Structural stability and model design

Structural stability and model design Donald A.R. George and Leslie ‘I’. Oxley

The paper mtroduces the concept of structural stab&y and proposes that tt should be consrdered a necessary property of sctenttfically valid models. Formalizatton of the concept ts considered m both hnear and non-lmear models A strong preference tn favour of the wtder use of non-linear models 1s supported by constderatton of the dangers of lineanzatton m dynamic models The importance of structural stability IS demonstrated wtth reference to dynamic rational expectattons models whrch exhrbrt the saddle-point property In such cases convergence to equtltbnum IS shown to be a structurally unstable property which can be forced by restncttve auxtllary assumpttons, whtch are htghltghted. Keywords

Structural stabdtty, Non-hnear models, Rational expectations

Macroeconomrc dynamics has, once agam, become a topical subJect, thts time m connection wrth the widespread application of the rational expectations (RE) hypothesis, (see for example Obstfeld [19], Begg [5], Eastwood and Venables [9]) However, unlike earlier discussions of dynamics, for example Baumol [4], Goodwm [9] and Samuelson [21], the ‘new’ generatron of modellers restnct then attention to linear models which exhrbrt the saddle-pomt property to the exclusion of other alternatives The clear impression given is that RE m some way makes knife-edge models acceptable, d not desirable For example Begg [5] states, ‘RE has turned the argument for global stability on Its head’ The first maJor arm of thus paper is to introduce the notion of structural stability and propose that tt be consrdered a necessary property of screnttfrcally valid models Sectron 1 explams the concept of structural stabthty and establishes its rmportance on The authors are with the Department of Economrcs, University of Edmburgh, William Robertson Building, George Square, Edinburgh EH8 9JY, UK The authors would hke to thank Stuart Sayer for stlmulatmg dlscussronsof ratlonal expectations dynamics and S:mon Clark for bnngmg MonshJma’s work on structural stab&y to their attentlon They are also grateful to Professor David Hendry for useful comments on an earher version of this paper grven at the 1983 Scottrsh Economic Society Conference Fmally, they are grateful to partlclpants m semmars given at Edmburgh, LSE and Uppsala for their constructive comments Any remammg errors are entirely stochastic and uncorrelated with the advice received Fmal manuscnpt received 30 August 1984

methodologrcal grounds In section 1 structural stability 1s formalized within the framework of the linear dynamic model, while sectron 2 extends the discussion to non-linear models Section 3 tackles the second maJor aim of the paper, namely the apphcatton of the concept of structural stabihty to dynamrc RE models Convergence to equrhbrmm IS shown to be structurally unstable m these models, though it is sometimes forced by appeal to extreme microeconomic assumptions These assumptrons are shown to be entirely independent of the RE hypothesis itself, and recourse to nonhnear models IS proposed as an alternative means of estabhshmg structural stability Section 4 concludes

1. Structural stabiiity and linearity It is widely accepted that scientific assertions, as distinct from say metaphysical ones, should refer to entitles which are, m prmciple, observable Otherwise, theoretrcal assertrons would be immune to emprrrcal testing It may be, however, that the underlymg assumptions of a theory are not themselves directly open to empnrcal test but that testable rmphcatrons can be drawn from them. Unfortunately though, the underlying assumptrons of a theory are unlikely to be exactly true but rather, it 1s to be hoped, are close upproxrmatzonsto reahty Under these crrcumstances then it is Important that the imphcatrons of a screntrfic theory are robust with respect to small variations m rts underlying assump

0264-9993/85/040307-10$03 00 @ 1985 Butterworth & Co (Publishers) Ltd

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Structural stabdtty and model desrgn D A R George and L T Oxley

tlons Small varlatlons m these assumptions should produce only small varlatlons m the theory’s lmphcatlons, not wdd and dramatlc ones Without this property, empirical testing of theories becomes impossible In fact, meaningful observations of any kmd may become impossible because of ‘random’ environmental perturbations m the condltlons under which observations are made Consider a chemical theory which predicts the outcome of a particular chemical reaction under condltlons of constant amblent temperature Whatever care the experimental chemist takes, he will not be able to hold the ambient temperature exactfy constant, it 1s bound to fluctuate slightly during the course of the expenment Suppose now that the outcome of the expenment 1s substantially different from what the theory predicted Is the theory refuted’ The theoretician can always reply that the ambient temperature was not exactly constant, as his theory required, and the experiment does not therefore constitute a refutation This would not be the case if the robustness property, discussed above, had been required of the theory ab tnttto All of this has important lmphcatlons for the sclenttflc modeller Many scientific assertions take the form of a dynamic principle together with some mltlal condltlons, which Jomtly imply some testable conclusions The theory of gravity, for example, asserts that a stone released near the earth’s surface will fall to the ground The mmal condltlons here are the mltlal position and velocity of the stone, while the dynamic 1s derived from Newton’s laws of motion and gravltltatlonal principle The mam lmphcatlon 1s the point on the earth’s surface at which the stone 1s expected to land, its terminal point To Aristotle, a teleologrcul explanation of the stone’s behavlour would have been adequate That is, one could simply assert that the stone falls because it 1s seekmg some termmal stare, namely resting on the earth’s surface Teleological explanations are not accepted m the natural sciences today but, as we shall see m section 3, they still seem to creep into economrcs Those sclentlfic assertions which take the form of mltlal condltlons plus dynamic can be neatly modelled using the theory of differential equations It 1s often possible to set up the appropriate model m the form, x = F(x, ar) x(0) = %

(1)

where x 1s a vector contammg some observable variables, (Y IS a vector of measurable parameters

308

and F IS contmuous with no forcing term ’ Within this framework the concept of robustness can be readily formalized A solution y(t) of (1) can be thought of as a parameterlzed family of functions of the form y(t, a9 $1

Such a solution will be called structurally stable If small perturbations m the parameters (Yand %do not cause the function y to transmute into another solution quahtatlvely different from it Otherwise, y will be called structurally unstable The defmltlon IS not complete, of course, since the meanings of ‘small perturbation’ and ‘quahtatlvely different’ have not been specified In any practical apphcatlon of the notion of structural stability. these terms require precise defmltlons appropriate to the context For example, if x contains a price x1, a path along which x1 tends to some equlhbnum XTshould be considered quahtatlvely different from a path along which x, explodes to mfmlty Note that ordinary stability, as discussed m economics, 1s concerned with the response of a model’s endogenous variables to small displacements from an equlhbrlum Structural stablllty, by contrast, 1s concerned with the behavlour of whole tlmepuths of endogenous vunubles m the face of perturbations m the model’s parameters Note also that structural stability concerns parameterlzed fumdres of functions By restricting the values which parameters may take, such a family may be reduced m size In certain contexts It may be possible to choose a restriction on parameter values m such a way as to force structural stability on a function which, m the wider family, was structurally unstable A particular case of this approach IS discussed m section 3 For interesting dlscusslons of structural stability see Thorn [23], Chlllmgworth [7], Monshlma [15] and Saunders [22] In any particular context, the concept of structural stability can provide a useful formahzatlon of the robustness property discussed above Given the methodological importance of this property, structural stability provides a criterion for Judging scienttfitally valid models Any model which purports to represent the actual dynamic behavlour of the economy m a structurally unstable way should be rejected A similar argument has been advanced by Baumol [3], among others, m connection with linear dlfference equation models of the trade cycle Here the problem 1s that persistent, regular cycles occur only ‘That IS, trme does not appear as an Independent argument of F

ECONOMIC MODELLING October 1985

Structural stab&y and model desrgn D A R George and L T Oxley

for certain exact parameter values Arbitrarily small perturbatrons m these parameters mduce a transmutation to erther damped or explosrve cycles Baumol’s [3] argument is as follows But our statlstlcs are never fine enough to dlstmgulsh between a umt root (of the characterlstrc equation of a linear difference equation) and one which takes values so It 1s usually possible to show that a slight close to it amendment m one of the simplifying assumptions ~111 eliminate the umt roots and so have profound quahtatlve effects on the system As Solow has pomted out, smce our premises are always necessarily more or less false, good theonsmg consists to a large extent m avoldmg assumptions like these, where a small change m what IS posited will seriously affect the conclusions

As 1s well known. the resolutron of these dtffrcultres was eventually found by Hicks [13], Goodwm [9] and Desar [8] among others m non-hnear models of the trade cycle We return to non-lmear models m sections 2 and 3 Thorn [23] has argued cogently that structural stabrhty 1s a central concept m most branches of science The concept of structural stablhty seems to me to be the key idea in the interpretation of phenomena of all branches of science (except perhaps quantum mechanics) forms that are subJectlvely ldentlfiable and are represented m our language by a substantive are necessanly structurally stable forms, any given object IS always under the dlsturbmg influence of Its environment, and these influences, however shght, will have some effect on its form Therefore there IS an open set consisting of the structurally stable forms, and the unstable forms, which can be changed by an arbltranly small perturbation, belong to its complement, which IS closed These unstable forms do not ment the name of forms and are strictly nonforms

Attention 1s now turned to a specral case of the system (1) m which the functron F takes the partrcular form mdrcated below Y =Ay-b Y(O) = Y

(2)

I

where y IS a variable n-vector, b and f are constant n-vectors and A ISan n x n matrix Particular models of thus form are analysed m section 3 of the paper In partrcular apphcatrons the elements of the vector y are often logs of economrc variables which cannot, by then nature, be negative Define an equrhbnum of (2) to be a vector y* such that Ay* - b = 0 Suppose A to be non-smgular,

ECONOMIC

MODELLING

then

October 1985

y* = A-$

and there 1s a unique equrhbnum variables.

By a change of

x=y-y* Equations generality,

(2) can be reduced, without to the homogeneous case

x = Ax

(3)

I

x(0) = 2

loss of

to which attention IS now turned Note that the ongm IS the only eqtnhbrrum of (3) It 1s well known (see Hnsch and Smale [14] and Arrowsmith and Place [2]) that the set of solutions of (3) depends on the ergenvalues of A and that n independent boundary condrtrons will force exncrly one on these solutrons Suppose A has n drstmct ergenvalues, of which nl have posrtrve real parts and the remauung n2 have negatrve real parts Then IR” may be split mto two subspaces. mtersectmg only at the ongm, of drmensron nl and n2 respectrvely The first subspace 1s spanned by the etgenvectors assocrated with the first nl ergenvalues and will henceforth be termed the unstable manrfold The second subspace IS spanned by the ergenvectors associated wrth the remammg n2 ergenvalues and will henceforth be termed the stable mumfold If % hes m the stable manifold then the unique solutron path to (3) converges to the equrhbnum (the ongm) Otherwise rt will diverge Any solutron path which tends towards an equrhbnum will be called convergent When 0
all,

a12

%n,

%,

$2

*

%I)

(4)

Of these parameters, the a,,‘~ are the elements of the matnx A, while the %k’sare elements of the mrtral vector f Two solutron paths x(t) and z(r), having the form of Equation (4) ~111 be defined as quahtattvely drfferent tff there extsts at least one I (1 s I d n) such that x,(t) and z,(t) have drfferent hmrts as t +a~ In fact, there are only three possible hmrts for the system (3), zero, + 03 and --03 Moreover, rf there exrsts one J such that x,(t) + 0 as t + 00 then x,{f) + 0 as f+m, for all] = 1 n Such a path IS clearly

309

Structuralstab&y and model desrgn D A R George and L T Oxley convergent, other paths will be called drvergent Remembering that the origin is the only eqmhbrmm of (3), this defimtion of ‘quahtatively different’ implies that two solution paths are qualitatively different d one converges to an eqmhbrmm while the other does not This seems reasonable given the economic context m which most models of the form of (3) actually arise ‘Small perturbatrons’ m the parameters of (3) will be defined m terms of the usual Euclidean norm Let B =

(all,

a12

arm, fl,

f2

f,>

(5)

be an (n’ + n) vector of parameters of (3) Then another (n’ + n) vector, y will be said to be derived from I3 via a small perturbafton lff 11f3 - y (1IS arbttranly small (where 11l3 - y I/ denotes the usual Euclidean metnc) It IS not difficult to show that, m the case of a saddlepomt system, if f lies m the stable mamfold then small perturbations of the parameters will cause the corresponding solution path x(t) to transmute mto a path quahtatively dtfferent from tt Solutions of (3) are convergent rf their mtttal points he m the stable manifold From this it follows that tn hear saddlepotnt models convergence to equtltbrtum IS structurally unstable It 1s also easy to show that tn hear saddlepotnt models dtvergence IS structurally stable On the arguments advanced above, any linear saddlepomt model which purports to model an economy converging to eqmhbrmm should be rejected As ~111be seen m sectton 3, these arguments provide the basis for a powerful crittque of many existing RE models

2. Non-linear models Convergence to equihbrmm can only be made a structurally stable property m hnear models by mststmg on global stabtltty If all etgenvalues of the matrix A have negative real parts then all solution paths converge and convergence 1s clearly structurally stable Gtven n independent boundary condtttons the path of the economy IS determined A much wider range of possible solution paths occurs m non-linear models however and structural stabthty 1s much more mterestmg In particular, a richer variety of hmitmg behaviour arises In addition to the simple divergence and convergence already dtscussed, there is, for example, the posstbthty of ltmtf cycle behaviour m non-hnear models Consider the (slightly) non-linear system

310

Figure 1. Stable limit cycle

(6) It has solution paths as illustrated m Ftgure 1 If Z = Y- = 0 the solution path is a smgle point, the ortgm itself Otherwise, the system spirals inwards or outwards towards the closed orbit represented by the bold curve m Figure 1 Thts closed orbit IS called a limit cycle The system of (6) shows considerable stability, unless it starts at the origin it always tends towards some regular cyclical behaviour It IS often the case that convergence to a hmit cycle occurs m non-linear models m a structurally stable way That is, arbitranly small perturbations m parameters (mcludmg mitral conditions) do not generally cause solutton paths which converge to a hmtt cycle to cease doing so (see Hirsch and Smale [14] and Arrowsmith and Place [2]) It is also mterestmg to note that limit cycles cannot occur m linear models Structurally stable limit cycle behaviour may well have mterestmg economic interpretations, for example, m the theory of the trade cycle (see eg Desai [6]) However, economists’ natural instincts, when faced with a dynamical system such as (6), are to

ECONOMIC MODELLING October 1985

Structural stab&y and model design

D A R George and L T Oxley

A much wtder range of structurally stable behavtour 1s possible m non-hnear models than m linear ones Moreover, hneartzation 1s to be treated with care since tt deals only with local homeomorphzc equzvalence of flows

3. Rational expectations models

Figure 2. Unstable spiral

locate tts eqmhbrta and then lmeartze the system about an equthbrmm by (usually tmphctt) appeal to Hartman’s theorem Thts theorem ensures that the flow of the hneanzed system 1s locally homeomorphtc to the flow of the original system (provtded the hneanzed system 1s not a centre) Applymg thts approach to the system (6), tt clearly has a single equthbnum at the origin and hneanzmg about the origin gives i=y

(7)

jJ=-x+y

fw = 2,

y(0) = y

The matrix of coefficients has two etgenvalues with postttve real parts and the solutton paths are unstable spirals (Figure 2) The model appears to diverge to mfimty and thus to be of httle interest Note that near the orzgzn the ongmal system and tts hneanzatton do have homeomorphtc flows, as Hartman’s theorem mdtcates The theorem only applres locally, however, and does not help m analysmg the global behavtour of the system Exammed globally, the system of Equation (7) may indeed be of economtc interest m mdtcatmg some form of perststent and regular cyclical behavtour ECONOMIC MODELLING

October 1985

In thts sectron we consider how most dynamic RE models are formulated, defmmg RE m the sense of Muth [17], te ‘the public’s sublecttve expectation will be equated with the true mathemattcal expectation implied by the model itself’ The crucial aspect of such an approach 1s that agents are ‘forward lookmg’ RE mdtvtduals may make errors as long as they are unsystematic One important problem with the forward looking nature of expectattons formatton, however, 1s that RE as a behavtoural hypothesis does not spectfy how far mto the future mdtvtduals are presumed to look This point will prove cructal when we consider how RE models are supposed to be ‘solved’. Imphctt m the Muth Rattonal defuntton of expectation formation 1s an inherent problem associated with stattsttcal dtstnbutton properties of the hypcthests. Most RE models circumvent the stattstrcal dtstnbutton problems by mvokmg the notton of Certainty Equtvalence (CE) Here we define CE m the sense of Begg [5], le When the solution of a stochastic model differs from the solution of a determmlstlc or non-random model only m the tnvtal respect that actual values of future vanables are replaced by current expectations of these future vanables, we say that the random model exhlblts certamly equlvalence

The use of such a CE construct has become a crucial element m standard RE models as tt allows the constderatton of a smgle market relevant expectation However, tt must be stressed that the CE simphftcatton 1s only posstble d the model 1s linear with zero mean addmve error terms, a pomt we wtll return to later If CE 1s invoked, any truly stochastic elements are ‘washed out’ of the system Stochastic perturbations will not have any effect on the determmtsttc elements of the model The crucial aspect of current dynamic RE macromodels, however, 1s that they mvanably exhtbrt the saddlepomt property to the excluston of any altemattves In order that we may mvestrgate how this restricted class of models emerge so frequently in the literature, for the moment we wtll accept the CE simphficatton, and constder the work of Butter and Miller [6], Eastwood and Venables [9] and Neary 311

Structural stab&y and model design D A R George and L T Oxley

Convergence here 1s a structurally stable phenomenon only tf the class of solution paths IS restricted solely to stable branch (see sectfon 1) If one admits, as one must m a full saddlepomt RE model that every salmon path 1s consistent with ratlonal expectations, then convergence IS a structurally unstable phenomenon Any small random perturbations of the state matrix A or the boundary

P t

condltlons ~111 cause a convergent path to transmute mto a divergent one In the Bulter/Mlller model. a factory burning down unexpectedly could lead to

ever expanding hquldlty and every falhng competltlveness Thoroughly undesirable, no doubt, but perfectly consistent with ratlonal expectations (m fact with perfect foresight) Such a posslblhty 1s ruled out m the Bulter/Mlllpr model because * e

Figure 3. Price and exchange Eastwood-Venables model

rate

dynamics

m the

Source Adapted from Eastwood and Venables, 1982

and Purvls [18] They present RE models2 havmg the reduced form of Equation (3) (in suitably transformed coordmates) In the first two models the matrix A is 2 X 2 while m the third It is 3 X 3 Solution paths for the Eastwood-Venables model are depicted m Figure 3 (adapted from Eastwood and Venables [9] p 288) The varrable p represents the domestic price level while e 1s the exchange rate The stable and unstable mamfolds are srmply straight lines, Indicated m Figure 3, and are therefore termed stable and unstable brunches respectlvely. The stable branch 1s the only convergent path Note that every solution path IS consistent with rational expectations (m fact with perfect foreaght) concerning the exchange rate Note also the followmg remarks of Eastwood and Venables The stable branch plays an important role m the analysis to follow, since we rule out by assumptton all paths which do The uniqueness of the not converge to a steady state path (actually followed by the economy) evidently depends crucially on the ussumphon that ratlonat agents anticipate convergence to a steady state (emphasis added) It 1s clear then that in addition to the RE (or perfect foreslght) hypothesis an extra ad hoc assumption IS required to force the desired result, that the economy follow the stable branch 2The authors of these three (typIcal) papers clearly assume that mdtviduals possess ratIonal expectations See Eastwood and Venables [9], p 286. Bulter and Miller [6]. p 146, Neary and Pums (181, pp 229-230 However, at times Bulter and Mdler, p 151, and Eastwood and Venables, p 287, substitute the term ‘perfect foreslght’ for ‘ratIonal expectations’

312

The assumption of the transversahty condltlon thdt rational agents ~111 not choose an unstable’ solution mean(s) that the Jump variable (e or c) ~111always assume the value required to put the system on the umque convergent solutlon traJectory (Bulter and Mdler (61. emphasis added)

As with the EastwoodWenables

model

an ad hoc

assumptron 1s brought rnto play to change a structurally unstable outcome mto a structurally stable one by restnctmg the class of solution paths to the stable branch alone This 1s precisely the same sort of assumption introduced by Neary and Purvls This gves rise to a typlcal saddlepoint structure the single positive root contributes a dtrectlon of mstablhty. but exchange rate speculators are assumed to choose dn uuttal value of e (exchange rate) and hence of n (‘real’ exchange rate) which ensures that the model converges towards a long-run equlhbrmm (Neary and Purv~s [18]. emphasrs added )

In their model however a further element of interest arlses because the model involves three dlfferentlal equations mstead of two The stable mamfold IS a plane, thus allowmg the posstblhty of cychcal convergence to the equlhbnum (see Figure 4) It should be noted however that perstsrent cyclical behavlour cannot occur m a structurally stable way m linear models, a point which has been exammed m sectron 2. We need to examme what ad hoc assumptions are required to force the economy to follow the stable branch remembering that UN solutron paths are consistent with RE The use of RE as a behavloural hypotheses alone cannot force the economy to follow the stable branch The best ‘explanation’ of the ad hoc assumption, IS found m Bulter and Miller op ctr The quote given above makes it clear that “Unstable here means ‘dIvergent’ Not to be confused with ‘structurally unstable’

ECONOMIC MODELLING October 1985

Structural stab&

and model design D A R George and L T Oxley

Equilibrium \

Stable mamfold

t-_

htial positlon

FIIre 5. The

x3

Figure 4. The

ltnear

model wrth three variables

X1 IS forward lookmg X2 and X, are backward lookmg The unstable mamfold has dlmenslon 1 equal to the number of forward lookmg vanables, and there IS a umque Jump onto the stable mamfold

to satisfy some transversahty condrtron In general terms RE IS being unrquely tied to a particular form of optrmrzmg behaviour At the mdivrdual utihty maxrmtzmg level mdrvrduals are presumed to maxrmtze some functtonal over the mfmite future. re agents are assumed

rational

m

max j U(C,, m,) e-“dt 0

6>0

(8)

where C represents consumption, m represents real money balances, 6 the discount rate, and typrcally we assume statronanty to grve an autonomous system. The necessary and suffrcrent condttrons for an optimum m such a model usmg the Hamiltoman approach, are well known (see Arrow and Kurz [l]). However, we wish to htghhght the restncttons that such condrtrons impose on RE mdrvrduals and particularly what role these condrtrons play m the optrmahty of convergent paths These issues are best demonstrated by consrdermg the opttmization problem m both the fume and mfimte horizon case In ECONOMIC MODELLING October 1985

ltnear model wtth three vat-tables

X, 1s forward lookmg. X, and X, are backward lookmg The unstable mantfold has dtmenslon 2 whtch exceeds the number of fomard lookmg vanables There IS no possible Jump onto the stable mamfold

the fimte honzon replaced by

case

(8) would

typrcally

be

T

max j u(C,, m,) eGb’dt 0

(9)

Here C, would presumably represent the control vanable and m, the state variable (although thus is often not made clear m the actual macromodels) Followmg the method of Pontryagm, mstruments (control vanables) would be chosen so as to maxtmaze (9) subject to some constraints on the chorce of instruments and mitral condrtrons on the state vanables The Pontryagm method then uses auxilhary variables4 whrch we wtll denote A which are functtons of ttme and whtch, at an opttmum, sattsfy. h(t) 3 0, h(7)

m(T) = 0

Thus is the transversahty

(10) condmon

whrch, m a

4The solution to the 2 x 2 maxlmuatlon problem can be represented dlagrammatlcally by plottmg state vanables agamst either control or auxtlhary variables The choice IS purely a matter of exposttlon

313

Structural stab&y and model design

D A R George and L T Oxley

[l], pp 6673 To recap, current RE models which exhtbit the saddlepomt property require utihty maximizmg mdividuals to identtfy the stable manifold, as such they need to solve a maximization problem such as (8) or (9) This requires a number of boundary conditions equal to the dimensions of the differential equation system Irutlal values of the state variables provide n of the 2n and satisfaction of the transversahty condition can provide the additional n However, m the fnnte horizon case convergent paths are not m general optima and m the mfmtte horizon case the transversahty condition is not always a necessary condition for an optimum 5 How does this optimizmg framework relate to RE as a behavioural hypotheses, particularly the reqmrement that mdivtduals maximize some mfunte horizon problem of the form of Equation (&X)7 We know RE allows mdividuals to be forward lookmg but as Hahn [II] states even if one believes that it (fhe saddlepomt property) holds, it only helps d one now demands RE over the

x3 Figure 6. The linear model with three variables X1 and X2

are forward lookmg. X3 IS backward looking The unstable mamfold has dlmenslon 1 which IS less than the number of forward looking variables There IS an mfmlte number of possible lumps onto the stable mamfold

differential equation system of size 2n, along with n nntial values of the state variables, provide the n additional conditions to determine completely the optimal solution to the maxtmtzation problem (see Arrow and Kurz for a full discussion of the necessary and sufficient condition for an optimum m the concave function case) in many fume horizon problems the stable branch does not satisfy the transversahty condition necessary for an optimum and therefore cannot be an optimum Moreover, the optimahty of certain non-convergent paths can often be demonstrated using turnpike arguments (see Intrilligater [ 151) When we turn to consider the mfunte horizon case the necessary conditions for an optimum remain vahd except perhaps for the transversahty condition The problem IS that no general theorem exists giving the mfunte horizon version of (10) as a necessary condition for an optimum, and several counter examples exist (see Halkm [12]) Satisfaction of some transversahty condition cannot, m general, be considered a necessary condition for an optimum This becomes clear when constdermg the arguments put forward m Arrow and Kurz 314

infinite future Now that IS askmg more than any sane person would be willing to grant I thmk the word ‘mfimtely should be inserted before REE to alert the unwary

Hahn [lo] also highhghts the necessity of mfmmes m RE models which mclude money He further demonstrates here that overlappmg generation models will not m general be the technical answer to the problem If a genuine RE hypothesis 1s embedded mto optlmlzmg models of the type dlscussed above further problems arise Indlvlduals are allowed to make (non-systematic) mistakes and once the CE pnnclple 1s abandoned RE models will contam stochastic elements Thus a convergent path may transmute mto a divergent one simply as a consequence of errors m expectation formation Moreover, parameter values cannot be measured exactly Shght errors m measurement should not lead to a predicted path of the economy dramatically different from the actual timepath RE theorists sometimes assert that these difficulties are taken account of by ‘Jump variables’ (see for example, Buiter and Miller [6]) Any movement away from the stable branch, caused by, for example, perturbations of the state matrix A (of Equation (3)) or the boundary conditions will be corrected by Jumping back onto the stable branch. ‘Obstfeld and Rogoff [20] consgder the role of the transversahty condltlons m mfmlte horizon utthty maxImlung RE models They suggest that satlsfactlon of such a condrtlon be considered necessary for an optimum, whereas, in fact, no such general conclusion can be drawn They do not, unfortunately, consider the central questlon of structural stab&y

ECONOMIC MODELLING October 1985

Structural stabzlztyand model design D A R George and L T Oxley

Jump vanables are elements of the vector x (in Equation (3)) which are able to move dtscontmuously In the Butter/Mtller model, for example, the exchange rate, e, 1s a Jump vanable and therefore so is competitiveness c However, there are technical requirements on Jump variables. In general, m linear saddlepomt models, the number of Jump vanables must equal the dtmenston of the unstable manifold If there are fewer lump variables then a suitable Jump may not exist, tf there are more, tt may not be unique These cases are illustrated m Figures 4, 5 and 6 These are simply technical requirements and m no way specify an economzc mechanzsm which ensures that the appropriate Jump takes place RE theorists frequently assume that such Jumps can and do occur However, what must be stressed IS that such Jumps have nothing to do with the RE hypothesis, since every solution path IS consistent wtth this hypothesis Jumps are required m such a model to force a structurally unstable outcome, to satisfy the transversahty condmon They are ad hoc assumpttons, necessary because of the hnear (or lmeanzed) saddlepomt structure mto whtch the RE hypothesis has been embedded It may be that some such Jump mechanism does provide a plausible explanation of the workings of the macroeconomy Until the operation of this type of mechamsm IS fully zntegrated mto the model and not tacked on as an unconvmcmg afterthought, the case must remam open A more convmcmg strategy would involve the modelhng of structurally stable outcomes As such Jumps would not be a necessary reqmrement to force structurally unstable outcomes We have exammed structural stability m sectton 1 and one important element relatmg to the concept is the preference for truly nor&rear models Two major outcomes emerge for the movement away from linear models Firstly, the possibility of persistent cychcal behaviour, a property that many economists regard as an important feature of actual macroeconomics, can be modelled m a structurally stable way Secondly, the switch to non-linear models means that the CE prmctple cannot be invoked Stochastic elements would, once again, become explicit m models which incorporate RE The problem IS, however, that, for example, the error term could enter multiphcattvely instead of addittvely This would complicate the solution of RE models (or at least recognize that expectation formation over the mfimte future 1s not a trivial matter) It may be that an assumption that the error term 1s small may have to be made. In general, the RE macromodels would have to be

ECONOMIC MODELLING

October 1985

formulated as non-linear stochasttc dtfferential equatton systems, which are technically dtfficult to solve, and may require the increased use of numencal solution methods However the maJor gam is that the multtphctty of solution paths would be structurally stable with no uncertainty as to whether a partzcular solutton is typtcal of the general form of the solution to the system as a whole What would be required however is some pnnctple of choosing which path to follow (given that many may converge) Some paths will have, for example, low inflation rates, others high, etc There would always be the knowledge, however, that ‘small mistakes’ would not drastically alter the final outcome, but merely alter slightly the time path of certain variables

4. Conclusions The basic arm of this paper was to argue that economtc phenomena should be modelled m a structurally stable way This property can be most easily satisfied m non&near dynamic models which also offer a wader class of possible solutton paths Persistent cyclical behaviour, for example, can only be modelled m a structurally stable way m non-linear models Lmeanzatron of inherently non-linear models IS a popular way of analysmg a dynamical system m the netghbourhood of some particular pomt, for example, on equthbnum We show, however, that thts can be mtsleadmg tf mtsmterpreted as provtdmg an analysts of the global behavtour of a system Linear, or lmeanzed RE models which exhibtt the saddlepomt property were used to tllustrate how structurally unstable solution paths can be forced, m thts case, by certain strong auxrlhary assumptions concernmg utility maxrmrzation We argue that such condmons should be considered separately from the rational expectattons hypothesis itself. These auxtlhary assumptions often rely on the satisfaction of some transversahty condttton to provrde boundary condmons We show that m the finrte horizon model convergent paths need not satisfy the transversahty condmon However m mfmite hortzon models the transversahty condrtton IS not, m all cases, a necessary condition for an opttmum In saddlepomt models therefore, the convergent paths are not only structurally unstable, but may also be s&optimal m finite horizon problems Moreover such a path may prove drfftcult to identtfy, using transversahty condmons, m infinite hortzon problems These ad hoc auxilhary assumptions are used by RE theorists to force structurally unstable outcomes mto structurally stable ones by restnctmg the

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Structural stab&y and model design D A R George and L T Oxley

class of solutions. However, they cannot, m general, be relied upon for such a purpose Lmearlty m RE models allows the CE prmclple to be invoked The use of non-linear models would mvolve the re-emergence of stochastic elements which would make the structural stablhty property more important m such models, but would also make expltctt solutions of such systems dlfflcult However the qualitative behavlour of such structurally stable models would be well understood and robust to small perturbations of parameters Linear saddlepomt models, by contrast, do not display the same structural stablhty The property of unique convergent paths m linear saddlepomt models would be replaced with a large number of structurally stable convergent paths, all of which sat&y the RE hypothesis The parrtculur path followed would depend on the values of boundary condltlons However such models would not rely on the ad hoc mechanical devices, such as jump variables, m order to force structurally unstable outcomes Small ‘mlstakes’ or perturbations m the system would lead, m structurally stable models, to paths quahtatlvely similar to those of the ongmal system

References K J Arrow and M Kurz, Pubhc Investment, the Rate of Return, and Optrmal Ftscal Pohcy, John Hopkms Press, London, 1970 D K Arrowsmith and C M Place, Ordmary Dtfferenttal Equattons, Chapman and Hall, London W J Baumol, ‘Topology of second-order hnear difference equations with constant coefflclents’. Econometrtca, Vol 26, 1958, pp 258-287 W J Baumol, Economtc Dynamics, 3 ed, Macmdlan. New York, 1970 D Begg, The Rational Expectattons Revolutton m Macroeconomrcs, Phdhp Allan, Oxford, 1982 W Buiter and M Miller, ‘Monetary pohcy and international competitiveness the problem of adlustment’, in W Eltis and P Sinclair, The Money Supply and the Exchange Rate, Oxford University Press, 1981

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D Chillingworth, Dtfferenttal Topology wrth a Vtew to Apphcattons, Pltman, London, 1976. M Desal, ‘Growth cycles and inflation m a model of the class struggle’, Journal of Economrc Theory, Vol 6, 1973, pp 527-545 R M Goodwin, ‘The non-linear accelerator and the persistence of business cycles’, Econometrtca. Vol 19. 1951, pp l-37 F Hahn, Money and lnflatton. Basil Blackwell, Oxford, 1981 F Hahn, Review of D K Begg’s, The Ratronal Expectattons Revolutton tn Macroeconomtcs, Economu Journal, Vol 93. 1983, pp 922-924 H Halkm, ‘Necessary conditions for optimal control problems with infinite horizons’. Econontetrtca. Vol 42, 1971, pp 267-272 J R Hicks, A Contrtbutton to the Theory of the Trade Cycle, Oxford University Press, Oxford. 1950 M Hirsch and S Smale, Dtfferenttal Equattons, Dynamical Systems and Linear Algebra. Academic Press, New York, 1974 M Intnlhgator, Mathemattcal Opttmtsatton and Economic Theory, Prentice Hall, New York, 1971 M Monshima. Dynamic Economic Theory, International Centre for Economics and Related Disciplines, London School of Economics, 1980 R Muth, ‘Rational expectations and the theory of price movements’, Econometnca, Vol 29, 1961, pp 315-335 J P Neary and D Purva, ‘Sectoral shocks in a dependent economy long-run adlustment and shortrun accommodation’, Scandmavtan Journal of Economrcs, Vol 84, 1982, pp 229-253 M Obstfeld, ‘Capital moblhty and devaluation m an optlmlsmg model with rational expectations’, Amertcan Economtc Review. Papers and Proceedmgs. Vol 71, 1981, pp 217-221 M Obstfeld and K Rogoff, ‘Speculative hypermflanon tn maximizing models can we rule them out’)‘, Journal of Polmcal Economy, Vol 91, 1983, pp 675-687 P Samuelson, Foundattons of Economtc Analysts, Harvard University Press, Cambridge. 1947 P T Saunders, An lntroductton to Catastrophe Theory, Cambridge University Press, Cambridge, 1980 R Thorn, Structural Stabduy and Morphogenesu, (English Translation), Benlamm, New York, 1976

ECONOMIC

MODELLING

October 1985