s=1}^{S}(1-\alpha_{s})\beta_{s}\right)d\log z_{0}\,+\,\sum_{s=1}^{S}\beta_{s}\,d\log z_{s}. \]

Relative to the horizontal economy, shocks to sector \(0\) now matter for aggregate output even though households never consume good \(0\) directly. What matters is that every final-good sector uses it as an input. The economy behaves like a horizontal economy augmented by one common productivity component, sector \(0\), whose aggregate weight is exactly the weighted expenditure share on the basic input.

Vertical economy.

Now consider a chain economy in which each sector is an input for the next, up until the last sector that produces the final good. There are two useful versions.

Version A: Labor only at the top of the chain.

flowchart LR
L((N))
s1((1))
s2((2))
s3((3))
H[HH]
L --> s1
s1 --> s2
s2 --> s3
s3 --> H

Sector \(1\) uses only labor, sector \(2\) uses only sector \(1\), sector \(3\) uses only sector \(2\), and so on. In a three-sector version,

\[ \Omega=\begin{bmatrix}0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{bmatrix},\qquad\Psi=\begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1 \end{bmatrix},\qquad\lambda=\begin{bmatrix}1\\ 1\\ 1 \end{bmatrix}. \]

Every unit of final output requires one unit of each upstream good, so each sector's gross sales equal GDP. Therefore every productivity shock receives weight one in aggregation:

\[ d\log Y=d\log z_{1}+d\log z_{2}+d\log z_{3}. \]

The aggregate technology is, of course,

\[ Y\;=\;\left(\prod_{s=1}^{S}z_{s}\right)N. \]

Version B: Labor in every stage.

flowchart LR
L((N))
s1((1))
s2((2))
s3((3))
H[HH]
L --> s1
s1 --> s2
s2 --> s3
s3 --> H
L -.-> s2
L -.-> s3

Suppose instead that every downstream sector combines labor and the previous stage with shares \(1-\omega\) and \(\omega\), respectively. Then for \(s\geq2\),

\[ \alpha_{s}=1-\omega,\qquad a_{s,s-1}=\omega, \]

with sector \(1\) still using only labor. In this case the chain still implies downstream propagation, but the effect decays geometrically:

\[ \psi_{sk}=\begin{cases} \omega^{s-k} & \text{if }s\geq k,\\ 0 & \text{if }s<k. \end{cases} \]

Hence the closer a sector is to final demand, the more heavily it loads on upstream productivity shocks.

Roundabout economy.

flowchart LR
L((N))
s1((1))
H[HH]
s1 --> s1
L --> s1
s1 --> H

Finally, consider the one-sector economy with self-use of intermediates. There is one sector that combines labor and its own output:

\[ y=z\,n^{1-\omega}x^{\omega},\qquad0<\omega<1. \]

Then

\[ \Omega=[\omega],\qquad\Psi=\frac{1}{1-\omega},\qquad\lambda=\frac{1}{1-\omega}. \]

The sectoral and aggregate formulas coincide:

\[ d\log y=d\log Y=\frac{1}{1-\omega}\,d\log z. \]

This is the cleanest example of amplification through intermediate inputs. The same productivity improvement is more important when the economy is more roundabout because the good is used repeatedly along the production chain.