Resilient-to-Fragile Transition and Excess Volatility in Supply Chain Networks

We study the disequilibrium dynamics of a stylised model of production networks in which firms use perishable and non-substitutable intermediate inputs, so that adverse idiosyncratic productivity shocks can trigger downstream shortages and output losses. To protect against such disruptions, firms hold precautionary inventories that act as buffer stocks. We show that, for a given dispersion of firm-level productivity shocks, there exists a critical level of inventories above which the economy remains in a stable stochastic steady state. Below this critical level, the system becomes fragile, i.e., it becomes prone to system-wide crises. As this resilience-fragility boundary is approached from above, aggregate output volatility rises sharply and diverges, even though shocks are purely idiosyncratic. Because inventories are costly, competitive pressures induce firms to economize on buffers. Although we do not explicitly model such costs, we argue that the resulting behaviour of individual firms drives the system close to criticality, generating persistent excess macroeconomic volatility – in other words, ``small shocks, large cycles’’ – in line with other settings where efficiency and resilience are in tension with each other. In the language of phase transitions, the resilient-to-fragile transition is continuous (supercritical): the economy exhibits a well-defined stochastic equilibrium with finite volatility on one side of the boundary, while beyond it the probability of a collapse in finite time tends to one. We characterize this transition primarily through numerical simulations and derive an analytical description in a high-perishability, high-connectivity limit.

💡 Research Summary

The paper develops a stylised dynamic model of production networks in which firms use perishable, non‑substitutable intermediate inputs and face idiosyncratic productivity shocks. Each firm holds precautionary inventories measured by a buffer parameter κ; κ = 0 corresponds to pure just‑in‑time ordering, while larger κ means enough stock to survive several periods of upstream disruption. Shocks have standard deviation σ and are purely firm‑specific. The authors show that the (σ, κ) space contains a well‑defined critical boundary separating a resilient regime—where the economy settles into a stochastic steady state with finite output volatility—from a fragile regime—where cascading shortages lead to a finite‑time collapse with probability one.

The model is built on Leontief (fixed‑coefficient) production functions, explicit perishability ω>0, and simple myopic adjustment rules for production targets and orders. By linearising the dynamics around the steady state, the state space is partitioned into cones corresponding to supply‑limited and demand‑limited regimes. Within each cone the dynamics are linear, allowing the authors to compute eigenvalues and derive local stability conditions. In the limit of vanishing perishability (ω→0) the analysis simplifies, and the authors obtain analytical expressions for the critical line.

Numerical simulations across a wide range of σ and κ confirm the analytical predictions. As κ approaches its critical value κ_c(σ) from above, aggregate output volatility rises sharply, far exceeding the 1/√N scaling expected under independent shocks. When κ falls below κ_c, even though the local linearised system remains formally stable, the network becomes vulnerable to cascading failures; a single firm’s production drop can propagate downstream, eventually causing a system‑wide collapse. This transition is continuous (super‑critical) in the language of phase transitions: the variance diverges as the boundary is approached, and the probability of collapse jumps to one beyond it.

A high‑connectivity, high‑perishability mean‑field limit is derived, collapsing the heterogeneous network into a single representative firm with average inventory and output. In this limit the critical condition reduces to a simple nonlinear equation linking κ and σ, providing a closed‑form estimate of the volatility divergence point. The analysis also shows that increasing the number of alternative suppliers (network degree) or enhancing the speed of demand re‑routing shifts the critical line outward, effectively enlarging the resilient region.

The paper situates its contribution within three strands of literature: (i) traditional input‑output and production‑network models that rely on price adjustments to clear markets, (ii) simulation‑based ARIO and disaster‑impact studies that capture inventory dynamics but lack analytical tractability, and (iii) recent work on self‑organized criticality in economic systems. By focusing on quantity constraints and inventory buffers rather than price flexibility, the authors demonstrate that even in a perfectly competitive setting, the physical availability of inputs can be a primary source of macro‑level volatility and crises.

Policy implications follow directly from the model’s mechanisms. Because inventory holding is costly, competitive pressures push firms toward low κ, moving the economy close to the critical boundary and generating “small shocks, large cycles.” Encouraging higher safety stocks—through strategic reserves, inventory subsidies, or regulation—can raise κ above κ_c and restore stability. Promoting supplier diversification and improving logistics to enable rapid demand reallocation also shift the critical line, reducing the likelihood of systemic collapse.

In sum, the study provides a rigorous theoretical framework that links micro‑level inventory management and network structure to macro‑economic volatility, offering a clear explanation for excess aggregate fluctuations observed in real economies and suggesting concrete avenues for resilience‑enhancing policy.