Banks, Credit Markets Frictions and Business Cycles

Transcription

1 Banks, Credit Markets Frictions and Business Cycles Ali Dib International Economic Analysis Department Bank of Canada July 11, 29 Abstract The current financial crisis highlights the need to develop DSGE models with real-financial linkages and an active banking sector. This paper proposes a fully micro-founded framework that incorporates optimizing banks, the interbank market, and the credit market into a DSGE model, and evaluates the role of banks and financial shocks in the U.S. business cycles. We assume two types of heterogenous banks that offer different banking services and interact in an interbank market. Loans are produced using interbank borrowing and bank capital, subject to the bank capital requirement condition. Banks, have monopoly power and set nominal deposit and lending prime rates, choose their portfolio compositions and their leverage ratio, measured by loan-to-bank-capital ratio, and may endogenously default on interbank borrowing and bank capital returns. The model also includes financial and quantitative and qualitative monetary easing shocks. The main findings are that: (1) The model captures the key features of the U.S. economy; (2) bank behavior substantially affects credit supply conditions, and the transmission of different shocks; (3) the banks leverage ratio is procyclical; and (4) financial and quantitative monetary easing shocks have significant effects on the U.S. business cycle fluctuations, while qualitative monetary easing shocks (swapping banks assets for bank capital injections) have no impacts on the real economy. JEL classification: E32, E44, G1 Keywords: Banks; Interbank market; Bank capital; Credit; Financial shocks; Monetary policy. I am grateful to Ron Alquist, Ricardo Caballero, Lawrence Christiano, Carlos de Resende, Mick Devereux, Philipp Maier, Virginia Queijo von Heideken, Julio Rotemberg, Lawrence Schembri, Moez Souissi, and seminar participants at the Bank of Canada, the workshop on Economic Modelling and the Financial Crisis jointly organized by the Bank of Canada and the IMF, the Canadian Economic Association annual meeting, MIT, IMF, and the Federal Reserve Bank of Richmond for their comments and discussions. The views expressed in this paper are those of the author and should be attributed to the Bank of Canada. International Economic Analysis Department, Bank of Canada. 234 Wellington St. Ottawa, ON. K1A G9, Canada. ADib@bankofcanada.ca, Phone: , Fax:

2 1. Introduction The ongoing global financial crisis underscores the need to develop DSGE models with realfinancial linkages and an active banking sector. Such a model would allow an empirical evaluation of banks role and behavior in the transmission and propagation of supply and demand shocks, and an assessment of the importance of financial shocks as a source of business cycles. The banking sector, however, has been ignored in most DSGE models used for policy purposes; whereas, in the literature, financial frictions are usually modeled only on the demand side of the credit market using either the Bernanke, Gertler and Gilchrist (1999) financial accelerator mechanism (BGG, hereafter) or the Iacovello (25) framework. 1 financial crisis, real-financial linkages have become the focus of attention. In light of the ongoing This paper proposes a microfounded framework that incorporates an active banking sector, interbank market, bank capital, and a credit market into a DSGE model with a financial accelerator à la BGG (1999). This model represents the first attempt to incorporate credit and interbank markets into a DSGE model to study real-financial linkages. The model is then used to evaluate empirically the role of profit-maximizing banks in business cycles and in the transmission and propagation of shocks to the real economy, to assess the importance of financial shocks in explaining macroeconomic fluctuations, and to examine the potential role of quantitative and qualitative monetary easing policies in offsetting the real impacts of the financial crisis. The paper is related to the following studies: Christiano, Motto and Rostagno (28), Cúrdia and Woodford (29a,b), de Walque, Pierrard and Rouabah (28), Gerali, Neri, Sessa and Signoretti (29), and Goodhart, Sunirand and Tsomocos (26). In contrast to the previous studies that examine the role of bank capital in the business cycle fluctuations, this paper introduces bank capital to satisfy the bank s capital requirement condition, which is a pre-condition to operate and make loans to entrepreneurs. 2 The proposed framework can address issues relevant to monetary policy and financial stability, such as: (1) the role of the banking sector in the economy and its real effects; (2) the impact of banks and credit markets on business cycle fluctuations; (3) the conduct of monetary policy during a financial crisis; and 1 For example, Carlstrom and Fuerst (1997), Cespedes, Chang and Velasco (24), Elekdag, Justiniano and Tchakarov (26), and Christensen and Dib (28). 2 For example, Holmstrom and Tirole (1997), Meh and Moran (24), Markovic (26), and others. 1

3 (4) the role of bank capital in determining credit supply conditions. The basic model is a DSGE model for a closed economy similar to Christensen and Dib (28), which is based on BGG (1999). The key additions to this model are the supply-side of credit market and an active banking sector. Thus, the model incorporates an optimizing banking sector with two different types of monopolistically competitive banks, an interbank market, endogenous credit supply, and bank capital. We consider an economy inhabited by two heterogenous households (workers and bankers); two heterogenous banks (saving and lending banks); three goods qproducers, entrepreneurs, capital producers, and retailers; a central bank; and the government. Households (workers and bankers) differ in their preferences, degree of risk aversion, and access to financial markets. Workers supply labor services to entrepreneurs, hold cash money, and save only in deposits at saving banks. Bankers, however, own the banks, accumulate bank capital, and save in government bonds. Banks supply different banking services, and the two types of banks interact in the interbank market. 3 Banks are monopolistically competitive. They, therefore, have the monopoly power to set nominal deposit and lending prime rates (subject to menu costs of adjusting these rates). To introduce heterogeneity in the the banking sector, we distinguish between two types of banks: Saving banks and lending banks. Saving banks collect deposits from workers, set the nominal deposit rates paid, and choose the composition of their portfolio (composed of risk-free assets and risky interbank lending) to maximize profits. 4 Lending banks borrow from saving banks on the interbank market and receive bank capital from bankers to satisfy the bank s capital requirement condition. This condition imposes a minimum level of bank capital should be held by lending banks in order to provide loans to entrepreneurs. Lending banks can receive, if needed, liquidity injections from the central bank and swap a fraction of their loans for bank capital injections. They use interbank borrowing and the total value of their bank capital to make loans according a Leontief technology that is subject to an intermediation process shock. Following Goodhart et al. (26), we assume endogenous strategic or necessary defaults on bank capital and interbank borrowing, optimally 3 We assume heterogenous banks to incorporate an interbank market where different banks can interact. 4 It is assumed that deposits at saving banks are fully insured, investing in risk-free assets involves payments of an insurance premium, while interbank lending are subject to a positive probability of default of lending banks. 2

4 chosen by lending banks; however, when defaulting, banks pay expected convex penalties during the next period. In addition, banks optimally choose their leverage ratio, that is, the ratio of loans to bank capital, subject to the maximum leverage ratio imposed by regulators. assume the presence of convex gains of holding bank capital in excess of the required level. This implies that variations in the banks leverage ratio directly affect the marginal cost of raising bank capital. 5 Entrepreneurs own the capital stock used in the production of whole sale goods, which are sold to retail firms on a perfectly competitive market. They are risk neutral and subject to idiosyncratic productivity shocks. They borrow from lending banks to finance partly their purchase of capital. In contrast to BGG (1999), the debt contracts are in nominal terms, which implies debt-deflation effects. The presence of asymmetric information between entrepreneurs and lenders creates financial frictions, which make entrepreneurial demands for capital depend on their financial position and on the lending prime rate set by lending banks. Capital producers build new capital and sell it to entrepreneurs, subject to investment adjustment costs. Retailers set nominal prices in a staggered fashion à la Calvo (1983) and Yun (1996). There is a government and a central bank that conducts monetary policy with three instruments. First, the policy rate, which is the interbank rate, conducted following a standard Taylor rule: The central bank adjusts short-term nominal interest rates in response to inflation and output changes. Second, quantitative monetary easing shocks that are liquidity the central bank can inject into lending banks (through the interbank market). This liquidity injection is associated with newly created money and, therefore, expands the balance sheets of the central bank and lending banks. Third, qualitative monetary easing shocks that the central bank can use to enhance credit supply conditions, by swapping a fraction of lending banks loans for bank capital injections. This policy aims to increase bank capital holdings by lending banks. 6 Liquidity injections from the central bank responds to banks emergency financing needs. Through these two last channels, the central bank can serve as the lender of last resort to lending banks in times of crisis. 5 The cost of bank capital depends on the bank s capital position. If banks hold excess bank capital, the marginal cost of raising bank capital on the market is lower, since banks are well capitalized. 6 Swapping banks assets for bank capital injection does not change the balance sheets of lending banks and the central bank; it changes only their assets compositions. Therefore, there is no newly created money associated with this liquidity injection. We 3

5 In the proposed framework, the banking sector affects credit market conditions and, thus, the real economy through the following channels: (1) variations in bank capital and bank capital price expectations; (2) monopoly power in setting nominal deposit and lending interest rates with nominal rigidities that imply moving interest rate spreads over business cycles; 7 (3) the optimal allocation of the banks portfolio between interbank lending and risk-free asset holdings; (4) the optimal choice of the banks leverage ratio that is subject to the bank capital requirement condition; (5) the default risk channels that arise from endogenous strategic or necessary defaults on interbank borrowing and bank capital returns; and (6) marginal costs of raising bank capital. In addition, central bank can inject liquidity into lending banks through the interbank market and swapping a fraction of lending banks loans for bank capital injections. The economy is subject to two supply shocks technology and investment-efficiency shocks; three demand shocks monetary policy, government spending, and preferences shocks, four financial shocks risk, financial intermediation process, and quantitative and qualitative monetary easing shocks. Supply and demand shocks are commonly used in the literature; however, financial shocks require some explanation. Risk shocks are modelled as shocks to the elasticity of the risk premium that affect the external finance costs of entrepreneurs. They are represented as shocks to the standard deviation of the entrepreneurial distribution, as Christiano et al. (29) argue, to agency costs paid by lending banks to monitor entrepreneurs output, and/or to entrepreneurs default threshold. 8 These shocks may be interpreted as exogenous changes in the confidence level of banks with credit risks of their borrowers and the health of the economy, thus affecting external costs of entrepreneurial borrowing. Shocks to financial intermediation process are exogenous events that affect loan production technology (credit supply) of lending banks. They may represent technological advances in the intermediation process and approximate perceived changes in creditworthiness. 9 Finally, quantitative and qualitative monetary easing shocks are used by the central bank to provide liquidity to the banking system and enhance banks conditions to provide credits to entrepreneurs and to offset the negative 7 See Cúrdia and Woodford (29) for the importance of moving spreads on monetary policy. 8 As shown in BGG (1999), the elasticity of the external finance premium to the entrepreneurs leverage ratio depends on the standard deviation of the entrepreneurial distribution, the agency cost parameter, and entrepreneurs default threshold. 9 Advances in financial engineering, credit rationing, and highly sophisticated methods for sharing risk are examples of intermediation process shocks. 4

6 impacts of the financial crisis. The main findings show that the model can reproduce most of the salient features of the U.S. economy: Key macroeconomic volatilities, autocorrelations, and correlations with output. Also, an interesting result is that the presence of an active banking sector with sticky deposit rates is a welfare improving. The welfare cost of uncertainty is lower in the model with the banking sector than without it. This finding results from the rigidity in setting deposit rates that reduce the volatility of the marginal rate of substitution of workers consumption, which leads to workers consumption smoothing. Bankers act, in this case, as insurers of workers consumption. This result is robust, even when simulating the models using different compositions of the shocks disturbing the economy. In addition, the presence of optimizing banks reduces the welfare cost associated with financial shocks to about one fourth of the model without the banking sector. Therefore, the main role of the banks in this economy is to reduce the negative impact of uncertainty, given by the presence of the different structural shocks, particularly financial shocks. Thus, banks offer some insurance against risks. The presence of the banking sector also affects the transmission and propagation of supply, demand, and financial shocks. In addition, financial shocks largely contribute to business cycle fluctuations: Financial shocks explain at least 4% of variations of the key macroeconomic variables. Thus, disturbances in the banking sector may be a substantial source of macroeconomic fluctuations and economic turmoils. We also find that the banks leverage ratio is procyclical, indicating that banks are willing to extend loans during booms and shrinking their credit supply during recessions. As well, the probabilities of bank defaults on interbank borrowing and bank capital are negatively correlated with output and hence counter-cyclical. Overall, the presence of an active banking sector, as proposed in this model, strongly affects the transmission and propagation of different shocks on output, investment, and capital prices. In particular, the banking sector reduces the effects of demand and financial shocks on all real variables. Impulse responses show that effects of risk shocks on output, investment, labor, and entrepreneurial net worth in the model without an active banking sector are almost twice as large as in the model with it. The paper proceeds as follows. Section 2 presents the model. Section 3 discusses the calibration procedure and data. Section 4 describes and discusses the empirical results. Section 5

7 5 presents the conclusions of the study. 2. The Model The economy is inhabited by two heterogenous households (workers and bankers) that differ in their preferences, degrees of risk aversion, and access to financial markets. The banking sector consists of two types of heterogenous monopolistically competitive banks (saving and lending banks) that offer different banking services and interact in an interbank market. Thus, the assumption of bank heterogeneity is to incorporate an interbank market where banks can exchange their received deposits. As in BGG (1999), the production sector consists of entrepreneurs, capital producers, and retailers. Finally, there is a central bank and a government. 2.1 Households Workers Workers derive utility from total consumption, C w t ; real money balances, M c t ; and leisure, 1 H t, where H t denotes hours worked. The workers preferences are described by the following expected utility function: The single-period utility is V w = E βwu t (Ct w, Mt c, H t ). (1) t= u( ) = e t 1 γ w ( C w t (C w t 1 )ϕ ) 1 γw + ϖ(m t c 1 υ ) 1 υ + η(1 H 1 ς t), (2) 1 ς where ϕ (, 1) is a habit formation parameter; γ w is a positive parameter denoting the workers risk aversion and the inverse of the elasticity of intertemporal substitution of consumption; υ denotes the money-interest elasticity; and ς is the inverse of the elasticity of intertemporal substitution of leisure. On the other hand, the parameters ϖ and η measure the weight on real cash balances and leisure in the utility function, respectively. e t is a taste shock that follows an AR(1) process. The representative worker enters period t with D t 1 units of real deposits in saving banks and Mt 1 c units of real money balances held outside of banks that do not earn interest.deposits 6

8 pay the gross nominal interest rate R D t set by saving banks between t and t During period t, workers supply labour to the entrepreneurs, for which they receive real labor payment W t H t, where W t is the economy-wide real wage. Furthermore, they receive dividend payments, Π R t, from retail firms, as well as a lump-sum transfer from the monetary authority, T t, and pay lump-sum taxes to government, T w t. Workers allocate their funds to private consumption C w t, real money holdings M c t, and real deposits, D t. Their budget constraint in real terms is Ct w + Mt c + D t W t H t + RD t 1 D t 1 π t where π t+1 = P t+1 /P t is the gross inflation rate. + M c t 1 π t + Π R t + T t T w t, (3) A representative worker household chooses C w t, M c t, H t, and D t to maximize its expected lifetime utility, Eq. (1), subject to the single-period utility function, Eq. (2), and the budget constraint, Eq. (3). The first-order conditions for this optimization problem are: ( ) [ C w 1 γw ( e t C w ) ] 1 γw t (Ct 1 w β w ϕe t e t+1 t+1 )ϕ (Ct w = C w )ϕ t λ w t ; (4) ϖ M c t R D t = λ w t ( 1 1 R D t = β w E t ( λ w t+1 π t+1 ) ; (5) η (1 H t ) ς = λw t W t ; (6) λ w ) t, (7) where λ w t is the Lagrangian multiplier associated with the budget constraint. Eq. (4) states that the marginal utility of consumption is a function of current and expected changes in consumption. This dynamic relation results from the habit formation assumption. Eq. (5) relates real cash demand to the current marginal utility of consumption and the nominal return on deposits. Eq. (6) equates the marginal rate of substitution between consumption and labour to the real wage. Finally, Eq. (7) relates the marginal rate of substitution to the real interest rate on deposits Bankers Bankers (bank owners) own the two types of banks from which they receive profits. They consume, have access to non-contingent government bond market, and accumulate bank capital 1 In this economy, Rt D is different from the return rate on government bonds. 7

9 supplied to lending banks to satisfy bank capital requirement for a contingent bank capital return. It is assumed that bankers preferences depend only on consumption and are given by V b = E βb (C t u t b t= ). (8) The single-period utility function is u( ) = e t 1 γ b ( C b t (C b t 1 )ϕ ) 1 γb, (9) where γ b is a positive structural parameter denoting bankers risk aversion and the inverse of the elasticity of intertemporal substitution. e t denotes the preference shock that follows an AR(1) process. Bankers enter period t with (1 δt 1 Z )Z t 1 units of bank capital stock, whose price is Q Z t in period t, where δt 1 Z is a probability of Banks default on bank capital occurred at the end of the period t 1. Bank capital pays a contingent nominal return rate Rt Z between t 1 and t. Bankers also enter period t with B t 1 units of real government bonds that pay the gross risk-free nominal interest rate R t between t and t + 1. During period t, bankers receive profit payments, Π sb t and Π lb t from saving and lending banks, and pay lump-sum taxes to government, T t b. They allocate these funds to consumption Ct b, real government bonds B t, and real bank capital acquisition Q Z t Z t. We assume quadratic adjustment costs to alter the bank s capital stock. 11 Bankers budget constraint in real terms is C b t + Q Z t Z t + B t = R t 1B t 1 π t + (1 δ Z t 1) RZ t Q Z t Z t 1 π t χ Z 2 ( πt Z t Z t 1 π ) 2 Q Z t Z t + Π sb t + Π lb t T b t. (1) A representative banker chooses Ct b, B t, and Z t in order to maximize its expected lifetime utility Eq. (8) subject to Eq. (9) and the budget constraint, Eq. (1). The first-order 11 We interpret these adjustment costs as costs paid to brokers or the costs of collecting information about the banks balance sheet. 8

10 conditions for this optimization problem are: ( ) Ct b 1 γb ( ) C b 1 γb e t β b ϕe t e t+1 t+1 = Ct b λ b t; (11) (C b t 1 )ϕ β b E t { λ w t+1 Q Z t+1 π t+1 (C b t )ϕ [ ] λ b t λ b = β b E t+1 t ; (12) R t π t+1 [ ( ) ( ) ]} (1 δt Z )Rt+1 Z πt+1 Z t+1 πt+1 Z 2 t+1 + χ Z π = λ w t Q Z t Z t [ 1 + χ Z ( πt Z t Z t 1 π ) πt Z t Z t 1 Z t ] ; (13) where λ b t is the Lagrangian multiplier associated with the bankers budget constraint. Eq. (11) determines the marginal utility of banker s consumption. Eq. (12) relates the marginal rate of substitution to the real interest rate on bonds. Finally, Eq. (13) corresponds to the optimal dynamic evolution of the bank capital stock. Combining conditions (12) and (13) yields the following condition relating return on bank capital Rt Z to the risk-free interest rate on government bonds R t : { [ Q Z ( ) ( ) ]} E t+1 t Q Z (1 δt Z )Rt+1 Z πt+1 Z t+1 πt+1 Z 2 t+1 + χ Z π t Z t Z t = R t [ 1 + χ Z ( πt Z t Z t 1 π ) πt Z t Z t 1 ]. (14) This condition implies three channels through which bank capital movements affect the real economy. First, the price expectation channel that arises from expectations of capital gains or losses from holding bank capital shares, due to expected changes in the prices of [ bank capital E t Q Z t+1 /Q Z ] t. This channel implies that the efficient market hypothesis does not hold in the short run. Second, the adjustment cost channel, a result of the information asymmetry between bankers and banks, implies changes in current and expected stocks of bank capital given by the terms χ Z ( ). The presence of adjustment costs are necessary to reduce the information asymmetry and are interpreted as costs to enter into the bank capital market. Finally, the default risk channel arises from the existence of the probability of default on bank capital repayment, δ Z t >, decided by the lending banks. This default probability is counter- 9

11 cyclical. Therefore, movements in bank capital, caused by macroeconomic fluctuations, have direct impacts on bank capital accumulation and consequently on credit supply conditions. 2.2 Banking sector The banking sector consists of two types of heterogenous profit-maximizing banks: Saving and lending banks. These two types of banks interact only in the interbank market Saving banks There is a continuum of saving banks, operating in a monopolistically competitive environment and collecting deposits D t from workers. We assume that all deposits are fully insured. Each bank j (, 1) sets the deposit interest rate Rj,t D paid on deposits and chooses the optimal allocation of its portfolio between lending a fraction s j,t of deposits on the interbank market, D j,t = s j,t D j,t, (interbank lending) to lending banks, and investing the fraction (1 s j,t ) in risk-free assets, Bt sb, (government bonds). Each period, there is a probability δt D that lending banks default on their interbank borrowing. When investing in non-risky assets, saving banks must pay an insurance premium (cost of holding risk-free assets). The interbank rate R t is set by the central bank. Table 1 displays the balance sheet of the j th saving bank. 12 Table 1: Saving bank s balance sheet Assets Liabilities Interbank lending: Dj,t Government bonds: B sb j,t Deposits: D j,t Given monopolistic competition and the imperfect substitution between deposits, the j th saving bank faces the following deposit supply function, that is increasing in the relative deposit interest rate across period. As in Gerali et al. (29), the individual deposit supply is D j,t = 12 Note that e D j,t = s j,t D j,t and B sb j,t = (1 s j,t )D j,t where s j,t (, 1). ( ) R D ϑd j,t D t, (15) R D t 1

12 where D j,t is deposits supplied to bank j, while D t denotes total deposits in the economy; and ϑ D > 1 is the elasticity of substitution between different types of deposits. 13 Also, there is a quadratic adjustment cost of intertemporally varying the deposit interest rate. This rigidity allows an interest rate spread that evolves over the cycle. We assume adjustment costs à la Rotemberg (1982), given by ( R D j,t Ad RD j,t = φ R D 2 R D j,t 1 1) 2 D t, (16) where φ R D bank is max {s j,t,r D j,t } E > is an adjustment cost parameter. The optimization problem of the j th saving t= β t b λb t { [(1 sj,tδt D )R t Rj,t D ] Dj,t χ } s ((1 s j,t )D j,t ) 1+ɛs Ad RD j,t, 1 + ɛ s subject to (15) and (16). Because bankers are the sole owners of banks, the discount factor is the stochastic process β t b λb t, where λ b t denotes the marginal utility of bankers consumption. 14 χ The terms s 1+ɛ s ((1 s j,t )D j,t ) 1+ɛ s represents the costs of holding risk-free assets and the payment of an insurance premium, where χ s > is a parameter determining the steady-state level of these costs. The parameter ɛ s denotes the elasticity of this insurance primium with respect to holding risk-free assets. In the symmetric equilibrium, the first-order conditions of this optimization problem, with respect to s t and R D t, are: s t = 1 1 ( ) δ D 1/ɛs t R t ; (17) D t 1 + ϑ D Rt D = ( 1 s t δt D ϑ D ( φ R D ϑ D R D t χ s ) )Rt χ s (1 s t ) 1+ɛ s Dt ɛs ) R D t 1 1 R D t R D t 1 + β bφ R D ϑ D ( ) R D t+1 R D 1 t+1. (18) 13 This supply function is derived from the definition of aggregate supply of deposits, D t, and the corresponding deposit interest rate, Rt D, in the monopolistic competition framework, as follows: D t = R 1 1+ϑD D ϑ D j,t dj! ϑ D ϑ D +1 and R D t = interest rate faced by each saving bank j (, 1). 14 Saving banks take R t and δ D t as given when maximizing their profits. R 1 1 RD 1+ϑ D 1+ϑ j,t dj D, where D j,t and Rj,t D are the supply and deposit R D t R D t 11

13 Condition (17) describes the interbank lending supplied by the saving banks; it states that the fraction s t of deposits allocated to interbank lending is decreasing in the probability of default on interbank lending and in the interbank rate, while it is increasing in total deposits. An increase in s t leads to an expansion in credit supply. Condition (18) defines the deposit interest rate, R D t, as a mark-down of the interbank rate. 15 Thus, increases in the riskiness of interbank lending, a higher δ D t, encourage saving banks to increase their risk-free holdings and reduce their interbank lending. Also, an increase in the interbank rate, the return rate on risk-free assets, reduces interbank lending supply. Nevertheless, an increase in total deposits expands interbank lending, leading to an expansion in credit supply conditions. This framework, therefore, adds two channels through which saving banks behavior affects credit supply conditions and the real economy. First, by setting deposit return rates in a monopolistically competitive market, combined with the nominal rigidity of deposit rates, saving banks influence the intertemporal substitution of consumption across periods and thus help in consumption smoothing. 16 Second, by optimally dividing deposits between interbank lending and risk-free asset holding, saving banks affect credit supply conditions by expanding or tightening credit market conditions Lending banks There is a continuum of lending banks, indexed by j (, 1), that operate in a monopolistically competitive market of providing loans to entrepreneurs. The j th lending banks borrows D j,t from saving banks on the interbank market and demand bank capital Z j,t from bankers, paying the bank capital price Q Z t and a non-contingent return rate R Z t+1.17 We assume that bank capital is financial assets held by the lending bank as government bonds that pay the riskfree rate R t. Each lending bank j can receive liquidity injections from the central bank, m j,t,(quantitative monetary easing shocks). Also, if needed, bank j may swap a fraction of its loans for bank capital injections, x j,t, from the central bank to respond to its emergency 15 This equation allows us to derive a New-Philips curve relating R D t to R D t 1, R D t+1, and R t. 16 Since the marginal rate of substitution equals the deposit rate, the sluggishness in this rate affects the intertemporal substitution between current and future consumption. 17 In this economy, interbank borrowing is always equal to interbank lending, so that e D t = s td t. 12

14 financing needs (qualitative monetary easing shocks). Through these channels, therefore, the central bank can serve as lender of last resort to lending banks in times of crisis. Each lending banks have monopoly power when setting their lending prime rate R L j,t subject to quadratic adjustment costs when intertemporally varying their lending rates. These banks also decide their defaults on their interbank borrowing and bank capital payments. Defaults can be either strategic or mandatary (when banks cannot afford to repay their debt). In addition, lending banks optimally choose their leverage ratio (ratio of loans to bank capital), taking into account the maximum ratio imposed by the regulators. We assume that having bank leverage ratio below the maximum required level implies quadratic gains. These gains directly affect the marginal cost of raising bank capital. To produce loans L j,t for entrepreneurs, the lending bank j uses interbank borrowing, D j,t, plus liquidity injection received from the central bank (quantitative monetary easing shocks), m j,t, and the total market value of its bank capital Q Z t Z j,t, plus capital injections received from the central bank, x j,t. We assume that banks use the following Leontief technology to produce loans: 18 L j,t = min { Dj,t + m j,t ; κ j,t ( Q Z t Z j,t + x j,t ) } Γ t, (19) where κ j,t < κ is the bank s j leverage ratio that is optimally chosen and κ is the maximum leverage ratio imposed by regulators. 19 When κ j,t < κ, the bank j accumulates bank capital beyond the required level. The variable Γ t represents a shock to the intermediation process affecting credit supply (loan production). 2 This shock represents the exogenous factors and approximates perceived changes in creditworthiness. Technological advances in the intermediation process can be considered another source of variation in Γ t. The process of loan evaluation certainly has evolved over time, through stochastic technological advances in information services. These variations may represent changes in total factor productivity in the intermediation process. Advances in computational finance and sophisticated methods of sharing risk are examples of this shock. 21 It is assumed that m t, x t, and Γ t exogenously evolve according to 18 Leontief technology implies perfect complementarity between deposits and bank capital when producing loans and satisfies the bank capital requirement condition. 19 Note that κ j,t is the ratio of bank s loans to its bank capital. Therefore, it is the inverse of the bank capital ratio. 2 Γ t is a shock to lending banks balance sheet. 21 This shock may reflect lending banks perception of the risk in the economy. Banks may underevaluate 13

15 AR(1) processes. 22 Using Leontief technology to produce loans implies perfect complementarity between interbank borrowing and bank capital. Furthermore, the marginal cost of producing loans is simply the sum of the marginal cost of interbank borrowing and that of raising bank capital. The latter is adjusted by the Bank s leverage ratio. Table 2 shows the j th lending bank s balance sheet in period t. Assets Table 2: Lending bank s balance sheet Liabilities Loans: L j,t x j,t Government bonds: B lb j,t = QZ t Z j,t + x j,t Interbank borrowing: Dj,t Bank capital: Q Z t Z j,t Central bank s money injection: m j,t Other terms: (Γ t 1)( D j,t + m j,t ) We note that a shock for swapping a fraction of loans for bank capital injection, x j,t, modifies only the composition of the bank lending assets. Nevertheless, shocks of liquidity injections, m j,t, and financial intermediation, Γ t, affect the total values of lending banks balance sheet, implying balance sheet expansion. At each period, the lending bank j sets the lending prime rate, Rj,t L, as a mark-up of the marginal cost of producing loans and the marginal costs of adjusting this nominal rate across periods. Bank j also optimally chooses κ j,t subject to the bank capital requirement condition, κ; and probabilities of default on bank capital and interbank borrowing, δ Z j,t and δd j,t, respectively. As in Gerali et al. (29), the adjustment costs associated with changes in lending prime rates are modelled à la Rotemberg (1982) and given by ( R L j,t Ad RL j,t = φ R L 2 R L j,t 1 1) 2 L t, (2) where φ R L > is an adjustment cost parameter. In addition, when choosing κ j,t < κ, there are quadratic gains since banks are well-capitalized. These gains are modelled using the following function: χ κ ( κ κj,t 2 κ Q Z t Z j,t ) 2, where χκ > is a parameter determining the steady-state value (overevaluate) the risk during booms (recessions). This exogenously increase (decrease) loan supply. During booms, Γ t > 1, so it is a credit easing shock, while during recession Γ t < 1 means a credit rationing shock. 22 The steady state values of m t and x t are zero, while that of Γ t is equal to unity. 14

16 of κ t. When κ j,t = κ, the bank s leverage ratio meets the required level exactly, and there is no gains associated with it. However, when κ j,t < κ, the bank leverage ratio is below the requirement and banks are well-capitalized, making raising bank capital cheaper. Wellcapitalized banks have lower costs of raising capital. Thus, the optimal choice of the banks leverage ratio affects the costs of lending directly through its impact on bank capital raising costs. 23 The lending bank optimization problem is to choose κ j,t, δj,t D, δz j,t, and RL j,t. The lending banks profit maximization problem is max {R L j,t,κ j,t,δ D j,t,δz j,t } E χ δ D t= β t b λb t 1 + ɛ δ D + χ ( κ κ κj,t 2 κ { R L j,tl j,t (1 δ D j,t)r t Dj,t R t m j,t [ (1 δ Z j,t)r Z t+1 R t ] Q Z t Z j,t ( δ D j,t 1 R t 1 Dj,t 1 π t ) 1+ɛδ D χ δ Z 1 + ɛ δ Z Q Z t Z j,t ) 2 (R L j,t R t )x j,t Ad RL j,t subject to (19), (2), and the following demand function for loans: ( δ Z j,t 1 R Z t Q Z t 1 Z j,t 1 }, π t ) 1+ɛδ Z L j,t = ( ) R L ϑl j,t L t, (21) R L t where ϑ L > 1 is the elasticity of substitution between different types of loans. 24 The discount factor is given by the stochastic process β t b λb t, where λ b t denotes the marginal utility of consumption of bankers the owners of the lending banks. The terms R t m j,t represents the cost of liquidity injections received from the central bank [ ] (quantitative monetary easing shocks), while (1 δj,t Z )RZ t+1 R t Q Z t Z j,t denotes the net cost of bank capital, which depends on payment of non-defaulted fraction net of the return from 23 Equation (27) hereafter displays the relation between the marginal cost of loans and the cost of raising bank capital. 24 This demand function is derived from the definition of aggregate demand of loans, L t, and the corresponding lending prime rate, Rt L, in the monopolistic competition framework, as follows: L t = R 1 1 ϑl L ϑ L j,t dj! ϑ L 1 ϑ L R and Rt L 1 = rate faced by each lending bank j (, 1). RL1 ϑ L j,t 1 1 ϑ dj L, where L j,t and Rj,t L are the loan demand and lending 15

17 ( ) holding bank capital as government bonds, Bt lb χ. The terms δ D δj,t 1 D R t 1 e 1+ɛ D δd j,t 1 1+ɛ δ D π t and ( ) 1+ɛδZ are increasing in the defaults on interbank borrowing and bank χ δ Z 1+ɛ δ Z δ Z j,t 1 RZ t QZ t 1 Z j,t 1 π t capital occurred during the previous period. The terms (R L j,t R t)x j,t denote the effects of qualitative monetary easing shocks on bank s profits, where R L j,t R t is the cost of swapping a fraction of loans for bank capital held as government bonds. In a symmetric equilibrium, where all banks take the same decisions, the first-order conditions of this optimization problem, with respect to κ t, δt D, δt Z, and Rt L ( κ t = κ 1 Γ t(rt L 1) χ κ Q Z t Z t [ δt D π t+1 = E t R t Dt [ δt Z = E t ( Rt χ δ D ( π t+1 Rt χ δ Z R Z t+1 QZ t Z t are: ) ; (22) ) 1 ɛ δ D ] ; (23) ) 1 ] ɛ δ Z ; (24) φ R L Rt L = 1 + ϑ L ϑ L 1 (ζ t 1) ϑ L 1 Rt 1 L 1 Rt 1 L (25) [( ) ] + β bφ R L R L ϑ L 1 E t+1 R L t 1 t+1, (26) R L t where [ ( ζ t = Γ 1 t R t + Rt+1 Z R t (Rt L 1) κ κ ) ] t Q Z t, (27) κ κ t is the marginal cost of producing loans. In addition, the Leontief technology implies the following implicit demand functions of interbank borrowing and bank capital: ( R L t R L t L t = Γ t ( D t + m t ); (28) L t = Γ t κ t ( Q Z t Z t + x t ). (29) Eq.(22) describes the banks leverage ratio as a function of different macroeconomic variables. It shows that κ t is decreasing in the return rate of loans, R L t, the required leverage ratio, κ t, and the loan technology shocks, Γ t ; whereas it is increasing in bank capital and bank capital prices. Eq. (23) indicates that the default on interbank borrowing increases in expected inflation, while it decreases in interbank rate and total interbank lending. An increase in expected 16 ) R L t

18 inflation reduces future default payments. In Eq. (24), the default on bank capital increases in expected inflation, policy rate, and return on bank capital, while it decreases in total value of bank capital held by the bank. Eq. (26) relates lending prime rates to the marginal cost of producing loans and to current costs and future gains of adjusting the lending prime rate. This equation allows us to derive a New-Keynesian Philips curve for the lending prime rate. Eq. (27) indicates that the marginal cost of producing loans depends on the cost of interbank borrowing, R t, and the shodow price of using capital to satisfy the capital requirement condition. In this case, the marginal cost of bank capital is equal to the difference between R Z t+1 and R t, the risky return paid on bank capital and the risk-free return on holding bank capital as government bonds, and the marginal benefit of holding bank capital in excess of required level Production sector Entrepreneurs The entrepreneurs behavior follows BGG (1999). Entrepreneurs manage firms that produce wholesale goods. Entrepreneurs are risk neutral and have a finite expected horizon for planning purposes. The probability that an entrepreneur will survive until the next period is ν, so the expected lifetime horizon is 1/(1 ν). This assumption ensures that entrepreneurs net worth (the firm equity) is never sufficient to self-finance new capital acquisitions, so they issue debt contracts to finance their desired investment expenditures in excess of net worth. At the end of each period, entrepreneurs purchase capital, K t+1, that will be used in the next period at the real price Q K t. Thus the cost of the purchased capital is Q K t K t+1. The capital acquisition is financed partly by their net worth, N t, and by borrowing L t = Q K t K t+1 N t from loan-making banks. These banks obtain funds from the interbank market and set the lending prime rate R L t, which is a mark-up of the marginal cost of producing loans. Therefore, in this framework, the reference rate is R L t, rather than the economy s nominal risk-free rate of bond return between t and t + 1, R t. The entrepreneurs demand for capital depends on the expected marginal return and the expected marginal external financing cost at t + 1, E t F t+1, which equals the real interest 25 If κ t = κ, ζ t = Γ 1 t ˆRt + κ 1 Q Z t `RZ t+1 R t. 17

19 rate on external (borrowed) funds. Consequently, the optimal entrepreneurs capital demand guarantees that E t F t+1 = E t [ r K t+1 + (1 δ)q K t+1 Q K t ], (3) where δ is the capital depreciation rate. The expected marginal return of capital is given by the right-side terms of (3), in which rt+1 K is the marginal productivity of capital at t + 1 and (1 δ)q K t+1 is the value of one unit of capital used in t + 1. BGG (1999) assume the existence of an agency problem that makes external finance more expensive than internal funds. The entrepreneurs observe their output costlessly, and output is subject to a random outcome. The lending banks incur an auditing cost to observe the output. After observing their project outcome, entrepreneurs decide whether to repay their debt or to default. If they default, the lending banks audit the loan and recover the project outcome, less monitoring costs. BGG solve a financial contract that maximizes the payoff to the entrepreneur, subject to the lender earning the required rate of return. BGG show that given parameter values associated with the cost of monitoring the borrower, characteristics of the distribution of entrepreneurial returns, and the expected life span of firms their contract implies an external finance premium, Ψ( ), that depends on the entrepreneur s leverage ratio. The underlying parameter values determine the elasticity of the external finance premium with respect to the firm leverage. Accordingly, the marginal external financing cost is equal to a gross premium for external funds plus the gross real opportunity costs equivalent to the risk-free interest rate. Thus, the demand for capital should satisfy the following optimality condition: [ ] R L E t F t+1 = E t t Ψ( ), (31) π t+1 ( R L where E t t is an expected real loan-prime rate (with Rt L set by the lending bank and depends on the marginal cost of making loans), and the external finance premium is given by π t+1 ) ( ) Q K rp t Ψ( ) = Ψ t K t+1 ; ψ t, (32) N t with Ψ ( ) < and Ψ(1) = 1, and ψ t represent an aggregate risk shock. 18

20 The external finance premium, Ψ( ), depends on the borrower s equity stake in a project (or, alternatively, the borrower s leverage ratio). As Q K t K t+1 /N t increases, the borrower increasingly relies on uncollateralized borrowing (higher leverage) to fund the project. Since this raises the incentive to misreport the outcome of the project, the loan becomes riskier, and the cost of borrowing rises. 26 following functional form In particular, the external finance premium is assumed to have the ( ) Q K ψt rp t Ψ( ) = t K t+1, (33) where ψ t is a time-varying elasticity of the external finance premium with respect to the entrepreneurs leverage ratio. Following Christiano et al. (28), we assume that ψ t is an aggregate risk shock that follows an AR(1) process. BGG (1999) shows that this elasticity, ψ >, depends on the standard deviation of the distribution of the entrepreneurs idiosyncratic shocks, the agency cost and the entrepreneurs default threshold. Therefore, a shock increasing ψ t may be risen by exogenous increases in the distribution of the entrepreneurs idiosyncratic shocks, the agency cost and the entrepreneurs default threshold. The result is a rise in, ψ t, the aggregate risk, and thus in external finance premium. 27 Aggregate entrepreneurial net worth evolves according to N t N t = νv t + (1 ν)g t, (34) where V t denotes the net worth of surviving entrepreneurs net of borrowing costs carried over from the previous period, 1 ν is the share of new entrepreneurs entering the economy, and g t is the transfer or seed money that new entrepreneurs receive from entrepreneurs who exit. 28 V t is given by V t = [ F t Q K t 1K t E t 1 F t (Q K t 1K t N t 1 ) ], (35) 26 When the riskiness of loans increases, the agency costs rise and the lender s expected losses increase. A higher external finance premium paid by successful entrepreneurs offsets these higher losses. 27 A positive shock to the standard deviation widens the entrepreneurs distribution, so lending banks are unable to distinguish the quality of the entrepreneurs, and increase their external finance premium for all borrowers. Similarly, increases in the agency costs or in the entrepreneurs default threshold rise the riskiness of lending out and results in rise in external finance premium. 28 The parameter ν will affect the persistence of changes in net worth. 19

21 where F t is the ex post real return on capital held in t, and [ ( )] R L E t 1 F t = E t 1 Q K t 1 Ψ t 1 K t ; ψ t 1 π t N t 1 is the cost of borrowing (the interest rate in the loan contract signed in time t 1). Earnings from operations in this period become next period s net worth. In our formulation, borrowers sign a debt contract that specifies a nominal interest rate. 29 The loan repayment (in real terms) will then depend on the ex post real interest rate. An unanticipated increase (decrease) in inflation will reduce (increase) the real cost of debt repayment and, therefore, will increase (decrease) the entrepreneurial net worth. To produce output Y t, the entrepreneurs use K t units of capital and H t units of labor following a constant-returns-to-scale technology: Y t A t K α t H 1 α t, α (, 1), (36) where A t is a technology shock common to all entrepreneurs and it assumed to follow a stationary an AR(1) process. Each entrepreneur sells its output in a perfectly competitive market for a price that equals its nominal marginal cost. The entrepreneur maximizes profits by choosing K t and H t subject to the production function (36). The first-order conditions for this optimization problem are: r K t = αξ t Y t K t ; (37) Y t W t = (1 α)ξ t ; H t (38) Y t = A t Kt α Ht 1 α, (39) where ξ t > is the Lagrangian multiplier associated with the production function (36) and denotes the real marginal cost Capital producers Capital producers use a linear technology, subject to an investment-specific shock Υ t, to produce capital goods K t+1, sold at the end of period t. 29 In BGG, the contract is specified in terms of the real interest rate. 3 We assume that entrepreneurial consumption is small and it drops out of the model. They use a fraction of final goods 2

22 purchased from retailers as investment goods I t, and the existing capital stock to produce new capital goods. The new capital goods replace depreciated capital and add to the capital stock. The disturbance Υ t is a shock to the marginal efficiency of investment. Since I t is expressed in consumption units, Υ t influences the amount of capital in efficiency units that can be purchased for one unit of consumption. Capital producers are also subject to quadratic investment adjustment costs specified as χ I 2 ( It I t 1 1) 2 It. The capital producers optimization problem, in real terms, consists of choosing the quantity of investment I t to maximize their profits, so that: { [ max E t βwλ t w t Q K t Υ t I t χ I I t 2 t= Thus, the optimal condition is 1 Q K t = Υ t χ I ( It I t 1 1 ) It I t 1 + β w χ I E t [ (It+1 ( ) ] } 2 It 1 I t I t. (4) I t 1 ) ( ) 2 It+1 Q K t+1 1 I t I t Q K t λ w t+1 λ w t ], (41) which is the standard Tobin s Q equation that relates the price of capital to the marginal adjustment costs. Note that, in the absence of investment adjustment costs, capital price Q K t is constant and equals 1. We introduce investment adjustment costs in the model to allow capital price variability, which contributes to volatility of entrepreneurial net worth. The quantity and price of capital are determined in the capital market. The entrepreneurial demand curve for capital is determined by equations (31) and (37), whereas the supply of capital is given by equation (41). The intersection of these curves gives the market-clearing quantity and price of capital. Capital adjustment costs slow down the response of investment to different shocks, which directly affects the price of capital. Furthermore, the aggregate capital stock evolves according to K t+1 = (1 δ)k t + Υ t I t χ ( ) 2 I It 1 I t, (42) 2 I t 1 where δ is the capital depreciation rate, and the shock Υ t follows an AR(1) process Retail firms Retail firms purchase the wholesale goods at a price equal to nominal marginal costs, P t ξ t, the marginal cost in the entrepreneurs sector, and differentiate them at no cost. The retail sector 21

23 is used only to introduce nominal rigidity into this economy. They then sell these differentiated retail goods in a monopolistically competitive market. Following Calvo (1983) and Yun (1996), we assume that each retailer cannot reoptimize its selling price, unless it receives a random signal. The constant probability of receiving such a signal is (1 φ p ); and, with probability φ p, the retailer j must charge the same price of the preceding period, indexed to the steady-state gross rate of inflation, π. At time t, if the retailer j receives the signal to reoptimize, it chooses a price P t (j) that maximizes discounted, expected real total profits for l periods, where it will not be allowed to reoptimize. The retailer s optimization problem is [ ] E (β w φ p ) l λ w t+l ΠR t+l (j), (43) max { e P t(j)} subject to the demand function 31 l= Y t+l (j) = ( ) θ Pt (j) Y t+l, (44) P t+l where the retailer s nominal profit function is Π R t+l (j) = ( π l Pt (j) P t+l ξ t+l ) Y t+l (j)/p t+l. (45) The first-order condition for P t (j) is P t (j) = The aggregate price is θ E t l= (β wφ p ) l λ w t+l Y t+l(j)ξ t+l θ 1 E t l= (β wφ p ) l λ w t+l Y t+l(j)π l. (46) /P t+l P 1 θ t = φ p (πp t 1 ) 1 θ 1 θ + (1 φ p ) P t. (47) These equations lead to the following New Keynesian Phillips curve: ˆπ t = β w E tˆπ t+1 + (1 β wφ p )(1 φ p ) φ p ˆξt, (48) where ξ t is the real marginal cost, and variables with hats are log deviations from the steadystate values (such as ˆπ t = log(π t /π)). 31 This demand function is derived from the definition of aggregate demand as the composite of individual final output (retail) goods and the corresponding price index in the monopolistic competition framework, as follows: R 1 Y t+l = Y t+l(j) θ 1 θ θ 1 R 1 1 θ dj and P t+l = P t+l(j) 1 θ 1 θ dj, where Y t+l (j) and P t+l (j) are the demand and price faced by each individual retailer j (, 1). 22