No 2014-18 November Working Paper A Searh-Theoreti Approah to Effiient Finanial Intermeiation Fabien Tripier Highlights The fi nanial risis has strengthene the role of non-fi nanial eposits as a stable soure of bank funing. Eviene suggests important swithing osts on eposit markets. Finanial intermeiation is stuie when househols pay searh osts to fi n aeuate fi nanial prouts an banks to attrat epositors an to selet borrowers. Finanial intermeiation is effi ient when interest rates are poste by banks or bargaine uner a speifi Hosios 1990 onition. Interbank market fritions are introue to show how an risis on this market leas to ineffi ient fi nanial intermeiation.
A Searh-Theoreti Approah to Eient Finanial Intermeiation Abstrat This artile evelops a searh-theoreti moel of fi nanial intermeiation to stuy the effi ieny onition of the banking setor. Competitive fi nanial intermeiation is etermine by the searh eisions of both househols to fi n aeuate fi nanial prouts an banks to attrat epositors through marketing an to selet borrowers through auiting an by the interest rate setting mehanism. The effi ieny of the ompetitive eonomy reuires that interest rates are poste by banks or are bargaine uner a speifi Hosios 1990 onition, whih aresses the hol-up problem inue by searh fritions on the reit an eposit markets. Interbank market fritions are introue to show how an interbank market risis leas to ineffi ient fi nanial intermeiation haraterize by reit rationing an high net interest margin. Keywors Banking; Searh, Mathing, Swithing Costs, Effi ieny. JEL C78, D83, G21. Working Paper CEPII Centre Etues Prospetives et Informations Internationales is a Frenh institute eiate to prouing inepenent, poliyoriente eonomi researh helpful to unerstan the international eonomi environment an hallenges in the areas of trae poliy, ompetitiveness, maroeonomis, international fi nane an growth. CEPII Working Paper Contributing to researh in international eonomis CEPII, PARIS, 2014 All rights reserve. Opinions expresse in this publiation are those of the authors alone. Eitorial Diretor: Sébastien Jean Proution: Laure Boivin No ISSN: 1293-2574 CEPII 113, rue e Grenelle 75007 Paris 33 1 53 68 55 00 www.epii.fr Press ontat: presse@epii.fr
A Searh-Theoreti Approah to Eient Finanial Intermeiation A Searh-Theoreti Approah to Eient Finanial Intermeiation 1 Fabien Tripier 1. Introution The nanial risis that began in 2007 has strengthene the role of non-nanial eposits as a soure of bank funing, whih has beome the new blak. 2 Deposit funing is part of "the urrent bak to basis poliy" formulate by the ECB 2010 3, an it goes bak to the bak to basis issue in nane: the eieny of nanial intermeiation, whih is the transformation 1 This is a revise version of a paper previously irulate uner the title "Eieny gains from narrowing banks: a searh-theoreti approah". I thank Davi Anolfatto, Aleksaner Berentsen, Benjamin Carton, Thomas Grjebine, Marlène Isoré, Philipp Kirher, François Langot, Etienne Lehman, Cyril Monnet, Fabien Postel-Vinay, Etienne Wasmer, an partiipants at the Cyles, Ajustment, an Poliy Conferene Fritions Sanbjerg Gos, Danemark, the Gerzensee Searh an Mathing in Finanial Markets Workshop Gerzensee, Switzerlan, the Searh an Mathing Annual Conferene University of Cyprus, the IRES Seminar Louvain, Belgium, the TEPP-CNRS Winter Shool Frane, the GRE Symposium on Money, Banking an Finane Nantes, Frane, the CEPII-PSE Maro-Finane Workshop Paris, Frane, the AFSE Congress Paris, Frane, an the T2M Conferene Lyon, Frane. Usual islaimers apply. Finanial support from the Chaire Finane of the University of Nantes Researh Founation is gratefully aknowlege. Univ. Lille 1 - CLERSE & CEPII fabien.tripier@univ-lille1.fr 2 This expression is borrowe from a 2012 report of the ompany Ernst & Young for Australian banks entitle "The rise of the eposits", in whih it is state that "Deposits are the new blak, lening playing seon le". 3 This episoe has been oumente wiely, notably by the ECB 2012: "bank funing strategies neee to be ajuste uikly in orer to expan the ustomer eposit base an reue the share of wholesale funing." Interestingly, the ECB 2010 makes a onnetion between the reversal from the interbank market to the retail eposit market an the ruial role playe by bank marketing in the proess: "As for other soures of funing, the risis has resulte in an inrease awareness of ierenes between banks, with banks with establishe brans gaining a ompetitive avantage vis-à-vis their weaker ompetitors." 3
A Searh-Theoreti Approah to Eient Finanial Intermeiation of non-nanial eposits into business loans by nanial intermeiaries. This paper revisits this issue. Its originality, whih forms its ontribution to the literature, is to assume that nanial servies to non-nanial ustomers are haraterize by relationship banking for both epositors an borrowers. 4 I provie empirial eviene that supports this assumption see Setion 2 an evelop a searh an mathing moel of nanial intermeiation base on this eviene. The optimality onitions are erive an the moel is use to stuy the transmission of interbank market fritions to retail banking markets. Eient nanial intermeiation is a onstraine-eient euilibrium that is the soial planner's solution to maximize steay-state welfare given searh osts. From the soial planner's point of view, the issue is to alloate an eient amount of resoures to searh ativities to ensure an eient level of nal goo proution. In a ompetitive eonomy, nanial intermeiation is etermine by the searh eisions of both househols to n aeuate nanial prouts an banks to attrat epositors through marketing an to selet borrowers through auiting. Even if markets are not fritionless, there are ways to reah eieny. The rst metho 5 was emonstrate by Hosios 1990 an reuires euality between the agent's bargaining power an the elastiity of the mathing funtion with respet to its searh eort. In this ase, the Hosios 1990 onition is satise an the searh externalities are internalize. However, this onition oes not hol for nanial intermeiation beause banks fae searh fritions in reruiting both epositors an borrowers. Agents eie, rst, to searh an then, if mathe, bargain interest rates. When bargaining on interest rates, a bank onsiers its gains an losses regarless of whether the bargain with the non-nanial agent suees. Without an agreement, the bank woul lose not only the value of one nanial relationship but also that of two relationships. 4 These relationships are known in the literature as relationship banking, ene by Goar et al. 2007 as follows: "One suh topi is relationship banking, whih an be ene simply as the provision of nanial servies repeately to the same ustomer". Relationship banking is not a new topi in nanial intermeiation. As explaine below, the novelty of this paper is to onsier relationship banking for both eposit an reit in a searh moel an to stuy the eieny onition of nanial intermeiation. 5 A seon metho, whih relies on prie posting an irete searh, is stuie thereafter. 4
A Searh-Theoreti Approah to Eient Finanial Intermeiation Inee, without an agreement with a epositor, the bank loses this epositor as well as a reitor who an no longer be nane. Likewise, without an agreement with a borrower, a bank loses this borrower as well as one epositor whose funs an no longer be investe. The bank faes a liuiity problem 6 whih leas to the hol up problem. The hol up problem onsiere herein proees from the neessity of two investments in searh an not of ex-ante investment in apital, tehnology or human apital as generally onsiere in the literature; e.g. Grout 1984, Malomson 1997 an Aemoglu an Shimer 1999. The Hosios 1990 onition for eieny is vali when there is no hol up problem. Otherwise, alternative bargaining powers are reuire to reah eieny. 7 Contrary to the ase stuie by Aemoglu an Shimer 1999, eieny an be ahieve herein with ex-post barganing. The iulty is that eient bargaining powers are ierent from reit an eposit markets even if mathing funtions are iential an are funtions of a large set of strutural parameters not only the elastiity parameter of the mathing funtion. These properties of eient bargaining powers hol whatever the negotiation is between two players the bank plus one non-nanial ustomers, a epositor or a borrower or three players the bank plus two non-nanial ustomers, a epositor an a borrower. Beause eient bargaining powers may be iult to implement, I onsier an alternative mehanism to set interest rates: prie posting. If banks post interest rates on markets instea of bargaining over them with non-nanial agents, the hol up problem ientie herein woul isappear. Banks jointly etermine the searh eorts an the interest rates an are not expose to the ouble loss in the ase of bargaining failure as esribe above. I show that the ompetitive eonomy is eient if banks post nanial ontrats on the market an if non-nanial agents iret their searh towar banks. As in Aemuglu an Shimer 1999, the interest of prie posting 6 It is a liuiity problem beause the bank annot buy or sell assets eposits or loans uikly on nanial markets. The ruial point here is that retail banking markets are sluggish as in Huang an Ratnovski 2011. 7 Petrosky-Naeau an Wasmer 2013 stuy the eieny of nanial intermeiation with searh on reit an labor markets. In their moel, the Hosios 1990's onition is suient to guarantee eieny beause rms o not have the problem of liuiity onsiere here for banks. 5
A Searh-Theoreti Approah to Eient Finanial Intermeiation is to avoi the hol up problem. This supplements its traitional interest in single market searh moels, thus making enogenous the Hosios 1990 onition; see Shimer 1996 an Moen 1997. I use this moel of eient nanial intermeiation to investigate the onseuenes of interbank market fritions in line with Due et al. 2005, Lagos an Roheteau 2009, an Lagos et al. 2011. To provie a role for interbank market, banks are speialize either in the eposit ativity or in the reit ativity. Two mathe banks negotiate a ontrat whih speies the mass of ustomers for eah bank an a payment from the reit bank to the eposit bank. There is an exogenous probability of math issolution on the interbank market that leas to the breakown of relationships with non-nanial ustomers. An inrease in this probability, whih an be interprete as interbank market risis, leas to an ineient nanial intermeiation haraterize by reit rationing an high net interest margin. 8 The remainer of the paper is struture as follows. Setion 2 gives the rationale for searh fritions in the banking setor. The issue of nanial intermeiation an its soially optimal solution are presente in Setion 3. The ompetitive euilibrium is ene in Setion 4 when interest rates are ex-post bargaine an in Setion 5 when interest rates are poste. The isussion an onluing remarks are presente in Setion 6. 2. Rationale For Searh Fritions The role of nanial intermeiation onsiere in the moel evelope herein oes not follow from strutural ierenes between eposits an loans 9, but from the existene of searh fritions in nanial markets. This Setion gives the rationale for searh fritions in the reit an eposit markets. 8 The net interest margin is the ierene between the reit interest rate an the eposit interest rate. 9 The traitional role of nanial intermeiation is to transform assets, whih are heterogeneous with respet to size, risk, or maturity. Here, there is a one-to-one orresponene between the per-perio eposit of one househol an the resoures neee to nane a one perio rm projet. 6
A Searh-Theoreti Approah to Eient Finanial Intermeiation 2.1. Creit Market Applying the searh moel to the reit market follows the literature initiate by Diamon 1990 an evelope by Den Haan et al. 2003, Wasmer an Weil 2004, an Dell'Arriia an Garibali 2005. Two rationales for reit searh fritions have been provie. The rst rationale is base on the existene of long-term relationships between leners an borrowers, whih are known as lening relationships. Berger an Uell 1998 report an average uration of lening relationship between small business rms an ommerial banks of 7.77 years. This was a robust observation in nanial markets at the beginning of the lening relationship literature, whih has been reviewe by Berger an Uell 1995 an Elyasiani an Golberg 2004. Both Den Haan et al. 2003 an Wasmer an Weil 2004 invoke this literature to motivate their reit market searh moel. 10 The seon rationale for reit searh fritions is provie by Dell'Ariia an Garibali 2005 an Craig an Haubrih 2013. They onstrut atabases of reit ows from banks an show that the reit market in the Unite States is haraterize by large, ylial ows of reit expansion an ontration that may be explaine in terms of the mathing frition. Base on these rationales, numerous theoretial moels inorporate the reit market searh moel to aress maroeonomi an nanial issues. 11 2.2. Deposit Market Applying the searh moel to the eposit market is a ontribution of this paper to the literature on fritional nanial markets. 12 Searh fritions have alreay been onsiere on the reit 10 Den Haan et al. 2003 argue that "... there is a mathing frition in the market to establish entrepreneur-lener relationships. This frition highlights the importane of long-term relationships". They evelop this argument in the Setion "Motivation for mathing frition" of their paper. 11 See, among others, Beaubrun-Diant an Tripier 2013, Besi et al. 2005, 2013, Chamley an Rohon 2001, Petrosky-Naeau an Wasmer 2013, an Petrosky-Naeau 2013. 12 Another stran of literature onsiers the role of nanial intermeiaries in searh-base moels of monetary exhange à la Kiyotaki an Wright 1989 to explain the use of bank liabilities as a meia of exhange, see He et al. 2005, 2008, an that banks improve welfare, see Berensten et al. 2007 an Gu et al. 2013. 7
A Searh-Theoreti Approah to Eient Finanial Intermeiation market as explaine just above, on over-the-ounter nanial markets, rst by Due et al. 2005 an then by Lagos an Roheteau 2009 an Lagos et al. 2011, among others, but not on the eposit market. 13 The rationale for eposit market searh fritions is, as for the reit market, the existene of long-term relationships between epositors an banks. I rst oument this fat, an I then explain it using the presene of swithing osts for househols an of relationship marketing by banks. The European Commission 2009 publishe a survey on onsumers' views regaring swithing servie proviers to ollet information about onsumers' experienes swithing proviers an their ability to ompare oers from various suppliers in several servie setors. The swithing rate in the last two years is 11% for the banking inustry as a whole, whih is notably lower than the rates observe in other servie setors, suh as ar insurane 25% or internet servie 22%. The moel evelope in this paper oes not apply to all nanial servies provie by banks, but to the remuneration of savings. It is therefore important to note that the swithing rate for savings or investment prouts only remains low, approximately 13% against 9%, for urrent bank aounts. Furthermore, this low swithing rate is not only observe in European ountries. Kiser 2002 reports a mean relationship uration of 13.3 years from a survey of Amerian onsumers in Mihigan. 14 Swithing osts is the most popular explanation for househols' behavior on the eposit market. When a househol eies to swith or when she enters the market, she must spen time an resoures to obtain information on servies oere by banks; this is the searh proess for househols on the eposit market. This searh proess woul be ostless an instantaneous without searh fritions. However, the omplexity of the retail nanial market makes this searh proess ostly an time-onsuming. Inee, Carlin 2009 argues that "[p]urhasing a 13 Isoré 2012 also introues searh fritions in funing soures of banks, but with stakeholers an not with househol epositors as onsiere here. 14 Beause the swithing rate is a proxy for the inverse of the uration of ustomer relationships, it orrespons to a swithing rate of approximately 7.5% per year or 15% every two years. 8
A Searh-Theoreti Approah to Eient Finanial Intermeiation retail nanial prout reuires eort. Beause pries in the market are omplex, onsumers must pay a ost time or money to ompare pries in the market." Similarly, aoring to Sirri an Tufano 1998, "[e]onomists aknowlege that onsumers' purhasing eisions whether for ars or funs are ompliate by the phenomenon of ostly searh". Aoringly, the European Commission 2009 reports that 43% of interviewe ustomers antiipate or experiene iulties swithing banking servies an 37% think that it is very an/or fairly iult to ompare oers in the banking setor. Consistent with this eviene, I assume in the moel evelope herein that househols pay searh osts on the eposit market an that these searh osts are interprete as swithing osts beause a househol must pay the osts to n another bank. 15 Swithing osts are intimately relate to the pratie of relationship marketing. The following observation about relationship marketing is mae by Chiu et al. 2005: "marketing ativities that attrat, evelop, maintain, an enhane ustomer relationships has hange the fous of a marketing orientation from attrating short-term, isrete transational ustomers to retaining long-lasting, intimate ustomer relationships." Relationship marketing is therefore preisely evote to inreasing the ost of swithing for ustomers. The unerlying motivation of banks is to inrease prots, as explaine by Degryse an Ongena 2008: "Swithing osts for bank ustomers represent an important soure of rents for banks, an an important motive for the evelopment of relationship as oppose to transation banking." Sharpe 1997, Shy 2002, an Martin-Oliver et al. 2008 have establishe both theoretially an empirially the impat of swithing osts on eposit interest rates using ata for the Unite States, Finlan, an Spain, respetively. In the moel evelope herein, bank searh osts are interprete as investment in relationship marketing beause they are neessary to reate long-term relationships with househols. 15 It is worth mentioning that if I ientify both searh osts an swithing osts, Wilson 2012 evelops a moel evote to istinguishing between searh osts an swithing osts. 9
A Searh-Theoreti Approah to Eient Finanial Intermeiation 3. The Issue of Finanial Intermeiation This Setion enes the issue of nanial intermeiation an presents the soially optimal solution. 3.1. Enowments an Tehnologies I onsier an eonomy with a raw goo that annot be onsume an a nal onsumption goo that is proue by using the raw goo as input. All agents househols, entrepreneurs, an banks share the same linear utility funtion an isount fator for the future, enote 2 ]0; 1[ with = 1= 1 r, where r is the assoiate interest rate: There are two proution tehnologies with ierent ualities. Househols possess the low-uality tehnology that proues h > 0 units of nal goo per unit of input. Entrepreneurs possess the high-uality tehnology that proues z > h units of nal goo per unit of input. Entrepreneurs have better tehnology, but all raw goos are initially given to househols. Eah househol hols an asset that elivers one unit of raw goo per perio. The eonomi issue is how to avoi autarky: how o we transfer raw goos from househols to entrepreneurs without a market for the raw goo e.g., without iret nane? This is the issue of nanial intermeiation, solve herein in the presene of searh fritions. 3.2. Searh Fritions I rst haraterize searh fritions on the reit market. Banks invest v in the searh to n entrepreneurs on the reit market, where is the searh ost per unit of eort an v is the banks' searh eort assuming a unit ontinuum of banks, it is eual to the searh eort of the representative bank. Searh is ostless for entrepreneurs, an the u unmathe entrepreneurs searh for a bank. 16 The per-perio ow of new lening relationships is given by the mathing funtion m v ; u, whih has onstant returns to sale an is inreasing in both arguments. 16 Searh is exogenous for entrepreneurs an enogenous for househols an banks. 10
A Searh-Theoreti Approah to Eient Finanial Intermeiation The n mathe entrepreneurs proue an remain mathe with a probability 1 ; where 2 ]0; 1[ is the probability of business failure. 17 The number of mathe entrepreneurs evolves as follows n = 1 n m u ; v 1 where the symbol is use to enote the next-perio value of state variables. The population of entrepreneurs is set to n an satises n = n u. Banks invest v in the searh to attrat househols to the eposit market, where is the searh ost per unit of eort an v is the banks' searh eort assuming a unit ontinuum of banks, it is eual to the searh eort of the representative bank. Unmathe househols proue low-uality tehnology; a part, u, of them eie to searh for a bank an to pay the per-perio ost h, whereas another part, o, of the househols prefer to remain outsie the banking setor. The per-perio ow of new eposit relationships is given by the mathing funtion m v ; u, whih has onstant returns to sale an is inreasing in both arguments. The n mathe househols remain mathe with a probability 1 ; where 2 ]0; 1[ is a preferene shok. 18 The number of mathe househols evolves as follows n = 1 n m u ; v 2 The population of househols is set to n an satises n = n u o. The number of proutive entrepreneurs annot exee the number of epositors n n 3 where n is also the amount of eposits eah househol eposits one inivisible unit of raw goo an n is the amount of reits eah entrepreneur borrows one inivisible unit of raw 17 After a failure, the entrepreneur buils a new business projet that shoul be auite by banks to be nane. 18 The househol eies to swith banks after a hange in its emographi omposition e.g., births, ivore or on the labor market e.g., job loss, promotion. The househol pays the searh osts to n the relevant nanial servie given the new situation. 11
A Searh-Theoreti Approah to Eient Finanial Intermeiation goo. Finally, the mathing tehnologies are Cobb-Douglas with the following properties m x u x ; v x = m x v x " u x 1 " 4 x = m x u x ; v x =v x = m x x " 1 = p x = x @m x u x ; v x =@u x = 1 " p x, @m x u x ; v x =@v x = " x where x = v x =u x is the market tightness, x an p x are the mathing probabilities, for x = f; g where stans for reit an for eposit. Without a loss of generality, the two mathing funtions share the same elastiity parameter ", but the sale parameter m x may be ierent. Lemma 1 The market tightness variables f x g x=f;g etermine the egree of nanial intermeiation an the soial welfare. Proof. See Appenix A. 3.3. The Soially Optimal Solution The soially optimal euilibrium is a onstraine-eient euilibrium. The soial planner hooses searh eorts to maximize steaystate welfare, taking the searh fritions as given. The value funtion assoiate with the problem of the soial planner is O n ; n = { max n z u ;fn x ;v g x x=f;g [ n 1 n m n n ; v [ n i n n n n u h u h h v v O n ; n 1 n m u ; v ] ] 5 } where the per-perio utility ow is ene as the nal goos proue by househols an entrepreneurs less the searh osts for househols an banks. f x g x=f;;ig are the Lagrangian 12
A Searh-Theoreti Approah to Eient Finanial Intermeiation multipliers assoiate with the onstraints 1, 2, an 3. The next proposition presents the solution of 5. Proposition 1 The soially optimal euilibrium exists an is uniue. Finanial intermeiation is soially optimal if the tehnology gap between househols an entrepreneurs is suiently high. Proof. The soially optimal alloation of resoures is ene by the market tightness variables f x g x=f;g that solve o = h " 1 " 6 r m o 1 " r 1 " m o = " z h 1 " o 7 The tehnology gap between househols an entrepreneurs is z h, an it shoul be suiently large to ensure a positive value for o. See Appenix B for etails. The soially optimal eposit market tightness o is given by euation 6 as a funtion of searh osts an elastiity parameters of the mathing funtions. 19 To inrease the amount of eposits, the soial planner an inrease the banks' searh eorts, at the ost for the marginal proutivity m 2 u ; v ; or the househols' partiipation, at the ost h for the marginal proutivity m 1 u ; v. Conition 6 makes eual the ost ratio, whih is h =, to the marginal proutivity ratio, whih is m 1 u ; v =m 2 u ; v = 1 " =", given the speiation 4 of the mathing funtion. The soially optimal reit market tightness o is then given by euation 7, where o is given by 6. The LHS of 7 measures the mathing osts of nanial intermeiation: how muh 19 This expression for the euilibrium tightness is ommon in searh moels with two enogenous partiipation rules, not one, as is usually assume. Inee, partiipation is enogenous for rms, but it is exogenous for workers in the stanar labor market searh moel. Wasmer an Weil 2004 obtain an expression for the reit market tightness similar to 6 beause they onsier the enogenous partiipation of both entrepreneurs an bankers but the exogenous partiipation of workers on the labor market. Here, partiipation is enogenous for banks on the eposit an reit markets, enogenous for househols on the eposit market, an exogenous for entrepreneurs on the reit market. 13
A Searh-Theoreti Approah to Eient Finanial Intermeiation it osts to ollet one unit of eposit an to selet one entrepreneur. The mathing osts are eual to x =m x x o 1 " : the per-perio searh ost x on market x = f; g is ivie by the probability of mathing, m x x o " 1. Mathing osts are isounte by r x, the sum of the rate of time preferene an of the separation probability. The RHS of 7 measures the soial benets of nanial intermeiation: the tehnology gap between househols an entrepreneurs, weighte by ", less the opportunity osts of being mathe for the entrepreneur, weighte by 1 ". Outsie the math, the entrepreneur woul have ontribute to soial welfare by searhing: with a marginal proutivity m 1 u ; v, it woul have reate the value of a new math, whih is eual to " =m 2 u ; v. The last term of the RHS of 7 orrespons to the prout m 1 u ; v " =m 2 u ; v, given the speiation 4 of the mathing funtion. 4. Competitive Finanial Intermeiation with Interest Rate Bargaining This Setion presents the ompetitive euilibrium with ex-post bargaining an unirete searh by non-nanial agents. 4.1. Searh 4.1.1. Non-Finanial Agents Househols' value funtions are enote D y, where y = fh; m; ug refers to the househol states: non-partiipating, mathe, an searhing, respetively. They are ene by D h = h D h 8 D m = 1 D m D u 9 D u = h h p D m [ 1 p ] D u 10 If the househol oes not partiipate, she proues h of nal goos. If she eies to searh for a bank, she still proues h but now pays h as a searh ost an has a probability p 14
A Searh-Theoreti Approah to Eient Finanial Intermeiation of forming a math with a bank. When she is mathe, the househol reeives units of the nal goo as eposit interests an remains in this state with a probability 1. Unmathe househols eie whether to searh. The free entry onition on the eposit market implies D h = D u or, euivalently, D h = D u! h = p [ D m D u] 11 h = p h r given 8, 9, an 10. The entry of househols is suh that the searh ost, h, is eual to searh payo: with a probability p, the househol earns the ierene between eposit interests an home proution h isounte by r. Entrepreneurs' value funtions are enote as C y, where y = fm; ug refers to the entrepreneur states: mathe an unmathe, respetively. They are ene by C u = p C m 1 p C u 12 C m = z 1 C m C u 13 The per-perio utility is zero when entrepreneurs searh an z when mathe, where is the amount of reit interests. The transition probabilities between states are p an : 4.1.2. Bank Searh Eorts The representative bank maximizes the isounte sum of prots ene as the reit interests less both the eposit interests an the searh osts subjet to the onstraints 1, 2, an 3. The bank value funtion is B n ; n { } = max n n v v B n fn x ;v x g ; n x=; [ n 1 n v ] [ n i n n 1 n v ] 14 15
A Searh-Theoreti Approah to Eient Finanial Intermeiation The bank hooses searh eorts fv x g x=f;g given the interest rates f x g x=f;g suh that r m 1 " r m 1 " = 15 see Appenix C.1 for etails. The optimality onition for banks, namely 15, an be ompare with its ounterpart for the soial planner, namely 7. The LHS terms of 7 an 15 are iential an orrespon to the mathing osts of nanial intermeiation. The RHS terms orrespon to benets of nanial intermeiation, whih may ier. The bank's benets are the interest margin, i.e., the ierene between reit an eposit interests in 15, whih may not oinie with the soial benets in 7. 4.2. Bargaining The bank is mathe with n epositors an n borrowers an therefore bargains with n n ustomers. Bargaining is iniviual an ex-post. When the bank bargains with a ustomer, she is onsiere as the marginal ustomer assuming that all other negotiations are terminate. Uner this assumption, there are only two players in the bargaining proess, namely the bank an the marginal ustomer an the Nash Solution an be viewe as the outome of a bilateral bargaining game in the strategi approah; see Rubinstein 1982 an Binmore et al. 1986. The ase of multilateral bargaining game is isusse in Setion 4.4. Interest rates satisfy x = arg max X m X u 1 x B x 16 for x = f; g an X = fc; Dg : The parameter 1 x measures the bargaining power of banks, an x measures the bargaining power of non-nanial ustomers for x = f; g. The surplus of non-nanial ustomers X m X u an be iretly ompute using the value funtion enitions 9 an 10 for househols an 12 an 13 for entrepreneurs. It is less iret for the bank's surplus B x, given the onstraint 3. Consier rst the ase where the marginal ustomer is a epositor. If bargaining fails, the bank is eprive of one epositor an one borrower annot proue beause the bank annot provie 16
A Searh-Theoreti Approah to Eient Finanial Intermeiation the raw goo, whih is neessary for the entrepreneur to proue. I assume that if the proution proess is interrupte, the entrepreneur shoul be auite one again to restart the proution proess. The same ours for borrowers. If bargaining fails, the bank is eprive of one borrower an one epositor beause the bank annot transform the raw goo into a nal goo to pay the eposit interest. In fat, the epositor withraws her unit of raw goo if the utility in the unmathe state is higher than the utility in the mathe state with no eposit interests, that is: D u > D m : In this ase, the epositor leaves the bank, onsumes the nal goo an =0 prospets for another bank. Using the value funtion enitions 9 an 10, it ours when or, euivalently, h > h D h > 1 D m D u 17 1 h =p given the free entry onition on the eposit market 11. Hereafter, I assume that the onition 17 hols. If the bargaining with a ustomer fails, the bank loses two ustomers an not only one. This situation results from the existene of searh fritions on the two markets an leas to the hol up problem.to formalize this point, I introue the funtions n y 1 nx whih satisfy 0 no hol up problem n y 1 nx = 1 hol up problem 18 for x = f; g, y = f; g an y 6= x: The loss of one epositor implies the loss of one borrower if n 1 n = 1 an not otherwise. Similarly, the loss of one borrower implies the loss of one epositor if n 1 n = 1 an not otherwise. Finally, the bank surplus is as follows B x = @B n ; n @n x n y =n y n x = 1 1 x [1 n y 1 nx ] 1 r x 19 for x = f; g, y = f; g an y 6= x; see the Appenix C.2 for etails. The bank's surplus on the market x is eual to the net interest margin plus the value of nanial relationships on the two markets, if these relationships are not estroye by the probability 1 x, less the 17
A Searh-Theoreti Approah to Eient Finanial Intermeiation isounte mathing osts. If n y 1 nx = 1, there is a hol up problem with the non-nanial ustomer x an the bank annot subtrat the mathing ost from its surplus. If n y 1 nx = 0, there is no hol up problem an the bank's surplus is lower than if n y 1 nx = 1. The hol up problem aets the bank's surplus an, therefore, impat the interest rates an searh eisions, as shown in the next Setion. 4.3. Competitive Euilibrium I ene the ompetitive euilibrium an then isuss its normative properties. Denition 1 The ompetitive nanial intermeiation with bargaine interest rates is haraterize by the interest rates f xg b whih satisfy x=f;g [ ] b = h 1 h m " 1 r 1 " 1 m b n 1 n m b 1 " b b = z r p b 1 [ m b 1 ] " n 1 n 1 " m b where the euilibrium market tightness variables f xg b are the solution of x=f;g [ ] h m " = 1 " 1 m b n 1 n b m b 1 " b 20 21 22 r m b 1 " r 1 1 m b 1 " 23 = 1 z h b r p b n 1 n b r 1 1 n 1 n b m b 1 " m Appenix C.3 provies the resolution etails for the Nash bargaining proess, an Appenix C.4 shows how to obtain the euilibrium onitions for market tightness. b 1 " The euilibrium eposit interest rate given by 20 is eual to i the value of self-proution, h, ii less the househol value of the nanial relationship, h =p, whih is preserve by the 18
A Searh-Theoreti Approah to Eient Finanial Intermeiation househol with the probability 1, iii plus the share = 1 of the bank values of nanial relationships, x = x, isounte by 1 r for x = f; g. When n 1 n is eual to zero, the househol reeives only a share of the bank value of the nanial relationship on the eposit market, as is ommonly the ase in moels that use searh fritions. Here, the novelty is that when n 1 n is eual to unity, the househol also reeives a share of the bank value of the nanial relationship on the reit market. This is the hol up problem: beause the bank is subjet to searh fritions on the reit market, the househol suees in appropriating a share of the lening relationship's value, even if the househol oes not partiipate in the searh externalities on the reit market. All other things being eual, the hol up problem makes eposit interests higher. The euilibrium reit interest rate given by 21 is eual to i the value of business proution, z, ii less the share = 1 of the bank value of nanial relationships, x = x, isounte by 20 r p b for x = f; g. When n 1 n is eual to zero, the entrepreneur reeives only a share of the bank value of the nanial relationship on the reit market, as is ommonly the ase in moels that use searh fritions. As previously, when n 1 n is eual to unity, the hol up problem ours: beause the bank is subjet to searh fritions on the eposit market, the entrepreneur suees in appropriating a share of the eposit relationship's value even if the entrepreneur oes not partiipate in the searh externalities on the eposit market. All other things being eual, the hol up problem makes reit interests lower. Euations 22 an 23 show how the hol up problem inuenes the euilibrium values of the tightness variables. When househols appropriate a share of the lening relationship's value, i.e., n 1 n b = 1 in the RHS term of euation 22, they are willing to pay a higher mathing ost, whih orrespons to the LHS term of euation 22. For a given value of the reit market tightness b, this mehanism tens towar a fall in the eposit market tightness b : 20 The isount rate an also be written as [1 r p 1 ]. It is ompose of the isount rate for time preferene, namely 1 r ; an of the ierene in probabilities of being mathe aoring to the urrent state, namely, p, if unmathe, an 1, if mathe. 19
A Searh-Theoreti Approah to Eient Finanial Intermeiation For banks, the hol up problem lowers the payo of nanial intermeiation, as in the RHS term of euation 23 for n 1 n b = n 1 n b = 1; onseuently, the mathing osts of nanial intermeiation shoul erease, i.e., the LHS term of euation 23. For a given value of the eposit market tightness b, this mehanism ats towar a fall in the reit market tightness b : The next proposition shows how the bargaining proess an oset the eets of the hol up problem. Proposition 2 There exist values for bargaining power that make the ompetitive nanial intermeiation eient. Proof. The values of bargaining powers f x og x=; imply that f x b = x og x=; where f x og x=; solve 6 an 7 an f x b g x=; solve 22 an 23. They are o = 1 " [1 "n 1 n b o o ] 1 1 " 24 o = 1 " z h o r z h o r o n 1n b o ""n1n b o o o n 1n b o ""n1n b o o r p o n 1 n b o 25 1 " where the soially optimal values for market tightness f x og x=; o not epen on the bargaining power. See Appenix C.5 for etails. Euations 24 an 25 generalize the Hosios 1990 onition for eieny. In stanar searh moels, the Hosios 1990 onition imposes euality between the agent's bargaining power an the elastiity of the mathing funtion with respet to its searh eort. In this ase, the searh externalities are internalize. This onition woul have been suient in the searh moel of nanial intermeiation propose in this paper without the hol problem: the onitions 24 20
A Searh-Theoreti Approah to Eient Finanial Intermeiation an 25 reue to o = o = 1 " 26 for n 1 n b = n 1 n b = 0: With the hol up problem, the Hosios 1990 onition is no longer suient to ensure eieny. The seon terms in euations 24 an 25, whih multiply the elastiity oeient 1 ", generalize the Hosios 1990 onition to aress the hol up problem. The next orollary explains the orretion. Corollary 1 Eieny of nanial intermeiation reuires that the banks' bargaining power inrease when the hol up problem ours. Otherwise, the eposit market tightness is lower than the soially optimal level an reit rationing may our. Proof. It follows from proposition 2, see Appenix C.6 for etails. If the bargaining power values are xe to the Hosios 1990 values given by the euation 26, ineient nanial intermeiation ours beause of the hol up problem. In Appenix C.6, I emonstrate that the ompetitive eposit market tightness is eual to or below its soially optimal level. The hol-up problem stimulates the entry of househols into the eposit market who are willing to pay higher mathing osts. The ompetitive reit market tightness an be lower than, eual to or higher than its soially optimal level beause of the existene of two eets. First, the hol up by entrepreneurs makes the ompetitive reit market tightness lower beause banks reue their searh eorts in response to a ut in reit interests. Seon, low tightness on the eposit market reues the mathing osts of banks on this market, whih are therefore willing to pay higher mathing osts on the reit market. Remember that banks onsier the total mathing osts to be the sum of the mathing osts on the eposit market an on the reit market, see the LHS of 15. For a given net interest margin, e.g., the RHS of 15, smaller mathing osts on one market imply higher mathing osts on the other one at euilibrium. If the rst eet ominates the seon, ineient nanial intermeiation ours with exessive reit rationing; otherwise, it ours with an exessive investment of sare resoures in the nanial setor. 21
A Searh-Theoreti Approah to Eient Finanial Intermeiation 4.4. The Case of Multilateral Bargaining In the ase of bilateral bargaining, all other negotiations are assume to be terminate. However, beause of the hol up problem, this assumption may not be onsiere appropriate. When a ustomer threatens to leave the bank, the other ustomer, whih risks losing her nanial relationship, may be intereste in renegotiating with the bank. In this ase, the bargaining is multilateral. Chae an Yang 1992, Krishna an Serrano 1996, an Suh an Wen 2006 show how reoniling the axiomati an strategi approahes to multilateral bargaining problem. 21 The Nash solution for three agents shoul be applie as follows { ; } } = arg max {D m D u C m C u B b 27 where b = 1. The bargaining problem is resolve in Setion D. I show the neessary onitions on the bargaining powers { ; } to ensure the eieny of the nanial intermeiation. One again, as in the ase of bilateral bargaining, the bargaining powers ier from the stanar Hosios onition 26 an are omplex funtions of the strutural parameters. Moreover, bargaining powers shoul be ierent for the eposit an the borrower, 6=, to ensure eieny - see Setion D. This result has important impliations for eonomi founations of 27. For example, in the Krishna an Serrano 1996 game with three players, the bargaining power is 1= 1 2! for the rst proposer an!= 1 2! for the two responers where! is the egree of impatiene see also Serrano 2008. If the bank is the rst proposer an the non-nanial agents the two responers, it implies: b = 1= 1 2! an = =!= 1 2! =! b. In this ase, there is no value for! that ensures the eieny of the ompetitive nanial intermeiation when bargaining is multilateral. 21 See Suton 1986 for an exposition of the iulty of extening the two-players Rubinstein 1982 moel to n-players game. 22
A Searh-Theoreti Approah to Eient Finanial Intermeiation 5. Competitive Finanial Intermeiation With Interest Rate Posting This Setion presents the ompetitive euilibrium when banks post interest rates an non- nanial agents iret their searh towar banks. The resolution of the searh moel of nanial intermeiation with prie posting is inspire by that of Kaas an Kirher 2013 evelope for the labor market with large rms. 5.1. Direte Searh 5.1.1. Househols The value funtion assoiate with the non-partiipating state is unhange an given by 8. The value funtion assoiate with the searhing state beomes D u = h [ ] h p D m 1 p D u 28 where D m is the value funtion assoiate with the mathe state 9 for the poste interest rate. Assuming that two interest rates are poste on the eposit market { } ; ; being the euilibrium rate an the rate poste by a bank that eviates from the euilibrium value. Househols an searh for a bank that oers the euilibrium rate or for the eviating bank. At the euilibrium, the searh payos shoul be eual or, euivalently, the value funtion 28 is the same for an h = h h p [ ] h p D m 1 p ] D m [ 1 p D u 29 D u Simpliations give [ p D m D u] [ = p D m D u] 30 I show in Appenix E.1 how to use the no-arbitrage onition ene by euation 30 to get the elastiity of the bank mathing probability with respet to the eposit interest rate @ 1 " = > 0 31 @ " h 23
A Searh-Theoreti Approah to Eient Finanial Intermeiation This expression will be neessary to etermine the bank's priing strategy. The sign of the partial erivative is positive: if the eviating bank inreases its poste interest rates, more househols searh towar this bank. Therefore, the probability to n new epositors for this bank is higher even if its searh eort v is onstant. 5.1.2. Entrepreneurs The value funtion assoiate with the mathe state is ene by 13. The value funtion assoiate with the searhing state beomes C u = p C m 1 p C u 32 Assuming that two interest rates are poste on the market { ; } ; being the euilibrium rate an the rate poste by a bank that eviates from the euilibrium value. Entrepreneurs an searh for a bank that oers the euilibrium rate or for the eviating bank. At the euilibrium, the searh payos shoul be eual or, euivalently, the value funtion 32 is the same for an Simpliations give = p p C m 1 p C m 1 p p = p C m C m C u C u 33 C u 34 C u 35 I show in Appenix E.2 how to use the no-arbitrage onition ene by euation 35 to get the elastiity of the bank mathing probability with respet to the reit interest rate [ ] @ 1 " 1 1 p @ = " 1 1 z < 0 36 This expression will be neessary to etermine the bank's priing strategy. The sign of the partial erivative is negative: if a bank ereases its poste interest rates, more entrepreneurs searh towar this eviating bank. Therefore, the probability to n new borrowers for this bank is higher even if its searh eort v is onstant. 24
A Searh-Theoreti Approah to Eient Finanial Intermeiation 5.2. Interest Rate Posting The state variable % x for the market x = f; g measures the amount of interests reeive or pai by the bank, whih evolves as follows % x = 1 x % x x v x x ; for x = f; g 37 This variable is a state variable beause the bank annot revise interest rates poste in the past. The representative bank maximizes P n ; n ; % ; % { = max % % v v P f% x ;nx ;v x ; x g x=f ;g [ n 1 n [ % i n n 1 % v ] [ n ] v } n ; n ; % ; % 1 n v ] [ % 1 % 38 v ] The prie posting strategy is etermine by the rst orer onition of program assoiate with the poste interest rate x on the market x = f; g: [ x : @ x x x @ x v x x @ x x @ x x x ] x v x = 0; x = f; g 39 see Setion E.3. The rst term aount for the impat of x on the reation of new nanial relationship, whih values are x, aoring to 1 x x v x, namely the reation of the mathing probability times the searh eort, v x. The sign of 1 x x epens on the market x as explaine in Setion 5.1. The seon term aount for the impat of x on the variation of the value funtion inue by the new amount of interests, whih is eual to x the erivative of the value funtion with respet to %. Varying x impats iretly the amount of interests for the ow of new ustomers, v x, an iniretly beause of the variation in this ow, 1 x v x. The euilibrium value of the two multipliers are = 1= r an = 1= r. The sign of is positive an that of negative, beause reit 25
A Searh-Theoreti Approah to Eient Finanial Intermeiation interests inrease the value funtion whereas eposit interests lower it. Variation in interests are isounte at the rate r x that takes into aount the preferene for the present an uration of the ommitment for poste interest rates. 5.3. Competitive Euilibrium I ene the ompetitive euilibrium an then isuss its normative properties. Denition 2 The ompetitive nanial intermeiation with poste interest rates is haraterize by the interest rates { } x p that satisfy x=f;g p = h 1 " " r m p 1 " 40 ] 1 " p [r = z " m p 1 " p where the euilibrium market tightness variables { } x p are the solution of p = h " 1 " x=f;g 41 42 r m p 1 " r 1 " m p = p p 43 Appenix E.3 shows how to obtain the euilibrium onitions for market tightness. The next proposition shows the eieny of the ompetitive euilibrium with this mehanism to set interest rates. Proposition 3 Interest rate posting makes the ompetitive nanial intermeiation eient. Proof. The ompetitive eonomy is haraterize by { } x p ; x with p x=; x p = x o See Appenix E.4. for x = ;. Finanial intermeiation is onstraint eient when non-nanial agents iret their searh eort an banks poste interest rates. If banks post interest rates on markets instea of bargaining 26
A Searh-Theoreti Approah to Eient Finanial Intermeiation over them with non-nanial agents, the potential market failure ientie above isappears. Banks jointly etermine the searh eorts an the interest rates an are not expose to the ouble loss in the ase of bargaining failure as esribe in Setion 4. 5.4. Interbank Market Fritions This Setion introues mathing fritions on the interbank market in line with Due et al. 2005, Lagos an Roheteau 2009, an Lagos et al. 2011. Banks are now speialize either in the eposit ativity or in the reit ativity. Deposit banks searh epositors on the eposit market but annot transform these eposits into nal goo onsumption. Creit banks searh borrowers on the reit market, whih are able to transform eposits into nal goo onsumption. The populations of eposit an reit bank are both eual to the unity. At eah perio, there are 0 < 1 ol mathes of reit an eposit banks that perform nanial intermeiation an 1 new reate mathes. In new mathes, a ontrat is written between the two banks whih stipulates the uantity of epositors ñ brought by the eposit bank to nane the ñ borrowers selete by the reit bank. Beause reit banks reeive interests from borrowers while eposit banks pay interests to epositor, the ontrat stipulates a payment % from the reit bank to the eposit bank. Thereafter, eah bank etermines inepenently the optimal strategy to reruit the ñ ustomers. With a probability an existing math between speialize banks is issolve, whih leas to the breakown of relationships with non-nanial ustomers. The parameter measures the size of fritions on the interbank market. 5.4.1. Finanial Contrat Speialize banks eie on the reruitment strategy ene by { % x ; v x ; } x but not on the masses of non-nanial ustomers n x, for x = f; g, that shoul be eual to ñ: Inee, the masses of non-nanial ustomers shoul be eie jointly to avoi waste in searh ativities epositors or borrowers woul be uselessly reruite. This eision is taken simultaneously 27
A Searh-Theoreti Approah to Eient Finanial Intermeiation with the interbank payment %. The Nash solution for this bargaining is { } fñ; %g = arg max P 0; 0 % P 0; 0 % 1 where P x ñ; % x is the value funtion of the bank speialize in market x 44 given the preetermine values of ñ an % x, for x = f; g : When speialize banks meet an bargain, both banks have no ustomers an therefore o not pay or reeive interests, hene ñ = % x = 0. If bargaining suees, speialize banks get the value funtion P x 0; 0 more or less the payment %. Otherwise, if bargaining breaks own, speialize banks get zero payo. The value funtion of the eposit bank is P ñ; % = [ ñ { max % ;v ; 1 ñ % v v ] [ % [ ]} P ñ; % 1 P 0; 0 1 ] % v 45 With a probability 1 the math is issolve an the eposit bank obtains the value funtion of the bank in a newly reate mathe, that is P 0; 0. Similarly, the value funtion of the reit bank is enote P ñ; % an satises P ñ; % = { [ ]} max % v P ñ; % % ;v ; 1 P 0; 0 [ ñ 1 ñ v ] [ % 1 % v 46 ] With a probability 1 the math is issolve an the reit bank obtaines the value funtion of the bank in a newly reate mathe, that is P 0; 0. The interbank payment solution is [ 1 r 1 % = [% r v 0 v ] 1 % r v 0 v ] 47 where 1 r = 1 1 r is the isount rate given the probability of separation. The resolution of the nanial ontrat 44 is etaille in Setion F. The interbank payment is the average of the isounte bank's prots weighte by the bargaining powers. For = 0, the eposit bank has no bargaining power an the payment overs the osts of nanial servies: 28
A Searh-Theoreti Approah to Eient Finanial Intermeiation the eposit interests % plus the searh osts per perio v an the isounte searh osts at the rst perio r v 0. For = 1, the reit bank has no bargaining power an the payment euals its prots: the reit interests % less the searh osts per perio v an the isounte searh osts at the rst perio r v 0. All other things being eual, the interbank market payment inreases with the eposit an reit interest rates an the searh osts on the eposit market while it ereases with the searh osts on the reit market. 5.4.2. Euilibrium Denition 3 The ompetitive nanial intermeiation with poste interest rates by speialize banks is haraterize by the interest rates f x s g x=f;g that satisfy 1 " s = h r 1 " " m s 48 1 " s [r = z " m s 1 " s where 1 r = 1 1 r : The euilibrium market tightness variables f x s g x=f;g are ] 49 the solution of s = h " 1 " 50 r m s 1 " r 1 " m s = s s 51 Appenix F shows how to obtain the euilibrium onitions for market tightness. The mass of nane entrepreneurs is n where the funtion n is ene in Setion A. The onseuenes of interbank market fritions are haraterize in the next Proposition. Proposition 4 When falls below the unity, interbank market fritions make ineient the ompetitive nanial intermeiation: reit rationing ours an interest rates are too high. Proof. It is straightforwar to see that for = 1 the ompetitive euilibrium ene by Denition 3 oinies with that of Denition 2, sine r = r, whih eieny is emonstrate 29