INTERNATIONAL ECONOMIC REVIEW

Vol. 22, No. 3, October, 1981

SOME NEW RESULTS ON THE DYNAMICS OF THE

GENERALIZED TOBIN MODEL*

BY JESS BENHABIB AND TAKAHIRO MIYAO'

The seminal article of Tobin [1965] on money and growth raised many

questions and led to a large and growing literature. In dealing with the unstable

character of the steady state and the explosive nature of almost all trajectories

that are generated by the Tobin model, Sidrauski [1967], Nagatani [1969], Stein

[1969], Hadjimichalakis [1971a, 1971b], Fischer [1972], and others "gener-

alized" the Tobin model by relaxing the short-run perfect foresight assumption

of the model and replacing it with the adaptive expectations hypothesis of Cagan

[1956]. Not surprisingly, it was found that the generalized Tobin model would

be stabilized provided that price expectations adjusted sufficiently slowly. If

expectations adjusted quickly, however, the instability problem remained.

More recently, Burmeister and Turnovsky [1978] constructed a model where

the speed of adjustment of short-run price expectations is characterized as a

parameter reflecting the degree of rationality, at one end beginning from an

irrational range, going through a range of adaptive expectations and to a final

limit of short-run perfect foresight. In our paper, we take a similar approach to

the dynamics of the generalized Tobin model and study exactly how the behavior

of the economic variables will be affected as the short-run price expectations of

the agents adjust more and more quickly. In particular, we demonstrate that

allowing expectations to adjust more quickly, as the agents start to predict faster,

does not necessarily lead to either the explosive, errant paths or the paths which

asymptotically converge to the steady state. Instead, these paths may very well

form trajectories of bounded and persistent oscillations.

First, we review the generalized Tobin model as formulated by Hadjimichalakis

[1971a]:

(1) k = sf(k)-(1 -s)(O-q)m-nk,

(2) ,i = m(O-p-n),

(3) y - - q),

(4) i=e[mn-L(k, q)] + q,

where the endogenous variables, k, mn, q and i are the capital-labor ratio, the

money stock per capita, the expected and the actual rates of inflation, respectively,

and the parameters, s, 0, n, y and E are the saving ratio, the rate of monetary

* Manuscript received May 12, 1980; revised January 12, 1981.

1 The authors are grateful to the editor and anonymous referees for comments and suggestions.

589

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590 J. BENHABIB AND T. MIYAO

expansion, the natural rate of growth, the speeds of adjustment of expectations

and of the price level, respectively. Note that the price adjustment equation (4)

includes the expected rate of inflation q, as suggested by Fischer [1972], and

Stein [1971], and others.

Substituting (4) into (2) and (3), we obtain three differential equations involv-

ing three variables k, m and q, whose steady state equilibrium values, k*, m* and

q*, can be found by setting k = == 0. Linearizing the dynamical system at

the equilibrium gives

~ k - ~ sf'-n -(1-s)n (1-s)m k-

(5) m =K smL1 -gm m(sL2-1) mm*j,

where all the variables are evaluated at the equilibrium. Let us define

a, - (sj' - n - em -yL

( sf '-n -(1-s)n | -gm m(sL2-1) sf '-n (1-s)m

(6) a2 ~~~+ +

smLI -gm I YE - ysL2 -ysL1 -ysL2

sf '-n -(1-s)n (1-s)m

a3 - gmLl -gm m(sL2-1)

-ysL1 Y8 - sL2

Then it is well known that a necessary and sufficient condition for local stability

of the linear system (5) is:

(7) a, > 0, a2> O, a1a2-a3 > O.

THEOREM 1. The steady state equilibrium is locally asymptotically stable

if the following conditions are satisfied.

(8) Y 0 - + 1m* + (1sk* <

and

(9) ark 1

where

(10) =s - _-q*L2/L, Sk*k*L1/L.

PROOF. (7) is satisfied under conditions (8) and (9) (See Appendix 1).

Since the stationary values k*, m* and q* are not affected by changes in y or a,

we can state from (8) that the smaller the value of y (the slower the adjustment of

expectations) or the greater the value of e (the faster the adjustment of the price

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DYNAMICS OF THE GENERALIZED TOBIN MODEL 591

level), the more likely is condition (8) satisfied. Also, the smaller the elasticity

of the money demand function with respect to q, or the greater the elasticity with

respect to k, the more likely is stability.2

In order to make the main point of this paper, let us go beyond the conventional

local stability analysis and examine what can happen if the steady state equilibri-

um becomes unstable due to certain changes in parameter values. The stability

conditions (8) and (9) indicate that increasing the speed of adjustment of expec-

tations beyond a certain value destabilizes the model. Thus it would seem that

as we move from adaptive expectations towards perfect foresight, we should obtain

"saddle-point" instability results analogous to those of the Tobin model. In

effect, the non-local situation is quite a bit more complex. As expectations

adjust faster, the emergence of instability at the steady state equilibrium seems to

be associated with persistent but bounded oscillations in the variables of the system.

For the sake of illustration, consider an increase in the stock of money at the

equilibrium. The immediate impact of this is to increase the price level and the

real money stock tends to fall back to its original level, but the initial increase in

money also tends to increase price expectations and reduce the capital stock.

The latter two effects reinforce the fall of the money supply, and depending on

their strength, they may cause the money stock to overshoot its long-run equilibri-

um level. As the money supply keeps falling beyond its equilibrium level, how-

ever, the effects on the other two variables are reversed: the capital stock rises

and expectations fall. Combined with the direct effect of the money stock on

the accumulation of money, the fall of the money stock will now be reversed.

This explanation, describing a possibility of divergent or convergent oscil-

lations, is based on local analysis in the neighborhood of the steady state. When

combined with the non-linearities of the system, oscillations which change from

being locally convergent to locally divergent as we vary the parameters, in general,

will also become non-trivial and persistent, forming closed orbits. We will now

prove the existence of closed orbits for the present system. We assume the

following:

(A.1) The functions L(k, q) and f(k) are Cs, where s?2 (continuous and at

least twice differentiable).

(A.2) There exists a set of parameter values such that the stationary point of

the dynamical system (I)-(4) is locally asymptotically stable.

LEMMA. Under (A.1) and (A.2), there exists a value of y=y*, where the

Jacobian of (5) has a pair of pure imaginary roots.

PROOF. By (A.2) there exist parameter values such that the necessary conditions

for the Jacobian of (5) to have roots with negative real parts, a,> 0, a2>0 and

2 Our stability conditions (8) and (9) are somewhat different from those obtained by

Hadjimichalakis [1971a, 1971b]. It is interesting to note that condition (9) guarantees the

uniqueness of the steady state equilibrium, as shown by Hadjimichalakis [1971b].

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592 J. BENHABIB AND T. MIYAO

a1a2-a3O0 (given by (7)), are satisfied. Since a1 = -(sf'-n)+cm+y8L2 and

the stationary values k*, m* and q* are independent of y, there exists a finite y=-

such that j > - (sf'- n - sm)/(cL2) so that a,<0. This implies that not all the

roots of the Jacobian can have negative real parts. Since the roots of the

Jacobian vary continuously with y, there must exist a value of y, say y*, where

either a real root or the real part of a pair of complex roots is equal to zero. If

a real root becomes zero, the determinant of the Jacobian must vanish, that is,

for y =y* we must have a3=0. But as can be seen from the proof of Theorem 1,

the sign of a3 is independent of y, as long as y does not change sign. Thus,

variations in y that do not alter its sign cannot make the determinant vanish, and

the Jacobian of (5) has a pair of pure imaginary roots at y =y*. Q. E. D.

THEOREM 2. Let (A.1) and (A.2) hold. Then there exists a continuous func-

tion y(b) with y(O)=y*, and for all sufficiently small values of 600, there exists

a continuous family of non-constant, positive periodic solutions [k(t, 6), in(t, 6),

q(t, 6)] for the dynamical system (1)-(4), which collapse to the stationary point

(k*, m*, q*) as 6-0.

PROOF. The proof follows from the Hopf Bifurcation Theorem (see Hopf

[1976] and Marsden and McCracken [1976, pp. 197-198]).3 We only have to

verify that the assumptions of the Hopf Theorem are satisfied. By (A.1) our

dynamical system is continuous and differentiable, and by (A.2) a stable stationary

point (k*, mn*, q*) exists and is independent of y. From the above lemma, the

Jacobian of the dynamical system has a pair of pure imaginary roots for y=Y*.

It is shown in Appendix 2 that the real part of the pure imaginary roots is not

stationary with respect to y. This covers the assumptions of the Hopf Theorem.

The existence of positive periodic solutions follows because the stationary point

is positive and the Hopf Theorem states that the periodic solutions collapse con-

tinuously onto the stationary point as b6-O. In other words, the amplitude of

the orbits approaches zero as b6-O. Q. E. D.

It is interesting to compare the above theorem with the early results on adaptive

expectations by Cagan [1956] and their application to the Tobin model by

Sidrauski [1967], Hadjimichalakis [1971a, 1971b], and others. Our results

show that the loss of stability that occurs as expectations adjust faster is associated

with the emergence of bounded, persistent oscillations in price, output and expec-

tations.4 This result holds no matter how quickly prices adjust, that is, no matter

how large the value of s is, since there always exists a value of y at which the

stability of the stationary point is lost. It should be noted that as y grows larger

8 Hopf states and proves the theorem for analytic functions. Howard and Koppel in thier

"editorial comments" in Hopf [1976] revise Hopf's proof and provide the C8 version used in

our proof.

I Similar techniques have been used by Benhabib and Nishimura [1979) to show the existence

of closed orbits in prices, outputs and stocks in a real rational expectations growth model with

long-run perfect foresight.

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DYNAMICS OF THE GENERALIZED TOBIN MODEL 593

we approach perfect foresight and instantaneous market clearing, and yet,

bounded, robust,5 and persistent orbits may exist globally beyond the local

neighborhood of the bifurcation value of y.6

The stability properties of the bifurcating orbits also require complex analysis,

but it is clear from the Hopf Theorem that the orbits will in general exist either in

a left (sub-critical) or a right (super-critical) neighborhood of y*. Given that the

third real root is negative, the bifurcating orbit will be locally attracting (locally

orbitally stable) if it is super-critical, and will be locally repelling (locally orbitally

unstable) if it is sub-critical. Whether the super-critical or sub-critical case holds

depends on the higher order non-linear terms in the Taylor expansion of the

dynamical system at the stationary point. For example, if the stationary point

is locally asymptotically stable at y* (at which point the Jacobian with pure

imaginary roots does not determine local stability), the bifurcating orbits will be

super-critical.7

The sub-critical case is also of special interest, and it corresponds to the "cor-

ridor stability" concept introduced by Leijonhufvud [1973]. For a left neighbor-

hood of y*, the economy will be locally stable for small perturbations around the

stationary point. The region of local stability in the neighborhood of the station-

ary point, however, is bounded by the closed orbit around it. A larger shock

may throw the economy out or beyond the orbit, in which case it does not have

a natural tendency to return to the stationary point. From an economic point

of view, sub-critical and super-critical orbits both seem plausible. Economic

considerations alone cannot restrict the..sign of the second, third and higher

derivatives of the functions appearing in the dynamical system (5) that determine

whether the orbits are sub-critical or super-critical.

New York University, U.S.A.

University of Southern California, U.S.A.

The orbits are "robust" in the sense that under any small perturbation to the parameters

of the system (to the underlying vector field) the orbits will not disappear, provided a certain

"characteristic multiplier" is not identically equal to unity. The case with the characteristic

multiplier identically equal to unity essentially corresponds to the linear case, and any small

nonlinearity will restore the robustness of the orbits. For a rigorous discussion, see Hirsch

and Smale [1976, pp. 309].

6 While a full global discussion of how the orbits change as theparameter A is varied beyond

the local neighborhood of its bifurcation value may be too technical to include here, some recent

results of Alexander and Yorke [1978] show that either (1) the bifurcation parameter of the

orbits or the period of the orbits will be pushed out of its defined domain (or go to infinity), or

(2) the orbit will close onto a stationary point distinct from the stationary point of the original

bifurcation.

See Marsden and McCracken [1976], and also Bruslinskaya [1961].

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594 J. BENHABIB AND T. MIYAO

APPENDIX 1

To prove Theorem 1, we find from (6) and (7)

(1-1) a, = -(sf'-n-ein-yeL2)

=- (sf '-s f + (1 -s)nm -sem-yeL2)

S f -f' + em + 'yL2-(1-s)nm >0?

= \k ,/k>0

because, from (8)

(1-2) 1+ y L2 (1 -s)n > 0.

(1-3) a2 =-(sf -n)em + (1 - s)nemL1 - emyeL2 - ysm(eL2 - 1)

-(sf -n)yeL2 + (1 - s)myeL,

=-(sf -n)(em + yeL2) + (1- s)emL,(n + y) + yeni

-(Sf -s + (1 - s)nM )(em + yeL2)

+ (1 - s)emL,(n + y) + ysm

=S({ -f')(em +ysL2) + (1 -s)ne k2 (n + ' Lk _I

-(1- s)n k yeL2 + ySm > 0,

from (8) (or (1-2)) and (9).

(1-4) a3 =-[(sf'- n)emyeL2 + (1 - s)nm(eL2 - 1)yeL1 + (1 - s)memL1ys

-(1- s)menyeL -(sf - n)m(eL2 - 1)yS - (1-s)neinLyeL2]

=-(1 -s)nm(eL2- 1)yeL + (1 - s)nemL1yeL2

-Sf'- fk + (I1-s)n MT mys

=S({ -f') mye+ (1 -s)nys m2 (L1 k _1)I

which leads to

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DYNAMICS OF THE GENERALIZED TOBIN MODEL 595

(1-5) ala2 - a3 = sQ -f') + gm + ysL2 - (1 - s)nmk !]Len

+ ( s)en (L k- I) + S -f ')(em + yeL2)

+ (1 -s)ycnL1 - (1- s)n!- yeL2j s(-{ -i)

yrnz+y(l - s)en (L1 I > O.

because, from (8)

(1-6) em + ysL2 - (1- s)nrnlk ? y.

APPENDIX 2

In this appendix we prove that the zero real parts of the pure imaginary roots

of the Jacobian of (5) are not stationary with respect to y. Let

(2-1) A3 + aA2 + a2A + a3 = 0

be the characteristic equation of the Jacobian. Then, by Orlando's formula (Sec

Gantmacher [1954, p. 197]),

(2-2) - 9l + A2)(Q'1 + X3)(2 + )3) = ala2- a3

must always hold. As stated in Gantmacher [1954], A and -A are both roots

of (2-1), if and only if ala2-a3=0. Note that a1, a2 and a3 are given by (6)

and also in the proof of Theorem 1. Now assume that x + yi and A are the roots

of (2-1) with x=O, which means that a~a2-a3=0. Differentiating both sides of

(2-2) with respect to y, we obtain

(2-3) -2 dx (A2 +y2) = d(ala2- a3)

dy dy

Then, from (1-1), (1-3) and (1-4) in the proof of Theorem 1, together with

a1a2-a3=O, it follows that

(2-4) /d(ala2 - a3) = da2 + a da2 - da3

dy -da +a dy dy

= EL2a ( + a, P)+ _9-- = a,(sL2 - P) < 0,

where

(2-5) p ( tg th fer+ (l - s)an d t /d 1) 0>

(2-3) anld (2-4) togetller lead to dxlxdy fO.

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596 J. BENHABIB AND T. MIYAO

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