Indeterminate growth paths and stability

*Corresponding author. E-mail: [email protected]

Journal of Economic Dynamics & Control24 (2000) 39}62

Indeterminate growth paths and stability

Thomas Russell!,*, Aleksandar Zecevic"! Department of Economics, Santa Clara University, Santa Clara, CA 95053, USA

" Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053, USA

Received 6 January 1997; accepted 10 August 1998

Abstract

Indeterminacy in an economic growth model arises whenever the stable manifold hasdimension greater than the number of predetermined initial conditions. The stability(indeterminacy) of transition paths in the Benhabib and Farmer (1996) model of growth isinvestigated, using both the Lyapunov method and numerical simulation techniques. Thesensitivity of transient dynamics is analyzed with respect to the choice of parametervalues and with respect to the choice of initial conditions. The likelihood that thebusiness cycle is a pure &sunspot' phenomenon is investigated. ( 2000 Elsevier ScienceB.V. All rights reserved.

JEL classixcation: E00; E3; O40

Keywords: Indeterminate growth; Lyapunov stability; Sunspot equilibrium

1. Introduction

Why does one economy grow faster than another? This remains one of the keyquestions of economic science, in large part because of the hope that anunderstanding of the causes of growth will point to policies which will enablecountries to achieve faster growth and therefore higher standards of living.

It seems very natural to begin the search for an explanation of why growthrates di!er by looking for di!erences across countries in those fundamentaleconomic attributes which might be expected to contribute to higher growth.For example, countries with higher growth rates may save more. Or, such

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 0 - 8

countries may have access to superior technology. Or perhaps citizens of highergrowth countries simply work harder.

This list of fundamental economic attributes could be expanded, but recentlya number of studies have called this &fundamentalist' approach into question. Ina variety of models of growth, there are empirically reasonable parameter valuesat which economic growth is indeterminate. That is to say, for given preferences,given technology, and a given initial capital stock, the future equilibrium growthpath of the economy is not unique, indeed is not even locally unique. This meansthat two &fundamentally' identical economies could evolve along quite di!erentgrowth paths.

The possibility of indeterminacy arises in a variety of economic growthmodels. For example, the Lucas (1988) model of economic growth with humancapital externalities is not uniquely determined by specifying initial physical andhuman capital, as shown in Xie (1994). Models with physical capital andexternalities have been investigated by, among others, Matsuyama (1991),Boldrin (1992), Boldrin and Rustichini (1994), Benhabib and Farmer (1994),Benhabib and Perli (1994), Gali (1994) and Benhabib and Farmer (1996).Indeterminacy in one sector models has also been considered by a number ofauthors, including Kehoe et al. (1991), Kehoe (1991) and Spear (1991).

In addition to its implications for growth theory, indeterminacy also hasimplications for the study of the business cycle. When the equilibrium isindeterminate, &animal spirits' can generate belief-driven cycles in output evenwhen the economy has no underlying fundamental (e.g. technological) uncer-tainty (see Farmer and Guo, 1994). A clear statement of these issues may befound in the symposium introduction by Benhabib and Rustichini (1994).

Many of the models of indeterminacy discussed in this symposium andelsewhere have a common mathematical structure. An economy's equilibriumgrowth path is represented as a dynamic system, typically a second-ordernonlinear di!erential (or di!erence) equation. Next, the steady state (or steadygrowth path) of this dynamic system is investigated. When the steady state ofsuch a system is locally stable, (i.e. when the eigenvalues of the Jacobian of thesystem at the steady state lie in the left half plane) indeterminacy follows. To seethis, suppose we specify one initial condition, say the initial capital stock, K

0.

Because of stability, there is now an open set of values of the second initialcondition, say initial consumption, such that any choice of initial consumptionin this set, taken in combination with K

0, converges to the steady state. Because

the model contains no explanation for which level of initial consumption will bechosen, growth is said to be &indeterminate'. Furthermore, when such a system issubject to an expectations shock, it produces a pure &animal spirits' cycle.

The fact that an equilibrium system is indeterminate is clearly of greattheoretical interest. What is not so clear is how much light this result sheds onthe diversity of experience of actual economies. Negative eigenvalues of theJacobian at the steady state give information about the behavior of the system

40 T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62

1This interesting interpretation of our results was kindly provided by a referee.

only in a neighborhood of its steady state. This neighborhood must be chosensmall enough that linearization approximations hold, and so may be very smallindeed.

Ideally, we seek information on the behavior of indeterminate economies overwide ranges of initial conditions and for long periods of time. Moreover, itwould be interesting to see how such economies behave when their dynamics isnot constrained to be linear. To obtain such information, di!erent mathematicaltechniques must be used.

In this paper we analyze the indeterminacy issue using two approaches.Firstly, we apply the classical stability analysis of Lyapunov to obtain ananalytical estimate of the set of stable (and therefore indeterminate) trajectories.By exhibiting a Lyapunov function for an indeterminate growth model, we canexplicitly evaluate the transitional dynamics for two economies with the sameinitial conditions. This allows us to examine whether or not stable modelsproduce dynamic paths consistent with the stylized facts of comparative growthexperience as set out in, say, Barro and Sala-i-Martin (1995). It also allows us tobound the range of behavior attributable to &animal spirits'. Paths produced bystochastic shocks in sunspot equilibrium models must converge to the models'steady state, so by describing a Lyapunov region we shed some light on thebehavior of such paths.1

Even before we begin this task, it is clear that there is some reason to doubtthe empirical importance of Lyapunov stable indeterminate systems. Supposewe have a two-dimensional nonlinear dynamic system, in which K(t) representsthe capital stock in the economy at time t and C(t) is the consumption level inthe economy at time t. In addition, let K* and C* be the common steady statevalues of capital and consumption in the economy, and let x

1(t),log (K(t)/K*)

and x2(t),log (C(t)/C*), respectively. Then, for Lyapunov stable indeterminate

systems, the following properties hold.

1. Given x(t),(x1(t)x

2(t))T, Lyapunov stability (by de"nition) ensures that for

any number e'0 there exists a small enough d'0 (generally depending one) such that Ex(t)E(e is guaranteed by Ex(0)E(d(e). In other words, econo-mies which start o! close together remain close together at all times.

2. There exists a set C such that x(t)3C for all t50 and for any trajectoryoriginating in this set.

3. Two economies whose initial conditions belong to the set C will convergeuniformly to their common steady state.

4. Although two economies which are fundamentally equivalent may exhibitdi!erent growth behavior, they spend an in"nite amount of time on equilib-rium growth paths which di!er from each other by less than g, where g is anypositive constant, no matter how small.

T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62 41

2 It is possible that the numerical techniques used by Mulligan and Sala-i-Martin (1993) could alsobe adapted to deal with this model, but we have not pursued this task.

Since economic measurement is always imprecise, it may even be the case thatthe indeterminacy predicted by Lyapunov's stability theory cannot be detectedempirically. Indeed, we will certainly be unable to detect indeterminacy forin"nite amounts of time if we use measuring instruments which are calibrated in"nite jumps.

This is not to say that indeterminacy in general is an empirically uninterestingphenomenon. It is simply to point out that using either local stability theoryaround the steady state or Lyapunov stability theorems as a way of detectingindeterminacy only allows us to predict a form of indeterminacy that is so localand so transient that we may never be able to observe it. Moreover, although the"nding of local Lyapunov stability is su$cient to guarantee indeterminacy, it iscertainly not necessary. Indeterminacy of the equilibrium growth path of aneconomic system will also follow if the steady state is an attractor (i.e. if a set ofpaths tends to the common steady state). The set of initial conditions for whichthe trajectories converge to the steady state will be referred to as the region ofattraction, and it is important to point out that this region need not coincide withthe set C.

This other form of indeterminacy (i.e. that associated with the region ofattraction) can yield much more varied growth experience for two funda-mentally identical economies than that generated by Lyapunov stability analy-sis. For example, two &attractive' economies with the same initial conditionscould move along sharply divergent paths before coming together, behaviorwhich is not possible if the economies originate in the set C.

For that reason, the second purpose of this paper is to revisit the issue ofindeterminacy using numerical techniques. The only paper we are aware ofwhich does this in the indeterminate case is the paper by Xie (1994). However,Xie was only able to solve the dynamic model explicitly by imposing restrictionson the parameters which simplify the underlying di!erential equations. Theserestrictions have no obvious empirical basis, so Xie could only speculate on thenature of the dynamics with economically reasonable parameter values. In thispaper we will examine the explicit dynamics of equilibrium growth paths usingempirically reasonable parameter values within the context of one model that isknown to generate indeterminacy, the interesting externality model of Benhabiband Farmer (1996) (which is an extension of the same authors' model from1994).2

Our two methods of analysis are complementary. In the "rst place we willexhibit a Lyapunov function for this model. For a given capital stock, thisfunction allows us to estimate a region of initial consumption levels for whichthe model is stable in the sense of Lyapunov. This, in turn, allows us to explicitly


bound both the transient equilibrium dynamics of economies which start in thisregion and the set of sunspot equilibria. Since this is the range of uniformconvergence, indeterminacy in this region may have less empirical interest.

With that in mind, we will also use numerical techniques to extend theLyapunov estimates of indeterminacy to the entire region of attraction. Usingthese techniques we try to answer a number of questions related to the empiricalimportance of the indeterminacy issue. In particular, we are interested inexamining the size of the region of attraction, the behavior of paths originatingin this region, and the sensitivity of these paths to the choice of initial conditionsand parameters. These questions will be answered within the context of a well-known &indeterminate' model which we now describe.

2. The Benhabib}Farmer model

The following model is due to Benhabib and Farmer (1996). It is a standardRamsay model of growth to which aggregate and sector speci"c externalitieshave been added. The model consists of two sectors } a consumption sectorC and an investment sector I, with sector output produced by the respectiveprivate technologies

C"A(kkK)a(k

L¸)b, I"B[(1!k

k)K]a[(1!k

L)¸]b. (1)

In Eq. (1) K represents the economy wide stock of capital, ¸ is the economywide stock of labor, and k

Kand k

Lare the respective fractions of K and ¸ used in

the consumption sector. Individual "rms take A and B to be constant, andconstant returns to scale hold at this level, implying a#b"1.

Externalities are introduced by assuming that

A"(kNkKM )ah(kN

LM̧ )bhKM ap M̧ bc, B"[(1!kN

K)KM ]ah[(1!kN

L) M̧ ]bhKM ap M̧ bc, (2)

where a bar over a variable represents average economy wide use, and averagesare taken as given by the individual "rm. The parameter h is a measure of sectorspeci"c externalities, while p and c are measures of aggregate capital and laborexternal e!ects, respectively. Given these parameters, it is convenient to de"nethree additional quantities: l,(1#h), a,a(1#h#p) and b,(1#h#c).

The consumer's preferences are given by a time separable utility functionalwhose instantaneous value is given by

;(t)"log C(t)!¸(t)1`s1#s

, (3)

where C(t) is consumption, ¸(t) is the labor supply and s represents the laborelasticity parameter. The consumer's optimization problem is then to maximize


the integral

J"P=

0

;(t)e~ot dt (4)

subject to

KQ (t)"I(t)!dK(t) (5)

and K(0)" K0, where I(t) denotes investment goods and d is the depreciation

rate of capital.The necessary conditions for optimizing integral can be formulated in terms

of its "rst variation. Namely, in order for pair MK(t), C(t)N to maximize it isnecessary that the "rst variation equals zero along this trajectory; it is alsorequired that the trajectory satis"es the transversality condition

limt?=

e~otK(t)K(t)"0, (6)

where K(t) is the standard co-state variable associated with the Hamiltonianformulation of the optimization problem.

In analyzing the "rst variation of Eq. (4), it is convenient to introduce a newvariable S de"ned as

S,Ka@l¸b@l

C1@l. (7)

The quantity S takes values between one and in"nity, and can be interpretedas the inverse of the factor share going to the consumption sector. Based onde"nition Eq. (7), we obtain relationships

bS"¸1`s, C(S!1)l~1"1

K, I"C(S!1)l (8)

and the solution to the optimization problem reduces to a pair of di!erentialequations

KQK"o#d!a

S

KK,

KQK"

S!1

KK!d. (9)

It is important to point out that variable S is in fact an implicit function ofK and K. Indeed, from Eqs. (7) and (8) one directly obtains

(S!1)1~lSl~(b@(1`s))"b(b@(1`s))KaK, (10)


which implies that the only independent variables in Eq. (9) are K and K. It isalso easily established that system (9) has a unique steady state, which can becomputed in the following sequence:

S*"o#d

o#d(1!a)N

K*"o#d

a(S*!1)(1~l)@(a~1)S**(l~1)@(a~1)~b@((a~1)(1`s))+bb@((a~1)(1`s))

N K*"aS*

(o#d)K*N C*"

1

K*(S*!1)1~l. (11)

For the purposes of stability analysis, it will be convenient to work withlogarithmic variables j,log K and k,log K, and use them to de"nex1,j!j* and x

2,k!k*. This produces a new system of di!erential equa-

tions

xR1"o#d!awSe~x1~x2,

xR2"wSe~x1~x2#we~x1~x2!o,

(12)

in which w,e~kH~jH"(o#d(1!a))/a, and the steady state is at the origin. Inother words, the problem can be formulated as a nonlinear dynamic system inthe form

xR "f (x), f (0)"0 (13)

which is suitable for a Lyapunov-type stability analysis. We should also mentionthat variable S can be expressed in terms of x

1and x

2as

(S!1)pSq"Meax2`x1, (14)

where p,1!l, q,l!b/(1#s) and M,bb@(1`s)eakH`jH.Eq. (13) can be simpli"ed by computing the Jacobian A(x) and linearizing

around the steady state, which produces

xR "A(0)x (15)

As is well known, the linear system (15) will be stable if and only if A(0) hasboth eigenvalues in the left half of the complex plane. The necessary andsu$cient conditions for this can be stated explicitly in terms of the trace anddeterminant of A(0)

Trace"o#o#d(1!a)

a(a!a)

RSRx

1

(0,

(16)

Det"[o#d(1!a)]2

a(a!1)

RSRx

1

'0,


where RS/Rx1

is evaluated at the steady state using Eq. (14). Such a formulationallows for an easy identi"cation of parameters for which indeterminacy canoccur. We should point out, however, that the stability of Eq. (15) only guaran-tees local stability of Eq. (13) around the origin. In other words, since Eq. (15) isa local approximation, all that can be said based on the eigenvalues of theJacobian is that indeterminacy exists in a neighborhood of the origin.

3. The Lyapunov method

To obtain an explicit bound on initial conditions that lead to indeterminacy,it is necessary to replace linearization with a more sophisticated approach. Inthis section we will use Lyapunov's method to estimate a region of stability forthis model. The following well known result provides a basis for our analysis(e.g. Rouche et al., 1977).

Theorem 1. Let x(t;x0) denote the solution of the nonlinear dynamic system (12)

that corresponds to initial condition x(0)"x0, and let XLRn be a set contain-

ing the origin. Assume also that there exists a continuously di!erentiablefunction < : XPR satisfying <(0)"0 and

<(x)'a(ExE), ∀x3X, (17)

<Q (x)(0, ∀x3X, (18)

where <Q (x) denotes the derivative along the solution of Eq. (12), anda : R

`PR

`is a monotonically increasing function. Then,

(i) There exists a region CLX such that x(t;x0)3C for all t50 and for all

x03C.

(ii) For any initial condition x03C

Ex(t; x0)EP0, tPR. (19)

The successful application of Theorem 1 hinges on the availability of a func-tion that satis"es Eqs. (17) and (18). If such a function can be found it will bereferred to as a ¸yapunov function, and the Lyapunov stability of the systemfollows automatically. In addition, Corollary 1 (e.g. Siljak, 1969) allows us toexplicitly estimate the region C.

Corollary 1. Assume that there exists a Lyapunov function < : XPR satisfyingthe conditions of Theorem 1 for system (12), and de"ne the set %(r) by

%(r),MxD<(x)4rN (20)

where r3R`. If r

0is the largest value of r satisfying %(r)LX, then the set C can

be estimated as C"%(r0).


For the nonlinear system in Eq. (12) we propose a Lyapunov function de"nedas

<(x)"f T(x)Hf (x), (21)

where H"(hij) is the unique symmetric positive de"nite solution of the matrix

equation

AT(0)H#HA(0)"!I. (22)

In Eq. (22) I denotes the identity matrix, and this equation is known to havea unique symmetric positive de"nite solution whenever the Jacobian A(0) has alleigenvalues in the left half plane (e.g. Gantmacher (1959)). Observing thatf (x)"0 if and only if x"0, it is easily seen that Eq. (16) directly implies Eq. (17)for the Lyapunov function in Eq. (21).

Eq. (16) also guarantees that Eq. (18) is satis"ed in some region XLRn

containing the origin. Indeed, it su$ces to observe that Eq. (18) can be writtenas

<Q (x)"f T(x)[AT(x)H#HA(x)] f (x) (23)

and that the Jacobian A(x) is a continuous function of x. Since Eq. (16) guaran-tees that A(0) has eigenvalues in the left half plane, from Eq. (22) and the conti-nuity of A(x) it follows that Eq. (18) is satis"ed in some neighborhood of theorigin.

In order to apply Theorem 1 and Corollary 1 to the Benhabib}Farmermodel (Eq. (12)), it is "rst necessary to explicitly determine a region X whereEq. (18) holds. To that e!ect, it is convenient to introduce the matrixM(x),AT(x)H#HA(x); the problem of identifying X now becomes equivalentto determining a region where M(x) is a negative de,nite matrix. Observing thatM(0) has a pair of negative real eigenvalues and that M(x) is a continuousfunction of x, it is easily seen that the boundary of X is de"ned by the set

B"MxDdet M(x)"0N. (24)

The main di$culty with this approach lies in the fact that we must deal withthe variable S"S(x

1, x

2), which is de"ned implicitly by Eq. (14). To resolve this

problem, we should "rst note that the partial derivatives of S with respect tox1

and x2

can be easily obtained from Eq. (14) as

RSRx

1

"

S(S!1)

pS#q(S!1),

RSRx

2

"

aS(S!1)

pS#q(S!1), (25)

using the implicit function theorem (which requires that pS#q(S!1)O0).Furthermore, de"ning u(x),w exp(!x

1!x

2) and e(S),(S!1)/(pS#

q(S!1)), the Jacobian A(x) can be expressed as a function of x and S

A(x, S)"Ca(1!e)Su a(1!ae)Su

[1!(1!e)S]u [1!(1!ae)S]uD. (26)


Fig. 1. The region X.

3A similar form for region X can be obtained for the earlier Benhabib}Farmer model (1994) (seeRussell and Zecevic, 1997; Russell and Zecevic, 1998).

It is important to observe that the variable u(x) appears in every element of theJacobian, and that by de"nition u(x)'0 for all x; consequently, the boundary ofX can be de"ned exclusively in terms of variable S. In other words, one can saythat the condition det M(x)"0 is equivalent to the condition

/(S)"4[a11

(S)h11#a

21(S)h

12] [a

12(S)h

12#a

22(S)h

22]

![a12

(S)h11#[a

11(S)#a

22(S)]h

12#a

21(S)h

22]2"0, (27)

where aij(S) are the elements of matrix u~1(x)A(x, S).

Given that M(0) is negative de"nite, it follows that at the steady stateU(S*)'0. As a result, it is necessary to compute only the two roots of equationU(S)"0 that encircle S*. These two roots (denoted in the following by S

1and

S2) de"ne two lines in the x

1, x

2plane which represent the boundary of region X

x2"!

1

ax1!

1

alogC

(Si!1)pSq

iM D, i"1, 2. (28)

Thus, we can conclude that region ) has the form shown in Fig. 1.3

4. The region of stability: Two test cases

To explicitly compute the region of stability, it is necessary to assignparameter values to the model. In the following, we will examine two separatecases. Case 1 uses a set of parameter values proposed by Benhabib and


Farmer (1996); these parameters are empirically reasonable and have the prop-erty that demand curves for labor slope down. We show, however, that theLyapunov stable region for these parameters is con"ned to an extremely smallset around the steady state.

For this reason, we examine a second case (referred to in the following as Case2). The parameter values chosen for Case 2 are an extension of those used byBenhabib and Farmer in an earlier one-sector model of indeterminacy, (1994). Itwas shown by the authors that in this one-sector model an upward slopingdemand curve for labor is a necessary condition for indeterminacy. At theparameter values used in Case 2, the aggregate demand curve for labor (that is,the curve which would be estimated by an economist who mistakenly viewedthis model as a one-sector model) slopes upward, and the size of the Lyapunovregion is several orders of magnitude larger. The comparison of these two caseswill therefore provide a "rst look at the question of parameter sensitivity.

Case 1. Following Benhabib and Farmer, in this case we assume that noaggregate externalities are present, implying that b/l"b and a/l"a. We set theparameter values to be: a"0.3, b"0.7, o"0.05, d"0.1, s"1, a"0.345,b"0.805 and l"1.15.

The unique symmetric positive de"nite solution of Eq. (22) is

H"C159.319 47.8938

47.8938 14.4556D. (29)

Recalling that S*"(o#d)/(o#d(1!a))"1.25, we now need to compute thetwo roots of U(S)"0 that encircle this point. A graph of U(S!S*) for therelevant region is shown in Fig. 2, indicating that the two roots of U(S)"0 arevery close to S*.

Using Newton's method, the actual values of the roots are found to beS1"1.249995745 and S

2"1.2503747. The boundary of region X is then given

by the following two lines

x2"!2.899x

1#2.4717]10~8,

(30)x2"2.899x

1!1.7816]10~6.

To identify the region of stability for this Lyapunov function, we need toimbed the largest set %(r) into the region X. For this case, we obtain

C,%(r0)"MxD<(x)42.4]10~11N. (31)

Given the de"nition of <(x), it now remains to compute the points in the x1,

x2

plane for which

f T(x)Hf (x)"2.4]10~11. (32)

The resulting region is shown in Fig. 3.


Fig. 2. The function U(S!S*) for test case 1.

As Fig. 3. makes clear, for this set of parameter values the Lyapunov stableregion is too small to be of economic interest. An economy whose initialconditions lie in this set generates growth and #uctuations many orders ofmagnitude smaller than those observed in the US economy. Capital stocks, forexample, must always be within 10~6% of their steady-state value, thus com-pletely ruling out observed rates of capital accumulation.

Of course, initial conditions could lie outside this Lyapunov set but insidea region of attraction, and still generate an equilibrium path. We compute theregion of attraction for this case in the next section, and now turn to examinethe Lyapunov stable region for Case 2.

Case 2. In this case it is assumed that aggregate capital and labor externalitiesare present, and the parameters are chosen as: a"1/3, b"2/3, o"0.02,d"0.07, s"0.25, a"0.83, b"1.66 and l"1.03. These parameter valuesare similar to the ones used in Benhabib and Farmer (1994), and they re-sult in a Jacobian A(0) with eigenvalues in the left half plane. When(b!1#(1#s)(l!1))'0 (which is clearly the case for this parameter con"g-uration), aggregate demand for labor is upward sloping. The unique symmetric


Fig. 3. The region C.

positive de"nite solution of Eq. (22) is given as

H"C200.872 86.1811

86.1811 38.937 D. (33)

In this case S*"1.35, and a graph of U(S) for the relevant region is shown inFig. 4. The roots are easily found to be S

1"1.333684 and S

2"2.904155, using

Newton's method.The boundary of region X is given by the following two lines

x2"!1.20482x

1#0.0061, x

2"1.20482x

1!0.93028, (34)

and the set C is found to be

C,%(r0)"MxD<(x)42]10~5N. (35)

Computing the points in the x1, x

2plane for which

f T(x)Hf (x)"2]10~5, (36)

we obtain the region of stability shown in Fig. 5.


Fig. 4. The function U(S) for test case 2.

Fig. 5. The region C for test case 2.


Fig. 5 indicates that capital stocks must always be within 3% of their steadystate value, which is six orders of magnitude larger than the variation observedin test case 1. Nevertheless, this set remains small for all practical purposes.

5. The region of attraction

Theorem 1 implies that solutions originating in set C are con"ned to this set atall times, so the maximum variation in K and K at any point in time cannotexceed the maximum variation of this region. Consequently, results obtained byLyapunov's method are typically conservative, and only very modest variationsin growth experience (or very mild cycles) can be detected. In this section, wepropose to investigate the potentially more interesting properties of attractivetrajectories originating outside of the region C.

In order to estimate a region of attraction for this model, it is necessary tosolve the di!erential equation (12) numerically. Conceptually, this can be doneby "xing the initial value of the capital stock, and incrementally varying theinitial consumption until the trajectory fails to converge to the steady state.However, before such a procedure can be implemented, there are two problemsthat need to be resolved. In the "rst place, the variable S is an implicit function ofx1

and x2, so Eq. (12) cannot be solved by direct numerical integration. Second-

ly, Eq. (12) is given in terms of x1and x

2, and consumption does not appear as an

initial condition.Regarding variable S"S(x

1, x

2), we propose to resolve the problem by

reformulating Eq. (12) as a system of di!erential-algebraic equations

xR1"o#d!awSe~x1~x2,

xR2"wSe~x1~x2#we~x1~x2!o, (37)

0"(S!1)pSq!Meax2`x1.

The numerical solution of such equations is a well known problem (e.g.Petzold, 1982), and the simulation can be performed using a variety of implicitnumerical integration schemes (the trapezoidal method being the most practicalapproach).

To explicitly incorporate consumption into the simulation, we propose to "xthe initial capital stock K

0and incrementally vary variable S

0instead of C

0. For

each chosen value S0, we can compute the corresponding x

1(0) and C

0using

Eqs. (14) and (8), respectively. In this way, one can easily monitor the incremen-tal change in initial consumption.

Case 1. The boundaries of the region of attraction for this case are summarizedin Table 1, in which x

20,log (K

0/K*) and x

30,log (C

0/C*), respectively.

This region is plotted in Fig. 6.


Tab

le1

Hig

han

dlo

wva

lues

for

initia

lco

nsu

mption

inth

ere

gion

ofat

trac

tion

x20

!0.

25]

10~

30

10~

31.

5]10

~3

2]10

~3

2.25

]10

~3

2.47

]10

~3

H5.

72]

10~

41.

98]

10~

33.

7]10

~3

3.93

]10

~3

3.69

]10

~3

3.23

]10

~3

1.42

]10

~3

x30

L!

6.9]

10~

4!

5.98

]10

~4

!3.

08]

10~

4!

1.35

]10

~4

3.79

]10

~5

1.24

]10

~4

1.98

]10

~4

Tab

le2

Hig

han

dlo

wva

lues

for

initia

lco

nsu

mption

inth

ere

gion

ofat

trac

tion

x10

!5

!3

!1

01

35

H1.

328

2.01

72.

858

3.23

53.

625

4.39

5.13

7

x20

L!

4.23

9!

2.57

9!

0.91

9!

0.08

90.

741

2.4

4.06

1


Fig. 6. The region of attraction in the x2, x

3plane.

The region of attraction for this case is larger than the correspondingLyapunov region, but is still clearly far too small to describe the growth andcyclical behavior of the US economy. For example, in the region of attractionthe capital stock can vary from !0.025% to 0.25% of its steady state value. Onthe other hand, in the region in which capital is growing (i.e. when capital isbelow its steady state value), consumption must lie between !0.06% and 0.2%of its steady state value.

We can conclude, therefore, that whether we view paths in the region ofattraction as describing diverse behavior across economies, or take the alterna-tive view that paths in this region provide the support for cycles driven byshocks to expectations, for these parameter values the dynamics are too concen-trated around the steady state to describe any observed behavior.

Case 2. The parameter values used in Case 2 provide a more plausible picture.The boundaries of the region of attraction in this case are summarized inTable 2 and are plotted in Fig. 7. As can be seen, this case allows for a morereasonable growth picture. For example, when the capital is at 40% of its steadystate value, the consumption can lie anywhere between 40% and 1,700% of itssteady state value. This allows for long periods of growth at observed growthrates, and in addition allows for a large range of possible future developmentacross economies.


Fig. 7. The region of attraction in the x2, x

3plane.

To demonstrate the dynamic behavior in this case, Figs. 8}10 plot possibleconsumption, capital and output paths for economies that originate near thelower boundary of the region of attraction. These "gures were generated for thecase where x

20"!0.288 and x

30"!0.328, which corresponds to initial

capital and consumption levels that are at 75% and 72% of their respectivesteady state values. As an additional illustration, in Fig. 11 we provide thetrajectory in the x

2and x

3plane.

The variation in possible trajectories becomes even more striking if weanalyze economies that start near the upper boundary of the region of attraction.In Figs. 12 and 13 we consider an extreme case for which x

20"!0.288 and

x30"3.1. This is equivalent to assuming that the initial capital is at 75% of its

steady state value, while the initial consumption is 22 times larger than its steadystate value.

The capital and consumption paths in these "gures exhibit very large vari-ations over time, which is not unexpected considering our choice of initialconsumption. Large variations also characterize the trajectory in the x

2,

x3

plane, which is shown in Fig. 14.As we can see, whether we view these paths as describing diverse growth

experience, or instead think of imposing a stochastic process over these paths togenerate a sunspot cycle, economically interesting behavior can be generated forreasonably long periods of time.


Fig. 8. Normalized consumption path C(t)/C*.

Fig. 9. Normalized capital path K(t)/K*.


Fig. 10. Normalized output path >(t)/>*.

Fig. 11. The trajectory in the x2, x

3plane.


Fig. 12. Normalized consumption path C(t)/C*.

Fig. 13. Normalized capital path K(t)/K*.


Fig. 14. The trajectory in the x2, x

3plane.

Finally we may ask what happens to paths which start outside the region ofattraction. To answer this question we solved the model numerically for a var-iety of initial conditions outside the region of attraction. In every case thesolution diverged to in"nity so rapidly that the transversality condition,(Eq. (6)), was violated. We have been unable to "nd any paths outside the regionof attraction which satisfy the necessary transversality condition of utilitymaximization.

6. Conclusion

This paper proposes a method for "nding explicit solutions to highly nonlin-ear indeterminate models of economic growth. The proposed method exploitsthe fact that the steady state of such models is stable. This allows us to "nda Lyapunov function for the system, and subsequently compute the region ofstability.

Since paths in the stability region converge uniformly to the steady state (byde"nition), behavior in this region will generally be uninteresting from aneconomic standpoint. Once a Lyapunov stable region has been computedanalytically, however, it is straightforward to extend it numerically to the region


of attraction. Some paths in this region can diverge signi"cantly before reachingthe steady state, and an examination of this region is potentially of more interest.

In this paper, the region of attraction was computed numerically for thetwo-sector model of indeterminacy developed by Benhabib and Farmer (1996).Two conclusions emerge.

1. At the parameter values proposed by Benhabib and Farmer (1996), the regionof attraction is so close to the steady state that the model cannot be used todescribe long run growth paths of developed economies (such as the USeconomy). As disappointing as this may be, the result is a testament to thepower of the proposed technique. Previous solution methods which linearizethe system around its steady state simply cannot address the question of thesize of the region of attraction.

2. If we choose di!erent parameter values (as it happens, values at whichaggregate demand slopes upward), the model generates far more reasonablegrowth behavior. Moreover, for this parameter con"guration equilibriumgrowth is very sensitive to the choice of the free initial condition. It istherefore possible that animal spirits could explain cycles in this case.

Our conclusions indicate that behavior in this model is highly parametersensitive. The parameters used in Case 1 were deliberately chosen by Benhabiband Farmer to make the model lie close to the boundary of indeterminacy.A systematic investigation of parameter con"gurations moving away from thisboundary may yield interesting insights. We hope to investigate the problem ofparametric stability in future work.

Acknowledgements

The authors wish to thank Professor Dragoslav Siljak for many helpfuldiscussions on this problem. All errors are the authors' own.

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