Optimal Monetary Poliy in a Model of the Credit Channel Fiorella De Fiore European Central Bank Oreste Tristani y European Central Bank 9 July 8 Preliminary and Inomplete Abstrat We onsider a simple extension of the basi New-Keynesian setup where nanial markets are imperfet. 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 show that our loglinearized model nests the ase with fritionless nanial markets. A key di erene is that marginal osts inrease with the output gap but also with the redit spread and the nominal interest rate. Seond, we nd that both tehnology and nanial market shoks generate a trade-o between output and in ation stabilisation. Third, we show that the presene of nanial market imperfetions and endogenous variations in redit spreads an be relevant for the haraterization of optimal monetary poliy. Keyworks: optimal monetary poliy, nanial markets, asymmetri information JEL odes: E5, E44 Diretorate General Researh, European Central Bank. Email: orella.de_ ore@eb.int. y Diretorate General Researh, European Central Bank. Email: oreste.tristani@eb.int.
Introdution Central banks devote muh e ort to evaluate the nanial position of households, rms and - nanial institutions, and to monitor the evolution of redit aggregates and interest rate spreads. One reason is that these fators a et the prospets for in ation and output and are therefore important for monetary poliy deisions. During several historial episodes, entral banks showed sharp reations to hanges in nanial onditions. This ourred in the US during the late 98s, when banks experiened large loan losses as 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 (6), this slowdown ontributed to the FOMC deision to redue the federal funds rate well below what suggested by an estimated Taylor rule. A seond example is provided by the nanial market turmoil, initiated in the rst half of 7 with the deterioration in the performane of nonprime mortgages in the US. In August 7, the FOMC justi ed a ut in the disount rate of 5 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. The importane of nanial onditions for the ondut of monetary poliy an hardly be rationalized in the ontext of the frameworks ommonly used for monetary poliy analysis, suh as the New-Keynesian model with nominal prie rigidities (see e.g. Woodford (3)). These frameworks typially rule out a role for redit aggregates and interest rate spreads by assuming fritionless nanial markets. We onsider the simplest possible extension of the basi New-Keynesian setup, where - nanial markets are imperfet. As in Bernanke, Gertler and Gilhrist (989) and Carltrom and Fuerst (997, 998), we model an environment where rms need to borrow funds in advane of prodution but they are redit onstrained. We deviate from those papers by assuming that loans are denominated in nominal - rather than real - terms. In our eonomy, 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 default risk, so that banks nd it optimal to harge a lending rate whih is above their marginal ost (the deposit 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 a redued form of our model is similar in struture to the one arising in the New-Keynesian setup with fritionless nanial markets. As in the standard ase, it is haraterized by an intertemporal IS equation, a New-Keynesian Phillips urve and a poliy rule. The main di erene is that marginal osts now inlude the output gap but also the redit spread (i.e. the di erene between the loan rate and the poliy instrument) and the nominal interest rate gap. These additional terms re et the existene of information asymmetries and redit onstraints. Flutuations in the redit spread and in the poliy interest rate a et in ation and output by hanging the availability of redit in the eonomy. Seond, we nd that both tehnology and nanial market shoks at as exogenous "ostpush" fators, i.e. they generate a trade-o between output and in ation stabilisation. Moreover, the existene of asymmetri information and the nominal denomination of debt introdue a role for the redit spread and the nominal interest rate as endogenous "ost-push" fators, whih do not appear in the benhmark model with fritionless nanial markets. Third, we analyze optimal monetary poliy when rms are redit onstrained and banks set loans and lending rates as a result of an optimal ontrat. We show that the presene of nanial market imperfetions and endogenous variations in redit spreads hange the haraterization of optimal monetary poliy. For a speial ase of the model, we obtain a simple quadrati approximation to the welfare funtion. We nd that welfare is redued by volatility of in ation and the output gap, as in the benhmark ase with fritionless nanial markets. In the presene of debt, however, welfare also tends to be redued by the volatility of the nominal interest rate and of the redit spread. In ongoing work, Curdia and Woodford (8) haraterize the 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 er in the underlying soure of nanial fritions and in the main ndings. While the presene of nanial market imperfetions does not a et the optimal target riterion for monetary poliy in their model, it does a et it in our environment. Other related works are Ravenna and Walsh (6) and Faia and Monaelli (6). The main di erenes are that the former paper haraterizes optimal monetary poliy when rms borrow in advane of prodution but, due to the absene of default risk, the ost of funds is the risk-free rate. The latter paper ompares the welfare losses of alternative simple interest rate rules in a model similar to ours, without hraterizing optimal monetary poliy. 3
The paper proeeds as follows. In setion, we desribe the environment. In setion, we derive the onditions haraterizing the equilibrium of the eonomy when nanial ontrats are written in nominal terms. In setion 3, we log-linearize the model, we derive its redued form and we ompare it to the redued form arising in the e ient equilibrium. 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. For a partiular ase of our model eonomy, we obtain a simple quadrati approximation of the soial welfare, whih we ompare to the one arising under fritionless nanial markets. We also 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. In setion 5, we haraterize numerially optimal monetary poliy under ommitment. In setion 6, we onlude. 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 arrangement between lenders and borrowers in the presene of asymmetri information and ostly state veri ation. 4
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.. 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 (; ) of di erentiated onsumption goods produed by retail rms, Z t = " t (j) " " " dj ; with " > : P t (j) denotes the prie of good j; and P t = CES aggregator. Nominal wealth at the beginning of period t + is given by h R i P t (j) " " dj is the prie of the 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 ; () where R m t denotes the interest paid on money holdings. 5
The household s problem is to maximize preferenes, de ned as ( ) X E o t [u ( t ) + (m t ) v (h t )] ; (3) where u > ; u < ; m ; mm < and v h > ; v hh > ; and m t M t =P t denotes real balanes. The problem is subjet to the budget onstraint De ne t Pt P t and m;t Rt Rm t R t M t + D t + E t [Q t;t+ A t+ ] W t ; (4) : 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 lead to the demand funtions where P t (j) is the prie of good j. Pt (j) " t (j) = t ; (8) P t. 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, using a linear tehnology y i;t = A t! i;t l i;t : (9) Here A t is an aggregate exogenous produtivity shok and! i;t is an iid produtivity shok with distribution funtion and density funtion. At the beginning of the period, eah rm reeives a nominal endowment P t that 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 has no impliation for the dynamis of the model, we assume for simpliity that the transfer is xed aross rms and time periods. 6
The assumption that rms reeive an endowment from the government at the beginning of the period is made for simpliity as it enables to provide an analytial haraterization of the optimal poliy in the presene of redit onstraints and information asymmetry. It is standard in this literature to assume that entrepreneurs 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 environment, the absene of apital aumulation implies that the persistene of the endogenous variables merely re ets the persistene of the exogenous shoks. Nonetheless, nanial fritions still a et the eonomy through the redit onstraint and the endogenous spread over the risk-free rate harged by nanial intermediaries to rms, whih re ets the existene of default risk... Fator 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. () 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 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 (). Here the expetation E[] is taken with respet to the idiosynrati shok unknown at the time of the fator 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 () 7
implying that E [y i;t ] = A t l i;t = A t x i;t w 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 () 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 nd it pro table to use the entire amount of funds and 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 output is onsumed in the monitoring ativity. If the realization of the idiosynrati 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 (; ) 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 standard ostly state veri ation (CSV) problem. The solution is a debt ontrat (see e.g. Gale and Hellwig (985)) that is haraterized by three properties. First, the repayment to the FI is onstant in states when monitoring does not our. Seond, the rm is delared bankrupt if and only if the xed repayment annot be honoured. Third, in ase of bankrupty, the FI ommits to monitor and ompletely seizes the output in the hands of the rm. 8
Reall that the presene of ageny osts implies that y i;t =! i;t t q t x i;t : De ne f (!) g (!; ) Z! Z!! (d!)! [ (!)]! (d!) (!) +! [ (!)] as the expeted shares of output aruing respetively to an entrepreneur and to a lender, after stipulating a ontrat that sets the xed repayment at P t t q t! it x i;t units of money. In ase of default, a stohasti fration t of the input osts x i;t ; measured in units of money, is used in monitoring. We assume that t follows a AR proess given by t = t + ;t : 3 At the individual rm level, total output and the government subsidy are split between the entrepreneur, the lender, and monitoring osts so that f (! t ) + g(! t ) = t (! t ) : The optimal ontrat is the pair (x i;t ;! i;t ) that solves the following CSV problem: max P t t q t f(! i;t )x i;t subjet to P t t q t g(! i;t )x i;t R d t P t (x i;t ) (4) P t [f (! i;t ) + g (! i;t ; t ) + t (!)] (5) P t t q t f(! i;t )x i;t P t (6) The optimal ontrat maximizes the entrepreneur s expeted pro ts subjet to the lender being willing to lend out funds, (4), the feasibility ondition, (5), and the entrepreneur being willing to sign the ontrat, (6). Notie that the intermediary needs to pay bak to the household a gross return equal to the safe interest on deposits, Rt d : Sine in equilibrium R t = Rt d ; the nanial intermediary s expeted return on eah unit of loans annot be lower than R t. The optimality onditions an be written as q t = x i;t = R t t (! i;t ) + t f(! ; (7) i;t)(! i;t ) f! (! i;t ) R t : (8) R t q t g (! i;t ; t ) 3 This assumption re ets the large time variation in bankrupty osts doumented by Natalui et al (4). 9
From equation (7), it follows that the terms of the ontrat depend on the state of the eonomy only through the aggregate mark-ups t and q t and the return R t. Hene, they are the same for all rms,! i;t =! t : Sine initial wealth is also the same aross rms, it follows from equation (8) that the size of the projet is the same aross rms. The onditions an thus be rewritten as R t q t = t (! t ) + t f(!t)(!t) (9) f! (! t) R t x t = : () R t q t g (! t ; t ) Notie 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. We an use the expression above to obtain the spread between the loan rate and the risk-free rate, t Rl t ; Rt d t =! t g(! t ; t ) : ()..3 Entrepreneurs 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. If they default, the bank ompletely seizes rm s prodution, sells it to the retail setor and pays an amount t q t x t in monitoring osts. Pro ts of entrepreneurs are entirely alloated to nal onsumption goods Z 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 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 get e t = f (! t ) q t x t ; ()
where e t = R e i;tdi is the aggregate entrepreneurial onsumption of the nal onsumption good. Using equations (9)-(), we an rewrite aggregate entrepreneurial onsumption as e t = R t H ( t ;! t ) (3) where.3 Retail rms H ( t ;! t ) + t (!t) f! (! t) : 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 = " Y t (j) " " " dj ; (4).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= 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) # ;
The rst-order onditions of the rm s pro t maximization problem imply that P Pt = " E t k= k P Q t+k t;t+k P P " t " Y t+k t+k P t " P E t k= k Pt Q " t;t+k Y t+k Now de ne ;t P t P t Y t + E t ;t Y t + E t ( X P " t+k ( 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 obtain the following onditions = " " t ;t + ( ) " ;t ;t = Y t + E t " t+ Q t;t+ ;t+ t ;t = Y t + E t " t+ Q t;t+ ;t+ : h Pt " + ( ) (P t ) "i " ; ".3. Prie dispersion Reall that the aggregate retail prie level is given by P t = h R i 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 (p t (j)) " dj: De ne also the relative prie dispersion term as s t Z (p t (j)) " dj: This equation an be written in reursive terms as.4 Market learing s t = ( ) Market learing onditions are listed below. Money: " " " t + " t s t : M s t = M t ;
Bonds: A t = Labor: h t = l t Loans: d t = x t Wholesale goods: Retail goods: y t = Z Y t (j) dj Y t (j) = t (j) + e t (j) ; for all j:.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. 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 Appendix A. 3
.6 The system in redued form After some algebra, the system of equilibrium onditions that haraterizes the evolution of the aggregate variables (together with an appropriate monetary poliy rule) an be written as + e b t = ( + ') a t ( + ) b t ( + ') Y b t Rt b + b t = + ' + Y 3 g b g t (5) by t e b R t ( + ') a t b t (6) by t = E tyt+ b brt E t b t+ + E bt+ t t b + e E t b t+ b t + g g E t b t+ b t (7) b t = b t + E t b t+ (8) where the oe ients ; ; 3 ; and are de ned in Appendix B. Notie that and are positive oe ients. In our eonomy, the mark-up 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 t = P t =P t : Equation (5) 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 of 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, any inrease in the poliy rate represents an additional ost whih is overed by harging a higher prie of wholesale goods. Equation (6) shows that the spread between the loan rate and the poliy rate inreases with 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 rate. An inrease in this latter generates a redution in the demand of nal goods and thus in 4
the demand of input for their prodution (wholesale goods). For a given amount of internal funds, leverage and the risk of default fall, reduing the spread. Equation (7) is the IS urve. The hange in the demand of nal onsumption goods reat to the real interest rate, as in the standard ase with fritionless nanial market. However, it also reats to hanges in the spread and in the mark-up. Equation (8) is the expetation-augmented Phillips urve, representing aggregate supply. It di ers from the formulation with fritionless nanial markets beause b t appears instead of aggregate demand, Y b t. However, we have already notied that the markup varies inversely with aggregate demand. A derease in the markup signals exess demand and generates in ationary pressures..6. Case with fritionless nanial markets We onsider the speial ase when monitoring osts are zero, i.e. t = ; 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. When t = ; for all t, f (! t ) + g (! t ; t ) = : Also, sine there are no monitoring osts, banks set! t as high as possible subjet to the onstraint that the rm is willing to sign the ontrat, i.e. f(! t ) = q t x t This maximizes banks pro ts, as they an size the prodution of all defaulting rms at no ost. In suh equilibrium, g (! t ; t ) = e t = t y t Y t = t + : Moreover, from the bank s zero pro t ondition, we have x t = R t R t q t t y t : b t = The log-linearized system an then be written as (R ) y ( + + ') + ( + ') by t + y RR q b (R t + ) y + ( + ') a t 5
by t = E tyt+ b brt E t b t+ b t = b t + E t b t+ In the limiting ase where ; R t = q t : The system then beomes b t = ( + ') b Y t b Rt + ( + ') a t by t = E tyt+ b brt b t = b t + E t b t+ E t b t+ The equations above oinide with the redued-form system of equilibrium onditions obtained by Ravenna and Walsh (6) 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. 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 (5)-(8) in terms of gaps from the e ient equilibrium, in whih t = and pries are exible. We the gap between atual and e ient output as e Y t = b Y t rewritten as b t = + ' + Y ey t e R e t + b ;t b Y e t : The system an then be ey t = E tyt+ e + e brt 'e + e 'e E bt+ t E t b t+ t b e 'e E t brt+ b Rt + t (9) b t = ( + ') e Y t + ( + ) b t + e R t + E t b t+ b ;t (3) where b ;t + ' + ' b ;t e + ' + ' e 'e E t b;t+ (E t a t+ a t ) + 3 (E t a t+ a t ) + + e b ;t g b g t (3) + ' a t b + ' t (3) + e g + 'e g E t b t+ b t (33) 6
Equation (9) 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 this model, however, the output gap depends also on the expeted hange in the nominal interest rate and in the redit spread, as well as on the omposite shok t. In the Phillips urve (3), the rst determinant of in ation is the output gap. This is a standard term, but enters here with a di erent oe ient re eting the presene of entrepreneurs in the eonomy. The term b ;t denotes a "ost-push" fator, whih is a ombination of exogenous shoks that do not generate variations in the e ient level of output. This fator is a linear ombination of tehnology shoks and nanial shoks. Intuitively, the tehnology shok has two separate e ets in our eonomy. On one hand, it generates an e ient variation in the e ient level of output, as in the standard model with fritionless nanial markets. This variation has no e et on the output gap and on in ation. On the other hand, however, an inrease in a t inreases the revenues of indebted rms and (for a given inidene of the monitoring osts) redues the ourrene of default. This variation a ets urrent output but not the e ient level of output, thus reating in ationary pressures. Finally, the seond and third terms in equation (3) re et endogenous "ost-push" e ets of the redit spread and the real interest rate, introdued by the existene of redit market imperfetions. 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, inluding an in ation response oe ient of.5 and an output gap response oe ient of.5. The strutural parameters are set in line with the literature. Following Levin, Natalui and Zakrajsek (4) we set long-run monitoring osts at 5% of the rm s output (i.e. = :5). We then alibrate the standard deviations of idiosynrati shoks (! ) and the subsity so that approximately % of rms go bankrupt eah quarter and that the steady state spread is equal to approximately % per year. 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-optimise pries = :66, implying that pries are hanged on average every 3 quarters. Finally, we set the persistene of tehnology and monitoring ost shoks to.9. 7
Figure displays the response of a few variables to a positive % tehnology shok under the Taylor rule. As is typially the ase, the shok produes downward pressure on in ation (denoted as "inf" in the gure). However, ontrary to what happens in the standard new- Keynesian setting with Calvo priing, the fall in in ation is not due to a negative output gap (y_gap). It is aused instead by a negative interest rate gap (r_gap), whih has an impat on in ation beause of the ost-hannel. At the same time, the shok tends to inrease the spread between loan rates and the poliy interest rate by the same amount as in the e ient equilibrium (delta_gap). Impulse responses to a positive % monitoring osts shok under the Taylor rule are presented in gure. The shok is ontrationary, but the onsumption and output gaps derease only slightly. Nevertheless, the shok has a diret impat on marginal osts, whih leads to a poliy tightening and an opening of the interest rate gap. 4 Seond order welfare approximation Following Woodford (3), 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, 3) as a speial ase. The appendix shows that the present disounted value of the soial welfare an be approximated by W t ' & "{ E t X t=t t where & is the weight assigned to households utility L t Y and & be t + & be t t L t # + t:i:p + O 3 ; (34) + b t + Y 'e Yt + e t Y Y by n t be t (35) be t = b R t + 3 b t 4 b t for parameters {,, 3, 4 de ned in the appendix. Intuitively, the entral bank s utility dereases with variations of in ation around its target, and of output around its e ient level. Unlike in the benhmark new-keynesian model, the 8
onsumption smoothing motive only applies to households onsumption, e t, rather than to total output, beause entrepreneurs are indi erent to shifting onsumption over time. 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. Welfare is also redued by utuations in the nominal interest rate b R t and in the redit spread b t (through the be t term). In spite of the fat that we have assumed the existene of a steady state subsidy to eliminate the rst order e ets of distortions on output, the general loss funtion (35) inludes linear terms due to the presene of asymmetri information. The osts of this distortion are of rst order and they only vanish when entrepreneurs disappears from the eonomy (note that the loss boils down to the usual b t + (' + ) = e Y t when steady state output is equal to households onsumption Y = ). In general, we will therefore need a seond order approximation of private agents deision rules in order to analyse welfare. Under the speial weight & = ( + ), however, rst order terms disappear and the loss funtion simpli es to L t b t + Y 'e Yt + Y Y e Y t brt + 3 b t 4 b t Y Y by t n brt + 3t b Expression (??) highlights the diret in uene on welfare of monitoring shoks b t. It also shows that inreases in the nominal interest rate and in the redit spread have also 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 ahieving the latter objetive. (36) 5 Optimal poliy 5. Disretion When the welfare funtion an be approximated as in (34) and (36), the problem of the entral bank is to maximize that objetive, subjet to the system of equilibrium onditions (??)-(3). Denote as ;t ; ;t and 3;t the Lagrangean multipliers assoiated to the onstraints. Taking rst-order onditions and solving for the multiplies, we an haraterise the disretionary equilibrium through one equation relating the evolution of in ation to the evolution of the 9
other variables. Suh equation takes the form b t = eyt ; b t ; R b t Notie that, in the fritionless ase, the optimality ondition amounts to b t = e Y t ; where > is an appropriately de ned oe ient. This is the ase, for instane, in Woodford (3, hapter 7, p. 47). It implies that the entral bank manages the output gap in order to ahieve the onstant in ation rate given by b t = ( ) + In our model, the target riterion (i.e. the relation between in ation and the output gap that ahieves the disretionary optimum) is a eted by the existene of nanial fritions. Notie that both the endogenous spread b t and the exogenous nanial shok b t limit the ability of the entral bank to use the output gap to ahieve the desired level of in ation. Moreover, the desired in ation level is not onstant. by e t : To realize this, plug the optimality ondition into the aggregate supply equation. Imposing rational expetations of private agents, i.e. b t = E t b t+ ; we obtain the optimal in ation rate for a entral bank that operates under disretion: b t = ( + ') Y b t e + er t + ( + ( + ') ) b t ( + ') g b t : ( )(+e) + ( + ') When nanial fritions reate an endogenous, time-varying spread between the loan rate and the risk-free rate, the optimal in ation rate under disretion re ets variations of the real interest rate gap, the redit spread and the exogenous shoks. b ;t 6 Optimal monetary poliy in the benhmark model We haraterize numerially the optimal monetary poliy under ommitment in the ase of the model desribed in setion, where deposits pay a nominal return Rt d : The optimal reation of monetary poliy an be derived for the general ase in whih linear terms are present in the quadrati approximation to the welfare funtion. Figure 3 displays impulse responses to a tehnology shok with optimal poliy under ommitment (and adopting a timeless perspetive, as in Woodford, 3). Compared to the results
in Figure, optimal poliy tends to lose welfare-relevant gaps muh more quikly and the in- ationary e ets of the shok are more ontained. After periods, the output gap and the spread between loan and poliy interest rates are losed. However, optimal poliy also produes a small positive interest rate gap, whih has an impat on marginal osts and therefore generates some positive in ationary pressures. This transmission hannel prevents a quik and omplete stabilisation of in ation in our model. 7 Conlusion [TO BE COMPLETED...] Appendix A. The log-linearized system of equilibrium onditions The equilibrium an then be haraterized as the solution to the following system of loglinearized equations in the variables nb t ; Y b t ; b h t ; bq t ; b t ; b t ; R b t ; b o t ; t b = Yt b (y ) b t + b t g g b t bq t = a t b t ' b h t b t b ht = b Y t a t br t = bq t b t + 3 b t by t = b t + b R t + 4 b t + 5 b t br t = E t b t+ + (E t b t+ b t ) b t = b t + E t b t+ where ( )( ), plus a monetary poliy rule.
B. The oe ients of the log-linearized equations The oe ients of the system (5)-(??) are given by = = 3 = 4 = 5 = C. Welfare approximation 4 f!! y g! > f! f! q! f! R > ( g! ) g f! g f!! f! + f ( g! ) f! f! f!! 4 g g + e f + f 3 g g : 3 5 q R f! +! (f! + ) > f! ( g! ) f! + 4 Our monetary poliy objetive is derived as the seond order approximation to a weighted average of the utilities of the household and of the entrepreneur, i.e. ( ) X E o t [&U t + ( &) Ut e ] where & is the weight of the utility of households in the poliy objetive. Households temporary utility an then be approximated as U t ' U + u b t + + u u b t v h h b ht + + v hhh b h t v h where hats denote log-deviations from the deterministi steady state and and h denote steady state levels. Similarly, entrepreneurial temporary utility Ut e an be expanded as Ut e ' e + be t + be t where e is the steady state level of entrepreneurial onsumption., households temporary utility an be rewritten as Under the funtional form U t = t U t ' h+' t +' h +' + ' + b t h +' b ht + ( ) b t h +' ( + ') b h t
We an express hours, households onsumption and entrepreneurial onsumption as h t = s ty t A t t = Y t e t h i where H ( t ;! t ) + t (!t) f! (! t) : e t = R t H ( t ;! t ) The period aggregate utility an be approximated as &U t + ( &) Ut e ' & h +' + ' + ( &) e +& b t + ( &) ebe t & h+' b h t + & ( ) b t h+' & ( + ') b h t + ( &) ebe t Now note that the resoure onstraint t = Y t e t an be approximated to seond order as b t = Y b Y t e be t + Y Y b t e be t b t while the prodution funtion implies simply b ht = a t + bs t + b Y t : where It follows that utility an be rewritten as &U t + ( &) Ut e { h +' h +' ' & bs Y t & by t e (& ( &) ) be t + be t h +' &b t & ( + ') Y h +' +& ( + ') a ty b t + tips Now onsider the FOC { & h ' t + ' = A t q t t h +' + ( &) e by t 3
Under perfet ompetition and fritionless redit markets, rms set the real wage at the marginal produt of labor, w t = A t : In our model, equation () implies that w t = At q t t : We provide households with a subsidy suh that h ' = w t ; and = q: Moreover, we assume that is small. It follows that in suh a steady state h +' = Ah = Y : In addition, we fous on the ase of a speial Pareto weight & = +, whih allows us to ignore rst order terms in entrepreneurial onsumption. It follows that the loss an be written as &U t + ( &) U e t { ' Y bs t + Y 'b Yt Y Y b t e be t + Y ( + ') a ty b t +t:i:s:p:+o 3 Now note that in the fully-e ient steady state we would have ( + ') a t = e t + 'Y e t whih an be used to substitute out the tehnology shok a t from the loss funtion. In addition, a rst order approximation to the equation for prie dispersion, of rst-order in bs t and seond-order in b t takes the form bs t ' This latter an be integrated forward to obtain " b t + bs t : bs t ' tx " t s=t s b s + t t+ bs t + O 3 : Multiplying this by t t and realizing that multiples of bs t are t.i.p., we get X t=t t t bs t ' ( ) ( ) " X t t=t b t t + t:i:p: + O 3 as long as bs t = O : Finally, note that entrepreneurial onsumption an be written as be t = b R t + 3 b t 4 b t 4
where 3 = 4 =!! f! (f! + ) ( g! ) > + 3g + f! g It follows that the approximated welfare funtion an be written as in the main text, for Referenes Y " ( ) ( ) > [] Bernanke, B.S., Gertler, M. and S. Gilhrist. The Finanial Aelerator in a Quantitative Business Cyle Framework. In: Taylor, John B. and Woodford, Mihael, eds. Handbook of maroeonomis. Volume C. Handbooks in Eonomis, vol. 5. Amsterdam; New York and Oxford: Elsevier Siene, North-Holland, 999, pp. 34-93. [] Carlstrom, C.T., and T. Fuerst. Ageny Costs, Net Worth, ad Business Flutuations: A Computable General Equilibrium Analysis. Amerian Eonomi Review, 997, 87, pp. 893-9. [3] Carlstrom, C.T., and T. Fuerst. Ageny Costs and Business Cyles. Eonomi Theory, 998,, pp. 583-597. [4] Curdia, V. and M. Woodford. Credit ritions and Optimal Monetary Poliy. Mimeo, 8. [5] Faia, E. and T. Monaelli. Optimal interest rate rules, asset pries, and redit fritions, Journal of Eonomi Dynamis and Control, 7, Vol. 3-, p. 38-354. [6] Levin, A.T., Natalui, F., and E. Zakrajsek. "The Magnitude and Cylial Behavior of Finanial Market Fritions." Sta WP 4-7, Board of Governors of the Federal Reserve System, 4. [7] Ravenna, F. and C. Walsh. "Optimal Monetary Poliy with the Cost Channel." Journal of Monetary Eonomis, 53, 6, pp.99-6. [8] Rudebush, Glenn (6). Monetary Poliy Inertia: Fat or Fition? International Journal of Central Banking, Vol. (Deember), 85-35. 5
Impulse responses to a positive tehnology shok under the Taylor rule x 3 _gap x 3 y_gap 4 6 8 4 6 8 x 3 e_gap x 3 inf 4 6 8 4 6 8 x 3 delta_gap x 3 r_gap 4 6 8 4 6 8
3 3 3 Impulse responses to a positive monitoring ost shok under the Taylor rule x 4 _gap x 4 y_gap 3 4 6 8 4 6 8 x 4 e_gap x 4 inf 3 4 6 8 4 6 8 x 4 delta_gap x 4 r_gap 3 4 6 8 4 6 8
Impulse responses to a positive tehnology shok under optimal poliy x 4 _gap x 4 y_gap 4 6 8 4 6 8 x 4 e_gap x 4 inf 4 6 8 4 6 8 x 4 delta_gap x 4 r_gap 4 6 8 4 6 8