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Financial Collateral and Macroeconomic Ampli…cation Federico Lubelloy Ivan Petrellaz Emiliano Santorox September 8, 2016 Abstract Financial institutions typically resort to collateralized debt to raise funds, providing …nancial assets as a guarantee in case of default on their debt obligations. This paper focuses on the connection between …nancial collateral and macroeconomic volatility. To this end, we design a credit economy á la Kiyotaki and Moore (1997) where bankers intermediate funds between savers and borrowers. Bankers’ability to borrow from savers is bounded by the limited enforceability of deposit contracts. If bankers default, savers acquire the right to liquidate bankers’real and …nancial asset-holdings. However, due to the vertically integrated structure of our credit economy, savers anticipate that liquidating …nancial assets is conditional on borrowers being themselves solvent on their debt obligations. This friction limits the collateralization of bankers’…nancial assets. In this context, decreasing the degree of …nancial collateralization exacerbates steady-state ine¢ ciencies — increasing the gap between borrowers’ and bankers’ marginal product of capital — re‡ecting into a procyclical bank leverage and thus amplifying macroeconomic ‡uctuations. In light of these properties, a banking regulator may help smoothing the business cycle through the introduction of a countercyclical capital bu¤er. JEL classi…cation: E32, E44, G21, G28 Keywords: Financial Collateral; Credit Chain; Liquidity; Macroprudential Policy. We thank Søren Hove Ravn for helpful comments and suggestions. The views expressed in this paper do not necessarily represent the views of the Banque Centrale du Luxembourg or the Eurosystem. y Banque Centrale du Luxembourg. Address: 2, Boulevard Royal, L-2983 Luxembourg. E-mail : [email protected]. z Birkbeck College, University of London and CEPR. Address: Malet Street, WCIE 7HX, London, UK. E-mail : [email protected]. x Department of Economics, University of Copenhagen. Address: Østerfarimagsgade 5, Building 26, 1353 Copenhagen, Denmark. E-mail : [email protected]. 1 1 Introduction Following the path-breaking contribution of Kiyotaki and Moore (1997) (KM, hereafter), a number of papers have incorporated collateral constraints into macroeconomic models to examine the role of limited enforceability of debt contracts in the transmission of various shocks (see Kocherlakota, 2000, Krishnamurthy, 2003, Iacoviello, 2005 and Liu et al., 2013; inter alia). In these models borrowers’ collateral is typically represented by real assets, such as physical capital or housing. In reality, a considerable amount of lending in developed economies is collateralized by …nancial assets, such as corporate or government bonds, mortgage-backed securities, warrants and credit claims. Financial institutions typically resort to collateralized debt to raise funds, providing …nancial assets as a guarantee in case of default on their debt obligations. This is the case for non-traditional banking activities – with sale and repurchase agreements (repos) employed as a main source of funding – as well as for commercial banks –where securitized-banking often supplements more traditional intermediation activities. In fact, banks employ …nancial collateral both for currency management purposes and, more recently, as part of non-standard monetary policy frameworks.1 A vast literature has focused on quantifying the dynamic multiplier emerging from limited enforceability in credit economies á la KM (see Cordoba and Ripoll, 2004, among others). The present paper takes a step aside from this tradition, examining instead the connection between …nancial collateral and macroeconomic volatility. To this end, we design a model where bankers intermediate funds between savers and borrowers. The baseline KM framework is extended to account for limited enforceability of deposit contracts between savers and bankers: as a result, deposits are bounded from above by bankers’ holdings of real and …nancial assets. However, due to the vertically integrated structure of our credit economy, savers anticipate that, in case of bankers’default, liquidating their …nancial assets is conditional on borrowers being themselves solvent on their debt obligations. If savers perceive …nancial assets to be relatively illiquid, they will be less prone to accept these as collateral. We …rst examine how …nancial intermediation and …nancial collateralization impact on the steadystate distribution of real and …nancial resources. In this respect, we report two main results. First, introducing …nancial intermediation in a KM economy produces a more even distribution of real assets, as compared with the original setting, where lenders face a higher steady-state user cost of capital and charge a higher loan rate. In light of this property, embedding …nancially constrained bankers into KM’s framework sets the steady-state equilibrium on a Pareto-superior allocation. Second, envisaging limited enforcement of both deposit and loan contracts induces a spread between the interest rate on loans and that on deposits, whose magnitude is negatively a¤ected by the degree of …nancial collateralization. In other words, when depositors perceive bankers’ …nancial assets to be relatively illiquid, this translates into an increase in the intermediation margin. In turn, this situation a¤ects the steady-state distribution of capital: increasing the pledgeability of …nancial assets reduces the gap between bankers’and borrowers’marginal product of capital – as real assets are redistributed from the latter to the former –thus alleviating the e¤ects of the debt enforcement problem. As for the equilibrium dynamics of the model economy, a low perceived liquidity of bankers’ …nancial assets is shown to amplify the response of gross output to productivity shocks. As in KM, a positive technology shift induces a net transfer of physical capital from the lenders to the borrowers, 1 The set of assets that central banks accept from commercial banks generally includes government bonds and other debt instruments issued by public sectors and international/supranational institutions. In some cases, also securities issued by the private sector can be accepted, such as covered bank bonds, uncovered bank bonds, asset-backed securities or corporate bonds. 2 the latter featuring a higher marginal product of capital. On one hand, this allows borrowers to expand their borrowing capacity. On the other hand, the decline of bankers’real assets is typically counteracted by the expansion of their …nancial assets. However, as these are perceived to be increasingly illiquid, the compensation e¤ect is gradually muted and the dynamics of deposits is eventually tied to that of bankers’ real assets. This implies that bankers have a further incentive to cut on their investment in physical capital to meet borrowers’ higher demand for credit, so that these can expand their capital investment further. In turn, the response of total production – which increases in borrowers’real assets, ceteris paribus –is ampli…ed, relative to situations in which bankers’feature a high degree of …nancial collateralization. In our framework a drop in the perceived liquidity of the …nancial assets held by the banking sector eventually re‡ects into a procyclical leverage, inducing relevant business cycle volatility. Reverting this procyclicality is of key importance to attenuate the magnitude of aggregate ‡uctuations. We discuss how the regulator may successfully attenuate the economy’s response to productivity shifts by devising a capital adequacy requirement on bankers’activity. The resulting constraint is isomorphic to the enforcement constraint arising in the decentralized solution of the model, which results from savers’ predicted outcome of the renegotiation in case bankers default on their debt obligations. In this context we show how the regulator may successfully attenuate the economy’s response to the productivity shock by devising a countercyclical capital bu¤er, as imposed by the Basel III regulatory framework in response to the 2007-2008 …nancial crisis. The number of studies focusing on …nancial collateral is surprisingly limited. The present paper relates to Oehmke (2014), who analyzes the dynamics of repo liquidations in the presence of …nancial intermediaries’default. Unlike our model, the liquidation strategies of repo lenders are driven both by strategic considerations and by lenders’ balance sheet constraints. Parlatore (2015) provides a microfoundation for the use of …nancial assets in the form of collateral, focusing on borrowers’optimal …nancing choice. In this context, borrowers and lenders assign di¤erent values to the collateral asset in equilibrium: in an environment with incomplete contracts, this asymmetry implies that collateralized debt contracts implement the optimal funding contract. Finally, Martin et al. (2012) envisage an overlapping generation model where collateral is represented by an intrinsically worthless asset, that is, a bubble that ‡uctuates in response to shocks to expectations. Despite the variety of setups, none of these works examines the role of …nancial collateralization in connection with the transmission and ampli…cation of shocks to the macroeconomy. The present paper also relates to a rapidly developing banking literature on the role of macroprudential policy-making. Some recent examples include Van den Heuvel (2008), Admati et. al (2010), Hellwig (2010), Martinez-Miera and Suarez (2012), Angeloni and Faia (2013), Harris et. al (2015), Clerc et. al (2015) and Begenau (2015). The common trait of these contributions is to rely on medium to large scale dynamic general equilibrium models. While an obvious advantage of this modeling approach is to allow for a variety of shocks, transmission channels and alternative policy settings, our framework allows for a neat interpretation of the interplay between bankers’balance sheet and their capital requirements. The rest of the paper is organized as follows: Section 2 presents the framework; Section 3 discusses the steady-state properties; Section 4 focuses on the equilibrium dynamics in the neighborhood of the steady state and the ampli…cation of shocks to productivity in connection with the degree of …nancial collateralization; Section 5 considers the role of macroprudential policy-making in smoothing macroeconomic ‡uctuations; Section 6 concludes. 3 2 Environment The economy is populated by three types of in…nitely-lived, unit-sized, agents: savers, bankers and borrowers.2 These are linked through a vertical credit chain:3 savers make deposits to the bankers, who act as …nancial intermediaries and extend credit to the borrowers. Two goods are traded in this economy: a durable asset, ‘capital’, and a non-durable good. Capital does not depreciate and is …xed in total supply to one. Capital is held by bankers as well as borrowers. All agents have linear preferences de…ned over non-durable consumption. The remainder of this section provides further details on the key characteristics of the actors populating the model economy and their decision rules. 2.1 Savers Savers are the most patient agents in the economy. In each period, they are endowed with an exogenous non-produced income. We assume that savers are neither capable of monitoring the activity of the borrowers, nor of enforcing direct …nancial contracts with them. As a result, savers make deposits at the …nancial intermediaries. The linearity of their preferences implies that savers are indi¤erent between consumption and deposits in equilibrium, so that gross interest rate on savings (deposits), RS , equals their rate of time preference, 1= S . Savers’budget constraint reads as: cSt + bSt = uS + RS bSt 1 ; (1) where cSt denotes the consumption of non-durables, bSt is the amount of savings and uS denotes the exogenous endowment. 2.2 Borrowers Borrowers’ ability to obtain external …nancial resources is bounded by the limited enforceability of their debt contracts. In line with Jermann and Quadrini (2012) we assume that, should borrowers default, bankers acquire the right to liquidate the stock of capital, ktB . Based on the predicted outcomes of the renegotiation, borrowers are subject to an enforcement constraint. Neither bankers nor borrowers are able to observe the liquidation value before the actual default, though borrowers have all the bargaining power in the liquidation process. With probability (1 !) (with ! 2 [0; 1]) bankers expect to recover no collateral asset after a default, while with probability ! bankers expect to be able to recover Et qt+1 ktB , where qt denotes the capital price at time t. To derive the renegotiation outcome, we consider the following default scenarios: 1. Bankers expect to recover Et qt+1 ktB . Since bankers can expropriate the whole stock of capital, borrowers have to make a payment that leaves bankers indi¤erent between liquidation and allowing borrowers to preserve the stock of collateral assets. This requires borrowers to make a payment at least equal to Et qt+1 ktB , so that the ex-post value of defaulting for the bankers is: RB bB t Et qt+1 ktB ; (2) where RB denotes the gross loan rate and bB t is the loan. 2 The model is a variation of the ‘Credit Cycles’framework of KM. The expression ‘credit chain’ is not to be intended as a network of …rms involved in trade credit relationships, as formalized by Kiyotaki and Moore (2004). 3 4 2. Bankers expect to recover no collateral. If the liquidation value is zero, liquidation is clearly not the best option for the borrowers. Therefore, borrowers have no incentive to pay the loan back. The ex-post default value in this case is: RB bB t : (3) Therefore, enforcement requires that the expected value of non defaulting is not smaller than the expected value of defaulting, so that the expected liquidation value is negative: ! RB bB t 0 Et qt+1 ktB + (1 !)RB bB t ; (4) which reduces to RB bB t !Et qt+1 ktB . (5) According to (5), the maximum amount of credit borrowers may access is such that the sum of principal and interest, RB bB t , equals a fraction of the value of borrowers’ capital in period t + 1. Borrowers also face a ‡ow-of-funds constraint: B B B cB t + R bt 1 + qt (kt B ktB 1 ) = bB t + yt ; (6) B where cB t and yt denote borrowers’consumption and production of perishable goods, respectively. As in KM, borrowers are assumed to combine capital and labor (which is supplied inelastically) through B t kt 1 , a linear production technology, ytB = where t is a multiplicative shock to productivity, whose dynamics is accounted for by the following process: log t = log t 1 + ut , where an iid shock. 2 [0; 1) and ut is Borrowers maximize their utility under the collateral and the ‡ow-of-funds constraints, taking RB as given. The resulting Lagrangian is: LB t = E0 1 X B t B B B B B #B t ct + R bt 1 + qt (kt ktB 1 ) bB t B t kt 1 (7) t=0 't bB t where cB t ! qt+1 ktB RB ; denotes borrowers’ discount factor, while #B t and 't are the multipliers associated with borrowers’budget and collateral constraint, respectively. The …rst-order conditions are: @LB t @bB t @LB t @ktB = 0) = 0) B B RB Et #B t+1 + #t #B t qt + B 't = 0; Et #B t+1 qt+1 + B (8) Et #B t+1 t+1 + !'t Et hq t+1 RB i = 0: (9) Condition (8) implies that a marginal decrease in borrowing today expands next period’s utility and relaxes the current period’s borrowing constraint. As to (9), acquiring an additional unit of capital today allows to expand future consumption not only through the conventional capital gain 5 and dividend channels, but also through the feedback e¤ect of the expected collateral value on the price of capital. As in KM, in the neighborhood of the steady state the collateral constraint turns out to be binding when ' > 0. This is the case when RB < 1= analysis. As we consider linear preferences (i.e., #B t B , which is imposed throughout the B = # = 1), (8) implies 't = ' = 1 B RB . As a result, (9) can be rewritten as B qt = 2.3 RB + ! 1 RB B RB Et qt+1 + B Et t+1 : (10) Bankers Bankers’ primary activity consists of intermediating funds between savers and borrowers, raising deposits from the former and extending credit to the latter. However, their ability to collect deposits is bounded by the limited enforceability of their deposit contracts with the savers. As in Gertler and Kiyotaki (2015) we assume that, upon bankers’default, savers acquire the right to liquidate bankers’ real and …nancial asset-holdings.4 At the time of contracting the amount of deposits, the liquidation value of bankers’assets is uncertain. In this respect, the enforcement problem is isomorphic to that characterizing bankers’lending relationship with the borrowers, in line with the arguments of Jermann and Quadrini (2012). However, in this case we assume that savers account for a cost (1 ) bB t they might have to bear in order to recover bankers’…nancial collateral. Implicitly, from the perspective of the savers the possibility to liquidate bB t in case of bankers’default is conditional on borrowers being themselves solvent on their debt obligations, so that …nancial resources located at the end of the credit chain can be recovered. In this respect, 2 [0; 1] indexes savers’perceived liquidity of bankers’ …nancial assets/lending. Therefore, in the extreme situation savers regard bankers’…nancial assets as completely illiquid and do not accept them as collateral we set = 0, while = 1 corresponds to a situation in which savers attach no risk to their ability of liquidating …nancial assets in case bankers’ default. To derive the renegotiation outcome, we assume that with probability 1 recover no collateral, while with probability savers expect to the expected recovery value is Et qt+1 ktI + bB t , where ktI denotes bankers’holdings of real assets and bB t represents the amount of …nancial collateral, net of the transaction cost. This implies the following default scenarios: 1. Savers expect to recover Et qt+1 ktI + bB t . Since savers expect to expropriate the stock real and …nancial assets after bearing a transaction cost (1 ) bB t , bankers have to make a payment that leaves savers indi¤erent between liquidation and allowing borrowers to preserve the stock of collateral assets. This requires bankers to make a payment at least equal to Et qt+1 ktI + bB t , so that the ex-post value of defaulting for the bankers is: RS bSt Et qt+1 ktI bB t : (11) 2. Savers expect to recover no collateral. If the liquidation value is zero, liquidation is clearly not the best option for the savers. Therefore, bankers have no incentive to pay deposits back. The 4 There are two main considerations why this assumption is a reasonable one: First, savers have no direct use of the collateral assets; second, even if collateral assets represent an attractive investment opportunity, savers have no experience in hedging. 6 ex-post default value in this case is: RS bSt : (12) Enforcement requires that the expected value of defaulting is not smaller than the expected value of defaulting, so that: RS bSt 0 Et qt+1 ktI bB t + (1 )RS bSt ; (13) which reduces to RS bSt Et qt+1 ktI + bB t ; (14) according to which the amount of deposits, together with the accrued interest, should be limited from above by a fraction of the total expected collateral value.5 The main role of the physical capital held by bankers is to serve as a bu¤er against which the intermediary is trusted to be able to meet its …nancial obligations. Yet, we assume that capital is not kept idling, so it can also be used as a production input. However, bankers lack the necessary expertise to pursue the production process, while featuring a specialization in intermediating funds between savers and borrowers. This implies that, for a given stock of real assets, borrowers are more productive than …nancial intermediaries. In light of this assumption, bankers’production technology is assumed to feature the following properties: ytI = I t I(kt 1 ); 0 00 (15) 0 0 with I > 0, I < 0, I (0) > % > I (1),6 and RB % where RS I B I RS 1 RB 1 I 1 I RS B ! 1 B RB ; (16) denotes bankers’discount factor and (16) is required to ensure an internal solution in which both bankers and borrowers demand physical capital.7 Bankers’‡ow-of-funds constraint reads as: S S cIt + bB t + R bt 1 + qt (ktI I ktI 1 ) = bSt + RB bB t 1 + yt ; 5 (17) This constraint embodies the notion that real and …nancial assets have di¤erent degrees of collateralization. This is because, in case of default, each type of asset needs to be collected at di¤erent levels of the credit chain. 6 Assuming a decreasing returns to scale (DRS) technology available to the borrowers would not alter our key results. As we will see in the next section, it is the relatively higher impatience of borrowers, combined with their collateral constraint, that endows them with a suboptimal stock of capital. This point is also discussed in KM. Introducing a DRS technology would only hinder the analytical tractability of the model. 7 The role of this property will be discussed further in Section 4.1. 7 where cIt denotes bankers’consumption. The Lagrangian for bankers’optimization reads as LIt = E0 1 X I t t #It [cIt + RS bSt 1 I + bB t + qt (kt ktI 1 ) (18) t=0 bSt where #It and cIt RB bB t 1 I t I(kt 1 )] t bSt qt+1 I k RS t bB t RS ; are the multipliers associated with bankers’ budget constraint and enforcement constraint, respectively. The …rst-order conditions are: @LIt @bSt @LIt @bB t @LIt @ktI = 0) RS = 0 ) RB = 0) I I Et #It+1 + #It Et #It+1 #It qt + I #It + t = 0; 1 RS t Et #It+1 qt+1 + (19) = 0; I h Et #It+1 (20) t+1 I 0 i (ktI ) + t Et [qt+1 ] = 0: RS (21) As we assume linear preferences, #It = #I = 1. Therefore, conditions (20) and (21) imply that the …nancial constraint holds with equality in the neighborhood of the steady state (i.e., long as (i) RS I < 1 and (ii) RB I t = > 0) as < 1.8 Speci…cally, condition (i) implies that bankers are relatively more impatient than savers,9 while condition (ii) implies that, unless either or equal zero, bankers charge a lending rate that is lower than their rate of time preference, as extending loans relaxes their collateral constraint. In light of these properties, a positive spread exists between the interest rate on loans and that on deposits: B R = Increasing RS 1 I and I RS : RS (22) compresses the wedge between RB and RS . Intuitively, an increase in the degree of real and/or …nancial collateralization increases the collateral value that savers expect to recover in case of bankers’default. This relaxes the …nancial constraint, eases more deposits and translates into a higher credit supply that compresses the lending rate. Finally, from (21) we can retrieve the Euler equation governing bankers’investment in real assets: qt = RS I + 1 RS I RS Et qt+1 + By relaxing (i) and allowing for I I Et h i 0 I I (k ) : t+1 t (23) RS = 1, (23) reduces to lenders’ euler equation in the conven- tional direct-credit economy á la KM.10 Under this circumstances, bankers are no longer …nancially constrained. As we shall see in the next section, this implies both a higher loan rate and a higher user cost of capital from the perspective of bankers/lenders, as compared with what observed when bankers face a binding collateral constraint. 8 Steady-state variables are reported without the time subscript. In this respect, imposing I RS = 1 reduces the model to the conventional KM economy. 10 In the original KM framework lenders are labelled as gatherers, while borrowers are called farmers. 9 8 2.4 Market Clearing To close the model, we need to state the market-clearing conditions. We know that the total supply of capital equals one: ktI + ktB = 1. As to the market for consumption goods, the aggregate resource constraint reads as: yt = ytI + ytB ; (24) where yt denotes the total demand of consumption goods. The aggregate demand and supply for credit are given by the two enforcement constraints (holding with equality) faced by borrowers and bankers, respectively: Et qt+1 ktB ; RB 1 RS bSt Et qt+1 ktI : bB = ! t (25) bB = t (26) Finally, as savers are indi¤erent between any path of consumption and savings, the amount of deposits depends on bankers’capitalization. Thus, the markets for deposits and …nal goods are cleared according to the Walras’Law. 3 Steady State Properties We …rst focus on the steady-state equilibrium. Financial frictions characterizing both the saversbankers relationship and the bankers-borrowers relationship deeply a¤ect the welfare properties of the model. Examining their interaction in the long-run is key for understanding their propagation of technology shocks. In the remainder we impose, without loss of generality, I(ktI 1 ) = ktI ating (10) in the non-stochastic steady state returns: q= RB 1 B RB 1 , with 2 [0; 1]. Evalu- B ! 1 B RB : (27) From (23) we retrieve the marginal product of bankers’capital, as a function of its price: 0 I (k I ) = kI 1 = RS 1 I 1 RS I I RS q; (28) so that Equations (27), (28) and k I + k B = 1 pin down borrowers’and bankers’holdings of capital. In turn, these allow us to characterize the steady-state ine¢ ciencies arising from the debt enforcement problems in place at di¤erent levels of the vertical credit chain. 9 Figure 1. Equilibrium in the steady state. Figure 1 provides a sketch of the long-run equilibrium of the economy. On the horizontal axis, borrowers’demand for capital is measured from the left, while bankers’demand from the right. The sum of the two equals one. On the vertical axis we report the marginal product of capital for both borrowers and bankers. Borrowers’marginal product of capital is indicated by the line ACE , while bankers’ marginal product is represented by the line DE 0 E . The …rst-best allocation would be attained at E 0 , where the product of capital owned by the bankers and borrowers is the same, at the margin. In our economy, however, the steady-state equilibrium is at E , where the marginal product of capital of the borrowers (mpk B = 1) exceeds that of the bankers (mpk I = %). That is, relative to the …rst-best, too little capital is used by the borrowers, due to their …nancial constraint. This implies a loss of output relative to the …rst-best, as indicated by the shaded area CE 0 E .11 Remark 1 elaborates further on the relationship between borrowers’and bankers’marginal product of capital: B Remark 1 As long as < I , bankers’marginal product of capital is lower than that of borrowers. 0 In fact, imposing I (k I ) < 1 returns the following inequality: B B I < As we assume given that also B I < RS 1 I I RS RS (RB RB + ! !) : (29) , the left-hand side of (29) is negative, while its right-hand side is positive, < 1 holds by assumption. Therefore, a de…ning feature of the equilibrium is that the marginal product of borrowers’ capital-holdings is higher than that of the bankers, given that the former cannot borrow as much as they want. As a result, any shift in capital usage from the borrowers to the bankers will lead to a …rst-order decline in aggregate output, as it will become clear when exploring the linearized economy. With respect to this property, the present economy is isomorphic to that put forward by KM, as the suboptimality of the steady-state equilibrium allocation only rests on borrowers’relatively higher impatience. 11 The area under the solid line, ACE D, is the steady-state output. 10 I Importantly, under RS = 1 (i.e., in a direct-credit economy à la KM, where savers and bankers have identical degrees of impatience) the productivity gap between bankers and borrowers would be even higher. This is due to bankers facing a higher steady-state user cost of capital and charging a higher loan rate, which exacerbates the steady-state ine¢ ciency in the allocation of capital. This is clearly indicated by the additional loss of output, relative to the …rst-best, as captured by the trapezoid CKM CE EKM (where EKM indicates the steady-state equilibrium in the KM-type setting). Therefore, a key result is that expanding KM’s framework with …nancially constrained bankers sets the steady-state equilibrium on a Pareto-superior allocation. Given this key property of the setting under examination, is important to understand how bankers’ collateral impacts on the steady-state ine¢ ciency in the allocation of capital. To this end, we de…ne the productivity gap between borrowers and bankers: mpk B mpk I =1 %: (30) The following summarizes the impact of …nancial collateralization on the productivity gap: Proposition 1 Increasing the degree of …nancial collateralization ( ) reduces the gap between bankers’ and borrowers’ marginal product of capital ( ). Proof. See Appendix A A higher degree of …nancial collateralization expands bankers’ lending capacity and compresses the spread charged over the deposit rate. In turn, lower lending rates allow borrowers to expand their borrowing capacity through a higher collateral value, ceteris paribus. The combination of these e¤ects is such that mpk I increases in the degree of …nancial collateralization, reducing the productivity gap with respect to the borrowers. This factor will play a key role in the ampli…cation of gross output in the face of a technology shock, as we will see in Section 4.1. 4 Equilibrium Dynamics To examine equilibrium dynamics, we log-linearize the Euler equations of both borrowers and bankers around the non-stochastic steady state.12 As for the borrowers: q^t = Et q^t+1 + (1 B where ) Et ^ t+1 ; B RB +! (1 RB RB ) q^t = Et q^t+1 + (1 where RS I + (1 RS I (31) . As for the bankers: ) Et ^ t+1 + RS ) and 1 1 k^tB ; (32) is the elasticity of the bankers’marginal product of capital times B 1 k the ratio of borrowers’to bankers’capital-holdings in the steady state (i.e., ). kB (1 ) B Once we obtain the solutions for q^t and k^t as linear functions of the technology shifter, we can determine closed-form expressions for the equilibrium path of the other variables in the model. Thus, we …rst focus on (31), whose forward-iteration leads to: q^t = ^ t ; 12 (33) Variables in log-deviation from their steady-state level are denoted by a "^". 11 1 1 where > 0. With this expression for q^t , we can resort to (32), obtaining k^tB = v ^ t ; where v 4.1 (34) ( )(1 1 ) 1 > 0. Financial Collateral and Macroeconomic Ampli…cation We have now lined up the elements necessary to examine the economy’s response to technology disturbances. Proposition 2 details the e¤ect induced by a marginal change in the degree of …nancial collateralization on borrowers’ capital-holdings and the capital price. Both variables are crucial to determine the size of the dynamic multiplier popularized by KM in this type of credit economies. Proposition 2 Increasing the degree of …nancial collateralization ( ) attenuates the impact of the technology shock on both borrowers’ holdings of capital and the capital price. Proof. See Appendix A Proposition 2 implies that the sensitivity of borrowers’capital-holdings to the technology shifter decreases in the degree of …nancial collateralization. The intuition for this is twofold: i) on the one hand, increasing determines a more even distribution of capital goods, as re‡ected by the drop in ; ii) on the other hand, being able to pledge a higher share of …nancial assets reinforces the sensitivity of the capital price to the capital gain component in borrowers’Euler equation, , through the drop in the loan rate, while reducing the sensitivity to the dividend component. The combination of these e¤ects mutes the dynamic multiplier embodied in this class of models, ultimately attenuating the overall degree of macroeconomic ampli…cation of the system, as captured by the response of gross output to the technology shock. To dig deeper on this aspect, we linearize total production in the neighborhood of the steady state: y B ^B k ; y t 1 y^t = ^ t + (35) According to (35), the dynamics of gross output is shaped by ^ t , as well as by borrowers’ capitalholdings at time t 1: the second e¤ect captures the endogenous persistence of gross output. In fact, y^t depends on the past history of shocks not only through the direct impact of ^ t , but also through the e¤ect of ^ t 1 on k^tB 1 , as implied by (34). In light of this, we can rewrite (35) as y^t = $^ t where $ 1 +v + ut yB y . (36) This result is important in that it shows how eliminating the key source of steady-state ine¢ ciency –i.e., attaining = 0 –implies that total output departures from its steady state would track the path of the technology shock, so that the model would feature no endogenous propagation of productivity shifts.13 13 This property echoes the role of the steady-state ine¢ ciency for short-run dynamics in the KM model. In their setting, closing the gap between the marginal products of capital of lenders and borrowers would imply no response at all to a productivity shift. In this respect, the key di¤erence between the two frameworks lies in that we assume an autoregressive shock, while they consider an undexpected temporary shift in technology. 12 There are three di¤erent channels through which an increase in savers’ perceived liquidity of bankers’ …nancial assets a¤ect the propagation of a technology shock. To see this, we compute the following derivative: @$ = @ y B @v yB @ +v y @ y @ +v @ y B =y : @ (37) Proposition 2 shows that @v=@ < 0, implying that …nancial collateralization reduces the impact of a technology shift on borrowers’capital-holdings, which we know exerting a …rst-order e¤ect on total output through (35). We also know from Proposition 1 that the productivity gap between borrowers and bankers shrinks as …nancial collateralization increases (i.e., @ =@ < 0). As to @ y B =y =@ : ! @ y B =y = @ 1 G RS G 1 kB RS RB y 2 (1 1+ kB 1 kB ) 1 RB RS B I 1 { 1 ; (38) which is positive, given that higher …nancial collateralization implies a net transfer of capital goods from bankers to borrowers. In turn, this implies both a …rst-order positive e¤ect on y B and a milder second-order positive impact on y (given that, concurrently, y I decreases at a lower pace than the increase in y B ), so that the overall e¤ect is positive. To sum up, an increase in causes competing e¤ects on $. As we already know, greater …nancial collateralization depresses the pass-through of ^ t 1 on borrowers’capital-holdings. Furthermore, raising exerts two e¤ects on the pass-through of k^B t 1 on y^t : (i) on the one hand, bankers’marginal product of capital increases, implying a reduction of the productivity gap; (ii) on the other hand, borrowers’contribution to total production increases, as the reduction in the productivity gap re‡ects higher capital accumulation in the hands of the borrowers. These competing forces potentially lead to mixed e¤ects on output ampli…cation, as captured by second-round e¤ects of technology disturbances. To address this point, Figure 2 plots $ as a function of and .14;15 As it emerges from Figure 2, increasing compresses $, at any level of . By contrast, increasing the income share of capital in bankers’production technology ampli…es the second-round response of output. This is because ampli…es the productivity gap through its positive e¤ect on .16 14 The aim of this exercise is to have an idea of the direction of the overall e¤ect exerted by …nancial collateralization on macroeconomic volatility, rather than quantifying an empirically plausible multiplier emerging from the interaction of bankers’and borrowers’…nancial constraints. We leave this task for future research employing a large scale dynamic general equilibrium model. 15 The discount factors are set in line with the conditions stating the relative degree of impatience of the three agents in the credit economy and are largely in line with existing (quarterly) calibrations involving heterogeneous agents economies: S = 0:99, I = 0:98, B = 0:97. We set = 0:95, in line with the empirical evidence showing that technology shocks are generally small, but highly persistent (see, e.g., Cooley and Prescott, 1995). As for and ! , their are set to 1 so as to ensure a wider set of admissible combinations of and that ensure positive holdings of capital for both bankers and borrowers. This is clearly displayed by the robustness evidence reported in Appendix C, which also shows that di¤erent combinations of and ! are close to irrelevant regarding the e¤ect of …nancial collateralization on macroeconomic ampli…cation. 0 0 16 It is also worth to highlight that increasing may violate the condition I (0) > % > I (1), which ensures an interior solution as for how much capital bankers should hold in the neighborhood of the steady state. To see why this is the case, recall that in the steady-state bankers’ marginal product of capital is tied to their user cost of capital through 1 kI = %. Increasing in‡ates bankers’marginal product of capital, while leaving their user cost una¤ected: Thus, as increases bankers are induced to hold an increasing stock of capital, so that the equality holds. An important aspect is that this e¤ect tends to kick in earlier as declines. This is because a drop in the degree of …nancial collateralization depresses bankers’ user cost of capital. Therefore, as declines and increases the set of steady-state allocations in 0 which both bankers and borrowers hold capital restricts, as the condition % > I (1) is eventually violated and borrowers’ may virtually end up with negative capital-holdings. 13 Figure 2. Business cycle ampli…cation. 0 1.6 0.2 ξ 1.5 1.4 0.4 1.3 0.6 1.2 1.1 0.8 1 1 0 0.2 0.4 µ 0.6 0.8 1 $ as a function of and , under the following parameterization: S = 0:99, I = 0:98, = ! = 1. The white area denotes inadmissible equilibria where bankers’capital-holdings are Notes. Figure 2 graphs B = 0:97, = 0:95, virtually negative. 4.2 The Role of Leverage To enlarge our perspective on the ampli…cation/attenuation induced by bankers’…nancial collateral, we take a closer look at their balance sheet. To this end, we de…ne bankers’equity as the di¤erence between the value of total assets (lending plus real assets) and liabilities (deposits): I eIt = bB t + qt kt bSt ; (39) I while leverage is de…ned as the ratio between loans and equity: levtI = bB t =et . Figure 3 reports the response of selected variables to a one-standard deviation shock to technology.17 As implied by (35), on impact output responds one-to-one with respect to the shock, regardless of the degree of …nancial collateralization. However, as increases the second-round response is gradually muted. To complement our analytical insight and provide further intuition on this attenuation, we examine the behavior of a set of variables involved in bankers’intermediation activity. In this respect, note that deposits tend to decline at low values of , while increasing as bankers can o¤er a higher share of their …nancial assets as collateral. The reason for this can be better understood by exploring the interaction between bankers’…nancial and real assets. Their interplay takes place on two levels: i) on the one hand, as embodied by (14), both assets have a positive e¤ect on savers’deposits; ii) on the other hand, assuming a …xed supply of physical capital implies a substitution e¤ect between real and B …nancial assets. In fact, according to borrowers’collateral constraint, bB t increases in kt . However, as 17 The baseline parameterization is the same as that employed in Figure 2. As for , we impose a rather conservative value of 0.4, which allows us to obtain a …nite distribution of capital in the steady state. 14 ktB = 1 ktI , increasing bankers’real asset-holdings exerts a negative force on lending.18 We also know that, due to the capital productivity gap between borrowers and bankers, an expansionary technology shock necessarily causes a decline of bankers’real assets, thus expanding borrowers’capital holdings and borrowing. Therefore, in equilibrium deposits are in‡uenced by two opposite forces, namely an expansion in the amount of bankers’…nancial assets and a contraction their stock of real assets. In light of this, it is important to understand that, as drops – re‡ecting a decline in the perceived liquidity of bankers’…nancial assets –the role of …nancial collateral is gradually muted and deposits eventually track the dynamics of bankers’ real assets. In this context, the contraction of bankers’ real asset-holdings overcomes the drop in deposits, so that lending expands in excess of bank equity, potentially leading to an increase in bankers’ leverage. In fact, a procyclical leverage ratio can be associated with a relevant degree of macroeconomic ampli…cation, when bankers’…nancial assets are judged to be relatively illiquid. Figure 3 shows this is the case for > 0:5. Figure 3. Impulse responses to a positive technology shock. Price of capital Output 1.3 Borrowers capital 0.3 3 1.2 1.1 2.5 0.25 1 2 0.9 0.2 0.8 1.5 0.7 0.15 1 0.6 0.5 0.1 0.5 0.4 0 5 10 15 20 0.05 0 5 Deposits 10 15 0 0 20 5 Lending 0.4 15 20 15 20 Lev erage 3 0.2 10 0.15 0.1 2.5 0 0.05 2 -0.2 0 1.5 -0.4 -0.05 1 -0.6 -0.1 0.5 -0.8 -1 0 5 10 15 20 0 0 -0.15 5 10 15 20 -0.2 0 5 10 Notes. Figure 3 graphs the response of selected variables to a one-standard-deviation shock to technology, under the following parameterization: S = 0:99, I = 0:98, B = 0:97, = 0:95, = ! = 1, = 0:4. 5 Capital Requirements and the Business Cycle The analysis of the previous section has shown a close connection between the cyclicality of bank leverage and macroeconomic ampli…cation. Therefore, attenuating the degree of procyclicality of bankers’ leverage is important to reduce the amplitude of ‡uctuations in gross output, especially when savers perceive bankers’ …nancial assets to be relatively illiquid. In recent years regulators 18 This is a distinctive feature of lender-borrower relationships involving the collateralization of a productive asset. In fact, KM show that the major reallocation of land from the lenders to the borrowers following a positive technology shock is only attenuated by relaxing the hypothesis of …xed supply of the real asset, while the direction of the transfer is not inverted. 15 have suggested to lean against credit imbalances and pursue macroeconomic stabilization through policy rules that set a countercyclical capital bu¤er. De facto, countercyclical capital regulation is a key block of the Basel III international regulatory framework for banks. Based on the analysis of the transmission mechanism in the previous section, we now examine the functioning of this type of policy tool within our framework. To this end, we assume that a hypothetical regulatory authority imposes a capital adequacy requirement, setting a minimum limit to the amount of equity: eIt where bB t ; (40) denotes the capital-to-asset ratio. Notably, combining (39) with (40) obtains bSt qt ktI + (1 ) bB t ; (41) which is remarkably similar to the enforcement constraint (14).19 In fact, a marginal increase (decrease) in the capital (leverage) ratio maps into a decrease in the degree of collateralization of …nancial assets. Intuitively, a higher leverage (lower capital) ratio implies a riskier exposure of the …nancial intermediary: this translates into a greater transaction cost savers would have to bear in the event of bankers’default, so as to seize their …nancial assets. We also allow for capital requirements to vary with the macroeconomic conditions (see, e.g., Angeloni and Faia, 2013, Nelson and Pinter, 2013 and Clerc et al., 2015): t = bB t bB ' ; ' 0: (42) For ' = 0 we implicitly impose a constant capital-to-asset ratio, while under ' > 0 the macroprudential rule implies a countercyclical capital bu¤er, which is a distinctive trait of the Basel III bank-capital regime.20 As a result of imposing (42), the loan rate is potentially time-varying, being a¤ected by an endogenous capital-to-asset ratio:21 RtB = RS (1 t) I 1 I RS : (43) Equation (43) can be linearized in the neighborhood of the steady state: ^ tB = ^t ; R (44) 19 A key di¤erence between the two constraints lies in the fact that (41) entails period-t capital value, while (14) contemplates the period-t + 1 expected capital value. 20 The regulatory framework evolved through three main waves. Basel I has introduced the basic capital adequacy ratio as the foundation for banking risk regulation. Basel II has reinforced it and allowed banks to use internal risk-based measure to weight the share of asset to be hold. Basel III has been brought in response to the 2007-2008 crisis, with the key innovation consisting of introducing countercyclical capital requirements, that is, imposing banks to build resilience in good times with higher capital requirements and relax them during bad times. According to this regime, capital regulation can respond to a wide range of macroeconomic indicators. Here we assume it to respond to deviations of bB t from its long-run equilibirum, bB . 21 Appendix B reports bankers’ optimization problem under the macroprudential rule (42). To focus on the role of macroprudential policy-making, and especially on the functioning of a countercyclical capital bu¤er that leans against the procyclicality of bank leverage, in the remainder we assume (41) to be more stringent than (14). Allowing for the co-existence of the two limits to bank deposits would impose to account for occasionally binding constraints, a task that goes beyond the scope of the present analysis. 16 where = 1 I I S R RB is positive, in light of assuming I RS < 1. We also linearize (42), obtaining: ^t = '^bB : t (45) After linearizing borrowers’…nancial constraint, we can substitute for ^bB t in (45) and plug the resulting expression into (44), so as to obtain: ^ tB = R ' 1+ ' Et q^t+1 + k^tB ; (46) so that we retrieve a connection between the loan rate and borrowers’ expected collateral value. Notably, increasing the responsiveness of the capital-to-asset ratio to changes in aggregate lending limits this e¤ect. In fact, increasing ' implies that marginal deviations of bB t from its steady state transmit more promptly to the capital-to-asset ratio and the loan rate, through the combined e¤ect captured by Equations (44) and (45). Therefore, higher sensitivity of the loan rate to variations in aggregate lending (i.e., a steeper loan supply function) imply a stronger (weaker) discounting of borrowers’expected collateral when this expands (contracts). Analogous implications can be drawn when considering an increase in the steady-state capital-to-asset ratio, through its e¤ect on . To assess the stabilization performance of the countercyclical capital bu¤er rule, we run two experiments: we …rst set the response coe¢ cient ' at a given level and vary the steady-state capitalto-asset ratio, ; in a second exercise we …x and vary '. Figure 4 reports the economy’s response to a technology shock in the …rst experiment: lowering the steady-state capital-to-asset ratio proves to be rather ine¤ective at mitigating the response of output (see Figure 5). This is not surprising, as we focus on a rather narrow range of values for , so as to consider capital-to-asset ratios in line with the full weight level of Basel I and the treatment of non-rated corporate loans in Basel II and III. In the second experiment the response coe¢ cient ' varies over the support [0; 1]. As expected, at ' = 0 (i.e., a capital-to-asset ratio at its steady state level) we observe the highest ampli…cation of the output response, while the lending rate and bank leverage are both constant. Increasing the degree of countercyclicality of the capital bu¤er proves to be e¤ective at attenuating the output response to the shock, by compressing bank leverage and raising the responsiveness of the lending rate, as implied (46). In the limit, as ' ! 1, the policy rule pushes leverage to display a strong degree of countercyclicality, so that gross output implies no endogenous propagation. 17 Figure 4. Impulse responses under di¤erent . Output Price of capital 1.4 0.3 1.2 0.25 1 0.2 0.8 0.15 0.6 0.1 0.4 0.05 Borrowers capital 5 4 3 0 5 10 15 20 2 0 Capital ratio 1.5 2.5 5 x 10 10 15 20 5 10 Lending rate -3 15 20 15 20 15 20 Lending 5 2 1 1 0 4 1.5 3 1 0.5 2 0.5 0 0 5 10 15 20 0 0 5 10 Equity 15 20 0 5 0.2 4 0.15 3 0.1 2 0.05 10 10 Lev erage 0.25 5 5 Deposits 6 1 0 1 0 15 20 -0.5 -1 0 0 5 10 15 20 -1.5 0 5 10 Notes. Figure 4 graphs the response of selected variables to a one-standard-deviation shock to technology, under the following parameterization: S = 0:99, I = 0:98, B = 0:97, = 0:95, = ! = 1, = 0:4, ' = 0:3. Figure 5. Impulse responses under di¤erent '. Price of capi tal Output 1.5 Borrowers capital 0.3 4 0.25 1 3 0.2 2 0.15 0.5 1 0.1 0 0 5 10 15 20 0.05 0 Capital ratio x 10 15 10 10 5 5 0 5 -3 10 15 20 0 0 5 Lendi ng rate 10 15 20 15 20 15 20 Lendi ng 5 4 3 2 1 0 0 5 10 15 20 -5 0 5 Equity 10 15 20 0 0 5 Deposits 15 10 Leverage 0.2 0 0.15 10 0.1 -5 0.05 5 0 0 0 5 10 15 20 -0.05 0 -10 5 10 15 20 0 5 10 Notes. Figure 5 graphs the response of selected variables to a one-standard-deviation shock to technology, under the following parameterization: S = 0:99, I = 0:98, B = 0:97, = 0:95, = ! = 1, = 0:4, = 0:08. 18 6 Concluding Remarks We have envisaged a credit economy where bankers intermediate funds between savers and borrowers. We have assumed that bankers’ ability to collect deposits is a¤ected by limited enforceability: as a result, if bankers default, savers acquire the right to liquidate bankers’ real and …nancial assetholdings. We have emphasized the use of bankers’ …nancial assets – which are represented by their loans to the borrowers –as a form of collateral in the deposit contracts. Due to the structure of our credit chain, which may well account for di¤erent forms of …nancial intermediation, savers anticipate that liquidating …nancial assets is conditional on borrowers being themselves solvent on their debt obligations. This friction limits the degree of collateralization of bankers’…nancial assets and, in turn, their borrowing capacity. In this context, we have demonstrated three main results: i) expanding KM’s framework with …nancially constrained …nancial intermediaries allows to attain a Pareto-superior steady-state allocation; ii) increasing …nancial collateralization dampens macroeconomic ‡uctuations by reducing the degree of procyclicality of bank leverage; iii) …nally, allowing for the presence of a banking regulator may help smoothing the business cycle through the introduction of a capital adequacy requirement on bankers’activity. Our model is necessarily stylized, though it can be generalized along a number of dimensions. For instance, a realistic extension consists of allowing bankers to issue equity (outside equity), so as to evaluate how a di¤erent debt-equity mix may a¤ect macroeconomic ampli…cation over expansions – when equity can be issued frictionlessly – and contractions, when equity issuance may be precluded due to tighter information frictions. 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