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

TL;DR

The paper introduces a stylised production-network model where firms rely on perishable, non-substitutable inputs and manage precautionary inventories. It demonstrates a continuous resilience–fragility transition: for a given shock amplitude $\boldsymbol{\sigma}$ there exists a critical inventory level $\boldsymbol{\kappa}_c(\boldsymbol{\sigma})$ separating a stable stochastic equilibrium from a regime prone to cascading shortages and finite-time crashes; as the transition is approached, aggregate volatility diverges, yielding excess volatility despite idiosyncratic shocks. The authors develop a cone-wise linear stability analysis and a high-perishability mean-field limit to obtain analytical expressions for the critical boundary $\sigma_c(\kappa)$, the crash-time scaling, and the excess-volatility behavior, while numerical simulations corroborate the phase diagram and finite-size scaling. They further show that supplier diversification and fast rewiring can raise or erase the critical boundary, illustrating a concrete efficiency–resilience trade-off with policy implications for inventory management and supply-chain diversification. Collectively, the work provides a dynamic, out-of-equilibrium mechanism for large macro fluctuations driven by network structure and buffers, offering a complement to equilibrium network models and informing strategies for robust, efficient production systems.

Abstract

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.

Figures (7)

• Figure 1: a. Colormap of the crash probability $P_{\rm crash}$ in the $(\kappa,\sigma)$ plane. Upon increasing $\sigma$, the economy transitions from resilient to fragile beyond a critical $\sigma_c(\kappa)$. For $(\sigma, \kappa)$ below the boundary line, the economy crashes almost surely. Note that there also appears to be a value $\sigma_{\max} \approx 1.$ beyond which the economy is always unstable, even when inventories are very high. b. Excess Volatility $EV$ as a function of $\sigma$ for a certain value $\kappa$, such that $\sigma_c \approx 0.8$ (black dashed line).
• Figure 2: Schematic representation of the supply-chain dynamics for firm $i$ in the case $K=2$ with the corresponding update rules and fluxes.
• Figure 3: Numerical stability analysis of the model in the $(\psi, \kappa)$ plane for different values of $\omega$ and $K/z=1/3$. The different colors indicate unstable (coral), stable (teal) and limit cycle (sand) states. We also show the four critical values of $\kappa$: $\kappa_{\min}, \kappa_c^\star, \kappa_c^-, \kappa_c^+$.
• Figure 4: a. Cumulative distribution $\mathbb P_N[\tau_c \leq T]$ as a function of the rescaled variable $\tilde{\sigma}=\ell^2(\sigma-\sigma_m(v))/w(v)$ for various values of $T$ and $N$. $\sigma_m(v)$ is determined numerically by inverting $\mathbb P_N[\tau_c \leq T](\sigma_m)=1/2$ while $\ell^{-2}w(v)$ is determined from the slope of $\mathbb P_N[\tau_c \leq T](\sigma)$ at $\sigma_m$. b. Numerically determined scaling function $s(v)=\ell^2(\sigma_m(v)-\sigma_c)$ as a function of $v$. c. Numerically determined scaling function $w(v)=\ell^2 W(v)$, with $W(v)$ the numerical width of $\mathbb P_N[\tau_c \leq T]$ at fixed $T$ and $N$. Note that outliers correspond to small values of $N,T$, for which deviations are expected. Parameters: $c=6$, $\omega=\psi=0.1$, $K=6$, $z=18$, $L_0=1$, $\kappa=2.6$.
• Figure 5: a. Collapse of the rescaled susceptibility $\chi(\sigma,T,N)T^{-\gamma}$ as a function of $\tilde{\sigma}=(\sigma-\sigma_m(v))/\ell^{-2}w(v)$ with exponent $\gamma=1.05$. b. Excess volatility $R$ (defined in Eq. \ref{['eq:excess_vol']}) a function of $\sigma-\sigma_c$ for the teal cut corresponding to $\kappa=2.6$ in Fig. \ref{['fig:phase_diagram']}-a. (Inset) Log-Log plot of $R$ vs. $\sigma - \sigma_c$, suggesting an inverse square-root divergence. The dashed line indicates a slope of $-0.54$. Parameters: $c=6$, $\psi=0.1$, $K=6$, $z=18$, $L_0=6$, $T=2000$, $N=750$, $\omega=0.1$, $\sigma_c=0.8$.