THE WELFARE COST OF BANK CAPITAL REQUIREMENTS

Transcription

1 TH WFAR COST OF BANK CAPITA RQUIRMNTS Skander Van den Heuvel The Wharon School Universiy of Pennsylvania Ocober, 2005 Absrac This paper measures he welfare cos of bank capial requiremens and finds ha i is surprisingly large. I presen a simple framework which embeds he role of liquidiy creaing banks in an oherwise sandard general equilibrium growh model. A capial requiremen plays a role, as i limis he moral hazard on he par of banks ha arises due o he presence of a deposi insurance scheme. However, his capial requiremen is also cosly because i reduces he abiliy of banks o creae liquidiy. A key resul is ha equilibrium asse reurns reveal he srengh of households preferences for liquidiy and his allows for he derivaion of a simple formula for he welfare cos of capial requiremens ha is a funcion of observable variables only. Using U.S. daa, he welfare cos of curren capial adequacy regulaion is found o be equivalen o a permanen loss in consumpion of beween 0. o nearly percen. Finance eparmen, Wharon School, 3620 ocus Walk, Philadelphia, PA 904, USA. mail: vdheuvel@wharon.upenn.edu. The auhor especially hanks Andy Abel and Joao Gomes for deailed commens and suggesions, as well as John Boyd, Mary ichenbaum, Gary Goron, Savros Panageas, Amir Yaron and seminar paricipans a he FIC, he Federal Reserve Board, he Federal Reserve Banks of Philadelphia and San Francisco, he NBR Summer Insiue, he Sociey for conomic ynamics and Wharon, for helpful suggesions. Sungbae An and Iamar rechsler provided excellen research assisance. All misakes are mine.

2 This paper asks, and provides an answer o, he following quesion: How large are he welfare coss of bank capial requiremens? While here are a number of papers on he heoreical benefis of capial adequacy regulaion, based on limiing he moral hazard involved wih deposi insurance 2 or exernaliies associaed wih bank failures, much less is known abou wheher here are also coss involved wih imposing resricions on he capial srucure of banks. Bu if here are only benefis o capial requiremens, why no raise hem o 00 percen and require all bank asses o be financed wih equiy? Clearly, o deermine he opimal level of capial requiremens, he quesion of heir social cos mus be addressed. In his paper, I argue ha capial adequacy regulaion may have an imporan cos because i reduces he abiliy of banks o creae liquidiy by acceping deposis. Afer all, a capial requiremen limis he fracion of bank asses ha can be financed by issuing deposi-ype liabiliies. The paper s main conribuion is o build a framework o analyze he resuling welfare cos, o derive a simple formula for is magniude and o use ha formula o measure he welfare cos. The framework embeds he role of liquidiy creaing banks in an oherwise sandard general equilibrium growh model. The welfare cos of capial requiremens depends crucially on he value of he banks liquidiy creaion. For his reason, households preferences for liquidiy are modeled in a flexible way. A key insigh from he model is ha equilibrium asse reurns reveal he srengh of hese preferences for liquidiy and his allows us o quanify he welfare cos wihou imposing resricive assumpions on preferences. Furhermore, he analysis shows ha capial requiremens can affec capial accumulaion and he aggregae level of bank asses. The formula for he welfare cos derived here akes hese general equilibrium feedbacks ino accoun. The model also incorporaes a raionale for he exisence of capial adequacy regulaion, based on a moral hazard problem creaed by deposi insurance. A capial requiremen is helpful in limiing his moral hazard problem, bu only in conjuncion wih bank supervision. This gives rise o a radeoff beween he level of he capial requiremen and he cos of supervision. The resuling welfare benefi of he capial requiremen is characerized. Wih he help of some addiional assumpions, a separae secion of he paper quanifies his benefi as well, and compares i o he welfare cos in order o examine wheher capial requiremens in he U.S. are currenly oo high or oo low. 2 See, for example, Giammarino, ewis, and Sappingon (993) and ewaripon and Tirole (994). Hellman e al. (2000) and Allen and Gale (2003) offer a more skepical view. iamond and ybvig (983) is ofen viewed as a heoreical jusificaion for deposi insurance.

3 In many counries, including he U.S., capial adequacy regulaion is based on he Basel Accords. In response o perceived shorcomings in he original Accord, praciioners have added more and more deailed refinemens, culminaing in he soon-o-be implemened Basel 2. One significan change is he increased aenion o bank supervision, formalized in he so-called Pillar 2 of he new Accord, which gives supervisors a range of new insrumens. In he language of Basel 2, he model in his paper sheds ligh on he relaion beween Pillar, he formal capial adequacy rules, and Pillar 2, and shows how his relaion gives rise o a radeoff beween he wo Pillars. A he same ime, in designing he new rules, regulaors have aemped o keep he required raio of capial o risk-weighed asses for a ypical bank approximaely he same. Bu is he 8% of he original Basel Accord a good number for he oal risk-based capial raio? This fundamenal quesion remains unaddressed. If we find ha he welfare cos of capial requiremens is rivial, his could be an argumen for creaing a simple, robus sysem of capial adequacy regulaion, wih low compliance and supervision coss, bu wih relaively high capial raios so as o make bank failure a sufficienly infrequen even. On he oher hand, if we find a high welfare cos of capial requiremens, his could be an argumen for lowering hem, by eiher acceping a higher chance of bank failure, or by designing a more risk-sensiive sysem wih he associaed increased supervision and compliance coss, which seems o be he rend in pracice. This paper is relaed o recen work by iamond and Rajan (2000) and Goron and Winon (2000), who also show capial requiremens may have an imporan social cos because hey reduce he abiliy of banks o creae liquidiy. Unforunaely, he models in hese papers do no easily lend hemselves o quanificaion of his cos, which is he main goal of his paper. xcep for he banking secor, he model presened here is closely relaed o Sidrauski (967). In using asse prices o learn abou preferences, he mehodology follows Alvarez and Jermann (2004). The res of he paper is organized as follows. Secion presens he model and analyzes agens decision problems, as well as equilibrium oucomes. Secion 2 derives a formula for he welfare cos of capial requiremens. Secion 3 uses his formula o measure he cos. Secion 4 analyzes an exension of he model ha incorporaes inermediaion fricions and he following secion presens addiional welfare cos measuremens, in par based on his exension. The welfare benefi of capial requiremens is discussed and measured in secion 6. Secion 7 addresses he effec of he capial requiremen on economic aciviy. The final secion concludes. 2

4 . The model The mos imporan respec in which he model deviaes from he sandard growh model is ha households have a need for liquidiy, and ha cerain agens, called banks, are able o creae financial asses, called deposis, which provide liquidiy services. Since a cenral goal of he model is o provide a framework no jus for illusraing, bu for acually measuring he welfare cos of capial requiremens, i is imporan o model he preferences for liquidiy in a way ha is no oo resricive. As much as possible, we would like he daa o provide he answer, no he specific modeling choices. To ha end, I follow Sidrauski (967) in adoping he modeling device of puing liquidiy services in he uiliy funcion. 3 This has wo disadvanages and one advanage. One disadvanage is ha i does no furher our undersanding of why households like liquid asses, bu his is no he opic of his paper, so his concern, in and of iself, can be dismissed. I is of course imporan o know ha he Sidrauski modeling device is consisen wih a range of more specialized, and arguably deeper, micro-foundaions. As shown by Feensra (986), a variey of models of liquidiy demand, such as hose wih a Baumol-Tobin ransacion echnology, are funcionally equivalen o problems wih money (or deposis)-in-he uiliy funcion. In ha equivalence, he laer is simply a derived uiliy funcion. Therefore, unless we impose resricions on ha derived uiliy funcion, all resuls will hold for any of hose more primiive models. A second disadvanage is ha if one needs o specify a paricular funcional form for he uiliy funcion, one is on loose grounds. For example, is he marginal uiliy of consumpion increasing or decreasing in deposis? Forunaely and his is he advanage of his approach here is no need o make unpalaable assumpions of his kind. I will show ha i is possible o derive a firs-order approximaion of he welfare cos of he capial requiremen wihou making any assumpions on he funcional form of he uiliy funcion, beyond he sandard assumpions ha i is increasing and concave. A rade-off involved wih modeling liquidiy in his flexible way, and embedding i in a general equilibrium analysis, is ha he modeling of he banks asses is no rich enough o incorporae many of he deails of risk-based capial requiremens. The environmen and he agens decision problems Time is discree and here are infiniely many periods. The economy consiss of households, banks, (nonfinancial) firms, and a governmen. Households own boh 3 ucas (2000) uses he framework of he Sidruaski model o measure he welfare cos of inflaion. 3

5 he banks and he nonfinancial firms. These firms combine capial and labor o produce he single good which households consume. I now discuss he assumpions for each of hese agens, and analyze heir decision problems in urn. For he reader s convenience, figure displays a imeline of he model. Households: There is a coninuum of idenical households wih mass one. Households are infiniely lived dynasies and value consumpion and liquidiy services. Households can obain hese liquidiy services by allocaing some of heir wealh o bank deposis, an asse creaed by banks for his purpose. As menioned, he liquidiy services of bank deposis are modeled by assuming ha he household s uiliy funcion is increasing in he amoun of deposis. Besides holding bank deposis, denoed d, households can sore heir wealh by buying and selling shares, or equiy, e. They supply a fixed quaniy of labor, normalized o one, for a wage, w. Taxes are lump-sum and equal o T. There is no aggregae uncerainy, so he represenaive household s problem is one of perfec foresigh: max β uc (, d) { c, d, e} = 0 = 0 s.. d + e + c = w+ R d + R e T + + T ( ) s= 0 s lim R ( d + e ) 0 T d + e a given T T where c is consumpion in period, R is he reurn on bank deposis, R is he reurn on (bank or firm) equiy, and β is he subjecive discoun facor. The reurns R and R, and he wage are deermined compeiively, so he household akes hese as given. The same applies for he axes. There is no disincion beween bank and nonbank equiy, since, in he absence of risk, hey are perfec subsiues for he household and will hus also command he same reurn. The second consrain is a no-ponzi game condiion, he hird an iniial condiion. The uiliy funcion is assumed o be concave, a leas once coninuously differeniable on 2 ++, increasing in boh argumens, and sricly increasing in consumpion: u (, c d) u(, c d) c> 0 and u (, c d) u(, c d) d 0 c d The firs-order condiions o he household s problem are easily simplified o β c c R = u ( c, d ) u ( c, d ) () 4

6 u ( c, d ) u ( c, d ) = R R (2) d c quaion (), which deermines he reurn on equiy, is he sandard uler equaion for he ineremporal consumpion-saving choice in a deerminisic seing, wih one difference: he marginal uiliy of consumpion may depend on he level of deposis. quaion (2) saes ha he marginal uiliy of he liquidiy services provided by deposis, expressed in unis of he consumpion good, should equal he spread beween he reurn on equiy and he reurn on bank deposis. This spread is he opporuniy cos of holding deposis raher han equiy. If ud ( c, d ) > 0, hen he reurn on equiy will be higher han he reurn on deposis o compensae for he fac ha equiy does no provide any liquidiy services. Banks: There is a coninuum of banks wih mass one, which make loans o nonfinancial firms and finance hese loans by acceping deposis from households and issuing equiy. The abiliy of banks o creae liquidiy hrough deposi conracs is heir defining feaure. Banks las for one period 4 and every period new banks are se up wih free enry ino banking. The balance shee, and he noaion, for he represenaive bank during period is: Asses iabiliies oans eposis Bank quiy Banks are subjec o regulaion, as well as supervision, by he governmen. One form of regulaion is deposi insurance. If a bank fails, he governmen (hrough a deposi insurance fund) ensures ha no deposior suffers a loss as a consequence of his failure. Tha is, all deposis are fully insured. quiy holders, as residual claimans, are lef wih nohing in he even of failure. The raionale for he deposi insurance is lef unmodeled. However, i has been argued ha deposi insurance improves he abiliy of banks o creae liquidiy. 5 Secondly, banks face a capial requiremen, which requires hem o have a minimum amoun of equiy as a fracion of (risk-weighed) asses. Since loans are he only ype of asse in his model, he capial requiremen simply saes ha equiy needs be a leas a fracion γ of loans for a bank o be able o operae: γ 4 This is wihou loss of generaliy, since here are no adjusmen coss, nor any agency problems beween banks and he oher opimizing agens, households and firms. 5 iamond and ybvig (983) provide a model of panic based bank runs, which can be seen as a raionale for deposi insurance. 5

7 For he momen, he capial requiremen is merely assumed. I will laer be shown how i can be socially desirable o have such a requiremen, as i miigaes he moral hazard problem ha arises due o he presence of deposi insurance. The bank can make safe or risky loans o nonfinancial firms, described below. Riskless 6 loans yield a rae of reurn R, which is deermined compeiively in equilibrium, so each bank akes i as given. Thus, a bank ha lends ou unis of he good o safe firms a he beginning of he period will receive nonrandom oal reurn of R unis a he end of he period. For now, i is assumed ha here are no ransacion coss involved in making loans and acceping deposis. Secion 4 will analyze an exension of he model wih cosly financial inermediaion. The presence of deposi insurance creaes a moral hazard problem: he bank has an incenive o engage in excessive risk-aking. Since his is he jusificaion for he capial requiremen, I inroduce a way for he bank o engage in excessive riskaking by assuming ha he bank has he opion of arificially raising he riskiness of is asses. Specifically, by direcing a fracion of is lending o a differen se of nonfinancial firms wih a risky echnology, described below 7, he bank can creae a loan porfolio wih riskiness σ ha pays off R + σ, where is a bank-specific shock wih posiive variance and negaive mean, equal o ξ ( ξ 0 ). Thus, he expeced reurn of he loan porfolio is decreasing in is risk. I is in his sense ha risk-aking is excessive: absen a moral hazard problem due o deposi insurance, he bank would always prefer σ = 0. While he bank can choose he riskiness of is loans, i is assumed ha bank supervision imposes an upper bound on he choice: σ [0, σ ]. This will be explained more fully in he discussion of he governmen. In he main ex I will work wih he following example disribuion for : wih probabiliy 0.5 = ( + 2 ξ ) wih probabiliy 0.5 (3) This very special example disribuion is used purely for exposiional reasons. As shown in he appendix C. o secion 6, all he resuls in his paper hold for an arbirary disribuion of wih bounded suppor and nonposiive mean. In addiion, i is imporan o keep in mind ha he assumpions regarding he deposi insurance, he excessive risk aking and is supervision maer only for he benefis of he capial requiremen, no for is welfare cos, nor for he measuremen of his cos. I am now in a posiion o sae he bank s problem. The objecive of he bank is o maximize shareholder value by deciding how many loans o make, how much risk o ake on, and how o finance is asses wih equiy and deposis. Alhough he 6 The asserion ha he bank can make riskless loans is a consequence of he echnology of he nonfinancial firms, as deailed below. 7 The echnology will be consisen wih he raes of reurn assumpions made here. 6

8 decision on how much equiy o issue will be endogenized, i is convenien o firs analyze he sub-problem of maximizing shareholder value righ afer he equiy has been issued and he bank has raised in equiy a he beginning of period. A ha poin he value of he bank s equiy is: 8 ( σ ) + V ( ) = max ( R ) R / R,, σ + s.. = + B γ σ [0, σ] (4) The noaion ( x ) + sands for max( x,0) and is he expecaions operaor. The firs consrain is he balance shee ideniy, he second is he capial requiremen, and he hird bounds σ. The erm ( R + σ ) R is he bank s ne cash flow a he end of he period. I consiss of ineres income from loans, minus any possible charge-offs on he loans, and minus he ineres owed o deposiors. If he ne cash-flow is posiive, shareholders are paid his full amoun in dividends. 9 If he ne cash flow is negaive, he bank fails and he deposi insurance fund mus cover he difference in order o indemnify deposiors, as limied liabiliy of shareholders rules ou negaive dividends. Shareholders receive zero in his even, so dividends equal (( + R + σ ) R ). A he beginning of period shareholders discoun he value of dividends, which are paid a he end of ha period, by heir opporuniy cos of holding his paricular bank s equiy. This opporuniy cos is R, he marke rae of reurn on equiy. Because dividends are eiher no subjec o risk, or, if σ > 0, heir risk is perfecly diversifiable, shareholders do no price he bank s risk. 0 Firs, I characerize he choice of σ condiional on and. Noe ha (( R + σ ) R ) + ( R σξ ) R if ( R σ (+ 2 ξ )) R 0 = 0.5(( ) R + σ R ) oherwise xpeced dividends are hus sricly decreasing in σ for low values of σ and sricly increasing in σ for sufficienly high values of σ. The reason is ha for high values 8 In wha follows, ime subscrips will be used only where necessary o avoid confusion. 9 There are no reained earnings since he bank lass for one period only. 0 Hence, he reamen of R as nonsochasic in he household problem is sill correc, since, even if banks are risky, households would no leave any such risk undiversified. Noe ha here is no disconinuiy a ( R σ(+ 2 ξ)) R = 0. 7

9 of σ, if he bank suffers a negaive shock, here is no enough equiy o absorb he loss and he excess loss is covered by he deposi insurance fund. Increasing risk furher a his poin increases he payoff o shareholders in he good sae ( = ) wihou lowering i in he bad sae. In oher words, he value of he pu opion associaed wih he deposi insurance fund increases wih σ. In conras, when σ is low, he value of his pu opion is zero and shareholders fully inernalize he reducion in ne presen value ha occurs when risk is increased. Because expeced dividends are a convex funcion of σ, here are only wo values o consider for he opimal choice of riskiness: σ = 0 or σ = σ. I is easy o show ha 0 iff σ = σ R R ( / ) 2 σ = σ oherwise (5) Because = γ, he following is a sufficien condiion for σ = 0 : σ R R ( γ ) (6) This is also a necessary condiion when he capial requiremen is binding. From now on, unless explicily saed oherwise, i is assumed ha (6) holds. The bank s sub-problem in (4) for shareholder value now simplifies o: ( ) V ( ) = max R R / R B, s.. = γ 0 The firs-order condiions are easily simplified o R R = γ R χ where χ is he Kuhn-Tucker muliplier associaed wih he capial requiremen: χ 0 and χ( γ ) = 0. The exisence of a (finie) soluion requires R R. Under ha condiion, he soluion is ( γ ) V B ( ) = R + ( )( R R ) R (7) The capial requiremen binds if and only if R > R. The inerpreaion is sraighforward: an exra uni of equiy can be len ou a he rae R. In addiion, he exra uni of capial allows he bank o make ( γ ) addiional loans and finance 2 When σ = R R ( / ), he bank is indifferen beween he wo choices. For convenience, i is assumed ha he bank chooses σ = 0 in ha case. 8

10 hose wih deposis, wihou violaing he capial requiremen, which requires γ. If R > R, he second opion has value, and he capial requiremen will be binding, oherwise no. I can now urn o he bank s decision on how much equiy o raise. The preissue value of he bank is V ( ). The bank maximizes his value when choosing B. Using (7), he firs-order condiion o ha problem is: 3 R R R R = + ( γ )( ) I is helpful o rewrie his as R = γr + ( γ) R (8) This has he inerpreaion of a zero-profi condiion: for a bank wih a binding capial requiremen, one uni of lending is financed by γ in equiy and ( γ ) in deposis. Thus, compeiion will equalize he rae of reurn o lending o he similarly weighed average of he required raes of reurn of equiy and deposis, whence (8). (The case of a nonbinding requiremen will be clear in a momen.) We have already esablished ha R R is necessary for a soluion o exis and ha he capial requiremen binds if and only if R < R. Hence, wo cases are possible:. If 2. If R = R = R, he capial requiremen is slack. R < R < R, he capial requiremen is binding, so = γ. B In eiher case, V ( ) = 0. Noe ha he sufficien condiion for σ = 0 o be opimal, given in (6), is seen o be equivalen o σ γ R (9) Again, his condiion is also necessary if he capial requiremen is binding. Firms: Nonfinancial firms canno creae liquidiy hrough deposis. They can, however, buy goods o use hem as capial, which can be combined wih labor inpu, o produce oupu of he good. Capial is purchased a he beginning of he period. To 3 As is common in problems wih consan reurns o scale, he firs-order condiion, raher han fully deermining he agen s choice, has he inerpreaion of a necessary condiion for he exisence of a finie soluion. If R < ( > ) R + ( γ )( R R ), hen ends o plus (minus) infiniy. If he firsorder condiion holds, is indeerminae, and hus so is he scale of he bank. 9

11 finance heir capial sock, firms can issue equiy o households, borrow from banks, or some combinaion of boh. The firm s balance shee, and noaion, for period is: Asses iabiliies K Physical Capial oans F Firm quiy Firms can employ a riskless or a risky producion echnology. 4 The riskless echnology is sandard. Oupu in period is F( K, H ), where H is hours of labor inpu and F( ) is a well-behaved producion funcion exhibiing consan reurns o scale. A fracion δ of he capial sock depreciaes during he period. There are no adjusmen coss and firms las for one period. 5 ach period, here is a coninuum of firms wih mass normalized o one, so each firm akes prices as given. As in he analysis of he bank s problem, i is convenien o sar wih he F firm s decision problem righ afer i has raised in equiy. A ha poin he value of he firm o is shareholders is 6 ( δ ) V ( ) = max F( K, H) + ( ) K wh R ( K ) / R F F F KH, Here I have subsiued ou loans using he balance shee ideniy. The firs-order condiions for he choices of labor and capial are sandard: (H) FH ( K, H) = w (0) (K) F ( K, H) + ( δ ) = R () K These opimaliy condiions, ogeher wih he consan reurns o scale assumpion, imply ha he soluion for he firm s shareholder value is: V ( ) = R R F F F F F F The pre-issue value of he firm is V ( ). I is assumed ha equiy canno be F F negaive: 0. Subjec o ha consrain, he firm chooses o maximize is pre-issue value. The firs-order condiion o his problem is: 4 I would be sraighforward o le all firms have risky producion, and herefore make all individual loans risky, even while keeping σ = 0 as feasible for banks, as long as he producion shocks are sufficienly imperfecly correlaed across firms, so ha he risk is perfecly diversifiable by lending o many firms. xcessive risk aking would hen correspond o no diversifying his risk. (ξ would equal zero in his case.) 5 The absence of adjusmen coss and agency problems implies ha his is wihou loss of generaliy. One can hink of ongoing firms as repurchasing heir capial sock each period. 6 Noe ha he absence of arbirage opporuniies implies ha nonfinancial firms have o offer shareholders he same reurn on equiy as banks, since here is no aggregae risk. 0

12 F F ( ) R / R = μ, μ 0, μ = 0 (2) where μ is he Kuhn-Tucker muliplier associaed wih he consrain ha firm equiy canno be less han zero. A finie soluion hus requires R R. F If R > R, hen = 0, so K =. In oher words, if bank loans are cheaper han equiy finance, he firm chooses o use only bank loans o finance is capial. If R = R, he firm s financial srucure is no deermined by individual opimaliy. In F F F eiher case economic profis, V ( ), equal zero. As he discussion of he economy s equilibrium will make clear, which case applies will depend on wheher households demand for liquidiy is saiaed or no. As an alernaive o his riskless echnology, firms can also choose o employ a risky echnology, in which case oupu is F( K, H) + σrf K, where is he same negaive mean, idiosyncraic shock as defined in (3) and σ RF is a parameer ( σ RF σ ). Risky firms provide a vehicle for banks o make he kind of risky loans described in he subsecion on banks. Alhough hese firms hus provide a raionale for he exisence of capial regulaion, as menioned, I will usually focus on he case ha he capial requiremen is sufficienly high, according o condiion (9), o preven banks from engaging in excessive risk aking. No risky firms will hen exis in equilibrium. For his reason analysis of hese firms is lef for appendix A, which shows how he opimal loan conrac wih a risky firm allows a bank o creae a loan porfolio wih riskiness σ by direcing a fracion σ / σ RF of lending o ha firm. Governmen: The governmen manages he deposi insurance fund, ses a capial requiremen γ [0,) and conducs bank supervision. The purpose of bank supervision is no only o enforce he capial requiremen, bu also o monior excessive risk aking by banks, σ. Supervisors can o some degree deec such behavior and sop any bank ha is caugh aemping o ake on excessive risk in order o proec he deposi insurance fund. I seems reasonable o assume ha a small amoun of risk aking is harder o deec han a large amoun. The larges level of risk-aking ha is sill jus undeecable is σ. σ is assumed o be a decreasing funcion of he resources spen on bank supervision: σ = ST ( ) wih S ( i ) 0 and 0 < S σ RF where T, a choice variable for he governmen, is he par of ax revenue spen on bank supervision. 7 The inerpreaion is ha, as more resources are devoed o bank 7 As in he sandard growh model wih governmen spending and lump sum axes, if T is se oo high, no equilibrium wih posiive consumpion exiss. I assume ha T is sufficienly low so ha a seady sae equilibrium wih posiive consumpion exiss. Appendix will make precise wha sufficienly low means for a paricular funcional form of he uiliy funcion, inroduced in secion 7.

13 supervision, banks are less able o engage in excessive risk aking wihou being deeced. The assumpion of imperfec observabiliy of excessive risk aking is imporan. If regulaors could perfecly observe each bank s riskiness, hey could simply adjus each bank s deposi insurance premium so as o make he bank pay for he ex ane expeced loss o he deposi insurance fund, hus eliminaing any moral hazard. Or hey could se each bank s capial requiremen as an increasing funcion of is riskiness, in such a way as o ensure ha he bank always inernalizes all he risks. Bu such perfec observabiliy is simply no realisic, so a moral hazard problem does exis. No allowing σ o exceed σ can be inerpreed as a risk-based capial requiremen or a risk-based deposi insurance premium, bu one based on observable risk. Under ha inerpreaion, regulaors deer deecable excessive risk aking (as hey should) by imposing a sufficienly high capial requiremen or a sufficienly high deposi insurance premium when such excessive risk aking is deeced. As long as i is sufficienly high, he precise value of he capial requiremen or premium when σ > σ is no imporan, as i will never be implemened in equilibrium. 8 The governmen has a balanced budge. ump-sum axes are se a wih ( { > 0} ψ ) T = T X + X (3) ( ( σ ( 2 ξ)) ) X = R R + (4) + If X > 0, X is he loss o deposi insurance fund due a bank failure, and 0.5 is he mass of banks ha fails if X > 0. In addiion, here may be a deadweigh cos of resolving bank failures, equal o ψ 0 per uni of deposis in failed banks. If (9) holds, we know ha σ = 0 and in ha case axes are simply: T = T. General quilibrium Given a governmen policy γ and T, an equilibrium is defined as a pah of consumpion, capial, employmen, and financial quaniies and reurns, for = 0,,2,, such ha:. Households, banks and firms all solve heir maximizaion problems, described above, wih σ = ST ( ) and axes se according o (3)-(4); 2. All markes clear, i.e. 8 I have assumed ha he bank pays a deposi insurance premium equal o zero when σ σ <. In he model, his is he acuarially fair deposi insurance premium when regulaion is successful in deerring excessive risk aking (i.e. when (9) holds he case I focus on). I also happens o be he deposi insurance premium ha virually all U.S. banks currenly pay. 2

14 and d F F = e = + = K H = F( K,) ξσ + ( δ) K = c + K + T + ( ψ /2) + { X > 0} I focus on he case ha (9) holds: ST ( ) γ R. Governmen policy can accomplish his by seing γ and/or T sufficienly high. In ha case, σ = 0, X = 0 and T = T. I will firs describe he resuling allocaion and hen provide explanaion. By combining he marke clearing condiions and equaions (), (2), (8), (0), (), and (2), i is possible o characerize he equilibrium in erms of a sysem in ( K, c ) wih R and d as auxiliary variables: K = F( K,) + ( δ ) K c T (5) β ( uc( c, d )/ uc( c, d)) = R (6) ud( c, d) FK( K,) + δ = R = R ( γ) (7) u ( c, d ) c where d is deermined according o one of he following wo cases:. If u ( c,( γ ) K ) = 0, he capial requiremen is no binding and d ud( c, d ) = 0, wih d ( γ ) K (8) 2. If u ( c,( γ ) K ) > 0, he capial requiremen is binding and d d = ( γ ) K (9) F Remark: In case 2, R < R, so = 0 (by (2)), = K, and e = = γk. In case F, which requires ha demand for liquidiy be saiaed a d = ( γ ) K, d,,, e and are no uniquely deermined. Noe ha if ud( c, d ) = 0, uc( c, d ) does no depend on d. In boh cases, remaining variables are deermined hrough (2) and (0) wih H =. Two resuls are key o undersanding how he equilibrium differs from he sandard growh model: Firs, uiliy maximizaion by households implies ha he pecuniary reurn on deposis is lower han he reurn on equiy by a spread equal o he marginal value of deposis liquidiy services expressed in unis of consumpion, u (, c d) u (, c d )(see equaion (2)). Second, he zero-profi condiion for banking d c 3

15 implies ha he rae on bank loans is he weighed average of he required reurns on equiy and deposis (equaion (8)). In he firs case, he level of deposis is so high ha he marginal value of liquidiy provision is zero, so he equiy-deposi spread is also zero. As a resul, he capial requiremen is no binding, and deposis, equiy and bank loans all command he same reurn. quiy and loans, and equiy and deposis, are perfec subsiues for firms and banks, respecively, so hese financial quaniies are no uniquely deermined in equilibrium. On he real side, here is no maerial difference in his case wih a sandard growh model (wih governmen spending and lump-sum axes). The reason is ha banks special abiliy o creae liquidiy has no marginal value. In he second, more ineresing case, he demand for liquidiy is no saiaed and he marginal value of liquidiy services is posiive. As a resul, he spread beween he reurn on equiy and deposis also exceeds zero and he capial requiremen is now binding, since banks wan o fund heir asses as much as possible wih he cheaper deposis. Because of perfec compeiion, banks fully pass on he lower cos of funding heir loans o heir borrowers. Thus, he loan rae declines, hough only by ( γ )( ud / uc), as banks sill have o finance a fracion γ of heir lending wih equiy. Nonfinancial firms now choose o finance all heir capial sock wih hese cheaper bank loans, raher han equiy. Because banks pass on he low pecuniary reurn on deposis o firms, in seady sae he capial sock is higher han wihou any preference for liquidiy. 9 An imporan relaed resul is ha he seady sae level of he capial sock is generally no invarian o he level of he capial requiremen. Secion 7 will explore his furher. As will be shown here, an increase in γ can increase or decrease he seady sae capial sock, depending on he ineres elasiciy of he demand for liquidiy. 2. The welfare cos of he capial requiremen The sraegy for quanifying he welfare cos of he capial requiremen is as follows. Firs, I presen a consrained social planner s problem. The qualificaion consrained means ha he social planner s problem shall respec he capial requiremen and devoe he same level of resources o bank supervision. This will ensure ha he allocaion ha solves he planner s problem is incenive compaible for he banks. Raher han solve for he firs-bes allocaion, his planner s problem is designed o replicae he decenralized equilibrium described above. Nex, I will 9 See equaion (7) and noe ha R = β in seady sae (see (6)) and F KK < 0. No preference for liquidiy refers o he special case ha, ucd (, ) = uc () for all c and d, and for some funcion u (so u d = 0). All oher funcions and parameers are kep he same in he comparison. 4

16 show ha he allocaion associaed wih he planner s problem is indeed idenical o he decenralized equilibrium s allocaion. Finally, I will exploi his equivalence o derive analyically a simple formula for he welfare cos of increasing he capial requiremen. The social planner s problem efine he following consrained social planner s problem: V ( θ) = max β u( c, d ) 0 { c, d, K+ } = 0 = 0 s.. F( K,) + ( δ ) K = c + K + T + ( γ ) K d 0 (20) where θ = ( γ, T, δ, β, K0 ). The firs consrain is he social resource consrain for σ = 0; 20 F he second consrain rewries he capial requiremen, imposing = 0. The firs-order condiions o his problem are: sp (c) uc( c, d) = λ sp (d) ud( c, d) = χ (K) sp λ [ F ( K,) + δ] β λ sp + χ sp ( γ) = 0 K sp sp where λ and χ are he agrange mulipliers on he social resource consrain and he capial requiremen, respecively. Combining hese condiions yields: β uc( c, d ) ud( c, d) FK( K,) + δ = ( γ) uc( c, d) uc( c, d) (2) sp In addiion, he complemenary slackness condiions d ( γ ) K, χ 0 and sp χ (( γ) K d) = 0, combined wih he firs-order condiion wih respec o deposis, and he concaviy of u (so udd ( c, d) 0 ), imply: if u ( c,( γ ) K ) = 0 hen d ( γ ) K, wih u ( c, d ) = 0 ; (22) d d if u ( c,( γ ) K ) > 0 hen d = ( γ ) K (23) d Combining equaions (2), (22) and (23) wih he social resource consrain (he firs consrain in problem (20)), i is apparen ha he allocaions of K, c and d are idenical o hose of he decenralized equilibrium summarized above in equaions (5) hrough (9). quaion (22) corresponds o an equilibrium wih a 20 The absence of excessive risk aking is simply par of he definiion of he planner s problem. 5

17 nonbinding capial requiremen ( case, equaion (8)), while (23) corresponds o he case of a binding capial requiremen ( case 2, (9)). Hence, he consrained social planner s problem replicaes he decenralized equilibrium when σ = 0 in he laer. As a resul, if σ = 0, welfare in he decenralized equilibrium is equal o V ( ) 0 θ, he value of he objecive funcion o he consrained social planner s problem. A formula for he marginal welfare cos The equivalence of he social planner s problem and he decenralized equilibrium can be used o measure he marginal effec on welfare of a change in he capial requiremen in he following way. Call he curren period period 0. Again, assume ha governmen policy is such ha (9) holds: ( ) γ for all 0, so ST R ha policy deers excessive risk aking in he decenralized equilibrium: σ = 0. Saring from his siuaion, I compue he marginal effec on welfare of raising γ, wihou alering T, using he envelope heorem, as follows: V ( θ ) 0 sp = βχ K = βud( c, d) K γ = 0 = 0 The las equaliy follows from he firs-order condiion (d) o he planner s problem. Since he allocaions of c, d and K are idenical o hose in he decenralized equilibrium, I can use he decenralized equilibrium values o evaluae he righ hand side of his equaion. Moreover, in he decenralized equilibrium, we have, using (2), u ( c, d ) K = u ( c, d )( R R ) K d c I compare his o he welfare effec of a permanen change in consumpion by a facor ( + ν ). Saring from he iniial equilibrium, he effec on welfare of changing consumpion from c o ( + ν ) c, for all, equals, o a firs-order approximaion, ( Σ (, ) ) = 0 β uc c d c ν. Nex, assume ha he economy is in seady sae in period 0. Then he firsorder approximaion of he welfare effec of an increase in γ by Δ γ simplifies hus: V0( θ ) ud( c0, d0) K0 uc( c0, d0)( R0 R0 ) d0 Δ γ = Δ γ = Δγ γ β ( β)( γ) Here I have also used he fac ha d = ( γ ) K if he capial requiremen binds, while ( R R ) = 0 oherwise. Similarly, wih period 0 a seady sae, uc( c0, d0) c0 ( u (, ) 0 c c d c = ) = β ν ν (24) β 6

18 quaing he righ-hand sides of hese las wo equaions yields he following resul. 2 Proposiion Assume ha he economy is in seady sae in he curren period and ha (9) holds. Consider permanenly increasing γ by Δ γ. A firs-order approximaion o he resuling welfare loss, expressed as he welfare-equivalen permanen relaive loss in consumpion, is ν Δ γ, where d ν = ( R R )( γ ) (25) c The above formula is empirically implemenable. Remarkably, i does no rely on any assumpions abou he funcional form of preferences, beyond he sandard assumpions of monooniciy, differeniabiliy and concaviy. Insead, he formula relies on asse reurns o reveal he srengh of he household s preference for liquidiy. An unnecessary increase in he capial requiremen reduces welfare by reducing he abiliy of banks o issue deposi-ype liabiliies. The firs facor in he formula for he welfare loss concerns he imporance of deposis in he economy. The second is he spread beween he reurn on bank equiy and he pecuniary reurn o deposis. This spread equals he amoun of consumpion households are willing o forgo in order o enjoy he liquidiy services of one addiional uni of deposis. Finally, ( γ ) Δ γ is he relaive change in deposis as a resul of changing he capial requiremen by Δ γ for a given level of bank asses. The formula is valid even if he capial requiremen happens no o bind. In ha case, R R is zero, so he welfare cos is also zero. Noe ha, while he proposiion assumes ha he economy is iniially in seady sae, he welfare loss akes ino accoun, o a firs-order approximaion, all he gains or losses associaed wih he ransiion o a new seady sae upon changing he capial requiremen. (Recall ha he seady sae capial sock depends on γ.) Tha is, simply comparing he welfare levels of differen seady saes associaed wih differen values of γ yields a differen (and wrong) answer because i does no ake ino accoun he welfare effecs of he ransiion beween he seady saes. Ineresingly, he formula can also be derived by incorrecly (!) assuming ha he equilibrium levels of he capial sock and bank asses are invarian o changes in γ. 22 The fac ha his is rue is a manifesaion of he envelope heorem: hese quaniies are consrained opimal in he sense of he social planner s problem, so 2 Solve for ν and muliply he resul by minus o ge he welfare-equivalen loss in consumpion. The formula in he proposiion omis ime (0) subscrips for readabiliy. 22 This is no inconsisen wih he previous saemens: he welfare effec of he ransiion o a new seady sae cancels wih he difference in welfare beween he wo seady saes ha is due o he differen level of he seady sae capial sock raher han he direc effec of Δ γ on deposis. 7

19 heir response o a change in γ has only a second-order effec on welfare. Of course, one would no have known his before going hrough he enire exercise. I may sill seem surprising ha no assumpions were needed on he funcional form he uiliy funcion. Afer all, o use he Sidrauski model o measure he welfare cos of inflaion, as ucas (2000) does, one needs o know he ineres elasiciy of money demand, which amouns o requiring more knowledge of he uiliy funcion. The difference is ha, while money in he Sidrauski model is creaed by a non-opimizing monopolis (he governmen), in his model he supply of liquidiy is creaed by compeiive banks. This addiional srucure in he model means we have some exra informaion on he welfare effec of he change in he quaniy of deposis. The mehodology used here may be of independen ineres. Recapiulaing, he seps are: () guess a consrained social planner s problem, inended o mimic he decenralized economy (raher han solve for he firs-bes); (2) verify ha i replicaes he decenralized equilibrium; and (3) differeniae he value of he planner s problem wih respec o he policy parameer, and exploi he equivalence wih he decenralized allocaion, o obain an analyic expression for he welfare effec. This mehod will generally only yield a firs-order approximaion and i may no always be possible o guess a workable planner s problem. Noneheless, when i works, his mehodology has a number of advanages compared wih a brue-force numerical approach. Firs, i is simpler. Second, he analyic expression may yield insigh ino he resul. Third, wih a brue-force numerical approach all funcional forms (e.g. he uiliy funcion) and parameers need o be specified. As can be seen from he formula in he proposiion, he informaional requiremens here are much weaker. The opimal capial requiremen The raionale for capial adequacy regulaion in he model is is role, join wih bank supervision, in prevening excessive risk aking. If bank supervision is imperfec ( ST ( ) > 0 for all T) and prevening excessive risk aking is socially opimal, hen he opimal capial requiremen will be sricly posiive. In he model, prevening excessive risk aking is socially opimal if eiher is direc cos, ξ, or is indirec cos due o cosly resoluion of bank failures, ψ, is sufficienly large. In conras, if boh hese coss are small, he social opimum is o have a zero capial requiremen and accep he resul ha half he banks will fail. The formula for he welfare cos of he capial requiremen is sill valid in his case, bu i expresses a cos ha is o be avoided, raher han o be compared o he benefi of he capial requiremen. Having said ha, he case for prevening excessive risk aking may be sronger han he model suggess, as he model (o economize on noaion) lacks one 8

20 poenial cos of bank failures: i implicily assumes ha he liquidiy services of deposis in failed banks are idenical o hose of solven banks, which seems hard o believe. For he remainder of his secion, I will focus on he case ha prevening excessive risk aking is socially opimal (due o high ψ and/or ξ), so ha here exiss a raionale for capial adequacy regulaion. Under ha hypohesis, in seady sae, he capial requiremen ha maximizes welfare is defined by: max V ( θ ) T, γ 0 γβ s.. ST ( ) The consrain is he incenive compaibiliy condiion (9), wih sae. The firs-order condiions o his problem imply V ( θ) V ( θ) dt + = γ T d γ ST ( ) = γβ valuaing his in seady sae yields R = β in seady dt cν = = (26) dγ βs'( T) ST ( ) = γβ Tha is, he marginal welfare cos of he capial requiremen (in unis of he good per period) should equal is marginal benefi in reducing bank supervision and compliance coss, given he incenive compaibiliy consrain. Making he reasonable assumpion ha here are diminishing reurns o bank supervision, so ha S > 0, a larger welfare cos demands higher supervision expendiures (a larger T) and hus a lower capial requiremen. Measuring he marginal benefi of he capial requiremen will require some addiional informaion on he supervision echnology S. This is no rue for he marginal welfare cos, and I now urn o quanifying his cos. 3. Measuremen of he welfare cos The main resul so far is an expression for he welfare cos of a bank capial requiremen. The expression lends iself o a calculaion of his cos based on daa. For his purpose, I employ annual aggregae balance shee and income saemen daa for all FIC-Insured Commercial Banks in he Unied Saes. These daa are 9

21 obained from he FIC s Hisorical Saisics on Banking (HSOB) and are based on regulaory filings. In mapping he heory o he daa, some choices need o be made. For deposis, ( = d), he HSOB s Toal eposis is used. The ne reurn on deposis ( R ) is calculaed as Ineres on Toal eposis divided by Toal eposis. 23 For consumpion, c, I use personal consumpion expendiures from he NIPA. As a measure of he capial requiremen γ he empirical counerpar of / is used. 24 This is compued as Toal quiy Capial plus Subordinaed Noes divided by Toal Asses. Subordinaed Noes are included because subordinaed deb couns, wihin cerain limis, owards regulaory ier 2 capial. Toal quiy Capial plus Subordinaed Noes does no exacly correspond o oal capial in he sense of he Basel Accord, on which curren capial adequacy regulaion in he US (and many oher counries) is based. However, daa on oal capial in he sense of he Basel Accord is only available saring in 996 and i seems more imporan o be able o use a longer ime span, especially since he formula for he marginal welfare cos in (25) is no very sensiive o he measuremen of γ. An alernaive would have been o use he acual regulaory numbers for he capial requiremen (eiher 0.08 for oal capial based on he Basel Accord or, more realisically, 0.0 based on he FICIA, he CAMS raings and he Gramm- each-bliley Ac). However, boh he daa and heory 25 sugges ha he vas majoriy of banks hold a buffer of equiy above he regulaory minimum so as o lower he risk of an adverse shock leading o capial inadequacy. Since he model absracs from his buffer sock behavior by assuming away any shocks, one would wan o include his buffer in he measuremen of γ as i is due o he capial adequacy regulaion in he firs place. 26 There is lile reason o expec he buffer iself would change dramaically in response o a change in he regulaory minimum capial raio. In any case, as menioned, he poin is no quaniaively very imporan. For example, as we change he measure of γ from an unreasonably low value, say, 0.04 o an unreasonably high value, say 0.5, holding consan he oher measuremens, he esimaed marginal welfare cos increases only by a facor.3 ( ( 0.5) = /( 0.04) ). Finally, a measure of he required reurn on (bank) equiy is needed. Since he model absracs from aggregae risk, a risk-adjused measure is needed. To avoid he 23 All daa are nominal. While he model is real, using nominal daa consisenly is fine, because he formula for he welfare cos in (25) conains only raios of quaniies and spreads of reurns. 24 This may seem incorrec if he capial requiremen is no binding. However, if ha is he case, he model implies ha R = R, so he welfare cos is zero regardless of how γ is measured. 25 See Van den Heuvel (2004) for a quaniaive model. 26 In addiion hese raios apply o risk weighed asses and off-balance shee iems, consideraions from which he model also absracs. 20

22 difficulies inheren in measuring he (ex ane) risk premium on regular equiy, 27 he measure I use is he average reurn on subordinaed bank deb. The reason for his choice is ha (a) subordinaed deb couns owards regulaory equiy capial, albei wihin cerain limis, and (b) defauls on his ype of deb have hisorically been very rare, so he deb is no very risky. As a measure for ( R ), he ne reurn on subordinaed deb is calculaed as Ineres on Subordinaed Noes and ebenures divided by Subordinaed Noes and ebenures. 28 The limis on he use of subordinaed deb for regulaory purposes imply ha his is a conservaive measure for he risk-adjused required reurn on bank equiy. Firs, because i is regarded as an inferior form of equiy, subordinaed deb can coun only owards ier 2 capial. Second, he amoun of subordinaed deb is limied o 50 percen of he bank s ier capial. Wha his means is ha if he ier capial raio is close o binding, subordinaed deb can coun for a mos approximaely 25 percen of oal capial. Since banks may use subordinaed deb o mee heir capial requiremens only up o hese limis (and hey do no have o use i), i is possible ha for many banks he required reurn on subordinaed deb is lower han he riskadjused reurn on regular equiy. To measure he welfare cos using he derived formula I compue long run averages for he deposi-consumpion raio, he spread beween subordinaed deb and deposis, and he capial asse raio. The sample period is se a , because The Basel Accord and he FIC Improvemen Ac enacing i were no fully implemened unil January, 993, and prior o Basel he use of subordinaed deb for regulaory purposes was raher limied. 29 For he mean deposi o consumpion raio is 0.6, he average ne reurns on deposis and subordinaed debs are, respecively, 3.08% and 6.26%, so he average spread is 3.8%, and he mean capial asse raio is 0.0. Hence, applying (25), a firs-order approximaion o he welfare cos of raising he capial requiremen by Δ γ is: νδ γ = ( d/ c) ( R R ) ( γ) Δγ ( 0.) γ = Δ = Δγ 27 For example, he hisorical average excess reurn on bank equiy would imply a high premium, bu does his equal he ex ane expeced premium? In addiion, depending on wha ineres rae is used o measure he excess reurn on equiy, his approach runs he risk of conaminaing he measured risk premium wih a liquidiy premium, which one would definiely wan o avoid in he presen conex. If on he oher hand one akes a model based measure of he ex ane risk premium based on reasonable sandard preferences, one would likely ge a much lower measure of his premium. (This is he well known equiy premium puzzle.) 28 Par of he HSOB s Subordinaed eb and ebenures does no qualify as regulaory capial. However, cross-checking wih he Repors on Condiion and Income ( call repors ) iem RCF560 indicaes ha he difference is minimal afer Secion 5 documens ha using a longer sample ( ) yields very similar resuls. 2

23 To inerpre his number, consider he welfare cos of he curren level of he capial requiremen, γ = 0., compared o a zero capial requiremen ( γ = 0). 30 This welfare cos is equivalen o a permanen loss in consumpion of ν 0. = % = 0.26%. This is no, in my view, a rivial welfare cos. Some well-known esimaes on he welfare coss of business cycles or he welfare gains of implemening he opimal moneary policy rule (aking as given average inflaion) are much smaller. Here is anoher way o inerpre his number. Consider lowering he effecive capial requiremen by percenage poin (o 0.09). And suppose regulaors can keep he probabiliy of bank failure as low as i is oday despie his change by spending more on bank supervision and imposing higher compliance coss. If he oal cos of keeping he probabiliy of bank failure he same is less han ν 0.0 c2002 = =.6 billion $ per year, hen his ough o be done: lowering he capial requiremen would be welfare improving in his way. 3 If no, he capial requiremen ough o be increased. I should be poined ou ha his esimae is conservaive in he sense ha, as menioned, he rue spread beween he required reurn on equiy and deposis may be higher han he one measured here due o he limis on he use of subordinaed deb for regulaory purposes. Secion 5 will provide some alernaive measures of he spread and associaed esimaes of he welfare cos. A differen objecion one migh have o he above calculaion of he welfare cos is ha i does no ake ino accoun any resource coss ha banks incur in servicing deposis or making loans. The former include he coss of ATM neworks, par of he cos of mainaining a nework of branches, ec. The laer include he coss of screening loan applicaions, collecing paymens, as well as par of he cos of mainaining a branch nework. These coss are no rivial. For he period ne nonineres coss of U.S. banks have averaged.29% of oal asses. The nex secion will address his concern by incorporaing ino he model resource coss associaed wih acceping deposis and/or making loans. Secion 5 will use he resuls of he model o show how his affecs he measured welfare cos of he capial requiremen. 30 This is, of course, a gross cos which ulimaely needs o be compared o he benefi of a 0% capial requiremen in reducing he risk of bank failures or in economizing on supervision cos. The number can also be inerpreed as (a firs-order approximaion o) he cos of a new regulaion ha increases γ by 0 percenage poins (doubling he effecive capial requiremen, from 0. o 0.2) wihou any change in bank supervision. 3 Of course, in realiy axaion is no lump-sum bu usually disorionary, so one would wan o make some allowance for ha. 22