# A note on uniqueness of clearing prices in financial systems

Abstract: The Eisenberg and Noe (2001) model of the financial system is general- ized to the case where default is solved by means of a bankruptcy rule. For regular financial networks a unique vector of clearing prices exists if only the bankruptcy rule is strongly monotonic. This shows uniqueness of the clear- ing prices on regular financial networks for the class of equal sacrifice rules by Young (1988), and many variations of the proportional rule as in Csoka and Herings (2018). This paper disentangles the role of network topology from the way defaults are solved.

## Summary (2 min read)

### 1 Introduction

- In the aftermath of the financial crisis in 2008 the delicate ways the players in the financial industries are intertwined is seen as the main source of the world wide spread of the shock caused by the subprime mortgage crises.
- The major lesson for the architecture of the financial network is that it cannot only be seen as a means by which institutions and firms may diversify their risk exposures, but that instead it may also be the main cause for the amplification of risk.
- More specifically, given such payment scheme, there are two types of nodes – the ones with a positive net value who will be able to pay all their liabilities and those with a negative net value that cannot.
- The analysis for other monotonic bankruptcy rules is similar as the induced games also show strategic complementarities.
- The question of uniqueness of clearing prices is also addressed by Csósak and Herings (2018), who present a discrete model that allows for decentralized clearing of the financial system.

### 2.1 Mathematical prerequisities

- Let Rn denote the n-dimensional Euclidean vector space.
- Special vector is the zero vector 0 with all zero coordinates.

### 2.2 Bankruptcy rules

- 1Here I chose to use the term bankruptcy problem, but in fact the rationing problems as in Moulin (2002) or taxation problems in Young (1988) are of the same mathematical structure.
- Well-known in the literature on taxation problems (see Young (1988), Lambert and Naughton (2009)) is the class of strictly monotonic rules which are referred to as equal sacrifice rules.
- Basically, strong monotonicity rules out a kind of exotic rules which are strictly monotonic and do allow for zero derivatives.

### 2.3 The economic model

- Where the connections or relations of agents within the network are shaped through the nominal liabilities an agent has to other agents in the system.the authors.
- Let τ ∈ Rn+ be the vector that summarizes the total nominal obligations of the agents in the system, i.e., for i ∈ N let τi := n∑ j=1 Lij. (4) This total obligation vector τ summarizes agent-wise the payment levels required to satisfy all the contractual liabilities in the network.
- The authors will assume that all liabilities have the same maturity date at which they become due and should be paid for.
- On the other hand, each agent i has some justified claim Lji on pj.
- This means that from payment pj by agent j, agent i obtains ri(Lj, pj).

### 2.4 Clearing payment vectors for a financial system

- Below the authors will focus on the question whether payment vectors exist, that see to a clearing of the financial system such that two minimal requirements are satisfied.
- First the authors will require from a payment vector that it expresses the idea of limited liability: no agent should pay more than the total of his cash inflow.
- This holds for the partially ordered set [0, τ ].
- For part (b) let p′ be any clearing vector.

### 3 Characterizing the clearing prices

- Eisenberg and Noe (2001) characterize vectors of clearing prices using the notion of a surplus set, i.e., a set of agents S with no external obligations and a positive aggregate operation cash flow: Definition 2 Then Lemma 1 shows O(i) should contain an agent with positive equity.
- If the set of defaulting agents is larger under pk+1 than under pk, then it must be that some agents pay their obligations in full under pk and default under pk+1, and the other agents either default or not both under pk+1 and pk.
- And this concludes their proof by induction as also the authors have shown that {pj} is a weakly decreasing sequence.

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##### References

2,743 citations

### "A note on uniqueness of clearing pr..." refers background in this paper

...Theorem 1 (Tarski (1955)) Let (A,≤) be any complete lattice(3) and suppose f : A → A is monotonically increasing, i....

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...Also ([0, τ ],≤) is a complete lattice, so that the implication of Tarski’s fixed point theorem (see Tarski (1955)) is that the set of fixed points of Φ is a complete lattice with respect to ≤....

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...(6) Theorem 1 (Tarski (1955)) Let (A,≤) be any complete lattice3 and suppose f : A → A is monotonically increasing, i.e., for all x, y ∈ A, x ≤ y implies f(x) ≤ f(y)....

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1,558 citations

1,099 citations

### "A note on uniqueness of clearing pr..." refers background or methods in this paper

...It is used by Eisenberg and Noe (2001) in order to define settlements after a firm defaults....

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...(12) This is the sequence that Eisenberg and Noe (2001) in the model with r = r refer to as the fictitious default sequence and the machinery producing the sequence is called the fictitious default algorithm....

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...In accordance with Eisenberg and Noe (2001) both aforementioned works stress the fact that clearing prices may not be unique – but the resulting allocation is....

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...The following definition is the Eisenberg and Noe (2001) version, only now for general bankrutpcy rules: Definition 1 A clearing payment vector for the financial system (L, e, r) is a vector p∗ ∈ [0, τ ] that satisfies (a) Limited Liability: p∗i ≤ n∑ j=1 ri(Lj, p ∗ j) + ei (b) Absolute Priority:…...

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...Eisenberg and Noe (2001) also propose an iterative procedure by which the clearing prices may be calculated, and in this process defaults may occur at different stages mimicking the indirect way financial institutions may be affected by earlier defaults....

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1,009 citations

### "A note on uniqueness of clearing pr..." refers background in this paper

...Other measures of financial instability and assessment of systemic risk are found in Elsinger et al. (2006), Acemoglu et al. (2015), Battiston et al. (2012)....

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...(2006), Acemoglu et al. (2015), Battiston et al. (2012). Crucial assumption in Eisenberg and Noe (2001) is the principle of proportionality; in case of a defaulting node, the corresponding clearing price is shared proportional to the liabilities of the node to the others....

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...(2006), Acemoglu et al. (2015), Battiston et al. (2012). Crucial assumption in Eisenberg and Noe (2001) is the principle of proportionality; in case of a defaulting node, the corresponding clearing price is shared proportional to the liabilities of the node to the others. Groote-Schaarsberg et al. (2018) and Csóka and Herings (2018) show in a continuous and discrete setting, respectively, that the assumption of proportionality in solving defaulting situa-...

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1,000 citations

### "A note on uniqueness of clearing pr..." refers background in this paper

...See for instance the overview of Glasserman and Young (2016) or Caccioli et al. (2018) which try to disentangle the problem by discussing various ways correlations between nodes in the financial system play a role....

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