Part 5: Input-Output Networks and Economic Activity Section 21

Beyond Hulten: Distortions

Once the initial allocation is inefficient, Hulten's theorem no longer holds even to first order. The reason is that the allocation changes in response to shocks, a consequence of the Envelope theorem not applying. To see this, we now reintroduce markups \(\{\mu_{s}\}\). Other wedges can be re-expressed as markups by adding fictional input suppliers with an appropriately defined markup.

Once \(\mu\neq1\), the revenue-based input-output matrix no longer coincides with the matrix of cost shares. Define the revenue-based input share matrix \(\Omega\) as before, with

\[ \omega_{sk}\;=\;\frac{p_{k}x_{sk}}{p_{s}y_{s}}. \]

The corresponding cost-based share is

\[ \widetilde{\omega}_{sk}\;\equiv\;\frac{p_{k}x_{sk}}{MC_{s}\,y_{s}}\;=\;\mu_{s}\omega_{sk}. \]

Let \(\widetilde{\Omega}=[\widetilde{\omega}_{sk}]\) denote the cost-based input-output matrix, \(\widetilde{\Psi}=(I-\widetilde{\Omega})^{-1}\) the associated Leontief inverse, and

\[ \widetilde{\lambda}\equiv\widetilde{\Psi}^{\prime}\beta \]

the vector of cost-based Domar weights.

It is also convenient to add labor as the last column of the Input-Output matrix, and the production of the final aggregate good as the first column. In this way, there is a single matrix that describes all the economy. The sectors can therefore go between \(0\) and \(S\), with \(s=0\) being the production of the fictional final output.

A convenient decomposition from Baqaee and Farhi (2020) is

\[ d\log Y\;=\;\underbrace{\widetilde{\lambda}^{\prime}d\log z}_{\text{pure technology}}\;+\;\underbrace{\left(-\widetilde{\lambda}^{\prime}d\log\mu-\widetilde{\Lambda}\,d\log\Lambda\right)}_{\text{change in allocative efficiency}}, \]

where \(\Lambda\) is vector of factors' revenue shares and \(\widetilde{\Lambda}\) is its cost-based counterpart. The first term says that technology shocks are now weighted by cost-based rather than revenue-based Domar weights. The second term says that reallocations matter at a first order because the initial equilibrium is not efficient.

With a single primary factor, labor, the change in real output is

\[\begin{align*} d\log Y\;=\; & \;\widetilde{\lambda}^{\prime}\left(d\log z-d\log\mu\right)\;-\;\sum_{s=0}^{S}\lambda_{s}(1-\theta_{s})\mathrm{Cov}_{\widetilde{\Omega}_{(s,:)}}\!\left(d\log p-d\log\Lambda,\Psi_{(:,S+2)}\right)\;. \end{align*}\]

where \(\Lambda\) is labor's revenue share and \(\Psi_{(s,S+2)}\) denotes the fraction of sector \(s\)'s revenue that is ultimately paid out to labor (from the last column of the \(\Psi\) matrix). The change in the relative prices (relative to labor) is

\[ d\log p-d\log\Lambda\;=\;\sum_{s=1}^{S}\widetilde{\Psi}_{(:,s)}\left(d\log\mu_{s}-d\log z_{s}\right). \]

We can use this to express the changes in output in terms of changes in technology, reallocation due to technological change, and reallocation due to changes in distortions:

\[\begin{align*} d\log Y\;=\; & \;\widetilde{\lambda}^{\prime}d\log z\;-\;\sum_{s=0}^{S}\frac{\lambda_{s}}{\mu_{s}}(\theta_{s}-1)\mathrm{Cov}_{\widetilde{\Omega}_{(s,:)}}\!\left(\sum_{k=1}^{S}\widetilde{\Psi}_{(:,k)}d\log z_{k},\Psi_{(:,S+2)}\right)\\ & \;+\;\sum_{s=0}^{S}\frac{\lambda_{s}}{\mu_{s}}\theta_{s}\mathrm{Cov}_{\widetilde{\Omega}_{(s,:)}}\!\left(\sum_{k=1}^{S}\widetilde{\Psi}_{(:,k)}d\log\mu_{k},\Psi_{(:,S+2)}\right)\;. \end{align*}\]

The first term is the direct effect of technologies and wedges through cost-based Domar weights. The covariance terms are the reallocation effects. They are positive when lower relative prices shift demand toward parts of the economy with high initial wedges, that is, toward supply chains with low values of \(\Psi_{(:,S+2)}\), as those sectors were too small to begin with.

Example: Horizontal economy.

Consider the horizontal CES economy in which each good is produced linearly from labor and there are no intermediate inputs. Let

\[ \lambda_{i}=\frac{p_{i}c_{i}}{Y}\qquad\text{and}\qquad\Lambda=\sum_{i=1}^{S}\lambda_{i}\mu_{i}^{-1} \]

denote, respectively, the household expenditure share on good \(i\) and labor's revenue share. Since the household is the only buyer, there is only one substitution elasticity, \(\theta_{C}\), and the relevant relative price is

\[ \log p_{i}-\log w=\log\mu_{i}-\log z_{i}. \]

Hence the expenditure share of good \(i\) satisfies

\[ d\log\lambda_{i}=(1-\theta_{C})\left[\left(d\log\mu_{i}-d\log z_{i}\right)-\sum_{j=1}^{S}\lambda_{j}\left(d\log\mu_{j}-d\log z_{j}\right)\right]. \]

Substituting this into the distorted-economy aggregation formula yields

\[ d\log Y=\sum_{i=1}^{S}\lambda_{i}\,d\log z_{i}\;+\;\frac{1-\theta_{C}}{\Lambda}\mathrm{Cov}_{\lambda}\!\left(d\log z,\mu^{-1}\right)\;+\;\frac{\theta_{C}}{\Lambda}\mathrm{Cov}_{\lambda}\!\left(d\log\mu,\mu^{-1}\right). \]

The first term is the direct effect of productivity shocks. The second and third terms are reallocation effects. If goods are substitutes, \(\theta_{C}>1\), then productivity improvements at initially high-markup producers raise aggregate output because they shift labor toward firms that were too small to begin with. Likewise, a fall in markups at initially high-markup producers raises output through the last covariance term, and this force is stronger the larger is \(\theta_{C}\). When \(\theta_{C}=0\), the markup covariance term vanishes, so markup changes have no reallocation effect.

The deadweight-loss of markups.

Near efficiency, Baqaee and Farhi (2019) show that the loss from wedges can be written as a Domar-weighted sum of Harberger triangles:

\[ d\log Y\;\approx\;-\frac{1}{2}\sum_{i=1}^{S}\lambda_{i}\,d\log y_{i}\,d\log\mu_{i}. \]

A wedge is more costly when it induces a larger quantity response and when it hits a sector with a larger Domar weight.

In a nested-CES economy we can further show that

\[ d\log Y\;\approx\;-\frac{1}{2}\sum_{s=0}^{S}\lambda_{s}\theta_{s}\text{Var}_{\Omega_{(s,:)}}\left(\sum_{k=0}^{S}\Psi_{(:,k)}d\log\mu_{k}\right). \]

From Baqaee and Rubbo (2023): "Unlike for productivity shocks, where the second-order effects originating from each producer j are positive if, and only if, its inputs are substitutes, for wedges the second-order terms are always (weakly) negative, and more so when goods are better substitutes. Intuitively, expenditure switch- ing by consumers and producers inefficiently reallocates quantities away from sectors with higher relative markup. This effect is stronger when elasticities of substitution, \(\theta_{s}\), are larger."

Hsieh-Klenow: A CES horizontal economy.

Consider now the CES horizontal economy in Hsieh and Klenow (2009). Firms use only labor and their output is aggregated into a final good with a CES technology with elasticity of substitution \(\theta>1\). A high-markup producer is too small relative to the efficient allocation. Dispersion in markups therefore misallocates labor across firms. In this case there are no intermediate inputs, so each firm's Domar weight is simply its equilibrium expenditure share \(\lambda_{i}=p_{i}c_{i}/Y\).

The loss from misallocation is proportional to the sales-share-weighted dispersion of markups, and the proportionality factor rises with the elasticity of substitution.

\[ \Delta\log Y\;\approx\;-\frac{1}{2}\theta_{C}\,\mathrm{Var}_{\lambda}(\Delta\log\mu). \]

Roundabout amplification of wedge distortions.

Now return to the roundabout economy with self-input share \(\omega\), and let \(\mu\) denote the markup wedge on the unique producer. Applying Equation 22 gives the analogue of Example 17 in Baqaee and Rubbo (2023):

\[ \Delta\log Y\;\approx\;-\frac{1}{2}\frac{\theta\omega}{(1-\omega)^{2}}(\Delta\log\mu)^{2}. \]

Equivalently, the deadweight-loss representation may be written as

\[ \Delta\log Y\;\approx\;-\frac{1}{2}\lambda\,\Delta\log y\,\Delta\log\mu,\qquad\lambda=\frac{1}{1-\omega},\qquad\Delta\log y=-\frac{\theta\omega}{1-\omega}\Delta\log\mu. \]

Losses are therefore increasing both in the elasticity of substitution \(\theta\) and in the intermediate-input share \(\omega\). A higher self-input share raises the Domar weight because the distorted good is used repeatedly along its own supply chain. At the same time, it makes the local Harberger triangle larger because the same wedge induces a larger reduction in total output.

A markup is not just a distortion on final demand. It also distorts the repeated choice between labor and materials inside production. The larger is \(\omega\), the more rounds of intermediate production there are, and the more times the same distortion is compounded. In this sense, intermediate inputs amplify misallocation losses twice: by enlarging the relevant deadweight-loss triangle itself and by increasing the Domar weight used to aggregate it. When \(\omega=0\), this amplification disappears, and the one-sector markup is a pure transfer.