Embed Size (px)
Citation preview
Financial Collateral and Macroeconomic Amplification∗
Federico Lubello† Ivan Petrella‡ Emiliano Santoro§
March 29, 2016
Abstract
We design an analytically tractable model 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 struc-
ture 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 col-
lateralization of bankers’financial assets. In this context, increasing the degree of financial
collateralization alleviates steady-state ineffi ciencies– reducing the productivity gap between
borrowers and bankers– and dampens macroeconomic fluctuations, whose amplitude rests
on the procyclicality of bank leverage. 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.†Banque Centrale du Luxembourg. Address : 2, Boulevard Royal, L-2983 Luxembourg. E-mail : fed-
[email protected].‡Bank of England, Birkbeck and CEPR. Addrress: Threadneedle St., London EC2R 8AH, 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 num-
ber 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 guarantee in case of default on their debt obligations. This is
the case for non-traditional banking activities—with sale and repurchase agreements (repos) em-
ployed 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
The present paper examines the connection between financial collateral and macroeconomic
amplification. 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 asset. However, due to the vertically integrated structure
of our credit economy, savers anticipate that liquidating bankers’financial 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.
In this context we examine how different degrees of financial collateralization impact on
both the steady-state distribution of real and financial resources, as well as on the transmission
and amplification of an aggregate technology shock, which is traditionally regarded as a key
driver of the business cycle. 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 spread between the loan and the deposit rate. In turn, this situation reflects into the
steady-state distribution of capital: increasing the pledgeability of financial assets reduces the
productivity gap between bankers and borrowers, as real assets are redistributed from the latter
to the former.
Increasing the degree of financial collateralization attenuates the response of gross output
to the productivity shock. This effect is intimately related to the role of bank leverage. When
savers’perceived liquidity of bankers’financial assets is relatively low, deposits drop in the face
of a positive productivity shock, while expanding as bankers are allowed to offer a higher share
1The set of assets that central banks accept from commercial banks generally includes government bondsand 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
of their financial assets as collateral. The combination of real and financial assets in the form
of bankers’collateral is crucial to this result: in the face of an expansionary technology shock,
the decline of real assets is typically counteracted by the expansion of financial assets (i.e.,
bank lending). However, as the degree of financial collateralization declines, this compensation
effect is gradually muted and the dynamics of deposits is eventually tied to that of bankers’real
assets, given that financial assets are perceived to be completely illiquid. In this context, the
countercyclical dynamics of deposits reflects into a procyclical leverage ratio.
Breaking the procyclicality of bank leverage is of key importance to attenuate the magnitude
of aggregate fluctuations in this model economy. However, this might be particularly challenging
when savers perceive bankers’financial assets to be relatively illiquid. To overcome this friction,
we assume that a macroprudential policy-maker might step in to set a capital adequacy require-
ment 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. Intuitively, this prop-
erty can be explained by noting that a higher leverage 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. 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 recently imposed by the Basel III regulatory
framework.
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 developing banking literature on the role of macropru-
dential policy in attenuating various sources of exogenous perturbation. Some recent examples
include Van den Heuvel (2008), Admati, DeMarzo, Hellwig, and Pfleiderer (2010), Hellwig
(2010), Martinez-Miera and Suarez (2012), Angeloni and Faia (2013), Harris, Opp, and Opp
(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 sources of exogenous perturbation, trans-
mission channels and alternative policy measures, the analytical intuition of their functioning
is severely limited.
3
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.
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 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
As in KM, borrowers’ability to obtain external financial resources is bounded by the limited
enforceability of their debt contracts. In light of this friction, bankers protect themselves against
the risk of borrowers’default by collateralizing their holdings of capital. Therefore, in case of
borrowers’ default bankers acquire the right to liquidate the stock of real assets. Based on
the predicted outcomes of the renegotiation (see derivations in Appendix A), borrowers will be
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
subject to the following enforcement constraint:
RBbBt ≤ ωEtqt+1kBt , ω ∈ [0, 1] , (2)
where kBt denotes borrowers’holdings of real assets, qt is the price of capital in non-durable
good units, bBt is the amount of credit to the borrowers and RB is the loan rate. According to
(2), 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 , (3)
where cBt and yBt denote borrowers’consumption and production of perishable goods, respec-
tively. As in KM, borrowers are assumed to combine capital and labor (which is supplied inelasti-
cally) 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, tak-
ing RB as given. The resulting Lagrangian is:
LBt = Et
∞∑t=0
(βB)t {
cBt − ϑBt[cBt +RBbBt−1 + qt(k
Bt − kBt−1)− bBt − αtkBt−1
](4)
−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; (5)
∂LBt∂kBt
= 0⇒ −ϑBt qt + βBEt[ϑBt+1qt+1
]+ βBEt
[ϑBt+1αt+1
]+ ωϕtEt
[qt+1RB
]= 0. (6)
Condition (5) implies that a marginal decrease in borrowing today expands next period’s utility
and relaxes the current period’s borrowing constraint. As to (6), acquiring an additional unit
of capital today allows to expand future consumption not only through the conventional capital
gain 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), (5)
implies ϕt = ϕ = 1− βBRB. As a result, (6) can be rewritten as
qt =βBRB + ω
(1− βBRB
)RB
Etqt+1 + βBEtαt+1. (7)
5
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 debt contracts with the savers. If
bankers default, savers acquire the right to liquidate bankers’real and financial asset-holdings.
As in Gertler and Kiyotaki (2011, 2015) we assume that, upon bankers’default, savers liquidate
the entire collateral asset position.4 However, at the time of contracting the amount of deposits,
the liquidation value of bankers’assets is uncertain. With probability 1− χ the recovery valueis zero, while with probability χ savers expect to recover the value Etqt+1kIt + ξbBt , where k
It
denotes bankers’holdings of real assets.5 We set ξ ∈ [0, 1], so as to account for the possibility
that debt enforcement is potentially ineffi cient when it comes to recover the book value of
bankers’financial assets. This assumption rests on the structure of our credit chain: 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. Thus, when formulating the
incentive compatibility constraint savers account for a cost (1− ξ) bBt they might have to bearin order to recover financial resources located at the bottom of the credit chain: therefore, in the
extreme situation in which ξ = 0 savers regard bankers’financial assets as completely illiquid
and do not accept them as collateral, while ξ = 1 corresponds to a situation in which savers
attach no risk to their ability of liquidating financial assets in case bankers default.
Based on the predicted outcomes of the renegotiation process, bankers will be subject to the
following enforcement constraint:
RSbSt ≤ χ(Etqt+1k
It + ξbBt
), (8)
according to which the amount of deposits, together with the accrued interest, should be limited
from above by a fraction of the expected collateral value.
Also bankers combine capital with labor so as to produce consumption goods. However, for
a given stock of real assets, we assume that borrowers are more productive than financial inter-
mediaries. The implicit assumption is that bankers lack the necessary expertise to pursue the
production process, while featuring deeper specialization in intermediating funds between savers
and borrowers. In light of this, bankers’production technology has the following properties:
yIt = αtI(kIt−1),
4There are two main considerations why this assumption is a reasonable one: First, savers have no direct useof the collateral assets; second, even if collateral assets represent an attractive investment opportunity, savershave no experience in hedging.
5Appendix B describes the renegotiation process between savers and bankers in case the latter default on theirobligations.
6
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
)] ,where βI denotes bankers’discount factor.
Bankers’flow-of-funds constraint reads as:
cIt + bBt +RSbSt−1 + qt(kIt − kIt−1) = bSt +RBbBt−1 + yIt , (9)
where cIt denotes bankers’consumption. The Lagrangian for bankers’optimization reads as
LIt = Et
∞∑t=0
(βI)t {
cIt − ϑIt [cIt +RSbSt−1 + bBt + qt(kIt − kIt−1) (10)
−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; (11)
∂LIt∂bBt
= 0⇒ RBβIEtϑIt+1 − ϑIt +
1
RSχξδt = 0; (12)
∂LIt∂kIt
= 0⇒ −ϑIt qt + βIEt[ϑIt+1qt+1
]+ βIEt
[ϑIt+1αt+1I
′(kIt )
]+ δtχ
Et [qt+1]
RS= 0. (13)
As we assume linear preferences, ϑIt = ϑI = 1. Therefore, conditions (12) and (13) 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 < 1.7 Specifically, condition (i) implies
that bankers are relatively more impatient than savers,8 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
. (14)
Notably, 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
6This condition is required for an internal solution in which both bankers and borrowers demand physicalcapital, as it will be discussed further in Section 4.1.
7Steady-state variables are reported without the time subscript.8 In this respect, imposing βIRS = 1 reduces the model to the conventional KM (1997) economy.
7
deposits and translates into a higher credit supply that compresses the lending rate.9
Finally, from (13) we can retrieve the Euler equation governing bankers’investment in real
assets:
qt =RSβI + χ
(1− βIRS
)RS
Etqt+1 + βIEt
[αt+1I
′(kIt )
]. (15)
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 ,
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, (16)
bBt =1
ξχ
(RSbSt − χEtqt+1kIt
). (17)
Finally, as savers are indifferent between any path of consumption and savings, the amount of
deposits (bSt ) depends on bankers’capitalization. Thus, the markets for deposits and final goods
are cleared according to the Walras’Law.
3 Steady State Properties
Prior to examing equilibrium dynamics, we focus on the steady-state properties of the economy.
Evaluating (7) in the non-stochastic steady state returns:
q =RBβB(
1− βB)RB − ω
(1− βBRB
) . (18)
From (15) we retrieve the marginal product of bankers’capital, as a function of the price of
capital:
I′(kI) = µ
(kI)µ−1
=RS(1− βI
)− χ
(1− βIRS
)RSβI
q, (19)
so that Equations (18), (19) and kI+kB = 1 allow us to pin down borrowers’holdings of capital.
The Pareto optimum for this model economy is attained at the point in which the product
of capital owned by the bankers and borrowers is the same at the margin. As borrowers have
a linear production technology and α = 1, their marginal productivity of capital equals 1.
9Note that (14) implies that we need not to impose (ii), as assuming (i) is enough for both conditions to hold.
8
However, 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 <χβB
(1− βIRS
)RS
+ωβI
(1− βBRB
)RB
. (20)
As we assume βB < βI , the left-hand side of (20) is negative, while its right-hand side is positive,
given that both βIRS < 1 and βBRB < 1 hold by assumption. Therefore, a defining feature
of the equilibrium is that the marginal productivity of borrowers’ is higher than that of the
bankers, given that the former cannot borrow as much as they want.10
In light of this steady-state suboptimality, it is important to study how the distribution of
real resources between bankers and borrowers is affected by the presence of financial collateral.
To this end, we first define the productivity gap between borrowers and bankers:
mpkB −mpkI ≡ ∆ = 1− %. (21)
wherempkB andmpkI denote the marginal productivity of borrowers and bankers, respectively.
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.As borrowers’marginal product of capital equals one in the steady state, we restrict our
analysis to the impact of ξ on mpkI :
∂mpkI
∂ξ=∂mpkI
∂RB∂RB
∂ξ. (22)
As to the partial derivative of bankers’marginal product of capital with respect to the loan
rate:
∂mpkI
∂RB= − κωβB
κ2RSβI. (23)
where κ ≡ RF(1− βF
)− ω
(1− βFRF
)> 0 and κ ≡ RS
(1− βI
)− χ
(1− βIRS
)> 0, so that
∂mpkI/∂RB < 0.
As for ∂RB/∂ξ < 0, this is negative, in light of assuming βIRS < 1:
∂RB
∂ξ= −
χ(1− βIRS
)βIRS
. (24)
Thus, both factors on the right-hand side of (22) are negative and, since ∂∆/∂ξ = −∂mpkI/∂ξ,10As 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.
9
increasing ξ inevitably reduces the productivity gap.�A higher degree of financial collateralization expands bankers’ lending capacity and com-
presses the spread charged over the deposit rate. In turn, lower lending rates imply a weaker
feedback effect of future collateral on qt, as embodied by (5): this translates into a higher steady
state price of capital. 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 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.11 As for the borrowers:
qt = φEtqt+1 + (1− φ)Etαt+1, (25)
where φ ≡ βBRB+ω(1−βBRB)RB
.
As for the bankers, we impose I(kIt−1) =(kIt−1
)µ, with µ ∈ [0, 1], so that:
qt = λEtqt+1 + (1− λ)Etαt+1 +1− λη
kBt , (26)
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 (25), whose forward-iteration leads to:
qt = γαt, (27)
where γ ≡ 1−φ1−φρρ > 0. With this solution for qt, we can resort to (26), obtaining
kBt = ναt, (28)
where ν ≡ 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.11Variables in log-deviation from their steady-state level are denoted by a "^".
10
Proposition 2 Increasing the degree of financial collateralization ( ξ) attenuates the impact
of the technology shock on both borrowers’holdings of capital and the capital price.
Proof.We first prove that increasing ξ attenuates the impact of the technology shock on borrowers’
capital-holdings. According to (28), ν quantifies the pass-through of αt on kBt . In turn, marginal
impact of ξ on ν can be computed as:
∂ν
∂ξ=
(λ− φ) (1− ρ) ρ
(1− λ) (1− φρ)
∂η
∂ξ+
(λρ− 1) (1− ρ) ρ
(1− φρ)2∂φ
∂ξ, (29)
where:
∂η
∂ξ=
∂η
∂kB∂kB
∂RB∂RB
∂ξand
∂φ
∂ξ=
∂φ
∂RB∂RB
∂ξ. (30)
Focusing on the second term on the right-hand side of (29), we can show this is negative, as:
(i) (λρ−1)(1−ρ)ρ(1−φρ)2 < 0, given that λρ < 1; (ii) ∂φ/∂RB = −ω/
(RB)2< 0; (iii) ∂RB/∂ξ < 0, as
implied by (24).
As to the first term on the right-hand side of (29): (λ−φ)(1−ρ)ρ(1−λ)(1−φρ) > 0. Furthermore:
∂η
∂kB= − 1
(1− µ) (kB)2< 0
and
∂kB
∂RB=
ω
κRB (µ− 1)
(1
µ
RBβBκRSβIκ
) 1µ−1
< 0, (31)
where κ ≡ RB(1− βB
)−ω
(1− βBRB
)and κ ≡ RS
(1− βI
)−χ
(1− βIRS
). As ∂RB/∂ξ < 0,
also the first term on the right-hand side of (29) is negative. Therefore, ν is a negative function
of ξ.
As to the impact of technology shocks on the capital price:
∂γ
∂ξ=∂γ
∂φ
∂φ
∂ξ. (32)
As for ∂γ/∂φ:
∂γ
∂φ= − 1− ρ
(1− φρ)2ρ < 0, (33)
while we already know that ∂φ/∂ξ > 0. Therefore, the overall effect of ξ on γ is negative.�Proposition 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
11
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, (34)
According to (34), 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 (28). In light of this, we can
rewrite (34) as
yt = $αt−1 + ut (35)
where $ ≡ ρ+ν∆yB/y. To evaluate the impact of financial collateralization on $, we compute
the following derivative
∂$
∂ξ= ∆
yB
y
∂v
∂ξ+ v
kB
y
∂∆
∂ξ+ v∆
∂(yB/y
)∂ξ
. (36)
Proposition 2 has already shown that ∂v/∂ξ < 0. 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− kF
)µ (1 + µ kF
1−kF
)κβGRSRBy2 (1− µ)
(1
µ
RBβBκRSβIκ
) 1µ−1
, (37)
which is positive. Thus, an increase in ξ causes competing effects on $. As we already know,
greater financial collateralization depresses the pass-through of αt−1 on borrowers’ capital-
holdings. Furthermore, raising ξ exerts two effects on the pass-through of kBt−1 on 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
1 plots $ as a function of ξ and µ. As it emerges from this numerical exercise, increasing ξ
compresses $, at any level of µ. By contrast, increasing the income share of capital in bankers’
production technology amplifies the second-round response of output. It is worth to highlight
that increasing µ may violate the condition I′(0) > % > I
′(1), which ensures an interior solution
as for how much capital bankers (and thus borrowers) should hold in the neighborhood of the
steady state. To see why this is the case, let us recall the steady-state expression that relates
12
bankers’marginal product to their user cost of capital:
µ(kI)µ−1
= %. (38)
Increasing µ inflates bankers’marginal product of capital, while leaving their user cost unaf-
fected: as a result, as µ increases bankers are induced to hold an increasing stock of capital, so
that (38) holds. An important aspect is that this effect tends to kick in earlier as ξ declines.
This is beause a drop in the degree of financial collateralization depresses bankers’user cost
of capital (i.e., the right-hand side of (38). Therefore, as ξ declines and µ increases the set
of steady-state allocations in which both bankers and borrowers hold capital restricts, as the
condition % > I′(1) is eventually violated and borrowers’may virtually end up with negative
capital-holdings.
Figure 1. Business cycle amplification.
µ
ξ
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1
1.1
1.2
1.3
1.4
1.5
1.6
Notes. Figure 1 graphs$ 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 inammissible equilibria where
bankers’capital-holdings are virtually negative.
To enlarge our perspective on the amplification/attenuation induced by bankers’financial
collateral, we take a closer look at bankers’ 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 (i.e., levIt = bBt /eIt ). Figure 2
13
reports the response of selected variables to a one-standard deviation shock to technology. As
implied by (34), 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 explained based on the formulation of bankers’enforcement constraint: in the face of an
expansionary technology shock, the decline of real assets (kIt ) is typically counteracted by an
expansion in the amount of financial assets entering as collateral (bBt ). However, as ξ declines this
compensation effect is gradually muted and deposits eventually track the dynamics of bankers’
real assets. In this context, the countercyclical dynamics of deposits necessarily traslates into a
procyclical leverage ratio.
Figure 2. Impulse responses to a positive technology shock.
0 5 10 15 200.2
0.4
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 15 200
0.5
1
1.5
2
2.5
3Borrowers capital
0 5 10 15 201
0.8
0.6
0.4
0.2
0
0.2Deposits
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
0.1Leverage
Notes. Figure 2 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,
µ = 0.4, χ = ω = 1.
5 Capital Requirements and the Business Cycle
The previous section has shown that attenuating the degree of procyclicality of banks’leverage
may be of key importance to reduce the amplitude of fluctuations in gross output. This might be
particularly challenging when savers perceive bankers’financial assets to be relatively illiquid,
so that leverage increases as a result of ξ being relatively low. To overcome this friction, we
implement a macroprudential policy tool aimed at leaning against credit imbalances, through
14
a policy rule that sets a countercyclical capital buffer.
We start by assuming that a hypothetical regulatory authority imposes a capital adequacy
requirement, which sets a minimum limit to the amount of equity:
eIt ≥ θbBt , (40)
where θ denotes the capital-to-asset ratio. Combining eIt = bBt + qtkIt − bSt with (40) we obtain:
bSt ≤ qtkIt + (1− θ) bBt . (41)
This constraint is similar to the enforcement constraint (8),12 which results from savers’pre-
dicted outcome of their renegotiation in the event bankers default on their debt obligations. In
fact, a marginal increase (decrease) 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 further assume that the regulator allows capital requirements to vary with the macro-
economic conditions (see, e.g., 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.13
As a result of imposing (42), the loan rate is potentially time-varying, being affected by an
endogenous capital-to-asset ratio:14
RBt =RS − (1− θt)
(1− βIRS
)βI
. (43)
Equation (43) can be linearized in the neighborhood of the steady state:
RBt = ψθt, (44)
12A key difference between the two constraints lies in the fact that (41) entails period-t capital value, while (8)contemplates the period-t+ 1 expected capital value.13The 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 useinternal risk-based measure to weight the share of asset to be hold. Basel III has been brought in response tothe 2007-2009 crisis, with the key innovation consisting of introducing countercyclical capital requirements, thatis, imposing banks to build resilience in good times with higher capital requirements and relax them during badtimes.14Appendix C reports the derivation of the model economy under the macroprudential rule (42).
15
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 sensisivity 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 macroprudential policy regime, 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 3 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 4). 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 the
capital-to-asset ratio are both constant. Implementing a countercyclical capital buffer proves
instead to be rather effective at attenuating the economy’s response to the shock, by rising in
the lending rate and compressing bank leverage.
16
Figure 3. Impulse responses under different θ.
0 5 10 15 200.5
1
1.5Output
0 5 10 15 200
0.1
0.2
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 20
103
0
1
2
3Lending rate
0 5 10 15 200
2
4
6Equity
0 5 10 15 201
2
3
4
5Lending
0 5 10 15 200
0.1
0.2
0.3Deposits
0 5 10 15 201.5
1
0.5
0Leverage
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,
µ = 0.4, χ = ω = 1, ϕ = 0.3.
Figure 4. Impulse responses under different ϕ.
0 5 10 15 200
0.5
1
1.5Output
0 5 10 15 200
0.1
0.2
0.3Price of capital
0 5 10 15 200
2
4
6Borrowers capital
0 5 10 15 201
0
1
2
3Capital ratio
0 5 10 15 20
103
2
0
2
4
6Lending rate
0 5 10 15 200
2
4
6Lending
0 5 10 15 201
1.5
2
2.5
3Equity
0 5 10 15 200
0.05
0.1
0.15
0.2Deposits
0 5 10 15 200
1
2
3Leverage
Notes. Figure 4 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,
17
µ = 0.4, χ = ω = 1, θ = 0.08.
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 inter-
mediation, 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 that increasing financial collateralization dampens macroeconomic fluctuations
by reducing the degree of procyclicality of bank leverage. Furthermore, we have shown that 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 necesssarily hinder
the analytical tractability of our framework, we leave them for future research projects based
on larger scale models.
18
References
[1] Admati, A. R., DeMarzo, P. M., Hellwig, M. F., and P. Pfleiderer, 2010, Fallacies, Irrelevant
Facts, and Myths in the Discussion of Capital Regulation: Why Bank Equity is Not Socially
Expensive, Working Paper Series of the Max Planck Institute for Research on Collective
Goods.
[2] Angeloni, I., and E. Faia, 2013, Capital Regulation and Monetary Policy with Fragile
Banks, Journal of Monetary Economics, 60(3):311—324.
[3] Begenau, J., 2015, Capital Requirements, Risk Choice, and Liquidity Provision in a Busi-
ness Cycle Model, SSRN Electronic Journal.
[4] Clerc, L., Derviz, A., Mendicino, C., Moyen, S., Nikolov, K., Stracca, L., Suarez, J., and
A. P. Vardoulakis, 2015, Capital Regulation in a Macroeconomic Model with Three Layers
of Default, International Journal of Central Banking, 11(3):9—63.
[5] Gertler, M., and N. Kiyotaki, 2015, Banking, Liquidity, and Bank Runs in an Infinite
Horizon Economy, American Economic Review, 105(7):2011—2043.
[6] Harris, M., Opp, C. C., and M. M. Opp, 2014, Macroprudential Bank Capital Regulation
in a Competitive Financial System, SSRN Electronic Journal.
[7] Hellwig, M., 2010, Capital Regulation After the Crisis: Business as Usual?, CESifo DICE
Report, 8(2):40—46.
[8] Iacoviello, M., 2005, House Prices, Borrowing Constraints, and Monetary Policy in the
Business Cycle, American Economic Review, 95(3):739—764.
[9] Jermann, U., and V. Quadrini, 2012, Macroeconomic Effects of Financial Shocks, American
Economic Review, 102(1): 238-71.
[10] Kiyotaki, N., and J. Moore, 1997, Credit Cycles, Journal of Political Economy, 105(2):211-
248.
[11] Kiyotaki, N., and J. Moore, 2004, Credit Chains, ESE Discussion Paper Series, No. 118.
[12] Kocherlakota, M. R., 2001, Risky Collateral and Deposit Insurance, The B.E. Journal of
Macroeconomics, 1(1):1—20.
[13] Krishnamurthy, A., 2003, Collateral Constraints and the Amplification Mechanism, Journal
of Economic Theory, 111(2):277—292.
[14] Liu, Z., Wang, P., and T. Zha, 2013, Land Price Dynamics and Macroeconomic Fluctua-
tions, Econometrica, 81(3):1147—1184.
[15] Martin, A., and J. Ventura, 2012, Economic Growth with Bubbles, American Economic
Review, 102(6):3033—3058.
19
[16] Martinez-Miera, D., and J. Suarez, 2012, A Macroeconomic Model of Endogenous Systemic
Risk Taking, CEPR Discussion Papers, No. 9134.
[17] Nelson, B. D., and G. Pinter, 2013, Macroprudential Capital Regulation in General Equi-
librium, mimeo, Bank of England.
[18] Oehmke, M., 2014, Liquidating Illiquid Collateral, Journal of Economic Theory, 149:183—
210.
[19] Parlatore, C., 2015, Collateralizing Liquidity, mimeo, NYU Stern.
[20] Van den Heuvel, S. J., 2008, The Welfare Cost of Bank Capital Requirements, Journal of
Monetary Economics, 55(2):298—320.
20
Appendix A. Derivation of Borrowers’Enforcement Constraint
In line with Jermann and Quadrini (2012), we assume that if borrowers default, bankers acquire
the right to liquidate the stock of capital. Based on the predicted outcomes of the renegotiation,
borrowers will be 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. Therefore, bankers expect to be able to recover
Etqt+1kBt only with probability ω ∈ [0, 1], while (1− ω) is the probability of not being able to
recover the asset value after a default.
To derive the renegotiation outcome, we have to 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 liq-
uidation 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:
RBt bBt − Etqt+1kBt .
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:
RBt bBt .
Therefore, when the financial contract is signed, the expected liquidation value is:
ω[RBt b
Bt − Etqt+1kBt
]+ (1− ω)RBt b
Bt . (47)
Enforcement requires that the expected value of non defaulting is not smaller than the expected
value of defaulting, so that:
0 ≥ ω[RBt b
Bt − Etqt+1kBt
]+ (1− ω)RBt b
Bt ,
which reduces to
RBt bBt ≤ ωEtqt+1kBt .
Appendix B. Derivation of Bankers’Enforcement Constraint
We assume that if bankers default, savers acquire the right to liquidate the stock of real and
financial assets. Based on the predicted outcomes of the renegotiation, bankers will be subject
to an enforcement constraint. Neither savers nor bankers are able to observe the liquidation
21
value before the actual default, though bankers have all the bargaining power in the liquidation
process. Therefore, savers expect to be able to recover Etqt+1kIt + ξbBt only with probability
χ ∈ [0, 1], while (1− χ) is the probability of not being able to recover the assets value after a
default. We impose ξ ∈ [0, 1] so as to allow for the possibility that savers cannot fully repossess
the resources they have extended to bankers, in case the latter default. This is rationalized by
assuming that savers account for a potential transaction cost (1− ξ) bBt they might have to bearto recover financial resources located at the bottom of the financial chain.
To derive the renegotiation outcome, we have to consider the following default scenarios:
1. Savers expect to recover Etqt+1kIt + ξbBt . Since savers can expropriate the whole stock of
real and financial assets, bankers have to make a payment that 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:
RSt bSt − Etqt+1kIt − ξbBt .
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 ex-post default value in this case is:
RSt bSt .
Therefore, when the deposits are made, the expected liquidation value is:
χ[RSt b
St − Etqt+1kIt − ξbBt
]+ (1− χ)RSt b
St . (48)
Enforcement requires that the expected value of defaulting is not smaller than the expected
value of defaulting, so that:
0 ≥ χ[RSt b
St − Etqt+1kIt − ξbBt
]+ (1− χ)RSt b
St ,
which reduces to
RSt bSt ≤ χ
(Etqt+1k
It + ξbBt
).
Appendix C. The Model with Capital Requirements
The Lagrangian for bankers’optimization reads as follows:
LIt = Et
∞∑t=0
(βI)t {
cIt − ϑIt [cIt +RSbSt−1 + bBt + qt(kIt − kIt−1) (49)
−bSt −RBbBt−1 − αtI(kIt−1)]− δt[bSt − qtkIt − (1− θt) bBt
]},
22
where ϑIt = 1 and δt are the multipliers associated with bankers’budget constraint and the
financial resource constraint, respectively. The first-order conditions are:
∂LIt∂bSt
= 0⇒ −RSβIEtϑIt+1 + ϑIt − δt = 0; (50)
∂LIt∂bBt
= 0⇒ RBβIEtϑIt+1 − ϑIt + (1− θt) δt = 0; (51)
∂LIt∂kIt
= 0⇒ −ϑIt qt + βIEt[ϑIt+1qt+1
]+ βIEt
[ϑIt+1αt+1I
′(kIt )
]+ δtqt = 0. (52)
In light of these conditions we can derive expressions for both RBt and qt in the presence of a
binding capital requirement constraint:
RBt =RS − (1− θt)
(1− βIRS
)βI
, (53)
qt =1
RSEtqt+1 +
1
RSEt
[αt+1I
′(kIt )
]. (54)
23
