Optimal Resilience in Multitier Supply Chains

Layer 1 — Overview

Research Question

Grossman, Helpman, and Sabal ask what market failures arise in vertical supply chains with multiple production tiers, limited (non-anonymous) supply networks, arms-length transactions, and recurrent risks of disruption at every node. They then ask what government policies would be required to implement the socially efficient (first-best) allocation as a decentralized equilibrium, and — in a second-best environment where subsidies to firm-to-firm transactions are politically infeasible — how optimal policies to promote resilience and network formation differ.

Model and Methodology

The paper develops a general-equilibrium model of a closed economy with an arbitrary number S+1 of vertical production tiers (tier 0 through tier S). A finite measure of “lead” firms in tier S produce differentiated consumer goods under monopolistic competition using labor and a CES bundle of intermediate inputs from tier S-1 suppliers. Firms in each intermediate tier combine labor and inputs from the tier above using a Cobb-Douglas production function. Tier 0 firms produce from labor alone.

Every firm faces an independent, non-zero probability of a catastrophic disruption (complete inability to produce). Firms may invest labor up front to moderate this risk — endogenous “resilience” — or may invest to forge relationships with a larger fraction of potential suppliers in the next upstream tier — endogenous “network thickness.” Each formed relationship costs k units of labor.

After disruption shocks are realized, surviving firms negotiate quantities and payments bilaterally. Bargaining is sequential (beginning with lead firms negotiating with tier S-1, then tier S-1 with tier S-2, and so on to tier 0), and within each round is governed by Nash-in-Nash equilibrium (Horn and Wolinsky, 1988): each firm takes as given the outcomes of its negotiations with all other partners. The Nash surplus is split with exogenous bargaining weight β_s for the downstream buyer in the s-to-s−1 negotiation.

The paper solves the planner’s direct-control problem and then characterizes the three sets of policy instruments needed to decentralize the first best: subsidies to input transactions between adjacent tiers, subsidies to investments in resilience (agility), and subsidies to network formation (redundancy). It then solves the second-best problem in which transaction subsidies are constrained to zero.

Main Findings

Transaction subsidies. In the competitive bargaining equilibrium, each pair of firms undervalues input transactions because the upstream firm anticipates paying a marked-up price when it bargains with its own suppliers. This cascading distortion means the private marginal cost of producing a tier-s good exceeds the social marginal cost. The optimal first-best transaction subsidy on sales by tier s firms (τ*s) equals [γ_s + (1−γ_s)μ{s−1}]^{−1}, where γ_s is the labor share in tier s production and μ_{s−1} is the endogenous markup factor from bargaining at the s-to-s−1 interface. This subsidy depends only on production function parameters and bargaining weights at the immediately adjacent tier. No subsidy is needed at tier 0 (the most upstream tier), and no subsidy is applied to final-good sales. Under Assumption 1 — inputs become weakly less substitutable as goods proceed downstream — the optimal purchase subsidies rise monotonically as one moves downstream along the supply chain.

Resilience subsidies (first best). Two offsetting forces govern the optimal subsidy to resilience investments θ*s at intermediate tiers: (i) firms capture only the fraction (1−β{s+1}) of the joint surplus that their resilience creates for downstream customers, creating underinvestment; (ii) optimal transaction subsidies inflate private profitability, creating a countervailing overinvestment incentive. The net optimal first-best subsidy for intermediate-tier firms is θ*s = (1−β{s+1}) / τ*_s. This formula depends only on technological and bargaining parameters of tier s and the tier immediately adjacent; it does not depend on conditions elsewhere in the chain. When production parameters and bargaining weights are uniform across tiers, the first-best resilience subsidy is the same at every interior tier. If goods become strictly less substitutable downstream, the first-best subsidy for resilience declines monotonically as one moves downstream, and may turn into an optimal tax for middle tiers where the transaction subsidy is large enough to over-incentivize resilience investment. The first-best resilience subsidy always applies at both extreme ends of the chain.

Network formation subsidies (first best). Despite firms’ private incentive to manipulate their number of upstream suppliers to improve bargaining position, the net strategic effect of network formation in general equilibrium exactly cancels the off-equilibrium spillovers to non-partners. As a result, the optimal first-best policy toward network formation at every tier is identical to the optimal policy toward resilience investment.

Second-best policies. When transaction subsidies are unavailable, uncorrected markups downstream from tier s depress demand for tier-s output, reducing profitability and incentives to invest in resilience below the first-best level. Second-best optimal subsidies for resilience and network formation therefore reflect production function parameters and bargaining weights throughout the entire downstream supply chain, not just at the immediately adjacent tier. Specifically, when buyer bargaining weights are non-increasing along the chain (β_{s+1} ≤ β_s for all s), the second-best subsidy to resilience falls monotonically as one moves downstream. This is the opposite pattern from what might be inferred from the first-best analysis when transaction subsidies are available: with non-increasing bargaining weights, second-best subsidies are larger for upstream producers than for downstream producers.

Scope Conditions

Results are derived for a closed economy. Welfare is measured by the CES utility of the representative consumer over differentiated final goods. The sequential bargaining structure assumes contracts are written after disruption shocks are realized. Assumption 1 (σ_1 ≥ σ_2 ≥ … ≥ σ_S > ε, where σ_s is the elasticity of substitution between inputs at tier s and ε is the demand elasticity for final goods) is maintained for sharper monotonicity results on the structure of optimal subsidies across tiers.

Layer 2 — Q&A

Q1: What is the precise structure of the supply chain in the model, and why does the bargaining take place sequentially rather than simultaneously across all tiers?

A: The economy has S+1 tiers. Tier 0 firms use only labor; tier s firms (s = 1,…,S−1) use labor and a CES bundle of tier s−1 inputs with elasticity of substitution σ_s > 1; tier S firms produce final differentiated goods using labor and tier S−1 inputs under Cobb-Douglas technology. Sequential bargaining is imposed because the vast number of simultaneous negotiations across all tiers makes a grand coalition impractical. The timing is that lead firms (tier S) first negotiate input quantities and payments with their tier S−1 suppliers; those suppliers, now contractually obligated to their downstream customers, then negotiate with tier S−2, and so on up the chain until tier 1 firms contract with tier 0 suppliers.

Q2: How is the markup factor defined, and what parameters determine it?

A: The markup factor μ_s is the ratio of the payment per unit made by tier s+1 firms to the production cost of tier s firms. It equals μ_s = (1−β_{s+1}) · [σ_{s+1}/(σ_{s+1}−1)] + β_{s+1}, where β_{s+1} is the exogenous bargaining weight of the downstream (tier s+1) buyer. When the downstream firm has all bargaining power (β_{s+1} = 1), the markup equals unity (competitive outcome). When the upstream firm has all bargaining power (β_{s+1} = 0), the markup equals the standard monopoly markup σ_{s+1}/(σ_{s+1}−1). For intermediate bargaining weights, the markup is a weighted average. The markup enters the optimal transaction subsidy formula by inflating the private marginal cost of producing tier-s inputs above the social marginal cost.

Q3: Why are no subsidies needed for the most upstream (tier 0) transactions or for final-good sales?

A: For tier 0 transactions: when tier 0 and tier 1 firms bargain, the negotiations occur last sequentially and so do not affect any prior agreements. There are no downstream cascading markup effects — tier 0 firms produce from labor alone, so their private marginal cost equals their social marginal cost. The joint surplus maximization by the pair thus aligns with the planner’s objective, yielding τ*_0 = 1 (no intervention needed). For final-good sales: final producers do mark up above marginal cost under monopolistic competition, but all varieties are symmetric, so the markup affects all goods equally and does not distort relative consumption choices. Hence τ*_S = 1.

Q4: What are the two offsetting forces that determine the optimal first-best subsidy to resilience investments at an intermediate tier?

A: First, a firm in tier s captures only the fraction (1−β_{s+1}) of the joint surplus that its survival creates for its downstream customers (the rest is appropriated through bargaining by those customers), leading to underinvestment relative to the social optimum. Second, the optimal transaction subsidy τ*_s < 1 raises the private profitability of firms in tier s above its social value, because public finances bear part of the cost of their input purchases. This inflated private profitability encourages resilience investment beyond what the planner desires. The net optimal policy is θ*s = (1−β{s+1}) / τ*_s, which may be a subsidy (θ*_s < 1) or a tax (θ*_s > 1) depending on which force dominates.

Q5: Why does the first-best subsidy for resilience at an intermediate tier depend only on local parameters (at tier s and its immediate neighbors), even though resilience investments generate spillovers to firms throughout the network?

A: When optimal transaction subsidies are in place at all tiers, a firm’s value becomes independent of the joint surplus in sales that occur between firms in tiers other than its own. That is, the positive spillovers to all firms farther upstream and downstream in a firm’s own network are exactly offset by the negative spillovers to firms in rival networks (including rival firms in the same tier). What remains after this general-equilibrium cancellation is only the benefit to the firm’s immediate downstream customers and the wedge created by the transaction subsidy. This result implies that the formula θ*s = (1−β{s+1}) / τ*_s does not involve conditions at tiers other than s and s−1.

Q6: Why does the optimal policy for network formation (supplier link investment) equal the optimal policy for resilience investment, despite the fact that network formation also strategically improves a firm’s bargaining position?

A: Firms in intermediate tiers do have a private incentive to form additional supplier links specifically to improve their bargaining position vis-à-vis their upstream suppliers (by improving their outside options) and vis-à-vis their downstream customers (by the same mechanism). However, the authors show by comparing the firm’s first-order condition for link formation with the planner’s first-order condition that this strategic motivation exactly balances the offsetting general-equilibrium effects from rival firms doing the same. After this cancellation, the residual wedge between private and social incentives for network formation is identical to that for resilience investment. Hence #_s = θ_s for all tiers.

Q7: How do second-best policies differ from first-best policies in terms of both the magnitude of subsidies and the information required to set them?

A: In the first best, the subsidy for resilience at tier s depends only on the bargaining weight β_{s+1} and the markup factor μ_{s−1} — parameters relevant to tier s and its immediate neighbors. In the second best, when transaction subsidies are unavailable, the optimal resilience subsidy at tier s is θ†s = J^{−1} · [1 − (cumulative distortion of all downstream tiers)] · (1−β{s+1}), where J captures aggregate labor-market effects of all markups throughout the chain. This formula requires knowledge of production function parameters (labor shares γ_j, markups μ_j, elasticities σ_j) for every tier j downstream from s. The second-best subsidy may be larger or smaller than the first-best subsidy; it is more likely to exceed the first-best subsidy for upstream tiers, where the cumulative downstream distortions (uncorrected markups contracting demand) produce a larger shortfall in private profitability and hence a larger underinvestment in resilience.

Q8: Under what condition do second-best subsidies fall monotonically as one moves downstream, and how does this compare to the first-best pattern?

A: The ratio of second-best subsidies at adjacent tiers (θ†_{s−1} / θ†s) equals [(1−β_s) / (1−β{s+1})] · [τ*s]^{−1}, where τ*s is the first-best transaction subsidy. If buyer bargaining weights are non-increasing along the chain — β{s+1} ≤ β_s for all s — then (1−β_s) ≤ (1−β{s+1}) and, combined with τ*s ≤ 1, the second-best subsidy is larger upstream than downstream (θ†{s−1} ≥ θ†_s). This contrasts with the first-best policy: when parameters are uniform across tiers, first-best resilience subsidies are the same at every interior tier, while second-best subsidies are strictly larger upstream than downstream.

Q9: What role does Assumption 1 (elasticities of substitution non-increasing as goods move downstream) play in the results?

A: Assumption 1 (σ_1 ≥ σ_2 ≥ … ≥ σ_S > ε) ensures that the operating profit function ~v_s(η) is concave in a firm’s network size η, which in turn ensures interior solutions to the network formation problem. It also delivers sharper monotonicity results: under this assumption, if other production parameters and bargaining weights are similar across tiers, the optimal purchase subsidies rise monotonically downstream, and the optimal first-best resilience subsidies decline monotonically downstream (potentially turning into taxes at some interior tiers). The assumption reflects the realistic view that inputs become more differentiated and specialized as they approach the final consumer good.

Q10: What are the limitations the authors identify regarding their model, and what extensions do they suggest?

A: Three main limitations are identified. First, the model assumes bargaining occurs after disruption shocks are realized, ruling out contingent contracts. Pre-disruption bargaining with contingent payments could mitigate double-marginalization inefficiencies and help internalize resilience externalities, though complex network-wide contingent contracts would likely be needed for full efficiency even in the second-best environment. Second, the model assumes symmetric firms within each tier, so downstream firms cannot sort on upstream firms’ observable resilience levels; if observable differences existed, downstream firms could seek out more reliable partners, partially internalizing the resilience externality. Third, the model covers only a closed economy with idiosyncratic (uncorrelated) shocks. Extensions to global supply chains, correlated (geographic) shocks, cross-country differences in wages and technologies, and optimal cooperative versus unilateral policy are identified as important directions for future research.

Key Concepts

Resilience (agility): In the paper’s usage, a firm’s endogenous investment in reducing the probability of a catastrophic disruption to its own operations. A firm in tier s hires r_s units of labor up front, which raises its survival probability φ_s(r_s), with φ’_s > 0 and φ’’_s < 0. Resilience is a relationship-specific investment in the sense that its payoff is realized only conditional on the firm surviving and then trading with its downstream customers.

Network thickness (redundancy): The fraction η_s of firms in the next upstream tier with whom a firm in tier s forms a supply relationship prior to the disruption shock. Forming k units of labor per link creates a thicker network that hedges against supplier disruption, increases input variety (and thus CES productivity), and improves bargaining positions vis-à-vis both upstream suppliers and downstream customers. Distinct from resilience: resilience reduces the firm’s own probability of disruption; network thickness provides substitutability across suppliers should some fail.

Markup factor (μ_s): The ratio of the per-unit payment made by tier s+1 firms to the production cost of tier s firms, as determined by Nash bargaining. Specifically, μ_s = (1−β_{s+1}) · [σ_{s+1}/(σ_{s+1}−1)] + β_{s+1}. The markup distorts private marginal costs above social marginal costs, causing underinvestment in transactions between firms and, transitively, in resilience and network formation.

Nash-in-Nash equilibrium: The bargaining solution concept used in the paper (following Horn and Wolinsky, 1988). Each pair of firms negotiates as if all other bilateral negotiations involving either party proceed at their equilibrium outcomes, both on and off the equilibrium path. This is the appropriate equilibrium concept when grand coalitions across all firms and all tiers are impractical.

Sequential bargaining: The specific timing structure in which negotiations proceed from the most downstream tier (lead firms bargaining with tier S−1 suppliers) sequentially upstream until tier 1 firms bargain with tier 0 suppliers. Each tier of firms, at the time they bargain with their own suppliers, are already contractually obligated to deliver specified quantities to their downstream customers. This obligation anchors the downstream firm’s outside option in any given bilateral negotiation.

First-best transaction subsidy (τ_s):* The fraction of the cost of a tier-s input that, under the optimal policy, the downstream (tier s+1) buyer must pay. Equals [γ_s + (1−γ_s) · μ_{s−1}]^{−1} < 1 for all intermediate tiers, i.e., it is always a subsidy. Designed to align private marginal cost in the bilateral negotiation with the social marginal cost by offsetting the distortion introduced by anticipated markups on the upstream firm’s own inputs.

Second-best subsidy: The optimal policy toward resilience and network formation when subsidizing firm-to-firm transactions is infeasible (constrained to τ_s = 1 for all s). Unlike first-best subsidies — which depend only on local tier parameters — second-best subsidies depend on production function parameters and bargaining weights throughout the entire downstream supply chain due to the uncorrected cumulative markup distortions.