Stability Properties of the Nominal Income
Targeting Rule in the Open Economy
Baotai Wang*
Economics Program
University of Northern British Columbia
3333 University Way
Prince George, British Columbia
CANADA V2N 4Z9
Tel: (250)960-6489, Fax: (250)960-5545
Email: wangb@unbc.ca
Abstract:
Results of this study show that stability for output and inflation prevails
under the nominal income targeting rule of monetary policy in a general open-economy
model when the asymmetric effects of the interest rate on output and inflation and the
different formations of inflation expectations are taken into the analysis.
Key Words: Nominal income targeting, Inflation expectations, Phillips curve, Open
economy, Economic stability
JEL Classification: E52
_________________________
*The author would like to thank Tomson Ogwang and Paul Bowles for their valuable
comments on this study. However, the usual disclaimer applies.
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1.
Introduction
There has been a general agreement among economists and policymakers that a monetary
policy should aim at stabilizing some nominal variables and that such a policy should be
based on a fixed rule, rather than on discretion. Over the last few decades, many central
banks have conducted monetary policy based on the rules that target either an interest rate
measure, or a monetary aggregate, an aggregate price index, such as the consumer price
index (CPI).
Since the 1980s, the nominal income targeting approach to monetary policy, one
of the possible policy rules advocated by some economists, has attracted considerable
attention. The appeal of the nominal income targeting rule in terms of delivering better
economic performance has been discussed in many studies. See, for example, Bean
(1983), Taylor (1985), McCallum (1988, 1990), Bryant, Hooper, and Mann (1993),
Feldstein and Stock (1994), Hall and Mankiw (1994), Frankel and Chinn (1995), and
Ratti (1997). However, the question of whether or not nominal income targeting would
result in dynamic stability for output and inflation remains in debate. Discussions on the
issue, so far, have been mainly conducted within the context of the closed economy and
special attention has been given to two factors: the stylized fact that the interest rate
affects output before inflation and the formation of backward-looking inflation
expectations, because they are found to be directly responsible for the instability results
in the closed economy. Detailed discussions in this regard can be found in Ball (1999a),
Svensson (1997), McCallum (1997), and Dennis (2001).
2
The objective of this study is to extend the discussions to the open-economy case.
To this end, Ball’s (1999b) open-economy model is modified and used to examine the
stability properties of the nominal income targeting rule under different conditions. The
main finding of this study is that the nominal income targeting rule does not necessarily
lead to instability for output and inflation in the open economy although both forementioned factors are taken into the analysis. Our results also show that different
formations of inflation expectations do not change the stability conditions for the policy
rule in such a model.
The rest of the paper is structured as follows. In Section 2, we discuss the openeconomy model. In Section 3, we examine the stability conditions for the nominal
income targeting rule with backward-looking inflation expectations. In Section 4, we
replace
backward-looking
inflation
expectations
by
forward-looking
inflation
expectations and discuss the stability conditions. In Section 5, we discuss some
implications from our results. Concluding remarks are made in the last section.
2.
The Model
The open-economy model, which is used to examine the stability properties of the
nominal income targeting rule, is specified as:
yt  1 yt 1   2 rt 1   3 et   t ,
1  1  0,  2  0,  3  0
(1)
 t   t 1  1 yt 1   2 et  t ,
1  0,  2  0
(2)
et   1rt 1   t ,
1  0
(3)
3
where y denotes the output gap (in logarithm);  , r , and e denote, respectively, the
inflation rate, the real interest rate, and the rate of real appreciation, all measured as
deviations from equilibrium levels;  ,  , and  are white-noise disturbances.
This model captures the basic assumptions made in Ball’s models (1999a, 1999b).
Equation (1) is the open-economy IS curve. It is assumed that current output depends on
its one-period lagged value and that the Marshall-Lerner condition is satisfied so that the
quantity of net exports, and thereby the quantity of equilibrium output, is decreasing in
the rate of real appreciation. Equation (2) is the open-economy accelerationist Phillips
curve where inflation expectations are backward-looking type. Equations (1) and (2)
together characterize the timing with which real interest rate affects output and inflation.
Output is determined by the one-period lagged interest rate while inflation is determined,
through output, by the two-period lagged interest rate. The important difference between
this model and the one specified in Ball (1999b), however, is that the current rate of real
appreciation, rather than a lagged one, is introduced in both demand and supply equations
as the prices seen by the consumers and producers are influenced directly by the current
prices of imported goods. Equation (3) closes the model by linking the rate of real
appreciation to the real interest rate and it is assumed that real appreciation takes time to
adjust.
3.
Stability in the Open-Economy Model with Backward-looking Expectations
Using the model presented in Section 2, we now examine the stability conditions for the
nominal income targeting rule.
4
Substituting (3) into (1) and (2) yields a two-equation system:
yt  1 yt 1  ( 2   3 1 )rt 1  ut
(4)
 t   t 1  1 yt 1   2 1rt 1  wt
(5)
where ut   t   3 t and wt  t   2 t are white noises since both are N (0, ) . The
monetary authority is assumed to conduct policies by manipulating the real interest rate
through its control over the nominal interest rate upon its backward-looking forecast on
future inflation to achieve its objective of a constant growth rate of nominal income. For
simplicity, this expected growth rate is set to be zero, i.e.,
Et 1 ( yt  yt 1   t )  0 ,
(6)
where E is the expectation operator. The Taylor-style optimal interest rate rule (Taylor,
1993) can then be derived by substituting (4) and (5) into (6),
rt 1 
1  1  1
1
yt 1 
 t 1 .
 2   3 1   2 1
 2   3 1   2 1
(7)
Substituting (7) back to (4) and (5), we derive the following reduced form of the model,
yt 
1  2 1  (1  1 )( 2   3 1 )
 2   3 1
yt 1 
 t 1  ut
 2   3 1   2 1
 2   3 1   2 1
(8)
t 
(1  1 )  2 1  1 ( 2   3 1 )
 2   3 1
yt 1 
 t 1  wt ,
 2   3 1   2 1
 2   3 1   2 1
(9)
which is clearly a VAR(1) system. Using capital letters A, B , and C to denote the
combined coefficients in (8) and (9), we can rewrite the VAR(1) system as
 yt   A  C   yt 1  ut 
    B C     w 
  t 1   t 
 t 
(10)
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From the restrictions on the values of the model parameters imposed in Equations (1),
(2), and (3), it can be easily verified that A  1 , A  B  1 , and 0  C  1 .
Stability of this system depends on the values of the characteristic roots of the
coefficient matrix.1 Given the coefficient matrix in (10), the characteristic roots, z , are
derived as:
z 2  ( A  C ) z  ( AC  BC )  0 .
(11)
According Young’s lemma (Young, 1971, pp.171-172)2, the necessary and sufficient
conditions for both roots being less than 1 in absolute value, which are equivalent to the
conditions for system stability, are AC  BC  1 and A  C  1  ( AC  BC ) .
Since A  B  1 and 0  C  1 , the first condition for stability, AC  BC  1, is
automatically satisfied regardless of what positive values these model parameters in C
take. The second condition, A  C  1  ( AC  BC ) , implies  (1  2C )  A  1 . Thus,
stability for both output and inflation under nominal income targeting prevails in this
open-economy model as long as the value of the combined coefficient A falls into this
range.
4.
Stability in the Open-Economy Model with Forward-looking Expectations
We now discuss the stability properties of nominal income targeting in the case of
forward-looking inflation expectations. To do so, the Phillips cure in the open-economy
1
See, Fomby, T.B., R.C. Hill, and S.R. Johnson (1984, p.536).
2
Young’s lemma states: For real equation x  sx  t  0 , the roots
2
t  1 and s  1  t .
6
x such that x  1 if and only if
model (Equation (2)) is revised by changing the backward-looking inflation expectations
to forward-looking inflation expectations. The revised Phillips curve is now written as
 t  Et 1 t 1  1 yt 1   2 et  t ,
1  0,  2  0
(12)
i.e., inflation in period t depends on expected inflation for period t  1 but expectations
are formed in period t  1. This specification is consistent with models discussed in
McCallum (1997) and Dennis (2001). In this case policy instrument, the Taylor-style
optimal interest rate, is derived from forward-looking expectations once the growth target
is set.
Again, substituting (3) into (1) and (12), imposing the growth target for nominal
income (6), and putting back the derived Taylor-style interest rate into the model, we
obtain the following two-equation system,
yt  Ayt 1  CEt 1 t 1  ut ,
(13)
 t  By t 1  CEt 1 t 1  wt ,
(14)
where combined coefficients A , B , and C are identical to those in the previous section.
Since the relevant state variables in the system of (13) and (14) are y t 1 , u t , and wt , we
may posit the model solution, based on the minimum-state-variable (MSV) approach
developed in McCallum (1983), as
yt  11 yt 1  12ut  13 wt ,
(15)
 t   21 yt 1   22ut   23 wt .
(16)
Equation (15) and (16) clearly show that stability for output and inflation depends only
on parameter 11 because if  1  11  1 holds then (15) implies output stability since u t
and wt are white-noise shocks, and if output is stable then (16) implies inflation stability.
7
Applying expectations operation to (16) and updating the result one period yields
Et 1 t 1   21 yt  11 21 yt 1  12 21ut  13 21wt . Substituting this into (13) and (14) and
equating them to (15) and (16) respectfully, we obtain
11 yt 1  12ut  13 wt  ( A  C11 21 ) yt 1  (C12 21  1)ut  C13 21wt ,
(17)
 21 yt 1   22ut   23 wt  ( B  C11 21 ) yt 1  C12 21ut  (C13 21  1)wt .
(18)
Equations (17) and (18) imply the undermined-coefficient requirements for the system:
11  A  C1121 ,
12  (C1221  1) ,
13  C13 21 ,
21  B  C1121 ,
22  C1221 ,
 23  C13 21  1 .
Since u t and wt the white-noise disturbances, their coefficients do not affect stability and
thus need not to be considered. Using 11  A  C1121 and 21  B  C1121 , we can
solve for 11 , the key parameter responsible for model stability,
11 
(1  C )  (1  C ) 2  4 AC
.
2C
(19)
Equation (19) provides an example of non-uniqueness in the rational expectations
models. For a unique solution, we follow McCallum (1997) and Dennis (2001) to choose
the negative term,  (1  C ) 2  4 AC , based upon the MSV criteria developed in
McCallum (1983, pp.146-7). The relevant condition for stability therefore requires
1 
(1  C )  (1  C ) 2  4 AC
 1.
2C
(20)
Upon simple re-arrangement of (20), it can be verified that for any C such that
0  C  1 , (20) will hold if  (1  2C )  A  1 , i.e., stability prevails and the stability
conditions are exactly the same as those for the previous case.
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5.
Policy Implications
The analysis presented in the last two sections indicate that, subject to certain conditions,
stability for output and inflation prevails in this open-economy model under a policy of
nominal income targeting. A careful inspection of these conditions provides some
interesting results.
Firstly, in the closed-economy case discussed in Ball (1999a), McCallum (1997),
and Dennis (2001) stability or instability depends solely on the supply-side parameters,
particularly on the slope of Phillips curve. In our open-economy case, however, stability
relies on both supply-side and demand-side parameters because stability requires
 (1  2C )  A  1 while A contains both demand-side and supply-side parameters.
Secondly, it is interesting to note that the model parameters in C are not directly
responsible for stability since C is always less than 1 but greater than 0. By examining
combined coefficient A and the stability conditions, we find that the model parameters
that are directly responsible for stability are  1 and  1 , which determine, respectively, the
effect of y t 1 on yt in the demand side and the effect of y t 1 on  t in the supply side,
because in the case of 0  1  1 and 0  1  1 stability for output and inflation is fully
guaranteed3 no matter what the positive values all other model parameters take.
Thirdly, since 0  C  1 implies  (1  2C )  3 , as the slope of the Phillips
curve,  1 , becomes larger, the risk of instability increases for the given values of other
3
1  2 1  (1  1 )( 2   3 1 )
, so 0  1  1 and 0  1  1 together
 2   3 1   2 1
0  A  1, but 0  A  1 is included in  (1  2C )  A  1 as required since C is positive.
Note that, since
guarantee
A
9
model parameters in A . Once  1 is large enough to cause A  3 , stability collapses
under the nominal income targeting. The important policy implication is that the flatter
the Phillips curve, the safer the policy of nominal income targeting in terms of stability.
5.
Concluding Remarks
In this paper, we have discussed the dynamic stability properties of the nominal income
targeting rule in an open economy when both the stylized fact that the interest rate affects
output before inflation and the formations of inflation expectations are taken into
account. Some interesting results have been found and reported in the previous section.
Although in theory monetary policy of nominal income targeting has attractive
properties in terms of delivering better economic performance as measured by stability of
output and inflation, this policy rule is indeed difficult to implement. For policy success,
it is required that the monetary authority must know all these economic relations and
behaviors correctly and must have accurate measures of macroeconomic variables, such
as the output gap, in order to choose a right interest rate measure. If mistakes are made, a
large efficiency loss in terms of more volatility of output and inflation may follow. These
difficulties explain, to an extent, why no central banks have so far adopted monetary
policy of nominal income targeting. However, this should not be used as the only factor
to justify whether this policy rule is better or worse than others.
The results of this study, therefore, suggest that while neither an absolutely
positive nor absolutely negative judgment should be given to the policy rule of nominal
10
income targeting, a comprehensive study on economic relations and behaviors is required
if implementation of such a monetary policy rule is in consideration.
11
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