Supply Network Formation and Fragility

Matthew Elliott; Benjamin Golub; Matthew V. Leduc

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Supply Network Formation and Fragility

Matthew Elliott, Benjamin Golub, Matthew V. Leduc

arXiv:2001.03853v7Jan 12, 2020econ.THcs.GTphysics.soc-ph

TL;DR

The paper asks how complex supply networks withstand failures when firms must source multiple essential inputs through risky relationships. It models multisourcing and strategic relationship investment, showing that decentralized equilibrium can place intermediate-productivity networks near a critical threshold where small systemic shocks are massively amplified.

Problem
The paper studies how complex supply networks with risky, essential sourcing relationships respond to idiosyncratic and aggregate disruptions.
Method
It models recursive supply reliability, compares centralized and decentralized relationship-strength investment, and analyzes equilibrium networks as production depth becomes large.
Results
For intermediate κ, equilibrium relationship strength lies near xcrit, while Theorem 1 establishes a unique symmetric undominated equilibrium for sufficiently large τ.
Takeaways & Limitations
A positive fraction of supply networks can be perched on the precipice in equilibrium, although a social planner never selects relationship strengths near xcrit.

Abstract

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We model the production of complex goods in a large supply network. Each firm sources several essential inputs through relationships with other firms. Individual supply relationships are at risk of idiosyncratic failure, which threatens to disrupt production. To protect against this, firms multisource inputs and strategically invest to make relationships stronger, trading off the cost of investment against the benefits of increased robustness. A supply network is called fragile if aggregate output is very sensitive to small aggregate shocks. We show that supply networks of intermediate productivity are fragile in equilibrium, even though this is always inefficient. The endogenous configuration of supply networks provides a new channel for the powerful amplification of shocks.

I. Model: The Supply Network

The model represents production as a network of specialized firms sourcing essential inputs through potential and operational relationships. Firms are functional when each required input has at least one operational supplier that is itself functional.

The network contains varieties produced by small firms, with each variety requiring specified inputs and occupying a supply-chain depth.Potential suppliers for each required input are drawn from compatible varieties one level upstream.
Each input has multiple potential suppliers, while relationship strength determines independently which potential links are operational.The potential network records compatibility; the realized network retains only operational sourcing links.
Depth-0 varieties are always functional, whereas deeper varieties function only when at least one functional supplier exists for every required input.This inductive rule determines the random set of functional varieties and hence aggregate production possibilities.
The framework captures multisourcing and relationship uncertainty, including failures related to delivery, misunderstandings, or unexpected costs.The same relationship-strength parameter can represent operational quality or investment-dependent success in maintaining supply relationships.
Supply-network reliability is the probability that a uniformly selected variety is functional and depends on relationship strength and the depth distribution.The model initially studies symmetric, exogenous relationship strength and links reliability to aggregate production possibilities.

B. A Discontinuity in Reliability

For deep supply networks, reliability changes discontinuously at a critical relationship strength. Below the threshold reliability vanishes, whereas above it remains bounded away from zero and becomes highly sensitive near the threshold.

Below xcrit, reliability converges to zero as network depth grows; above xcrit, it remains bounded away from zero.This threshold behavior holds for any n ≥ 2 and m ≥ 2 under the stated deep-network limit.
Figure 3 illustrates the transition from zero successful production below xcrit to more than 70 percent above the threshold in the m = 2, n = 4 example.Panel A shows a finite-depth curve, while panel B shows its limiting shape as depth tends to infinity.
The analysis focuses on large typical depths, while bounded-depth networks are treated separately in the online appendix.Thus, the discontinuity result is principally a deep-network result rather than a universal finite-depth claim.
The reliability curve becomes steep near xcrit, with small increases in relationship strength producing large changes in aggregate reliability.The right-hand derivative becomes arbitrarily large as x approaches xcrit from above in the deep-network limit.

C. The Reasons for the Shape of the Reliability Function

The reliability shape follows from recursive production requirements: each input needs at least one operational functional supplier, and all required inputs must succeed. In deep networks, this recursion generates a sharp transition whose location shifts with multisourcing and complexity.

A variety’s reliability is computed recursively from supplier reliability, starting with certainty for depth-0 varieties.The map R_x(r) gives the probability that a firm is functional when each supplier is functional with probability r.
For each input, at least one of n suppliers must have both an operational link and a functional supplier, and this must hold across all m inputs.This logic yields R_x(r) = (1 − (1 − xr)^n)^m.
In deep networks, fixed points of the recursive map determine limiting reliability, producing a sharp transition for every m ≥ 2 and n ≥ 2.Below the critical value only the zero fixed point remains; above it a positive fixed point exists.
Increasing multisourcing shifts the reliability curve upward and lowers the relationship-strength threshold for the discontinuity.Increasing production complexity shifts the curve downward and raises that threshold.
The paper connects reliability to aggregate output through an increasing concave production function and then studies endogenous relationship-strength choices.The endogenous-strength analysis compares planner investment with decentralized investment in supply-network robustness.

A. A Planner’s Problem

The planner chooses a common relationship-strength level by weighing supply-network output against investment costs. For sufficiently deep networks, optimal investment jumps from low reliability to a level above the critical threshold, never stopping near that threshold.

A. A Planner’s Problem: The planner maximizes network output net of scaled relationship-strength costs over a common strength level x.Reliability depends on x and the distribution of supply-chain depths, while the planner’s cost is cP(x)/κ.
A. A Planner’s Problem: For sufficiently deep networks, Proposition 2 establishes a threshold κcrit separating low- and high-investment planner solutions.The proposition applies for fixed n and m and sufficiently large τ, with solutions characterized relative to xcrit.
A. A Planner’s Problem: When κ is sufficiently low, institutional investment is too costly, so the planner chooses low relationship strength and reliability.The planner’s solution remains below xcrit − ϵ in this region.
A. A Planner’s Problem: As κ crosses κcrit, optimal institutional investment increases discontinuously to a level strictly above xcrit and remains above it thereafter.The planner’s investment does not settle near the critical level because marginal social benefits become arbitrarily large near xcrit while marginal costs remain bounded.
A. A Planner’s Problem: The decentralized model has firms simultaneously choosing nonnegative investments before the supply-network realization, with relationship strength xif = x̄ + yif.The baseline x̄ represents institutional relationship quality, and firms’ payoffs trade expected gross profit against investment cost.
A. A Planner’s Problem: Firms’ expected profits equal their production probability times conditional gross profit, minus the cost of investment, while social welfare sums production value and total costs.Conditional gross profit decreases with the fraction of functioning firms, reflecting stronger profits under less competition.

IV. Equilibrium Supply Networks and Their Fragility

As production networks become deep, decentralized equilibrium has three regimes: unproductive, critical, and noncritical. Intermediate-cost environments place relationship strength near the critical threshold, making arbitrarily small shocks capable of causing discontinuous production losses.

IV. Equilibrium Supply Networks and Their Fragility: Deep supply networks exhibit unproductive, critical, and noncritical equilibrium regimes as κ varies.Low κ yields arbitrarily low reliability; intermediate κ places strength near xcrit; high κ yields strength above xcrit and robustness to small shocks.

A. Equilibrium Reliability

The equilibrium analysis characterizes firms’ symmetric investment choices through best responses and network reliability. For deep networks, equilibrium moves from no investment to a critical band and then to a robust noncritical region as κ increases.

A. Equilibrium Reliability: The analysis uses symmetric equilibria defined by mutual best responses and selects the welfare-maximizing equilibrium among them.This selection corresponds to the highest-investment, highest-reliability symmetric equilibrium.
A. Equilibrium Reliability: Theorem 1 gives a unique symmetric undominated equilibrium for sufficiently deep networks under the maintained assumptions.The equilibrium relationship strength is denoted xτ*(κ), with thresholds depending on n and m.
A. Equilibrium Reliability: For κ below the lower threshold, equilibrium has no investment and relationship strength equals the baseline x̄.In the large-depth limit, baseline strength below xcrit implies reliability approaches zero, supporting the zero-investment equilibrium.
A. Equilibrium Reliability: For intermediate κ, equilibrium strength lies within [xcrit − ϵ, xcrit + ϵ], defining the critical regime.These equilibria place the network near the threshold where reliability changes sharply.
A. Equilibrium Reliability: For κ above the upper threshold, equilibrium strength exceeds xcrit, defining a noncritical regime; equilibrium strength also increases with κ.The resulting supply network is robust to small shocks in the deep-network limit.
A. Equilibrium Reliability: Across a finite interval of κ values, a positive fraction of networks converges to xcrit in equilibrium, unlike the planner’s solution, which avoids that region.Figure 6 illustrates the intersections of the reliability and best-response curves underlying these regimes.

B. Fragility

Critical equilibria are fragile because small shocks to baseline relationship quality can collapse production while firms’ investments remain fixed. Proposition 3 assigns fragility to the critical κ interval and robustness to the higher-κ region.

B. Fragility: Critical equilibria create this fragility because small unanticipated shocks to baseline institutional quality can trigger production collapse.The shock is applied to x̄ while firms’ investment and entry decisions remain fixed.
B. Fragility: Equilibrium fragility is defined by an arbitrarily small reduction in relationship strength causing reliability to approach zero as network depth grows.Equilibrium robustness is the absence of this property.
B. Fragility: Under Theorem 1’s conditions, fragility occurs for κ in the critical interval [κ̄, κ̄], whereas κ above the upper threshold is robust.This conclusion follows from the equilibrium characterization of critical and noncritical relationship strengths.

C. Some Comments on Interpretation

The model’s interpretation emphasizes systemic relationship shocks, timing, and three essential features: specific disrupted relationships, endogenous investment, and complex production. These features connect the theory to observed supply-chain cascades and decentralized robustness decisions.

C. Some Comments on Interpretation: Systemic shocks can arise from weaker contract enforcement, credit-market disruptions, trade regulations, court backlogs, or macroeconomic supply stresses.These shocks make many relationships slightly more prone to failure, potentially disrupting networks broadly.
C. Some Comments on Interpretation: The model’s short run freezes relationship strengths, whereas the medium run permits firms to adjust investment and potentially restore productivity.The analysis selects the most productive recovery equilibrium, limiting modeled losses after the shock.
C. Some Comments on Interpretation: The essential model features are specific disruption-prone relationships, endogenous strengthening investments, and production requiring multiple essential inputs across sufficiently many layers.These features jointly support the model’s supply-network fragility mechanism.
C. Some Comments on Interpretation: Specific sourcing relationships facilitate cascades because firms cannot quickly replace failure-prone suppliers, as illustrated by documented supply-chain disruptions.The interpretation is consistent with evidence from banking, trading, apparel, food, diamonds, and manufacturing networks.
C. Some Comments on Interpretation: Relationship investments can be soft practices that improve understanding and cooperation, while noncooperative incentives generally prevent firms from funding others’ relationships.A firm invests up to its own marginal private benefit, so heterogeneous returns typically concentrate investment responsibility.
C. Some Comments on Interpretation: The framework can also represent robustness investments among processes or inputs within a firm when agency frictions decentralize those decisions.This reinterpretation preserves the model’s relevance beyond interfirm supply networks.

B. Contrasts with Models without the Essential Features

Benchmark models show that the paper’s sharp fragility depends on relationship-based sourcing combined with complex production. Spot markets remove discontinuities, while simple production retains a continuous transition rather than a sharp jump.

B. Contrasts with Models without the Essential Features: The benchmarks are designed to isolate the necessity of relationship sourcing and multiple essential inputs for the paper’s main mechanism.The section explicitly presents spot-market and simple-production models as contrasts to the essential features.
B. Contrasts with Models without the Essential Features: Reliability increases smoothly with sourcing success in spot markets, showing that perfect spot markets remove the main model’s discontinuities.The benchmark considers the probability that a random firm can produce as individual sourcing attempts become more successful.
B. Contrasts with Models without the Essential Features: With simple production requiring one relationship-sourced input, production becomes positive above x = 0.5 but changes continuously rather than through a sharp jump.The derivative is discontinuous at the threshold, unlike the discontinuous transition in complex production.
B. Contrasts with Models without the Essential Features: In the simple-production benchmark with n = 2, production is zero for x < 0.5 and strictly positive for x > 0.5.The threshold reflects the branching condition that expected operational links reach one supplier.
B. Contrasts with Models without the Essential Features: The simple-production transition is continuous because its network growth threshold differs from the discontinuous phase transition driving the main complex-production results.The comparison distinguishes the benchmark’s continuous transition from the main model’s sharp equilibrium fragility.

C. Robustness: Extensions Relaxing Simplifying Assumptions

The paper’s main conclusions survive several extensions, including extensive-margin investment, anticipated shocks, firm-level shocks, and endogenous entry. Entry can itself generate a nonempty range of equilibrium fragility.

C. Robustness: Extensions Relaxing Simplifying Assumptions: Relationship investment can operate on the extensive margin by stochastically discovering capable suppliers, while the same framework also permits intensive-margin quality improvements.The model assumes a fixed set of n potential suppliers, each found independently with probability x_if.
C. Robustness: Extensions Relaxing Simplifying Assumptions: The fragility analysis extends to anticipated shocks, so unanticipated shocks are not essential to the result.The extended profit calculation incorporates the expected arrival probability of the shock.
C. Robustness: Extensions Relaxing Simplifying Assumptions: The key precipice conclusions remain robust when shocks directly hit firms rather than supply links.The model focuses on relationship shocks for simplicity, but the same mathematical forces operate with node-level disruptions.
C. Robustness: Extensions Relaxing Simplifying Assumptions: The extensions show that the main insights do not depend on fixed varieties, unanticipated shocks, or relationship-level shocks alone.These robustness checks relax entry, shock-timing, and shock-location assumptions.
C. Robustness: Extensions Relaxing Simplifying Assumptions: Endogenous entry preserves equilibrium fragility over a nonempty open set of productivity parameters.When reliable production attracts entry, competition reduces profits and weakens incentives to maintain strong relationships.

D. Robustness to Heterogeneity

Heterogeneous firm types preserve sharp production transitions and equilibrium fragility under stated positive-investment conditions. In product networks, critical products transmit fragility upstream through a weakest-link structure.

D. Robustness to Heterogeneity: The heterogeneous extension partitions firms by depth and allows type-dependent symmetric equilibrium strategies and reliability functions.Low- and high-depth firms may have distinct gross profits, costs, and relationship strengths.
D. Robustness to Heterogeneity: Equilibrium fragility also persists for high-depth firms when their equilibrium investment remains positive for sufficiently large depth.Under the proposition’s condition, the heterogeneous model inherits baseline fragility.
D. Robustness to Heterogeneity: Under the heterogeneous model, low-depth firms can remain productive even when high-depth firms are disrupted because the low-depth group operates independently of the high-depth group.This conclusion reflects the model’s upstream-only dependence structure.
D. Robustness to Heterogeneity: Sharp production transitions persist under heterogeneity, with arbitrarily small relationship-strength shocks causing large reliability drops at some parameter values.This establishes that homogeneous network positions are not necessary for the transition.
D. Robustness to Heterogeneity: When one product is critical, every product that depends on it directly or indirectly is also critical.This weakest-link property applies along directed paths in the product dependence graph.
D. Robustness to Heterogeneity: All producers within a strongly connected component of the product dependence graph are either critical together or not critical together.The result extends the weakest-link logic to mutually connected product groups.

VI. Related Literature

The paper connects its model to research on production networks, complementarities, endogenous fragility, and network phase transitions. Its distinctive contribution is to show that multisourcing and endogenous relationship investment do not eliminate severe fragility.

The model extends production-network research by endogenizing input-output relationships and firms’ investments in relationship strength under shocks.
Strong complementarities remain capable of generating severe equilibrium fragility even when firms can mitigate supply risk through multisourcing.
The paper introduces discontinuous phase transitions into a production-network model with customized multisourcing relationships.
Unlike related network studies with exogenous networks or shocks, the paper endogenizes investments that affect link-operation probabilities and can place equilibria on a precipice.
The paper’s fragility mechanism complements earlier work on endogenous amplification of apparently diversifiable idiosyncratic volatility.

VII. Concluding Discussions

The concluding welfare analysis finds that decentralized investment is insufficient relative to the social optimum. The resulting underinvestment can leave productive equilibria fragile, because small reliability improvements may produce very large welfare gains.

A. Welfare Implications: Fragile equilibrium networks are inefficient because a uniform reliability improvement can generate arbitrarily large marginal benefits.
A. Welfare Implications: There is always underinvestment in relationship strengths in a productive equilibrium, with equilibrium strength below the planner’s selected efficient level.
A. Welfare Implications: The welfare analysis evaluates how changing relationship strength affects total surplus, including producer surplus net of investment costs.
A. Welfare Implications: The welfare wedge includes consumer-surplus nonappropriability, business stealing, and a reliability externality affecting firms’ investment incentives.

B. Implications for Policies Supporting Investment and Robustness

The policy discussion distinguishes medium-run interventions affecting investment incentives from short-run responses to aggregate disruptions. Marginal measures often leave fragile equilibria intact, whereas sufficiently large interventions can move networks into a nonfragile regime.

Marginal Interventions on Investment Incentives: Marginal subsidies or modest institutional improvements typically fail to resolve fragility because they crowd out private investment nearly one-for-one.
Larger Interventions to Improve Investment: Nonmarginal subsidies or profitability improvements can shift equilibrium from the fragile regime into the nonfragile regime.
Policy Scope: In the fragile regime, all productive equilibria are fragile, so temporary interventions cannot remove fragility; permanent cost reductions or related changes are required.
Targeting Interventions: Reshoring may improve robustness when long-distance or weak-institution links are more prone to disruption.
Targeting Interventions: Targeting should prioritize upstream, central complex industries because downstream firms inherit fragility from fragile products.
Reactive Interventions: The Short Run: Short-run reactive policies are relevant because fragility’s extreme effects occur before firms can adjust medium-run investment decisions.

C. Embedding Precipices in a Macroeconomic Model: Aggregate Volatility

Embedding heterogeneous supply networks in a larger economy preserves severe shock amplification while limiting the aggregate decline to the output share of affected networks. Macroeconomic fragility arises when some complex networks have productive equilibria.

The larger economy is modeled as many heterogeneous supply networks whose complexity, multisourcing, and productivity parameters are drawn from a full-support distribution.
Macroeconomic fragility requires complex networks with positive equilibria; if only simple networks can produce, aggregate fragility does not arise.
If the economy contains complex supply networks with positive equilibria, a small relationship-strength shock causes a discontinuous loss of a positive fraction of aggregate output.
Unlike a single-network collapse, aggregate output need not fall to zero because unaffected networks continue producing.
D. Some Implications for Industrial Development: The industrial-development application compares complex and simple products using reliability multiplied by product value, with complex products assigned higher value.
D. Some Implications for Industrial Development: The development discussion links discontinuities to rapid industrialization, rising intermediate-input shares, institutional quality, and disruption rates, while requiring more detailed modeling for a full study.