-W7-

possible for plausible systemic "frictions" to invalidate the instability
often associated with the use of myopic perfect foresight in these models.

The contrast between findings in this case and the two previous, however,
suggests that some compromise with both the "myopic" and the "perfect"

aspects of this assumption are needed. It is noteworthy that neither making
exchange-rate forecasts non-instantaneous (Case II) nor systematically
obsolete (Case I) by itself is sufficient to bring on stability. In fact,
these two cases are associated with a higher order of instability than the
original model -- in the sense that a larger number of free variables is needed
to place the system, once disturbed, on its stable dynamic path.

It is also evident that, from a technical point of view, the manner
by which the particular constellation of assumptions in Case III brings about
stability is quite similar to that associated with an adaptive rule. Although
it is imbedded in a macro structure, the source of instability in these models
is in some fundamental sense in the overly rapid adjustment of the exchange-rate
forecasting process. Hence, in order to stabilize the system it is natural to
look first at mechanisms that either slow the forecasting process or the adjust-
ment of the exchange rate itself. Both an adaptive expectations rule and the
lag developed in Case III perform this function. The assumptions of Case III,
in effect, allow the substitution of the left-hand derivative for the right-
hand derivative of the current expected exchange rate for a given cohort of
transactcrs in the foreign-exchange market../ In episodes when the exchange rate

would otherwise tend to diverge at an accelerating pace away from equilibrium, it

1/ Actually, for any cohort of transactors, their common forecasts of the rate of
change of the exchange rate are taken only at separated points. These rates of
change can be applied to line segments that can then be joined in the obvious way
to form a continuous path for the expected exchange rate for this cohort, with
kinks at each time of market entry. It is the derivatives on the left-hand side
and right-hand sides of these kinked points that are referred to above. For a dis-
cussion of when such an expectations structure may be consistent with perfect fore-
sight in a similar example, see Gray and Turnovsky (1979b) and Burmeister and Turn-
ovsky (1976).


-12-

III. Intermittant participation with expectations based on data from
previous entry.

As a third alternative, consider next the following specification,
which combines features of the previous two: We continue to assume that
transactors in the foreign-exchange market enter only at intervals (defined
by © , as before), but now we assume that their projection of the rate of change
of the exchange rate for the next interval is the same as its actual rate of

change cver the most recent interval. Accordingly, equation (3) is revised to
ky

(3.c) E (t) = > (E(t) - E(t-0)) + ko V(t) + ky

Proceeding as before, (3.c) can be joined with (1) to obtain

(4.c) m E(t + 0) +m, E(t +0) + ny E (t) +n, E(t) + n=0

where

n= loro,
k, a

= 271. >

My =~ T= Ko ay <0 ?,

2
k,/0

1 202,

n= >
1 T= k,/0

7 ay k,/® 2 0?,
0 - k,/e

s
"

agky = kay

A=
1 - k,/0

Since the homogeneous part of the corresponding characteristic function has

exactly the same form as (5.b), corresponding equations for this case that are
identical in form to (6.b.1), (6.b.2), and (7.b) can be derived, with the only
difference lying in the interpretation of coefficients. In particular, in this

case the ratio m/ny is given by


-4-

system of linear difference-differential equations. Attention is limited
here to systems of order no larger than two in both differences and deriva-
tives, but similar conditions apply to higher-order systens /

First, it is convenient as a preliminary step to rearrange the
system's equations to form a single second-order, mixed equation in one of

the variables of the following form:
(i) L(x) + kK = 0,

where L(x) = q) x (t+ 2w) +a, x (t+ 2m) +q) x(t +2)

+

my x (tw) +m x (t+ wm) + mx(t + w)

+

ny x (t) +n, x (t) +n, x (t),
K is a constant,and w is an arbitrary finite lead. Substitution of a salution of

the form x = e** into L(x) allows us to express (i) as

(ii) L(e®*) + K = a(a) eh + k,

where J(A), the characteristic function corresponding to L, is

2)w 2 dw

(444) dA) = (ay ve qa + ale

+ (ny +n

+ (my wt mA + m,)e

y+ N,)-

As with simpler, non-mixed dynamic systems, the signs of the roots to the

characteristic equation,

(iv) JQA) =

1/ A useful summary of techniques for evaluating stability in mixed systems
is found in Gray and Turnovsky (1979b). For a more detailed and comprehen-
sive treatment, see Bellman and Cooke (1963).


- 18 -

is this feature which exercises a degree of control on the system and, if
strong enough, permits the return of the system to its equilibrium. As with
an adaptive rule, however, there is a clear tension between this specification
and strict rationality, since the potential for stability depends on agents
sustaining forecasts that are incorrect over finite intervals.
It is somewhat reassuring to be able to add one more argument to
what is still a fairly short list that indicates that the foreign exchange market
may be more stable than some recent models would have us believe. At this point,
it is a rather small comfort, however, since the conditions for stability are
still fairly specific and difficult to quantify. Until the model's structural
parameters are better known and the effects of some of its special assumptions
are better explored, it is difficult to say whether the critical lag is more on
the order of months or minutes. If it is the latter, however, it suggests that the
lags and frictions associated with the normal course of business may help to keep the
foreign exchange market under control, rather than contribute to its instability.
For similar reasons one has to be wary at this point of proposals
which would deliberately introduce or enlarge a lag of this type in the system
so as to achieve greater stability. Even putting the question of enforcement of
the lag aside for the moment, to be certain that greater stability would be the
result we would have to be assured that expectations are formed in the fashion de-
scribed in the model. If they were not -- and instead closer to the strictly
rational standard of Case II, for example -- then less rather than greater
stability might be the consequence. Also, sight should not be lost of the fact
that attempting to stabilize the system by this device might require the intro-
duction of a very large lag with its own attendant economic distortions on the real

side of the economy that we have not even considered here.


R.

- Dornbusch, "Exchange-Rate Economics:

BIBLIOGRAPHY

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

Press, New York, 1963.

Burmeister and A.R. Dobell (1970), Mathematical Theories of Economic Growth,

Macmillan, London, 1970.

. Cagan (1956), "The Monetary Dynamics of Hyperinflation," in M. Friedman, ed.,

Studies in the Quantity Theory of Money, Chicago University Press,
Chicago, 1956.

. Dornbusch (1976), "Exchange Rate Expectation and Monetary Policy,"

Journal of Political Economy, December 1976.

Where Do We Stand?", Brookings Papers
on Economic Activity, 1980:1.

. Freeman (1982), "J-Curves and Stability of the Foreign-Exchange Market,"

International Finance Discussion Paper No.198, Board of Governors

of the Federal Reserve System, Washington, D.C., January 1982.

. Gray and S. Turnovsky (1979a), "The Stability of Exchange Rate Dynamics Under

Perfect Myopic Foresight," Journal of International Economics,

October 1979.

. Gray and S. Turnovsky (1979b), “Expectational Consistency, Information 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, April, 1976.

Tobin (1978), "A Proposal for International Monetary Reform," Cowles Founda-

tion Discussion Paper #506, Yale University, October 1978.


-15-

however, it is evident that for these roots we seek to establish conditions
(i.e., conditions on underlying parameters of the system) such that both
F(A) > 0 and G'(A) > 0 for all even-labeled roots ./ By using (9.c.1) and
(9.c.2) together with G(A) = 0, one can show that this task is equivalent to

establishing conditions under which

2
-m. mM. nom.
- 1 A sind ao San A A} 4 a
(10.c) mingb 1° 1 Yo) 1,
[x m, sind m, sind
—— ) Sin A] - | —— | - J
pm 1 A wT nN

for all even-labeled roots. Taking note of the fact that sin <0 for all even-
labeled roots, it is apparent that all terms on the left-hand side of (10.c) are
positive. It also can be confirmed that increases in 0 will cause each of the
three major terms in each line on the left-hand side of (10.c) to increase,

with the exception of the third term in the lower line (which declines but always
remains positive).

Although condition (10.c) is exact, because it involves endogeneously
determined values of X (and an infinite set of them, as well) it fails to shed much:
light on the circumstances under which (10.c) holds and, thereby, under which
C.(2) car be met for all roots. Our main interest here, however, is on the effect
of introcucing the lag © on stability. Focusing on © and holding other parameters
constant, it is readily shown that for sufficiently small values of 0 (i.e., ve lies
of © that approach 2k, from above) inequality (10.c) cannot be satisfied since
sin A and, therefore, the left-hand side of the second line in (10.c) can be made
arbitrarily close to zero.

T/7 The distinction being made here is that we can reject the other possible way to
satisfy G'F>0 for alla -- i.e., both G'<0 and F<0.


m, = 1/120,

mi => (ky a, + ay) >0,
_ 2

Ny => k/t <0,

ny = a) ky/t <0,

n= (ay ky - ke ay) -

The homogeneous part of the corresponding characteristic function is,

there“ore,
(5a) Ja) = (ma +m) eX + (nga® + nya).

It is evident by inspection that this polynomial has no principal term;
hence, on the basis of condition B., we can determine immediately that an

infornation lag in the formation of exchange-rate expectations -- of any

V/

duration -- cannot serve to stabilize the original instantaneous system.—
In fact, when there is no principal term, the characteristic equation has
an infinity of roots with a positive real part. This profusion of unstable

roots means that even the usual device of an initial jump in the exchange rate

will be inadequate to bring the system to a stable trajectory.2/

Il. Intermittant participation in the asset market

As an alternative approach, let us suppose instead that individual
transactors in the asset market do not or cannot participate on a continuous
basis. Instead, they enter the market at intervals no shorter than @ periods
in duration, say, either because of a minimum holding-period requirement on
assets or because, as a practical matter, the bidding and performance on con-

tracts can only be done at discrete intervals, In fact, let us assume that the

1/ In some other versions of this case (not formally reported here) in which
additional leads and lags to other variables were added, this characteristic
indicating instability in the system was found to persist. For example, a

version in which the level of wealth in equation (3) was lagged as well -- pro-
ducing an effect not unlike a transmission lag in the foreign-exchange market --
had 10 principal term.

2/ One possible approach to resolving this dilemma is to suspend equation (3.a) for
a period of length t following a shock and assume that in this interval the system
finds a segment of a continuous (possibly unique), stable trajectory by, in effect,
making shifts in E(t + Y) for all real y in [0,t] » an infinite number of
momentary shifts.



~6-
Next, let us write J(iA) as
J(iA) = F(A) + i GA).
Then, if J(A) has a principal term (of order (r,s)), necessary and
sufficient conditions for A. to hold are that V
C. (1) All the roots of G(A) = 0 are real, and

(2) G'(A) + F(A) >0 for all these roots.

Finally, a useful procedure for establishing C.(1) above involves application
of the following property:
For the function G(A) above to have only real roots, it is necessary anc

sufficient that

D. In the interval [> (2nk -6), (2nk +6)] where 67s an

arbitrary constant, and k is a positive integer,

G(A) must have exactly 4sk+r real roots, starting with an

interval defined by a sufficiently large k.

Various combinations of these conditions are used below to diagnose
the stability properties of several variations on a basic perfect-foresight

model. This basic model is presented next.

The Basic Open-Economy Model

A fairly broad class of models of an open economy with a continuously
adjusting exchange rate can be represented in linear form by the following two

semi-reduced-form, dynamic equations:

1/ This statement of necessary and sufficient conditions, originally due to Pont-
ryagin, is but one of several alternative versions. For more detail, see Bellman
and Cooke (1963), Chapter 13. Depending on the problem, other versions may be
more convenient to apply.


-5-

are crucial to determining stability./

Necessary and sufficient conditions that equation (iv) and the

system from which it is derived be globally stable are

A. (1) All characteristic roots (i.e., the roots of (iv)) have
non-positive real parts, and
(2) All characteristic roots with a zero real part be non-

complex.

A useful concept for assessing whether or not condition A. is met is that of a
principal term. In this context the latter is a term, Cog arTe SMW of the

polynomial
v s :
ma,eTM)= 2 2 cal el,

such that c, . 40 and for any other term oem with C4510» either r>i, s>j
’

or r>i. $= j orr=i, s>j. It has been shown that condition A. is met only

if

B. Polynomial J(A) in (iii) has a principal term.

Otherwise, J(A) = 0 has an infinity of roots with a positive real part. Hence,
the presence of a principal term is a necessary condition for stability; its

absence is indicative of endogenous instability in the original system.

1/ Note that the characteristic equation is typically a transcendental equation
which has an infinite number of roots -- a fact which often compounds considerably
the difficulties encountered in solution and analysis of stability.


- 16 -

Thus, as. might be expected on the basis of the original basic model, this
system is not globally stable for very short lags.

Furthermore, it is relatively easy to establish that the system
can be made globally stable by a sufficiently large value of © . To confirm
this important property, we offer the following argument: Let us define x

as the smallest even-labeled root (for a given ©) such that

- m2
AsinA >]
mM
is satisfied for all A>’. (The existence of such a X is guaranteed by a

non-zero lower bound on |sin A|.) Since all these roots satisfy (10.c) for
this or any larger value of © , we now concentrate on the finite set of even-
labeled roots for which A <i. It is easily shown that increases in 0 will
expand the amplitude of both L(A) and R(A) in Figure 2. As can be seen, this
will tend to increase the value of all even-labeled roots and, more importantly,

raise R(X) for these roots. Since

m. +
R(x) = - <2 SA,

it is always possible to find a sufficiently large © (the least of which we
designate as 6 ) such that R(A) >1 for all A<i, thereby satisfying (10.c).
Accordingly, for a> condition C.(2) is satisfied for all roots, condition A.

is met as well, and the system must be globally stable./

Additional Comments

Several points that bear on stability in international macromodels are
illustrated by the three examples above. First, Case III shows that it is
1/ In practice, global stability may be achieved by values of © much below 6.

The value of © is of interest only in that it guarantees that, for a sufficiently
large © , the system can be made stable.


system is dislodged from equilibrium, unless there is some mechanism (such
as a jump in a variable) to bring the system to its unique, stable trajectory,

jt will diverge from equilibrium. /

Model Variations
I. Information lags in expectations formation

Consider now the effect on the basic model if current expectations
of the rate of change of the exchange rate are always formed on the basis of
obsolete information -- either because the market provides information to
participants with a lag or because there is some systematic delay in market
participants' processing of information. In either case, if tis a measure of this

lag, we must rewrite equation (3) as
.a) =k E(t-
(3-4) E(t) kK, E (t-t) + ky V(t) + ko:

Equations (1) and (3-a) now constitute a mixed differential-difference equation
system. To determine how stability properties are influenced by this lag, we
make use of the procedures and conditions discussed earlier.

First, by differentiation of (3-4) and suitable algebraic manipula-

tions, we can obtain the following second-order equation in E:
(4.a) mE (t +t) +m, (t+) + ny E(t) +n, E(t) +H = 0,

where

1/ The system in (1) and (2) can be stabilized by introducing an adaptive rule of
the following type for the rate of change of the expected exchange rate:

a(t) = a (E(t) - e(t)] a > 0

It can be shown that, when a is sufficiently small, all characteristic roots
of the system formed by (1), (2), and the adaptive rule above are negative,
making the system globally stable.


International Finance Discussion Papers
Number 199

January 1982

STRUCTURAL LAGS AND STABILITY IN INTERNATIONAL MACROMODELS

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 .e cleared with the author or authors.


-ll-
The homogeneous part of the corresponding characteristic function is,
therefore,
(5.b) d(A) = (mr +m )er®s (nna +n),
° 1 0
which clearly has a principal term. Proceeding as before, when we write

J(iA) as F(A) + i GA), we obtain

(6.b.1) F(A) = m, cos 2 - m Asin A +n);

(6.b.2) G(A)

m, d cos At m, sin A+ ny he

Condition C. above includes the requirement that all the zeros of G(A) be real,
which may be established in turn via condition D. It is evident by inspection
that A = 0 is one solution to G(\) = 0. To help in evaluating the other zeros

of G(A), we divide (6.b.4 by X and write it as

m. mM cs

1 =. 9 sink
(7b) Vt Ry cos A= - 7 yO
The ‘eft and right-hand sides of (7.b) are depicted in Figure 1 as L(A) and R(A),
respectively. Since sin /A diminishes toward zero as A gets large, L(A) will
continue to intersect R(A) for large values of only if |m,/n4 | >1. But, re-
ferring back to (4.b), in this case

6/k

- 1
Im/ml = | rere bs)

Accordingly , condition D. can never be satisfied by a positive value of 6 of
any size, and therefore, ‘this system cannot be made globally stable by the intro-

duction of a lead in this form.


- 10 -

intervals between entry are exactly of size 0. Also, to finesse problems that
might arise in aggregation across individuals, we assume that the timing of
transactors' participation is staggered so that active transactors are dis-
tributed evenly over any interval of length ©, and that each transactor holds
a proportionate share of total wealth./ Accordingly, if we maintain a strict
perfect-foresight rule (though no longer a myopic perfect-foresight rule), the
appropriate modification of (3) is

ky
(3.b) E(t) = (E(t +0) = E(t) + kp V(t) + ky.
Thus, the average per-period rate of change of the exchange rate (E(t + 9) - E(t))/0,
replaces e(t) in equation (3). In effect, in (3.b) it is assumed that any market
participant at a given time, t, forecasts E for the time of his next entry into the
market, t + 0, and that this forecast is exactly equal to the actual outturn,
“E(t + 0).

Together with (1), this equation can be used to obtain the following

first-order equation in E

(4.b) m, E(t + ©) + m, E(t + ©) +n, E(t) + ny E(t) + n=0,

where
k 2
1/0
m= _ 39
1 1 + k,/9 ?
-a
_ 2
"oo" T¥kK7e 7 o>
ny = 1/a>0,
ka
- 21
"o - T¥e7e * 32 <0>
a= kpa ~ aak,
T+ k,/0 .

1/ A model which deals with this issue more elegantly is found in Gray and
Turnovsky (1979b). The extra detail related to this point appears to make
little difference to the qualitative results, however.


(1) V(t)

ay E(t) + ay V(t) + a5»

(2) E(t) = ky eft) + kp V(t) + ky,

where V_ is the level of home-country wealth,

E is the exchange rate (in units of home/foreign currency) ,

e is the expected instantaneous rate of change of the exchange rate,
and ay> ag» a» ky» Kos and ky are constants. The following sign conventions

also apply:

>0, k,< 9.

a.>0, a, <0, k 2

1

Equation (1), which expresses goods-market equilibrium, reflects the
fact that the current account (measured by v) will tend to be raised by a depre-
ciation of the exchange rate (higher E) or by reduced domestic wealth (lower V).
Equilibrium in the asset market, as shown in (2) implies that a higher current
exchange rate (i.e., a lower valued home currency) is required when there is an
expectation of more rapid depreciation (e higher) or when domestic wealth (V)
is smaller.

It is conventional to introduce myopic perfect foresight into this

model by substituting E(t) for e(t) in (2) so that
(3) E(t) = ky E(t) + ky W(t) + ky.

Equations (1) and (3) then constitute a well defined system in two variables
which displays the conditional stability property referred to earlier, Under
the sign conventions indicated, one of the roots of the corresponding character-

istic equation for this system is assured to be positive. Accordingly, if the


-19 -

Despite these qualifications I feel that it is of more than theoret-
ical interest that global stability can be induced through a structural lag,

since in many contexts this approach is more easily supported than some other
competing stabilizing devices. Furthermore, for reasons mentioned above, we

have deliberately confined attention to lags related to equation (2), the asset-
market equilibrium equation. It is interesting to speculate whether some form

of lag in the goods-market relationship -- where lags have a long tradition and
are, if anything, likely to be more pronounced -- might be an additional source of
stability. This, however, is an open question that must be left for later

research._/

1/ Some of these possibilities have been explored in another paper (Freeman, 1982)

in which it was found that the introduction of "J-curve" effects into the current
account tended, if anything, to’increase the instability of the system.


-3-

has been suggested by more than a few analysts as a means of reducing the
market's instability, and, as we shall see, not all types of "sand" will do
for this task.1/ Some may even give rise to greater variability in the for-
eign exchanges.

From a technical point of view, the introduction of a finite lead
or lag into an otherwise continuous dynamic model gives rise to a system of
mixed differential-difference equations. Since the formal analysis of these
mixed systems is considerably more involved than it is for their simpler, non-
mixed counterparts, in the next section we present the essentials for determ-
ining stability. These conditions are then used to evaluate the long-run
properties of variations on a basic model. Following the classical format
of most well-constructed fairy tales and many humorous aneccoi.., a trio of
cases are described in detail -- the first two of which are false leads that
are shown to fail the stability tests. The point of including these cases is
not only the conventional one of building the requisite suspense, but also to
illustrate by examples that fail the nature of the conditions that must be met.
The impetuous reader is welcome to jump ahead to the case that works, but
should be forewarned that the findings there are limited at this point to a
particular, somewhat special case. How far they can be generalized is not clear

yet, but the preceding cases at least suggest the character of those limits.

Stability in Mixed Systems

In this section, we shall spell out briefly and without formal proof

conditions that are necessary and sufficient for global stability of a mixed

1/ For example, Tobin has recently suggested that this might be done through
a tax on foreign-exchange conversions (Tobin, 1978), a view which is discussed
(and oppesed) by Dornbusch (1980).


-13-

&

m/ny = ky -1

Evidently, for relatively small values of 0, i.e., when ~2< 2, the ratio

1
m/n, will be no greater than unity in absolute value, and condition D.

cannot be met for the same reason as in the previous case. However, when the lag
measured by © is sufficiently large, i.e., when naa 2, then L(A) will continue
to intersect R(A) at indefinitely large positive (and negative) values for ) .
An illustration is provided in Figure 2

To see whether or not condition D. can be satisfied when c > 2, let
us determine first if the system's characteristic equation has the correct number
of roots. Since the principal term of (5.b) is of order (1,1), condition D.
requires there to be 4k + 1 real roots over intervals of size (-2kr-5), (2kn+6 ) |
for a sufficiently large value of k. We have already taken note of the fact that
zero is one of those real roots. Also, since L(A) and R(A) are symmetric about
d = 0, there are the same number of negative and positive roots, and we can restrict
our attention to counting the latter. If we iabel the positive roots successively
starting from ro = 0 as AgeAq Ag etc., it may be seen that the odd-labeled roots
h ;55°1,3,5,7-..| are found, one each, in the segments [in (j+ 2) | -- i.e., one
root in each occurance of the "second quadrant", when A is interpreted as a radial
measure. Similarly, even-labeled roots ( .3

J
the segments (i-1)1,50 | -- i.e, in either the "third" or "fourth quadrants" in

j=2,4,6,8...) are located, one each, in

each "cycle" of X. Furthermore, the occurances in the "fourth quadrant", if relevant
at all, are limited to relatively small A's in the sequence of even-labeled roots.

Hence, if we take the arbitrary constant 6 to be zero, it is clear that G(A) will

1/ When the latter condition is met, coefficients Mg> M+ No» and ny in (3.c) are
all positive.


-14-

have exactly 4k+1 zeros for any integer k on an interval of size [-2kn, 2kr |.
Condition D. is met, and this establishes in turn condition C.(1).

Next, we need to determine the conditions under which C.(2) is met,
i.e., conditions such that G'(A)-F(A)>1 for all zeros of G(A). The function
G'(A) is found by differentiating (6.b.2) to get

(8.c) G'(A) = n+ (m + m,) cos A-m AsindA,
while F(A), given earlier, is
(6.b.1) F(A) = ny +m, cos A-mAsin ad.

First, for Age we have

&'(X)
F(A)

1+m) +m, > 0,

+ >
my 4 0,

and condition C.(2) is clearly met for this root. Next, consider the odd-labeled
roots ../ Let us use G(A) = 0 to rewrite (6.b.1) and (8.c) as

2
nym . mi
: Lo sin i 0 2
(9.c.1) F(A) = my - 7, - con eae | +m "| >

2
nym - TM
' _ To _ ;sink () 2 .
(9.c.2) G'(A) = - TM or ame ) fe +m, +m 2 ]

For odd-labeled roots (all of which lie in the "second quadrant"), sin A/A is

mM

positive. Also, it is readily shown that m - is negative. Hence, boti:
F(A) and G'(A) must be negative, and C.(2) is always met for odd-labeled roots.
Conditions that relate to the even-labeled roots are more involved,
largely because, for roots located the "third quadrant", sin A and cos ) carry the
same algebraic sign. Inasmuch as the even-labeled roots increase without bound,

T/ Since F(A), G(A) and G'(\) are all symmetric around A=0, we need only examine
properties of the positive roots.


-2-

limiting cases of a more general specification with stability conditions
that are more forgiving -- then the burden is shifted to the modeler to
spell out more exactly the character of this more general case.

In this connection, it is known that models of the type described
above sometimes can be transformed into models that are globally stable by
inserting an adaptive-expectations rule in place of strict myopic perfect
foresight. This modification has the unappealing feature, however, that
it replaces an assumption that forecasters are always right with a mechanical
rule that says, in effect, that forecasters are almost always wrong. Not
only does this run counter to the spirit of rational expectations that per-
vades most of modern macroeconomic theory, but also in such a modified model
global stability occurs only if certain limiting restrictions are met. When
they are not, the original dilemma remains /

In the exercises below, we take a somewhat different approach. In-
stead, we look at the effect of introducing various discrete lags and leads
into a basic international model to see which, if any, can give rise to global
stability in a system which otherwise would be only conditionally stable.

Although this modeling procedure is subject to some of the same shortcomings as

is the use of an adaptive-expectations rule, the cases considered below have the
attraction that their leads and lags can be related to plausible structural
features of a real system. The findings here are aimed primarily at assessing

a particular technical issue, but they are not without some policy implications
as well. "Throwing some sand into the wheels" of the foreign exchange market

J/ Slowly adjusting adaptive expectations. stabilize the simple Cagan model (Cagan,
1956), for example, and the same device has been used in other somewhat more elab-
orate frameworks. (See, for instance, Burmeister and Dobell, (1970), p. 186-189).

The use of this approach is also discussed in footnote 1 > page 8 below. Gray

and Turnovsky (1979a) consider an alternative approa i
exchange-rate adjustment itself. Pproachs the sTowing down of the


One of the features of many popular international macromodels that
is troublesome is their apparent sensitivity to very slight changes in
assumptions and initial conditions. This is especially the case for models
with an asset sector in which holdings depend on the instantaneous expected
rate of change of a variable, such as the exchange rate, and in which asset
holders are taken to exhibit "myopic perfect foresight" in making their fore-
casts. Typically these models are only conditionally stable -- i.e., if they
are disturbed from equilibrium, some free variable (often the current exchange
rate) must accomplish a discrete, exactly correct jump so as to put the system
on its unique stable trajectory to the new equilibrium. In the strict version
of these models, even the slightest deviation from this path or the smallest
error in this jump will lead -- in the absence of intervention or other bounds
on the system -- to its collapse through an explosive movement of the exchange
rate and other related variables ./

The question of how best to interpret this theoretical tendency toward
instability presents a dilemma. On one hand, to accept exactly correct jumps
in variables as an empirical proposition seems to place an unduly heavy burden
on the market's power to achieve intertemporal equilibrium on its own, partic-
ularly when forecasting skills in actual practice surely fall short of the
perfect--foresight ideal. This would suggest that the actual foreign exchange
market has a very high degree of endogenous instability and, therefore, must
in practice be limited by some other constraints. On the other hand, if we

regard such continuous models as no more than convenient approximations --

1/ Among the many models of this type now in print, two -- those by Dornbusch
(1976) and Kouri (1976) -- have been especially influential. A fairly compre-
hensive treatment of conditional stability, or the saddlepath poroperty as it is
often raferred to, is found in Gray and Turnovsky (1979a). The problem is not
confined to international models; it is a potential issue in almost any model
with an asset sector.


T aanBry

(x)1

(x)u *(x)1



LZ -

Lg-

(xa *(¥)1

vA @anBLy