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Monetary reaction functions in a small open economy
Journal of Economic Dynamics and Control 9 (1985) l-24. North-Holland
MONETARY
REACIION
FUNCTIONS ECONOMY
IN A SMALL, OPEN
Masanao AOKI* university
of cali/ornia.
hs
Angeles,
CA 90024,
UsA
Received May 1983, final version received March 1985 Monetary reaction functions are constructed in an intertemporal optimization framework. The time-varying coefficients to be put on the variations of the exchange rate and the asset stock are then derived by solving a matrix Riccati equation. As the planning horizon extends to infinity, a related algebraic Biccati equation coincides with those derived from minimization of stationary state variances only under certain conditions. Under the assumed full information by the government, reaction functions using realized values of exogenous shocks result in dynamics that do not enjoy the same stability property as the ones obtained by explicit inter-temporal optimization.
1. Introduction Rules for changing the money stock in conducting exchange market intervention, i.e., choices of reaction functions, have received much attention in the literature. See Turnovsky (1983), Canzoneri (1982), Henderson (1980), or Boyer (1978) for example. Roughly speaking, three distinct choices must be made in designing a reaction function. One is the set of variables that are the arguments of the reaction function. These variables may be endogenous, such as the exchange rate, or exogenous if policy makers do not suffer from information deficiency and can observe or deduce without error from available information the realized values of exogenous shocks. The second is the choice of target variables, i.e., the variables that enter the criterion. Typically, the criterion is optimized with respect to weights assigned to the arguments of the reaction function. The third is the form of the criterion itself. A typical criterion might be expressed in the form of p steady state variance, a (discounted) time average of a variance of the target variables, or a weighted sum of the variances of two (or more) variables, and so forth. The existing literature on this general topic can be improved in at least three ways. One has to do with the choice of the criterion. Instead of minimizing the variance of some economic variable at a point in time or its steady state value, a more explicit, dynamic (i.e., intertemporal) criterion and analysis of monetary reaction functions are desirable. This is because stochastic disturbances *I thank the anonymous referees of this journal for their useful comments. I am grateful to one of them, in particular, who made many constructive suggestions that lead to this improved version. 01651889/8S/S3.3001985.
Elsevier Science Publishers B.V. (North-Holland)
2
U. Aoki,
Monetary
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functions
typically are not limited to one-time shocks, and we should, therefore, treat them explicitly as ongoing disturbances, i.e., as stochastic processes. Second, in any study one should show the relation if any between the steady-state and dynamic component of the reaction function coefficients. Third, since no continuous-time, stochastic formulation of models in this area seems to be available in the literature, an analysis using this framework is needed. This paper attempts to advance the state of knowledge in these three ways. As in Henderson (1980) we keep our focus on the effects of a trade surplus on the adjustment time path of the economy, but in this paper we examine not just a one-time shock but rather the effects of on-going disturbances by using a continuous-time, linear, stochastic differential equation model under rational expectations for expected exchange rate changes. As in Boyer (1978), we abstract from ‘the issue of limited availability of information, and in section 5 of this paper assume that the government knows the realized values of disturbances to any variables it wishes to use in the reaction function. Boyer’s analysis, however, is essentially static because his criterion function is time-free. We compare the reaction functions resulting from optimization of four related objective functions: (i) the variance of a single endogenous variable or a weighted sum of two such variances, (ii) the time average of such variance expressions, corrected for a terminal time contribution, (iii) the limit of the criterion in (ii) as the time interval over which the average is taken goes to infinity, and (iv) the discounted time integral of some variance expression. We solve Riccati equations to express dynamic reaction function’s coefficients as functions of the planning horizon. Although known in the systems literature, Riccati equations have not been used before to examine how fast the reaction coefficients in class (ii) approach steady state values, nor the question has been answered whether the steady state values in class (iii) are the same as the constant reaction coefficients obtained for class (i). Indeed, one might expect that steady state criterion and dynamic criterion, i.e., criteria (i) and (ii) define, in effect, the same reaction function’s coefficients as the relevant time horizon goes to infinity. This is shown to be true only under certain technical conditions discussed in section 4. The solution of an algebraic Riccati equation becomes non-zero when the condition is not met, This implies that a non-zero correction term to the reaction coefficients of class (i) must be added to obtain the coefficients for class (iii). This has not been reported in the literature before. In addition, when the dynamic reaction function asymptotically approaches that derived in (i), we examine how fast this dynamic correction term goes to zero as the length of the planning horizon goes to infinity. Our method is quite general and can be used to draw some policy implications about how the weights assigned to the exchange rate variations and changes in the asset holdings vary as the length of the planning horizon of the government changes. In the simple model used in this paper, dynamic corrections to the reaction coefficients on variations of the exchange rate and of the holdings of foreign money stock.approach zero at the same rate of the inverse of the length of the
M. Aoki,
Monetary
reaction
functions
3
planning horizon, while variation of the holdings of the foreign money stock can be more heavily weighted than that of the exchange rate near the beginning of the planning horizon. In view of the conditional stability of the dynamics of rational expectations models, i.e., since the equilibrium exhibits a saddle-point property, we also examine the implications of adopting reaction rules tied to exogenous shocks for dynamic system stability. To conduct this comparison in a simple framework, we use a small open-economy model which is described in section 2. We show that reaction functions composed from the state variables of the economy tends to improve the stability property of the model, while those constructed using realized values of the exogenous shocks to the economy do not affect the stability at all. In the model, we focus on two sources of dynamics: One is due to rational expectations, and the other is due to accumulation of foreign currency by domestic residents through trade. We use primarily the variance of the disposable income as the argument of the various criterion functions. In addition, we also discuss the implications of using a weighted sum of the variances of the disposable income and the domestic output price. Section 3 considers a stationary-state reaction function. Dynamic reaction functions for classes (ii) through (iv) are discussed in section 4. We then return in section 5 to a stationary reaction function on the conditionally stable manifold that reflects reactions to combinations of exogenous shocks. The paper concludes with some final ,remarks in section 6. 2. Model
Our intention is to examine the effects of several alternative reaction functions on the dynamic behavior of a small open economy near a stationary equilibrium as the economy is disturbed by monetary and real shocks. These shocks are modeled as mean-zero, covariance-stationary, stochastic processes. We do not include securities in the model so as to provide as simple a benchmark analysis as possible. While maintaining our main focus on the effects of trade surplus on the path of economy as in Henderson (1980), this paper deals with the effects of ongoing disturbances rather than one-time shocks. We use a slightly modified, flexible price version of the small open-economy model recently used by Obstfeld and Rogoff (1984).’ Our model is specified in ‘In Obstfeld and RogotT (1984), there is an equation for aggregate demand yd - -+ra( r - 0) + yy + U, where T -p - e is the terms of trade, and u is a shock to foreign demand for domestic output, and an equation for real supply y’ which is assumed to be zero. Noting that the real interest rate equals -a+. the good market clearing condition can be used to generate the adjustment equation for the terms of trade, i - (l#./ua) 7 - o/m. In the absence of foreign shock, u - 0 (and no domestic or foreign supply shock), the terms of trade remains at the value of the initial equilibrium value which is taken to be zero. Then the model dynamics are two-dimensional as described in the main body of the paper.
4
M. Aoki,
Monetary
reaction
functions
deviational log-linear form as follows: (money demand),
(1)
(CPI index),
(2)
T=p-e-p*
(terms of trade),
r=r*+r?+Tj
(uncovered interest parity),
(3) (4)
yL
(aggregate demand),
(5)
(current account),
(6)
(good market equilibrium),
(7)
m-q=
-lr+J/y+[
where q=ap+(l-cr)e=e+ar and
f=
--cpT- +-4)+YY Kg
-
Klf+
K2r
+
“,
Yd =Y
where we set y of home country, r* and p* of the center country variables to zero. Appendix 1 summarizes the steps involved in these derivations. In later sections, we discuss consequences of replacing the demand for real money balances by a portfolio balance equation,* m-e-f=
-lr+(.
0’)
This small open economy produces a single composite good. Here p is the logarithm of the money price of domestically produced good, P, measured from its initial equilibrium level, i.e., p equals In(P/P,). Throughout this paper subscript 0 denotes the value of a variable prevailing in the initial equilibrium; e is the logarithm of the exchange rate, i.e., ln( E/E,) where E is the domestic currency price of foreign currency; r is the implicit domestic nominal interest rate given by the uncovered interest parity relation; f is the stock of foreign money held by the private sector of this small open economy, ‘Use of (1’) rather than (1) modifies the exchange rate dynamics to be (recalling that we assume that y is zero) The
dynamic
6-(e+f-nt)/l+[/I-7j. matrix in
(6’)
(9) changes into
@jK$
!;I.
However, the matrix Q,= @ - bw: still has the same form given by (15), i.e., @-bwl-
[
;
’ -0
1
where
o = x2 +
K,/I
>
0.
For this reason, the analysis of this paper remains basically the same for this respecified model as well.
hf. Aoki,
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reaction
functions
5
i.e., f stands for ln(F/F,). The remaining variables are exogenous: The world price is normalized to be one and we set p* =p$ = 0; M is the logarithm of the nominal money supply measured from the initial equilibrium level, i.e., ln(M/M,), where M is the nominal money stock; (Y is the share of home goods in the domestic consumer price index; r* is the nominal interest rate in the world; [ is a shock to the portfolio; t is a shock to the world interest rate; and { is a shock to the domestic import demand which is a part of the trade (current account) equation (6), i.e., the disturbance term in (6) is made up of a linear combination of all three stochastic processes. See (9). The uncovered interest parity relation yields the differential equation for e when r is substituted out from the relations (1) and (4), i.e., ct=(-m+e+()/l-77,
(8)
where r defined in (3) has been set to zero. Since, in the absence of real demand or supply shocks, the terms of trade remains as its initial equilibrium value, i.e., r is identically zero. Foreigners are assumed not to hold domestic money. Eq. (6) is already in a form of adjustment for f. A third dynamics exists for the terms of trade that comes out of the good market clearing condition. In this paper we restrict our attention to the two-dimensional submodel that describes the adjustment time paths for the exchange rate and f, keeping r at ze!o. 3. Adjustment
dynamics
We write (6) and (8) jointly after setting r to zero to record the adjustment dynamics for future reference: dz=(@z-bm)dt+dn,
(9)
where the state vector z has e and variables, where
@= [ ;;,
-oKl],
f as the components, i.e., as the state
b=+[;2],
z= [;I,
and the noise n is a mean-zero, covariance stationary stochastic process with components
See appendix 1 for definitions.
M. Aoki,
6
Monetary
reaction
functions
Because ~2 and K~ are both positive, one eigenvahte of the dynamic matrix is stable, and the other unstable, corresponding to the fact that f is predetermined but e is not. Accordingly, the origin (e = 0 and f= 0), i.e., the equilibrium point of the economy is a saddle point. We regard all shocks to be independent mean-zero, finite variance, white noise processes, so that the noise n in (9) is also a white noise process with VariallCe
02= n
iJ 1 1 ee
ad
a4
aff
’
where
Here, at etc., denotes the variances of the elementary noise stochastic process 5, etc. Eq. (9) also implies that z is uncorrelated with the contemporaneous value of the noise process n. 4. Reaction functions 4.1. Stationary reaction functions composed of state variables Eq. (9) governs the time evolution of e and f given a realization of the noise n and a time path of m. We now examine the implications of adopting a policy reaction function of the form m, = wee + u,f = o’z I’
(10)
where w’ = (w,, o,), on the adjustment behavior of the economy. Later in section 5 we contrast this class of reaction functions which are made up of linear combinations of the (state) variables with an alternative class of reaction functions constructed from the realized values of the disturbances such as m, = orn c + w2nJ, or when m is made up as a weighted sum of the realized values of the elementary noise processes such as 5 or 7 themselves. The reaction function defined in (10) mod&es the dynamics shown in (9) into dz = (@ - bo’)zdt
+ dn.
01)
M, Aoki,
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7
As a concrete example and as a vehicle of demonstration, we consider home real disposable income, denoted by h, as a target variable. The disposable income, in its log-linear form, can be given by h = -h,e-(h,+ah,)i,
where h, through h, are some positive constants. (Appendix 1 gives the detail of this expression.) Since we assume no exogenous real demand or supply shock, the terms of trade remains constant, hence i is zero. We substitute C out using (8) to express the target variable as h,= --71e + rrn - hlne= p’z + rrn - hlne,
(12)
where p= ~(10)’
and
q=(h,)/l>O.
When m specitied by (10) is substituted in h, the real disposable income is related to e and f by h, = (p + ITW)‘Z~- hln,.
Since the vari&ce subscript 1)
03)
of the real disposable income is expressible as (dropping
var(h)=E[z’(p+no)(p+ao)‘z]
-t-hfu,,,
because ne is uncorrelated with the contemporaneous w=w”=
-p/s=(lO)
z, the choice of w by (14)
produces var( h) = hfu,,. Two things should be noted about this reaction function’s coefficients. First, they are independent of exogenous noise statistics. The second noteworthy fact about (14) is that this reaction function does not stabilize the dynamics given in (9), i.e., the use of this reaction coefficient in (10) does not move the unstable eigenvalue of the dynamic matrix to the stable region of eigenvalues, because the reaction coefficients specified by (14) transforms the dynamic matrix into @-bw;=
10 01 0
-I$
’
which leaves one eigenvalue at zero.
8
M. Aoki,
Monetary
reaction
functions
In effect we note from (11) that the use of the reaction coefficients given by (14) render the exchange rate a white noise process, de = dn.3 Dynamic reaction functions we consider in section 4.2 stabilize the dynamics as we soon see. To provide better control of the exchange rate adjustment behavior, one may wish to switch to an objective function such as the time average T- ‘IarE h f dt or something similar; in section 4.2 below we shall turn to criteria (ii)mentioned in the introduction. 4.2. Dynamic reaction function
To ensure that our subsequent manipulations take an objective function of class (ii),
are mathematically
valid, we
06)
for some finite T and some symmetric, positive semi-definite matrix S subject to the dynamics (9). [Later we divide V, by T and let T go to infinity to consider criteria of class (iii).] The second term depending on the state vector at T may be thought of as a proxy to represent the contribution from the integral from T to inlkity, or the expected value of some other time integral. Consider the reaction function of the form (lo), but now allow w to change with time. Because of the linearity of the dynamics (9), and the quadratic nature of the objective function, it is known that V, of the form z;r,z, + y, solves the stochastic dynamic progr armning functional equation (also known as Bellman or Hamilton-Jacobi equation). [See Davis (1977, ch. 5).] Heuristically, letting A be a small positive time increment the functional equation becomes G-z, + Y, = mir@dh,)A
+ z:+J,+,,z,+r,
+ Y,+A] 1
where from (12) var(h,)=z;Z,z,+o,,
z, = pp’+ 7r(o,pr+ pw;)+ w,w;, a,, = h&,. From (11) z ,+,, equals z, + (@ - bw:)z,A + An, where An denotes the incre‘This corresponds to the discrete-time random walk model of the exchange rate such as the one discussed in Nelson and Plosser (1982).
M. Aoki,
Monetary
reaction
functions
ments (i.e., the change) in n over this small interval A. assumption on the exogenous elementary noises we note mean zero, and the variance equals u,‘d. Carrying out respect to w,, and dividing both sides by A and letting derive the optimal reaction coefficient vector
9
From our white noise that the increment has the minimization with A approach zero, we
u,=(--p+r,k)/lr=w”+r,k/a,
(17)
where k = b/a, and the matrix r, and the term y, which are independent of z, are determined as the respective solutions of - dr/dt
= lY + ‘P’r - TKr,
K = kk’,
‘P A II’ + kp’,
I-(T) = s,
08)
where
and -d~Jdt=a~+tr(@~),
yT= 0.
(19)
Appendix 2 contains the details of this derivation. Compared with (14), the reaction function’s coefficients consist of two parts, the first being the same as (14). Since we show later that r, approaches zero as T approaches infinity, the first term in (17) represents steady state coefficients, and the second dynamic adjustment or dynamic correction part. The differential equation for r is known as the Riccati equation. Note the absence of a constant term on the right-hand side of the Riccati equation. This accounts for the fact that a steady state solution is zero. This last observation is important when we compare reaction functions for class (i) and (iii). We return to this point later. We discuss the solution of the Riccati equation in appendix 3 and proceed here on the basis that we have its solution. In appendix 3, we also establish that the leading terms of the dynamic correction expression in r,k = (&,, a,,) of (17) approaches zero as T becomes large. More spedifically we obtain
4, where
= S,h,/(T-
t)
and
G?*,= -c/(
T- t)erl(T-‘),
10
For
hf. Aoki,
T - t
Monetary
reaction
funclions
near zero,
4,
and
=;I S,/h,
a,, =
S&hl.
From (14) and (17), we note that the weight on the exchange rate is initially while that on the holding of foreign money is L?z,= 1+ 4, = 1 f S,/h,, S2fc2/hl. The elast ici t y K~ can be greater than one if the Wedth elasticity of demands for imports is large.4 As T - t becomes larger, both dynamic correction terms approach zero at least at the rate l/(T - t). It is interesting to observe that as T gets large, the reaction coefficient for f becomes negative and vanishes faster than 9,,. Thus, for smalI values of T - t, the dynamic correction conceivably puts more weight on the holdings of foreign exchange while for large values of T - t, its weight is very small and negative. We also observe that the dynamics are stable because the matrix
has stable eigenvalues for any finite T. By dividing V, by T - t, we can calculate an objective function of class (iii). For this purpose, we need an expression for y,/( T - t) where y, is obtained by integrating (19). Its time average may be approximated by
where r, is the solution of the algebraic Riccati equation which is obtained by setting the right-hand side to zero in (18), o=lv+Pr-rKr, and the optimal reaction function is approximated by m,
= w;z,
= ( - p + r,k)‘z,/a.
4The inequality x2 > h, holds if and only if (QX/EF)oh ’ (A/QWo[l - (QX/WoM]~ where j/= (6’lnJ/~YlnJT)~ and jz= [L3InJ/aln(A/Q)],. By definition 1(2 = KI + KY= (QX/WO{ I(M/A)oh +A(M/QH)o/I}. Hence
~2’1 -i2+il(A/Q%/~2> (EF/QX)O(A/W/~~ where j1 is the disposable income elasticity of demand for imports and j2 is the wealth elasticity of demand for imports. If the holding of foreign money is smaller than the receipts from exports, QX, and if A/M is not too small, this inequality can conceivably be satisfied.
M. Aoki, Monetay reaction functions
11
This differs from (14) if and only if I’,k is not zero. In appendix 3 we show that this expression happens to be zero for the criterion function in (16), and the limit of V,/( T - t ) as T approaches in6nity is equal to IJ~.The value of this criterion function is the same as the one associated with the reaction function coefficient vector w, given in (14). It is important to emphasize that this asymptotic coincidence of the reaction functions, i.e., the fact that r, approaches zero as T becomes large, is the consequence of using h: as an integrand in the criterion function of class (ii). Restricting the functional form of the integrand and the terminal cost expression to be quadratic, other choices of the integrand, if they produce constant terms in the Riccati equations corresponding to (18), then cause their r, to be non-zero. For example, if we use a weighted sum of the disposable income and the exchange rate, such as E(hf + /.tef) for some positive constant CL,then I” is a positive semi-definite solution of the algebraic Riccati equation
o=~ [ t 8I +r*+w-rm, which is not zero. This fact is established in appendix 4. The dynamic components of the reaction coefficients then become
r,k/n+
ffi01’
as T approaches inhnity. The reaction coefficient vector then is
This vector is different from w, which stiB minimizes the value of the criterion function at a point in time, i.e., the variance at a point in time
Next, consider a class (iv) objective function such as V, = 1-e 0
-~lE(h~+~e~)ds
where
PLO.
It is slightly more convenient to use the current-value cost function, replacing V, by W, = V,e@ [see Davis (1977, p. 191)]. Then, the Belhnan equation has the solution IV= z’Gz, where G is symmetric and positive semi-definite, and
hf. Aoki.
12
Monetary
reaction
functions
satisfied the algebraic Riccati equation
Hence, V, - z’Gze-8’ --) 0 as t approaches infinity. Accordingly, the optimal reaction function becomes m,=(-p+Gk)‘t,/s.
(22)
This policy reaction function emphasizes the near future, while the averagevariance policy puts relatively more weight on the long run. Because (21) is quadratic, it can be solved explicitly to ‘elds the reaction function’s coefficients that turn out to be (- /3 + ep2 + 4~ )/2, and 0. Thus the reaction coefficient on the exchange rate is strictly larger than that in (20). The average cost criterion neglects the behavior of the system over any initial interval. Discounted criteria of class (iv) have the opposite effects of emphasizing the initial performance. 5. Stationary
reaction
functions
We now relate the dynamic reaction functions of the previous subsection to those more familiar in the economic literature, i.e., reaction functions of the type m, = 0~6, + u2q, + w&
or
m, = w3n, + w,n,,
(23)
and briefly discuss the consequences of assuming that the government knows the realized values of all exogenous shocks. We conform to the well-established custom in the literature of confining attentions to solutions of (9) on the conditionally stable manifold in this section. We can write the conditionally stable solution of the exchange rate [see Bellman (1953, p. 89)] as e,= (l/l)~me(f-s)/t[m,-ln,ol
ds.
(24
Once the stochastic process (m,} is specified by (23), say, then (23) and (24) jointly with the exogenous noise stochastic processes determine the stochastic process for the exchange rate, or real disposable income. We put a hat to denote the unanticipated portion of the variable. From (12) we can write the unanticipated real disposable income as h,=n(B,+riz,-lit,).
(25)
hf. Aoki,
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reaction
functions
13
As is well known, any anticipated future change in the time paths associated with m and n produce immediate responses in the exchange rate and in the holdings of foreign money. This fact is denoted by putting bars over m and n, indicating the anticipated parts of these processes. Recall that we regard {m,} and { n, } as mean-zero, weakly stationary, finite variance, white-noise processes. Therefore, E, and is are zero for s SEt. We can, thus, regard (24) as the expression for the unexpected portion of the exchange rate in response to the unexpected changes in m and ns, hence will drop the hats from now on. Note that the variance of the exchange rate is given by var( e,) = ( l//)2l”
dslm ds e(2f-s-u)/’
by the covariance stationarity of the stochastic process m - In,. The variance of the real disposable income is proportional to that of m - In,, hence the reaction coefficients o,=l
and w,=l . reduce the variance of the disposable income to zero. An important fact to observe is that the reaction function’s coefficients are independent of the noise covariances. This is not so if we use the reaction function m, = wnf. Then var(m, - lfl,) = ~JJ”u,,- 2&,,+
12aep.
Hence the value of w given by lu,//u,, minimizes the variance. To set the coefficient to this value we need the ratio of Qu,,. Using only n/, the knowledge of the covariance uc/ helps to recover the part of n, from nj. A simplifying fact about (23) and (24) is that both stochastic processes {e} and { h ) depend on the same combination of the basic stochastic process in the form of m - In,. Hence the same reaction function can simultaneously null the variances of e and h to zero, i.e., (in the absence of real shocks as we have assumed) a fixed exchange rate regime produces zero variation of the real disposable income. This fact is not robust, however. When the asset sector equation (1) is replaced by (l’), then the dynamic matrix 0 in (9) changes into
where a stands for aI - uZ/l. Let X denote the unstable eigenvalue of this
M. Aoki,
14
modified matrix. e,=
K,
Then /
I
Monetary
reaction
functions
(24) must be changed into
meA(r-u)Nl( u) du,
where K, = I/[/@
+ u)],
and
while the stochastic process in (25) becomes h,=r(-e,+N,(t))
where
N,(r)=m,-In,.
Therefore, unless [ ~Jl]m( U) - n,(u) is chosen to be either zero or proportional to m - In,, Nr and N, are two independent stochastic processes with different variances. The reaction coefficients that minimizes the variance of the real disposable income are given as the solution of
where a-'=(K;/2x)(h+
Kl),
u,,+(h
+ K,)-‘UC,,
as and
CO‘,=(A+
K1)-‘.
Note, again that w3 and w, are functions of the model parameters only and not of noise statistics, because the realization of all exogenous noise were assumed known. The class of reaction functions of this section does not modify the dynamics of the economy. This type of reaction functions will become unsatisfactory when stocks are not weakly stationary but contain some deterministic components. To dampen this, the dynamic matrix should be stable as in section 4.2.
M. Aoki,
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15
6. Concluding remarks
Using a simple model of a small open economy to minimize deviation of real disposable income from its stationary equilibrium value, we have compared reaction functions obtained by explicit intertemporal optimization over a finite horizon with their limiting expressions over an infinite horizon or with timeless reaction functions dealing with steady state values. The reaction function’s coefficients obtained by dynamic optimization over a finite horizon are shown to converge to their stationary counterpart only when the associated algebraic Riccati equation does not contain a constant term. When constant terms are present in the algebraic Riccati equations, non-zero correction terms need be added to the stationary coefficients in general. By means of an example with a quadratic function of real disposable income and the exchange rate, we indicate that non-zero constant terms will be present in the Riccati equations if the arguments of the criterion functions of the intertemporal optimizations contain more than one independent linear combinations of the exogenous noise stochastic processes. By examining a variety of criterion functions, we have illustrated that initial phase of adjustments or long-run aspects of adjustments can alternatively be emphasized with obvious implications for choices of planning horizon for the government. The example of this paper illustrates a possibility that deviation of foreign financial asset may appear in the reaction function with any significance only towards the end of the planning horizon for a government with a finite but a long planning horizon, or with a short planning horizon. Finally, the paper points out the fundamental differences in the stability of the economy employing reaction functions made up of the state variables and of realized values of the exogenous noises. When the government utilizes the latter type of reaction functions, the government is essentially cancelling out the exogenous noises while leaving the economy unaltered. With the use of the former type of reaction functions, the government actually improves stability of the economy. This distinction will become important if the model leaves out some sources of dynamics or the exogenous noises that tend to destabilize the economy, or if the government can not observe or exactly reconstruct the realized values of exogenous noises.
Appendix 1: Derivation of the log-linear model
We follow the procedure detailed in Aoki (1981). Eqs. (1) through (4) are standard, except possibly for the noise n introduced into (4). This noise may be regarded as a foreign interest rate shock, or a proxy for risk premium. The exchange rate is denoted by E, which is the domestic currency price of a unit of foreign exchange. Eq. (1) comes from demand for
M. Aoki,
16
Moueray
reacriotr
fwrcriotu
real money balances M/Q = L(r, Y), where the nominal stock of domestic money is deflated by the consumer price index, Q, which positively depends on the price of the domestic good, P, and on that of the foreign good, EP*. In this paper P* is exogenous and is set to one. The parameter (Y in (2) is the elasticity of Q with respect P evaluated at the initial equilibrium state and denotes the share of domestic goods in CPI. We denote values of variables in the initial equilibrium state by subscript 0. Demand for domestic goods consists of two components, domestic and foreign demand. The foreign component which is the exports from home, X, is taken to depend only on the relative price, P/E, while the domestic component is specified to be a function of real saving, the level of real economic activity which is measured by Y, and relative price P/E. Real saving is a function of disposable income, H, and real wealth, (M + EF)/Q, where F is the stock of foreign money held by the private sector of this economy. Therefore, expenditure on domestic goods by domestic residents is specified to be given by
where A stands for the total financial asset, h4 + EF. Real imports are specified by J=J(H,Y,
P/E, A/Q).
(A.1)
The current account equation is given by P= (Q/E>(x-J).
(A.21
The variables D and X are related to Y by Y = D + X by definition. In a stationary equilibrium i/E and Q/Q are both zero. Real disposable income is given by
H= Y+(~/E)(EF/Q)-T,-(A/Q)(~/Q),
(A.3)
where TX is real taxes. By assumption, both Y and r, stay at their respective equilibrium levels. Thus, to the first order of smallness, the domestic component of the aggregate demand d = ln( D/D,) is expressible as d=d,h-dp+d3(a-q),
(A.4)
where a = ln( A/A,), d, = (aln D/aln H),, d, = -(a In D/aln d,=(alnD/alnA),. From (A.3), the disposable income h = ln( H/H,) is given by
A= -u+w-Q)=
-ll,e-(h,+ah*)i,
P/E),,
and
(A-5)
M. Aoki.
Motrerrm,
reaction
functions
17
where (2) has been used, and h,=(M/QH),
and
h,=(EF/QH),.
Substitute (AS) into (A.4) to derive d= -d,h,e-d,(h,+cllh,)i-d*T.
(‘4.6)
Here, we drop d,(a - q) as being small. The relation Y = D + X becomes
yd= (WY),d+(X/Y)x,
(A.71
in the log-linear variables where x stands for ln( X/X,). It equals -X’T where the symbol x’ is the relative price elasticity of demand for imports, i.e., x’ = - ( 8 In X/c? In P/E),. On the assumption that the ratio hi//( hi + cubz) is small and substituting (A.6) into (A.7), after * is also substituted out by (1) and (4) the log-linear expression for the aggregate demand becomes yd = -+T - uai, where the coefficients Cpand u are both positive. The goods market clearing condition implies that the terms of trade remains at zero, in the absence of real demand or supply shocks. Hence we can set the terms of trade variable to zero in our model. In particular, (A.5) simplifies to h= -/I,&=
-/I,(-m+e+[)/l-?J.
This is eq. (12) in the main body of the paper. To derive (6) we start from (A.2) and write it for the log-linear variable, f = ln( F/F,), as f=(QX/EF),(q+x--e-j), where the equality of (QX/EF), and (QJ/E)O has been used, and the definition of j as the log-linear variable of J. We recall that the terms of trade is zero, hence the expression, q + x - e, is zero. From (A.l), j=j,h+j,(a-q)+l’, where the disposable introduced we obtain
coefficients ji and j, are the elasticity of J with respect to the income and real wealth, respectively, and 5’ is a random shock term to real imports. Combining these two expressions with that in (AS), the adjustment equation for the current account, L4.8)
hf. A oki, Moneraty
18
reaction
functions
where Kl= (1 - e)(QX/EF)&, K; = [e/(1 - @I (x/fh.h~,/~> K;
(QX/Ef’),-&e.
=
Here, 8 is the ratio, (M/A),, of the stock of domestic money in the domestic wealth. After r is substituted out, the adjustment equation for j becomes
fmancial
(A-9) where the symbol K~ (A.9) is defined by
to
is equal
K; + K;I
by definition.
where { is (QX/EF),J’ by definition. When the asset sector is specSed by a portfolio than by (l), the interest rate is related to m by
The disturbance “/ in
balance equation (1’) rather
r=(-m+e+f+t)/l.
Substitute this into (A.7) to see that the trade balance equation remains the same as (A-9) if the parameter K~ is suitably redefined by K~ - K;J/I, still assumed positive to preserve the stability property of the total dynamics. Appendix 2: Derivation of reaction c&3cients From the adjustment dynamic equation (ll), = z, + (Qi - ho:)z,A
=,+A
The right-hand min 0,
+ An.
side of the dynamic progr amming functional equation becomes
[z;Z,z,A
+
u;A
+
E,(Z:+A~+A~,+A)
+yr+A]~
where E,(Z:+~~+~Z,+~)
=
z;l-,z, + z;T, + A ( @ - ho;)z,A +z;( @ - h~:)‘r,+Az,A
+ u( r,+ipnz)A.
M. Aoki,
Monemy
reaction
19
jimctiom
Therefore the functional equation becomes z + z: rt-Tt+A A ’
Yt -;r+A
= u; + tr( r,+fl”)
+min(z;[z,+T,+,(@-h;) @I
+ (@-
hw:)‘r,+A]
II)
Recalling that Z, depends on w,, the differentiation bracket yields, after A + 0, o, = ( - p + r,k)‘/r
where
-
(A-10)
of the expression in
(A-11)
k = b/lr,
after noting that T is continuous in t. Substituting (A.ll) into (A.lO), letting A + 0, equating terms independent of z, on the RHS with -dy/dt, those quadratic in z, with -z;(dT/dt)z,, and noting that z, is arbitrary, we can set - dy/dt - drjdt
= a, + tr r,a,,?, =
Ik=@+kp’
and
r* + 9r - rKr,
YT=
0,
r,= s,
where
K=kk’.
Appendix 3: Solution of the Riccati equation We derive the solution of the Riccati equation - dr/dt
=
r* + wr - rKr,
r,=s20,
(A.12)
where
K = kk’,
k = b/r.
Consider the solutions of the two related linear matrix differential equations f=
-9’x
>
x,=s
jl=‘ky-ICY,
yr=I.
Then, it is easy to verify that r is given by xy-‘, by direct differentiation. The technicaJ conditions to ensure the existence of the solution of (A-12) are
20
M. Aoki,
Monerq
reaction
functions
satisfied here since the inverse of y exists. See Aoki (1976, p. 164) or Bucy and Joseph (1968, p. 67) for example. The matrix ck of (18) happens to be symmetric, and is given by
[ 11
\k= ; -OK.
The solution matrix of the difTerential equation for x is given by x(f) = eY(T-r)S, where
err =
[
1 0
0 e-‘I’
1 *
Next, solve for the matrix y substituting the solution for x obtained above. It is given by yf=e w-n+
/I
re*.(‘-‘)Ke-
+(J- r)S&.
Denote the element of the matrix y, by Y,~(I). Carrying out the integral we derive
Ydd
= (‘%/KI)(~
-X-*)%/h:,
Yn(d
= (‘d’dS,(X
Yzz(d
= X + (K%‘Q)(X
- 1)/h:, - X-‘&/h:,
where x=e r,(T-0
and
S=diag(S,,S,).
The dynamic correction terms are
hf. Aoki.
Monetaty
reacfion&r&ms
For large (T - t) values, Y&)
IyI=I
= (T-
tT;:)sl
W,/h:,
[l
+(K~/2K,)&/h;]X.
1 ThUS
42*, = S,h,/( T- I)
and
S2bA)
52,, = -
(T-z)X
For T - I near zero,
Y120)
=
‘d#=-
II/h:
= 0,
Y2,tr)=K2S,(T-t)/h:50,
y22(1)
=
1 +(K1
+
S,K#)(T-
I)
=
1,
hence
and 52,, = S,/h,
and
52,, = S2KZ/hl.
In L$,, the term y&t) is dominant both for small T- t and large T- t. In L?,,, however, ~~~(1) is dominant for large T- t but yll(r) becomes dominant for small T - I, hence the change in the sign of L?,,.
M. Aoki,
22
Monetaty
reaction
functions
The algebraic Riccati equation that corresponds to (A.12) is given by where
O=A+r,\k+‘kr,-r,Kr,
Writing out the components of the matrix equations for the three elements of r,:
r,
A=diag(p,l). explicitly, we obtain three
0 = p -(u + K3b)2/(f7r)2,
0 = 2CK2
+ (b + K,C)“/(
h)2,
where
We select the solutions to yield a positive semi-definite r,, i.e., a = filq, b = c = 0. Therefore, the steady state expressions of the reaction coefficient r,k become (u + bK3)/(d) = fi and (b + CK3)/( d) = 0. Appendix 4: Alternative objective functions Suppose we replace h f in the objective function by another expression involving two or more different combinations of exogenous stochastic processes. For example, as an integrand, we take hf + pef for some positive value of p. This change generally produces a constant term in the Riccati equation causing r, not to vanish. First, observe that the expression
replaces E,hf in appendix 1. We deduce that the new reaction function coefficient vector is still of the same form, w,=
++r,ky.
The Riccati equation changes, however, into
- dr/dt = A + r\k + wr - ru,
(A.13)
M. Aoki,
Monetary
reaction
functions
23
where
Appendix 5: Solution of (21)
x = (st + wd/~~
and Y= (8, + wd/ln.
Then (21) can be restated for the components as
o=pg,+x*-p, 0 = (P + K&h + XYv 0 =
t P +
%)&
+Y2.
A positive semi-definite solution with g, > 0 and g, = g, = 0 is obtained as the positive root of the quadratic polynomial equation
i.e.,
This solution produces the reaction coefficients (- j3 + d-)1/2 and 0, respectively, as correction terms to wU in (15). Note that as p approaches 0, it produces the same correction term as class (ii). References Aoki, M., 1981, Dynamic analysis of open economies (Academic Press, New York). Aoki, M., 1976, Optimal control and system theory in dynamic economic analysis (North-Holland, Amsterdam). Bellman, R.E.. 1953. Stability theory of differential equations (McGraw-Hill, New York). Boyer, R.S., 1978. Optimal foreign exchange market intervention, Journal of Political Economy 86, 1045-1055. Bucy. R.S. and P.D. Joseph, 1968, Filtering for stochastic processes with application to guidance (Wiley, New York). Canzoneri, M.B., 1982, Exchange intervention policy in a multiple country world, Journal of International Economics 13,267-289.
24
M. Aoki,
Monetary
reaction
functions
Davis, M.H.A., 1977, Linear estimation and stochastic control (Chapman and Hall. London). Henderson, D.W., 1980, The dynamic effects of exchange market intervention policy: Two extreme views and a synthesis, Kredit und Kapital6, Suppi. Nelson, C.R. and C.I. Plosser, 1982, Trends and random walks in macroeconomic time series, Journal of Monetary Economics 10, 139-162. Obstfeld, M. and K. Rogoff, 1984, Exchange rate dynamics with sluggish prices under alternative price-adiustment rules, International Economic Review 25. 159-174. T&ovsky,-S.J., 1983, Exchange market intervention policies in a small open economy, in: J.S. Bhandari and B.H. Putnam, eds., Economic interdependence and flexible exchange rates (MIT Press, Cambridge, MA).
MONETARY
REACIION
FUNCTIONS ECONOMY
IN A SMALL, OPEN
Masanao AOKI* university
of cali/ornia.
hs
Angeles,
CA 90024,
UsA
Received May 1983, final version received March 1985 Monetary reaction functions are constructed in an intertemporal optimization framework. The time-varying coefficients to be put on the variations of the exchange rate and the asset stock are then derived by solving a matrix Riccati equation. As the planning horizon extends to infinity, a related algebraic Biccati equation coincides with those derived from minimization of stationary state variances only under certain conditions. Under the assumed full information by the government, reaction functions using realized values of exogenous shocks result in dynamics that do not enjoy the same stability property as the ones obtained by explicit inter-temporal optimization.
1. Introduction Rules for changing the money stock in conducting exchange market intervention, i.e., choices of reaction functions, have received much attention in the literature. See Turnovsky (1983), Canzoneri (1982), Henderson (1980), or Boyer (1978) for example. Roughly speaking, three distinct choices must be made in designing a reaction function. One is the set of variables that are the arguments of the reaction function. These variables may be endogenous, such as the exchange rate, or exogenous if policy makers do not suffer from information deficiency and can observe or deduce without error from available information the realized values of exogenous shocks. The second is the choice of target variables, i.e., the variables that enter the criterion. Typically, the criterion is optimized with respect to weights assigned to the arguments of the reaction function. The third is the form of the criterion itself. A typical criterion might be expressed in the form of p steady state variance, a (discounted) time average of a variance of the target variables, or a weighted sum of the variances of two (or more) variables, and so forth. The existing literature on this general topic can be improved in at least three ways. One has to do with the choice of the criterion. Instead of minimizing the variance of some economic variable at a point in time or its steady state value, a more explicit, dynamic (i.e., intertemporal) criterion and analysis of monetary reaction functions are desirable. This is because stochastic disturbances *I thank the anonymous referees of this journal for their useful comments. I am grateful to one of them, in particular, who made many constructive suggestions that lead to this improved version. 01651889/8S/S3.3001985.
Elsevier Science Publishers B.V. (North-Holland)
2
U. Aoki,
Monetary
reuction
functions
typically are not limited to one-time shocks, and we should, therefore, treat them explicitly as ongoing disturbances, i.e., as stochastic processes. Second, in any study one should show the relation if any between the steady-state and dynamic component of the reaction function coefficients. Third, since no continuous-time, stochastic formulation of models in this area seems to be available in the literature, an analysis using this framework is needed. This paper attempts to advance the state of knowledge in these three ways. As in Henderson (1980) we keep our focus on the effects of a trade surplus on the adjustment time path of the economy, but in this paper we examine not just a one-time shock but rather the effects of on-going disturbances by using a continuous-time, linear, stochastic differential equation model under rational expectations for expected exchange rate changes. As in Boyer (1978), we abstract from ‘the issue of limited availability of information, and in section 5 of this paper assume that the government knows the realized values of disturbances to any variables it wishes to use in the reaction function. Boyer’s analysis, however, is essentially static because his criterion function is time-free. We compare the reaction functions resulting from optimization of four related objective functions: (i) the variance of a single endogenous variable or a weighted sum of two such variances, (ii) the time average of such variance expressions, corrected for a terminal time contribution, (iii) the limit of the criterion in (ii) as the time interval over which the average is taken goes to infinity, and (iv) the discounted time integral of some variance expression. We solve Riccati equations to express dynamic reaction function’s coefficients as functions of the planning horizon. Although known in the systems literature, Riccati equations have not been used before to examine how fast the reaction coefficients in class (ii) approach steady state values, nor the question has been answered whether the steady state values in class (iii) are the same as the constant reaction coefficients obtained for class (i). Indeed, one might expect that steady state criterion and dynamic criterion, i.e., criteria (i) and (ii) define, in effect, the same reaction function’s coefficients as the relevant time horizon goes to infinity. This is shown to be true only under certain technical conditions discussed in section 4. The solution of an algebraic Riccati equation becomes non-zero when the condition is not met, This implies that a non-zero correction term to the reaction coefficients of class (i) must be added to obtain the coefficients for class (iii). This has not been reported in the literature before. In addition, when the dynamic reaction function asymptotically approaches that derived in (i), we examine how fast this dynamic correction term goes to zero as the length of the planning horizon goes to infinity. Our method is quite general and can be used to draw some policy implications about how the weights assigned to the exchange rate variations and changes in the asset holdings vary as the length of the planning horizon of the government changes. In the simple model used in this paper, dynamic corrections to the reaction coefficients on variations of the exchange rate and of the holdings of foreign money stock.approach zero at the same rate of the inverse of the length of the
M. Aoki,
Monetary
reaction
functions
3
planning horizon, while variation of the holdings of the foreign money stock can be more heavily weighted than that of the exchange rate near the beginning of the planning horizon. In view of the conditional stability of the dynamics of rational expectations models, i.e., since the equilibrium exhibits a saddle-point property, we also examine the implications of adopting reaction rules tied to exogenous shocks for dynamic system stability. To conduct this comparison in a simple framework, we use a small open-economy model which is described in section 2. We show that reaction functions composed from the state variables of the economy tends to improve the stability property of the model, while those constructed using realized values of the exogenous shocks to the economy do not affect the stability at all. In the model, we focus on two sources of dynamics: One is due to rational expectations, and the other is due to accumulation of foreign currency by domestic residents through trade. We use primarily the variance of the disposable income as the argument of the various criterion functions. In addition, we also discuss the implications of using a weighted sum of the variances of the disposable income and the domestic output price. Section 3 considers a stationary-state reaction function. Dynamic reaction functions for classes (ii) through (iv) are discussed in section 4. We then return in section 5 to a stationary reaction function on the conditionally stable manifold that reflects reactions to combinations of exogenous shocks. The paper concludes with some final ,remarks in section 6. 2. Model
Our intention is to examine the effects of several alternative reaction functions on the dynamic behavior of a small open economy near a stationary equilibrium as the economy is disturbed by monetary and real shocks. These shocks are modeled as mean-zero, covariance-stationary, stochastic processes. We do not include securities in the model so as to provide as simple a benchmark analysis as possible. While maintaining our main focus on the effects of trade surplus on the path of economy as in Henderson (1980), this paper deals with the effects of ongoing disturbances rather than one-time shocks. We use a slightly modified, flexible price version of the small open-economy model recently used by Obstfeld and Rogoff (1984).’ Our model is specified in ‘In Obstfeld and RogotT (1984), there is an equation for aggregate demand yd - -+ra( r - 0) + yy + U, where T -p - e is the terms of trade, and u is a shock to foreign demand for domestic output, and an equation for real supply y’ which is assumed to be zero. Noting that the real interest rate equals -a+. the good market clearing condition can be used to generate the adjustment equation for the terms of trade, i - (l#./ua) 7 - o/m. In the absence of foreign shock, u - 0 (and no domestic or foreign supply shock), the terms of trade remains at the value of the initial equilibrium value which is taken to be zero. Then the model dynamics are two-dimensional as described in the main body of the paper.
4
M. Aoki,
Monetary
reaction
functions
deviational log-linear form as follows: (money demand),
(1)
(CPI index),
(2)
T=p-e-p*
(terms of trade),
r=r*+r?+Tj
(uncovered interest parity),
(3) (4)
yL
(aggregate demand),
(5)
(current account),
(6)
(good market equilibrium),
(7)
m-q=
-lr+J/y+[
where q=ap+(l-cr)e=e+ar and
f=
--cpT- +-4)+YY Kg
-
Klf+
K2r
+
“,
Yd =Y
where we set y of home country, r* and p* of the center country variables to zero. Appendix 1 summarizes the steps involved in these derivations. In later sections, we discuss consequences of replacing the demand for real money balances by a portfolio balance equation,* m-e-f=
-lr+(.
0’)
This small open economy produces a single composite good. Here p is the logarithm of the money price of domestically produced good, P, measured from its initial equilibrium level, i.e., p equals In(P/P,). Throughout this paper subscript 0 denotes the value of a variable prevailing in the initial equilibrium; e is the logarithm of the exchange rate, i.e., ln( E/E,) where E is the domestic currency price of foreign currency; r is the implicit domestic nominal interest rate given by the uncovered interest parity relation; f is the stock of foreign money held by the private sector of this small open economy, ‘Use of (1’) rather than (1) modifies the exchange rate dynamics to be (recalling that we assume that y is zero) The
dynamic
6-(e+f-nt)/l+[/I-7j. matrix in
(6’)
(9) changes into
@jK$
!;I.
However, the matrix Q,= @ - bw: still has the same form given by (15), i.e., @-bwl-
[
;
’ -0
1
where
o = x2 +
K,/I
>
0.
For this reason, the analysis of this paper remains basically the same for this respecified model as well.
hf. Aoki,
Monetary
reaction
functions
5
i.e., f stands for ln(F/F,). The remaining variables are exogenous: The world price is normalized to be one and we set p* =p$ = 0; M is the logarithm of the nominal money supply measured from the initial equilibrium level, i.e., ln(M/M,), where M is the nominal money stock; (Y is the share of home goods in the domestic consumer price index; r* is the nominal interest rate in the world; [ is a shock to the portfolio; t is a shock to the world interest rate; and { is a shock to the domestic import demand which is a part of the trade (current account) equation (6), i.e., the disturbance term in (6) is made up of a linear combination of all three stochastic processes. See (9). The uncovered interest parity relation yields the differential equation for e when r is substituted out from the relations (1) and (4), i.e., ct=(-m+e+()/l-77,
(8)
where r defined in (3) has been set to zero. Since, in the absence of real demand or supply shocks, the terms of trade remains as its initial equilibrium value, i.e., r is identically zero. Foreigners are assumed not to hold domestic money. Eq. (6) is already in a form of adjustment for f. A third dynamics exists for the terms of trade that comes out of the good market clearing condition. In this paper we restrict our attention to the two-dimensional submodel that describes the adjustment time paths for the exchange rate and f, keeping r at ze!o. 3. Adjustment
dynamics
We write (6) and (8) jointly after setting r to zero to record the adjustment dynamics for future reference: dz=(@z-bm)dt+dn,
(9)
where the state vector z has e and variables, where
@= [ ;;,
-oKl],
f as the components, i.e., as the state
b=+[;2],
z= [;I,
and the noise n is a mean-zero, covariance stationary stochastic process with components
See appendix 1 for definitions.
M. Aoki,
6
Monetary
reaction
functions
Because ~2 and K~ are both positive, one eigenvahte of the dynamic matrix is stable, and the other unstable, corresponding to the fact that f is predetermined but e is not. Accordingly, the origin (e = 0 and f= 0), i.e., the equilibrium point of the economy is a saddle point. We regard all shocks to be independent mean-zero, finite variance, white noise processes, so that the noise n in (9) is also a white noise process with VariallCe
02= n
iJ 1 1 ee
ad
a4
aff
’
where
Here, at etc., denotes the variances of the elementary noise stochastic process 5, etc. Eq. (9) also implies that z is uncorrelated with the contemporaneous value of the noise process n. 4. Reaction functions 4.1. Stationary reaction functions composed of state variables Eq. (9) governs the time evolution of e and f given a realization of the noise n and a time path of m. We now examine the implications of adopting a policy reaction function of the form m, = wee + u,f = o’z I’
(10)
where w’ = (w,, o,), on the adjustment behavior of the economy. Later in section 5 we contrast this class of reaction functions which are made up of linear combinations of the (state) variables with an alternative class of reaction functions constructed from the realized values of the disturbances such as m, = orn c + w2nJ, or when m is made up as a weighted sum of the realized values of the elementary noise processes such as 5 or 7 themselves. The reaction function defined in (10) mod&es the dynamics shown in (9) into dz = (@ - bo’)zdt
+ dn.
01)
M, Aoki,
Momray
reacfion
functions
7
As a concrete example and as a vehicle of demonstration, we consider home real disposable income, denoted by h, as a target variable. The disposable income, in its log-linear form, can be given by h = -h,e-(h,+ah,)i,
where h, through h, are some positive constants. (Appendix 1 gives the detail of this expression.) Since we assume no exogenous real demand or supply shock, the terms of trade remains constant, hence i is zero. We substitute C out using (8) to express the target variable as h,= --71e + rrn - hlne= p’z + rrn - hlne,
(12)
where p= ~(10)’
and
q=(h,)/l>O.
When m specitied by (10) is substituted in h, the real disposable income is related to e and f by h, = (p + ITW)‘Z~- hln,.
Since the vari&ce subscript 1)
03)
of the real disposable income is expressible as (dropping
var(h)=E[z’(p+no)(p+ao)‘z]
-t-hfu,,,
because ne is uncorrelated with the contemporaneous w=w”=
-p/s=(lO)
z, the choice of w by (14)
produces var( h) = hfu,,. Two things should be noted about this reaction function’s coefficients. First, they are independent of exogenous noise statistics. The second noteworthy fact about (14) is that this reaction function does not stabilize the dynamics given in (9), i.e., the use of this reaction coefficient in (10) does not move the unstable eigenvalue of the dynamic matrix to the stable region of eigenvalues, because the reaction coefficients specified by (14) transforms the dynamic matrix into @-bw;=
10 01 0
-I$
’
which leaves one eigenvalue at zero.
8
M. Aoki,
Monetary
reaction
functions
In effect we note from (11) that the use of the reaction coefficients given by (14) render the exchange rate a white noise process, de = dn.3 Dynamic reaction functions we consider in section 4.2 stabilize the dynamics as we soon see. To provide better control of the exchange rate adjustment behavior, one may wish to switch to an objective function such as the time average T- ‘IarE h f dt or something similar; in section 4.2 below we shall turn to criteria (ii)mentioned in the introduction. 4.2. Dynamic reaction function
To ensure that our subsequent manipulations take an objective function of class (ii),
are mathematically
valid, we
06)
for some finite T and some symmetric, positive semi-definite matrix S subject to the dynamics (9). [Later we divide V, by T and let T go to infinity to consider criteria of class (iii).] The second term depending on the state vector at T may be thought of as a proxy to represent the contribution from the integral from T to inlkity, or the expected value of some other time integral. Consider the reaction function of the form (lo), but now allow w to change with time. Because of the linearity of the dynamics (9), and the quadratic nature of the objective function, it is known that V, of the form z;r,z, + y, solves the stochastic dynamic progr armning functional equation (also known as Bellman or Hamilton-Jacobi equation). [See Davis (1977, ch. 5).] Heuristically, letting A be a small positive time increment the functional equation becomes G-z, + Y, = mir@dh,)A
+ z:+J,+,,z,+r,
+ Y,+A] 1
where from (12) var(h,)=z;Z,z,+o,,
z, = pp’+ 7r(o,pr+ pw;)+ w,w;, a,, = h&,. From (11) z ,+,, equals z, + (@ - bw:)z,A + An, where An denotes the incre‘This corresponds to the discrete-time random walk model of the exchange rate such as the one discussed in Nelson and Plosser (1982).
M. Aoki,
Monetary
reaction
functions
ments (i.e., the change) in n over this small interval A. assumption on the exogenous elementary noises we note mean zero, and the variance equals u,‘d. Carrying out respect to w,, and dividing both sides by A and letting derive the optimal reaction coefficient vector
9
From our white noise that the increment has the minimization with A approach zero, we
u,=(--p+r,k)/lr=w”+r,k/a,
(17)
where k = b/a, and the matrix r, and the term y, which are independent of z, are determined as the respective solutions of - dr/dt
= lY + ‘P’r - TKr,
K = kk’,
‘P A II’ + kp’,
I-(T) = s,
08)
where
and -d~Jdt=a~+tr(@~),
yT= 0.
(19)
Appendix 2 contains the details of this derivation. Compared with (14), the reaction function’s coefficients consist of two parts, the first being the same as (14). Since we show later that r, approaches zero as T approaches infinity, the first term in (17) represents steady state coefficients, and the second dynamic adjustment or dynamic correction part. The differential equation for r is known as the Riccati equation. Note the absence of a constant term on the right-hand side of the Riccati equation. This accounts for the fact that a steady state solution is zero. This last observation is important when we compare reaction functions for class (i) and (iii). We return to this point later. We discuss the solution of the Riccati equation in appendix 3 and proceed here on the basis that we have its solution. In appendix 3, we also establish that the leading terms of the dynamic correction expression in r,k = (&,, a,,) of (17) approaches zero as T becomes large. More spedifically we obtain
4, where
= S,h,/(T-
t)
and
G?*,= -c/(
T- t)erl(T-‘),
10
For
hf. Aoki,
T - t
Monetary
reaction
funclions
near zero,
4,
and
=;I S,/h,
a,, =
S&hl.
From (14) and (17), we note that the weight on the exchange rate is initially while that on the holding of foreign money is L?z,= 1+ 4, = 1 f S,/h,, S2fc2/hl. The elast ici t y K~ can be greater than one if the Wedth elasticity of demands for imports is large.4 As T - t becomes larger, both dynamic correction terms approach zero at least at the rate l/(T - t). It is interesting to observe that as T gets large, the reaction coefficient for f becomes negative and vanishes faster than 9,,. Thus, for smalI values of T - t, the dynamic correction conceivably puts more weight on the holdings of foreign exchange while for large values of T - t, its weight is very small and negative. We also observe that the dynamics are stable because the matrix
has stable eigenvalues for any finite T. By dividing V, by T - t, we can calculate an objective function of class (iii). For this purpose, we need an expression for y,/( T - t) where y, is obtained by integrating (19). Its time average may be approximated by
where r, is the solution of the algebraic Riccati equation which is obtained by setting the right-hand side to zero in (18), o=lv+Pr-rKr, and the optimal reaction function is approximated by m,
= w;z,
= ( - p + r,k)‘z,/a.
4The inequality x2 > h, holds if and only if (QX/EF)oh ’ (A/QWo[l - (QX/WoM]~ where j/= (6’lnJ/~YlnJT)~ and jz= [L3InJ/aln(A/Q)],. By definition 1(2 = KI + KY= (QX/WO{ I(M/A)oh +A(M/QH)o/I}. Hence
~2’1 -i2+il(A/Q%/~2> (EF/QX)O(A/W/~~ where j1 is the disposable income elasticity of demand for imports and j2 is the wealth elasticity of demand for imports. If the holding of foreign money is smaller than the receipts from exports, QX, and if A/M is not too small, this inequality can conceivably be satisfied.
M. Aoki, Monetay reaction functions
11
This differs from (14) if and only if I’,k is not zero. In appendix 3 we show that this expression happens to be zero for the criterion function in (16), and the limit of V,/( T - t ) as T approaches in6nity is equal to IJ~.The value of this criterion function is the same as the one associated with the reaction function coefficient vector w, given in (14). It is important to emphasize that this asymptotic coincidence of the reaction functions, i.e., the fact that r, approaches zero as T becomes large, is the consequence of using h: as an integrand in the criterion function of class (ii). Restricting the functional form of the integrand and the terminal cost expression to be quadratic, other choices of the integrand, if they produce constant terms in the Riccati equations corresponding to (18), then cause their r, to be non-zero. For example, if we use a weighted sum of the disposable income and the exchange rate, such as E(hf + /.tef) for some positive constant CL,then I” is a positive semi-definite solution of the algebraic Riccati equation
o=~ [ t 8I +r*+w-rm, which is not zero. This fact is established in appendix 4. The dynamic components of the reaction coefficients then become
r,k/n+
ffi01’
as T approaches inhnity. The reaction coefficient vector then is
This vector is different from w, which stiB minimizes the value of the criterion function at a point in time, i.e., the variance at a point in time
Next, consider a class (iv) objective function such as V, = 1-e 0
-~lE(h~+~e~)ds
where
PLO.
It is slightly more convenient to use the current-value cost function, replacing V, by W, = V,e@ [see Davis (1977, p. 191)]. Then, the Belhnan equation has the solution IV= z’Gz, where G is symmetric and positive semi-definite, and
hf. Aoki.
12
Monetary
reaction
functions
satisfied the algebraic Riccati equation
Hence, V, - z’Gze-8’ --) 0 as t approaches infinity. Accordingly, the optimal reaction function becomes m,=(-p+Gk)‘t,/s.
(22)
This policy reaction function emphasizes the near future, while the averagevariance policy puts relatively more weight on the long run. Because (21) is quadratic, it can be solved explicitly to ‘elds the reaction function’s coefficients that turn out to be (- /3 + ep2 + 4~ )/2, and 0. Thus the reaction coefficient on the exchange rate is strictly larger than that in (20). The average cost criterion neglects the behavior of the system over any initial interval. Discounted criteria of class (iv) have the opposite effects of emphasizing the initial performance. 5. Stationary
reaction
functions
We now relate the dynamic reaction functions of the previous subsection to those more familiar in the economic literature, i.e., reaction functions of the type m, = 0~6, + u2q, + w&
or
m, = w3n, + w,n,,
(23)
and briefly discuss the consequences of assuming that the government knows the realized values of all exogenous shocks. We conform to the well-established custom in the literature of confining attentions to solutions of (9) on the conditionally stable manifold in this section. We can write the conditionally stable solution of the exchange rate [see Bellman (1953, p. 89)] as e,= (l/l)~me(f-s)/t[m,-ln,ol
ds.
(24
Once the stochastic process (m,} is specified by (23), say, then (23) and (24) jointly with the exogenous noise stochastic processes determine the stochastic process for the exchange rate, or real disposable income. We put a hat to denote the unanticipated portion of the variable. From (12) we can write the unanticipated real disposable income as h,=n(B,+riz,-lit,).
(25)
hf. Aoki,
Monetary
reaction
functions
13
As is well known, any anticipated future change in the time paths associated with m and n produce immediate responses in the exchange rate and in the holdings of foreign money. This fact is denoted by putting bars over m and n, indicating the anticipated parts of these processes. Recall that we regard {m,} and { n, } as mean-zero, weakly stationary, finite variance, white-noise processes. Therefore, E, and is are zero for s SEt. We can, thus, regard (24) as the expression for the unexpected portion of the exchange rate in response to the unexpected changes in m and ns, hence will drop the hats from now on. Note that the variance of the exchange rate is given by var( e,) = ( l//)2l”
dslm ds e(2f-s-u)/’
by the covariance stationarity of the stochastic process m - In,. The variance of the real disposable income is proportional to that of m - In,, hence the reaction coefficients o,=l
and w,=l . reduce the variance of the disposable income to zero. An important fact to observe is that the reaction function’s coefficients are independent of the noise covariances. This is not so if we use the reaction function m, = wnf. Then var(m, - lfl,) = ~JJ”u,,- 2&,,+
12aep.
Hence the value of w given by lu,//u,, minimizes the variance. To set the coefficient to this value we need the ratio of Qu,,. Using only n/, the knowledge of the covariance uc/ helps to recover the part of n, from nj. A simplifying fact about (23) and (24) is that both stochastic processes {e} and { h ) depend on the same combination of the basic stochastic process in the form of m - In,. Hence the same reaction function can simultaneously null the variances of e and h to zero, i.e., (in the absence of real shocks as we have assumed) a fixed exchange rate regime produces zero variation of the real disposable income. This fact is not robust, however. When the asset sector equation (1) is replaced by (l’), then the dynamic matrix 0 in (9) changes into
where a stands for aI - uZ/l. Let X denote the unstable eigenvalue of this
M. Aoki,
14
modified matrix. e,=
K,
Then /
I
Monetary
reaction
functions
(24) must be changed into
meA(r-u)Nl( u) du,
where K, = I/[/@
+ u)],
and
while the stochastic process in (25) becomes h,=r(-e,+N,(t))
where
N,(r)=m,-In,.
Therefore, unless [ ~Jl]m( U) - n,(u) is chosen to be either zero or proportional to m - In,, Nr and N, are two independent stochastic processes with different variances. The reaction coefficients that minimizes the variance of the real disposable income are given as the solution of
where a-'=(K;/2x)(h+
Kl),
u,,+(h
+ K,)-‘UC,,
as and
CO‘,=(A+
K1)-‘.
Note, again that w3 and w, are functions of the model parameters only and not of noise statistics, because the realization of all exogenous noise were assumed known. The class of reaction functions of this section does not modify the dynamics of the economy. This type of reaction functions will become unsatisfactory when stocks are not weakly stationary but contain some deterministic components. To dampen this, the dynamic matrix should be stable as in section 4.2.
M. Aoki,
Monetary
reaction
functions
15
6. Concluding remarks
Using a simple model of a small open economy to minimize deviation of real disposable income from its stationary equilibrium value, we have compared reaction functions obtained by explicit intertemporal optimization over a finite horizon with their limiting expressions over an infinite horizon or with timeless reaction functions dealing with steady state values. The reaction function’s coefficients obtained by dynamic optimization over a finite horizon are shown to converge to their stationary counterpart only when the associated algebraic Riccati equation does not contain a constant term. When constant terms are present in the algebraic Riccati equations, non-zero correction terms need be added to the stationary coefficients in general. By means of an example with a quadratic function of real disposable income and the exchange rate, we indicate that non-zero constant terms will be present in the Riccati equations if the arguments of the criterion functions of the intertemporal optimizations contain more than one independent linear combinations of the exogenous noise stochastic processes. By examining a variety of criterion functions, we have illustrated that initial phase of adjustments or long-run aspects of adjustments can alternatively be emphasized with obvious implications for choices of planning horizon for the government. The example of this paper illustrates a possibility that deviation of foreign financial asset may appear in the reaction function with any significance only towards the end of the planning horizon for a government with a finite but a long planning horizon, or with a short planning horizon. Finally, the paper points out the fundamental differences in the stability of the economy employing reaction functions made up of the state variables and of realized values of the exogenous noises. When the government utilizes the latter type of reaction functions, the government is essentially cancelling out the exogenous noises while leaving the economy unaltered. With the use of the former type of reaction functions, the government actually improves stability of the economy. This distinction will become important if the model leaves out some sources of dynamics or the exogenous noises that tend to destabilize the economy, or if the government can not observe or exactly reconstruct the realized values of exogenous noises.
Appendix 1: Derivation of the log-linear model
We follow the procedure detailed in Aoki (1981). Eqs. (1) through (4) are standard, except possibly for the noise n introduced into (4). This noise may be regarded as a foreign interest rate shock, or a proxy for risk premium. The exchange rate is denoted by E, which is the domestic currency price of a unit of foreign exchange. Eq. (1) comes from demand for
M. Aoki,
16
Moueray
reacriotr
fwrcriotu
real money balances M/Q = L(r, Y), where the nominal stock of domestic money is deflated by the consumer price index, Q, which positively depends on the price of the domestic good, P, and on that of the foreign good, EP*. In this paper P* is exogenous and is set to one. The parameter (Y in (2) is the elasticity of Q with respect P evaluated at the initial equilibrium state and denotes the share of domestic goods in CPI. We denote values of variables in the initial equilibrium state by subscript 0. Demand for domestic goods consists of two components, domestic and foreign demand. The foreign component which is the exports from home, X, is taken to depend only on the relative price, P/E, while the domestic component is specified to be a function of real saving, the level of real economic activity which is measured by Y, and relative price P/E. Real saving is a function of disposable income, H, and real wealth, (M + EF)/Q, where F is the stock of foreign money held by the private sector of this economy. Therefore, expenditure on domestic goods by domestic residents is specified to be given by
where A stands for the total financial asset, h4 + EF. Real imports are specified by J=J(H,Y,
P/E, A/Q).
(A.1)
The current account equation is given by P= (Q/E>(x-J).
(A.21
The variables D and X are related to Y by Y = D + X by definition. In a stationary equilibrium i/E and Q/Q are both zero. Real disposable income is given by
H= Y+(~/E)(EF/Q)-T,-(A/Q)(~/Q),
(A.3)
where TX is real taxes. By assumption, both Y and r, stay at their respective equilibrium levels. Thus, to the first order of smallness, the domestic component of the aggregate demand d = ln( D/D,) is expressible as d=d,h-dp+d3(a-q),
(A.4)
where a = ln( A/A,), d, = (aln D/aln H),, d, = -(a In D/aln d,=(alnD/alnA),. From (A.3), the disposable income h = ln( H/H,) is given by
A= -u+w-Q)=
-ll,e-(h,+ah*)i,
P/E),,
and
(A-5)
M. Aoki.
Motrerrm,
reaction
functions
17
where (2) has been used, and h,=(M/QH),
and
h,=(EF/QH),.
Substitute (AS) into (A.4) to derive d= -d,h,e-d,(h,+cllh,)i-d*T.
(‘4.6)
Here, we drop d,(a - q) as being small. The relation Y = D + X becomes
yd= (WY),d+(X/Y)x,
(A.71
in the log-linear variables where x stands for ln( X/X,). It equals -X’T where the symbol x’ is the relative price elasticity of demand for imports, i.e., x’ = - ( 8 In X/c? In P/E),. On the assumption that the ratio hi//( hi + cubz) is small and substituting (A.6) into (A.7), after * is also substituted out by (1) and (4) the log-linear expression for the aggregate demand becomes yd = -+T - uai, where the coefficients Cpand u are both positive. The goods market clearing condition implies that the terms of trade remains at zero, in the absence of real demand or supply shocks. Hence we can set the terms of trade variable to zero in our model. In particular, (A.5) simplifies to h= -/I,&=
-/I,(-m+e+[)/l-?J.
This is eq. (12) in the main body of the paper. To derive (6) we start from (A.2) and write it for the log-linear variable, f = ln( F/F,), as f=(QX/EF),(q+x--e-j), where the equality of (QX/EF), and (QJ/E)O has been used, and the definition of j as the log-linear variable of J. We recall that the terms of trade is zero, hence the expression, q + x - e, is zero. From (A.l), j=j,h+j,(a-q)+l’, where the disposable introduced we obtain
coefficients ji and j, are the elasticity of J with respect to the income and real wealth, respectively, and 5’ is a random shock term to real imports. Combining these two expressions with that in (AS), the adjustment equation for the current account, L4.8)
hf. A oki, Moneraty
18
reaction
functions
where Kl= (1 - e)(QX/EF)&, K; = [e/(1 - @I (x/fh.h~,/~> K;
(QX/Ef’),-&e.
=
Here, 8 is the ratio, (M/A),, of the stock of domestic money in the domestic wealth. After r is substituted out, the adjustment equation for j becomes
fmancial
(A-9) where the symbol K~ (A.9) is defined by
to
is equal
K; + K;I
by definition.
where { is (QX/EF),J’ by definition. When the asset sector is specSed by a portfolio than by (l), the interest rate is related to m by
The disturbance “/ in
balance equation (1’) rather
r=(-m+e+f+t)/l.
Substitute this into (A.7) to see that the trade balance equation remains the same as (A-9) if the parameter K~ is suitably redefined by K~ - K;J/I, still assumed positive to preserve the stability property of the total dynamics. Appendix 2: Derivation of reaction c&3cients From the adjustment dynamic equation (ll), = z, + (Qi - ho:)z,A
=,+A
The right-hand min 0,
+ An.
side of the dynamic progr amming functional equation becomes
[z;Z,z,A
+
u;A
+
E,(Z:+A~+A~,+A)
+yr+A]~
where E,(Z:+~~+~Z,+~)
=
z;l-,z, + z;T, + A ( @ - ho;)z,A +z;( @ - h~:)‘r,+Az,A
+ u( r,+ipnz)A.
M. Aoki,
Monemy
reaction
19
jimctiom
Therefore the functional equation becomes z + z: rt-Tt+A A ’
Yt -;r+A
= u; + tr( r,+fl”)
+min(z;[z,+T,+,(@-h;) @I
+ (@-
hw:)‘r,+A]
II)
Recalling that Z, depends on w,, the differentiation bracket yields, after A + 0, o, = ( - p + r,k)‘/r
where
-
(A-10)
of the expression in
(A-11)
k = b/lr,
after noting that T is continuous in t. Substituting (A.ll) into (A.lO), letting A + 0, equating terms independent of z, on the RHS with -dy/dt, those quadratic in z, with -z;(dT/dt)z,, and noting that z, is arbitrary, we can set - dy/dt - drjdt
= a, + tr r,a,,?, =
Ik=@+kp’
and
r* + 9r - rKr,
YT=
0,
r,= s,
where
K=kk’.
Appendix 3: Solution of the Riccati equation We derive the solution of the Riccati equation - dr/dt
=
r* + wr - rKr,
r,=s20,
(A.12)
where
K = kk’,
k = b/r.
Consider the solutions of the two related linear matrix differential equations f=
-9’x
>
x,=s
jl=‘ky-ICY,
yr=I.
Then, it is easy to verify that r is given by xy-‘, by direct differentiation. The technicaJ conditions to ensure the existence of the solution of (A-12) are
20
M. Aoki,
Monerq
reaction
functions
satisfied here since the inverse of y exists. See Aoki (1976, p. 164) or Bucy and Joseph (1968, p. 67) for example. The matrix ck of (18) happens to be symmetric, and is given by
[ 11
\k= ; -OK.
The solution matrix of the difTerential equation for x is given by x(f) = eY(T-r)S, where
err =
[
1 0
0 e-‘I’
1 *
Next, solve for the matrix y substituting the solution for x obtained above. It is given by yf=e w-n+
/I
re*.(‘-‘)Ke-
+(J- r)S&.
Denote the element of the matrix y, by Y,~(I). Carrying out the integral we derive
Ydd
= (‘%/KI)(~
-X-*)%/h:,
Yn(d
= (‘d’dS,(X
Yzz(d
= X + (K%‘Q)(X
- 1)/h:, - X-‘&/h:,
where x=e r,(T-0
and
S=diag(S,,S,).
The dynamic correction terms are
hf. Aoki.
Monetaty
reacfion&r&ms
For large (T - t) values, Y&)
IyI=I
= (T-
tT;:)sl
W,/h:,
[l
+(K~/2K,)&/h;]X.
1 ThUS
42*, = S,h,/( T- I)
and
S2bA)
52,, = -
(T-z)X
For T - I near zero,
Y120)
=
‘d#=-
II/h:
= 0,
Y2,tr)=K2S,(T-t)/h:50,
y22(1)
=
1 +(K1
+
S,K#)(T-
I)
=
1,
hence
and 52,, = S,/h,
and
52,, = S2KZ/hl.
In L$,, the term y&t) is dominant both for small T- t and large T- t. In L?,,, however, ~~~(1) is dominant for large T- t but yll(r) becomes dominant for small T - I, hence the change in the sign of L?,,.
M. Aoki,
22
Monetaty
reaction
functions
The algebraic Riccati equation that corresponds to (A.12) is given by where
O=A+r,\k+‘kr,-r,Kr,
Writing out the components of the matrix equations for the three elements of r,:
r,
A=diag(p,l). explicitly, we obtain three
0 = p -(u + K3b)2/(f7r)2,
0 = 2CK2
+ (b + K,C)“/(
h)2,
where
We select the solutions to yield a positive semi-definite r,, i.e., a = filq, b = c = 0. Therefore, the steady state expressions of the reaction coefficient r,k become (u + bK3)/(d) = fi and (b + CK3)/( d) = 0. Appendix 4: Alternative objective functions Suppose we replace h f in the objective function by another expression involving two or more different combinations of exogenous stochastic processes. For example, as an integrand, we take hf + pef for some positive value of p. This change generally produces a constant term in the Riccati equation causing r, not to vanish. First, observe that the expression
replaces E,hf in appendix 1. We deduce that the new reaction function coefficient vector is still of the same form, w,=
++r,ky.
The Riccati equation changes, however, into
- dr/dt = A + r\k + wr - ru,
(A.13)
M. Aoki,
Monetary
reaction
functions
23
where
Appendix 5: Solution of (21)
x = (st + wd/~~
and Y= (8, + wd/ln.
Then (21) can be restated for the components as
o=pg,+x*-p, 0 = (P + K&h + XYv 0 =
t P +
%)&
+Y2.
A positive semi-definite solution with g, > 0 and g, = g, = 0 is obtained as the positive root of the quadratic polynomial equation
i.e.,
This solution produces the reaction coefficients (- j3 + d-)1/2 and 0, respectively, as correction terms to wU in (15). Note that as p approaches 0, it produces the same correction term as class (ii). References Aoki, M., 1981, Dynamic analysis of open economies (Academic Press, New York). Aoki, M., 1976, Optimal control and system theory in dynamic economic analysis (North-Holland, Amsterdam). Bellman, R.E.. 1953. Stability theory of differential equations (McGraw-Hill, New York). Boyer, R.S., 1978. Optimal foreign exchange market intervention, Journal of Political Economy 86, 1045-1055. Bucy. R.S. and P.D. Joseph, 1968, Filtering for stochastic processes with application to guidance (Wiley, New York). Canzoneri, M.B., 1982, Exchange intervention policy in a multiple country world, Journal of International Economics 13,267-289.
24
M. Aoki,
Monetary
reaction
functions
Davis, M.H.A., 1977, Linear estimation and stochastic control (Chapman and Hall. London). Henderson, D.W., 1980, The dynamic effects of exchange market intervention policy: Two extreme views and a synthesis, Kredit und Kapital6, Suppi. Nelson, C.R. and C.I. Plosser, 1982, Trends and random walks in macroeconomic time series, Journal of Monetary Economics 10, 139-162. Obstfeld, M. and K. Rogoff, 1984, Exchange rate dynamics with sluggish prices under alternative price-adiustment rules, International Economic Review 25. 159-174. T&ovsky,-S.J., 1983, Exchange market intervention policies in a small open economy, in: J.S. Bhandari and B.H. Putnam, eds., Economic interdependence and flexible exchange rates (MIT Press, Cambridge, MA).