Liquidity and Bank Capital Requirements

Transcription

1 Liquidiy and Bank Capial Requiremens Hajime Tomura Bank of Canada November 3, 2009 Preliminary draf Absrac A dynamic compeiive equilibrium model in his paper incorporaes illiquidiy of asses due o asymmeric informaion abou asse qualiy. In he model, boh a negaive produciviy shock and an increase in he degree of asymmeric informaion can cause a simulaneous deerioraion of illiquidiy of asses and he marke price of asses. Illiquidiy of asses leads o liquidiy ransformaion by banks, and banks finance par of heir asses hrough public equiies (bank capial) o preven a bank run in equilibrium. The capial-asse raio of banks increases in illiquidiy of bank asses and he volailiy of he marke price of bank asses. homura@bankofcanada.ca. The views expressed herein are hose of he auhor, and should no be inerpreed as hose of he Bank of Canada. 1

2 1 Inroducion This paper presens a dynamic compeiive equilibrium model in which illiquidiy of asses arises due o asymmeric informaion abou asse qualiy. There are wo main resuls. Firs, i is shown ha boh a negaive produciviy shock and a rise in he degree of asymmeric informaion can cause an increase in illiquidiy of asses and a drop in he marke price of asses, as occurred during he financial crisis since Second, he model shows ha illiquidiy of asses leads o financial inermediaion for liquidiy ransformaion. I is found ha liquidiy-ransforming banks are necessarily suscepible o a self-fulfilling bank run due o illiquidiy of bank asses and need o finance par of heir asses hrough public equiies (bank capial) o preven a bank run. The equilibrium capial-asse raio of banks increases in illiquidiy of bank asses and he volailiy of he marke price of bank asses. The model is a version of he AK model, where only a par of agens can produce new capial from goods due o an idiosyncraic shock o each agen. Call he agens who can produce new capial as producive and hose who canno as unproducive. Agens are anonymous and canno borrow agains fuure income. In he model, he number of he producive is so small ha he amoun of goods produced from he producive s own capial is shor of he efficien level of aggregae invesmen in new capial. The producive sell heir used capial o obain goods from he unproducive for maximizing heir invesmens in new capial. The compeiive marke for capial, however, is conaminaed by adverse selecion, since every uni of used capial depreciaes a is own rae and he depreciaion rae is privae informaion for he holder of he uni of used capial. The adverse selecion lowers he marke price of capial, leading o a marke s undervaluaion of he average qualiy of used capial held by each agen. Thus, used capial held by each agen is illiquid as a whole. In his paper, illiquidiy is defined as undervaluaion in he marke. 2

3 The degree of illiquidiy of capial flucuaes in response o shocks. There are wo ypes of shocks in he model; a produciviy shock and a change in he range of possible depreciaion raes of capial. I is shown ha boh ypes of shocks can cause an increase in illiquidiy of capial and a decline in he marke price of capial, as occurred during he financial crisis since Firs, a negaive produciviy shock reduces he marke price of capial, since a decline in agens income lowers aggregae spending on capial. A decline in he marke price of capial in urn discourages agens from selling high-qualiy capial, which increases illiquidiy of capial due o worsened adverse selecion. Second, an expansion of he range of possible depreciaion raes of capial increases he degree of asymmeric informaion in he marke for capial. This effec of he shock worsens adverse selecion, and a resuling increase in illiquidiy of capial lowers he marke price of capial. Illiquidiy of capial also explains why financial inermediaion is necessary for he economy. Banks can supply liquid securiies o agens by holding illiquid capial and financing he cos hrough bank deposis. Since idiosyncraic depreciaion raes of capial held by banks cancel each oher ou, he average qualiy of each bank s asses becomes public informaion, which makes bank deposis issued agains bank asses free from illiquidiy. Agens can increase invesmens in new capial by soring wealh hrough bank deposis when hey are unproducive and selling hem when hey are producive. I is found ha liquidiy-ransforming banks would be suscepible o a self-fulfilling bank run if hey issued bank deposis up o he rue value of bank asses, since repayable bank deposis would exceed he liquidaion value of bank asses due o illiquidiy of capial. 1 To preven a bank run, banks need o finance he difference beween he rue value and he liquidaion value of heir asses hrough public equiies (bank capial). The dynamic analysis of he model idenifies wo facors ha deermine he minimum capial-asse raio for banks o preven a bank run. Firs, o eliminae he possibiliy of a 1 This is he same ype of he panic-based bank run as analyzed by Diamond and Dybvig (1983). 3

4 bank run, banks mus limi he repayable amoun of bank deposis o he liquidaion value of heir asses in he nex possible recession. This facor drives he minimum capial-asse raio of banks o be pro-cyclical, since he rue value of bank asses increases during booms, while he limi on bank deposis remains equal o he liquidaion value of bank asses during recessions. Second, when negaive shocks hi he economy, illiquidiy of capial increases due o worsened adverse selecion, as explained above. This facor enlarges he difference beween he rue value and he liquidaion value of bank asses, which drives he minimum capial-asse raio of banks o be couner-cyclical. Overall, he capial-asse raio of banks increases in illiquidiy of bank asses and he volailiy of he marke price of bank asses, and he relaive balance beween hese wo facors deermines he cyclicaliy of he capial-asse raio of banks. In he numerical examples of he dynamics of he model, he capial-asse raio of banks is pro-cyclical when business cycles are driven by produciviy shocks and i is couner-cyclical when business cycles are driven by changes in he degree of asymmeric informaion (i.e., changes in he range of possible depreciaion raes of capial). The mos relaed paper o his paper is Kiyoaki and Moore (2005). They inroduce a resaleabiliy consrain on capial in a dynamic compeiive equilibrium model. This consrain is inerpreed as conrolling for he effeciveness of financial inermediaion ha alleviaes asymmeric informaion by bunching asses wih idiosyncraic qualiies. This paper formalizes his inerpreaion and endogenizes he dynamics of liquidiy of asses and bank capial requiremens. Also, his paper conribues o he lieraure on dynamic compeiive equilibrium models of banking, in which banks are usually eiher enrepreneurs whose bank capial is heir inernal ne-worh or firms ha earn zero profi and do no need o mainain any bank capial. 2 The model in his paper incorporaes publicly-owned banks subjec o bank capial requiremens, which arise from asymmeric informaion endogenously. 2 For example, see Williamson (1987), Bernanke and Gerler (1989), Holmsröm and Tirole (1997), and Chen (2001). 4

5 The remainder of he paper is organized as follows. Secion 2 describes he model. Secion 3 derives aggregae equilibrium condiions. Secion 4 analyically solves he model wihou a banking secor. Secion 5 analyzes liquidiy ransformaion by banks in equilibrium and he dynamics of he capial-asse raio of banks. Secion 6 concludes. 2 The model 2.1 The agen s problem There is a discree-ime economy wih a coninuum of infinie-lived agens and a represenaive bank in a compeiive banking secor. Denoe he se of agens in he economy by I and he measure on he coninuum of agens by µ. Each agen can produce homogeneous goods from capial in he beginning of every period. The producion funcion for goods is: y i, = α k i, 1, α {ᾱ, α}, (1) where i is an index for each agen, denoes a ime period, y i, is oupu, k i, 1 is capial held a he end of he previous period, and α is he produciviy of capial common o all he agens. Capial is divisible and each infiniesimal uni of capial depreciaes a is own rae afer producion. Denoe by k i,δ, 1 he densiy of capial ha is used by agen i and depreciaes a he rae of δ afer he producion in period. Depreciaion raes are independenly and idenically disribued by a uniform disribuion, such ha: δ U[ δ, δ + ], {, }, (2) k i,δ, 1 = k i, 1, (3) 5

6 where δ (0, 1). Noe ha ( ) 1 is he densiy of he uniform disribuion. Each value of α and is deermined by a Markov process. For x = α, and for all, he condiional probabiliy ha x +1 = x = x is η x ( [0, 1]) and he condiional probabiliy ha x +1 = x = x is η x ( [0, 1]). Assume ᾱ, α > 0 and ha, (0, 1 δ). Assume ha he depreciaion rae of each infiniesimal uni of capial is privae informaion for he agen who uses he uni of capial for producion in he beginning of each period. The depreciaion rae becomes public informaion when he capial is used for producion again in he nex period, which reveals he depreciaion rae hrough he amoun of goods produced by he capial. 3 Call depreciaed capial afer producion as used capial. Agens can rade used capial in a compeiive marke, where a price is se o each infiniesimal uni of capial. Assume ha agens are anonymous and ha he price of used capial in he marke canno be coningen on he characerisics of he buyer or he seller. As a consequence, every uni of capial is raded a an idenical price in each period. 4 By he law of large numbers, he realized average depreciaion rae of used capial bough by each buyer equals he average depreciaion rae of used capial sold in he marke, which is denoed by ˆδ. 5 Only a par of agens can produce new capial from goods. The producion funcion for new capial is: i i, = φ i, x i,, φ i, {0, φ}, (4) 3 Afer he revelaion of depreciaion raes, capial ne depreciaion becomes homogeneous once and hen each uni of capial depreciaes a is own rae. 4 If here are muliple compeiive markes sored by he amoun of capial sold by each seller, hen he quaniy of sold capial could signal he qualiy of he capial. Even in his case, anonymiy of sellers would le each seller spli her sold capial in muliple los and sell hem in differen markes o maximize he oal revenue from he sales. This paper absracs from he ineracion beween compeiive marke prices and his ype of seller s sraegic behaviour. 5 This is a common feaure of compeiive equilibrium models wih adverse selecion. See Gale (1992) and Eisfeld (2004) for example. 6

7 where i i, is newly produced capial and x i, is he amoun of goods invesed in capial. The produciviy of invesmen, φ i,, is deermined by an idiosyncraic Markov process for each agen. The condiional probabiliy ha φ i,+1 = φ i, = φ is ρ P ( [0, 1]) and he condiional probabiliy ha φ i,+1 = φ i, = 0 is ρ U ( [0, 1]) for all i and. The new capial only maerializes in he beginning of he nex period before he iming of producion and canno be raded oday. Each agen maximizes expeced uiliy from consumpion every period. The agen s maximizaion problem is: max E β s ln c i,s {c i,s, x i,s, ki,s o, l i,δ,s} s=0 s=0 s.. c i,s + x i,s + Q s k o i,s + b i,s + (1 + ζ)v s s i,s δ+ s = α s k i,s 1 + Q s l i,δ,s dδ + R s b i,s 1 + (D s + V s )s i,s 1, δ s δ+ s k i,s = φ i,s x i,s + (1 ˆδ s )ki,s o + (1 δ) (k i,δ,s 1 l i,δ,s ) dδ, δ s (5) l i,δ,s [0, k i,δ,s 1 ], c i,s, x i,s, k o i,s b i,s, s i,s 0, where β (0, 1) and ζ > 0. The firs consrain of he maximizaion problem (5) is a flow-of-fund consrain, where c i,s is consumpion, ki, o is he amoun of used capial bough from he marke, l i,δ, is he densiy of used capial wih a depreciaion rae δ sold by he agen in he marke, b i,s is he amoun of one-period bank deposis, s i,s is he number of bank equiies, Q s is he marke price of used capial, R s is he ex-pos deposi ineres rae, V s is he ex-dividend price of bank equiies, D s is he amoun of bank dividends per equiy, and ζ is a marginal cos of holding bank equiies. The bank-equiy holding cos is a reduced-form represenaion of coss of managing equiies, such as a ransacion cos and a cos of monioring ha is necessary 7

8 o make he bank o pay dividends. 6 The exisence of his cos will make agens require a higher rae of reurns on bank equiies han bank deposis. Thus, equiy financing becomes cosly for he bank. The flow-of-fund consrain also incorporaes an assumpion ha agens canno borrow agains heir fuure income due o heir anonymiy ha makes i difficul o enforce heir commimens. The second consrain is he law of moion of capial. The hird consrain means ha he sales of used capial mus be non-negaive and ha he agen canno sell more han he amoun of used capial he agen owns. The fourh consrain is a non-negaiviy consrain on choice variables. Each agen akes as given he probabiliy disribuion of {Q s, D s, V s, R s, ˆδ s, α s, s, φ i,s } s= The bank s problem The represenaive bank in he compeiive banking secor can buy used capial from he marke and sell bank equiies and one-period bank deposis o agens. The implici assumpion behind he difference in he abiliy o borrow (including equiy financing) beween agens and he bank is ha he bank is no anonymous, which makes i feasible o enforce is commimens. In conras, agens are anonymous and i is difficul o enforce heir commimens. Assume ha wriing coningen conracs ha are enforceable by he cour is sill cosly, so ha he bank can only issue deposis and equiies. 7 If here are equiy holders of he bank, hen he bank maximizes he value of he bank for equiy holders. In his case, he bank maximizes he value of (D + V )s i, 1 for each agen every period, since he maximum of each agen s uiliy funcion increases in he agen s wealh, given he probabiliy disribuion of exogenous variables for agens. Since 6 The predeermined amouns of repaymens o deposis are proeced by laws. In conras, equiy holders need o find he oal cash flows for he bank and negoiae wih he bank on he amoun of dividends. 7 Noe ha equiies are no coningen conracs ha specify coningen reurns ex-ane. Insead, ex-pos negoiaion on dividends mus ake place as if defaul on deb occurs every period. See Har and Moore (1994) for more deails on he feaure of equiies as a financial conrac. 8

9 {s i, 1 i I} is predeermined, maximizing (D + V )s i, 1 for all i I is equivalen o maximizing he oal value of he bank for equiy holders, (D +V )S B, 1, where S B, denoes he number of bank equiies issued by he bank. The oal value of he bank for equiy holders is derived from a flow-of-fund consrain on he bank: D S B, 1 + R B B, 1 + Q (K o B, L B,) = α K B, 1 + B B, + V (S B, S B, 1 ), (6) where K B, 1 is he amoun of capial held a he end of he previous period, L B, is he amoun of used capial sold by he bank, K o B, is he amoun of used capial bough from he marke, and B B, is he amoun of bank deposis issued by he bank. Noe ha he las erm on he righ-hand side of he equaion is he revenue from newly issued equiies or he expendiure on equiy repurchase. If he bank fulfills deposi conracs, hen he ex-pos deposi ineres rae, R, is he ex-ane non-coningen ineres rae specified by deposi conracs in he previous period, which is denoed by R 1. If he bank defauls, hen he ex-pos deposi ineres rae equals he recovery rae of deposis. Assume ha a bank run occurs if he repayable amoun of deposis exceeds he liquidaion value of capial held by he bank. 8 In his case, he bank canno roll over is deposis and mus maximize he repaymen o deposiors by liquidaing all he capial i owns. Since he liquidaion value of he bank s capial is less han he 8 As shown below, he presen discouned value of fuure income generaed by he bank s capial exceeds he liquidaion value of he capial. Thus, if he bank can roll over deposis, he bank can avoid defaul. Bu if all he deposiors expec ha he bank canno roll over deposis in his case, hen heir expecaions are self-fulfilling. 9

10 repayable amoun of deposis, he bank mus defaul and bank equiies lose value. Thus: B B,, D, V, KB, o = 0, L B, = K B, 1, R = (α+q)k B, 1 B B, 1 if R 1 B B, 1 > (α + Q )K B, 1, R = R 1 if R 1 B B, 1 (α + Q )K B, 1. (7) Noe ha he recovery rae of deposis, R, on he firs line is deermined by he flow-of-fund consrain (6). When maximizing (D +V )S B, 1, he bank inernalizes he price of bank equiies, V, and he ex-ane deposi ineres rae, R, he laer of which responds o he level of bank deposis hrough he probabiliy of a bank run in he nex period. These prices are deermined by he firs-order condiions for bank securiies in he agen s maximizaion problem (5): (1 + ζ)v E [ βci, (D +1 + V +1 ) 1 E [ βci, R +1 c i,+1 c i,+1 ], (8) ]. (9) The firs line is for bank equiies and he second line for bank deposis. On each line, he lefhand side of he weak inequaliy is he marginal cos of he bank securiy in erms of curren consumpion and he righ-hand side is he marginal reurn from he bank securiy. If he sric inequaliy holds, hen he agen does no hold he bank securiy. The equaliy holds for he agen who has he larges value of he righ-hand side erm among agens. Denoe he indices of hese agens for bank equiies and bank deposis by i S, and i B,, respecively. 10

11 The oal value of he bank is deermined by a recursive maximizaion problem such ha: (D + V )S B, 1 = Ω (K B, 1, B B, 1, R 1 ) max α K B, 1 + Q (L B, K {KB, o,l B,,B B,, R B, o ) R B B, 1 + B B, } [ βc i, Ω +1 (K B,, B B,, + E R ] ) (1 + ζ)c i,+1 i = i S,, { } s.. 1 = E βc i, min R, (α +1 + Q +1 )K B, (B B, ) 1 c i,+1 i = i B,, (10) K B, = (1 ˆδ )K o B, + (1 δ)(k B, 1 L B, ), L B, [0, K B, 1 ], K o B,, B B, 0, he bank-run condiion (7). The bank akes as given he probabiliy disribuion of {Q s, ˆδ s, α s, βc i,s (c i,s+1 ) 1 i {i S,s, i B,s }} s=. The value funcion, Ω, is ime-dependen since hese exogenous variables for he bank are ime-varying. Noe ha he objecive funcion is he flow-of-fund consrain (6), where V is replaced wih he firs-order condiion for bank equiies held by he agen i S,. The firs consrain of he bank s maximizaion problem (10) is he firs-order condiion for he agen i B, s bank deposis, in which he definiion of he ex-pos ineres rae in Equaion (7) is subsiued. 9 The bank inernalizes he ex-ane deposi ineres rae, R, hrough his consrain. The second consrain is he law of moion of capial for he bank. I is assumed ha bank does no know he depreciaion rae of each infiniesimal uni of capial he bank holds. 10 Thus, he average depreciaion rae of capial sold by he bank 9 In he firs consrain, K B, (B B, ) 1 is replaced wih infiniy if B B, = This assumpion will ensure ha he bank does no sell a low-qualiy fracion of used capial selecively in equilibrium. If he bank also had privae informaion, hen he adverse selecion problem could be worsened by he exisence of he bank since he bank does no have an opporuniy of invesmen in new capial and would sell only a low-qualiy fracion of used capial. Even in his case, he average qualiy of bank asses 11

12 equals δ by he law of large numbers. On he hird line are a consrain ha he bank canno sell used capial more han i owns and a non-negaiviy consrain on bank s choice variables. The las line is he bank-run condiion described above. If here is no equiy holder for he bank (i.e., S B, 1 = 0), hen he bank maximizes he profi from iniial public offering of is equiies and consumes he profi righ away. The profi equals he value of Ω. Thus he maximizaion problem (10) covers his case Definiion of an equilibrium For exposiion purpose, his paper firs shows he endogenous deerminaion of illiquidiy of capial analyically by using he model wihou he bank and hen inroduce he bank o he model o show ha illiquidiy of capial leads o liquidiy ransformaion by he bank. Each exercise needs o define an equilibrium. Call a se of endogenous variables, {c i,s, x i,s, k i,s, ki,s o, l i,s, b i,s, s i,s, K B,s, KB,s o, L B,s, B B,s, S B,s, Q s, ˆδ s, D s, V s, R s i I} s=0, coningen on he realizaion of {α s, s, φ i,s i I} s=0 as a coningen plan. Given he se of parameer values, { δ, φ, β, ζ, ρ P, ρ U, ᾱ, α, η α, η α,,, η, η }, and he iniial condiion on {k i, 1, b i, 1, s i, 1, K B, 1, B B, 1, S B, 1, R 1 i I}, an equilibrium for he model wih a banking secor is defined as a coningen plan characerized by: he maximizaion problems (5) and (10) are solved; agens and he bank hold raional expecaions; he average depreciaion rae of used capial sold in he marke is deermined by ˆδ = I δ+ I δ δ l i,δ, dδ µ(di) + δ L B, δ+ ; (11) δ l i,δ, dδ µ(di) + L B, and he markes for used capial, bank deposis and bank equiies clear every period, such would be public informaion, given ha agens have raional expecaions of bank behaviour. 11 I can be shown ha he bank s profi from iniial public offering becomes zero in equilibrium. 12

13 ha I δ+ δ+ ki,δ, o dδ µ(di) + KB, o = l i,δ, dδ µ(di) + L B,, (12) δ I δ b i, µ(di) = B B,, (13) I s i, µ(di) = S B,. (14) I An equilibrium for he model wihou a banking secor is a coningen plan characerized by: he maximizaion problem (5) is solved wih b i, = s i, = 0 for all i and ; agens hold raional expecaions; and Equaions (11) and (12) are saisfied wih KB, o = L B, = 0 for all. 3 Aggregae equilibrium condiions 3.1 Shock processes The dynamic analysis of he model wih a banking secor will invesigae business cycles driven by each ype of he wo shocks, α and. Se = when analyzing produciviydriven business cycles, and se ᾱ = α, when analyzing informaion-driven business cycles. Wih hese assumpions, he number of possible saes in he nex period becomes wo every period, which will simplify he bank s problem abou wheher he bank should ake he risk of a bank-run in he nex period, or no. 13

14 3.2 Agen s behaviour Call agens wih φ i, = φ as producive and hose wih φ i, = 0 as unproducive. Suppose ha he following condiions hold: φ > (1 ˆδ )Q 1, (15) [ ] βci, R +1 1 > E c i,+1 φ i, = φ, (16) [ ] βci, (D +1 + V +1 ) (1 + ζ)v > E c i,+1 φ i, = φ, (17) [ ] βci, R +1 1 = E c i,+1 φ i, = 0, (18) [ ] βci, (D +1 + V +1 ) (1 + ζ)v = E c i,+1 φ i, = 0. (19) The lef-hand side of he firs condiion is he produciviy of invesmen in new capial, and he righ-hand side is he quaniy of capial ne depreciaion ha an agen can purchase from he marke wih a uni of goods. This condiion implies ha invesmen in new capial is more profiable han purchasing used capial from he marke. The oher condiions imply ha he rae of reurns on he producive s invesmen ino new capial dominaes he raes of reurns on bank securiies and ha he unproducive are indifferen beween consumpion and holding bank securiies. Wih hese condiions, i is possible o show ha x i, > 0 and ki, o = b i, = s i, = 0 for he producive, ha x i, = 0 for he unproducive, and ha agens i S, and i B, are unproducive. These condiions will be verified in he numerical examples of equilibria considered below. Each agen sells used capial if selling used capial has a higher rae of reurns han 14

15 keeping he used capial unil he nex period. Thus: k i,δ, 1, if Q λ i, (1 δ), l i,δ, = 0, oherwise, (20) where λ i, is he shadow value of capial ne depreciaion a he end of period (i.e., k i, ), which is given by he Lagrange muliplier for he law of moion of capial in he maximizaion problem (5). I can be shown ha, given x i, > 0 for he producive, he producive s shadow value of capial ne depreciaion equals he marginal cos of producing new capial. Thus: λ P, = φ 1, (21) where λ P, denoes he shadow value of capial ne depreciaion, λ i,, for he producive. On he oher hand, he unproducive s shadow value of capial ne depreciaion does no necessarily equal he marginal acquisiion cos of capial ne depreciaion from he marke, Q (1 ˆδ ) 1, since k o i, can be zero if he unproducive choose o sore heir wealh only hrough bank securiies. I can be shown ha: λ U, = Q (1 ˆδ ) 1, if ki, o > 0 for he unproducive, ki, o = 0 for he unproducive, if λ U, < Q (1 ˆδ ) 1, (22) in equilibrium, where λ U, denoes he shadow value of capial ne depreciaion, λ i,, for he unproducive. The second line implies ha he unproducive choose ki, o = 0 if heir shadow value of capial ne depreciaion is lower han he marginal acquisiion cos of capial ne depreciaion. Given Equaion (20), he lower bound for he depreciaion rae of used capial sold by 15

16 each agen is deermined by: { { δ i, = max δ, min δ +, 1 Q }}. (23) λ i, The maximum and he minimum operaors in Equaion (23) ensure ha δ i, is wihin he range of he uniform disribuion of δ. Denoe he values of δ i, for he producive and he unproducive by δ P, and δ U,, respecively. Given Equaions (21)-(23), hese values are defined as: δ P, = max { δ, min { δ }} +, 1 φq, (24) { { δ U, = max δ, min δ +, 1 Q }}. (25) λ U, Apply he envelop heorem o he maximizaion problem (5) o find ha: λ i, = E [ βc i, c i,+1 ( α +1 + λ i,+1 δi,+1 δ +1 1 δ +1 dδ + Q +1 )] δ dδ. (26) δ i,+1 +1 This dynamic opimizaion condiion implies he following decision rule for each agen: c i, = (1 β)w i,, (27) λ i, k i, + b i, + (1 + ζ)v s i, = βw i,, (28) where w i, is he agen s ne-worh defined by ( δi, 1 δ w i, α + λ i, δ dδ + Q ) δ+ 1 dδ k i, 1 + R b i, 1 + (D + V )s i, 1. δ i, (29) In he definiion of ne-worh, he fracion of used capial sold by he agen is evaluaed by 16

17 he marke price of used capial, Q, while he fracion of used capial kep by he agen is evaluaed by he shadow value of capial ne depreciaion, λ i,, for he agen. In Equaion (28), capial ne depreciaion a he end of he period, k i,, is also evaluaed by he shadow value of capial ne depreciaion for he agen. Aggregaion of Equaion (28) for each ype of agen leads o he following aggregae decision rules: K P, φ = β {[ α + 1 φ δp, δ 1 δ dδ + Q ] δ+ 1 dδ [ρ P K P, 1 + (1 ρ U )K U, 1 ] δ P, +(1 ρ U )[R B U, 1 + (D + V )S U, 1 ]}, (30) {[ δu, 1 δ λ U, K U, + B U, + (1 + ζ)v S U, = β α + λ U, δ dδ + Q ] δ+ 1 dδ δ U, [(1 ρ P )K P, 1 + ρ U K U, 1 ] + ρ U [R B U, 1 + (D + V )S U, 1 ]}, (31) where K P, = {i φ i, =φ} k i, µ(di), K U, = {i φ i, =0} k i, µ(di), B U, = {i φ i, =0} b i, µ(di), and S U, = {i φ i, =0} s i, µ(di). Also, he law of moion of capial in he maximizaion problem (5) implies ha: φx P, = K P, (1 ˆδ )K o U, = K U, δp, 1 δ dδ [ρ P K P, 1 + (1 ρ U )K U, 1 ], (32) δ δu, δ 1 δ dδ [(1 ρ P )K P, 1 + ρ U K U, 1 ], (33) where X P, = {i φ i, =φ} x i, µ(di) and K o U, = {i φ i, =0} ko i, µ(di). 17

18 3.3 Bank s behaviour The represenaive bank in he compeiive banking secor solves he maximizaion problem (10), given ha he number of possible saes in he nex period is wo every period as assumed above. Denoe he lower value of α +1 +Q +1 by ω +1 and he higher value by ω +1. The soluion o he maximizaion problem (10) implies he following proposiion. Proposiion 3.1. Prob( ω +1 ) denoes he condiional probabiliy ha α +1 + Q +1 = ω +1, given he values of period- variables. In equilibrium wih Inequaliies (15)-(17), he oal value of he bank for equiy holders is given by: Ω (K B, 1, B B, 1, R 1 ) [ α + λ B, (1 δ) ] K B, 1 R 1 B B, 1, if R 1 B B, 1 (α + Q )K B, 1, = 0, if R 1 B B, 1 > (α + Q )K B, 1, (34) where λ B, = max{λ B,, λ B, }, (35) βc i, [α +1 + Q +1 (1 δ) ] ω λ B, = E 1 ˆδ [ ] (1 + ζ)c i,+1 φ i, = 0 + E βci, ω +1 c i,+1 φ i, = 0, (36) βc i, [α +1 + Q +1 (1 δ) ] ω λ B, = Prob( ω 1 ˆδ )E (1 + ζ)c i,+1 φ i, = 0, α +1 + Q +1 = ω +1 [ ] βci, (α +1 + Q +1 ) + E c i,+1 φ i, = 0. (37) 18

19 Also: R B B, = ω +1 K B,, if λ B, > λ B,, (38) R B B, = ω +1 K B,, if λ B, < λ B,. (39) Proof: See Appendix A. Equaion (34) implies ha he oal value of he bank for equiy holders is deermined by he shadow value of capial ne depreciaion for he bank, which is denoed by λ B,. Equaions (35)-(39) indicae ha, o maximize he shadow value of capial ne depreciaion, he bank compares he payoffs from he wo levels of bank deposis, R B B, = ω +1 K B, and R B B, = ω +1 K B,, in equilibrium. The bank focuses on hese wo opions, since i mus pay a higher rae of reurns on bank equiies han bank deposis due o he bank-equiy holding cos for agens, ζ, and prefers o finance is asses hrough bank deposis as much as possible. The bank chooses R B B, = ω +1 K B,, if increasing B B, above his level would reduce he price of bank equiies oo much by making a bank run possible in he nex period. The value of λ B, equals he larger value beween λ B, and λ B,, which are he shadow values of capial ne depreciaion when R B B, = ω +1 K B, and when R B B, = ω +1 K B,, respecively. I is possible o show ha λ B, = Q (1 ˆδ ) 1 if K o B, > 0 and ha L B, = 0 if ˆδ > δ and K o B, > 0 in equilibrium. See Appendix A for he proof of hese resuls. The firs resul implies ha he shadow value of capial ne depreciaion for he bank mus equal he marginal acquisiion cos of capial ne depreciaion if he bank buys used capial. The second resul implies ha i is no profiable for he bank o sell he bank s own used capial wihou knowing he rue qualiy of each uni of capial when he bank buys low-qualiy used capial from he marke. If L B, = 0, hen he law of moion of capial for he bank 19

20 becomes: K B, = (1 ˆδ )K o B, + (1 δ)k B, 1. (40) 3.4 Aggregae equilibrium condiions Hereafer, suppose ha ˆδ > δ, λ B, > λ B,, and λ B, = Q (1 ˆδ ) 1, so ha L B, = 0, KB, o > 0 and R B B, = ω +1 K B, in equilibrium. Thus, he bank conducs liquidiy ransformaion and prevens a bank run by conrolling he supply of bank deposis. These condiions will be verified in equilibria considered below. Given L B, = 0, Equaion (11) implies ha he average depreciaion rae of used capial sold in he marke, ˆδ, is deermined by: ˆδ = δ+ δ P, δ dδ [ρ P K P, 1 + (1 ρ U )K U, 1 ] + δ+ δ U, δ dδ [(1 ρ P )K P, 1 + ρ U K U, 1 ] δ+ δ P, 1 dδ [ρ P K P, 1 + (1 ρ U )K U, 1 ] + δ+ δ U, 1 dδ [(1 ρ P )K P, 1 + ρ U K U, 1 ]. (41) Also, Equaions (32)-(33) and (40) (he laws of moion of capial for he producive, he unproducive, and he bank, in order) and Equaion (12) (he marke clearing condiion for used capial) imply ha he aggregae law of moion of capial for he economy is: K P, + K U, + K B, = φx P, + (1 δ)(k P, 1 + K U, 1 + K B, 1 ). (42) Given Inequaliies (15)-(17), λ B, > λ B, and ˆδ > δ, he equilibrium dynamics of {K P,, X P,, K U,, K o U,, B U,, λ U,, (D + V )S U, 1, V S U,, K B,, K o B,, Q, R, R, ˆδ, δ P,, δ U, } is 20

21 sequenially deermined by Equaions (18)-(19), (24)-(25), (30)-(33), and (40)-(42), and: Q 1 ˆδ = E [ α + Q (1 δ) ] 1 ˆδ K B, 1 R 1 B U, 1, (43) βc i, [α +1 + Q +1 (1 δ) ] ω 1 ˆδ βc i, ω +1 (1 + ζ)c i,+1 c i,+1 φ i, = 0, (44) (D + V )S U, 1 = R B U, = ω +1 K B,, (45) R = R 1. (46) Equaions (43), (44), (45), and (46) are derived from: Equaion (34) and λ B, = Q (1 ˆδ ) 1 ; Equaions (35)-(36) and λ B, = Q (1 ˆδ ) 1 ; Equaion (38); and Equaion (7), in order. The marke clearing condiions for bank equiies and bank deposis, S B, = S U, and B B, = B U,, respecively, are subsiued in Equaions (43) and (45). 4 Endogenous illiquidiy of asses This secion shows he closed form for he equilibrium dynamics of he model wihou a banking secor o show endogenous deerminaion of illiquidiy of used capial analyically. This is for exposiion purpose, since he closed form for he dynamics of he model wih a banking secor canno be obained. The inuiion behind he resuls of he model wihou a banking secor is shared wih he model wih a banking secor. In his secion, x i, = 0 and k o i, > 0 for he unproducive, since here is no supply of bank securiies and he unproducive can only sore heir wealh hrough buying used capial. Given Inequaliy (15), x i, > 0 and k o i, = 0 for he producive. Aggregae equilibrium condiions are idenical wih he model wih a banking secor excep ha K B, = B U, = S U, = 0 for all. Given k o i, > 0 for he unproducive, Equaion 21

22 (22) implies ha λ U, = Q (1 ˆδ ) 1. Then, Equaions (24)-(25) lead o: δ P, = max { δ, 1 φq } < δu, = ˆδ, (47) given Inequaliy (15). 12 Given he values of α,, K P, 1 and K U, 1 and ha K B, = B U, = S U, = 0 for all, Equaions (31), (32) and (42) imply ha he equilibrium values of ˆδ and Q in each period are deermined by Equaion (41) and: g(ˆδ, Q, α,, θ K, 1 ) [ Q ] δp, 1 ˆδ (1 δ)(1 1 δ + θ K, 1 ) θ K, 1 dδ δ ( β α + Q δu, ) 1 δ δ+ 1 1 ˆδ dδ + Q dδ = 0, (48) δ δ U, where θ K, 1 ρ PK P, 1 + (1 ρ U )K U, 1 (1 ρ P )K P, 1 + ρ U K U, 1, (49) and he values of δ P, and δ U, are as shown in Equaion (47). Then, he values of K P, and K U, are deermined by Equaions (30)-(31), given λ U, = Q (1 ˆδ ) 1 and B U, = S U, = 0 for all. The firs erm of he funcion g is he shadow value of aggregae capial ne depreciaion ha he unproducive mus hold a he end of he period in equilibrium and he second erm is he fracion of he unproducive s aggregae ne-worh ha is spen on used capial. Boh erms are normalized by he aggregae ne-worh of he unproducive. 12 Noe ha δ P, < ˆδ δ +. 22

23 4.1 Illiquidiy of capial due o adverse selecion The resul ha δ U, = ˆδ indicaes ha he qualiy of used capial sold by he unproducive is always worse han he average qualiy of used capial sold in he marke. This is adverse selecion. In equilibrium, he adverse selecion raises he average depreciaion rae of used capial sold in he marke, ˆδ, which leads o ˆδ > δ. This resul can be confirmed by subsiuing δ U, = ˆδ in Equaion (41). The resul ha ˆδ > δ implies ha each agen s used capial as a whole is undervalued in he marke, which can be shown by subsiuing λ U, = Q (1 ˆδ ) 1 ino Equaion (26): Q = (1 ˆδ )E [ βc i, c i,+1 ( α +1 + λ i,+1 δi,+1 δ +1 1 δ +1 dδ + Q +1 ) ] δ dδ δ i,+1 +1 φ i, = 0. (50) This equaion shows ha he marke price of used capial, Q, depends on ˆδ. The rue average value of each agen s used capial is obained by replacing ˆδ wih δ on he righhand side of he equaion, given λ i,+1 and Q +1 in he nex period. Thus, he resul ha ˆδ > δ implies ha he marke value of used capial is lower han he rue average value of used capial held by each agen. In his paper, define illiquidiy of an asse as undervaluaion of he asse in he marke. The degree of illiquidiy of each agen s used capial as a whole is measured by he difference beween ˆδ and δ. Hereafer, ake ˆδ as he indicaor of illiquidiy of used capial, since δ is fixed. 4.2 Response of illiquidiy of capial o produciviy shocks Secions 4.2 and 4.3 will show ha boh a negaive produciviy shock and an increase in he degree of asymmeric informaion can cause an increase in illiquidiy of asses (used capial) 23

24 represened by ˆδ and a decline in he marke price of asses, Q, as occurred during he financial crisis since Equaions (41) and (48) have he following characerisics: Lemma 4.1. Equaion (48) is downward-sloping curves on he (Q, ˆδ ) plane, given he values of α, and θ K, 1. Equaion (41) is also downward-sloping, if δ P, = 1 φq, and is a fla line, if δ P, = δ. Equaion (48) has a seeper slope han Equaion (41) a he inersecion of he wo curves, if β is sufficienly close o 1. Proof: See Appendix B. Figure 1 draws Equaions (41) and (48) on he (Q, ˆδ ) plane and shows how a decline in α makes hem shif. 13 I is obvious ha Equaion (41) does no shif. Equaion (48) shifs inward, since a decline in α reduces he aggregae income of he unproducive, which lowers Q hrough a decreased aggregae spending on used capial. If δ P, = 1 φq in he new equilibrium, hen he equilibrium shifs along a downward-sloping par of Equaions (41), as shown in he figure. In his case, a decline in he marke price of used capial, Q, due o a negaive produciviy shock discourages he producive from selling high-qualiy used capial in he marke (i.e., a rise in δ P, ), which increases he average depreciaion rae of used capial sold in he marke, ˆδ. 4.3 Response of illiquidiy of capial o shocks o he degree of asymmeric informaion Figure 2 shows he effec of a rise in, which increases he degree of asymmeric informaion. An increase in makes Equaion (41) shif upward unambiguously, since an expanded range of depreciaion raes of used capial les each agen sell he increased low-qualiy fracion of used capial while keeping he increased high-qualiy fracion of used capial, which raises ˆδ hrough worsened adverse selecion. 13 Due o he log uiliy funcion, Equaions (41) and (48) are valid irrespecive of he sochasic process of he shocks in he model. 24

25 On he oher hand, he following lemma implies ha he direcion of he shif in Equaion (48) is ambiguous: Lemma 4.2. Proof: See Appendix C. g > 0, if δ P, is sufficienly close o ˆδ. g < 0, if δ 14 P, δ. Since g Q > 0, Equaion (48) shifs inward in he firs case ( g > 0) and ouward in he second case ( g < 0), in response o an increase in. 15 The op panel of Figure 2 shows he firs case and he boom panel shows he second case. In he firs case, ˆδ increases and Q decreases. Since Equaion (41) implies ha ˆδ is close o δ + if δ P, is close o ˆδ, his resul indicaes ha an increase in he degree of asymmeric informaion causes a simulaneous deerioraion of illiquidiy of used capial and he marke price of used capial if adverse selecion in he asse marke is so severe ha he volume of rade in he marke is small. 5 Liquidiy ransformaion and bank capial requiremens 5.1 Liquidiy ransformaion by he bank This secion analyzes he feaures of he model wih a banking secor. The aggregae decision rules specified by Equaions (30)-(31) are useful o explain why agens hold bank securiies. 14 On he balanced growh pah, δ P, is close o ˆδ if ρ P and 1 ρ U are high, and δ P, δ if ρ P and 1 ρ U are low. See Appendix D for more deails. 15 The inuiion for his resul is ha, when g > 0, a rise in reduces he fracion of used capial kep by he producive, δ P, δ (1 δ)/( ) dδ, in he firs erm of he funcion g in Equaion (48). As a consequence, he unproducive mus absorb a larger amoun of used capial, which reduces he price of used capial, Q, given he value of ˆδ. When g < 0, a rise in increases he fracion of used capial kep by he producive. Less supply of used capial leads o an increase in Q, given he value of ˆδ. Also, in boh cases, a rise in increases he second erm of he funcion g in Equaion (48) (i.e., he aggregae ne-worh of he unproducive), since he unproducive benefi from more opporuniies for adverse selecion. This effec would increase Q. When g > 0, his effec is dominaed by he effec of an increased supply of used capial by he producive. 25

26 By subsiuing Equaions (43) and (46), Equaions (30)-(31) can be rewrien as: K P, φ = β {[ α + 1 φ δp, δ 1 δ dδ + Q ] δ+ 1 dδ [ρ P K P, 1 + (1 ρ U )K U, 1 ] δ P, [ +(1 ρ U ) α + Q (1 δ) } ]K 1 ˆδ B, 1, (51) λ U, K U, + B U, + (1 + ζ)v S U, = β {[ δ+ δ U, 1 δu, 1 δ α + λ U, dδ + Q δ [ [(1 ρ P )K P, 1 + ρ U K U, 1 ] + ρ U α + Q (1 δ) 1 ˆδ ] dδ } ]K B, 1. (52) Noe ha K U, 1 is replaced wih K B, 1 as he unproducive shif heir porfolio from used capial o bank securiies. Thus, comparing he coefficiens o K U, 1 and K B, 1 clarifies he benefi of holding bank securiies. The following proposiion holds. Proposiion 5.1. If Inequaliy (15) holds and ˆδ > δ, hen: Q (1 δ) 1 ˆδ > 1 δp, 1 δ φ δ dδ + Q δ+ δ P, 1 dδ. (53) If ˆδ > δ and λ U, = Q (1 ˆδ ) 1, hen: Q (1 δ) δu, 1 δ 1 ˆδ < λ U, δ dδ + Q δ+ δ U, 1 dδ. (54) Proof: See Appendix E. Inequaliy (53) implies ha he value of he producive s ne-worh increases as K U, 1 is replaced wih K B, 1. Thus, agens can increase invesmens in new capial by soring wealh hrough bank securiies when hey are unproducive and selling hem when hey are producive. Noe ha, when agens sell bank securiies, hey ransfer a share of he whole 26

27 used capial he bank holds. Since idiosyncraic depreciaion raes of he bank s whole used capial cancel each oher ou, he value of he bank s used capial ha backs bank securiies becomes public informaion, which makes bank securiies free from adverse selecion. Hence, as indicaed by Equaions (43), he value of bank securiies reflecs he shadow value of bank s whole used capial, i.e., [α + Q (1 ˆδ ) 1 (1 δ)]k B, 1, insead of he liquidaion value of he used capial, (α +Q )K B, 1. Sellers of bank securiies can obain a fair amoun of goods for bank securiies. On he oher hand, Inequaliy (54) indicaes ha here is a case where holding bank securiies is ex-pos cosly for he unproducive if hey remain unproducive in he nex period, since hey lose he opporuniy o sell a low-qualiy fracion of used capial a an overvalued marke price. Overall, he unproducive hold bank securiies if he expeced benefi of holding liquid bank securiies for increasing invesmen ino new capial dominaes he expeced cos of losing he opporuniy o sell low-qualiy used capial a an overvalued marke price Comparaive saics analysis of inroducion of he bank o he economy Figure 3 compares balanced growh pahs wih and wihou he bank. The figure is a numerical example of comparaive saics around a se of benchmark parameer values ha approximaely replicaes he pos-war sample average of US daa on he balanced growh pah wih he bank. 17 In he figure, he ime index,, is omied from he noaion of each 16 If he probabiliy for he unproducive o be producive in he nex period, 1 ρ U, is sufficienly low, hen he unproducive do no hold bank securiies and financial inermediaion does no arise in equilibrium. 17 The benchmark parameer values are ( δ, φ, β, ζ, ρ P, ρ U ) = (0.1, 4.75, 0.99, 0.02, 0.45, 0.55), ᾱ = α = 0.03, and = = Suppose he lengh of a period in he model is a year. For in U.S., he average real GDP growh rae is 3.4%, he average real ineres rae on 3-monh reasury bills is 3.9%, and he average raio of he bank credi of commercial banks o he fixed asses in he economy is 15.0%. These numbers are approximaely replicaed by he growh rae of aggregae oupu (G 1), R 1, and K B /(K P +K U + K B ), in order. The capial-asse raio of he bank is around 8% on he balanced growh pah in he model, which is he minimum requiremen by he Basel agreemen. The 10% annual depreciaion rae of capial implied by δ is a sandard assumpion. Rouwenhous (1995) repors ha he equiy premium on S&P 500 was 1.99% on average for The equiy premium on bank equiies in he model akes a similar value. The 27

28 variable. The deposi ineres rae, R, for he model wihou he bank is a hypoheical rae wih no supply of bank deposis. The figure shows balanced growh pahs under various values of ζ. Figure 3 illusraes ha, given parameer values, inroducion of he bank o he economy increases R. This resul is consisen wih he analyical resul shown in Secion 5.1, ha agens can increase invesmens in new capial by soring heir wealh hrough bank securiies when hey are unproducive and selling hem when hey are producive. Since bank securiies le agens suffer less from illiquidiy of heir asses when hey are producive, he expeced consumpion in he case of becoming producive increases, which leads o a decline in he sochasic discoun facor, βc i, (c i,+1 ) 1, for he unproducive and hus a rise in R. Despie his posiive effec of bank securiies on he producive who used o be unproducive, Figure 3 shows ha he gross rae of growh of aggregae oupu, which is denoed by G Y Y 1, where Y I y i, µ(di), (55) does no necessarily increase wih inroducion of he bank o he economy in he long run. Noe ha inroducion of he bank o he economy leads o a decline in he marke price of used capial, Q, hrough a drop in he sochasic discoun facor for he unproducive. A decline in Q discourages agens from selling high-qualiy capial, which leads o a rise in δ P and δ U hrough Equaion (23). This effec raises ˆδ. A resuling increase in illiquidiy of used capial reduces invesmens in new capial by he producive who coninue o be producive from he previous period, since hese agens do no hold bank securiies and have o suffer from worsened undervaluaion of used capial hey hold. The figure shows ha his negaive daa sources for he firs hree sample averages are NIPA daa from he BEA and financial daa from he Federal Reserve Board. Noe ha ρ P = 1 ρ U, which implies ha he arrival of he opporuniy o produce new capial is i.i.d. for each agen. This assumpion is se o reduce he dimension of he parameer space. 28

29 effec on he producive who coninue o be producive from he previous period dominaes he posiive effec of bank securiies on he producive who used o be unproducive, if he marginal bank-equiy holding cos, ζ, is sufficienly high. This is because he bank-equiy holding cos is incurred only by he unproducive and his cos reduces he ne posiive effec of bank securiies on he producive who used o be unproducive. 5.3 Bank capial requiremens: dynamic analysis Equaions (19) and (43)-(45) imply ha he capial-asse raio of he bank is given by: V S B, B B, + V S B, = = 1 ˆδ E Q E { [ βci, (1+ζ)c i,+1 ] } α +1 + Q +1(1 δ) ω 1 ˆδ φi, = 0 K B, Q (1 ˆδ ) 1 K B, { [ βc i, Q +1 (ˆδ +1 δ) ] } (1 + ζ)c i,+1 1 ˆδ + (α +1 + Q +1 ω +1 ) +1 φ i, = 0. (56) Noe ha he denominaor of he raio, he value of bank asses, equals he oal value of liabiliies, B B, + V S B,, in he balance shee of he bank. The firs line of Equaion (56) shows ha he capial-asse raio of he bank depends on he presen discouned value of he difference beween he shadow value of he bank s used capial, [α +1 + Q +1 (1 ˆδ +1 ) 1 (1 δ)]k B,, and he borrowing limi on bank deposis, i.e., he wors possible liquidaion value of he bank s used capial in he nex period, ω +1 K B,. The presen discouned value of his difference mus be financed hrough public equiies. The second line of Equaion (56) implies ha he difference is posiive and can be decomposed ino wo facors. Firs, illiquidiy of capial causes a gap beween he shadow value and he realized liquidaion value of he bank s used capial, which appears as Q +1 (ˆδ +1 δ)(1 ˆδ +1 ) 1 in Equaion (56). Noe ha his is posiive by ˆδ +1 > δ. Second, since he liquidaion value of he bank s used capial flucuaes, he realized liquidaion value of he bank s used capial can be more han he wors possible liquidaion value. The difference 29

30 beween he wo appears as α +1 + Q +1 ω +1 in Equaion (56). The presen discouned value of he difference is posiive by he definiion of ω +1. Call he firs facor as illiquidiy facor and he second facor as marke-price facor. To illusrae he effecs of hese facors on equilibrium dynamics of he capial-asse raio of he bank, Figures 4 and 5 show sample pahs of he dynamics of he model driven by changes in α and, respecively. See Appendix F for he numerical soluion mehod. The sochasic process of α is se so ha he growh rae of oupu is around 4% in booms and around 2% in recessions, on average. The sochasic process of is se so ha flucuaes symmerically around he benchmark value specified in Secion 5.2 and is upper value ( ) akes he maximum value ha makes he lower bound of he range of depreciaion raes equal o zero. For boh processes, he ransiion probabiliies of he shocks are se so ha he expeced duraions of booms and recessions are 4 years, given ha he lengh of a period in he model is inerpreed as a year. 18 The parameers excep for he shock parameer in each figure ake he benchmark values specified in Secion 5.2. Each figure shows he sample pah when he shock parameer keeps changing is value every 4 periods for a sufficienly long ime. Figure 4 indicaes ha he capial-asse raio of he bank is pro-cyclical when business cycles are driven by produciviy shocks. This resul is due o he marke-price facor. When a posiive produciviy shock his he economy, he expeced income from used capial increases since he shock is persisen. This effec raises he expeced value of Q +1 hrough Equaion (44) while posiive produciviy shocks hi he economy. This effec in urn increases he gap beween he expeced realized liquidaion value of bank s used capial and is wors possible value in he nex period. Thus, he capial-asse raio of he bank rises during booms. I falls during recessions by he same mechanism ha works in he opposie 18 In Figure 4, ᾱ = , α = and = = In Figure 5, = 0.1, = 0.08 and ᾱ = α = For boh figures, η x = η x = 0.75 for x = α,. 30

31 direcion. Hence, he capial-asse raio of he bank becomes pro-cyclical. Noe ha he indicaor of illiquidiy of used capial, ˆδ, is couner-cyclical in Figure 4. This is because he pro-cyclical flucuaions in Q induce he producive o sell highqualiy capial (i.e., a decline in δ P, ) during booms and o keep i (i.e., a rise in δ P, ) during recessions. Even hough his illiquidiy facor drives he capial-asse raio of he bank o be couner-cyclical, he marke-price facor dominaes he illiquidiy facor in he numerical example shown in he figure. In conras, Figure 5 indicaes ha he capial-asse raio of he bank is couner-cyclical when business cycles are driven by changes in he degree of asymmeric informaion (changes in ). In his case, a decline in reduces adverse selecion in he marke for used capial, which lowers ˆδ and hus illiquidiy of used capial. As a consequence, he capial-asse raio of he bank drops. A he same ime, a decline in illiquidiy of used capial faciliaes he ransfer of goods from he unproducive o he producive in exchange for used capial, raising he growh rae of oupu. By a similar mechanism, an increase in causes an increase in he capial-asse raio of he bank and a decline in he growh rae of oupu. Hence, he capial-asse raio of he bank becomes couner-cyclical. While he counercyclical movemens of ˆδ cause pro-cyclical movemens of Q hrough Equaion (44), which drives he capial-asse raio of banks o be pro-cyclical, he illiquidiy facor dominaes he marke-price facor in he numerical example shown in he figure. 6 Conclusion This paper presens a dynamic compeiive equilibrium model in which illiquidiy of asses arises endogenously due o asymmeric informaion abou asse qualiy. I is shown ha boh a negaive produciviy shock and an increase in he degree of asymmeric informaion can cause a simulaneous deerioraion of illiquidiy of asses and he marke price of asses, 31

32 as occurred during he financial crisis since The model also shows ha illiquidiy of asses leads o liquidiy ransformaion by banks and ha banks mus mainain posiive bank capial o preven a self-fulfilling bank run due o illiquidiy of bank asses. The dynamic analysis of he model indicaes ha, o preven a bank run, he capial-asse raio of banks should be linked o illiquidiy of bank asses and he volailiy of he marke price of bank asses. The numerical examples sugges ha he equilibrium capial-asse raio of banks is pro-cyclical during regular business cycles driven by produciviy shocks and ha i is couner-cyclical when business cycles are driven by changes in he degree of asymmeric informaion. While he equilibrium capial-asse raio of banks in he model is marke discipline imposed by raional agens, i can be seen as a benchmark for dynamic bank-capial regulaion, since one of he purposes of he regulaion is o achieve financial sabiliy by prevening bank runs. Formal analysis of opimal dynamic bank-capial regulaion, including he opimal balance beween marke discipline and regulaion, is lef for fuure research. 32