Transfers and Resilience in Economic Networks

Alejandro Montecinos-Pearce; Francisco Parro

doi:10.1515/bejte-2021-0152

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Transfers and Resilience in Economic Networks

Alejandro Montecinos-Pearce and Francisco Parro

Published/Copyright: June 26, 2023

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From the journal The B.E. Journal of Theoretical Economics Volume 24 Issue 1

Abstract

We study how decentralized transfers, originated by a selfish motive to preserve direct links, affect the resilience of a network against a shock. Well-off agents transfer resources to other agents in order to prevent the shock from reaching their neighbors. We show that the connectivity of a well-defined portion of the network, specifically, the propagation network, determines the resilience of the entire network. We also show how, although transfers are allocated to a subset of agents, the resilience effect of these transfers is amplified. Lastly, we show that the wealth distribution impacts resilience by determining the incentives of the agents to transfers resources.

Keywords: cascade effects; networks; resilience; transfers

JEL Classification: D20; D85; E32; Z13

Corresponding author: Alejandro Montecinos-Pearce, Universidad Adolfo Ibáñez, School of Business, and Centro de Estudios de Conflicto y Cohesión Social (COES), Santiago, Chile, E-mail: amontecinos@uai.cl

Appendix A: Existence of an Equilibrium: Discussion

This appendix studies the existence of an equilibrium transfer vector in three families of social structures. Specifically, for each of these general social structures, we argue that an equilibrium exists. The analysis performed in this section can be applied to study any conjectured equilibrium transfer vector.^[55] We begin by providing some definitions that we use in the discussion that follows.

The set of networks that can be sustained in (α, ϵ) by some transfer vector t ∈ R + P α , ϵ is X ̃ ( α , ϵ ) = g ∈ G ( N ) : g = H α , ϵ , t and t ∈ R + P α , ϵ . Let g ^∅ denote the empty network. Let c p ( α , ϵ , g ) = max ϵ p − Π p g , Y , 0 if η _p(g) ≠ ∅ and c _p(α, ϵ, g) = 0 if η _p(g) = ∅, for all p ∈ P ( α , ϵ ) and all g ∈ X ̃ ( α , ϵ ) . Then, c _p(α, ϵ, g) are the transfers to p needed to sustain g ∈ X ̃ ( α , ϵ ) . Let P ̃ g be the set of poor agents that must receive strictly positive transfers to sustain g ∈ X ̃ ( α , ϵ ) , and K ̃ g be the set of rich agents that have productive relations with poor agents in g ∈ X ̃ ( α , ϵ ) . Then, P ̃ ( g ) = { p ∈ P ( α , ϵ ) : c p ( α , ϵ , g ) > 0 } and K ̃ ( g ) = { k ∈ K ( α , ϵ ) : η k ( g ) ∩ P ( α , ϵ ) ≠ ∅ } , for g ∈ X ̃ ( α , ϵ ) . The total transfers needed to sustain g ∈ X ̃ ( α , ϵ ) are C ( g ) = ∑ p ∈ P ̃ g c p ( α , ϵ , g ) . In addition, B k ( g ) = ∑ p ∈ η k ( g ) ∩ P ( α , ϵ ) y k p is the value of the links of k with poor agents in g ∈ X ̃ ( α , ϵ ) . Finally, the amount of resources that are feasible to transfer for k in (α, ϵ) are

E k = ∑ k ′ ∈ η k ( g 0 ) y k k ′ + y k k − y ̲ − ϵ k , for k ′ ∈ K ( α , ϵ ) . Consider the following cases:

Suppose # K ̃ ( g ) = 1 for all g ∈ X ̃ ( α , ϵ ) − g ∅ . In this case, there are no strategic interactions between rich agents; only one rich agent has productive links with poor agents in each network of X ̃ ( α , ϵ ) − g ∅ . A solution to problem (1) exists for t _−k = 0. Then, an equilibrium transfer vector exists.
Suppose # K ̃ ( g ) ≥ 2 and E _k < C(g) for all k ∈ K ̃ g and all g ∈ X ̃ ( α , ϵ ) − g ∅ . In this case, there are strategic interactions between rich agents. However, it is not feasible for a single rich agent to sustain any of the networks in which the agent has some productive links. Pick a transfer vector such that t ^p = 0 for all p ∈ P ( α , ϵ ) . It is not feasible for any rich agent to deviate from that strategy. Then, an equilibrium transfer vector exists.
Suppose # K ̃ ( g ) ≥ 2 for all g ∈ X ̃ ( α , ϵ ) − g ∅ . In addition, suppose there is some k ∈ K ( α , ϵ ) such that B _k(g) − C(g) ≥ 0 and E _k ≥ C(g) for some g ∈ X ̃ ( α , ϵ ) . In this case, there are strategic interactions between rich agents. Furthermore, there is some rich agent for whom it is feasible to sustain a network of the set X ̃ ( α , ϵ ) that preserves productive links (weakly) valued above the cost of sustaining it. Pick any k ′ ∈ K ( α , ϵ ) and g ̄ ∈ X ̃ ( α , ϵ ) such that B k ′ ( g ̄ ) − C ( g ̄ ) ≥ 0 and B k ′ ( g ̄ ) − C ( g ̄ ) ≥ B k ′ ( g ) − C ( g ) for all g ∈ X ̃ ( α , ϵ ) . Let t k ′ p = c p ( α , ϵ , g ̄ ) for all p ∈ P ( α , ϵ ) and t _k = 0 for all k ≠ k ′ ∈ K ( α , ϵ ) . Then, by construction of the transfer vector, agent k′ has no incentive to deviate from that strategy. If other rich agents also have no incentive to deviate from the zero-transfer strategy, then an equilibrium transfer vector exists. Suppose that there is a rich agent who has incentives to deviate from this strategy by making positive transfers. These transfers are directed to poor agents p ∉ P ̃ g ̄ . Then, more links are preserved. Therefore, other rich agents have incentives not to deviate from the zero-transfer strategy or to make positive transfers in order to preserve a network with even more links. This type of reasoning and the fact that set X ̃ ( α , ϵ ) is finite and discrete implies that an equilibrium transfer vector exists.

Appendix B: Two-step Procedure to Solve the Problem (1)

We solve problem (1) in two-steps. First, we compute an efficient transfer vector to sustain a fixed network. Second, we solve for the profit maximizing ex post network subject to transfer efficiency. Lemma B2 formally states that a solution of this two step procedure is a solution to problem (1).

B.1 First Step

Let X ( α , ϵ , t − k ) = g ∈ G ( N ) : g = H α , ϵ , ϕ ( t k , t − k ) and t k ∈ R + P α , ϵ for t − k ∈ R + P α , ϵ . Then, for fixed (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , t − k ∈ R + P α , ϵ , and g ̄ ∈ X ( α , ϵ , t − k ) , each element of k’s set of efficient transfer vectors that sustains g ̄ solves

(B1) max t k ∈ R + P α , ϵ π k α , ϵ , ϕ ( t k , t − k ) = Π k H α , ϵ , ϕ ( t k , t − k ) , Y − ϵ k − ∑ p ∈ P α , ϵ t k p s . t . H α , ϵ , ϕ ( t k , t − k ) = g ̄

By the definition of H, for all g ̄ ∈ X ( α , ϵ , t − k ) there exists t ̄ k ∈ R + P α , ϵ such that H α , ϵ , ϕ ( t k , t − k ) = g ̄ implies t k p ∈ [ t ̄ k p , ∞ ) for each p ∈ P ( α , ϵ ) . Therefore, a solution to problem (A1) exists for every g ̄ ∈ X ( α , ϵ , t − k ) . This occurs because π _k is linear and strictly decreasing in each t k p ∈ [ t ̄ k p , ∞ ) . Let t ̃ k ( α , ϵ , t − k , g ̄ ) be a solution to problem (B1).^[56]

B.2 Second Step

Let X f ( α , ϵ , t − k ) = g ∈ X ( α , ϵ , t − k ) : t ̃ k ( α , ϵ , t − k , g ) ∈ T k ( α , ϵ ) for t − k ∈ R + P α , ϵ . Then, for fixed (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , and t − k ∈ R + P α , ϵ , the profit maximizing ex post network subject to transfer efficiency solves

(B2) max g ∈ X f α , ϵ , t − k π ̃ k α , ϵ , t − k , g = Π k g , Y − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p ( α , ϵ , t − k , g ) .

A solution to problem (B2) exists because the set X f ( α , ϵ , t − k ) is finite and there exists π ̃ k ∈ R for each g ∈ X f ( α , ϵ , t − k ) . A solution to problem (B2) is g*.

B.3 The Two Step Procedure Result

We start by stating and proving an auxiliary result (Lemma B1), which is then used to prove the main result of this appendix (Lemma B2). Let r p ( α , ϵ , t ̄ , g ) = max ϵ p − Π p g , Y − t ̄ , 0 . That is, r p ( α , ϵ , t ̄ , g ) are the resources that p needs to survive in g when she receives transfers t ̄ .

Lemma B1

Fix (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , t − k ∈ R + P α , ϵ , and g ̄ ∈ X ( α , ϵ , t − k ) . Suppose t ̃ k ( α , ϵ , t − k , g ̄ ) solves problem (A1) for k. Then,

t ̃ k p ( α , ϵ , t − k , g ̄ ) = 0 if η p g ̄ = ∅ r p α , ϵ , t − k p , g ̄ if η p g ̄ ≠ ∅

for all p ∈ P ( α , ϵ ) .

Proof

Fix (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , t − k ∈ R + P α , ϵ , and g ̄ ∈ X ( α , ϵ , t − k ) . Suppose t ̃ k ( α , ϵ , t − k , g ̄ ) solves problem (A1) for k. Then, H ( α , ϵ , ϕ ( t ̃ k ( α , ϵ , t − k , g ̄ ) , t − k ) ) = g ̄ . For fix g ′ ∈ X ( α , ϵ , t − k ) , the function Π k ( g ′ , Y ) − ϵ k − ∑ p ∈ P α , ϵ t k p and the function Π p ( g ′ , Y ) − ϵ p + t − k p + t k p are additively separable in t k p . Pick any p ∈ P ( α , ϵ ) and let t k ′ ∈ R + P α , ϵ be such that t k ′ l = t ̃ k l ( α , ϵ , t − k , g ̄ ) for all l ≠ p and t k ′ p = t k p such that t k p ∈ R + . First, we show that η p ( g ̄ ) = ∅ implies t ̃ k p ( α , ϵ , t − k , g ̄ ) = 0 . Then, we show that η p ( g ̄ ) ≠ ∅ implies t ̃ k p ( α , ϵ , t − k , g ̄ ) = r p α , ϵ , t − k p , g ̄ .

Suppose that η p g ̄ = ∅ . By definition, r p α , ϵ , t − k p , g ̄ ≥ 0 . Assume r p α , ϵ , t − k p , g ̄ = 0 . Then, the definition of r _p implies that Π p ( g ̄ , Y ) − ϵ p + t − k p + t k p ≥ 0 for all t k p ∈ [ 0 , ∞ ) . Therefore, the definition of H and the construction of t k ′ imply that H ( α , ϵ , ϕ t k ′ , t − k ) = g ̄ for all t k p ∈ 0 , ∞ . Hence, by the definition of π _l, π k ( α , ϵ , ϕ t k ′ , t − k ) is linear and strictly decreasing in t k p ∈ 0 , ∞ , which implies that t ̃ k p ( α , g ̄ , t − k ) = 0 . Now assume that r p α , ϵ , t − k p , g ̄ > 0 . Then, the definition of r _p implies that Π p ( g ̄ , Y ) − ϵ p + t − k p + t k p < 0 for all t k p ∈ 0 , r p α , ϵ , t − k p , g ̄ and that Π p ( g ̄ , Y ) − ϵ p + t − k p + t k p ≥ 0 for all t k p ∈ [ r p α , ϵ , t − k p , g ̄ , ∞ ) . Therefore, the definition of H implies that H ( α , ϵ , ϕ t k ′ , t − k ) = g ̂ for all t k p ∈ 0 , r p α , ϵ , t − k p , g ̄ and that H ( α , ϵ , ϕ t k ′ , t − k ) = g ̃ for all t k p ∈ r p α , ϵ , t − k p , g ̄ , ∞ . Hence, by the definition of π _l, π k ( α , ϵ , ϕ t k ′ , t − k ) is linear and strictly decreasing in t k p ∈ 0 , r p α , ϵ , t − k p , g ̄ and in t k p ∈ r p α , ϵ , t − k p , g ̄ , ∞ , which implies that t ̃ k p ( α , ϵ , t − k , g ̄ ) ∈ 0 , r p α , ϵ , t − k p , g ̄ . If η p ( g ̃ ) ≠ ∅ , then g ̃ ≠ g ̄ . Therefore, t k ′ does not solve problem (A1). Let a = t k ′ if t k ′ p = 0 , and let b = t k ′ if t k ′ p = r p α , ϵ , t − k p , g ̄ . Suppose η p ( g ̃ ) = ∅ . Then, by construction of t k ′ , g ̃ = g ̂ = g ̄ , which implies that π _k(α, ϵ, ϕ(a, t _−k)) > π _k(α, ϵ, ϕ(b, t _−k)). Therefore, a is the unique solution to problem (B1).

Finally, suppose that η p g ̄ ≠ ∅ . By definition r p α , ϵ , t − k p , g ̄ ≥ 0 . Suppose r p α , ϵ , t − k p , g ̄ = 0 . By construction t ̃ k p ( α , ϵ , t − k , g ̄ ) ≥ 0 . Therefore t ̃ k p ( α , ϵ , t − k , g ̄ ) ≥ r p α , ϵ , t − k p , g ̄ . Now, suppose r p α , ϵ , t − k p , g ̄ > 0 . By the definition of r _p, t k p < r p α , ϵ , t − k p , g ̄ implies Π p ( g ̄ , Y ) − ϵ p + t − k p + t k p < 0 , which contradicts η p g ̄ ≠ ∅ . Therefore, t ̃ k p ( α , ϵ , t − k , g ̄ ) ≥ r p α , ϵ , t − k p , g ̄ . Now we complete the proof by showing that t ̃ k p ( α , ϵ , t − k , g ̄ ) = r p α , ϵ , t − k p , g ̄ . The definitions of π _l and r _p imply that Π p ( g ̄ , Y ) − ϵ p + t − k p + t k p ≥ 0 for all t k p ∈ r p α , ϵ , t − k p , g ̄ , ∞ . Hence, the definition of H and the construction of t k ′ imply that H ( α , ϵ , ϕ t k ′ , t − k ) = g ̄ for all t k p ∈ r p α , ϵ , t − k p , g ̄ , ∞ . Therefore, the definition of π _l implies that π k ( α , ϵ , ϕ t k ′ , t − k ) is linear and strictly decreasing for t k p ∈ r p α , ϵ , t − k p , g ̄ , ∞ . Thus, by optimality t ̃ k p ( α , ϵ , t − k , g ̄ ) = r p α , ϵ , t − k p , g ̄ . □

Lemma B2

Fix (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , and t − k ∈ R + P α , ϵ , t ̃ k α , ϵ , t − k , g * solves problem (B1) and g* solves problem (B2) if, and only if, t k * = t ̃ k α , ϵ , t − k , g * solves problem (1).

Proof

Fix (α, ϵ) ∈ A, k ∈ K ( α , ϵ ) , and t − k ∈ R + P α , ϵ . First we prove the only if part. Let t ̃ k ( α , ϵ , t − k , g ̃ ) be a solution to problem (B1) for g ̄ = g ̃ . Let t k * = t ̃ k α , ϵ , t − k , g * . Then, H α , ϵ , ϕ t k * , t − k = g * . Suppose g* solves problem (B2). By construction, Π k H α , ϵ , ϕ t k * , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k * p = Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g * . Suppose t k * does not solves problem (1). Then, Π k H α , ϵ , ϕ t k ′ , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k ′ p > Π k H α , ϵ , ϕ t k * , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k * p for some t k ′ ∈ R + P α , ϵ . Let H α , ϵ , ϕ t k ′ , t − k = g ′ . Let t ̃ k α , ϵ , t − k , g ′ be the solution of problem (A1) for g ̄ = g ′ . Then, Π k ( g ′ , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g ′ ≥ Π k ( g ′ , Y ) − ϵ k − ∑ p ∈ P α , ϵ t k ′ p . Suppose g′ = g*. Then, Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g * > Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g * , which is a contradiction. Now, suppose that g′ ≠ g*. Then, Π k ( g ′ , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g ′ > Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g * , which contradicts that g* solves problem (B2). Therefore π k α , ϵ , ϕ t k * , t − k ≥ π k α , ϵ , ϕ t k ′ , t − k for all t k ′ ∈ T k α , ϵ .

Now we prove the if part. Suppose t k * is such that π k α , ϵ , ϕ t k * , t − k ≥ π k α , ϵ , ϕ t k ′ , t − k for all t k ′ ∈ T k α , ϵ , i.e. t k * solves problem (1). Suppose t k * does not solve problem (B1) for g ̄ = g * . Then, there exists t k ′ ∈ R + P α , ϵ such that H α , ϵ , ϕ t k ′ , t − k = g * and Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t k ′ p > Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t k * p . Therefore,

Π k H α , ϵ , ϕ t k ′ , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k ′ p > Π k H α , ϵ , ϕ t k * , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k * p , which contradicts π k α , ϵ , ϕ t k * , t − k ≥ π k α , ϵ , ϕ t k ′ , t − k for all t k ′ ∈ T k α , ϵ . Thus, t k * solves problem (B1) for g ̄ = g * , i.e. t k * = t ̃ k p α , ϵ , t − k , g * . Suppose g* does not solve problem (B2). Then, there exists g ′ ∈ X f ( α , ϵ , t − k ) such that Π k ( g ′ , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g ′ > Π k ( g * , Y ) − ϵ k − ∑ p ∈ P α , ϵ t ̃ k p α , ϵ , t − k , g * . Let t k ′ = t ̃ k p α , ϵ , t − k , g ′ and H α , ϵ , ϕ t k ′ , t − k = g ′ . Therefore, Π k H α , ϵ , ϕ t k ′ , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k ′ p > Π k α , ϵ , H ϕ t k * , t − k , Y − ϵ k − ∑ p ∈ P α , ϵ t k * p , which contradicts π k α , ϵ , ϕ t k * , t − k ≥ π k α , ϵ , ϕ t k ′ , t − k for all t k ′ ∈ T k α , ϵ . □

Algorithm

Now we sketch an algorithm to solve the two-step problem. First, we fix t − k ∈ R + P α , ϵ . Second, given t _−k, we construct the set of networks that can be sustained through agent k’s transfers, that is, the set X ( α , ϵ , t − k ) . To build this set, we use the algorithm described in Section 2.3. Specifically, we construct 2 P α , ϵ transfer vectors, each of which contains transfers of 0 or t ̄ for each poor agent. We set t ̄ arbitrarily large so that the poor agent receiving these transfers survives. Then, for each of these transfer vectors, we compute the sequence of layers until obtain H ( α , ϵ , t ̄ ) , that is, the ex post network under t ̄ . The set of ex post networks resulting from this process is X ( α , ϵ , t − k ) . Notice that the cardinality of this set is less than 2 P α , ϵ because some ex post networks result from different transfer vectors; e.g. transfers to all agents in the first layer, t ̄ ′ , and transfers to all poor agents, t ̄ ″ , imply H ( α , ϵ , t ̄ ′ ) = H ( α , ϵ , t ̄ ″ ) = g 0 . We store all networks of X ( α , ϵ , t − k ) in a set of adjacent matrices, A , which describes the links that are active/inactive under each network of the set X ( α , ϵ , t − k ) . Third, we compute c p α , ϵ , t − k p , g = max ϵ p − Π p g , Y − t − k p , 0 if η _p(g) ≠ ∅ and c p α , ϵ , t − k p , g = 0 if η _p(g) = ∅, for all p ∈ P ( α , ϵ ) and all g ∈ X ( α , ϵ , t − k ) . Then, c p α , ϵ , t − k p , g are the transfers to p needed to sustain g ∈ X ( α , ϵ , t − k ) . This computation is straightforward using the revenue matrix Y and the set of adjacent matrices A . Using these matrices, we can also compute

C k ( t − k , g ) = ∑ p ∈ P ( α , ϵ ) c p α , ϵ , t − k p , g and B k ( g ) = ∑ p ∈ η k ( g ) ∩ P ( α , ϵ ) y k p ; that is, the minimum amount of resources that k needs to transfer to sustain each g ∈ X ( α , ϵ , t − k ) , and the values of the links that k can preserve by sustaining g ∈ X ( α , ϵ , t − k ) . Fourth, we remove from the set X ( α , ϵ , t − k ) the networks for which C k ( t − k , g ) < ∑ k ′ ∈ η k ( g 0 ) y k k ′ + y k k − y ̲ − ϵ k , for k ′ ∈ K ( α , ϵ ) ; that is networks that are not feasible to sustain for k. The resulting set is X f ( α , ϵ , t − k ) . Fifth, we compute B _k(g) − C _k(t _−k, g) for all g ∈ X f ( α , ϵ , t − k ) and choose g* which maximizes this expression; the dimensionality of this problem is just # X f ( α , ϵ , t − k ) . Then, t k p = c p α , ϵ , t − k p , g * is a solution to problem (1).

Appendix C: Proofs

Lemma C1

Fix (α, ϵ) ∈ A. If p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ), then p∉R _f′(α, ϵ) for any f′ ∈ F(α, ϵ) such that f′ ≠ f.

Proof

Fix (α, ϵ) ∈ A. Pick any p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ). Suppose p ∈ R _f′(α, ϵ) for some f′ ∈ F(α, ϵ) such that f′ ≠ f. Then, the definition of R _f(α, ϵ) implies that Θ_pf(g ⁰) ≠ ∅ for f ∈ N g ̃ ( α , ϵ ) and that N θ ∩ N g ̃ ( α , ϵ ) = { f } for some θ ∈ Θ_pf(g ⁰). Analogously, the definition of R _f′(α, ϵ) implies and Θ_pf′(g ⁰) ≠ ∅ for f ′ ∈ N g ̃ ( α , ϵ ) and that N θ ∩ N g ̃ ( α , ϵ ) = { f ′ } for some θ ∈ Θ_pf′(g ⁰). Then, there exists p ̂ ∈ R f ( α , ϵ ) and p ̂ ∈ R f ′ ( α , ϵ ) such that p ̂ ∈ N θ and N θ ∩ N g ∼ ( α , ϵ ) = { f , f ′ } for some θ ∈ Θ_ff′(g ⁰). By definition, f ∈ F(α, ϵ) implies that Θ_jf(g ⁰) ≠ ∅ for some j ∈ S 1 ( α , ϵ ) and Θ_fk(g ⁰) ≠ ∅ for some k ∈ K ( α , ϵ ) . Moreover, f ∈ N g ̃ ( α , ϵ ) implies that N ^θ ∩ R(α, ϵ) = ∅ for some θ ∈ Θ_jf(g ⁰), and N ^θ′ ∩ R(α, ϵ) = ∅ for some θ′ ∈ Θ_fk(g ⁰). Analogously, f′ ∈ F(α, ϵ) implies that Θ_jf′(g ⁰) ≠ ∅ for some j ∈ S 1 ( α , ϵ ) and Θ_f′k(g ⁰) ≠ ∅ for some k ∈ K ( α , ϵ ) . Moreover, f ′ ∈ N g ̃ ( α , ϵ ) implies that N ^θ ∩ R(α, ϵ) = ∅ for some θ ∈ Θ_jf′(g ⁰), and N ^θ′ ∩ R(α, ϵ) = ∅ for some θ′ ∈ Θ_f′k(g ⁰). Hence, p ̂ ∈ N θ for some θ ∈ Θ_jk(g ⁰) such that j ∈ S ¹(α, ϵ) and k ∈ K ( α , ϵ ) . Therefore, p ̂ ∈ N g ̃ ( α , ϵ ) which contradicts p ̂ ∈ R f ( α , ϵ ) and p ̂ ∈ R f ′ ( α , ϵ ) . □

Lemma C2

Fix (α, ϵ) ∈ A. If p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ), then f ∈ S m ( α , ϵ ) implies that p ∈ S l ( α , ϵ ) such that l > m.

Proof

Fix (α, ϵ) ∈ A. Pick any p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ).

Let f ∈ S m ( α , ϵ ) . Suppose p ∈ S l ( α , ϵ ) such that l ≤ m. Then, there exists Θ_pj(g ⁰) ≠ ∅ such that j ∈ S 1 ( α , ϵ ) and f∉N ^θ for some θ ∈ Θ_pj(g ⁰). Then, there exists f′ ∈ F(α, ϵ) such that f′ ≠ f and p ∈ R _f′(α, ϵ), which contradicts Lemma C1. □

Proposition 1

A poor agent receives strictly positive transfers in equilibrium only if the agent is in the propagation network. That is, for a fixed (α, ϵ) ∈ A, if t k * p > 0 for some k ∈ K ( α , ϵ ) and some p ∈ P ( α , ϵ ) , then p ∈ N g ̃ ( α , ϵ ) .

Proof

Fix (α, ϵ) ∈ A. Let t k * p > 0 for some k ∈ K ( α , ϵ ) and some p ∈ P ( α , ϵ ) . The proof is trivial for the case R(α, ϵ) = ∅ since g ̃ ( α , ϵ ) = g 0 . Then, suppose R(α, ϵ) ≠ ∅. Suppose that t k * p > 0 for some p ∈ P ( α , ϵ ) such that Θ_pk(g ⁰) = ∅ for all k ∈ K ( α , ϵ ) . Let t′ be a transfer vector such that t′^p = 0 for all p ∈ P ( α , ϵ ) such that Θ_pk(g ⁰) = ∅ for all k ∈ K ( α , ϵ ) , and t ′ p = t * p otherwise.

Then, η k H ( α , ϵ , t ′ ) = η k H ( α , ϵ , t * ) for all k ∈ K ( α , ϵ ) . Therefore, π k α , ϵ , ϕ t k ′ , t − k * > π k α , ϵ , ϕ t k * , t − k * for some k ∈ K ( α , ϵ ) . Then, t* does not solve problem (1) for some k ∈ K ( α , ϵ ) . Suppose now that t k * p > 0 for some p ∈ P ( α , ϵ ) such that Θ_pk(g ⁰) ≠ ∅ for some k ∈ K ( α , ϵ ) . Suppose p ∉ N g ̃ ( α , ϵ ) . Then, Θ_pk(g ⁰) ≠ ∅ for some k ∈ K ( α , ϵ ) implies that p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ). Let now t′ be a transfer vector such that t′^p = 0 for all p ∈ ⋃ f ∈ F ( α , ϵ ) R f ( α , ϵ ) , and t ′ p = t * p otherwise. By Lemma C1, η _p(g ⁰) ∩ F(α, ϵ) = {f} or η _p(g ⁰) ∩ F(α, ϵ) = ∅ for all p ∈ R _f(α, ϵ). By Lemma C2, if p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ), then f ∈ S m ( α , ϵ ) implies that p ∈ S l ( α , ϵ ) such that l > m. Hence, π _f(α, ϵ, t′) = π _f(α, ϵ, t*) for all f ∈ F(α, ϵ). Therefore, π _p(α, ϵ, t′) = π _p(α, ϵ, t*) for all p ∈ P ( α , ϵ ) − R ( α , ϵ ) . Then, η k H ( α , ϵ , t ′ ) = η k H ( α , ϵ , t * ) for all k ∈ K ( α , ϵ ) . Therefore, π k α , ϵ , ϕ t k ′ , t − k * > π k α , ϵ , ϕ t k * , t − k * for some k ∈ K ( α , ϵ ) . Then, t* does not solve problem (1) for some k ∈ K ( α , ϵ ) . Hence, t* implies that t * p = 0 for all p ∈ P ( α , ϵ ) such that (i) Θ_pk(g ⁰) = ∅ for all k ∈ K ( α , ϵ ) or (ii) p ∈ R _f(α, ϵ) for some f ∈ F(α, ϵ). Therefore, t k * p > 0 for some k ∈ K ( α , ϵ ) and some p ∈ P ( α , ϵ ) implies that p ∈ N g ̃ ( α , ϵ ) . □

Proposition 2

The survival of a frontier agent implies the survival of its entire ramification. That is, for a fixed (α, ϵ) ∈ A, if t* is such that π f α , ϵ , t * ≥ 0 for f ∈ F(α, ϵ), then π p α , ϵ , t * ≥ 0 for all p ∈ R _f(α, ϵ).

Proof

Fix (α, ϵ) ∈ A. Pick any f ∈ F(α, ϵ). Without loss of generality, suppose f ∈ S m ( α , ϵ ) . Let t* be such that π f α , ϵ , t * ≥ 0 . Pick any p ∈ R _f(α, ϵ). By Lemma C2, p ∈ S l ( α , ϵ ) such that l > m. Suppose p ∈ S m + 1 ( α , ϵ ) . By definition, g m α , ϵ = N , G 0 − L ⋃ q ∈ { 1 , … , m } S q α , ϵ . Then, Π p ( g m − 1 α , ϵ , Y ) − ϵ p ≥ 0 and Π p ( g m α , ϵ , Y ) − ϵ p < 0 . Moreover, G g m − 1 α , ϵ − G g m α , ϵ = L ⋃ q ∈ { 1 , … , m } S q α , ϵ − L ⋃ q ∈ { 1 , … , m − 1 } S q α , ϵ . By the definition of R _f(α, ϵ) and Lemmas C1 and C2, η p ( g 0 ) ∩ S m ( α , ϵ ) = { f } for all p ∈ R f ( α , ϵ ) ∩ S m + 1 ( α , ϵ ) . Let g ̂ m α , ϵ = N , G 0 − L ⋃ q ∈ { 1 , … , m } S q α , ϵ − { f } . Then, Π p ( g m − 1 α , ϵ , Y ) − ϵ p = Π p ( g ̂ m α , ϵ , Y ) − ϵ p ≥ 0 for all p ∈ R f ( α , ϵ ) ∩ S m + 1 . Suppose now p ∈ S m + 2 ( α , ϵ ) . Then, Π p ( g m α , ϵ , Y ) − ϵ p ≥ 0 and Π p ( g m + 1 α , ϵ , Y ) − ϵ p < 0 . Moreover, G g m α , ϵ − G g m + 1 α , ϵ = L ⋃ q ∈ { 1 , … , m + 1 } S q α , ϵ − L ⋃ q ∈ { 1 , … , m } S q α , ϵ . The definition of R _f(α, ϵ) and lemmas C1 and C2 also imply that η p ( g 0 ) ∩ S m + 1 ( α , ϵ ) = R f ( α , ϵ ) ∩ S m + 1 ( α , ϵ ) for all p ∈ R f ( α , ϵ ) ∩ S m + 2 ( α , ϵ ) . Let g ̂ m + 1 α , ϵ = N , G 0 − L ⋃ q ∈ { 1 , … , m + 1 } S q α , ϵ − R f ( α , ϵ ) ∩ S m + 1 ( α , ϵ ) . Then, Π p ( g m α , ϵ , Y ) − ϵ p = Π p ( g ̂ m + 1 α , ϵ , Y ) − ϵ p ≥ 0 for all p ∈ R f ( α , ϵ ) ∩ S m + 2 . Following analogous steps for the cases p ∈ S l ( α , ϵ ) such that l > m + 2 we conclude that t* be such that π f α , ϵ , t * ≥ 0 implies that π p α , ϵ , t * ≥ 0 for all p ∈ R _f(α, ϵ). □

Proposition 3

For a fixed ϵ ∈ R n , in fully ramified social structure with positive equilibrium transfers (t* > 0), the lower bound of resilience of the network converges to 1 b as the number of agents in each ramification (m − 1) increases (m → ∞).

Proof

The proposition is trivially proved by taking the limit of q ̲ = m + K b m + K as m → ∞. □

Proposition 4

Suppose ( α , ϵ ) ∈ A ̂ such that Assumption 1 is satisfied, ϵ _i = 0 for all i ∉ S 1 ( α , ϵ ) , and I ≥ 2.There exist at least I distributions of wealth such that (i) they trigger the same level of resilience in the absence of transfers, and (ii) they trigger I different equilibrium levels of resilience in the presence of transfers. That is, for each m ∈ {1, … , I}, there exists w_m ∈ W(α, ϵ) such that q ( α w m , ϵ , 0 ) = K α , ϵ n and q α w m , ϵ , t m * = K α , ϵ + # S ̃ m ( α , ϵ ) + # R f , m ( α , ϵ ) n for some t m * .

Proof

Pick ( α , ϵ ) ∈ A ̂ such that Assumption 1 is satisfied, ϵ _i = 0 for all i ∉ S 1 ( α , ϵ ) , and I ≥ 2. Let r p ( α , ϵ , t ̄ , g ) = max { ϵ p − Π p g , Y − t ̄ , 0 } . By Assumption 1, there exist w m ′ ∈ W ( α , ϵ ) such that S i ( α , ϵ ) = S i α w m ′ , ϵ for all i ∈ {1, … , I} and r p α w m ′ , ϵ , 0 , g m − 1 = δ for all p ∈ S m α w m ′ , ϵ , some m ∈ {1, … , I}, and some δ > 0 arbitrarily close to zero. Hence, q ( α w m , ϵ , 0 ) = K α , ϵ n . Fix m ∈ {1, … , I}. Let t ̂ be such that t ̂ k p = δ # K ̂ p α w m ′ , ϵ for all p ∈ S m α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ and K ̂ p α w m ′ , ϵ ⊆ K α w m ′ , ϵ is such that there exists θ ∈ Θ_kp(g ⁰) and N θ ∩ K α w m ′ , ϵ = { k } for all k ∈ K ̂ p α w m ′ , ϵ , and t ̂ k p = 0 otherwise. Then, t ̂ p = δ for all p ∈ S m α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ . Let H ( α , ϵ , t ̂ ) = g ̂ . Then, by the definition of A ̂ and t ̂ , η k ( g ̂ ) = η k ( g 0 ) for all k ∈ K α w m ′ , ϵ . Moreover, the definition of δ implies that t ̂ k ∈ T k α w m ′ , ϵ for all k ∈ K α w m ′ , ϵ . Suppose t′ such that t k ′ p > t ̂ k p for some p ∈ S m α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ and some k ∈ K α w m ′ , ϵ , and t k ′ p = t ̂ k p otherwise. The definition of t ̂ implies that H α w m ′ , ϵ , t ′ = g ̂ . Hence, π k α w m ′ , ϵ , t ′ < π k α w m ′ , ϵ , t ̂ for some k ∈ K α w m ′ , ϵ .

Suppose now t′ such that t k ′ p < t ̂ k p for some p ∈ S m α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ and some k ∈ K α w m ′ , ϵ , and t k ′ p = t ̂ k p otherwise. The definitions of t ̂ and A ̂ implies that η k H ( α , ϵ , t ′ ) ⊂ η k ( g ̂ ) or t k ′ p < 0 for some k ∈ K α w m ′ , ϵ . Then, the definition of δ implies that π k α w m ′ , ϵ , t ′ < π k α w m ′ , ϵ , t ̂ or t k ′ ∉ T k α w m ′ , ϵ for some k ∈ K α w m ′ , ϵ . Lastly, suppose t′ such that t k ′ p > 0 for some p ∉ S m α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ and some k ∈ K α w m ′ , ϵ and t k ′ p = t ̂ k p otherwise. We have two cases. First, t k ′ p > 0 for p ∈ R α w m ′ , ϵ , which contradicts Proposition 1. Second, by the definitions of t ̂ and A ̂ , t k ′ p > 0 for p ∈ S l α w m ′ , ϵ ∩ g ̃ α w m ′ , ϵ and l ≠ m implies that H α w m ′ , ϵ , t ′ = g ̂ . Hence, π k α w m ′ , ϵ , t ′ < π k α w m ′ , ϵ , t ̂ for some k ∈ K α w m ′ , ϵ . Thus, t ̂ is an equilibrium of Γ. Moreover, the definition of A ̂ and the definition of layer imply that π p α w m ′ , ϵ , t ̂ ≥ 0 for all p ∈ S ̃ m ( α , ϵ ) , which is the set of all the agents in the propagation network in layers m through I.

Proposition 2 implies that π p α w m ′ , ϵ , t ̂ ≥ 0 for all p ∈ R _f,m(α, ϵ), which is the set of all the agents that belong to some ramification that stems from some frontier agent in layers m through I. Thus, q α w m ′ , ϵ , t m * = K α , ϵ + # S ̃ m ( α , ϵ ) + # R f , m ( α , ϵ ) n for t m * = t ̂ . Analogous steps can be followed to prove the result for any m ∈ {1, … , I}. □

References

Acemoglu, D., V. M. Carvalho, A. Ozdaglar, and A. Thabaz-Salehi. 2012. “The Network Origin of Aggregate Fluctuations.” Econometrica 85 (5): 1977–2016.Search in Google Scholar

Acemoglu, D., A. Ozdaglar, and A. Thabaz-Salehi. 2015. “Systemic Risk and Stability in Financial Networks.” The American Economic Review 105 (2): 564–608. https://doi.org/10.1257/aer.20130456.Search in Google Scholar

Acemoglu, D., A. Ozdaglar, and A. Thabaz-Salehi. 2017. “Microeconomic Origins of Macroeconomic Tail Risks.” The American Economic Review 107 (1): 54–108. https://doi.org/10.1257/aer.20151086.Search in Google Scholar

Acharya, V. V., and T. Yorulmazer. 2007. “Too Many to Fail–An Analysis of Time-Inconsistency in Bank Closure Policies.” Journal of Financial Intermediation 16 (1): 1–31. https://doi.org/10.1016/j.jfi.2006.06.001.Search in Google Scholar

Acharya, V. V., and T. Yorulmazer. 2008. “Cash-in-the-market Pricing and Optimal Resolution of Bank Failures.” Review of Financial Studies 21 (6): 2705–42. https://doi.org/10.1093/rfs/hhm078.Search in Google Scholar

Acharya, V. V., H. S. Shin, and T. Yorulmazer. 2011. “Crisis Resolution and Bank Liquidity.” Review of Financial Studies 24 (6): 1773–81. https://doi.org/10.1093/rfs/hhq073.Search in Google Scholar

Agca, S., V. Babich, J. R. Birge, and J. Wu. 2022. “Credit Shock Propagation along Supply Chains: Evidence from the CDS Market.” Management Science 68 (9): 6355–7064.10.1287/mnsc.2021.4174Search in Google Scholar

Aghion, P., P. Bolton, and S. Fries. 1999. “Optimal Design of Bank Bailouts: The Case of Transition Economies.” Journal of Institutional and Theoretical Economics 155 (1): 51–79.Search in Google Scholar

Allen, F., and D. Gale. 2000. “Financial Contagion.” Journal of Political Economy 108 (1): 1–33. https://doi.org/10.1086/262109.Search in Google Scholar

Ambrus, A., M. Mobius, and A. Szeidl. 2014. “Consumption Risk-Sharing in Social Networks.” The American Economic Review 104 (1): 149–82. https://doi.org/10.1257/aer.104.1.149.Search in Google Scholar

Bakshi, N., and S. Mohan. 2016. “Mitigating Disruption Cascades in Supply Networks.” Working Paper.Search in Google Scholar

Banerjee, T., and Z. Feinstein. 2021. “Price Mediated Contagion through Capital Ratio Requirements with VWAP Liquidation Prices.” European Journal of Operational Research 295 (3): 1147–60. https://doi.org/10.1016/j.ejor.2021.03.053.Search in Google Scholar

Baqaee, D. R. 2018. “Cascading Failures in Production Networks.” Econometrica 86 (5): 1819–38. https://doi.org/10.3982/ecta15280.Search in Google Scholar

Barrot, J.-N., and J. Sauvagnat. 2016. “Input Specificity and the Propagation of Idiosyncratic Shocks in Production Networks.” Quarterly Journal of Economics 131 (3): 1543–92. https://doi.org/10.1093/qje/qjw018.Search in Google Scholar

Becker, G. S. 1976. “Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology.” Journal of Economic Literature 14 (3): 817–26.Search in Google Scholar

Becker, G. S. 1981. “Altruism in the Family and Selfishness in the Market Place.” Economica 48 (189): 1–15. https://doi.org/10.2307/2552939.Search in Google Scholar

Becker, G. S., and N. Tomes. 1986. “Human Capital and the Rise and Fall of Families.” Journal of Labor Economics 4 (3): 1–39. https://doi.org/10.1086/298118.Search in Google Scholar

Becker, G. S., and R. J. Barro. 1988. “A Reformulation of the Economic Theory of Fertility.” Quarterly Journal of Economics 103 (1): 1–25. https://doi.org/10.2307/1882640.Search in Google Scholar

Berle, A. A., and G. C. Means. 1932. The Modern Corporation and Private Property. New York: Harcourt, Brace & World, Inc.Search in Google Scholar

Bernard, B., A. Capponi, and J. E. Stiglitz. 2022. “Bail-ins and Bail-Outs: Incentives, Connectivity, and Systemic Stability.” Journal of Political Economy 130 (7): 1805–59. https://doi.org/10.1086/719758.Search in Google Scholar

Bigio, S., and J. La’O. 2016. Financial Frictions in Production Networks. NBER Working Paper No. 22212.Search in Google Scholar

Birge, J. R., and J. Wu. 2014. “Supply Chain Network Structure and Firm Returns.” Working Paper.Search in Google Scholar

Boehm, C., A. Flaaen, and N. Pandalai-Nayar. 2019. “Input Linkages and the Transmission of Shocks: Firm-Level Evidence from the 2011 Tohoku Earthquake.” The Review of Economics and Statistics 101 (1): 60–75. https://doi.org/10.1162/rest_a_00750.Search in Google Scholar

Bourlés, R., Y. Bramoullé, and E. Perez-Richet. 2017. “Altruism in Networks.” Econometrica 85 (2): 675–89.10.3982/ECTA13533Search in Google Scholar

Breiger, R. L., and P. E. Pattinson. 1986. “Comulated Social Roles: The Duality of Persons and Their Algebras.” Social Networks 8: 215–56. https://doi.org/10.1016/0378-8733(86)90006-7.Search in Google Scholar

Cachon, G. P., T. Randall, and G. M. Schmidt. 2007. “In Search of the Bullwhip Effect.” Manufacturing & Service Operations Management 9 (4): 457–79. https://doi.org/10.1287/msom.1060.0149.Search in Google Scholar

Cantillo, M. 1998. “The Rise and Fall of Bank Control in the United States: 1890-1939.” The American Economic Review 88 (5): 1077–93.Search in Google Scholar

Capponi, A., P.-C. Chen, and D. Yao. 2016. “Liability Concentration and Systemic Losses in Financial Networks.” Operations Research 64 (5): 1121–34. https://doi.org/10.1287/opre.2015.1402.Search in Google Scholar

Carpenter, J., J. Holmes, and P. H. Matthews. 2008. “Charity Auctions: A Field Experiment.” Economic Journal 118: 92–113. https://doi.org/10.1111/j.1468-0297.2007.02105.x.Search in Google Scholar

Carvalho, V. M., M. Nirei, Y. U. Saito, and A. Tahbaz-Salehi. 2021. “Supply Chain Disruptions: Evidence from the Great East Japan Earthquake.” Quarterly Journal of Economics 136 (2): 1255–321. https://doi.org/10.1093/qje/qjaa044.Search in Google Scholar

Cordella, T., and E. Levy Yeyati. 2003. “Bank Bailouts: Moral Hazard vs. Value Effect.” Journal of Financial Intermediation 12 (4): 300–30. https://doi.org/10.1016/s1042-9573(03)00046-9.Search in Google Scholar

DellaVigna, S., J. A. List, and U. Malmendier. 2012. “Testing for Altruism and Social Pressure in Charitable Giving.” Quarterly Journal of Economics 127 (1): 1–56. https://doi.org/10.1093/qje/qjr050.Search in Google Scholar

Demsetz, H., and K. Lehn. 1985. “The Structure of Corporate Ownership: Causes and Consequences.” Journal of Political Economy 93 (6): 1155–77. https://doi.org/10.1086/261354.Search in Google Scholar

Dixit, A. 1983. “Vertical Integration in a Monopolistically Competitive Industry.” International Journal of Industrial Organization 1 (1): 63–78. https://doi.org/10.1016/0167-7187(83)90023-1.Search in Google Scholar

Eisenberg, L., and T. H. Noe. 2001. “Systemic Risk in Financial Systems.” Management Science 47 (2): 236–49. https://doi.org/10.1287/mnsc.47.2.236.9835.Search in Google Scholar

Elliot, M., B. Golub, and M. O. Jackson. 2014. “Financial Network and Contagion.” The American Economic Review 104 (10): 3115–53. https://doi.org/10.1257/aer.104.10.3115.Search in Google Scholar

Fama, E. F., and M. C. Jensen. 1983. “Separation of Ownership and Control.” The Journal of Law and Economics 26 (2): 301–25. https://doi.org/10.1086/467037.Search in Google Scholar

Franks, J., and C. Mayer. 1997. “Corporate Ownership and Control in the UK, Germany, and France.” The Journal of Applied Corporate Finance 9 (4): 30–45. https://doi.org/10.1111/j.1745-6622.1997.tb00622.x.Search in Google Scholar

Frey, B. S., and S. Meier. 2004. “Social Comparison and Pro-social Behavior: Testing ’Conditional Cooperation’ in a Field Experiment.” The American Economic Review 94 (5): 1717–22. https://doi.org/10.1257/0002828043052187.Search in Google Scholar

Gao, S. Y., D. Simchi-Levi, C.-P. Teo, and Z. Yan. 2019. “Disruption Risk Mitigation in Supply Chains: The Risk Exposure Index Revisited.” Management Science 67 (3): 831–52. https://doi.org/10.1287/opre.2018.1776.Search in Google Scholar

Glattfelder, J. B. 2010. “Ownership Networks and Corporate Control Mapping Economic Power in a Globalized World.” Doctoral Thesis.Search in Google Scholar

Glasserman, P., and H. Peyton Young. 2016. “Contagion in Financial Networks.” Journal of Economic Literature 54 (3): 779–831. https://doi.org/10.1257/jel.20151228.Search in Google Scholar

Grossman, S. J., and O. D. Hart. 1986. “The Cost and Benefits of Ownership: A. Theory of Vertical and Lateral Integration.” Journal of Political Economy 94 (4): 691–719. https://doi.org/10.1086/261404.Search in Google Scholar

Hart, O. D., and J. Moore. 1990. “Property Rights and the Nature of the Firm.” Journal of Political Economy 98 (6): 1119–58. https://doi.org/10.1086/261729.Search in Google Scholar

Hunneus, F. 2020. “Production Network Dynamic and the Propagation of Shocks.” Manuscript.Search in Google Scholar

Jackson, M. O. 2008. Social and Economic Networks. Princeton: Princeton University Press.Search in Google Scholar

Jain, N., K. Girotra, and S. Netessine. 2016. “Recovering from Disruptions: The Role of Sourcing Strategy.” Working Paper.10.2139/ssrn.2682522Search in Google Scholar

Jensen, M. C., and W. H. Meckling. 1976. “Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure.” Journal of Financial Economics 3 (4): 305–60. https://doi.org/10.1016/0304-405x(76)90026-x.Search in Google Scholar

Klein, B., R. Crawford, and A. Alchian. 1978. “Vertical Integration, Appropriable Rents and the Competitive Contracting Process.” The Journal of Law and Economics 21 (2): 297–326. https://doi.org/10.1086/466922.Search in Google Scholar

Kolm, S.-C. 2006. “Introduction to the Economics of Giving, Altruism and Reciprocity.” In Handbook of the Economics of Giving, Altruism and Reciprocity, Vol. 1, edited by S.-C. Kolm, and J. M. Ythier, 4–122. Amsterdam: North Holland.Search in Google Scholar

Kolm, S.-C., and J. Mercier Ythier. 2006. Handbook of the Economics of Giving, Altruism and Reciprocity. Amsterdam: North Holland.10.1016/S1574-0714(06)01001-3Search in Google Scholar

Meier, S. 2007. “Do Subsidies Increase Charitable Giving in the Long Run? Matching Donations in a Field Experiment.” Journal of the European Economic Association 5 (6): 1203–22. https://doi.org/10.1162/jeea.2007.5.6.1203.Search in Google Scholar

Osadchiy, N., V. L. Gaur, and S. Seshadri. 2016. “Systematic Risk in Supply Chain Networks.” Management Science 62 (6): 1755–77. https://doi.org/10.1287/mnsc.2015.2187.Search in Google Scholar

La Porta, R., F. Lopez-De-Silanes, and A. Shleifer. 1999. “Corporate Ownership Around the World.” The Journal of Finance 54 (2): 471–517. https://doi.org/10.1111/0022-1082.00115.Search in Google Scholar

Perotti, E. C., and J. Suarez. 2002. “Last Bank Standing: What Do I Gain if You Fail?” European Economic Review 46 (9): 1599–622. https://doi.org/10.1016/s0014-2921(02)00241-6.Search in Google Scholar

Rogers, L. C. G., and L. A. M. Veraart. 2013. “Failure and Rescue in an Interbank Network.” Management Science 59 (4): 882–98. https://doi.org/10.1287/mnsc.1120.1569.Search in Google Scholar

Shang, J., and R. Croson. 2009. “A Field Experiment in Charitable Contribution: The Impact of Social Information on the Voluntary Provision of Public Goods.” Economic Journal 199 (540): 1422–39. https://doi.org/10.1111/j.1468-0297.2009.02267.x.Search in Google Scholar

Shleifer, A., and R. W. Vishny. 1986. “Large Shareholders and Corporate Control.” Journal of Political Economy 94 (3): 461–88. https://doi.org/10.1086/261385.Search in Google Scholar

Simchi-Levi, D., H. Wang, and Y. Wei. 2018. “Increasing Supply Chain Robustness through Process Flexibility and Inventory.” Production and Operations Management 27 (8): 1476–91. https://doi.org/10.1111/poms.12887.Search in Google Scholar

Tomlin, B. 2006. “On the Value of Mitigation and Contingency Strategies for Managing Supply Chain Disruption Risks.” Management Science 52 (5): 639–57. https://doi.org/10.1287/mnsc.1060.0515.Search in Google Scholar

Vitali, S., J. B. Glattfelder, and S. Barttison. 2011. “The Network of Global Control.” PLoS One 6 (10): e25995. https://doi.org/10.1371/journal.pone.0025995.Search in Google Scholar

Vivier-Lirimont, S. 2006. “Contagion in Interbank Debt Networks.” Manuscript.Search in Google Scholar

Yang, Z., G. Aydin, V. Babich, and D. R. Beil. 2009. “Supply Disruptions, Asymmetric Information, and a Backup Production Option.” Management Science 55 (2): 192–209. https://doi.org/10.1287/mnsc.1080.0943.Search in Google Scholar

Zarghamee, H. S., K. D. Messer, J. R. Fooks, W. D. Schulze, S. Wu, and J. Yan. 2017. “Nudging Charitable Giving: Three Field Experiments.” Journal of Behavioral and Experimental Economics 66: 137–49. https://doi.org/10.1016/j.socec.2016.04.008.Search in Google Scholar

Zenou, Y. 2016. “Key Players.” In The Oxford Handbook of the Economics of Networks, edited by Y. Bramoullé, A. Galeotti, and B. W. Rogers, Chap. 11, 4–122. Oxford: Oxford University Press.10.1093/oxfordhb/9780199948277.013.14Search in Google Scholar

Received: 2021-11-17

Accepted: 2023-06-07

Published Online: 2023-06-26

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Articles in the same Issue

https://doi.org/10.1515/bejte-2021-0152

Keywords for this article

cascade effects; networks; resilience; transfers