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Systemic credit freezes in financial lending networks

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Abstract

This paper develops a network model of interbank lending, in which banks decide to extend credit to their potential borrowers. Borrowers are subject to shocks that may force them to default on their loans. In contrast to much of the previous literature on financial networks, we focus on how anticipation of future defaults may result in ex ante “credit freezes,” whereby banks refuse to extend credit to one another. We first characterize the terms of the interbank contracts and the patterns of interbank lending that emerge in equilibrium. We then study how shifts in the distribution of shocks can result in complex credit freezes that travel throughout the network. We use this framework to analyze the effects of various policy interventions on systemic credit freezes.

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Notes

  1. For example, see Acemoglu et al. [2], Cabrales et al. [17], Elliott et al. [25], Gai and Kapadia [29], Jorion and Zhang [34], and Allen and Gale [9].

  2. Brunnermeier [14] and Duffie [22] for general discussions.

  3. Also see Zawadowski [40], Farboodi [27], and Erol [26], who study how endogenous formation of financial networks can shape systemic risk.

  4. This stage is introduced to rule out equilibria that may arise due to coordination failures: banks may refuse to extend credit to others if they worry that no bank will subsequently extend them a credit line with sufficiently favorable terms. The withdrawal stage in the model rules out the possibility of such miscoordinations. See Di Maggio and Tahbaz-Salehi [20] for a discussion.

  5. This assumption thus rules out the possibility of “fractional defaults” as in Eisenberg and Noe [23] and Acemoglu et al. [2], whereby banks may only default on a fraction of their obligations to their creditors.

  6. This statement assumes that, given financial network \(({{\,\mathrm{{\mathbf {R}}}\,}},{{\,\mathrm{{\mathbf {x}}}\,}})\), the repayment equilibrium at \(t=2\) is unique for all realizations of \({{\,\mathrm{{\mathbf {z}}}\,}}\). We show in the Appendix that, for all \({{\,\mathrm{{\mathbf {z}}}\,}}\), the repayment equilibrium is indeed unique for any financial network emerging in equilibrium.

  7. This restriction is introduced in order to ensure that the best response of banks when offering interest rates to their potential borrowers converge to the equilibrium point in question as we take the limit of the trembles towards zero.

  8. See Appendix A for a formal definition of strong equilibrium and more details on its implications for equilibrium refinement. This concept is closely related to “trembling-hand perfect equilibrium” in extensive-form games.

  9. Because the standard Lebesgue measure is not well-defined over the space of continuous probability distributions, we use the notion of generic probability distribution from [39]. This notion is based on the use of “probes,” such as polynomial functions of order k as approximations to smooth probability distributions. Generic properties are those that hold for almost all order k polynomials. See Appendix C for more details.

  10. Throughout we refer to credit freezes in order to emphasize that following a change in the distribution of shocks \({\mathcal {Q}}\), the decision not to lend by some banks leads to stoppages in credit flows.

  11. This result only holds for economies with a single entrepreneur. As we show in Sect. 5, the impact of increased competition on lending is ambiguous when there are multiple entrepreneurs in the network.

  12. Note, however, that the chain subnetwork along which lending takes place is endogenously determined, as it depends on the structure of \({{\,\mathrm{{\mathbf {G}}}\,}}\) and the shock distribution \({\mathcal {Q}}\). Hence, limiting attention to arbitrary chain networks is not without loss of generality.

  13. Formally, there is always a strong equilibrium where either (i) $1 flows from the depositor to the entrepreneur or (ii) there is a systemic credit freeze. However, there may be other equivalent strong equilibria, for instance, where bank 1 offers a prohibitively large interest rate to bank 2, but with no flow of funds anywhere in the chain.

  14. Notice the contrast with Corollary 1: while the corollary considers the addition of a link to a network (with a given set of banks), this theorem considers adding a new bank to a chain network (which thus removes a link and adds two new links to the new bank).

  15. We restrict \(\zeta \) to be in (0, 1) so that no bank can fully absorb a counterparty loss. This assumption guarantees that any default cascade that begins at some agent j propagates upstream to all its direct and indirect lenders.

  16. See Chateauneuf et al. [18].

  17. These observations also imply that if we allow banks to choose the riskiness of their outside investments, limited liability may push them towards riskier assets, but with significant negative systemic implications.

  18. A high bankruptcy cost F encourages banks to diversify in order to avoid costly default. Our assumption that \(F < {\underline{F}}\) ensures that the lack of diversification as shocks become more correlated does not dominate the increase in expected profits from making the loans.

  19. Note that, in this proposition, we assume large values of F to control for risk attitudes, as in Proposition 2 (see the discussion in Sect. 4.2).

  20. Recall from Theorem 3 that while, in equilibrium, interbank borrowing and lending always occurs in a tree structure, the opportunity network \({\mathbf {G}}\) need not be a tree. We now separately consider the implications of a tree opportunity network.

  21. We require sufficiently large F for the same reason as in Propositions 2 and 5: bank \(\Omega \) with high risk-bearing capacity is more risk-averse to lending; this guarantees that there is no shift in risk attitudes by channeling funds through the additional intermediary.

  22. In the context of Example 1, inserting a risk-bearing bank resets the compensating interest rate differential between the borrower and lender back to 0. Hence, if a fraction of the banks have risk-bearing capacity, then these differentials do not grow unboundedly as the chain gets longer.

  23. Note that this effect is distinct from the one emphasized by Farboodi [27] . In Farboodi [27], core banks have higher-return but riskier projects, allowing peripheral banks to obtain intermediation rents using their own source of funds, which in turn creates inefficient levels of systemic risk. In our case, we obtain essentially the opposite result: voluntary intermediation comes from the fact that peripheral banks can insulate themselves and reduce potential default cascades by channeling funds through larger “safer” intermediaries who are unlikely to default.

  24. For discussions of optimal policies in models based on ex post contagion, see Bernard et al. [12] and Kanik [35].

  25. For simplicity, we are modeling this policy intervention as a direct liquidity injection or transfer. It is equivalent to a subsidized loan from the central bank. In particular, if the bank has to repay the central bank an amount \(r_i \epsilon _i\) (where \(r_i\) is the discount interest rate from the central bank) at time \(t=2\), provided that doing so does not put the bank in default, then all of our results apply identically.

  26. While in reality asset purchases do not target a single bank, we think of such a policy as targeting the distressed assets composing this bank’s balance sheet. For instance, the Fed’s policy to purchase mortgage-backed securities (MBS) during the crisis was in-part designed to target large dealer banks whose balance sheets comprised of sizable MBS positions.

  27. Providing the depositor with liquidity does not change her incentives for lending, so the central bank must condition these funds on their use for interbank lending. The policy is equivalent to one where the central bank acts as a “depositor” itself, and directly lends to banks connected to the depositor in \({\mathbf {G}}\) (but not others).

  28. This adverse shift corresponds to a leftward shift of the distribution function \({\mathcal {Q}}(z_j)\). The amount of the shift, \(\delta \), is the anticipated liquidity shock bank j now faces.

  29. This result is in the same spirit as Jackson and Pernoud [33], but relates to ex ante rescue policies (before the realization of liquidity shocks) to ensure lending markets continue to function when future solvencies are in-question.

  30. See https://www.bis.org/publ/bcbs283.pdf.

  31. Note that we cite Theorem 2 in the proof of Theorem 3 to show it is a directed tree, but we are leveraging only uniqueness of the interest rate and borrowing stages, and not the unique repayment equilibrium, which is the only time we use Theorem 3 in this proof.

  32. Note that these expectations depend on the offer order \({\mathcal {O}}\), but are simply integrals over realizations of liquidity shocks, as is the form in Ott and Yorke [39], given that banks are not indifferent between making multiple offers for \(R_{i \rightarrow j}\) when \(x_{i \rightarrow j} > 0\) as shown in the following paragraph.

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Funding

Multidisciplinary University Research Initiative (Grant No. W911NF-12-1-0509).

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Correspondence to James Siderius.

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We are grateful to Celso Brunetti, Sebastian Infante, Borghan Narajabad, and Carlos Ramirez for helpful suggestions.

Appendices

Appendix: Strong equilibrium

In this appendix, we provide a refinement of the economy’s (subgame perfect) equilibrium in Definition 3 by considering a variant of agent-form trembling-hand perfect equilibrium, according to which banks may tremble around the interest rates offered in equilibrium, with the set of trembles restricted to thick-tailed distributions.

To formalize this concept, let \(\epsilon _{m} = (\epsilon _{m,ij})_{(i,j)\in {\mathbf {G}}}\) denote a vector of random variables with distribution \(H_m\), where each \(\epsilon _{m,ij}\) is drawn independently from an atomless distribution with full support over \({\mathbb {R}}_+\) and cumulative distribution function \(H_{m,ij}\). We say the sequence \(\{H_m\}_{m=1}^\infty \) generates a sequence of thick-tailed trembles if (i) \(\lim _{m\rightarrow \infty }\epsilon _{m, ij}= 0\) almost surely for all i, (ii) \(\lim _{m\rightarrow \infty } (1-H_m(x))/H_m'(x) = 0\), and (iii) \(\lim _{m\rightarrow \infty } H''_m(x)/H_m'(x) < \infty \) for all \(x >0\).

Definition 9

Let \(\{H_m\}_{m=1}^n\) denote any sequence of distribution functions generating a sequence of thick-tailed trembles \(\{\epsilon _m\}_{m=1}^\infty \). A strong equilibrium is a collection of interest rate offers \(\bar{{\mathbf {R}}}\), borrowing decisions \({{\,\mathrm{{\mathbf {x}}}\,}}({{\,\mathrm{{\mathbf {R}}}\,}})\), and repayments \({\varvec{y}}({{\,\mathrm{{\mathbf {R}}}\,}},{{\,\mathrm{{\mathbf {x}}}\,}}, {\mathbf {z}})\), such that there exists a sequence \((\bar{\mathbf{R}}_m, {{\,\mathrm{{\mathbf {x}}}\,}}_m, {\mathbf {y}}_m)\) where (i) \((\bar{\mathbf{R}}_m, {{\,\mathrm{{\mathbf {x}}}\,}}_m, {\mathbf {y}}_m)\) is a subgame perfect equilibrium subject to the trembles \({\tilde{R}}_{m, ij} = \bar{R}_{m, ij} + \epsilon _{m, ij}\) for all m, and (ii) \(\lim _{m \rightarrow \infty } || (\bar{\mathbf{R}}_m, {{\,\mathrm{{\mathbf {x}}}\,}}_m, {\mathbf {y}}_m) - (\bar{\mathbf{R}}, {{\,\mathrm{{\mathbf {x}}}\,}}, {\varvec{y}})||_{\infty } = 0\).

Recall from our discussion in Sect. 2 that there may be multiple subgame perfect equilibria, as banks could play weakly dominated strategies as best responses. Allowing for trembles in the strong equilibrium then rules out such equilibria. To see the role of thick-tailed trembles, note that, in general, banks face a tradeoff whenever they offer a higher interest rate to a potential borrower. On the one hand, conditional on being the most competitive lender, a higher rate ensures a higher profit margin for the bank. On the other hand, the higher rate also increases the likelihood that the bank is undercut by any of its competitors. Fat-tailed trembles ensure that the latter effect always dominates the former. As a result, less competitive banks (i.e., those with higher borrowing costs themselves) elect to charge just enough of a premium to break-even in expectation (accounting for the the risk of lending).

Appendix: Proofs

1.1 Auxiliary Lemmas

Lemma 1

In every (strong) borrowing equilibrium:

$$\begin{aligned} \{i \in {\mathcal {N}}_{in}(j): x_{i \rightarrow j} > 0\} = \arg \min _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j} \end{aligned}$$

and \(\sum _{i \in {\mathcal {N}}_{in}(j)} x_{i \rightarrow j} = \sum _{k \in {\mathcal {N}}_{out}(j)} x_{j \rightarrow k}\).

Proof

We prove this by backward induction on j according to order. For the last bank j according to order \(\mathcal {L}\) to borrow, suppose that bank j borrows some amount \(x_{i \rightarrow j}^* > 0\) from a lender i with \(R_{i \rightarrow j} > \min _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j}\). Since there are no restrictions on borrowing, bank j could borrow \(x_{i \rightarrow j}^*\) from some bank \(i^* = \arg \min _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j}\) and increase its profit by \(x_{i \rightarrow j}^*(R_{i \rightarrow j} -R_{i^* \rightarrow j})\). This dominates borrowing from a more expensive lender, so this cannot be the case in equilibrium. Similarly, if \(\sum _{i \in {\mathcal {N}}_{in}(j)} x_{i \rightarrow j} < \sum _{k \in {\mathcal {N}}_{out}(j)} x_{j \rightarrow k}\), then bank j pays the prohibitive shortfall cost, whereas if \(\sum _{i \in {\mathcal {N}}_{in}(j)} x_{i \rightarrow j} +\delta = \sum _{k \in {\mathcal {N}}_{out}(j)} x_{j \rightarrow k}\) for some \(\delta > 0\), bank j loses \((R_{i^* \rightarrow j}-1)\delta \) whenever it does not default and nothing when it does. Since the former occurs with positive probability (see Lemma 2), doing such is not profitable.

Now consider some bank j borrowing at time \(\tau \) in \({\mathcal {L}}\). By the inductive hypothesis, it is clear that no bank borrowing after j conditions its borrowing decision on who bank j borrows from. Via the same logic as before, it is clear then that bank j borrows entirely from bank \(i = \arg \min _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j}\). Similarly, by the inductive hypothesis, the borrowing decisions of any banks \(k \in {\mathcal {N}}_{out}(j)\) are not affected by bank j’s borrowing decision, except possibly if both \(j \rightarrow k\) and \(k \rightarrow j\). Since in the perturbed game we have \(R_{j \rightarrow k} \ne R_{k \rightarrow j}\) almost surely, it cannot be that both \(j = \arg \min R_{j \rightarrow k}\) and \(k = \arg \min R_{k \rightarrow j}\), so either \(x_{j \rightarrow k} = 0\) always or k does not condition its borrowing on the decision of j. Therefore, just by the same reasoning as before, we must have \(\sum _{i \in {\mathcal {N}}_{in}(j)} x_{i \rightarrow j} = \sum _{k \in {\mathcal {N}}_{out}(j)} x_{j \rightarrow k}\) for bank j, completing the inductive step. \(\square \)

Lemma 2

If \({\mathcal {Q}}(\mathbf{z})\) is generic then for any \({\mathcal {K}} \subset {\mathcal {B}}\), the probability the set of banks \({\mathcal {K}}\) default and the set of banks \({\mathcal {B}} \backslash {\mathcal {K}}\) do not default is always positive and never equal to 1.

Proof

By Example 3.9 in Ott and Yorke [39], \({\mathcal {Q}}(z_i)\) must be unbounded for all \(z_i\). Since profits from interbank lending for bank j, \(\sum _{k \in {\mathcal {N}}_{out}(j)} y_{k \rightarrow j}\) are bounded above by \((n+|{\mathcal {E}}|) r^*\), for every bank j we know there exists probability \(p_j > 0\) such that \(z_j < (n+|{\mathcal {E}}|) r^*\), and so bank j defaults. By independence, the probability banks \({\mathcal {K}}\) default is at least \((\min p_j)^{|{\mathcal {K}}|}\). Similarly, the most bank j could owe (even without repayments) is \((n+|{\mathcal {E}}|)r^*\), and for every bank j we know there exists probability \(p_j > 0\) such that \(z_j > (n+|{\mathcal {E}}|) r^*\), and so bank j does not default. By independence, the probability some bank \(i \in {\mathcal {K}}\) does not default is \(p_i\), so the set of banks \({\mathcal {K}}\) do not default with probability at least \(p_i > 0\). \(\square \)

Lemma 3

In any single-entrepreneur network \(\mathbf{G}\), there is a systemic freeze if and only if there exists no path \(P = 0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_k \rightarrow E\) (where E is an entrepreneur) with interest rates \(\{R_{0 \rightarrow i_1}, R_{i_1 \rightarrow i_2}, \cdots , R_{i_k \rightarrow E}\} \equiv \mathbf{R}_P\) such that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all agents j on P, given \(R_{k \rightarrow \ell } = \varnothing \) for all \(k \rightarrow \ell \) not on P (where \({\mathbb {E}}\) is over the realizations of \(\mathbf{z}\)).

Informally, this condition says there is a systemic freeze (i.e., no interbank lending) if and only if we cannot construct a path from the depositor to the entrepreneur, such that all banks prefer to lend at these interest rates than not engage in interbank lending at all.

Proof

For the “if” direction, we prove the contrapositive: if there is no systemic freeze, then there must exist a path \(P = 0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_k \rightarrow E\) where the interest rates \(\mathbf{R}_P\) give us \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\). By Theorem 3, we know the financial network \(\mathbf{x}_*\) is an intermediation path P from the depositor to entrepreneur. Assume, however, this path has at least one bank j with \({\mathbb {E}}[\pi _j] < {\mathbb {E}}[(z_j)_+]\). By definition of the equilibrium, bank j is aware that no other bank in P acting later will withdraw its offer conditional on j not withdrawing, and moreover, all banks will borrow and lend so that P is the financial network. Therefore, bank j has a profitable one-shot deviation to withdraw its offer, contradicting that this is an equilibrium.

For the “only if” direction, suppose there exists a path \(P = 0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_k \rightarrow E\) where some interest rates \(\mathbf{R}_P\) give us \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all \(j \in P\). By means of contradiction, suppose there is a systemic freeze. Consider the last agent \(i^*\) to act in \({\mathcal {O}}\) on the path P. Conditional on \(R_{k \rightarrow \ell } = \varnothing \) for all \(k \rightarrow \ell \), and given the interest rates \(\mathbf{R}^*_P\) up until agent \(i^*\) (not necessarily equal to \(\mathbf{R}_P\)) such that agent \(i^*\) can offer some \(R_{i^*}\) and satisfy \({\mathbb {E}}[(z_{i^*})_+] \le {\mathbb {E}}[\pi _{i^*}]\) and \({\mathbb {E}}[(z_{j})_+] \le {\mathbb {E}}[\pi _{j}]\) for all banks j on P. Then it is a best-response for bank \(i^*\) to offer some \(R_{i^*}\), and no bank on P to withdraw (which does at least as well as \({\mathbb {E}}[(z_{i^*})_+]\)). By backward induction, we see that every bank i on this path P can offer some \(R_i\) such that \({\mathbb {E}}[(z_j)_+] \le {\mathbb {E}}[\pi _j]\) for all banks \(j \in P\), and that conditional on offering \(R_i\), bank j does (weakly) better than offering \(\varnothing \). Since these offers do affect those banks outside of P, it is still an equilibrium for these banks to offer \(\varnothing \). However, repeating this argument, we see that the first bank to offer according to \({\mathcal {O}}\) in P would prefer to offer to the next bank in P as opposed to not offer (i.e., offer \(\varnothing \)), and then not withdraw the contract. This contradicts the assumption that a systemic freeze was the equilibrium. \(\square \)

1.2 Proof of Theorem 1

To prove part (a), we construct a repayment equilibrium for every realization of \(\mathbf{z}\), iteratively (let \(\tau \) be the \(\tau \)th iteration). Let \({\mathcal {D}}_\tau \subset {\mathcal {B}} \cup {\mathcal {E}}\) be the set of entrepreneurs in default at iteration \(\tau \). At \(\tau = 0\), assume \({\mathcal {D}}_\tau ={\mathcal {B}} \cup {\mathcal {E}}\). At each \(\tau \ge 1\), for each bank j, if \(z_j + \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau -1}} R_{j \rightarrow k} x_{j \rightarrow k} \ge \sum _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j} x_{i \rightarrow j}\), then do not include j in \({\mathcal {D}}_{\tau }\), otherwise do. For entrepreneur k, at each \(\tau \ge 1\), if \(r^* \ge \sum _{j \in {\mathcal {N}}_{in}(k)} R_{j \rightarrow k} x_{j \rightarrow k}\), then do not include k in \({\mathcal {D}}_{\tau }\), otherwise do.

We prove this algorithm constructs a repayment equilibrium. First notice that if \(j \not \in {\mathcal {D}}_{\tau }\) then \(j \not \in {\mathcal {D}}_{\tau '}\) for all \(\tau ' \ge \tau \). This can be shown by induction: in the base case, the set of non-defaulting banks is empty, so this set can only increase. At the inductive step, we note that \(\sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau -1}} R_{j \rightarrow k} x_{j \rightarrow k} \ge \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau -2}} R_{j \rightarrow k} x_{j \rightarrow k}\), so each bank (or entrepreneur) will be able to meet its obligations in all \(\tau ' \ge \tau \) if it can at \(\tau \). Since \({\mathcal {D}}_{\tau }\) is a decreasing set, and there are finitely many banks, we are guaranteed this algorithm terminates at some \(\tau ^*\) with either \({\mathcal {D}}_{\tau ^*} = {\mathcal {D}}_{\tau ^*-1}\) or \({\mathcal {D}}_{\tau ^*} = \emptyset \). In the latter case, we know \({\mathcal {D}}_{\tau ^*+1} = {\mathcal {D}}_{\tau ^*}\) so it is without loss of generality to consider only the former. We claim this admits a repayment equilibrium. For each bank \(j \not \in {\mathcal {D}}_{\tau ^*}\) we have:

$$\begin{aligned} z_j + \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau ^*}} R_{j \rightarrow k} x_{j \rightarrow k} = z_j + \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau ^*-1}} R_{j \rightarrow k} x_{j \rightarrow k} \ge \sum _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j} x_{i \rightarrow j} \end{aligned}$$

and for each bank \(j \in {\mathcal {D}}_{\tau ^*}\) we have:

$$\begin{aligned} z_j + \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau ^*}} R_{j \rightarrow k} x_{j \rightarrow k} = z_j + \sum _{k \in {\mathcal {N}}_{out}(j) \backslash {\mathcal {D}}_{\tau ^*-1}} R_{j \rightarrow k} x_{j \rightarrow k} < \sum _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j} x_{i \rightarrow j} \end{aligned}$$

(and similar for entrepreneurs), which proves the claim.

For any \(\mathbf{R}\), and given the existence of a repayment equilibrium, the borrowing stage is a finite extensive-form game with perfect information [where the terminal nodes represent “random” payoffs, but where the banks maximize according to expected utility of Eq. 2]. By Zermelo’s theorem, there exists a pure strategy borrowing equilibrium that can always be derived through backward induction, which establishes part (b).

Taking the borrowing equilibrium as given, the weak (subgame) perfect equilibrium also exists in pure strategies by Zermelo’s theorem. In the strong equilibrium, for every perturbed game, after each offer, nature makes a move which perturbs the offer randomly (see Appendix A). To show this, we amend Zermelo’s theorem for every node in the game-tree. For agent \(j = {\mathcal {O}}(n+1)\) offering last, she simply chooses:

$$\begin{aligned} \mathbf{R}_j^* \in \arg \max _{\mathbf{R} \in {\mathbb {R}}^{|{\mathcal {N}}_{out}(j)|}} {\mathbb {E}}[\pi _j(\mathbf{R}) | {\mathcal {H}}_j] \end{aligned}$$

where utility \(\pi _j\) is given in Eq. 2, \({\mathcal {H}}_j\) is the entire history of offers, and expectation is over the interest rate trembles of agent j. For agent \(i = {\mathcal {O}}(t)\) offering at time t, she simply chooses:

$$\begin{aligned} \mathbf{R}_i^* \in \arg \max _{\mathbf{R} \in {\mathbb {R}}^{|{\mathcal {N}}_{out}(i)|}} {\mathbb {E}}[\pi _i(\mathbf{R},\mathbf{R}_{-i}) | {\mathcal {H}}_i ] \end{aligned}$$

where utility \(\pi _i\) is given in Eq. 2 taking the actions \(\mathbf{R}_{-i}\) of all future agents \(\{k: {\mathcal {O}}^{-1}(k) > {\mathcal {O}}^{-1}(i)\}\) as given by backward-induction, \(h_i\) is the history of offers for agents \(\{k: {\mathcal {O}}^{-1}(k) < {\mathcal {O}}^{-1}(i)\}\), and the expectation is over the interest rate trembles of agent i and all future offering agents \(\{k: {\mathcal {O}}^{-1}(k) \ge {\mathcal {O}}^{-1}(i)\}\). Both the interest rate offers and borrowing decisions are in pure strategies. Therefore, every perturbed game has a (subgame perfect) equilibrium in pure strategies. By the convergence and uniqueness of these equilibria in Theorem 2 with trembles given in Appendix A, we see there exists a strong equilibrium in pure strategies. \(\square \)

1.3 Proof of Theorem 2

By Theorem 3,Footnote 31 we know that the financial network \((\mathbf{R}_*, \mathbf{x}_*)\) is a directed tree. Let \({\mathcal {T}}\) be a strong topological order on this network. Then working from the agents closest to the depositor, we can solve for the unique repayment equilibrium via backward induction. In particular, for bank j at topological index \({\mathcal {T}}(j)\), we know that \(y_{j \rightarrow i} = 0\) for some bank \(i \in {\mathcal {N}}_{in}(j)\) if \(z_j + \sum _{k \in {\mathcal {N}}_{out}(j)} y_{k \rightarrow j} - \sum _{i \in {\mathcal {N}}_{in}(j)} R_{i \rightarrow j}x_{i \rightarrow j} < 0\), otherwise \(y_{j \rightarrow i} = R_{i \rightarrow j}x_{i \rightarrow j}\), where \(\sum _{k \in {\mathcal {N}}_{out}(j)} y_{k \rightarrow j}\) is known because \({\mathcal {T}}(k) > {\mathcal {T}}(j)\) for all \(k \in {\mathcal {N}}_{out}(j)\). Therefore, we can iteratively compute the repayment equilibrium for any \(\mathbf{z}\), which is uniquely determined.

For any set of interest rates \(\mathbf{R}\) in a perturbed game, we know that with probability 1 no two interest rates are identical, so borrowing takes the form given in Lemma 1 (i.e., a directed tree) . Let us consider the set \(\mathbf{X}_*\), the \(\limsup _{n \rightarrow \infty }\) of borrowing networks (i.e., the set of all equilibrium borrowing networks which appear infinitely often as \(n \rightarrow \infty \)). Such a set \(\mathbf{X}_*\) is necessarily non-empty. Suppose there are two distinct lending trees \(T, T'\) appearing in \(\mathbf{X}_*\). Consider some bank j that lies at the intersection of these trees but borrows from different lenders i and \(i'\) in T and \(T'\), respectively. By construction of \(\mathbf{X}_*\), as \({\varvec{\epsilon }}_{m} {\mathop {\rightarrow }\limits ^{a.s.}} 0\), bank i lends to bank \(j^*\) with positive probability and bank \(i'\) lends to bank \(j^*\) also with positive probability. It clearly cannot be the case that i and \(i'\) make positive profits as \({\varvec{\epsilon }}_m {\mathop {\rightarrow }\limits ^{a.s.}} 0\), given the (strong) equilibrium interest rates \(\mathbf{R}_* , \mathbf{R}'_*\), respectively. Otherwise, whichever bank makes positive profits can reduce its interest rate by an arbitrarily small amount, which as \({\varvec{\epsilon }}_m {\mathop {\rightarrow }\limits ^{a.s.}} 0\) would guarantee that it has the unique lowest interest rate and makes arbitrarily close to the same (positive) profits. Let \(\partial \pi _{mi}\) (resp. \(\partial \pi _{mi'}\)) denote the marginal profit of lending to bank \(j^*\) for bank i (resp. bank \(i'\)). Therefore, \(\lim _{m \rightarrow \infty } {\mathbb {E}}^{{\mathcal {Q}}}[\partial \pi _{mi'}'] = {\mathbb {E}}^{{\mathcal {Q}}}[\partial \pi _{mi}] = 0\) is a necessary condition.Footnote 32 For a generic \({\mathcal {Q}}\) and a generic tuple of interest rates \(R_{i\rightarrow j}, R'_{i' \rightarrow j}\) (holding others constant) this will not be satisfied (see Appendix C, Proposition 12(c)). Thus, the tuple of \(R_{i\rightarrow j}, R'_{i' \rightarrow j}\) where (marginal) profits for bank i and bank \(i'\) are zero, lie on a set of measure zero. Let us induct on the path from \(j^*\) to the depositor in both T and \(T'\). To do so, replace i with the unique lender of i in T (call it \(\ell \)) and replace \(i'\) with the unique lender of \(i'\) in \(T'\) (call it \(\ell '\)). If these agents are not distinct, then we can replace \(j^*\) from before with \(\ell ^* =\ell =\ell '\) and repeat the above argument. Otherwise, by the same reasoning as the above, it must be the case that \(\ell \) and \(\ell '\) do not make positive (marginal) profits when lending to i and \(i'\), respectively. We can repeat this argument as needed until we either: (i) reach the depositor or (ii) reach an intermediation chain from the depositor to some bank \(\beta ^*\) which is the same in both T and \(T'\). In the former case, for generic \({\mathcal {Q}}\) and a given risk-free rate \(r_0\), the only interest rates charged to banks in T and \(T'\) that allow both to be strong equilibria lie on a set of measure zero. By similar reasoning as before (using Ott and Yorke [39]), the expected profit of the depositor is generically larger either under T or \(T'\), and by charging an arbitrarily small difference in interest rate, can change the equilibrium to either T or \(T'\) with probability 1. Similarly, for the intermediation chain from the depositor to bank \(\beta ^*\) can be replaced by an “equivalent depositor” with a different risk-free rate \({\tilde{r}}_0\) of the outside technology. Therefore, both \({\mathcal {Q}}\) (on the rest of the network) and \({\tilde{r}}_0\) are generic, so the same argument applies. Finally, either \(\beta ^*\) or the depositor is better of deviating to a marginally different offer (which has an arbitrarily small impact on profits in either T or \(T'\)), but necessarily induces either T or \(T'\) to never be the borrowing network. This means in a strong equilibrium, there will be a unique lending network \(\mathbf{x}_*\) (i.e., the set \(\mathbf{X}_*\) is a singleton).

To show \(\mathbf{R}_*\) is essentially unique in the strong equilibrium, it is enough to prove that no bank is indifferent between offering any two interest rates whenever \(x_{i \rightarrow j} > 0\) (the result then follows from Zermelo’s theorem and that other offers do not affect payoffs). We do this by backward induction on the offer order \({\mathcal {O}}\). Consider some bank j who takes as given its interest rate offers and chooses \(\bar{\mathbf{R}}_j\). Agent j maximizes its (marginal) profit of lending to bank \(k \in {\mathcal {N}}_{out}(j)\), taking as given offers to banks \(k' \in {\mathcal {N}}_{out}(j) \backslash \{k\}\). Then, it chooses \(\bar{R}_{j \rightarrow k}^* \in \arg \max _{{\tilde{R}}_{j \rightarrow k}}{\mathbb {E}}[\partial \pi _{j \rightarrow k}({\tilde{R}}_{j \rightarrow k})]\). If \(x^*_{j \rightarrow k} > 0\) with positive probability (bounded away from zero) as \(\varvec{\epsilon }_m \rightarrow \mathbf{0}\) it is clear by Lemma 1 that \(\bar{R}_{j \rightarrow k}^* \rightarrow \min _{j'} R_{j' \rightarrow k}\) (where the min includes competing banks \(j'\) over k who do not immediately withdraw in the following stage). Otherwise, as we concluded before, bank j’s offer to bank k does not affect the essential uniqueness of \(\mathbf{R}_*\). For the inductive step, consider some other bank \(j'\) that offers, taking as given the history all interest rate offers, and all (relevant) future offers as known with certainty, given \(\mathbf{R}_j\) (by the inductive assumption, since no bank is indifferent when its offer is relevant). As before, agent j maximizes its (marginal) profit of lending to bank \(k \in {\mathcal {N}}_{out}(j)\), taking as given offers to banks \(k' \in {\mathcal {N}}_{out}(j) \backslash \{k\}\). If \(x^*_{j \rightarrow k} > 0\) with positive probability (bounded away from zero) as \(\varvec{\epsilon }_m \rightarrow \mathbf{0}\), then \(\bar{R}_{j \rightarrow k}^* \rightarrow R_{j\rightarrow k}^* = \arg \max _{R_{j \rightarrow k}}{\mathbb {E}}^{{\mathcal {Q}}}[\partial \pi _{j \rightarrow k}(R_{j \rightarrow k})]\), which is unique by genericity of \({\mathcal {Q}}\) (and uniqueness of future “relevant” offers), see Proposition 12(d) in Appendix C. Otherwise, bank j’s offer solves:

$$\begin{aligned} \bar{R}_{j \rightarrow k}^*&= \arg \max _{\bar{R}_{j \rightarrow k}} {\mathbb {E}}^{{\mathcal {Q}}}\left[ \partial \pi _{j \rightarrow k}\left( \bar{R}_{j \rightarrow k}+ \epsilon _{j \rightarrow k, m}\right) \right] \\&= \arg \max _{\bar{R}_{j \rightarrow k}} {\mathbb {E}}^{{\mathcal {Q}}}\left[ \partial \pi _{j \rightarrow k}\left( \bar{R}_{j \rightarrow k}+ \epsilon _{j \rightarrow k, m}\right) \Big | \bar{R}_{j \rightarrow k} + \epsilon _{j \rightarrow k, m} \right. \\&\left. \le \min _{j', j''} \{{\tilde{R}}_{j' \rightarrow k}, \bar{R}_{j'' \rightarrow k}\left( {\tilde{R}}_{j \rightarrow k}\right) +\epsilon _{j' \rightarrow k, m}\} \right] \\&\quad \cdot {\mathbb {P}}\left[ \bar{R}_{j \rightarrow k} + \epsilon _{j \rightarrow k, m} \le \min _{j', j''} \left\{ {\tilde{R}}_{j' \rightarrow k}, \bar{R}_{j'' \rightarrow k}\left( {\tilde{R}}_{j \rightarrow k}\right) +\epsilon _{j' \rightarrow k, m}\right\} \right] \end{aligned}$$

As \(\varvec{\epsilon }_m \rightarrow \mathbf{0}\), the above converges to:

$$\begin{aligned} \bar{R}_{j \rightarrow k}^*&= \arg \max _{\bar{R}_{j \rightarrow k}} {\mathbb {E}}^{{\mathcal {Q}}}\left[ \partial \pi _{j \rightarrow k}(\bar{R}_{j \rightarrow k}) \Big | \bar{R}_{j \rightarrow k} \le \min _{j', j''} \left\{ {\tilde{R}}_{j' \rightarrow k}, \bar{R}_{j'' \rightarrow k}({\tilde{R}}_{j \rightarrow k})\right\} \right] \\&\quad \cdot {\mathbb {P}}\left[ \bar{R}_{j \rightarrow k} \le \min _{j', j''} \left\{ {\tilde{R}}_{j' \rightarrow k}, \bar{R}_{j'' \rightarrow k}(\bar{R}_{j \rightarrow k})\right\} \right] \\&=\arg \max _{\bar{R}_{j \rightarrow k}} \int _{{\mathcal {Q}}} \partial \pi _{j \rightarrow k}(\bar{R}_{j \rightarrow k}) \prod _{j''} \left( 1-H_m\left( \bar{R}_{j'' \rightarrow k}(\bar{R}_{j \rightarrow k}) - \bar{R}_{j \rightarrow k}\right) \right) \, \, d{\mathcal {Q}} \\&\Longrightarrow \int _{{\mathcal {Q}}} \frac{\partial \, \, \partial \pi _{j \rightarrow k}}{\partial \bar{R}_{j \rightarrow k}} \prod _{j''} \left( 1-H_m(\bar{R}_{j'' \rightarrow k} - \bar{R}_{j \rightarrow k})\right) \\&\quad + \partial \pi _{j \rightarrow k}(\bar{R}_{j \rightarrow k})\sum _{j''} \left[ 1-\frac{\partial \bar{R}_{j'' \rightarrow k}}{\partial \bar{R}_{j \rightarrow k}} \right] H_m'\left( \bar{R}_{j'' \rightarrow k}(\bar{R}_{j \rightarrow k}) \right. \\&\quad \left. - \bar{R}_{j \rightarrow k}\right) \prod _{j''' \ne j''} \left( 1-H_m\left( R_{j''' \rightarrow k} - \bar{R}_{j \rightarrow k}\right) \right) \, \, d {\mathcal {Q}} = 0 \end{aligned}$$

By the assumption on \(H_m\) in Appendix A, it is clear that \({\mathbb {E}}^{{\mathcal {Q}}}[\partial \pi _{j \rightarrow k}(\bar{R}_{j \rightarrow k})]\rightarrow 0\) as \(m \rightarrow \infty \) in equilibrium if \(\partial \bar{R}_{j'' \rightarrow k}/\partial \bar{R}_{j \rightarrow k}\) remains (sufficiently) bounded away from 1. For this consider \(j''\)’s problem:

$$\begin{aligned} \bar{R}_{j'' \rightarrow k}^*({\tilde{R}}_{j' \rightarrow k})&= \arg \max _{\bar{R}_{j'' \rightarrow k}} {\mathbb {E}}^{{\mathcal {Q}}}\left[ \partial \pi _{j'' \rightarrow k}(\bar{R}_{j'' \rightarrow k}+ \epsilon _{j \rightarrow k, m})\Big | \bar{R}_{j'' \rightarrow k} + \epsilon _{j'' \rightarrow k, m} \le \min _{j'} {\tilde{R}}_{j', k}\right] \\&\quad \cdot {\mathbb {P}}\left[ \bar{R}_{j'' \rightarrow k} + \epsilon _{j'' \rightarrow k, m} \le \min _{j'} {\tilde{R}}_{j', k}\right] \\&=\arg \max _{\bar{R}_{j'' \rightarrow k}} \int _{{\mathcal {Q}}}\int _{-\infty }^{\min _{j'} {\tilde{R}}_{j'\rightarrow k}- \bar{R}_{j'' \rightarrow k}} \partial \pi _{j'' \rightarrow k}(\bar{R}_{j'' \rightarrow k}+ \alpha ) \, \, dH(\alpha ) \,\, d{\mathcal {Q}} \end{aligned}$$

By the fundamental theorem of calculus, our first-order condition reduces to:

$$\begin{aligned} \Longrightarrow&\int _{{\mathcal {Q}}} H'\left( \min _{j'} {\tilde{R}}_{j'\rightarrow k}- \bar{R}^*_{j'' \rightarrow k}\right) \left( \partial \pi _{j'' \rightarrow k}\left( \min _{j'} {\tilde{R}}_{j' \rightarrow k}\right) \right) = 0 \end{aligned}$$

By the implicit function theorem, we observe that:

$$\begin{aligned}&\int _{{\mathcal {Q}}} \Big [H''\left( \min _{j'} {\tilde{R}}_{j'\rightarrow k}- \bar{R}^*_{j'' \rightarrow k}\right) \left( \partial \pi _{j'' \rightarrow k}\left( \min _{j'} {\tilde{R}}_{j' \rightarrow k}\right) \right) \left( 1-\frac{\partial \bar{R}^*_{j'' \rightarrow k}}{\partial \min _{j'} {\tilde{R}}_{j'\rightarrow k}}\right) \\&\quad +H'\left( \min _{j'} {\tilde{R}}_{j'\rightarrow k}- \bar{R}^*_{j'' \rightarrow k}\right) \frac{\partial \, \, \partial \pi _{j'' \rightarrow k}}{\partial \min _{j'} {\tilde{R}}_{j' \rightarrow k}}\Big ] \, \, d{\mathcal {Q}}= 0 \end{aligned}$$

which implies \(\partial \bar{R}_{j'' \rightarrow k}/\partial \min _{j'} {\tilde{R}}_{j'\rightarrow k}\) is (sufficiently) bounded away from 1 as \(m \rightarrow \infty \), given that \(\lim _{m \rightarrow \infty } H_m''/H_m' < \infty \), as assumed in Appendix A.

Finally, as we saw before, this implies by genericity there is a unique offer \(\bar{R}_{j\rightarrow k}\) that gives bank j zero (expected) profits, via the inductive step and given the history of offers. Therefore, the interest rates \(\mathbf{R}\) are unique in the strong equilibrium. \(\square \)

1.4 Proof of Theorem 3

(i): Clearly if \(x_{i \rightarrow j} > 0\) in the borrowing equilibrium, then \(R_{i \rightarrow j} \ne \varnothing \). Otherwise if \(x_{i \rightarrow j} = 0\) and \(R_{i \rightarrow j} = R^* \ne \varnothing \) (otherwise, we are done), consider the withdrawal decision of bank i in the offer stage. Because the remaining subgame is perfect information, bank i’s information set assigns probability 1 to bank i choosing \(x_{i \rightarrow j} = 0\) in the borrowing stage. This means bank i is indifferent to offering \(R^*\) and offering \(R_{i \rightarrow j} = \varnothing \) (i.e., withdrawing). Moreover, by Lemma 1 this deviation does not affect the future withdrawal decisions of banks \(k \ne i\) or the borrowing decisions of banks \(k \ne j\). By induction, it can therefore be established there exists a strong equilibrium where \(R_{i \rightarrow j} = \varnothing \) if and only if \(x_{i \rightarrow j} = 0\) (which is the contrapositive of the statement).

(ii): We first claim the financial network cannot contain any directed cycles. Suppose to the contrary we have a cycle of banks \(i_0 \rightarrow i_2 \rightarrow i_k \rightarrow i_0\) such that \(x_{i_{\alpha } \rightarrow i_{\alpha +1}} > 0\) (with mod k). Take \({\underline{x}} = \min _{\alpha } x_{i_{\alpha } \rightarrow i_{\alpha +1}} > 0\). Consider the case where \(R_{i_{\alpha } \rightarrow i_{\alpha +1}} \ge R_{i_{\alpha +1} \rightarrow i_{\alpha +2}}\) for some \(\alpha \). Then after observing all interest rate offers, bank \(i_{\alpha +1}\)’s decision to not withdraw the offer to \(i_{\alpha +2}\) is dominated by withdrawing. If bank \(i_{\alpha +1}\) withdraws, by Lemma 1 it can borrow \(x_{i_{\alpha } \rightarrow i_{\alpha +1}} > {\underline{x}}\) less from its lenders and lend less to bank \(i_{\alpha +1}\) by the same amount. In the event that bank \(i_{\alpha +1}\) is insolvent, both give the same payoff; in the event that bank \(i_{\alpha +1}\) is solvent, bank \(i_{\alpha +1}\) gets at least as much payoff when bank \(i_{\alpha +2}\) is solvent (and strictly more when \(R_{i_{\alpha } \rightarrow i_{\alpha +1}} > R_{i_{\alpha +1} \rightarrow i_{\alpha +2}}\)), and gains at least \(R_{i_{\alpha } \rightarrow i_{\alpha +1}} \cdot {\underline{x}} > 0\) when bank \(i_{\alpha +2}\) is insolvent. Since the latter event occurs with positive probability by Lemma 2, withdrawing dominates not withdrawing bank \(i_{\alpha +2}\)’s offer, so in equilibrium we must have \(R_{i_{\alpha } \rightarrow i_{\alpha +1}} < R_{i_{\alpha +1} \rightarrow i_{\alpha +2}}\) for all \(\alpha \). But because this is a cycle starting and ending at the same bank \(i_0\), this cannot be.

Now by definition of strong equilibrium (Appendix A), for any perturbed game, no distinct interest rate offers are identical with probability 1. By Lemma 1, the borrowing equilibrium consists of every bank and entrepreneur borrowing from its cheapest lender. Therefore, every bank borrows from at most one other bank, which implies \(\mathbf{x}_*\) (and by part (a), \(\mathbf{R}_*\) as well), is a directed tree. By Theorem 2, the (unique) financial network of the strong equilibrium (which is the limit of perturbed games) must also be a directed tree. \(\square \)

1.5 Proof of Theorem 4

In a single-entrepreneur network, this is a direct consequence of Lemma 3, since the existence of a systemic freeze depends only the risk profile \({\mathcal {Q}}\) and the network \(\mathbf{G}\) and not the order of actions \(({\mathcal {O}}, {\mathcal {L}})\). For multiple entrepreneurs, identical reasoning as Lemma 3 can be extended to the case of trees, which are guaranteed to be the structure of the financial network in Theorem 3, except where we replace the profitable path P in Lemma 3 with profitable tree T. \(\square \)

1.6 Proof of Proposition 1

First, we show that if the entrepreneur has a credit freeze in \(\mathbf{G}\), then it has a credit freeze in every chain subnetwork \(\mathbf{H} \subset \mathbf{G}\). We prove the contrapositive: if there is lending to the entrepreneur in some chain \(\mathbf{H}\), then there must be lending in \(\mathbf{G}\). By Lemma 3, we know there exist some interest rates \(\mathbf{R}_P\) along the path \(P = \mathbf{H}\) with \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all j on this path. In \(\mathbf{G}\), because \(P \subset \mathbf{G}\), the same set of interest rates \(\mathbf{R}_P\) along P does not change \({\mathbb {E}}[\pi _j]\) (because \(R_{k \rightarrow \ell } = \varnothing \) for all \((k \rightarrow \ell ) \in \mathbf{G}\backslash \mathbf{H}\)). Therefore, applying Lemma 3 again, we see there is no systemic credit freeze in \(\mathbf{G}\), so the sole entrepreneur does not experience a credit freeze in \(\mathbf{G}\).

Next, we show if the entrepreneur has no credit freeze in \(\mathbf{G}\), then there exists some chain subnetwork \(\mathbf{H} \subset \mathbf{G}\) where the entrepreneur does not experience a credit freeze. Consider the path P guaranteed by Lemma 3 such that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for some set of interest rates along this path. Taking \(\mathbf{H} = P\), we note that these inequalities still hold in \(\mathbf{H}\) for the same set of interest rates (neither \({\mathbb {E}}[\pi _j]\) nor \({\mathbb {E}}[(z_j)_+]\) change), so by Lemma 3, there is no credit freeze in \(\mathbf{H}\). \(\square \)

1.7 Proof of Corollary 1

By Proposition 1, if the entrepreneur experiences a credit freeze in \(\bar{\mathbf{G}}\), then it experiences a credit freeze for every chain subnetwork. Since every chain subnetwork in \(\underline{\mathbf{G}}\) is present in \(\bar{\mathbf{G}}\) because \(\underline{\mathbf{G}} \subset \bar{\mathbf{G}}\), there is a credit freeze for every chain subnetwork of \(\underline{\mathbf{G}}\), so once again by Proposition 1, there is a (systemic) credit freeze in \(\underline{\mathbf{G}}\). \(\square \)

1.8 Proof of Theorem 5

For (a), we show that if \(\mathbf{G}\) has no credit freeze with \(r_0\), then it has no credit freeze in \(\mathbf{G}'\) with \(1 \le r'_0 \le r_0\). Again, by Lemma 3 we have interest rates \(R_{0 \rightarrow 1}, R_{1 \rightarrow 2}, \ldots , R_{n \rightarrow (n+1)}\) in \(\mathbf{G}\) such that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all \(j \in \{0, \ldots , n+1\}\). If we consider this same set of interest rates in \(\mathbf{G}'\), then it is clear that \({\mathbb {E}}[\pi '_j] = {\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+] = {\mathbb {E}}[(z_j)_+]\) for all \(j \in \{1, \ldots , n\}\). Then:

$$\begin{aligned} 0 = {\mathbb {E}}[(z_0)_+] = {\mathbb {E}}[(z_0)_+] \le {\mathbb {E}}[\pi _0]&= {\mathbb {E}}\left[ (z_0 + y_{1 \rightarrow 0} - r_0)_+ \right] \\&\le {\mathbb {E}}\left[ (z_0 + y_{1 \rightarrow 0} - r'_0)_+ \right] = {\mathbb {E}}[\pi _0'] \end{aligned}$$

By Lemma 3, there is no credit freeze in \(\mathbf{G}'\) with \(r_0' \le r_0\). Of course, setting \(r_0 = r^*\) leads to a credit freeze, so therefore there exists some \(\bar{r}_0\) where \(r_0 > \bar{r}_0\) leads to credit freeze. Finally, to note that \(\bar{r}_0 < r^*\), by Lemma 2 bank 1 defaults and bank 2 survives with positive probability, so bank 2 must make positive rents.

For (b), we show that if \(\mathbf{G}\) has no credit freeze with \(r^*\), then there is no credit freeze in \(\mathbf{G}'\) with \(r^{*'} \ge r^*\). We utilize Lemma 3 again; we have interest rates \(R_{0 \rightarrow 1}, R_{1 \rightarrow 2}, \ldots , R_{n \rightarrow (n+1)}\) in \(\mathbf{G}\) such that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all \(j \in \{0, \ldots , n+1\}\). Because \(R_{n \rightarrow (n+1)} \le r^*\) in equilibrium, we know that \(R_{n \rightarrow (n+1)} \le r^{*'}\), so the entrepreneur is still solvent with probability 1 and has \({\mathbb {E}}[\pi _{n+1}] = r^{*'} - R_{n \rightarrow (n+1)} \ge 0 = {\mathbb {E}}[(z_{n+1})_+]\). Therefore, it is easy to see \({\mathbb {E}}[\pi '_j] = {\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+] = {\mathbb {E}}[(z_j)_+']\) for all \(j \in \{0, \ldots , n+1\}\). By Lemma 3, there is no credit freeze in \(\mathbf{G}'\) with \(r^{*'} \ge r^*\). For the same reason as (a), it is clear that \({\underline{r}}^* > r_0\), as bank 2 must make positive rents from lending to bank 1.

For part (c), consider a chain of length n with no credit freeze. We first show that the chain of length \(n-1\) will also not experience a credit freeze. By Lemma 3, in the n-bank chain \(\mathbf{G}\), there exist interest rates \(R_{0 \rightarrow 1}, R_{1 \rightarrow 2}, \ldots , R_{n \rightarrow (n+1)}\) such that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for all \(j \in \{0, \ldots , n+1\}\). In the \((n-1)\)-bank chain \(\mathbf{G}'\), let us consider the same set of interest rates \(R_{0 \rightarrow 1}, R_{1 \rightarrow 2}, \ldots , R_{(n-1) \rightarrow n}\) (less the final offer, which does not exist in the shorter chain). We first claim \(y'_{j \rightarrow j-1}\) (under \(\mathbf{G}'\)) FOSD \(y_{j \rightarrow j-1}\) (under \(\mathbf{G}\)) for all \(j \in \{1, \ldots , n+1\}\) for these interest rates. It is sufficient to show probability of repayment in \(\mathbf{G}'\) exceeds that in \(\mathbf{G}\). We prove this by induction. Since the entrepreneur repays with probability 1 when \(R_{(n-1) \rightarrow n} \le r^*\), the probability bank \(n-1\) repays to bank \(n-2\) is:

$$\begin{aligned} {\mathbb {P}}[z_{n-1} + R_{(n-1) \rightarrow n} \ge R_{(n-2)\rightarrow (n-1)}] \ge {\mathbb {P}}[z_{n-1} + y_{n \rightarrow (n-1)} \ge R_{(n-2) \rightarrow (n-1)}] \end{aligned}$$

Suppose that \(y'_{j \rightarrow j-1}\) FOSD \(y_{j \rightarrow j-1}\). Then:

$$\begin{aligned} {\mathbb {P}}[z_{j-1} + y'_{j \rightarrow j-1} \ge R_{(j-2)\rightarrow (j-1)}] \ge {\mathbb {P}}[z_{j-1} + y_{j \rightarrow j-1} \ge R_{(j-2)\rightarrow (j-1)}] \end{aligned}$$

so \(y'_{j-1 \rightarrow j-2}\) FOSD \(y_{j-1 \rightarrow j-2}\). Finally, we see that for all \(j \in \{0, \ldots , n\}\):

$$\begin{aligned}&{\mathbb {E}}[(z_j)_+']={\mathbb {E}}[(z_j)_+] \le {\mathbb {E}}[\pi _j] \\&\qquad = {\mathbb {E}}\left[ \left( z_j+y_{(j+1) \rightarrow j}-R_{(j-1)\rightarrow j}\right) _+\right] - F \cdot {\mathbb {P}}\left[ z_j< R_{(j-1)\rightarrow j} - y_{(j+1) \rightarrow j}\right] \\&\qquad \le {\mathbb {E}}\left[ \left( z_j+y'_{(j+1) \rightarrow j}-R_{(j-1)\rightarrow j}\right) _+\right] - F \cdot {\mathbb {P}}\left[ z_j < R_{(j-1)\rightarrow j} - y'_{(j+1) \rightarrow j}\right] = {\mathbb {E}}[\pi _j'] \end{aligned}$$

Therefore, by Lemma 3, there is no credit freeze in \(\mathbf{G}'\), the \((n-1)\)-bank chain. Finally, by Lemma 2 note there exist \(p, q > 0\) (independent of i) such that probability that any bank \((i-1)\) on this chain defaults is at least \(p > 0\) and the probability bank i does not default is at least \(q > 0\) (by symmetry). The probability that both of these events occur simultaneously is at least pq by independence. Therefore, risk premia in the chain must satisfy \(R_{i \rightarrow (i+1)} \ge R_{(i-1) \rightarrow i}/(1-pq)\) to make nonnegative profits. Therefore, for large enough \(\bar{n}\), given the depositor is lending at least \(r_0\), it is clear the (minimum) interest rate needed to charge the entrepreneur exceeds \(r^*\), which implies by Lemma 3 there will be a credit freeze for all \(n \ge \bar{n}\). \(\square \)

1.9 Proof of Proposition 2

Suppose there is no systemic credit freeze in \({\mathcal {Q}}\), so by Lemma 3 there exist \(R_{0 \rightarrow 1}, \ldots , R_{n \rightarrow (n+1)}\) such that:

$$\begin{aligned} {\mathbb {E}}[(z_j)_+]&\le {\mathbb {E}}\left[ \left( z_j + y_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+ \right] - F \cdot {\mathbb {P}}\left[ z_j < R_{(j-1) \rightarrow j} - y_{(j+1) \rightarrow j} \right] \end{aligned}$$

for all j. If \({\mathcal {Q}}'\) FOSD \({\mathcal {Q}}\), we prove that \(y'_{j+1\rightarrow j}\) FOSD \(y_{j+1 \rightarrow j}\). We do so by induction. Note the entrepreneur always repays in equilibrium regardless of the risk profile. Bank j repays if and only if \(z_j \ge R_{(j-1) \rightarrow j} - y_{(j+1) \rightarrow j}\). It is straightforward to see \(\mathbf{z}_{-j}\) is a sufficient statistic for \(y_{(j+1) \rightarrow j}\), and \(y'_{(j+1)\rightarrow j}\) FOSD \(y_{(j+1)\rightarrow j}\) (by assumption) so we know that:

$$\begin{aligned} {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} - y_{(j+1) \rightarrow j}\right] \le {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1) \rightarrow j}\right] \end{aligned}$$
(5)

which implies that \(y'_{j \rightarrow (j-1)}\) FOSD \(y_{j \rightarrow (j-1)}\) by rearranging. It is clear the inequality is strict if the conditional distribution \(\mathbf{z}_j | \mathbf{z}_{-j}\) under \({\mathcal {Q}}'\) is different than under \({\mathcal {Q}}\) for some \(\mathbf{z}_{-j}\) (i.e., if bank j experiences an adverse shift). For all banks j without an adverse shift, we have:

$$\begin{aligned}&{\mathbb {E}}[(z'_j)_+] = {\mathbb {E}}[(z_j)_+] \\&\qquad \le {\mathbb {E}}\left[ \left( z_j + y_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+ \right] - F \cdot {\mathbb {P}}\left[ z_j< R_{(j-1) \rightarrow j} - y_{(j+1) \rightarrow j} \right] \\&\qquad \le {\mathbb {E}}\left[ \left( z'_j + y'_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+ \right] - F \cdot {\mathbb {P}}\left[ z'_j < R_{(j-1) \rightarrow j} - y'_{(j+1) \rightarrow j} \right] \end{aligned}$$

For all banks j with an adverse shift, the inequality in (5) is strict, so we have for some \(\varepsilon > 0\):

$$\begin{aligned}&{\mathbb {E}}[(z'_j)_+] -\varepsilon \le {\mathbb {E}}[(z_j)_+] \\&\quad \le {\mathbb {E}}\left[ \left( z_j + y_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+ \right] - F \cdot {\mathbb {P}}\left[ z_j< R_{(j-1) \rightarrow j} - y_{(j+1) \rightarrow j} \right] \\&\quad< {\mathbb {E}}\left[ \left( z'_j + y'_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+ \right] - F \cdot {\mathbb {P}}\left[ z'_j < R_{(j-1) \rightarrow j} - y'_{(j+1) \rightarrow j} \right] - \varepsilon \end{aligned}$$

for sufficiently large F. Killing the \(\varepsilon \) expressions on both sides of the inequality, we see by Lemma 3, we see there is no credit freeze under \({\mathcal {Q}}'\). \(\square \)

1.10 Proof of Proposition 3

First, consider the case of \(F = 0\). Consider some risk profile \({\mathcal {Q}}\) that has more tail risks than \({\mathcal {Q}}'\). If \({\mathcal {Q}}\) has no credit freeze, then by Lemma 3, there exist \(\{R_{0\rightarrow 1}, \ldots , R_{n \rightarrow (n+1)}\}\) such that for all \(j \in \{0,\ldots , n\}\):

$$\begin{aligned} {\mathbb {E}}[(z_j)_+] \le {\mathbb {E}}\left[ \left( z_j + y_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j} \right) _+\right] \end{aligned}$$

Let us consider the same set of interest rates in the network with \({\mathcal {Q}}'\). First, we show by induction that the single-crossing property means \(y'_{(j+1) \rightarrow j}\) FOSD \(y_{(j+1) \rightarrow j}\). Again, in equilibrium, the entrepreneur always repays. Assuming \(y'_{(j+1) \rightarrow j}\) FOSD \(y_{(j+1) \rightarrow j}\), then:

$$\begin{aligned} {\mathbb {E}}[y'_{j \rightarrow (j-1)}]&= R_{(j-1) \rightarrow j}{\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1)\rightarrow j}\right] \\&= R_{(j-1) \rightarrow j}\Big ( {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1)\rightarrow j} \Big | z'_j \ge r^* \right] {\mathbb {P}}[z'_j \ge r^*] \\&\quad + {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1)\rightarrow j} \Big | z'_j< r^* \right] {\mathbb {P}}[z'_j< r^*]\Big ) \\&= R_{(j-1) \rightarrow j}\left( {\mathbb {P}}[z'_j \ge r^*] + {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1)\rightarrow j} \Big | z'_j< r^* \right] {\mathbb {P}}[z'_j< r^*]\right) \\&\ge R_{(j-1) \rightarrow j} \left( {\mathbb {P}}[z_j \ge r^*] + {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} - y'_{(j+1)\rightarrow j} \Big | z_j< r^* \right] {\mathbb {P}}[z_j< r^*]\right) \\&\ge R_{(j-1) \rightarrow j}\left( {\mathbb {P}}[z_j \ge r^*] + {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} - y_{(j+1)\rightarrow j} \Big | z_j< r^* \right] {\mathbb {P}}[z_j < r^*]\right) \\&= {\mathbb {E}}[y_{j \rightarrow (j-1)}] \end{aligned}$$

where the first inequality follows from single-crossing at \(\lambda \ge r^*\) and the second inequality follows from the inductive hypothesis. Because \(y_{j \rightarrow (j-1)}\) is binary, this is sufficient for FOSD. We have the following realized values for \((z_j + y_{(j+1)\rightarrow j} - R_{(j-1) \rightarrow j})_+ - (z_j)_+\) for bank j:

$$\begin{aligned} {\left\{ \begin{array}{ll} R_{j \rightarrow (j+1)} - R_{(j-1)\rightarrow j}, \text { if } z_j \ge 0; y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)} \\ R_{j \rightarrow (j+1)} - R_{(j-1)\rightarrow j} + z_j, \text { if } R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z_j< 0; y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)} \\ -R_{(j-1)\rightarrow j}, \text { if } z_j \ge R_{(j-1)\rightarrow j}; y_{(j+1) \rightarrow j} = 0 \\ -z_j, \text { if } 0 \le z_j < R_{(j-1)\rightarrow j}; y_{(j+1) \rightarrow j} = 0 \end{array}\right. } \end{aligned}$$

Note \(\mathbf{z}_{-j}\) is a sufficient statistic for \(y_{(j+1)\rightarrow j}\) and \(y'_{(j+1) \rightarrow j}\). We can break the above into three regions: (i) \(z_j \ge R_{(j-1) \rightarrow j}\), (ii) \(0 \le z_j \le R_{(j-1) \rightarrow j}\), and (iii) \(R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z_j < 0\). In the first region, we have:

$$\begin{aligned}&(R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}) {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} \Big | y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \\&\qquad - R_{(j-1) \rightarrow j} {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} \Big | y_{(j+1) \rightarrow j} = 0\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \\&\quad = {\mathbb {P}}\left[ z_j \ge R_{(j-1) \rightarrow j} \right] \left\{ (R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}) {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \right. \\&\qquad \left. - R_{(j-1) \rightarrow j} {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \right\} \\&\quad \le {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} \right] \left\{ (R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}) {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \right. \\&\qquad \left. - R_{(j-1) \rightarrow j} {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \right\} \\&\quad \le {\mathbb {P}}\left[ z'_j \ge R_{(j-1) \rightarrow j} \right] \left\{ (R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}) {\mathbb {P}}\left[ y'_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \right. \\&\qquad \left. - R_{(j-1) \rightarrow j} {\mathbb {P}}\left[ y'_{(j+1) \rightarrow j} = 0\right] \right\} \end{aligned}$$

where the equality is from independence, the first inequality is from single-crossing, and the second inequality is from the previous intermediate result about repayment. For the second region, we have:

$$\begin{aligned}&(R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}){\mathbb {P}}\left[ 0 \le z_j \le R_{(j-1) \rightarrow j} \Big | y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \\&\qquad - {\mathbb {E}}\left[ z_j \Big | 0 \le z_j \le R_{(j-1) \rightarrow j}; y_{(j+1)\rightarrow j} = 0\right] {\mathbb {P}}\left[ 0 \le z_j \le R_{(j-1) \rightarrow j} \Big | y_{(j+1) \rightarrow j} \right. \\&\quad \left. = 0\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \\&\quad = {\mathbb {P}}\left[ 0 \le z_j \le R_{(j-1) \rightarrow j}\right] \Big \{(R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}){\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \\&\qquad - {\mathbb {E}}\left[ z_j \Big | 0 \le z_j \le R_{(j-1) \rightarrow j}\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \Big \} \\&\quad = {\mathbb {P}}\left[ 0 \le z'_j \le R_{(j-1) \rightarrow j}\right] \Big \{(R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}){\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \\&\qquad - {\mathbb {E}}\left[ z'_j \Big | 0 \le z'_j \le R_{(j-1) \rightarrow j}\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0\right] \Big \}\\&\quad \le {\mathbb {P}}\left[ 0 \le z'_j \le R_{(j-1) \rightarrow j}\right] \Big \{(R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}){\mathbb {P}}\left[ y'_{(j+1) \rightarrow j} = R_{j \rightarrow j+1}\right] \\&\qquad - {\mathbb {E}}\left[ z'_j \Big | 0 \le z'_j \le R_{(j-1) \rightarrow j}\right] {\mathbb {P}}\left[ y'_{(j+1) \rightarrow j} = 0\right] \Big \} \end{aligned}$$

where the first equality follows from independence, the second equality follows from condition (i), and the inequality follows from the intermediate result. Finally, in the third region:

$$\begin{aligned}&\left( R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}+{\mathbb {E}}\left[ z_j \Big | R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z_j< 0\right] \right) \\&\qquad \cdot {\mathbb {P}}\left[ R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z_j< 0\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)}\right] \\&\quad = \left( R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}+{\mathbb {E}}\left[ z'_j \Big | R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z'_j< 0\right] \right) \\&\qquad \cdot {\mathbb {P}}\left[ R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z'_j< 0\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)}\right] \\&\quad \le \left( R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j}+{\mathbb {E}}\left[ z'_j \Big | R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z'_j< 0\right] \right) \\&\qquad \cdot {\mathbb {P}}\left[ R_{(j-1) \rightarrow j} - R_{j \rightarrow (j+1)} \le z'_j < 0\right] {\mathbb {P}}\left[ y'_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)}\right] \end{aligned}$$

These together imply that \({\mathbb {E}}[\pi _j'] - {\mathbb {E}}[(z_j')_+] \ge {\mathbb {E}}[\pi _j] - {\mathbb {E}}[(z_j)_+] \ge 0\), so by Lemma 3, there is no credit freeze under \({\mathcal {Q}}'\). To generalize to any F, simply note that because \(y'_{(j+1) \rightarrow j}\) FOSD \(y_{(j+1) \rightarrow j}\), the default probability of any bank j is less with risk profile \({\mathcal {Q}}'\) (less tail risks), so there continues to be no systemic freeze even when \(F > 0\). \(\square \)

1.11 Proof of Proposition 4

Consider some risk profile \({\mathcal {Q}}\) that is a normal distribution with common mean \(\mu > 0\), variance \(\sigma > 0\), and correlation \(\rho \) for all banks. It is sufficient by continuity in the default cost F to take \(F = 0\) and note the result will still hold for all small values of F. Let us consider interest rates \(R_{i \rightarrow (i+1)} = r_0 + (i+1)\cdot \frac{r^*-r_0}{n+1}\) for all \(i \in \{0, \ldots , n\}\). Then the payoff of bank i is given by:

$$\begin{aligned} {\mathbb {E}}[\pi _j]&= {\mathbb {E}}[(z_j + y_{(j+1) \rightarrow j} - R_{(j-1) \rightarrow j})_+] \\&= {\mathbb {E}}\left[ (z_j + R_{j \rightarrow (j+1)} - R_{(j-1) \rightarrow j})_+ \Big | y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)}\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)} \right] \\&\quad +{\mathbb {E}}\left[ (z_j - R_{(j-1) \rightarrow j})_+ \Big | y_{(j+1) \rightarrow j} = 0\right] {\mathbb {P}}\left[ y_{(j+1) \rightarrow j} = 0 \right] \end{aligned}$$

For every \(\epsilon > 0\), there exists \(\rho ^* < 1\) such that with correlation \(\rho > \rho ^*\), \({\mathbb {P}}[\min _j z_j> 0 | z_1 > 0] \ge 1-\epsilon \) and \({\mathbb {P}}[\max _j z_j> 0 | z_1 > 0] \ge 1-\epsilon \). It is clear that when \(\min _j z_j > 0\), then \(y_{(j+1) \rightarrow j} = R_{j \rightarrow (j+1)}\) and when \(\max _j z_j < 0\), then \(y_{(j+1) \rightarrow j} = 0\) for all \(j \in \{0, \ldots , n\}\). Therefore, the above expression reduces to:

$$\begin{aligned} {\mathbb {E}}[\pi _j]&\ge (1-\epsilon ) {\mathbb {P}}[z_1> 0] \left( {\mathbb {E}}[z_j | z_j > 0] + \frac{r^*-r_0}{n+1}\right) \end{aligned}$$

whereas

$$\begin{aligned} {\mathbb {E}}[(z_j)_+]&\ge {\mathbb {E}}[z_j | z_j> 0]{\mathbb {P}}[z_j> 0] = {\mathbb {E}}[z_j | z_j> 0]{\mathbb {P}}[z_1 > 0] \end{aligned}$$

Taking \(\epsilon \) close enough to zero (by taking \(\rho ^*\) close enough to 1), we obtain that \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\). By Lemma 3, there is no credit freeze with \({\mathcal {Q}}\) with \(\rho > \rho ^*\). \(\square \)

1.12 Proof of Proposition 5

For part (a), suppose lending path P gives us the borrowing network \(\mathbf{x}_*\) before the adverse shifts. By assumption of (a), \({\mathcal {Q}}(z_i) = {\mathcal {Q}}'(z_i)\) for all banks i along the path P, as all banks experiencing an adverse shift experienced a credit freeze before the shift. We know the current lending path P satisfies the conditions of Lemma 3 both before and after the adverse shift, in that \({\mathbb {E}}[\pi ^{P}_j] - F \cdot {\mathbb {P}}[\pi ^{P}_j < 0]\) is the same before and after the adverse shifts for all banks \(j \in P\), for any interest rates \(\mathbf{R}_P\) (as is \({\mathbb {E}}[(z_j)_+]\) for all \(j \in P\)). For any other path \(P'\), the same logic as in Proposition 2 shows that \({\mathbb {E}}[\pi ^{P'}_j] - F \cdot {\mathbb {P}}[\pi ^{P'}_j < 0]\) is no greater than before the adverse shifts for all \(j \in P'\), and that \({\mathbb {E}}[\pi ^{P'}_j] - F \cdot {\mathbb {P}}[\pi ^{P'}_j < 0] - {\mathbb {E}}[(z_j)_+]\) does not increase after the adverse shifts, given sufficiently large F. This establishes that \(\mathbf{x}_*\) is the same before and after the adverse shifts; in particular, no bank on P loses access to credit after the adverse shift.

For (b), if the entrepreneur does not experience a credit freeze after the adverse shift, then by Lemma 3, there exists a path in P isomorphic to the chain network with interest rates \(\mathbf{R}\) such that \({\mathbb {E}}[(z_j)_+] \le {\mathbb {E}}[\pi _j]\) along this chain. Note that the chain network \(\mathbf{H} \subset \mathbf{G}\) thus does not experience a credit freeze. By Proposition 2, when F is sufficiently large, there is no credit freeze in \(\mathbf{H}\) before the adverse shifts. By Proposition 1, this implies there is no systemic freeze in \(\mathbf{G}\) after the adverse shifts, and in particular Theorem 3 guarantees the entrepreneur still borrows $1. So lending does not decrease before the adverse shifts. \(\square \)

1.13 Proof of Proposition 6

We prove this result by induction. Suppose there is just a single bank in \(j \in {\mathcal {R}}\). Let \(j^*\) be a borrower of the depositor who also lends (directly or indirectly) to j. The choice of \(j^*\) is unique because \(\mathbf{G}\) is a tree; to see this, if there were multiple \(j_1^*, j_2^*\), consider two paths \(P_1, P_2\) that both have bank j on it, and by taking the first (topologically from the depositor) bank \(k \in P_1 \cap P_2\), we see that k has at least two (potential) lenders, which is a contradiction. Consider all banks \({\mathcal {B}}^*\) borrowing from the depositor. Let \(d_j\) denote the (random) binary variable of whether bank j defaults. For all \(k \in {\mathcal {B}}\) we have that:

$$\begin{aligned} {\mathbb {E}}[\pi _0] = {\left\{ \begin{array}{ll} \max _{\{R_{0 \rightarrow k}\}_{k\in {\mathcal {B}}^*}}{\mathbb {E}}\left[ \sum _{\{k \in {\mathcal {B}}^*: R_{0 \rightarrow k} \ne \varnothing \}} (R_{0 \rightarrow k} (1-d_k(\{R_{0 \rightarrow \ell }\}_{\ell \in {\mathcal {B}}^*})) - r_0) x_{0 \rightarrow k} \right] \\ \text { subject to }\, \, \, {\mathbb {E}}\left[ \sum _{\{k \in {\mathcal {B}}^*: R'_{0 \rightarrow k} \ne \varnothing \}} (R'_{0 \rightarrow k} (1-d_k(\{R'_{0 \rightarrow \ell }\}_{\ell \in {\mathcal {B}}^*})) - r_0) x'_{0 \rightarrow k} \right] \\ \quad \le 0 \, \,\forall \, R'_{0 \rightarrow k} \le R_{0 \rightarrow k} \text { for all } k \end{array}\right. } \end{aligned}$$

Note that because \(\mathbf{G}\) is a tree, \(d_{k}(\{R_{0 \rightarrow \ell }\}_{\ell \in {\mathcal {B}}^*}) = d_{k}(R_{0 \rightarrow k})\). By linearity of expectation, we have:

$$\begin{aligned} {\mathbb {E}}[\pi _0] = {\left\{ \begin{array}{ll} \max _{\{R_{0 \rightarrow k}\}_{k\in {\mathcal {B}}^*}}\sum _{\{k \in {\mathcal {B}}^*: R_{0 \rightarrow k} \ne \varnothing \}} (R_{0 \rightarrow k} (1-{\mathbb {E}}[d_k(R_{0 \rightarrow k})]) - r_0) x_{0 \rightarrow k} \\ \text { subject to }\, \, \, \sum _{\{k \in {\mathcal {B}}^*: R'_{0 \rightarrow k} \ne \varnothing \}} (R'_{0 \rightarrow k}(1-{\mathbb {E}}[d_k(R'_{0 \rightarrow k})]) - r_0)x'_{0 \rightarrow k} \le 0 \, \, \forall \, R'_{0 \rightarrow k}\\ \quad \le R_{0 \rightarrow k} \text { for all } k \end{array}\right. } \end{aligned}$$

This is a separable problem because removing the depositor would disconnect the graph, and so the interest rates charged to one bank have no bearing on the payoffs of the other banks linked to the depositor. So, in particular:

$$\begin{aligned} {\mathbb {E}}[\pi _0] = {\left\{ \begin{array}{ll} \sum _{\{k \in {\mathcal {B}}^*: R_{0 \rightarrow k} \ne \varnothing \}} \max _{R_{0 \rightarrow k}} (R_{0 \rightarrow k} (1-{\mathbb {E}}[d_k(R_{0 \rightarrow k})]) - r_0) x_{0 \rightarrow k} \\ \text { subject to }\, \, \, (R'_{0 \rightarrow k}(1-{\mathbb {E}}[d_k(R'_{0 \rightarrow k})]) - r_0)x'_{0 \rightarrow k} \le 0 \, \, \forall \, R'_{0 \rightarrow k} \le R_{0 \rightarrow k} \text { for all } k \end{array}\right. } \end{aligned}$$

Since no adverse shifts occurred for any banks in the subtrees of \(k \ne j^*\), we know that \({\mathbb {E}}[d_k(R_{0 \rightarrow k})]\) is the same before and after the adverse shift at bank j (for all \(R_{0 \rightarrow k}\)). Because all of the above problems are separable over k, it is clear the financial network \((\mathbf{R}_*, \mathbf{x}_*)\) in all subtrees except possibly the one at bank \(j^*\) remains the same. In particular, any of these banks experience a credit freeze if and only if they did so before the adverse shift. Iteratively adding any banks j to \({\mathcal {R}}\) who experience an adverse shift, and repeating the above argument gives the desired result. \(\square \)

1.14 Proof of Proposition 7

By Lemma 2, consider some set of contracts \(R_{0 \rightarrow 1}, \ldots , R_{n \rightarrow n+1}\) such that \({\mathbb {E}}[\pi _k] \ge {\mathbb {E}}[(z_k)_+]\) for all banks k before the addition of the risk-bearing bank. Note that because \(z_j \ge r^*\), then \(z_j + y_{(i+1) \rightarrow j} - R_{i \rightarrow j} \ge 0\), so bank j never defaults, even if \((i+1)\) does not repay j. Consider the set of contracts \(R'_{0 \rightarrow 1}, \ldots , R'_{(i-1) \rightarrow i}, R'_{i \rightarrow j}, R'_{j \rightarrow (i+1)}, \ldots , R'_{n \rightarrow (n+1)}\) with \(R'_{k \rightarrow k+1} = R_{k \rightarrow k+1}\) for all \(k \ne i\) and \(R'_{i \rightarrow j} = R_{(i-1) \rightarrow i}, R'_{j \rightarrow (i+1)} = R_{i \rightarrow (i+1)}\). As in Proposition 2, when F is large it is sufficient to check default probabilities under these contracts are lower with risk-bearing bank i, then Lemma 2 guarantees there will no systemic freeze.

We prove \(y'_{(k+1) \rightarrow k}\) FOSD \(y'_{(k+1) \rightarrow k}\) for all \(k \ne i\), that \(y'_{j \rightarrow i}\) FOSD \(y_{(i+1) \rightarrow i}\), and that \({\mathbb {E}}[(z_j)_+] \le {\mathbb {E}}[\pi _j]\). It is clear that \(y_{(k+1) \rightarrow k} = y'_{(k+1) \rightarrow k}\) for all \(k \in \{1, \ldots , i-1\}\). Since \(y'_{j \rightarrow i} = R'_{i \rightarrow j}\) almost surely, it FOSD all other y, including \(y_{(i+1) \rightarrow i}\). We prove for \(k \in \{i+1, \ldots , 0\}\) by induction. We see that:

$$\begin{aligned} {\mathbb {E}}[y_{(k+1) \rightarrow k}]&= R_{k \rightarrow (k+1)} {\mathbb {P}}[ z_{k+1} + y_{(k+2) \rightarrow (k+1)} - R_{k \rightarrow (k+1)} \ge 0] \\&\le R_{k \rightarrow (k+1)} {\mathbb {P}}[ z_{k+1} + y'_{(k+2) \rightarrow (k+1)} - R_{k \rightarrow (k+1)} \ge 0] \\&= {\mathbb {E}}[y'_{(k+1) \rightarrow k}] \end{aligned}$$

where the inequality follows from the inductive step. Lastly, we know that \({\mathbb {E}}[(z_i + y_{(i+1) \rightarrow i} - R_{(i-1) \rightarrow i})_+] - F\cdot {\mathbb {P}}[z_i + y_{(i+1) \rightarrow i} - R_{(i-1) \rightarrow i} < 0] \ge 0\). This implies that for large enough F, we have:

$$\begin{aligned} {\mathbb {E}}[\pi _j]&= {\mathbb {E}}[z_j + y_{(i+1) \rightarrow j} - R'_{i \rightarrow j}] \ge {\mathbb {E}}[z_j + y_{(i+1) \rightarrow j} - R_{(i-1)\rightarrow i}] \ge {\mathbb {E}}[z_j] \end{aligned}$$

Thus, \({\mathbb {E}}[\pi _j] \ge {\mathbb {E}}[(z_j)_+]\) for bank j, and we have showed there is no systemic freeze. \(\square \)

1.15 Proof of Proposition 8

Part (a) of the result follows by the exact same reasoning as Theorem 3 for why in the original economy, the network cannot contain directed cycles. For part (b), note that the contracts offered with quantity-restrictions are a superset of those offered without them. Therefore, extending Lemma 3, if there exists a path P and interest rates \(\mathbf{R}_P\) along this path such that the conditions of Lemma 3 hold, then for the same path there exist a set of interest rates and quantity-restrictions given by \((\mathbf{R}_P, |{\mathcal {E}}|+1)\) such that the same conditions (i.e., willingness to lend) hold as well, as none of quantity restrictions bind. Thus, there cannot be systemic freezes with quantity restrictions if there is not a systemic freeze in the original economy. \(\square \)

1.16 Proof of Proposition 9

It is enough to prove that if there exists a budget B that restores lending, then giving B to the depositor restores lending. Without loss of generality, suppose \(\mathbf{G}\) is a chain. If B restores lending, then by Lemma 3 there exists \(\sum _{i=0}^n \epsilon _i \le B\) and \((R_{0 \rightarrow 1}, \ldots , R_{n \rightarrow (n+1)})\) such that \({\mathbb {E}}[(z_i + \epsilon _i)_+] \le {\mathbb {E}}[(z_i + \epsilon _i + y_{(i+1) \rightarrow i} - R_{(i-1) \rightarrow i})_+]\) for all \(i \in \{1, \ldots ,n+1\}\), with \({\mathbb {E}}[\epsilon _0 + y_{1 \rightarrow 0} -r_0] \ge 0\). Instead, consider giving B entirely to the depositor. Similarly, consider interest rates \(R'_{i \rightarrow (i+1)} = R_{i \rightarrow (i+1)} - \sum _{k=0}^{n-i-1}\epsilon _{n-k}\) for all \(i \in \{0, \ldots , n-1\}\). Then:

$$\begin{aligned} {\mathbb {E}}[(z_i)_+] \le {\mathbb {E}}[(z_i+\epsilon _i)_+]&\le {\mathbb {E}}[(z_i + \epsilon _i + y_{(i+1) \rightarrow i} - R_{(i-1) \rightarrow i})_+] \\&= {\mathbb {E}}[(z_i + y'_{(i+1) \rightarrow i} - R'_{(i-1) \rightarrow i})_+] \end{aligned}$$

for all \(i \in \{1, \ldots , n\}\), where the equality follows from the fact that \((y'_{(i+1) \rightarrow i} - R'_{(i-1) \rightarrow i}) - (y_{(i+1) \rightarrow i} - R_{(i-1) \rightarrow i}) = \epsilon _i\) (and by simple induction, i.e., \({\mathbb {P}}[y_{(i+1) \rightarrow i} = 0] = {\mathbb {P}}[y'_{(i+1) \rightarrow i} = 0]\)). Finally, note that for the depositor:

$$\begin{aligned} 0\le {\mathbb {E}}[\epsilon _0 + y_{1 \rightarrow 0} - r_0 ] \le {\mathbb {E}}[B + y'_{1 \rightarrow 0} -r_0] \end{aligned}$$

which then implies by Lemma 3 there is no systemic freeze. \(\square \)

1.17 Proof of Proposition 10

By Definition 8, we know if bank j is hit with an adverse shift and the freeze is simple, there exists a (direct or indirect) lender \(j^*\) of j such that all banks with frozen credit are a (direct or indirect) borrower of bank \(j^*\). Consider the distribution \(z_j' - z_j\), where \(z_j'\) is the (random) liquidity shock at bank j after the adverse shift and \(z_j\) is the liquidity shock before the distribution shift. Then setting \(\epsilon _j = z_j' - z_j\) (which requires budget \(B^* = \epsilon _j\)) reverses the effects of the shock and restores full lending.

Consider \(B^*\) to be the smallest budget needed to restore full lending in a targeted policy of the form from (a). Let \(j^*\) be the only bank the depositor lends to with (direct or indirect) borrowers whose credit is frozen. Finally, let \(x^* > 0\) be the amount lent to all other banks connected to the depositor, other than \(j^*\). We claim that \(B^{**} \ge B^* + x^* > B^*\) is the minimum budget required to restore lending in the untargeted policy (if it is possible). Because the freeze is simple, the depositor still uses funds \(x^*\) from the central bank to lend to banks other than \(j^*\) (i.e., the depositor does not change its lending decisions after the intervention). Thus, for any \(B < B^* + x^*\), the depositor would set \(R'_{0 \rightarrow j^* } x'_{0 \rightarrow j^*} < \max \{0, B-x^*\} + R_{0 \rightarrow j^*}x_{0 \rightarrow j^*}\), where \('\) denotes quantities after the rescue policy. By assumption, \(\sum _{k \in {\mathcal {B}}^*(j^*)} \epsilon _k \ge B^*\) is a necessary condition to restore full lending to \(j^*\)’s (direct or indirect) borrowers, where \({\mathcal {B}}^*(j^*)\) is the set of (direct or indirect) borrowers of \(j^*\). By the same reasoning as in Proposition 9, there exists no set of interest rates \(\mathbf{R}_{{\mathcal {B}}^*(j^*)}\) in \({\mathcal {B}}^*(j^*)\) that mimic such a policy given that \(\epsilon _{j^*} = \max \{0, B-x^*\} < B^*\). Thus, no untargeted policy that restores full lending with budget \(B^*\) exists. \(\square \)

Appendix: Prevalence Theory

This section is dedicated to explaining the relevant details of Ott and Yorke [39] needed for our work. This is of importance when we discuss generic risk profiles \({\mathcal {Q}}\), because the usual definition of “genericity” does not extend well to infinite-dimensional spaces (such as probability distribution functions). The rich theory of Ott and Yorke [39] allows us to handle a wide range of risk profiles (both discrete and continuous) throughout this paper.

We begin with the following discussion from Ott and Yorke [39] on the desirable properties of geniricity. If X is a topological vector space, a sound theory of genericity for topological vector spaces should satisfy the following genericity axioms.

  1. (i)

    A generic subset of X is dense in X.

  2. (ii)

    If \(L \supset G\) and G is generic, then L is generic.

  3. (iii)

    A countable intersection of generic sets is generic.

  4. (iv)

    Every translate of a generic set is generic.

  5. (v)

    A subset G of \({\mathbb {R}}^n\) is generic if and only if G is a set of full Lebesgue measure in \({\mathbb {R}}^n\).

In standard measure-theoretic terms, a subset \(G \subset {\mathbb {R}}^n\) is said to be generic if \(\mathbf{R}^n \backslash G\) has zero Lebesgue measure. This has problems in infinite-dimensional spaces: every separable Banach space with a translation-invariant Boreal measure (which is not identically zero) must assign infinite measure to all open sets. The example provided is the following: take the open ball \(B(x, \epsilon )\). We can construct infinitely many disjoint open balls of radius \(\epsilon /4\) containing with \(B(x, \epsilon )\). Each of the balls has the same measure, and if the measure of \(B(x, \epsilon )\) is finite, these balls of radius \(\epsilon /4\) must have zero measure. But then the entire space can be covered by \((\epsilon /4)\)-radius balls, so the space must have measure 0 (which fails to satisfy Axiom 5).

Definition 10

(Definition 3.1 in Ott and Yorke [39]). Let X be a completely metrizable topological vector space. A Borel set \(E \subset X\) is said to be prevalent if there exists a Borel measure \(\mu \) on X such that:

  1. (a)

    \(0< \mu (C) < \infty \) for some compact subset C of X, and

  2. (b)

    the set \(E+x\) has full \(\mu \)-measure (that is, the complement of \(E+x\) has measure zero) for all \(x \in X\).

More generally, a subset \(F \subset X\) is prevalent if F contains a prevalent Borel set; we say that almost every element of X lies in F or that F is generic.

Proposition 11

(Proposition 3.3 in Ott and Yorke [39]) The theory of prevalence satisfies Axioms 1–5.

Therefore, when we refer to a property holding for a generic risk profile \({\mathcal {Q}}\), we mean the set of \({\mathcal {Q}}\) where this property holds is prevalent in the space of probability distribution functions. We present some useful facts which are useful and leveraged throughout the paper:

Proposition 12

The following are true:

  1. (a)

    For any constant c, for almost all discrete probability distributions \({\mathcal {Q}}\), \({\mathbb {E}}^{{\mathcal {Q}}}[z_j] \ne c\).

  2. (b)

    For any constant c, for almost all continuous (and differentiable) probability distributions \({\mathcal {Q}}\), \({\mathbb {E}}^{{\mathcal {Q}}}[z_j] \ne c\) (consequence of Example 3.6).

  3. (c)

    For almost all continuous (or countably discrete) probability distributions,\({\mathcal {Q}}\) is unbounded above and below (consequence of Example 3.9).

  4. (d)

    For almost all continuous probability distributions \({\mathcal {Q}}\) and continuous (and sufficiently differentiable) functions f, \({\mathbb {E}}^{{\mathcal {Q}}}[f(\alpha )]\) has a unique global maximum in \(\alpha \).

For each of these, the trick is to find a finite-dimensional subspace \(P \subset X\) which is known as probe for a set \(F \subset X\). This holds whenever there exists a Borel set \(E \subset F\) such that \(E+x\) has full \(\lambda _P\)-measure for all \(x \in X\). This is a sufficient condition for a set F to be prevalent. Many genericity conditions in infinite-dimensional spaces (such as those probability distributions) can be proven using prevalence. See the paper Ott and Yorke [39] for examples.

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Acemoglu, D., Ozdaglar, A., Siderius, J. et al. Systemic credit freezes in financial lending networks. Math Finan Econ 15, 185–232 (2021). https://doi.org/10.1007/s11579-020-00272-z

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