Part 5: Input-Output Networks and Economic Activity Section 16

A Domar Economy

Consider a static economy with sectors \(s=1,\dots,S\). Each sector produces a distinct good that can be consumed by the representative household or used as an intermediate input by other sectors. All sectors have Cobb-Douglas technology, so that sectoral output is an index of inputs aggregated using a weighted geometric average as in Domar (1961). Sector \(s\) has productivity \(z_{s}>0\) and technology

\[ y_{s}=z_{s}n_{s}^{\alpha_{s}}\prod_{k=1}^{S}x_{sk}^{a_{sk}},\qquad\alpha_{s}+\sum_{k=1}^{S}a_{sk}=1; \]

where \(n_{s}\) is labor, \(x_{sk}\) is the quantity of good \(k\) used by sector \(s\), and

Preferences over final consumption are also Cobb-Douglas:

\[ Y=\prod_{s=1}^{S}c_{s}^{\beta_{s}},\qquad\beta_{s}\geq0,\qquad\sum_{s=1}^{S}\beta_{s}=1. \]

The variable \(Y\) is both real GDP and final utility in money-metric units. The representative household supplies \(N\) units of labor inelastically.

This notation allows goods to play different roles in the economy. If \(\beta_{k}>0\) and \(a_{sk}>0\) for some \(s\), then good \(k\) is both a final good and an intermediate input, like electricity. If \(\beta_{k}=0\), then good \(k\) is a pure intermediate input, like steel rods. If \(\beta_{k}>0\) and \(a_{sk}=0\) for every \(s\), then good \(k\) is a pure final good.

This is the static portion of the multisector real business cycle environment in Long and Plosser (1983). This same setup has been used in other applications such as the role of production networks in Acemoglu et al. (2012).

Competitive equilibrium.

The equilibrium consists of a wage \(w\), prices for sectoral goods \(\{p_{s}\}\), outputs \(\{y_{s}\}\), consumption allocations \(\{c_{s}\}\), and input allocations \(\{n_{s},\{x_{sk}\}\}\), given the labor endowment \(N\) and productivities \(\{z_{s}\}\).

The demand for sectoral consumption comes from the household. Because preferences are Cobb-Douglas, we have

\[ c_{s}=\beta_{s}\frac{w\,N}{p_{s}}. \]

Input allocations solve the firms' cost minimization problem given prices and productivities. For any target output \(y_{s}\), firm \(s\) solves

\[ \min_{n_{s},\{x_{sk}\}_{k=1}^{S}}wn_{s}+\sum_{k=1}^{S}p_{k}x_{sk}\qquad\text{s.t.}\qquad y_{s}\leq z_{s}n_{s}^{\alpha_{s}}\prod_{k=1}^{S}x_{sk}^{a_{sk}}. \]

Under perfect competition and constant returns to scale, zero profits then imply \(p_{s}y_{s}=wn_{s}+\sum_{k=1}^{S}p_{k}x_{sk}\). Cost minimization delivers the familiar cost-share equations:

\[ x_{sk}=a_{sk}\frac{p_{s}y_{s}}{p_{k}},\qquad n_{s}=\alpha_{s}\frac{p_{s}y_{s}}{w}. \]

Substituting these choices back into the production function and taking logs gives

\[ \log\left(\frac{p_{s}}{w}\right)=\kappa_{s}-\log z_{s}+\sum_{k=1}^{S}a_{sk}\log\left(\frac{p_{k}}{w}\right), \]

where

\[ \kappa_{s}\equiv-\alpha_{s}\log\alpha_{s}-\sum_{k=1}^{S}a_{sk}\log a_{sk}, \]

with the usual convention that a zero share contributes zero to the sum.

If we let \(\widehat{p}\) denote the vector of relative log prices, \(\kappa\) the vector of constants, \(\log z\) the vector whose \(s\) th element is \(\log z_{s}\), and \(A=[a_{sk}]\) the matrix of technological coefficients, this system can be written as

\[ \widehat{p}=A\widehat{p}+\kappa-\log z, \]

so that

\[ \widehat{p}=(I-A)^{-1}(\kappa-\log z). \]

For comparative statics, the constants drop out and we obtain

\[ d\widehat{p}=-(I-A)^{-1}d\log z. \]

The matrix \((I-A)^{-1}\) will turn out to play a central role in the analysis of input-output networks, and so we give it a name,

\[ \Psi\,\equiv\,(I-A)^{-1}, \]

and denote its \((s,k)\) element \(\psi_{sk}\).

Market clearing and sales shares.

The market-clearing condition for sectoral good \(s\) is

\[ y_{s}=c_{s}+\sum_{j=1}^{S}x_{js}. \]

Multiplying by \(p_{s}\) and using the firms' first-order conditions gives

\[ p_{s}y_{s}=\beta_{s}w\,N+\sum_{j=1}^{S}a_{js}p_{j}y_{j}. \]

With the final-good price index normalized to one, nominal GDP and real GDP both equal labor income, so \(Y=w\,N\). It is therefore natural to define the sales share of sector \(s\) in GDP by

\[ \lambda_{s}\;\equiv\;\frac{p_{s}y_{s}}{w\,N}\;=\;\frac{p_{s}y_{s}}{Y}. \]

The following is an important result that links the sales shares \(\{\lambda_{s}\}\) to the matrix \(\Psi\) defined above. We return to this later when we relate the results of this economy to concepts from network and graph theory.

Proposition 16.1

The vector of sales shares then satisfies

\[ \lambda\;=\;\Psi^{\prime}\beta\;=\;(I-A^{\prime})^{-1}\beta, \]

or equivalently,

\[ \lambda^{\prime}\;=\;\beta^{\prime}\Psi. \]

Proof

Divide the nominal market-clearing condition by \(Y=w\,N\) to get

\[ \lambda_{s}\;\equiv\;\frac{p_{s}y_{s}}{Y}\;=\;\beta_{s}\,+\,\sum_{j=1}^{S}a_{js}\frac{p_{j}y_{j}}{Y}\;=\;\beta_{s}\,+\,\sum_{j=1}^{S}a_{js}\lambda_{j}. \]

Stacking these equations across sectors yields

\[ \lambda=\beta+A^{\prime}\lambda. \]

Rearranging gives the result.

With this, we can solve for the output of each sector in terms of productivity. A sector's output depends on its own productivity shock and on the productivity of the sectors that are upstream from it in production.

Proposition 16.2: Sectoral Output

Sectoral output can be written as

\[ \log y_{s}\;=\;\delta_{s}+\sum_{k=1}^{S}\psi_{sk}\log z_{k}, \]

where \(\delta_{s}\) is a constant independent of the shocks.

Proof

From the definition of the sales share \(\lambda_{s}\), we have \(y_{s}\,=\,\lambda_{s}N\left(\frac{p_{s}}{w}\right)^{-1}\), where the relative prices satisfy \(\log\left(\frac{p}{w}\right)\,=\,\Psi(\kappa-\log z)\). Hence, \(\log\left(\frac{p_{s}}{w}\right)\) is

\[ \hat{p_{s}}\;=\;\sum_{k=1}^{S}\psi_{sk}\kappa_{k}\,-\,\sum_{k=1}^{S}\psi_{sk}\log z_{k}. \]

Taking logs and replacing for the relative price,

\[ \log y_{s}\;=\;\log\lambda_{s}\,+\,\log N\,-\,\sum_{k=1}^{S}\psi_{sk}\kappa_{k}\,+\,\sum_{k=1}^{S}\psi_{sk}\log z_{k}. \]

All terms except the last summation are constant with respect to the productivity vector, so we collect them as

\[ \delta_{s}\equiv\log\lambda_{s}+\log N-\sum_{k=1}^{S}\psi_{sk}\kappa_{k}. \]

This gives the result. Crucial for this is that \(\lambda_{s}\) depends only on parameters from the previous proposition.

Having solved for the competitive equilibrium allocation we can now aggregate and solve for \(Y\). Changes in aggregate output are weighted averages of sectoral productivity changes, where the weights are gross sales relative to GDP rather than value-added shares. In this Cobb-Douglas economy the weights \(\lambda_{s}\) are constant.

Proposition 16.3: Aggregation

Aggregate output satisfies

\[ \log Y=\delta_{Y}+\sum_{s=1}^{S}\lambda_{s}\log z_{s}, \]

where \(\delta_{Y}\) is a constant independent of the shocks and we use the Cobb-Douglas price index as numeraire.

Proof

Aggregate log output satisfies

\[ \log Y\,=\,\sum_{s=1}^{S}\beta_{s}\log c_{s}\,=\,\sum_{s=1}^{S}\beta_{s}\log\left(\beta_{s}\frac{w\,N}{p_{s}}\right)\,=\,\sum_{s=1}^{S}\beta_{s}\log\beta_{s}\,+\,\log(N)\,-\,\sum_{s=1}^{S}\beta_{s}\log\frac{p_{s}}{w}, \]

The last term can be written in vector notation as

\[ \sum_{s=1}^{S}\beta_{s}\log\frac{p_{s}}{w}\;=\;\beta^{\prime}\widehat{p}\;=\;\beta^{\prime}\Psi(\kappa-\log z)\;=\;\lambda^{\prime}(\kappa-\log z). \]

Define \(\delta_{Y}\) as

\[ \delta_{Y}\;\equiv\;\sum_{s=1}^{S}\beta_{s}\log\beta_{s}\;+\;\log N\;-\;\lambda^{\prime}\kappa. \]

This gives the result.

Our objective now is to provide a compelling explanation for the results we just derived. Why is the sales share the right object to use when aggregating sectoral shocks? What lies behind the relationship between sales shares and the weights of inputs in production?