Endogenous Production Networks Under Supply Chain Uncertainty

Q1. What is the model’s core setup and how does technique choice generate an endogenous network?

Each of n sectors chooses input shares αi = (αi0, αi1, …, αin) — where αi0 is the labor share — before observing sectoral TFP realizations εt, subject to a convex cost function Ai(αi) that penalizes deviation from fixed-proportion baseline techniques; the equilibrium network α is then the solution to a social planner’s problem that maximizes expected welfare W = E[y] − (ρ/2)V[y], where y is log GDP and ρ is the coefficient of relative risk aversion.* Because cost functions Ai are jointly determined by the input shares chosen and by Hessian matrices Hi that govern substitutability/complementarity of inputs, sectors can substitute or complement in the production of any given good, and the equilibrium network balances expected log GDP gains from choosing more productive suppliers against the variance reduction from choosing safer suppliers.

Q2. What role do Domar weights play, and how do they generalize to the endogenous-network case?

In the standard fixed-network case (Hulten’s theorem), each sector i’s contribution to aggregate log GDP equals its Domar weight ωi = (expenditure on sector i’s output)/(total GDP) — a sufficient statistic for first-order productivity effects. The paper extends this: with an endogenous network, the social planner’s optimality conditions imply that equilibrium Domar weights equal the shadow value of relaxing each sector’s resource constraint, and these shadow values respond to changes in beliefs (μ, Σ) through the induced changes in α.* Lemmas 3–5 characterize these responses: a sector’s Domar weight increases in its expected log TFP (μi), and changes in variance Σij propagate through the network via the adjustment terms in the first-order conditions, so that a single sector’s volatility change affects the Domar weights of all connected sectors.

Q3. What are the key propositions about how beliefs affect aggregate welfare and GDP?

Proposition 6 (monotone welfare response to mean beliefs): welfare W is increasing in each sector’s mean log TFP μi, and the marginal effect equals the sector’s Domar weight; this holds even though expected log GDP E[y] may non-monotonically respond to μi when inputs are gross substitutes, because the variance-reduction benefit of adjusting away from the now-more-productive but higher-variance sector can temporarily dominate. Proposition 7 (variance increases hurt expected log GDP): for substitutable inputs, a rise in Σii decreases E[y] because firms shift away from the more volatile sector toward less productive alternatives; for complementary inputs, the same shift also reduces E[y] because complementary inputs move together. Corollary 4 shows that welfare W always falls when uncertainty rises, combining these effects.

Q4. How does the flight-to-safety mechanism work in a multi-sector economy?

When uncertainty about sector i’s productivity rises, the optimal technique response is to reduce αji for all downstream sectors j that use sector i as an input substitute, and increase labor shares or shares in less volatile inputs; since sectors with lower μ but lower Σ become relatively more attractive on a risk-adjusted basis, the network reconfigures toward “safer” but typically less productive suppliers. The cascading link-destruction example (Section 7) illustrates this: when an industry’s production becomes uncertain, the endogenous deletion of risky links propagates across the network as complementary and substitute linkages amplify or dampen the flight to safety, with the direction depending on whether inputs are gross complements or substitutes in the Hessian Hi.

Q5. What does the calibration to US data find about the quantitative importance of the endogenous network?

The calibrated model with 37 BEA sectors (1948–2020) achieves a cross-sectional correlation between model and data Domar weights of 0.96 (though the model average Domar weight of 0.03 is below the data’s 0.05) and matches the data correlations Corr(ωjt, μjt) = 0.1 and Corr(ωjt, Σjjt) = −0.4 closely (model delivers 0.1 and −0.3 respectively). Comparing the flexible-network baseline to a fixed-network alternative, expected log GDP is 2.1% lower in the fixed-network economy, and welfare differs by a similar 2.1%. This suggests the endogenous reallocation of input shares over the sample period — as some sectors became persistently more productive — delivered substantial gains relative to a static network.

Q6. What happens during high-uncertainty episodes such as the Great Recession?

During the Great Recession (2007–2009), the estimated uncertainty measure Σt spiked sharply; firms responded by shifting techniques toward safer suppliers, resulting in expected log GDP that is about 0.25% lower in the baseline than in a “no uncertainty” (Σ = 0) economy and GDP variance that is about 2.4% lower. The insurance paid off in terms of realized outcomes: realized log GDP in the baseline economy was approximately 2.7% higher than in the “as-if Σ = 0” economy in 2009, because firms had taken out insurance against exactly the kind of bad TFP draws that materialized during the crisis. The perfect-foresight economy (where εt is known before technique choice) outperforms the baseline by up to 3% in realized GDP during the Great Recession — the maximum value of uncertainty resolution.

Q7. What is the key distinction between the effects of mean and variance changes for welfare vs. expected GDP?

Changes in mean beliefs μi and welfare W are co-monotone (Proposition 6), but changes in μi and expected log GDP E[y] can be non-monotone when inputs are substitutes: a small increase in μi for a less productive sector can actually lower E[y] in the short run because firms shift toward that sector at the expense of more productive alternatives, even though this shift reduces variance and raises welfare. The divergence between E[y] and W is the key mechanism: when ρ > 0 (risk-averse households), reducing variance has positive welfare value even when it lowers the level of expected GDP, so the production network adjusts in directions that appear sub-optimal for average productivity but are optimal for welfare. The calibrated relative risk aversion parameter ρ̂ = 4.3 indicates meaningful risk aversion that makes these variance-mean trade-offs quantitatively relevant.

Q8. What is the role of input complementarity versus substitutability in determining network responses?

When inputs i and j are gross substitutes (Hessian element [Hi]ij < 0), an increase in sector j’s uncertainty Σjj induces sectors that use both i and j to shift away from j and toward i, reducing j’s Domar weight and increasing i’s — the network becomes more concentrated in safer inputs. When inputs are gross complements ([Hi]ij > 0), an increase in Σjj also reduces the demand for the complementary input i, because both inputs must be used together and the safe input i becomes jointly less attractive when paired with volatile j; this can cause both E[y] and V[y] to fall simultaneously, resulting in an ambiguous welfare effect that depends on the magnitude of ρ relative to the E[y]-V[y] trade-off (Corollary 4 ensures welfare falls, but the split across E[y] and V[y] depends on complementarity structure).

Key concepts

technique choice : a sector’s endogenous selection of input shares αij prior to observing TFP realizations; the key margin of adjustment in the model through which uncertainty shapes the production network; characterized by convex cost functions Ai that favor balanced input use around baseline shares α°.

endogenous Domar weight : the share of total expenditure on a sector’s output in aggregate nominal GDP, computed in the model’s equilibrium; equals the shadow value of the sector’s resource constraint and responds to changes in beliefs (μ, Σ); in the fixed-network case reduces to the standard Hulten-theorem Domar weight.

flight to safety : the equilibrium response in which firms shift their input shares away from high-mean, high-variance suppliers toward lower-mean, lower-variance alternatives when aggregate uncertainty rises; generates the prediction that network restructuring during recessions reduces GDP volatility while raising expected production costs.

risk-adjusted expected welfare (W) : the social planner’s objective, defined as E[y] − (ρ/2)V[y] where y is log GDP and ρ is the coefficient of relative risk aversion; this non-separable objective function generates the trade-off between expected productivity and risk reduction that drives endogenous network formation.

cascading link destruction : the propagation of reduced sectoral linkages through the network when one sector’s uncertainty rises; in examples with complementary inputs, the reduced demand for a volatile sector also reduces demand for its complements, potentially amplifying the flight to safety beyond the directly affected sector.