Suspension in a Global-Games version of theDiamond-Dybvig model�y

Pidong Huang

November 10, 2011

Abstract

This work builds on the model in Goldstein and Pauzner (GP)(2005), a global-games version of the Diamond-Dybvig (DD) (1983)model in which there is uncertainty about the long-term return andin which agents observe noisy signals about that return. GP limitedtheir investigation to a banking contract that makes a noncontingentpromised payo¤to those who withdraw early until the bank�s resourcesare exhausted. We amend the contract and permit suspension. As weshow, there is a class of suspension policies that gives rise to unique-ness without requiring the new assumption introduced in a proof inGP; namely, that the short-term return is also random. In general,both the GP policy and my generalization of it to allow suspensionseem not to be best banking contracts. However, if the return un-certainty is su¢ ciently small, then there are policies in the class westudy that imply ex ante welfare close to the �rst-best outcome in DD,which, itself, is an upper bound on welfare in the model with returnuncertainty.

1 Introduction

This papers builds on the model in Goldstein and Pauzner (GP) (2005), aglobal-games version of the Diamond-Dybvig (DD) (1983) model in which

�Department of Economics, Penn State University, Email:pzh110@psu.eduyI am especially grateful to Neil Wallace for helpful guidance and encouragement. All

errors are my own.

1

there is uncertainty about the long-term return and in which agents observenoisy signals about that return.GP limited their investigation to a banking contract that makes a non-

contingent promised payo¤ to those who withdraw early until the bank�sresources are exhausted. Here, that contract is amended to permit suspen-sion. There are two reasons to do this. First, suspension works perfectlyin the no aggregate-risk version of the DD model. (It uniquely implementsthe �rst-best outcome.) Because versions of GP are close to that model,it is plausible that suspension would also work well in such versions. Sec-ond, as shown below, there is a class of suspension policies that gives rise touniqueness without requiring the new assumption introduced in a proof inGP� namely, that the short-term return is also random.In general, both the GP policy and the generalization of it that allows

suspension seem not to be best banking contracts in a model with returnuncertainty and signals about it. Viewed as mechanisms, such contractsmake no attempt to elicit information about the signals that agents receiveand to make suspension and payo¤s to at least some depositors contingenton that information. However, if the return uncertainty is su¢ ciently small,then there are policies in the class I study that imply ex ante welfare close tothe �rst-best outcome in DD, which, itself, is an upper bound on the welfarein the model with return uncertainty. Moreover, for one such policy, noisysignals are not necessary for uniqueness. In other words, the ex ante welfareproperties of the DD suspension policy are robust to the kind of uncertaintyintroduced by GP, provided it is small.The virtue of such uncertainty is that in a long sequence of .i.i.d. realiza-

tions of the model, all of the following four outcomes occur with positive prob-ability: fgood long� term return; poor long� term returng �fsuspensionnot invoked; suspension invokedg. Put di¤erently, the model with a smallamount of GP uncertainty combined with a banking contract that permitssuspension gives a unique equilibrium, does well in terms of ex ante welfare,and is able to account for a rich history of banking-system outcomes, includ-ing ones with a variety of banking-system di¢ culties� di¢ culties that arerare, but do occur.Of the above four ex post outcomes, all but one is undesirable from an ex

post perspective. One of the undesirable expost outcomes is a realization witha good long-term return, but one in which all agents attempt to withdrawearly because they received signals that make a good long-term return veryunlikely. In that case, suspension is invoked and the ex post outcome does

2

badly in matching the pattern of payments over time with the pattern of type(patient, impatient) realizations across people. The other ex post undesirableoutcomes are those with a poor long-term returns. Whether or not thisoutcome is accompanied by a realized suspension, from an ex post perspectivepeople will wonder why they deposited and why the banking system o¤eredterms that induced them to deposit.The remainder of the paper is organized as follows. The model is set out

in Section 2. Section 3 has the uniqueness results. Section 4 deals with exante welfare. Section 5 contains the proofs, while section 6 o¤ers concludingremarks.

2 Model

Our model is essentially GP, but without uncertainty about the short-termreturn. There are three dates, one good and a nonatomic unit measure ofagents. Each agent is endowed with 1 unit of good at date 0. The agentbecomes impatient with probability � and the agent becomes patient withprobability 1��. Impatient agents can consume only at date 1. They obtainutility u(c1). Patient agents can consume at either date 1 or date 2. Theyderive utility u(c1+c2). Type is i.i.d. across agents and there is no aggregateuncertainty about type. The function u(:) satis�es the following properties: itis twice continuously di¤erentiable, increasing, strictly concave, and satis�esu(0) = 0, u0(0) =1, and �cu00(c)=u0(c) > 1.A risky production technology is available to everyone. One unit of date-

0 investment yields R > 1 units of date-2 output with probability p(�) and0 with probability 1 � p(�), where �, labeled the fundamental, is uniformlydistributed over [0; 1] and p : [0; 1] ! [0; 1] and is continuous, strictly in-creasing, and satis�es p(0) = 0 and p(1) = 1. If liquidated at date 1, theinvestment pays return 1. The technology has a higher long-term return inthe sense that E�(p(�))u(R) > u(1).Figure 1 shows two examples of possible p(�) functions. Limiting cases

of each, indicated by the arrows, serve di¤erent purposes later.

3

Figure 1: p(�)

The fundamental �, which is not observed, determines a private signalreceived by each agent. The private signal is �0 = �+ "0, where "0 is indepen-dently and uniformly distributed over [�"; "]. It follows that given signal �0,� is uniformly distributed over [max(0; �0 � ");min(1; �0 + ")]1.The timeline is as follows. At date 0, the bank o¤ers a deposit contract

(x; �) 2 [1;+1) � [�; 1] with x� � 1. It gives each agent who is able towithdraw at date 1, x units of output per unit of deposit, while allowing nomore than � proportion of the agents to withdraw at date 1. For the moment,I assume that agents deposit in the bank and that all available resources areinvested2.Just prior to date 1, the fundamental, the agent types and their signals

are realized. Then agents simultaneously choose between withdraw and wait.The date-1 payo¤ from playing withdraw is

c1 =

�x with probability �=max(�; n)0 with probability 1� �=max(�; n) ;

1Because agents know that � � 1, the right end of the interval ismin(1; �0+"). Similarly,the left end is max(0; �0 � ").

2There exists a contract that is consistent with a unique equilibrium and under whichagents want to deposit rather than remain in autarky.

4

where n is the proportion who choose to withdraw. At date 2, the long-termreturn is realized. Each agent who played wait and each agent who playedwithdraw but received 0 at date 1 get the date 2 payo¤

c2 =

�R[1� xmin(�; n)]=[1�min(�; n)] if R is realized0 otherwise

:

Because each impatient agent necessarily plays withdraw, it is su¢ cientto focus on the play of the patient. I focus on symmetric strategies and allowrandomization. Thus, a strategy is y : [�"; 1 + "]! [0; 1], where y(�0) is theprobability that a patient agent with signal �0 plays withdraw. The solutionconcept is Nash equilibrium.

De�nition 1 The function y is a symmetric equilibrium if it is a best re-sponse to the play of y by all other agents.

Following the global game literature, a threshold equilibrium plays animportant role in what follows.

De�nition 2 The function y is a threshold equilibrium if there exists b� suchthat y(�0) = 1 for �0 < b� and y(�0) = 0 for �0 > b�.In what follows, I distinguish between two kinds of threshold equilibria.

Following most of the literature, I reserve the term run for a threshholdequilibrium in which b� � 1 + " and in which, therefore, y(�) � 1. Any otherthreshhold equilibrium is called interior.

3 Uniqueness

Several properties are necessary forGlobal Game results to hold in this model.The somewhat problematic property is an interval of the fundamental forwhich wait is a dominant strategy. GP get this interval by making the short-term return depend on the fundamental as follows: they assume that thereexists �̂ 2 (0; 1) such that if � � �̂, then the date 1 return is R. I get thisproperty from suspension contracts that reserve su¢ cient resources for date2 payo¤s.Let

A = f(x; �)j(1� x�)R1� � > xg, (1)

which is displayed in Figure 2.

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Figure2: Set A

If the returnR is realized, then the date-2 payo¤is no less than (1�x�)R=(1��). Therefore, if the contract is in set A, then choosing wait is a dominantstrategy for an agent who realizes a su¢ ciently high signal. That is su¢ cientto assure that the Global Game uniqueness results holds.

Proposition 1 (Uniqueness) Let (x; �) 2 A. There exists ", which candepend on (x; �), such that if " 2 (0; "], then there is a unique symmetricequilibrium and it is an interior threshold equilibrium.

To avoid the dependence of " on (x; �) (in fact, "! 0 as (x; �) approachesthe upper boundary of A), I can restrict contracts to a slightly smaller set.

Corollary 1 (" independent of (x; �)) For any k 2 (0; R� 1), let

Ak = f(x; �)j1� x�1� � R > x+ kg. (2)

There exists "k > 0, such that for any (x; �) 2 Ak, if " 2 (0; "k], then thereis a unique equilibrium and it is an interior threshold equilibrium.

Contracts within A give rise to a unique equilibrium. Outside of A, themodel always has a run equilibrium.

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Proposition 2 (Existence of run) If (x; �) =2 A, then a run equilibrium(y(�0) � 1) exists.

In general, I are not sure about uniqueness outside of set A. However,when (x; �) is near the upper boundary of A, then there exists an interiorthreshold equilibrium.

Proposition 3 (Existence of threshold) If x > 1, � > � and (x; �) isnear A, then an interior threshold equilibrium exists for a su¢ ciently smallpositive ".

The last two propositions imply that the model is consistent with multipleequilibria for contracts outside of A.Contracts in A with � = � have a unique equilibrium even without noise;

namely, even if " = 0. All other contracts in A require noise for uniqueness.

Proposition 4 (Necessity of noise) Suppose " = 0. If (x; �) 2 A and� > �, then multiple equilibria exist.

4 Optimality

Although contracts inA have a unique equilibrium, in general, our contracts�and, hence, the GP contracts� seem not be the best implementable arrang-ments. They preclude giving almost everyone some date 1 payo¤when signalsimply that 0 is a very likely date 2 return. In particular, our contract doesnot permit the bank to elicit information about the signals from agents andto make the suspension parameter and the date-1 payo¤ of at least someagents contingent on such information. Despite that, a bit can be said aboutex ante (representative-agent) welfare under our class of contracts.The �rst result says that limiting the contracts further to those with � = �

can be costly.

Proposition 5 (Desirability of � > �) There exist economies (p(:); R) forwhich the best contract in A has � > �.

As shown in the proof, one way to get such ecomomies is by using alimiting case of the p(�) function depicted by the solid curve in Figure 1. Forsuch p(�) functions, the optimal contract within set A has � > �, and, byway of proposition 4, has a unique equilibrium only in the presence of noise.

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The economies in the last proposition have a lot of uncertainty and, there-fore, are ones for which the gain from enriching the class of contracts is pre-sumably large. If I interpret the DD model to have p(�) � 1, then they arealso economies that are far from the DD economy. I next turn to economiesthat are close to the DD economy in the sense that they have p(�) functionsthat are close to p(�) � 1 (see the dotted p(�) function in Figure 1). Noticethat DD �rst-best welfare is an upper bound on ex ante welfare in our envi-ronment. The next result says that for economies close to DD in that sense,our contract has ex ante welfare close to that upper bound and, hence, isclose to being ex ante optimal.Let x� be the �rst-best date-1 consumption in DD. The assumption

�cu00(c)=u0(c) > 1 implies that (1 � �x�)R=(1 � �) � x� > 0. Hence, thereexists k 2 (0; (1 � �x�)R=(1 � �) � x�). Consider a sequence of economiesfpl(:), "lg1l=1 such that liml!1 pl(�) = 1 for � > 0 and liml!1 "l ! 0, where"l satis�es Corollary 1 for the function pl(�).

Proposition 6 (Perturbation of DD) Let k 2 (0; (1 � �x�)R=(1 � �) �x�). For any (x�; �) 2 Ak (see (2)), the limit of ex ante welfare is the�rst-best welfare in DD.

Notice that Ak contains the DD suspension contract (x�; �). Under thiscontract, I obtain uniqueness even if there is no noise� even if agents observethe fundamental directly. It follows that the DD suspension contract is robustto the introduction of a small amount of date-2 return uncertainty withoutthe introduction of noisy signals.

5 Proof

Some notations are prepared �rst. Then the proofs are presented accordingto their order in previous sections. The proof for Proposition 1 contains 3lemmas. Lemmas 1-2 show that (x; �) 2 A gives rise to properties necessaryfor Global Game results to hold in our environment. Lemma 3 applies GPGlobal Game technique.Following the Global game literature, I consider a threshold strategy, de-

noted by its cut-o¤value b�. The proportion of agents who choose to withdraw

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depends on the fundamental and the strategy:

n(�;b�) =8><>:

1 if � � b� � "� if � > b� + "

1� 1��2"(� � b� + ") if otherwise

. (3)

If an agent plays wait, she receives the lottery that gives expected utilityu([1�xmin(�; n)]=[1�min(�; n)]R)p(�) given �. If an agent plays withdraw,she receives x units of good with probability �=max(�; n), and receives thelottery with probability 1� �=max(�; n). Therefore the conditional expectedgain from choosing wait given � is

v(�; n; x; �) = fu[1� xmin(�; n)1�min(�; n) R]p(�)� u(x)g

�

max(�; n): (4)

De�ne � : A! [0; 1] and � : A! [0; 1]:

�(x; �) = p�1(u(x)

u(1�x�1�� R)

) (5)

�(x; �) = p�1(u(x)

u(1�x�1�� R)

). (6)

Note that �(:) and �(:) satisfy

v(�(x; �); �; x; �) = 0 (7)

v(�(x; �); 1; x; �) = 0. (8)

Therefore wait is the dominant action for � > �(x; �), while withdraw is thedominant action for � < �(x; �).De�ne " : A! R+ so that

"(x; �) = minf(1� �(x; �))=3; �(x; �)=3; 1=4g. (9)

Note that �(x; �), �(x; �) 2 (0; 1) for any (x; �) 2 A. Thus I have "(x; �) > 0.Let " 2 (0; "(x; �)]. The conditional distribution of � given �0 2 [��"; �+"]

is uniform over [�0 � "; �0 + "], and thus the conditional expected gain fromplaying wait is

U(�0; y; x; �) =1

2"

Z �0+"

�0�"v(�; n(�; y); x; �)d�. (10)

Lemma 1 collects some properties of our generalized contracts.

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Lemma 1 For any threshold b�, the function v(�; n(�;b�); x; �) satis�es thesingle crossing property: There exists at most one � such that v(�; n(�;b�); x; �) =0.The function U(b�;b�; x; �) is strictly increasing and continuous in b� 2

["; 1� "].If there exists b� 2 ["; 1 � "] such that U(b�;b�; x; �) = 0, then a threshold

equilibrium exists and its cut-o¤ value is b�.Proof. For any threshold strategy b�, n(�;b�) is decreasing in � and p isstrictly increasing in �. Therefore

u[1� xmin(�; n(�;b�))1�min(�; n(�;b�)) R]p(�)� u(x) (11)

is strictly increasing in � and is equal to 0 for at most one �. Then v(�; n(�;b�); x; �)is equal to 0 for at most one �.U(b�;b�; x; �) is is strictly increasing in b� 2 ["; 1 � "], because when both

the private signal and the threshold strategy increase by the same amount,an agent�s belief about how many other agents withdraw is unchanged, butthe return from waiting is higher.U(b�;b�; x; �) = 0 implies that v(�; n(�;b�); x; �) crosses zero once. Observ-

ing signal �0 below b� shifts probability from positive values of v to negativevalues. Thus U(�0;b�; x; �) < U(b�;b�; x; �) = 0. Similarly, signals above b�implies U(�0;b�; x; �) > U(b�;b�; x; �) = 0.Lemma 1 will be used to show the existence of a threshold equilibrium

both in A and out of A, while lemmas 2-3 contain properties unique to A.

Lemma 2 Let (x; �) 2 A and " 2 (0; "(x; �)]. The function U has twodominant intervals: If �0 � �(x; �)+" (�0 � �(x; �)�"), then U(�0; y; x; �) > 0(U(�0; y; x; �) < 0) for any strategy y.

Proof. If �0 � �(x; �) � ", then it is almost surely the fundamental � <�(x; �). Note that u[1�xn

1�n R]p(�)�u(x) is strictly decreasing in n and strictlyincreasing in �. Then it is almost surely

u[1� xmin(�; n)1�min(�; n) R]p(�)� u(x) (12)

< u[1� x�1� � R]p(�)� u(x) = 0, (13)

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which implies v(�; n; x; �) < 0 for all n 2 [�; 1]. Hence U(�0; y; x; �) < 0 forany strategy y. Similarly if �0 � �(x; �) + ", I have U(�0; y; x; �) > 0.Lemmas 1-2 together imply the existence of a unique threshold equilib-

rium in A, while lemma 3 precludes other possibilities.

Lemma 3 Let (x; �) 2 A and " 2 (0; "(x; �)]. Then any equilibrium is athreshold equilibrium.

Proof. Suppose by contradiction that y is an equilibrium, but not a thresholdstrategy. De�ne

�0B = supf�0 : U(�0; y; x; �) < 0g: (14)

By Lemma 2, if �0 � �(x; �) � ", then U(�0; y; x; �) < 0 for any strategy y.Therefore set f�0 : U(�0; y; x; �) < 0g is not empty, and hence �0B is well-de�ned. Furthermore, if �0 � �(x; �)+", then U(�0; y; x; �) > 0. Hence I have�0B � �(x; �) + " � 1� 2". Since �(x; �) + " � 1� 2", I have �0B � 1� 2".Since y is not a threshold equilibrium, there exists �0 < �0B such that

U(�0; y; x; �) > 0. Hence �0A is well de�ned:

�0A = supf�0 < �0B : U(�0; y; x; �) > 0g. (15)

By Lemma 2, if �0 � �(x; �) � ", then U(�0; y; x; �) < 0. Hence I have�0A � �(x; �) � ". Because �(x; �) � " � 2", I have �0A � 2". Because U iscontinuous at �0A and �

0B, I have

U(�0A; y; x; �) = U(�0B; y; x; �) = 0. (16)

I decompose the intervals over which the above two integrals are com-puted into a common part, two disjoint parts. Since �0A 2 [2"; 1 � 2"], thefundamental � is uniformly distributed over [�0A � "; �0A + "] given signal�0A. Similarly, � is uniform over [�0B � "; �0B + "] given �0B. Then I decompose[�0A�"; �0A+"] and [�0B�"; �0B+"] and let c = [�0A�"; �0A+"]\ [�0B�"; �0B+"],[�01; �

02] = [�

0B�"; �0B+"]nc and [

��02 ; ��01 ] = [�

0A�"; �0A+"]nc. De�ne the "mirror

image" transformation �� :[�01; �

02]! [

��02 ; ��01 ] so that

�� = �0A + �

0B � �. (17)

I can rewrite (16)Z �02

�01

v(�; n(�); x; �)d� =

Z �02

�01

v( �� ; n(

�� ); x; �)d�. (18)

I assume �0B � �0A 2 ("; 2") in Figure 3. But the proof is general.

11

Figure 3

I will show that (18) cannot hold. But before, I collect some intermediateresults about both sides of (18).Let �(:) be inverse of the restriction of n(:) in [�01; �

02]. De�ne the "inverse"

of n( �� ):�!� (n) = inff� : n( �� ) < n or � = �02g.

Claim 1 n(:) decreases in [�01; �02] at the rate of (1��)=2"; n(:) increases in

[ ��02 ; ��01 ] at the rate less than (1� �)=2"; For any �0 2 [

��01 ; �

01], n(�

0) � n(�01);n(�02) � n(

��02) � n(�01); For all �0 2 [�01; �02],

��0 < �01 and n(

��0 ) � n(

��02); �(:)

is strictly decreasing; n( ����!� (n)) = n and

�!� (n) is strictly decreasing.

The proof of claim 1 is as follows. As for the LHS of (18), (14) and (15)imply that U(�0; y; x; �) < 0 for �0 2 (�0A; �0B) and that U(�0; y; x; �) > 0 for�0 > �0B. Thus, as the fundamental � increases from �01 to �

02, patient agents

who observe signals �0 > �0B and choose wait replace patient agents whoobserve signals �0 2 (�0A; �0B) and choose withdraw. Hence n(:) decreases in[�01; �

02] at the rate of (1��)=2". As for the RHS of (18), I have U(�0; y; x; �) <

0 if �0 2 (�0A; �0B), but the sign of U(�0; y; x; �) in [�0A� "; �0A) is undeterminedin Figure 3. Therefore, as the fundamental � decreases from

��01 to

��02 , patient

agents who choose withdraw are replaced by patient agents who may or may

not choose wait. Hence n(:) increases in [ ��02 ; ��01 ] at the rate less than (1 �

�)=2". If I move � from ��01 to �

01 and follow the similar argument, then for

any �0 2 [ ��01 ; �

01], n(�

0) � n(�01). If n( ��02) > n(�

01), then I have n(�

0) > n(�01)

for all �0 2 [ ��02 ; ��01 ] and (18) does not hold, a contradiction. Because n(:) is

12

strictly increasing in [�01; �02], �(:) is strictly decreasing. Note that n(

�� ) is

decreasing in [�01; �02].�!� (n) is strictly decreasing.

I now turn to show that (18) does not hold. I rewrite the LHS of (18):Z �02

�01

v(�; n(�); x; �)d� =

Z n( ��02)

n(�01)

v(�(n); n; x; �)d�(n) (19)

+

Z n(�02)

n( ��02)

v(�(n); n; x; �)d�(n).

Note that ����!� (n) � �01 < �(n) for all n 2 [n(

��02); n(

��01)). Because v(�; n; x; �)

is strictly increasing in �, I haveZ n( ��02)

n(�01)

v(�(n); n; x; �)d�(n) >

Z n( ��02)

n(�01)

v( ����!� (n); n; x; �)d�(n). (20)

I rewrite the RHS of (18) asZ �!� (n(�01))�01

v( �� ; n(

�� ); x; �)d� +

Z �02

�!� (n(�01))

v( �� ; n(

�� ); x; �)A(�)d� (21)

+

Z �02

�!� (n(�01))

v( �� ; n(

�� ); x; �)(1� A(�))d�,

where

A(�) =

(1 if @n(

�� )

@�9 and < 0

0 otherwise. (22)

If�!� (:) is di¤erentiable with @

�!� (:)@n

< 0 at n, by de�nition of�!� (n), then n( �� )

is also di¤erentiable with @n( �� )

@�< 0 at

�!� (n). Note that

�!� (:) is strictly

decreasing and, hence, is almost everywhere di¤erentiable with @�!� (:)@n

< 0.

Thus I have A(�!� (n)) = 1 almost everywhere.

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The second integral in (21) isZ �02

�!� (n(�01))

v( �� ; n(

�� ); x; �)A(�)d�

=

Z n( ��02)

n(�01)

v( ����!� (n); n; x; �)A(

�!� (n))d[

�!� (n)� �(n)] (23)

+

Z n( ��02)

n(�01)

v( ����!� (n); n; x; �)d�(n).

(23) holds because A(�!� (n)) = 1 almost everywhere.

Combining (19), (21), (23)and (20), I haveZ n(�02)

n( ��02)

v(�(n); n; x; �)d�(n) <

Z �!� (n(�01))�01

v( �� ; n(

�� ); x; �)d� (24)

+

Z n( ��02)

n(�01)

v( ����!� (n); n; x; �)A(

�!� (n))d[

�!� (n)� �(n)]

+

Z �02

�!� (n(�01))

v( �� ; n(

�� ); x; �)(1� A(�))d�.

I will reach a contradiction against (24) in what follows.The integrals in RHS of (24) have the same length as the integral in the

LHS of (24):Z �!� (n(�01))�01

d� +

Z n( ��02)

n(�01)

A(�!� (n))d[

�!� (n)� �(n)] +

Z �02

�!� (n(�01))

(1� A(�))d�

=

Z �02

�01

d� �Z �02

�!� (n(�01))

A(�)d� (25)

+

Z n( ��02)

n(�01)

A(�!� (n))d

�!� (n)�

Z n( ��02)

n(�01)

A(�!� (n))d�(n)

=

Z �02

�01

d� �Z n(

��02)

n(�01)

A(�!� (n))d�(n) (26)

=

Z n(�02)

n( ��02)

d�(n). (27)

14

Integration by substitution and�!� (n(

��02)) = �

02 implyZ �02

�!� (n(�1))

A(�)d� =

Z n( ��02)

n(�1)

A(�!� (n))d

�!� (n),

and hence (26).R �02�01d� =

R n(�02)n(�01)

d�(n) and A(�!� (n)) = 1 imply (27).

The weights on value of v in both sides of (24) are nonnegative. By

claim 1, I have n( ��02) � n(�02). Because �(n) is decreasing in n, the LHS of

(24) can be treated as a regular integral with nonnegative weight on value

v(�(n); n; x; �). By claim 1, n(�01) � n( ��02) and

�!� (n)��(n) is decreasing in n,

the second integral in the RHS of (24) is a regular integral with nonnegative

weights on value v( ����!� (n); n; x; �). Also n(�01) � n(

��02) implies

�!� (n(�01)) <

�02. Hence the third integral in the RHS of (24) is a regular integral withnonnegative weight on value v(

�� ; n(

�� ); x; �). Similar statement holds for

the �rst integral.

The remaining is divided into two cases: v(�(n( ��02)); n(

��02); x; �) � 0 and

v(�(n( ��02)); n(

��02); x; �) < 0. Suppose v(�(n(

��02)); n(

��02); x; �) � 0. Because

n( ��02) > n(�

02)3, [n(�02); n(

��02)) is well-de�ned. Note that �(n) is strictly de-

creasing in n. Because v(�; n; x; �) is strictly increasing in � and strictly de-creasing in n when v(�; n; x; �) � 0, the values of function v in the LHS of (24)is strictly greater than v(�(n(

��02)); n(

��02); x; �): v(�(n); n; x; �) > v(�(n(

��02)); n(

��02); x; �)

for all n 2 [n(�02); n( ��02)):

Note that ��0 < �01 and n(

��0 ) � n(

��02) for all �

0 2 [�01; �02]. Therefore thevalues of function v in the RHS of (24) is strictly less than v(�(n(

��02)); n(

��02); x; �):

v( ��0 ; n(

��0 ); x; �) < v(�(n(

��02)); n(

��02); x; �) for �

0 2 [�01; �02), and v( ����!� (n); n; x; �) <

v(�(n( ��02)); n(

��02); x; �) for n 2 (n(

��02); n(�

01)].

3If n( ��02) = n(�

02), both sides of inequality (24) are equal to zero, a contradiction.

15

Then I have a contradiction to (24):Z n(�02)

n( ��02)

v(�(n); n; x; �)d�(n)

> v(�(n( ��02)); n(

��02); x; �)

Z n(�02)

n( ��02)

d�(n) (28)

= v(�(n( ��02)); n(

��02); x; �)f

Z �!� (n(�01))�01

d� (29)

+

Z n( ��02)

n(�01)

A(�!� (n))d[

�!� (n)� �(n)]

+

Z �02

�!� (n(�01))

(1� A(�))d�g

>

Z �!� (n(�01))�01

v( �� ; n(

�� ); x; �)d� (30)

+

Z n( ��02)

n(�01)

v( ����!� (n); n; x; �)A(

�!� (n))d[

�!� (n)� �(n)]

+

Z �02

�!� (n(�01))

v( �� ; n(

�� ); x; �)(1� A(�))d�.

(28) holds because v in the LHS of (24) is strictly greater than v(�(n( ��02)); n(

��02); x; �),

while (30) holds because v in the RHS is strictly less than v(�(n( ��02)); n(

��02); x; �)

and has nonnegative weights.

Now suppose v(�(n( ��02)); n(

��02); x; �) < 0, which implies

u[1� xmin(�; n(

��02))

1�min(�; n( ��02))

R]p(�(n( ��02)))� u(x) < 0.

Because ��0 < �01 and n(

��0 ) � n(

��02) for all �

0 2 [�01; �02] in claim 1, I have

u[1� xmin(�; n(

��0 ))

1�min(�; n( ��0 ))

R]p( ��0 )� u(x)

< u[1� xmin(�; n(

��02))

1�min(�; n( ��02))

R]p(�(n( ��02)))� u(x),

16

and, thus, v( ��0 ; n(

��0 ); x; �) < 0 for �0 2 [�01; �02]. Similarly, I have v(

����!� (n); n; x; �) <

0 for n 2 [n( ��02); n(�

01)]. Hence the RHS of (24) is strictly negative, which

implies Z n(�02)

n( ��02)

v(�(n); n; x; �)d�(n) < 0: (31)

Hence v(�(n( ��02)); n(

��02); x; �) < 0. Similarly, I haveZ n(

��02)

n(�01)

v(�(n); n; x; �)d�(n) < 0: (32)

If c is empty, I haveRcv(�; n(�); x; �)d� = 0; If c is nonempty, by claim 1,

I have for � 2 c = [ ��01 ; �

01], n(�) � n(�01) � n(

��02). By the above argument,

v(�; n(�); x; �) < 0. Therefore I haveZc

v(�; n(�); x; �)d� � 0: (33)

Hence I have a contradiction to 16:

U(�B; y; x; �)

=

Zc

v(�; n(�); x; �)d� (34)

+

Z n( ��02)

n(�01)

v(�(n); n; x; �)d�(n)

+

Z n(�02)

n( ��02)

v(�(n); n; x; �)d�(n)

< 0. (35)

Then the three lemmas lead to the following proof.Proof of Prop 1. By lemma 2, U(b�;b�; x; �) > 0 if b� = �(x; �) + ";U(b�;b�; x; �) < 0 if b� = �(x; �) � ". By lemma 1, U(b�;b�; x; �) is continuousand strictly increasing in b� 2 ["; 1 � "]. By (9), [�(x; �) � "; �(x; �) + "] is asubset of ["; 1 � "]. Then, by the intermediate value theorem, there existsa unique b� 2 [�(x; �) � "; �(x; �) + "] such that U(b�;b�; x; �) = 0. Lemma 1

17

implies that b� is a threshold equilibrium. The model does not have otherthreshold equilibria because b� such that U(b�;b�; x; �) = 0 is unique. Lemma3 implies that there is no other types of equilibrium.Corollary 1 �nds a positive lower bound of the function "(x; �) in a subset

of A:Proof of Coro 1. I have

�(x; �) � p�1(u(x)

u(1��x1�� R)

) � p�1( u(1)u(R)

) > 0 (36)

�(x; �) � p�1(u(x)

u(1��x1�� R)

) � p�1( u(x)

u(x+ k)) � p�1(u(R� k)

u(R)) < 1. (37)

The �rst inequality of (36) is due to x � 1. The �rst inequality of (37) holdsbecause (x; �) 2 Ak and, hence, 1��x1�� R > x + k, while the second one holdsbecause x+ k � R.Let

"�k = minf(1� p�1(u(R� k)u(R)

))=3; p�1(u(1)

u(R))=3; 1=4g, (38)

(36)-(38) imply0 < "�k � "(x; �) (39)

for all (x; �) 2 Ak. By Proposition 1, when 0 < " < "�k, any (x; �) 2 Ak givesrise to uniqueness.Proof of Prop 2. For any (x; �) 62 Ak, I have

1� x�1� � R � x. (40)

For strategy y � 1, I have n(�; y) � 1 and

v(�; n(�; y); x; �) � fu[1� x�1� � R]p(�)� u(x)g� < 0. (41)

Each agent receives no positve expected gain from playing wait given any� 2 [0; 1]. Thus y � 1 is a equilibrium.Proof of Prop 3. Note that any contract at the upper boundary of set A

18

with suspension parameter greater than �4, satis�es x = 1�x�1�� R and, hence,

0 <

Z �

�

u[1� xn1� n R]� u(x)dn (42)

+

Z 1

�

fu[1� x�1� � R]� u(x)g

�

ndn.

I choose suspension parameter � slightly bigger so that (x; �) is not in A andsatis�es (42).Note that lim�!0[��minf�=3; (1� �)g] = 0 and lim�!1[��minf�=3; (1�

�)g] = 1. By the intermediate value theorem, I can �nd � 2 (0; 1) so that

0 =

Z �

�

u[1� xn1� n R]p(� �minf�=3; (1� �)g)� u(x)dn (43)

+

Z 1

�

fu[1� x�1� � R]p(� �minf�=3; (1� �)g)� u(x)g

�

ndn.

In what follows, I want to show that when " 2 (0;minf�=3; (1 � �)g], thereexists a threshold equilibrium.Note that 0 < 2�=3 < � � minf�=3; (1 � �)g < � � " < � + " < � +

minf�=3; (1� �)g � 1. Then the fundamental � is uniform over [�� "; �+ "]given �. Hence I have the following

U(�; �; x; �) � 1

2"

Z �+"

��"v(� � "; n(�; �); x; �)d� (44)

=

Z �

�

u[1� xn1� n R]p(� � ")� u(x)dn (45)

+

Z 1

�

fu[1� x�1� � R]p(� � ")� u(x)g

�

ndn

>

Z �

�

u[1� xn1� n R]p(� �minf�=3; (1� �)g)� u(x)dn (46)

+

Z 1

�

fu[1� x�1� � R]p(� �minf�=3; (1� �)g)� u(x)g

�

ndn

= 0. (47)

4If suspension parameter is equal to �, we cannot raise suspension parameter so that42 is satis�ed.

19

Let � = ��minf�=3; (1��)g�". It satis�es �+" < ��minf�=3; (1��)g �1 and �� " = ��minf�=3; (1� �)g� 2" � 0. The fundamental � is uniformover [� � "; � + "] given �. Hence I have the following

U(�; �; x; �) � 1

2"

Z �+"

��"v(� + "; n(�; �); x; �)d� (48)

=

Z �

�

u[1� xn1� n R]p(� + ")� u(x)dn (49)

+

Z 1

�

fu[1� x�1� � R]p(� + ")� u(x)g

�

ndn

=

Z �

�

u[1� xn1� n R]p(� �minf�=3; (1� �)g)� u(x)dn (50)

+

Z 1

�

fu[1� x�1� � R]p(� �minf�=3; (1� �)g)� u(x)g

�

ndn

= 0. (51)

By the intermediate value theorem, I can �nd b� 2 [�; �] so that U(b�;b�; x; �) =0. Furthermore, since 1 � � + " > � � " � 0, I has b� 2 ["; 1� "]. By lemma1, b� is an equilibrium.Proof of Prop 4. Note that v(1; 1; x; �) > 0 and v(0; 1; x; �) < 0. By theintermediate value theorem, there exists � 2 (0; 1) so that v(�; 1; x; �) = 0.Similarly, I can �nd � 2 (0; 1) so that v(�; �; x; �) = 0.Because � > �, I have

0 = v(�; �; x; �) = v(�; 1; x; �) < v(�; �; x; �), (52)

where the inequality holds because v(�; n; x; �) is strictly decreasing in nwhen it is nonnegative. (52) implies � < �. Hence (�; �) is nonempty.For any � 2 (�; �), I have

v(�; 1; x; �) < v(�; 1; x; �) = 0 = v(�; �; x; �) < v(�; �; x; �), (53)

where the inequalities hold because v(�; n; x; �) is strictly increasing in �. Ifobserving signal � 2 (�; �), each patient agent obtains gain v(�; �; x; �) > 0from playing wait, given that other patient agents also play wait. There existsan equilibrium y such that y(�) = 0 for � 2 (�; �). Similarly, there exists anequilibrium y0 such that y0(�) = 1 for � 2 (�; �).

20

Proof of Prop 5. Within this proof, I assume that R is su¢ cientlybig so that [Ru(1)]=u(R) > 1. Construct a sequence fpl(:)g such thatliml!1 pl(�) = 1 if � � 1� u(1)

u(R), and liml!1 pl(�) = 0 if � < 1� u(1)

u(R)5.

For any (x; �) 2 A, if the fundamental is � < p�1l (u(1)u(R)

) � 2", then allagents observe signals �0 < p�1l (

u(1)u(R)

) � ". Observing such signal, agentsknow � < p�1l (

u(1)u(R)

). They choose withdraw because for any � < p�1l (u(1)u(R)

),

u(1� x�1� � R)pl(�) <

u(1)

u(R)u(1� x�1� � R) � u(x), (54)

where the second inequality holds because x � 1 and, hence, 1�x�1�� R < R.

Therefore, for any l, if the fundamental is � < p�1l (u(1)u(R)

) � 2", withdraw isthe best response for all agents under any (x; �) 2 A.For any k 2 (0; R� 1), I consider (x; �) 2 Ak (see 2). If � � 1� u(1)

u(R)+2",

all agents observe signals �0 � 1 � u(1)u(R)

+ " and know that the fundamental

� � 1� u(1)u(R)

. They choose wait because

u(1� x�1� � R)pl(�) > u(x), (55)

for an arbitrarily large l6, independent of (x; �). To be precise, there existsl� independent of (x; �) such that if l > l�, then when the fundamental is� � 1� u(1)

u(R)+2", wait is the best response for all agents for any (x; �) 2 Ak.

When � 2 (p�1l (u(1)u(R)

)� 2"; 1� u(1)u(R)

+ 2"), the best response is not crucial inwhat follows.Now I consider the limiting welfare. Assuming that the limiting best

5We assume that pl converges to 1 when � � 1 � u(1)u(R) faster than it does to 0 when

� < 1� u(1)u(R) so that pl is productive in long run for all l.

6u(x)=u( 1�x�1�� R) inAk has a upper bound strictly less than 1. Since liml!1 pl(�) = 1 for

� � 1� u(1)u(R) , for any su¢ ciently large l, (55) holds for all contracts in Ak.

21

response is as above, I have the following limiting welfare for any (x; �) 2 A:

liml!1;"!0

jZ 1

1� u(1)u(R)

+2"

f�u(x) + (1� �)u(1� x�1� � R)pl(�)gd�

+

Z 1� u(1)u(R)

+2"

p�1l (u(1)u(R)

)�2"fmin(n; �)u(x)

+[1�min(n; �)� �max(0; 1� �n)]u(

1� x�1� � R)pl(�)gd�

+

Z p�1l (u(1)u(R)

)�2"

0

f�u(x) + (1� �)(1� �)u(1� x�1� � R)pl(�)gd�

�[�u(x) + (1� �)u(1� x�1� � R)]

u(1)

u(R)

��u(x)(1� u(1)

u(R))j

= liml!1;"!0

j � 2"�u(x) + �u(x)[p�1l (u(1)

u(R))� 2"� (1� u(1)

u(R))] (56)

(1� �)u(1� x�1� � R)[

Z 1

1� u(1)u(R)

+2"

pl(�)d� �u(1)

u(R)]

(1� �)(1� �)u(1� x�1� � R)

Z p�1l (u(1)u(R)

)�2"

0

pl(�)d�

+

Z 1� u(1)u(R)

+2"

p�1l (u(1)u(R)

)�2"fmin(n; �)u(x)

+[1�min(n; �)� �max(0; 1� �n)]u(

1� x�1� � R)pl(�)gd�j

= 0. (57)

Since the function u is bounded, the convergence is uniform7.For all (x; �) 2 AnAk, when the fundamental is � � 1 � u(1)

u(R)+ 2", wait

may or may not be the best response for all agents for any su¢ ciently large lindepentent of (x; �). But the above limiting welfare of (x; �), which assumesthat wait is the best response, is an upper bound on the welfare of equilibrium

7To be precise, for 8� > 0, 9 l� > 0& "� > 0 independent of (x; �) s.t. if l > l� and" 2 (0; "�), then the di¤erence between the welfare of (x; �) and its limit is less than � forall (x; �) 2 A.

22

under (x; �) in the limit. Because the partial derivative of the limiting welfarewith respect to x at (R=(1� �+ �R); �) is

�u0(R=(1� �+ �R))� �u0(R=(1� �+ �R))Ru(1)u(R)

< 0. (58)

Therefore, for a su¢ ciently small k, I can �nd (x; �) 2 Ak so that the limitingwelfare di¤erence between (x; �) and (x0; �) is greater than some small � > 0,for all (x0; �) 2 AnAk. Note that (x; �) 2 Ak has the above response in thelimit and that the convergence is uniform. Then for an arbitrarily big l andan arbitrarily small " > 0, independent of x0, (x; �) gives rise to welfare inequilibrium strictly higher than any (x0; �) 2 AnAk does.There exist �0 > � and k0 < k so that (x; �0) 2 Ak0 for any x such that

(x; �) 2 Ak. For such k0, (x; �0) has the above limiting response. The limitingwelfare di¤erence between (x; �0) and (x; �) is greater than (�0 � �)u(1)[1 �u(1)=u(R)] > 0 for all x such that (x; �) 2 A (Check the LHS of (56)). Thenfor an arbitrarily big l and an arbitrarily small " > 0, independent of x,(x; �0) gives rise to welfare in equilibrium strictly higher than (x; �) does forany x such that (x; �) 2 Ak.Overall, I can �nd l� & "� > 0 independent of x such that if l > l� and

" 2 (0; "�), any (x; �) 2 A has welfare strictly lower than another contract inA with suspension parameter �0 > � does.Proof of Prop 6. For each l, there exists a unique threshold equilibriumand denote its cut-o¤ value by b�l. In the proof of proposition 1, I haveb�l 2 ["�l ; 1 � "�l ]. Hence the fundamental � is uniform over [b�l � "�l ;b�l + "�l ]given the signal b�l. The value of function U at b�l is

0 = liml!1

U(b�l;b�l; x; �) (59)

= [ liml!1

pl(b�l)](Z �

�

u[1� xn1� n R]dn+

Z 1

�

fu[1� x�1� � R]

�

ndn) (60)

�Z �

�

u(x)dn�Z 1

�

u(x)�

ndn.

Integration by substitution implies (60).By (59)-(60), liml!1 pl(b�l) exists and belongs to (0; 1). Because liml!1 pl(�) =

1 for all � 2 (0; 1], I have liml!1 b�l = 0. Also Note that liml!1 "�l = 0. Then

23

I have8

liml!+1

Z 1

b�l+"�l (pl(�)� 1)d� = 0. (66)

The limiting welfare di¤erence between (x�; �) in DD and (x�; �) in the

8Suppose � > 0 is chosen arbitrarily small. Then for a su¢ ciently big l, �0l + "

�l is

su¢ ciently close to 0 and, thus, is smaller than �. Hence we have

0 >

Z 1

�0l+"

�l

(pl(�)� 1)d� (61)

>

Z 1

�

(pl(�)� 1)d� +Z �

�0l+"

�l

(pl(�)� 1)d� (62)

> (1� �)(pl(�)� 1)� (�� �0l � "�l ). (63)

Because pl(�) is nonnegative and strictly increasing in �, inequality (63) holds. Becauseliml!1 pl(�) = 1 and liml!1(�

0l + "

�l ) = 0, inequalities (61)-(63) become

0 � liml!+1

Z 1

�0l+"

�l

(pl(�)� 1)d� � ��. (64)

Since � > 0 is chosen arbitrarily small, we have

liml!+1

Z 1

�0l+"

�l

(pl(�)� 1)d� = 0. (65)

24

perturbed model is equal to zero as follows:

liml!1fZ 1

b�l+"�l [�u(x�) + (1� �)u(1� x

��

1� � R)pl(�)]d�

+

Z b�l�"�l0

[�u(x�) + (1� �)(1� �)u(1� x��

1� � R)pl(�)]d�

+

Z b�l+"�lb�l�"�l [

�

max(n(�;b�l); �)�u(x�)+u[

1� x�min(�; n(�;b�l))1�min(�; n(�;b�l)) R]pl(�)[(1� �)� (n(�;b�l)� �)�

max(n(�;b�l); �) ]+u(x�)

(n(�;b�l)� �)�max(�; n(�;b�l)) ]d�g

��u(x�)� (1� �)u(1� x��

1� � R)

= liml!1fZ 1

b�l+"�l (1� �)u(1� x��1� � R)(pl(�)� 1)d�g (67)

= 0. (68)

Because liml!1 b�l = 0 and liml!1 "�l = 0, (67) holds. (68) follows from (66).

6 Concluding remark

Suspension is widely used during �nancial crises and has attracted attentionin the literature. This paper introduces another potential role of suspensionwith respect to Global Game literature. Suspension generates the upperinterval property as an endogenous result, while previous Global Game lit-erature directly assume such property9. In this paper, by reserving enoughresources for future withdrawals, suspension replaces GP new assumptionabout the short-term return that appears for the �rst time in their proof.Suspension might be understated, because common knowledge is widely

assumed in the literature. In fact, such feature might be too strong to assume.

9For the purpose of demonstrating the technique, Morris and Shin (1998) directlyassumes that the government prefers, and is able to defend the target exchange rate.

25

This motivates the construction of Global Game environment, where di¤erentagents can have di¤erent signals, and hence do not have common knowledgeabout the fundamental. In this paper, the absence of such environmentdiminishes, but does not eliminate the power of suspension. Some contractsin our subset still have uniqueness and, thus intrinsic coordination10 evenwithout noise, by �xing the proportion early withdrawals at the proportionof the impatient. Overall, Global Game environment, which departs fromcommon knowledge assumption, enlarges the set of suspension contracts thathave intrinsic coordination.As regards welfare, DD suspension is robust to the introduction of a

small amount of uncertainty and a small amount of noise. As the environ-ment departs further from DD, their suspension is almost surely not thebest banking arrangement. DD suggested one possible departure where theproportion of the impatient is uncertain, and the literature responded to it.Another departure introduces uncertain return. In our example with a lot ofreturn uncertainty, there almost surely would be substantial social gain fromeliciting information from agents and conditioning payo¤ and suspension pa-rameter on it. But even if I ignore such gain, I show that DD suspensionparameter is not the best within our class of contracts.

References

[1] Diamond, D. W., and Dybvig, P. H. (1983):" Bank runs, depositinsurance, and liquidity" Journal of Political Economy

[2] Ennis, H., and Keister, T. (2010): "Bank Runs and Institutions: ThePerils of Intervention" American Economics Review

[3] Ennis, H., and Keister, T. (2010): "Banking Panics and Policy Re-sponses" Journal of Monetary Economy

[4] Goldstein, I., and Pauzner, A. (2005): " Demand- deposit contractsand the probability of bank runs" Journal of Finance

[5] Green, E.,J., and Lin, P. (2003): " Implementing e¢ cient mechanismsin a model of �nancial intermediation" Journal of Economic Theory

10The coordination for the sun-spot device in Peck and Shell (2003) is exogenous, whileour noisy signal device has the coordination as a endogenous outcome.

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[6] Peck, J., and Shell, K. (2003):"Equilibrium bank runs" Journal ofPolitical Economy

[7] Morris, S., and Shin, H. S. (1998):"Unique equilibrium in a model ofself-ful�lling currency attacks" American Economic Review

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