Stock-out avoidance as a bank motive for holding liquidity reserves *

Transcription

1 Sock-ou avoidance as a bank moive for holding liquidiy reserves * Jochen H. F. Günner Firs draf: May 8, 2 Absrac This paper reinerpres he precauionary sock-ou avoidance moive, usually associaed wih invenory holding of inermediae or final goods producers, in he conex of financial inermediaries opimal liquidiy reserves. Each bank faces idiosyncraic uncerainy abou is volume of deposi inflows when choosing he opimal supply of credi o firms, and is hus subjec o an occasionally binding liquidiy consrain. Assuming a funcional form for he probabiliy disribuion of sochasic deposi realizaions, we are able o analyze he implicaions of bank-specific uncerainy for he aggregae saic and dynamic equilibrium, respecively. The underlying DSGE model endogenizes boh he demand for and he coss of financial inermediaion. Individual uncerainy impairs he banking secor s efficiency and ranslaes ino a lower seady-sae level of economic aciviy. The higher he variance of deposi inflows, he severer are financial fricions and he higher is he cos of inermediaion. A main characerisic of he model s dynamic behavior is ha, in line wih inuiion, banks respond o a ransiory decrease in uncerainy by lending more and holding fewer liquidiy reserves. Keywords: Bank liquidiy reserves; General equilibrium; Financial fricions; Sock-ou avoidance JEL Classificaion: C6; E32; E5 * This paper represens work in progress and is incomplee! I would like o hank Chrisopher House for very helpful discussions and advice. The paper was wrien while he auhor was a visiing scholar a he Universiy of Michigan, Ann Arbor. Financial suppor by he German Academic Exchange Service (DAAD) is graefully acknowledged. Jochen Günner is Visiing Scholar a he Deparmen of Economics of he Universiy of Michigan, Ann Arbor. Address: 37 Lorch Hall, 6 Tappan Sree, Ann Arbor, MI guenner@umich.edu, elephone: Furhermore Research Associae a he Chair of Economic Theory a he Oo-von-Guericke- Universiy Magdeburg. Address: PO Box 42, 396 Magdeburg, Germany. jochen.guenner@ovgu.de, elephone: , fax:

2 Inroducion This paper implemens he precauionary sock-ou avoidance moive, usually associaed wih invenory holding and invesmen in inermediae or final goods producion, in a profi maximizing banking secor. Individual banks are uncerain abou he realizaion of a sochasic sae variable he inflow of household deposis and hus subjec o an occasionally binding liquidiy consrain. Real banks incur coss when running shor of liquidiy, e. g. in erms of penaly ineres on emergency borrowing or forgoing profiable invesmen opporuniies. For his reason, precauionary liquidiy-ou avoidance represens a meaningful moive o explain he excessive reserve holding of corporae banks. An individual bank s choice of is opimal level of liquidiy is no fundamenally differen from ha of any oher profi maximizing enerprise and can be considered a special case of invenory managemen. A he same ime, liquidiy problems of a single bank migh propagae o oher financial inermediaries and give rise o sysemic risk. 2,3 We focus on he effec of uncerainy in cash flows on he opimal expansion of credi by individual banks as well as on he saic and dynamic implicaions in aggregae general equilibrium. This idea goes back o Edgeworh (988) who proposed o deermine he opimal volume of bank lending by operaional gaming. Orr and Mellon (96) were he firs o formalize i in a saic, sochasic framework. In his paper, he banking secor is incorporaed in a closed-economy dynamic sochasic general equilibrium (DSGE) model. We endogenize he demand for financial inermediaion by assuming ha bank loans are a direc inpu facor ino final goods producion. Banks incur real resource coss of credi provision as proposed by Balensperger (98). As a consequence, he loan ineres rae corresponds o an endogenous markup over he ineres rae paid on household deposis. Our model economy comprises a represenaive consumer, a represenaive final goodsproducing firm, and a coninuum of profi-maximizing banks wih mass one. The household eners period wih previously accumulaed physical capial and ineres-earning For parial equilibrium models of he sock-ou avoidance invenory heory, see e. g. Abel (985), Kahn (987), and Bils and Kahn (2); for a general equilibrium analysis see e. g. Fisher and Hornsein (2), Khan and Thomas (27), and Iacoviello e al. (29). 2 The seminal conribuion on he real economic damage of a bank run and he associaed liquidiy gap is found in Diamond and Dybvig (983). 3 See Freixas and Roche (28), chapers 7 o 9, for an exensive reamen of bank runs, liquidiy risk a he level of he banking firm, and in he banking secor as a whole.

3 deposis. I supplies he capial sock o he goods producer, and homogenous labor services o boh he firm and he banking secor. Moreover, he household redeposis an opimal share of is wealh in banks where hey yield a risk-free reurn. Besides physical capial and labor, producion of he final good requires anoher inpu: The banking secor provides exernal finance from deposis and iniiaed credi conracs using a Leonief-ype ransformaion funcion. In order o iniiae a loan, bank i mus employ labor services for screening and monioring purposes. 4 When he bank receives sufficien deposis, i can recover hese real resource coss hrough a mark-up above he deposi ineres rae. If i runs ou of liquidiy, however, he coss are sunk. Uncerainy arises from he assumpion ha he deposi inflows in period are randomly disribued across individual banks. While he level of aggregae deposis is endogenously deermined by he general equilibrium and common knowledge, each bank i receives a share of household deposis ha is drawn from a probabiliy disribuion wih mean equal o he aggregae level of deposis. Free enry ino financial inermediaion implies ha each bank makes zero expeced profis in equilibrium. By he law of large numbers, aggregae banking secor profis are zero boh in expecaion and ex pos. Assuming a funcional form for he disribuion of idiosyncraic sae variables, we can aggregae he decision rules of individual banks along he lines of Wen (28). Aggregae banking secor behavior is hen fully characerized by he momens of he underlying probabiliy disribuion. Furhermore, he liquidiy consrain of bank i which binds in some saes of he world and does no bind in oher saes will never be acive for he banking secor, as a whole, allowing for a log linear approximaion of he dynamic model around he non-sochasic seady sae. A comparaive saics exercise illusraes he relaion beween he shape parameer of he Pareo disribuion he idiosyncraic sochasic realizaions are drawn from and he model s saionary equilibrium. We find ha wih increasing σ, i. e. wih less variable bank-specific deposi inflows and hus lower uncerainy, banks iniiae and acually give more loans o he goods-producing firm, while holding fewer liquidiy reserves, alhough he expeced level of deposis decreases wih he shape parameer. The seady-sae resuls also illusrae ha he nonnegaiviy consrain on aggregae reserves never binds 4 The laer assumpion represens a shorcu o guaranee ha he firm will always honor is deb. I was incorporaed in a more elaborae loan producion funcion in Goodfriend and McCallum (27). 2

4 for he banking secor as a whole. Insead, lending is bounded above by he volume of iniiaed loans. Eliminaing uncerainy (σ ), he moive of each individual bank and of he aggregae banking secor o hold liquidiy reserves disappears. The main virue of aggregaion is ha we can analyze he dynamic behavior of an inegraed general equilibrium model wih kinked firs-order condiions. We find ha he impulse responses o sandard supply and demand shocks are qualiaively and quaniaively similar o hose well-esablished in he real business cycle (RBC) lieraure. Ye, here is clear feedback from financial inermediaion o he real economy. Shocks emanaing from he banking secor, e. g. a change in he uncerainy abou bank-specific deposi inflows or he produciviy of labor services in banking, have real consequences. In paricular, banks respond o a ransiory reducion in he uncerainy defined above by expanding he provision of loans and reducing liquidiy reserves. The res of he paper is organized as follows. Secion 2 describes he DSGE model and derives he agens individual decision rules as well as aggregae equilibrium condiions. We calibrae he parameer values and compue a benchmark seady sae in secion 3. Secion 4 examines he comparaive saics, while secion 5 conains an analysis of he model s dynamics. Secion 6 concludes. 2 The Model The model economy is se up in infinie discree ime and comprises hree economic eniies: A represenaive household, a represenaive final goods-producing firm, and a decenralized banking secor. The household accumulaes physical capial and saves in erms of bank deposis. For he former o be indifferen beween invesing and saving, deposis mus pay a risk-free ineres rae equal o he ne marginal produc of capial. The household supplies labor services o banks and labor as well as capial o he goods-producing firm. The laer combines hree inpu facors o produce a homogeneous good which can be consumed by he household, invesed o increases he producive capial sock available in he nex period, or saved as deposis. Besides labor and physical capial, producion of he final good requires exernal finance, i. e. bank loans, as a hird inpu. Banks ac as financial inermediaries beween he represenaive consumer and he firm, 3

5 convering household deposis ino firm loans. We assume ha only banks have access o a screening and monioring echnology ha requires labor services in order o guaranee ha a firm will no defaul on is loans, while he household iself is unable o provide hose loans direcly. Due o he cos of labor inpus, banks charge he borrowing firm a markup above he deposi rae. A he beginning of each period, when bank i iniiaes loans, is informaion se conains he aggregae amoun he household wans o deposi, bu no is idiosyncraic realizaion. The bank-specific level of deposis is drawn from a ime-varying probabiliy disribuion wih commonly observed parameers. Accordingly, he bank faces individual uncerainy ha mus be aken ino accoun when deermining is opimal expansion of loans. Unless he banking secor can fully diversify he idiosyncraic risk hrough a fricionless inerbank lending marke, his uncerainy will have subsanial aggregae consequences. 2. The Household The infiniely-lived represenaive consumer derives posiive uiliy from consumpion, C, of he final good and negaive uiliy from hours spen working in eiher he firm, N, or he banking secor, M. She accumulaes producive capial, K, or can alernaively ransfer consumpion across periods by means of bank deposis, D. The laer yield a risk-free reurn ρ. Assuming ha labor services are a homogeneous good, work in eiher secor pays he same wage rae, w. The represenaive household chooses {C, N, M, K, D } o maximize is discouned lifeime uiliy, E β ν ζ +ν ν= γ ( C γ +ν C+ν h ) a N +γn +ν, () + γ n subjec o he aggregae budge consrain in period, C + I + D + φ k 2 ( K 2 ) K w (N + M ) + r k K + ( + ρ )D, (2) K and subjec o he equaion of moion of he capial sock, K = ( δ)k + I. As usual, δ is he per-period rae of capial depreciaion, while increasing or decreasing he capial sock implies quadraic adjusmen coss. Moreover, he household s uiliy funcion displays habi formaion in consumpion. In (), β is he subjecive discoun 4

6 facor, γ and γ n capure he degree of ineremporal subsiuabiliy of consumpion and labor ime, respecively, and a is a scaling parameer. ln (ζ ) = ρ ζ ln (ζ )+ε ζ represens an auocorrelaed preference shock. The aggregae amoun of bank deposis in (2) is D [ d (i) η η di], (3) where η perains o he elasiciy of subsiuion across deposi accouns in differen banks and d (i) is drawn from a probabiliy disribuion specified below. While aggregae D is a decision variable of he represenaive consumer, he amoun of deposis she sores in bank i is hus sochasic, ex ane. Wihou loss of generaliy, one can assume ha, while he amoun of deposis a bank i is deermined by he realizaion of d (i), deposis in differen banks remain perfec subsiues, seing η = in he following. The household s firs order condiions (FOCs) wih respec o C, N, M, K, and D, respecively, are: C λ C = ζ ( C C h γ ) βhe ζ + ( C + C h γ ) N, M λ w = a (N + M ) γn (5) K λ [ + φ k ( K K )] = βe λ + [( + r k + δ) +φ k ( K + ) K + + φ k K K 2 ( K ) ] K (6) D λ = βe λ + ( + ρ ) (7) 2 (4) The household s equilibrium condiions are complemened by he budge consrain and he capial accumulaion equaion. 2.2 The Firm The represenaive goods-producing firm hires labor and rens physical capial from he household, while borrowing exernal funds, L, from he banking secor. Is homogeneous oupu is sold in a perfecly compeiive marke, i. e. he firm has no price-seing power and earns zero profis, in equilibrium. In fac, he final good serves as a numéraire and we normalize is price o. The profi maximizaion of he represenaive firm corresponds o a sraighforward saic decision problem: max π = Y r k K w N r L (8) K,N,L 5

7 subjec o he consan-reurns-o-scale (CRS) Cobb-Douglas producion funcion, Y = A K α N α 2 L α α 2. (9) A represens an economy-wide echnology innovaion which follows an auoregressive sochasic process, ln (A ) = ρ A ln (A ) + ε A. The firm s FOCs wih respec o K, N, and L, respecively, are: K r k K = α Y () N w N = α 2 Y () L r L = ( α α 2 )Y, (2) where r k denoes he user cos of capial and r is he ineres rae on bank loans. 2.3 The Banking Secor The banking secor comprises a coninuum of financial inermediaries indexed i [, ]. In line wih mos of he lieraure on he microeconomic heory of he banking firm, we assume ha bank i mus accep any amoun of deposis he household wans o sore in is accouns. A he beginning of each period, banks learn he aggregae level of deposis, D, and hus he parameers of he probabiliy disribuion heir sochasic individual realizaion, d (i), will be drawn from. Based on his informaion, bank i iniiaes an opimal amoun of deposis, l (i), i mus sick o. The laer represens an upper bound o he acual amoun of loans, l (i), he bank can gran afer d (i) realizes. In order o lend he enire iniiaed amoun, bank i mus receive sufficien deposis. As menioned before, he fracion of D commied o each bank by he represenaive household is he oucome of a preference shock ha realizes only afer banks choice of l (i). For his reason, d (i) represens an exogenous sae variable from he perspecive of bank i. The preference shock can be inerpreed as a posiive or negaive innovaion o he household s confidence in a paricular bank ha leaves he aggregae level of deposis in he banking secor unaffeced. 5 Each individual bank has access o a common CRS screening or monioring echnology, l (i) = A b m (i), (3) 5 We ignore he possibiliy ha he household looses confidence in all agens of he banking secor. 6

8 ha ensures full repaymen of an approved borrower. The provision of loans requires cosly labor inpu, m (i), which is priced in by banks and implies an ineres rae above he risk-free deposi rae, ρ. Noe ha (i) he credi inermediaion cos arises for he volume of iniiaed loans, l (i), which is necessarily larger or equal o he volume of acual lending, l (i), and (ii) he simplisic funcional form in (3) implies a consan marginal cos of credi provision. The laer assumpion bears he convenien propery ha he disribuion of lending among banks has no impac on he marginal cos of loan provision and here is an economy-wide ineres rae, r. In every sae of he model economy, bank i mus saisfy he balance-shee ideniy, l (i) + rr (i) d (i), (4) and a non-negaiviy consrain on is reserves or liquidiy holdings, rr (i). (5) The risk-neural manager of bank i chooses l (i) in order o maximize expeced profis, E π b (i) = ( + r )E l (i) + E rr (i) ( + ρ )E d (i) w m (i). (6) Noe ha in (4) and (6), l (i) is he acual volume of credi, while m (i) is an implici funcion of he volume of iniiaed loans, l (i). Subsiuing for d (i) ino (6) from he balance shee ideniy and making use of he monioring funcion (3) he bank s profi maximizaion problem simplifies o max l(i) E π b (i) = (r ρ ) E l (i) ρ E rr (i) w A b l (i). (7) The firs-order condiion w. r.. he level of iniiaed loans, l (i), is given by E l (i) l (i) = w A b (r ρ ), (8) i. e., he ne marginal revenue from an addiional uni of iniiaed lending, (r ρ ) El(i) l (i), mus equal he marginal cos of credi provision, w. A b The iming of sochasic evens in he model implies ha any aggregae uncerainy is resolved a he beginning of period, and each bank i iniiaes is loans based on a common informaion se ha includes D bu excludes d (i). As a consequence, all banks choose 7

9 he same opimal l condiional on he aggregae sae of he economy and, assuming mass, he banking secor as a whole iniiaes L l = l (i). We close he banking secor by assuming ha deposis are homogeneous from he banks perspecive. Free enry ino financial inermediaion will hen drive down expeced bank profis o zero. 6 In his conex, remember ha acual lending is bounded above by he minimum of deposis and iniiaed loans. Bank i supplies loans o he goods-producing firms using a Leonief-ype producion funcion: l (i) = min { l, d (i)} = min { L, d (i)}, (9) where d (i) is he bank-specific realizaion of sochasic deposi inflows, while l is he uniform amoun of lending iniiaed according o (3). Since d (i) is beyond conrol of bank i, rr (i) is pinned down by he balance shee ideniy for any given amoun of credi. From (4), i is sraighforward o derive he level of reserves as rr (i) = max { d (i) l, } = max { d (i) L, }. (2) Equaions (9) and (2) ranslae ino he following wo possible bank-specific scenarios: Case A: If d (i) > l, acual lending is bounded above by iniiaed loans and he nonnegaiviy consrain on liquidiy is no binding. l (i) = l and rr (i) = d (i) l Case B: If d (i) l, acual lending is bounded above by bank-specific deposis and he non-negaiviy consrain on liquidiy is binding. l (i) = d (i) and rr (i) = By he law of large numbers, aggregae bank behavior is herefore characerized by he following equaions: D E d (i) = d (i)di = d(i) d (i)df ( d) (2) L E l (i) = l (i)di = d(i) l d (i)df ( d) + d(i)> l l df ( d) (22) R E rr (i) = rr (i)di = d(i) l ()df ( d) + d(i)> l ( d (i) l ) df ( d), (23) and he free enry condiion which implies zero aggregae profis in he banking secor: Π b = πb (i)di = r L ρ D w! L =. (24) 6 In he presen framework wih only idiosyncraic uncerainy lef afer banks make heir decisions, zero expeced profis are equivalen o zero aggregae profis. A b 8

10 2.4 Analyical Soluion Assuming a suiable probabiliy disribuion for he realizaions of bank-specific deposis, equaions (2) (23) ranslae o convenien closed-form represenaions. Suppose d (i) is drawn from a ime-varying Pareo disribuion wih cumulaive disribuion funcion d σ F ( d) = b σ and probabiliy densiy funcion f ( d) = σ b σ d (b, ). We hen have d σ wih suppor D = d(i) d (i)df ( d) = b σ b σ d σ d( d) D = σ σ b 7 L = b l σ b σ R = d σ d( d) + l σ b σ d σ l l σ b σ d σ d( d) L = σ (σ b b σ σ b σ l d σ d( d) R = σ bσ l σ. l σ ) For a given level of aggregae deposis, D, and an exogenous process of he shape parameer, σ, which deermines he dispersion of he Pareo disribuion, he corresponding scale parameer in period, b, is pinned down by b = σ σ D. Moreover, i is sraighforward o derive he response of expeced acual lending, E l (i) o a marginal increase in iniiaed lending, l : E l (i) l L L = ( b σ ) >. (25) L Subsiuing his parial derivaive ino (8), we can solve for he secor-wide opimal level of iniiaed deposis as ( b σ w ) = L A b (r ρ ) L = ( r ρ w /A b σ ) b. (26) The opimal amoun of iniiaed invesmen projecs decreases wih he marginal cos of loan provision, w /A b, and increases wih he ineres rae spread earned in financial inermediaion as well as wih he scale parameer of he Pareo disribuion, b. When expecing higher deposi inflows, a bank always wans o iniiae and hus acually give more loans. 8 Of course, parial equilibrium consideraions of his kind mus be regarded wih cauion, as every parameer change migh rigger poenially offseing general equilibrium effecs in his framework. 7 The mean of a Pareo disribuion wih minimum value b and shape parameer σ is given by σ σ b. 8 Since l (i) = l = L, his conclusion holds for bank i as well as for all oher banks, and he banking secor as a whole. 9

11 Bearing his in mind, consider he parial derivaive of expeced and aggregae reserves, R, w. r.. iniiaed lending: E rr (i) l R L = ( b σ ) <, (27) L i. e. aggregae reserves are a decreasing funcion of L, ceeris paribus. 3 Calibraion and Seady Sae The calibraion of parameer values follows he RBC lieraure s sandard for simulaion of quarerly daa. We herefore merely summarize hem in Table, ogeher wih deliberaely chosen seady-sae values of he oal facor produciviy in goods producion and banking, A and A b, respecively, he preference or demand shock, ζ, and he Pareo disribuion s shape parameer, σ. 9 While σ, characerizing he dispersion of he Pareo disribuion, is assumed o follow an exogenous sochasic process and σ is hus explicily calibraed, he scale parameer b and is seady-sae value are deermined endogenously wihin general equilibrium. Remember ha for a given seady-sae volume of deposis, D, and shape parameer, σ, he Pareo disribuion s scale parameer is pinned down by b = σ σ D. Noe furher ha here is nohing dynamic abou a bank s idiosyncraic deposi shock. The laer prevails in he seady sae, as well. Ye, he disribuion he individual sochasic realizaions, d (i), are drawn from becomes saionary wih parameers b and σ. Benchmark Parameer Values α α 2 β γ γ n δ a h φ k A A b ζ σ Table : Calibraion of real economy parameers for he simulaion of quarerly ime series Given he se of parameer values in Table, i is sraighforward o solve numerically for he seady-sae values lised in Table 2. Boh ρ and r k are a direc consequence of 9 A b and σ are calibraed o produce reasonable seady-sae values of he loan ineres rae and he share of employmen in he banking secor. Given he highly nonlinear naure of he model and he large number of equilibrium condiions, a closed-form analyical soluion of he seady sae i no admissible. Insead, he saionary equilibrium values in Table 2 have been obained by means of he Gauss-Newon mehod in MATLAB.

12 Seady-Sae Values Y C I K N M D L R L b w ρ r k r λ Table 2: Seady-sae values of he real economy and banking secor variables for σ = 5 he parameers β and δ. The calibraion also ensures ha he represenaive consumer spends approximaely one hird of her ime endowmen working in eiher banking or goods producion. Seady-sae employmen in he banking secor, M, accouns for 6% of oal employmen, a value in line wih he share of employees in financial aciviies in oal non-farm employmen, as repored by he Bureau of Labor Saisics. Opimal iniiaion of credi implies a quarerly markup of he loan ineres rae above he deposi ineres rae of.4 percenage poins. Noe ha he banking secor, as a whole, holds a subsanial amoun of seady-sae reserves which resul from an opimal choice of L according o he saionary version of (26) and he implied loan oucome, L. Necessarily, acual lending is bounded below by b and bounded above by he minimum of D and L. Moreover, aggregae reserves are deermined by he banking secor-wide balance shee ideniy, R D L. The following secion examines he sensiiviy of he model s saionary general equilibrium w. r.. he shape coefficien, σ, of he Pareo disribuion funcion. 4 Comparaive Saics Figure plos he Pareo disribuion s probabiliy densiy funcion (pdf ) for differen values of σ. For illusraive purposes, we have arbirarily se he scale parameer o, here. The graph shows ha he mass of he pdf a is lower bound, b, increases wih he shape parameer, i. e. he dispersion of he Pareo disribuion decreases wih higher σ. In general equilibrium, σ also affecs he mean of he disribuion or, o be precise, he spread beween he mean, D, and he lower bound, b. Throughou he res of he paper, σ is reaed as an exogenous and possibly ime-dependen variable, whereas b Remember ha he k h momen of a Pareo disribuion exiss and is finie if and only if σ > k. As a consequence, he disribuion has infinie variance for a shape parameer of 2.

13 9 8 σ=2 σ=5 σ=4 σ= 7 6 pdf f X (x) x Figure : Pareo probabiliy densiy funcion for differen values of he shape parameer σ is fully endogenized. The reader migh hus imagine he Pareo disribuion underlying he idiosyncraic realizaions of d (i) as shifing endogenously over ime, wih occasional disurbances o is shape parameer. The former variaion arises from any disurbance of he general equilibrium, e. g. a supply or demand shock, whereas he laer represens an exogenous deviaion of σ from σ. 4. The Banking Secor Figure 2 plos D, L, L, b, and R as funcions of he Pareo shape parameer, σ. The seady-sae value of aggregae household deposis corresponds o he mean of d(i) and is decreasing wih σ. Consider for example a shape coefficien of 2, implying infinie variance. In his case, he banking secor as a whole iniiaes significanly less credi. L undercus he benchmark value in Table by abou 4%. Alhough he aggregae level of deposis and hus he expeced realizaion of d(i) is higher for each bank i, he higher variance of he Pareo disribuion raises he risk of a deposi draw ha falls shor of he volume of iniiaed loans and causes sunk coss. I. e., he expeced marginal revenue from an addiional uni of l (i) decreases, while banks face a (largely) consan marginal cos in erms of screening and monioring poenial borrowers. Due o he symmery of he opimal cuoff, his holds equally for all banks. Figure 2 also illusraes ha he gap beween he disribuion s mean, D, and is lower bound, b, is a falling funcion of σ. Moreover, he opimal volume of iniiaed credi conracs is sricly lower han expeced deposis. Acual lending is herefore bounded 2

14 Household Deposis Iniiaed Fin. Invesmen Acual Fin. Invesmen Pareo lower bound Liquidiy Reserves.6.4 Loans/Iniiaed Loans/Deposis Iniiaed/Deposis Reserves/Iniiaed Reserves/Deposis σ * Figure 2: Sensiiviy of he banking secor s seady-sae equilibrium values w. r.. σ σ * Figure 3: Sensiiviy of he banking secor s seady-sae equilibrium raios w. r.. σ from below by b and bounded from above by L for all values of he shape parameer. Wih increasing σ, his band seadily narrows because he banking secor s asses ne of reserves, L, and is liabiliies, D, converge. Due o he fac ha he variaion in d(i) reduces, banks symmerically expand heir iniiaed credi provision. As a consequence, aggregae acual lending increases, as well, while he banking secor as a whole reduces is holdings of liquidiy reserves. Figure 3 redraws he banking secor s seady sae in erms of raios. Obviously, iniiaed loans represen a binding upper bound o acual lending for mos values of σ, whereas he raio of aggregae loans over deposis is sricly smaller han L / L. Noe ha his is an immediae consequence of (2), i. e. he nonnegaiviy consrain on reserves is binding for bank i, while bank j is unable o exhaus is reserves due o an insufficien sock of iniiaed credi. In line wih Figure 2, he fracion of reserves in deposis as well as in iniiaed loans decreases simulaneously wih he variabiliy of d(i) draws. The severiy of financial fricions and hus he cos of financial inermediaion obviously increases wih he degree of uncerainy regarding bank-specific deposi inflows. 4.2 The Real Economy Figure 4 plos he seady sae of seleced real and financial variables as a percenage of heir benchmark values from Table 2 agains he Pareo disribuion s shape parameer. Apparenly, he variance of idiosyncraic realizaions has a sizeable impac on he model economy s saionary general equilibrium. 3

15 Oupu Household Consumpion %age of benchmark benchmark %age of benchmark σ * Invesmen σ * Employmen in Firms %age of benchmark %age of benchmark σ * σ * 5 Employmen in Banks 2 Real Wage %age of benchmark benchmark %age of benchmark %age poins p.a σ * Loan Ineres Rae 25 %age of benchmark σ * Banking Secor Variables Aggregae Deposis Acual Loans Aggregae Reserves σ * σ * Figure 4: Banking and he real economy. Sensiiviy of seady-sae values w. r.. σ For σ = 2, abou 2.5% of oupu, consumpion, and invesmen are los relaive o he benchmark seady sae. A he same ime, he opimal employmen in goods producion is.6% higher han for σ = 5. Severer financial fricions imply a subsiuion of inpu facors, i. e. an increase in N ha compensaes, o some exen, for he lower capial sock and a reduced volume of loans. On he oher hand, increasing he shape parameer furher enails a poenial rise in oupu, consumpion, and invesmen by up o.24% relaive o he benchmark, while employmen falls by less han half a percen. Due o he fac ha A b is consan in seady sae, employmen in he banking secor is direcly proporional o he amoun of iniiaed loans and hus increasing in σ. 4

16 5 Model Dynamics Making a funcional form assumpion for he probabiliy disribuion of he bank-specific random variable d (i), we can aggregae he coninuum of banks by he law of large numbers. For small deviaions from seady sae, i. e. for small exogenous disurbances o produciviy in goods producion, preferences, he variance of deposi realizaions or he labor produciviy in financial inermediaion, he occasionally binding nonnegaiviy consrain on individual banks reserves becomes irrelevan for he dynamic behavior of he aggregae banking secor as well as of aggregae real variables. I is hus sraighforward o log linearize he sysem of equilibrium condiions around he non-sochasic seady sae and o simulae he impulse responses o srucural shocks. The log linear equaions are summarized in Appendix A. 5. Real Economy Shocks Figure 5 plos he dynamic behavior of seleced variables in response o a sandard supply and demand shock, respecively. The former refers o a posiive one-percen innovaion in oal facor produciviy of he goods-producing firm, A, while he demand shock is modeled as a change in he preference parameer ζ. Boh disurbances occur in period, are ransiory and serially correlaed wih coefficiens ρ A =.95 and ρ ζ =.8. Following a supply shock, he marginal produc of physical capial, labor, and exernal finance increases. While he firm responds by demanding more loans, a pronounced rise in he real wage induces i o subsiue away from labor services. In order o expand he producive capial sock, capial invesmen mus increase, on impac. Due o he fac ha bank credi represens a direc inpu ino final goods producion, here is higher demand for liquidiy by he banking secor, as well. Accordingly, he rise in household consumpion is less pronounced han in sandard RBC models. The hump shape arises from he assumpion of habi formaion in consumpion. Noe ha he expansion of iniiaed loans by each bank and in aggregae ( l = L ) exceeds he increase in aggregae and expeced deposi inflows, D. Alhough he ineres rae on loans falls anoher reason for he firm s subsiuion away from labor, he spread beween ρ and r widens. This more han compensaes for he real wage increase ha applies o he labor services banks mus hire in order o provide credi. The impulse 5

17 .8 Oupu Y Supply Shock Demand Shock.5 Consumpion C.5 Invesmen I Firm Employmen N.5 Bank Employmen M Real Wage w Household Deposis D.5 Acual Loans L 2 Bank Reserves R Deposi Ineres Rae ρ.3 Loan Ineres Rae r User Cos of Capial r k Figure 5: Impulse responses o srucural supply, demand, and global ineres rae shocks; x-axis: Quarers afer he shock; y-axis: %age deviaions from seady sae (%age poins for ρ). response of acual bank lending is qualiaively equivalen o ha of L. The quaniaive difference corresponds o he spread beween M and L in Figure 5, as labor produciviy in he banking secor is no affeced by a TFP shock in goods producion. A posiive one-percen preference shock raises household consumpion by up o.45% relaive o seady sae wih lag of wo quarers. Habi formaion implies a hump-shaped impulse response, and C has reurned o is iniial level afer approximaely 2 quarers. 6

18 In order o afford a higher consumpion level, he represenaive household is willing o work more in goods producion, even a a lower real wage. Neverheless, he rise in GDP falls shor of he consumpion increase, and he household mus compensae by cuing back on capial invesmen and, in paricular, deposi savings. Facing a lower expeced volume of deposi inflows and a reducion in he r ρ spread, banks iniiae less credi. L (consider again he corresponding impulse response of M in Figure 5) and L fall by up o.86 and.78% of heir seady-sae values, respecively. Noe ha, in his model, real disurbances affec he banking secor via wo channels, i. e. direcly hrough changes in he demand for financial inermediaion which represens an inpu ino goods producion, and indirecly hrough flucuaions in he ne marginal reurn of producive capial and hus in he deposi ineres rae. 5.2 Banking Secor Shocks The model also incorporaes wo sochasic disurbances ha arise direcly from financial inermediaion. The erm uncerainy shock refers o an exogenous change in he shape parameer, σ, of he Pareo disribuion underlying he bank-specific d (i) realizaions. Banking echnology shocks are innovaions in he labor produciviy A b of screening and monioring services. Like in he real economy case, boh shocks occur in period, are ransiory and serially correlaed wih coefficien values ρ σ =.8 and ρ A b =.95. While he laer implies ha echnological innovaion in goods producion and banking are equally persisen, ρ σ is calibraed more or less arbirarily. An increase in he shape parameer affecs boh he model s seady sae and dynamics. For small-enough innovaion, however, we can neglec he saionary equilibrium effecs. Figure 6 plos seleced impulse responses o a 2% increase in he exogenous variable σ relaive o is seady-sae value. Noe ha his corresponds o an insananeous rise from 5 o 6 in quarer, followed by a seady convergence back o he benchmark value. σ is aained afer approximaely five years (no ploed). 2 In response o he shock, he disance beween D and b decreases, and iniiaing more loans becomes less risky for individual banks. While he aggregae volume of household deposis rises, iniiaed and, as a consequence, acual lending increases by even more. 2 One migh argue ha periods of excepionally high or low uncerainy in financial inermediaion end o las for less han 2 quarers. By increasing or decreasing ρ σ, he effecs of an uncerainy shock emerging from he banking secor, as modeled here, can be prolonged or abbreviaed. 7

19 2.5.5 Oupu Y Uncerainy Shock Bank Technology Shock Consumpion C 2 Invesmen I Firm Employmen N Bank Employmen M Real Wage w Household Deposis D 6 Acual Loans L Bank Reserves R Deposi Ineres Rae ρ Loan Ineres Rae r User Cos of Capial r k Figure 6: Impulse responses o srucural supply, demand, and global ineres rae shocks; x-axis: Quarers afer he shock; y-axis: %age deviaions from seady sae (%age poins for ρ). The predicion ha banks opimal sock of liquidiy reserves decreases when uncerainy abou he bank-specific realizaion of d (i) decreases and vice versa represens one of he model s key virues. The reducion in R is consisen wih profi-maximizing behavior of individual banks and of he banking secor as a whole. Noe also ha he volume of reserves remains below is seady-sae value even afer he shock has died ou. Lower uncerainy riggers flucuaions in real economy variables hrough he changes in 8

20 aggregae household deposis, banks demand for labor services, and he response of he loan ineres rae. Because of he relaively lower inpu facor coss, he goods-producing firm subsiues labor and exernal funds for physical capial. However, he pronounced drop in equilibrium invesmen quickly leads o an erosion of he capial sock. Oupu and employmen in goods producion fall shor of heir seady-sae values approximaely one year afer he shock occurs. The banking echnology shock corresponds o a wo-percen increase in A b relaive o is seady-sae value of 275. Unforunaely, his posiive innovaion has a perver effec on general equilibrium, in our model. While banks become more producive in ransforming screening and monioring services ino iniiaed loans and aggregae deposis rise, each bank hires less labor and lends fewer funds, on impac. Insead, he banking secor ends up holding an exra % of liquidiy reserves relaive o he saionary equilibrium. This counerinuiive aggregae behavior seems o arise from he assumpion of zero expeced profis or, equivalenly, he free-enry condiion in he banking secor. We would have expeced a posiive effec of more efficien financial inermediaion on he real economy. Ye, he model predics a relaively shor-lived expansion of oupu, driven by higher employmen, whereas he ne inpu of he oher wo facors in goods producion capial and exernal funds drops relaive o he seady sae. The changes in K, N, and L merely reflec he decrease and increases, respecively, in he corresponding facor coss w, r k, and r. 6 Conclusion The presen paper reinerpres he sock-ou avoidance moive of invenory holding and invesmen in goods producion in he conex of he opimal liquidiy reserves decision of individual banks which are subjec o occasionally binding liquidiy consrains. Bank i lacks informaion on he amoun of deposis i receives in any given period. Assuming a funcional form for he disribuion of idiosyncraic sae variables, he kinked FOCs of a coninuum of individual banks can be aggregaed along he lines of Wen (28). We find ha wih increasing shape parameer, σ, of he underlying Pareo disribuion, i. e. wih lower variance of he realizaions, banks iniiae a higher volume of credi boh in absolue erms and relaive o expeced deposi inflows. Due o decreasing uncerainy, 9

21 he raio of acually underaken o iniiaed lending is rising in σ. As a consequence, individual banks and he banking secor as a whole will hold fewer reserves, financial fricions are less severe, and he cos of financial inermediaion decreases. The virue of his mehod of aggregaion is ha i faciliaes he derivaion of dynamic responses in a DSGE model wih occasionally binding and hus kinked equilibrium condiions. The impulse responses o sandard demand and supply shocks are qualiaively and quaniaively similar o hose well-esablished in he RBC lieraure. A he same ime, here is clear feedback from he banking secor o he real economy. Disurbances o financial inermediaion, e. g. a change in he uncerainy abou deposi inflows faced by banks or in he produciviy of labor services in banking, have real consequences. In paricular, we find ha banks respond o a ransiory reducion in uncerainy by expanding he provision of loans and by reducing liquidiy reserves. The presened model can be elaboraed in several dimensions. While we endogenize he demand for and he cos of financial inermediaion, his is done in a raher ad hoc way. Loans are assumed o be a direc inpu facor in final goods producion and banks face real resource coss when iniiaing credi. These modeling choices allow us o circumven he inroducion of an enrepreneurial secor like in Bernanke e al. (999) or Chrisiano e al. (25). Alernaively, we could have imposed a working capial requiremen on goods producers as in Neumeyer and Perri (25) and Günner (2). Moreover, he issue of liquidiy-ou avoidance is only ackled in a sylized special case. Individual banks are suck wih he chosen volume of iniiaed credi, when heir idiosyncraic deposi realizaions are drawn. Given he sae of he economy, he non-negaiviy consrain on reserves will hus be binding for some banks bu no for all. There is ample room for model refinemens, here; e. g. he possibiliy of inerbank lending or liquidiy pooling ha individual banks can op-in and op-ou of a a premium, condiional on he realizaion of heir idiosyncraic sae variables. In paricular, he presen model does no ye incorporae moneary policy. In addiion o is adherence o a sandard ineres rae rule, he cenral bank migh funcion as a lender of las resor o banks in dire sraighs (see Freixas and Roche (28), chaper 9.5) and represen a subsiue for inerbank lending. 2

22 Appendix A. The Log Linearized Model Below, ˆx represens he percenage deviaion of variable x from is saionary equilibrium in period. Variables wihou a ime subscrip correspond o heir seady sae. 3 Noe ha he denoaions ˆρ and ˆr have a slighly differen meaning: The deposi ineres rae and he loan ineres rae ener he loglinear sysem in erms of absolue, i. e. percenage poin deviaions from seady saes. Consider firs he equilibrium condiions for he represenaive household and he goodsproducing secor of he economy: λc (ˆλ + Ĉ) = C ( h)( γ) [ˆζ + ( γ) (Ĉ hĉ )] βc ( h)( γ) E [ˆζ + + ( γ) (Ĉ + hĉ)] (A.) γ n ˆλ + ŵ = (N + M) (N ˆN + M ˆM ) (A.2) ˆλ = E [ˆλ + + βr kˆr + k + βφ k ˆK+ ] φ k ( + β) ˆK + φ k ˆK (A.3) ˆλ = E ˆλ+ + + ρ ˆρ (A.4) CĈ + D ˆD + IÎ = wn (ŵ + ˆN) + wm (ŵ + ˆM) + r k K (ˆr k + ˆK ) +( + ρ)d ˆD + ρdˆρ (A.5) Ŷ = Â + α ˆK + α 2 ˆN + ( α α 2 ) ˆL (A.6) ˆK = δî + ( δ) ˆK (A.7) Ŷ = ˆr k + ˆK (A.8) Ŷ = ŵ + ˆN (A.9) Ŷ = ˆr r + ˆL (A.) Â = ρ A Â + ε A (A.) ˆζ = ρ ζ ˆζ + ε ζ (A.2) Moreover, he log linear equaions for he aggregae banking sysem are as lised below: ˆ L = Â b + ˆM (A.3) LˆL = σ [σbˆb σb σ ˆσ b σ L σ (σˆb + ( σ)ˆ L σ σ ˆσ )](A.4) rl (ˆr + ˆL ) = ρd (ˆρ + ˆD ) + w A b L (ŵ + ˆ L Âb ) (A.5) 3 The superscrip marking seady-sae values in he main ex is dropped for noaional convenience. 2

23 ˆb ˆ L + σˆσ = ŵ Âb r ρ (ˆr ˆρ ) ˆD = ˆb σ ˆσ (A.6) (A.7) rr ˆ = D rr ˆD L rr ˆL (A.8) ˆσ = ρ σ ˆσ + ε σ (A.9) Â b = ρ b AÂb + ε Ab (A.2) 22

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