Optimal Monetary Policy in a Model of the Credit Channel

Transcription

1 Optimal Monetary Poliy in a Model of the Credit Channel Fiorella De Fiore y European Central Bank Oreste Tristani z European Central Bank 9 September 2008 First draft Abstrat We onsider a simple extension of the basi new-keynesian setup in whih we relax the assumption of fritionless nanial markets. In our eonomy, asymmetri information and default risk lead banks to optimally harge a lending rate above the risk-free rate. Our ontribution is threefold. First, we derive analytially the loglinearised equations whih haraterise aggregate dynamis in our model and show that they nest those of the new- Keynesian model. A key di erene is that marginal osts inrease not only with the output gap, but also with the redit spread and the nominal interest rate. Seond, we nd that nanial market imperfetions imply that exogenous disturbanes, inluding tehnology shoks, generate a trade-o between output and in ation stabilisation. Third, we show that, in our model, an aggressive easing of poliy is optimal in response to adverse nanial market shoks. Keyworks: optimal monetary poliy, nanial markets, asymmetri information JEL odes: E52, E44 We wish to thank Mihael Woodford for many interesting disussions, Giovanni Lombardo for useful omments and suggestions, and Krzysztof Zalewski for exellent researh assistane. y DG Researh. orella.de_ ore@eb.int. z DG Researh. oreste.tristani@eb.int.

2 Introdution Central banks devote muh e ort to the analysis of the nanial positions of households, rms and nanial institutions, and to monitor the evolution of redit aggregates and interest rate spreads. One reason is that nanial market onditions are pereived to be fators whih ontribute to shape the performane of the eonomy and to a et its in ationary prospets. This pereption appears to be onsistent with the time series properties of some nanial variables. For example, redit spreads in both the US and the euro area display a positive orrelation with in ation (see Figure ). The orrelation is suggestive of a possible role for spreads in a eting rms marginal osts. Higher spreads would then be re eted in a ausal way into higher pries. In several historial episodes, entral banks have also reated sharply to hanges in nanial onditions. One example are the US developments during the late 980s, when banks experiened large loan losses as a onsequene of the bust in the real estate market. Due to weak nanial onditions, banks ould not raise new apital and, beause of the requirement to omply with the Basel Aord, they were fored to ut bak on loans. This led to a slowdown in redit growth and aggregate spending. Aording to Rudebush (2006), this slowdown ontributed to the FOMC deision to redue the Federal funds rate well below what suggested by an estimated Taylor rule. A more reent example is provided by the nanial market turmoil initiated in 2007 with the deterioration in the performane of nonprime mortgages in the US. In August 2007, the FOMC justi ed a ut in the disount rate of 50 basis points by expressing onerns about the ongoing deterioration of nanial market onditions and tightening of redit onditions, whih inreased appreiably the downside risks to growth. To analyze whether nanial market onditions ought to have a bearing on monetary poliy deisions, we onsider a simple extension of the basi New-Keynesian setup, where rms need to borrow funds in advane of prodution and informational fritions haraterize nanial markets. As in Bernanke, Gertler and Gilhrist (989) and Carlstrom and Fuerst (997, 998), we assume that rms have private information about the realization of an idiosynrati produtivity shok, whih banks an only monitor ex-post at a ost. The presene of asymmetri information introdues the risk of default on loans, so that banks nd it optimal to harge a lending rate whih is above their marginal ost (the deposit rate). One deviation from the 2

3 set-up adopted in Bernanke, Gertler and Gilhrist (989) and Carlstrom and Fuerst (997, 998) is that we assume that loans are denominated in nominal rather than real terms. Figure : Credit spreads and in ation US Euro area Soure: Fred database, ECB and AWM database (see Fagan et. al, 999). Legend: SCPNF: spread between the 3-month ommerial paper rate (to non- nanial orporations sine 997) and the 3-month Treasury Bill rate; PCEINF4: year-on-year PCE in ation; Spread: spread between the interest rate on short-term loans to non- nanial orporations and the 3-month interbank rate. The main appealing feature of our model is its analytial simpliity and the possibility to disentangle the role of nanial fritions for in ation and output dynamis. We obtain three main sets of results. First, we show that the loglinear approximation of the aggregate strutural equations of our model is similar in struture to the one arising in the new-keynesian setup with fritionless nanial markets. As in the new-keynesian ase, private setor deisions an be haraterized by an intertemporal IS equation and a Phillips urve. These relationships, however, inlude additional terms to re et the existene of informational asymmetries. The main di erene is that rms marginal osts re et, on top of the osts of labour input, also the redit spread and the nominal interest rate. The latter two variables matter beause they determine the ost of redit in the eonomy. Thus, they at as endogenous "ost-push" fators. Seond, we nd that both tehnology and nanial market shoks operate as exogenous ost-push fators in the model. This is notieable for tehnology shoks, whih in the benhmark model with fritionless nanial markets generate fully e ient utuations in output 3

4 and onsumption. In our model, however, these utuations produe orresponding variations in rms exposure to external nane. The ensuing volatility in redit spreads and bankrupty ratios represent the ine ient impliations of tehnology shoks in the presene of redit fritions. Third, we show that the presene of nanial market imperfetions and endogenous variations in redit spreads hange the haraterization of optimal monetary poliy. Using an analyti, seond-order approximation of the welfare funtion, we demonstrate that welfare is diretly a eted not just by the volatility of in ation and the output gap, as in the benhmark ase with fritionless nanial markets, but also by the volatility of the nominal interest rate and of the redit spread. In quantitative terms, the optimal poliy reation to tehnology shoks is not dramatially di erent from the ase with fritionless nanial markets. More spei ally, near omplete in ation stabilization remains optimal. In reation to a nanial market shok whih inreases the redit spread, however, a more signi ant short-term trade-o arises between in ation and output stabilization. In order to help households smooth their onsumption path in reation to the shok, optimal monetary poliy is aggressively expansionary for one year. Our paper is not the rst attempt to analyze monetary poliy in models with redit fritions. Ravenna and Walsh (2006) haraterizes optimal monetary poliy when rms need to borrow in advane to nane prodution. However, there is no default risk in that model and the ost of naning for rms is the risk-free rate. We show that our model nests that of Ravenna and Walsh (2006) in the speial ase in whih the osts of asymmetri information disappear. Faia and Monaelli (2006) ompares the welfare losses of optimized simple interest rate rules in models with a struture similar to ours, but it does not haraterize fully optimal (Ramsey) monetary poliy. Similarly, Christiano, Motto and Rostagno (2003) fous on how monetary aggregates ould provide useful information for the entral bank in a medium sale DSGE model with several fritions, inluding redit fritions. Christiano, Motto and Rostagno (2006) argue that the monetary poliy reation to a stok market boom/bust yle would be superior, in terms of welfare, if liquidity developments were taken into aount. Our paper is losest to reent work by Curdía and Woodford (2008), whih also haraterizes optimal monetary poliy in a model where nanial fritions matter, beause of heterogeneity in the spending opportunities available to di erent households. Our work di ers in the underlying soure of nanial fritions. Finanial fritions are mirofounded in our model and redit 4

5 spreads arise from an expliit haraterization of loan ontrats. Curdía and Woodford (2008) assume instead a exible, redued-form funtion linking the redit spread to maroeonomi onditions. The paper proeeds as follows. In setion 2, we desribe the environment and derive the onditions haraterizing the equilibrium of the eonomy when nanial ontrats are written in nominal terms. In setion 3, we disuss the log-linearized version of our model, in omparison to the new-keynesian benhmark. This enables us to highlight the e et of nanial market fritions on in ation and output dynamis. In setion 4, we haraterize the optimal monetary poliy. We derive a simple quadrati approximation of the soial welfare, whih we ompare to the one arising under fritionless nanial markets. In setion 5, we derive the rst-order onditions of the soial planner problem under disretion and we disuss the role of nanial fritions for the optimal ondut of monetary poliy. We then haraterize numerially optimal monetary poliy under ommitment. Setion 6 onludes. 2 The environment The eonomy is inhabited by a representative in nitely-lived household, a ontinuum of wholesale rms produing a homogeneous good and owned by risk-neutral entrepreneurs, a ontinuum of monopolistially ompetitive retail rms produing di erentiated goods and owned by the households, a zero-pro t nanial intermediary, and a entral bank. In the rst part of the period, households deide how to alloate their nominal wealth among existing nominal assets, namely money, a portfolio of nominal state-ontingent bonds, and one-period deposits. In the seond part of the period, they reeive wage inome and purhase onsumption goods. Wholesale rms are endowed with a linear prodution tehnology that uses labor as the only input. They need to pay workers in advane of prodution. At the beginning of the period, eah rm reeives from the government an exogenous nominal endowment that is used as the rm s internal funds. However, these funds are not su ient to nane the wage bill, so rms need to raise external nane. In their produtive ativity, wholesale rms fae idiosynrati produtivity shoks and thus default risk. Lending ours through perfetly ompetitive banks, whih are able to ensure a safe return by providing funds to the ontinuum of rms faing idiosynrati shoks. Bank loans take the form of risky debt, whih is the optimal ontratual 5

6 arrangement between lenders and borrowers in the presene of asymmetri information and ostly state veri ation. Firms in the retail setor buy the homogeneous good from wholesale rms in a ompetitive market and use them to produe di erentiated goods at no osts. Beause of this produt di erentiation, retail rms aquire some market power and beome prie makers. In their prie-setting ativity, however, they are not free to hange their prie at will, beause pries are subjet to Calvo ontrats. Retail rms are owned by the households, who reeive their pro ts. 2. Households At the beginning of period t; the nanial market opens. First, the interest on nominal nanial assets aquired at time t is paid. The households, holding an amount W t of nominal wealth, hoose to alloate it among existing nominal assets, namely money M t ; a portfolio of nominal state-ontingent bonds A t+ eah paying a unit of urreny in a partiular state in period t+; and one-period deposits denominated in units of urreny D t paying bak Rt d D t at the end of the period. In the seond part of the period, the goods market opens. Households money balanes are inreased by the nominal amount of their revenues and dereased by the value of their expenses. Taxes are also paid or transfers reeived. The amount of nominal balanes brought into period t + is equal to M t + P t w t h t + Z t P t t T t ; () where h t is hours worked, w t is the real wage, Z t are nominal pro ts transferred from retail produers to households, and T t are lump-sum nominal taxes olleted by the government. t denote a CES aggregator of a ontinuum j 2 (0; ) of di erentiated onsumption goods produed by retail rms, Z t = 0 " t (j) " " " dj ; with " > : P t (j) denotes the prie of good j; and P t = CES aggregator. h R i 0 P t (j) " " dj is the prie of the 6

7 Nominal wealth at the beginning of period t + is given by W t+ = A t+ + R d t D t + R m t fm t + P t w t h t + Z t P t t T t g ; (2) where Rt m denotes the interest paid on money holdings. The household s problem is to maximize preferenes, de ned as ( ) X E o t [u ( t ) + (m t ) v (h t )] ; (3) 0 where u > 0; u < 0; m 0; mm < 0 and v h > 0; v hh > 0; and m t M t =P t denotes real balanes. The problem is subjet to the budget onstraint M t + D t + E t [Q t;t+ A t+ ] W t ; (4) De ne t Pt P t and m;t Rt Rm t R t : The optimality onditions an be written as v h (h t ) u ( t ) = w t (5) R t = E t [Q t;t+ ] (6) R t = R d t u ( t ) + m (m t ) = R t E t u ( t+ ) + m (m t+ ) t+ m (m t ) u ( t ) = m;t m;t : (7) Moreover, the optimal alloation of expenditure between the di erent types of goods leads to the demand funtions Pt (j) " t (j) = t ; (8) P t where P t (j) is the prie of good j. 2.2 Wholesale rms The wholesale setor onsists of a ontinuum of ompetitive rms, indexed by i; owned by in nitely lived entrepreneurs. Eah rm produes the amount y i;t of a homogeneous good, 7

8 using a linear tehnology y i;t = A t! i;t l i;t : (9) Here A t is an aggregate, serially orrelated produtivity shok and! i;t is an idiosynrati, iid produtivity shok with distribution funtion and density funtion. The prodution funtion (9) re ets our hoie to abstrat from apital aumulation. This is in ontrast with the rest of the literature that introdues redit fritions in maromodels, where entrepreneurs are assumed to deide in period t how to alloate their pro ts to onsumption and investment expenditures (see e.g. Carlstrom and Fuerst (997) and Bernanke, Gertler and Gilhrist (999)). The value of the stok of apital available to rms in period t+ provides the rm with a ertain net worth (internal funds) that an be used in that period prodution. In that environment, aggregate shoks a et the evolution of rms net worth, thus reating endogenous persistene. In our model, we assume instead that eah rm reeives a onstant endowment at the beginning of eah period, whih an be used as internal funds. Sine these funds are not su ient to nane the rm s desired level of prodution, rms need to raise external nane. As a result, nanial fritions a et the eonomy as in models with apital so that, for example, a spread arises endogenously between the loan rate harged by nanial intermediaries to rms and the risk-free rate, to re et the existene of default risk. At the same time, our simpler set-up enables us to provide an analytial haraterization of eonomi dynamis and of optimal poliy in the presene of redit onstraints and information asymmetry Labor demand Firms need to raise external nane to pay for labor servies. Before observing the idiosynrati produtivity shok! i;t ; rms sign a ontrat with the nanial intermediary to raise the amount P t (x i;t ) ; for total funds at hand P t x i;t ; where x i;t w t l i;t. (0) We assume that entrepreneurs sell output only to retailers. Let P t be the prie of the wholesale homogenous good, and P P t t = t the relative prie of wholesale goods to the aggregate prie 8

9 of retail goods. Eah rm i s demand for labor is derived by solving the problem max P t P t E [A t! i;t l i;t ] w t l i;t subjet to the naning onstraint (0), where the expetation E[] is taken with respet to the idiosynrati shok unknown at the time of labor hiring deision, and w t denotes the payment of labor servies measured in terms of the nal onsumption good: Denote the Lagrange multiplier on the naning onstraint as (q i;t ). Optimality requires that q i;t = q t = A t w t t () x i;t = w t l i;t (2) implying that E [y i;t ] = t q t x i;t : (3) Equation (3) states that, as the prodution funtion is CRS, wholesale rms must sell at a mark-up t q t over rms prodution osts. This allows them to over for the presene of redit fritions and for the monopolisti distortion in the retail setor. This latter matters for rms in the wholesale setor beause P t is the de ator of the nominal wage, and thus a ets the real marginal ost faed by wholesale produers. Equation (2) states that the naning onstraint is always binding. Given the ontrat stipulated by the rm with the nanial intermediary (whih sets the amount of funds x i;t and the repayment on these funds), the rm always nds it pro table to use the entire amount of funds and to produe, also when expeted produtivity is low. This way, it an minimize the probability of default The nanial ontrat Loans are stipulated in units of urreny after all aggregate shoks have ourred, and repaid at the end of the same period. Lending ours through the nanial intermediary, whih ollets deposits from households and use them to nane loans to rms. Firms fae an idiosynrati produtivity shok, whose realization is observed at no osts only by the entrepreneur. The nanial intermediary an monitor its realization but a fration of the rm s input is onsumed in the monitoring ativity. If the realization of the idiosynrati 9

10 shok is su iently low, the value of the rm s prodution is not su ient to repay the loan and the rm defaults. Households lend to rms through a nanial intermediary, whih is able to ensure a safe return. This is possible beause by lending to the ontinuum of rms i 2 (0; ) produing the wholesale good, the nanial intermediary an di erentiate the risk due to the presene of idiosynrati shoks. The informational struture orresponds to a ostly state veri ation (CSV) problem. The solution is a standard debt ontrat (see e.g. Gale and Hellwig, 985) whih is derived in the appendix. The optimality onditions an be written as q t = R t t (! t ) + t f(!t)(!t) f! (! t) x t = R t R t q t g (! t ; t ) (4) : (5) where! t is a threshold for the distribution of the idiosynrati produtivity shok below whih rms go bankrupt, and f (! t ) and g (! t ; t ) are the expeted shares of output aruing to the entrepreneur and the bank, respetively. Moreover, t denotes the monetary osts of loan monitoring, as a funtion of the rm s input osts. Given the large time-variation in bankrupty osts doumented by Natalui et al. (2004), t is assumed to be subjet to serially orrelated shoks. Substituting equation (4) into equation (5), it an be notied that a hange in the nominal interest rate has no diret impat on the amount of real funds borrowed by entrepreneurs; the amount of funds is only modi ed in general equilibrium, to the extent that it indues hanges in the threshold! t. An inrease in the nominal interest rate, however, tends to lead to an inrease in the mark-up q t due to the presene of ageny osts. Notie also that the gross interest rate on the loan extended to rm i, Ri;t l ; an be baked up from the debt repayment. It is given by P t! t t q t x t = R l i;tp t (x t ) implying that R l i;t = Rl t; for all i. 0

11 We an use the expression above to obtain the spread between the loan rate and the risk-free rate, t R l t=r d t, an be written as an endogenous funtion of the! t threshold Entrepreneurs t =! t g(! t ; t ) : (6) Entrepreneurs have linear preferenes on onsumption and are in nitely lived. They onsume a CES basket of di erentiated goods similar to that of households. At the end of eah period, entrepreneurs sell their output to the retail setor and, if they do not default, repay the debt. Pro ts of entrepreneurs are entirely alloated to nal onsumption goods Z 0 P t (j) e i;t (j) dj = P t (! i;t! t ) t q t x t ; where e i;t (j) is rm i s onsumption of good j. Notie that R 0 P t (j) e i;t (j) = P t e i;t ;where e i;t is the demand of the nal onsumption good of entrepreneur i. Aggregating aross rms, we obtain e t = f (! t ) q t x t, where e t = R 0 e i;tdi is the aggregate entrepreneurial onsumption of the nal onsumption good. Using equations (4)-(5), we an rewrite aggregate entrepreneurial onsumption as where H ( t ;! t ) + t (!t) f! (! t). e t = R t H ( t ;! t ) (7) Equation (7) shows that entrepreneurial onsumption depends only on the nominal interest rate, on the bankrupty threshold! t, and on the exogenous shok t. The two endogenous variables, however, a et entrepreneurial onsumption through di erent hannels. As mentioned above, an inrease in the nominal interest rate has no diret e et on loans and a ets nanial onditions mainly by induing an inrease in the mark-up q t. paribus, however, a higher R t leads to an inrease in entrepreneurial onsumption. Ceteris Changes in the threshold! t at instead by modifying the output share f (! t ) (together with g ( t ;! t )). Sine entrepreneurs pro ts are dereasing in the threshold, an inrease in bankrupty rates tends to depress entrepreneurial onsumption. Finally, the impat of t on entrepreneurial onsumption is relatively omplex. An inrease in monitoring osts t implies a larger waste of resoures for the eonomy. For given! t, this tends to redue the amount of prodution available for onsumption, inluding entrepreneurial

12 onsumption. At the same time, a higher t will produe hanges in the threshold! t. If total prodution hanges little, rms will have to pay a higher interest rate spread and! t will tend to inrease, whih implies a redution in entrepreneurial onsumption. If however the shok is su iently ontrationary, the demand for redit will fall and! t will derease. 2.3 Retail rms As in Bernanke, Gertler and Gilhrist (999), monopolisti ompetition ours at the "retail" level. More spei ally, a ontinuum of monopolistially ompetitive retailers buy wholesale output from entrepreneurs in a ompetitive market and then di erentiate it at no ost. Beause of produt di erentiation, eah retailer has some market power. Pro ts are distributed to the households, who own rms in the retail setor. Let Y t (j) be the quantity of output sold by retailer j. This quantity an be used for households onsumption, t (j) ; and for entrepreneurs onsumption, e t (j). Hene, Y t (j) = t (j) + e t (j) : The nal good Y t is a CES omposite of individual retail goods with " > : Z Y t = 0 " Y t (j) " " " dj ; (8) 2.3. Prie setting We assume that eah retailer an hange its prie with probability ; following Calvo (983). Let P t (j) denote the prie for good j set by retailers that an hange the prie at time t; and Y t (j) the demand faed given this prie. Then eah retailer hooses its prie to maximize expeted disounted pro ts, given by " X E t k Pt Q t;t+k k=0 where Q t;t+k = u( t+)+ m(m t+ ) u ( t)+ m(m t) : P t+k P t+k Y t+k (j) # ; 2

13 The rst-order onditions of the rm s pro t maximization problem imply that Pt = " P t " E t P k=0 k Q t;t+k P t+k P " t+k P E t k=0 k Pt Q " t;t+k P t " Y t+k Y t+k P " t+k Now de ne ;t P t P t Y t + E t 2;t Y t + E t ( X ( X k P t+k Q t;t+k k= k= k Pt " Q t;t+k P " t+k P " t+k P " t Y t+k ) Y t+k ) Using the expression for the aggregate prie index, P t = and substituting out P t P t, we an reursify the rst order ondition as = " " t ;t + ( ) " 2;t ;t = Y t + E t " t+ Q t;t+ ;t+ t 2;t = Y t + E t " t+ Q t;t+ 2;t+ : h Pt " + ( ) (P t ) "i " ; " Prie dispersion Reall that the aggregate retail prie level is given by P t = h R i 0 P t (j) " " dj : De ne the relative prie of di erentiated good j as p t (j) Pt(j) P t and divide both sides by P t to express everything in terms of relative pries, = R 0 (p t (j)) " dj: De ne also the relative prie dispersion term as s t Z 0 (p t (j)) " dj: This equation an be written in reursive terms as s t = ( ) " " " t + " t s t : 3

14 2.4 Market learing Market learing onditions are listed below. Money: Mt s = M t ; Bonds: A t = 0 Labor: h t = l t Loans: d t = x t Wholesale goods: y t = Z 0 Y t (j) dj Retail goods: Y t (j) = t (j) + e t (j) ; for all j: 2.5 Competitive equilibrium The entral bank needs to speify an additional rule for either Rt m or Mt s : It is onvenient to express this rule in terms of m;t. In order to failitate the omparison of our model with the standard New-Keynesian setup, we assume a monetary poliy suh that m;t = m ; for all i: Then, and we an de ne m (m t ) = m m u ( t ) U ( t ; m;t ) u ( t ) + m : m Under a poliy of onstant m;t ; money demand beomes reursive and an therefore be negleted for the solution of the system. 4

15 We assume a funtional form U ( t ; m ) v (h t ) = t log t+ ; bp t (j) = log p t (j) ; a t = log A t ; and b t = log t : h+' t +' and we de ne b t+ The system of equilibrium onditions an be written in log-linearized form as reported in the appendix. 2.6 The system in redued form The system of equilibrium onditions that haraterizes the evolution of the aggregate variables (one a monetary poliy rule is spei ed) an be linearized around a zero-in ation steady state as + e b t = ( + ') a t + bt 2 + ' Yt b Rt b (9) + 3 g b t = + ' + Y by t = E t b Yt+ brt b g t by t e R b t ( + ') a t 2 b t (20) E t b t+ + E t bt+ b t (2) + e E g t b t+ b t + g E t b t+ b t b t = + e b t + E t b t+ (22) where the oe ients ; 2 ; 3 ;, 2 and are de ned in the appendix. Notie that and 2 are positive oe ients. In our eonomy, the mark-up t = P t =P t is inversely related to the marginal osts of retail rms. Indeed, an inrease in the input ost of retail prodution, i.e. a higher prie of wholesale goods, generates a fall in the mark-up. Equation (9) shows that the markup is negatively related to three fators. The rst is the spread between the loan rate and the poliy rate. An inrease in the spread implies a higher ost of external nane for wholesale rms, whih then need to inrease the prie of intermediate goods, P t : The seond is the demand for nal goods. In the presene of higher demand for retail goods, and orrespondingly of intermediate goods to be used as prodution inputs, wholesale rms need to pay a higher real wage to workers to indue them to supply the required labor servies. This inreases the prie of wholesale goods, P t ; relative to the prie of retail goods, P t. The third is the nominal interest rate. Wholesale rms must borrow funds to nane prodution through nominal loans. As a result, 5

16 any inrease in the poliy rate represents an additional ost whih is overed by harging a higher prie of wholesale goods. Equation (20) summarizes equilibrium onditions in the redit market. Notie that beause of the assumption that debt ontrats take plae period-by-period, only ontemporaneous variables enter this equation. It is reasonable to expet that this feature of the redit market equilibrium would hange in a riher model. In setion 3, we will express the system (9)-(22) in terms of gaps from the e ient equilibrium and disuss in more details the onditions orresponding to equations (20)-(22). In the rest of this setion, we show that our model nests both the ost-hannel model of Ravenna and Walsh (2006) and the standard new-keynesian model. We onsider the speial ase when monitoring osts are zero, i.e. t = 0; for all t: In this ase, rms still need to borrow in advane of prodution. However, the information asymmetry onerning wholesale rms produtivity disappears beause banks an monitor at no ost. The system then beomes b t = + ' b Yt b Rt + ( + ') a t by t = E t b Yt+ b t = brt E t b t+ + e b t + E t b t+ The equations above oinide with the redued-form system of equilibrium onditions obtained by Ravenna and Walsh (2006) in their model of the "ost-hannel," where rms borrow in advane of prodution but, sine there is no asymmetri information nor default risk, they simply pay the risk-free rate on these funds. Finally, the system would boil down to the new-keynesian model in the absene of a nominal interest rate e et on marginal osts. 3 The system in deviation from the e ient equilibrium In order to haraterize the optimal response of monetary poliy, it is onvenient to write the redued form of the system (9)-(22) in terms of gaps from the e ient equilibrium, in whih t = 0, 0 and pries are exible. We denote a variable with a hat and a supersript e as the log-deviation of the variable from its steady state in the e ient equilibrium, whih is 6

17 haraterized by by e t = E t b Y e t+ br e t + ' b Y e t = ( + ') a t ; and where br t e denotes the real interest rate. We nd it useful to de ne the output gap, Y e t ; as atual output in deviation from e ient output, when both variables are linearized around the atual steady state Y. Note that under this de nition the output gap will not be zero in steady state, but equal to the di erene between the two steady states y log Y log Y e. We an now rewrite the system as b t = + ' + Y e e Yt Rt b + b 2;t (23) ey t = E tyt+ e + e brt ' e E t b t+ R be t e e 2 bt ' e E tt+ b + brt ' e E t b Rt+ + t (24) b t = + ' e Yt + b R t bt + E t b t+ b ;t (25) where b ;t + ' + ' (E ta t+ a t ) ' + ' (E ta t+ a t ) + g g + e b 2;t 2 e e ' e E b;t+ t b + e g ;t + ' e g E t b t+ b t and g denotes the partial derivative of g (! t ; t ) with respet to. b t (26) + ' + ' a t 2 b t (27) Equation (23) shows that the spread between the loan rate and the poliy rate inreases with exess aggregate demand. An inrease in the demand of retail (and thus also of wholesale) goods implies an impliit tightening of the redit onstraint, sine the exogenously given amount of internal funds must now be used to nane a higher level of debt. The inreased default risk generates a larger spread. For the same reasons, the spread dereases with the nominal interest 7 (28)

18 rate. An inrease in the latter variable generates a redution in the demand for nal goods and thus in the demand for input in their prodution (wholesale goods). For a given amount of internal funds, leverage and the risk of default fall, reduing the spread. Equation (24) is a forward-looking IS-urve desribing the determinants of the gap between atual output and its e ient level. The rst line of the expression shows that, as in the standard new-keynesian model, the gap is a eted by its expeted future value and by the real interest rate. In our model, however, the output gap also depends on the expeted hange in the nominal interest rate and in the redit spread, as well as on the shok t. Note that this dependene is not present in a ost hannel model: it would disappear in the absene of monitoring osts. A higher spread between loan and deposit rates is ontrationary in our model, beause it indues an inrease in bankrupty rates and a fall in entrepreneurial onsumption. In our alibration, an expeted inrease in the spread between periods t and t + tends instead to be expansionary, in spite of the fat that entrepreneurs are myopi in their onsumption patterns. The transmission of this e et operates through households onsumption. Through the aggregate resoure onstraint, the redution in t + entrepreneurial onsumption, whih is due to the higher expeted spread, also tends to imply an inrease in future households onsumption. Sine households are forward looking, this e et will feed through to urrent households onsumption, thereby leading to an expansionary e et on output. On top of the standard real interest rate e et, hanges in the nominal interest rate have an impat on output whih operates through similar, but opposite, hannels to those of the spread. A higher nominal interest rate will in fat have a small expansionary e et, as it will inrease the nanial mark-up and entrepreneurial onsumption. However, an expeted inrease in the nominal interest rate will be ontrationary, as it will lead to an expeted fall in households future onsumption. Equation (25) represents an extended Phillips urve. The rst determinant of in ation in this equation is an output gap term. This term is standard, but it enters here with a di erent oe ient re eting the presene of entrepreneurs in the eonomy. As in the ost hannel model, equation (25) also inludes a nominal interest rate term, whose inrease pushes up marginal osts. Finally, the novel feature of our model is the presene of a redit spread in 8

19 the equation. A higher redit spread also inreases rms osts and therefore exerts independent pressure on in ation. The redit spread and the nominal interest rate therefore at as endogenous "ost-push" terms on in ation. Finally, the three equations are a eted by all exogenous disturbanes, whih therefore at as exogenous "ost-push" fators in the Phillips urve. More spei ally, tehnology shoks are also partly ine ient through their e et on the redit market. This is in ontrast with the standard new-keynesian model, in whih they only generate e ient variations in output. The reason is that the output expansion whih will typially follow a positive tehnology shok generates the need for an inrease in external nane and in leverage, hene leading to an inrease in the redit spread. In turn, the higher redit spread will a et output and in ation through the hannels desribed above. 3. Impulse responses As a benhmark for omparison with the optimal poliy ase, we provide some evidene on the quantitative impliations of the model through an impulse response analysis. For this purpose, we lose the model with a simple monetary poliy rule of the Taylor-type with interest rate smoothing. The parameters of the rule are hosen in line with the values estimated in Smets and Wouters (2007) for the US, namely an in ation response oe ient of 2.0, an output gap response oe ient of 0. and an interest rate smoothing equal to 0.8. The strutural parameters are set in line with the literature. Following Levin, Natalui and Zakrajsek (2004) we set long-run monitoring osts at 5% of the rm s output (i.e. = 0:5). We then alibrate the standard deviations of idiosynrati shoks (! ) and the subsidy so that that the annualized steady state spread is equal to 2% and roughly % of rms go bankrupt eah quarter. As to monopolisti ompetition and retail priing, we assume " = 7, leading to a steady-state mark-up of 7%, and a probability of not being able to re-optimize pries = 0:66, implying that pries are hanged on average every 3 quarters. Finally, we set the persistene of tehnology and monitoring ost shoks to 0.9. Figure 2 displays the impulse responses of output, the output gap (de ned as the di erene between atual output and the e ient level of output), the poliy interest rate and in ation to a positive % tehnology shok under the Taylor rule. The impulse responses under our Sine the steady states of output y and of the e ient level of output y e are di erent, the output gap term in the Taylor rule is written as gap = by t by e t y. 9

20 model denoted as "redit hannel model" are ompared to two well-known benhmarks: a model with the ost hannel, whih is obtained when t = 0 and = 0; and a standard new-keynesian model. The most notable feature of Figure 2 is that the three models with nominal rigidities produe extremely similar impulse responses under the Taylor rule. As is typially the ase, a tehnology shok exerts downward pressure on in ation (denoted as "inf") and on the interest rate on deposits ("i_dep"). The fall in in ation orresponds to almost the same negative output gap ("ygap") in our model and in the standard new-keynesian model. It is slightly less pronouned, and turns positive after a few quarters, in the model with the ost hannel. In the latter model, the fall in the poliy interest rate has an expansionary e et through the ensuing redution in marginal osts. In our model, the same e et is ounterated by an inrease in the redit spread so that the output gap remains negative as in the new-keynesian model. The responses of households onsumption ("ons_h") are equally very similar. Of ourse, our model also has impliations for the stok of redit and the spread between loan and deposit rates. Credit expands almost one-to-one with households onsumption, but this also implies an inrease in leverage, as rms net worth is onstant. As a result, the bankrupty rate in the eonomy inreases and so does the redit spread. A pro-ylial response of the redit spread to tehnology shoks is standard in models adopting the Carlstrom and Fuerst (997) set-up, but the data show that spreads tend to inrease during reessions this is the ase, for example, for the di erene between lowest and highest rates on orporate bond yields in the US (see e.g. Figure in Levin et al., 2004). This is a problem in terms of the ability of our model to repliate a key feature of the redit market through utuations in tehnology shoks. Nevertheless, it is not ruial for our results in two important respets. The rst is that the spei ylial properties of redit spreads should not hange the inentives of the entral bank to smooth out utuations in suh spreads. Our general haraterization of the entral bank s objetives should therefore remain valid also within a model in whih spreads were antiylial in response to tehnology shoks. The seond reason is that we are partiularly interested in haraterizing optimal poliy in the fae of shoks whih originate in the nanial market. We show below that these shoks do give rise to a realisti, ounterylial response of the redit spread. Figure 3 presents impulse responses to a poliy shok. The similarity of three models is even more striking in this gure. The ontration in the output gap and the orresponding fall 20

21 in in ation is virtually indistinguishable in the three models, and so is the monetary poliy response. As in the ase of tehnology shoks, the quantity of redit, leverage, and the spread between loan and deposit rates all move downwards with output, after a poliy tightening. It should be emphasized that the spei results in Figures 2 and 3 depend on the exat spei ation of the poliy rule. With the original Taylor rule (with response oe ient of.5 on in ation and 0.5 to the output gap), for example, the responses of some variables notably the output gap and in ation would be more di erent aross models. Other features whih are often employed to inrease the realism of models with nominal rigidities, e.g. habit formation, ould generate further di erenes aross models. Nevertheless, Figures 2 and 3 suggest that it may be very di ult to disriminate empirially aross models without looking also at nanial variables, suh as interest rate spreads or the stok of loans. They also suggest that the existene of redit fritions is not a su - ient ingredient for nanial variables to play a quantitatively important role in shaping the monetary poliy transmission mehanism. At least in our set-up, even if nanial variables do reat endogenously to eonomi developments and do play a diret role in the way shoks are transmitted through the eonomy, they modify little the reation of output and in ation to "standard" maroeonomi shoks. In spite of the results in Figures 2 and 3, however, redit fritions turn out to be important in two respets. First, they modify the objetive of monetary poliy ompared to the ase of fritionless nanial markets. Seond, they beome important when shoks whih a et the maroeonomy originate in nanial markets. We analyze these two impliations of redit fritions in the remainder of the paper. 4 Seond order welfare approximation Following Woodford (2003), we obtain a poliy objetive funtion by taking a seond order approximation to the utility of the eonomy s representative agents. Sine our eonomy is populated by households and entrepreneurs, the poliy objetive funtion will be a weighted average of the (approximate) utility funtions of these two agents. The approximation to the objetive funtion takes a form whih nests the one in the benhmark new-keynesian model (see Woordford, 2003) as a speial ase. 2

22 The appendix shows that the present disounted value of soial welfare an be approximated by W t0 ' & "{ 2 E t 0 X t=t 0 t t0 L t # + t:i:p: (29) where & is the weight assigned to households utility, t:i:p: denotes terms independent of poliy and L t b 2 t + Y 2 ' eyt y (e t y ) 2 e by e t + y be t + e & be t + & 2 be2 t (30) where be t is log-entrepreneurial onsumption (in deviation from the steady state) and {,, 3, 4 are parameters de ned in the appendix. The rst three terms in equation (29) are ommon to the new-keynesian model. Intuitively, soial welfare dereases with variations of in ation around its target, and of the output gap around its (non-zero) steady state level. The rst reason for disliking variations in the output gap is that households wish to smooth their supply labour. The seond reason is that households also wish to have a smooth onsumption pattern over time. Unlike in the benhmark new-keynesian model, the onsumption smoothing motive only applies to households onsumption, e 2 t, rather than to total output, beause entrepreneurs are indi erent about the timing of their onsumption. The main di erene relative to the benhmark New-Keynesian model with fritionless - nanial markets is in the additional terms now appearing in the welfare approximation. The rst of the additional terms, whih is proportional to by e t + y be t, ontributes positively to welfare. The presene of this term is again related to households onsumption smoothing motive (this term would disappear if households utility were linear, i.e. when = 0). Under an output expansion indued by a tehnology shok, an inrease in entrepreneurial onsumption absorbs aggregate resoures and thus ontributes to smooth the path of households onsumption over time. The last two terms in equation (30), whih are proportional to be t and be 2 t, have an ambiguous impat on welfare, depending on whether the weight of households in soial welfare is larger or smaller than a ertain threshold & = +. The quadrati term in entrepreneurial onsumption is due to two reasons. On the one hand, from the aggregate resoure onstraint households onsumption is a onave funtion of 22

23 entrepreneurial onsumption, so that households dislike utuations in be t. In a seond order approximation of the resoure onstraint, this shows up in a negative quadrati term in be t. On the other hand, entrepreneurial utility is, by onstrution, a onvex funtion of the logarithm of entrepreneurial onsumption. This implies that entrepreneurs will have a preferene for volatile log-onsumption. The sign of the term proportional to be 2 t on soial welfare depends on whih one of these two e ets prevails, whih is in turn determined by the relative importane of households in soial utility. The linear term in entrepreneurial onsumption is also the result of ontrasting fores. A higher value of entrepreneurial onsumption is obviously bene ial for entrepreneurial welfare. At the same time, any entrepreneurial onsumption is detrimental for households welfare, as it subtrats from the eonomy resoures whih ould be onsumed by households. The net e et of this term on welfare is again determined by the relative weight of households in soial utility. The linear term in entrepreneurial onsumption an have strong e ets on optimal poliy. Their dependene on the relative weight of households and entrepreneurs in soial welfare is, however, unappealing. For this reason, in the rest of our derivations we selet the partiular weight & = +, suh that the ontrasting motives of households and entrepreneurs anel out and rst order terms disappear entirely from soial welfare. Under this speial weight &, the loss funtion simpli es to L t b 2 t + Y 2 + e Y ' eyt y 2 Y + 2 by e t + y Yt e Rt b 3t b eyt y e 2 brt + 3t b 4 b t (3) where equation (7) was used to write entrepreneurial onsumption in terms of the nominal interest rate R b t, the redit spread b t and the exogenous shok b t. This expression allows us to perform a omplete derivation of optimal poliy using the linearized poliy equations (23)-(25). Compared to the ase of the standard new-keynesian model, the novel terms in expression (3) are those with a oe ient proportional to e= (notie that these terms vanish when entrepreneurs disappear from the eonomy and Y = ). These novel terms inlude rst, within square brakets, elements proportional to the squared nominal interest rate and the squared loan-deposit rate spread. Hene, the presene 23

24 of asymmetri information in the eonomy introdues diretly both an interest rate smoothing and a "spread smoothing" motive for optimal poliy. At the same time, these terms are relatively small in our alibration, where households onsumption takes up the lion share of output. Under normal irumstanes, therefore, the interest rate smoothing onern is unlikely to be predominant ompared to the objetive of maintaining prie stability. The term within square brakets in equation (3) also inlude a number of ross produts between endogenous variables. More spei ally, a planner would be averse to a positive ovariane between the nominal interest rate and the loan-deposit rates spread. A high ovariane would inrease the volatility of entrepreneurial onsumption, with negative spillovers on households onsumption-smoothing motive. The planner would however not be averse to a positive ovariane between, on the one side, the output gap, on the other side, either the interest rate or the loan-deposit rates spread. These ovarianes would be welome to the extent that they help to smooth utuations in the output gap. Finally, the last term in equation (3) shows that inreases in the nominal interest rate and in the redit spread have a positive e et on welfare, if they are aompanied by an inrease in the e ient level of output i.e. an inrease in produtivity. The reason is that households are willing to reap the bene ts of the higher produtivity on real wages, but wish to smooth their onsumption pattern over time. A higher inidene of monitoring osts in the eonomy at a time of high produtivity helps to ahieve the latter objetive. 5 Optimal poliy 5. Disretion When the welfare funtion an be approximated as in (29) and (3), the problem of the entral bank is to maximize that objetive, subjet to the system of equilibrium onditions (23)-(25). The appendix shows that, in the speial ase in whih ' = 0 and =, the target rule whih haraterizes the disretionary equilibrium takes the simple form b t = e by e t + y h eyt y e Y brt + 3 b t i e Y 4b t for parameters and e de ned in the appendix. 24

25 Note that, in the fritionless ase, = beomes b t = eyt and e = 0 so that the optimality ondition whih orresponds to standard results in the new-keynesian ase see, for instane, Woodford (2003, hapter 7, p. 47). This target riterion implies that the entral bank will hoose to engineer a onstant, positive in ation rate, given the output-in ation trade-o impliit in the Phillips urve. Any rise in in ation above that level would be met by a poliy response suh as to produe a negative output gap. In our model, the target riterion whih would be followed by a entral bank under disretion is a eted by the existene of nanial fritions. While the output gap remains important, both other endogenous variables and shoks limit the ability of the entral bank to use the output gap to ahieve the desired level of in ation. A surge in in ation ould also be ountered through ations whih a et the spread b t and the nominal interest rate. In addition exogenous shoks, inluding both the nanial shok b t and tehnology shoks (through the e ient level of output b Y e t ), a et the target riterion. This implies that the optimal in ation rate varies in the fae of these shoks. Di erently from the benhmark new- Keynesian ase, some temporary deviations from the entral bank s objetive may oasionally be desirable. y 5.2 Optimal monetary poliy under ommitment We haraterize numerially the optimal monetary poliy under ommitment in the speial ase haraterized above in whih linear terms disappear from the quadrati approximation of the welfare funtion. Under this assumption, steady state in ation is zero and the linear system in equations (23)-(25) is a orret approximation. To haraterize more general ases, we need a full seond-order approximation of the poliy equations, whih we perform using Dynare. 2 In all ases, we onentrate on optimal poliy under a timeless perspetive, as in Woodford (2003). Figure 4 displays impulse responses to a tehnology shok when monetary poliy is set optimally. One again, for the tehnology shok we ontrast optimal poliy in the redit 2 We ompute the rst order onditions of the welfare maximisation problem of the poliy maker using Giovanni Lombardo s lq_solution routine available at 25

26 hannel model with the optimal poliies whih would arise in a model with the ost-hannel and in the standard new-keynesian model. 3 Compared to the results in Figure 2, optimal poliy leads to more signi ant di erenes in the three models onsidered here. As is well-known, optimal poliy would ensure omplete prie stability and full stabilization of the output gap in the new-keynesian model. The poliy interest rate would fall on impat and then return slowly to the baseline. Under the redit hannel, near-full in ation stabilization remains optimal in response to tehnology shoks. However, the path of the poliy interest rate whih ahieves this outome would be somewhat di erent. The poliy rate is kept onstant for one period, before reahing levels roughly onsistent with those in the new-keynesian model. Partly as a result of the slowed monetary easing, output inreases less than in the e ient equilibrium and a negative output gap ensues. The impulse response of the output gap highlights the di erenes between our model and the ost hannel model. In the latter ase, the impat redution in the nominal interest rate is more marked even if not as aggressive as in the new-keynesian benhmark and strongly expansionary, so that the output gap inreases after the tehnology shok. This is not the ase in our model beause of the inrease in the spread, whih remains proylial as in the simple rule benhmark. Figure 5 displays impulse responses to a positive shok to b t. This shok is representative of a broader array of " nanial shoks" whih ould be de ned in our model, notably shoks to the subsidy, or to the standard deviation of idiosynrati shoks. All these shoks would produe similar impulse responses. The inrease in b t generates immediately an inrease in the loan-deposit rate spread the shok is in fat normalized to produe a perentage point inrease in the spread. The larger spread ats like a lassial ost-push shok. It depresses households onsumption and output while reating in ationary pressures. Under a Taylor rule, the net e et of the shok is a perentage point inrease in in ation, in spite of a progressive poliy tightening in nominal terms. At the same time, there is a pronouned fall in households onsumption, beause of the output loss due to bankrupties. The amount of redit also falls after the shok. 3 In all ases, we assume the existene of a steady state subsidy whih eliminates rst-order terms in output from the seond-order expansion of individuals utility. The subsidy is slightly di erent in the three ases: it is equal to = ( ) in the new-keynesian model, R= ( ) in the ost-hannel model, and q= ( ) in our model. 26