*Corresponding author. E-mail: [email protected] 24 (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 has dimension greater than the number of predetermined initial conditions. The stability (indeterminacy) of transition paths in the Benhabib and Farmer (1996) model of growth is investigated, using both the Lyapunov method and numerical simulation techniques. The sensitivity of transient dynamics is analyzed with respect to the choice of parameter values and with respect to the choice of initial conditions. The likelihood that the business cycle is a pure&sunspot'phenomenon is investigated. ( 2000 Elsevier Science B.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 key questions of economic science, in large part because of the hope that an understanding of the causes of growth will point to policies which will enable countries to achieve faster growth and therefore higher standards of living.
It seems very natural to begin the search for an explanation of why growth rates di!er by looking for di!erences across countries in those fundamental economic attributes which might be expected to contribute to higher growth. For example, countries with higher growth rates may save more. Or, such
countries may have access to superior technology. Or perhaps citizens of higher growth countries simply work harder.
This list of fundamental economic attributes could be expanded, but recently a number of studies have called this&fundamentalist'approach into question. In a variety of models of growth, there are empirically reasonable parameter values at which economic growth isindeterminate. That is to say, for given preferences, given technology, and a given initial capital stock, the future equilibrium growth path of the economy is not unique, indeed is not even locally unique. This means that two&fundamentally'identical economies could evolve along quite di!erent growth paths.
The possibility of indeterminacy arises in a variety of economic growth models. For example, the Lucas (1988) model of economic growth with human capital externalities is not uniquely determined by specifying initial physical and human capital, as shown in Xie (1994). Models with physical capital and externalities 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 of authors, including Kehoe et al. (1991), Kehoe (1991) and Spear (1991).
In addition to its implications for growth theory, indeterminacy also has implications for the study of the business cycle. When the equilibrium is indeterminate,&animal spirits'can generate belief-driven cycles in output even when the economy has no underlying fundamental (e.g. technological) uncer-tainty (see Farmer and Guo, 1994). A clear statement of these issues may be found in the symposium introduction by Benhabib and Rustichini (1994).
Many of the models of indeterminacy discussed in this symposium and elsewhere have a common mathematical structure. An economy's equilibrium growth path is represented as a dynamic system, typically a second-order nonlinear di!erential (or di!erence) equation. Next, the steady state (or steady growth path) of this dynamic system is investigated. When the steady state of such a system is locally stable, (i.e. when the eigenvalues of the Jacobian of the system at the steady state lie in the left half plane) indeterminacy follows. To see this, 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 initial condition, say initial consumption, such thatanychoice of initial consumption in this set, taken in combination withK
0, converges to the steady state. Because
the model contains no explanation for which level of initial consumption will be chosen, growth is said to be&indeterminate'. Furthermore, when such a system is subject to an expectations shock, it produces a pure&animal spirits'cycle.
1This interesting interpretation of our results was kindly provided by a referee.
only in a neighborhood of its steady state. This neighborhood must be chosen small enough that linearization approximations hold, and so may be very small indeed.
Ideally, we seek information on the behavior of indeterminate economies over wide ranges of initial conditions and for long periods of time. Moreover, it would be interesting to see how such economies behave when their dynamics is notconstrained to be linear. To obtain such information, di!erent mathematical techniques 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 an analytical estimate of the set of stable (and therefore indeterminate) trajectories. By exhibiting a Lyapunov function for an indeterminate growth model, we can explicitly evaluate the transitional dynamics for two economies with the same initial conditions. This allows us to examine whether or not stable models produce dynamic paths consistent with the stylized facts of comparative growth experience as set out in, say, Barro and Sala-i-Martin (1995). It also allows us to bound the range of behavior attributable to&animal spirits'. Paths produced by stochastic shocks in sunspot equilibrium models must converge to the models'
steady state, so by describing a Lyapunov region we shed some light on the behavior of such paths.1
Even before we begin this task, it is clear that there is some reason to doubt the empirical importance of Lyapunov stable indeterminate systems. Suppose we have a two-dimensional nonlinear dynamic system, in whichK(t) represents the capital stock in the economy at timetandC(t) is the consumption level in the economy at timet. In addition, letK*andC*be the common steady state values of capital and consumption in the economy, and letx
1(t),log (K(t)/K*)
any numbere'0 there exists a small enoughd'0 (generally depending on e) such thatEx(t)E(eis guaranteed byEx(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 trajectory
originating in this set.
3. Two economies whose initial conditions belong to the set Cwill converge uniformly to their common steady state.
2It is possible that the numerical techniques used by Mulligan and Sala-i-Martin (1993) could also be 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 that the indeterminacy predicted by Lyapunov's stability theory cannot be detected empirically. Indeed, we will certainly be unable to detect indeterminacy for in"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 uninteresting phenomenon. It is simply to point out that using either local stability theory around the steady state or Lyapunov stability theorems as a way of detecting indeterminacy only allows us to predict a form of indeterminacy that is so local and 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 is certainly not necessary. Indeterminacy of the equilibrium growth path of an economic system will also follow if the steady state is anattractor(i.e. if a set of paths tends to the common steady state). The set of initial conditions for which the trajectories converge to the steady state will be referred to as theregion of attraction, and it is important to point out that this regionneed notcoincide with the setC.
This other form of indeterminacy (i.e. that associated with the region of attraction) 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 conditions could move along sharply divergent paths before coming together, behavior which is not possible if the economies originate in the setC.
For that reason, the second purpose of this paper is to revisit the issue of indeterminacy using numerical techniques. The only paper we are aware of which 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 restrictions on the parameters which simplify the underlying di!erential equations. These restrictions have no obvious empirical basis, so Xie could only speculate on the nature of the dynamics with economically reasonable parameter values. In this paper we will examine the explicit dynamics of equilibrium growth paths using empirically reasonable parameter values within the context of one model that is known to generate indeterminacy, the interesting externality model of Benhabib and Farmer (1996) (which is an extension of the same authors' model from 1994).2
bound both the transient equilibrium dynamics of economies which start in this region and the set of sunspot equilibria. Since this is the range of uniform convergence, indeterminacy in this region may have less empirical interest.
With that in mind, we will also use numerical techniques to extend the Lyapunov estimates of indeterminacy to the entire region of attraction. Using these techniques we try to answer a number of questions related to the empirical importance of the indeterminacy issue. In particular, we are interested in examining the size of the region of attraction, the behavior of paths originating in this region, and the sensitivity of these paths to the choice of initial conditions and 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 standard Ramsay model of growth to which aggregate and sector speci"c externalities have been added. The model consists of two sectors } a consumption sector Cand an investment sector I, with sector output produced by the respective private technologies
C"A(k
kK)a(kL¸)b, I"B[(1!kk)K]a[(1!kL)¸]b. (1)
In Eq. (1)Krepresents the economy wide stock of capital,¸is the economy wide stock of labor, andkKandkLare the respective fractions ofKand¸used in the consumption sector. Individual "rms take A and B to be constant, and constant returns to scale hold at this level, implyinga#b"1.
Externalities are introduced by assuming that
A"(kNkKM )ah(kNL¸M)bhKMap¸Mbc, B"[(1!kNK)KM ]ah[(1!kNL)¸M]bhKM ap¸Mbc, (2) where a bar over a variable represents average economy wide use, and averages are taken as given by the individual"rm. The parameterhis a measure of sector speci"c externalities, whilepandcare measures of aggregate capital and labor external e!ects, respectively. Given these parameters, it is convenient to de"ne three additional quantities:l,(1#h),a,a(1#h#p) andb,(1#h#c).
The consumer's preferences are given by a time separable utility functional whose instantaneous value is given by
;(t)"logC(t)!¸(t)1`s
1#s, (3)
the integral
0, whereI(t) denotes investment goods anddis 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 is necessary that the "rst variation equals zero along this trajectory; it is also required that the trajectory satis"es the transversality condition
lim
t?=
e~otK(t)K(t)"0, (6)
where K(t) is the standard co-state variable associated with the Hamiltonian formulation of the optimization problem.
In analyzing the"rst variation of Eq. (4), it is convenient to introduce a new variableSde"ned as
S,Ka@l¸b@l
C1@l . (7)
The quantityStakes values between one and in"nity, and can be interpreted as the inverse of the factor share going to the consumption sector. Based on de"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!erential equations
It is important to point out that variableSis in fact an implicitfunction of KandK. Indeed, from Eqs. (7) and (8) one directly obtains
which implies that the only independent variables in Eq. (9) areKandK. It is also easily established that system (9) has a unique steady state, which can be computed in the following sequence:
For the purposes of stability analysis, it will be convenient to work with logarithmic variables j,logK and k,logK, and use them to de"ne
other words, the problem can be formulated as a nonlinear dynamic system in the form
xR"f(x), f(0)"0 (13)
which is suitable for a Lyapunov-type stability analysis. We should also mention that variableScan be expressed in terms ofx
1andx2as
(S!1)pSq"Meax2`x1, (14)
wherep,1!l,q,l!b/(1#s) andM,bb@(1`s)eakH`jH.
Eq. (13) can be simpli"ed by computing the JacobianA(x) and linearizing around the steady state, which produces
xR"A(0)x (15)
whereRS/Rx
1is evaluated at the steady state using Eq. (14). Such a formulation
allows for an easy identi"cation of parameters for which indeterminacy can occur. We should point out, however, that the stability of Eq. (15) only guaran-teeslocalstability of Eq. (13) around the origin. In other words, since Eq. (15) is a local approximation, all that can be said based on the eigenvalues of the Jacobian is that indeterminacyexistsin 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. In this section we will use Lyapunov's method to estimate aregion of stabilityfor this model. The following well known result provides a basis for our analysis (e.g. Rouche et al., 1977).
Theorem 1.Letx(t;x
0) denote the solution of the nonlinear dynamic system (12)
that corresponds to initial conditionx(0)"x
0, and letXLRnbe a set
contain-ing the origin. Assume also that there exists a continuously di!erentiable function<:XPRsatisfying<(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), and a:R`PR
`is a monotonically increasing function. Then,
(i) There exists a regionCLXsuch thatx(t;x
0)3Cfor allt50 and for all x
03C.
(ii) Foranyinitial conditionx
03C
Ex(t;x
0)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 be referred to as a ¸yapunov function, and the Lyapunov stability of the system follows automatically. In addition, Corollary 1 (e.g. Siljak, 1969) allows us to explicitly estimate the regionC.
For the nonlinear system in Eq. (12) we propose a Lyapunov function de"ned as
<(x)"fT(x)Hf(x), (21)
whereH"(h
ij) is the unique symmetric positive de"nite solution of the matrix
equation
AT(0)H#HA(0)"!I. (22)
In Eq. (22)Idenotes the identity matrix, and this equation is known to have a unique symmetric positive de"nite solution whenever the JacobianA(0) has all eigenvalues in the left half plane (e.g. Gantmacher (1959)). Observing that f(x)"0 if and only ifx"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 written as
<Q (x)"fT(x)[AT(x)H#HA(x)]f(x) (23)
and that the JacobianA(x) is a continuous function ofx. Since Eq. (16) guaran-tees thatA(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 the origin.
In order to apply Theorem 1 and Corollary 1 to the Benhabib}Farmer model (Eq. (12)), it is "rst necessary to explicitly determine a region Xwhere Eq. (18) holds. To that e!ect, it is convenient to introduce the matrix M(x),AT(x)H#HA(x); the problem of identifyingXnow becomes equivalent to determining a region whereM(x) is anegative de,nite matrix. Observing that M(0) has a pair of negative real eigenvalues and that M(x) is a continuous function ofx, it is easily seen that the boundary ofXis de"ned by the set
B"MxDdetM(x)"0N. (24)
The main di$culty with this approach lies in the fact that we must deal with the variableS"S(x
1,x2), which is de"ned implicitly by Eq. (14). To resolve this
problem, we should"rst note that the partial derivatives of Swith respect to x
using the implicit function theorem (which requires that pS#q(S!1)O0). Furthermore, de"ning u(x),wexp(!x
1!x2) and e(S),(S!1)/(pS# q(S!1)), the JacobianA(x) can be expressed as a function ofx andS
A(x,S)"
C
a(1!e)Su a(1!ae)SuFig. 1. The regionX.
3A similar form for regionXcan be obtained for the earlier Benhabib}Farmer model (1994) (see Russell and Zecevic, 1997; Russell and Zecevic, 1998).
It is important to observe that the variableu(x) appears ineveryelement of the Jacobian, and that by de"nitionu(x)'0 for allx; consequently, the boundary of Xcan be de"ned exclusively in terms of variableS. In other words, one can say that the condition detM(x)"0 is equivalent to the condition
/(S)"4[a
Given that M(0) is negative de"nite, it follows that at the steady state U(S*)'0. As a result, it is necessary to compute only the two roots of equation U(S)"0 thatencircle S*. These two roots (denoted in the following byS
1and
Thus, we can conclude that region)has the form shown in Fig. 1.3
4. The region of stability: Two test cases
Farmer (1996); these parameters are empirically reasonable and have the prop-erty that demand curves for labor slope down. We show, however, that the Lyapunov stable region for these parameters is con"ned to an extremely small set around the steady state.
For this reason, we examine a second case (referred to in the following as Case 2). The parameter values chosen for Case 2 are an extension of those used by Benhabib and Farmer in an earlier one-sector model of indeterminacy, (1994). It was shown by the authors that in this one-sector model an upward sloping demand curve for labor is a necessary condition for indeterminacy. At the parameter values used in Case 2, the aggregate demand curve for labor (that is, the curve which would be estimated by an economist who mistakenly viewed this model as a one-sector model) slopes upward, and the size of the Lyapunov region is several orders of magnitude larger. The comparison of these two cases will therefore provide a"rst look at the question of parameter sensitivity.
Case 1. Following Benhabib and Farmer, in this case we assume that no aggregate externalities are present, implying thatb/l"banda/l"a. We set the parameter values to be: a"0.3,b"0.7,o"0.05,d"0.1,s"1,a"0.345, b"0.805 andl"1.15.
The unique symmetric positive de"nite solution of Eq. (22) is
H"
C
159.319 47.893847.8938 14.4556
D
. (29)Recalling thatS*"(o#d)/(o#d(1!a))"1.25, we now need to compute the two roots of U(S)"0 that encircle this point. A graph of U(S!S*) for the relevant region is shown in Fig. 2, indicating that the two roots ofU(S)"0 are very close to S*.
Using Newton's method, the actual values of the roots are found to be S
1"1.249995745 andS2"1.2503747. The boundary of regionXis then given
by the following two lines
To identify the region of stability for this Lyapunov function, we need to imbed the largest set%(r) into the regionX. For this case, we obtain
C,%(r
Fig. 2. The functionU(S!S*) for test case 1.
As Fig. 3. makes clear, for this set of parameter values the Lyapunov stable region is too small to be of economic interest. An economy whose initial conditions lie in this set generates growth and #uctuations many orders of magnitude smaller than those observed in the US economy. Capital stocks, for example, 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 inside a region of attraction, and still generate an equilibrium path. We compute the region of attraction for this case in the next section, and now turn to examine the Lyapunov stable region for Case 2.
Fig. 3. The regionC.
positive de"nite solution of Eq. (22) is given as
H"
C
200.872 86.181186.1811 38.937
D
. (33)In this caseS*"1.35, and a graph ofU(S) for the relevant region is shown in Fig. 4. The roots are easily found to beS
1"1.333684 andS2"2.904155, using
Newton's method.
The boundary of regionXis given by the following two lines
x
2"!1.20482x1#0.0061, x2"1.20482x1!0.93028, (34)
and the setCis found to be
C,%(r
0)"MxD<(x)42]10~5N. (35)
Computing the points in thex1,x2plane for which
fT(x)Hf(x)"2]10~5, (36)
Fig. 4. The functionU(S) for test case 2.
Fig. 5 indicates that capital stocks must always be within 3% of their steady state value, which is six orders of magnitude larger than the variation observed in test case 1. Nevertheless, this set remains small for all practical purposes.
5. The region of attraction
Theorem 1 implies that solutions originating in setCare con"ned to this set at all times, so the maximum variation inK andK at any point in time cannot exceed the maximum variation of this region. Consequently, results obtained by Lyapunov's method are typically conservative, and only very modest variations in growth experience (or very mild cycles) can be detected. In this section, we propose to investigate the potentially more interesting properties of attractive trajectories originating outside of the regionC.
In order to estimate a region of attractionfor this model, it is necessary to solve the di!erential equation (12) numerically. Conceptually, this can be done by "xing the initial value of the capital stock, and incrementally varying the initial consumption until the trajectory fails to converge to the steady state. However, before such a procedure can be implemented, there are two problems that need to be resolved. In the"rst place, the variableSis an implicit function of x
1andx2, so Eq. (12) cannot be solved by direct numerical integration.
Second-ly, Eq. (12) is given in terms ofx
1andx2, and consumptiondoes notappear as an
initial condition.
Regarding variable S"S(x
1,x2), 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 implicit numerical integration schemes (the trapezoidal method being the most practical approach).
To explicitly incorporate consumption into the simulation, we propose to"x the initial capital stockK
0and incrementally vary variableS0instead ofC0. For
each chosen value S
0, we can compute the corresponding x1(0) andC0using
Eqs. (14) and (8), respectively. In this way, one can easily monitor the incremen-tal change in initial consumption.
Case1.The boundaries of the region of attraction for this case are summarized in Table 1, in which x
20,log (K0/K*) andx30,log (C0/C*), respectively.
Table 1
High and low values for initial consumption in the region of attraction
x
20 !0.25]10~3 0 10~3 1.5]10~3 2]10~3 2.25]10~3 2.47]10~3
H 5.72]10~4 1.98]10~3 3.7]10~3 3.93]10~3 3.69]10~3 3.23]10~3 1.42]10~3
x
30 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
Table 2
High and low values for initial consumption in the region of attraction
x
10 !5 !3 !1 0 1 3 5
H 1.328 2.017 2.858 3.235 3.625 4.39 5.137
x
20 L !4.239 !2.579 !0.919 !0.089 0.741 2.4 4.061
T.
Russell,
A.
Zece
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ic
/
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of
Economic
Dynamics
&
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}
Fig. 6. The region of attraction in thex
2,x3plane.
The region of attraction for this case is larger than the corresponding Lyapunov region, but is still clearly far too small to describe the growth and cyclical behavior of the US economy. For example, in the region of attraction the capital stock can vary from!0.025% to 0.25% of its steady state value. On the other hand, in the region in which capital is growing (i.e. when capital is below 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 of attraction as describing diverse behavior across economies, or take the alterna-tive view that paths in this region provide the support for cycles driven by shocks to expectations, for these parameter values the dynamics are too concen-trated around the steady state to describe any observed behavior.
Fig. 7. The region of attraction in thex
2,x3plane.
To demonstrate the dynamic behavior in this case, Figs. 8}10 plot possible consumption, capital and output paths for economies that originate near the lowerboundary of the region of attraction. These"gures were generated for the case where x
20"!0.288 and x30"!0.328, which corresponds to initial
capital and consumption levels that are at 75% and 72% of their respective steady state values. As an additional illustration, in Fig. 11 we provide the trajectory in thex
2andx3plane.
The variation in possible trajectories becomes even more striking if we analyze economies that start near theupperboundary of the region of attraction. In Figs. 12 and 13 we consider an extreme case for whichx
20"!0.288 and x
30"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 steady state value.
The capital and consumption paths in these "gures exhibit very large vari-ations over time, which is not unexpected considering our choice of initial consumption. Large variations also characterize the trajectory in the x
2,
x3plane, which is shown in Fig. 14.
Fig. 8. Normalized consumption pathC(t)/C*.
Fig. 10. Normalized output path>(t)/>*.
Fig. 11. The trajectory in thex
Fig. 12. Normalized consumption pathC(t)/C*.
Fig. 14. The trajectory in thex
2,x3plane.
Finally we may ask what happens to paths which start outside the region of attraction. To answer this question we solved the model numerically for a var-iety of initial conditions outside the region of attraction. In every case the solution diverged to in"nity so rapidly that the transversality condition, (Eq. (6)), was violated. We have been unable to"nd any paths outside the region of attraction which satisfy the necessary transversality condition of utility maximization.
6. Conclusion
This paper proposes a method for"nding explicit solutions to highly nonlin-ear indeterminate models of economic growth. The proposed method exploits the fact that the steady state of such models is stable. This allows us to "nd a Lyapunov function for the system, and subsequently compute the region of stability.
of attraction. Some paths in this region can diverge signi"cantly before reaching the steady state, and an examination of this region is potentially of more interest. In this paper, the region of attraction was computed numerically for the two-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 region of attraction is so close to the steady state that the model cannot be used to describe long run growth paths of developed economies (such as the US economy). As disappointing as this may be, the result is a testament to the power of the proposed technique. Previous solution methods which linearize the system around its steady state simply cannot address the question of the size of the region of attraction.
2. If we choose di!erent parameter values (as it happens, values at which aggregate demand slopes upward), the model generates far more reasonable growth behavior. Moreover, for this parameter con"guration equilibrium growth is very sensitive to the choice of the free initial condition. It is therefore possible that animal spirits could explain cycles in this case.
Our conclusions indicate that behavior in this model is highly parameter sensitive. The parameters used in Case 1 were deliberately chosen by Benhabib and Farmer to make the model lie close to the boundary of indeterminacy. A systematic investigation of parameter con"gurations moving away from this boundary may yield interesting insights. We hope to investigate the problem of parametric stability in future work.
Acknowledgements
The authors wish to thank Professor Dragoslav Siljak for many helpful discussions on this problem. All errors are the authors'own.
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