DP2020-05

F i n a n c i a l M a r k e t I n c o m p le t e n e s s
and International Cooperation
on Capital Controls
Shigeto KITANO
Kenya TAKAKU

Revised October 25, 2021

Financial Market Incompleteness and International Cooperation on
Capital Controls
Shigeto Kitanoa and Kenya Takakuba

RIEB, Kobe University, Japan; b Faculty of International Studies, Hiroshima City

University, Japan
ABSTRACT
We examine how the degree of ﬁnancial market incompleteness aﬀects welfare gains
from international cooperation on capital controls. When ﬁnancial markets are incomplete, international risk sharing is disturbed. However, the optimal global policy
signiﬁcantly reverses the welfare deterioration due to ineﬃcient risk-sharing. We
show that when ﬁnancial markets are more incomplete, the welfare gap between the
optimal global policy and the Nash equilibrium increases, and the welfare gains frominternational cooperation on capital controls then become larger.
KEYWORDS
ﬁnancial markets; incomplete markets; policy cooperation; capital controls;
optimal policy; welfare; Ramsey policy; open-loop Nash game
JEL CLASSIFICATION
D52; E61; F32; F38; F42; G15This work was supported by KAKENHI (20K01744, 20H05633). Earlier versions of the paper were presented at 25th CEF conference at Carleton University, the JSME annual meeting at Gakushuin University,
the seminars at Xiamen University, and Kobe University. We sincerely appreciate the referee’s helpful comments and constructive suggestions that have improved our paper signiﬁcantly. We are also grateful to Hao
Jin, Daisuke Ida, Gilles Dufrénot, Jun Nagayasu, Yushi Yoshida, Kimiko Sugimoto, and meeting participants
for their comments.

1

1.

Introduction

Many studies discuss the gains from monetary policy coordination. Among them, several employ two-country models with complete ﬁnancial markets, which implies thatinternational risk sharing is perfect (e.g., Clarida et al. 2001, 2002; Benigno and Benigno 2003, 2006; Corsetti and Pesenti 2001, 2005; Fujiwara and Teranishi 2017). On
the other hand, some studies employ two-country models with incomplete ﬁnancial
markets, and recent studies extend this literature in several directions. Corsetti et al.
(2010) show that incomplete markets break an open-economy version of divine coincidence and cause policy trade-oﬀs. Benigno (2009) shows that gains from the optimalmonetary policy increase with cross-country asymmetries in initial net international
positions. Rabitsch (2012) studies welfare gains from monetary policy cooperation under three types of international ﬁnancial market structures: complete markets, ﬁnancial
autarky, and incomplete markets.
Although many studies examine gains from monetary policy coordination frommany diﬀerent perspectives, as mentioned above, few authors analyze the gains frompolicy coordination in capital controls.1 Noteworthy exceptions are De Paoli and Lipinska (2013) and Heathcote and Perri (2016). De Paoli and Lipinska (2013) show that
capital controls can be beggar-thy-neighbor policies, and policy coordination in capital controls can yield gains. This is because individual countries have incentives tomanage their terms of trade to stabilize their own output ﬂuctuations, but the uncoordinated use of capital controls disturbs international consumption risk sharing
and deteriorates global welfare. Using a two-country model augmented with capital
accumulation, Heathcote and Perri (2016) show that for certain parameterizations,capital controls can lead to better international risk sharing. Through this improved
risk sharing, symmetric capital controls can be welfare improving for both countries
compared to free international capital mobility, which implies that capital controls canbe Pareto-improving.
In this study, we also examine the welfare gains from policy coordination on capi-

1 If we do not restrict our attention to policy coordination, there is a vast literature on theoretical studies ofcapital controls. See, for example, Kitano and Takaku (2018), Kitano and Takaku (2020), and the references
therein. For empirical studies of capital controls, see, for example, Boero et al. (2019) and the references therein.

2

tal controls. However, we focus on how the degree of ﬁnancial market incompletenessaﬀects welfare gains from international cooperation on capital controls. Incomplete
markets disturb international risk-sharing, while the optimal global policy mitigatesrisk-sharing ineﬃciency. We ﬁnd that when ﬁnancial markets are more incomplete,global welfare deteriorates under the Nash equilibrium; however, the optimal global
policy mitigates the welfare loss due to ﬁnancial market ineﬃciencies. When ﬁnancial
markets are more incomplete, we ﬁnd that the welfare gap between the optimal global
policy and the Nash equilibrium increases, and welfare gains from international cooperation on capital controls then become larger. To the best of our knowledge, noprevious work addresses this relationship between the gains from policy coordinationin capital controls and the degree of ﬁnancial market incompleteness in the related
literature.
The remainder of the paper proceeds as follows. In Section 2, we present a standard two-country model with incomplete asset markets. In Section 3, we characterize
optimal global policy and Nash equilibrium. In Sectoin 4, we perform a comparative
analysis of welfare for the two cases. In Secton 5, we conduct robustness check. We
present our conclusions in Section 6.

2.

Model

We consider a familiar two-country model populated with a continuum of agents ofunit mass, in which the population in segment [0, n] belongs to country H and that insegment (n, 1] belongs to country F . Our model is a standard two-country model withincomplete asset markets, which is similar to those in previous studies (e.g., Benigno
2009; Corsetti et al. 2010; Rabitsch 2012; De Paoli and Lipinska 2013; Kim and Kim2018).2

2 The two-country models in Benigno (2009), Corsetti et al. (2010), and Rabitsch (2012) incorporate nominal
rigidities into their models. The two-country model in Kim and Kim (2018) incorporates capital.

3

2.1.

Model setup

The household in country H maximizes the following expected life utility:E0∞∑

β t {u(Ct ) − v(Lt )},

(1)

t =0

where E0 denotes the mathematical expectations operator conditional on the information available at time 0, and β ∈ (0, 1) is the discount factor. The functions u and v
are increasing in the composite consumption index Ct and the labor supply Lt , respectively. We assume that the household’s utility functions u(Ct ) and v(Lt ) are iso-elastic
functions:u(Ct ) =Ct1−σ − 1
,
1−σ

(2)

and
L1+χ
v(Lt ) = t .
1+χ

(3)

We deﬁne the home composite consumption index Ct as

[ 1
] θ
θ−1
θ−1
1
θ−1
Ct = ν θ (CH,t ) θ + (1 − ν) θ (CF,t ) θ,(4)where 1 − ν = (1 − n)λ. The parameter λ indicates the degree of openness. The
corresponding home consumption-based price index is

] 1
[
1−θ
Pt = ν(PH,t )1−θ + (1 − ν)(PF,t )1−θ.

(5)

Similarly, the foreign composite consumption index Ct∗ is

] θ
[ 1
θ−1
1
θ−1
∗
∗ θ−1
) θ + (1 − ν ∗ ) θ (CF,t,Ct∗ = ν ∗ θ (CH,t
) θ

(6)

where ν ∗ = nλ. The corresponding foreign consumption-based price index is

[
] 1
∗ 1−θ
∗ 1−θ 1−θ.
Pt∗ = ν ∗ (PH,t)+ (1 − ν ∗ )(PF,t)(7)

All goods are traded and the law of one price holds:∗
PH,t = St PH,t
,

4

(8)

and
∗
PF,t = St PF,t,(9)where St denotes the nominal exchange rate. However, purchasing power parity does
not hold due to the home bias speciﬁcation given by (5) and (7). We deﬁne the realexchange rate asqt =St Pt∗,
Pt(10)

which can deviate from one. It follows from Eqs. (5), (7), and (10) that the international
relative prices satisfy
θ−1pH= ν + (1 − ν)t( F )1−θ
pt
,
pH
t

(11)

and

( F )1−θ( H )θ−1ptpt∗∗= ν + (1 − ν )
,
qt
pH
t
H

(12)

F

Pt
Pt
F
where pHt ≡ Pt and pt ≡ Pt . We deﬁne the terms of trade as

pFt
.
pH
t

(13)

Yt = At1+χ Lt ,(14)

T oT t =The production technology is given by
χ

and
χ

Yt∗ = A∗t 1+χ L∗t .

(15)

The budget constraint for a household in country H is∗
Ct + (1 + τt−1 ) Rt−1

qt
qt−1

Dt−1 +

δ 2
D t = D t + pHt Y t + Tt ,2(16)

where Dt denotes the external debt position, Rt∗ denotes the gross interest rate on
external debt, τt denotes the tax rate of the external debt, and Tt denotes a lump-sumtransfer from the government. Following many previous two-country studies (Benigno5

2009; De Paoli and Lipinska 2013; Kim and Kim 2018), we introduce quadratic adjustment costs, 2δ Dt 2 , to characterize incomplete ﬁnancial markets.
The tax revenue from capital controls in Eq. (16) are transferred to the household∗
Tt = τt−1 Rt−1

qt
qt−1

Dt−1 ,(17)such that capital controls have no eﬀect on the economy’s resource constraint.
Maximizing the lifetime utility (1) with respect to Dt (and Ct ) subject to the budget
constraint (16) yields the Euler equation in country H:
1 − δ Dt = β (1 + τt ) Rt∗

qt+1 uC (Ct+1 ).
qt uC (Ct )(18)

Maximizing the lifetime utility (1) with respect to Yt (and Ct ) subject to the budget
constraint (16) (and considering (2), (3), and (14)), we also obtain−σ
pH= At −χ Yt χ .
t Ct

(19)

The household in country F maximizes the following expected life utility:∞∑E0

β t {u(Ct∗ ) − v(L∗t )}.

(20)

t =0

The budget constraint for a household in country F is
(
) ∗
∗
∗
Ct∗ + Bt∗ = 1 − τt−1
Rt−1 Bt−1+ p∗Ft Yt∗ + Tt∗ ,(21)

where Bt∗ denotes the external asset position in country F , τt∗ denotes the tax rateof the external asset, and Tt denotes a lump-sum transfer from the government. The
transfer policy in country F , which implies that the government returns the tax oninterest income to households, is∗∗∗Tt∗ = τt−1
Rt−1
Bt−1
.

(22)

Maximizing the lifetime utility (20) with respect to Bt∗ (and Ct∗ ) subject to the budget
constraint (21) yields the Euler equation in country F :1 = β ∗ (1 − τt∗ ) Rt∗∗ )uC (Ct+1.
uC (Ct∗ )(23)

Maximizing the lifetime utility (20) with respect to Yt∗ (and Ct∗ ) subject to the budget

6

constraint (21) (and considering the equivalent functional forms to (2), (3), Eqs. (10),and (15)), we also obtainpFt ∗ −σ= A∗t −χ Yt∗ χ .
C
qt t
2.2.

(24)

Risk-sharing gap

Following Viani (2010), we deﬁne the “risk-sharing gap,” which is a key variable inour study, as(
gapt = log

SDF t
SDFt∗

)
.

(25)

Herein, SDF t denotes the “stochastic discount factor” of country H:
SDF t = βuC (Ct+1 ) qt+1.
uC (Ct ) qt

(26)

SDFt∗ denotes the “stochastic discount factor” of country F :SDFt∗ = β ∗∗ )uC (Ct+1.
uC (Ct∗ )(27)

In complete asset markets, agents have access to complete array of state-contingent
claims. Under complete markets, the risk-sharing gap is zero, since perfect cross-borderrisk-sharing makes the stochastic discount factors of the two countries equal. However,in this model, households of two counties have access to only a single risk-free bondtraded internationally, which implies that asset markets are incomplete. Under incomplete markets, the gap deviates from zero. Since the gap indicates a deviation fromthe allocation under complete markets, we can measure the lack of risk sharing (i.e.,
risk-sharing ineﬃciency) with the risk-sharing gap in Eq.(25). In other words, therisk-sharing gap indicates the degree of market incompleteness.

2.3.

Equilibrium and exogenous shocks

From the households’ preferences, we can derive the demand for domestic and foreign
goods:( )−θ [
1−n ∗Y t = pHν Ct +
ν
t
n
7

( )−θ
]
1
Ct∗ ,
qt

(28)

and

( )−θ [ n (1 − ν)Yt∗ = pFt
Ct + (1 − ν ∗ )
1−n

( )−θ
]
1
Ct∗ .
qt

(29)

The equilibrium in the asset market requires that
Dt = Bt∗ .

(30)

The productivity shocks in countries H and F are, respectively,
log At = ρ log At−1 + ϵt ,(31)

log A∗t = ρ log A∗t−1 + ϵ∗t .

(32)

and

The equilibrium of this economy is a set of stationary stochastic processes: { Ct , Ct∗ ,F∗∗ ∞Yt , Yt∗ , Lt , L∗t , Rt∗ , Dt , Bt∗ , qt , pHt , pt , T oTt , Tt , Tt , gapt , SDFt , SDFt }t=0 satisfying

Eqs. (10), (11), (12), (13), (14), (15), (17), (18), (19), (22), (23), (24), (25), (26), (27),(28), (29), (30), given the exogenous stochastic processes At , A∗t , and an initial value
for D−1 .

2.4.

WelfareWe perform numerical experiments using a second-order approximation of the modelwith the perturbation method following Schmitt-Grohé and Uribe (2004). 3 Following
Schmitt-Grohé and Uribe (2006), we measure the conditional welfare of country H bycomputing the expected welfare conditional on the initial non-stochastic steady state:W 0 ≡ E0
= E0

∞
∑
t=0
∞
∑

β t U (Ct , Lt )(33)
β t U ((1 + η)C, L),

t=0

where
U (Ct , Lt ) = u(Ct ) − v(Lt ),(34)

3 Kim and Kim (2003) show that second-order solutions are necessary because conventional linearization may
generate spurious welfare reversals when long-run distortions exist in the model.

8and C and L denote their non-stochastic steady state levels. With η, we evaluate andcompare conditional welfare levels under diﬀerent parameters or policies.
Similarly, we compute conditional global welfare by deﬁning

W0W ≡ E0
= E0

∞
∑
t=0
∞
∑

β t U W (Ct , Ct∗ , Lt , L∗t )(35)
β t U W ((1 + η)C, L),

t=0

where U W (Ct , Ct∗ , Lt , L∗t ) = nU (Ct , Lt ) + (1 − n)U (Ct∗ , L∗t ). The transition from theﬁrst line to the second follows from the fact that C = C ∗ and L = L∗ in their nonstochastic steady states by structure.

3.

Optimal global policy and Nash equilibrium

3.1.

Optimal global policy

We characterize the optimal global policy by analyzing welfare-maximizing Ramsey
policies with commitment. Under the optimal global policy, a global policy maker
optimizes capital controls to maximize global welfare, considering the private sector’s
response to the implemented policies. Setting up a Lagrangian problem in which theglobal policy maker maximizes global welfare subject to the ﬁrst-order conditions of
the private agents and the market-clearing conditions of the two economies, we obtainthe ﬁrst-order conditions for the global policy maker. The ﬁrst-order conditions for the
global policy maker, the ﬁrst-order conditions of the private agents, and the marketclearing conditions of the two economies characterize the cooperation equilibrium.
In Section 4.2, we examine the optimal global capital controls policy. Under the
optimal global policy, we include τt (or τt∗ ) as a policy instrument.4 We let xt denote theN × 1 vector of endogenous variables. Except for the policy instrument, the remaining
N − 1 endogenous variables in xt satisfy the N − 1 structural conditions, which are

Et f (xt , xt+1 , ζt ) = 0,

4 Following De Paoli and Lipinska (2013), we include only one policy instrument.

9

(36)

where the vector ζt denotes the exogenous variables. We obtain the global policy
maker’s ﬁrst-order conditions by setting up a Lagrangian problem in which the global
policy maker maximizes global welfare subject to the ﬁrst-order conditions of the private agents and the market-clearing conditions of the two economies. More speciﬁcally,
we derive the optimal Ramsey policy from the maximization problem:

max
E0
∞

{xt }t=0

s.t.

∞
∑

β t U W (xt , ζt )t=0

Et f (xt , xt+1 , ζt ) = 0.

(37)

We set up the Lagrangian problem:

L 0 = E0∞∑β t {U W (xt , ζt ) + λ′t f (xt , xt+1 , ζt )},

(38)

t=0

where λt denotes the Lagrange multipliers associated with the ﬁrst-order conditions of
the private agents and the market-clearing conditions of the economy in (36). Taking
the derivatives of L0 with respect to the N endogenous variables, we obtain the Nﬁrst-order conditions, which are characterized by the following equation: 5

U1W (xt , ζt ) + Et λ′t f1 (xt , xt+1 , ζt ) + β −1 λ′t−1 f2 (xt−1 , xt , ζt−1 ) = 0.

(39)

Taking the derivatives of L0 with respect to λt , we obtain the N − 1 equilibrium conditions in the private sector in (36). The ﬁrst-order conditions of the private agents, themarket-clearing conditions of the two economies, and the global policy maker’s ﬁrstorder conditions characterize the cooperation equilibrium. The Ramsey equilibrium
process is therefore characterized by the N − 1 equations (36) and the N equations(39). For the variables, we have N elements of x and N − 1 multipliers of λ. In total,
we therefore have 2N − 1 variables and 2N − 1 equations.

5 The ﬁrst-order necessary condition for optimality at t = 0 is (39) with λ10−1 = 0.

3.2.

Nash equilibrium

Under a Nash equilibrium, the Home policy maker optimizes capital controls in country
H to maximize country H’s welfare level, while the Foreign policy maker optimizes
capital controls in country F to maximize country F’s welfare level. Thus, the strategicinteraction between the two countries’ policy makers leads to diﬀerent outcomes fromthose under the welfare-maximizing cooperative policy. Speciﬁcally, we consider an
open-loop Nash equilibrium in which the actions of each country’s policy maker are
best response to the other policy maker’s best response.
Setting up two Lagrangian problems in which the Home and Foreign policy makers
maximize their own country’s welfare subject to the ﬁrst-order conditions of the private
agents and the market-clearing conditions of the two economies, we obtain the ﬁrstorder conditions for each country’s policy maker. The ﬁrst-order conditions for eachcountry’s policy maker, the ﬁrst-order conditions of the private agents, and the marketclearing conditions of the two economies characterize an open-loop Nash equilibrium.
In Section 4.2, we also examine a Nash equilibrium in which the Home policy maker
optimizes τt to maximize country H’s welfare level, while the Foreign policy maker optimizes τt∗ to maximize country F’s welfare level. Speciﬁcally, we consider an open-loop
Nash equilibrium in which each country’s policy maker chooses the optimal allocation
given the evolution of the other country’s policy instrument. We let xt denote the N ×1
vector of endogenous variables in this case.6 Except for the two policy instruments,
the remaining N − 2 endogenous variables in xt satisfy the N − 2 structural conditions,which we again express with

Et f (xt , xt+1 , ζt ) = 0.

(40)

We obtain the ﬁrst-order conditions for the each country’s policy maker by setting up
two Lagrangian problems in which the Home and Foreign policy makers maximize theirown welfare subject to the ﬁrst-order conditions of the private agents and the marketclearing conditions of the two economies. More speciﬁcally, we set up a Lagrangian

6 N in 3.1 is not identical to N in 3.2.

11

problem for country i (1 or 2):

Li,0 = E0∞∑

β t {U i (xt , ζt ) + λ′i,t f (xt , xt+1 , ζt )}.

(41)

t=0

Taking the derivatives of Li,0 with respect to the N − 1 variables (except for thepolicy instrument of the other country), we obtain the 2N − 2 ﬁrst-order conditions
characterized by

U1i (xt , ζt ) + Et λ′i,t f1 (xt , xt+1 , ζt ) + β −1 λ′i,t−1 f2 (xt−1 , xt , ζt−1 ) = 0.

(42)

Taking the derivatives of Li,0 with respect to λi,t , we obtain the N − 2 equilibrium
conditions in the private sector in (40), which are common for both countries. The
ﬁrst-order conditions for each country’s policy maker, the ﬁrst-order conditions ofeach country’s private agents, and the market-clearing conditions of the two economies
characterize an open-loop Nash equilibrium. The Nash equilibrium process is therefore
characterized by the N − 2 equations (40) and the 2N − 2 equations (42). For thevariables, we have N − 2 elements of x and 2N − 2 multipliers of λ. In total, wetherefore have 3N − 4 equations and 3N − 4 variables.
We obtain the relevant ﬁrst-order conditions that characterize the optimal global
policy and Nash equilibrium problems using Bodenstein et al. (2018)’s Matlab procedures.7

4.
4.1.

Parameterization and numerical analysis
Parameterization

We choose standard parameter values from the relevant literature, which we summarize
in Table 1. However, we vary the parameter value of δ to examine how the asset
market incompleteness aﬀects the dynamics of the two economies in Section 4.2. We
also choose a diﬀerent value for the elasticity of substitution between domestic and
7 Bodenstein et al. (2018)’s program reads a Dynare model ﬁle and generates the ﬁrst-order conditions ofpolicymakers under the optimal global policy and the Nash equilibrium. See Adjemian et al. (2011) for details
on Dynare.

12imported goods θ to show the robustness of our analysis results in Setction 5.
Table 1. Parameterization

β
σ
χ
δ
λ
n
θ
ρ
σϵ

Description
Discount factor
Inverse intertemporal elasticity of substitution
Inverse elasticity of labor supply
Bond adjustment cost parameter
Degree of openness
Home country size
Elasticity of substitution between domestic and imported goods
Persistence of productivity shockStandard deviation of productivity shock

4.2.

Results

Value
0.99
1
0.47
0.01; [0.001, 0.1]
0.5
0.5
3
0.95
0.0071Figure 1 compares the impulse responses of the main variables to a negative home
productivity shock. Figure 1(a) is the case when θ = 0.5, which implies that the homeand foreign goods are complements. Figure 1(b) is the case when θ = 3, which implies
that the home and foreign goods are imperfect substitutes. In both cases, the solidand dotted curves indicate the case with high (δ = 0.1) and low frictions (δ = 0.001),
respectively. As Corsetti et al. (2010) and De Paoli and Lipinska (2013) argue, theelasticity of substitution between the home and foreign goods, θ, is a critical parameterto dynamics in the two-country models. In Figures 1(a) and (b), the position of thesolid and dotted curves critically depends on the parameter θ. For example, in thecase of “Home output,” the solid curve is lower than the dotted curve in (a), but thesolid curve is higher than the dotted curve in (b). However, in the “Risk-sharing gap”
case, although the position of the two curves is still reversed between (a) and (b),we should note that the solid curve deviates from zero further than the dotted curve
does in either case. In the high-friction case (δ = 0.1), the “Risk-sharing gap” deviates
further from zero compared to the low-friction case (δ = 0.001). In other words, in thehigh-friction case, international risk sharing is more disturbed and the international
ﬁnancial markets are more incomplete compared to the low-friction case. Related with
this, in the “External debt holdings” case, the solid curve remains closer to zero than13

-0.221

2

3

4

Percent Dev.from SS

Percent Dev.from SS

0

-0.05

-0.1

2

3

4

2
0
1

5

Home consumption

-0.15
1

4

5

= 0.1
= 0.001

2

3

4

-0.22
-0.24
-0.26
1

5

3

4

1

2

3

4

5

Risk sharing gap

-2

= 0.1
= 0.001

-4
1

0.3

-0.18

5

Percent Dev.from SS

Percent Dev.from SS

-0.6
4

5

10 -3

2

3

4

5

External debt holdings (Home)

0.2
0.1
0
1

2

3

4

5

Terms of trade (Home)

-0.5

3

4

-0.2
2

-0.4

2

3

-0.16

Terms of trade (Home)

1

2

Home consumption

External debt holdings (Home)

-0.1

-0.2
1

0

-0.2

Percent Dev.from SS-0.2

Home output

Risk sharing gap
Percent Dev.from SS

-0.18

-3

Percent Dev.from SS

Percent Dev.from SS

Percent Dev.from SS

10
6Percent Dev.from SS

Home output

-0.16

-0.06
-0.08
-0.1

5

1

2

3

4

(a) θ = 0.5.

5(b) θ = 3.

Figure 1. Responses to a negative (one standard deviation) home productivity shock.

the dotted curve does in either case. This is again because in the high-friction case,the international ﬁnancial markets are more incomplete and international risk sharing
is more disrupted compared to the low-friction case. The disruption of internationalrisk sharing restricts changes in external debt holdings.
In Figure 1, we ﬁnd that both in case (a) and (b), a higher degree of friction causes
the impulse response of the “Risk-sharing gap” to deviate further. Figure 2 (a) plots
the standard deviation of the risk-sharing gap deﬁned by Equation (25) under diﬀerent
degrees of friction. The solid curve in Figure 2 (a) depicts the case without capital
controls. Consistent with Figure 1, the solid curve in Figure 2 (a) indicates that thestandard deviation of the risk-sharing gap increases as the degree of friction increases.
Figure 2 (b) depicts the associated conditional global welfare level, which we measure
as a percentage of steady state consumption. The solid curve in Figure 2 (b) depicts
the case without capital controls, which indicates that the conditional global welfare
decreases as the degree of friction increases because the two countries suﬀer from more
insuﬃcient risk-sharing as the degree of friction increases.
In Figure 2 (a), the dashed curve plots the standard deviations of the risk-sharing
gap under the Nash equilibrium, in which the standard deviation of the risk-sharing
gap increases when the degree of friction increases, as well as in the case without capital
controls. However, comparing the dashed and solid curves, we see that the standard
deviation of the risk-sharing gap under the Nash equilibrium is higher than that in
the case without capital controls. In Figure 2 (b), the dashed curve plots the global

14

0.04

0.09

0.035

0.08

% SS cons. eq.

0.025

Global policy
Without capital controls
Nash equilibrium

0.07

Global policy
Without capital controls
Nash equilibrium

0.03

0.02

0.015

0.06

0.05

0.04

0.01

0.03

0.005

0.02

0.01

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Conditional global welfare.

(a) Standard deviation of the risk-sharing gap(%).

Figure 2. Standard deviations of the risk-sharing gap and global welfare levels under the optimal global
policy, the Nash equilibrium, and no capital controls for diﬀerent values of δ.

welfare level under the Nash equilibrium. Consistent with the dashed curve in Figure
2 (a), the dashed curve in Figure 2 (b) indicates that global welfare deteriorates as thedegree of friction increases due to the resulting insuﬃcient risk-sharing. Comparing
the dashed and solid curves in Figure 2 (b), we see that the global welfare level under
the Nash equilibrium is lower than that in the case without capital controls, which isconsistent with the comparison between the solid and dashed curves in Figure 2 (a).
The dashed dotted curve in Figure 2 (a) plots the standard deviations of the risksharing gap under the optimal global policy. Compared with the Nash equilibrium caseand the no capital controls case, the optimal global policy keeps the standard deviationof the risk-sharing gap close to zero. In Figure 2 (b), we see that compared to the othertwo cases, the optimal global policy prevents global welfare from deteriorating, evenwhen the degree of friction is high, which is consistent with the dashed dotted curvein Figure 2 (a).
Figure 3 depicts how the tax rates of the external debt and asset (i.e., capital controls), which are diﬀerent under diﬀerent regimes (optimal global policy and Nashequilibrium), respond to a negative home productivity shock. In Figure 3, the solid
curve, the dashed curve, and the dashed dotted curve depict the case without capital
controls, the Nash equilibrium case, and the optimal global policy case, respectively.
The optimal global policy reduces the rate of “Home tax (on external debt)” (left upper panel) and encourages Home’s external borrowing (right lower panel) in response15

to a Home negative productivity shock. Since the optimal global policy improves international risk sharing, the “Risk sharing gap” becomes zero (right upper panel). 8
In contrast, in the Nash equilibrium case, the Home policy maker raises the rate of“Home tax (on external debt)” (left upper panel) and discourages Home’s externalborrowing (right lower panel) compared to the other two cases. At the same time, theForeign policy maker raises the rate of “Foreign tax (on external asset)” (left lower
panel), which also disturbs international risk sharing. As international risk sharing is
disturbed under the Nash equilibrium, the “Risk sharing gap” deviates further fromzero (right upper panel) even compared to the case without capital controls.
10-3

4

10-3

1
0
Percent Dev.from SS

2
0
-2
-4

-1
-2
-3
-4

-6

-5
1

2

3

4

5

1

Home tax (on external debt)

3

4

5

Risk sharing gap

10-3

4

2

0.5

Percent Dev.from SS

2
0
-2
-4

Global policy
Without capital controls
Nash equilibrium

0.4
0.3
0.2
0.1

Without capital controls
Nash equilibrium

-6

0
1

2

3

4

5

1

Foreign tax (on external asset)

2

3

4

5

External debt holdings (Home)

Figure 3. Responses of tax rates, risk-sharing gap, and external debt holdings to a negative home productivity
shock under the optimal global policy, the Nash equilibrium, and no capital controls (δ = 0.01).

Related to Figure 3, Figure 4 (a) depicts how the responses of tax rates, risk-sharing
gap, and external debt holdings to a negative home productivity shock diﬀer underdiﬀerent degrees of friction (δ = 0.001, 0.01, and 0.1) in the optimal global policycase. As the degree of friction is larger, the optimal global policy reduces the rate of“Home tax (on external debt)” more aggressively (left upper panel) so as to correct
the international risk sharing ineﬃciency. For any degree of friction, as a result, the
8 In the optimal global policy case, only one policy instrument is included (De Paoli and Lipinska 2013).

16

-0.01-0.03= 0.001= 0.01
= 0.1

-0.04

1.8
1.6

0

1.4

-0.5

1

2

3

4

5

-4
-4.5
-5
-5.5

-1
1

Home tax (on external debt)

2

3

4

1

5

2

3

4

5

1

Home tax (on external debt)

Risk sharing gap

2

3

4

5

Risk sharing gap

10-3

0.2

0.5

Percent Dev.from SS

2
1.8

0.4

Percent Dev.from SS1= 0.001= 0.01
= 0.1

1.2

-0.05

10-3

-3.5

2

0.5

Percent Dev.from SS

-0.02

10-3

10-3

1

Percent Dev.from SS

0

1.6
0.3

1.4
1.2

0.2

0.15
0.1
0.05

1
0.1

0
1

2

3

4

5

External debt holdings (Home)

1

2

3

4

5

Foreign tax (on external asset)(a) Global policy.

1

2

3

4

5

External debt holdings (Home)(b) Nash equilibrium.

Figure 4. Responses of tax rates, risk-sharing gap, and external debt holdings to a negative home productivity
shock for diﬀerent values of δ (= 0.001, 0.01, and 0.1).

optimal global policy succeeds in making the “Risk sharing gap” zero (right upper
panel), and the paths of “External debt holdings (Home)” turn out identical (rightlower panel). Similarly, Figure 4 (b) depicts how the responses of the same variables
to the same shock diﬀer under diﬀerent degrees of friction but in the Nash equilibrium
case. Under the Nash equilibrium, the “Risk sharing gap” is not zero, and deviates
further from zero as the degree of friction is higher (right upper panel). As the degreeof friction is larger, international risk sharing is more disturbed, and then the pathof “External debt holdings (Home)” is closer to zero (right lower panel). In contrast
to the optimal global policy case, the Home policy maker raises the rate of “Hometax (on external debt)” in response to a negative home productivity shock (left upper
panel). At the same time, the Foreign policy maker raises the rate of “Foreign tax (onexternal asset)” (left lower panel), which also disturbs international risk sharing. It isnoteworthy that as the degree of friction is lower, both of the policy makers impose
a higher rate of tax (left upper and lower panels) and disturbs international risk
sharing further, which is in contrast to the optimal global policy’s role of encouraging
international risk sharing.
Figure 5 (a) plots the diﬀerence in the standard deviations of the risk-sharing gap
between the Nash equilibrium and the optimal global policy (i.e., the diﬀerence between the dashed curve and the dashed dotted curve in Figure 2 (a)). Figure 5 (a)shows that as the degree of friction increases, the diﬀerence in the standard deviations
17

0.038

0.08

0.036

0.075

0.07

0.032

0.065

% SS cons. eq.

0.034

0.03

0.028

0.06

0.055

0.026

0.05

0.024

0.045

0.022

0.04

0.02

0.035

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

δ

(a) Nash equilibrium − Global policy: Diﬀerencein the standard deviations of the risk-sharing gap
(%).

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

(b) Global policy − Nash equilibrium: Diﬀerence inconditional global welfare levels.

Figure 5. (a) Diﬀerence in the standard deviations of the risk-sharing gap between the Nash equilibrium andoptimal global policy for diﬀerent values of δ.(b) Diﬀerence in conditional global welfare levels between the
optimal global policy and the Nash equilibrium for diﬀerent values of δ.

of the risk-sharing gap between the Nash equilibrium and the optimal global policy
increases. Figure 5 (b) plots the diﬀerence in the (conditional) global welfare levels
between the optimal global policy and the Nash equilibrium, in which the diﬀerence
in global welfare levels increases when ﬁnancial markets are less eﬃcient, which is consistent with the results in Figure 5 (a). This implies that the gain from international
cooperation on capital controls is larger when ﬁnancial markets are less eﬃcient.
In Figure 5, we set the elasticity of substitution between the home and foreign
goods, θ, to 3. We next consider cases with diﬀerent values of θ based on Rabitsch(2012)’s ﬁnding that welfare gains from monetary coordination critically depend on
the elasticity of substitution between the home and foreign goods. Similar to Figure 5(a), Figure 6 (a) plots the diﬀerences in the standard deviations of the risk-sharing gap
between the Nash equilibrium and the optimal global policy, but for diﬀerent valuesof δ and θ. As Figure 6 (a) shows, as θ decreases to 1, the diﬀerence in the standard
deviation of the risk-sharing gap decreases, but it starts to increase for lower values ofθ(< 1) (for any value of δ).9
Similar to Figure 5 (b), Figure 6 (b) plots the diﬀerence in the global welfare levels
between the optimal global policy and the Nash equilibrium, but for diﬀerent values ofδ and θ. As Figure 6 (b) shows, as θ decreases to 1, the diﬀerence in the global welfare
9 The case where θ = 1 corresponds to a special case in which an automatic form of risk insurance is provided(Cole and Obstfeld 1991).

18

levels decreases, but it starts to increase for lower values of θ(< 1) (for any value of
δ). In Figure 5 (b), we have already shown that the diﬀerence in global welfare levels
between the optimal global policy and the Nash equilibrium increases as the degreeof friction increases. However, Figure 6 (b) shows that for lower values of θ(< 1),
the welfare gains from policy cooperation in capital controls, which we measure as
the diﬀerence in global welfare levels between the optimal global policy and the Nashequilibrium, can be much larger than that in the benchmark case with θ = 3.

(a) Nash equilibrium − Global policy: Diﬀerence inthe standard deviations of the risk-sharing gap (%).

(b) Global policy − Nash equilibrium: Diﬀerence inthe conditional global welfare levels.

Figure 6. (a) Diﬀerence in the standard deviations of the risk-sharing gap between the Nash equilibrium andoptimal global policy for diﬀerent values of δ and θ. (b) Diﬀerence in conditional global welfare levels between
the optimal global policy and the Nash equilibrium for diﬀerent values of δ and θ.

19

5.

Robustness

In Section 4.2, we show our results for the case in which the home and foreign goods
are imperfect substitutes (θ = 3), except for Figure 1 (a). In this section, we show that
our main results remain unchanged when the home and foreign goods are complements(θ = 0.5).
As Figure 7 shows, we obtain similar ﬁgures of the standard deviation of the risksharing gap and conditional global welfare level for diﬀerent values of δ corresponding
to Figure 2 in Section 4.2.
0.14

-0.25
-0.3

0.12

-0.35

Global policy
Without capital controls
Nash equilibrium

0.08% SS cons. eq.

0.1

Global policy
Without capital controls
Nash equilibrium

0.06

-0.4
-0.45
-0.5
-0.55

0.04
-0.6

0.02
-0.65

0

-0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Conditional global welfare.

(a) Standard deviation of the risk-sharing gap(%).

Figure 7. Standard deviations of the risk-sharing gap and global welfare levels under the optimal global
policy, the Nash equilibrium, and no capital controls for diﬀerent values of δ (θ = 0.5).

We also obtain similar ﬁgures in Figure 8 corresponding to Figure 5. We shouldnote that when θ = 0.5, the diﬀerence in the conditional global welfare levels under
the optimal global policy and the Nash equilibrium can reach more than 0.14%, up to
0.43%, which is higher than that in Figure 5.

20

0.13

0.45

0.12

0.4

0.35

% SS cons. eq.

0.11

0.1

0.09

0.3

0.25

0.2

0.08

0.15

0.07

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Nash equilibrium − Global policy: Diﬀerencein the standard deviations of the risk-sharing gap
(%).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Global policy − Nash equilibrium: Diﬀerence inconditional global welfare levels.

Figure 8. (a) Diﬀerence in the standard deviations of the risk-sharing gap under the Nash equilibrium andthe optimal global policy for diﬀerent values of δ.(b) Diﬀerence in the conditional global welfare levels under
the optimal global policy and the Nash equilibrium for diﬀerent values of δ (θ = 0.5).

21

6.

Conclusions

We have investigated how the degree of ﬁnancial market incompleteness aﬀects thewelfare gains from international cooperation on capital controls. Under higher degrees
of incompleteness, international cooperation on capital controls becomes more welfare
improving compared to the Nash equilibrium case. The intuition is straightforward.
When ﬁnancial markets are less eﬃcient, international risk sharing is more disturbed.
However, since policy cooperation signiﬁcantly reverses the welfare deterioration due
to ineﬃcient risk-sharing, international cooperation on capital controls becomes morecritical when ﬁnancial markets are more incomplete.

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