SONDERFORSCHUNGSBEREICH 504

Transcription

1 SONDERFORSCHUNGSBEREICH 504 Rationalitätskonzepte, Entsceidungsveralten und ökonomisce Modellierung No Liquidity and Ambiguity: Banks or Asset Markets? Jürgen Eicberger and Willy Spanjers June 007 We would like to tank seminar participants at te Norwegian Scool of Management (Sandvika), te Sonderforscungsbereic 504, Department of Economics, Kingston University, Universität Manneim L 13, Manneim

2 Liquidity and Ambiguity: Banks or Asset Markets? 1 Jurgen Eicberger and Willy Spanjers 3 Tis version June 007 Abstract We study te impact of ambiguity on two alternative institutions of nancial intermediation in an economy were consumers face uncertain liquidity needs. Te ambiguity te consumers experience is modeled by te degree of condence in teir additive beliefs. We analyze te optimal liquidity allocation and two institutional settings for implementing tis allocation: a secondary asset market and a bank deposit contract. For full condence we obtain te well-known result tat consumers prefer te bank deposit contract over te asset market, since te former can provide te optimal cross subsidy for consumers wit ig liquidity needs. Wit increasing ambiguity tis preference will be reversed: te asset market is preferred, since it avoids inecient liquidation if te bank reserve oldings turn out to be suboptimal. JEL Classication Codes: D8, G1, G. Keywords: Financial institutions, Liquidity, Ambiguity, Coquet Expected Utility. 1 We would like to tank seminar participants at te Norwegian Scool of Management (Sandvika), te University of Oslo, te Univeristy of Birmingam, te Bundesbank (Frankfurt a.m.), te Free University in Berlin, te University of Frankfurt a.m., Keele University, te University of Hoeneim, Cemnitz University of Tecnology and Yale University, as well as participants of te 004 Annual Meeting of te Verein for Socialpolitik in Dresden and te 005 Anual Meeting of te European Economic Association in Amsterdam for elpful comments and suggestions on earlier versions of te paper. Postal address: Jurgen Eicberger, Alfred Weber Institute, Department of Economics, University of Heidelberg, Grabengasse 14, D-69117, Germany. 3 Corresponding Autor. Postal address: Willy Spanjers, Scool of Economics, Kingston University, Penryn Road, Kingston-upon-Tames, Surrey KT1 EE, United Kingdom. 1

3 Liquidity and Ambiguity: Banks or Asset Markets? 1. Introduction In a globalizing world wit its rapid tecnological progress and environmental degradation, ever larger parts of our lives are exposed to non-calculable risk or, as it is often referred to, ambiguity. Suc risks range from natural disasters and te spread of unknown diseases to instable regimes and terrorist attacks. Financial decisions are exposed to te eects of ambiguity as well, toug it may not always be recognized as suc. In tis paper, we relate te prevalence of specic forms of nancial intermediation to te degree of ambiguity faced by investors. Tis allows us to sed a new ligt on two familiar observations. Firstly, te consensus-oriented economies of Germany and Japan operate bank-based nancial systems, as opposed to te market-based systems of te UK and te USA 4. Secondly, it proves dicult to establis private sector nancial intermediation in countries wit a weak rule of law and signicant (incalculable) political risks 5. In te setting of our model, we nd tat wen investors face small amounts of ambiguity, as in consensus-oriented economies, competitive banks are preferred to nancial markets. For intermediate degrees of ambiguity, as may be found in te UK and te USA, tis preference reverses. For ig levels of incalculable risk, wic prevail in developing countries, neiter banks nor markets can improve on te outcome obtained in te absence of nancial intermediation. Many economic activities depend crucially on facilities enabling economic agents to raise liquid funds against claims on teir future income. Future income streams are by nature uncertain and, terefore, dicult to contract on. Hence, we nd a pletora of nancial institutions serving te purpose of providing liquidity, ranging from credit contracts, wic allow customers to raise liquid funds against claims on future payments, to secondary asset markets, in wic organized trade of illiquid assets can take place. In te case of credit contracts, loans require sucient collateral or a more or less complicated assessment procedure of future payments. Secondary asset markets can be establised for omogeneous asset categories wit suciently regular demand and supply tat justify te costs of an organized market. Bank deposit contracts are a special instrument of liquidity provision. Banks accept deposits of liquid funds and promise to repay liquid funds in te future at any time te depositor claims tem back. Deposits oer banks te opportunity to invest at least part of tese funds in illiquid assets, since on average only a fraction of deposits will be called upon in any period. In tis paper, we contrast two institutional arrangements for liquidity provision: a bank deposit 4 See, e.g. Edwards and Fiscer (1994). 5 See, e.g. Levine (1997).

4 Liquidity and Ambiguity: Banks or Asset Markets? 3 contract as in Diamond and Dybvig (1983) and a secondary asset market as in Diamond (1997) and Allen and Gale (004). Diculties wit liquidity provision arise from te inerent uncertainty about asset returns, but also from uncertainty about individual and aggregate liquidity needs. We will focus ere on uncertainty about individual and aggregate liquidity needs and will disregard uncertainty about asset returns 6. In most of te existing economic literature, uncertainty is viewed as ignorance about te outcome of a random draw from a known probability distribution, were te known probability distribution is identied wit te actual frequency of tese outcomes in a population. For example, consumers assume tat preference parameters, wic determine teir private liquidity needs in te future, are randomly drawn wit probabilities equal to te actual frequencies of tese preference parameters in te population. We will sow tat ambiguity about tese distributions can cange well-known results of te literature on nancial intermediation. In particular, we will argue tat institutional arrangements, suc as bank deposit contracts and secondary asset markets, are quite distinct in teir robustness wit respect to investors' ambiguity about individual and aggregate liquidity needs Modelling ambiguity. Uncertainty as long been recognized as an important factor determining economic activities. Te distinction between risk, i.e., situations were te probabilities of events are known, and ambiguity, i.e., situations were tis is not te case, as served Knigt (191) as te foremost explanation of economic penomena suc as prot and entrepreneurial activity. For several decades te beaviorist teory of subjective expected utility by Savage (1954) appeared to ave rendered tis distinction obsolete. If individuals faced wit uncertainty beave as if tey eld a subjective probability distribution over events, ten, from an analytical point of view, beavior under risk and under uncertainty can be treated in te same way. Yet early evidence by Ellsberg (1961) suggested tat te ypotesis of a well-dened subjective probability distribution cannot be maintained empirically. Systematic laboratory experiments ave conrmed Ellsberg's conjecture, see Camerer and Weber (199). It appears to be wellestablised now tat certain aspects of uncertainty cannot be captured by te assumption of a subjective probability distribution. Despite tese inconsistencies wit actual beavior under uncertainty, expected utility teory as proved to be a very successful modelling tool. Important economic insigts were obtained 6 Te impact of uncertain asset returns is addressed in Jacklin and Battacarya (1988).

5 Liquidity and Ambiguity: Banks or Asset Markets? 4 from te distinction between risk preferences and beliefs. We believe, owever, tat tere are economic situations were te ambiguity of agents aects economic performance in a way wic can be tested empirically and were te success of economic policies depend on weter one takes te ambiguity of economic agents appropriately into account 7. In recent years, substantial progress as been made in modelling decision-making under uncertainty witout subjective probabilities. Scmeidler (1989) and Gilboa (1987) proposed a teory were decision makers' beliefs are represented by non-additive probabilities (or capacities). Coquet expected utility (CEU) teory, a generalization of subjective expected utility, can accommodate many dierent weigting scemes for events wile maintaining some separation of beliefs and outcome evaluation, wic is important in economic applications in order to identify risk preferences 8. Generalizing additive beliefs to non-additive beliefs allows one to accommodate empirically observed anomalies like, e.g., te Ellsberg-paradox 9. Witout imposing additional restrictions on capacities, owever, predictions about economic beavior are typically less precise. Eicberger and Kelsey (1999) and Cateauneuf, Eicberger and Grant (004) study special classes of capacities wic restrict te number of free parameters 10 and provide economic interpretations for tem. In tis paper, we will restrict attention to beliefs tat can be represented by simple capacities. Simple capacities are convex combinations of an additive probability distribution and te capacity of complete ignorance. Tey reduce te large number of free parameters, wic are typical for general capacities. Moreover, since a simple capacity is convex, a CEU decision maker can also be viewed as a pessimist wit beliefs described by a set of multiple additive prior distributions. Simple capacities are a special case of E-capacities wic are studied in Eicberger and Kelsey (1999) and of NEO-additive capacities wic are analyzed and axiomatized in Cateauneuf, Eicberger and Grant (004). Wit a simple capacity, te Coquet expected utility of an act is a weigted average of te expected utility of tis act wit respect to an additive probability distribution and te min- 7 See, e.g. Kelsey and Spanjers (004) on small rms, Spanjers (006) and Gatak and Spanjers (007) on monetary policy and Spanjers (007) on currency crises. 8 Furter developments include te unication of te approaces of Scmeidler (1989) and Gilboa (1987) (Sarin and Wakker (199)), applications to portfolio coice (Dow and Werlang (199)) and game teory (Marinacci (000)), and a concept of ambiguity aversion (Girardato and Marinacci (00)). 9 See Ellsberg (1961). 10 Tese capacities satisfy furter desiderata like compatibility wit te multiple prior approac (Cateauneuf, Eicberger and Grant (004)), a reasonable support notion (Eicberger and Kelsey (003)), and consistent updating rules (Eicberger, Grant and Kelsey (004)).

6 Liquidity and Ambiguity: Banks or Asset Markets? 5 imum utility obtainable wit te act. Te weigt attaced to te expected utility part can be interpreted as te degree of condence of te decision maker in te additive probability. Hence, one can give ambiguity and condence a parametric interpretation. 1.. Liquidity and nancial intermediation. According to Diamond and Dybvig's seminal article, liquidity problems arise because consumers, wo do not know teir private liquidity needs in future periods, ave to decide on investments wic require a long-term commitment of teir funds 11. Since liquidity needs are private information, direct contracting is impossible and an agency problem arises. In tis context, one can raise and answer important questions about te institutional design of nancial intermediation and its regulation. Te earlier literature assumes tat te illiquid asset can be liquidated at par 1. Wit tis assumption, tere is no liquidity problem but an incompleteness of contract due to unrealized insurance opportunities wic can be accommodated by a bank deposit contract. Te more recent literature assumes tat te illiquid asset cannot be liquidated at all. Hence, a combined problem of insucient liquidity and incomplete insurance contracts arises 13. Diamond (1997) sows tat secondary asset markets can resolve te liquidity problem but cannot provide te necessary cross subsidy in order to deal wit te incomplete insurance issue. In contrast, a bank deposit contract can solve bot problems and implement te optimal contract. A central question in tis literature concerns te precarious coexistence of banks and secondary asset markets. Diamond (1997) and Allen and Gale (004) assume restricted access of consumers to te secondary asset market. Wit restricted market participation consumers can secure liquidity and obtain, at least partially, te cross subsidy required by te optimal contract. In tis paper, we will assume tat te illiquid asset can be liquidated at some cost. Compared to te literature, tis is an intermediate case. Tere is a liquidity problem, but tere can be no perfect commitment not to liquidate early, wic is implied in te assumption of Diamond (1997) tat te illiquid asset cannot be liquidated at all. Toug we assume tat consumers ave risk neutral preferences, as in Cari and Jagannatan (1988), a cross subsidy problem occurs if some consumers' return from olding te liquid asset exceeds te return on te illiquid asset. As in Diamond and Dybvig (1983) te optimal contract requires a cross subsidy from consumers wit low liquidity needs to consumers wit ig liquidity needs, yet not for insurance reasons. Hence, disregarding ambiguity, we obtain te results of Diamond (1997). 11 See Diamond and Dybvig (1983). 1 See Diamond and Dybvig (1983) and Jacklin (1987). 13 See Diamond (1997) and Allen and Gale (004).

7 Liquidity and Ambiguity: Banks or Asset Markets? 6 Te main contribution of tis paper consists in its analysis of te role of ambiguity and con- dence for intermediary institutions. Comparing bank deposits and secondary asset markets, we obtain te following results. In te presence of ambiguity, neiter bank deposit contracts nor te asset market implement te ex-ante optimal allocation of liquidity. Te evaluation of institutions depends on te degree of consumers' condence regarding te probability distribution over private and aggregate liquidity needs. Wit low condence, neiter bank deposits nor asset trading can improve upon te allocation witout intermediation. For middle levels of condence, a secondary asset market is te preferred institution. If te level of ambiguity is low, and ence condence is ig, bank deposit contracts oer te ex-ante preferred metod of liquidity provision Organization of te paper. In te following section, we dene simple capacities and embed te decision criteria used in tis paper in te more general teory of capacities and te Coquet integral. Section 3 describes te economic model, analyzes individual beavior witout intermediation, and studies te optimal incentive-compatible allocation. Te following two sections deal wit te secondary asset market and te bank deposit contract, respectively. Section 6 compares te performance of tese institutions under ambiguity, Section 7 contains concluding remarks. Longer proofs are gatered in an appendix.. Ambiguous beliefs Simple capacities form a special class of capacities. Consider a state space S wic is a compact subset of R n and let p be an additive probability distribution on S: A simple capacity ; based on te additive probability distribution p; is te set function dened by 8 < p(e) for E S (E) = : 1 for E = S for any p-measurable set E and some [0; 1]: For = 1 te simple capacity coincides wit te additive probability distribution p: Te parameter can be interpreted as te degree of condence in te additive probability distribution p: We call := 1 te degree of ambiguity. Simple capacities maintain many properties of additive probability distributions and ave a natural interpretation in terms of beliefs. Te degree of ambiguity reects te deviation from te additive probability distribution p. In tis interpretation, ambiguity is simply te counterpart of te decision maker's condence in a probabilistic assessment. Consider a p-measurable function f : S! R: Te Coquet integral of f wit respect to a simple capacity ; te Coquet expected value of f; is te convex combination wit weigt

8 Liquidity and Ambiguity: Banks or Asset Markets? 7 of te expected utility of f wit respect to te probability distribution p and te worst outcome of f on S. Te following result is proved in Eicberger and Kelsey (1999, Proposition.1). Proposition 1. Coquet integral of a simple capacity Consider a simple capacity wit te additive probability distribution p: Te Coquet integral CEU(f; ) of a p-measurable function f wit respect to te simple capacity as te following form: Z CEU(f; ) = S f dp + min f(s): (1) ss Proposition 1 oers an intuitive and parsimonious preference representation 14 of a decision maker facing ambiguity about p. For te case of full condence, = 1; one as te familiar expected utility form. As ambiguity increases,! 1; and condence in p falls,! 0; more weigt is given to te worst outcome of f on S: For = 0; te maximin decision rule obtains. Te Coquet integral of a simple capacity models ambiguity in a special way. We feel, owever, tat tis payo description captures at least some aspects of Knigt's ideas Te economy We consider an economy wit many, relative to te market, small consumers 16. Tis is modelled by te assumption of a continuum set of consumers, te interval [0; 1]: Te economy extends over tree periods. In Period 0; eac consumer is endowed wit one unit of wealt (money) and faces te following investment opportunities: Payo in Assets Period 0 Period 1 Period 1. Asset matured 1 0 Asset liquidated Money 0 to Money 1 to We assume > 1 > 1 : Tis payo structure justies calling te asset illiquid. Te asset oers a long-term investment possibility wit a better return tan money, if eld to maturity in Period : Compared to money it is illiquid, owever, since te liquidation payo 1 in Period 1 falls sort of te return from olding money. Uncertainty about liquidity needs is te focus 14 A simple capacity is a special case of a NEO-additive capacity wic as been axiomatised in Cateauneuf, Eicberger and Grant (004). 15 See Knigt (191). 16 Te basic structure of te model (disregarding ambigous beliefs), follows Eicberger (199).

9 Liquidity and Ambiguity: Banks or Asset Markets? 8 of tis paper. We abstract, terefore, from uncertainty about te payos of te assets 17. Tere are two types of ex-ante identical consumers. In Period 1; consumers privately learn teir type t f; g: Te type of a consumer determines is preference for liquidity in Period 1: Type-dependent preferences are represented by a risk-neutral von Neumann-Morgenstern utility index u(z 1 ; z ; t) = t z 1 + z were z 1 and z denote consumption in Period 1 and Period ; respectively. Trougout te paper, te following assumption about te parameter values of our model is maintained. Assumption 1. Liquidity preference (i) > > 1; (ii) > 1 : According to Assumption 1 (i), consumers of type strictly prefer to old money, wile consumers of type prefer an investment in te illiquid asset. Assumption 1 (ii) guarantees tat te liquidation value of te illiquid asset 1 is so low tat it does not pay for a consumer wit ig liquidity needs to liquidate an investment in te illiquid asset in Period 1: Tis part of Assumption 1 is not strictly necessary for our analysis. It is owever useful for te exposition since it allows us to skip discussing several cases wic are of little interest. Te liquidation value 1 (0; 1) falls between two extreme cases. In Diamond and Dybvig (1983) te long-term asset as te same liquidity as money, 1 = 1; wile in Diamond (1997) te illiquid asset as no payo in Period 1, 1 = 0: In our paper, consumers are not riskaverse. Hence, we need to assume some illiquidity 1 < 1; for, oterwise, te payos of te illiquid asset would simply dominate olding money. A liquidation rate 1 = 0 forms te oter extreme. Diamond (1997) makes tis assumption because e considers a secondary market for claims to te illiquid asset 18. Te market price in te secondary market can be viewed as an endogenously determined liquidation rate 1. As in Diamond and Dybvig (1983), it is assumed tat te probability of a consumer being assigned a particular type equals te proportion of tis type in te economy. In addition, we 17 Spanjers (1999, Capter 5) extends te comparison between banks and asset markets to te case of ambiguity about te illiquid asset's return. 18 For 1 = 0; liquidation is impossible. Hence, any sceme wic collects funds in Period 0 and redistributes tem in Period 1 can perfectly commit to not liquidating early in favor of early consumers at te expense of late customers. Tis assumption will be important below, wen we consider te deposit contract, and will be discussed in more detail tere.

10 Liquidity and Ambiguity: Banks or Asset Markets? 9 will assume tat tere is also uncertainty about te proportion of consumers of eiter type. Hence, in Period 0; bot individual and aggregate liquidity needs are unknown. Denote by te unknown proportion of consumers wit ig liquidity needs : In Period 0; consumers know neiter teir type t nor te proportion of -types in te population. Wile te proportion of consumers wit ig liquidity needs becomes common knowledge in Period 1; consumers learn only privately te information about teir type t: Beliefs of consumers are represented by a subjective joint probability distribution P over te unknown parameters (t; ) f; g [0; 1]. Ambiguity is modelled as lack of condence in tis additive probability distribution P. Te following assumption caracterizes te beliefs of consumers. Assumption. Beliefs 1. Population sares: Conditional on te population sare te probability distribution over types t equals te proportions of types in te economy, wenever tis conditional probability is welldened,. Correct beliefs: P (j) = and P (j) = 1 : Consumers' marginal beliefs about te population sare of -types are concentrated on te true proportion, p() = 1 for 0 oterwise were p() denotes a cumulative distribution function on te set of population sares for te -types, [0; 1]: In order to make our results comparable wit te literature, e.g. wit Diamond and Dybvig (1983), Jacklin (1987), Diamond (1997) and Allen and Gale (004), beliefs about te proportion of consumers wit ig liquidity needs are identied wit teir actual population sare : Toug consumers ave point predictions for te population sares of eac type, wic will turn out to be correct in Period 1; tey may still experience ambiguity about tese predictions in Period 0: ; Assumption implies te following joint probability distribution: 8 8 < if = < 1 if = P (; ) = ; P (; ) = : : 0 oterwise 0 oterwise :

11 Liquidity and Ambiguity: Banks or Asset Markets? Investment witout intermediation. If tere are no intermediary institutions, ten eac consumer simply decides on te fraction of wealt m to be eld as money and on te fraction to be invested in te illiquid asset, 1 m. Tis decision yields consumption (z 1 ; z ) = (m; (1 m)) and, for type t; te utility u(m; (1 m); t) = t m+ (1 m): By Assumption 1 (ii), > 1 ; we need not consider te case of consumers wanting to liquidate teir long-term investment. In Period 0; wen consumers ave to coose teir investment strategy m; tey are uncertain bot about teir type t and te proportion of types in te population,. If te proportion of type consumers were known, te ex-ante expected utility of a consumer would be [ + (1 ) ] m + (1 m): If consumers lack condence in te probability distribution P (t; ); i.e., < 1; ten te ex-ante Coquet expected utility of a consumer is CEU(m; ) = [( + (1 ) ) m + (1 m)] +(1 ) min [ t m + (1 m)] (t;)f;g[0;1] = [ + (1 ) ] m + (1 m): Maximizing CEU(m; ) over m [0; 1] yields te maximal ex-ante Coquet expected utility as a function of te degree of condence ; a result wic we summarize in Proposition. Proposition. In an economy witout intermediation, te optimal investment policy 8 0 for < >< m n() = [0; 1] for = ; >: 1 for > yields te maximal ex-ante Coquet expected utility V n () = CEU n (m n(); ) = maxf ; + (1 ) g: Consumers of type prefer to old money in Period 1. Hence, money olding is te optimal investment policy if bot te assessed probability of becoming a consumer wit ig liquidity needs, ; and te degree of condence in tis belief,, are suciently ig. Oterwise, all money is invested in te illiquid asset. Since te illiquid asset as a certain return ; it is te preferred coice if te low type is realized. Hence, it becomes te default option for consumers wit a ig degree of ambiguity. Clearly, money would become te default option for low condence if te illiquid asset ad an uncertain return.

12 Liquidity and Ambiguity: Banks or Asset Markets? Optimal contract. Te allocation wic consumers can generate individually in tis economy is suboptimal. Even taking into account te informational constraints, tere is scope for Pareto improvements by pooling resources in Period 0 and investing tem jointly. If tere was no uncertainty, = 1; consumers could pool teir funds in Period 0 and invest te proportion (1 ) in te illiquid asset, olding te rest as money. Tis investment strategy would allow tem to pay out a unit of money to te consumers wit ig liquidity need in Period 1 and te amount to consumers wit low liquidity needs in Period : Tis would yield an expected utility exceeding te expected utility wic consumers can guarantee temselves, + (1 ) > Vn (1): As Diamond (1997) points out tis pure liquidity provision, toug improving upon autarky, is not optimal. Since te type of a consumer is private information and since we consider uncertainty about individual and aggregate liquidity needs, suc an asset pooling sceme needs to be studied carefully. For an optimal allocation, te resources of all consumers are pooled in Period 0 and optimally invested in money and te illiquid asset. In Period 0; before types t f; g are privately known and before te proportion of types is common knowledge, all consumers are identical. In Period 1 all uncertainty is resolved, owever, consumers' types are not publicly known. Hence, te optimal contract will assign type-contingent payouts, z = (z 1 ; z ) and z = (z 1; z ); subject to self-selection constraints, wic reect te private information of consumers. Moreover, an optimal allocation must maximize individual utility witout ignoring te informational asymmetry. In Period 1; te optimal type-contingent payout sceme (z ; z) will maximize te average utility of consumers given te, ten known, public information about te population sare. Te average utility for a population wit a fraction of type consumers is U(z ; z; ) = u(z ; ) + (1 ) u(z; ) = [ z 1 + z ] + (1 ) [ z 1 + z ]: () Te coice problem of te optimal contract as two stages. In Period 0; te fraction M of aggregate wealt eld as money is determined. Te remaining wealt is invested in te illiquid asset. In Period 1, once te investment decision M as been taken and once te proportion of type consumers, [0; 1]; as become public knowledge, te type-contingent payouts (z ; z) are determined. We analyze tese two stages in turn.

13 Liquidity and Ambiguity: Banks or Asset Markets? 1 Te optimal payout sceme. Given te aggregate money oldings M and a realized proportion of type consumers, te optimal payout sceme (z ; z) must maximize consumers' average utility subject to self-selection and feasibility constraints: max z ;z U(z ; z; ) s.t. z 1 + z z 1 + z ; S z 1 + z z 1 + z ; S z 1 + (1 ) z 1 = M; F 1 z + (1 ) z = (1 M); F z 1 0; z 1 0; z 0; z 0: (3) Te feasibility constraints, F 1 and F ; guarantee tat aggregate payouts in periods 1 and can be nanced given te investment policy M: According to Assumption 1 (ii), it can never be optimal to liquidate te long-term investment in Period 1; ence tis possibility is disregarded in decision problem (3). Incentive-compatibility of te payout sceme follows from te two self selection constraints S and S : Disregarding te self-selection constraints S and S ; one optimal allocation would be ez 1 = 0; ez 1 = M ; ez = 0 and ez = (1 M) 1 ; yielding te optimal average utility U(M; ) = M + (1 M): (4) For given initial money oldings M, owever, it depends on weter tis solution satises te self-selection constraints. Te constraint S will be binding for M < (1 M) 1 ; i.e., for > M M + (1 M) = (M): Te oter incentive constraint S will bind, if (1 M) 1 < M olds, i.e., if < M M + (1 M) = (M): (5) Te optimal solution from Equation (4) is valid for all [(M); (M)]: From te linearity of te average utility function U(z ; z; ) in Equation () it is immediately clear tat te optimal allocation fails to be unique at an optimum were te self-selection constraints are not binding. Any second-period consumption allocation (z ; z ) satisfying te self-selection constraints and te constraint z + (1 ) z = (1 M) would also be optimal. It is terefore possible to transfer second-period consumption from -type to -type consumers at no cost in terms of te average utility. Hence, te optimal average utility of Equation (4) remains uncanged for > (M):

14 Liquidity and Ambiguity: Banks or Asset Markets? 13 In contrast, if te constraint S is binding, ten one as to decrease rst-period consumption of consumers wit ig liquidity needs and increase rst-period consumption of -type consumers in order to satisfy te self-selection constraints. For < (M); one obtains te optimal allocation ez 1 = M + (1 M); ez 1 = M 1 (1 M); ez = 0 and ez = 1 (1 M) yielding an average utility of U(M; ) = M + (1 M) +(1 ) M = ( + (1 ) ) 1 (1 M) + (1 M) 1 [ M + (1 M)] : In summary, te maximal average utility obtainable wit an optimal contract is a function of te population sare of -type consumers and te investment M made in Period 0: 8 ( >< +(1 )) [ M + (1 M)] for < (M) U(M; ) = : (7) >: M + (1 M) oterwise (6) Te optimal investment policy. In Period 0; wen consumers are still uncertain about teir types and te proportion of types ; one obtains te Coquet expected value of an investment decision M by taking te Coquet integral of te average utility U(M; ) in Equation (7). Notice tat information about te types of consumers is not necessary for te optimal investment coice. Tis information becomes relevant in Period 1; wen consumers privately know teir types. Te payout sceme derived in te previous subsection will guarantee trutful revelation of tis information. For any number " [0; 1]; denote by (") = " + (1 ") te expected return on money oldings in Period 1: By Assumption te marginal probability distribution over population sares of te -type consumers is concentrated on te true proportion : Hence, U(M; ) is te expected average utility of a consumer in Period 0: Te ex-ante worst case is obtained for te combination t = L and = 0 and yields a utility of U(M; 0): Given a degree of condence ; te te ex-ante Coquet expected utility function is = CEU o (M; ) = U(M; ) + (1 ) U(M; 0) (8) 8 >< [ M + (1 M)] for < (M) >: () () M + (1 M) oterwise : Te optimal investment policy M will be cosen to maximize CEU o (M; ) over all M [0; 1]:

15 Liquidity and Ambiguity: Banks or Asset Markets? 14 Condition < (M) in Equation (5) is equivalent to te condition M > + (1 ) = M(): Figure 1 sows te Coquet expected utility function of Equation (8). CEU () +(1 )... ()... CEU o (M; ) 0. M() 1 M Figure 1: Optimal reserves It is immediately obvious tat te optimal fraction of wealt eld as money is 8 < M() for () > Mo () = [0; M()] for () = : : 0 for () < Substituting te optimal investment coice in te Coquet expected utility function, CEU o (M o (); ); yields te optimal ex-ante Coquet expected utility as a function of te degree of condence : We summarize tis result in te following proposition. Proposition 3. Te maximal Coquet expected utility from an optimal contract is V o () = CEU o (Mo (); ) ( ) = max + (1 ) 8 >< = >: ( ) + (1 ) ; 1 if () > if () :

16 Liquidity and Ambiguity: Banks or Asset Markets? 15 Te maximal ex-ante Coquet expected utility V n () obtainable for a consumer in te absence of intermediary institutions was derived in Proposition. Tis value can be compared wit te Coquet expected utility V o () of an optimal contract from Proposition 3. Figure sows te ex-ante Coquet expected utility levels for tese two cases. CEU V o () V n () 0. o. n 1 Figure : Optimal contract vs. no intermediation Te critical degrees of condence o = and n = ( ) are obtained were ( o) = + (1 ) and ( n ) = old, respectively. Te Coquet expected utility in te case of no intermediation forms a lower bound for te ex-ante Coquet expected utility wit any kind of voluntary intermediation. Te optimal contract, on te oter and, provides an upper bound for wat any intermediary institution can acieve. In te following sections, we investigate dierent institutional settings. We compare an asset market for te illiquid asset in Period 1 wit a deposit contract oered by a competitive bank. 4. Asset market As a rst institutional environment, suppose tat in Period 1 consumers can sell claims to teir investment in te illiquid asset. Te possibility to trade claims on te asset makes tis investment more liquid and provides an extra incentive to invest in it. As before, consumers decide in Period 0 ow muc of teir wealt to invest in te asset and ow muc to keep as liquid money oldings.

17 Liquidity and Ambiguity: Banks or Asset Markets? Market for claims in Period 1. Let us consider rst te market for claims to te illiquid asset in Period 1: At tis stage, te aggregate money olding M of consumers is given, types are private knowledge and te actual proportion of -types is common knowledge. Since all consumers are identical in Period 0; individual investment policies can be assumed to equal teir aggregates. Denote by q te price of a claim to one unit of te illiquid asset in terms of money. If te price is ig enoug, consumers of type, wo old some illiquid asset, will try to sell it order to benet from teir ig value for liquidity. Te aggregate supply of suc claims is: 8 1 M for q > >< [ (1 M); 1 M] for q = S(q) = (1 M) for > q > : [0; (1 M)] for q = >: 0 for > q For ig prices, q > ; bot types of consumers would like to sell teir claims. For low prices, > q; no one wants to sell tem. In te price range ( ; ) only -type consumers want to sell teir claims to te illiquid asset. Similarly, consumers of type t want to buy securities for prices below t. Hence, one obtains te following aggregate demand: 8 >< D(q) = >: 0 for q > [0; (1 ) M q ] for q = Figure 3 sows tese demand and supply curves. (1 ) M q for > q > [(1 ) M q ; M q ] for q = M q for q < Te market for claims clears for a price in te range [ ; ]: Te equilibrium price q E depends on te proportion of types and te aggregate investment policy M: 8 M >< if < M+(1 M) q E 1 M (; M) = 1 M >: if [ M M +(1 M) ; if > : M M+(1 M) ] : (9) M M +(1 M) At an equilibrium price q E (; M) ( ; ); all consumers of type sell teir claims and all consumers of type use teir money oldings to buy claims. 4.. Investment decision in Period 0. We now turn to te investment decision in Period 0: Since tere is a continuum of consumers, a single consumer's sare in te aggregate investment is negligible. Hence, a consumer will take te market price of claims in Period 1; q E (; M); as given.

18 Liquidity and Ambiguity: Banks or Asset Markets? 17 q S(q)... q E (1 M) D(q) quantity Figure 3: Market for claims to te illiquid asset Given a price for claims on te illiquid asset of q in Period 1; denote by R m (q; t) = maxf t ; q g and R a (q; t) = maxf t q; g te implicit returns in utility on olding one unit of money or one unit of te illiquid asset, respectively. Te indirect utility of a type t consumer wo olds m units of money and wo expects a price of q for claims to te illiquid asset in Period 1, bv a (m; q; t); is bv a (m; q; t) = m R m (q; t) + (1 m) R a (q; t): Te subscript a of te indirect utility function refers to te institutional framework of an asset market. A consumer's prediction of te equilibrium asset price q E (; M) depends on te aggregate money oldings M and te proportion of -types : Hence, indirect utility depends also on tese variables. In order to simplify notation, we write v a (m; M; ; t) = bv a (m; q E (; M); t): Uncertainty about type and proportion of types is modelled again by te degree of condence wic consumers old in te point expectation : Hence, one obtains te following Coquet expected indirect utility: CEU a (m; M; ) = [ v a (m; M; ; ) + (1 ) v a (m; M; ; )] + (1 ) min (t;)f;g[0;1] v a(m; M; ; t): Consumers coose teir initial investment m to maximize CEU a (m; M; ) given aggregate money oldings M: In an equilibrium, aggregate money oldings M must be consistent wit

19 Liquidity and Ambiguity: Banks or Asset Markets? 18 individual decisions m, M = Z 1 0 m di = m : Tis consistency is equivalent to te assumption tat tere exists a price q for claims in Period 1 wic clears te market and produces returns in utility on te two assets (R m (q ; t); R a (q ; t)); wic makes consumers indierent about teir initial investment given te uncertainty about (t; ): Te following teorem sows tat an equilibrium exists for any degree of condence [0; 1]: Proposition 4. Equilibrium in te asset market Tere exists a unique equilibrium (q a(); M a ()) in te asset market satisfying qa() = 1 and Ma () = ; for = 1; qa() ( ; 1) and M a () ( +(1 ) ; ) for ( o ; 1); qa() = and M a () [0; +(1 ) ]: for [0; o ]; yielding an ex-ante expected utility of V a () = q a() + (1 ) : (10) Proof. See te appendix. In an asset market equilibrium, te price q of claims to te illiquid asset must full a dual role: it must clear te market in Period 1 for given oldings of money and te illiquid asset, and it must yield equal Coquet expected returns from olding money and from investing in te illiquid asset in Period 0. If te latter condition were not satised, consumers would eiter old only money or only te illiquid asset, and no trade would occur in Period 1: We will demonstrate in Section 6 tat tis dual task impairs te asset market's potential to acieve te optimal allocation in all but te trivial case of an equilibrium witout trade. As an institutional arrangement, owever, te asset market may dominate te oter intermediary institutions. If tere is no ambiguity, for = 1; te asset market price will be q a() = 1 and te ex-ante expected utility is V a (1) = + (1 ) : As in Diamond (1997) a secondary market for te illiquid asset can only provide liquidity services but not te optimal cross-subsidy from investors wit low liquidity needs to investors

20 Liquidity and Ambiguity: Banks or Asset Markets? 19 wit ig liquidity needs. Clearly, tis provision of liquidity via te secondary claims market improves upon te allocation wic a consumer could provide in isolation, V a (1) = + (1 ) > maxf ; + (1 ) g = V n (1): but it falls sort of te potential payo possible according to te optimal contract, V o (1) = + (1 ) > > + (1 5. Banks ) = V a (1): In an alternative institutional arrangement, liquidity is provided by competing banks. Banks collect funds from consumers, invest tem jointly and, tus, can provide alternative payouts in te two periods. Te instrument to acieve tis intertemporal allocation is te deposit contract. Bank deposit contracts can provide te cross-subsidy required by te optimal contract. In contrast to te secondary asset market, owever, deposit contracts are exposed to a risk of coordination failure. If more depositors witdraw teir deposits in Period 1 tan provided for by te bank, illiquid assets ave to be liquidated at te unfavorable rate 1 in order to fulll te deposit contract. Excess witdrawals in Period 1 diminis payouts on deposits in Period ; wic may induce long-term depositors to witdraw teir funds early. In tis section, we will study ow ambiguity about aggregate witdrawals aects a consumer's evaluation of te deposit contract. For a bank deposit contract, te liquidation possibility, 1 > 0; becomes essential. Wit 1 = 0; te bank would be unable to liquidate long-term investment in favor of early witdrawals. Hence, long-term payos would not be aected. Tere would be no incentive for long-term depositors to witdraw early, even if early witdrawers were to suer losses on teir deposits. Wit a secondary market for te illiquid asset, te equilibrium price would determine te liquidation rate endogenously. Studying a secondary market for te illiquid asset in te presence of te bank deposit contract, as in Diamond (1997), would exceed te scope of one paper Te deposit contract. A deposit contract species repayments for bot periods according to te following rules: 1. Witdrawals in Period 1 are made on demand. Tey are treated as senior to witdrawals in Period : If witdrawals in Period 1 exceed a bank's reserves, te bank will liquidate

21 Liquidity and Ambiguity: Banks or Asset Markets? 0 part or all of its long-term investment in te illiquid asset in order to satisfy depositors' demand for liquid funds in Period 1:. Witin eac period, consumers ave te same priority. If witdrawals exceed te resources of te bank, ten consumers calling back teir deposits obtain a repayment proportional to teir initial deposit. 3. In Period banks distribute teir remaining wealt to depositors, wo did not witdraw in Period 1. In Period 0; consumers deposit teir wealt wit banks. Banks decide on ow to invest tese funds. Based on teir prediction of witdrawals in Period 1; banks old part of teir deposits as reserves in te form of money and invest te remainder in te illiquid asset. Tis policy guarantees te contracted repayments in bot periods, provided te bank predicts witdrawals correctly and does not ave to resort to te liquidation of illiquid assets. Free entry and competition among banks about te terms of deposit contracts ensures zero prots. It also guarantees an investment policy in te interest of depositors 19. Tese assumptions allow us to portray te competing banks by a representative bank. Formally, te deposit contract of a bank is caracterized by te interest rates (i 1 ; i ) promised for Periods 1 and, respectively. Since liquidation of funds invested in te illiquid asset is costly, te bank olds reserves R equal to its payments predicted for Period 1: If te fraction W 0 of depositors witdraws teir funds in Period 1; te bank as to pay out (1 + i 1 ) W 0 : Hence, te bank must old reserves in terms of money equal to R = (1 + i 1 ) W 0 : Remaining deposits, 1 R; will be invested in te illiquid asset: In Period ; competition forces te bank to pay out all its returns from investment, (1 R); to depositors wo did not witdraw in Period 1: Wit an initial amount of deposits equal to 1; te zero-prot condition, (1 + i ) (1 W 0 ) = (1 R); determines te interest rate i : Hence, interest rates (i 1 ; i ) are functions of te bank's reserve policy R and predicted witdrawals W 0, i 1 (R; W 0 ) = R W 0 1; i (R; W 0 ) = 1 R 1 W 0 1: (11) It is, owever, te actual fraction of witdrawals in Period 1; W, togeter wit te priority rules specied above, wic determine te actual payos of deposits, 1 (W ; R; W 0 ) and 19 Competition among banks is in te spirit of Allen and Gale (1998). For a more extensive discussion in a similar context we refer to Spanjers (1999, Capter 3).

22 Liquidity and Ambiguity: Banks or Asset Markets? 1 (W ; R; W 0 ); as and 1 (W ; R; W 0 ) = minf R ; R + 1 (1 R) g (1) W 0 W (W ; R; W 0 ) = max 0; 1 W (1 R) + 1 maxfw 0 W; 0g 1 R maxfw 1 W 0 Equation (1) reects te priority rule tat returns on deposits in Period 1 will be maintained W 0 ; 0g : (13) as long as possible, i.e., as long as [1 + i 1 (R; W 0 )] W = R W 0 W is less tan R + 1 (1 R), te maximal amount of liquidity a bank can raise in Period 1: Te actual return in Equation (13) follows from te assumption tat banks will distribute all teir funds in Period : 1 + i... (W ; R; W 0 ) 1 (W ; R; W 0 ) i 1.. W 0 = R 1+i 1 fw. W 1 W Figure 4: Actual returns of deposits Figure 4 sows te actual returns of deposits as a function of te witdrawals in Period 1. Notice tat, for W = W 0 ; actual returns equal te promised returns, 1 (W 0 ; R; W 0 ) = 1 + i 1 (R; W 0 ) and (W 0 ; R; W 0 ) = 1 + i (R; W 0 ): Moreover, let W be te aggregate level of witdrawals for wic all assets ave to be liquidated in order to maintain te return on deposits in Period 1; W = R + 1 (1 R) W 0 ; ten R (W ; R; W 0 ) = 0 olds. According to te assumptions on asset returns made in Assumption 1, banks will not oer interest rates on a deposit contract suc tat neiter consumers wit ig liquidity needs nor consumers wit low liquidity demand will witdraw teir deposits in Period 1: Hence, banks will coose a reserve policy R; and associated interest rates, suc tat only consumers wit

23 Liquidity and Ambiguity: Banks or Asset Markets? ig liquidity demand will witdraw teir deposits in Period 1: Wit suc a deposit contract banks must prepare for witdrawals equal to te proportion of consumers wit ig liquidity needs. We assume tat banks ave rational expectations regarding te proportion of consumers wit ig liquidity needs. By Assumption, tis implies Z W 0 = dp() = : Given believes about aggregate witdrawals in Period 1; W 0 = ; te bank's reserve policy R, and te implied deposit interest rates (i 1 (R; ); i (R; )) according to Equation (11), must guarantee tat only consumers wit ig liquidity needs will want to witdraw teir funds in Period 1; (1 + i 1 (R; )) (1 + i (R; )) (1 + i 1 (R; )) : (14) Equation (14) sows te incentive compatibility constraints wic a bank's reserve policy R must satisfy. 5.. Te depositor's problem. Consider a consumer wo as deposited all funds wit te bank in Period 0: In Period 1 consumers learn teir types, and te aggregate demand for liquidity becomes known as well. Type- consumers will witdraw teir funds if (W ; R; ) 1 (W ; R; ) olds. Oterwise tey will leave teir deposits in te bank. Similarly, consumers of type will not witdraw teir deposits for 1 (W ; R; ) (W ; R; ): Hence, one can summarize te aggregate witdrawal beavior by 8 1 if (W ; R; ) < 1 (W ; R; ) >< [; 1] if (W ; R; ) = 1 (W ; R; ) W(W ; R) = if 1 (W ; R; ) < (W ; R; ) < 1 (W ; R; ) [0; ] if >: 1 (W ; R; ) = (W ; R; ) 0 if 1 (W ; R; ) < (W ; R; ) : Aggregate witdrawals W are a Nas equilibrium if tey are a xed point of W(W ; R), i.e. W(W ; R) = W. Figure 4 sows te return functions 1 (W ; R; ) and (W ; R; ): For te special case of = 1; one can use tis diagram to ceck for wic levels of W tere is an equilibrium. Given our assumptions on te asset payouts, te return functions intersect just once at te level of witdrawals W f : In general, te critical level of witdrawals W f occurs if te proportion of consumers wit ig liquidity needs exceeds te bank's reserves suc tat second-period returns on deposits fall

24 Liquidity and Ambiguity: Banks or Asset Markets? fw... ṣ s W(W ; R) s 1 equilibrium correspondence P PPP... Pq s 0. fw 1 W 0. fw 1 Figure 5: Witdrawal equilibria to a level were consumers wit low liquidity needs become indierent between witdrawing and leaving teir deposits in te bank, 1 ( W f ; R; ) = ( W f ; R; ): Hence, it is obvious tat tere are two types of equilibria 0 : (i) W(; R) = ; regular equilibrium (ii) W(1; R) = 1: bank-run equilibrium Given a bank's reserve policy R; it is clear from Figure 4 tat W = is a Nas equilibrium if W f olds, oterwise W = 1 is te unique equilibrium. Figure 5 sows in its left part a typical equilibrium constellation. Te critical value f W is a function of te bank's reserve policy R. In te rigt part of Figure 5, te equilibrium correspondence for varying proportions of consumers wit ig liquidity needs is displayed. We will demonstrate in te next section, tat te bank will always coose a reserve policy wic guarantees tat te rationally expected witdrawals W 0 = are less tan f W : In a regular equilibrium, only consumers wit ig liquidity needs will witdraw teir deposits in Period 1; wile in te bank-run equilibrium all depositors will witdraw teir funds. Te bank-run equilibrium always exists. In contrast, a regular equilibrium exists only if aggregate liquidity needs are not too ig. In line wit te literature 1, we will assume tat te regular equilibrium W = obtains wenever it exists. 0 Strictly speaking, W(W ; R; W 0) is a correspondence and tere is also a mixed strategy equilibrium f W ; were all consumers wit ig liquidity needs and some consumers wit low liquidity needs witdraw teir deposits. Tis mixed equilibrium, wic we disregard, is obtained for 1 ( f W ; R; W 0) = ( f W ; R; W 0): 1 See, for example, Diamond (1997) or Allen and Gale (004).

25 Liquidity and Ambiguity: Banks or Asset Markets? 4 By coosing to witdraw teir deposits or to leave tem wit te bank in Period 1, consumers of type t can obtain te utility v b (W ; R; t) = maxf t 1 (W ; R; ); (W ; R; )g: (15) In Period 0; consumers face uncertainty about teir type and te aggregate demand for liquidity : Te Coquet expected utility of a deposit contract in Period 0 is given in te following lemma. Lemma 5. For a bank coosing reserve policy R in Period 0; te Coquet expected utility of a consumer from a deposit contract is CEU b (R; ) = [ + (1 ) ] R + [ + (1 ) 1 ] (1 R): (16) Proof. See te appendix Banks' reserve policy. Wen coosing teir reserve policy R banks implicitly also determine te interest rates on deposits (Equation (11)). Competition forces banks to make tis coice in te interest of consumers. Hence, banks will coose R suc tat te consumers' ex-ante Coquet expected utility CEU b (R; ); derived in Lemma 5, is maximized, subject to te constraint tat consumers wit low liquidity needs do not witdraw funds in Period 1; Equation (14). As solution of te decision problem, coose R to maximize CEU b (R; ) subject to 1 + i (R; ) [1 + i 1 (R; )] ; one obtains te optimal reserve policy R : From Equation (16) and Assumption 1, it is immediately clear tat CEU b (R; ) is a strictly increasing function of R: Since i (R; ) is strictly decreasing and i 1 (R; ) strictly increasing in R, te constraint 1+i (R ; ) = [1 + i 1 (R ; )] must be binding. Substituting from Equation (11), one obtains te following Lemma. Lemma 6. If all consumers (voluntarily) deposit teir wealt wit te bank, ten te optimal reserve oldings are R = + (1 : (17) ) Notice tat te optimal reserve oldings R do not depend on te degree of condence : Moreover, te optimal reserve oldings R equal te aggregate money oldings of te optimal contract M o (); derived in Section 3..