Welfare Analysis of Policy Measures for Financial Stability

Transcription

1 Bank of Japan Working Paper Series Welfare Analysis of Policy Measures for Financial Sabiliy Ko Munakaa * kou.munakaa@boj.or.jp Koji Nakamura ** kouji.nakamura@boj.or.jp Yuki Teranishi ** yuuki.eranishi@boj.or.jp No.3-E- March 03 Bank of Japan -- Nihonbashi-Hongokucho, huo-ku, Tokyo 03-00, Japan * Oia Branch ** Financial Sysem and Bank Examinaion Deparmen Papers in he Bank of Japan Working Paper Series are circulaed in order o simulae discussion and commens. Views expressed are hose of auhors and do no necessarily reflec hose of he Bank. If you have any commen or quesion on he working paper series, please conac each auhor. When making a copy or reproducion of he conen for commercial purposes, please conac he Public Relaions Deparmen (pos.prd8@boj.or.jp) a he Bank in advance o reques permission. When making a copy or reproducion, he source, Bank of Japan Working Paper Series, should explicily be credied.

2 Welfare Analysis of Policy Measures for Financial Sabiliy Ko Munakaa y, Koji Nakamura z, and Yuki Teranishi x Absrac We inroduce he nancial marke fricion hrough he search and maching in he loan marke ino a dynamic sochasic general equilibrium (DSGE) model. We reveal ha he second order approximaion of social welfare includes he erms relaing credi, such as credi marke ighness, he volume of credi, and a loan separaion rae, in addiion o he in aion rae and he oupu gap under he nancial marke fricion. Our analyical resul jusi es he reason why he opimal policy should ake he credi variaion ino accoun. We inroduce a moneary policy and oher policy measures for he nancial sabiliy ino he model. The opimal oucome is achieved hrough he moneary and oher policy measures by aking ino accoun no only price sabiliy bu also nancial sabiliy. JEL lassi caion: Keywords: E44; E5; E6 Opimal macroprudenial policy; opimal moneary policy; nancial marke fricion We hank Kosuke Aoki for valuable suggesions and commens. Views expressed in his paper are hose of he auhors and do no necessarily re ec he o cial views of he Bank of Japan. y Oia Branch, Bank of Japan. kou.munakaa@boj.or.jp z Financial Sysem and Bank Examinaion Deparmen, Bank of Japan. kouji.nakamura@boj.or.jp x Financial Sysem and Bank Examinaion Deparmen, Bank of Japan. yuuki.eranishi@boj.or.jp

3 Inroducion Financial crises reveal criical roles of nancial and credi markes by inducing economic disrupions. Under such a siuaion, accompanied wih a lack of room of he moneary policy, policy makers shed ligh on he policy measures including macroprudenial policy for nancial sabiliy as a new fronier of he policy. Some empirical and heoreical sudies clarify he need of policy measures for nancial sabiliy. Borio (0) empirically shows a di erence beween nancial cycles and business cycles and hen jusi es necessiy of coexisence of he moneary policy and he macroprudenial policy. Theoreical sudies for opimal policy under models wih nancial fricions have sared o be conduced. For example, Quin and Rabanal (0) sudy he opimal combinaion of he moneary and macroprudenial policies in he DSGE model. They show ha he macroprudenial policies become quaniaively imporan if he credi erm is in he policy maker s objecive funcion. Suh (0) shows ha he macroprudenial policy should respond o credi o improve welfare apar from he moneary policy responses o he oupu gap and he in aion rae. Kannan, Rabanal, and Sco (0) show ha he moneary policy reacion o credi growh can help macroeconomic sabilizaion. These sudies, however, do no provide he answers for why couning he credi ino he policy can improve welfare. In his paper, we inroduce he nancial marke fricion hrough he search and maching in he loan marke ino a DSGE model, following Wasmer and Weil (000) and Den Haan, Ramey, and Wason (003). 34 We hen reveal ha his policy objecive funcion Drehmann, Borio, and Tsasaronis (0) also show such empirical resuls. Blanch ower and Oswald (998) and Peerson and Rajan (00) empirically show ha search and maching fricions play an imporan role in he loan marke. 3 This ype of search and maching fricion is also assumed in he labor marke as in Morensen and Pissarides (994) and Rogerson, Shimer, and Wrigh (005). Ravenna and Walsh (0) derive he quadraic welfare funcion in he model wih he search and maching fricion in he labor marke. 4 Di eren ypes of nancial fricions are assumed in oher former sudies. Bernanke, Gerler, and Gilchris (999, henceforh BGG), which is he rs o sress ha credi marke imperfecions have a significan in uence on business cycle dynamics. In he BGG model, his nancial marke wedge is deermined by ime-varying leverage, in ha endogenous mechanisms in credi markes work o amplify and propagae shocks o he economy. In Bianchi (00), he nancial marke fricion is inroduced by exernaliies in which privae agens undervalue he dynamics of ne worh since he agens fail o inernalize spillover e ec among hem.

4 includes a erm closely relaed o he volume of credi. In addiion, we inroduce a moneary policy and policy measures for he nancial marke fricion ino he model. The policy measures for he nancial marke fricion aim o achieve nancial sabiliy. We show ha opimal policy measures change according o he ype of policies. When he degree of compeiion beween a rm and a bank in a loan ineres rae seing hrough Nash bargaining is endogenized and he policy measure, which is inerpreed as a regulaion policy, inervenes in such a bargaining, accompanied wih he opimal moneary policy, he opimal policy measure perfecly sabilizes he in aion rae. This is because he policy measure for nancial sabiliy leads o price sabiliy hrough he cos channel. Anoher policy measure ha is inerpreed as a macroprudenial policy inervenes in a endogenous credi separaion. In his case, a welfare funcion addiionally includes erms for credi separaion. Thus, his opimal macroprudenial policy adjuss he separaion rae by aking ino accoun rade-o among he separaion rae, credi marke ighness, he in aion rae, and consumpion. In any case, he policy measures for nancial sabiliy have a close relaion wih price sabiliy and consequenly wih he moneary policy. The res of he paper is organized as follows. The following secion ses up a model. In Secion 3, we derive he second order approximaion from he consumer s uiliy funcion and commen on is properies. In Secion 4, we derive an opimal moneary policy and show he properies of his opimal moneary policy. In Secion 5, we derive opimal policy measures for nancial sabiliy under various seings and reveal he ineracion beween he opimal policy measures for nancial sabiliy and he opimal moneary policy. In Secion 6, we conclude he paper. The Model. Household A he beginning of ime, a represenaive household wih money M deposis D in is bank accoun, receives as a lump-sum pro from rms and a bank in real erms, and hen eners he goods marke o consume in real erm wih price P. The cash-in-advance (IA) consrain of he household is 3

5 5 M P + D P. The household ges he deposi back wih he nominal ineres rae R D ime, leading o he budge consrain a he end of M + P = M P + (RD )D P + : Assuming ha he uiliy of he household depends only on he consumpion, he IA consrain is always binding if R D >, and he household s problem is expressed as subjec o max ;D E i u( +i ); () We assume he following uiliy funcion. = + RD D D P : u( ) ; where is a coe cien of relaive risk aversion of he household. This opimizaion problem leads o = ; = E + P P + R D ; () where is he Lagrangian muliplier in he household s opimizaion problem. I is assumed ha he consumpion consiss of goods labeled by j [0; ]. The consumpion of each good c (j) is relaed o he oal consumpion by Z = 0 " c (j) " " " dj where " (0; ) is an exogenous sochasic variable relaed o he elasiciy of subsiuion. The households choose he level of consumpion of each good c (j) in order o minimize 4

6 he cos R 0 p (j)c (j)dj, given he level of oal consumpion and he price of each good p (j). This minimizaion yields where p (j) " c (j) =, P Z P p (j) 0 " " dj. (3). Wholesale Firms Each rm seeks for a credi line in he credi marke a ime if he rm is a credi seeker. The number of credi seekers is u, he oal number of credi lines is L, and he amoun of each credi line in real erms is a. If a rm has a credi conrac a ime bu fails o receive an updae on he credi line wih he probabiliy a ime, he rm becomes a credi seeker. On he oher hand, if a rm does no have a credi conrac a ime, i auomaically becomes a credi seeker a ime. Under his seing, he number of credi seekers a ime is obained by u = ( )L : (4) redi seeking rms paricipae in he bargaining of credi lines wih a represenaive bank. As a resul of he maching process, L (he number of credi lines a ime ) is relaed o L by L = ( )L + p F u ; where p F is he probabiliy of geing a credi line for each credi seeking rm. When a rm has a credi line, he rm uses he amoun of credi a o buy reail goods a(j), which are hen used as an inpu or a swea cos o produce wholesale goods wih produciviy Z. A rm s gain when i has a credi line is hen f = Z ar L + E + ( )f + + (p F +f + + ( p F +)f 0 +) ; where R L is he real ineres rae for he credi, P P w is he price markup by reail rms, and P w is he price of a wholesale good. The rs wo erms show he ne pro 5

7 from producion a ime and he hird erm is he discouned presen value of he fuure pro. On he oher hand, a rm s gain when i fails o receive a credi line is f 0 + = E p F + f+ + ( p F +)f 0 + : Since he rm canno obain a credi line, i does no produce a ime and i only has he discouned fuure values. These equaions imply ha he gain from a credi line for a credi seeking rm is.3 Bank f f f 0 = Z ar L + E + ( )( p F +)f + : (5) A represenaive bank decides on he number of credi vacancies V o pos in order o maximize is pro B wih respec o L subjec o B = (R L )al (R D )D P V + E + B + ; L = ( )L + q B V ; where is a cos for posing each credi vacancy and q B is he vacancy lling rae. In realiy, includes he cos of collecing borrowers informaion and examining he loan applicaions. This maximizaion problem yields q B = (R L + )a + E ( ) q+ B : (6) The bank s expeced gain from one credi line should be, wih free enry, he same as he cos of a vacancy pos. This condiion implies ha he bank s gain for a credi line is J = q B : (7) We assume ha he cos of posing credi vacancies V is again paid by consumpion of reail goods v (j), in a similar way as he household consumpion and he rms inpu for producion a. Therefore, he demand for good j and he oal demand are given by y d (j) c (j) + a(j)l + v (j), 6

8 and Y d + al + V, respecively, where y d P (j) " (j) = Y d. (8) P.4 Reail Firms A reail rm producing goods j faces he demand given by he equaion (8), and i has o pay P w o buy wholesale goods as an inpu o produce he nal goods. In order o inroduce he price sickiness, i is assumed ha a rm can adjus is price each period wih probabiliy! as in he model by alvo (983). The pro maximizaion problem of a reail rm when i has a chance o adjus is price P max P " +i ( + )P (!) i E P +i P w +i becomes P P +i "+i Y d +i where he demand equaion (8) is used. We here assume ha he subsidy for rms is se o ensure ha he price exibiliy is achieved a he e cien seady-sae equilibrium discussed below. Noe ha he average price level P is given by #, P " = (!) (P ) " +!P "..5 Loan Marke Maching The number of new credi maches is p F u = q B V = u V in a ob-douglas form where is a subsiuion for u and V, and is a consan parameer for maching. To make he calculaion clearer, we use he credi marke ighness so ha = V u ; (9) 7

9 V = u ; p F = ; (0) q B = ; () L = ( )L + u : () In he loan marke, he oal gain from loan exension is he sum of credi seeking rms gain f and he bank s gain J as follows max f b R L J b ; where b is he bargaining power of credi seeking rms. This is an asymmeric Nash bargaining condiion. By aking he rs order condiion wih respec o R L, ( b)f = bj. (3) For convenience, we simplify he equaions (5) and (6) by using he equaions (0), (), (3), and (7) o eliminae p F, q B, f, and J, yielding and b b = Z ar L + + E ( ) b + b + = (R L + )a + E +, ( ) respecively. If we furher eliminae R L from hese equaions, we obain he following condiion ha relaes o he markup o he credi marke ighness : Z = a + b (4) + E ( ) b + b +. 8

10 .6 Marke learing ondiion Since one uni of he wholesale goods is needed as an inpu o produce one uni of each reail goods j, he marke clearing condiion for he wholesale goods is ZL = Z 0 y d (j)dj. Togeher wih he demand equaion for he reail goods (8), he following marke clearing condiion is obained: where ZL Q = + al + V : (5) Q Z P (j) " dj (6) 0 P represens he dispersion of he prices of reail goods due o he price sickiness for reail rms. 3 Welfare rieria 3. E cien Seady-Sae Equilibrium The second order expansion of he household s uiliy funcion is se around he e cien seady-sae equilibrium, which is he equilibrium designed by a social planner wihou credi maching ine ciency or price dispersion. Such a siuaion can only be achieved () when he Hosios condiion b = for he bargaining power b and he maching share is sais ed and () when he subsidy for reail rms is chosen o make sure = " (" )(+) planner is expressed as =. Under hese assumpions, he opimizaion problem for he social max ;L;u; E i f +i + +i [ZL +i al +i +i u +i +i ] + +i ( )L+i + +iu +i L +i + s+i [u +i + ( )L +i ]g; where,, and s are Lagrangian mulipliers for he consrains. 9

11 The soluion o his opimizaion problem yields he following condiion for he e cien seady-sae equilibrium: Z a v L which is simpli ed for convenience as v = ( ) ( ) L L u ; = : (7) Noe ha he bar above each variable (e.g., v) implies he e cien seady-sae value of he variable (v ). 3. Policy Objecive Funcion The second order expansion of he household s uiliy funcion around he e cien seady sae yields u( ) = u() + u c bc + bc + u cc bc = u() + u c bc + bc u cbc ; where bc is he log-deviaion of from he e cien seady-sae value. The goal of he following calculaion in his subsecion is o express his uiliy u( ) by only b l, b l, and he in aion rae bp bp. This calculaion closely follows he derivaion of a policy objecive funcion by Ravenna and Walsh (0) for he search and maching in he labor marke. By using he marke clearing condiion of he equaion (5), bc + bc = ZL bq + (Z a)l b l + b v l bv + bv : (8) Noe ha he e cien seady-sae value of he price dispersion erm Q is Q =, and he log-deviaion of his erm bq is already in he second order, as is shown below. Using he equaion (9) bv = bu + b ; and he equaion (4) bu + bu = b l + b l ; where ( ) L u : 0

12 Then, we obain up o he second order bv + bv = b + b b l + b l b b l : (9) By using his equaion and he equaion (), we can eliminae b and ge where u( ) = u() u c ZLbq L + u c L u c b l + b l + L b l + b l ; b l + b l u : v bl () u b l Thus, he uiliy of he equaion () is rewrien as i u( +i ) = u() u c ZL i bq +i + u c L i b l+i + b l+i + u cl i b l +i + b l +i u c v () u c i b l+i u b l+i L X i b l+i + b l+i : By using he e cien seady-sae condiion given by he equaion (7), he wo erms on he righ-hand side of he equaion above are shown o depend only on b l becomes, and he uiliy i u( +i ) = u c ZL i bq +i (0) u c v () i b l+i u b l+i L u c X i b l +i + b l +i + :i:p:;

13 where :i:p: denoes "erms independen of policy." Nex, we consider he sum over he price dispersion erms bq +i. By he de niion of he equaion (6), bq = Z 0 dj exp [ " (bp (j) bp )] ' "( E bp )( + b" ) + " h V + E bp i, where E E j bp (j) = R 0 bp (j)dj and V V ar j bp (j) = E j bp (j) (E j bp (j)). Since he de niion of he aggregae price P given by he equaion (3) can be used o show ha E bp ' ( ")V : Up o he second order in bp, we can hus rewrie bq as bq ' "V, leading o i bq +i = " X i V +i. () On he oher hand, he equaion o obain V is wrien as V = V ar j bp (j) = V ar j bp (j) E = E j (bp (j) E ) ( E E ). Here, we remember ha only he fracion! of all rms adjus heir prices o P, while oher rms do no change heir prices p (j). bp, he log-deviaion of P, can in urn be expressed by bp and bp. Hence, V '!E j (bp (j) E ) + (!)E j bp E E E '! V + (!)E j! bp '! V + (!)E j! bp!! bp E ( E E )!! bp bp (bp bp ),

14 up o he second order again in bp. Using he in aion rae bp bp, we hus have The sum of V becomes V '! V +!!. We herefore obain i V +i = V + i V +i i V +i = = V + = V + i= = V +! i V +i+! (!)(!) i! V +i +!! +i i V +i +!! i +i. i +i + :i:p:. ombining his equaion wih he equaions (0) and (), where i u( +i ) = u czl " i +i u c v () i b l+i u b l+i L u c X i b l +i + b l +i + :i:p:; (!)(!).! We also use he approximaions up o he second order and b +i = () bl+i u b l+i = () (bu +i u bu +i ) ; 3

15 L bc +i = b l +i + b l +i ; and nally, we have he following second order expansion of he household s uiliy funcion i u( +i ) = i +i + b +i + c bc +i + :i:p:; where u c ZL ", u c v( ) and c u c. We noe ha, even if we inroduce exogenous produciviy shock Z in he model, we can derive a mahemaically-similar formula for he uiliy by aking he di erence from he e cien sochasic sae. This poin is discussed in he appendix. The opimal policy faces he rade-o among he in aion rae, consumpion, and credi marke ighness. 5 The credi marke ighness is a new feaure for he opimal policy under he loan marke fricion. Moreover, he approximaed welfare funcion can be ransformed as i u( +i ) = i +i + bl+i () u b l+i + c bc +i : Thus, he opimal policy should respond o he changes in credi. This resul suppors he papers of Quin and Rabanal (0), Suh (0), and Kannan, Rabanal, and Sco (0), claiming ha he policy should respond o he changes in credi, in addiion o he in aion rae and he oupu gap. 3.3 Linearizaion We show he closed linearized sysem of he economy around he e cien equilibrium. For general sochasic non-e cien sae, he alvo-ype sickiness inroduced in he reail secor leads o he sandard Phillips curve wih a cos-push shock b" as = E + b" + b. () 5 úrdia and Woodford (009) show ha an approximaed welfare funcion includes he credi spread erm under he model where households face a nancial marke fricion. Also, Teranishi (008) shows ha under he saggered cos channel model an approximaed welfare funcion includes he growh of he loan ineres rae. 4

16 The reail price markup erm b in his equaion can be obained by he linearized equaion (4), The IS relaion is given by Zb = ( ) v b L (E bc + bc ) : u E b + (3) We call bc as he oupu gap. The credi marke ighness is given by bc = E bc + brd E + : (4) b = bl u b l The relaion beween consumpion and credi is given by bc = L b l + b l : (5) : (6) I is also noeworhy ha he log-linearized deviaion of he loan ineres rae br L relaed o he credi marke ighness b and he deposi ineres rae br D equaion: is by he following ar L br L = ( ) b ( )( ) E b + + ( ) br D E +. The credi marke ighness and he real deposi rae deermine he loan ineres rae. This equaion can be ransformed as ar L br L = ( ) ( ) + ( ) + ( )( ) u ( )u + ( )L + L E b l+ (7) + ( )L ( + ) b l b l : 5

17 Thus, he loan ineres rae and credi have a close relaion. 6 4 Moneary Policy We assume ha a cenral bank conrol he nominal ineres rae on deposis o maximize he social welfare, following Woodford (003). The opimal precommimen policy rule for he cenral bank under he imeless perspecive should herefore be obained by solving he following minimizaion problem: X min E i ;c;;r D ;l +i + b +i + c bc +i subjec o he Phillips curve of equaion (), he markup of equaion (3), he IS relaion of equaion (4), and he expressions for he credi marke ighness of equaion (5), and he consumpion of equaion (6). Then, he opimal moneary policy, which is he nominal ineres rae, is given by he rs order condiions for, b, bc, b l, and br D, respecively as: c bc + ' ' ' = 0; (8) b v ( ) ZL (' u ' ) + ' 3 = 0; (9) Z (' ' ) + ' ' + ' 4 = 0; (30) ' 3 + u E ' 3+ + L ' 4 L E ' 4+ = 0; (3) ' = 0; (3) where ', ', ' 3, and ' 4 are Lagrangian mulipliers for he Phillips curve of he equaion (), he IS relaion of he equaion (4), and he expressions for he credi marke ighness of he equaion (5), and he consumpion of he equaion (6), respecively. 6 By using he equaion (7), i is possible o include he loan rae erm in he approximaed welfare funcion. This resul is consisen wih he one in Teranishi (008). 6

18 5 Policy Measures for Financial Sabiliy 5. Inervenion for Nash Bargaining We inroduce a policy measure ha inervenes in he seing of he loan ineres rae beween a rm and a bank hrough Nash bargaining. The moneary auhoriy conrol he parameer b of Nash bargaining as max f b R L J b ; where b is he policy variable of he moneary auhoriy. In realiy, he moneary auhoriy conrols he degree of compeiion in he loan marke by changing regulaions. In his case, he reail price markup erm b in he equaion (3) changes as Zb = ( ) v L (b u E b + ) (E bc + bc ) b v bb ( b) L u E b b+. The welfare funcion and oher pars of he model, however, do no change. Then, he opimal policy measure for nancial sabiliy is given by ' u ' = 0. Accompanied wih he opimal moneary policy, under he opimal policy measure for nancial sabiliy, he in aion rae is xed o zero, and oher variables are also zero agains he cos-push shock. This is because nancial sabiliy leads o price sabiliy via he cos channel. This case shows ha he policy measure for nancial sabiliy can hold a close relaion wih price sabiliy and ulimaely wih he moneary policy. 5. Inervenion for redi Separaion Anoher way o inervene in he credi marke could be limiing or conrolling he credi separaion rae. 7 In his policy measure, he moneary auhoriy conrols he volume of credi, and his can be inerpreed as he macroprudenial policy. If he change in 7 We can also make he credi vacancies or he maching parameer as policy variables. 7

19 a ecs no oher exogenous parameers, hen i can be shown in our model ha he uiliy is maximized by always seing o zero. In he real economy, however, consraining naural separaion of credi beween banks and rms may resul in a deerioraion of produciviy in a long run, and a similar feaure for he separaion beween workers and rms is endogenously capured by he maching model by Morensen and Pissarides (994). In addiion o allowing he policy makers o conrol he separaion rae, we herefore inroduce he following phenomenological relaion beween he average produciviy of wholesale rms Z and he separaion rae : Z = f( ); where f is a monoonically-increasing concave funcion. We hen assume ha he expansion of he above relaion up o he second order from a seady-sae value can be expressed as Z bz + bz = f b + (f + f ) b, (33) where he rs and he second derivaive of he funcion f under he e cien seady-sae equilibrium saisfy f > 0 and f 0, respecively. In addiion o he equaion (7), he e cien seady-sae condiion becomes, ( )f =, (34) where can be nonzero. In his case, he welfare funcion becomes i u( +i ) = i +i + b +i + c bc +i + b +i (35) + i b bl+i +i b l+i + :i:p:, where u c L jf j and u c Lf. See he appendix for he deails of he calculaion. The implicaion of his equaion is clear. The fac ha he las erm is linear in b suggess ha, when he number of credi increases, he sociey is beer o by seing he separaion rae higher han he e cien equilibrium value. The second-order erm b +i is hen he cos incurred by excessive conrol of b and re ec he concaviy of he funcion f. The ime-variaion in he separaion rae modi es he relaion beween he credi ighness and he number of credi as 8

20 b = b l u b l + In addiion, he markup equaion is modi ed as Zb = ( ) v b L (E bc + bc ) + (b E b + ). L b. (36) u u E b + On he oher hand, he e cien seady-sae condiion (34) ensures ha he equaion (6) beween consumpion and credi is virually unchanged up o he rs order. The opimal policy can hus be obained by maximizing he above uiliy subjec o he Phillips curve of he equaion (), he modi ed markup of he equaion (37), he IS relaion of he equaion (4), and he expressions for he credi marke ighness of he equaion (36), and he consumpion of he equaion (6). This maximizaion replaces one of he rs order condiions (3) by (37) (b E b + ) ' 3 + u E ' 3+ + L ' 4 L E ' 4+ = 0. (38) In addiion, he following rs order condiion for b is obained: b ( b l b l ) + Z (' ' ) or by using he equaions (8) and (3), u ( ) ' 3 = 0, (39) ' 3 = u c L h ( ) jf jb " + b i l b l. (40) u The opimal macroprudenial policy adjuss he separaion rae by aking accoun of he rade-o among he separaion rae, credi marke ighness, consumpion, and he in aion rae, while all of he separaion rae, credi marke ighness, and consumpion are closely relaed o he amoun of credi b l by he equaions (6) and (36). In fac, we can explicily derive he following policy by subsiuing he equaions (30) and (40) ino he equaion (38) o eliminae ' 3 and ' 4 : A b A E b + = A 3 " + b l b l A 4 E " + + b l + b l, (4) 9

21 where A + ( )jf j, u A + ( )jf j, A 3 L + u, A 4 L +. Noe ha he equaion (6) is used o eliminae bc. The form of he equaion (4) clearly suggess ha he opimal choice of he separaion rae is deermined by he in aion rae and he increase in he credi " + b l b l. 6 oncluding Remarks We inroduce he search and maching ype of fricion ino he loan marke in a DSGE model. In his model, he second order approximaion of social welfare includes he erm relaing o credi, such as credi marke ighness, he volume of credi, and a loan separaion rae, in addiion o he in aion rae and he oupu gap. This is a new nding in a eld of he opimal policy. The oucome of he opimal policy measure changes in accordance wih he ype of policy measures for nancial sabiliy. For he fuure research, he following poins could be of ineres. The model considered in his paper is resriced o he case where he cenral bank and moneary auhoriy coordinae heir policy choices so as o maximize social welfare. An alernaive assumpion would be ha wo policy makers se heir respecive policies in a non-cooperaive way. Moreover, we can assume a siuaion where he macroprudenial policy can have impac on disurbances, even hough he moneary policy canno a ec hem. They can be disurbances from credi spreads on he policy ineres rae. Anoher issue is a quesion for wha kinds of policies are simple and implemenable o replicae he opimal policy measure for nancial sabiliy, is as in he case wih he Taylor rule in he opimal moneary policy analysis. In such an analysis, i would be ineresing o quaniaively evaluae prioriy among sabilizaion of credi, he in aion rae, and he oupu gap. 0

22 References [] Bernanke, B., Gerler, M., Gilchris, S., 996. The Financial Acceleraor in a Quaniaive Business ycle Framework. Handbook of Macroeconomics, edied by J.B. Taylor and M. Woodford, [] Bianchi, J, 00. redi Exernaliies: Macroeconomic E ecs and Policy Implicaions. American Economic Review: Paper & Proceedings 00, pp [3] Blanch ower, D. and A. Oswald Wha Makes an Enrepreneur? Journal of Labor Economics 6-, pp [4] Borio,., 0. Rediscovering he Macroeconomic Roos of Finanical Sabiliy Policy: Journey hallenges and a Way Forward. BIS Working Papers No [5] alvo, G.A., 983. Saggered Prices in a Uiliy-Maximizing Framework. Journal of Moneary Economics -3, pp [6] úrdia, V. and M. Woodford, 009. redi Fricion and Opimal Moneary Policy. Mimeo. [7] Den Haan, W.J., G. Ramey, and J. Wason, 003. Liquidiy Flows and Fragiliy of Business Enerprises. Journal of Moneary Economics 50, pp [8] Drehmann, M.,. Borio, and K. Tsasaronis, 0. haracerising he Financial ycle: Don Lose Sigh of he Medium Term! BIS Working Papers No [9] Kannan, P., P. Rabanal, and A. Sco, 0. Moneary and Macroprudenial Policy Rules in a Model wih House Price Booms. The B.E. Journal of Macroeconomics, Issue, pp [0] Morensen, D.L. and.a. Pissarides, 994. Job reaion and Job Desrucion in he Theory of Unemploymen. Review of Economic Sudies 6, pp [] Peerson, M. and R. Rajan, 00. Does Disance Sill Maer? The Informaion Revoluion in Small Business Lending. Journal of Finance 57, pp [] Quin, D. and P. Rabanal, 0. Moneary and Macroprudenial Policy in an Esimaed DSGE Model of he Euro Area. Mimeo.

23 [3] Ravenna, F. and.e. Walsh, 0. Welfare-Based Opimal Policy wih Unemploymen and Sicky Prices: A Linear-Quadraic Framework. Americal Economic Journal: Macroeconomics 3, pp [4] Rogerson, R., R. Shimer, and R. Wrigh, 005. Search-Theoreic Models of he Labor Marke: Survey. Journal of Economic Lieraure XLIII, pp [5] Suh, H., 0. Macroprudenial Policy: Is E ecs and Relaionship o Moneary Policy. Federal Reserve Bank of Philadelphia Working Paper No. -8. [6] Teranishi, Y., 008. Opiaml Moneary Policy under Saggered Loan onracs. IMES Discussion Paper Series E-08, Bank of Japan. [7] Wasmer, E. and P. Weil, 000. The Macroeconomics of Labor and redi Marke Imperfecions. American Economic Review 94(4), pp [8] Woodford, M., 003. Ineres and Prices: Foundaion of a Theory of Moneary Policy. Princeon Universiy Press, Princeon, NJ.

24 Appendix A Policy Objecive Funcion wih Produciviy Shock The models in he main ex do no consider he e ec of he produciviy shock Z. This is because, even if Z is aken ino accoun as a shock, he mahemaical forms of he model would remain he same by aking he di erence from an e cien sochasic sae equilibrium. In his subsecion, we show ha he produciviy shock alers neiher he policy objecive funcion nor any of he linearized srucural equaions. When we have a sochasic exogenous produciviy Z, he second order expansion of he household s uiliy funcion around he e cien seady-sae equilibrium becomes i u( +i ) = i +i + b +i + c bc +i +u c ZL bz i +i + bz +i + u c ZL i bz +i b l+i + :i:p:. Here, alhough he erm u c ZL bz i +i + bz +i is clearly :i:p:, he cross erm beween bz +i and b l +i seems relevan. In order o eliminae his erm, we consider he log-linearized deviaion from he dynamics of e cien sochasic sae. This e cien sochasic sae is obained by imposing he Hosios condiion b = and no price markup =, alhough allowing he produciviy shock Z o move. We wrie he log-linearized value of a variable X a he e cien sochasic sae as x e and he deviaion from he e cien sochasic sae as ex bx x e. A he e cien sochasic sae, he consumpion up o he rs order is c e = L Zbz l e + l e. (4) On he oher hand, a he e cien sochasic equilibrium, he Euler equaion () becomes c e = E c e + re, where r e is he real ineres rae. By subsiuing he equaion (4) ino his equaion, we obain

25 bz = E bz + + Z ( E l e + + l e + l e l e ) The policy objecive funcion can be divided as ZL re. (43) where i u( +i ) = A + B + :i:p:, (44) A = u c ZL i bz +i b l+i is he collecion of he erms ha include bz, and L X u c Z i bz +i b l+i bz +i b l+i B = i +i u c v i b L +i u c X i b l +i + b l +i represens he oher erms. By using he equaion (43) only for he las erm in A, A = u c ZL i bz +i b l+i L X + u c Z i Z = u c ZL i bz +i b l+i + :i:p: + u c Z L X L X u c Z i bz +i b l+i bz +i+ b l+i i Z l+i+ e + l+i e + l+i e l+i e l+i+ e + l+i e + l+i e l+i e ZL re b +i l+i ZL re b +i l+i. On he oher hand, he equaion (4) implies ha, a he e cien sochasic equilibrium, we have Zbz = ( ) v L e u E + e + r e. By subsiuing his, A can be furher simpli ed as

26 A = u c ( ) v i e b +il +i u e b +i+l +i L X + u c i l+i+ e + l+i e + l+i e l e bl+i +i + :i:p: = u c ( ) v i u e b +il +i u e b +i+l +i + u c ( )v i +i e b +i L X + u c i ( l+i+ e + l+i e + l+i e l+i e ) b l +i + :i:p:, where b = (b l u b l ) is used. The wo erms in he rs summaion on he righ-hand side are again :i:p:, and he las summaion can be rewrien as i ( l+i+ e + l+i) e b l +i + i (l+i e l+i e ) b l +i X = i ( l+i e + l+i e ) b l +i ( l e + l e ) b l + i (l+i e l+i e ) b l +i = X We hus obain i ( l+i e + l+i e ) b l +i + b l +i + :i:p:. L A = u c X i ( l+i e + l+i e ) + u c ( )v i +i e b +i + :i:p:. b l +i + b l +i This expression for A is subsiued ino he policy objecive funcion of equaion (44), yielding 3

27 i u( +i ) = u c + u c ( )v L X i +i e b +i i ( l+i e + l+i e ) b l +i + b l +i i +i u c v i b +i L u c X i b l +i + b l +i + :i:p: = i +i u c u c v = u c v which can be simpli ed as: L X h i b l +i + b l +i i +i b +i e + :i:p: i +i l+i e + l+i e i L u c X i e l +i + e l +i i e +i + :i:p:, i u( +i ) = i +i + e +i + c ec +i. We herefore con rm ha he form of he uiliy-based policy objecive funcion remains he same even when we inroduce he produciviy shock. In addiion, we can easily see ha all he relevan srucural equaions (, 3, 4, 5, and 6) can be wrien idenically if we replace bx by ex for all he variables X. B Policy Objecive Funcion wih Time-Dependen Separaion Rae In his secion, we show he derivaion of he equaions (34) and (35). The e cien seady-sae condiion is obained by he following maximizaion problem: 4

28 max ;L;u;; E i f +i + +i[f( +i )L +i al +i +i u +i +i ] + +i [( +i )L +i + +iu +i L +i ] + s +i [u +i + ( +i )L +i ]g. By aking he rs-order condiions for he ve variables and rearranging he equaions, we obain and f( ) a = E + ( + ) f 0 ( ) L = L +.! + + +, A he e cien seady-sae equilibrium, he former condiion becomes idenical o he equaion (7), while he laer can be rearranged o he condiion (34). We noe ha he laer equaion, when linearized around he e cien seady-sae equilibrium, is wrien as f b + f bl b l = ; and could be obained from he opimal policy in he main ex, if here were no in aion. This poin can be easily con rmed by subsiuing he equaion (9) ino he equaion (39) o eliminae ' 3 and by remembering ha ' is zero wihou in aion from he equaion (8). As for he second-order expansion of he uiliy, boh he ime-dependence of he separaion rae and he produciviy Z make he calculaion slighly complicaed, alhough he basic procedure of he derivaion is sraighforward. For example, he produciviy eners he expansion of consumpion, and he equaion corresponding o he equaion (8) becomes: bc + bc = ZL (bz + bz + bz b (Z a)l l bq ) + b l + b l v bv + bv. On he oher hand, he expansion of he credi vacancy (9) is modi ed by he imedependence of he separaion rae as: 5

29 bv + bv = b + b + L u b l + b l b + b + b b l + b b. b b l Afer eliminaing he credi marke ighness b by using L = ( )L + u, we obain he following expansion of he uiliy: u( ) = u() + u c ZL bz + bz + bz b l bq v u c L + u c L u c ( ) L u b l + b l + L b + b + b b l b l + b l Zbz + b l + b l v L v bl () u b l b. ( ) L u A his sage, by using he equaions (33) and (34), we observe ha he rs-order di erence beween bz and b is cancelled, and we nally obain he equaion (35). 6