Essays on banking theory

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DOCTORAL

THESIS

Essays on Banking Theory

Author:

Diego Martins Silva

Supervisor: Ricardo de Oliveira Cavalcanti

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy

Escola de Pós-Graduação em Economia

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Silva, Diego Martins

Essays on banking theory / Diego Martins Silva. – 2019. 121 f.

Tese (doutorado) - Fundação Getulio Vargas, Escola de Pós-Graduação em Economia.

Orientador: Ricardo de Oliveira Cavalcanti. Inclui bibliografia.

1.Bancos. 2. Crise financeira. 3. Seguro de depósitos. I. Cavalcanti, Ricardo de Oliveira. II. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. III. Título.

CDD – 332.1

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FUNDAÇÃO GETÚLIO VARGAS

Abstract

Escola de Pós-Graduação em Economia

Doctor of Philosophy

Essays on Banking Theory

by Diego Martins Silva

In this thesis, we study the bank-run literature following the mechanism-design approach. In first chapter, we study the impact of wariness on optimal payment contract offered by bank, and we study the conditions to multiple equilibria in the model. In chapter two, we study interbank de-posit as a mechanism to avoid bank-run. We look for the most resilient network without exposing banks to the risk of contagion. The last chapter is about deposit insurance. We discuss about the effects of deposit insurance on payment contract.

Keywords: Bank Run; Mechanism Design; Wariness

Resumo

Nesta tese, nós estudamos a literatura de corrida bancária seguindo a abordagem de desenho de mecanismo. No primeiro capítulo, nós estudamos o impacto de depositantes cautelosos no contrato ótimo oferecido pelo banco. Além disso, nós estudamos as condições para a existência de múltiplo equilílibrio no modelo. No capítulo dois, nós estudamos depósito interbancário como um mecanismo para evitar corrida bancária. Nós procuramos a rede financeira mais resiliente que não expõe os bancos ao risco de contágio. O último capítulo é sobre seguro-depósito. Nós discutimos sobre os efeitos do seguro-depósito no contrato de pagamento.

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Acknowledgements

First and foremost, I would like to thank God for his blessings, giving me the mental and physical strength to accomplish this research work.

I am indebted to my advisor Ricardo de Oliveira Cavalcanti for his research guidance and insightful comments. I thank him for developing my potentialities, always believing in my work. I also would like to thank Professor Jan Pieter Krahnen for receiving me in Goethe University.

I would like to express my gratitude and appreciation to my family for all support during this long step. I also thank my wife for all the support throughout my academic life.

I am grateful to many colleges for comments and stimulating discussions. I have learned a lot from them and many other students, professors and co-workers at EPGE/FGV.

Finally, I thankfully acknowledge the financial support of Conselho Nacional de Desenvolvi-mento Científico e tecnológico (CNPQ),Coordenação de AperfeiçoaDesenvolvi-mento de Pessoal de Nível Superior (CAPES) and Graduate School of Economics at Fundação Getúlio Vargas - EPGE/FGV.

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3.3.2 Feasibility Constraint. . . 66 3.3.3 Truth-Telling Constraint . . . 67 3.3.4 Expected Utility. . . 67 3.4 Equilibrium . . . 67 3.4.1 Pairwise Banking . . . 69 3.4.2 Numerical Exercise . . . 74 3.5 Final Remarks . . . 77 A Chapter 1 Appendix 79 A.1 Proofs . . . 79 A.2 Properties of gn . . . 88 A.3 Properties of fn . . . 95 B Chapter 2 Appendix 97 B.1 Proofs . . . 97 C Chapter 3 Appendix 101 C.1 Proofs . . . 101 C.2 Algorithm to N>2 . . . 104 Bibliography 107

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List of Figures

1.1 Wariness effect with 5 depositors. . . 22

1.2 Expected utility for different level of wariness. . . 26

1.3 Consumption by position for a give wariness level (ωN −1_{= θ}N −1_). _{. . . 28}

2.1 Ring network . . . 34

2.2 Complete network . . . 34

2.3 Optimal payment contract for N=3.. . . 45

2.4 Optimal payment contract for N=4.. . . 46

2.5 Maximum utility for each n ∈ N . . . 49

2.6 Effect of wariness on consumption plan. . . 50

2.7 Effect of φ on equilibrium. . . 53

3.1 Sequence of actions . . . 65

3.2 Effect of DIS on payment allocation with p = 0.5. . . 73

3.3 Effect of DIS on payment allocation with p = 0.1. . . 73

3.4 Expected utility in a bank-run . . . 75

3.5 Effect of τ on second-date payment . . . 76

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List of Tables

1.1 Sequence of payments in bank run with different levels of wariness.. . . 21

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Introduction

In this thesis, we study the bank-run literature following the mechanism-design approach. In chapter 1, we re-evaluate the Diamond and Dybvig (1983) model with agents looking at the bank-run equilibrium. We label agents as wary depositors since they place great emphasis on the lower possible payment. We find that they are less likely to run once the worst second-date payment is higher than the worst first-date payment in a sequential-service mechanism. We also see that an extreme environment with a high level of wariness implies a truncated economy which all depositors receive the initial deposit back until the first patient depositor shows up. In such a situation without disclosure about the position, we find one single equilibrium which no one runs. If the agent knows his position, he waits to consume at t = 2 when he is the first patient type to shows up.

In chapter 2, we discuss the optimal network when depositors overemphasize the event of a bank-run in a multiple queue environment. We analyse the decision of different queues intercon-necting themselves using interbank deposit insurance. We find that a density network is desirable when banks believe that they are able to mitigate a negative effect and an empty network if the risk of contagion is high enough. We also discuss the effect of network on the liquidity level of op-timal payment contract. Once queues protect themselves with deposit insurance, in equilibrium they intensify the liquidity risk discussed in Diamond and Dybvig (1983).

In chapter 3, we discuss the effect of a deposit insurance scheme (DIS) in an economy with multiple isolated banks. The scheme is funded by participants and it follows a pre-determined insurance payment scheme. An external player transfers insurance benefit to all banks that (N-1) depositors are running. The total insurance payment depends on resources collected by the external authority and the number of eligible queues to receive the insurance benefit. We discuss the effect of DIS on the optimal payment contract. More specifically, we analyse the existence of bank-run equilibria and if the optimal payment contract is incentive compatible. We find that DIS prevents bank-run equilibria at the same time that it may expose the environment to contagion. We also see that the insurance policy relaxes the truth-telling condition.

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Chapter 1

Bank Runs With Wary Depositors

1.1 Introduction

Diamond and Dybvig (1983, henceforth DD) argue that banks are fragile due to the liquidity prob-lem. In good equilibrium, depositors correctly announce their type while in the bad one, some of them misrepresent. The multiple equilibria are due to lack of knowledge about the other an-nouncements. The belief that all depositors are running has a better reply to run too. Despite reaching a bad equilibrium, this is a rational action since depositors try to receive the highest pos-sible payment. In our work, we argue that depositors may behave differently as a function of the event they emphasize.

Guiso et al. (2018) suggest that fear may affect financial decision. The same individual may behave as risk-averse and risk-seeking in a short period. In their article, they see that agents changed from risk-lover behavior to risk-averse after the 2008 financial crises. The fear or the panic caused by the financial crises may change the focus of the individuals. Bordalo et al. (2012b) argue that the decision of agents depends on the events the investor are focusing. They label such event as salience event. Individuals give a disproportional weight for salience events when they take a decision. This theory applied in different fields1 explain why the same individual shows risk-love characteristic in some decisions and risk aversion in others.

In our work, we consider that depositor emphasize the lowest possible outcome whey he faces a choice under uncertainty. We define these individuals as wary. Araujo et al. (2011) use the con-cept of wariness to describe infinite-lived individuals who overlook the earns but not the looses at far away dates. Specifically, they tend to neglect gains at distance dates but not losses. We ap-ply wary agents theory to look for consequences in the decision to run or not, for the best of our 1_{See Bordalo et al. (}_2012a_{), Bordalo et al. (}_2013a_{), Bordalo et al. (}_2013b_{) and Bordalo et al. (}₂₀₁₅_{) for some application}

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knowledge, we are the first to do so.

Instead of only looking at the expected payment conditional to all announcements, agent anal-yses the worst outcome contingent to his announcement. There is no improvement in DD struc-ture with suspended payment once the ex-ante worst event is to be an impatient depositor after the payment is suspended, in other words, the worst event is "receiving nothing". In this work, we use the notion of sequential-service as in Wallace (1988), where depositors do not all contact the bank at the same time. The bank must provide payment to depositors with the available in-formation.

In a similar Green and Lin (2003, henceforth GL) structure, the decision to run is a function of the difference between the expected utility of tell the truth or not plus the difference between the worst possible outcome received when he lies their type and not. In such a model, the first payments at t = 1 are bigger than the last ones. However, the depositor does not choose the position. They only decide to announce their type truthfully or not. We assume throughout that an agent who is indifferent between declaring honestly and not doing so tells the truth.

The choice depends if they know in which position they are. If individuals have no disclo-sure about their position, as in Peck and Shell (2003, henceforth PS), agents consider lying if the expected utility to do so is bigger than to tell the truth. Once we consider wary depositors, they worry about the worst payment conditional to their announcement. Otherwise, if depositors have disclosure about their type, they look at the difference in expected utility between tell the truth or not and the difference between the worst payment conditional to the announcement in their position.

As demonstrated by Green and Lin (2003), with disclosure about the position, the last in the queue has no incentive to lie about his type. We find in our model that this is still true when de-positors are wary. However, the difference between both expected utility reduces when wariness increases. Using an algorithm inspired by Ennis and Keister (2009), we re-evaluate the bank-run theorem in low return economy by Bertolai et al. (2014).

The rationale behind our findings is that the fear to be the last in a bank run encourage pa-tient depositor to wait until second-date, once the worst payment in the second period is not lower than the worst payment in the first-date payment during a bank-run event with sequential-service. Wary depositor faces the trade-off between the lowest first-date payment and the lowest second-date payment. In the mechanism which banks update payment as a function of informa-tion, patient depositor anticipates that misrepresenting his type implies the worst possible ex-ante

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payment. To avoid this, he announces his correct type more likely.

We analyse the impact of wariness on bank-run contract payment when the level of wariness is low and when it is high. In the low level, there is one strictly lower payment which increases, in equilibrium, as the level of wariness increases. In the extreme case of a high level, the fear to be the last one is so high that, in equilibrium, the payment mechanism is truncated. Depositors only receive a payment bigger than initial deposit if at least one patient depositor already shows up (and tells the truth)2.

In the event of a bank run, all depositors receive the initial deposit back, which means that there is no loss in being the last one in a bank run. The main result here is that the first patient depositor to show up does not have the incentive to misrepresent himself, once in this case, he receives only the initial deposit instead of receiving something at least equal to the initial deposit. Moreover, without disclosure about the position, the patient depositor has no incentive to make a false announcement.

Other related works are Andolfatto et al. (2007) and Bertolai et al. (2018). The first one argues that independence in type is a crucial hypothesis to guarantee a no-run equilibrium in GL work while the last ask about the consequences of bank-run equilibrium when many small banks make a mutual arrangement. Following Andolfatto et al. (2007), we also look for the effect of wariness when depositor knows the previous announcements.

The remainder of the paper is organized as follows. The second section introduces the envi-ronment and the concept of wariness. The third derives the efficient allocation. The fourth and fifth describe the insights with a low and high level of wariness, respectively. And finally, the last section offers some concluding remarks.

1.2 Model

The economy consists of one single bank and N ex-ante homogeneous individuals. There are two periods, in the first period each depositor receives an initial endowment of e > 0 and a type preference. If depositor receives the type ω = 0, he is defined as an impatient depositor who has utility at consumption only at t = 1. However, if he receives the type ω = 1, we define him as a patient depositor who may wait to consume at t = 2. The type preference follows a Bernoulli distribution, with probability p > 0 the agent desires to consume at t = 1 only and with probability

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(1 − p)he can wait to consume at t = 2.

vi(c1, c2) = u(c1+ ωic2) (1.1)

where ctis the consumption level at t = {1, 2}.

Assumption 1. The utility function u: R+ ⇒ [−∞, ∞) is twice differentiable, strictly increasing and

concave.

Agents announce their type to the bank in sequence, which provides the payment as a function of the announced type, the available resources and the history of announcements. This sequential-service constraint imposes that consumption depends only on the information provided until that moment. Resources not consumed at t = 1 are reinvested at the rate R > 1.

Bank knows the position that an individual is at when he announces his type. PS assume that the bank does not disclose the position to depositors while GL assume disclosure. We consider both cases separately. Also, we discuss the case when depositors know the previous announce-ment.

The row of the N announcements form the event ω = (ω1, ω2, ..., ωN), an element of the event

space Ω = {0, 1}N _{where ω}

iis the type of depositor at position i on the queue. The event ω has the

following probability, P (ω) = pN −|ω|_{(1 − p)}|ω|_{, where |ω|=}PN

i=1ωi.

At i-th position, bank only knows the i − 1 previous announcements. If depositor at position iis impatient, his payment is pinned down by the history ωi−1 _{∈ Ω = {0, 1}}i−1_{, where ω}i−1 ₌

(ω1, ω2, ..., ωi−1). On the other hand, bank already knows all announcements when it pays the

patient depositors, once all depositor announces their type at t = 1. Depositors know that their payment is pinned down by the history of announcements and of his announcement. They realize that there is a specific event ω such that they achieve the lowest possible utility. If they do not know their position, they emphasize the event and position that provides the infimum of utility.

Assumption 2. Wary depositors emphasize the worst event when they face a choice under uncertainty.

We assume that depositors are wary. In other words, they emphasize the event they have the lowest utility. The utility of a wary depositor is,

˜

v(c1, c2) = Eωu(c1, c2; ω) + β inf

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where β is the coefficient of wariness. We may rewrite equation above as

˜

v(c1, c2) = pu(c10) + (1 − p)u(c11+ c21) + βM in{u(c10), u(c11+ c21)} (1.3)

where cti is the consumption of type-i depositor in period t. Assume a payment contract where

the depositor derives more utility being a patient depositor. In other words, u(c11+ c21) > u(c10).

The ex-ante expected utility of wary depositor is represented as,

˜

v(c1, c2) = (p + β)u(c10) + (1 − p)u(c11+ c21) (1.4)

which means that depositor gives an additional weight to the lowest utility. He emphasises the event where he is a type-0 depositor.

The planner, or the bank, offers a payment contract (xi, yi)Ni=1that pays xi(ωi−1, ωi)at t=1 and

yi(ω−i, ωi)at t=2 to the i-th depositor who announces a type ωi, where ωi−1indicates the set of

an-nouncements made before the i-th announcement and ω−iindicates the set of all announcements

except the i-th. Since type-0 depositor derives utility only at t=1, bank offers yi(ωN −1, 0) = 0. For

the type-1 depositor, we assume that bank offers no payment at t=1 once the resources grow at rate R>1 on second-period, xi(ωi, 1) = 0.

Bank defines an optimal contract (xi, yi)Ni=1that maximizes the expected utility of depositors

subject to a feasibility constraint and a truth-telling (TT) constraint.

M ax {xi(ω),yi(ω)}Ni=1 X ω∈Ω P (ω) N X i=1 1 N[(1−ωi)u(xi(ω i−1_))+ω iu(yi(ω))]+β inf (ω,j)u((1−ωj)xj(ω j−1_)+ω jyj(ω)) (1.5) s.t N X i=1 (1 − ωi)xi(ω) + ωiR−1yi(ω) ≤ N e ∀ ω ∈ Ω (1.6) Truth-Telling Conditions

The feasibility constraint indicates that the sum of first-date payments, xi, and the value

present of second-date payments, R−1yi, must be lower or equal to the total resource of the

econ-omy, N e.

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their type to the bank. Remember that impatient depositor never lies his type; he would have zero utility otherwise. We shall observe only the patient individual announcement. The structure of the TT constraints depends on the information that patient depositor has. If he knows his position, we say that (x, y) are incentive compatible if

Eω[u(yi(ω−i, 1))] + β inf

ω u(yi(ω−i, 1)) ≥ Eω[u(xi(ω

i−1_{, 0)] + β inf}

ω u(xi(ω i−1_{, 0))}

(1.7)

and if the depositor has no disclosure about his position,

Eω 1 N N X i=1 u(yi(ω−i, 1)) + β inf (ω,j)u(yj(ω−j, 1)) ≥ Eω 1 N N X i=1 u(xi(ωi−1, 0) + β inf (ω,j)u(xj(ω j−1_{, 0))} (1.8) The TT constraints have an additional term when depositors are wary. Note that depositor updates the worst possible outcome as he receives new information. When he knows his position, he gives special attention to the worst possible event in his position. However, when depositor does not know his position, he emphasizes the worst event for all possible positions. Considering a strictly increasing utility function, the individual has the lowest value of utility when he receives the lowest payment.

Definition 1. An optimal allocation is defined as the sequence (xi, yi)Ni=1that maximizes objective function

(1.9) subject to feasibility constraint (1.10) and truth-telling constraints.

Take an optimal allocation (xi, yi)Ni=1without wariness and two events ω ∈ Ω and ˆω ∈ Ωsuch

that the payment to the k-th depositor in the history of events ω and ˆωprovides the lowest and the second lowest utility, respectively3. Note that it might be optimal to bank increase the payment on the worst event since depositors are wary. The implicit cost is to deviate from the optimal allocation (xi, yi)N_i=1reducing the first part of expected utility and increasing the second one.

X ω∈Ω P (ω) N X i=1 1 N[(1 − ωi)u(xi(ω i−1_{)) + ω} iu(yi(ω))] | {z } Reduce + βu(xyk(ω)) | {z } Increase

Note, however, that bank has no incentive to increase the payment xyk(ω)more than xyk(ˆω)since

the lowest utility does not change in this case. The worst event becomes ˆω instead of ω. If bank wants to improve even more u(xyk(ω)), it shall increase the payment at ˆωas well. It implies that

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the worst event remains the worst event even with wary depositors, the difference is that the utility in the "bad" event is slightly higher with wariness.

Lemma 1. In equilibrium, the worst event is the same in an environment with and without wariness.

Proof. See "AppendixA.1".

By the lemma above, we guarantee that the objective function is continuous. We add two constraints to the original problem in order to illustrate the result of the lemma above,

M ax {xi(ω),yi(ω)}Ni=1 Eω N X i=1 1 N[(1 − ωi)u(xi(ω i−1_{)) + ω} iu(yi(ω)) + β inf (ω,j)u((1 − ωj)xj(ω j−1_{) + ω} jyj(ω)) (1.9) s.t N X i=1 (1 − ωi)xi(ω) + ωiR−1yi(ω) ≤ N e ∀ ω ∈ Ω (1.10) u(xi(ω)) ≥ u(xy(ω)) (1.11) u(yi(ω)) ≥ u(xy(ω)) (1.12) Truth-Telling Conditions

Following the Assumption 1 on Green and Lin (2003), bank solves the relaxed problem where the TT constraint is ignored. Note that the second-date payment is a function of the whole history of announcements ω. By the concavity of the utility function yi(ω) = yj(ω) = y(ω) whenever

ωi = ωj = 1. Bank pays the exact amount of resources to all type-1 depositors. Assuming ω 6= θ,

where θ is a null vector with N zeros, the feasibility constraint implies the following function to second-date payment. y(ω) = R |ω| N e − N X i=1 (1 − ωi)xi (1.13) where |ω|=PN i=1ωi.

Without loss of generality, assume that the lowest possible utility occurs on the first period at position k with the history of events ωk_{. Applying the equation (}_1.13_{) on objective function, the}

First Order Condition (F.C.O) for any xi(ωi) 6= xk(ωk)such as ωi6= θi,

P (ωi−1)u0(xi(ωi−1)) + λi(ωi−1)u0(xi(ωi−1)) − R

X ˜ ω6=θ ˜ ωi−1=ωi−1

P (˜ω)u0(y(˜ω)) + λN +1(ω)u0(y(˜ω))

= 0

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where λi(ωi−1)is the Lagrange multiplier associated by a constraint showing that the payment

xi(ωi−1)provides an at least bigger utility than payment xk(ωk−1)for i ≤ N . The λN +1(ω)is the

Lagrange multiplier associated to the second-date payment in the history of events ω. We specify a Lagrange multiplier for the second-date payment because the worst second-date payment might be equal to the worst first-date payment for some level of wariness. Now, considering ωi _{= θ}i_,

∀i < N , [P (θi) + λi(θi)]u0(xi(θi)) − [P (θN) + λN(θN)]u0(xN(θN)) − R X ˜ ωk_=ω ˜ ω6=θ [P (˜ω) + λN +1(¯ω)]u0(y(˜ω)) = 0 (1.15) Considering the FCO to the payment associated to the worst event, when ωk _{6= θ}k_is,

P (ωk)u0(xk(ωk)) + β −X ω∈Ω λ(ω) + N +1 X i=1 ωk 6=ωk λi(ω) u0(xk(ωk)) − R X ˜ ωk_=ωk P (˜ω)u0(y(˜ω)) = 0 (1.16) and for ωk_{= θ}k_{, ∀k < N ,} P (θk)u0(xk(θk)) + β −X ω∈Ω N +1 X i=1 ωk 6=ωk λi(ω) u0(xk(θk)) − P (θ)u0(xN(θN)) − R X ˜ ωk_=ωk ˜ ω6=θ P (˜ω)u0(y(˜ω)) = 0 (1.17) We may represent equations (1.14), (1.15),(1.16) and (1.17) together as one,

[P (ωk) + ¯λk(ωk)]u0(xk(ωk)) − Iωk_=θk[P (θ) + ¯λ_N(θ)]u0(x_N(θN)) = R X ˜ ω6=θ ˜ ωk =ωk [P (˜ω) +λN +1¯ (˜ω)]u0(y(˜ω)) (1.18) where Iωk_=θk is an indicator function of ωk = θk; and

¯ λi(ωi) =          λi(ωi) if ωi6= ωk β −P ω∈Ω PN +1 i=1 ωk 6=ωk λi(ω) if ωi_{= ω}k (1.19)

Defining a historic dependent function αk:{0, 1} → R+as a historic dependent function,

αi(ωi) ≡ 1 +

¯ λi(ω)

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the F.O.C of the planner problem is represented as: P (ωk)αk(ωk)u0(xk(ωk)) − Iωk_=θkP (θ)α_N(θN)u0(x_N(θN)) = R X ˜ ω6=θ ˜ ωk =ωk P (˜ω)αN +1(ω)u0(y(˜ω)) (1.21)

The historic dependent function replaces the original wariness term for β > 0. For all events which the associate utility is the infimum, the history dependent term is directly dependent on the wariness term. For the other terms, the Lagrange multiplier indicates if the associated payment is bigger or equal to the lowest one. The importance of this term is that the central planner may choose to create new worst events as a trade-off to improve the worst payment. In other words, as the central planner increases the lower payment, more events become the worst event. Suppose two events ω and ˆω such that the associated payment for an individual at position i is x and y, respectively, such that y − x = > 0. Originality, ω provides the infimum of utility. However, if the central planner increases the lowest payment x in to increase the expected utility of the individual, the event ˆωbecomes the worst event as well. In such a case, the historic dependent term αi(ˆω)is equal to the first line of equation (1.20) with λi(ˆω) > 0. Note that if bank chooses to

increase x in 2, it improves the worst payment in only, since the payment y will be the newest payment associated to the worst event. In order to increase the worst payment more than , bank must increases the payment y as well. It means that the worst event in an economy without wary depositors is the same in an economy with wary depositor, as stated by lemma1.

We rewrite the optimization problem using the historic dependent function. The problem has a similar structure as in related literature, which allows us to solve the problem recursively.

Proposition 1. The problem with wary depositors is equivalent to a problem with a modified utility

func-tion vi(c1, c2) = α(ω)u(c1, c2)where α(ω) are defined in definition ??.

M ax {xi,yi}Ni=1 X ω∈Ω P (ω) N X i−1 1 N[(1 − ωi)αi(ω i−1_)u(x

i(ωi−1)) + ωiαi(ω−i)u(yi(ω−i))] (1.22)

s.t N X i=1 (1 − ωi)xi(ω) + ωiR−1yi(ω) ≤ N e (1.23) T ruth − T ellingCondition

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The same First Order Condition verifies the proof of the proposition above on problem (1.9) and the problem with history-dependent utility. We see on the next pages that for high β, the central planner chooses to increase the number of worst cases in exchange to improve the worst outcome. In such a case, the new worst scenarios have weight α as a function of the wariness condition. A high level of wariness implies a high value of α.

The first order condition of the problem with historic dependent function is the same as equa-tion (1.21). Taking the derivative of the FCO with respect to R in the neighbourhood of 1,

P (ωk)αk(ωk)u00(e)x0k(ωk) − Iωk_=θkP (θ)α_N(θN)u00(e)x0_N(θN) = X ˜ ωk_6=θk ˜ ωk =ωk P (˜ω)αN +1(ω) u0(e) + u00(e) e − 1 |ω| N X i=1 (1 − ωi)x0(ωi)

since for R = 1 xi(ω) = y(ω) = efor all i = 1, ..., N and ω ∈ Ω, we have that αi(ωi−1) = 1 + β for

all i = 1, ..., N + 1 and ω ∈ Ω. We may rewrite the function as,

P (ωk)(1 + β)u00(e)x0_k(ωk) + Iωk_=θkP (θ)(1 + β)u00(e)x0_N(θN) = X ˜ ωk_6=θk ˜ ωk =ωk P (˜ω)(1 + β) u0(e) + u00(e) e − 1 |ω| N X i=1 (1 − ωi)x0(ωi)

defining µ = _uu000(e)_(e)+ e,

P (ωk)x0_k(ωk) + Iωk_=θkP (θ)x0_N(θN) = X ˜ ωk_6=θk ˜ ωk =ωk P (˜ω) µ − 1 |ω| N X i=1 (1 − ωi)x0(ωi) ∀ ωk_{6= θ}

The above system of linear equation has a unique solution for all β > 0 as stated by Bertolai et al. (2014) on the following lemma.

Lemma 2. (Bertolai, Cavancanti and Monteiro(2014)) The derivatives x0_i(ωi), ωi _{∈ {0, 1}}i_{, ω}

i = 0 if

1 ≤ i ≤ N and ω 6= θ if k = N , satisfy the system of linear equations

P (ωk)x0_k(ωk) + Iωk_=θkP (θ)x0_N(θN) = X ˜ ωk_6=θk ˜ ωk =ωk P (˜ω) µ − 1 |ω| N X i=1 (1 − ωi)x0(ωi) for ¯ωk∈ {0, 1}k_{and ¯}_ω

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By the uniqueness of the solution, we may find solutions for a specific utility function reaching a general solution using µ. Let x∗(ωi₎N

i=1 denote the optimal allocation for an utility function

where µ = 1, then x(ωi) = ˜µx∗(ωi)is the solution to an economy with an utility function where µ = ˜µ. Besides the level change, note that the ratio between two distinct payment x(ωi) and x(ωk) is the same whenever the utility function for all ωi _{6= ω}k_{, once the utility function respects}

the assumption1. On the following sections, we consider homothetic economies since they have closed form derivatives.

1.2.1 Worst outcome

By lemma1, we may find the worst event in a wary environment economy looking at the economy without wariness. Consider an economy with homothetic utility function u(x) = x_1−σ1−σ with σ > 1. Taking β = 0, the optimization problem presented in the last section is identical to the problem in Ennis and Keister (2009) and Bertolai et al. (2014). It is possible to find payment for each depositor for all ω ∈ Ω solving the problem recursively.

By equation (1.13), the second-date payment is a function of R, the quantity of type-1 an-nouncements and the remaining resources after the event ω, z(ω) ≡

N e −PN

i=1xi(ωi−1)

. Note that every patient depositor receives the same payment for a given ω ∈ Ω, yi(ω) = y(ω). The value

function after all announcements is the utility in consuming y(ω) times the number of patient depositor.

VN +1(ωN) ≡ |ω|u(y(ω)) = |ω|σR(1−σ)u(z(ω)) (1.24)

where f1(ω) ≡ |ω|R

(1−σ) σ .

The value function on the last position may be represented by the history of events until posi-tion N , ωN −1. VN(ωN −1) = M ax c≥0 {p[u(c) + f1(ω N −1_{, 0)}σ_u(z(ωN −1_{) − c)] + (1 − p)f} 1(ωN −1, 1)σu(z(ωN −1))} (1.25) where, z(ωN −1) = N e − N −1 X i=1 (1 − ωi)xi(ωi−1) (1.26)

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The optimal payment for an impatient depositor on last position, c, is a function of reserve z(ωN −1_).4

c = 1

1 + f1(ωN −1, 0)

z(ωN −1) (1.27)

Replacing the last equation on (1.25), we may represent the value function for the last position as a function of the term f1.

VN(ωN −1) = [p(1 + f1(ωN −1, 0))σ+ (1 − p)f1(ωN −1, 1)σ]z(ωN −1) = f2(ωN −1)σz(ωN −1) (1.28)

where f2(ωN −2)σ = [p(1 + f1(ωN −1, 0))σ + (1 − p)f1(ωN −1, 1)σ]. The value function on position

N − 1can be represented in a similar form as equation (1.25) and so on until the value function at first position. In a similar way, we find the term fnlooking forward,

fn(ω) = [p(1 + fn−1(ω, 0))σ+ (1 − p)fn−1(ω, 1)σ]

1

σ (1.29)

and the outcome for impatient depositor.

xi(ωi−1) =

1

1 + fN −i+1(ωi−1, 0)

z(ωi−1) (1.30)

By equation (1.26), we may define the first-date payments and the second-date payments as a function of the total resources of the economy and the terms f (ω) ∀ ω ∈ Ω defined above. Knowing each payment, it is possible to find which event provides the lowest payment and consequently the infimum of the utility.

The first impatient depositor to show up receives a proportion of the total resources of the economy, Y = N e. The proportion depends of the number of type-1 announcements so far, xi(ωi−1) = _1+f 1

N +1−i(ωi−1)Y where i denote the position the first impatient showed up. The subse-quent payment for an impatient depositor is a function of the number of patient depositor and the bank’s reserve, z(ωj−1_{) = Y − x}

i, where j denotes the position the second impatient showed up.

Replacing the first payment in the reserve function, we may represent the payment to the second impatient depositor to shows up as a function of the total resources of the economy as well.

xj(ωj−1) = 1 1 + fN +1−j(ωj−1) z(ωj−1) = 1 1 + fN +1−j(ωj−1) fN +1−i(ωi−1) 1 + fN +1−i(ωi−1) Y 4_{In order to see the algebra in details, please, see Bertolai et al. (}₂₀₁₄_).

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for all j > i such that only ωi and ωj are equal to zero. We may represent the second payment to

an impatient depositor as a function of the payment to the first type-0 depositor.

xj(ωj−1) =

fN +1−i(ωi−1)

1 + fN +1−j(ωj−1)

xi(ωi−1) (1.31)

Note that the ratio of any two consecutive payments to impatient depositors, with or without type-1 announcement between them, depends on the ratio of the proportion of the associated term f. We demonstrate in Lemma6in the appendixA.1that this ratio, fN +1−i(ωi−1)

1+fN +1−j(ωj−1), is always lower than one if there is no type-1 announcement between them. In other words, if there is no patient depositor between two type-0 announcement, the second impatient depositor always receives less than the first one to show up. A candidate of worst payment to impatient depositor is the last payment on the queue. The following proposition states that the worst event to an individual who claims to be impatient is to receive the last payment during a bank run.

Proposition 2. The lowest payment to a type-0 depositor is the N -th payment in a history of events with

only type-0 announcements.

Knowing the result from the proposition above, the subsequent question is what event is the worst for a patient announcement. To answer this question, we shall write the second-date pay-ment as a function of N e and the term f (ω) as well. Without loss of generality, we write the second-date payment above considering that there is only one patient depositor.

y(ei) = RN e i−1 Y j=1 fj(1) 1 + fj(1) N Y j=i+1 fj(0) 1 + fj(0) (1.32)

Where eiis a vector of the canonical base of RN.

Bank pays a value greater than the initial deposit to type-0 depositor due to the belief of the existence of patient depositor. Patient depositors accept to receive less than Re on the second period in order to receive something greater than e in the case of being impatient. In other words, patient depositors "sacrifices" consumption in order for impatient ones receive more.

The above function shows us that saving decreases as an individual announces to be type-1, since the term f increases in the number of type-1 announcements. Note that, as sooner the first patient depositor shows up, sooner the saving decreases which imply a lower payment to

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the patient depositor. However, if more than one type-1 depositor shows up, the second-date payment increases because more resources are applied at the technology R. The worst payment for a type-1 depositor is received when he is the first on the queue followed only by impatient depositors.

Proposition 3. The patient depositor receives the lowest payment when he is the first on queue followed by

impatient depositors only.

To find the worst ex-ante event to an individual, we must compare the worst payment for the type-0 and type-1 depositors. The proposition below shows us that the payment on the worst event for an impatient depositor is always lower than the worst payment for a patient depositor.

Proposition 4. In an economy with sequential-service and without wariness, the payment for the last

depositor in a bank run is strictly the lowest payment.

We easily conclude the proposition above comparing the second-date payment to the first depositor on queue with the first-date payment to the last one on the queue, both payments in history e1. Once R > 1, y(e1) > xN(e1), and by proposition2, xN(e1) > xN(θ).

1.3 Wariness Effect

Knowing the infimum of the utility, we may solve the problem following the same algorithm in the last section, recursively. By feasibility constraint, the second-date payment is a function of the reserve z(ω) = Y −PN

i=1(1 − ωi)xi(ωi)and the number of depositor who wait to consume in the

second period, |ω|=PN

i=1ωi.

y(ω) = R

|ω|z(ω) (1.33)

The value function on date two is pinned down by the sequence of announcements. Using a Constant Relative Risk Aversion (CRRA) utility function, the value function is,

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Considering g(ω)σ _{= |ω|}σ_α

N −1(ω)R1−σ, the value function for the last depositor in the queue

is represented by the following equation,

VN(ωN −1, β) = M ax

c≥0 {p[αN +1(ω

N −1_)u(c)+g

1(ωN −1, 0)σu(z(ωN −1)−c))]+(1−p)g1(ωN −1, 1)σu(z(ωN −1))}

(1.35) Where c is the payment for the last impatient depositor on the queue and z(ωN −1) is the bank reserve after the history of announcements ωN −1 _{as defined by equation (}_1.26_{). Resolving the}

above maximization, we find a similar expression as Ennis and Keister (2009) and Bertolai et al. (2014). c = 1 1 + f1(ωN −1) z(ωN −1) (1.36) where f1(ωN −1) = g1(ω N −1₎ αN +1(ωN −1) 1 σ.

Note that the solution is identical as the one found on last section with β = 0. Recursively, we find the value function for impatient in i-th position as well.

Vi(ωi−1, β) = M ax

c≥0 {p[u(c) + gN −i+1(ω

i−1_{, 0)}σ_u(z(ωi−1_{) − c)] + (1 − p)g}

N −i+1(ωi−1, 1)σu(z(ωi−1))}

(1.37) where gN −i+1(ωi−1)σ = {p[α(ωi−1)

1

σ + g_{N −i}(ωn−1, 0)]σ + (1 − p)g_{N −i}(ωn−1, 1)σ}.

The withdraw rule has two main terms. The term g defined by the last equation represents the savings with wary agents, while the term f represents the upper bound of the term g, it is defined as Ennis and Keister (2009).

By equation (1.36), we can state the reserve function z(θ) after each position i, 1 ≤ i ≤ N , in a bank-run event. zi(θi−1) = Y N Y j=N −i+1 fj(θj−1) 1 + fj(θj−1) (1.38)

For an economy with CRRA utility with an aversion risk coefficient bigger than 1, the last agent in a bank run receives the lowest possible payment, proposition4. Once the last payment in a bank run is strictly lower, fj(θj−1) = gj(θj−1)for all j 6= N or ω 6= θ for j = 1, ..., N which

implies the following reserve function at position i.

zi(θi−1) = Y N Y j=N −i+1 gj(θj−1) 1 + gj(θj−1) (1.39)

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Once, by definition, gn(ω) ≥ fn(ω)for all n ≤ N , the above equation represents the increase of

savings by banks when depositors are wary.

Lemma 3. Wariness has a positive effect on term gn(θ)for n>1 in the neighbourhood of β = 0.

∂gn(θ) ∂β β=0 = n Y j=2 gj(θ) 1 + gj−1(θ) (1−σ) pn−1 pN 1 σ > 0 (1.40)

For ω 6= θ the effect of wariness is null for all positions.

By lemma 3above, banks increase saving when depositors are wary, which implies that the last one in a bank-run event receives more in a wary environment than without wariness. Another result is that bank pays less for all other depositors who announce to be type-0.

Proposition 5. In an environment of low level of wariness, all First-date payments reduce in the worst

event except for the lowest one.

Proof. By equation (1.36), the i-th impatient depositor receives in the worst event.

xi(θ) =

1 1 + fN −i+1(θ)

z(θi−1) (1.41)

Taking the partial derivative of xi in relation to wariness coefficient, it is possible to see that it

has a conflicting effect on bank-run payment. ∂xi(θ) ∂β = − ∂fN −i+1(θ) ∂β 1 (1 + fN −i+1(θ))2 z(θi−1) +∂z(θ i−1₎ ∂β 1 1 + fN −i+1(θ) (1.42)

The first term represents the effect of wariness on proportional payment in respect to z(θi−1_),

while the second term represents the effect on reserve z(θi−1). The first term is negative since the partial derivative of f (θ) in respect to wariness coefficient is positive5. Bank pays a smaller proportion of the total reserve z(θi−1₎_{to the i-th depositor to increase savings.}

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It is immediate that wariness has a positive effect on the reserve z(θi−1_{). The second term}

confirms the idea since the partial derivative of z(θi−1₎ _{in respect to β is positive (in the}

neigh-bourhood of β = 0). ∂z(θi−1) ∂β β=0 = z(θi−1) N X j=i+1 ∂fj ∂β 1 fj(θi−1) − 1 1 + fj(θi−1) > 0 (1.43)

where the inequality occurs due to the effect of wariness on term f, by proposition16on appendix

A.3.

We can state that the negative effect on the rate of total reserve is greater than the positive effect on total reserve. Note that the first payment is a function of the economy’s total resource, Y . The partial derivative of x1 in respect to β is negative since he has only the first term of equation

(1.42). ∂x1(θ) ∂β β=0 = −∂fN(θ) ∂β β=0 1 (1 + fN(θ))2 Y < 0 (1.44) We may find the effect of wariness on first-date payment at any position as a function of the payment to the immediately before type-0 depositor. By equation (1.31),

∂xn(ωn−1) ∂β = ∂fn−2 ∂β (1+fN −n+1)− ∂fN −n+1 ∂β fn−2 xn−1(ωn−1) (1 + fN −n+1)2 + fN −n+2(ω N −j₎ 1 + fN −n+1(ωn−1) ∂xn−1(ωn−1) ∂β in the neighbourhood of β = 0, there is only one lowest payment, to the last one on the queue in a Bank-Run. For any θn−1_{6= θ}N −1_{, we have the following expression,}

∂xn(θn−1) ∂β = ∂gn−2 ∂β (1+gN −n+1)− ∂gN −n+1 ∂β gn−2 xn−1(θn−1) (1 + fN −n+1)2 + gN −n+2(ω N −j₎ 1 + gN −n+1(ωn−1) ∂xn−1(θn−1) ∂β note that the signal of the last term on the right side of the equation depends on the effect of wariness on previously payment. As we have seen, the first payment is negatively affected by wariness, which implies that the last term applied to n = 2 is negative. In order to find the signal of the term inside brackets, we must use lemma3.

∂gN −n+2 ∂β β=0 (1 + gN −n+1) − ∂gN −n+1 ∂β β=0 gN −n+2 = p(1 + gN −n+1) σ gσ_{N −n+2} − 1 gN −n+2 ∂gN −n+1 ∂β β=0 = − (1 − p)g2(1) σ gσ₃ gN −n+2 ∂gN −n+1 ∂β β=0 < 0

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where the last equality is due the definition of term g. We find that the second First-date payment is negatively affected by wariness as well, which implies that the second term on the correspon-dent function for n=3 is negative and so on. We guarantee by induction that all first-date payments in a Bank-Run decreases in the neighbourhood of β = 0.

Knowing that the last on the queue receives the remaining resource, we can affirm that partial derivative of xN in respect to β is positive.

∂xn(θ) ∂β β=0 = ∂z(θ N −1₎ ∂β β=0 > 0

For any ω 6= θ, as the event is not the worst possible, wariness does not affect the share of total reserve that a specific depositor receives. The effect on total payment is non-negative since bank saves when it thinks it is in a Bank-Run event. The bigger saving in the history of event ω 6= θ where the first n depositors are impatient, ωn= 0induces a higher payment for type-1 depositors.

Proposition 6. Wariness has a non-negative effect on second-date payment.

Proof. Assume without loss of generality ω ∈ Ω such that ωj = 1and ωi = 0for all i = 1, ..., N and

i 6= j. We may represent the second-date payment by equation (1.33),

y(ω) = R 2 Y −X i6=j xi = RY j−1 Y l=1 fl(1) 1 + fl(1) N Y l=j+1 fl(0) 1 + fl(0) (1.45)

which implies the following effect of wariness on y(ω),

∂y(ω) ∂β = RY j−1 X k=1 ∂fk(1) ∂β 1 (1 + fk(1))2 j−1 Y l=1 l6=k fl(1) 1 + f1(1) N Y l=j+1 fl(0) 1 + fl(0) +RY N X k=j+1 ∂fk(0) ∂β 1 (1 + fk(0))2 j−1 Y l=j+1 l6=k fl(0) 1 + f1(0) j−1 Y l=1 fl(1) 1 + fl(1) (1.46)

the effect of wariness on second-date payment depends on the effect of wariness on term fk(1)

and fk(0). By proposition16, wariness has no effect on term f when at least one patient depositor

shows up. It represents that banks stop to save for the last on queue once bank knows it is not happening a Bank-Run event. Otherwise, wariness has a positive effect on term f when no patient depositor shows up, ∂f (0)_∂β > 0. It implies that wariness has a non-negative effect on second-date

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payment. Note that for all ω ∈ Ω such ω1 6= 1 the effect is positive. However, for ω ∈ Ω such that

ω1 = 1the effect is null.

The following tables exemplify the effect of wariness on payment for all depositors. Note on table 1 that last type-0 agent has a bigger payment with a bigger wariness coefficient. The second and third depositor have their payment decreased at the first moment while the first on queue always receives less as wariness level increases. About the second-date payment, note that it increases with wariness as well which is true since banks save more in a wary environment. The same reason why the last on bank run receives more. The first on queue "sacrifices" consumption to provide liquidity to the bank.

ω = {0, 0, 0, 0, 1}& {R, δ, e} = {1.05, 2, 1} β x1 x2 x3 x4 y5 0.000 1.009 1.006 1.002 0.996 1.033 0.005 1.007 1.002 0.996 0.996 1.046 0.010 1.004 0.998 0.998 0.998 1.048 0.015 1.000 0.999 0.999 0.999 1.049 0.020 1.000 1.000 1.000 1.000 1.050

TABLE 1.1: Sequence of payments in bank run with different levels of wariness.

ω = {0, 0, 1, 0, 0}& {R, δ, e} = {1.05, 2, 1} β x1 x2 y3 x4 x5 0.000 1.009 1.006 1.025 1.006 1.000 0.005 1.007 1.002 1.027 1.008 1.002 0.010 1.004 0.998 1.029 1.011 1.004 0.015 1.000 0.999 1.030 1.012 1.005 0.020 1.000 1.000 1.030 1.012 1.006

TABLE 1.2: Sequence of payments in selected event with different levels of wariness.

In the extreme case, with a sufficiently high β, all payments in a bank-run event converge to the initial deposit, e. We can see that in the last row of table 2. Note that only impatient depositors who show up after the type-1 announcement receives a payment bigger than e. The worst payment increases as wariness rises once bank tries to save resources for the last on the queue. In some point, the worst and second worst payment become equal. In such point, the correspondent historic dependent function, α, is equal to the corresponding term of g.

xN −1(θN −2) = xN(θN −1) 1 1 + g2(θN −2)/αN(θ) 1 σ z(θN −2) = g2(θ N −2_)/α N(θ) 1 σ 1 + g2(θN −2)/αN(θ) 1 σ z(θN −2) g2(θN −2) = αN(θN −2) (1.47)

With a even higher β, both payments increase until xN −2(θN −3) = xN −1(θN −2). We can see

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FIGURE1.1: Wariness effect with 5 depositors.

By equation (1.47), we have that,

1 1 + g3(θN −3)/αN −1(θ) 1 σ z(θN −3) = 1 2 g3(θN −3)/αN −1(θ) 1 σ 1 + g3(θN −3)/αN −1(θ) 1 σ z(θN −3) 2αN −1(θN −3) 1 σ = g₃(θN −3)

With a straightforward calculation, we find that in the limit exist β such as the first payment is equal to the second payment and so on. In this case, we have that,

(N − 1)α1(θ1)

1

σ = g_N(θ1)

which implies that,

x1(θ1) = 1 1 + gN(θ1)/α1(θ1) 1 σ N e = 1 1 + (N − 1)N e = e

In conclusion, there are two interest neighborhoods of β to analyse the behaviour of depositors. The first one when β is sufficiently small such that the last participant in a bank-run receives the strictly lower payment and the second one when β is big enough such that everyone receives the initial deposit in a bank-run event and a payment at least equal to the initial deposit in any other event.

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1.4 Low level of Wariness

An economy with low level of wariness has the property that strictly lowest payment remains as the strictly lowest payment. In order words, the level of wariness is not enough to increase the worst outcome to the same level as the second worst. Applying this concept in the economy with homothetic preferences, we conclude that the last agent on bank-run still receives the strictly lowest payment in equilibrium.

1.4.1 Disclosure of position and low level of wariness.

If depositors know about their position, the worst event for a type-0 announcement is the event which all agents before him announce to be impatient as well. For a type-1 preference, the worst event is being followed by impatient depositors only.

Eω[u(y(ω−i))] + βu(y(ei)) ≥ Eω[u(xi(ωi−1))] + βu(xi(θi−1)) (1.48)

The effect of wariness on Truth-Telling condition with disclosure is pinned down by the po-sition. Define a function Ti(R, β) that represents the difference between the expected utility to

announce correctly or not the type at position i when depositor is wary.

T (R, β) = X

ω∈Ω

P (ω)[u(xi(ωi)) − u(y(ω−i, 1))] + β[u(xi(θi) − u(yi(e1)))] (1.49)

The function evaluated at point (1, β) is equal to zero since all payments are equal to the initial deposit. The effect of wariness on Truth-Telling condition in a low return economy with disclosure about positions is evaluated by the signal of the derivate of T (R, β) in respect to R and β. If the signal is negative, we conclude that wariness relax the TT in a low return economy, on the other side, we say that wariness tights the truth-telling condition.

Note that the incentive compatible condition depends on the worst event given the announce-ment. It indicates that for the last positions, where the worst payment occurs with a type-0 an-nouncement, the TT is relaxed. However, for the first positions, the effect may be the opposite.

Proposition 7. Wariness relaxes Truth-Telling condition at the last positions.

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When the depositor in i-th position believes that everyone else is running, he runs if the first-date payment in position i is greater than the second-first-date payment, which is precisely the worst case for type-0 and type-1, respectively. Visually, the condition to not run is identical to the related literature.

u(yi(ei)) + βu(y(ei)) ≥ u(xi(θi−1)) + βu(xi(θi−1))

u(yi(ei)) ≥ u(xi(θi−1)) (1.50)

Due to assumption 1, the above inequality holds whenever the payment at t = 2 is greater or equal to the payment at t = 1 for the i-th depositor. Define a function Fi : R2 → R that

represents the difference between second-date and first-date payments for the i-th depositor, Fi(R, β) = yi(ei) − xi(θi−1). We say that wariness reduces parameters with bank-run equilibrium

if ∂Fi(R,β) ∂β > 0due to assumption 1. ∂Fi(R, β) ∂β = ∂yi(ei) ∂β − ∂xi(θi−1) ∂β > 0 (1.51)

for all i = 1, ...N . The strict inequality is an immediately result of the fact that wariness has a non-negative effect on second-date payment and a negative effect on first-date payment for all depositor except the last one, propositions5and6. For the last one is easy to see that the effect of wariness on y(.) is bigger than the effect on x(.) once R > 1. For the first one, although wariness has no effect on the second-date payment the effect of wariness is negative on first-date payment.

Theorem 1. With disclosure about the position, wariness does not incentive Bank-Run equilibria at

posi-tion i for all i = 1, ..., N .

For the last on bank run, the wariness intensifies the gain to wait. It reinforces the conclusion of Green and Lin (2003) that there are no bank-run equilibria when bank disclosure position to depositors and their types are independent.

1.4.2 No disclosure of position and low level of wariness.

Without disclosure about their type, the TT condition when depositors are wary has two addi-tional terms. The worst outcome when depositor tells the truth, M in

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payment if he lies his type, M in ω∈Ω x(ω N −1_{, 0).} Eω 1 N N X i=1 u(yi(ω−i, 1)) +βu M in ω∈Ω yi(ω−i, 1) ≥ E_ω 1 N N X i=1 u(xi(ωi−1, 0) +βu M in ω∈Ω xi(ω i−1_{, 0)} (1.52) By proposition 2and proposition 3, the worst event for a patient depositor is to be on the first position followed by impatient depositors only, and for an impatient depositor, the worst payment is to be the last on queue with type-0 announcements only.

Theorem 2. Wariness relaxes Truth-Telling Condition in a low return economy when depositors do not

know their positions.

Proof. In an analogous way in the case with disclosure of positions, the combined effect of R and wariness on truth-telling condition is represented by,

∂2T (1, 0) ∂R∂β = u 0 (e) N X i=1 −∂ 2_f N −i+1(0) ∂R∂β 1 1 + fN −i+1(0) e + ∂xN(θ N₎ ∂R − ∂y(e1) ∂R (1.53)

where by proposition 18 the first term inside the brackets is negative, and by proposition 4the second term inside the brackets is negative as well. These implies that wariness relaxes the Truth-Telling Condition without disclosure in a low return economy.

We may find the effect of wariness on Bank-Run equilibria as well. Define that a bank run exist if and only if F (R, β) > 0, where:

F (R, β) = 1 N N X i=1 u(xi(θi−1)) − N X i=1 u(y(ei)) + β[u(xN(θN −1)) − u(y(e1))] (1.54)

Bertolai et al. (2014) find that there are bank-run equilibria in a low return economy if the set of parameters implies a positive partial derivative of function F in respect to R = 1. In order to see the effect of wariness in bank-run equilibrium, we analyse if for the same parameter set in Bertolai et al. (2014), there are bank-run equilibria with wary depositors. In other words, we analyse the partial derivative of function F in respect to R and β at the point (R, β) = (1, 0).

Theorem 3. In a low return economy, the set of parameters with bank-run equilibria is smaller with wary

depositors.

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The effect of wariness on the bank-run condition can be summarized as the effect of R on the worst event for both types of agents in a low return economy plus the combined effect of R and β on second-date payments. Once depositor does not know his position, he compares the payment if he lies in the last position on the queue and if he tells the truth on the first position followed by a bank run. The effect of the return in the neighborhood of 1 is negatively greater in first-date payment since it exposes the economy to an illiquid problem. With a sufficiently small β, by proposition4, the lowest payment if depositor truthfully reveals his type is always bigger than the lowest payment in case of misrepresenting. The effect is intensified by the non-negative effect of wariness on the second-date payment for any ω ∈ Ω.

FIGURE1.2: Expected utility for different level of wariness.

Figure 1.2represents the effect of wariness on the decision to wait. The continuous red line represents the utility to consume at t = 1 when everyone else is running while the continuous black line represents the agent who chooses to wait until t = 2 to consume. The effect on last payments is more visible than the first ones. The expected utility to wait on the last position increases more than the expected utility to run. This difference is due to R being greater than 1,

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once both have the same level of reserve. The effect on second-date payment is not well visible in these graphs.

The dash lines represent the expected utility to run and wait when depositor has no disclosure about his position, color red and black respectively. We can see the effect stated by theorem3, as the level of wariness increases, the difference between both lines reduces.

1.5 High level of wariness

An economy with a high level of wariness has the property that no one receives an amount lower than the initial deposit in a bank-run event. As discussed at the end of section 3, the lowest payment is more significant in economies with higher level of wariness. The upper bound of the lowest payment is equal to the initial deposit, e. Note, by feasibility constraint, that in the event of a bank run someone can withdraw an amount bigger than the initial deposit if and only if someone withdraws less than e. Knowing that, with a sufficiently high β, we have a truncated economy where everybody receives the initial deposit until some patient depositor shows up.

1.5.1 Disclosure of position and high level of wariness.

If depositors know their position, they choose not to run if the expected utility when telling the truth is not lower than to misrepresent their type, considering that all other players are running.

u(y(ei)) > u(xi(θi−1)) (1.55)

As we see in the last section, the wariness term does not appear here because depositor is already considering the worst scenario, where everybody runs. In a truncated economy, the best reply when everybody is running is wait once the second-date payment is not lower than the lowest payment by lemma1. It implies that the second-date payment in an environment with a high level of wariness is not lower than the initial deposit.

Theorem 4. In a high-level wariness environment with disclosure about the position, the second-date

consumer always tells the truth if he has not known about previous announcements.

In other words, the theorem above reaffirms the GL model conclusion of no bank-run equilibria when bank discloses the position to depositors and the type is independent. The difference here

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is that we do not need to use backward induction to find the non-existence of bank-run equilibria. It occurs due to the disproportional huge afraid of receiving the worst ex-ante possible payment.

Using the proposed idea from Andolfatto et al. (2007), assume that all depositors are informed about each announcement, they choose to truthfully or not announce their type knowing the exact payment in case of lie and estimating the payment he would receive if he tells the truth followed by type-0 announcements only.

u(yi(ωi−1, 1, θN −i)) > u(xi(ωi−1, 1)) (1.56)

where ωi−1 _{is the history of announcements up to position i and θ}N −i_{is the sequence of type-0}

announcements after the i-th depositor. Note that if only type-0 announcement was made, the truth-telling condition is identical to the case where depositors do not know the past. In such a case, we have the following proposition.

Proposition 8. In a high-level wariness environment, the first type-1 consumer on queue always tells the

truth if he knows he is the first one, independently of the position.

The same cannot be said about other patient depositors, especially the second one. As Ennis and Keister (2009) argue, the type-1 agent on the last position always reveals the correct type while the penultimate only does in some set of parameters which means that we can find a partial bank-run equilibrium.

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The above graph exemplifies that there exist a set of parameters which the second type-1 agent chooses to misrepresent his type when he believes all depositors after him lie about their type. Even knowing that the first depositor correctly reveals be patient, the second one chooses to with-draw at t=1.

1.5.2 No disclosure of position and high level of wariness.

We argue that the fear to receive the lowest payment plus the knowledge about the position and, in some case, the past helps to prevent the bad equilibrium in a high level of wariness environment. In a PS model without disclosure about the position, agents look to the expected utility when everyone else runs and the utility in the worst case conditional to his type. By proposition2and proposition3, the ex-ante worst payment for a first-date consumer is to be the last in the bank-run event while for a second-date consumer is to be the first on the queue followed by only type-0 announcements. The condition of no run is represented by the difference between these four elements. 1 N N X i=1 u(y(ei)) + βu(y(e1)) ≥ 1 N N X i=1

u(xi(θi−1)) + βu(xN(θN −1)) (1.57)

Once again, in a high wariness level economy depositors receive the initial deposit back in the bad equilibrium. They choose not to run if their expected utility to consume on second-period is at least equal to the initial deposit.

1 N

N

X

i=1

u(y(ei)) + βu(y(e1)) ≥ (1 + β)u(e) (1.58)

The condition above is always respected since there is no payment inferior to the initial deposit in such economy and the condition is not biding since there is at least one event where second-date depositor withdraws more than e. As stated by Green and Lin (2003), a patient depositor in the last position always receives more if he tells the truth.

y(eN) = R(N e − N −1

X

i=1

xi(θ)) = R(N e − (N − 1)e) = Re > e (1.59)

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Theorem 5. In a high-level wariness environment without disclosure about the position, the second-date

consumer always tells the truth.

1.6 Final Remarks

In this work, we review some important results on related literature when depositors emphasize events which they have the lowest utility. We have seen that the model can be solved recursively, using the algorithm from Ennis and Keister (2009). The introduction of a position dependent term on utility allows us to find a closed form to all payments and value function. The last on queue consumption, during a bank run, increases as the level of wariness rises. On the other side, all other depositors receive a lower payment in a bank-run. It occurs as an attempt by banks to increase saving in order to pay more in the worst ex-ante event.

The truth-telling condition has an additional term when depositors fear about the worst event. Impatient and patient depositors emphasize their worst possible payment. Agents update the expected lowest outcome as they receive new information about their position and previous an-nouncement. For a type-1 depositor in position i, for example, the lower payment occurs when he is followed by announcements 0 only. For type-0, it occurs when there is no type-1 announcement before him. The condition to run with wariness when depositor has disclosure about his position is visually identical as in related literature. The no bank-run equilibria result stated by Green and Lin (2003) remains with wary depositors. Actually, once wariness has a non-negative effect on second-date payment and a negative effect on first-date payment for i < N , we conclude that wariness increases the benefits to tell the truth when others lie. The conclusion is independent of the level of wariness.

If bank does not provide information of position to depositors, wariness reduces the set of parameters with multiple equilibria. We review the finds of Bertolai et al. (2014) in the neighbour-hood of R = 1 and β = 0. The optimal allocation must attend a modified Truth-Telling condition where depositor gives an additional relevance to the worst event conditional to his announce-ment. If he claims to be an impatient depositor, the worst event is to be the last one in a history of type-0 type only. On the other side, for a type-1 announcement, the worst event is to be at the first position in a queue full of impatient depositors. Once the last is always lower than the former, the Truth-Telling Condition is relaxed when depositors are wary. The non-negative effect of wariness on second-date payment contributes to relax the TT condition as well.

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In a high-level wariness environment, we find that to correctly announce the type is the only equilibrium. Given a high β, bank offers a payment contract that pays the initial deposit back to all depositors until the first depositor claims to be patient. After that, the bank guarantees a payment at least equal to the initial deposit.

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Chapter 2

Endogenous Financial Network in a

Wary Environment

2.1 Introduction

The network is a relevant structure nowadays on an agent’s decision process. Due to the network, agents can obtain positive payoffs on many examples such as political alliances, trading relation-ships, citations, and others. We say that an individual is in a network when he has a link with someone from that network. An important kind of interconnection is the financial one. Banks or financial institutions interconnect themselves to improve welfare, profit or even to diversify investments. However, a network can be a channel to spread negative shocks. Individuals must find the optimum exposure on a network in order to improve the gain without exposing them-selves to systemic risk. The individual’s decision together with other individuals’ decisions gives us the network equilibrium.

In this paper, we seek an optimum network given depositors’ preference. We consider that agents know that there is a positive probability of some bankruptcy spreading negative effects on everyone inside the network. The planner chooses a payment allocation to the depositor and a level of exposure on other banks considering such a case. In a multiple bank environment, individuals worry about the case of bankruptcy in their bank (or queue) due to a sunspot or due to contagion. We label such individuals as wary depositors. They overemphasize the event they receive the lowest payment, which is the event where their queue bankrupts. The worst event is a salient event for the individual in the concept defined in Bordalo et al. (2013b)1_{. Araujo et al.}

(2011) defines wary depositor as an individual who overlooks gains at distant dates but not the 1_{Other related articles are Bordalo et al. (}_2012a_{), Bordalo et al. (}_2012b_{), Bordalo et al. (}_2013a_{), Bordalo et al. (}₂₀₁₅₎

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losses. Different from us, their paper has infinity-lived agents. We, otherwise, use an environment close to Diamond and Dybvig (1983) with two periods. The wary individual worries about the worst possibility of outcome in both periods.

The network structure, in our work, is formed by an interbank deposit between the queues that resemble the repo market. We work in an environment where different queues do not pool resources as in Bertolai et al. (2018). In order to exchange resources, queues (or banks) make mutual deposit insurance contracts. The interbank connections increase the payment of a bank that suffers a bad event or equilibrium at the cost of other banks’ depositors. As we discuss in this paper, the network may spread a negative shock toward all other banks. This event is defined as contagion.

There is a set of papers that talk about network and contagion, the main example is Allen and Gale (2000) and Freixas et al. (2000) but fewer subset concerns about contagion on network formation2. However, we have not seen a paper that considers wary agent on network formation with contagion.

We try to answer if a full network, denominated as a complete network, is the optimum choice with the risk of contagion. Alternatively, if a network with the lowest number of interlinkages, labeled as a ring network, is the best one to increase depositors’ expected utility or even some-thing between them. Allen and Gale (2000) try to answer the question in a Diamond and Dybvig’s environment. Given a negative shock with null probability on a ring structure network, and in a complete network structure, the authors find that a complete network is more resilient to conta-gion. FIGURE 2.1: Ring network FIGURE 2.2: Complete network

A highly interconnected financial system is more resilient to contagion as argued by Allen and Gale (2000). However, this does not mean that contagion does not occur in the complete network. In a case where a negative shock is sufficiently high to bankrupt a bank and its partners, the

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