-12-

to (13) above) for (12) to hold is

(14) P,(0) <0 and Pp(0) >0.

Finally, to show when (14) is met, let us write (8), when 8 =0, as
sin a(0)/a(0) = m,/m :

and substitute into (11) to get

(15) P(0)=a(0)m tan” !a(0)+(a(0)/m)1

It is easy to confirm that as the value of (m,/m) approaches zero, the values
of (0) and a4 (0) approach m and 2m, respectively, and that . -tan Xo)
approaches, respectively, -® and +TM. Consequently, for a given value of m>
it is always possible to find a value of m large enough in absolute size (and
not necessarily infinite) so as to satisfy both (13) and (14), and thereby give
rise to at least second-order conditional stability in (7).

A particularly interesting example of this occurs if we consider simply
extending the lag in (7) (i.e., making@ larger) without changing any other
parameters. For the "L-curve" example above, this means that both m, and m must
increase, but the latter more rapidly; hence, the ratio (m,/m) must decline in
absolute value as 6 is extended, and a4 (0) and a,(0) approach 7 and 27 as before.

In this case, the expression equivalent to (15) above is

. (
(16) P(o)=a(0) |-(c4/e) -cosu(0)_},,
1-cos“a(0) J

Again, it is easy to show that, aso gets large, [cosa (0)/(1-cosa(0) J approaches
-eand +°, for a4(0)= TM and a (0) =21, respectively. Hence, with unchanged

elasticities, a sufficiently long lag, by itself, is enough to raise a


-4-

as interest rates, are assumed to be determined endogenously.) Equation (2)
assumes that the present rate of change in wealth, vs is equal to the current-
account position, which is determined, in turn, by the present levels of both
the exchange rate and wealth itselt./

If we now impose the requirement that transactors in the foreign
exchange market have perfect foresight (i.e., thate = E), we can rearrange

(1) and linearize both (1) and (2) to obtain
(3) E= a, + ay E+ ay Vv, a,>0, a, >05

(4) V= by + by E+ bo Vv, by>0, by <0.

For purposes of later comparisons, it will be expedient to write this pair of
first-order differential equations as a single second-order equation in one of the
variables. Arbitrarily selecting E as the variable of interest here, we can

differentiate (3) and substitute to obtain

(5) E +c, E+E E+K=0,

wherethe constants Cys ¢, and K are given by

39,2/
- (a, + by) >0,

"

4

c

"

(bo a) - by ay) <0,

K = (by a - ay by).

‘7 Clearly a number of variations on a basic international portfolio model can
fit this deliberately general characterization. The literature in this area is
extensive, but a recent, representative paper that synthesizes several approaches
and provides a useful bibliography is Rodriguez (1980).

To avoid unnecessary complications, interest payments have been ignored in
specifying equation (2).
2/ It is usual to assume in these models that c,>0 -- that is, that the stabilizing
effects of wealth on its own rate of change (by <0) dominate the potentially de-
stabilizing effect of the exchange rate on its“expected (and actual) rate of change
(a,>0). This assumption is made, it appears, largely because, if the opposite were
the case and the current-account balance responded perversely to an exchange-rate
change, a (first-order conditionally stable) rational expectations solution might
not exist. We shall continue to honor this tradition in this analysis but without
great conviction.


-16 -

Although these findings suggest that an unstable system is a legitimate
theoretical and empirical possibility, the implicit threat of an explosion or collapse
of the foreign-exchange market (or macro economy, for that matter) is made a little
less urgent if we recall some of the limitations of this exercise. For one thing,
official intervention has been ruled out by assumption; for another, the model has been
made strictly linear. In view of the latter simplification, the model is strictly valid
only in the neighborhood of its long-run equilibrium. If it were to veer too far away
from this point, its basic relationships would be altered by non-linearities ( due
to risk aversion in portfolios, for example) -- possibly so as to maintain the system
within bounds. When instability of the type described above is present, however, this
point might not be reached until an unacceptably large swing in the exchange rate has
occurred.

The findings above also raise some unsettling questions about the use and
interpretation of jumping variables as a device to ensure stability. We have focussed
here on Jags and leads in trade, but similar instability problems could well arise, with
even more force, in models with lags from other sources of which there are many candi-
dates. Since there is usually only a small number of jump-variables available to accom-
modate shocks in more complex frameworks, this suggests that a high degree of instability
may be endemic -- at least from a theoretical point of view.

Clearly the need for discrete jumps in variables is bound closely to the con-
venient analytical device of distinguishing markets according to whether they adjust
slowly or very rapidly (i.e., instantaneously). In such a framework, instantaneous
jumps may be understood as a compressed, stylized version of a more protracted process,
the details of which are usually not shown. It may be, therefore, that sufficient slow-
ing of the rate of adjustment of such a market or of the expectations formation process
(by introducing a learning process, an adaptive mechanism, or some other “friction")
could rid these models of some of their less appealing features ,l/ but whether or not
this is the case is an open question.

T7 For an examale in which a conditionally stable system can be made globally stable

{at a higher order) by introducing an adaptive rule consistent with rational expectations,
see Mussa (1975).


-9-

Equation (7) is a transcendental equation in z which typically has an
infinite number of roots. Some of these may be real; the others are complex and in
conjugate pairs. The properties of the roots of (7) can be analyzed somewhat more

easily by considering its quadratic and exponential parts separately. If we define

>

Uz)ez2 + mz +m,

and

-Z
(z)= -noe >

then a solution of (7) satisfies

Yz)=P(z)

The two functions, Q(z) and P(z), are graphed in Figure 1. Since P(z) is negatively
sloped throughout, it is evident that it can intersect Q(z) at no more than one point
to the right of the minimunof Q(z); to the left of the minimum, there may be two
intersections, a tangency, or none at all, depending on parameter values. The inter-
cepts of P(z) and Q(z) are Ny and my respectively; since we require, (m,tn,)<0,
evidently there must be exactly one positive real root of (7), and the system is no
less than first-degree conditionally stable.

Whether or not equation (7) also has complex roots with positive real parts
remains to be answered. To gain some insight into when this is the case, let us

write a representative complex root in the form
P=B8t+ai ’

where both a and B are real. Substituting this expression into (7), we find that

any complex root of (7) must meet the following pair of conditions:


-6-

Introducing J-curve Effects

Equations (2), (4), and (5) presume that a depreciated exchange rate
(a numerically higher value of E) is associated with a more positive level of
the contemporary current account. That is, holding wealth constant, the
counterpart in this model of the extended Marshall-Lerner condition is satisfied,
and no J-curve effects are present, To introduce a workable approximation to a

J-curve, let us modify equations (2) and (4) to

(2.a.) V(t) = G*(E(t), E(t-8),V(t)), aG*/3E( t)<0,3G*/dE( t-6)>0,9G*/aV(t)<05

(4.a.) V(t) = bg tb] E(t)+bpV(t)+b5e(t-8)» b,<0,b5<0,b,705

1

where the argument (t-9) indicates values of a variable taken 3 time units ago.

Although there are obvious limitations to such a simple two-period version of the

J-curve l/ nonethelcss it Goes allow us to adjusteé, by and by to generate a rather wide

range of "J" shapes. In this notation, a conventional J-curve is produced when the present

exchange rate is negatively related to the present current-account position (b,<0),

while the exchange rate of one period previous is positively related to the present

current-account position.2/ In order to make fair comparisons, however, we shéll

V/ ideally, we should like to express the present current account as functionally
dependent on all the exchange-rate levels over some relevant time interval, rather than
at only two specific points. Even the simplest specification of this sort, however,
leads to equations somewhat similar to so-called "renewal" equations and unmanage-

able technical complexities. For a discussion of equations of this type, see Bellman
and Cooke (1963), Chapter 7.

2/ As is pointed out by Magee (1973), there is no theoretical reason that the response
curve must have a J- shape. The response of the measured current account can be decomposed
into "currency contract effects (due to valuation effects on contracts already in place),
"pass-through" effects (related to effects on the pricing of traded goeds), and eventual
volume adjustments. For any given exchange-rate change, these elements can cverlap in a
complex tashion giving averitable "alphabet soup" (to quote Magee) of potential patterns.

There is some evidence, however, that the simple J-shape is commonly found Ss
example, Spitaller, (1980).) P p y nd. (See, for


Stability properties of (5) can be determined by investigating the signs
of the real parts of the roots to the homogeneous portion of (5). As is

well known, in a simple second-order system of this type, the corresponding

roots are

Inasmuch as ¢ is negative, it is evident that in this type of model equation

(5) must always have two real roots, one negative and one positive;

the system exhibits, therefore, first-order conditional stability. A closer look

at the components of ¢ reveals that the problem in this case (problem, in the sense
that the presence of a positive root prevents the system from being globally

stable, as would be the case if all roots were negative in their real parts)

arises from the positive sign on ay: This observation should not be surprising,
since a positive value for ay merely indicates that accurately anticipated ex-
pectations of an exchange-rate change tend to put pressure on the rate in the

same direction as the expected change itself -- a property that is well known to be
destabilizing in much simpler, earlier models of exchange-rate dynamics 2/ In the
version at hand, however, a Qualified weak form of stability is attained by
introducing the device of an exactly-correct jump in the current exchange rate

after any exogenous shock, as described earlier.

J/ For an early discussion of some of these issues see Baumol (1957). Although most
of the modern papers that deal with conditional stability in this context seem to
trace their intellectual origins to Dornbusch (1976) and Kouri (1976), the pedigree
of this concept is really much longer. Earlier, partial models of the foreign-ex-
change market were also conditionally stable in the obvious sense that an exactly-
correct jump to the equilibrium rate would keep the market stable following any exogen-
ous shock. Ina certain sense, the more modern approach has simply imbedded this basic
property in a higher-order system which includes a feedback into the foreign exchange
market from other sectors. One cannot help but notice, however, that most earlier

writers avoided the temptation to simply assert that such a stability-ensuring jump
in the exchange rate would take place automatically.


Bibliography

. Baumol (1957), "Speculation Profitability, and Stability,” Review of
Economics and Statistics, August, 1957.

. Bellman and K. Cooke (1963), Differential-Difference Equations, Academic
Press, New York, 1963.

. Britton (1970), "The Dynamic Stability of the Foreign Exchange Market,"
Economic Journal, March, 1970.

. Dornbush (1976) "Expectations and Exchange Rate Dynamics?" Journal of
Political Economy, December, 1976.

. Dornbush and S. Fischer (1980), "Exchange Rates and the Current Account,"
American Economic Review, December, 1980.

. Driskill and S. McCafferty (1980), "Speculation, Rational Expectations, and
Stability of the Foreign Exchange Market," Journal of International
Economics, February, 1980.

. Gray and S. Turnovsky (1979), "Expectational Consistency, Informational Lags, and
the Formulation of Expectations in Continuous Time Models," Econometrica,
November, 1979.

. Kouri (1976), "The Exchange Rate and the Balance of Payments in the Short Run

and in the Long Run: A Monetary Approach," Scandinavian Journal of Economics,
Vol. 78, No. 2, 1976.

- Magee (1973), “Currency Contracts, Passthrough and Devaluation," Brookings
Papers on Economic Activity, No. 1, 1973.

« Mussa, "Sticky Prices and Disequilibrium Adjustment in a Rational Model of
the Inflationary Process," (processed), University of Rochester, 1976.

. Rodriguez (1980), "The Role of Trade Flows in Exchange Rate Determination:
A Rational Expectations Approach," Journal of Political Economy, December,
1980.

. Rogoff (1980), Essays on Expectations and Exchange Rate Variability, unpub-
lished doctoral dissertation, M.I.T., 1980.

- Spitaller, "Short-Run Effects of Exchange Rate Changes on the Terms of Trade
and Trade Balance," I.M.F. Staff Papers, June, 1980. :

+ Williamson (1972) "Another Case of Profitable Destabilizing Speculation,"
Journal of International Economics, February, 1973.

. Wilson (1979), "Anticipated Shocks and Exchange Rate Dynamics," Journal
of Political Economy, June, 1979.

. Wilson and W. Takacs (1980), "Expectations and the Adjustment of Trade Flows
Under Floating Exchange Rates: Leads, Lags, and the J-Curve," International
Finance Discussion Paper No. 160, Board of Governors of the Federal Reserve
System, April, 1980.


V/

case.’ The stability properties of (5.a.) can be determined by examining the sign of

the real parts of the roots of the,characteristic equation associated with “he homogen-

ous part of (5.a.). Since solutions to (5.a.) take the exponential form,
E(t)=e"t :
the characteristic equation for (5.a.) can be written as

2

-r0_
(6) H(r) =r’ + cr tc, + doe "= 0.

Furthermore, if we let z=r9, we can transform (6) to the more convenient form,

2 “Ze
(7) H(z) = 2 + mz +m + noe “= oO,
where
m=C,6>0 >
= 2
M=Cy Cary
-d ae
n=4d,9 <0,
and

ey 4
+n =in = ct
mtn =m sco" <0

i7 An extensive treatment of mixed-type equations is found in Bellman and Cooke (1963).
Some simple, practical guidelines on determining their properties is provided in the
appendix to Gray and Turnovsky (1979). The analysis in this paper fcecuses mainly on

the strongest conjecture from the popular accounts -- namely, the presence of a funda-
mental (endogenous) instability. To deal with the weaker conjectures -- i.e., the possi-
bilities of initial perverse exchange-rate movements and overshooting -- one needs to
consider initial conditions and the corresponding particular solution to the system. For
even the simplest examples, it is not easy to obtain a concise closed-form solution to a
mixed-type equation. In general, however, initial conditions on both e ard V must be
specified over an interval of length @ (or conditions that are equivalent); the solution
will usually take the form of a series of smooth functions, joined at every integral mult-
iple of @ at a point where the first derivative is discontinuous. Properties of the solu-
tion in its early stages can be inferred for elementary examples by building up the solu-
tion by a simple step-by-step "continuation" technique. (See Bellman and Cooke (1963),
Chapter 3.) In the long-run, for a stable system, the trajectory is essentially indepen-
dent of the intial conditions. For the system in this paper, it is fairly easy to confirm
that in many circumstances, depending in part on initial conditions, exactly the prop-
erties described in popular accounts are seen -- namely, perverse movements and overshoot-
ing -- but a detailed treatment is beyond the scope of this paper


International Finance Discussion Papers
Number 198

January 1982

J-CURVES AND STABILITY OF THE FOREIGN-EXCHANGE MARKET

by

Richard Freeman

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished material)
should be cleared with the author or authors.


- 10 -

(8) 206 + myo - nge"Psin a=0 ,

(9) ge + mb +m, = -nje"Peos a ta”

So as to be somewhat more systematic, consider first the special case
in which the present current account is positively related to the lagged
exchange rate but unrelated to the present exchange rate -- a situation that
might be described as a (reverse) "L-curve". In such a case, we specify m,=0 and
nym <0... Equation (8) defines implicitly a (non-unique) relationship between

a and 8, which we designate by
(10) a = $(8):

Using (10), we can now express the right-hand side of (9), for any particular

a that satisfies (10), as
(11) P(8)= -me"Peos(o(s))+ 628),
where the notation, P(B), is meant to indicate that a is now endogenous.

The existence of a complex root to equation (7) with a positive real part

requires that
(12) Q(g)=P(B), B>0,

where $(8) stands for one of the values of athat satisfies (10) (the samea in

both its appearances in (11)).

To establish sufficient conditions for (12) to hold, we shall consider

only the relevant values of ain the first full cycle -- i.e., where m<o<2n. V

‘T/ There may well be other solutions for a>2 7; we confine the analysis to the
range where a< m because (sina/a) has the greatest amplitude in this range.


Figure 1

Quadratic and exponential part of:

=72. -Z,
H(z)=z +m ztmytne °5

= en eZ
P(z)= noe : ny <0,

Q(z)= 2m zm, m >95

+n_=m <0.
mtng=m 0


customarily keep the long-run exchange-rate elasticity constant during any orth-
ographic contcrtions -- i.e., we require the sum of by and bg in (4.a.) to be constant,
positive, and equal to by in (2).

Following the same general pattern as before, we can now combine (3) and

(4.a.) to obtain a second-order equation ing, i.e.,
(5.a.) E(x) +e,E(t) +, E(t) +d, E(t-6)+K=0,

where
c= ~(a,+b,)>0,
Coz (b,a,-bya9) 5
d= ~(b3a4)<0,
K=(b9a,-aob,)-
Notice that according to our conventions the sum of Cy and qd, above is negative
and equal to the value of © in equation (5). Furthermore, although qd, is certainly
negative, there is no clear presumption as to the sign of Cy:

Stability Properties of the Model with a J-turve

Equation (5.a.) differs from its predecessor, equation (5), primarily by
the inclusion of the lagged term and is, thereby, a mixed differential-difference
equation. As such, finding and expressing its solution is considerably more compli-
cated than in the previous case, but the general procedures for determining stability

and other basic properties follow lines parallel to those of the simpler, non-mixed


As we shall see from the findings below, under certain conditions,
the introduction of J-curve effects can raise the order of conditional
stability of a system from first to second order and beyond. Since in these
cases there is an insufficiency of jumping variables to place the system on
its stable trajectory, such a system must be inherently unstable. This finding
is of some interest in its own right, since it tends to confirm the conjecture drawn
from the popular view, but it also has broader implications for this general
class of dynamic model as well. Inasmuch as lags and leads are pervasive in

any realistic macro system, it suggests either that stability conditions

in macro systems may be less easily met than had been thought previously
or, at least, that there are some important shortcomings to models with jump-

ing variaples wnicn are not yet fully appreciated.

A model of exchange-rate adjustment

At the risk of greatly oversimplifying matters, I shall take the
following two semi-reduced-form relationships to be a fair representation of a

broad class of international macro models:
(1) E= Fle.V), aF/de >0, oF/aV <0;
(2) V=G(EV), 36/26 >0, 3G/aV <0,

where E is the exchange rate (in home currency per unit of foreign exchange) ,

€ is the expected rate of change of the exchange rate, and V is the nominal

value of home-country wealth when all assets are expressed in home currency._/
Equation (1) expresses the relationship between domestic wealth, the

exchange rate and its expected rate of change, when asset markets are in equilib-

rium. (Asset stocks are taken to be fixed, and other financial variables, such

T/ The underlying model could just as well be expressed in terms of real wealth
without any important changes in the conclusions.


-13-

system such as that in equation (7) to conditional stability of at least
second-order. Unless some variable in addition to the present exchange
rate is allowed to make an appropriate exact jump, such a system cannot
automatically return to a stable equilibrium following an exogenous shock.

The remarks above deal with the effects of a change in only one
dimension of the J-curve, its duration. We can also look, however, at the
effect on stability of altering the degree of "stagger" in the J by strengthening
the near-term, perverse effects, while at the same time adjusting lagged
effects to keep constant the long-run cumulative effect of an exchange-rate

change. When m, #0, the expression equivalent to (15) is
(17) P(0)= = iita2(0) + ampsin®'a(0)-tan”'a(0)1
where a(0) is given by

sino(0) /o(0) = m/m, .

Again, as m, gets sufficiently large (with m and m constant), a(0) approaches 7
and 27, and{sinTM!a(0)-tan”!a(0)1 approaches -~ and 0, respectively. Since
(-itto2(0)) is positive and finite, it is evident that absolutely larger values
of my will tend to raise the level of an otherwise first-order stable system to
second order and beyond. Hence, the greater the "stagger" in the J-curve, the
greater is the likelihood of an unstable system.

Finally, it is worth taking a brief look at one other manipulation of
the J-curve -- specifically, a movement forward. There is persuasive evidence that
some traders may accelerate or delay current-account transactions in anticipation of
a future exchange-rate change!/ If such a tendency is widespread, the consequence
T7 Anticipation effects in trade among the industrialized countries have been

Fnvestigated empirically in a recent paper by J. Wilson and Takacs (1980) .
Additional discussion of this issue is found in Magee (1978).


- 1 -

Inasmuch as the analysis of this equation is very similar to that of the
preceeding section, we shall not repeat it here. Suffice it to say that
(7.b.) always has exactly one positive real root; in addition, if fy is
sufficiently large, there are also complex roots with positive real parts,
and the economic system cannot be stabilized by a single discrete jump in the
exchange rate.
Concluding Remarks

Based on the findings above, I have to conclude that the popular view --
namely, that the presence of a J-curve response can lead to endogenous instabil-
ity -- has been largely confirmed. In fact, it appears that the mere presence
of a lag in the current-account reaction to exchange-rate changes, even without
the extra destablizing kick provided by near-term perverse effects, can be
destabilizing . Moreover, this can be the case even if the trade response

satisfies the conventional Marshall-Lerner stability conditions. Furthermore,

the findings also suggest that forward-looking trade effects related to anticipated
changes in exchange rates can produce a similar instability in the exchange-rate
adjustment mechanism.

Nor do these findings appear to be special cases without empirical rele-
vance. Although we cannot very well attach numbers to the key parameters above with-
out specifying more details of a particular model, it is apparent that the
unstable result is more likely in every instance when my is a small numter.

The coefficients that measure own-variable effects on @ and V (i.e. ay and by)
are of opposite sign and enter into m additively, and there is no apparent
reason to exclude the possibility that they might nearly offset one another to

produce a sufficiently small (but still positive) value for m.


-2-

assumption that transactors make forecasts based on full knowledge of the
structure of the economy -- including, in this case, knowledge of present and
future J-curve effects. / In this framework, rather than being "misled" by
incoming data, agents are assumed to incorporate it, along with all other rele-
vant available data, into forecasts that are consistent (i.e., identical) with
actual realized outcomes.

International macro models that presuppose rational expectations (or its
close relative, short-run perfect foresight) with respect to exchange rates

are now in wide use. Many of these models exhibit a property that I shall

refer to here as first-order, conditional stability. This term, when applied to a

model, indicates that the system of differential equations by which the model

can be represented has exactly one root with a positive real part. The presence

of this positive root means that the corresponding economic system is unstable
unless at least one variable can make a discrete jump at some point to place the
system exactly on its unique stable trajectory. Without such a jump, given enough
time the system will spontaneously diverge from equilibrium. In many applications,
the present level of the exchange rate provides a convenient jump-variable, although

stories to explain how the foreign-exchange market can produce just such an exact

shift are tortuous and, in my view, improbable if taken literally.2/

J/ The approach in this paper bears a superficial resemblance to a branch of the litera-
ture in this area in which market participants fully anticipate the effects of a future
shock to the system. (See for example, Dornbusch and Fischer (1980), C. Wilson (1979),
or Rogoff (1979).) The "rules of the game" in these papers dictate that a future, fully
anticipated event results in a discrete jump in the present exchange rate. Jumps at
other (future) moments are excluded because they imply infinite capital gains. Here,
lagged responses in the current account play a similar role in that they are fully antici-
pated (and, because of the specification of the lag as discrete, they impact at a single
future moment). They differ, however in that they are generated continuously so that
they imply a continuous adjustment of the present exchange rate, rather than a jump. As
we shall see below, the discrete specification of the lag is primarily a convenience.

2/ We need not take these explanations literally, of course, and a good deal of work has
gone into showing how a different expectations-formation process (such as an adaptive
expectations rule) or some endogenous learning process can soften the dynamics and
enhance a model's stability.


This paper considers certain aspects of the interaction of the exchange
rate and current account during the adjustment of an open economy to an exogenous
shock. Its special focus is on the character of the adjustment path when the
current account reacts to an exchange-rate change in a delayed and staggered
fashion - i.e., with a so-called J-curve response. The study is motivated by the
frequent observation in recent popular accounts of exchange-market developments
that some currencies -- such as the yen, for example -- have been especially volatile be-
cause of the presence of pronounced J-curve effects. According to these accounts, trans-
actors in the foreign-exchange market are "misled" by the near-term, perverse effects of
an excharge-rate change on the current account. Their response gives rise, in turn,
to further pressures on the exchange rate in the same direction as that of the
original shift. As a result, this perverse feedback between the exchange rate and
current account will generate a movement of the exchange rate in the
"wrong" direction, at least until lagged,stabilizing current-account reactions come
into play. Some accounts have gone further to suggest that such a process may generate
endogenous cycles that overshoot the ultimate equilibrium and may even give rise to a
fundamentally unstable dynamic. In the latter case, some sort of intervention or
other braking device is required to maintain the system within acceptable bounds.

One purpose of this paper is to assess the extent to which these properties
are found in a more formal model of exchange-rate adjustment both in which J-curve effects
are present and which also spells out a more complete (and more supportable) view of
expectations formation./ To be specific, I adopt here the now conventional
“There have been a limited number of papers that deal with the relationship
between lagged trade responses and stability of the foreign-exchange market; leading ex-
amples are Williamson (1972) and Britton (1970). In general, these earlier models
of the foreign-exchange market are quite partial in nature and regard the determina-

tion of the exchange rate and its equilibrium as flow phenomena, A more recent paper
that takes a slightly different approach from this one is Driskill and McCafferty (1980) .


-lle-

First, for there to be any values of ain this (or any subsequent cycles)
that satisfy (8), it is easy to determine (from (8)) that the following

necessary condition on my and m must be met:

wo
sina

<
a*

(13) ie

where a* is the value of a in the interval, [1,27], for which
astan a .

Condition (13) simply says that a positive complex root requires that the
algebraic sum of own-variable effects in (3) and (4) not be of too large a
scale./
Consider now the two intercepts of P(B). (again referring only to the
interval,[1,27]). We designate these as P(0) and P9(0) corresyonding to
P(g) at 8=0, witha taken at the smaller and larger values that satisfy (10)
in the indicated range, respectively. ‘These values of a, I shall designate as
a, (0) and ap(0). Since m<a, (0) <oy(0)<2m, it can be confirmed that

P, (0)<P,(0).

The inverse of the relationship P(g) is continuous in the domain

1P,(0),P9(0)) , positive infor at least some P in this domain, and has at

most two zeros. Accordingly, a simply expressed sufficient conditicn (in addition

T/ Though it is rather restrictive, condition (13) is by no means impossible

to satisfy in a practical example. Recall from earlier discussions that a, and

bo, two components of m, that enter additively, are of opposite sign. Accordingly.
tney could very nearly Offset one another and, thereby, keep m, very small.


-14-
could be a negative association between the current account and the expected
exchange rate of, say, one period forward. These anticipation effects also
would have their counterpart in a more than offsetting positive relationship
between the contemporary levels of the current account and exchange rate as
anticipation effects are made up for in the subsequent period. In our nota-
tion, if we now ignore lagged terms, the presence of anticipation effects could

be shown in a modified version of (5.a.) as

(5.b.) E(t)+ cy E(t) + cgE(t) + FoE(t+0) + K = 0

where
c170 >
c <0,
F250 ,
0
and
<p V
cytf =c<0

Following the same procedures as before, the characteristic equation for (5.b.)

can be written as

(7.b.) H(z)=22 +mz+m, + soe =0.

where
mrocyo >
m=o°c<c 0 5
0°20
S,=6 £70 s
and

z a.
Sim e°c<0 .

T/ This specification assumes, in effect, short-run perfect foresight on the part
of traders in goods markets -- an assumption which seem reasonable in view of its
application to foreign-exchange market transactions elsewhere in the model.