Optimal Bank Reserve Remuneration
and Capital Control Policy∗
Chun-Che Chi†

Stephanie Schmitt-Grohé‡Martı́n Uribe§

November 4, 2023

Abstract
A central prediction of open economy models with a pecuniary externality due toa collateral constraint is that the unregulated economy overborrows relative to what
occurs under optimal policy. A maintained assumption in this literature is that households borrow directly from foreign lenders. This paper shows that if foreign lending isintermediated by domestic banks and the government can pay interest on bank reserves
and impose capital controls, the unregulated economy underborrows. The optimal bankreserve policy is countercyclical. By increasing bank reserves during contractions, thegovernment acts as a lender of last resort to collateral-constrained households.
JEL Classification codes: E58, F38, F41. Keywords: Open Economy, Overborrowing, Banking Channel, Pecuniary Externality, Capital Controls, Interest on Bank
Reserves.

∗
We thank for helpful comments Nina Biljanovska, Isabel Correia, Sebastián Fanelli, Cristina Manea,
and seminar participants at the Bank of Italy, the U. of Chile, CUNY, the Toulouse School of Economics,
IMF, the 2021 Copenhagen Macro Days, the XIII Workshop in International Economics and Finance heldin Quito, the University of Surrey, UC Davis, NYU, Yale, Maryland, Wisconsin, LSE, the Bank of Israel,the 12th ifo Conference, UBC, Simon Fraser, CREI, PSE, Bocconi, the Chicago Fed, the Richmond Fed, the
FRBSF, Georgia State, the Florida Macroeconomics Series, the 2022 CEMLA-FRBNY-ECB Conference on
Monetary Policy Challenges, and the 2nd WE ARE IN Macroeconomics and Finance Conference.
†
Institute of Economics, Academia Sinica. E-mail: ccchi@econ.sinica.edu.tw.
‡
Columbia University, NBER, and CEPR. E-mail: ss3501@columbia.edu.
§
Columbia University and NBER. E-mail: martin.uribe@columbia.edu.

1Introduction

A central question in open economy macroeconomics is whether, when left to their own
devices, countries overborrow. This question has been largely analyzed in the context ofmodels in which households are subject to a collateral constraint, whereby debt is limited
by a fraction of income or the value of an asset (Bianchi, 2011; Korinek, 2011; Benigno etal. 2013; Dávila and Korinek, 2018; Jeanne and Korinek, 2019; among others). Collateral
constraints of this type create a pecuniary externality because individual agents take asgiven the prices of objects that they pledge as collateral but in the aggregate these prices are
determined endogenously. A key result in this literature is that the unregulated economy
overborrows from the rest of the world relative to what it would borrow under optimal capital
control policy.
A common feature of this class of model is the assumption that private agents borrowdirectly from foreign lenders. In reality, however, individual agents seldom borrow directlyfrom foreign lenders. Instead, capital inflows are intermediated by banks operating in domestic markets. A natural question is whether this simplification has consequences for themain predictions of this class of model. This paper revisits the question of overborrowing
in the context of a model that builds on the collateral-constraint framework by adding abank-intermediation channel. The formulation of the banking channel follows Cúrdia andWoodford (2011).
The paper studies an open economy with a collateral constraint by which household debt
is limited by a fraction of income. A banking sector receives deposits from foreign investors
and lends them to households. This intermediation activity is costly. Banks can mitigate
the cost of originating loans by holding reserves at the central bank.
As in the related literature on macroprudential policy in open-economy models with collateral constraints, the government can impose capital control taxes. With the introductionof a banking sector the interest rate on reserves emerges as an additional policy instrumentthat the government may use jointly with capital controls to achieve an allocation that improves upon the one associated with the unregulated competitive equilibrium. Thus, relative
to the standard overborrowing model, the present environment features an additional friction, bank intermediation, and an additional policy instrument, interest on bank reserves.
The fiscal expenditures (revenues) stemming from capital controls, interest payments onbank reserves, and operating a reserve facility are assumed to be financed by income taxes(transfers).
We show that a social planner with access to capital controls and interest on reserves
as policy instruments can implement an allocation in which households internalize the pe-

1cuniary externality and banks internalize the social costs and benefits of reserve provision.
Specifically, the social planner maximizes the household’s lifetime utility subject to the economy’s resource constraint, the bank’s balance sheet, and the household’s collateral constraint
evaluated at the equilibrium price of collateral.
The central result of the paper is that under plausible parameterizations the economy
with a bank intermediation channel underborrows. That is, the distribution of external debt
in the competitive equilibrium in which the government neither remunerates reserves nor
taxes capital flows lies to the left of the one associated with the equilibrium in which theinterest rate on bank reserves and capital control taxes are set optimally.
To understand the intuition behind this result note first that in a small open economy
consumption smoothing requires that negative income shocks be financed by increases in theexternal debt. If households borrow directly from foreign lenders, a binding collateral constraint that forces them to deleverage after the negative income shock can prevent them fromsmoothing consumption. That is, household deleveraging implies economy wide deleveraging.
By contrast, in the economy with banks, by the bank’s balance sheet, foreign deposits (i.e.,the country’s external debt) are allocated either to loans to households or to bank reservesat the central bank. When the economy suffers a negative shock that causes the household’s
collateral constraint to bind, the government finds it optimal to step in and raise the interest
rate on bank reserves. In turn, the increase in the interest rate on reserves induces banks todeposit at the central bank the funds that are no longer demanded by financially constrainedhouseholds. The increased holdings of bank reserves by the central bank make their way into
the budget constraints of households via fiscal policy (reductions in income taxes) providing
liquidity when households need it. With this intervention the economy continues to borrowfrom abroad despite a binding constraint at the household level. In this way, the consolidated
government acts as a lender of last resort to the non-financial sector. The prediction of themodel that during a crisis financial intermediation by the central bank increases is in linewith the observed behavior of central bank balance sheets during the global financial crisis(see, for example, Allen and Moessner, 2013, for evidence from the euro area).
The presence of a collateral constraint and a banking channel opens the question ofwhether the collateral constraint should be placed at the level of the bank or at the level ofthe non-bank private sector. Following the literature on overborrowing cited above, we adopt
the latter option (the collateral constraint limits household debt). This modeling choice has
consequences for the predicted behavior of the bank lending spread around financial crises.
We show that when the collateral constraint is imposed at the level of the household, thelending spread on bank loans tends to fall during financial crises because a binding collateral
constraint represents a decline in the demand for bank credit. We also show that when the2collateral constraint is imposed at the bank level, the lending spread tends to rise because inthis case a financial crisis represents a fall in the supply of credit. However, the equilibrium
real allocation (including the volume of borrowing) is independent of these two alternative
formulations.
A macroprudential instrument closely related to reserve remuneration is reserve requirements. A question of interest is whether the equilibrium outcomes a policymaker can attainwith one of these instruments can also be achieved or improved upon with the other. The
paper shows that there is a clear ranking between these two policy tools: bank reserve remuneration strictly welfare dominates reserve requirements. Intuitively, by paying intereston reserves, the central bank controls the price of this component of the bank’s asset side
but allows the quantity (bank reserves) to be determined endogenously. On the other hand,a reserve requirement with no interest on reserves represents a restriction on both the quantity and the price of bank reserves, therefore reducing the set of real allocations that it can
support as competitive equilibria.
This paper is related to two strands of literature, one on overborrowing in open-economy
models with collateral constraints in the nonfinancial sector and one on closed-economy
models with a banking channel. Open economy models with collateral constraints at thehousehold level are studied in Mendoza (2002), Uribe (2006), Korinek (2011), Bianchi (2011),
Benigno et al. (2013), Dávila and Korinek (2018), Jeanne and Korinek (2019), Schmitt-Grohéand Uribe (2021), Arce, Bengui, and Bianchi (2022), and Davis, Devereux, and Yu (2022),
among others. The banking model follows Cúrdia and Woodford (2011) and Eggertsson etal. (2019). Uribe and Yue (2006) introduce bank intermediation along the lines of the model
considered in this paper to create a spread between the domestic and the world interest rates.
However, their formulation does not contemplate a role for bank reserves. The present paperbuilds upon these two bodies of work by combining a collateral constraint at the household
level and a banking sector in the context of an open economy.
Exceptions to the overborrowing result in the related literature are Benigno et al. (2013),
Schmitt-Grohé and Uribe (2021), and Drechsel and Kim (2022). In these papers, the unregulated economy underborrows. However, the reasons for underborrowing in these papers
are different from the ones stressed here. In Benigno et al. underborrowing stems from introducing production in the nontradable sector or distortionary sectoral taxation. In thepresent environment the bank intermediation channel produces underborrowing even without production or distortionary taxation in the nontradable sector. Schmitt-Grohé and Uribe(2021) show that the canonical open-economy model with a flow collateral constraint exhibits
multiple equilibria under plausible calibrations. In this environment, underborrowing is theconsequence of excess saving caused by the possibility of self-fulfilling financial crises. In the3present paper the banking channel leads to underborrowing even under parameterizations
for which the equilibrium is unique. Drechsel and Kim (2022) consider earnings-based collateral constraints on firms. In their framework, underborrowing arises because firms fail tointernalize the effect of their own borrowing on equilibrium wages and hence earnings. In thepresent model, the introduction of a banking channel causes the economy to underborrow
even in the absence of labor or capital as factor inputs in production. Finally, the paper
is related to a class of models in which a collateral constraint is placed on banks rather
than on households. Bocola and Lorenzoni (2020) and Céspedes and Chang (2020) represent
examples of such formulations in the context of an open economy.
The remainder of the paper is organized as follows. Section 2 develops the model, introduces the policy instruments, and derives the constrained optimal allocation. Section 3
performs the quantitative analysis and presents the main results. Section 4 studies two
extensions, placing the collateral constraint at the level of banks and replacing interest on
reserves with reserve requirements. Section 5 concludes.

2

The Model

In this section, we present a model of an open economy in which banks serve as intermediaries
between foreign investors, who supply funds, domestic households, who demand bank loans,and the domestic government, who operates a reserve facility. The specification of the banklending channel follows Cúrdia and Woodford (2011). The banking sector is embedded
into a standard open economy model with a flow collateral constraint, whereby household
debt is limited by a fraction of income, along the lines of Mendoza (2002), Bianchi (2011),
and Korinek (2011). The collateral constraint introduces a pecuniary externality because
the relative price of nontradable goods, which determines the value of collateral, is taken
as exogenous by individual borrowers but is endogenous to the economy. After presenting
the model and the definition of a competitive equilibrium, the section characterizes theconstrained optimal allocation attainable by a government that uses as policy instrumentsinterest on bank reserves and capital controls.

2.1

Banks

We assume that the economy has a large number of identical, perfectly competitive financial
intermediaries, which we will refer to as banks. Each period, banks issue loans, lt, holdreserves, rt, capture deposits, dt , and distribute dividends, πt . Banks face intermediation
costs, denoted Γt . This cost is meant to capture expenses such as those related to loan4monitoring and management. The sequential budget constraint of a bank is
πt + lt + rt + (1 + idt−1 )dt−1 + Γt = (1 + ilt−1 )lt−1 + (1 + irt−1)rt−1 + dt ,(1)

where idt−1 is the interest rate paid by the bank on deposits held from period t − 1 to period t,
ilt−1 is the interest rate charged by the bank on loans made in period t − 1 and due in periodt, and irt−1 is the interest rate the bank earns on reserves deposited at the central bank fromperiod t − 1 to period t. The right-hand side of the sequential budget constraint represents
the bank’s sources of funds and the left-hand side the uses of funds.
The intermediation cost is assumed to depend on the volume of loans and bank reserves,
Γt = Γ(lt , rt ).

(2)

We introduce the following assumptions about this function:
Assumption 1 (Intermediation Cost Function). The function Γ(·, ·) satisfies: (i) Γ(·, ·) ≥ 0;(ii) Γl (·, ·) ≥ 0 and Γr (·, ·) ≤ 0; (iii) Γll ≥ 0, Γrr ≥ 0, Γlr < 0, and Γll Γrr − Γ2rl ≥ 0; (iv)
Γ(0, ·) = Γl (0, ·) = Γr (0, ·) = 0 and Γ(l, ·) > 0 for l > 0, and (v) there exists a finite level ofreserves, r̄ > 0, such that Γr (·, r) = 0 for all r ≥ r̄.
Assumptions (i)-(iii) are standard. In particular, the assumption that the cost function
is nondecreasing in loans is meant to capture administrative and default costs of originatingbank credit to the private sector, the assumption that it is nonincreasing in bank reserves
is meant to capture that bank reserves contribute to reducing default risk and possible
maturity mismatches between bank liabilities and assets, and the assumption of a negative
cross derivative makes reserves complementary in the production of loans. Assumption (iv)
is meant to capture the idea that the central bank has zero default risk, so, aside frominterest differentials, it is costless for banks to park funds there in the form of reserves. As
will become apparent shortly, this assumption will play a role in determining the optimal
interest-on-reserve policy. Assumption (v) is common in models with a formulation of thebanking sector of the type studied here (Cúrdia and Woodford, 2011; Eggertsson et al.,2019). It says that there exists a satiation level of reserves above which reserves cease tolower the intermediation costs of loans. At the satiation point, however, intermediation costs
need not vanish.
Following the related literature, we assume that banks distribute as dividends their
beginning-of-period net worth,πt = (1 + ilt−1 )lt−1 + (1 + irt−1 )rt−1 − (1 + idt−1 )dt−1 .
5(3)

In the context of the literature on the banking channel, the rationale for assuming an adhoc dividend rule is that if banks were assumed to choose the dividend stream to maximize
the lifetime welfare of the representative household, they could fully neutralize the financial
frictions. The assumption that banks do not act on behalf of the representative household isjustified by the observation that in reality banks are owned by a small fraction of householdswith potentially different pricing kernels than the rest of the population. Modeling thissource of heterogeneity explicitly is beyond the scope of this paper.
Combining the bank’s sequential budget constraint (1), the intermediation cost function (2), and the dividend policy function (3) evaluated in periods t and t + 1, one can
write

πt+1
ilt − idt
irt − idt=lt +rt − Γ(lt , rt ).
1 + idt
1 + idt
1 + idt(4)This expression provides an alternative interpretation of the dividend policy. Banks distribute at the beginning of period t + 1 all of the operating profits of period t. Thus, werefer to πt+1 as profits or dividends interchangeably. By Assumption 1 profits vanish at
lt = rt = 0. Thus, a profit maximizing bank would never distribute negative dividends.
Banks choose lt ≥ 0 and rt ≥ 0 to maximize (4), taking as given ilt, idt , and irt . Profit
maximization implies the following first-order conditions with respect to lt and rt:
ilt − idt
≤ Γl (lt , rt),
1 + idtandirt − idt
≤ Γr (lt, rt ),1 + idt

lt ≥ 0, lit − idt− Γl (lt , rt) lt = 0,
1 + idt(5)

rt ≥ 0, r

it − idt− Γr (lt, rt ) rt = 0.
1 + idt(6)

Optimality condition (5) says that when the volume of loans is positive, lt > 0, the marginal
net revenue of originating a loan, given by the lending spread (ilt − idt)/(1 + idt ), must equal
the marginal cost of originating a loan Γl (lt, rt ). When bank reserves are positive, optimality
condition (6) says that the bank holds bank reserves so as to equate their marginal benefit,−Γr (lt , rt), to their marginal cost, −(irt − idt)/(1 + idt ). Since Γl (·, ·) is nonnegative, optimality
condition (5) implies that when the volume of loans is positive, the deposit rate is the lower
bound of the loan rate. Similarly, because Γr (·, ·) is nonpositive, optimality condition (6)implies that the deposit rate is the upper bound of the reserve rate.
The bank’s sequential budget constraint (1) and the dividend rule (3) imply thatlt + rt + Γ(lt, rt ) − dt = 0,(7)which says that deposits are used to fund loans and reserves and to cover intermediation6

costs. With some abuse of terminology, we refer to this expression as the balance sheet ofthe bank at the end of the period.

2.2

Households

Households have preferences for consumption described by the utility function
E0∞X

β tu(ct ),

(8)

t=0

where ct denotes consumption, β ∈ (0, 1) is the subjective discount factor, and u(·) is an increasing and strictly concave period utility function. Consumption is a composite of tradable
and nontradable goods,
ct = A(cTt , cN
t ),

(9)

where cTt and cNt denote consumption of tradables and nontradables and A(·, ·) is an increasing, concave, and linearly homogeneous aggregator function. Each period, households
are endowed with ytT units of tradable goods and ytN units of nontradable goods, receive
dividends πt from the ownership of banks, pay income taxes at the rate τt , and can borrowfrom banks at the rate ilt. Their sequential budget constraint islTN
cTt + pt cN
t + (1 + it−1 )lt−1 = (1 − τt )[yt + pt yt + πt ] + lt ,(10)

where pt is the relative price of nontradables in terms of tradables.
Loans face a collateral constraint, which depends on the value of income in units of
tradable goods as follows,lt ≤ κ(ytT + pt ytN ),(11)

where κ > 0 is a parameter. We use this specification of collateral to be in line with therelated literature (e.g., Bianchi, 2011) and for analytical tractability. An alternative plausible
but less tractable specification is one in which collateral is proportional to disposable income,including after-tax profits.
Households choose processes ct , cTt , cNt , and lt to maximize the lifetime utility function (8)subject to the aggregation technology (9), the sequential budget constraint (10), and thecollateral constraint (11), taking as given pt , ilt , πt, τt , ytT , and ytN . The first-order conditions
associated with this problem are
T N
u0(A(cTt , cNt ))A1 (ct , ct ) = λt ,

7

(12)

A2(cTt , cNt )= pt ,T
A1(ct , cN
t )

(13)

λt (1 − µt ) = β(1 + ilt )Et λt+1 ,(14)µt ≥ 0,(15)µt [κ(ytT + pt ytN ) − lt] = 0,

(16)

and

where β tλt and β tλt µt are the Lagrange multipliers associated with the sequential budget
constraint (10) and the collateral constraint (11), respectively.

2.3

Foreign Lenders

Banks capture deposits from foreign lenders at the world interest rate i∗t and pay capital
control taxes at the rate τtc , which introduces a wedge between the world interest rate andthe interest rate effectively paid by domestic banks on deposits. Specifically, idt and i∗t arelinked by the relationship1 + idt = (1 + τtc )(1 + i∗t ).

(17)

The capital control tax rate can take positive or negative values. When τtc < 0, the government subsidizes capital inflows and when τtc > 0, it taxes them.

2.4

The Government

The government levies income taxes and capital control taxes, τt and τtc . It also operates abank reserve facility by setting the interest rate on bank reserves, irt , and standing ready toaccept any amount of reserves, rt , offered by banks. As in Cúrdia and Woodford (2011), we
assume that operating the bank-reserve facility is costly. Specifically, we assume that thecentral bank faces the cost Γr (rt ) with the following properties:
Assumption 2 (Bank-Reserve Cost Function). The function Γr (·) satisfies: (i) Γr (·) is
increasing and convex; and (ii) Γr (0) = 0 and Γr (r) > 0 for r > 0.
The need to introduce a cost of providing bank reserves is twofold. First, it is empiricallyrealistic. In section 3.1, we document that central banks’ unit operating costs are nonnegligible and in fact higher than those of private banks. For emerging countries, the costfunction could also capture expenses of a financial nature, for example, if public borrowing(in this case in the form of bank reserves) raises the risk of sovereign default (Arce, 2023).

8

Second, central bank operating costs have real consequences. In the absence of reserve provision costs, the government can circumvent the banking friction and the household collateral
constraint. The proof of this claim is in Appendix A.1
The government’s budget constraint is thencτt (ytT + pt ytN + πt ) + τt−1
(1 + i∗t−1 )dt−1 + rt = (1 + irt−1 )rt−1 + Γr (rt ).

(18)

We assume that the government does not play Ponzi games.

2.5

Competitive Equilibrium

In equilibrium, the market for nontradable goods must clear,NcN
t = yt .

(19)

The budget constraint of the bank (1), the budget constraint of the household (10), theinterest-rate parity condition (17), the budget constraint of the government (18), and themarket clearing condition in the nontraded sector (19) imply the following economy-wide
resource constraint:
cTt + (1 + i∗t−1 )dt−1 = ytT − Γ(lt, rt ) − Γr (rt ) + dt .

(20)

We are now ready to define a competitive equilibrium.
Definition 1 (Competitive Equilibrium). A competitive equilibrium is a set of processes lt,
rt , dt , ilt, idt , cTt , pt , λt , and µt satisfying (5)-(7), (11)-(17), and (20), for t ≥ 0, given a
reserve remuneration policy irt , a capital control tax rate τtc , exogenous processes i∗t , ytT , andytN , and the initial condition (1 + i∗−1 )d−1 > 0.
Following the open economy literature with collateral constraints, we consider an environment in which agents are impatient in the sense that they discount future period utilities
at a higher rate than the one at which the world financial market discounts future payments.
Formally, we assume thatβ(1 + i∗t ) < 1.
1

The result that in the absence of costly reserve provision the central bank can neutralize the banking
channel friction appears in Cúrdia and Woodford (2011) in the context of a closed economy. The proof in
the context of the present open-economy framework has value added for two reasons. First, in addition to
the banking channel, the model features an occasionally binding collateral constraint. Second, the proof
shows that capital controls are redundant (τtc = 0) and that the optimal interest rate on reserves is the worldinterest rate (irt = i∗t ).

9

In the related literature this condition is assumed to be strong enough to ensure an equilibrium in which the country is a net external debtor at all times. Throughout the present
analysis, we assume that this is indeed the case.

2.6

The Constrained Optimal Allocation

Consider a social planner who aims to achieve the following real allocation:
Definition 2 (Constrained Optimal Allocation). The constrained optimal allocation is a setof processes cTt , dt , lt ≥ 0, and rt ≥ 0 that solves the problem
max E0∞X

β tu(A(cTt , ytN ))t=0

subject to the bank’s balance sheet (7), the economy-wide resource constraint (20), and to
the loan constraint
A2 (cTt , ytN ) N
T
y ,
(21)
l t ≤ κ yt +A1 (cTt , ytN ) ttaking as given the processes i∗t , ytT , and ytN , and the initial condition (1 + i∗−1 )d−1 > 0.
The three constraints of the social planner’s problem are a subset of the complete setof equilibrium conditions listed in Definition 1. It follows immediately that the constrained
optimal allocation delivers at least as much welfare as any competitive equilibrium.
The constrained optimal allocation can be supported by an appropriate combination ofbank-reserve remuneration policy irt and capital control policy τtc . To see this, in the definitionof a competitive equilibrium (Definition 1) set µt = 0, which ensures that conditions (15)and (16) hold, λt to satisfy (12), pt to satisfy (13), ilt to satisfy the Euler equation (14), idt tosatisfy (5), irt to satisfy (6), and τtc to satisfy (17). The paths of bank profits (πt ) and of theincome tax rate (τt ) associated with this equilibrium can be constructed residually to satisfy
equations (3) and (18), respectively. We summarize this result in the following proposition.
Proposition 1 (Interest on Reserves, Capital Controls, and the Constrained Optimal Allocation). There exists a pair of processes irt and τtc that support the constrained optimal
allocation described in Definition 2 as the competitive equilibrium described in Definition 1.
In words, the optimal policy addresses the two externalities present in the model: (i) It
makes households internalize the pecuniary externality created by the collateral constraint.
That is, households act as if they understood that the relative price of nontradables equals
their own marginal rate of substitution of tradable for nontradable goods. This property isalso present in a version of the model without a banking sector (e.g., Bianchi, 2011). And (ii)10the optimal policy makes banks internalize that bank reserves provide liquidity to households
and that they are costly to produce. That is, banks act as if they understood that when
they increase their bank reserves they allow the government to make transfers to households,
which can be particularly welfare increasing when the collateral constraint is binding, andthat the cost of doing so is dictated by the function Γr (rt). Both this externality and its
treatment by the social planner represent a novel contribution to the open-economy related
literature on macroprudential policy and financial frictions.
An important theme of the quantitative analysis we conduct in what follows is that under
the optimal policy, when the collateral constraint binds, domestic loans and bank reserves
move in opposite directions. That is, bank reserves provide liquidity when households areborrowing constrained. Here, we show analytically that when the collateral constraint does
not bind, under the optimal policy loans and bank reserves move in the same direction.
To see this, note that the optimization problem in Definition 2 implies that when the loan
constraint (21) holds with strict inequality, for any value of dt , the social planner chooses
lt and rt to minimize the resource cost of loan and reserve provision subject to the bank’sbalance sheet. Specifically, when the collateral constraint is slack, lt and rt solve the problem
min Γ(lt, rt ) + Γr (rt){lt ,rt }subject to (7), given dt . When lt and rt are both positive, the first-order condition associatedwith this problem isΓl (lt, rt )Γr (lt, rt ) + Γr0 (rt )=.
(22)
1 + Γl (lt, rt )1 + Γr (lt, rt )Roughly speaking, this optimality condition says that when the collateral constraint is slack,the social planner equates the private marginal cost of originating loans to the central bank’s
marginal cost of reserve provision net of the private bank’s marginal benefit of holding
reserves. Under relatively weak conditions, namely, 1 + Γr (lt, rt ) > 0, and Γr0 (rt ) < 1, thisoptimality condition implies that lt and rt move in the same direction. This means thatwhen the collateral constraint is slack under the optimal policy, movements in the desired
level of external debt, dt , are achieved by moving the volume of loans and bank reserves inthe same direction, so that dt , lt, and rt comove positively. The economic significance of thisprediction is that when the collateral constraint is slack the planner does not find it optimal
to compensate credit contractions (falls in lt) with liquidity injections (increases in rt ). As it
will become apparent in the quantitative analysis that follows, the picture is quite different
when the collateral constraint binds for the planner. In such circumstances, lt and rt movein opposite directions, reflecting the fact that the central bank substitutes reserves for loans

11to preserve liquidity.

3

Quantitative Analysis

The question we wish to address next is how the economy behaves on average and around
financial crises under the constrained optimal equilibrium and under the laissez-faire equilibrium, where the latter is defined as the equilibrium resulting under the policy irt = τtc = 0, forall t. To this end, we turn to a quantitative characterization of the model’s dynamics underthese two policy regimes. Appendix B describes the numerical methods used to approximate
the equilibrium dynamics with and without government intervention.

3.1

Calibration by Simulated Method of Moments

The period utility function takes the CRRA formu(ct ) =ct1−σ − 1,1−σwith σ > 0. The aggregator function of tradable and nontradable consumption takes the
CES formhi1/(1−1/ξ)T 1−1/ξ
N 1−1/ξ
A(cTt , cN
)
=
ac
+
(1
−
a)c
,
t
t
t

(23)

with ξ > 0 and a ∈ (0, 1).

The financial intermediation cost function of banks takes the form
Γ(lt , rt ) = Alt1+α[1 + φ(rt − r̄)2 I(rt < r̄)],(24)

with A, α, φ, r̄ > 0. The operating cost function of the central bank takes the formΓr (rt ) = Brt1+α.

(25)

The specification of the two cost functions assumes that the volume elasticity, 1 + α, isthe same for the central bank and the commercial bank. The purpose of this assumption
is to economize on parameters. It implies that the administrative and monitoring costs ofloans and of reserve provision are similarly sensitive to the scale of operation. Because thecoefficients A and B can in principle be different from each other and are determined tomatch actual data, the assumed parameterization allows for the total, the average, and themarginal intermediation costs to differ across the two types of bank. The specifications of
the cost functions Γ(l, r) and Γr (r) satisfy Assumptions 1 and 2, respectively.
12

Table 1: Calibration
Parameter

Value

σ
a
ξ
i∗
β
κ
A
α
φ
r̄
B

2
0.31
0.83
0.04
0.86670.28920.02581.538523.6933
0.11161.7488

nyT
13
nyN
13
800
 Tnd T y[-0.1093,
0.1093]
ln
y
,
ln
 N

N
ln y , ln y[-0.1328, 0.1328][d, d][0.4, 1.05]

Description
Structural Parameters
Inverse of intertemporal elasticity of consumption
Parameter of CES aggregator
Elasticity of substitution between tradables and nontradables
World interest rate
Subjective discount factor
Parameter of collateral constraint
Parameter of intermediation cost function Γ(l, r)Parameter of the intermediation cost functions Γ(l, r) and Γr (r)Parameter of intermediation cost function Γ(l, r)Parameter of intermediation cost function Γ(l, r)Parameter of intermediation cost function Γr (r)Discretization of State Space
Number of grid points for ln ytT , equally spaced
Number of grid points for ln ytN , equally spaced
Number of grid points for dt , equally spaced
Range for logarithm of tradable output
Range for logarithm of nontradable output
Debt range unregulated economy

Note. The time unit is a year.

13

Table 2: Empirical and Predicted Targeted Moments

(1)
(2)
(3)
(4)
(5)
(6)

MomentFormula Observed Predicted
il −id
0.04990.0501
Lending spread
1+idrReserve-to-deposit ratio
0.06440.0629dd
Debt-to-output ratio
0.29000.2835yT +pyNΓ(l,r)
0.01750.0181
Intermediation-cost-to-deposit ratiodΓr (r)
Central-bank-operating-cost-to-reserve ratio
0.02050.0209r
Frequency of binding collateral constraint
0.05000.0548Notes. Lines 1, 2, and 5 are cross-country medians of cross-time medians. The
definition of an emerging country and the countries included follow Uribe andSchmitt-Grohé (2017). Data Sources: Lending spread, International Financial
Statistics (IFS); reserve-to-deposit ratio, Bankscope data for commercial banks;
debt-to-output ratio, Bianchi (2011); intermediation-cost-to-deposit ratio, Philippon (2015) and Bazot (2018); central-bank-operating-cost-to-bank-reserve ratio,
Bankscope data for central banks. Appendix C provides details on data sources,
sample periods, and country coverage.

The calibration of the parameters of the model is summarized in Table 1. The time
unit is meant to be one year. Following Bianchi (2011), we set σ = 2, a = 0.31, ξ = 0.83,
and the world interest rate i∗t at a constant value of 4 percent per annum. We also follow
Bianchi (2011) in setting the degree of relative impatience, β(1 + il ), where il denotes theaverage interest rate on bank loans in the laissez-faire equilibrium. Bianchi’s model does notinclude bank intermediation, so loans to households originate directly from foreign lenders,
and relative impatience equals β B (1 + i∗), where β B = 0.91 denotes the subjective discountfactor in Bianchi (2011). We therefore impose the restriction β(1 + il ) = β B (1 + i∗), which
implies that β = β B [(1 + i∗ )/(1 + il )]. The object (1 + il )/(1 + i∗ ) is the average gross lending
spread in the laissez-faire economy, which we set to 1.0499 to match the median value
observed in a sample of 38 emerging economies over the period 1985-2016 (see Table 2).
This yields β = 0.8667.
The sources of uncertainty are the endowments of tradable and nontradable goods, ytT
and ytN . The natural logarithms of these two variables are assumed to follow a bivariate
AR(1) process. The parameters of this process are set to the values used in Bianchi (2011).
The remaining parameters, A, α, φ, r̄, B, and κ, pertain to the financial side of theeconomy. They are calibrated by simulated method of moments (SMM) to jointly match
six moments for which the unregulated model economy can produce precise predictions: (1)14The lending spread ((il − id )/(1 + id )). Using data from the IMF’s International Financial
Statistics (IFS), we estimate the average value of this spread across emerging countries to be
4.99 percent per year. Recalling that the world interest rate is set at 4 percent, this estimate
implies that in the model economy banks lend to the domestic private sector at a rate morethan twice as high as the rate at which they borrow from international lenders. (2) The
reserve-to-deposit ratio (r/d). We estimate this ratio using data on commercial banks from
Bankscope. On average, banks in emerging countries hold 6.44 percent of their deposits inthe form of reserves at the central bank. This ratio is more than three times the one observed
in rich countries (1.98 percent). (3) The debt-to-output ratio (d/(y T + py N )). In the model,
dt is both the amount of bank deposits and the country’s net foreign debt position. Thisis because, being a representative-agent economy, the model does not feature deposits bydomestic agents (all households are borrowers). For this reason, one must take a stance on
whether to calibrate the ratio d/(y T + py N ) to match the observed deposit-to-output ratioor to match the observed net-foreign-debt-to-output ratio. We pick the latter option to keep
in line with calibrations in the related literature. Specifically, following Bianchi (2011), we
set d to be 29 percent of output.2 (4) The bank-operating-cost-to-deposit ratio (Γ(l, r)/d).
We set this ratio to 0.0175. This calibration lies in the middle of the range estimated byPhilippon (2015) and Bazot (2018), who estimate that bank unit costs range from 1.5 and2 percent in the United States (first author) and in a set of 20 countries (second author).
The sample in Bazot (2018) contains mostly developed countries. However, it includes four
countries that during the sample period of his study, 1970-2014, can be considered emerging
economies, namely, China, Portugal, South Korea, and Spain. The average unit cost acrossthese four countries is 1.99 percent, which is close to the value assigned in the calibration.
(5) The central-bank-operating-cost-to-reserve ratio (Γr (r)/r). Using data from Bankscopefor central banks, we estimate this ratio to be on average 2.05 percent in emerging countries.
This value is slightly lower than the one observed across rich countries, 2.55 percent in thesame database. (6) The probability of a binding collateral constraint. We set this moment
to 5 percent (or one sudden stop every 20 years on average), which is a value within therange used in the related literature.3 Appendix C provides more information on the data
sources for the empirical moments related to the banking channel, namely, (1), (2), and (5).
2This argument does not apply to the calibration of the reserve-to-deposit ratio, r/d, because both r andd are expected to be proportionally affected by the presence of domestic deposits.
3
In some of the related literature the definition of a sudden stop includes the additional requirementof a concurrent current account improvement of a certain size. For example, Bianchi (2011) requires animprovement in the current account of at least one standard deviation. As it turns out, in the present
model eighty percent of the episodes of a binding collateral constraint in the unregulated economy satisfy
this additional criterion.

15

Figure 1: Unconditional Distributions of Debt With and Without a Bank Intermediation
Channel

25

25

20

20

15

15

10

10

5

5

0
0.7

0.8

0.9

1

0
0.7

1.1

0.8

0.9

1

1.1

Notes. The left panel corresponds to an economy without a bank intermediation channel and theright panel to an economy with a bank intermediation channel. Parameters take the values shownin Table 1 when applicable. The debt densities associated with constrained optimal allocations areshown with solid lines and debt densities of unregulated economies with broken lines. The figure
shows that in the absence of a banking channel there is overborrowing, whereas in the presence ofthe bank intermediation friction there is underborrowing.

Table 2 reports the six targeted moments and the corresponding predictions of the unregulated economy. The relevant steady state for the calibration is the stochastic steadystate rather than the deterministic one, as in the latter the collateral constraint binds at
all times. For this reason the calibration is computationally demanding, and exact matches
are not possible in general. However, as a comparison of the last two columns of Table 2
suggests, the match is quite close.

3.2

Underborrowing

A well-known result that arises in a version of the present model without a bank intermediation channel is that the unregulated economy overborrows (e.g., Bianchi, 2011). Specifically,
when households can borrow directly from foreign lenders, the equilibrium density functionof external debt is located to the right of the one associated with the constrained optimal
allocation that can be supported with capital control taxes. The economy without banks
is a special case of the one analyzed here in which r = 0 and Γ(l, r) = Γr (r) = 0, for all l.
The left panel of Figure 1 plots the equilibrium distribution of debt in the economy withoutbanks for the unregulated and constrained optimal cases. All parameters of the model otherthan those pertaining to the bank and central bank cost functions take the values shown in
Table 1. The resulting economy is identical to the one analyzed in Bianchi (2011) except
16

for the values of β and κ, which are slightly different (0.8667 versus 0.91 and 0.2892 versus
0.32, respectively), and a finer grid of the exogenous state variables (146 versus 16 distinct
pairs (y T , y N )). The debt density under optimal capital control policy lies to the left of theone associated with the unregulated economy. Thus, the plot shows that the model withoutbanks reproduces the standard overborrowing result.
The picture is quite different when household borrowing from foreign lenders is intermediated by banks. The result is shown in the right panel of Figure 1. Now the distributionof debt in the unregulated economy lies to the left of the corresponding distribution under
the constrained optimal equilibrium, suggesting that in the economy with a banking channel
the unregulated economy underborrows. On average, external debt is 10 percent larger inthe regulated economy than in the unregulated economy. Section 4.4 shows that the underborrowing result is robust to perturbing the parameters pertaining to the banking channel.
The intuition for why the economy with a banking channel underborrows is that intereston reserves appropriately applied dampen the negative macroeconomic consequences of credit
crunches at the household level. Essentially, bank reserves introduce a cushion between debt
and private loans. This can be seen from the balance sheet of the bank, equation (7), which,
up to the resource cost of producing loans, says that lt + rt = dt . So when lt is restricted by
the collateral constraint, the government can achieve the desired level of external debt bymaking up the shortfall in loans with reserve creation. The increase in bank reserves finds
its way into the budget constraint of households through a relaxation of fiscal policy.
Under the optimal policy, the government finds it particularly useful to expand bankreserves when the private sector is borrowing constrained. Suppose, for example, that theeconomy faces a negative tradable endowment shock that results in a binding collateral
constraint. Since the endowment process is mean reverting, the intertemporal approach tothe current account dictates that the economy should finance the negative shock by borrowing
from international lenders (i.e., by increasing dt ). However, the fall in output tightens thecollateral constraint, making banks reluctant to extend loans to households. In this case, thecentral bank can induce an increase in bank reserves by offering a higher interest rate on thistype of financial asset. Thus, through an increase in bank reserves, the economy as a whole
can increase external debt, dt , even though lt is impeded to expand by the binding collateral
constraint. The government assists households in smoothing consumption by transferring
via a reduction in the income tax rate, τt , the resources it raises in the form of bank reserves.
Figure 2 displays the distributions of loans and reserves in the economy with banks in
the unregulated and constrained optimal equilibria. Under optimal policy the economy has
a larger volume of both loans and bank reserves. The reason why the equilibrium volume
17

Figure 2: Unconditional Distributions of Loans and Reserves

20
100
15

80
60

10

40
5
0

20
0.7

0.8

0

0.9

0

0.05

0.1

0.15

0.2

Note. The densities associated with the constrained optimal allocation are shown with a solid line
and the densities of the unregulated economy (irt = τtc = 0) with a broken line.

of loans is higher under the optimal policy than under laissez-faire is that in the former thegovernment plays the role of lender of last resort to households, which induces households
to reduce precautionary saving (i.e., operate closer to the debt limit). Consistent with thisintuition, the optimal distribution of bank reserves has a fat right tail, which serves to limit
the magnitude of macroeconomic deleveraging (falls in dt ) when the household’s collateral
constraint binds. By contrast, in the unregulated economy the distribution of bank reserves
not only lacks a fat right tail but displays a mass concentration at zero. The mass at zeroarises because there are sudden stops that imply so severe a decline in the demand for loansthat banks have no use for reserves. Recall that in the laissez-faire economy bank reserves
are not remunerated, so their sole purpose is to reduce the marginal cost of initiating loans.
The lower the volume of loans is, the smaller the marginal benefit of reserves will be becauseΓlr < 0.
Table 3 reports the unconditional correlation of reserves with loans and output. Consistent with the above narrative, the correlations of bank reserves with loans and output
are positive under laissez-faire but negative under optimal policy. This difference in comovement is a reflection of the fact that in the laissez-faire economy the demand for reserves
by banks is dictated only by the amount of loans the bank can originate. On the other
hand, in the constrained optimal allocation the government raises the interest rate on reserves during bad times and in this way induces banks to hold more reserves even when they18

Table 3: Unconditional Correlations of Reserves with Loans and Output
Laissez-faire Constrained optimalcorr(rt, lt )
0.99
−0.48
corr(rt, ytT + pt ytN )
0.54
−0.79
Note. The column labeled laissez-faire corresponds to the competitive equilibrium with irt = τtc = 0 and the column labeled constrained optimal to the competitive equilibrium with irt and τtc chosen optimally.

make fewer loans. In those circumstances, the correlation between loans and reserves canbecome negative, which is the case as the table shows under the present parameterization of
the model. Section 4.4 shows that this result is robust to perturbations in the values of theparameters defining the banking channel.

3.3

Welfare Gains of Regulation With and Without Banks

In the economy with a banking channel, the welfare gains of optimal regulation through bankreserve remuneration and capital controls are larger than in the economy without banks. We
calculate the welfare cost of living in the unregulated economy instead of in the economy with
optimal regulation as the percentage change in the stream of consumption in the former thatmakes households indifferent between living in one or the other. This welfare cost measure,which we denote λ(ytT , ytN , dt−1 ), is state dependent and therefore takes transitional dynamics
into account. Formally, λ(ytT , ytN , dt−1 ) is defined as

Et

∞
X
k=0

β

k

h

λ(yT ,ytN ,dt−1 )1 + t 100



1−σ

cut+k

i1−σ

−1
= Et

∞
X

co 1−σ − 1
β k t+k,1−σk=0where cut and cot denote consumption in the unregulated and regulated economies in period t.
P∞P∞
co 1−σ −1
cu 1−σ −1
and v o(ytT , ytN , dt−1 ) ≡ Et k=0 β k t+k1−σLetting v u (ytT , ytN , dt−1 ) ≡ Et k=0 β k t+k1−σdenote lifetime welfare in the unregulated and regulated economies conditional on the current
state, we can solve the above expression for λ(ytT , ytN , dt−1 ) to get

(
)
1
 1−σ
o T
N
v
(y
,
y
,
d
)(1
−
σ)(1
−
β)
+
1
t−1
t
t
− 1 × 100.
λ(ytT , ytN , dt−1 ) =v u (ytT , ytN , dt−1 )(1 − σ)(1 − β) + 1
To obtain a summary measure of the welfare cost of no regulation, we compute the expectedvalue of λ(ytT , ytN , dt−1 ) using the ergodic density of (ytT , ytN , dt−1 ) induced by the unregulated19

economy.
We find that the average welfare cost of living in the unregulated economy is 0.31 percent
of lifetime consumption. When we eliminate the banking channel, we find that the welfare
cost is only 0.10 percent of lifetime consumption. Thus, the domestic banking channel
triples the welfare cost of laissez-faire.4 The banking channel amplifies the welfare cost of
no intervention because it adds a second externality to the model, namely, a discrepancy
between the private and social costs of bank reserve provision. This externality is absent
when domestic households are allowed to borrow directly from foreign lenders. The findingthat incorporating a banking sector triples the welfare cost of no policy intervention suggeststhat the banking friction is at least as relevant as the collateral constraint stressed in the
overborrowing literature.

3.4

Optimal Bank Reserve Policy During Sudden Stops

To understand how the social planner manages a sudden stop in the presence of the bankintermediation friction, we examine equilibrium dynamics in the unregulated economy andin the constrained optimal equilibrium around a typical episode in which the collateral constraint binds in the unregulated economy. To this end, we simulate the unregulated economy
for 1 million periods and extract all windows of eleven years containing a binding collateral
constraint in the middle. This yields 54,842 sudden stop episodes, which is consistent withthe targeted frequency of a binding collateral constraint of 5 percent (see Table 2). For
each variable, we compute the average across the sudden stop episodes. The result is shownwith broken lines in Figures 3 and 4. The period in which the collateral constraint binds isnormalized to 0, so time runs from period -5 to period 5.
To compare the sudden stop dynamics in the unregulated economy with those in theeconomy with optimal capital-control and reserve-remuneration policies, for each sudden
stop episode in the former economy, we compute the equilibrium dynamics implied by theconstrained optimal equilibrium assuming that in period -5 (five years prior to the suddenTNstop) the unregulated and regulated economies are in the same state (y−5
, y−5
, d−6 ). We

then hit the regulated economy with the same sequence of endowment shocks that buffeted
the unregulated economy between periods -5 and 5. The results are shown with solid lines
in Figures 3 and 4.
In the unregulated economy, the typical sudden stop occurs when the economy suffersa string of negative shocks to the endowments of tradable and nontradable goods (top row
4

The welfare cost in the absence of banks, 0.10 percent, is similar to the one reported in Bianchi (2011),
0.14 percent. The discrepancy is due to the slight differences in the calibration of β and κ and in thediscretization of the exogenous state variables discussed in section 3.2.

20

Figure 3: The Typical Sudden Stop Episode

1.02

1.02

1

1

0.98
0.98
0.96
0.96
0.94
0.94

0.92
0.9
-5

-4

-3

-2

-1

0

1

2

3

4

0.92
-5

5

1.05

0.25

1

0.2

0.95

-4

-3

-2

-1

0

1

2

3

4

5

-4

-3

-2

-1

0

1

2

3

4

5

-4

-3

-2

-1

0

1

2

3

4

5

0.15

0.9
0.1
0.85
0.05
0.8
0

0.75

-0.05

0.7
0.65
-5

-4

-3

-2

-1

0

1

2

3

4

-0.1
-5

5

2.5

0.95
0.9
0.85

2

0.8
0.75
0.7

1.5
-5

-4

-3

-2

-1

0

1

2

3

4

0.65
-5

5

Notes. The sudden stop associated with the constrained optimal allocation is shown with a solidline and the sudden stop associated with the unregulated economy (irt = τtc = 0) with a broken
line. In the bottom right panel the value of collateral is shown with the same line type as the levelof loans. The two variables are identifiable because the former is never below the latter.

21

Figure 4: The Typical Sudden Stop Episode (continued)

0.9

5.4
5.2

0.85

percent

percent

5
0.8

0.75

4.8
4.6
4.4

0.7
4.2
0.65
-5

-4

-3

-2

-1

0

1

2

3

4

4
-5

5

0.16

0

0.14

-0.5

-4

-3

-2

-1

0

1

2

3

4

5

-4

-3

-2

-1

0

1

2

3

4

5

-4

-3

-2

-1

0

1

2

3

4

5

-1

percent

0.12
0.1
0.08

-1.5
-2
-2.5

0.06

-3

0.04
0.02
-5

-3.5
-4

-3

-2

-1

0

1

2

3

4

-4
-5

5

1

1

0.95

0.8

0.9
0.6
0.85
0.4
0.8
0.2

0.75
0.7
-5

-4

-3

-2

-1

0

1

2

3

4

0
-5

5

Note. The sudden stop associated with the constrained optimal allocation is shown with a solidline and the sudden stop associated with the unregulated economy (irt = τtc = 0) with a broken
line.

22

of Figure 3). Both endowments fall by more than 8 percent between periods -5 and 0. By
construction, the path of the two endowments is the same in the regulated and unregulated
economies. Given the relative price of nontradables, the fall in output causes a decline in
the value of collateral. When the collateral constraint binds in the unregulated economy,
consumption of tradables falls by more than tradable output (middle left panel). This isbecause the economy, forced to deleverage, must run a trade balance surplus (middle rightpanel). The fall in aggregate demand depresses the relative price of nontraded goods, thatis, the real exchange rate depreciates (bottom left panel). This happens in spite of the factthat the endowment of nontraded goods also experiences a large contraction. The fall in
the relative price of nontradables further tightens the collateral constraint, a phenomenon
known as a Fisherian deflation (bottom right panel).
By contrast, in the economy with optimal capital controls and optimal bank-reserve
remuneration, the contraction in the demand for nontradables and the real depreciation are
milder. The reason is not that in the constrained optimal equilibrium the collateral constraint
does not bind. In fact, in the regulated economy households are often borrowing constrained
before, during, and after the sudden stop (see the bottom right panel of Figure 4). Instead,the reason why in the regulated economy aggregate demand and the real exchange rate areless affected by the contraction in the endowments is that under the optimal macroprudential
policy the economy as a whole continues to have access to international credit. This isapparent from the bottom left panel of Figure 4, which shows that debt falls sharply in theunregulated economy but is little changed in the regulated one. In fact, in the regulated
economy debt increases slightly during the entire episode. This is because, although byconstruction the level of debt at the beginning of period -5 (i.e., d−6 , not shown) is the samein the unregulated and regulated economies, the unconditional average level of debt is higher
in the latter than in the former. So over the entire time window, the regulated economy istransitioning to a higher level of debt.
The key difference between the sudden stop in the unregulated and regulated economies
lies in the behavior of bank reserves. The sudden stop causes household deleveraging (adecline in loans) in both, the unregulated and the regulated economies (top left panel of
Figure 4). However, the social planner manages to avoid aggregate deleveraging (a decline
in deposits) by raising bank reserves (middle left panel of Figure 4). By contrast, in theunregulated economy, the decline in loans is accompanied by a decline in bank reserves,which exacerbates macroeconomic deleveraging. As explained earlier, the reason why banks
have less use for (unremunerated) reserves is that they are complementary in the productionof loans, which in turn experience a sharp decline during the sudden stop.
For the constrained optimal allocation to be supported as a competitive equilibrium, the
23

4

4

3

3

percent

percent

Figure 5: The Typical Sudden Stop Episode (concluded)

2

1

0
-5

2

1

-4

-3

-2

-1

0

1

2

3

4

0
-5

5

-4

-3

-2

-1

0

1

2

3

4

5

-4

-3

-2

-1

0

1

2

3

4

5

0.05

2
1

0

percent

0
-1

-0.05

-2
-3
-4
-5

-4

-3

-2

-1

0

1

2

3

4

-0.1
-5

5

Note. The sudden stop associated with the constrained optimal allocation is shown with a solidline and the sudden stop associated with the unregulated economy (irt = τtc = 0) with a broken
line.

social planner must create incentives to ensure that banks choose the optimal quantities
of loans and bank reserves. The middle right panel of Figure 4 displays the behavior of
the reserve spread, (irt − idt )/(1 + idt ), during the typical sudden stop. In the unregulated
equilibrium the reserve spread is constant at all times because central bank reserves are
unremunerated and because the deposit rate equals the (constant) world interest rate, as thegovernment does not impose capital controls. Under the optimal policy, the reserve spreadincreases during the sudden stop, which incentivizes banks to elevate their reserve holdings.
The top right panel of Figure 4 displays the behavior of the lending spread, (ilt−idt )/(1+idt ).
When the collateral constraint binds (period 0), the lending spread falls in the unregulated
and the regulated economies. The reason is that the sudden stop represents a decline in the
demand for loans rather than a decline in the supply of loans. (Section 4.1 expands thisintuition by studying the dynamics of the lending spread when the collateral constraint isplaced on the bank’s side, in which case deleveraging disturbs the supply of loans.)
24

Figure 5 displays the levels of the interest rate on bank reserves (irt ), the capital control
tax rate (τtc ), the income tax rate (τt ), and income tax revenue, τt (ytT + pt ytN + πt). As is wellknown, in models with a collateral constraint of the type considered in this paper, the levelof the capital control tax rate τtc is not uniquely pinned down when the collateral constraint
is binding (see, for example, Schmitt-Grohé and Uribe, 2017). In this case there is a whole
range of capital control tax rates consistent with the optimal competitive equilibrium. This
indeterminacy, which does not spill over to the real allocation or to interest rate spreads, does
extend to the levels of interest rates and the income tax rate (irt , ilt, idt , and τt ). Consequently,
the figure does not display values for these variables when they are indeterminate. The top
left panel of the figure shows that in the run-up to the crisis, the government raises theinterest rate it pays on bank reserves. It does so to induce banks to become more liquid.
In turn, the increased stock of reserves held by banks at the central bank finds its way into
the households’ budget constraints through income tax cuts (more precisely, through anincrease in income-based transfers, see the bottom panels of the figure). Thus, both bankreserve remuneration and transfers are countercyclical around sudden stops. By contrast, asin the related literature without a bank intermediation channel, the capital control tax rate
is procyclical (i.e., τtc increases as the economy approaches the crisis). The reason why thesocial planner tightens capital controls near the crisis is that, although reserve remuneration
softens the consequences of sudden stops, they do not make them costless, so, in an uncertain
environment, it still pays for the benevolent government to try to avoid a binding collateral
constraint by discouraging external borrowing.

3.5

A Graphical Explanation of the Mechanism

To understand the behavior of quantities and prices of loans and bank reserves during asudden stop consider the following graphical explanation. The left panel of Figure 6 depicts
the loan market. The supply of loans is given by the marginal cost of bank intermediation,Γl (l, r) (efficiency condition (5)). Holding bank reserves constant, the loan supply schedule
is increasing in l. When the collateral constraint is slack, the demand for loans is downward
sloping and stems from the household’s Euler equation. The higher is the interest on loans,the lower the demand for loans will be, as households have an increased incentive to postpone
consumption. The initial equilibrium is at point a, where the supply and demand for loansintersect. The volume of loans is l0 and the lending spread is (il0 − id )/(1 + id ) (recall thatin the unregulated equilibrium id is constant and equal to i∗).
Consider now the market for bank reserves, which is depicted in the right panel of Figure 6. The supply of reserves is perfectly elastic at the constant spread (ir − id )/(1 + id ) =25

Figure 6: The Loan and Reserve Markets During a Sudden Stop in the Unregulated Economy
ir −id1+id

il −id1+id

0Γl (l, r1)Γl (l, r0)

il0 −id
1+id

−i∗
1+i∗

a

b

r1

r0

r̄
r

b
Γr (l1 , r)

a

D
l1

l0

Γr (l0 , r)l−i∗/(1 + i∗), as the central bank stands ready to supply any amount of reserves to privatebanks at a zero interest rate (ir = 0). Holding the volume of loans constant, the demand forbank reserves is given by the marginal benefit of reserve holdings by private banks, Γr (l0, r)(efficiency condition (6)). The demand schedule is upward sloping in the range 0 < r < r̄,
which is the relevant one for the present analysis. Equilibrium in the bank reserve market
occurs at point a, where the demand for reserves meets the (horizontal) supply of reserves.
Consider now the effect of a sudden stop on the markets for loans and reserves. Supposethat the economy suffers a negative endowment shock that makes the collateral constraint
bind, forcing households to deleverage. Suppose that the volume of loans demanded after thenegative shock is l1 < l0. In the market for loans (left panel of Figure 6), this is represented
by a kink in the demand for loans. For simplicity, we assume that the new demand for loans
is given by the original one for l < l1. At l = l1 , the new demand schedule is vertical. In thereserve market (right panel of Figure 6), the fall in the volume of loans shifts the demandschedule up and to the left from Γr (l0, r) to Γr (l1, r) (recall that Γrl < 0). The supply
schedule of reserves is unchanged. The new equilibrium is at point b. The equilibrium levelof reserves falls from r0 to r1 < r0 . In turn, the fall in the stock of bank reserves shifts
the loan supply schedule up and to the left from Γl (l, r0) to Γl (l, r1). The new equilibriumin the loan market is at point b. (For expositional convenience we describe these effects as
occurring sequentially but in fact they occur simultaneously.)
Comparing the initial equilibrium, points a in both panels, with the equilibrium after theshock, points b, suggests that the sudden stop causes a fall in both the volume of loans andthe stock of bank reserves. The effect on the lending spread is the result of two opposing
forces. The contraction in the demand for loans pushes the lending spread down, whereas
the contraction in bank reserve holdings pushes it up. This second effect does not dominate26

because it is second order (triggered by the fall in loans itself), but it does attenuate the fallin the loan spread. This intuition is consistent with the relatively small decline displayed by
the lending spread in the unregulated economy over the sudden stop episode in the calibrated
model (broken line in the top right panel of Figure 4).
The intuitive explanation for why the lending spread does not display a hike during asudden stop is that the decline in the equilibrium volume of loans is a consequence of thecontraction in the demand for loans by private households. By contrast, in models in which
the collateral constraint is placed at the level of the bank as opposed to at the level ofthe household, a sudden stop represents a contraction in the supply of loans and hence isassociated with an increase in the lending spread (as, for example, in the model of Céspedes
and Chang, 2020).
The situation is quite different when the government intervenes. The adjustment to anegative endowment shock is illustrated in Figure 7. Initially, the markets for loans and bankFigure 7: The Loan and Reserve Markets of the Regulated Economy During a Sudden Stopir −id1+id

il −id1+idail0 −id0
1+id0
il1 −id1
1+id1

r0 r1

0

b
l1

r

ir1 −id1
1+id1

Γl (l, r0)Γl (l, r1)

ir0 −id0
1+id0

r̄

b
Γr (l1 , r)

a

D
l0

Γr (l0 , r)lreserves are in equilibrium at point a. The equilibrium levels of loans and bank reserves are
l0 and r0 and the lending and reserve spreads are (il0 − id0 )/(1 + id0 ) and (ir0 − id0 )/(1 + id0).
As in the unregulated economy, the sudden stop causes a kink in the demand schedule forloans at l1 < l0 (left panel), and a shift up and to the left in the demand schedule for bankreserves from Γr (l0, r) to Γr (l1, r) (right panel). Now, unlike in the unregulated economy, toavoid a collapse in the bank-reserve market, the government increases the banks’ incentive
to hold reserves by raising the reserve spread from (ir0 − id0 )/(1 + id0 ) to (ir1 − id1)/(1 + id1) >(ir0 − id0)/(1 + id0 ). Thus, the horizontal supply of reserves shifts up in a parallel fashion (rightpanel of Figure 7). If the increase in the reserve spread is large enough, the new equilibrium
level of reserves can be larger than before the sudden stop. This is the case illustrated in
the right panel of Figure 7, where at the new equilibrium, given by point b, the level of bank27

reserves is r1 > r0 . The central bank has an incentive to act aggressively because, to avoid acontraction in the level of external debt, the fall in the volume of loans must be compensatedby an increase in the holdings of bank reserves. In the loan market (left panel of Figure 7),
the increase in the stock of reserves shifts the loan supply schedule down and to the rightfrom Γl (l, r0) to Γl (l, r1). The new equilibrium is at point b, where the lending spread has
fallen from (il0 − id0 )/(1 + id0 ) to (il1 − id1 )/(1 + id1 ).
In sum, the intuition derived from Figure 7 is that if the government intervention raises
the reserve spread sufficiently, then the sudden stop is associated with a fall in the volume
of loans, an increase in the stock of bank reserves, and a fall in the lending spread. These
qualitative effects are consistent with the predictions of the calibrated model under optimalreserve remuneration and capital control policies shown in Figure 4.

4

Extensions and Sensitivity

This section presents three extensions: First, an analysis of how the lending spread behavesduring sudden stops depending on whether disruptions in the loan market stem from thedemand for loans as in the baseline model or from the supply of loans. To this end it starts
by studying a variation of the model in which the collateral constraint is placed on the sideof the bank. In this case, contrary to what happens when the collateral constraint is placedon the side of the household, the lending spread rises during sudden stops. However, inthe context of the model studied in this paper, the real allocation (including the volume ofloans) is independent of where the collateral constraint is placed. The second extension isa variation of the model with reserve requirements instead of interest on bank reserves. It
establishes that the latter welfare dominates the former as a policy tool. The third extension
establishes that absent a collateral constraint (i.e., in a version of the model featuring thebanking channel as the sole financial friction) reserve remuneration suffices to induce theconstrained optimal allocation and capital controls are redundant.

4.1

The Lending Spread During Financial Crises: Demand or Supply Driven?

The baseline model assumes a collateral constraint at the household level. Under this formulation a binding collateral constraint represents a contraction in the demand for loans.
An implication of this specification is that during a sudden stop the lending spread falls (topright panel of Figure 4). If instead the model were to feature a lending limit at the banklevel, a financial crisis would trigger an increase in the lending spread.
28

To see this, consider a variation of the model in which the collateral constraint continues
to be given by equation (11) but assume that it is a constraint of the bank and not of
the household. As before the problem of the bank consists in maximizing profits, given inequation (4). However, now the bank must satisfy the loan limit constraint (11). Formally,banks pick lt and rt so as to maximize
ilt − idt
irt − idtl+rt − Γ(lt, rt )t
1 + idt
1 + idt
subject to
lt ≤ κ ytT + pt ytN ,taking as given ilt , idt , irt , ytT , pt , and ytN . Let µBt ≥ 0 denote the Lagrange multiplier on the
bank’s loan limit. Then the first-order conditions associated with the optimal choice of lt
and µBt are
ilt − idt
≤ Γl (lt, rt ) + µBt ,1 + idt

lt ≥ 0, lit − idt
B− Γl (lt, rt ) − µt lt = 0,
1 + idtµB
t ≥ 0,

and

(26)
(27)


lt ≤ κ ytT + pt ytN ,



T
N
µB
− lt = 0.
t κ yt + pt yt(28)

Comparing the present setting with the one in which the collateral constraint is placed at
the household level, we have that equation (26) replaces equilibrium condition (5) and equations (27) and (28) replace equilibrium conditions (15) and (16). The first-order condition
with respect to rt continues to be equation (6), and the balance sheet constraint of the bankcontinues to be (7).
Now the household does not face the collateral constraint (11). Thus, its problem consists
in choosing ct , cTt , cNt , and lt to maximize the utility function (8) subject to the aggregator (9), the sequential budget constraint (10), and some borrowing limit that prevents itfrom engaging in Ponzi schemes, taking as given pt , ilt, πt , τt , ytT , and ytN . The first-orderconditions of this problem are (12), (13), and
λt = β(1 + ilt)Et λt+1 .

(29)

The difference between this optimization problem and that in which the household does face29

a collateral constraint is that condition (29) replaces condition (14).
The assumption that the borrowing limit is placed at the level of the bank rather thanat the level of the household affects neither the interest rate parity condition (17) nor the
economy wide resource constraint (20). A competitive equilibrium in the economy with thecollateral constraint at the bank level can then be defined as follows:
Definition 3 (Competitive Equilibrium in the Economy with the Collateral Constraint at
the Bank Level). A competitive equilibrium in the economy with a collateral constraint at thelevel of the bank is a set of processes lt, rt , dt , ilt , idt , cTt , pt , λt , and µB
t satisfying (6), (7),(11), (12), (13), (17), (20), and (26)-(29), for t ≥ 0, given a reserve remuneration policy irt ,a capital control tax rate τtc , exogenous processes i∗t , ytT , and ytN , and the initial condition(1 + i∗−1 )d−1 > 0.
Appendix D proves the following proposition:
Proposition 2 (Equivalence of Equilibrium with Collateral Constraints at the Bank orHousehold Level). Suppose the set of processes lt, rt , dt , ilt, idt , cTt , pt , and λt is a competitive
equilibrium in the economy with a collateral constraint at the household level (i.e., satisfies
Definition 1). Then this set of processes also represents a competitive equilibrium of theeconomy with the collateral constraint at the bank level (i.e., satisfies Definition 3) except for
the loan rate, ilt , in states in which the collateral constraint binds. In these states the loan
rate is strictly larger in the economy with the collateral constraint at the bank level than in
the economy with the collateral constraint at the household level.
According to this proposition all variables except for the lending rate behave identicallyin the economies with the collateral constraint at the level of the household and at the levelof the bank. The key difference between the two formulations is that when the collateral
constraint binds the lending spread increases in the economy with the collateral constraint
at the bank level but falls in the economy with the collateral constraint at the household
level.
This difference can be quantitatively significant. Figure 8 displays the behavior of thelending rate around the typical sudden stop under laissez-faire in the economies with thecollateral constraint at the bank and household levels. The figure is produced using theparameter values shown in Table 1. When the collateral constraint binds (period 0) the loan
rate skyrockets to 73 percent per year in the economy with a collateral constraint at thebank level, but remains flat (in fact falls by 0.45 percentage points) in the economy with thecollateral constraint at the household level.
The main message conveyed by Figure 8 is that the observed behavior of the lending rate
can provide information on whether disruptions in financial markets stem from the financial
30

Figure 8: Behavior of the Lending Rate Around a Sudden Stop in the Unregulated Economy90
CC at household level
CC at bank level80

percent per year

70
60
50
40
30
20
10
0
-5

-4

-3

-2

-1

0

1

2

3

4

5

Notes. CC stands for collateral constraint. The figure shows that in a sudden stop, the lending rateincreases sharply when the collateral constraint enters at the bank level, but falls slightly when thecollateral constraint enters at the household level.

or the nonfinancial sector. Being able to make this distinction is not inconsequential forpolicymakers. Historically, the presumption that a key symptom of a financial crisis is a
sharp increase in the lending rate has led to the misdiagnosis of major financial crises. A
case in point is the Great Depression of 1929 to 1933. This episode featured a lack of a
spike in the lending rate. Rockoff (2021) shows that the call money rate—the interest rate
charged by banks to stock brokers on collateralized loans—actually fell between June 1930
and June 1931. According to Rockoff, the lack of a hike in this interest rate was a key
reason why Oliver M. W. Sprague, the major authority on financial crises at the time andan economic advisor of the Bank of England and the Roosevelt administration, failed torecognize a financial crisis in the economic developments that unfolded during this period.
Rockoff further speculates that “had he [Sprague] diagnosed a banking panic and called foran aggressive response by the Federal Reserve, it might have made a difference; but he did
not.”
There is also some evidence that during the great recession of 2008 demand for loans fellin emerging countries. For example, De la Torre, Perı́a, and Schmukler (2010) examine theeffects of the great recession on bank lending to small and medium enterprises in Argentina,
Chile, and Colombia using the World Bank’s on-site survey of bank business and risk managers as well as other survey sources. They find that no bank reported having a shortage offunds to lend. Further, they document that a sizable share of banks (70 percent in Argentina,31

33 percent in Chile, and 63 percent in Colombia) reported experiencing a significant decline
in the demand for loan products.

4.2

Non-Equivalence of Reserve Remuneration and Reserve Requirements

A macroprudential instrument that is sometimes used in emerging countries is reserve requirements. Here we ask whether the constrained optimal allocation with reserve remuneration and capital control taxes can also be supported by an appropriate combination of reserve
requirements and capital controls. We also ask whether reserve remuneration welfare dominates reserve requirements as a macroprudential tool. The analysis that follows establishes
that in the present theoretical framework the policy maker can achieve a better outcome byusing a combination of interest on reserves and capital controls than by using a combinationof reserve requirements and capital controls.
Suppose the central bank does not pay interest on bank reserves, irt = 0, but imposes a
reserve requirementrt ≥ δtdt ,where δt ∈ [0, 1) is a policy instrument. In addition, the government continues to have accessto capital control taxes. Combining the reserve requirement with the bank’s balance sheet
constraint (7) yields
rt ≥ δt[lt + rt + Γ(lt , rt)].

(30)

Then the problem of a bank consists in choosing lt ≥ 0 and rt ≥ 0 so as to maximize profits,ilt − idt
irt − idtl+rt − Γ(lt , rt),t
1 + idt
1 + idt
subject to the reserve requirement (30), taking as given ilt, irt , idt , and δt . Letting ηt denotethe Lagrange multiplier on (30), the first-order conditions of the bank’s problem arert ≥ δt[lt + rt + Γ(lt , rt)],ilt − idt
≤ Γl (lt, rt )+ηt δt[1+Γl (lt, rt )],1 + idtandηt ≥ 0,ηt {rt − δt[lt + rt + Γ(lt , rt)]} = 0

lt ≥ 0, lit − idt
lt− Γl (lt, rt ) − ηt δt[1 + Γl (lt, rt )] = 0
1 + idt(32)−idt− Γr (lt , rt) − ηt [1 − δt − δt Γr (lt , rt)] = 0,
1 + idt

32

(31)

(33)

where the last first-order condition uses the fact that reserves are unremunerated (irt = 0).
Definition 4 (Competitive Equilibrium with Reserve Requirements and Capital Controls).
A competitive equilibrium with reserve requirements and capital controls is a set of processes
cTt , dt , pt , ilt, idt , λt , µt , lt, rt , and ηt satisfying (7), (11)-(17), (20), and (31)-(33) for t ≥ 0,
given a reserve requirement policy δt , a capital control tax rate τtc , exogenous processes i∗t ,ytT , and ytN , and the initial condition (1 + i∗−1 )d−1 > 0.
All constraints in Definition 2 are equilibrium conditions of the economy with reserve
requirements (Definition 4). It follows immediately that the best competitive equilibriumwith interest on reserves and capital controls yields at least as much welfare as the best competitive equilibrium attainable with reserve requirements and capital controls. Furthermore,
Appendix E proves the following claim:
Claim 1. [Non-Equivalence of Interest on Reserves and Reserve Requirements] In general,the constrained optimal allocation of Definition 2 does not satisfy the competitive equilibrium
conditions of the economy with reserve requirements and capital controls listed in Definition 4.
Claim 1 shows that the constrained optimal allocation attainable with interest on reservesand capital controls defined in Definition 2 cannot be supported as a competitive equilibriumwith reserve requirements, Definition 4. We thus have that the best competitive equilibriumwith reserve remuneration and capital controls strictly welfare dominates the best competitive equilibrium with reserve requirements and capital controls. We summarize this resultin the following proposition.
Proposition 3 (Welfare Dominance of Reserve Remuneration Over Reserve Requirements).
The constrained optimal allocation attainable with a combination of interest on reservesand capital controls (irt , τtc ) strictly dominates in welfare the constrained optimal allocation
attainable with a combination of reserve requirements and capital controls (δt , τtc ).

4.3

Redundancy of Capital Controls in the Absence of a Collateral
Constraint

Here, we consider an economy in which household borrowing from banks is not subject toa collateral constraint. We show that in this case the optimal level of capital control taxes
is zero (τtc = 0). The intuition behind this result is that the lack of a collateral constraint
eliminates the pecuniary externality that the presence of a relative price in the collateral
constraint creates. As a result, the social planner does not need a policy instrument dedicated to making households internalize this externality. The remaining policy instrument,
33

namely, reserve remuneration, is still necessary because it serves the purpose of making banks
internalize the net social benefits (and costs) of holding reserves.
A competitive equilibrium in the economy without the collateral constraint (11) is defined
as follows:
Definition 5 (Competitive Equilibrium in the Economy without a Collateral Constraint).
A competitive equilibrium in the economy without a collateral constraint on loans is a set ofprocesses lt , rt, dt , ilt , idt , cTt , cNt , pt , and λt , satisfying (5)-(7), (12), (13), (17), (19), (20),and
λt = β(1 + ilt)Et λt+1 ,(34)

for t ≥ 0, given a reserve remuneration policy irt , a capital control tax rate τtc , exogenous
processes i∗t , ytT , and ytN , and the initial condition (1 + i∗−1 )d−1 > 0.
The constrained optimal allocation of a social planner with access to interest on bankreserves (irt ) and capital control taxes (τtc ) now takes the form:
Definition 6 (Constrained Optimal Allocation in the Economy without a Collateral Constraint). The constrained optimal allocation is a set of processes cTt , dt , lt ≥ 0, and rt ≥ 0
that solves the problem
max E0∞X

β tu(A(cTt , ytN ))t=0

subject to the bank’s balance sheet (7) and the economy-wide resource constraint (20), taking
as given the processes i∗t , ytT , and ytN , and the initial condition (1 + i∗−1 )d−1 > 0.
The Lagrangian associated with the social planner’s optimization problem is
L = E0∞X
β t u(A(cTt , ytN )) + λrt ytT + dt − cTt − (1 + i∗t−1 )dt−1 − Γ(lt , rt) − Γr (rt)t=0r r
+λt µt [lt + rt + Γ(lt , rt) − dt ]} .

It can be shown that in the constrained optimal allocation loans are positive, lt > 0 (seeAppendix F). Then, the first-order conditions with respect to cTt , dt , and lt areu0 (A(cTt , ytN ))A1(cTt , ytN ) = λrt ,

(35)

λrt+1
r
∗
1 − µt = (1 + it )βEt r ,
λt

(36)

and
µrt =

Γl (lt, rt ).
1 + Γl (lt , rt)
34

(37)

Suppose {cTt , dt , lt, rt } is the constrained optimal allocation. To see that this allocation
satisfies the competitive equilibrium conditions listed in Definition 5, proceed as follows. SetrlcNt to satisfy (19), set λt = λt , so it satisfies (12), and set pt to satisfy (13). Set it so that(34) holds. This yields (1 + ilt ) = λrt /β/Et λrt+1 . Combing this expression with the social
planner’s first-order condition (36), we have
1 + ilt =1 + i∗t.
1 − µrt

Next set idt so that it satisfies (5) holding with equality. Combining the resulting expression
with the social planner’s first-order condition (37) yields
1 + ilt =1 + idt.
1 − µrt

The above two expressions imply that
idt = i∗t .
Set irt to satisfy (6). Finally, choose τtc to satisfy (17). Because idt = i∗t , the capital control
taxes that support the constrained optimal equilibrium must be zero at all times (τtc = 0).
We summarize this result in the following proposition:
Proposition 4 (Redundancy of Capital Controls in the Absence of a Collateral Constraint).
The constrained optimal allocation in the absence of a collateral constraint given in Definition 6 can be supported as a competitive equilibrium by an appropriate reserve-remuneration
policy irt and the capital control policy τtc = 0.

4.4

Sensitivity

This section presents an analysis of the sensitivity of three central results of the paper tovariations in the parameters defining the banking friction. Table 4 shows how predicted
moments change as the parameters A, B, α, φ, and r̄ take values 25 percent higher or lower
than in the baseline calibration. The first column of the table compares the mean debt-tooutput ratio in the unregulated and constrained optimal economies. For all parameter values
considered, the constrained optimal economy holds on average a higher level of debt per unitof output than the unregulated economy, suggesting that the underborrowing result stressedin the paper is a robust prediction of the model. We were able to detect overborrowing for
extreme values (not shown) of the parameter B governing the central bank’s cost of reserve
provision. Specifically, the debt-to-output ratio becomes higher in the unregulated economy
35

Table 4: Sensitivity to Changes in Parameters of the Banking Channel

Parameter
Name
Value
Baseline–
High A
0.03
Low A
0.02
High B
2.19
Low B
1.31
High α
1.92
Low α
1.15
High φ
29.62
Low φ
17.77
High r̄
0.14
Low r̄
0.08(dt /yt )o(dt /yt )u

1.10
1.08
1.12
1.09
1.13
1.16
1.04
1.09
1.12
1.08
1.13

corr(rt , lt)
Unregulated Regulated
0.99
-0.48
0.97
-0.26
0.98
-0.65
0.99
-0.45
0.99
-0.52
0.99
-0.50
0.98
-0.46
0.98
-0.47
0.98
-0.49
0.98
-0.43
0.99
-0.53

corr(rt , ytT + pt ytN )Unregulated Regulated
0.54
-0.79
0.45
-0.76
0.57
-0.82
0.54
-0.77
0.54
-0.81
0.50
-0.81
0.57
-0.75
0.55
-0.78
0.48
-0.79
0.52
-0.78
0.51
-0.80

Notes. All moments are computed unconditionally. The objects (dt/yt)u and (dt/yt )o denote themean debt-to-output ratio in the unregulated and constrained optimal equilibria. The first line in
the body of the table, labeled Baseline, corresponds to the baseline parameterization of the model
given in Table 1.

than in the regulated economy for values of B above 7 times the baseline value. This range
is empirically implausible because it induces values of the cost of reserve provision per unitof bank reserve, Γr (r)/r, of over 14.5 percent, whereas the observed value—targeted in thecalibration—is only 2.2 percent.
Columns 3 and 4 of the table compare the correlation between bank reserves and bankloans in the unregulated and constrained optimal equilibria. In the former, this correlationis consistently positive, whereas in the latter it is consistently negative. This result points to
the robustness of the prediction of the model that through the optimal reserve remuneration
policy the central bank acts as a lender of last resort by boosting bank reserves at times
when bank loans are low. Columns 5 and 6 compare the correlation between bank reserves
and output in the unregulated and constrained optimal equilibria. For all parameterizations considered the same pattern as observed in the baseline calibration emerges: in theunregulated economy reserves are procyclical (they increase with output) whereas in the constrained optimal equilibrium reserves are countercyclical. This suggests that the result thatthe government finds it optimal to raise reserves in contractions to ensure that deleveraging
at the household level does not spill over to the aggregate level is also a robust prediction of
the model.

365Conclusion

This paper contributes to a literature on open economy models with a pecuniary externality
due to an occasionally binding collateral constraint faced by private agents. A central resultin this literature is that the economy overborrows from the rest of the world. The innovationof the present study is to replace the assumption that households borrow directly fromforeign lenders with the more realistic assumption that foreign lending is intermediated bybanks operating in domestic financial markets. The paper characterizes constrained optimalequilibria attainable by a policymaker with access to capital controls and interest on bankreserves.
An important finding of the paper is that in the model with a collateral constraint anda bank lending channel, the unregulated equilibrium displays underborrowing. External
debt, private borrowing, and bank reserves are all lower in the laissez-faire equilibrium
than in the constrained optimal equilibrium. In the constrained optimal allocation the
country can borrow more than under laissez-faire because in the former the government
acts as a lender of last resort to borrowing-constrained households. During contractions inwhich households are collateral constrained, the government raises the interest rate on bankreserves to induce banks to deposit more reserves at the central bank. The government
then channels these resources to households through a more relaxed fiscal policy, thereby
alleviating their liquidity needs. In this way, deleveraging at the household level does not spillover into economy wide deleveraging. By contrast, in the laissez-faire economy household
deleveraging causes economy wide deleveraging. This is so because when the demand forloans collapses, so does the demand for bank reserves. In turn, bank reserves fall because
they are a complementary input in the production of bank loans. Without government
intervention, the banking channel amplifies macroeconomic deleveraging because the demand
for foreign deposits declines not only because households demand fewer loans but also because
banks demand fewer reserves.
An important question in macroprudential banking policy is whether reserve requirements and interest on bank reserves are equivalent policy instruments. In the context of theopen economy model studied in this paper, this is not the case. As it turns out, reserve
remuneration strictly dominates reserve requirements in a welfare sense. The reason is that
reserve remuneration controls the price of reserves but lets its quantity be determined endogenously. By contrast, reserve requirements without interest on reserves amounts to fixing
both the quantity and the price of reserves.
The present analysis focuses on variations in household income as the shifter of the household demand for loans. This choice was made to keep the results of the paper comparable37

with key papers in the related literature. It is an interesting avenue for future research toinvestigate the consequences of introducing additional sources of uncertainty affecting thedemand for or the supply of loans. Another relevant shock affecting the demand for loans isvariations in the upper bound of the leverage ratio of households (κ in the notation of thepaper). This shock is of interest because it can capture contractions stemming from failuresin the domestic financial sector. A shock that affects the supply of loans to emerging countries is disturbances to the world interest rate or to the country interest rate premium (i∗tin the notation of the paper). Introducing this shock is of interest in light of the extensive
literature suggesting that variations in the cost of external credit represent an importantdriver of aggregate fluctuations in emerging countries.
In the paper, the government is assumed to raise liquidity from the domestic banking
system. The implicit assumption is that the government maintains a more fluid relationshipwith domestic banks than with foreign lenders. So in responding to short-run funding needsof liquidity constrained households, the government finds it easier to go through the domesticbanking system. It is an interesting question for future research to relax this assumption
and explore an environment in which the government uses a mix of domestic and foreignsources of funds, by, for example, introducing differentiated transactions costs. Finally,
because the present study considers an endowment economy, financial frictions do not affect
employment, investment, or aggregate activity. It would be of interest to extend the analysis
to a production economy to shed light on these important dimensions.

38

Appendix
A

Optimal Policy When Bank Reserve Provision Is
Costless

This appendix shows that when Γr (r) = 0, the optimal interest-on-reserve policy achieves thefirst-best allocation. It further shows that in this case capital controls are redundant, that is,
the first best allocation can be supported with the policy τtc = 0 for all t. Define the first-best
allocation as the one that solves the problem of a social planner who is constrained only by
the sequential resource constraint and the prohibition to play Ponzi schemes. That is, thesocial planner is neither subject to the collateral constraint nor to the bank intermediation
friction. The following definition provides a formal statement:
Definition A1 (First-Best Allocation). The first-best allocation is a pair of processes of
tradable consumption and foreign deposits, c̃Tt and d˜t , that solves the problem
max E0∞X

β tu(A(cTt , ytN ))t=0

subject to the sequential resource constraint
cTt + (1 + i∗t−1 )dt−1 = ytT + dt

(A1)

and to a no-Ponzi-game constraint, taking as given the processes i∗t , ytT , and ytN , and theinitial condition (1 + i∗−1)d−1 > 0.
The first-order condition of this maximization problem with respect to holdings of foreigndeposits, dt , is the Euler equation
λ̃t = β(1 + i∗t )Etλ̃t+1 ,withλ̃t = u0(A(c̃Tt , ytN ))A1(c̃Tt , ytN ).
To show that the first-best allocation can be supported as a competitive equilibrium by
an appropriate interest-on-reserve policy and no capital controls, one must show that there
exists a process for irt such that all conditions in Definition 1 are satisfied with τtc = 0. aresatisfied when evaluated at the processes for consumption and foreign deposits associatedwith the first-best allocation. To this end, start by setting cT = c̃T and dt = d˜t . Set lt = 0.
t

39

t

Then the sequential resource constraint in the competitive equilibrium (20) becomes identicalto the resource constraint of the social planner (A1), since Γ(0, rt ) = 0 by Assumption 1.
Next, set µt = 0. It follows immediately that equilibrium conditions (15) and (16) aresatisfied. To satisfy (12), set λt = λ̃t . Next, set ilt = i∗t , which guarantees that equilibrium
condition (14) is satisfied. Pick pt residually to satisfy (13). Because lt = 0 for all t, thecollateral constraint (11) holds. Set rt = d̃t . This guarantees the satisfaction of equilibrium
condition (7). Set idt = i∗t . This ensures that equilibrium condition (17) is satisfied because
τtc = 0. Equilibrium condition (5) holds because lt = 0, ilt = idt = i∗t , and Γl (·, ·) ≥ 0. To
ensure that (6) is satisfied, set the policy rate irt equal to idt (= i∗t ) and invoke Assumption 1.
This completes the proof that when the provision of bank reserves is costless (Γr (rt ) = 0)
the first-best allocation can be supported as a competitive equilibrium by an appropriate
interest-on-reserve policy (irt = i∗t ) and no capital controls (τtc = 0). The following propositionsummarizes this result.
Proposition A1 (Optimal Policy When Reserve Provision is Costless). Suppose that Γr (·) =0 and that Assumption 1 holds. In an economy with equilibrium conditions given by Definition 1, the first best allocation, given in Definition A1, can be supported as a competitive
equilibrium by the interest-on-reserve policy irt = i∗t and no capital control taxes, τtc = 0.
The optimal bank reserve remuneration policy irt = i∗t eliminates households’ need toborrow from banks, lt = 0. All of the financial intermediation occurs between banks and thegovernment. Banks take deposits from international lenders and deposit them entirely at thecentral bank in the form of reserves. In effect, the government borrows from banks at theworld interest rate and transfers resources to the private sector via income taxes or subsidies,
as needed. From the point of view of the household, these taxes or subsidies are exogenous.
In equilibrium households endogenously become hand-to-mouth agents. They have accessto bank loans at the world interest rate and have collateral to back them, but neverthelesschoose not to use this credit facility. Cúrdia and Woodford (2011) show, in the context ofa closed economy, that, as in the present environment, the optimal reserve remuneration
policy allows agents to completely circumvent the bank intermediation friction. Here, theoptimal policy also allows agents to completely circumvent the financial friction arising fromthe collateral constraint, which makes it feasible to attain the first-best allocation.

B

Computation

The computation employs global methods over a discretized state space. Specifically, it uses
13 equally spaced points for each of the exogenous driving forces, ln y T and ln y N . The
40

transition probability matrix of the vector (ln y T , ln y N ) is computed using the simulation
approach described in Schmitt-Grohé and Uribe (2014). The endogenous state, external
debt d, is discretized using 800 equally spaced points. The lower panel of Table 1 provides
more details.
The unregulated or laissez-faire equilibrium (irt = τtc = 0) is approximated using an
Euler equation iteration procedure over the discretized state space. The constrained optimalcompetitive equilibrium, that is, the equilibrium in which the policymaker optimally sets
irt and τtc , is computed using a value function iteration approach over the discretized state
space.
Unlike in a version of the present model without a banking channel (e.g., Bianchi, 2011;
Schmitt-Grohé and Uribe, 2021), the computation of the constrained optimal equilibriumturns out to be more involved than that of the unregulated economy. The reason is that in
the unregulated economy, given a value for dt , the volume of loans and reserves, lt and rt,
are determined by solving equations (6) and (7), evaluated at irt = 0 and idt = i∗t , which is arelatively simple numerical problem. With rt and lt in hand, consumption of tradables, cTt , isfound residually by solving the sequential resource constraint (20). This consumption level
and the associated candidate value for dt are feasible if the equilibrium collateral constraint(21) is satisfied, which is also a simple condition to check. By contrast, the social planner is
not constrained by equation (6), since she can pick irt . As a result, given dt , the values of lt,
rt , and cTt are jointly determined as the solution to the problem of maximizing cTt subject to(7), (20), and (21). Thus, the social planner solves a static optimization problem for eachcandidate choice of dt and for each state (ytT , ytN , dt−1 ). In turn, this static optimization
problem is nested in the dynamic optimization problem of choosing the debt policy function,
dt .

C

Data Sources

In this appendix, we provide an overview of the data sources used to analyze targeted
moments in Table 2. These moments include the lending spread, the reserve-to-deposit
ratio, and the central-bank-operating-cost-to-reserve ratio. The classification of rich andemerging countries adheres to the criteria set forth in Uribe and Schmitt-Grohé (2017).
1. [Lending Spread](a) Date source: IMF International Financial Statistics December 2017 and August2021 archive.

41

(b) Definition: The lending spread is defined as the deposit rate minus the lending
rate. The deposit rate is Financial, Interest Rates, Deposit, Percent per annum(series identifier: FIDR PA). The lending rate is Financial, Interest Rates, Lending Rate, Percent per annum (series identifier: FILR PA).
(c) Data frequency and coverage: annual data from 1985 to 2016.
(d) Country list (38 emerging countries and 16 rich countries)
i. Emerging countries: Albania, Algeria, Argentina, Bahrain, Bolivia, Botswana,
Brazil, Bulgaria, Chile, Colombia, Costa Rica, Cyprus, Dominican Republic,
Ecuador, Egypt, El Salvador, Gabon, Greece, Guatemala, Hungary, Iran,
Jordan, South Korea, Malaysia, Mexico, Morocco, Namibia, New Zealand,
Panama, Paraguay, Peru, Portugal, Spain, Syrian Arab Republic, Thailand,
Trinidad and Tobago, Uruguay, and Venezuela.
ii. Rich countries: Austria, Belgium, Canada, Finland, France, Germany, Hong
Kong, Ireland, Italy, Japan, Netherlands, Singapore, Sweden, Switzerland,
United Kingdom, and United States.
2. [Reserve-to-deposit ratio](a) Date source: Bankscope.
(b) Definition: The reserve-to-deposit ratio is defined as bank reserves in USD divided by bank deposit in USD. Bank reserves is Due From Central Banks (series
identifier: data38390) and bank deposit is Total Deposits, Money Market andShort-term Funding (series identifier: data11580).
(c) Data frequency and coverage: annual data from 1985 to 2016.
(d) Country list (34 emerging countries and 17 rich countries)
i. Emerging countries: Albania, Argentina, Bahrain, Bolivia, Botswana, Brazil,
Bulgaria, Chile, Colombia, Costa Rica, Cyprus, Dominican Republic, Ecuador,
Egypt, El Salvador, Greece, Guatemala, Hungary, Jordan, South Korea, Mexico, Namibia, New Zealand, Panama, Paraguay, Peru, Portugal, South Africa,
Spain, Syrian Arab Republic, Trinidad and Tobago, Turkey, Uruguay, andVenezuela.
ii. Rich countries: Australia, Austria, Belgium, Canada, Denmark, Finland,
France, Germany, Hong Kong, Ireland, Italy, Netherlands, Norway, Singapore, Sweden, Switzerland, and United Kingdom.
3. [Central-bank-operating-cost-to-reserve ratio]
42

(a) Date source: Bankscope(b) Definition: The central-bank-operating-cost-to-reserve ratio is defined as the central bank’s operating cost in USD divided by reserves held in central banks in
USD. The central bank’s operating cost is calculated as the sum of Total NonInterest Expenses (series identifier: data10170), Loan Impairment Charge (series
identifier: data10200), and Equity-accounted Profit/ Loss - Operating (series identifier: data10180). Reserves held in central banks is Total Deposits, Money Market
and Short-term Funding (series identifier: data11580).
(c) Data frequency and coverage: annual data from 1985 to 2016.
(d) Country list (36 emerging countries and 18 rich countries)
i. Emerging countries: Albania, Argentina, Bahrain, Bolivia, Botswana, Brazil,
Bulgaria, Chile, Colombia, Costa Rica, Cyprus, Dominican Republic, Ecuador,
El Salvador, Greece, Guatemala, Hungary, Iran, Israel, Jordan, South Korea,
Malaysia, Morocco, Namibia, New Zealand, Paraguay, Peru, Portugal, South
Africa, Spain, Thailand, Trinidad and Tobago, Tunisia, Turkey, Uruguay, andVenezuela.
ii. Rich countries: Australia, Austria, Belgium, Canada, Denmark, Finland,
France, Germany, Hong Kong, Ireland, Italy, Japan, Netherlands, Norway,
Singapore, Sweden, Switzerland, and United Kingdom.

D

Proof of Proposition 2: Equivalence of Equilibriumwith Collateral Constraints at the Bank or Household Level

Consider processes for lt, rt , dt , ilt, idt , cTt , pt , λt , and µt that constitute a competitive
equilibrium of the economy with the collateral constraint at the household level, that is,processes that satisfy Definition 1. We wish to show that this allocation, with the exceptionof ilt, also satisfies Definition 3. Since (6), (7), (11), (12), (13), (17), and (20), belong toboth Definition 1 and Definition 3, and do not feature ilt, what needs to be shown is thatequilibrium conditions (26)-(29) hold.
Let il,B
denote the equilibrium lending rate in the economy with the collateral constraintt
at the bank level. Suppose in a given date and state the collateral constraint is slack,l,Blt < κ(ytT + pt ytN ), so that µt = 0. Set µB= ilt. Then (26)-(29)t = µt = 0 and it
are satisfied. Next, consider the case that the collateral constraint binds in the baseline
43

allocation, lt = κ(ytT + pt ytN ). Note that in this case lt > 0. SetµBt =µt (1 + ilt ),1 − µt (1 + idt )Bwhich implies that µB
t is strictly positive. The facts that µt is positive and that the collateral

constraint holds with equality ensure that (27)-(28) hold. It remains to show that (26) and(29) are satisfied. Pick il,B
so that it satisfies the left-hand side expression of (26) withtequality. This then implies that the middle and the right-hand side expression of (26) also
hold and yields
1 + il,B= (1 + idt )(1 + Γl (lt, rt )) + µttλt.
βEtλt+1

Now use the left expression of (5) holding with equality to replace (1 + idt )(1 + Γl (lt, rt ))with 1 + ilt , and replace (1 + ilt ) in turn with (14), which yields (1 + ilt ) = (1 − µt ) βEtλλtt+1 .
Substituting these expressions in the above displayed equation, we obtain(1 + il,B
t ) =

λt
.
βEt λt+1

It follows that (29) is satisfied, which is what we set out to show. Finally, to see that il,B
≥ iltt
use the facts that βEtλλtt+1 = (1 + ilt)/(1 − µt ) and that 0 ≤ µt < 1

E

Proof of Claim 1: Non-Equivalence of Interest on
Reserves and Reserve Requirements

Consider the processes cTt , rt , lt , and dt that solve the optimization problem in Definition 2.
By construction, these processes satisfy the bank’s balance sheet (7) and the economy’ssequential resource constraint (20). Set λt and pt to satisfy equilibrium conditions (12)and (13). The collateral constraint (11) is then satisfied by construction. Consider a date
in which lt > 0 and the collateral constraint is slack.5 Then, µt = 0, which implies thatequilibrium conditions (15) and (16) are satisfied. The interest rate on loans, ilt, is thendetermined residually by the Euler equation (14). It remains to check whether (17) and(31)-(33) also hold. Because lt > 0, the left expression of (32) holds with equality. If ηt = 0,then this expression and (33) form a system of two equations in one unknown, idt , which is in5

One can show that lt > 0 at all times under the relatively weak assumption Γl (0, rt) = 0, which is satisfied
by the functional forms used in the quantitative analysis (see Section 3.1). The quantitative analysis further
shows that under the assumed calibration in the constrained optimal allocation the collateral constraint does
not bind at all times.

44

general inconsistent. If, on the other hand, ηt is different from zero, then the left expressionin (31) holds with equality. This expression, the left expression in (32) holding with equality,and (33) represent a system of three equations in three unknowns, δt , idt , and ηt . There areno guarantees, however, that the solution to this system will yield a non-negative value forthe Lagrange multiplier on the reserve requirement, ηt .6

F

Optimality of Positive Loans in an Economy Without a Collateral Constraint

This appendix presents the proof of the following claim used in the proof of Proposition 4:
Claim F1. In the absence of a collateral constraint, the constrained optimal allocation given
in Definition 6 features positive loans, lt > 0.
To establish this claim, note from Definition 6 that in the absence of the collateral
constraint, given dt , the social planner chooses the variables lt and rt to minimize the sumof the private resource cost of bank intermediation and the public resource cost of reserve
management, subject to the balance sheet constraint. Formally, given dt , the planner solves
min C(lt, rt ) ≡ Γ(lt , rt ) + Γr (rt ){lt ,rt }subject to (7). Given our focus on equilibria in which dt > 0 for all t, it is clear from (7) thateither lt or rt must be positive. Differentiating the objective function and the constraint, weget
0dC(lt, rt) = Γl (lt, rt)dlt + Γr (lt, rt )drt + Γr (rt )drt
anddlt + drt + Γl (lt , rt)dlt + Γr (lt , rt)drt = 0.
Suppose, contrary to the claim, that lt = 0, so that rt > 0. In this case, by Assumption 1,
Γl (lt , rt) = Γr (lt, rt ) = 0. Then, using the differentiated constraint to get rid of drt in the
differentiated cost, we get0dC(0, rt) = −Γr (rt )dlt ,which is negative for dlt > 0. This shows that lt = 0 cannot be optimal, as the planner could
reduce the resource cost by increasing loans.
6
For example, the real allocation in the calibrated economy of Section 3.2 implies values of ηt ranging
from -0.03 to 0.05.

45

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