International Reserves and Monetary Policy
Avner Bar-Ilan*
Department of Economics
University of Haifa
31905 Haifa, Israel
Dan Lederman
Department of Economics
University of British Columbia
Vancouver, BC, V6T 1Z1, Canada
November 2006
Abstract: We present a model that determines both monetary policy and international
reserves. To that end we introduce a reserves constraint into a standard monetary model.
Myopic central bank might raise its current interest rate in order to satisfy the constraint.
However, optimizing, forward looking bank would probably raise interest rate more. This
strict monetary policy today would build precautionary reserves that serve two purposes:
allow future monetary policy to better stabilize inflation and output and lower the
likelihood of forthcoming financial crisis.
JEL Classification code: E58, F33
Keywords: international reserves, drift control
*Corresponding author. bar-ilan@econ.haifa.ac.il, fax: 972-4-8240059, phone: 972-48240021
1
1. Introduction
A major line of research that attempts to explain the amount of international reserves held
by a central bank (CB) is the buffer stock model.1 Another model that was developed
recently to explain the level of reserves is the drift control model.2 According to both
models, the objective of the CB is to minimize the cost associated with holding and
managing reserves. Other objectives that the CB might have are not explicitly modeled.
In this paper we analyze foreign reserves within the context of a standard dynamic
monetary model. The CB uses monetary policy to stabilize inflation and output, its
standard goals. However, the bank is also aware of the effect that monetary policy might
have on the accumulation or depletion of reserves. The outcome is joint determination of
monetary policy and level of foreign reserves.
We introduce a constraint on the minimum amount of reserves and allow reserves to be
positively correlated with interest rates. A myopic CB sets the interest rate at the optimal,
unconstrained rate that ignores the reserves constraint, if this rate satisfies the constraint;
otherwise, it sets the interest rate at the minimum level that satisfies the constraint.
Forward-looking CB, however, sets higher interest rate than the myopic bank, and
correspondingly accumulates more reserves. On one hand, this strict monetary policy is
costly in terms of contemporaneous stabilization. On the other hand, the additional
precautionary reserves allow future monetary policy to be less costly and better stabilizer.
Moreover, these additional reserves reduce the probability of a future financial crisis
when the level of reserves will be insufficient.
The structure of the paper is as follows. Section 2 presents the model, while the analytic
solution is outlined in section 3. Section 4 presents the numerical solution, while section
5 concludes.
1
2
Examples are Edwards (1985) and Aizenman and Marion (2004).
See Ata et al. (2005) and Bar-Ilan et al. (2006).
2
2. The Model
We adopt here the basic monetary model of Clarida et al. (1999), augmented by an
international reserves constraint as introduced by Romer (2002). The CB maximizes the
following two-period objective function
(1)
 1  2 t 1

m ax  E1     xt2   t2    .

 2  t 1
The variable xt denotes the deviation of output from potential level and  t is the
inflation rate, both in period t. The discount rate is denoted by  , Et denotes
expectations as of period t, and  is a parameter that measures the loss of output relative
to inflation. The economy is characterized by the following two equations
(2)
xt   it  Et t 1   gt , E ( gt )  0, Var ( gt )   g2
(3)
 t   xt   Et t 1  ut ,
E (ut )  0, Var (ut )   u2
Equation (2) is an “IS” curve that relates the output gap inversely to the real interest rate,
where it is the nominal interest rate and  is positive parameter. Equation (3) is a
Phillips curve that relates inflation positively to output and expected inflation. The output
and inflation shocks, gt and ut respectively, are both white noise error terms.
A standard monetary model is composed of maximizing the objective function (1) subject
to the two constraints, (2) and (3). We introduce international reserves, Rt , into this
model. Reserves cannot fall below a certain lower bound, denoted by B, B  0 ,
(4)
Rt  B .
3
The dynamics of reserves is,
(5)
Rt  Rt 1  dt .
The drift term, dt , increases with the interest rate, as in Romer (2002),
(6)
dt     it  vt , E ( t )  0, Var ( t )   v2 .
The parameters  and  are positive, and  t is white noise term that reflects shocks to
the current account or the capital account.
3. Analytic Solution
The CB chooses nominal interest rate in order to maximize the objective function (1)
subject to the constraints (2)-(6). We solve this problem recursively.
Period 2
Period 2 is the terminal period, when intertemporal considerations do not play a role.
Ignoring the foreign reserves constraint (4), the solution is the discretionary solution
presented in Clarida et al. (1999),
(7)
it 
(8)
t 
(9)
gt


q
u,
 t

  2


q
1
,
  2
ut ,
xt    t  

  2
ut .
4
Note that the interest rate is determined after the realization of the shocks. The output
shock gt is fully offset by the monetary policy, where the CB leans against the wind of
the inflation shock and optimally splits it between the inflation and output terms.3
We denote the solution (7)-(9) the unconstrained solution. We now take into account the
reserves constraint. The minimum level of interest rate that satisfies the reserves
*
constraint, i2 , from equations (4)-(6), is
i2* 
(10)
1

( B  R1    v2 ) .
*
We denote i2 as the constrained interest rate. The optimal interest rate i2 is the
unconstrained level (7) if this level satisfies the reserves constraint; that is, at least as
large as the constrained level (10). The interest rate i2 is therefore
(11)
 g q

i2  max  2 
u2 , i2*  .



The interest rate takes the constrained level when the three shocks satisfy the following
inequality
(12)
 q

u2  g 2  v2  B  R1  .


Note that the lower the reserves level R1 at the end of period 1 (start of period 2), the
higher the probability that the interest rate will be constrained.
3
Since 0 

  2
 1 , then  t  ut , at a cost of xt  0 .
5
The constrained monetary policy is costly relative to the unconstrained one. This cost is
composed of two terms. First, output and inflation are not at their optimal, unconstrained
level. Second, we assume that there is a fixed cost C whenever the monetary policy is
constrained and the reserves constraint is binding. This cost captures a financial crisis
when the level of foreign reserves is insufficient and the CB must alter its monetary
policy accordingly. The expected cost of the latter term, expectations taken at the end of
period 1, is
(13)
C  Pr[
 q

u2  g 2  v2  B  R1  ] ,


where Pr denotes probability.
In order to evaluate the first term of the cost, notice from equations (2) and (3) that when
the interest rate changes by an increment  , xt and  t change as
(14)
xt   (  Et  t 1 ) ,
(15)
 t  xt   (  Et  t 1 ) ,
where Et  t 1 is the expected value of the change in
the change of it by
 t 1 that takes place because of
 . Using second order Taylor approximation, the welfare cost of
changing the nominal rate by  relative to the unconstrained level is,
(16)
L 
1
 (xt )2  ( t )2 

2
Since t=2 is the terminal condition, we get
(17)
x2   2 ,
6
(18)
 2   2 ,
(19)
1
L2 ( R1 )   2 2 2 (   2 ) .
2
The expected increase in the loss in period 2 because of the deviation of output and
inflation from their unconstrained levels, E L2 , is:

(20)
1
E L2 ( R1 )   2 (   2 )   22 f ( 2 )d 2 .
2
 2 0
The integral is over the values of (u2 , g 2 , v2 ) that satisfy the inequality (12),
and  2 , 0   2   , is the increase in the interest rate because of the reserves constraint.
That is,
(21)
1
g q
 2  ( B  R1   v2 )  2  u2 .

 
The total additional expected cost of the constrained monetary policy relative to the
unconstrained policy is the sum of equations (13) and (20). Both terms fall with the level
of reserves R1 .
Period 1
Period 1 starts with an exogenous, given level of Reserves R0 . Then the shocks
( g1 , u1 , v1 ) are realized and a decision about the interest rate i1 takes place. The
interesting tradeoff is as follows. The CB might want to raise the interest rate i1 in order
to increase the reserves level R1 and to reduce the expected loss in the second period due
7
to insufficient reserves, financial crisis and constrained monetary policy (equations (13)
and (20)). On the other hand, raising i1 changes x1 and
 1 and this might be costly as
well.
If for period t=1 the CB does not consider the intertemporal considerations explained
above, then the solution is similar to period-2 solution. However, this solution that was
optimal for the terminal period is now suboptimal and myopic. This myopic interest rate,
similar to equations (10) and (11), is
(22)
 g q

i1 (myopic)  max  t 
u1 ,i1*  ,
 

(23)
i1* 
1

( B  R0    v1 ) .
Define 1 as 1  i1  i1 ( myopic) . Equations (14) and (15) yield
 1
,
1  
(24)
 1  
(25)
x 1   1
1  2
.
1  
Plugging these values into (16) yields
(26)
 2 12
 (1  2 )2   2  .
L1 ( R0 ) 
2 
2(1   )
The CB chooses the interest rate i1  0 that minimizes the expected NPV of the welfare
loss given by equations (13), (20), and (26).
8
4. Numerical Solution
To reiterate, the optimization problem to be solved is,
(27)



q
 1 2
 g2
2
Min  C  Pr 
u2 
 v 2  B  R1   +  (   )   22f ( 2 )d  2 
1

 
 2

0

 2 12
 (1  2 )2   2  ,
+
2 
2(1   )
where the integral is over values that satisfy inequality (12), and 1 is the increase in
interest rate above the myopic value in order to reduce the probability of a future
financial crisis.
The numerical solution is solved as follows. Given certain initial reserves R0 and
realization of first period shocks, we draw randomly 100,000 values of second period
shocks ( g 2 , u2 , v2 ) . The shocks are drawn from jointly normal density function with zero
mean, variances  g , u , v , and zero cross correlations. The optimal 1 minimizes the
2
2
2
function (27). 4
We start by looking at the interest rate as a function of the initial reserves R0 . Figure 1
shows the interest rate i1 , the myopic interest rate i1 (myopic) , and the difference between
the two rates, 1 . The level of reserves R1 accumulated as a result of the chosen interest
4
Unless stated otherwise, the parameter values used in the numerical solution are
and
g1  u1  v1  0
R0

B



u
g
v

C

5
100
5
0
80
0.01
0.01
1
0.1
0.5
200
0.95
9
rate is presented in Figure 2. The CB optimally increases the interest rate above the
myopic level in order to accumulate reserves as a precaution against future financial
crisis. Although this is especially pronounced for low levels of initial reserves, only when
the initial reserves are close to double the required level then the myopic and optimal
interest rates coincide. Although a CB that ignores intertemporal considerations will raise
the reserves only up to the required level B=5, the minimum level of reserves held in
figure 2 is over 40% higher. The precautionary reserves = R ( i 1 )  R  i 1  myopic   are
also depicted in figure 2. These reserves, together with 1 , are constant when R0  B as
the myopic interest rate builds reserves up to B. When R0  B ,the need for precautionary
reserves, and correspondingly 1 , is lower.
We turn now to the effect of the parameter C, the cost of a financial crisis, on monetary
policy. Figure 3 shows how the interest rate depends on C, while figure 4 presents a
similar graph for foreign reserves R1 and C. As expected, the larger is C the higher are
both the optimal interest rate and the resulting reserves level that reduces the probability
of a future crisis. The interesting observation from these figures is that the optimal
interest rate is above the myopic level and reserves are above the required level even
when C=0. The myopic CB wishes to smooth output and inflation in the first period only
and thus sets R1  R0  B  5 . However, a forward-looking bank, even if a future financial
crisis is not an issue, wishes to hold additional reserves in order to smooth future output
and inflation. This allows us to break in figure 4 the precautionary reserves to smoothing
reserves (precautionary reserves when C=0), and net precautionary reserves
(precautionary minus smoothing reserves).
We study also the effect of the parameter  on the interest rate i1 and the level of
reserves R1 . The parameter  , defined in equation (1), measures the welfare cost of
output relative to inflation deviations. Small levels of  indicate CBs that are concerned
about inflation stabilization rather than output stabilization. In the terminology of the
literature, this defines independent CB. We find that monetary policy conducted by
10
independent banks is stricter than the policy run by "soft" banks. This is reflected in
higher interest rates and levels of foreign reserves, as demonstrated in Figure 5.
5. Conclusion
This paper introduces a foreign reserves constraint into a standard macroeconomic model.
The positive correlation between reserves inflow and interest rate forces the CB to have a
minimal interest rate that supports a threshold level of reserves.
However, even when the CB has currently sufficient reserves it would optimally choose a
higher interest rate than the myopic interest rate that ignores or just meets the reserves
constraint. This strict monetary policy today increases the level of reserves and serves as
a precaution against adverse circumstances that might take place in the future. The
additional reserves assist in future monetary stabilization and lower the chance of a
financial crisis. Our numerical solution demonstrates that these precautionary reserves
could be quantitatively significant. Our model contributes therefore to the understanding
of why CBs, in particular those which are more vulnerable to a financial crisis,
accumulate large amounts of foreign reserves.
11
References
Aizenman, J. and N. Marion, 2004, International Reserve Holdings with Sovereign Risk
and Costly Tax Collection, Economic Journal 114, 569-591.
Ata, B., M. Harrison and L. Shepp, 2005, Drift Rate Control of a Brownian Processing
System, Annals of Applied Probability 15. 1145-1160.
Bar-Ilan, A., N. Marion and D. Perry, 2006, Drift Control of International Reserves,
forthcoming Journal of Economic Dynamics and Control.
Clarida, R., J. Gali and M. Gertler, 1999, The Science of Monetary Policy:
A New Keynesian Perspective, Journal of Economic Literature 37, 1661-1707.
Edwards, S., 1985, On the Interest-Rate Elasticity of the Demand for International
Reserves, Journal of International Money and Finance 4, 287-95.
Romer, D., 2002, Short-Run Fluctuations, UC Berkeley.
12
0.04
0.06
optimal  1
0.04
1
0
i1(myopic)
0.02
0
-1
-0.8
-0.6
optimal i1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.02
(R0-B)/B
Figure 1
11
3
10
2
9
1
8
0
optimal reserves
7
-1
-0.8
-0.6
-0.4
-0.2
0
(R0-B)/B
Figure 2
13
0.2
0.4
0.6
0.8
-1
precautionary reserves
precautionary reserves
optimal reserves
interest rate
0.02
-4
6
x 10
5
interest rate
4
3
optimal i1
2
1
i1(myopic)
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8
0.9
1
cost
Figure 3
5.06
optimal reserves
5.05
5.04
5.03
net precautionary reserves
5.02
5.01
5
smoothing reserves
0
0.1
0.2
0.3
0.4
0.5
cost
Figure 4
14
0.6
0.7
10
0.08
interest rate
0.06
optimal i1
8
0.05
optimal reserves
0.04
7
0.03
0.02
i1(myopic)
0
0.5
1

Figure 5
15
6
1.5
optimal reserves
9
0.07