-17-

allocates its (fixed) income to consumption, transactions costs, and
increments (positive or negative) to its money balances, as shown in
equation (16).

Following the procedures outlined in section II for Brock's problem,
we can solve for the set of price and consumption paths that satisfy both
the usual necessary conditions for an optimum and the requirements of goods
and money market equilibrium. This set consists of al] price and
consumption paths that satisfy both the Euler equation and the budge

constraint for this problem -- that is, equations (17) and (18) below.

Pe
(17) P, = -a'(m,) -ps for all t.
(18) CRY - n(m,)s for all t.

It. will prove useful later, in our discussion of transversality
condit“ons, to have established an arbitrage interpretation of equation
(17). The Euler equation can be usefully regarded as a restriction on the
path of prices that must hold if various finite horizon arbitrage schemes
(or "perturbations") are to be unprofitable along that path. This
restriction must be met if the path is optimal; accordingly, it is a
necessary (but not sufficient) condition for an optimum. For the purposes
of illustration, we will be considering a very specific arbitrage scheme.
The Euler equation can, however, be developed from any feasible finite
horizon perturbation.

Consider a candidate equilibrium solution to the problem posed in

equations (15) and (16) above -- an arbitrary price path and an associated


-~ 35 -

all type A paths -- can not be ruled out on the same grounds. Along these
paths, the time rate of change of real balances slows as the level of real
balances falls, nearing zero as real balances go to zero. As time goes to
infinity, then, the level of real balances asymtotically approaches zero.
From the foregoing discussion, it is evident that we can eliminate as
an equilibrium solution any implosive real balance path of type B or C.
Accordingly, it is possible to rule out all paths that satisfy the following

condition:

(39) lim m.n'(m,) < 0.
m0

Equation (14) of section II gives the analogous condition for the Brock
model.

The preceding discussion suggests an obvious strategy for those
interested in insuring uniqueness of equilibrium in the models they work
with: Employ only models whose dynamic properties satisfy equation (39) or
its equivalent. The problem with this strategy is that the conditions under
which equation (39) holds are not economically relevant conditions. The
same is true of equation (14) for Brock's model. This argument is made by
Obstfeld and Rogoff in a recent paper on hyperinflationary
equilibria 4/ Using Brock's model, they show that the conditions
under which implosive real balance paths can be excluded also imply that the
utility of real balances necessarily goes to negative infinity as real
balances go to zero. This, it may argued, does not reflect a reasonable

view of the economic role of real money balances.


- 31 -

money balances. Accordingly explosive real balance paths -- or,
alternatively, implosive price paths -- offer the household at least one
profitable arbitrage opportunity.

Differentiation of equation (34) with respect to T demonstrates an
additional point: The discounted gains associated with the decumulation of
u units of money are independent of when the decumulation takes place. The
derivative calculated in equation (36) does not vary with T. Thus, if such
an arbitrage is profitable at any one point in time, it is equally
profitable at all points in time. In the case of explosive ‘real balance
paths, then, the household has an incentive to reduce its money holdings,
but is indifferent with regard to timing. This result is also implicit in
the Euler equation given by equation (17). The relationship shown there
could equally well have been derived by writing down the conditions under
which the household would be indifferent between exchanging a unit of money
for goods today, and the same action carried out ata later date.

Finally, from equation (36) we can obtain a transversality condition
that is a necessary condition for an optimum in this model. Consider once
again the expression on the left-hand-side of equation (36). It is
well-defined for any finite value of T+] along any path of prices and real
balances that imply feasible (non-negative) levels of consumption. It's
limiting value as T+] goes to infinity is a positive constant for all
explosive real balance paths and is zero for all implosive real balance
paths and the steady state. For equation (36) to hold, then, the integral
on its right-hand-side must have a finite value -- that is, it must
converge. This means that as T+] becomes arbitrarily large, the value of

the integral must approach zero. This, in turn, implies that the leftt-hand-


- 40 -

Thus, identifying the specific economic role of money is shown to be
important for some questions.

A critical feature of Brock's model is the assumption of separable
utility. Suppose instead that the household's utility function takes the

more gereral form given by equation (3')

1 _* -pt
(3') 2 *f e U(cy m,)dt,
where

ue >0, Vee § 0,
uy > 0, am < 0.

If equation (3) of section II is replaced by equation (3'), the resulting

optimization: problem produces an Euler equation of the form

cae) fee alte 2
t tam “c

In the case of separable utility, the cross-term Vom is equal to zero
and equation (42) reduces to equation (10) of section II. The dynamic
properties of this more general model will be the same as those of the
simpler model as long as Vom is not negative and large enough to make
the denominator of the right side of equation (42) negative. Thus, a
condition sufficient to guarantee a saddle-point in the general model is

(43) Ugg > 0»


-ll-

. v'(m,) m
(13) im m, = lim m,[p - J=lim -—-———.

nro m0 u'(y) m,+0 u'(y)

If this limit is zero, my goes to zero as Mm, goes to zero,
resulting in path A. If the limit is a constant, path B results. And if
the limit doesn't exist (it is negative infinity), path C results.

All of the real balance paths described above may be regarded as
candidate equilibrium solutions. Following Brock, we will divide them into
three groups and study then further. Consider first all the paths along
which real balances are growing over time. It is possible to show that none
of these paths are true equilibrium solutions. Formally, they may be ruled
out because they violate a transversality condition. Intuitively, it can be
shown that households can increase their utility by running dow their money
balances along such. paths. These arguments are examined in greater detail
in section IV. For now we simply note that it is possible to exclude as
true eqilibria explosive real balance paths (or, equivalently, implosive
price paths). This is true in Brock's model as well as the transactions
cost framework of the next section.

Consider next all the paths along which real balances are shrinking
over time. Some of these paths can be rule out because they imply that real
balances turn negative at some (finite) point in time. This is true of all
paths of type B and type C. Along these paths, the time rate of change of
real balances goes to a negative constant or negative infinity as the level
of real balance approaches zero. At some point, then, the equations of

motion that generate these paths imply that real balances pass through zero


- 46 -

References

Brock, William A. "A Note on Hyper-inflationary Equilibria in Long Run
Perfect Foresight Monetary Models: A Correction." (mimeograph) .

"Money and Growth: The Case of Long Run Perfect Foresight.”
“International Economic Review 15, No. 3 (October 1974): 750-77.

"A Simple Perfect Foresight Monetary Model." Journal of Monetary
“Economics 1, (April 1975): 133-50.

Calvo, Guillermo A. "On Models of Money and Perfect Foresight."
International Economic Review 20, No. 1 (February 1979): 83-103.

Dornbusch, Rudiger. “Expectations and Exchange Rate Dynamics." Journal of
Political Economy 84, No. 6 (December 1976): 1161-76.

Gray, Jo Anna and Salant, Stephen. Transversality Conditions in Infinite
Horizon Models." International Finance Discussion Paper 4172 (January,
1981).

Intriligator, Michael D. Mathematical Optimization and Economic Theory.
Prentice-Hall, Inc. (Englewood Cliffs, N.J.), 1971.

Kouri, Pentis "The Exchange Rate and the Balance of Payments in the Short
Run and in the Long Run: A Monetary Approach." Scandinavian Journal
of Economics (1976): 281-304.

Lucas, Robert. "An Equilibrium Model of the Business Cycle." Journal of
Political Economy 83, No. 6 (December 1975): 1113-44.

Obstfeld, Maruice and Rogoff, Kenneth. "Speculative Hyperinflations in

Maximizing Models: Can We Rule Them Outr" Mimeograph, Federal Reserve
Board (November 1981).

Sargent, Thomas J. and Wallace, Neil. "The Stability of Models of MOney and
Growth with Perfect Foresight." Econometrica 41, No. 6 (November
1973): 1043-48.


- 20 -

where
0 0 fort < T.
t-T . 1 forT<t<T+1.
n(t) = and n(t) =
T+2-t -1 forT+1<¢t¢T+2.
0) 0 fort >T+ 2.

The corresponding path of consumption is given by

aA 0 .
M M, - un(t)
(21) ch sy = w(t) - t= y- (4) + mat
PPP P P'
t t t t

Note that (19) through (21) are correct for any proposed solution, not
just the constant price and consumption paths depicted in figure 3.

It is now possible to evaluate the change in household utility
associated with this arbitrage scheme. Begin by substituting equations
(19) through (21) into the expression for lifetime household utility given
by equation (15). This produces a characterization of lifetime utility

involving u, prices, the time T, and other exogenous variables:

0

T a
(22) 2(u) =F ULy - 1(G) Je Prat
0 po
T42 Ho - un(t) lt) sot
+S ULy - (E——) + walt ye Ptat
f

Pe Pe
@ yo
+f ULy - w(t) Je Pat.
T+2 Pe


- 36 -

It is, however, easier to discuss the economic role of money in a
model in which money is assigned an explicit economic function. The
Obstfeld-Rogoff argument can, of course, be made in the context of the
transactions cost framework introduced in this paper. It can be shown that
if equation (39) holds, it must be true that transactions costs become
infinite as real balances go to zero. This is infeasible for an economy in
which resources are limited since it implies negative levels of consumption.

It has been established that the only economically relevant
specifications of transactions costs (or utility, for Brock's model) are
those that produce implosive real balance paths of type A. As already shown
these paths can not be ruled out on the grounds that they lead to infeasible
(negative) values of the price level. Nor do they provide unexploited
arbitrage opportunities. They can, nevertheless, be ruled out as
equilibrium solutions if they lead to infeasible (negative) levels of
consumption. This can not occur in Brock's model, but it can occur in a
transactions cost model. In particular, it will occur for any specification
of the transactions technology that generates transactions costs in excess
of household income as real balances become arbitrarily small. It is
possible, then, to rule out implosive real balance paths for specifications

under which it is true that

(40) lim n(m,) > Ye

me 0

Accordingly, for the transactions cost model of this paper, there are
conditions under which it is possible to rule out implosive real balance

path of type A. Again, however, these conditions are not economically


In the last section of the paper, more general versions of both
Brock's model and the transactions cost model are presented. These more
general models are not necessarily. dynamically unstable. It is argued that
saddle-point instability is a desirable property of models intended tc
explain the behavior of variables (such as exchange rates or interest rates)
that are observed to "jump" in response to exogenous shocks. A condition is
developed that guarantees saddle-point dynamics in the general versions of
the two models of this paper. In Brock's model, there is no a priori
presumption as to whether or not this condition should be net. In the
transactions cost model of this paper, the condition is necessarily
violated. However, in an alternative transactions cost framework the
condition is necessarily satisfied. The conclusion of the section is that
the answers to some questions may depend critically on the specific story
one chooses to tell about why households hold money. Models in which money
enters the utility function directly can not adequately answer these

questions.


- 29 -

Along any optimal path of prices and consumption it must be true

that
(35) at = 0 for all T.

u=0

Since all the paths under consideration in this section satisfy the Euler
equation (equation (17)), the first integral on the right hand side of
equation (34) is equal to zero. Thus, for equation (35) to hold, it must be

true that

ured, yere(T*)
(36) TH rut (cy at (md) (1/P0ye dt for att Te
Prat T+

Along any optimal solution, then, equation (36) must hold. The
interpretation of this condition is closely related to the intuitive
arguments set out earlier. The left-hand-side of equation (36) gives the
present discounted value (in terms of utility) of the additional consumption
the household can obtain by giving up one unit of nominal money balances at
time T+]. The right-hand-side of the equation gives the present discounted
value (in terms of utility )of the stream of increased transactions costs
incurred if money balances are lowered by one unit from time T+] on. If the
two sides are equal, the proposed arbitrage is unprofitable.

In order to establish the existence of profitable arbitrage
opportunities along explosive real balance paths, it is sufficient to

demonstrate that equation (36) does not hold along such paths. To do this,


- 18 -

consumption path that satisfy the budget constraint given by equation (16).
Given the consumption path, the price path is an equilibrium price path if
i, = 0 at every instant in time along the path. The household

must be satisfied with its money holdings at every point in time. Thus, for
equilibrium price and consumption paths, consumption must be equal to real
income less real transactions costs. That is, equation (18) must be
satisfied. In addition, however, the proposed consumption path must be
optimal for the household, given the proposed price path. The consumption
path will be optimal if all perturbations of the path that satisfy the
household's budget constraint produce no increase in the household's
utility. An example of such a perturbation would be a decumulation of the
household's nominal money balances for some period of time, followed by a
subsequent accumu] ation that just restores the household's nominal balances
to their initial level. For the household's budget constraint to be
satisfied, this requires a corresponding initial increase in the household's
rate of consumption followed by a decrease, all relative to the original
proposed path of consumption. Because this perturbation involves an
"exchange" of consumption during one time period for consumption during
another, it is referred to as an arbitrage scheme, or simply an arbitrage,
in what follows.

If an arbitrage scheme like the one proposed above raised the
household's utility, the household would exploit that opportunity and
attempt to decumulate money for a period of time. This, however, would mean
that the household is not satisfied with its money holdings at every point
in time along the proposed price and consumption paths. Accordingly, the

Proposed paths could not constitute an equilibrium solution.


m* my,
A ae
ta
=
& ta
A
for A: lim mm (in) =0
m,+0
for B: lim mr! (m,) = constant
m>0
t
for C: lim mr'(m,) = - ©
m,>0 t t
Figure 4

The Dynamics of the Transactions
Cost Model


An Example of a Finite Horizon Perturbation

t
u(5-)
Pe
[ee it
_ 1 . wet
| 1 1 | “pve
a ' t
'
1 ' !
4 t t
0 T THI TH2 0 T TH TH t
o
~~~
0 | ! 0
M
re T Pa t T 7 C.
1 ~s 1 _—"M 1 | | ‘A
panne : Lich
1
\ \ \ t i H 1 t
0 T T+ T+2 0 T T+ T+2
0 _ A
Pea Pe
t
. 0
Figure 3


- 30 -

consider the following conceptual exercise. Suppose that for some time T+]
equal to to equation (36) holds. Now think about values of T+1 greater
than to: As T+1 increases, the left side of equation (36) decreases,

but approaches a positive constant (a lower limit) as T+] becomes
arbitrarily large. This follows from the fact that U' is a constant and
Pray decreases with time at a rate that is less than p at any finite

point in time, but approaches p as T+] becomes arbitrarily large. (See
equation (17).) As T+1 becomes arbitrarily large, the rate of change of
ee (T#1) and Pray both have limiting values of p. Consequently,

their ratio approaches a positive constant.

By contrast, the right side of equation (36) approaches zero as T+]
becomes large. This is shown by evaluating the integrand as time (and real
balances) become infinite. It has just been established that the first term
in the integrand is a constant and that the product of the last two terms in
the integrand goes to a constant as time becomes infinite. That leaves the
term n(n) » Which, under any economically rel evant
specification // must approach zero as no becomes large. The
limiting value of the integrand, then, is also zero along any explosive real
balance path. It follows that as T+] becomes arbitrarily large, the
integral on the right side of equation (36) becomes arbitrarily small.

From the preceding arguments, the following may be concluded: If
equation (36) holds for some T+] equal to to along an explosive real
balance path, there will exist a time ty greater than to for which
it will be true that equation (36) is volated. Thus, along any explosive
real balance path, there is at least one point in time at which it will be

profitable for the household to permanently decrease it holdings of nominal


-15-

transactions cost model of the present section. In Brock's model, this
condition (which is sufficient, but not necessary) will be met under one set
of (apparently) reasonable restrictions on the household's utility function
but viclated under another. In the transactions cost model discussed below,
this ccndition is violated under any reasonable specification of the
househcld's utility function. It is pointed out, however, that the
ambiguity associated with Brock's framework can be duplicated jn an
alternative transactions cost framework. Thus, the answers to some
questicns may depend critically on the specific story one chooses to tell
about why households hold money. Models in which money enters the utility
function directly cannot adequately address these questions.

One of the tasks undertaken in this section is an intuitive
development of the Euler equation using an “arbitrage” argument. This
argument is, in essence, a simple application of calculus of variations.
Readers already familiar with this ground may wish to skip the bulk of the
material appearing between equations (18) and (27). They are advised,
however, to acquant themselves with the particular arbitrage argument (or
perturbation) employed, since it will play a role in the discussion of
transverality conditions in section IV.

We turn now to the specification of a model in which money is held
for the explicit purpose of reducing transactions costs. There are variety
of ways in which transactions costs might be modelled. The approach taken
in this section was choosen for its simplicity. Some alternate approaches
are touched on in section V. Again, we begin by describing the
representative household's problem, which is to maximize a utility function

of the form


(3) i=

ote

eP*Lu(e,) + v(m )Idt,

subject to the budget constraint

(4) Poy = Pace +My

Following conventional notation, p is an internal rate of discount, Cy

and m, are real household consumption and money balances at time t,

Pe is the price level, and ih is the instantaneous time rate of

change of nominal money balances. Real income rains from the heavens on

households at a fixed rate of y units per period. Consumption and the price

level are contrained to be non-negative. Note that money yields utility

directly in this formulation. " Further, the utility function is additive in

consumption and real balances. This is critical for Brock's results.
Formally, the solution to the household's problem is found by

maximizing the Hamiltonian function, H, with respect to the household's

control variable, Cee where H is given by

(5) H = [u(c,) + v(m,) Je®* + a, [Py - Peg]

It is inportant to note that the household treats the path of the price
level, Pee as exogenous. From the maximum principle, we know that any
interior consumption plan that maximizes H must satisfy equations (6)
through (8) below.2/

(6) gH. ut (c, Je Pt ~ APE = 0, for all t.

cE


- 27 -

identifies the paths of variables associated with the proposed optimal
solution, and the superscript "A" identifies paths associated with the
arbitrage scheme. As before the household takes the path of prices as
given. To facilitate comparison with the results of section III, the
transactions costs model of that section is used. The paths of ws

it, and ct may be written as follows:

(29) Wh = M2 - un(t),

(30) Mf = -un(t),

vv owh wo ~ un(t)
(31) Aw y- (ty -teey- (Qt —) + ald ,
P P P.
t t t t
where
0 0 ‘for t<T,
n(t) = Jt-T and n(t) = 1 for T<t<T+l,
1 0 for t>T+1.

Substituting equation (31) into the equation (15) produces an expression
for household utility involving u, prices, the time T, and other exogenous

variables:


-~ 25 -

unbounded level of transactions costs, (or infinitely negative utility) at
some sinall but finite level of real balances. Implosive real balance paths
can also be ruled out if they lead to negative levels of consumption.

This can not occur in Brock's model but can occur in the particular
transactions costs model described in the previous section. The major
conclusion of the section is that uniqueness of equilibrium can not be
guaranteed in either model under any reasonable set of assumptions about the
nature and role of real money balances.

A. Explosive Real Balance Paths

Along real balance paths of this kind, prices decrease, asymtotically
approaching zero, and real money balances increase, becoming arbitrarily
large. Transactions costs decrease and consumption increases. If these
paths are optimal, the household must be unable to increase its lifetime
utility by varying its consumption plan or money holdings in any way. There
can exist no profitable arbitrage opportunities of any kind along an optimal
solution. All the paths under consideration in this section satisfy the
Euler equation and the household's budget constraint. Accordingly, we are
guaranteed the absence of profitable arbitrage opportunities along any
finite segment of these paths. The arbitrage schemes already precluded,
then, are those that are eventually "reversed". At some (finite) point in
time, real money balances and consumption are returned to their original
paths. There is no assurance, however, that "unreversed" arbitrage schemes
are unprofitable along these paths. In fact, they are profitable. In
demonstrating this point, we will use the unreversed version of the
arbitrage scheme that was employed in the preceding section to establish the

Euler equation.


Under the assumption of rational expectations (which, in a
deterministic model, is equivalent to the assumption of perfect foresight),
the expected rate of inflation is equal to the realized rate of in“lation.
Under this assumption, the model is dynamically unstable. It has a unique
steady state in which the value of real balances is one, the price level is
equal to the nominal money stock, and the rate of inflation is zero. If the
price level assumes any value other than it's steady state value, P*, prices
will diverge ever further from their steady state value as time passes. The
price path will be explosive if the price level initially exceeds P* and
"implosive" (Py goes to zero) if the price level is initially less
than P*.

It will be convenient to study the dynamics of this model and
subsequent ones in terms of the path of real balances rather than “he price
level. If the nominal money stock is fixed, the rate of change of real

balances is simply the negative of the rate of change of the price level.

In this model, then,

(2) m/m, = -(1/a)1n(m,), where m, = (My/P,)«

This relationship is depicted in figure 1.

It is common in dynamically unstable macro-models to arbitrarily
select the "stable arm" of the model in calculating equilibrium. When the
dynamics are one-dimensional, as they are in the Sargent and Wallace
framework, this means admitting only the steady state (or states) as an
equilibrium. As long as the admissable equilibria of a model are “estricted

to steady state paths, the solution of the model is typically


II. A Selective Review of the State-of-the Art

The purpose of the present paper is to explore the conditions under
which the common practice of specifying a dynamically unstable macro-model
and then arbitrarily selecting the stable arm of that model may be
justified. This section provides a brief review of two commonly cited
papers. The first of the two serves to motivate the very general question
posed above. It provides an extremely simple example of a dynamically
unstable rational expectations macro-model. The second addresses the
question of the non-uniqueness of equilibrium in this model by exploring the
same question in a maximizing framework that generates similar dynamics.

As noted in the introduction, examples abound of both open and closed
economy macro-models that “fly off the handle" unless precisely correct
inital conditions on prices and/or exchange rates are satisfied.

Sargent and Wallace (1973) illustrate this phenomenon neatly in a simple
model in which real growth is zero, income and the nominal money supply are
fixed, and the price level is assumed to be consistent with equilibrium in

the money market, satisfying at each moment in time a Cagan money demand

function of the form.

(1) Inipe) = alp)®.

As usual M, P, and (P,/Ps) denote the nominal money
stock, tne price level, and the expected rate of inflation, respectively.
The real rate of interest is omitted from this formulation since it is

assumed to be a constant.


-16-

(15) Z=S ePtu(cy)at,

ous

subject. to the budget constraint

(16) Pry = Pycy - Pyn(m,) - Mes

and the restrictions

u'> 0, u" = 0,

n' <0, 7" > 0.
Notation is the same as in the previous section. Note that real money
balances do not appear in the utility function. Lifetime utility is a
function only of the household's consumption stream. Instantaneous utility,
Ulcy) » is a linear function of the level of consumption. This
assumption is critical to the conclusions of this section in exactly the
same way that the assumption of separable utility is critical to the
conclusions drawn from Brock's model in the preceding section. Both
assumptions are relaxed in section V. Real balances influence lifetime
utility through their affect on the level of real resources available for
consumption. Specifically, the conversion of income into consumption
involves a transactions technology in which real resources and real money
balances are substitutable. Real resource transactions costs per household
are given by the function n(m,)- As real balances increase,
transactions costs decrease. But the reduction in transactions costs is
smaller, the higher the initial level of real balances; the technology is

subject to diminishing returns. The representative household, then,


straightforward, yielding a unique equilibrium as long as the steady state
is unique. If stability is imposed on the Sargent and Wallace model, for
example, the only admissable equilibrium solution to the model is the steady
state, in which the value of real balances and the price level are constant.
An unanticipated increase in the money supply on this model simply leads to
a matching instantaneous jump in the price level -- real balances never
deviate from their steady state value of unity. In the absence of the
assumption of stability, one is left with a staggering infinity of
admissable equilibria -- all but one of which involve ever increasing or
ever decreasing holdings of money balances.

The arbitrary selection of the stable arm in the kind of model
described above is sometimes justified by reference to an underlying
framework involving intertemporal optimizing behavior. A frequently cited
example of such a framework is provided by Brock (1975). However, contrary
to the impression these citations generate, Brock's work does hot Justify
the arbitrary selection of the stable arm of a dynamically unstable model.
To the contrary, it is possible to delineate clearly in his model the
conditions under which explosive price paths can not be ruled out as
equilibrium solutions. Further, it is argued in section III ‘that these
conditions are the only economically relevant ones. We turn now to a brief
summary of Brock's model and the conclusions it produces.

Consider an economy composed of a number of identical, infinitely
lived households, each of which maximizes a utility function of the

formt/ :


-21-

Given an arbitrary path of prices and an initial money stock My that
satisfy equation (18), we wish to know if household utility can be increased
by a small perturbation of the original path -- that is, by increasing u
from a value of zero to some small poitive value. The change in household
utility associated with such an exercise is just the derivative of Z(u! with

respect to u, evaluated at y equal to zero. It is given by equation (23).

T+2 n
(23) Stu) 7! CU" (cee (me LGD TPE + uricey (ALG) Pt.
i”

t t

Integration by parts of the second term in the integral allows us to

rewrite equation (23) in the following way:3/

0. 50
T+2 a'(m P.
(24) SOD) ee” ur (c)n(tye— + = + Ere tat.
du =0 T t pd po pO 2
“ t t (Pe)

If the originally proposed price and consumption paths are optimal, it

must be true that

dZ

u=0

This, in turn, will be true if the expression in brackets on the

right-hand-side of equation (24) is equal to zero, or

(26) P/P, = -n'(m?) - p forT<t<T+2.
Pt t <t<


International Finance Discussion Papers
Number 197

January 1982

DYNAMIC INSTABILITY IN RATIONAL EXPECTATIONS MODELS:
AN ATTEMPT TO CLARIFY

by

Jo Anna Gray

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 author or authors.


- 42 -

model this effect is necessarily negative. An increase in real balances
lowers transactions costs, increasing consumption. Higher consumption, in
turn, generates a lower marginal utility of consumption. Thus real balances
and cosumption are necessarily substitutes in this model.

It would be incorrect to assume, on the basis of the foregoing
discussion, that consumption and money act as substitutes in all
transactions cost models of money demand. Consider, for example, the
following problem: The household maximizes a utility function that includes

both consumption goods and leisure,

This model specifies a transactions technology in which leisure and real
money balances are substitutable. An increase in money lowers transactions
costs, wich increases the household's leisure time. If consumption goods
and leisure time are complements (as assumed here), the increase in leisure
raises the marginal utility of consumption goods. In this transactions cost
model, then, money and consumption act as complements.

The Euler equation for this problem is given by


-10-

(11) Ce = Ys for all t.

Equation (10) is often referrred to as the Euler equation. It summarizes
the dynamic characteristics of the model's candidate solution paths.
Intuitively, it is the condition that insures the absence of profitable
arbitrage opportunities along any finite segment of an optimal consumption
plan. This interpretation of the Euler equation is discussed in greater
detail in section III below.

Again, it will be convenient to study the dynamics of this model in
terms of the path of real balances rather than the pricel level. Given a

fixed stock of nominal money, equations (10) and (11) imply equation (12).

fh, v' (m,)

ct

(12)

"
°
'

my, u'(y)
This relationship is depicted in figure 2. The steady state level of real
money balances, m*, is found by setting my equal to zero in
equation (12). If real balances initially exceed m*, equation (12) states
that Mm, will increase indefinitely over time. Under the assumption of
diminishing marginal utility of money, it is also true that the rate of
increase of m, will increase as Mm, rises. The case in which real
balances initially fall short of m* is somewhat more complicated.
Equation (12) can produce, under different specifications of the function
v(m), any one of the three implosive real balance paths shown. At issue is
the limiting value of in, as m approaches zero. From (12),

this limit can be written as


aH ye ' -pt

(7) En = omAg GY (m,)(1/P, Je > for all t.
OH Ly

(8) 3 ; = M, = Pry - PrCes for all t.

Given an arbitrary price path, equations (6) through (8) can be solved for
the set of consumption plans that meet the necessary conditions for an
optimum. However, we are not interested in all price paths and their
associated optimal consumption plans. We are interested only in equilibrium
price and consumption paths. Equilibrium in the markets for goods and money
requires that the fixed stock of nominal money balances be demanded by
households at each point in time. For the representative household this
means that planned net increments to its nominal money balances must: be

zero, or

(9) M, =0, forall t.

Combining equation (19) with equations (6) through (8) allows us to solve
for the set of equilibrium price and consumption paths that meet the
necessary conditons for an opt imum :2/ This set consists of all price
and consumption paths that satisfy both equation (10) and equation (11)

below.

P, vi(m,)
(10) tt. tp, for all t.
P, u(y)


- 38 -

V. Stability Versus Instability

Both of model's studied so far have been dynamically unstable; the
unique steady-state of each has been a saddle-point. For neither of these
models have we been able to prove that the steady-state is its only
equilibrium solution. Even so, there is an advantage to working with such
models relative to working with models that are stable in the neighborhood
of the steady-state.

As discussed earlier, it is common practice to specify a model with
saddle-point properties and to then arbitrarily select the stable arm of
that model. This procedure rules out implosive and explosive solutions by
assumption -- and, as show above, may not be justifiable. The advantage to
such a modelling strategy is that it leads to unique solution paths for all
the variables of the model. This is of particular interest when analyzing
the effects of exogenous disturbances. Consider, for example, the effects
of an unanticipated increase in the nominal supply of money in either of the
models studied so far. If the steady-state is assumed to be the only
equilibrium solution in these models, it follows that the price level must
instantaneously increase in the same proportion as the money supply, leaving
real balances and consumption unchanged. The only effect of an
unanticipated change in the money supply is a jump in the price level.

Of course, not all models are dynamically unstable. More general
versions of both of the models of this paper can be shown to be stable
under some conditions. However, stability is not necessarily a desirable
property of such a model. It can lead to a multiplicity of solution paths,
all of which converge to the steady-state. Consider again an unanticipated

increase in the nominal money supply, this time in a stable version of one


- 26 -

Consider a particular candidate solution along which real money
balances are increasing through time. Suppose that at some arbitrary time
T, the household begins to decumulate nominal money balances at a rate u,
continues to decumulate money at this rate until time T+], and then
maintains the resulting (lower) level of nominal money balances thereafter.
The essential feature of this arbitrage scheme is that money balances are
never returned to their original path. It is an unreversed arbitrage. It
can be shown that the gains to this arbitrage are positive along any
explosive real balance path. Intuitively, the net gain to converting a
small amount of money into consumption along such paths is positive and
increases with time. (Net gains, here, are discounted to the time at which
the arbitrage is initiated.) As real money balances become arbitrarily
large, the discounted stream of increased transactions costs that would be
incurred if nominal money balances were decreased by a unit becomes
arbitrarily small. But the benefit to reducing nominal balances by a unit,
measured by the amount of consumption that could be purchased by a unit of
money, or VPs becomes arbitrarily large. It follows that the net
gain, discounted to the time the arbitrage is initiated, must increase over
time and must be positive for sufficiently distant future points in time.
As the analysis below will demonstrate, this net gain, when discounted back
to the beginning of the household's planning horizon, is not only positive
but is independent of when the arbitrage is initiated.

Formally, these results can be obtained by explicitly evaluating the
change in household utility resulting from the arbitrage scheme outlined
above. For clarity, the scheme under consideration is illustrated in figure

5, Notation is the same as in the previous section. The superscript "oO"


- B-

UT
. -AD) -»
ee ae
P Uv *
Vem Ce)
7 c

Under the assumptions outlined above, this model is necessarily

dynamically unstable.


M
t age
t
0
0 T T Mt
i} ey
sup ao eM
|
' 1 t
0 T T+1 0
My, as
~~
0 ' 5
= T Me +
~ ! | '
0 ! 7S an
Way |---—4—- -= Mm ! a Cc
| 1 | '
| n wt ! !
0 T T+1 0 T T+1
O
PEE
t
0
Figure 5

An Example of an Infinite Horizon Perturbation


(¢]

Figure 1

The Dynamicsof the Sargent and
Wallace Model


-19-

Formalizing the arguments outlined above is relatively
straightforward. We begin with an arbitrary price path and consumption
stream along which households are satisfied holding the (fixed) nominal
stock of money existence. An example of such a proposed solution,
involving constant prices and consumption, is depicted in figure 3. The
paths of variables corresponding to this candidate optimal solution are
identified by the superscript "0". The arbitrage scheme to be evaluated is
also shown in figure 3, the associated paths identified by the superscript
"A". Note, again, that the household takes the path of prices as given.

The arbitrage scheme shown in figure 3 may be characterized as follows. The
household consumes according to the proposed solution, neither adding to nor
subtracting from its initial holdings of nominal money, up to some arbitrary
time T. At time T, it begins to decumulate nominal money at a constant rate
u, Where » is small. The decumulation continues until time T+ 1, giving a
total change in nominal money balances at that point in time of y units.
From T + 1 to T + 2, the household accumulates nominal money at the same
rate, u, returning the household to its initial money stock at time T+ 2.
From time T + 2 on, the household again consumes according to its original
plan and hclds a constant stock of nominal money. Thus, the paths of

" and it may be written as

(19) ME = MP = un(t)

and

(20) Mf = -un(t),


- 33 -

The preceding analysis shows that along any explosive real balance
path households can make themselves better off by reducing their money
balances and increasing their consumption at some point in their lives. At
the moment the decumulation of money begins, there will be an excess demand
for goods and an excess supply of money. Therefore for no explosive real
balance path -- or, alternatively, no implosive price path -- can be an
equilibrium path. Of the total set of price paths consistent with equations
(17) and (18), we may now exclude from further consideration all real
balance paths that increase over time. It is left to the interested reader
to proove that the same arguments can be used to exclude explosive real
balance paths in Brock's model.

Two classes of candidates for equilibrium real balance paths remain
-- the steady state path and the set of paths that diverge from the steady
state toward zero. We turn now to the second of these.

B. Implosive Real Balance Paths

Along real balance paths of this kind, prices increase, becoming
arbitrarily large, and real money balances decrease. Transactions costs
increase with time, implying continual decreases in the level of household
consumption. If these paths are optimal, there can be no profitable
arbitrage opportunities available to the household along them. Since they
all satisfy the Euler equation, we are guaranteed the absence of profitable
arbtirage opportunities along any finite segment of these paths. Further,
the transversality condition given by equation (37) is automatically
satisfied by any implosive real balance path. Thus, households can not make

themselves better off along such paths by permanently increasing their


- 34 -

nominal money balances. Unreversed arbitrage schemes of the sort examined
in the preceding subsection are unprofitable along implosive real balance
paths. This does not mean, however, that such paths are necessarily
admissable equilibrium solutions. They can be ruled out if they lead to
infeasible (negative) values of the price level or consumption.

From the results of section III, we know implosive real balance paths
can be divided into three classes. Examples of each are shown in figure 3.
The Euler equation can produce, under different specifications of the
transactions technology, any one of the three types of paths show there.
At issue is the limiting value of iy as m, goes to zero.
This limit is given by equation (28) of section III and is repeated here for

convenience

. = Vi 1
(28) nro mM = nro mm (m,)-
If this limit is zero, paths of type A result. If the limit is a
constant, type B paths result. And if the limit is negative infinity, type
C paths result.

Paths of types B and C -- those for which the time rate of change of
real balances goes to a negative constant or negative infinity as real
balances approach zero -- can be ruled out as equilirium solutions. The
equation of motion that generates these paths (the Euler equation) implies
that at some finite point in time real balances pass through zero and become
negative. Given a fixed (positive) nominal money stock this, in turn,

implies a negative price level, which is infeasible. The remaining paths --


- 23 -

consumption good to the internal rate of discount, p. If real balances
initially exceed m*, they will increase at an increasing rate over time. If
real balances initially fall short of m*, one of three implosive real
balance paths will result. Which of three occurrs depends on the limiting
value of ay as m, approaches. In the transactions costs

model of the present section, this limit is given by

(28) lim m = lim m,n! (m,).
m,.+0 m0

The economic interpretation of the term m, m'(m,) is identical to

t
the interpretation of the term -m,v' (m,)/u' (y) in Brock's model.

Both represent the stock of real balances times the marginal value of real
balances in terms of the consumption good. If, as m, goes to zero, the
limitirg value of this term is zero, path A results. If the limit is a
constart, path B results. And if the limit is negative infinity, path C
results.

We have developed, in this section, a model in which money demand is
motivated by transactions costs. This model produces price and real balance
dynamics that are identical to those of Brock's model. As before, we are
confrorited with a multiplicity of candidate solutions to the model, all of
which satisfy the Euler equation and the household's budget constraint. The
Euler equation guarantees the absence of profitable arbitrage opportunities

along any finite segment of these solutions.


the Brock model. Second, an intuitive development of the Euler equation
associated with the model is undertaken. It is shown that the Euler
equation may be regarded as a restriction on the path of prices that must
hold if various finite horizon arbitrage schemes are to be unprofitable
along the path.

A more careful study of explosive and implosive real balance paths is
undertaken in section IV. It is verified that explosive real balance paths
are not equilibrium solutions. Along such paths, the household can increase
its lifetime utility by exploiting an open-ended arbitrage opportunity --
one not ruled out by the Euler equation, which all these paths necessarily
satisfy. This arbitrage scheme is shown to produce a transversality
condition which must be satisfied by all optimal real balance paths. It is
violatec by all the explosive real balance paths examined in this paper.
Thus, we are able to provide an arbitrage interpretation of the use of
transversality conditions in infinite horizon maximization problems.
Implosive real balance paths are shown to satisfy the transversality
condition appropriate for the two models discussed in this paper. They can,
nevertheless, be ruled out if they lead to infeasible (negative) levels of
prices or consumption. This can occur in either model if certain
restriczions are satisfied. However, applying a result recently developed
by Obstfeld and Rogoff (1981), these restrictions are shown to be
economically irrelevant. The major conclusion of the section is that
uniqueness of equilibrium can not be guaranteed in either model under any
reasonable set of assumptions about the economic role of real money

balances.


- 41 -

That is, consumption and money balances must be regarded by the household as
comp! ements 22/ A priori, there is little to suggest whether such a
restriction is reasonable or not.

Consider next the transactions cost model developed in section III.
In that model it is assumed that the household's utility at a point in time
js a linear function of the level of its consumption -- that the marginal
utility of consumption is a constant. Suppose instead that the more common
assumption of diminishing marginal utility is adopted. In that case, the
correct Euler equation for the model is given by

Ph oat (m) = 0

(44) — = ——__+—___.

Ph 1- m{n' (m,)U"/U']

In the special case of constant marginal utility of consumption, U" is

equal to zero and equation (44) reduces to equation (17) of section III.
The dynamics of the more general model will be the same as the dynamics of
the simpler model as long as the denominator of the right side of equation
(44) is positive. A condition sufficient to insure this result would be the
analog of equation (43) -- the requirement that aa! (m,)U" be positive.

By assumption, this condition is not be met. Both a! (m) and U" are
negative. It is interesting to note the interpretation of the negative of
the product of these two terms. It gives the effect on the marginal utility
of consumption of the change in consumption generated by a change in real
money balances. More concisely, it represents the effect on the marginal
utility of consumption of a change in real balances. It is the conceptual

equivalent of the term Vom in Brock's model. In our transactions cost


- 39 -

of the models discussed above. Without additional assumptions, thee is no
way to tell what will happen to the price level at the time of the increase.
It could rise, fall, or remain unchanged. Regardless of what happens to
the price level in the first instant, it will proceed to converge
monotonically to its new steady-state value thereafter. As a result, there
will be a multiplicity of solution paths following such shocks, one
corresponding to each feasible level of the price level at the instant of
the shock.

In some stable models this problem is solved by assuming that
the price in question -- here, the price level -- can not jump. This
assumption identifies the solution path leading from the pre-shock price to
the new steady-state price as the unique equilibrium solution to the model.
In models in which the price in question is the price level, the assumption
of sticky prices may be appropriate. In models where the relevant price is
the exchange rate or an interest rate such an assumption is likely to be
inappropriate. If, then, one is interested in explaining the behavior of a
variable that is observed to "jump" in response to some shocks, a model with
saddle-point properties is preferable to a model that is stable in the
neighborhood of the steady-state.

In the remainder of this section, we will examine more general
versions of Brock's model and the transactions cost model of section III.
For both models there is a condition that is sufficient (but not necessary)
to insure that the model has saddle-point properties. That condition can be
met in Brock's model but can not be met in the transactions costs “ramework.
It is argued, however, that an alternative transactions technology

can produce results analogous to those of the generalized Brock model.


-12-

and become negative. Given a fixed (positive) nominal money stock, this, in
turn, implies a negative price level, which is infeasible. The remaining
paths -- all type A paths -- can not be ruled out. Along these paths, the
time rate of change of real balance slows as the level of real balances
falls, nearing zero as real balances go to zero. As time goes to infinity,
the level of real balances asymtotical ly approaches zero. Such path can not
be ruled out on the grounds that they are infeasible. Nor do they -

present profitable arbitrage opportunities as explosive real balance paths
do (alternatively, they violate no know transversality condition.) It
follows, then, that it is possible to eliminate as equilibrium solutions
only those implosive real balance paths that satisfy the following

condition:

v' (m,)

(14) Tim [+ m(——4)] < 02!
m0 u'(y)

Finally, we turn to the steady state solution -- m = m* for all t.
This solution is feasible and violates no known transversality condition.
Therefore it can not be ruled out as an equilibrium path of real balances.

The results of Brock's model may be summarized as follows: If

equation (14) holds, the steady state is the unique equilibrium solution to
his model. If equation (14) does not hold (the limiting value of
mv! (TM,) is equal to zero), there exists a multiplicity of
equilibrium solutions. These consist of the steady state and an infinite
number of implosive real balance paths -- one corresponding to each possible

level of initial real balances. This model, then, does not provide


-14-

III. A Transactions Cost Approach

In this section we develop and analyze a framework in which money is
held because it reduces transactions costs. Money does not enter the
household's utility function directly, as it does in Brock's model. There
are several reasons for choosing to introduce such an alternative to Brock's
model. First, it will serve to demonstrate that the dynamics of Brock's
model are not special to his model and, in particular, are not due to the
fact that money enters the utility function directly. Explicit modelling of
the household's incentives to hold money balances does not lead to price or
real balance dynamics that differ in any important way from those produced
by Brock. Specifically, in the transactions cost model studied be ow it is
generally not possible to rule out a multiplicity of equilibrium solutions,
all but one of which involves an ever decreasing level of real money
balances. Further, the conditions under which uniqueness of equilibrium can
be insured in the model are precisely analogous to those of the Brock model.

A second reason for exploring a transactions cost approach is that it
makes it easier to judge the “economic relevance" of the kind of restriction
discussed above. We will see in section IV, for example, that the
conditions which gurantee a unique equilibrium in the transactions cost
model of this section also imply that transactions costs become unbounded as
real money balances approach zero. This is not an economically relevant
specification since it implies negative consumption at some point along any
implosive real balance path.

“Finally, this approach will prove useful in section V, which develops
a condition that guarantees saddle-point dynamics (as opposed to local

stability) in more general versions of both Brock's model and the


render them inaccessible to the wide audience which needs to use them. It
is hoped that this paper will help to rectify that situation.

The paper begins, in section II, with a brief review of two commonly
cited papers. The first, by Sargent and Wallace (1973), provides an
extremely simple example of a dynamically unstable rational expectations
macro-model. The second, by Brock (1975), addresses the question of the
non-uniqueness of equilibrium in this model by exploring the same question
in a maximizing framework that generates similar dynamics. The results of
the section do not provide unqualified support for the practice of
arbitrarily selecting the stable arm of dynamically unstable models. In
Brock's model, the steady-state is show to be the unique equilibrium
solution to the model only if certain restrictions on the household's
utility function are met. If these restrictions are not met, there exist a
multiplicity of equilibrium solutions to Brock's problem -- the steady-state
and an infinite number of implosive real balance paths. The question of the
economic relevance of these restrictions is postponed to a later section.

In section III, a transactions cost model of money demand is
developed and analyzed. In this framework money is held because it reduces
transactions costs; money does not enter the household's utility function
directly, as it does in Brock's model. Among the results of the section are
the following: First, the price and real balance dynamics of this
transactions cost model are identical to those of Brock's model. It is
generally not possible to rule out a multiplicity of equilibrium sol stions
to the model, all but one of which involves an ever decreasing level of real
money balances. Further, the conditions under which uniqueness of

equilibrium can be insured in the model are precisely analogous to those of


~ 32 -

side of equation (36) must go to zero as T+1 goes to infinity. Thus, a

necessary condition for an optimum in this problem is

u! (cpyee*
(37) lim — TF 0.
tre pt

This can be rewritten in a more familiar form.

(38) Vin age®t = 0, where a4 = U'(CO)/PE.

This concition is often imposed in infinite horizon optimization problems,
frequently without justification. It requires that the discounted utility
value of an additional unit of nominal money balances go to zero as time
goes to infinity -- that the cost of reversing our proposed arbitrage go to
zero as the time of reversal goes to infinity 12/ As demonstrated

above, it is a necessary condition for an optimum in the transactions cost
model explored in this section because it insures the absence of profitable
arbitrage opportunities that are not excluded by the Euler equation. An
analogous condition is also necessary in Brock's model. It does not
follow that the transversality condition developed in this séction is
appropriate for all problems. It is not. It is possible, however, to
develop the correct transversality condition (if it exists) for most

problems by using arbitrage arguments similar to the ones employed

here.13/

Explosive real balance paths do, of course, violate the
transversality conditon set out in equation (37). As discussed earlier, the

limit in question is a positive constant for such paths, not zero.


-~ 22 -

To show that equation (26) is a necessary condition for an optimum

requires one more step. For a particular arbitrage scheme, denoted by a
particular function of time, n(t), the derivative dZ(u)/du could, in
princ:ple, equal zero even if equation (26) did not hold. While this could
occur for a particular function, n(t), it could not, however, occur for all
possible functions, n(t), that met the boundary conditions n(T) = 0 and

n(T + 2) = 0. That is, if the change in utility associated with all
arbitrage schemes on the interval (T, T+2) is to be zero, equation (26) must
hold. Finally, the interval (T, T+2) was selected arbitrarily. The
arguments that produced equation (26) could be made for any choice of T.
Accordingly, equation (26) must hold at every point in time along any
equilibrium solution. This completes our proof that the Euler equation is a
necessary condition for an opt imun.2/

The Euler equation given by equation (17) describes the set of price
paths that are candidates for equilibrium in the model of this section. As
before, however, it will be convenient to study the dynamics of the model in
terms of the path of real balances rather than the price level. Given a

fixed nominal stock of money, equation (17) implies equation (27).

me
(27) ater a" (m,) for all t.

+

This relationship is depicted in figure 4. Comparison of figure 4 with
figure.2 shows that the dynamics of this model are identical to the dynamics
of Brock's model. In both models the steady state level of real balances,

m*, is found by equating marginal value of real balances in terms of the


Dynamic Instability in Rational Expectations Models:
An Attempt to Clarify

by
Jo Anna Gray*

I. Introduction and Summary

A prevalent feature of deterministic rational expectations
macro-models is dynamic instability. Examples abound of both open and
closed economy macro-models that "fly off the handle" unless precisely
correct initial conditions on prices and/or exchange rates are
satisfied _/ A common practice in calculating the equilibrium of these
models is to arbitrarily select the stable arm of the model. This practice,
in effect, rules out explosive and implosive solutions by assumption. It is
sometimes justified by reference to an underlying framework involving
intertemporal opt imi zing behavior 2! A frequently cited example of
such a framework is provided by Brock (1975) 2/

The primary purpose of the present paper is to explore the
conditions under which the popular procedure of specifying a dynamically
unstable macro-model and then arbitrarily selecting the stable arm of that
model may be justified. Many of results of this paper can be found

elsewhere. Most of them, however, come from a literature so technical as to

*The author is a member of the staff of the Board of Governors of the
Federal Reserve System. I am grateful to Matt Canzoneri, Bob Flood, and Ken
Rogoff for helpful discussions during the writing of this paper. I am also
grateful to William Brock, Dale Henderson and George Moore for their help
and enccuragement on a previous draft of this paper. The views expressed in
this paper are mine alone. They do not represent the official views of the
Federal Reserve System, its Board of Governors, or its staff.


-~ 44 -
Footnotes

Vsee Sargent and Wallace (1973), Penti Kouri (1976), Rudiger Dornbush

(1976), to mention only a few.
2/ see Lucas (1975).
3/see also Brock (1974) and Brock (mimeo.)

Al Brock's original work is formulated in discrete time. The

continuous time analog of the simplest of his models is presented here.
5! see Intriligator (1971), Chapter 14.

S/equation (11) is obtained by setting he = 0 in (8). To
derive equation (10), first differentiate (6) totally with respect to time

and manipulate the resulting expression to get

Substituting (7) and (11) into this expression gives (10).

2/grock overlooked this condition in his original work (Brock (1974)
and (1975)). This omission is corrected in a later note (Brock (mimeo.)).
I am indebted to Brock for pointing this condition out to me in an early

conversation pertaining to his original papers.

8/ysing integration by parts, we have

T+2 °

fur (co) (MtdyePtat =
9

7 °°


-~ 2 -

0

T m
Zu) = J uly - x(-t)JePtat

0 p°
TH woe un(t)

(32) +S Uy ~ by + mje Pat
T t Pe
° wy

+f uly- tp yk Ptat

TH Pt

Given the proposed solution, we wish to know if the household's utility can
be increased by a small perturbation of its consumption plan -- that is, by
increasing uy from zero to some small value in equation (32). The resulting
change in household utility is given by equation (33).

F T+1 .
(3) GOD] = cur cefar (apy eLa ye PF + u" (chy MG re PF
we t . t

. 0 0)/1_),-et
+ Sf  UN(CE)n' (mz) (—)eP "dt.
Tey tO tpt

Integration by parts of the second term in the first integral allows us to

rewrite equation (33) as follows:

50

T+] P

(34) S2GD) =f ur(cOym(ty (1/72) Ent (me) + 9 +b -JePtat
de |e t t t 70

t

17e0 _y,-e(T#1)
U'(CrL, Je °
T+] 0, 0 0, .-pt
i + su (Ch)n(m,) (1/P,)ePvdt.
Pral T+1 t t t

+


-37-

relevant. Any economically relevant specification of transactions costs
would, presumably, include the restriction that transactions costs never

exceed the household's resources, which implies that

(41) lim n(m,) Ky

mo

If equation (41) holds, consumption remains non-negative at all points along
any implosive real balance path.

We may conclude, then, that implosive real balance paths -- or,
alternatively, explosive price paths -- can not be ruled out as equilibrium
solutions in either of the models studied in this paper. Under any
reasonable set of assumptions concerning the economic role of real money
balances, these paths fail to violate any known necessary condition for an
optimum or feasibility requirement.

We turn, finally, to the steady state path of real balances. This
path satisfies the transversality condition given by equation (37). It
violates no feasibility requirements. It is an admissable equilibrium
solution.

The results of this section may now be summarized as follows:
Uniqueness of equilibrium can not be guaranteed in either of the models
studied in this paper so far under any reasonable set of assumptions

concerning the economic role of real money balances.


- 24 -

IV. A Closer Look at Explosive and Implosive Paths

The transactions cost model of the preceding section produced a
multiplicity of candidate solution paths for prices and real balances. The
characteristics of these paths are identical to those of the candidate
solutions produced by Brock's model. As before, they can be divicied into
three groups -- those along which real balances increase over time, those
along which real balances decrease over time, and those along which real
balances are constant over time.

In this section, we substantiate and provide the intuition behind the
claims made in section II concerning explosive and implosive real balance
paths. It is verified that explosive real balance paths are not equilibrium
solutions. Along such paths, the household can increase it's lifetime
utility by exploiting an open-ended arbitrage opportunity -- one not ruled
out by the Euler equation, which all these paths necessarily satisfy. This
arbitrage scheme is shown to produce a transversality condition wiich must
be satisfied by all optimal real balance paths. It is violated by all the
explosive real balances paths examined in this paper. Thus, we are able to
provide an arbitrage interpretation of the use of transversality conditions
in infinite horizon maximization problems 0/

Implosive real balance paths are shom to automatically satisfy the
transversality condition appropriate for the models discussed in this paper.
They can, nevertheless, be ruled out as equilibrium solutions if they lead
to infeasible (negative) values of the price level. This will occurr if
certain restrictions on the transactions cost function, (mm), (or, in
Brock's model, on the function v(m, )) are met. However, these

restrictions are shown to be economically irrelevant in that they imply an


m.

it = mle -

v'(m,)

t
Tor

"
°o

for A: lim -m,Vv' (m,) =
m> 0

for B: lim -mv'(m,) = constant

m, +0

for C: lim -mv'(m,) = - @
m, +0 t t

Figure 2

The Dynamics of Brock's Model


- 45 -

1-0, -pt 0 50
u'Cr)e Prn(t) | T+2 T+2 ot PPE =
ae - SF yt (cOyn(tye te —t__t rat.

po T T t (poy?
t t

The first term on the right-hand-side of this equation is zero since
n(T+2) = n(T) = 0. Substituting the integrand of the remaining term into

the right-hand-side of equation (23) produces equation (24).

our proof follows closely the development of the Euler equation

presented in Chapter 12 of Intriligator (1971).

10/this is not to say that the transversality condition developed in

this paper -- which is the usual condition imposed in such problems -- is
appropriate for all problems. It is not. For a detailed discussion of how
to develop the transversality condition appropriate for particular classes

of problems, see Gray and Salant (1981).

Lf op 1 (me) to approach zero as ne becomes
arbitrarily large, it is sufficient to assume that transactions costs have a
lower bound. To assume otherwise would imply that transactions costs become

infinitely negative as real balances increase.

12/se¢ Gray and Salant (1981) for a more detailed discussion of this

interpretation of the standard transversality condition.
13/again, see Gray and Salant (1981).
14/see Obsfeldt and Rogoff (1981).

1/the importance of this restriction for more general versions of

Brock's model is discussed in Calvo (1979).

18/this model, according to Bob Flood (who first described it to me),

is known to some as the “Virginia monetary model."


-13-

unqualified support for the practice of arbitrarily selecting the stable arm
of macro-models with similar dynamic characteristics.

It is interesting to note that the dynamics of the Sargent and
Wallace model (pictured in figure 1) are qualitatively identical to the
dynamics of the Brock model for the case in which equation (14) does not
hold (shown by path A of figure 2). This is, of course, exactly the case in
which it is not possible to identify the steady state as the unique
equilibrium of the model.

For those interested in insuring uniqueness of equilibrium in their
models, the preceding discussion suggests an obvious strategy: Employ only
models whose dynamic properties satisfy equation (14) or its equivalent.
Unfortunately, this strategy is flawed. In section IV we will show that the
conditions under which equation (14) holds are not economically relevant
conditions. Under any reasonable set of assumptions about the nature and
role of real money bal ances, the limit set out in equation (14) is equal to
zero. The practice of selecting the stable arm of dynamically unstable
models remains, in general, a practice that can claim no theoretical

support. Price bubbles are presently alive and well, in theory at least.