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Financial Collateral and Macroeconomic Amplification∗
Federico Lubello† Ivan Petrella‡ Emiliano Santoro§
September 8, 2016
Abstract
Financial institutions typically resort to collateralized debt to raise funds, providing financial
assets as a guarantee in case of default on their debt obligations. This paper focuses on the
connection between financial 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 financial
asset-holdings. However, due to the vertically integrated structure of our credit economy, savers
anticipate that liquidating financial assets is conditional on borrowers being themselves solvent on
their debt obligations. This friction limits the collateralization of bankers’financial assets. In this
context, decreasing the degree of financial collateralization exacerbates steady-state ineffi ciencies
– increasing the gap between borrowers’and bankers’marginal product of capital – reflecting
into a procyclical bank leverage and thus amplifying macroeconomic fluctuations. In light of these
properties, a banking regulator may help smoothing the business cycle through the introduction
of a countercyclical capital buffer.
JEL classification: 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 notnecessarily represent the views of the Banque Centrale du Luxembourg or the Eurosystem.†Banque Centrale du Luxembourg. Address : 2, Boulevard Royal, L-2983 Luxembourg. E-mail : fed-
[email protected].‡Birkbeck College, University of London and CEPR. Address: Malet Street, WCIE 7HX, London, UK. E-mail :
[email protected].§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 financial 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 financial assets as a guaran-
tee 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 financial 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 en-
forceability 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 financial col-
lateral 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 financial assets. However, due to the vertically integrated
structure of our credit economy, savers anticipate that, in case of bankers’default, liquidating their fi-
nancial assets is conditional on borrowers being themselves solvent on their debt obligations. If savers
perceive financial assets to be relatively illiquid, they will be less prone to accept these as collateral.
We first examine how financial intermediation and financial collateralization impact on the steady-
state distribution of real and financial resources. In this respect, we report two main results. First,
introducing financial 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 financially 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 affected by the degree of financial
collateralization. In other words, when depositors perceive bankers’financial assets to be relatively
illiquid, this translates into an increase in the intermediation margin. In turn, this situation affects
the steady-state distribution of capital: increasing the pledgeability of financial 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 effects of the debt enforcement problem.
As for the equilibrium dynamics of the model economy, a low perceived liquidity of bankers’fi-
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,
1The set of assets that central banks accept from commercial banks generally includes government bonds and otherdebt instruments issued by public sectors and international/supranational institutions. In some cases, also securitiesissued by the private sector can be accepted, such as covered bank bonds, uncovered bank bonds, asset-backed securitiesor 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 financial assets. However, as these are perceived to be increas-
ingly illiquid, the compensation effect 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 amplified, relative to situations in which bankers’feature
a high degree of financial collateralization.
In our framework a drop in the perceived liquidity of the financial assets held by the banking sector
eventually reflects into a procyclical leverage, inducing relevant business cycle volatility. Reverting
this procyclicality is of key importance to attenuate the magnitude of aggregate fluctuations. 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 buffer, as imposed by the Basel III regulatory
framework in response to the 2007-2008 financial crisis.
The number of studies focusing on financial collateral is surprisingly limited. The present paper
relates to Oehmke (2014), who analyzes the dynamics of repo liquidations in the presence of financial
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 financial assets in the form of collateral, focusing on borrowers’optimal
financing choice. In this context, borrowers and lenders assign different 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 fluctuates in response to shocks to expectations. Despite the variety of setups, none
of these works examines the role of financial collateralization in connection with the transmission and
amplification of shocks to the macroeconomy.
The present paper also relates to a rapidly developing banking literature on the role of macropru-
dential 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 ap-
proach 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 amplification of shocks to productivity in connection with the degree of
financial collateralization; Section 5 considers the role of macroprudential policy-making in smoothing
macroeconomic fluctuations; Section 6 concludes.
3
2 Environment
The economy is populated by three types of infinitely-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 financial 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
fixed in total supply to one. Capital is held by bankers as well as borrowers. All agents have linear
preferences defined 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 financial contracts with them. As a result, savers make deposits
at the financial intermediaries. The linearity of their preferences implies that savers are indifferent
between consumption and deposits in equilibrium, so that gross interest rate on savings (deposits),
RS , equals their rate of time preference, 1/βS . Savers’budget constraint reads as:
cSt + bSt = uS +RSbSt−1, (1)
where cSt denotes the consumption of non-durables, bSt is the amount of savings and u
S denotes the
exogenous endowment.
2.2 Borrowers
Borrowers’ability to obtain external financial 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, kBt . 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− ω) (with ω ∈ [0, 1])
bankers expect to recover no collateral asset after a default, while with probability ω bankers expect
to be able to recover Etqt+1kBt , 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 Etqt+1kBt . Since bankers can expropriate the whole stock of capital,
borrowers have to make a payment that leaves bankers indifferent between liquidation and
allowing borrowers to preserve the stock of collateral assets. This requires borrowers to make a
payment at least equal to Etqt+1kBt , so that the ex-post value of defaulting for the bankers is:
RBbBt − Etqt+1kBt , (2)
where RB denotes the gross loan rate and bBt is the loan.2The model is a variation of the ‘Credit Cycles’framework of KM.3The expression ‘credit chain’ is not to be intended as a network of firms involved in trade credit relationships, as
formalized by Kiyotaki and Moore (2004).
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:
RBbBt . (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:
0 ≥ ω[RBbBt − Etqt+1kBt
]+ (1− ω)RBbBt , (4)
which reduces to
RBbBt ≤ ωEtqt+1kBt . (5)
According to (5), the maximum amount of credit borrowers may access is such that the sum of
principal and interest, RBbBt , equals a fraction of the value of borrowers’ capital in period t + 1.
Borrowers also face a flow-of-funds constraint:
cBt +RBbBt−1 + qt(kBt − kBt−1) = bBt + yBt , (6)
where cBt and yBt 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
a linear production technology, yBt = αtkBt−1, where αt is a multiplicative shock to productivity, whose
dynamics is accounted for by the following process: logαt = ρ logαt−1 + ut, where ρ ∈ [0, 1) and ut is
an iid shock.
Borrowers maximize their utility under the collateral and the flow-of-funds constraints, taking RB
as given. The resulting Lagrangian is:
LBt = E0
∞∑t=0
(βB)t {
cBt − ϑBt[cBt +RBbBt−1 + qt(k
Bt − kBt−1)− bBt − αtkBt−1
](7)
−ϕt(bBt − ω
qt+1kBt
RB
)},
where βB denotes borrowers’discount factor, while ϑBt and ϕt are the multipliers associated with
borrowers’budget and collateral constraint, respectively. The first-order conditions are:
∂LBt∂bBt
= 0⇒ −βBRBEtϑBt+1 + ϑBt − ϕt = 0; (8)
∂LBt∂kBt
= 0⇒ −ϑBt qt + βBEt[ϑBt+1qt+1
]+ βBEt
[ϑBt+1αt+1
]+ ωϕtEt
[qt+1RB
]= 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 effect 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 ϕ > 0. This is the case when RB < 1/βB, which is imposed throughout the
analysis. As we consider linear preferences (i.e., ϑBt = ϑB = 1), (8) implies ϕt = ϕ = 1 − βBRB. Asa result, (9) can be rewritten as
qt =βBRB + ω
(1− βBRB
)RB
Etqt+1 + βBEtαt+1. (10)
2.3 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 financial 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− ξ) bBt theymight have to bear in order to recover bankers’financial collateral. Implicitly, from the perspective of
the savers the possibility to liquidate bBt in case of bankers’default is conditional on borrowers being
themselves solvent on their debt obligations, so that financial resources located at the end of the
credit chain can be recovered. In this respect, ξ ∈ [0, 1] indexes savers’perceived liquidity of bankers’
financial assets/lending. Therefore, in the extreme situation savers regard bankers’financial assets as
completely illiquid and do not accept them as collateral we set ξ = 0, while ξ = 1 corresponds to a
situation in which savers attach no risk to their ability of liquidating financial assets in case bankers’
default.
To derive the renegotiation outcome, we assume that with probability 1 − χ savers expect to
recover no collateral, while with probability χ the expected recovery value is Etqt+1kIt + ξbBt , where
kIt denotes bankers’holdings of real assets and ξbBt represents the amount of financial collateral, net
of the transaction cost. This implies the following default scenarios:
1. Savers expect to recover Etqt+1kIt + ξbBt . Since savers expect to expropriate the stock real and
financial assets after bearing a transaction cost (1− ξ) bBt , bankers have to make a paymentthat leaves savers indifferent between liquidation and allowing borrowers to preserve the stock
of collateral assets. This requires bankers to make a payment at least equal to Etqt+1kIt + ξbBt ,
so that the ex-post value of defaulting for the bankers is:
RSbSt − Etqt+1kIt − ξbBt . (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
4There are two main considerations why this assumption is a reasonable one: First, savers have no direct use ofthe collateral assets; second, even if collateral assets represent an attractive investment opportunity, savers have noexperience in hedging.
6
ex-post default value in this case is:
RSbSt . (12)
Enforcement requires that the expected value of defaulting is not smaller than the expected value
of defaulting, so that:
0 ≥ χ[RSbSt − Etqt+1kIt − ξbBt
]+ (1− χ)RSbSt , (13)
which reduces to
RSbSt ≤ χ(Etqt+1k
It + ξbBt
), (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 buffer against which the
intermediary is trusted to be able to meet its financial 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 financial intermediaries. In light of this assumption, bankers’production technology
is assumed to feature the following properties:
yIt = αtI(kIt−1), (15)
with I′> 0, I
′′< 0, I
′(0) > % > I
′(1),6 and
% ≡RBβB
[RS(1− βI
)− χ
(1− βIRS
)]RSβI
[RB
(1− βB
)− ω
(1− βBRB
)] , (16)
where βI denotes bankers’discount factor and (16) is required to ensure an internal solution in which
both bankers and borrowers demand physical capital.7
Bankers’flow-of-funds constraint reads as:
cIt + bBt +RSbSt−1 + qt(kIt − kIt−1) = bSt +RBbBt−1 + yIt , (17)
5This constraint embodies the notion that real and financial assets have different degrees of collateralization. This isbecause, in case of default, each type of asset needs to be collected at different levels of the credit chain.
6Assuming 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 collateralconstraint, that endows them with a suboptimal stock of capital. This point is also discussed in KM. Introducing a DRStechnology would only hinder the analytical tractability of the model.
7The 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
∞∑t=0
(βI)t {
cIt − ϑIt [cIt +RSbSt−1 + bBt + qt(kIt − kIt−1) (18)
−bSt −RBbBt−1 − αtI(kIt−1)]− δt(bSt − χ
qt+1RS
kIt − χξbBtRS
)},
where ϑIt and δt are the multipliers associated with bankers’ budget constraint and enforcement
constraint, respectively. The first-order conditions are:
∂LIt∂bSt
= 0⇒ −RSβIEtϑIt+1 + ϑIt − δt = 0; (19)
∂LIt∂bBt
= 0⇒ RBβIEtϑIt+1 − ϑIt +
1
RSχξδt = 0; (20)
∂LIt∂kIt
= 0⇒ −ϑIt qt + βIEt[ϑIt+1qt+1
]+ βIEt
[ϑIt+1αt+1I
′(kIt )
]+ δtχ
Et [qt+1]
RS= 0. (21)
As we assume linear preferences, ϑIt = ϑI = 1. Therefore, conditions (20) and (21) imply that the
financial constraint holds with equality in the neighborhood of the steady state (i.e., δt = δ > 0) as
long as (i) RSβI < 1 and (ii) RBβI < 1.8 Specifically, condition (i) implies that bankers are relatively
more impatient than savers,9 while condition (ii) implies that, unless either χ or ξ 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:
RB =RS − χξ
(1− βIRS
)βIRS
. (22)
Increasing χ and ξ compresses the wedge between RB and RS . Intuitively, an increase in the degree
of real and/or financial collateralization increases the collateral value that savers expect to recover in
case of bankers’default. This relaxes the financial 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 =RSβI + χ
(1− βIRS
)RS
Etqt+1 + βIEt
[αt+1I
′(kIt )
]. (23)
By relaxing (i) and allowing for βIRS = 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 financially
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.
8Steady-state variables are reported without the time subscript.9 In this respect, imposing βIRS = 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.
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: kIt + kBt = 1. As to the market for consumption goods, the aggregate resource
constraint reads as:
yt = yIt + yBt , (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:
bBt = ωEtqt+1k
Bt
RB, (25)
bBt =1
ξχ
(RSbSt − χEtqt+1kIt
). (26)
Finally, as savers are indifferent between any path of consumption and savings, the amount of de-
posits depends on bankers’capitalization. Thus, the markets for deposits and final goods are cleared
according to the Walras’Law.
3 Steady State Properties
We first focus on the steady-state equilibrium. Financial frictions characterizing both the savers-
bankers relationship and the bankers-borrowers relationship deeply affect 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(kIt−1) =(kIt−1
)µ, with µ ∈ [0, 1]. Evalu-
ating (10) in the non-stochastic steady state returns:
q =RBβB(
1− βB)RB − ω
(1− βBRB
) . (27)
From (23) we retrieve the marginal product of bankers’capital, as a function of its price:
I′(kI) = µ
(kI)µ−1
=RS(1− βI
)− χ
(1− βIRS
)RSβI
q, (28)
so that Equations (27), (28) and kI +kB = 1 pin down borrowers’and bankers’holdings of capital. In
turn, these allow us to characterize the steady-state ineffi ciencies arising from the debt enforcement
problems in place at different 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 DE0E∗. The first-best allocation would be
attained at E0, 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 (mpkB = 1) exceeds that of the bankers (mpkI = %). That is, relative
to the first-best, too little capital is used by the borrowers, due to their financial constraint. This
implies a loss of output relative to the first-best, as indicated by the shaded area CE0E∗.11 Remark 1
elaborates further on the relationship between borrowers’and bankers’marginal product of capital:
Remark 1 As long as βB < βI , bankers’marginal product of capital is lower than that of borrowers.
In fact, imposing I′(kI) < 1 returns the following inequality:
βB − βI <βBχ
(1− βIRS
) (RB + ωξ
)RS (RB − ω)
. (29)
As we assume βB < βI , the left-hand side of (29) is negative, while its right-hand side is positive,
given that also βIRS < 1 holds by assumption. Therefore, a defining 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 first-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.11The area under the solid line, ACE∗D, is the steady-state output.
10
Importantly, under βIRS = 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 ineffi ciency in the allocation of capital. This
is clearly indicated by the additional loss of output, relative to the first-best, as captured by the
trapezoid CKMCE∗E∗KM (where E∗KM indicates the steady-state equilibrium in the KM-type setting).
Therefore, a key result is that expanding KM’s framework with financially 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 ineffi ciency in the allocation of capital. To this end, we define
the productivity gap between borrowers and bankers:
mpkB −mpkI ≡ ∆ = 1− %. (30)
The following summarizes the impact of financial collateralization on the productivity gap:
Proposition 1 Increasing the degree of financial collateralization (ξ) reduces the gap between bankers’and borrowers’marginal product of capital (∆).
Proof. See Appendix AA higher degree of financial 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 effects
is such that mpkI increases in the degree of financial collateralization, reducing the productivity gap
with respect to the borrowers. This factor will play a key role in the amplification 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:
qt = φEtqt+1 + (1− φ)Etαt+1, (31)
where φ ≡ βBRB+ω(1−βBRB)RB
. As for the bankers:
qt = λEtqt+1 + (1− λ)Etαt+1 +1− λη
kBt , (32)
where λ ≡ RSβI+χ(1−βIRS)RS
and η−1 is the elasticity of the bankers’marginal product of capital times
the ratio of borrowers’to bankers’capital-holdings in the steady state (i.e., η ≡ 1−kBkB(1−µ)).
Once we obtain the solutions for qt and kBt 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 first focus on (31), whose forward-iteration leads to:
qt = γαt, (33)12Variables in log-deviation from their steady-state level are denoted by a "^".
11
where γ ≡ 1−φ1−φρρ > 0. With this expression for qt, we can resort to (32), obtaining
kBt = vαt, (34)
where v ≡ η1−λ
(λ−φ)(1−ρ)ρ1−φρ > 0.
4.1 Financial Collateral and Macroeconomic Amplification
We have now lined up the elements necessary to examine the economy’s response to technology
disturbances. Proposition 2 details the effect induced by a marginal change in the degree of financial
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 financial collateralization (ξ) attenuates the impact of thetechnology shock on both borrowers’holdings of capital and the capital price.
Proof. See Appendix AProposition 2 implies that the sensitivity of borrowers’capital-holdings to the technology shifter
decreases in the degree of financial collateralization. The intuition for this is twofold: i) on the one
hand, increasing ξ determines a more even distribution of capital goods, as reflected by the drop in η;
ii) on the other hand, being able to pledge a higher share of financial assets reinforces the sensitivity
of the capital price to the capital gain component in borrowers’Euler equation, φ, through the drop
in the loan rate, while reducing the sensitivity to the dividend component. The combination of these
effects mutes the dynamic multiplier embodied in this class of models, ultimately attenuating the
overall degree of macroeconomic amplification 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:
yt = αt + ∆yB
ykBt−1, (35)
According to (35), the dynamics of gross output is shaped by αt, as well as by borrowers’capital-
holdings at time t− 1: the second effect captures the endogenous persistence of gross output. In fact,
yt depends on the past history of shocks not only through the direct impact of αt, but also through
the effect of αt−1 on kBt−1, as implied by (34). In light of this, we can rewrite (35) as
yt = $αt−1 + ut (36)
where $ ≡ ρ + v∆yB
y . This result is important in that it shows how eliminating the key source of
steady-state ineffi 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
13This property echoes the role of the steady-state ineffi ciency for short-run dynamics in the KM model. In theirsetting, closing the gap between the marginal products of capital of lenders and borrowers would imply no response atall to a productivity shift. In this respect, the key difference between the two frameworks lies in that we assume anautoregressive shock, while they consider an undexpected temporary shift in technology.
12
There are three different channels through which an increase in savers’ perceived liquidity of
bankers’financial assets affect the propagation of a technology shock. To see this, we compute the
following derivative:
∂$
∂ξ= ∆
yB
y
∂v
∂ξ+ v
yB
y
∂∆
∂ξ+ v∆
∂(yB/y
)∂ξ
. (37)
Proposition 2 shows that ∂v/∂ξ < 0, implying that financial collateralization reduces the impact of
a technology shift on borrowers’capital-holdings, which we know exerting a first-order effect on total
output through (35). We also know from Proposition 1 that the productivity gap between borrowers
and bankers shrinks as financial collateralization increases (i.e., ∂∆/∂ξ < 0). As to ∂(yB/y
)/∂ξ:
∂(yB/y
)∂ξ
=ωχ(1− βGRS
) (1− kB
)µ (1 + µ kB
1−kB
)κβGRSRBy2 (1− µ)
(1
µ
RBβBκRSβIκ
) 1µ−1
, (38)
which is positive, given that higher financial collateralization implies a net transfer of capital goodsfrom bankers to borrowers. In turn, this implies both a first-order positive effect on yB and a milder
second-order positive impact on y (given that, concurrently, yI decreases at a lower pace than the
increase in yB), so that the overall effect is positive. To sum up, an increase in ξ causes competing
effects on$. As we already know, greater financial collateralization depresses the pass-through of αt−1on borrowers’capital-holdings. Furthermore, raising ξ exerts two effects on the pass-through of kBt−1on yt: (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 reflects higher capital accumulation in the hands of the borrowers.
These competing forces potentially lead to mixed effects on output amplification, as captured by
second-round effects of technology disturbances. To address this point, Figure 2 plots $ as a function
of ξ and µ.14 ,15
As it emerges from Figure 2, increasing ξ compresses $, at any level of µ. By contrast, increasing
the income share of capital in bankers’production technology amplifies the second-round response of
output. This is because µ amplifies the productivity gap through its positive effect on η.16
14The aim of this exercise is to have an idea of the direction of the overall effect exerted by financial collateralizationon macroeconomic volatility, rather than quantifying an empirically plausible multiplier emerging from the interactionof bankers’and borrowers’financial constraints. We leave this task for future research employing a large scale dynamicgeneral equilibrium model.15The 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:βS = 0.99, βI = 0.98, βB = 0.97. We set ρ = 0.95, in line with the empirical evidence showing that technology shocksare generally small, but highly persistent (see, e.g., Cooley and Prescott, 1995). As for χ and ω , their are set to 1 so asto ensure a wider set of admissible combinations of ξ and µ that ensure positive holdings of capital for both bankers andborrowers. This is clearly displayed by the robustness evidence reported in Appendix C, which also shows that differentcombinations of χ and ω are close to irrelevant regarding the effect of financial collateralization on macroeconomicamplification.16 It is also worth to highlight that increasing µ 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 thecase, recall that in the steady-state bankers’marginal product of capital is tied to their user cost of capital throughµ(kI)µ−1
= %. Increasing µ inflates bankers’marginal product of capital, while leaving their user cost unaffected: Thus,as µ increases bankers are induced to hold an increasing stock of capital, so that the equality holds. An important aspectis that this effect tends to kick in earlier as ξ declines. This is because a drop in the degree of financial collateralizationdepresses bankers’user cost of capital. Therefore, as ξ declines and µ increases the set of steady-state allocations inwhich 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 amplification.
µ
ξ
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1
1.1
1.2
1.3
1.4
1.5
1.6
Notes. Figure 2 graphs $ as a function of ξ and µ, under the following parameterization: βS= 0.99, βI= 0.98,βB= 0.97, ρ = 0.95, χ = ω = 1. The white area denotes inadmissible equilibria where bankers’capital-holdings arevirtually negative.
4.2 The Role of Leverage
To enlarge our perspective on the amplification/attenuation induced by bankers’financial collateral,
we take a closer look at their balance sheet. To this end, we define bankers’equity as the difference
between the value of total assets (lending plus real assets) and liabilities (deposits):
eIt = bBt + qtkIt − bSt , (39)
while leverage is defined as the ratio between loans and equity: levIt = bBt /eIt . 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
financial collateralization. However, as ξ 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 ξ, while increasing as bankers can offer a higher share of
their financial assets as collateral. The reason for this can be better understood by exploring the
interaction between bankers’financial and real assets. Their interplay takes place on two levels: i) on
the one hand, as embodied by (14), both assets have a positive effect on savers’deposits; ii) on the
other hand, assuming a fixed supply of physical capital implies a substitution effect between real and
financial assets. In fact, according to borrowers’collateral constraint, bBt increases in kBt . However, as
17The baseline parameterization is the same as that employed in Figure 2. As for µ, we impose a rather conservativevalue of 0.4, which allows us to obtain a finite distribution of capital in the steady state.
14
kBt = 1−kIt , increasing bankers’real asset-holdings exerts a negative force on lending.18 We also knowthat, 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 influenced by two opposite forces, namely an
expansion in the amount of bankers’financial assets and a contraction their stock of real assets. In
light of this, it is important to understand that, as ξ drops — reflecting a decline in the perceived
liquidity of bankers’financial assets —the role of financial 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 amplification, when bankers’financial assets are
judged to be relatively illiquid. Figure 3 shows this is the case for ξ > 0.5.
Figure 3. Impulse responses to a positive technology shock.
0 5 10 15 20
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3Output
0 5 10 15 200.05
0.1
0.15
0.2
0.25
0.3Price of capital
0 5 10 15 200
0.5
1
1.5
2
2.5
3Borrowers capital
0 5 10 15 201
0.8
0.6
0.4
0.2
0
0.2
0.4Deposits
0 5 10 15 200
0.5
1
1.5
2
2.5
3Lending
0 5 10 15 200.2
0.15
0.1
0.05
0
0.05
0.1
0.15Lev erage
Notes. Figure 3 graphs the response of selected variables to a one-standard-deviation shock to technology,under the following parameterization: βS= 0.99, βI= 0.98, βB= 0.97, ρ = 0.95, χ = ω = 1, µ = 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 amplification. Therefore, attenuating the degree of procyclicality of
bankers’ leverage is important to reduce the amplitude of fluctuations in gross output, especially
when savers perceive bankers’ financial assets to be relatively illiquid. In recent years regulators
18This is a distinctive feature of lender-borrower relationships involving the collateralization of a productive asset. Infact, KM show that the major reallocation of land from the lenders to the borrowers following a positive technologyshock is only attenuated by relaxing the hypothesis of fixed supply of the real asset, while the direction of the transferis not inverted.
15
have suggested to lean against credit imbalances and pursue macroeconomic stabilization through
policy rules that set a countercyclical capital buffer. 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 ≥ θbBt , (40)
where θ denotes the capital-to-asset ratio. Notably, combining (39) with (40) obtains
bSt ≤ qtkIt + (1− θ) bBt , (41)
which is remarkably similar to the enforcement constraint (14).19 In fact, a marginal increase (de-
crease) in the capital (leverage) ratio maps into a decrease in the degree of collateralization of financial
assets. Intuitively, a higher leverage (lower capital) ratio implies a riskier exposure of the financial
intermediary: this translates into a greater transaction cost savers would have to bear in the event of
bankers’default, so as to seize their financial 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):
θtθ
=
(bBtbB
)ϕ, ϕ ≥ 0. (42)
For ϕ = 0 we implicitly impose a constant capital-to-asset ratio, while under ϕ > 0 the macro-
prudential rule implies a countercyclical capital buffer, 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 affected by an en-
dogenous capital-to-asset ratio:21
RBt =RS − (1− θt)
(1− βIRS
)βI
. (43)
Equation (43) can be linearized in the neighborhood of the steady state:
RBt = ψθt, (44)
19A key difference 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.20The 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-basedmeasure to weight the share of asset to be hold. Basel III has been brought in response to the 2007-2008 crisis, with thekey innovation consisting of introducing countercyclical capital requirements, that is, imposing banks to build resiliencein good times with higher capital requirements and relax them during bad times. According to this regime, capitalregulation can respond to a wide range of macroeconomic indicators. Here we assume it to respond to deviations of bBtfrom its long-run equilibirum, bB .21Appendix 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 buffer that leans againstthe procyclicality of bank leverage, in the remainder we assume (41) to be more stringent than (14). Allowing for theco-existence of the two limits to bank deposits would impose to account for occasionally binding constraints, a task thatgoes beyond the scope of the present analysis.
16
where ψ = 1−βIRSβIRB
θ is positive, in light of assuming βIRS < 1. We also linearize (42), obtaining:
θt = ϕbBt . (45)
After linearizing borrowers’financial constraint, we can substitute for bBt in (45) and plug the resulting
expression into (44), so as to obtain:
RBt =ψϕ
1 + ψϕ
(Etqt+1 + kBt
), (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 effect. In fact, increasing ϕ implies that marginal deviations of bBt from its steady state
transmit more promptly to the capital-to-asset ratio and the loan rate, through the combined effect
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 effect on ψ.
To assess the stabilization performance of the countercyclical capital buffer rule, we run two
experiments: we first set the response coeffi cient ϕ at a given level and vary the steady-state capital-
to-asset ratio, θ; in a second exercise we fix θ and vary ϕ. Figure 4 reports the economy’s response
to a technology shock in the first experiment: lowering the steady-state capital-to-asset ratio proves
to be rather ineffective at mitigating the response of output (see Figure 5). This is not surprising,
as we focus on a rather narrow range of values for θ, 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 coeffi cient ϕ varies over the support [0, 1]. As expected, at
ϕ = 0 (i.e., a capital-to-asset ratio at its steady state level) we observe the highest amplification of the
output response, while the lending rate and bank leverage are both constant. Increasing the degree
of countercyclicality of the capital buffer proves to be effective 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 ϕ → ∞, the policy rule pushes leverage to display a strong degree ofcountercyclicality, so that gross output implies no endogenous propagation.
17
Figure 4. Impulse responses under different θ.
0 5 10 15 200.4
0.6
0.8
1
1.2
1.4Output
0 5 10 15 200.05
0.1
0.15
0.2
0.25
0.3Price of capital
0 5 10 15 201
2
3
4
5Borrowers capital
0 5 10 15 200
0.5
1
1.5Capital ratio
0 5 10 15 200
0.5
1
1.5
2
2.5x 10 3 Lending rate
0 5 10 15 201
2
3
4
5
6Equity
0 5 10 15 201
2
3
4
5Lending
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25Deposits
0 5 10 15 201.5
1
0.5
0Lev erage
Notes. Figure 4 graphs the response of selected variables to a one-standard-deviation shock to technol-ogy, under the following parameterization: βS= 0.99, βI= 0.98, βB= 0.97, ρ = 0.95, χ = ω = 1, µ = 0.4,ϕ = 0.3.
Figure 5. Impulse responses under different ϕ.
0 5 10 15 200
0.5
1
1.5Output
0 5 10 15 200.05
0.1
0.15
0.2
0.25
0.3Price of capi tal
0 5 10 15 200
1
2
3
4Borrowers capital
0 5 10 15 200
5
10
15Capital ratio
0 5 10 15 205
0
5
10x 10
3 Lending rate
0 5 10 15 200
1
2
3
4
5Lending
0 5 10 15 200
5
10
15Equity
0 5 10 15 200.05
0
0.05
0.1
0.15
0.2Deposits
0 5 10 15 20
10
5
0
Leverage
Notes. Figure 5 graphs the response of selected variables to a one-standard-deviation shock to technol-ogy, under the following parameterization: βS= 0.99, βI= 0.98, βB= 0.97, ρ = 0.95, χ = ω = 1, µ = 0.4,θ = 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 affected by limited enforceability: as
a result, if bankers default, savers acquire the right to liquidate bankers’ real and financial asset-
holdings. We have emphasized the use of bankers’financial 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 different forms of financial intermediation, savers anticipate
that liquidating financial assets is conditional on borrowers being themselves solvent on their debt
obligations. This friction limits the degree of collateralization of bankers’financial assets and, in turn,
their borrowing capacity. In this context, we have demonstrated three main results: i) expanding KM’s
framework with financially constrained financial intermediaries allows to attain a Pareto-superior
steady-state allocation; ii) increasing financial collateralization dampens macroeconomic fluctuations
by reducing the degree of procyclicality of bank leverage; iii) finally, 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 different debt-equity mix may affect macroeconomic amplification over expansions —
when equity can be issued frictionlessly —and contractions, when equity issuance may be precluded
due to tighter information frictions. This factor should counteract the role of financial assets and help
obtaining a countercyclical leverage. In connection with this point, we could also allow for occasionally
binding financial constraints, so as to evaluate how the policy-maker should behave across contractions
—when constraints tighten —and expansions, when constraints may become non-binding. However,
as this type of extensions necessarily hinder the analytical tractability of our problem, we leave them
for future research projects based on large scale models.
19
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21
