Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

Prudential Capital Controls? The Impact of Different
Collateral Constraint Assumptions
Mitsuru Katagiri, Ryo Kato, and Takayuki Tsuruga

Discussion Paper No. E-12-014

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

March, 2013 (revised March, 2015)

Prudential Capital Controls or Bailouts? The Impact
of Di¤erent Collateral Constraint Assumptions
Mitsuru Katagiriy, Ryo Katoz, and Takayuki Tsurugax
March 2015

Abstract
A fast growing literature on small open economy models with pecuniary externalities has provided the theoretical grounds for the policy analysis of macro prudential
regulations. Using the framework of Jeanne and Korinek (2010), we investigate whether
a subsidy on debt during crises as a form of bailout can outperform prudential capital
controls. We show that the result depends on the functional form of the collateral constraint faced by households. If households collateralize their assets that they purchaseat the same time as their borrowing, subsidizing debt during crises is preferable. If, on
the other hand, the maximum borrowing is constrained by the value of their assets thatthey have purchased before they borrow, a stronger case can be made for prudential
capital controls.
JEL Classi…cation: E32, F38, G01, G18
Keywords: Financial crises, Credit externalities, Bailouts, Macroprudential policies
We would like to Gianluca Benigno, Timothy Kam, Takashi Kamihigashi, Shigeto Kitano, Keiichiro
Kobayashi, Teruyoshi Kobayashi, Césaire Meh, Eric R. Young and the sta¤ of Bank of Japan for helpfulcomments. We also thank seminar and conference participants at Hitotsubashi University, Kobe University,
National Graduate Institute for Policy Studies, Victoria University of Wellington, the 14th Annual Macro
Conference, the 10th Biennial Paci…c Rim Conference, the 18th Australasian Macroeconomic Workshop, theSociety for Economic Dynamics Meeting, the 19th International Conference of Computing in Economics andFinance, the Asian Meeting of the Econometric Society, and the 2013 Annual Meeting of Japan Society ofMonetary Economics for helpful discussions. Takayuki Tsuruga gratefully acknowledges the …nancial support
of a Grant-in-Aid for Scienti…c Research (24730269). Views expressed in this paper are those of the authorsand do not necessarily re‡ect the o¢ cial views of the Bank of Japan.
yBank of Japan; e-mail: mitsuru.katagiri@boj.or.jpzBank of Japan; e-mail: ryou.katou@boj.or.jpx
Graduate School of Economics, Kyoto University and the Centre for Applied Macroeconomic Analysis;e-mail: tsuruga@econ.kyoto-u.ac.jp1

1

Introduction

Should policymakers in emerging market economies rely on macro prudential policies during
normal times or bail out borrowers at the time of a …nancial crisis? Recent studies havehighlighted the importance of prudential controls on cross-border capital ‡ows and macro
prudential regulations to prevent ine¢ cient boom-bust cycles. Among others, Jeanne andKorinek (2010, hereafter JK), Bianchi (2011), and Bianchi and Mendoza (2012) emphasize that market-determined asset prices can generate pecuniary externalities that distort…nancing decisions of economic agents. A key ingredient in their models is an occasionally binding collateral constraint that depends on the market value of the collateral. Basedon the framework with this constraint, these early studies advocate that the government
should preemptively impose a Pigouvian tax on debt during normal times and internalize
the externalities.1
This paper extends the JK model to consider policy prescriptions for coping with crises.
In particular, we consider two assumptions on the collateral constraint and investigate how
the di¤erence in the assumptions a¤ects policy prescriptions for coping with crises. The
two collateral constraints di¤er in the timing in which households’assets are collateralized.
To emphasize the timing, we call the collateral constraints either a “beginning-of-period”
or an “end-of-period” collateral constraint. The “beginning-of-period” collateral constraint
is assumed in JK. Under this constraint, the households collateralize the assets that they
have purchased before they borrow. On the other hand, under the “end-of-period” collateral constraints, households collateralize the assets that they purchase at the same time as
borrowing. This end-of-period collateral constraint has also been employed by other studies(e.g., Kiyotaki and Moore, 1997 and Bianchi and Mendoza, 2012).
We …nd that the choice between the two assumptions is not innocuous. We show that,
under the beginning-of-period collateral constraint, subsidizing debt during crises cannot
achieve better allocation than that under the laissez-faire economy. In other words, thebailout is neutral. By contrast, subsidizing debt during crises can achieve the …rst-best
allocation if households are subject to an “end-of-period”collateral constraint.
Our results suggest that policymakers should have knowledge of the structure of the collateral constraint that the households are faced with. As JK and others suggested, prudential
1

More recent papers include Benigno, Chen, Otrok, Rebucci, and Young (2012a, 2012b, 2013), Bianchi(2013), Bianchi and Mendoza (2013), Dávilla (2011), Jeanne and Korinek (2012), Korinek and Simsek (2013).
Farhi and Werning (2012, 2013) and Schmitt-Grohé and Uribe (2012, 2013) argue for the prudential capital
controls under the nominal price or wage rigidities.

2

capital controls can achieve the second-best allocation. If the beginning-of-period collateral
constraint is a more plausible assumption, a policy prescription is that policymakers should
rely on the prudential capital controls via the Pigouvian tax on debt. This is because abailout through subsidizing debt is neutral in terms of welfare. Conversely, under the endof-period collateral constraint, the bailout can achieve the …rst-best allocation and thus is
more desirable than the prudential capital controls.
In the literature, early studies have noted that the government can restore the …rst-best
allocation in some models with an occasionally binding collateral constraint. For example,
using a model similar to Bianchi (2011), Benigno, Chen, Otrok, Rebucci, and Young (2012a,
hereafter BCORY) show that, if the government can use additional distortionary policy
instruments on top of capital controls (e.g., a price support policy in the form of a subsidy
on collateral or collateralized nontradable goods), then the collateral constraint can always
be removed and the government can achieve the …rst-best allocation.2 Our …nding starkly
contrasts with BCORY. In this paper, “bailout” refers to the subsidy on foreign debt. To
keep the comparison between prudential capital controls and bailouts fair, we assume thatthe government is given a single policy instrument that a¤ects the cost of borrowing fromabroad (i.e., intervention in the credit market). In our model, the government can achieve
the …rst-best allocation without relying on additional distortionary policy measures such as
outright purchase of the collateral and subsidy on nontradable good consumption.
If bailouts are theoretically desirable, the next question for policymakers would be whether
the bailouts are practically feasible. To answer this question, the present paper also performs
numerical experiments and assesses the size and frequency of policy interventions in credit
markets. Our assessment suggests that strong expectations for bailouts akin to moral hazard
lead to an unrealistically large bailout size and a high frequency of intervention. In fact, on
average, a large lump-sum tax equivalent to as much as 31.6 percent of annual householdincome must be imposed to achieve the …rst-best allocation. On the frequency, the government needs to intervene almost every year. These results point to a large gap compared withthe actual observation because very few – perhaps no – governments in emerging market
economies have embarked on such large bailouts with such high frequency. Our experiments
2

See also Jeanne and Korinek (2012) and Benigno, Chen, Otrok, Rebucci, and Young (2012b) for the casewhere multiple policy instruments are available for crisis management. Schmitt-Grohé and Uribe (2013)
show that the open economy model with downward nominal wage rigidities can give rise to pecuniary
externalities. They argue that bailouts by devaluation of the country’s currency could restore the …rst-best
allocation. Likewise, using a model of banking, Green (2010) argues that bailouts can lead to a sociallye¢ cient outcome.

3

suggest that further research be called for to explore what can …ll the gap.
This paper is organized as follows. Section 2 introduces the JK model with the twocollateral constraints used for our comparisons. Section 3 extends the JK model to consider
the optimal debt subsidy as a bailout. Section 4 performs numerical experiments. Section 5
concludes.

2

Two collateral constraints

We begin with a small open economy model developed by JK. Suppose that the utility ofidentical atomistic households is given by u (c0 ) + u (c1 ) + c2 , where u (c) = c1 = (1),> 0, and ct represents consumption in period t. Domestic households’budget constraints
for each period are given by
c0 + p0 1 = d1 + p0

(1)

c1 + d1 + p1 2 = e + d2 + p1 1(2)

c2 + d2 =

2 y;

(3)

where dt is the debt to be repaid at the beginning of the period t and t represents thedomestic collateral held by the households at the beginning of period t. Here, pt is the priceof collateral traded in a competitive market. Throughout all periods, the world interest
rate is set to zero for simplicity. At the beginning of period 0, the households hold one
unit of collateral. In this period, they borrow d1 from abroad and purchase 1 as well as
the consumption goods.3 In period 1, the households have three sources of in‡ow: sales ofcollateral p1 1 , new borrowing d2 , and an endowment e that is not pledgeable to foreign
lenders. They use them for consumption c1 , repaying d1 , and purchasing collateral 2 . In the…nal period, the households must repay d2 after receiving returns on collateral y. Following
JK, we assume that the return on collateral can be acquired only by domestic agents and thevalue of collateral in period 2 is lost after the households receive y. We also assume that the
supply of collateral assets is inelastic and normalized to one. In this JK model, the linear
utility in period 2 implies that, if there is no collateral constraint, consumption in periods 0
and 1 is unity (i.e., the …rst-best level of consumption).
The model introduces a collateral constraint for d2 . As JK discuss, a low value of e
may result in the binding collateral constraint and precipitate a crisis (e.g., sudden stop in
3

Without loss of generality, the initial value of foreign debt is set to zero.

4

capital in‡ows). With the binding collateral constraint in period 1, the desired borrowing is
generally impossible and the households must accept a large reduction in c1 . As such, period
1 corresponds to the period of a crisis.

2.1

The beginning-of-period collateral constraint

Each household faces a collateral constraint of the form
d2

1 p1 :(4)Note that, following JK, the borrowing capacity is constrained by the market value of collateral at the beginning of period 1. The parameter 2 (0; 1] represents the ceiling on theleverage in the collateral constraint.4 In a symmetric equilibrium, 1 must be unity because
the supply of collateral asset is one. Not surprisingly, pecuniary externalities arise from thefeedback loop between the collateral price and borrowing. When a su¢ ciently low e takes
place, the collateral constraint binds. The households try to prevent consumption reduction by decreasing net demand for their collateral. The households’deleveraging results indeclines in collateral prices, and the decline in p1 further tightens their collateral constraints.
While each atomistic household takes p1 as given, the households’decision as a whole has thegeneral equilibrium e¤ect on p1 . As a result, the general equilibrium e¤ect cannot be internalized by price-taking households and the laissez-faire equilibrium is not generally Paretoe¢ cient. This result calls for the “macro prudential policies”that have been widely discussed
in the literature.5
JK show that, in their stochastic model where e is random, a Pigouvian tax can replicate
the second-best allocation solved by the constrained social planner.6 Replace the period-1
budget constraint (2) by
c1 + (1 + ) d1

T + p1 2 = e + d2 + p1 1 ;(5)where the government runs a balanced budget: the lump-sum transfers T equal tax on debt
4

We consider a slightly more general constraint because JK assume = 1.
Examples of this research include Bianchi (2011), Jeanne and Korinek (2011), Korinek (2011), Bianchi
and Mendoza (2012) and Benigno et al. (2012a, 2012b, 2013).
6
See JK for details on the constrained social planner’s problem in maximizing the households’ utility
subject to the resource constraints and the same collateral constraint as that of households.
55d1 . In JK, the following macro prudential tax is proposed:=

E0 [ sp p0 (m1 )];E0 [u0 (c1 )](6)

where E0 denotes the expectations operator conditional on the information at t = 0.7 In(6), sp is the Lagrange multiplier for the collateral constraint that the social planner faces,u0 (c1 ) denotes the marginal utility of consumption, and p0 (m1 ) > 0 is the derivative of p1
with respect to the level of the liquid net worth m1 e d1 . The shadow price of holding debt
sp and the asset pricing function p (m1 ) are obtained from the constrained social planner’s
problem in which she internalizes the general equilibrium e¤ect of m1 on p1 .8 This prudential
tax on debt reduces borrowing in the pre-crisis period and can mitigate reductions in asset
prices in the crisis period.

2.2

The end-of-period collateral constraint

As a variant of the collateral constraint (4), consider
d2

2 p1 ;(7)

where the value of collateral is evaluated by the end-of-period holding of collateral 2 , rather
than 1 . As discussed in JK, their key results on prudential taxes remain the same across thetwo collateral constraints:9 (i) the same collateral constraint in equilibrium (i.e., d2
p1 );(ii) the same feedback loop between the collateral price and borrowing; and (iii) the same
form of the Pigouvian tax. These results remain essentially unaltered even under the in…nitehorizon setting.10
7More speci…cally, the information set does not include the realization of e.
The asset pricing function is given by p (m1 ) = y=u0 (c1 ) = y=u0 (d2 + m1 ) and di¤ers from the asset
pricing function under the laissez-faire economy. In other words, JK follow the “constrained e¢ ciency”
de…nition of Kehoe and Levine (1993) in their social planner’s problem.
9
See footnote 4 in JK.
10Jeanne and Korinek (2011) numerically recon…rm the robustness to the assumptions on collateral constraints in Appendix A.2.
8

6

3

The optimal debt subsidy as a bailout

We consider how changes in the assumption regarding the collateral constraint a¤ect bailouts.
For simplicity, we assume in this section that e is deterministic rather than stochastic. This
simpli…cation allows us to obtain an explicit solution for the optimal debt subsidy, but has
no e¤ect on our argument.
To introduce bailouts, we replace (2) by
c1 + d1 + p1 2 = e + (1 + s) d2

S + p1 1 ;(8)where s 0 is a subsidy on debt and S is the lump-sum tax, satisfying S = sd2 .11 A balanced
government budget ensures that household resources are kept unchanged both intra- andinter-temporally. Thus, the only distinction between the prudential capital controls andbailouts is the question of whether to raise the cost of debt before a crisis or to reduce it
during one.
In the following two subsections, we will present propositions on the optimal subsidy
under the two di¤ering collateral constraints, (4) and (7). Then, we will interpret the two
propositions in the context of policy implications.

3.1

The beginning-of-period collateral constraint

The households maximize their utility u (c0 ) + u (c1 ) + c2 , subject to the budget constraints(1), (8), (3) and the beginning-of-period collateral constraint (4). The …rst-order conditionsareu0 (c0 ) = u0 (c1 )(1 + s) u0 (c1 ) = 1 +p0 = p1
p1 =

y

(9)

m
0

u (c1 ) + m
u0 (c0 )u0 (c1 )

:

(10)
(11)
(12)

Here m represents the Lagrange multiplier for (4). In (10), the households choose d2 by comparing the marginal cost 1+ m on the right-hand side with the marginal bene…t (1 + s) u0 (c1 )11

Following Jeanne and Korinek (2012), our policy analysis rules out the possibility that the government
uses the non-distortionary inter-temporal lump-sum taxes and transfers to fully relax the collateral constraint.

7

on the left-hand side. Other things being equal, a higher subsidy on debt encourages a household to hold more debt during a crisis. The asset pricing equations in each period are givenby (11) and (12). In (11), m p1 represents the extra bene…t of holding more collateral
under the beginning-of-period collateral constraint. This extra bene…t increases only p0 but
has no e¤ect on p1 .
The …rst proposition establishes that, under the beginning-of-period collateral constraint,
subsidizing debt during a crisis does not improve the welfare, compared to the laissez-faireeconomy.
Proposition 1 Suppose that the household maximizes the utility of u (c0 )+u (c1 )+c2 subject
to (1), (8), (3) and the beginning-of-period collateral constraint (4). Then, the optimal
subsidy on debt s is zero if (2 e) =y
1. If 0 < < (2 e) =y, on the other hand, theallocation is fully independent of s and is equivalent to the allocation under the laissez-faireeconomy.
Proof. It is straightforward to obtain the unconstrained …rst-best allocation: cF B;0 =cF B;1 = dF B;1 = 1, cF B;2 = y 2 + e, and dF B;2 = 2 e. Likewise, it can be easily
shown that the price of collateral under the …rst-best allocation is pF B;0 = pF B;1 = y. For(2 e) =y1, the collateral constraint does not bind under the laissez-faire economy(dF B;2 < pF B;1 ). Hence, the optimal subsidy s is trivially zero. We next consider the caseof 0 < < (2 e) =y, which means that the collateral constraint binds under the laissezfaire economy. In this case, together with (12), (4) implies (i) d2 = y=u0 (c1 ). Because of
the market clearing conditions for collateral (i.e., 1 = 2 = 1) and the government budget
constraint, (1), (8), and (3) can be simpli…ed to (ii) c0 = d1 , (iii) c1 + d1 = e + d2 , and (iv)c2 + d2 = y. Furthermore, (9) implies that (v) c0 = c1 . The allocation in this decentralized
economy, fc0 ; c1 ; c2 ; d1 ; d2 g, can be fully determined by (i) - (v) if the unique equilibrium
exists. Because (i) - (v) do not include s, the resulting allocation is fully independent of s.
Therefore, the allocation must be equivalent to that under the laissez-faire economy.

3.2

The end-of-period collateral constraint

We next consider the same bailout under the end-of-period collateral constraint (7). While
the …rst-order conditions (9) and (10) remain the same as before, (11) and (12) must be8

replaced by(13)

p0 = p1
p1

y= 0
u (c1 )

;

(14)

m

respectively. In contrast to (11) and (12), the extra bene…t of holding the collateral ( m p1 )a¤ects both p0 and p1 under the end-of-period collateral constraint.
The next proposition states that the optimal s can replicate the unconstrained …rst-best
allocation. Proposition 2 summarizes our second main result.
Proposition 2 Suppose that the household maximizes the utility of u (c0 )+u (c1 )+c2 subject
to (1), (8), (3), and the end-of-period collateral constraint (7). Then, there exists timeconsistent s with which the decentralized equilibrium achieves the unconstrained …rst-best
allocation, and s is equal to the equilibrium shadow price of holding debt in the decentralized
economy:s = m (s ) 0:
(15)
If 0 << (2e) =y, s is given by
s =

y

1
2

e

> 0:

(16)

Otherwise, s = 0:
Proof. As in the proof of Proposition 1, the optimal subsidy for (2 e) =y1
is trivial: s = m = 0 and the collateral constraint does not bind under s . For 0 < <(2 e) =y, dF B;2 > pF B;1 . If a positive s exists, it must be the case that m (s ) > 0 anddF B;2 = p1 . Here, from (10), we note that m depends on s. Substituting the unconstrained…rst-best allocation into (10) and (14) yields s = m (s) and p1 = y= (1 s ), respectively.
Therefore, s in (16) is obtained by eliminating p1 from p1 = y= (1 s ) and dF B;2 =p1 . Because dF B;2 is feasible under p1 satisfying p1 = y= (1 s ), s achieves the …rstbest allocation fcF B;0 ; cF B;1 ; cF B;2 ; dF B;1 ; dF B;2 g. The solution s is a function of exogenous
variables and hence time-consistent.12
12

In the appendix that is available upon request, we show that Proposition 2 can be extended to thecontinuous time version of the model.

9

3.3

Interpretation

Proposition 1 indicates that, if the collateral constraint is given by (4) as in JK, subsidizing
debt during crises does not improve the welfare. This result implies that the prudential capital controls that can achieve the second-best allocation are strictly preferred to the bailout.
By contrast, however, Proposition 2 shows that, if the collateral constraint is replaced by (7),the optimal subsidy s can prevent …re sales of collateral and achieve the …rst-best allocation.
In terms of policy prescription, the bailout outperforms the prudential capital controls under
the assumption of the end-of-period collateral constraint.
The key to understanding our result is the price of collateral p1 at a time of crisis. Supposethat the collateral constraint binds in period 1 (i.e., 0 < < (2 e) =y). In this case, thegovernment may wish to intervene in the credit market with s > 0 if it can in‡ate the priceof collateral. Moreover, the households can enjoy even the …rst-best level of consumption if
the asset price is in‡ated to ensure that dF B;2 = p1 . Therefore, dF B;2 uniquely determines
the target level of the asset price for the government:
p1 =dF B;2

=

2

e

> y;

(17)

where the strict inequality is ensured by the assumption of < (2 e) =y.
The question is whether the government can in‡ate asset prices. Under (4), the subsidy
on new borrowing has no e¤ect on households’decisions because their borrowing capacity
is predetermined by 1 p1 . Households’demand for collateral ( 2 ) is not stimulated by thesubsidy. As a result, p1 remains una¤ected by the subsidy and the only option for thehouseholds is to demand less collateral ( 2 ) to compensate for consumption. After all, theallocation turns out to be the same as that under the laissez-faire economy with a low priceof collateral. By contrast, under (7), households know that if they buy more collateral ( 2 ),then they can borrow more, because 2 a¤ects their borrowing capacity. The lower costof new debt owing to the subsidy provides them with …nancing for the purchase of new
collateral ( 2 ). This …nancing for the new collateral purchase stimulates the demand, which
results in higher p1 .
From the viewpoint of the government, p1 under (14) can be seen as a function of thepolicy instrument s. In particular, along with (10), the asset pricing equation (14) can be
expressed asyp1 =:[1 (1 + s) ] u0 (c1 ) +10This equation indicates that the government can control the price of collateral by choosings. If the government sets s = s = m (s ),
p1 =

y
1 s

:

(18)

This s is consistent with the target price given by (17), while satisfying all the …rst-orderconditions and the constraints. Hence, under the end-of-period collateral constraint, thegovernment can achieve the …rst-best allocations.
In the literature, BCORY propose a di¤erent policy prescription that can restore the…rst-best allocation. In this regard, Proposition 2 in our paper is related to their …ndings.
The optimal subsidy in our model, however, starkly di¤ers from the BCORY’s prescription
in three respects. First, the policy instrument in this paper di¤ers from BCORY’s. The
government in BCORY subsidizes the household collateral purchase, while that in our model
subsidizes household borrowing. Intervening in collateral markets may require additional
capacity (i.e., additional policy instruments) of the government.13 Our argument is applicable
to any government that has the capacity to implement capital controls (i.e., the capacity toa¤ect the cost of borrowing). The second di¤erence is slightly technical. Under the BCORY’sprescription, the collateral constraint never binds. In the context of our model, this meansthat dF B;2 < p1 and m = 0. To ensure the e¤ectiveness of the optimal subsidy in ourmodel, m needs to take strictly positive values, because, as indicated by (14), the non-zero
Lagrange multiplier enables asset price in‡ation. Third, the BCORY’s prescription can also
be interpreted as crisis prevention rather than bailouts, because crises never take place in
BCORY. By contrast, the optimal subsidy in our model requires actual …scal expenditure
and, in fact, needs to be activated as a tangible intervention.

4

Quantitative evaluation of the optimal subsidy

Proposition 2 in the previous section indicates that theoretically there exists an “optimal
bailout” in the form of subsidy on debt during a crisis. However, a natural question forpolicymakers would be whether the optimal subsidy is practically feasible. To answer thisquestion, we extend the three-period model to a stochastic in…nite-horizon model that can
13

See Propositions 3 and 4 in BCORY. They employ the collateral constraint in which the income fromtradable and non tradable endowments can be pledged as collateral as in Bianchi (2011). Using this setupof the model, the government commits to supporting the relative price of non tradables to tradables duringcrises by either a subsidy on nontradable good consumption or a tax on tradable good consumption.

11

be calibrated to the data. Using the extended model, we examine (i) whether the policy
intervention for implementing the “optimal bailout”is realistic in terms of the size; and (ii)how frequently the policy intervention needs to be made to restore the …rst-best allocation.
As a preparatory step, we …rst show that the result of Proposition 2 continues to hold evenin the case of a stochastic in…nite-horizon model and that the optimal subsidy on debt
becomes state-contingent. We then explore how s ‡uctuates over time to assess the sizeand frequency of bailouts in a calibrated model.

4.1

The in…nite-horizon model

We consider the stochastic in…nite-horizon model, similar to Jeanne and Korinek (2011) andBianchi and Mendoza (2012). The households choose dt+1 and t+1 to maximize
E0

"1
X

t

t=0

where is the discount factor satisfying
period budget constraint:
ct + dt + pt t+1 =

#

(19)

u(ct ) ;2 (0; 1). Each household faces the period-by-t et + (1 + st )dt+1
RSt + p t ;(20)

and the end-of-period occasionally binding collateral constraint:
dt+1
R

pt t+1 :(21)

In this maximization problem, dt+1 is non-state-contingent one-period debt. The real interest
rate on the non-state-contingent debt is R > 1 rather than unity. Every period, eachhousehold receives the exogenous endowment of collateral , which is normalized to one. He
receives a stochastic income et based on the predetermined share of collateral assets t (i.e.,
dividends). As before, the value of collateral is lost after receiving the return on collateral.
The budget constraint here is basically the same as (8), but is also similar to (3) in terms of
the returns on collateral. Finally, the collateral constraint is the same as (7) except for R.

12

The …rst-order conditions are standard:(1 + st ) u0 (ct ) =
pt =
0 =

REt u0 (ct+1 ) + m;t
Et [u0 (ct+1 ) et+1 ]u0 (ct )m;tdt+1
pt t+1
m;t ,
R

(22)
(23)
m;t

0; and pt t+1

dt+1
R

0:

(24)

In equilibrium, the markets for collateral and consumption goods clear: t+1 = and ct +dt =et +dt+1 =R for all t, respectively. As before, we assume a balanced budget of the government:
St = st dt+1 =R.
The …rst-best allocation must satisfy the following …rst-order conditions:u0 (cF B;t ) =pF B;t =REt [u0 (cF B;t+1 )]Et [u0 (cF B;t+1 ) et+1 ];u0 (cF B;t )

(25)
(26)

which yield the policy functions cF B;t = cF B (dt ; et ) and dF B;t+1 = dF B (dt ; et ) as functions ofstate variables. Likewise, the asset pricing function is obtained from the model without thecollateral constraint: pF B;t = pF B (dt ; et ). The following proposition shows that there exists
a state-contingent optimal subsidy st = s (dt ; et ) consistent with the …rst-best allocation in
the stochastic in…nite-horizon model.
Proposition 3 Suppose that the household maximizes (19) subject to (20) and the end-ofperiod collateral constraint (21). Then, there exists a price function ps (dt ; et ) and a timeconsistent subsidy s (dt ; et ), with which the decentralized equilibrium characterized by (20)(24) achieves the unconstrained …rst-best allocation fcF B;t ; dF B;t+1 g1
t=0 . Furthermore, thesubsidy st = s (dt ; et ) is proportional to the Lagrange multiplier for (21):s (dt ; et ) =m (dt ; et ; st )0
u [cF B (dt ; et )]

0;

(27)

where m (dt ; et ; st ) represents the Lagrange multiplier for the collateral constraint, given st .
This subsidy st is time-consistent. Furthermore, if dF B (dt ; et ) =R > pF B (dt ; et ), s (dt ; et )is given by1
pF B (dt ; et )
> 0:
(28)
s (dt ; et ) =dF B (dt ; et ) =R
Otherwise, s (dt ; et ) = 0:13

Proof. We consider two cases for the states of the economy (dt ; et ): (i) dF B (dt ; et ) =R
pF B (dt ; et ) and (ii) dF B (dt ; et ) =R > pF B (dt ; et ). Here, the conditions distinguish whether
or not the …rst-best level of debt is feasible in the laissez-faire economy. For each case, wewill con…rm that the …rst-order conditions (22)–(24) are satis…ed under s (dt ; et ) when they
are evaluated at the …rst-best allocation cF B (dt ; et ) and dF B (dt ; et ).
Consider the states satisfying (i). If s (dt ; et ) = 0, m [dt ; et ; s (dt ; et )] = 0. Then, the…rst-order conditions of (22) and (23) are the same as (25) and (26), and the allocation is
the …rst-best. Therefore, ps (dt ; et ) = pF B (dt ; et ) characterized by (26). The optimality ofs (dt ; et ) = 0 is con…rmed for the states satisfying (i).
Next, for the states satisfying (ii), consider the price of collateral that achieves dF B (dt ; et )with the binding collateral constraint. If
ps (dt ; et ) =dF B (dt ; et )
;
R

(29)

then (24) is satis…ed together with m (dt ; et ; s (dt ; et )) 0. Combining (27), (23), and (29)yieldsEt (u0 fcF B [dF B (dt ; et ) ; et+1 ] ; et+1 g et+1 )dF B (dt ; et )
;
(30)
=
R
u0 [cF B (dt ; et )] (1 st )for all states (dt ; et ) satisfying (ii). Using the above equation and (26), we can solve forst = s (dt ; et ) and the solution turns out to be (28). Because st is chosen to satisfy (30),(23) and (24) are obviously satis…ed at the …rst-best allocation. Finally, using (27), (25) canbe rewritten as[1 + s (dt ; et )] u0 [cF B (dt ; et )] = REt (u0 fcF B [dF B (dt ; et ) ; et+1 ]g) +m [dt ; et ; s(dt ; et )] ;which is exactly (22) under st = s (dt ; et ). Therefore, all the …rst-order conditions (22)–(24)are satis…ed under s (dt ; et ). This s is completely determined by the current state of theeconomy and hence is time-consistent.
The proposition con…rms that, as in the case of the three-period model, the optimal
subsidy s (dt ; et ) can achieve the …rst-best allocation. The mechanism behind the result isthe same as that in the three-period model. The government can fully control the asset priceto avoid …re sales by choosing st .

14

4.2

The size and frequency of bailouts

4.2.1

Calibration

The parameters are mainly taken from Bianchi and Mendoza (2012). For the household’s
preference, u (ct ) = c1t = (1), with= 2:0. The discount factor
is set to 0:96,
calibrating the model to the annual frequency. The ceiling on the household borrowing percollateral asset is set at 0.36. We assume that the total factor productivity in Bianchi and
Mendoza (2012) can be translated into the stochastic process of the dividend et in our model.
The stochastic process of the total factor productivity in Bianchi and Mendoza (2012) follows
a log-normal AR(1) process. We thus employ the same stochastic process for et as theirs:log(et+1 ) = log(et ) + t+1 , where t
N (0; " ) for all t. Here and " are set to 0.53
and 0.014, respectively. To ensure stationary dF B (dt ; et ), we assume that households facea small risk premium on their foreign debt.14 Speci…cally, we replace the budget constraint(20) with a slightly di¤erent form:
ct + dt + pt t+1 =t et + (1 + st )

dt+1
Rt+1

St + p t ;where Rt+1 = R + [exp(dt+1 ) 1]; R =1.028 and =0.01.15
To con…rm that parameters are reasonably calibrated, we simulate the laissez-faire economy with the occasionally binding collateral constraint. We interpret GDP in our model as
et . In the simulation, the mean of the debt-to-GDP ratio is 33.5 percent. This ratio is inclose proximity to the calibration target in Benigno et al. (2013), who calibrate their modelto the Mexican economy based on the updated dataset of Lane and Milesi-Ferretti (2007).
We also compute the probability of binding collateral constraints in the laissez-faire economy
as a proxy of the crisis probability. The probability is 6.57 percent, which is broadly in linewith the literature.16
14

See Schmitt-Grohé and Uribe (2003).
It is straightforward to show that Proposition 3 can be extended to the case with a risk premium on
foreign debt.
16
For reasonable values of the crisis probability, Benigno et al. (2013) target the crisis probability at 8
percent per year. In the empirical studies, Basel Committee on Banking Supervision (2010) reports two
empirical crisis probabilities for 11 advanced economies over 1985-2009, based on the datasets of Reinhart
and Rogo¤ (2008) and Laeven and Valencia (2008). The estimated crisis probabilities are 5.2 percent in
Reinhart and Rogo¤ (2008) and 4.1 percent in Laeven and Valencia (2008).
15

15

4.2.2

Simulation results

Table 1 compares the moments generated by the model without intervention (i.e., the laissezfaire economy) with those generated under the optimal subsidy. Overall, the optimal subsidy
on debt takes extremely large values, meaning that massive bailouts are required to restore
the …rst-best allocation. On average, st is 67.0 percent. To …nance this subsidy, the government needs to impose a large lump-sum tax equivalent to as much as 31.6 percent of annual
household income (e ). Figure 1 plots the simulated path of the optimal subsidy over 300
periods. The …gure indicates that the government activates bailouts very frequently. In the…gure, st takes a value of zero only in period 13. This means that, over 300 periods, bailouts
are activated 299 times.
The reason for the massive and frequent bailouts is straightforward. Expectations ofbailouts strongly incentivize households to hold a large amount of debt. As long as the debt
in the …rst-best allocation is su¢ ciently large compared to that in the laissez-faire economy,
the collateral constraint is likely to bind. Whenever the collateral constraint binds, thegovernment bails out households by encouraging new borrowing that is used to repay thelarge amount of existing debt. Given their expectations that they will be bailed out, it is
optimal for households to roll over the large amount of new borrowing. This new borrowing
substantially increases the probability that the collateral constraint will continue to bind in
the next period. As a result, the government almost always needs to bail out households.
We argue that there is a gap between theory and data. For comparison, we take anempirical estimate reported by Frydl (1999). Based on Frydl (1999), an empirically realistic
size of a bailout would amount to somewhere between 1 to 3 percent of GDP for a single year
in the aftermath of a crisis.17 Comparisons of the sizes of intervention between the modeland the practice suggest that the optimal subsidy would be di¢ cult to implement, perhaps
because of political con‡icts which are not taken into account in our model.
17

Frydl (1999) discusses two empirical works estimating (i) …scal costs (resolution cost) and (ii) the lengthof …nancial crises as his baseline estimates (Caprio and Klingebiel, 1996 and Lindgren, Garcia, and Saal,
1996). In the two empirical papers, the average …scal costs are 13.6 percent of GDP in the former and 7.2
percent in the latter, whereas the average duration of crises is 4.5 years and 6.2 years, respectively. We
compute the average …scal cost for a single year by dividing average total …scal costs by the average lengthof …nancial crises, suggesting that the average …scal cost for a single year ranges from 1.16 (7.2/6.2) to 3.02
(13.6/4.5) percent of GDP.

16

4.2.3

Sensitivity of simulation results

Although unrealistically large and frequent bailouts seem di¢ cult to implement, the size andfrequency depend on parameters in the model. Therefore, one could argue that it is possible
to attain a realistic size and frequency of bailouts by changing parameters in the model.
To consider this argument, this subsection discusses whether the di¢ culty of the optimal
subsidy is robust to changes in the model parameters.
Proposition 3 suggests that st can take lower values under looser collateral constraint(i.e., a larger ). Under a larger , the government may activate bailouts less often, because
the collateral constraint is less likely to bind. Hence, in the economy with a large , theoptimal subsidy can be practically feasible. In Figure 2, we plot the optimal subsidy in theeconomy with = 0:60. With this parameterization, the maximum value of st over 300
periods is 2.90 percent and the size of policy intervention measured by the lump-sum tax
relative to GDP is 1.71 percent. On the frequency, the incidence of the policy activationis two out of 300 periods. Thus, bailouts appear to be easier to implement in an economy
with = 0:60. However, the model fails to explain crisis probabilities under the laissezfaire economy: in our simulations, the probability of binding collateral constraints in thelaissez-faire economy is only 0.25 percent, much lower than that reported in the previous
studies. Therefore, it is sensible to conclude that, although an “optimal bailout” may be
implementable in an economy resilient to negative shocks, it would not be feasible in mosteconomies that are fragile to shocks.
The optimal subsidy st can be also a¤ected by the risk-premium parameter . In ourbenchmark simulation, is set to a somewhat large value of 0.01. In the literature on thesmall open economy real business cycle model, this parameter is usually set to a very small
number to calibrate the model without the collateral constraint to match the volatility ofthe debt-to-GDP ratio. For example, Schmitt-Grohé and Uribe (2003) set this parameter to
0.0007. We could have used their parameterization, but the level of the debt in this case ismuch higher than what we obtained in the benchmark simulation. Because (28) indicatesthat a larger amount of foreign debt increases the size and frequency of bailouts, this low
makes the optimal subsidy more unrealistic.

17

5

Concluding remarks

This paper analyzed the roles of bailouts in managing …nancial crises. Using the simple
framework employed by Jeanne and Korinek (2010), we showed that the timing assumptionof the collateral constraint has a non-negligible impact on the debate regarding macro prudential policies and bailouts. If we employ Jeanne and Korinek’s (2010) original functional
form of the collateral constraint, the policy that subsidizes debt during crises does not increase the welfare, and thus this bailout fails to outperform the prudential capital controls.
If households can collateralize their assets that they purchase at the same time as theirborrowing, the same policy can improve the welfare much more than the prudential capital
controls. Our analytical results in the deterministic three-period model can be extended
into a more general stochastic in…nite-horizon framework. Using the extended model, we
examined the practical feasibility of the optimal subsidy on debt. In terms of the practicalfeasibility, however, the optimal subsidy would be massive in size and require very frequent
activation. We conclude that, while an “optimal bailout”is theoretically feasible, in practice
it would be di¢ cult to implement.
We presented our results in the simpli…ed models to obtain the results analytically. As
in the seminal works by Jeanne and Korinek (2010) and Bianchi (2011), we considered an
endowment economy without an endogenous labor supply. To extend the analysis into a
more general framework would be an important step for future research.

References[1] Basel Committee on Banking Supervision, 2010, “An Assessment of theLong-Term Economic Impact of Stronger Capital and Liquidity Requirements,”
http://www.bis.org/publ/bcbs173.htm.
[2] Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E. R. Young, 2012a, “Capital Controlsor Exchange Rate Policy? A Pecuniary Externality Perspective,”Federal Reserve Bankof St. Louis Working Paper No. 2012-025A.
[3] Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E. R. Young, 2012b, “Optimal Policy
for Macro-Financial Stability,” Federal Reserve Bank of St. Louis Working Paper No.
2012-041.

18

[4] Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E. R. Young, 2013, “Financial Crises
and Macro-Prudential Policies,” Journal of International Economics, 89 (2), pp. 453470.
[5] Bianchi, J., 2011, “Overborrowing and Systemic Externalities in the Business Cycle,”
American Economic Review, 101 (7), pp. 3400-3426.
[6] Bianchi, J., 2013, “E¢ cient Bailouts?”unpublished manuscript.
[7] Bianchi, J., and E. G. Mendoza, 2012, “Overborrowing, Financial Crises, and ‘Macroprudential’Policy,”unpublished manuscript.
[8] Bianchi, J. and E. G. Mendoza, 2013, “Optimal Time-consistent Macroprudential Policy,”NBER working paper No. 19704.
[9] Caprio, G., and D. Klingebiel, 1996, “Bank Insolvencies: Cross-Country Experience,”
Policy Research Working Paper No. 1620, World Bank.
[10] Dávilla, E., 2011, “Dissecting Fire Sales Externalities,”unpublished manuscript.
[11] Farhi, E. and I. Werning, 2012, “Dealing with Trilemma: Optimal Capital Controls
with Fixed Exchange Rates,”NBER working paper No. 18199.
[12] Farhi, E. and I. Werning, 2013, “A Theory of Macroprudential Policies in the Presenceof Nominal Rigidities,”unpublished manuscript.
[13] Frydl, E. J., 1999, “The Length and Cost of Banking Crises,” IMF Working Paper,
WP/99/30.
[14] Green, E. J., 2010, “Bailouts,”Economic Quarterly , 96 (1), pp. 1–32.
[15] Jeanne, O., and A. Korinek, 2010, “Excessive Volatility in Capital Flows: A Pigouvian
Taxation Approach,”American Economic Review, 100 (2), pp. 403–407.
[16] Jeanne, O., and A. Korinek, 2011, “Managing Credit Booms and Busts: A Pigouvian
Taxation Approach,”unpublished manuscript.
[17] Jeanne, O., and A. Korinek, 2012, “Macroprudential Regulation versus Mopping Up
after the Crash,”unpublished manuscript.

19

[18] Kehoe, T. J., and D. Levine, 1993, “Debt-Constrained Asset Markets,”Review of Economic Studies, 60 (4), pp. 865-888.
[19] Kiyotaki N. and J. Moore, 1997, “Credit Cycles,” Journal of Political Economy, 105(2), pp. 211-248.
[20] Korinek, A., 2011, “The New Economics of Prudential Capital Controls: A Research
Agenda,”IMF Economic Review, 59 (3), pp. 523-561.
[21] Korinek A. and A. Simsek, 2013, “Liquidity trap and Excessive Leverage,”unpublishedmanuscript.
[22] Laeven, L., and F. Valencia, 2008, “Systemic Banking Crises: A New Database,” IMF
Working Paper No. WP/08/224.
[23] Lane, P. R., and G. M. Milesi-Ferretti, 2007, “The External Wealth of Nations Mark II:
Revised and Extended Estimates of Foreign Assets and Liabilities, 1970-2004,”Journalof International Economics, 73 (2), pp. 223-250.
[24] Lindgren, C. J., G. G. Garcia, and M. I. Saal, 1996, Bank Soundness and Macroeconomic
Policy, International Monetary Fund.
[25] Reinhart, C., and K. Rogo¤, 2008, “Banking Crises: An Equal Opportunity Menace,”
NBER Working Paper No. 14587.
[26] Schmitt-Grohé, S., and M. Uribe, 2003, “Closing Small Open Economy Models,”Journalof International Economics, 61 (1), pp. 163–185.
[27] Schmitt-Grohé, S. and M. Uribe, 2012, “Prudential Policy for Peggers,”NBER working
paper No. 18031.
[28] Schmitt-Grohé, S. and M. Uribe, 2013, “Downward Nominal Wage Rigidity, Currency
Pegs, and Involuntary Unemployment,”unpublished manuscript.

20

A

Appendix

This appendix describes a continuous-time version of the model in Section 3.

A.1

Setup

Suppose that the households live only between 0 and 1, and maximize their life-time utility.
The households’maximization problem ismax

fct ;xt ; t ;dt g1t=0

Z 1u(ct )dt + c+
1;s.t. d_t = pt xt + ct"

_ t = xtd_t
pt ( t + xt )c+
1 + d1 =(31)

0

st dt + d_t + St

(32)
(33)

dt for t 2 [0; 1]

(34)
(35)

1 y;

where u (c) = c1 = (1) and > 0. To be consistent with the setup in the discrete
time model, we introduce the linear (instantaneous) utility at an instant after t = 1. This
consumption is denoted by c+
1 . If there were no collateral constraint, the marginal utility
is one and all resources remaining at t = 1 are consumed at this instant. Because thehouseholds obtain returns on collateral y at t = 1, the remaining resources are 1 y d1 . Thisis represented by (35). Regarding other constraints, (32) is the budget constraint, where " isan endowment that is not pledgeable to foreign lenders. In (33), we introduce xt to representchanges in the demand for collateral. The households are also faced with the end-of-periodcollateral constraint (34).18 In this problem, we assume that the initial conditions are d0 = 0
and 0 = 1.
Before deriving the …rst-order conditions, we rewrite the constraints. First, (32) can be
written as
1(pt xt + ct " st dt + St ):
(36)
d_t =
1 + st18

The beginning-of-period collateral constraint can be written as d_t

21

pt t

dt

Using (36), we eliminate d_t from (34) and rewrite the collateral constraint as
1(pt xt + ct
1 + stpt ( t + xt )" + dt + St )

0

Z 1

h

(37)

The Lagrangian is given by
L =

Z 1
0

+

u(ct )dt + ( 1 y

Z 1

t

0

Z 1

=

+
0
Z 1

d1 )

1
(pt xt + ct
1 + sttpt ( t + xt )"d_t dt +st dt + St )

t xt

0

1
(pt xt + ct
1 + sti_ t dt" + dt + St ) dt

H (ct ; xt ; dt ; t ; t ) + t pt ( t + xt )
0
Z 1
_
_
+ ( 1 y d1 )t dt + t t dt;

1(pt xt + ct
1 + st" + dt + St )

dt
(38)

0

where H (ct ; xt ; dt ;t;

H (ct ; xt ; dt ;t ) is given byt;t ) = u(ct ) +tFurther, applying the integral by parts to

L =

Z 1
0

fH (ct ; xt ; dt ;t;1
(pt xt + ct
1 + stR1__

t dt +t tt ) + dt _ t +

t_t

0

1
(pt xt + ct
1 + st1 d1
1 1 + 0:+ t

pt ( t + xt )+ ( 1y

d1 )

22

"

st dt + St ) +t xt :

dt in (38), we have

" + dt + St )

dt

(39)

The …rst-order conditions for ct , xt , dt , andt areu0 (ct ) =( tt ) pt+1 + st

t

_t =
t and

t

(40)

1 + st=_t =respectively. The terminal conditions forfor d1 and 1 :

t

t

(41)

pt

t + st t(42)

1 + stt pt ;(43)

t are obtained from the …rst-order condition

1

=

1

= y

(44)

1

(45)

The government budget is assumed to be balanced at every t: st dt + d_t = St and the
supply of collateral is constant (i.e., t = 1 for t 2 [0; 1]). Using the assumptions of thebalanced government budget and the constant supply of collateral (i.e., _ t = xt = 0), thelaws of motion for dt and the complementary slackness condition can be simpli…ed. In
particular, the equilibrium law of motion for dt is
d_t = ct(46)";

and the complementary slackness condition is
( pt

dt

ct + ")

t = 0;

t

0, pt

dt

ct + "

where the last inequality can be obtained from (37) as follows:0pt ( t + x t )
=

pt

=

pt

=

pt

1
(pt xt + ct
1 + st" + dt + St )i_dt + dt

1 h
ct " + dt + st
1 + st1[ct " + st (ct ") + (1 + st ) dt ]1 + stdt ct + ":

23

0;

(47)

Here the second equality uses the government budget constraint and the third equality comesfrom (46).
It is straightforward to obtain the unconstrained …rst-best allocation in this model. Suppose that t = 0 and st = 0. Let variables with subscript FB be the variables holding under
the unconstrained economy. Then (42) reduces to _ F B;t = 0. This implies that, together
with (44), F B;t = 1 for t 2 [0; 1] since F B;t is constant. Under the assumption of the
CRRA utility, it immediately follows from (40) and (46) that cF B;t = 1 and d_F B;t = 1 "for t 2 [0; 1]. The di¤erential equation for dF B;t yields dF B;t = (1 ") t. To summarize, the…rst-best allocation isdF B;t = (1") tcF B;t = 1

(48)
(49)

for t 2 [0; 1] and c+(1 "). To obtain the price of collateral in the unconstrained
F B;1 = yeconomy, (41) and (43) become pF B;t = F B;t and _ F B;t = 0, respectively. Because 1 = yfrom (45), it immediately follows
pF B;t = y for t 2 [0; 1] :

A.2

(50)

The bailouts under the end-of-period collateral constraint

In the continuous time model, we have the following proposition similar to Proposition 2 inthe paper.
R1
Proposition A.1. Suppose that the household maximizes the utility of 0 u(ct )dt+c+
1 subject
to (32), (33), (35) and the end-of-period collateral constraint (34). Then, there exists timeconsistent subsidy st with which the decentralized equilibrium achieves the unconstrained…rst-best allocation and st is equal to the equilibrium shadow price of holding debt in thedecentralized economy:st = t 0:
Depending on the tightness of the collateral constraint, the optimal subsidy st is characterized
in the following three cases.

24Case (i): If 0 << (11] =[y exp( 1)], st > 0 for t 2 [0; 1] and given by") [3 exp( 1)st =3 exp(t 1)(t + 1)

1

y exp(t 1)(t + 1)(1 ")(51)Case (ii): If (1 ") [3 exp( 1) 1] =[y exp( 1)]< 2 (1 ") =y, st = 0 for t 2 [0; ]but (51) holds for t 2 ( ; 1], where = 1 log [3 y = (1 ")].
Case (iii): If 2 (1 ") =y, st = 0 for t 2 [0; 1].
To prove Proposition A.1, we need to show that, for all cases, we can construct a set of continuously di¤erentiable price, Lagrange multiplier, and costate variables fpt ; t ; t ; t gt2[0;1] ,where fcF B;t ; dF B;t gt2[0;1] and fpt ; t ; t ; t gt2[0;1] satisfy the …rst-order conditions (40)-(43)and the complementary slackness condition (47) with the optimal subsidy fst gt2[0;1] .
In the proof, we …rst consider Case (iii). In this case, the collateral constraint does notbind and thus there is no need to activate the bailouts (i.e., st = 0 for all t 2 [0; 1]) toachieve the …rst-best allocation. Next, we turn to Cases (i) and (ii). Case (i) represents thatthe collateral constraint always binds for all t 2 [0; 1] while Case (ii) represents that it binds
only for t 2 [ ; 1]. In both cases, we show that all …rst-order conditions are satis…ed under
the …rst-best allocation and the optimal subsidy.
The …rst step is to show that, under the condition for in Case (iii), the …rst-best
2 (1 ") =y is
allocation is achieved under st = t = 0 for t 2 [0; 1]. The condition
equivalent to
0

y

=

2 (1

")

pF B;t(1

pF B;tdF B;t

") t

(1

")

cF B;t + ";for all t 2 [0; 1]. Thus, dF B;t + cF B;t "pF B;t for all t 2 [0; 1] if
2 (1 ") =y. From(47), t = 0 for all t 2 [0; 1], and F B;t = pF B;t = y trivially satisfy the …rst order conditions.
Next, we consider Cases (i) and (ii), where st can be positive for some t. Under st = t ,(40) can be written as
t

t

1+

t

= u0 (cF B;t ) = 1 )t =1 for t 2 [0; 1]:(52)

This implies _ t = 0 from (42) and is consistent with (44). Note that cF B;t = 1 and (48) are25

consistent with c+
F B;1 = y") due to (46). Also, (52) simpli…es (41) to

(1

= pt

t

(53)

pt :

t

For Case (i), we suppose that t > 0 for all t 2 [0; 1] and construct prices under the
supposition. We then con…rm that the supposition of t > 0 is consistent with all …rst-best
allocation and the constructed t and pt . To compute t , we solve the di¤erential equation
for t under st = t > 0:dF B;t + cF B;t "
= 0;
_t
t+
which can be obtained from (43), (53), and the complementary slackness condition implying
pt = (dF B;t + cF B;t ") = . The above equation immediately follows from (48) and (49) that_tt+(t + 1)(1

")

= 0:

(54)

+ C exp(t);(55)

The general solution to this di¤erential equation ist =(t + 2)(1")

where C denotes an arbitrary constant. By the terminal condition (45), (55) ist =(t + 2)(1

")

+ y

3(1

")

exp(t

(56)

1):

Taking derivative of this equation with respect to t yields
_t =

1

"

+ y

3(1

")

exp(t

(57)

1):

Now, we can …nd the path of t and the optimal subsidy shown in the proposition.
Substituting (57) and pt = (dF B;t + cF B;t ") = = (t + 1) (1 ") = into (43) yields
1

"

+ y

3(1

")

exp(t

26

1) =

t (t + 1) (1") :Solving fort , we havet

1
(t + 1)
3 exp(t 1)=(t + 1)

=

y
3
exp(t
(1 ") (t + 1)
(t + 1)
y exp(t 1)
1;(t + 1)(1 ")

1)
(58)

Note that st = t leads to (51) in the proposition.
To complete the proof for Case (i), we must con…rm the supposition t > 0 as well as
t > 0 for all t 2 [0; 1], consistent with the interpretation as the Lagrange multiplier. Here,
if we eliminate y from (58) using the condition < (1 ") [3 exp ( 1) 1] = [y exp ( 1)] in
Case (i), we can con…rm that t > 0 for all t 2 [0; 1]. Moreover, (43) implies that _ t < 0 for
all t 2 [0; 1] because t > 0 and pt = (dF B;t + cF B;t ") = > 0 for all t 2 [0; 1]. Note that
evaluating (56) at t = 0, we have
0 =

2 (1

")

+ y

3 (1

")

exp ( 1) > 0

because 2 3 exp ( 1) > 0. From the terminal condition (15), 1 = y > 0. Therefore, giventhat 0 > 0, 1 > 0, and _ t < 0 for all t 2 [0; 1], t must be positive for t 2 [0; 1], consistent
with the interpretation as the Lagrange multiplier for _ t = xt .
For Case (ii), we need to consider the time periods t 2 [0; ] and t 2 ( ; 1] separately. In
particular, we suppose that t = 0 for t 2 [0; ) and t > 0 for t 2 [ ; 1], and show that theconstructed t and t are consistent with all …rst-order conditions and the supposition.
For t 2 ( ; 1], we construct t by solving the di¤erential equation for t under t > 0
as in Case (i). From (57), • t = [y 3 (1 ") = ] exp (t 1), and • t < 0 because Case (ii)satis…es < 3 (1 ") =y. Since t = 0 at t = where = 1 log [3 y = (1 ")], we have
_ t = 0 at t = . Therefore, • t < 0 implies that _ < 0 for all t 2 ( ; 1], con…rming the
supposition of t > 0. Furthermore, the terminal condition for 1 = y and _ t < 0 for allt 2 ( ; 1] ensure that t must be positive for all t 2 ( ; 1].
For t 2 [0; ), the supposition of st = t = 0 implies that t = pt from (53) and _ t = 0
from (43). Hence, pt and t are constant at pt = t = 0 for t 2 [0; ]. This 0 is obtained
from (56) at t ! . That is,0==( + 1)(1")=

dF B; + cF B;

27"= p > 0;

because t must be continuously di¤erentiable for all t 2 [0; 1] including t = . This positivet is consistent with the interpretation as the Lagrange multiplier. Finally, we con…rm the
supposition of t = 0 for t 2 [0; ). The obtained
impliesdF B;t + cF B;t"<dF B; + cF B;"= p = pt ;for any t 2 [0; ), leading to pt > dF B;t +cF B;t ". This is consistent with the complementaryslackness condition under t = 0 and thus con…rms the supposition of t = 0.

28

Table 1: Average and standard deviations of the variables
Laissez-faire economy Bailouts
Debt

0.335
(0.016)0.962(0.018)0.990(0.015)0.471(0.045)1.273(0.121)
0.985(0.009)Subsidy (percent)-Fiscal cost (percentage of GDP)-

67.0
(22.4)
31.6(13.2)

Asset priceConsumption

Note: The numbers in each entry report the average and the standard deviations of the debt position, asset
price, and consumption in the laissez-faire economy and the economy with bailouts. The numbers withoutparentheses refer to the average of variables, and the numbers in parentheses below the average are thestandard deviations of the corresponding variables. These moments are from the numerically approximatedergodic distributions with 1,000 periods and 1,000 simulations. The second column also reports the averageand standard deviation of the optimal subsidy and the …scal cost of bailouts measured by the lump-sum tax
relative to the GDP.

29

Figure 1: Simulated path of st : Benchmark calibration
Simulated path of subsidy
1.4
subsidy st*
1.2

1

st

*

0.8

0.6

0.4

0.2

0

0

50

100

150
year

200

250

Figure 2: Simulated path of st : Economy with300

= 0:60

Simulated path of subsidy
0.03
subsidy st*
0.025

st*

0.02

0.015

0.01

0.005

0

0

50

100

150
year

30

200

250

300